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abstract: 'We use the Higgs mechanism to investigate connections between higher-rank symmetric $U(1)$ gauge theories and gapped fracton phases. We define two classes of rank-2 symmetric $U(1)$ gauge theories: the $(m,n)$ scalar and vector charge theories, for integer $m$ and $n$, which respect the symmetry of the square (cubic) lattice in two (three) spatial dimensions. We further provide local lattice rotor models whose low energy dynamics are described by these theories. We then describe in detail the Higgs phases obtained when the $U(1)$ gauge symmetry is spontaneously broken to a discrete subgroup. A subset of the scalar charge theories indeed have X-cube fracton order as their Higgs phase, although we find that this can only occur if the continuum higher rank gauge theory breaks continuous spatial rotational symmetry. However, not all higher rank gauge theories have fractonic Higgs phases; other Higgs phases possess conventional topological order. Nevertheless, they yield interesting novel exactly solvable models of conventional topological order, somewhat reminiscent of the color code models in both two and three spatial dimensions. We also investigate phase transitions in these models and find a possible direct phase transition between four copies of $\mathbb{Z}_2$ gauge theory in three spatial dimensions and X-cube fracton order.'
author:
- Daniel Bulmash
- Maissam Barkeshli
bibliography:
- 'references.bib'
title: 'The Higgs Mechanism in Higher-Rank Symmetric $U(1)$ Gauge Theories'
---
Introduction {#sec:intro}
============
The recent discovery of “fracton" phases of matter[@ChamonGlass; @BravyiChamonModel; @HaahsCode; @YoshidaFractal; @VijayFractons; @VijayNonAbelianFractons] has led to considerable recent research provided a new class of three-dimensonal gapped phases of matter beyond conventional topological order, and which apparently cannot be described using standard gauge field theories. In addition to subextensive ground state degeneracy on topologically nontrivial manifolds, fracton phases are defined by possessing excitations whose motion is restricted to subdimensional manifolds. These theories have prompted a great deal of recent excitement[@BravyiHaahSelfCorrection; @SivaBravyiMemory; @EmergentPhasesFractons; @ShiFractonEntanglement; @FractonEntanglement; @RecoverableInformation; @FractonCorrFunctions].
Recently, certain gapless versions of fracton phases have also been found in terms of higher-rank symmetric $U(1)$ gauge theory[@XuFractons1; @XuFractons2; @RasmussenFractons; @PretkoSubdimensional; @PretkoElectromagnetism; @PretkoWitten]. These theories possess gapless “photon” modes, together with “matter” whose motion is confined to subdimensional manifolds.[@PretkoSubdimensional] These higher-rank symmetric gauge theories have received little attention in the field theory literature, as they inherently break Lorentz symmetry, although some of them are closely related to studies of Lifshitz gravity.[@XuFractons1; @XuFractons2; @XuHorava]
While fracton phases share some features of conventional topological order, they appear to require geometric data in addition to topological data[@PretkoElasticity; @GromovElasticity; @SlagleGenericLattices; @ShirleyXCubeFoliations]. Constructions involving gauging subsystem symmetries[@VijayGaugedSubsystem; @Williamson2016], layering two-dimensional conventional topological order[@VijayLayer; @MaLayer; @RegnaultLayer], coupled chains[@HalaszFractons], or partons[@HsiehFractonsPartons] are known to produce fracton phases, but it is unclear whether these relationships are essential or incidental. In particular, the general relationships between gapped fracton phases, gapless higher-rank gauge theories, and conventional topological order are not well-understood. In order to understand these relationships, one natural approach is to consider the Higgs mechanism in the higher rank $U(1)$ theories, since the Higgs mechanism is known to relate gapless phases of conventional gauge theories to gapped, topologically ordered ones.
An intriguing feature of gapped fracton phases is that it is also unclear whether, and to what extent, they can be described using continuum field theories. A natural guess is that a Higgs phase of a continuum higher rank gauge theory might provide such a continuum field theoretic description. Recently, Ref. found a field theoretic representation of the X-Cube model[@VijayGaugedSubsystem], which is a an example of a gapped fracton model. This further raises the question of whether such a field theory can be related to a Higgs phase of a gapless higher rank gauge theory.
In this paper, we present a general analysis of a wide class of higher-rank symmetric $U(1)$ lattice gauge theories and their Higgs phases. This helps us elucidate the relation between fracton orders, conventional topological order, and gapless higher-rank symmetric gauge theories. Our results are briefly summarized below.
Structure of paper and summary of main results
----------------------------------------------
We begin in Sec. \[sec:U1theories\] by defining a set of rank-2 symmetric lattice gauge theories which have the symmetry of the square (cubic) lattice in $d=2$ ($d=3$). These theories are defined by a set of rotor variables $A_{ij}({\bf r}) \sim A_{ij}({\bf r}) + 2\pi$ on the sites and faces of a square (cubic) lattice. The conjugate momenta are the “electric fields” $E_{ij}({\bf r})$. There are two classes of theories, the $(m,n)$ scalar and $(m,n)$ vector charge theories, where $m$ and $n$ are relatively prime integers. These theories are defined by their Gauss’ Law constraints, which induce a set of gauge transformations, summarized in Table \[tab:theoryList\]. The scalar and vector charge theories studied in Refs. , whose continuum limit is invariant under continuous spatial rotations, correspond in our notation to the $(1,1)$ scalar charge and $(2,1)$ vector charge theories.
[@lclcl]{} **Theory** & & **Gauss’ Law** & & **Gauge transformation**\
$(m,n)$ scalar && $m \sum_i \Delta_i^2 E_{ii} + n \sum_{i\neq j} \Delta_i \Delta_j E_{ij} = \rho$ && $A_{ij} \rightarrow A_{ij} - \begin{cases}
m \Delta_i^2 \alpha & i=j\\
n \Delta_i \Delta_j \alpha & i \neq j
\end{cases} $\
\
$(m,n)$ vector && $m\Delta_j E_{jj} + 2n\sum_{i\neq j}\Delta_i E_{ij} = \rho_j$ && $A_{ij} \rightarrow A_{ij}- \begin{cases}
m\Delta_i \alpha_i & i=j\\
n(\Delta_i \alpha_j + \Delta_j \alpha_i) & i \neq j
\end{cases}$\
Section \[sec:U1theories\] also provides details of these theories and presents local lattice rotor models whose low-energy subspace is the gauge theory coupled to charge-$p$ matter fields. With the exception of the $(0,1)$ scalar charge theory in $d=2$, the $(1,0)$ and $(0,1)$ vector charge theories in $d = 2$ and $d = 3$, these models yield a well-defined class of field theories.
An important question is the extent to which these gapless theories are stable to arbitrary perturbations. The models that we consider correspond to compact $U(1)$ higher-rank gauge theories, and thus an understanding of stability requires ruling out non-perturbative instanton processes. We expect that the $d = 2$ theories that we consider are unstable to proliferation of instantons, and are thus not stable phases. As such, they may be thought of as multi-critical points, and it is an interesting question to understand how many relevant operators exist at their respective fixed points. In $d = 3$, most of the theories do appear to correspond to stable phases. In particular, the $(m,n)$ vector charge models are self-dual for $m,n>0$, which, using extensions of arguments for the $(2,1)$ theory[@XuFractons1; @XuFractons2; @RasmussenFractons], implies stability of the theory. Similarly, the $(1,0)$ scalar charge theory also has a self-duality and is stable. On the other hand, the other $(m,n)$ scalar charge theories do not have a clear self-duality. Nevertheless, consideration of the magnetic sector of these theories suggests that they are stable for $m,n > 0$, although we leave a detailed analysis for future work. The stability of the $(0,1)$ scalar charge theory in $d= 3$ also requires further study, as it does possess non-trivial instanton process whose relevance must be carefully analyzed.
In Section \[sec:intuition\] we explain the physical intuition for how the Higgs mechanism affects subdimensional charges in the $U(1)$ theories. Specifically, we demonstrate how condensation of charge $p$ excitations in the $(1,1)$ scalar and $(2,1)$ vector charge theories necessarily renders the charged excitations fully mobile in all directions. Thus the Higgs phases of such theories possess conventional topological order, described by a conventional discrete gauge theory.
We then explicitly study the Higgs transitions and phase diagrams for general $(m,n)$ scalar and vector charge theories in Secs. \[sec:Scalar2DHiggs\]-\[sec:evenOddVector\_d3\]. Our main results for the topological order of the Higgs phases are summarized in Table \[tab:HiggsPhases\]. In particular, we find that in $d=3$, the $(2r, 2s+1)$ scalar charge theories, for $r, s \geq 0$, yield the X-cube fracton phase[@VijayGaugedSubsystem] upon condensation of charge 2 particles. This is a remarkable result, as we see that the $(2r, 2s+1)$ scalar charge theories form a class of gapless higher rank gauge theories that appear to be stable phases of matter (at least for $r > 0$), and thus can be thought of as gapless “parent” phases of the X-cube fracton phases. As seen in Table \[tab:HiggsPhases\], the Higgs phases of many of the theories that we study simply reduce to conventional topological order; nevertheless, in several cases they correspond to exactly solvable models of topological order that are quite different from the usual well-known toric code models, and thus they may be of independent interest. In particular, these models are reminiscent of the color codes.[@ColorCodes]
We further find that the $(2r+1,2s+1)$ scalar and vector Higgsed charge models, for integer $r$ and $s$, in $d=3$ have particularly interesting phase diagrams; we study them in Sections \[sec:oddOddScalar\_d3\] and \[sec:oddOddVector\_d3\] respectively. The $(2r+1,2s+1)$ scalar charge theory has $\mathbb{Z}_2^4$ topological order in its Higgs phase. Upon adding a certain strong Zeeman field, the effective Hamiltonian is the X-cube model; this raises the interesting possibility of a direct confinement-like transition between conventional topological order and fracton order in three spatial dimensions. The Higgs phase of the $(2r+1,2s+1)$ vector charge theory has $\mathbb{Z}_2^7$ topological order and can be driven to $\mathbb{Z}_2$ topological order in a suitable strong Zeeman field limit.
$U(1)$ **Charge Type** $(m,n)$ **Higgs Phase**
------------------------ -- ------------------ -- ------------------------------------ -- -- -- --
$d=2$ scalar
($2r+1$, $2s+1$) $\mathbb{Z}_2^3$ topological order
$(2r,2s+1)$ Trivial
$(2r+1, 2s+2)$ Trivial
$(1,0)$ $\mathbb{Z}_2^4$ topological order
$d=2$ vector
($2r+1$, $2s+1$) $\mathbb{Z}_2^3$ topological order
($2r+2$, $2s+1$) $\mathbb{Z}_2^4$ topological order
($2r+1$, $2s$) Trivial
$(0,1)$ Trivial
$d=3$ scalar
($2r+1$, $2s+1$) $\mathbb{Z}_2^4$ topological order
$(2r,2s+1)$ X-Cube fracton order
$(2r+1, 2s+2)$ Trivial
$(1,0)$ $\mathbb{Z}_2^8$ topological order
$d=3$ vector
($2r+1$, $2s+1$) $\mathbb{Z}_2^7$ topological order
($4r+2$, $2s+1$) $\mathbb{Z}_2$ topological order
($4r$, $2s+1$) Trivial
($2r+1$, $2s$) Trivial
Higher-Rank Symmetric $U(1)$ Lattice Gauge Theories {#sec:U1theories}
===================================================
In this section, we carefully define the set of rank-2 symmetric $U(1)$ lattice gauge theories. The set of theories with continuous rotational symmetry is known[@RasmussenFractons], but for later purposes we will also write down the taxonomy of rank-2 theories respecting the symmetries of the square (cubic) lattice in $d=2$ ($d=3$).
General Setup {#subsec:ClassesOfTheories}
-------------
The starting point of these gauge theories is a set of gauge field variables $A_{ij}$, where we define $A$ to be symmetric so that $A_{xy}$ and $A_{yx}$ are just relabelings of the same degree of freedom. The construction may be defined on a lattice, in which case the diagonal components $A_{ii}$ live on the sites of the lattice and the off-diagonal components $A_{ij}$ for $i \neq j$ live on the faces of the square (cubic) lattice in $d=2$ ($d=3$), as shown in Fig. \[fig:latticeSetups\]. Alternatively, the theory may be constructed in the continuum. We will generally use lattice notation, but we will specify when the lattice is important.
We then define a symmetric tensor $E_{ij}$ of momenta such that $$\begin{aligned}
[A_{ij}({\mathbf{r}}),E_{kl}({\mathbf{r}}')] &= \frac{i}{2}\delta_{{\mathbf{r}},{\mathbf{r}}'}{\left(}\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}{\right)}\label{eqn:comms}\end{aligned}$$
$A_{ij}$ and $E_{ij}$ to both transform as tensors under spatial rotations, that is, (for example) $E_{ij} \rightarrow \sum_{k,l} R_{ik}R_{jl}E_{kl}$ where $R$ is a rotation matrix. The form of Eq. guarantees rotational invariance of the commutation relations. The factor of $1/2$ ensures that $A_{ii}$ is canonically conjugate to $E_{ii}$: $[A_{ii},E_{ii}] = i$. However, for $i \neq j$, we have $[A_{ij},E_{ij}] = i/2$. This non-standard normalization for canonical conjugates must be tracked carefully in what follows.
Next, we demand the existence of local Gauss’ Law constraints, which take the form: $$G_a(E;{\mathbf{r}}) = \rho_a({\mathbf{r}}),$$ where $G_a$ is some local linear differential operator acting on the $E_{ij}$ tensor and $\rho_a$ is the corresponding matter field. The subscript $a$ allows for multiple Gauss’ Laws corresponding to multiple species of charges.
Finally, we compute the gauge transformation rule arising from the Gauss’ Law constraint. This is done most clearly in the zero-charge sector, where we demand that a state ${|A_{ij}\rangle}$ which is an eigenstate of all the $A_{ij}$ operators obey $$e^{i\alpha_a({\mathbf{r}}) G_a(E;{\mathbf{r}})}{|A_{ij}\rangle} = {|A_{ij}\rangle}$$ for each $a$, each ${\mathbf{r}}$, and arbitrary $\alpha_a$. But $e^{i\alpha_a({\mathbf{r}})E_{ij}({\mathbf{r}})}$ shifts the eigenvalue of $A_{ij}({\mathbf{r}})$ according to the commutation relations Eq. , which means that the two configurations of $A_{ij}$ related by $e^{i\alpha G_a}$ must represent the same state, i.e. these states are gauge-equivalent.
Hamiltonians
------------
The general structure of the Hilbert space and the Hamiltonians we consider is as follows. The Hilbert space of the theory consists of two pieces. First, we have the rotor variables $A_{ij} \sim A_{ij} + 2\pi$ and their conjugates, as discussed in Sec. \[subsec:ClassesOfTheories\]. Second, we have a set of compact rotor variables $\theta_a \sim \theta_a + 2\pi$ and their canonical conjugates $L_a$ with commutation relations $$[\theta_a({\mathbf{r}}), L_b({\mathbf{r}}')] = i\delta_{{\mathbf{r}},{\mathbf{r}}'}\delta_{ab}$$ The values of $a$ and the locations of the $\theta$ variables will depend on the theory, but in general there is one value of $a$ per Gauss’ Law constraint. We defer further discussion until talking about the individual theories.
The Hamiltonian for every theory will break down into the same basic structure, identical to the structure of the usual lattice $U(1)$ gauge theory, but the form of each term will depend on the theory. $$H = H_{Maxwell}+H_{Higgs}+H_{Gauss}
\label{eqn:HStructure}$$
The Maxwell term provides dynamics for the gauge field $A_{ij}$ and electric field $E_{ij}$ degrees of freedom. Its form is $$\begin{aligned}
H_{Maxwell} = \sum_{{\mathbf{r}},i} &{\left(}\tilde{h}_{s} E_{ii}^2 - \frac{1}{g_s^2}\cos(B_{ii}){\right)}\nonumber \\
&+ \sum_{{\mathbf{r}},i<j} {\left(}\tilde{h}_{f} E_{ij}^2 - \frac{1}{g_{f}^2}\cos(B_{ij}){\right)}\label{eqn:MaxwellTerm}\end{aligned}$$ where $B_{ij}$ is the simplest (i.e. contains the fewest derivatives) combination of the $A_{ij}$ which is gauge invariant, and $\tilde{h}_s$, $\tilde{h}_f$, $g_s$, and $g_f$ are coupling constants. The form and symmetry of $B_{ij}$ will depend on the gauge transformation rules of the theory.
The Higgs term contains the dynamics of the matter field(s) $\theta$ and $L$. Most notably, it couples the charge-$p$ matter field $\theta$ to the gauge field $A_{ij}$. The form of this coupling depends on the theory but is dictated by gauge invariance. This coupling term will be taken strong in order to enter a Higgs phase.
Finally, the Gauss’ Law term dynamically identifies $pL$ with the right-hand side of Gauss’ Law. This term has the schematic form $$H_{Gauss} = \tilde{U}\sum_{a,{\mathbf{r}}} {\left(}G(E) - pL_a{\right)}^2
\label{eqn:GaussTerm}$$ There is one term in the sum for each $L$ variable, that is, one per site in the scalar charge theories and one per link in the vector charge theories. At large $\tilde{U}$, the low-energy subspace is the gauge theory restricted to charge-$p$ particles (since, by hand, we only put charge-$p$ matter into the theory). Implementing Gauss’ law as an energetic constraint rather than an exact constraint allows the gauge theory to emerge at low energies from a local lattice rotor model whose Hilbert space has a local tensor product structure. What remains for each theory is to use Gauss’ Law to specify the structure of the matter field and to use gauge invariance to specify the form of the magnetic field $B_{ij}$ and the form of $H_{Higgs}$.
Scalar Charge Theories
----------------------
We now define the full set of theories that we consider and the relations between them. These are summarized in Table \[tab:theoryList\]; there are scalar and vector charge theories. A trace constraint may also be added to these theories[@RasmussenFractons]; we leave investigation of such theories to future work.
We begin with the scalar charge theories.
### Set of Theories
The scalar charge theory with continuous rotation invariance has Gauss’ Law $$\sum_{ij} \Delta_i \Delta_j E_{ij} = \rho
\label{eqn:11ScalarCharge}$$ which generates the gauge transformation $$A_{ij} \rightarrow A_{ij} + \Delta_i \Delta_j \alpha.$$ Here $\Delta_i$ is a lattice derivative in the $i$ direction.
If we demand only the symmetries of a square (cubic) lattice, the diagonal components of $E_{ij}$ are not symmetry-related to the off-diagonal components, and we may modify Gauss’ Law to the following $$m\sum_i \Delta_i^2 E_{ii} + n \sum_{i\neq j}\Delta_i \Delta_j E_{ij} = \rho$$ which we call the $(m,n)$ scalar charge theory. Keeping careful track of the factors of 2 in the commutators, this leads to the gauge transformation rule $$A_{ij} \rightarrow A_{ij} + \begin{cases}
m\Delta_i^2 \alpha & i = j\\
n\Delta_i \Delta_j \alpha & i\neq j
\end{cases}
\label{eqn:ScalarGaugeTransform}$$ In all these theories, there is a single matter field which, on the lattice, lives on the sites.
Clearly the scalar charge theory with continuous rotational invariance, Eq. , is the $(1,1)$ scalar charge theory in our nomenclature. How many distinct theories are there? If the gauge group is noncompact, that is, if we do not enforce $A_{ij} \sim A_{ij}+2\pi$, then $E_{ij}$ is not quantized and may be rescaled freely. Therefore all nonzero $m$ and $n$ generate the same theory. However, if the gauge group is compact, $E_{ij}$ is quantized and may only be rescaled by a sign while maintaining the quantization conditions. Therefore, in general different values of $m$ and $n$ are *not* equivalent. We generally require $m$ and $n$ to be relatively prime integers (common factors may be removed by rescaling the matter field $\rho$)[^1].
In each theory, charges are created by the local operators $e^{\pm iA_{ij}}$, which are raising and lowering operators for the $E_{ij}$. The charge configurations created by these local operators in the $(m,n)$ theories are determined by inspection of Gauss’ Law, and are shown in Fig. \[fig:scalarU1Operators\].
We now examine different theories individually.
### $(1,0)$ Scalar Charge Theory {#subsubsec:10ScalarSetup}
The $(1,0)$ theory is special in that the off-diagonal components of $A_{ij}$ are gauge-invariant and decouple from the diagonal components. Since the off-diagonal components are decoupled and have no gauge transformation associated with them, we discard them as a trivial sector of the theory.
The magnetic field is $$B_{i} = \sum_{jk}\epsilon_{ijk} \Delta_j^2 A_{kk}$$ for $i=z$ in $d=2$ and $i=x,y,z$ in $d=3$. Here $\epsilon$ is the Levi-Civita symbol. The Higgs term of the Hamiltonian is $$H_{Higgs} = \sum_{{\mathbf{r}}} \frac{L({\mathbf{r}})^2}{2M} - V \sum_i \cos(\Delta_i^2 \theta + p A_{ii})$$
In $d=2$ local operators create point-like magnetic excitations, so we expect the theory to be unstable to instanton proliferation. In $d=3$, this theory is self-dual at $V=0$; if we write $$E_{ii} = \sum_{jk}\epsilon_{ijk}\Delta_j^2 h_{kk}$$ where $h$ is a diagonal rank-2 tensor, then $[h_{jj},B_{k}] = i\delta_{jk}$ reproduces the correct commutation relations between $A_{jj}$ and $E_{kk}$. The self-duality of the Hamiltonian can be checked explicitly. The duality and gauge invariance is enough to show, following similar arguments to Refs. , that the $(1,0)$ theory in $d=3$ is stable to confinement. Its photon mode has the soft dispersion $\omega \sim k^2$.
Electric charges in this theory are immobile, while dipoles can move only in one dimension, along the direction of their dipole moment. To show this, we adapt the arguments of Ref. . In this theory, dipole moments are conserved: $$\int x_i \rho({\mathbf{r}}) d^d{\mathbf{r}} = \int x_i \sum_j \Delta_j^2 E_{jj} d^d{\mathbf{r}} = \int \Delta_i E_{ii} d^d{\mathbf{r}} = 0$$ where we integrated by parts and used the fact that the $E_{ii}$ are single-valued. Moving an isolated charge would violate this conservation law and is therefore not allowed.
Another way to see this is to examine the local operators which create charge; these operators are $e^{iA_{ii}}$. Using the form of Gauss’ Law, it is straightforward to see that $e^{iA_{xx}}$, for example, creates the charge configuration in Fig. \[fig:scalarAxx\] (with $m=1$). Alternatively, this operator is a hopping operator which moves an $x$-directed dipole by one unit in the $x$ direction; dipoles may move in the direction of their dipole moment. However, because $e^{iA_{xy}}$ is not present in this theory, any operator that moves an $x$-directed dipole in the $y$ direction creates additional charges. Hence, the dipoles are mobile only along their dipole moment. There is no local hopping operator for a single charge.
### $(0,1)$ Scalar Charge Theory {#scalar-charge-theory}
In the $(0,1)$ theory the diagonal components of $A_{ij}$ are gauge-invariant and decouple from the off-diagonal components. The diagonal components may then be discarded as a trivial sector of the theory.
In $d=2$, $A_{xy}$ is the only nontrivial degree of freedom remaining. It is not possible to create a gauge-invariant operator solely out of $A_{xy}$; accordingly, no magnetic field can be defined in $d=2$. We therefore expect that the $(0,1)$ theory in $d=2$ is highly degenerate; its Maxwell theory has no $B^2$ term, which corresponds to a speed of light equal to zero. This theory is therefore highly unstable to perturbations that take the system out of the gauge theory subspace, and thus not well-defined.
In $d=3$, we can construct a magnetic field $$B_{ii} = \sum_{ab}\epsilon_{iab}\Delta_a A_{bi}$$ The labeling is for later comparison with other theories. Treating $B$ as a (diagonal) rank-2 tensor, we see that it is traceless. The Higgs term of the Hamiltonian is $$H_{Higgs} = \sum_{{\mathbf{r}}} \frac{L({\mathbf{r}})^2}{2M} - V \sum_{i < j} \cos(\Delta_i\Delta_j \theta + p A_{ij})$$
Unlike the $(1,0)$ theory, the $d=3$ $(0,1)$ theory is not self-dual. Local operators create point-like magnetic excitations, so we expect that instanton processes induce confinement. Ref. showed that the theory is indeed unstable to confinement.
The photon mode in this $(0,1)$ theory has dispersion $\omega \sim k$. Following the same arguments as in Sec. \[subsubsec:10ScalarSetup\], one can check that isolated electric charges are immobile, and that dipoles can move in any direction perpendicular to their dipole moment. Isolated charges are immobile because, as before, there is a dipole conservation law $\int x_i \rho({\mathbf{r}})d^3{{\mathbf{r}}} = 0$. The only local operators which create charge, $e^{iA_{ij}}$ for $i \neq j$, create charges in the set of four shown in Fig. \[fig:scalarAxy\]. This operator is exactly a transverse hopping operator for a dipole, but no longitudinal hopping operator exists because the diagonal components of the electric field do not exist.
This motion is reminiscent of the restricted mobility of excitations in the X-cube model; we will show later that Higgsing the $d=3$ $(0,1)$ model indeed produces the X-cube model.
### $(m,n)$ Scalar Charge Theory
We now consider the rest of the $(m,n)$ theories, i.e. $m,n$ relatively prime positive integers.
$$H_{Higgs} = \sum_{{\mathbf{r}}} \frac{L({\mathbf{r}})^2}{2M} - V_1 \sum_{{\mathbf{r}},i < j} \cos(n\Delta_i\Delta_j \theta + p A_{ij}) - V_2 \sum_{{\mathbf{r}},i}\cos(m\Delta_i^2 \theta + p A_{ii})
\label{eqn:mnScalarHiggsTerm}$$
For the later part of the paper, we will want to take a single strong-coupling limit which Higgses the entire gauge field. We therefore want $V_1$ and $V_2$ to scale to strong coupling at the same rate; for these purposes, it suffices to take them equal.
In $d=2$, the magnetic field has two components $$\begin{aligned}
B_{zx} &= m\Delta_x A_{xy}- n\Delta_y A_{xx}\\
B_{zy} &= m\Delta_y A_{xy}- n\Delta_x A_{yy}
\label{eqn:11ScalarBd2}\end{aligned}$$ The notation is for consistency with $d=3$, where the magnetic field is a traceless, non-symmetric tensor $$B_{ij} = \begin{cases}
\sum_{ab} \epsilon_{iab}\Delta_a A_{bi} & i = j\\
\sum_{a\neq i,j} {\left(}m\Delta_j A_{aj} - n \Delta_a A_{jj}{\right)}& i \neq j
\end{cases}
\label{eqn:scalarB}$$
In neither $d=2$ nor $d=3$ does this model have a clear self-duality. Local operators create point-like magnetic excitations in $d=2$ but not in $d=3$, so we expect that the theory is unstable to confinement in $d=2$ but is stable in $d=3$, consistent with expectations[@PretkoSubdimensional] for the $(1,1)$ model.
The photon mode has dispersion $\omega \sim k$. These theories all have the dipole conservation law $\int x_i \rho({\mathbf{r}}) d^d{\mathbf{r}}=0$, so electric charges are immobile. Dipoles can propagate in any direction, but on the lattice they have the curious property that the distance by which they can move depends on $m$ and $n$. For example, in the $(1,1)$ theory, $e^{iA_{ii}}$ is a longitudinal hopping operator for a unit dipole and $e^{iA_{ij}}$ for $i\neq j$ is a transverse hopping operator, as can be deduced from Fig. \[fig:scalarU1Operators\]. In the $(1,2)$ theory, $e^{iA_{ij}}$ for $i \neq j$ is a transverse hopping operator for dipoles with moment 2 (see Fig. \[fig:scalarAxy\]); the simplest transverse hopping operator for a unit dipole is shown in Fig. \[fig:dipoleHop\] and moves the unit dipole by *two* lattice units.
![Transverse hopping operator for a unit dipole in the $(1,2)$ scalar charge theory. The $e^{iA_{ij}}$ operators act on the black rotors and create the charge configuration shown. Because $e^{iA_{xy}}$ only creates charge-2 objects in the $(1,2)$ theory, it cannot cause a unit dipole to hop, unlike the $(1,1)$ theory.[]{data-label="fig:dipoleHop"}](dipoleHop.pdf){width="4cm"}
As a further note, it can be checked explicitly that the $(m,n)$ theory for $m,n \neq 0$ can be produced by condensing a bound state of charge $n$ in the $(1,0)$ theory and charge $-m$ in the $(0,1)$ theory. This means that in some sense one may think of the $(1,0)$ and $(0,1)$ theories as the fundamental scalar charge theories.
Vector Charge Theories
----------------------
We next examine the possible vector charge theories, proceeding similarly to the scalar charge theories. Here, the matter field consists of rotor variables $\theta_a({\mathbf{r}} )$ that are defined on links that are oriented in the $a$ direction, and which transform as a vector under rotations. Consequently, there are $d$ different types of charges, which are subject to $d$ Gauss’ Laws.
### Set of Theories
The vector charge theory with continuous rotational invariance has the following $d$ Gauss’ Laws: $$2\sum_{i} \Delta_i E_{ij} = \rho_j .
\label{eqn:21VectorCharge}$$ The factor of $2$ is present to cause $\rho_j$ to take on integer values; this is needed because the off-diagonal components of $E_{ij}$ may be half-integers. This generates the $d$ gauge transformations $$A_{ij} \rightarrow A_{ij} + \Delta_i \alpha_j + \Delta_j \alpha_i$$ where the $\alpha_i$ are $d$ independent gauge transformations.
As before, demanding only the symmetries of a square (cubic) lattice allows modifications $$m \Delta_i E_{ii} + 2n \sum_{i\neq j}\Delta_i E_{ij} = \rho_j$$ which we call the $(m, n)$ vector charge theory. The factor of $2$ is again present to make $\rho_j$ integer-valued when $m$ and $n$ are integers. This leads to the gauge transformation rule $$A_{ij} \rightarrow A_{ij} + \begin{cases}
m\Delta_i \alpha_i & i = j\\
n{\left(}\Delta_i \alpha_j + \Delta_j \alpha_i{\right)}& i\neq j
\end{cases}
\label{eqn:VectorGaugeTransform}$$ In these theories, there are $d$ matter fields $\theta_i$ which, on the lattice, live on the $i$-directed links.
Clearly the rotationally invariant scalar charge theory Eq. is the $(2,1)$ vector charge theory in our nomenclature. We proceed as before to classify the set of possible theories. Again, if the gauge field is noncompact, all nonzero $m$ and $n$ generate the same theory because $E_{ij}$ can be rescaled. For the same reasons as before, with a compact gauge group, relatively prime integers $m$ and $n$ generate distinct theories.
The charge configurations created by local operators in the $(m,n)$ theories are again determined by inspection of Gauss’ Law, and are shown in Fig. \[fig:vectorU1Operators\].
We now examine different theories individually.
### $(1,0)$ and $(0,1)$ Vector Charge Theories
As in the scalar charge case, the $(1,0)$ vector charge theory has decoupled, gauge-invariant off-diagonal components of $A_{ij}$ which can be discarded as trivial. Likewise, the diagonal components of $A_{ij}$ may be discarded in the $(0,1)$ theory. However, it can be checked that in neither of these theories can a magnetic field be defined; no gauge-invariant linear combination of the remaining $A_{ij}$ exist. These theories effectively have zero “speed of light" and are thus highly degenerate and unstable to perturbations that take the theory out of the gauge invariant subspace. Although a Higgs mechanism can still be defined, it will turn out to produce trivial Higgs phases in these theories.
### $(m,n)$ Vector Charge Theory
Although the $(1,0)$ and $(0,1)$ vector charge theories are unstable, the theories we now consider, i.e. $(m,n)$ theories with $m,n$ relatively prime positive integers, are much better behaved.
The Higgs term in the Hamiltonian is of the form $$\begin{aligned}
H_{Higgs} = &\sum_{{\mathbf{r}},i} \frac{L_i({\mathbf{r}})^2}{2M} - V_1 \sum_{{\mathbf{r}},i} \cos(m\Delta_i \theta_i + p A_{ii}) \nonumber \\
&- V_2 \sum_{{\mathbf{r}},i<j}\cos(n(\Delta_i \theta_j + \Delta_j \theta_i) + p A_{ij})\end{aligned}$$ For the same reasons as in the scalar charge theories, we will choose to take $V_1=V_2$ for simplicity.
In $d=2$, the magnetic field has one component $$\begin{aligned}
B_{zz} = n \sum_{a \neq b} \Delta_a^2 A_{bb} - m\Delta_x\Delta_yA_{xy} \end{aligned}$$ The notation is for consistency with $d=3$, where the magnetic field is a symmetric tensor
$$B_{ij} = \begin{cases}
\frac{1}{2}\sum_{a \neq b \neq i} {\left(}2n\Delta_a^2 A_{bb}-m\Delta_a\Delta_b A_{ab}{\right)}& i=j\\
\frac{2}{3-(-1)^m}\sum_{k \neq i,j}\left[m{\left(}\Delta_i \Delta_k A_{jk} + \Delta_j \Delta_k A_{ik} - \Delta_k^2 A_{ij} {\right)}- 2n \Delta_i \Delta_j A_{kk}\right] & i \neq j
\end{cases}
\label{eqn:vectorB}$$
The peculiar factor in front of the off-diagonal terms, which is $1$ when $m$ is odd and $1/2$ when $m$ is even, merits explanation. In the expression for $B_{ij}$ for $i \neq j$, $A_{kk}$ appears with an even coefficient $2n$. Therefore, under $A_{kk}({\bf r}) \rightarrow A_{kk}({\bf r}) + 2\pi$ at a specific single site ${\bf r}$, we have $B_{ij}({\bf r}) \rightarrow B_{ij}({\bf r})+4n\pi$. Under $A_{ij}({\bf r}) \sim A_{ij}({\bf r}) +2\pi$ at a single specific site ${\bf r}$, we have $B_{ij}({\bf r}) \rightarrow B_{ij}({\bf r})+2m\pi$. If $m$ is odd, the fact that $m$ and $2n$ are relatively prime implies that $B_{ij} \sim B_{ij} + 2\pi$, and $\cos(B_{ij})$ is the minimal term in the Hamiltonian which respects this identification. If $m$ is even, then the factor of $1/2$ maintains $B_{ij} \sim B_{ij} + 2\pi$; were the factor of $1/2$ not present, a term $\cos(B_{ij}/2)$ would be allowed in the Hamiltonian. The prefactor just absorbs that factor of $1/2$ into the definition of the magnetic field.
The $(2,1)$ model at $V=0$ has been shown[@XuFractons2] to confine in $d=2$ and to be self-dual and stable in $d=3$. These arguments carry through with minimal modification for the general $(m,n)$ theories ($m,n\neq 0$): these theories are also confining in $d=2$ and self-dual and stable in $d=3$.
The photon mode has dispersion $\omega \sim k^2$. The motion of charges depends on the values of $m$ and $n$. The operator $e^{iA_{ii}}$ is a longitudinal hopping operator for particles with vector charge $m\hat{x}_i$, (here $\hat{x}_i$ is the elementary vector charge on an $i$-directed link) as shown in Fig. \[fig:vectorAxx\], while $e^{iA_{ij}}$ for $i \neq j$ creates a loop of charges of magnitude $n$, as shown in Fig. \[fig:vectorAxy\]. Charges whose charge components are all multiples of $m$ may only move in one dimension, along the direction of the charge, while other charges are confined.
Similarly to the scalar charge theories, the $(m,n)$ theory for $m,n \neq 0$ can be produced starting from the decoupled $(1,0)$ and $(0,1)$ theories. One does this by condensing the bound states of charge $n\hat{x}_i$ in the $(1,0)$ theory with charge $-m\hat{x}_i$ in the $(0,1)$ theory.
General Comments on Higgs Phases {#sec:intuition}
================================
Before explicitly solving the models that arise upon spontaneously breaking the higher rank $U(1)$ gauge symmetry, we will provide in this section some general comments and intuition for what to expect for the properties of the resulting Higgs phases.
Scalar Charge Theories
----------------------
Let us first discuss the $(1,1)$ scalar charge theory. Recall that the operators that create electric charges in the $U(1)$ theories are $e^{iA_{ij}}$, which are raising operators for the $E_{ij}$ and accordingly modify the eigenvalues of the charge density on various sites. In particular, $e^{iA_{xx}}$ creates a line of charges of value $1, -2,$ and $1$ on neighboring lattice sites as shown in Fig. \[fig:scalarAxx\], and $e^{iA_{xy}}$ creates a square of charge-$1$ particles as shown in Fig. \[fig:scalarAxy\].
The Higgs procedure condenses charge $p$ particles; for the moment, we specialize to $p=2$. Hence only the parity of charges is well-defined after Higgsing. In particular, the charge $-2$ particle created by $e^{iA_{xx}}$ may be absorbed into the condensate, and $e^{iA_{xx}}$ becomes the distance-2 hopping operator $Z_{xx}$ for $\mathbb{Z}_2$ charges (the reason for the notation will be made clear later). This process is shown in Fig. \[fig:dipoleHopHiggs\]. The only change to the action of $e^{iA_{xy}}$ (see Fig. \[fig:scalarAxy\]) is that $+1$ and $-1$ charges are now equivalent since they differ by a condensed charge. Therefore, we see that in the Higgs phase, individual charges are now free to propagate and are no longer immobile. We expect, then, that the resulting Higgs phase will possess some form of conventional topological order.
![Effect of the Higgs mechanism on the local operator $e^{iA_{xx}}$ in the $(1,1)$ scalar charge theory. The original charge configuration (top) is modified when charge-2 particles are condensed; the charge-2 particle is absorbed into the condensate, and charges $+1$ and $-1$ become equivalent.[]{data-label="fig:dipoleHopHiggs"}](dipoleHopHiggs.pdf){width="5cm"}
The above line of reasoning extends to general $(m,n)$ scalar charge theories. By examining Fig. \[fig:scalarU1Operators\], it is clear that if $m$ is odd, $e^{iA_{ii}}$ becomes a hopping operator for $\mathbb{Z}_2$ charge by 2 units in the $i$ direction, and charges become mobile. If $m$ is even, $e^{iA_{xx}}$ acts trivially because all the charges that it creates can be absorbed into the condensate. Likewise, if $n$ is odd, the Higgsed version of $e^{iA_{xy}}$ simply creates four (identical) $\mathbb{Z}_2$ charges. If $n$ is even, then $e^{iA_{xy}}$ acts trivially.
The behavior of the local operators in the electric sector of the scalar charge theory after Higgsing therefore depend entirely on the parities of $m$ and $n$ (which are relatively prime). We label the classes of Higgsed theories using representations for $m$ and $n$ which make their parities clear. The distinct classes of Higgsed scalar charge theories are those arising from the $(1,0)$, $(0,1)$, $(2r+1,2s+1)$, $(2r+2,2x+1)$, $(2r+1,2s+2)$ scalar charge theories with $r,s$ nonnegative integers. ($(0,1)$ and $(1,0)$ are distinguished separately because their magnetic fields behave differently from other values of $(m,n)$.)
In the Higgsed ($2r+1,2s$) theory, electric excitations can hop two lattice sites in any direction. Since $n$ is even, there is no one-site hopping operator, so we expect $2^d$ decoupled copies of the resulting theory, each living on a different sublattice of lattice constant 2. This will turn out to be $2^d$ copies of the $\mathbb{Z}_2$ toric code.
The difference between the Higgsed $(2r+1,2s)$ and $(2r+1,2s+1)$ scalar charge theories is that in the latter the operator $e^{iA_{xy}}$ (considered after Higgsing) acts nontrivially. In $d = 2$, it couples the four “copies" by locally creating or annihilating a bound state of the charges on all four sublattices. That is, the $d= 2$ Higgsed $(2r+1,2s+1)$ scalar charge theory should be produced from the $(2r+1,2s)$ theory by condensing the four-charge bound state. The topological order turns out to be three copies of the $\mathbb{Z}_2$ toric code. The analogous consideration in $d = 3$ leads to four copies of the $\mathbb{Z}_2$ toric code, as explained in detail in the subsequent sections.
Finally, the Higgsed $(2r,2s+1)$ theory only allows electric particles to be created in sets of four, which is reminiscent of fracton phases. In $d = 3$, we find that indeed these $(2r, 2s+1)$ scalar charge theories yield the X-cube model upon breaking the $U(1)$ higher rank gauge symmetry to its $\mathbb{Z}_2$ subgroup. In $d = 2$ however, careful examination reveals that we obtain the trivial gapped phase.
The above results can be readily generalized to the condensation of charge $p$, with $p>2$, although the analysis is slightly more complicated. For example, in a $U(1)$ theory with $m=1$, consider the operator shown in Fig. \[fig:dipoleHopHiggsp3\]. Before Higgsing, it creates four charges, $+1$, $-3$, $+3$, and $-1$ in a line. If charge $p=3$ condenses, then the $\pm 3$ charges can be absorbed into the condensate and this operator becomes a hopping operator for $\mathbb{Z}_3$ charges. It is a straightforward generalization to show if charge $p$ is condensed, then there is a distance-$p$ hopping operator in the Higgsed theory when $m$ and $p$ are relatively prime. Accordingly, if $m$ and $p$ are relatively prime, then the Higgsed $(m,0)$ theory decouples into sublattices of lattice constant $p$, and the topological order turns out to be $p^2$ copies of the $\mathbb{Z}_p$ toric code. More generally, the resulting theory depends only on the values of $m$ and $n$ modulo $p$. We will not consider $p>2$ in much further detail.
![Operator generating a $\mathbb{Z}_3$ charge hopping operator in the $p=3$ Higgsed $(1,1)$ scalar charge theory.[]{data-label="fig:dipoleHopHiggsp3"}](dipoleHopHiggsp3.pdf){width="6cm"}
Vector Charge Theories
----------------------
Similar logic may be used on the vector charge theories. When $m$ is odd, in the $\mathbb{Z}_2$ Higgs phase, $e^{iA_{ii}}$ becomes a hopping operator for charges on the $i$ links (see Fig. \[fig:vectorAxx\]); therefore such $i$-directed charges are mobile in the $i$ direction. If $m$ and $n$ are both odd, then in the $\mathbb{Z}_2$ Higgs phase charges become mobile in all $d$ dimensions. The operator which moves a particle in a direction transverse to the link it lives on has a non-obvious form, shown in Fig. \[fig:vectorTransverseHopHiggs\]. If $m$ is even, then charges are confined because $e^{iA_{ij}}$ ($i \neq j$) can only create closed strings of charge and $e^{iA_{ii}}$ acts trivially.
![Operator in the $(1,1)$ vector charge theory that, after $p=2$ Higgsing, allows $\mathbb{Z}_2$ charges on $x$-directed links to hop in the $y$ direction.[]{data-label="fig:vectorTransverseHopHiggs"}](vectorTransverseHopHiggs.pdf){width="8cm"}
As such, the behavior depends mostly on the parities of $m$ and $n$. However, there are some subtleties in the magnetic sector in $d=3$. The $\mathbb{Z}_2$ Higgsed $(m,n)$ vector charge models in $d=3$ are actually labeled not just by the parity of $m$ and $n$. Specifically, if $m$ is even and nonzero, then there is a distinction between the $m \equiv 0$ mod 4 and $m \equiv 2$ mod 4 theories. We will discuss this further in Sec. \[subsec:vectorHiggsChanges\].
The distinct classes of $\mathbb{Z}_2$ Higgsed vector charge theories in $d=3$ thus arise from the $(1,0)$, $(0,1)$, $(2r+1,2s+1)$, $(2r+1, 2s+2)$, $(4r+2,2s+1)$, and $(4r+4,2s+1)$ vector charge theories, where $r$ and $s$ are nonnegative integers.
Scalar Charge Higgs in $d=2$ {#sec:Scalar2DHiggs}
============================
Here we will discuss the Higgs phases of the scalar charge theory in $d= 2$. For a review of relevant aspects of the Higgs mechanism in standard (rank-1) compact $U(1)$ gauge theory, see Appendix \[app:HiggsReview\].
Our focus here is on explaining the details of theories for which the $\mathbb{Z}_2$ Higgs phases are non-trivial. This occurs for the $(2r+1,2s+1)$ and $(1,0)$ scalar charge theories. Other cases, where the $\mathbb{Z}_2$ Higgs phases are trivial theories, are discussed in Appendix \[app:trivialHiggs\]. All the Higgs phases are summarized in Table \[tab:HiggsPhases\].
Higgsing Procedure {#subsec:HiggsingProcedure}
------------------
We begin by describing the models that we obtain by taking the gauge-matter coupling in the Hamiltonian Eq. to be large. These induce condensation of the charge $p$ matter fields, inducing a $\mathbb{Z}_p$ Higgs transition. We first explain the example of the $(1,1)$ scalar charge theory, and subsequently discuss the generalization to $(m,n)$.
We recall the general form of the Hamiltonian Eq. , and take $V$ (see Eq. ) much larger than all other scales in the problem, which freezes $$\Delta_i \Delta_j \theta + p A_{ij} = 2\pi n$$ Given any initial gauge choice, $\theta$ may be set uniformly to zero by choosing the gauge transformation $\alpha({\mathbf{r}}) = -\theta({\mathbf{r}})/p$ where the $\theta$ on the right-hand side is defined using the initial gauge choice. In this gauge, $$A_{ij} = \frac{2\pi}{p} n
\label{eqn:PinningA}$$ for $n \in \mathbb{Z}$. For simplicity, we specialize to $p=2$. Then $e^{iA_{ij}} = \pm 1$ on each site or plaquette. Furthermore, since $e^{iA_{ij}}$ is a raising operator for $E_{ij}$, its action flips the sign of $(-1)^{(2-\delta_{ij})E_{ij}} = \pm 1$. The factor of 2 for the off-diagonal piece is present because of the factor of 1/2 in its commutation relations, see Eq. (\[eqn:comms\]). Therefore, the spectrum and the commutation relations of $e^{iA_{ij}}$ and $(-1)^{(2-\delta_{ij})E_{ij}}$ in the low-energy subspace are reproduced by the identification $e^{iA_{ij}} = Z_{ij}$ and $(-1)^{E_{ij}} = X_{ij}$ where $X_{ij} = X_{ji}$ and $Z_{ij}=Z_{ji}$ are Pauli matrices and have $$\begin{aligned}
[X_{ab},Z_{cd}] = 0 & \text{ if } (a,b) \neq (c,d) \nonumber \\
\lbrace X_{ab},Z_{cd} \rbrace = 0 & \text { if } (a,b)=(c,d)\end{aligned}$$ We emphasize that the indices $a,b,c,d$ label which spin the operator is associated with. That is, $X_{xx}$ is a $2\times 2$ matrix, not a matrix element. The arrangement and labeling of the spin degrees of freedom is inherited from the parent rotor variables shown in Fig. \[fig:2DLatticeSetup\]. The generalization to $p>2$ simply replaces the spin-1/2 particles by $p$-state clock variables and $X_{ij}$ and $Z_{ij}$ by generalized Pauli matrices. We specialize to $p=2$ for the rest of the paper; the generalizations are mostly straightforward.
The magnetic terms $\cos{B_{ij}} \equiv b_{ij}$ in the Hamiltonian are precisely products of $Z$ operators after Higgsing. Their forms, which can be straightforwardly deduced from the form of $B_{ij}$ in Eq. , are shown in Fig. \[fig:2DScalarHam\].
The operator in the Gauss’ Law term Eq. of the Hamiltonian is strongly fluctuating because charge is condensed. However, since only charge-2 matter fields are condensed, charge parity is still a good quantum number. The $U(1)$ version of $H_{Gauss}$ should then be replaced by a mod 2 Gauss’ law, which is enforced by the term $$-U\sum_{{\mathbf{r}}} a({\mathbf{r}}) \equiv -U\sum_{{\mathbf{r}}} (-1)^{\sum_{i, j} \Delta_i \Delta_j E_{ij}},
\label{eqn:aterm}$$ with $U \propto \tilde{U}$. Above we have suppressed the site labels for $E_{ij}$. In the gauge theory ($U \rightarrow \infty$) language, $a$ is constrained to equal 1, which just says that $\sum_{i,j} \Delta_i \Delta_j E_{ij}$ is even, in accordance with Gauss’ Law. Note that charge 1 excitations in this model cost an energy on the order of $U \propto \tilde{U}$, which is due to the fact only charge $p$ excitations exist below the energy scale $\tilde{U}$ in the $U(1)$ theory.
Note that several diagonal terms drop out of the expression of Gauss’ Law: $$\begin{aligned}
(-1)^{\Delta_i^2 E_{ii}({\mathbf{r}})} &= X_{ii}({\mathbf{r}}+\hat{x}_i )(X_{xx}({\mathbf{r}}))^{-2}X_{xx}({\mathbf{r}}-\hat{x}_i)
\nonumber \\
&= X_{xx}({\mathbf{r}}+\hat{x}_i )X_{xx}({\mathbf{r}}-\hat{x}_i)\end{aligned}$$ This is precisely the manifestation of the intuition we saw in Sec. \[sec:intuition\]. The $E_{xx}({\mathbf{r}})$ terms were responsible for the charge-2 particles created by $e^{iA_{xx}}$; these terms drop out of the expression for $a$ because charge-2 particles are condensed.
From this expression, we find that the operator $a({\mathbf{r}})$ (which lives on sites) involves 8 spins in $d=2$. Its form is shown in Fig. \[fig:2DScalarHam\]. The final Hamiltonian is $$\begin{aligned}
H_{2D} = -\frac{1}{g^2}&\sum_{\text{links}}(b_{zx}+b_{zy}) - h_s \sum_{\text{sites},i}X_{ii} \nonumber \\
&- h_p \sum_{\text{plaquettes}} X_{xy} - U \sum_{\text{sites}} a \label{eqn:2DScalarHiggs}\end{aligned}$$ Compared to Eq. , we have set $g_f = g_s = g$ for simplicity and renamed $h_f$ ($f$ for “face") to $h_p$ ($p$ for “plaquette") to be more appropriate for $d=2$. The $h_s$ and $h_p$ terms induce fluctuations in the gauge field, analogous to the $\tilde{h}_s$ and $\tilde{h}_f$ terms in the parent $U(1)$ theory.
The Higgs procedure for general $(m,n)$ scalar charge theories is exactly the same as for the $(1,1)$ theory. The large-$V$ limit is still well-defined, $\theta$ can be gauged away, and the condition Eq. still results, independent of $m$ and $n$. The operator identifications are therefore identical. The only differences come in the form of Gauss’ Law and the magnetic field operators. In particular, only the $m$th power of site operators $X_{ii}$ and the $n$th power of the plaquette operator $X_{xy}$ appear in the $(m,n)$ version of Eq. . Therefore, the form of $a$ is determined only by the parity of $m$ and $n$. In particular, if $m$ ($n$) is even, the site (plaquette) operators are absent in $a$.
Likewise, the form of the magnetic field term depends on $m$ and $n$. For $m,n \neq 0$, only the $m$th power of $Z_{xy}$ and the $n$th power of $Z_{ii}$ appear in the expressions for $b_{zi}$, and again the form of the $b_{zi}$ depend only on the parity of $m$ and $n$ if both are nonzero. The $(0,1)$ theory has no magnetic field, while the $(1,0)$ theory has a somewhat different form of the magnetic field; these cases must be treated separately but analogously.
In accordance with the intuition from Sec. \[sec:intuition\], we have found that the Higgsed theory depends entirely on the parity of $m$ and $n$ (for $m,n \neq 0$). There are therefore five scalar charge theories to consider: the $(1,0)$, $(0,1)$, $(2r+1,2s+1)$, $(2r+1,2s)$, and $(2r,2s+1)$ theories. (Recall that $m$ and $n$ can always be defined to be relatively prime, so there is no $(2r,2s)$ theory.) More generally, for $p > 2$, the Higgsed theory depends on $m$ and $n$ modulo $p$.
$(2r+1,2s+1)$ Scalar Charge Theory
----------------------------------
The central claim of this section is that the Higgsed model Eq. , which describes the $\mathbb{Z}_2$ Higgs phase of the $(2r+1, 2s+1)$ scalar charge theories, has the schematic phase diagram shown in Fig. \[fig:PhaseDiagram2DScalarA\]. We will show that the Higgs phase has $\mathbb{Z}_2^3$ topological order, which can be driven either into $\mathbb{Z}_2^4$ topological order or to a confined (paramagnetic) phase by tuning the “Zeeman fields” $h_s$ and $h_p$.
![Schematic phase diagram at $V = \infty$ for the $d=2$ $(1,1)$ scalar charge theory. The phases at large and small $h$ are accurate, but the phase boundaries and behavior at intermediate coupling are schematic. The direct transition from $\mathbb{Z}_2^3$ to paramagnetism is protected by $C_4$ rotation symmetry.[]{data-label="fig:PhaseDiagram2DScalarA"}](PhaseDiagram_2DScalarA.pdf){width="5cm"}
### Explicit Solution at $h_s = h_p = 0$
We first show explicitly that at $h_s=h_p=0$, the Higgsed $(1,1)$ scalar charge model in $d=2$ has $\mathbb{Z}_2^3$ topological order (three copies of the toric code). This will be shown by computing the ground state degeneracy, the excitations, and their fusion and braiding rules.
At $h_s = h_p = 0$, Eq. is a commuting projector model and thus is exactly soluble. Ground states ${|G\rangle}$ must obey the simultaneous but not necessarily independent constraints $(a-1){|G\rangle} = 0$ and $(b_{zi}-1){|G\rangle} = 0$. The number of ground states is simply $2^{N-C}$ where $N$ is the number of spins and $C$ is the number of independent constraints. For commuting projector models of spin-1/2 particles, each constraint can be encoded as a binary vector such that independent constraints produce linearly independent (over $\mathbb{Z}_2$) vectors; see Ref. for the details. Therefore $C$ is equal to the rank over $\mathbb{Z}_2$ of the matrix consisting of all these binary vectors. Using this method we checked numerically (for even $L\leq 50$) that the ground state degeneracy of Eq. on an $L \times L$ torus is $2^{6}$, which is the correct degeneracy for $\mathbb{Z}_2^3$ topological order.
One such ground state is constructed in the string-net picture by starting from the state ${|\lbrace X = +1 \rbrace\rangle}$ in which all spins are in the $X=+1$ eigenstate. Obviously this satisfies all the $a=1$ constraints but not the $b_{zi}=1$ constraints. One ground state ${|0\rangle}$ is formed as the superposition $${|0\rangle} = \sum_{\lbrace n_i({\mathbf{r}}) \rbrace \in \lbrace 0, 1 \rbrace^{2L^2}}\prod_{{\mathbf{r}},i} (b_{zi}({\mathbf{r}}))^{n_i({\mathbf{r}})}{|\lbrace X = +1 \rbrace\rangle}$$ That is, ${|0\rangle}$ is a superposition of all possible products of $b_{zi}$ applied to the spin-polarized state ${|\lbrace X = +1 \rbrace\rangle}$.
The other ground states are, as usual, created by acting on ${|0\rangle}$ with Wilson loop operators wrapping around the handles of the torus. To understand the ground states, it suffices to understand the excitations of the model, as the Wilson loops can be constructed from the string operators which create pairs of anyons.
Consider the state $Z_{ii}({\mathbf{r}}){|0\rangle}$; it is still an eigenstate of all the terms in the Hamiltonian, but since $Z_{ii}$ anticommutes with $a$, the eigenvalue of $a({\mathbf{r}}\pm \hat{i})$ is $-1$. That is, $Z_{ii}$ creates a pair of electric excitations separated by two sites in the $i$ direction. If, for the moment, we disregard the action of $Z_{xy}$, this motivates a guess that there are up to four topologically distinct single-electric-charge excitations in the model, one on each of the four sublattices of lattice constant 2. A pair of such excitations is created by a string of $Z_{ii}$ operators separated by two sites; three types of excitations are shown on the left-hand side of Fig. \[fig:2DExcitationPairs\] .
Considering only the action of the site operators $Z_{ii}$, it would seem that there are four topologically distinct electric excitations. However, $Z_{xy}$ anticommutes with the four $a$ operators which touch its plaquette. That is, the local action of $Z_{xy}$ converts the bound state of three charges on a single plaquette into a single-charge state on the fourth sublattice, shown in Fig. \[fig:ZxyLocalAction\]. Therefore there are only three independent electric anyons, but each one carries a local degree of freedom which is modified by $Z_{xy}$.
The story is similar in the magnetic sector; the independent excitations are slightly more complicated but produced similarly, and are shown on the right-hand side of Fig. \[fig:2DExcitationPairs\]. Again there are three distinct excitations, and it can be easily checked that $X_{xy}$ applied to the end of a string modifies local degrees of freedom.
It is straightforward to check from the string operators that, as labeled in Fig. \[fig:2DExcitationPairs\], $e_i$ and $m_i$ braid as $e$ and $m$ in the toric code and $e_i$ and $m_j$ braid trivially for $i \neq j$. Therefore, this model indeed has $\mathbb{Z}_2^3$ topological order.
### Condensation Transition From Large $h_p$
The local degree of freedom associated with the action of $X_{xy}$ or $Z_{xy}$ is important in that it allows for transitions to other nontrivial phases. To illustrate the point, we will show that at large $h_p$, two excitations which differ only by a local operator become topologically inequivalent, driving the model to $\mathbb{Z}_2^4$ order. Equivalently, starting at large $h_p$ and reducing it condenses the four-electric-charge bound state of $\mathbb{Z}_2^4$ topological order. A related mechanism will occur in several other models that we study.
We begin by simply finding the effective Hamiltonian at large $h_p$. At zeroth order, the low-energy subspace consists of any spin configuration on the sites and all the plaquette spins pinned to the eigenstate $X_{xy} = +1$. This leads to extensive ground state degeneracy, with low-energy states labeled entirely by the site spin configuration. This degeneracy is split in degenerate perturbation theory by $U, 1/g^2 >0 $. The lowest-order contributions are first-order in $U/h_p$ and fourth-order in $1/g^2h_p$; the effective Hamiltonian is $$H_{eff} = -\sum_{\text{sites}}{\left(}U\tilde{a} + K \tilde{b} {\right)}\label{eqn:2DScalarLargeHpHamiltonian}$$ where $\tilde{a}$ and $\tilde{b}$ are the operators shown in Fig. \[fig:2DDecoupledTC\] and $K \sim 1/(g^8h_p^3)$
We claim that this effective Hamiltonian is precisely *four* copies of the toric code. To see this, consider the sublattice of spins shown in Fig. \[fig:2DSublattices\]. If we interpret these spins as living on the links of a lattice of length 2, as indicated by the set of darkened bonds in Fig. \[fig:2DSublattices\], by inspection each $\tilde{a}$ acts on exactly one such sublattice. On that sublattice, it acts exactly as a toric code star operator. Likewise, each $\tilde{b}$ acts on a single sublattice as a toric code plaquette operator. Hence each of the four distinct sublattices is one copy of the toric code.
Notice that all the operators that create electric excitations at large $h_p$ also create electric excitations on the same sites at $h_p=0$. However, as we saw previously, at $h_p=0$ the bound state of three of those excitations is equivalent to the fourth *up to a local application of* $Z_{xy}$. By contrast, at large $h_p$, $Z_{xy}$ takes the system out of the low-energy subspace; accordingly, the process in which a bound state of three types of charge is fused into the fourth type of charge is no longer allowed. That is, at small $h_p$, this fourth type of electric charge is equivalent to the fusion of the other three types, but at large $h_p$ it is a topologically distinct excitation.
### Large-$h_s$ limit
Finally, we may ask about the topological order of the large $h_s$ limit. In this case, the electric sector is confined because electric strings have finite tension, and one may expect a trivial paramagnetic limit. We demonstrate this using degenerate perturbation theory.
The low-energy subspace consists of all states with the site spins in the $X_{ii} = +1$ eigenstate. The $a$ and $h_p$ terms both contribute at first order in perturbation theory; $a$ is a product of $X_{xy}$ around a plaquette of the dual lattice, and $h_p$ is a longitudinal magnetic field. To first order, the model is classical and fully gapped. Importantly, the $b_{zi}$ terms contribute in degenerate perturbation theory only at $L$th order in $1/(g^2h_s)$, with $L$ the linear system size (At this order, the string of $b_{zi}$ operators consists only of face spins and thus commutes with the $h_s$ terms). In this limit, $1/g^2$ needs to be larger than $U$ and $h_p$ by an amount exponentially large in the system size in order to have an effect comparable to the gap in the first-order model. Hence, in the thermodynamic limit the system is indeed a trivial paramagnet.
As can be seen from Fig. \[fig:2DExcitationPairs\], $C_4$ rotation symmetry rotates electric particles into each other. If this symmetry is preserved, then all of the electric particles should condense at the same time and $\mathbb{Z}_2^3$ transitions directly to a paramagnetic phase. Breaking rotational symmetry generally leads to intermediate phases.
$(1,0)$ Scalar Charge Theory {#scalar-charge-theory-1}
----------------------------
Recall that the $(1,0)$ scalar charge theory has no (nontrivial) plaquette degrees of freedom. The Higgsed Hamiltonian can be checked straightforwardly to be equal to the effective Hamiltonian Eq. of the large-$h_p$ limit of the Higgsed $(2r+1,2s+1)$ scalar charge theory, with $K$ replaced by $1/g^2$ (see Fig. \[fig:2DDecoupledTC\] for operator definitions). This Higgsed theory is therefore exactly four decoupled copies of the $\mathbb{Z}_2$ toric code. It confines at large $h_s$.
Vector Charge Higgs in $d=2$ {#sec:HiggsOtherd2}
============================
The Higgsing procedure for the vector charge theories has only minor differences from the scalar charge theories. We will comment on those differences, then discuss the theories that produce nontrivial Higgs phases; see Appendix \[app:trivialHiggs\] for discussion of trivial Higgs phases. All the Higgs phases are summarized in Table \[tab:HiggsPhases\].
Changes to the Higgsing Procedure {#subsec:vectorHiggsChanges}
---------------------------------
As in the scalar charge theories, the matter field can still be gauged away, and the operator identifications all go through; if $m$ ($n$) is even, then the site (plaquette) spins drop out of Gauss’ Laws and the plaquette (site) spins drop out of the magnetic field. In $d=2$, by the same arguments as for the scalar charge theories, the Higgsed $(m,n)$ vector charge theory for nonzero $m$ and $n$ is determined entirely by the parity of $m$ and $n$ (or, for $p>2$, the values of $m$ and $n$ modulo $p$). There are therefore five total theories to consider. This changes slightly in $d=3$, as will be discussed in Sec. \[sec:oddOddVector\_d3\].
$(2r,2s+1)$ Vector Charge
-------------------------
Note that this case includes the $(2,1)$ theory, which is the theory with continuous rotational invariance.
The Higgsed model has Hamiltonian $$\begin{aligned}
H = -\frac{1}{g^2}&\sum_{\text{sites}} b - U \sum_i \sum_{i-\text{links}}a_i \nonumber \\
&- h_s \sum_{\text{sites},i}X_{ii} - h_p \sum_{\text{plaquettes}}X_{xy}
\label{eqn:vectorA2DHam}\end{aligned}$$ where the forms of the operators $a_i$ and $b$ are shown in Fig. \[fig:2DVectorAHam\]. The site and plaquette spins are decoupled.
By inspection the plaquette sector of this model is the classical Ising model in a magnetic field, which has a unique ground state for all $h_p \neq 0$.
As for the site sector, for $h_s \ll 1/g^2$, i.e. in the Higgs phase, we can perform degenerate perturbation theory. The $h_s$ term contributes at fourth order in $1/g^2h_s$ and produces the same model as the large-$h_p$ limit of the $(2r+1,2s+1)$ scalar charge model Eq. (compare $b$ in Fig. \[fig:2DVectorAHam\] to $\tilde{b}$ in Fig. \[fig:2DDecoupledTC\]). Its topological order is four copies of the toric code, i.e. $\mathbb{Z}_2^4$ lattice gauge theory. At large $h_s$, we simply have a trivial paramagnet. The phase diagram is shown in Fig. \[fig:PhaseDiagram2DVectorA\]; the direct transition to a paramagnet is protected by $C_4$ rotation symmetry, which permutes the decoupled copies of toric code.
$(2r+1,2s+1)$ Vector Charge
---------------------------
The Hamiltonian for the $(2r+1,2s+1)$ vector charge theory takes the same form as the Hamiltonian Eq. for the $(2r,2s+1)$ theory, but $a_i$ and $b$ take different forms because of the difference in factors of 2 in Gauss’ Law and the magnetic field. Their forms are shown in Fig. \[fig:VectorEHam2D\].
At $h_s = h_p=0$, the Higgsed $(2r+1,2s+1)$ vector charge theory is equivalent to the model obtained in the Higgsed $(2r+1,2s+1)$ scalar charge theory, as can be seen by comparing Figs. \[fig:VectorEHam2D\] and \[fig:2DScalarHam\] and performing a global spin rotation. The Higgs phase of this model therefore also has $\mathbb{Z}_2^3$ topological order.
However, the “electric field" terms $hX_{ij}$ are not dual to those in the $(2r+1,2s+1)$ scalar charge theory, so their effects should be studied as well. Since each magnetic excitation is created by an $X_{ii}$ operator, large $h_s$ should condense the entire magnetic sector, confining the electric sector and leading to a trivial paramagnet. In degenerate perturbation theory, this arises from the fact that $X_{xy}$ and the $a_i$ contribute at first order (the site spins simply drop out of the $a_i$) and $b$ only contributes at order $L^2$ in $1/g^2h_s$, where $L$ is the linear system size. Therefore, up to exponentially small corrections in the system size, the first-order effective Hamiltonian, which describes a classical paramagnet, describes the system accurately.
At large $h_p$, the story is similar; the $X_{ii}$ and $a_i$ contribute at first order in perturbation theory and describe a classical paramagnet of the site spins, while $b$ contributes at $L$th order and can be neglected.
The phase diagram is summarized in Fig. \[fig:PhaseDiagram2DVectorE\]. Much like the Higgsed $(2r+1,2s+1)$ scalar charge theory, the direct transition to a paramagnet is protected by the $C_4$ rotational symmetry of the square lattice, which permutes magnetic particles.
$(2r,2s+1)$ Scalar Charge Higgs in $d = 3$ and Fracton Order
============================================================
We now turn to the $d=3$ models. For the scalar charge models, the Higgsing procedures are all identical to the $d=2$ case, as is the argument that the resulting model depends only on the parity of $m$ and $n$. Accordingly, we will simply state the resulting Hamiltonian and then analyze the phase diagram.
We begin by studying the models which have fractonic Higgs phases.
$(0,1)$ Scalar Charge
---------------------
The Hamiltonian of the $d=3$ Higgsed $(0,1)$ scalar charge model is $$H = -U\sum_{\text{sites}} a - \frac{1}{g^2}\sum_{\text{cubes},i} b_{ii} - h_f \sum_{\text{faces},i<j} X_{ij}
\label{eqn:XCube}$$ where $\tilde{a}$ and $\tilde{b}_{ii}$, shown in Fig. \[fig:XCube\], are just $a$ and $b_{ii}$ with the site operators frozen out. This is precisely the X-cube model[@VijayGaugedSubsystem] on the dual cubic lattice. Accordingly, this phase has fracton order at small $h_f$. At large $h_f$, the system becomes paramagnetic.
![Terms in the Hamiltonian Eq. for the Higgsed $(0,1)$ scalar charge model in $d=3$. $a$ is a product of Pauli $X$ operators on the twelve green spins, while the $b_{ii}$ are products of four Pauli $Z$ operators on the orange spins.[]{data-label="fig:XCube"}](XCube.pdf){width="7cm"}
The $(0,1)$ scalar charge model is the natural $U(1)$ generalization of the X-cube model, so the relationship between them is not surprising. However, it can be checked that the $(0,1)$ scalar charge model has pointlike magnetic monopoles and may therefore suffer from confinement; the relevance of the monopole creation operators must be studied in more detail. The X-cube phase, on the other hand, is known to be stable.
$(2r+2,2s+1)$ Scalar Charge
---------------------------
The $(2r+2,2s+1)$ scalar charge model (for $r,s \geq 0$) must be considered separately from the $(0,1)$ theory because of the presence of the $A_{ii}$ degrees of freedom in the $U(1)$ theory.
The Hamiltonian for the $d=3$ Higgsed $(2r+2,2s+1)$ scalar charge theory is $$\begin{aligned}
H = -U &\sum_{\text{sites}} a - \frac{1}{g^2}\sum_{\text{cubes}, i}b_{ii} - \frac{1}{g^2}\sum_{\text{links},i,j}b_{ij}\nonumber \\
& - h_s \sum_{\text{sites},i}X_{ii} - h_f \sum_{\text{faces},i<j}X_{ij}
\label{eqn:3DScalarHam}\end{aligned}$$ with the operators $a$ and $b_{ij}$ defined in Fig. \[fig:ScalarHamEvenOdd3D\].
![Terms in the Hamiltonian for the Higgsed $(2r,2s+1)$ scalar charge theory in $d=3$. The $a$ operators are associated with the sites and are products of twelve $X_{ij}$ operators acting on the green spins. The $b_{ij}$ operators are products of two (off-diagonal terms, associated with links) or four (diagonal terms, associated with cubes) Pauli $Z_{ij}$ operators acting on the orange spins.[]{data-label="fig:ScalarHamEvenOdd3D"}](ScalarHamEvenOdd_3D.pdf){width="6cm"}
By inspection, the face spins form the X-cube model Eq. (c.f. Fig. \[fig:XCube\]), and the site spins form decoupled, interpenetrating planes of transverse field 2+1-dimensional Ising models. The various decouplings in this model are, of course, fine-tuned. In particular, there are operators in the $U(1)$ theory which are irrelevant (in the renormalization group sense) in the photon phase but which will couple the site spins to the face spins (as well coupling the planes of Ising model to each other). However, the topological stability of the X-cube model ensures that these operators do not destroy the fracton order.
The $2+1$-dimensional Ising order of the site degrees of freedom, on the other hand, is fine-tuned, with $3L$ degenerate ground states at $h_s=0$. The fate of this degeneracy is non-universal and depends on what operators are added to the theory.
We have found an infinite class of $U(1)$ scalar charge models with a fractonic Higgs phase. Note that among the scalar charge theories, the only theory with continuous rotational invariance is the $(1,1)$ theory, which is not in the class we consider here. We will find that this is a general feature of the models we consider - X-cube order is never obtained by Higgsing a model with continuous rotational symmetry.
$(2r+1,2s+1)$ Scalar Charge Higgs in $d=3$ {#sec:oddOddScalar_d3}
==========================================
For the next two sections, we will focus on the $(2r+1,2s+1)$ scalar and vector charge models. Their Higgs phases are described by interesting models for multiple copies of $d=3$ toric code topological order, and they have rich phase diagrams.
The $\mathbb{Z}_2$ Higgs phase of the $(2r+1,2s+1)$ scalar charge model produces an interesting model with $\mathbb{Z}_2^4$ topological order, and upon breaking rotational invariance it can be driven to a fracton phase – the X-cube model. Its phase diagram is shown schematically in Fig. \[fig:PhaseDiagram3DScalarA\].
The Hamiltonian for the $d=3$ Higgsed $(2r+1,2s+1)$ scalar charge theory is the same form Eq. as for the $(2r+2,2s+1)$ models we considered in the previous section, but the operators $a$ and $b_{ij}$ are defined differently - they are shown in Fig. \[fig:3DScalarHam\]. The operator $a$ is an 18-spin operator, while all the $b_{ij}$ are 4-spin operators. We now analyze this model in various limits.
Higgs Phase
-----------
We show explicitly that at $h_f=h_s=0$, this model realizes $\mathbb{Z}_2^4$ topological order, i.e. it is equivalent to four copies of the $\mathbb{Z}_2$ toric code.
At $h_f=h_s=0$ we again have a commuting projector model and the ground state degeneracy can be computed as in the $d=2$ models; we find that it is $2^{12}$ on the $L\times L \times L$ $3-$torus for $L$ even. This is consistent with $\mathbb{Z}_2^4$ topological order.
As usual, one ground state is obtained by starting from the state with all spins in the $X_{ij}=+1$ eigenstate and superposing over all applications of $b_{ij}$ operators on this reference state. The other ground states are obtained by applying Wilson loops around handles of the 3-torus, so we turn to the excitations.
The behavior of this model in the electric sector is similar to the $d=2$ $(2r+1,2s+1)$ scalar charge theory; $Z_{ii}$ creates a pair of excitations which can hop by two sites in any direction. Temporarily ignoring the face spins, there are eight obvious electric excitations, created by acting with $Z_{ii}$ on any of the eight sublattices with lattice constant 2. Four are shown on the left side of Fig. \[fig:ExcitationPairs3D\].
{width="9cm"}
As in the $d=2$ $(2r+1,2s+1)$ scalar charge theory, these excitations are not all independent. The local action of $Z_{ij}$ is the same as in $d=2$ (see Fig. \[fig:ZxyLocalAction\]); it locally turns any three excitations on the sites of a single face into a single excitation on the fourth site of that face. Straightforward counting shows that this leaves four independent electric excitations, shown in Fig. \[fig:ExcitationPairs3D\], and that $Z_{ij}$ toggles local degrees of freedom for these excitations.
As expected, there are four independent magnetic string excitations, created by membrane operators, shown in Fig. \[fig:ExcitationPairs3D\]. Using the labeling in Fig. \[fig:ExcitationPairs3D\], it can be checked by inspection that a Wilson loop for $e_i$ anticommutes with the membrane operator creating an $m_i$ string and commutes with the operators creating an $m_j$ string for $i \neq j$. Accordingly the topological order is $\mathbb{Z}_2^4$.
Large $h_f$ Limit {#subsec:scalarToEightCopies}
-----------------
The large $h_f$ limit is treated analogously to the $d=2$ case; the face spins are pinned to the $X_{ij} = +1$ eigenstate. The operator $Z_{ij}$ which transmutes a three-charge bound state of the $h_f=0$ model to the fourth charge on the face now leaves the low-energy subspace; accordingly, the fourth charge on a face is now an independent excitation from the other three charges on that face. Therefore, all the eight electric excitations discussed previously become topologically distinct, and the system should have $\mathbb{Z}_2^8$ topological order.
This conclusion can be checked explicitly by computing the effective Hamiltonian with degenerate perturbation theory. The electric terms contribute at first order in $U/h_f$ and the off-diagonal magnetic terms contribute at fourth order in $1/g^2h_f$; the effective Hamiltonian is $$H_{eff} = - \sum_{\text{sites}}{\left(}U \tilde{a} + K \sum_i \tilde{b}_i{\right)}\label{eqn:EightToricCodesHam}$$ where $\tilde{a}$ and $\tilde{b}_i$ are shown in Fig. \[fig:3DDecoupledTC\] and $K \sim 1/g^8h_f^3$. (The diagonal magnetic terms only contribute at higher order in $1/g^2h_f$.)
Similarly to the $d=2$ case, this model is precisely eight copies of the $d=3$ toric code. The proof is similar to $d=2$; consider only the spins shown in Fig. \[fig:3DSublattice\]. There are eight such sublattices of spins (translate the sublattice by one unit in any set of lattice basis directions), and each term in the Hamiltonian acts on only a single sublattice; the $a$ terms as a toric code star operator and the $b_i$ terms as a toric code plaquette operator. Collecting all the terms which act on each of the eight possible sublattices produces exactly one copy of the toric code for each of the eight sublattices.
The transition from large $h_f$ to small $h_f$ is therefore a condensation transition of four-charge bound states, where the four charges live on a single face of the lattice. Equivalently, as $h_f$ increases, a local degree of freedom gets frozen out, causing excitations which differed only by that local degree of freedom to become topologically distinct.
In the absence of any symmetries, a direct transition from $\mathbb{Z}_2^8$ to $\mathbb{Z}_2^4$ would either be first order, or it would correspond to a multicritical point. Here the lattice symmetries, which permute the different electric charges, can protect the critical point, leaving only one relevant perturbation that tunes between the $\mathbb{Z}_2^4$ and $\mathbb{Z}_2^8$ phases. Analogous statements hold also for the transition to the trivial paramagnetic phase.
As an aside, the Hamiltonian for the Higgsed $(1,0)$ scalar charge model happens to be exactly Eq. (with operator definitions in Fig. \[fig:3DDecoupledTC\]). This model therefore also has $\mathbb{Z}_2^8$ topological order and confines at large $h_s$.
Large $h_s$ Limit - X-Cube Phase
--------------------------------
The large $h_s$ limit can also be treated in degenerate perturbation theory, much like the large $h_f$ limit. In $d=2$, this limit produced a trivial paramagnet. In $d=3$, however, the resulting model is very different. The $a$ operator again contributes at first order (as does $h_f$), but so do $b_{ii}$; these operators did not exist in $d=2$. The remaining spins live on the faces of the lattice, and the effective Hamiltonian is the X-cube model Eq. .
Remarkably, then, as $h_s$ increases this model may have a direct transition from $\mathbb{Z}_2^4$ conventional topological order to X-cube order. Such a transition is of considerable interest; we defer it to future work.
Recall that the $(1,1)$ model, which Higgses to the model under consideration, has continuous rotational invariance. In taking $h_s$ large with $h_f$ small, we have broken the continuous rotational invariance down to the symmetry of the cubic lattice. Thus we see that in this model, the appearance of the X-cube model required terms whose continuum limit breaks continuous rotational symmetry down to discrete rotational symmetry of the cubic lattice.
$(2r+1,2s+1)$ Vector Charge Higgs in $d=3$ {#sec:oddOddVector_d3}
==========================================
Before discussing the $(2r+1,2s+1)$ vector charge theory, we briefly comment on a small modification of the results of the Higgs procedure in $d=3$ for vector charge theories. The factor of $2/(3-(-1)^m)$ in the off-diagonal components of $B_{ij}$ in Eq. leads to a difference between $m \equiv 0$ mod 4 and $m \equiv 2$ mod 4. Specifically, if $m = 2m_0$ with $m_0$ an integer, then for $i \neq j$
$$B_{ij} = \sum_{k \neq i,j} \left[ m_0 {\left(}\Delta_i \Delta_k A_{jk} + \Delta_j \Delta_k A_{ik} - \Delta_k^2 A_{ij}{\right)}- n \Delta_i \Delta_j A_{kk}\right]$$
Upon Higgsing, this operator becomes, in the low-energy subspace, a product of $Z_{ij} = e^{iA_{ij}}$. In particular, the face spins $Z_{ij}$ for $i \neq j$ always appear raised to the power of $m_0$. If $m_0$ is even, then the face spins will all drop out. If $m_0$ is odd, then the face spins are present. Therefore the Higgsed theories with different parities of $m_0 = m/2$ are distinct.
The Hamiltonian for the $d=3$ $(2r+1,2s+1)$ vector charge model is
$$H = -U \sum_i \sum_{i-\text{links}} a_i - \frac{1}{g^2}\sum_{\text{sites}, i}b_{ii} - \frac{1}{g^2}\sum_{\text{faces},i<j}b_{ij} - h_s \sum_{\text{sites},i}X_{ii} - h_f \sum_{\text{faces},i<j}X_{ij}
\label{eqn:OddOddVectorHam3D}$$
The forms of $a_i$ and $b_{ij}$ are shown in Fig. \[fig:3DVectorOddOddHam\], and its phase diagram (a summary of this section) is given in Fig. \[fig:PhaseDiagram3DVectorOddOdd\].
Higgs Phase
-----------
The Higgs phase is, as usual, understood from the $h_f=h_s=0$ commuting projector model. Its ground state degeneracy is computed to be $2^{22}$. The local operator $c = \prod_{\text{cube}} Z_{ij}$, for any elementary cube, commutes with the Hamiltonian. There is only one such independent operator (the others are generated by multiplying $c$ by various $b_{ij}$ for $i \neq j$). The simplest operator which commutes with $H$ but anticommutes with $c$ is the product $C$ of all $X_{xy}$ in planes with even values of $z$, all $X_{xz}$ in planes with even values of $y$, and all $X_{yz}$ in planes with even values of $x$. We can therefore think of $C$ as generating an Ising ($\mathbb{Z}_2$) global symmetry, but such a symmetry is spontaneously broken in 3+1D. After accounting for the symmetry enrichment, the topological degeneracy is at most $2^{21}$; we claim that the remaining degeneracy is topological, and that the topological order is $\mathbb{Z}_2^7$.
It is simplest to analyze the magnetic sector. Breaking the site degrees of freedom into eight sublattices as usual, the action of the $b_{ii}$ on the site degrees of freedom looks like toric code plaquette operators on lattice constant 2 sublattices. Accordingly, membranes of $X_{ii}$ operators living on the sublattices create eight types of simple magnetic strings, an example of which is shown in Fig. \[fig:VectorESingleMagnetic3D\]. As usual, though, we must account for action of $X_{ij}$. Applying a string of $X_{ij}$ creates the set of excitations shown in Fig. \[fig:VectorEEightMagnetic3D\]. It is straightforward to check that this pattern of excitations is precisely the bound state of all eight simple strings. Therefore, there are only seven independent magnetic strings, as a bound state of seven simple strings may be converted into the eighth simple string using the aforementioned string operator. It is straightforward but laborious to show that there are also seven independent electric excitations which braid appropriately for $\mathbb{Z}_2^7$ topological order; two examples are shown in Fig. \[fig:ElectricStringsVectorE3D\].
![Two types of string operators ($Z$ operators acting on orange spins) and their associated electric particles (green links) in the Higgsed $d=3$ $(2r+1,2s+1)$ vector charge model. Eight strings total are obtained by shifting these operators by zero or one unit in one or both lattice directions perpendicular to the green links, but only seven of are topologically distinct because bound states of eight excitations can be destroyed with a local operator.[]{data-label="fig:ElectricStringsVectorE3D"}](ElectricStringsVectorE3D.pdf){width="3.5cm"}
Large-field limits
------------------
### Large $h_s$
At $h_s=0$, every magnetic string excitation in the $\mathbb{Z}_2^7$ topological order can be created exclusively with $X_{ii}$. When $h_s$ is taken large, one expects all the magnetic strings to condense and the electric particles to confine. Remarkably, this process simultaneously causes a point-like magnetic excitation to *deconfine*, and the large-$h_s$ model has $\mathbb{Z}_2$ topological order.
To understand what has happened from an intuitive confinement picture, consider the action of a string of $X_{ij}$ operators on the ground state at $h_s=h_f=0$, shown in Fig. \[fig:OpenFaceStringVectorE3D\]. This string has tension because it anticommutes with magnetic site terms $b_{ii}$ along the string, shown as orange spheres in the figure, and also anticommutes with a set of off-diagonal magnetic terms at each end of the string, shown as the blue faces in the figure. At large $h_s$, the site excitations (rather, strings of them) are condensed; this causes the string to lose tension and the collection of magnetic face excitations at each end of the string becomes a deconfined anyon.
![Open string of $X_{xy}$ acting on the green faces and its associated excitations in the Higgs phase of the $d=3$ vector charge $(2r+1,2s+1)$ model. The blue faces have magnetic excitations; off-diagonal $b_{ij}$ terms have eigenvalue $-1$ on those phases. The orange spheres are site excitations ($b_{ii}$ which have eigenvalue $-1$) and give the string tension; at large $h_s$, they condense, deconfining the collection of blue face excitations.[]{data-label="fig:OpenFaceStringVectorE3D"}](OpenFaceStringVectorE3D.pdf){width="4cm"}
More explicitly, we can perform the usual degenerate perturbation theory. In this limit, the site spins freeze out of the $a_i$ and the off-diagonal magnetic terms contribute at first order. The diagonal magnetic terms contribute at sixth order, but they generate operators equal to products of four off-diagonal terms and thus do not affect the analysis. The effective Hamiltonian is $$H_{eff} = -U\sum_{i \text{ links}}\tilde{a}_i - \frac{1}{g^2}\sum_{\text{faces},i < j}b_{ij} - h_f\sum_{\text{faces}}X_{ij}
\label{eqn:effective3DTC}$$ where $\tilde{a}_i$ are $a_i$ with the site operators removed. This is a commuting projector model at $h_f=0$.
To understand the $h_f=0$ model, note that the $\mathbb{Z}_2$ symmetry generator $C$ still commutes with the Hamiltonian. To make the discussion clearer, let us imagine breaking this symmetry explicitly by adding $-\lambda \sum_{\text{cubes}} c$ to the Hamiltonian for $\lambda$ small. Noting that $a_i$ and $c$ are the plaquette and star operators of the $d=3$ toric code on the dual lattice, we see that this symmetry-broken model is exactly the $d=3$ toric code. At $\lambda = 0$, then, the system is in a $\mathbb{Z}_2$ toric code phase with additional spontaneous breaking of an Ising symmetry. The topological order must therefore be stable to a small but nonzero $h_f$.
### Large $h_f$
At large $h_f$, the system has a unique, approximately classical ground state. As usual, we check this by degenerate perturbation theory. The $a_i$ and $h_s$ contribute at first order. If $1/g^2=0$, then this first-order effective model is immediately seen to be classical and have a unique ground state.
The lowest-order contribution of the $b_{ij}$ is at tenth order in $1/g^2h_f$ (six $b_{ii}$ form a closed 2x2x2 cube and four $b_{ij}$ for $i \neq j$ remove the rest of the face spins), but it just contributes an overall constant. The lowest-order nontrivial contribution occurs at $L$th order in degenerate perturbation theory. Therefore, in the thermodynamic limit, with any $1/g^2 \ll h_f$, the classical effective model remains valid.
$(4r+2, 2s+1)$ Vector Charge Higgs in $d=3$ {#sec:evenOddVector_d3}
===========================================
Recall that the $(2,1)$ vector charge model has continuous rotational invariance. In this section we analyze the Higgs mechanism for the class of theories that include said $(2, 1)$ vector charge model.
The Hamiltonian for the $d=3$ Higgsed $(4r+2,2s+1)$ vector charge theory is
$$H = -U \sum_i \sum_{i-\text{links}} a_i - \frac{1}{g^2}\sum_{\text{sites}, i}b_{ii} - \frac{1}{g^2}\sum_{\text{faces},i<j}b_{ij} - h_s \sum_{\text{sites},i}X_{ii} - h_f \sum_{\text{faces},i<j}X_{ij}
\label{eqn:3DVectorAHam}$$
with the operators shown in Fig. \[fig:3DVectorAHam\].
![Terms in the $d=3$ Higgsed $(4r+2,2s+1)$ vector charge theory Hamiltonian. The $a_i$ are associated with the center bonds and are the product of Pauli $X$ operators on the four green spins. The diagonal (off-diagonal) $b_{ij}$ operators are associated with the center site (face) and are products of Pauli $Z$ operators on the four (fourteen) orange spins.[]{data-label="fig:3DVectorAHam"}](3DVectorAHam.pdf){width="7cm"}
We analyze the Higgs phase by taking $h_s=h_f=0$. In this limit, it is obvious that $Z_{ii}$ on every site commutes with the Hamiltonian. The simplest product of site $X$ operators which commutes with the Hamiltonian is the product of all $X_{ii}$ operators in a plane perpendicular to the $i$ direction; these membrane operators anticommute with the $Z_{ii}$. Hence this model has a non-topological degeneracy which scales as the linear system size $L$. Adding a small $h_f$ obviously does not split this degeneracy, and a small $h_s$ only contributes at order $L^2$ in degenerate perturbation theory, splitting the ground state degeneracy in an exponentially small fashion.
This limit of the model is therefore fine-tuned, as adding $-\gamma \sum_i Z_{ii}$ to the Hamiltonian with $\gamma$ arbitrarily small splits the degeneracy associated with the site degrees of freedom. Although $\sum_i Z_{ii}$ does not obviously appear from Higgsing a gauge-invariant operator, it commutes with the residual $\mathbb{Z}_2$ gauge symmetry generated by the $a_i$. On general field-theoretic grounds, a nonzero $\gamma$ should therefore be generated.
More generally, any operator involving $X_{ii}$ splits the non-topological degeneracy by a energy exponentially small in the system size so we can ignore all such operators. Any operator involving only the $Z_{ii}$ that splits the non-topological degeneracy simply freezes the site spins to some particular product state configuration, although the particular configuration depends on what operators we add. As such, we are justified in calling the topological order of the resulting phase the “Higgs phase" so long as the phase associated with the plaquette sector is gapped.
If the site spins are frozen to $Z_{ii}$ eigenstates, then it is straightforward to see that the effective model becomes that of the large-$h_s$ limit of the Higgsed $(2r+1,2s+1)$ vector charge model Eq. . That model has $\mathbb{Z}_2$ topological order, so the Higgs phase of the present $(4r+2,2s+1)$ model also has $\mathbb{Z}_2$ topological order.
Several other theories behave similarly to the $(4r+2,2s+1)$ vector charge model discussed here, except in those cases splitting the non-topological degeneracy leads to topologically trivial phases. We discuss them in Appendix \[subsec:illDefined\].
Discussion
==========
We have defined a large class of rank-2 $U(1)$ gauge theories, which we refer to as the $(m,n)$ scalar and vector charge theories. These are invariant under the discrete rotational symmetries of the square (cubic) lattices in $d = 2$ ($d = 3$). The previously studied rank-2 theories whose continuum limit possesses continuous rotational symmetry correspond to the $(1,1)$ scalar and $(2,1)$ vector charge theories. Remarkably, we find most of the $(m,n)$ scalar and vector theories correspond to stable gapless phases of matter in $d = 3$, and to critical or multi-critical points in $d = 2$, and thus form an interesting class of field theories that are worthy of further study. The matter field in these theories is constrained to move along subdimensional manifolds, in a manner dictated by $(m,n)$ and whether it is a scalar or vector charge theory.
Breaking the $U(1)$ gauge symmetry down to a discrete subgroup, such as $\mathbb{Z}_2$, gives rise to a large class of exactly solvable models. We find that in most cases, the $\mathbb{Z}_2$ Higgs phases describe either topologically trivial phases of matter, or possess conventional topological order and correspond to multiple copies of a conventional $\mathbb{Z}_2$ toric code phase. Nevertheless, the exactly solvable models that arise are new and reminiscent of the color code models[@ColorCodes]; it might thus be interesting to consider these models from the perspective of quantum error correction.
Our results provide a number of general lessons regarding fractons and higher rank gauge theories:
First, we have expanded the set of gapless higher rank field theories, whose matter fields have restricted, subdimensional dynamics. In $d=2$, while many or all of these theories may not correspond to stable phases of matter, it appears they can at least correspond to (multi)-critical points. However the Higgs phases of all these theories possess either trivial or conventional topological order, which confirms the expectation that gapped fracton phases cannot exist in $d = 2$.
Moreover, we found that the Higgs phases of models with continuous rotational symmetry, such as the $(1,1)$ scalar and $(2,1)$ vector charge theories do not give rise to gapped fracton phases. However the $\mathbb{Z}_2$ Higgs phase of the $(1,1)$ scalar charge theory in $d = 3$ does admit a transition to X-cube fracton order in the limit of a strong “Zeeman” field that breaks the continuous rotational symmetry of the continuum theory down to a discrete subgroup. The existence of a possible transition between the conventional $\mathbb{Z}_2^4$ topological order and X-cube fracton order in $d= 3$ may be of a qualitatively new type of quantum phase transition that requires further study.
We also found that the X-cube fracton order can emerge as the Higgs phase of the $(2r, 2s+1)$ scalar charge theories in $d = 3$. It is not clear whether the $(0,1)$ scalar charge theory corresponds to a stable gapless phase of matter. However, it appears that the $(2r, 2s+1)$ theories with $r > 0$ do correspond to stable gapless phases. Remarkably, this suggests the existence of stable gapless higher rank gauge theories whose Higgs phases yield fracton order.
Notably, the above results suggest that X-cube fracton order cannot arise from a theory that is invariant under continuous rotations.
Our results demonstrate that the gapped X-cube fracton order can indeed be described within the framework of quantum field theory, albeit with a novel type of gauge theory. In particular, we can consider a continuum version of the $(m,n)$ scalar charge theories, coupled to a charge $p$ complex scalar field, whose condensation breaks the $U(1)$ rank-2 gauge symmetry down to $\mathbb{Z}_p$. It would be interesting to understand the relation between this continuum Higgs theory and an alternative continuum field theory description of the X-cube phase, presented in Ref. .
The considerations presented here raise the question of whether all gapped fracton phases can emerge as Higgs phases of stable gapless higher rank gauge theories. For example, there is a natural generalization of the higher rank scalar and vector charge theories presented so far, defined by the Gauss Law terms: $$\begin{aligned}
\sum_{\{i\},\{\alpha\}} M_{i_1 \cdots i_n}^{a; \alpha_1 \cdots \alpha_k} \partial_{i_1} \partial_{i_2} \cdots \partial_{i_n} E_{\alpha_1, \alpha_2, \cdots, \alpha_k}^a = \rho_a ,\end{aligned}$$ where $a$ labels distinct flavors of charges and the constraint is on a rank-k electric field $E_{\alpha_1, \alpha_2, \cdots, \alpha_k}^a $. Any discrete or continuous rotational invariance imposes constraints on the types of tensors $M$ that can appear in the above. In fact, although for brevity we do not consider them in this paper, cubic symmetry allows another term $\sum_{i\neq j} \Delta_i^2 E_{jj}$ in Gauss’ Law. A natural question is whether Higgs phases of such generalized higher rank gauge theories can describe all consistent types of subdimensional dynamics for particles that arise in gapped fracton phases. In particular it would be interesting to understand whether the Chamon model[@ChamonGlass] and Haah’s code[@HaahsCode] can possibly arise from such a construction.
*Note:* During the completion of this paper, we learned of closely related work by Ma *et al*[@MaHiggs].
DB is supported by the Laboratory for Physical Sciences and Microsoft. MB is supported by NSF CAREER (DMR-1753240) and JQI-PFC-UMD.
Review of the Standard Higgs Mechanism {#app:HiggsReview}
======================================
We briefly define the lattice theory of a compact (rank-1) $U(1)$ lattice gauge field coupled to charge-$p$ bosonic matter. We place canonically conjugate rotor variables $\theta$ and $L$ on the sites of a cubic lattice; this will be our matter field. Define $\theta \sim \theta +2\pi$ so that $L$ has integer eigenvalues and $e^{i\theta}$ is a raising operator for $L$. Next, orient the links of the lattice and place canonically conjugate rotor variables $A_i$ and $E_i$ on the links, where $i$ labels the direction of the link. Here $$[A_i({\mathbf{r}}), E_j({\mathbf{r'}})] = i\delta_{ij}\delta({\mathbf{r}}-{\mathbf{r}}')$$ Compactness means we identify $A_i \sim A_i+2\pi$, which quantizes $E_i$ to integer eigenvalues. The standard $U(1)$ gauge transformation is $A_i \rightarrow A_i - \Delta_i \alpha$, leading to the Gauss’ Law operator $$G(E) = \sum_i \Delta_i E_i$$
The gauge transformation leads to a Hamiltonian of the usual form given by Eq. . The only difference is that $\tilde{h}_s E_{ii}^2$ and $\tilde{h}_p E_{ij}^2$ are replaced by a single term $\tilde{h}E_i^2$ on the links of the lattice. Note that we are enforcing Gauss’ Law energetically on a local rotor model rather than as a strict constraint on the Hilbert space. The magnetic field takes its usual form $$B_i = \sum_{j,k}\epsilon_{ijk}\Delta_j A_k$$ for $i=z$ in $d=2$ and $i=x,y,z$ in $d=3$. The Higgs coupling is $$H_{Higgs} = \sum_{{\mathbf{r}}}\frac{L({\mathbf{r}})^2}{2M} - V\sum_{i,\text{sites}} \cos(\Delta_i \theta + p A_i)$$
We take $V \rightarrow \infty$ and restrict our attention to the low-energy subspace from now on.
In said subspace, we require $$\Delta_i \theta + p A_i = 2\pi n$$ for $n \in \mathbb{Z}$. We may choose a gauge where $\theta = 0$ (mod $2\pi$) at every point; in this gauge, $$A_i = \frac{2\pi}{p} n$$ for $n \in \mathbb{Z}$. For simplicity, we specialize to $p=2$; the larger $p$ case is a straightforward generalization. Then $e^{iA_i} = \pm 1$ on each link. Furthermore, since $e^{iA_i}$ is a raising operator for $E_i$, its action flips the eigenvalue of $(-1)^{E_i}$. Therefore, the spectrum and the commutation relations of $e^{iA_i}$ and $(-1)^{E_i}$ in the low-energy subspace are reproduced by the identification $e^{iA_i} = Z$ and $(-1)^{E_i} = X$ where $X,Z$ are the usual Pauli matrices.
In this language, $\cos(B_i)$ is exactly the operator $\prod_{\square} Z$, where the product is around a plaquette perpendicular to the $i$ direction. The $E_i^2$ term of $H_{Maxwell}$ penalizes fluctuations in $A_i$; after Higgsing, this means we should be penalizing fluctuations in $Z$. Hence the Hamiltonian should have a term $-h X$, where $h \propto \tilde{h}$.
Finally, because charge is condensed, Gauss’ Law only a well-defined constraint modulo 2. In the constraint language (i.e. in the $\tilde{U} \rightarrow \infty$ limit) there is a strict constraint $$1 = (-1)^{\Delta_i E_i - 2L} = (-1)^{\Delta_i E_i} = \prod_{\text{star}}X$$ where the product is around the usual star operator. This is the $\mathbb{Z}_2$ gauge constraint. Taking $\tilde{U}$ finite gives an energetic penalty to states violating this condition. The Hamiltonian is therefore $$H = -U\sum_{\text{sites}} \prod_{\text{star}}X - \frac{1}{g^2}\sum_{\text{plaquettes}} \prod_{\square} Z - h \sum_{\text{links}} X$$ where $U \propto \tilde{U}$.
We have thus produced $\mathbb{Z}_2$ gauge theory by Higgsing the $U(1)$ lattice gauge theory. The generalization to $p>2$ is straightfoward and produces $\mathbb{Z}_p$ lattice gauge theory. The model with finite charge gap and $h=0$ is also known as the $d$-dimensional $\mathbb{Z}_p$ toric code, and has topological order; its ground state degeneracy is $p^d$ on the $d$-torus.
Theories With Trivial Higgs Phases {#app:trivialHiggs}
==================================
Several of the theories produce topologically trivial Higgs phases, which we study in this appendix. We use the word “trivial" to describe three different physical cases. First, the model can have a unique classical ground state (i.e. the Hamiltonian is a sum of trivial, commuting terms such that the exact ground state is a product state). Second, it can be mapped a transverse field quantum Ising model, such that the ground state spontaneously breaks a global symmetry. Third, the precise model for the Higgs phase that we study can have a sub-extensive ground state degeneracy, which must be split by additional perturbations; in these cases the perturbations drive the system to a topologically trivial phase, while the patterns of any possible global symmetry breaking will depend on the type of perturbations added.
We will examine each of these cases in turn.
Classical Ground States
-----------------------
Whenever the parent $U(1)$ theory does not admit a magnetic field, its Higgs phase has a classical ground state. To see how this occurs, consider (for example) the $(0,1)$ scalar charge theory in $d=2$. Its Higgsed Hamiltonian is $$H = -U \sum_{\text{sites}} a - h_p \sum_{\text{plaquettes}} X_{xy}$$ where $a$ is a four-spin product of $X_{xy}$ operators shown in Fig. \[fig:Scalar01d2\]. Since only $X$ operators appear in this theory, at all $h_s \neq 0$ the model has a unique, classical ground state consisting of all spins in the $X=+1$ eigenstate (assuming $U,h_p>0$).
![$a$ operator for the Higgsed $(0,1)$ scalar charge model in $d=2$.[]{data-label="fig:Scalar01d2"}](Scalar01d2.pdf){width="2cm"}
Clearly whenever a theory fails to admit a magnetic field, the Hamiltonian will only involve $X$ operators and will be a sum of Gauss’ Law operators and onsite $X_{ij}$ terms. In all such cases, the model is trivial in the sense that it is fully classical and has a unique ground state.
The other theories in which this occurs are the $(1,0)$ and $(0,1)$ vector charge theories in both $d=2$ and $d=3$.
Mappings to Ising Models
------------------------
Several Higgsed theories map onto standard models of spontaneous symmetry breaking. We now list each such theory.
### $d=2$ $(2r+1,2s)$ Scalar Charge
The Higgsed $(2r+1,2s)$ scalar charge Hamiltonian has the same form Eq. as the Higgsed $(2r+1,2s+1)$ scalar theory in $d=2$, but $a$ and $b_{zi}$ take new forms, shown in Fig. \[fig:2DScalarEHam\].
We see that the site and plaquette spins have decoupled. Both sectors are topologically trivial. The plaquette spin Hamiltonian is, by inspection, the transverse field Ising model on the dual lattice. The site spins are trivial; they are classical and have a unique ground state at all $h_s \neq 0$.
### $d=3$ $(2r+1,2s+2)$ Scalar Charge
The Hamiltonian of the $d=3$ $(2r+1,2s+2)$ scalar charge theory takes the same schematic form as the $d=3$ $(2r+1,2s+1)$ scalar charge theory Eq. but with different operators, shown in Fig. \[fig:ScalarEHam3D\].
![Terms in the Hamiltonian of the Higgsed $(2r+1,2s+2)$ scalar charge theory. The $a$ operators are associated with the sites and are products of six $X_{ii}$ operators acting on the green spins. The $b_{ij}$ operators are products of two (off-diagonal terms, associated with links) or four (diagonal terms, associated with cubes) Pauli $Z_{ij}$ operators acting on the orange spins.[]{data-label="fig:ScalarEHam3D"}](ScalarEHam_3D.pdf){width="7cm"}
The site and face spins decouple and the site spins form a classical model with a unique ground state at $h_s \neq 0$. The face spins have a 3D transverse field Ising model Hamiltonian with additional four-body terms (the $b_{ii}$) which maintain the Ising symmetry. At $h_p=0$, Ising ferromagnetic states are favored by all the $b_{ij}$; the four-body terms simply modify details of the excitations. This Higgs phase is therefore stable to small $h_p$.
Although the Higgs phase is topologically trivial, this model does have a topologically nontrivial phase in its phase diagram, in the regime $1/g_{ij}^2 \ll h_S \ll g_{ii}^2$, for $i \neq j$. To see this, we set $1/g_{ij}^2 = 0$ for $i \neq j$ and take $h_s \ll 1/g_{ii}^2$. The effect of $h_s$ is incorporated by degenerate perturbation theory; the resulting model is precisely the $d=3$ $\mathbb{Z}_2$ toric code Hamiltonian, which is stable to weak $1/g_{ij}^2 > 0$.
Theories With Non-Universal Trivial Higgs Phases {#subsec:illDefined}
------------------------------------------------
As discussed in Sec. \[sec:intuition\], our operational definition of “Higgs phase" is the regime $h_s,h_f \ll 1/g_{ij}^2$ for all $i,j$. As we saw in the $(4r+2,2s+1)$ vector charge model in $d=3$, examined in Sec. \[sec:evenOddVector\_d3\], in this regime our treatment of the Higgs mechanism can produce a fine-tuned Hamiltonian. That is, there is non-topological degeneracy which is split by local operators which are not present in the Hamiltonian we derive but which are allowed by the residual $\mathbb{Z}_2$ gauge invariance. The precise ground state, including any possible patterns of global symmetry breaking, depends on what operators are added, but the topological order of any such ground state can still be determined.
In this appendix, we list the theories which have this fine-tuned property but which have trivial topological order. Many of our arguments from Sec. \[sec:evenOddVector\_d3\] carry over to these theories.
### $d=2$ $(2r+2,2s+1)$ Scalar Charge
![Terms in the Hamiltonian and for the $d=2$ Higgsed $(2r+2,2s+1)$ scalar charge theory. The electric term $a$ is a four-spin interaction on the sites and the magnetic terms $b_{zi}$ are Ising interactions on the $i$-directed bonds.[]{data-label="fig:ScalarEvenOdd2DHam"}](ScalarEvenOdd2D.pdf){width="7cm"}
The Hamiltonian has the same structure Eq. as the other scalar charge models, but with $a$ and $b_{zi}$ as shown in Fig. \[fig:ScalarEvenOdd2DHam\]. The site and plaquette spins decouple. The plaquette spins are classical and have a unique ground state at all $h_p \neq 0$. The site spin sector consists of decoupled transverse field Ising chains, one for each row (involving only the $yy$ spins) and each column (involving only the $xx$ spins) of the lattice.
At $h_s=0$, every $Z_{ii}$ commutes with the Hamiltonian, as do the strings $\prod_{x} X_{yy}(x,y_0)$ and $\prod_y X_{xx}(x_0,y)$ for each $x_0,y_0$. At $h_s \neq 0$, the strings still commute with the Hamiltonian, and $h_s$ only contributes at $L$th order in perturbation theory. Therefore, $h_s$ only splits the degeneracy by an amount exponentially small in the system size; the same argument holds for any local operator involving $X_{ii}$ operators. Hence only products of $Z_{ii}$ operators can split the degeneracy in the thermodynamic limit, but adding any such term to the $h_s=0$ model simply chooses some particular ground state for each Ising chain. Different operators can lead to different ground states. If any degeneracy remains, it still can be split completely by adding $\sum_i Z_{ii}$ to the Hamiltonian, so no topological order can be present; therefore, any possible ground state is topologically trivial.
### $d=2$ $(2r+1,2s+2)$ Vector Charge
![ Terms in the Hamiltonian of the $d=2$ Higgsed $(2r+1,2s+2$) vector charge theory Hamiltonian. $a_i$ are associated with the bonds and form Ising interactions acting on the green spins. The $b$ operator is associated with the center site and is a product of the indicated Pauli $Z$ operators on the four orange plaquette spins.[]{data-label="fig:VectorOddEven2D"}](VectorOddEven2D.pdf){width="7cm"}
The Hamiltonian has the usual form Eq. for vector charge models, but with the forms of $a_i$ and $b$ are given in Fig. \[fig:VectorOddEven2D\].
Again the site and plaquette spins decouple. The site spins form decoupled classical Ising chains. At $h_p=0$, all the $Z_{xy}$ commute with the Hamiltonian, as do the string operators $\prod_{x}Z_{xy}(x,y_0)$ and $\prod_{y}Z_{xy}(x_0,y)$ for fixed positions $x_0$ and $y_0$. The $Z_{xy}$ and the string operators anticommute, so there is non-topological ground state degeneracy. Making $h_p$ nonzero but weak will only contribute at $L$th order in degenerate perturbation theory, so $h_p$ only splits the degeneracy by an exponentially small amount. Similarly to the $d=2$ $(2r+2,2s+1)$ scalar charge theory discussed above, the degeneracy is fine-tuned, as local operators can split it. For the same reasons as in the $d=2$ $(2r+2,2s+1)$ scalar charge model discussed above, the resulting ground state depends on what operators are chosen, but said state will always be topologically trivial.
### $d=3$ $(4r+4,2s+1)$ Vector Charge
The Hamiltonian of the $d=3$ $(4r+4,2s+1)$ vector charge theory takes the same schematic form as the $d=3$ $(2r+1,2s+1)$ vector charge theory Eq. but with different operators, shown in Fig. \[fig:Vector41Ham3D\].
![Terms in the Hamiltonian of the Higgsed $(4r+4,2s+1)$ vector charge theory. The $a$ operators are associated with the links and are products of four $X_{ij}$ operators acting on the green spins. The $b_{ij}$ operators are products of four Pauli $Z_{ij}$ operators acting on the orange spins.[]{data-label="fig:Vector41Ham3D"}](Vector41Ham3D.pdf){width="6.5cm"}
The sites and faces decouple. The face sector is a classical model with a unique ground state.
In the site sector at $h_s=0$, obviously every $Z_{ii}$ commutes with the Hamiltonian. There are also $\mathcal{O}(L^2)$ independent string operators which commute with the Hamiltonian and anticommute with the $Z_{ii}$; an example is $\prod_y X_{xx}(x_0+1,y,z_0) X_{xx}(x_0+1,y,z_0+2)X_{zz}(x_0,y,z_0+1)X_{zz}(x_0+2,y,z_0+1)$ where $x_0$ and $z_0$ are fixed positions. Rotations of this operator are also valid. These string operators are the lowest-order operators that enter in the effective Hamiltonian when $h_s$ is taken weak but nonzero; they appear at order $4L$ in degenerate perturbation theory. Therefore, $h_s$ nonzero only splits the ground state degeneracy by an exponentially small amount, but there are local operators (e.g. $Z_{ii}$) which commute with the $a_i$ and split the degeneracy at first order in perturbation theory. For the same reasons as in the other models in this section, the resulting ground state depends on what local operators are chosen, but said state will always be topologically trivial.
Much like the $d=3$ $(2r+1,2s+2)$ scalar charge model, there is also nontrivial topological order in the phase diagram when $1/g_{ij}^2 \ll h_s \ll 1/g_{ii}^2$. Using similar arguments, the effective model at $1/g_{ij}^2 = 0$ obtained from degenerate perturbation theory is the Hamiltonian Eq. , which is eight decoupled copies of the toric code.
### $d=3$ $(2r+1,2s+2)$ Vector Charge
The Hamiltonian of the $d=3$ $(2r+1,2s+2)$ vector charge theory also takes the same schematic form as the $d=3$ $(2r+1,2s+1)$ vector charge theory Eq. but with different operators, shown in Fig. \[fig:VectorOddEven3D\].
![Terms in the Hamiltonian of the Higgsed $(2r+1,2s+2)$ vector charge theory. The $a$ operators are associated with the links and are Ising-like products of $X_{ii}$ operators acting on the green spins. The $b_{ij}$ operators are products of ten (off-diagonal terms, associated with faces) or four (diagonal terms, associated with sites) Pauli $Z_{ij}$ operators acting on the orange spins.[]{data-label="fig:VectorOddEven3D"}](VectorOddEven3D.pdf){width="6.5cm"}
Again, the sites and faces decouple. The site sector has a unique classical ground state. The face sector is fine-tuned in a similar fashion to the Higgsed $d=3$ $(4r+4,2s+1)$ vector charge theory. This can be seen by examining the face sector at $h_f = 0$. In this limit, every $Z_{ij}$ commutes with the Hamiltonian, but there are also $\mathcal{O}(L^2)$ independent string operators which commute with the Hamiltonian but anticommute with various $Z_{ij}$. An example is the operator $\prod_{x}X_{xy}(x,y_0,z_0)X_{xz}(x,y_0,z_0)$, where $y_0$ and $z_0$ are fixed spatial positions. By standard degenerate perturbation theory arguments, setting $h_f$ small but nonzero only contributes at $2L$th order in perturbation theory, which leads to an exponentially small splitting of the degeneracy. As in the rest of this section, the resulting ground state depends on what local operators are chosen, but said state will always be topologically trivial.
[^1]: In fact, exactly one (up to rescalings of the matter field) of $m$ or $n$ could in principle be irrational. However, the compactness conditions of the $A_{ij}$ lead to the magnetic field obeying $B_{ij} \sim B_{ij}+2\pi m$, $B_{ij}+2\pi n$ for $i\neq j$. If one of $m$ or $n$ is irrational, the resulting magnetic field cannot be incorporated into the Hamiltonian in a straightforward way while respecting the compactness condition. We generally expect such theories to be at best highly unstable.
|
---
abstract: 'It is pointed out that there exists an interesting strong and weak duality in the Landau-Zener-Stueckelberg potential curve crossing. A reliable perturbation theory can thus be formulated in the both limits of weak and strong interactions. It is shown that main characteristics of the potential crossing phenomena such as the Landau-Zener formula including its numerical coefficient are well-described by simple (time-independent) perturbation theory without referring to Stokes phenomena. A kink-like topological object appears in the “magnetic” picture, which is responsible for the absence of the coupling constant in the prefactor of the Landau-Zener formula. It is also shown that quantum coherence in a double well potential is generally suppressed by the effect of potential curve crossing, which is analogous to the effect of Ohmic dissipation on quantum coherence.'
address:
- |
Department of Physics, University of Tokyo,\
Bunkyo-ku, Tokyo 113, Japan
- |
Department of Physics, Ibaraki University,\
Mito 310, Japan
author:
- 'Kazuo Fujikawa[^1] and Hiroshi Suzuki'
title: 'Duality in Landau-Zener-Stueckelberg potential curve crossing'
---
Introduction {#sec:one}
============
The potential curve crossing is related to a wide range of physical and chemical processes, and the celebrated Landau-Zener formula [@1; @2; @3] correctly describes the qualitative features of those processes [@4; @5; @6; @7; @8; @9; @10]. It has been recently shown [@11] that the potential crossing problem contains interesting modern field theoretical ideas, namely, the duality and gauge transformation.
The adiabatic and diabatic pictures in potential curve crossing problem are related to each other by a field dependent $su(2)$ gauge transformation [@5; @8], and we point out that this transformation leads to an interchange of strong and weak potential curve crossing interactions, which is analogous to the electric and magnetic duality in conventional gauge theory [@12]. This strong and weak duality allows a reliable perturbative treatment of potential curve crossing phenomena at the both limits of very weak (adiabatic picture) and very strong (diabatic picture) potential crossing interactions.
A model Hamiltonian of potential curve crossing and duality {#sec:two}
===========================================================
To analyze the potential curve crossing, we start with a model Hamiltonian defined in the so-called diabatic picture [@5; @8] $$\begin{aligned}
H&=&{1\over2m}\hat p^2+{V_1(x)+V_2(x)\over2}
+{V_1(x)-V_2(x)\over2}\sigma_3
\nonumber\\
&+&{1\over g}\sigma_1
\label{eq:two.one}\end{aligned}$$ where $\sigma_3$ and $\sigma_1$ stand for the Pauli matrices. We assume throughout this article that the potential crossing occurs at the origin, $V_1(0)=V_2(0)=0$ (see Fig. 1).
If one neglects the last term in the above Hamiltonian, one obtains the unperturbed Hamiltonian in the diabatic picture $$H_0\equiv{1\over 2m}\hat p^2+{V_1(x)+V_2(x)\over2}
+{V_1(x)-V_2(x)\over2}\sigma_3.
\label{eq:two.two}$$ This Hamiltonian $H_0$ describes two potentials, which are decoupled from each other. The last term in (\[eq:two.one\]), $H_I\equiv\sigma_1/g$ with a constant $g$, causes the transition between these two otherwise independent potential curves. In other words, if one takes $g\rightarrow{\rm large}$, this case physically corresponds to a [*complete*]{} potential crossing from a view point of [*adiabatic*]{} two-potential crossing in Fig. 2. Namely, $g$ stands for the strength of potential crossing interaction, and $g\rightarrow{\rm large}$ corresponds to a very strong potential crossing interaction. On the other hand, if one lets $g$ smaller, the effects of the last term in (\[eq:two.one\]) become substantial and the Hamiltonian $H_0$ (\[eq:two.two\]) does not present a sensible zeroth order Hamiltonian.
To deal with the case of a small $g$, we perform the non-Abelian “gauge transformation,” $$\begin{aligned}
\Phi(x)&=&e^{i\theta(x)\sigma_2/2}\Psi(x),
\nonumber\\
H'&=&e^{i\theta(x)\sigma_2/2}He^{-i\theta(x)\sigma_2/2},
\label{eq:two.three}\end{aligned}$$ where $\sigma_2$ is a Pauli matrix. The Hamiltonian in the new picture is given by $$\begin{aligned}
H'&=&{1\over2m}
\left[\hat p-{\hbar\over2}\partial_x\theta(x)\sigma_2\right]^2
+{V_1(x)+V_2(x)\over2}
\nonumber\\
&+&\left[{V_1(x)-V_2(x)\over2}\cos\theta(x)
+{1\over g}\sin\theta(x)\right]\sigma_3
\\
&+&\left[-{V_1(x)-V_2(x)\over2}\sin\theta(x)
+{1\over g}\cos\theta(x)\right]\sigma_1.
\nonumber
\label{eq:two.four}\end{aligned}$$ To eliminate the potential curve mixing, the last term of (\[eq:two.four\]), we choose the gauge parameter $\theta(x)$ as [@8] $$\cot\theta(x)=g{V_1(x)-V_2(x)\over2}\equiv f(x).
\label{eq:two.five}$$ We then obtain the Hamiltonian in the [*adiabatic*]{} picture $$H'=H_0'+H_I',
\label{eq:two.six}$$ where $$\begin{aligned}
H_0'&\equiv&{1\over2m}\hat p^2+{U_1(x)+U_2(x)\over2}
\nonumber\\
&+&{U_1(x)-U_2(x)\over2}\sigma_3,
\label{eq:two.seven}\end{aligned}$$ and $$\begin{aligned}
H_I'&\equiv&-{\hbar\over4m}
\left[\hat p\partial_x\theta(x)
+\partial_x\theta(x)\hat p\right]\sigma_2
\nonumber\\
&+&{\hbar^2\over8m}\left[\partial_x\theta(x)\right]^2.
\label{eq:two.eight}\end{aligned}$$ The potential energies in the adiabatic picture are related to those in the diabatic picture as (Fig. 2) $$\begin{aligned}
U_{1,2}(x)&\equiv&{V_1(x)+V_2(x)\over2}
\nonumber\\
&\pm&\sqrt{\left[{V_1(x)-V_2(x)\over2}\right]^2+{1\over g^2}}.
\label{eq:two.nine}\end{aligned}$$ From the definition of the gauge parameter in (\[eq:two.five\]), the “gauge field” $\partial_x\theta(x)$ is expressed as $$\partial_x\theta(x)=-{f'(x)\over1+f(x)^2}.
\label{eq:two.ten}$$
The transition from the diabatic picture to the adiabatic picture is a field dependent transformation.
In the adiabatic picture, the $\sigma_2$ dependent term in the interaction $H_I'$ (\[eq:two.eight\]) causes the potential crossing. If one neglects $H_I'$, the two potentials characterized by $U_1(x)$ and $U_2(x)$ do not mix with each other: Physically, this means [*no*]{} potential crossing. This suggests that $H_I'$ is proportional to the coupling constant $g$, since a small $g$ corresponds to [*weak*]{} potential crossing by definition. This is in fact the case as is clear from (\[eq:two.ten\]) and (\[eq:two.five\]).
We thus conclude that the two extreme limits of potential crossing interaction should be reliably handled in perturbation theory; namely, the strong potential crossing interaction in the [*diabatic*]{} picture, and the weak potential crossing interaction in the gauge transformed [*adiabatic*]{} picture. This is analogous to the electric-magnetic duality in conventional gauge theory [@12]: The diabatic picture may correspond to the electric picture with a coupling constant $e=1/g$, and the adiabatic picture to the magnetic picture with a coupling constant $g$.
A general criterion for the validity of perturbation theory in the adiabatic picture (\[eq:two.six\]) is $${\hbar\over2}|\partial_x\theta(x)|\ll|p(x)|,
\label{eq:two.eleven}$$ which is expected to be satisfied when the coupling constant $g$ is small and the incident particle is sufficiently energetic.
Landau-Zener formula {#sec:three}
====================
As an illustration of the duality discussed in Section \[sec:two\], we re-examine a perturbative derivation of the Landau-Zener formula in both of the adiabatic and diabatic pictures [@1; @5; @8]. For definiteness, we shall assume $V_1'(0)>V_2'(0)$ as in Fig. 1.
Let us start with the adiabatic picture with weak potential crossing interaction. Since the gauge field generally vanishes, $\partial_x\theta(x)\rightarrow0$ for $|x|\rightarrow\infty$, we can define the [*asymptotic*]{} states in terms of the eigenstates of $H_0'$ (\[eq:two.seven\]). We define the initial and final states $\Phi_i$ and $\Phi_f$ by $$\Phi_i(x)=\pmatrix{\varphi_1(x)\cr 0\cr},\quad
\Phi_f(x)=\pmatrix{0\cr \varphi_2(x)\cr},
\label{eq:three.one}$$ which satisfy $$\begin{aligned}
\left[{1\over2m}\hat p^2+U_1(x)\right]\varphi_1(x)
&=&E\varphi_1(x),
\nonumber\\
\left[{1\over2m}\hat p^2+U_2(x)\right]\varphi_2(x)
&=&E\varphi_2(x).
\label{eq:three.two}\end{aligned}$$ We then obtain the potential curve crossing probability due to the perturbation $H_I'$ (\[eq:two.eight\]) $$w(i\rightarrow f)
={2\pi\over\hbar}|\langle\Phi_f|H_I'|\Phi_i\rangle|^2.
\label{eq:three.three}$$ The transition matrix element is given by $$\begin{aligned}
\langle\Phi_f|H_I'|\Phi_i\rangle&=&
-{\hbar^2\over4m}\int_{-\infty}^\infty dx\,\partial_x\theta(x)
\\
&\times&\left[-\varphi_2'(x)\varphi_1(x)
+\varphi_2(x)\varphi_1'(x)\right].
\nonumber
\label{eq:three.four}\end{aligned}$$
To evaluate the matrix element, we use the WKB wave functions [@4]: $$\varphi_1(x)=\cases{
\displaystyle
{C_1\over2\sqrt{|p_1(x)|}}
\exp\left[-{1\over\hbar}\int_{a_1}^xdx\,|p_1(x)|\right],\cr
\displaystyle
{C_1\over\sqrt{p_1(x)}}
\cos\left[{1\over\hbar}\int_x^{a_1}dx\,p_1(x)
-{\pi\over4}\right]\cr}
\label{eq:three.five}$$ for $x>a_1$ and $x<a_1$, respectively, and $$\varphi_2(x)=\cases{
\displaystyle
{C_2\over2\sqrt{|p_2(x)|}}
\exp\left[-{1\over\hbar}\int_{a_2}^xdx\,|p_2(x)|\right],\cr
\displaystyle
{C_2\over\sqrt{p_2(x)}}
\cos\left[{1\over\hbar}\int_x^{a_2}dx\,p_2(x)
-{\pi\over4}\right]\cr}
\label{eq:three.six}$$ for $x>a_2$ and $x<a_2$, respectively. The semi-classical momenta in the adiabatic picture are defined by $$p_{1,2}(x)\equiv\sqrt{2m[E-U_{1,2}(x)]},
\label{eq:three.seven}$$ and $a_1$ and $a_2$ denote the classical turning points (see Fig. 2). The normalization of $\varphi_1(x)$ is chosen as $C_1=2\sqrt{m}$ to make the probability flux of the incident wave unity. On the other hand, the final state wave function in (\[eq:three.three\]) has to be normalized by the delta function with respect to the energy, $\langle\Phi_2'|\Phi_2\rangle=\delta(E_2'-E_2)$ and this specifies $C_2=2\sqrt{m}/\sqrt{2\pi\hbar}$.
We estimate the matrix element (15) by using the oscillating parts of the wave functions (\[eq:three.five\]) and (\[eq:three.six\]). This treatment is justified if the following conditions are satisfied:\
(i) $|p(0)|\rightarrow{\rm large}$ and $m\rightarrow{\rm large}$ with $v=|p(0)|/m$ kept fixed such that non-relativistic treatment is valid in the physically relevant region.\
(ii) $g\rightarrow{\rm small}$, but with $${1\over g}\ll{1\over2m}|p(0)|^2
\label{eq:three.eleven}$$ to ensure (\[eq:two.eleven\]) and the condition $$\beta\ll a,
\label{eq:three.twelve}$$ where $a$ is an average turning point and $\beta$ is a typical geometrical extension of $\partial_x\theta(x)$. If (\[eq:three.twelve\]) is satisfied, we can estimate the matrix element by using the oscillating parts of wave functions only since $\partial_{x}\theta(x)$ rapidly goes to zero for $|x|\gg\beta$ on the real axis.
The integral (15) is then written as $$\begin{aligned}
&&\langle\Phi_f|H_I'|\Phi_i\rangle
\simeq-{i\hbar C_1C_2\over 8m}\int dx\,\partial_x\theta(x)
\\
&&\times\biggl\{
\exp\left[{i\over\hbar}\int_{a_1}^xdx\,p_1(x)
-{i\over\hbar}\int_{a_2}^xdx\,p_2(x)\right]
\nonumber\\
&&-\exp\left[-{i\over\hbar}\int_{a_1}^xdx\,p_1(x)
+{i\over\hbar}\int_{a_2}^xdx\,p_2(x)\right]\biggr\},
\nonumber
\label{eq:three.nine}\end{aligned}$$ where we have set $p_1(x)/p_2(x)=1$ in the prefactors. This is justified if $\hbar/|p(0)|\ll\beta$, the characteristic length scale of the present problem, by letting $p(0)$ large as is specified in (i). Therefore we need to evaluate an integral of the form $$\begin{aligned}
I&\equiv&\int_{-\infty}^\infty dx\,\partial_x\theta(x)
\\
&\times&\exp\left[{i\over\hbar}\int_{a_1}^xdx\,p_1(x)
-{i\over\hbar}\int_{a_2}^xdx\,p_2(x)\right].
\nonumber
\label{eq:three.ten}\end{aligned}$$
We here present an explicit evaluation of (22) for the linear potential crossing problem, $V_1(x)=V_1'(0)x$ and $V_2(x)=V_2'(0)x$, on the basis of local data without referring to Stokes phenomena. For sufficiently large energy, $E-(U_1+U_2)/2\gg(U_1-U_2)/2$, the difference of momenta can be approximated as \[see (\[eq:two.nine\])\], $$\begin{aligned}
&&\int_0^xdx\,[p_1(x)-p_2(x)]
\\
&&\simeq-\int_0^xdx\,{2\over v(x)g\beta}\,\sqrt{x^2+\beta^2}
\nonumber
\label{eq:three.thirteen}\end{aligned}$$ where we used $$\begin{aligned}
f(x)&=&g{V_1'(0)-V_2'(0)\over2}x\equiv{x\over\beta},
\\
v(x)&\equiv&{1\over m}
\sqrt{2m\left[E-{U_1(x)+U_2(x)\over2}\right]}
\nonumber
\label{eq:three.fourteen}\end{aligned}$$ and $v(x)$ is approximated to be a constant $v=v(0)$ in the following. We also have from (\[eq:two.ten\]) $$\partial_x\theta(x)=-{\beta\over x^2+\beta^2}
\label{eq:three.fifteen}$$ and thus $$\begin{aligned}
&&I\simeq
-\exp\left[{i\over\hbar}\int_{a_1}^0dx\,p_1(x)
-{i\over\hbar}\int_{a_2}^0dx\,p_2(x)\right]
\nonumber\\
&&\times\int_{-\infty}^\infty dx\,{\beta\over x^2+\beta^2}
\exp\left(-{2i\over\hbar vg\beta}
\int_0^xdx\,\sqrt{x^2+\beta^2}\right)
\nonumber\\
&&=-\exp\left[{i\over\hbar}\int_{a_1}^0dx\,p_1(x)
-{i\over\hbar}\int_{a_2}^0dx\,p_2(x)\right]
\nonumber\\
&&\times\int_{-\infty}^\infty dx\,\exp[-i\alpha F(x)]
\label{eq:three.sixteen}\end{aligned}$$ where $$\begin{aligned}
F(x)&\equiv&\int_0^xdx\,\sqrt{x^2+1}+{1\over i\alpha}\ln(x^2+1),
\nonumber\\
\alpha&\equiv&{2\beta\over\hbar vg}
={4\over\hbar vg^2[V_1'(0)-V_2'(0)]}>0.
\label{eq:three.seventeen}\end{aligned}$$ We evaluate the integral (\[eq:three.sixteen\]) by a saddle point approximation with respect to $\alpha$. We thus seek the saddle point $$F'(x)=\sqrt{x^2+1}+{1\over i\alpha}{2x\over x^2+1}=0,
\label{eq:three.eighteen}$$ which is located between the real axis and the pole positions $x=\pm i$ of $\partial_x\theta(\beta x)$ so that we can smoothly deform the integration contour; these poles also coincide with the complex potential crossing points. If one sets $x=iy$ in (\[eq:three.eighteen\]) for $-1<y<1$, one has $$\sqrt{1-y^2}=-{1\over\alpha}{2y\over1-y^2}
\label{eq:three.nineteen}$$ which has a [*unique*]{} solution $$x_s=iy_s\simeq-i+{i\over2}\left({2\over\alpha}\right)^{2/3}
\label{eq:three.twenty}$$ for [*large*]{} $\alpha$. (The complex conjugate of $x_s$ is located in the second Riemann sheet.) For this value of the saddle point $$\begin{aligned}
F(x_s)&=&\int_0^{x_s}dx\,\sqrt{x^2+1}
+{1\over i\alpha}\ln\left({2\over\alpha}\right)^{2/3}
\nonumber\\
&\simeq&-{\pi i\over4}+{2\over3}{i\over\alpha}
+{1\over i\alpha}\ln\left({2\over\alpha}\right)^{2/3},
\nonumber\\
F''(x_s)&\simeq&-3i\left({\alpha\over2}\right)^{1/3}.
\label{eq:three.twentyone}\end{aligned}$$ We thus have a Gaussian integral which decreases in the direction [*parallel*]{} to the real axis $$\begin{aligned}
I&\simeq&-\left({\alpha\over2}\right)^{2/3}
e^{2/3}\exp\left(-{\pi\alpha\over4}\right)
\\
&\times&\exp\left[{i\over\hbar}\int_{a_1}^0dx\,p_1(x)
-{i\over\hbar}\int_{a_2}^0dx\,p_2(x)\right]
\nonumber\\
&\times&
\int_{-\infty}^\infty dx\,
\exp\left[-3\left(\alpha\over2\right)^{4/3}(x-x_s)^2\right]
\nonumber\\
&=&
-\sqrt{\pi\over3}e^{2/3}\exp\left(-{\pi\alpha\over4}\right)
\nonumber\\
&\times&\exp\left[{i\over\hbar}\int_{a_1}^0dx\,p_1(x)
-{i\over\hbar}\int_{a_2}^0dx\,p_2(x)\right]
\nonumber
\label{eq:three.twentytwo}\end{aligned}$$ From (\[eq:three.nine\]) we obtain $$\begin{aligned}
&&\langle\Phi_f|H_I'|\Phi_i\rangle\simeq
-\sqrt{\pi\over3}e^{2/3}\sqrt{\hbar\over2\pi}
\nonumber\\
&&\times\sin\left\{{1\over\hbar}
\left[\int_{a_1}^{0}dx\,p_1(x)-\int_{a_2}^{0}dx\,p_2(x)\right]
\right\}
\nonumber\\
&&\times
\exp\left\{-{\pi\over\hbar vg^2[V_1'(0)-V_2'(0)]}
\right\}.
\label{eq:three.twentythree}\end{aligned}$$ It is interesting that the numerical value of the coefficient of the above expression, $\sqrt{\pi}e^{2/3}/\sqrt{3}=1.99317$, is very close to the canonical value $2$ [@4], and we replace it by $2$ in the following. As for the past analysis of the prefactor in the time-dependent perturbation theory, see papers in [@13]. We thus have the transition probability from (\[eq:three.three\]) $$\begin{aligned}
&&w(i\rightarrow f)
\nonumber\\
&&\simeq
4\sin^2\left\{{1\over\hbar}\,
\left[\int_{a_1}^0dx\,p_1(x)-\int_{a_2}^0dx\,p_2(x)\right]\right\}
\nonumber\\
&&\times
\exp\left\{-{2\pi\over\hbar vg^2[V_1'(0)-V_2'(0)]}\right\}
\nonumber\\
&&\simeq2\exp\left\{-{2\pi\over\hbar vg^2[V_1'(0)-V_2'(0)]}
\right\}
\label{eq:three.twentyfour}\end{aligned}$$ where we replaced the square of sine function by its average $1/2$ in the final expression. We emphasize that the numerical coefficient of $w(i\rightarrow f)$ is fixed by time-independent perturbation theory and the local data without referring to global Stokes phenomena; this is satisfactory since linear potential crossing is a locally valid idealization.
We interpret that $w(i\rightarrow f)$ in (\[eq:three.twentyfour\]) expresses [*twice*]{} the non-adiabatic transition probability. Notice that our initial state wave function contains the reflection wave as well as the incident wave. Therefore the transition probability per [*one*]{} crossing is given by the half of (\[eq:three.twentyfour\]), $$P(1\rightarrow2)\simeq
\exp\left\{-{2\pi\over\hbar vg^2[V_1'(0)-V_2'(0)]}\right\},
\label{eq:three.twentyseven}$$ which is the celebrated Landau-Zener formula [@4; @5]. Our perturbative derivation presented here is conceptually much simpler than the past works [@1; @4; @5; @8], and it should be useful for a pedagogical purpose also.
It is interesting to study the same problem in the diabatic picture in Fig. 1 with $H_I=\sigma_1/g$ for large $g$. The evaluation of the matrix element is the standard one described in the textbook of Landau and Lifshitz [@4], for example. We have for $E>0$, $$\begin{aligned}
&&w(i\rightarrow f)\simeq
{8\pi\over\hbar v(0)g^2[V_1'(0)-V_2'(0)]}
\nonumber\\
&&\times\cos^2\left[{1\over\hbar}\int_0^{a_2}dx\,p_2(x)
-{1\over\hbar}\int_0^{a_1}dx\,p_1(x)
-{\pi\over4}\right]
\nonumber\\
&&\simeq{4\pi\over\hbar vg^2[V_1'(0)-V_2'(0)]}.
\label{eq:three.thirtyfour}\end{aligned}$$ \[$v(0)$ is the velocity at the crossing point, $v(0)=\sqrt{2E/m}$.\] We again interpret (\[eq:three.thirtyfour\]) as twice the potential crossing probability because our initial state wave function contains the reflection wave as well as the incident wave. The transition probability per one potential crossing is given by the half of (\[eq:three.thirtyfour\]).
A simple [*interpolating*]{} formula, which reproduces (\[eq:three.twentyfour\]) in the weak coupling limit and (\[eq:three.thirtyfour\]) in the strong coupling limit, is given by $$\begin{aligned}
&&w(i\rightarrow f)\simeq
2\exp\left\{-{2\pi\over\hbar vg^2[V_1'(0)-V_2'(0)]}\right\}
\nonumber\\
&&\times\left(1-
\exp\left\{-{2\pi\over\hbar vg^2[V_1'(0)-V_2'(0)]}\right\}
\right)
\label{eq:three.thirtysix}\end{aligned}$$ This expression is also consistent with the (semi-classical) conservation of probability [@4].
Motivated by duality, we re-examined a perturbative derivation of the Landau-Zener formula, and we re-derived the formula (\[eq:three.twentyfour\]) including its numerical coefficient on the basis of perturbation theory. However, our final result (\[eq:three.twentyfour\]) in the adiabatic picture does not contain the coupling constant as a prefactor. This is related to an interesting topological object in the present formulation. From the definition of (\[eq:two.ten\]), the “gauge field” satisfies the relation $$\begin{aligned}
\int_{-\infty}^\infty dx\,\partial_x\theta(x)
&=&\theta(\infty)-\theta(-\infty)
\nonumber\\
&=&-\pi,
\label{eq:three.thirtyseven}\end{aligned}$$ which is [*independent*]{} of the coupling constant $g$; we assume $f(x)\rightarrow\pm\infty$ for $x\rightarrow\pm\infty$, respectively. Because of this kink-like topological behavior of $\theta(x)$, the coupling constant does not appear as a prefactor of the matrix element in perturbation theory if the wave functions spread over the range which well covers the geometrical size of $\partial_x\theta(x)$. The precise criterion of the validity of perturbation theory is thus given by (\[eq:two.eleven\]): This condition is in fact satisfied if the conditions (\[eq:three.eleven\])–(\[eq:three.twelve\]) are satisfied. For small values of $x$, the small coupling $g$ helps to satisfy (\[eq:two.eleven\]). Even for the values of $x$ near the average turning point $a$, we have $${\hbar\over2}|\partial_x\theta(a)|
\simeq{1\over2}\left({\beta\over a}\right){\hbar\over a}\ll
{\hbar\over a}\simeq|p(a)|,
\label{eq:three.thirtyeight}$$ where $\beta$ stands for the typical geometrical size of $\partial_x\theta(x)$. The estimate in the left hand side is based on linear potentials (\[eq:three.fifteen\]), but we expect that the condition is satisfied for more general potentials as well. We thus clarified the basic mechanism why the prefactor of the Landau-Zener formula (\[eq:three.twentyseven\]) should come out to be very close to unity in time-independent perturbation theory.
Discussions {#sec:four}
===========
Motivated by the presence of interesting weak and strong duality in the model Hamiltonian (\[eq:two.one\]) of potential curve crossing, we re-examined a perturbative approach to potential crossing phenomena. We have shown that straightforward [*time-independent*]{} perturbation theory combined with the zeroth order WKB wave functions provides a reliable description of general potential crossing phenomena. Our analysis is based on the local data as much as possible without referring to global Stokes phenomena. Formulated in this manner, perturbation theory becomes more flexible to cover a wide range of problems.
The effects of dissipative interactions on macroscopic quantum tunneling have been extensively analyzed in the path integral formalism [@14] and also in the canonical (field theoretical) formalism [@15]. It is generally accepted that the Ohmic dissipation suppresses the macroscopic quantum coherence; in fact, an attractive idea of a dissipative phase transition has been suggested [@14].
It is plausible that the effects of potential curve crossing with nearby potentials influence the quantum coherence of the two degenerate ground states. One can in fact confirm that the potential curve crossing generally suppress the quantum coherence by using the perturbation theory for both limits of strong and weak curve crossing interactions [@11]. From a view point of symmetry, the lowest order perturbation in the present problem and the dissipative interaction in the Caldeira-Legget model [@14] both correspond to a dipole approximation. However, a perturbative analysis of basically non-perturbative tunneling phenomena requires a great care. In Ref. [@11], an explicit calculation of rather limited configurations has been performed, which in fact indicates the general suppression of quantum coherence by potential curve crossing. This suppression phenomenon of quantum coherence may become important in the future when one takes the effects of the environment into account in the analysis of potential curve crossing processes.
From a view point of general gauge theory, it is not unlikely that the electric-magnetic duality in conventional gauge theory is also related to some generalized form of potential crossing in the so-called moduli space [@12]. We hope that our work may turn out to be relevant from this view point also.
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[^1]: Invited talk presented at 7th Asia Pacific Physics Conference, August 19-23, 1997, Beijing, China (to be published in the Proceedings)
|
---
abstract: 'In this paper, we will show that $\RT^{2}+\WKLo$ is a $\Pi^{1}_{1}$-conservative extension of $\BN[3]$.'
author:
- 'Theodore A. Slaman'
- Keita Yokoyama
bibliography:
- 'bib-rt2inf.bib'
date: 'July 5, 2018 '
title: |
The strength of Ramsey’s theorem for\
pairs and arbitrarily many colors
---
Introduction
============
The strength of Ramsey’s theorem is well-studied in the setting of reverse mathematics. In this paper, we will focus on the first-order consequences of Ramsey’s theorem for pairs over the base system $\RCAo$. On the first-order part of Ramsey’s theorem for pairs and two colors ($\RT^{2}_{2}$), Hirst[@Hirst-PhD] showed that it implies $\BN[2]$ and then Cholak/Jockusch/Slaman[@CJS] proved that $\RT^{2}_{2}+\WKLo+\IN[2]$ is a $\Pi^{1}_{1}$-conservative extension of $\IN[2]$. Thus, its first-order part is in between $\BN[2]$ and $\IN[2]$. There are many studies to determine the exact strength, and recently Chong/Slaman/Yang[@CSY2017] showed that $\RCAo+\RT^{2}_{2}$ does not imply $\IN[2]$, and Patey/Yokoyama[@PY2018] showed that $\WKLo+\RT^{2}_{2}$ is a $\Pi^{0}_{3}$-conservative extension of $\BN[2]$, which means that the first-order part of $\RT^{2}_{2}$ is closer to $\BN[2]$.
How about the strength of Ramsey’s theorem for pairs and arbitrarily many colors ($\RT^{2}$)? Over $\RCAo$, one may easily see that $\RT^{2}_{k}$ implies $\RT^{2}_{k+1}$, but that does not mean $\RT^{2}_{2}$ implies $\RT^{2}$ since the induction available within $\RCAo$ is not strong enough. Indeed, the case for $\RT^{2}$ is very similar to the case for $\RT^{2}_{2}$ and the following are known.
$\RT^{2}+\RCAo$ implies $\BN[3]$.
\[thm:CJS-RT2\] $\RT^{2}+\WKLo+\IN[3]$ is a $\Pi^{1}_{1}$-conservative extension of $\IN[3]$.
Hence, the first-order part of $\RT^{2}$ is between $\BN[3]$ and $\IN[3]$. Here, we will sharpen the proof of this theorem, and determine the exact first-order part of $\RT^{2}$, namely it is $\BN[3]$. For the basic notions of this area, see [@CJS; @Slicing-the-truth; @SOSOA].
The first-order part of $\RT^{2}$
=================================
Our main conservation theorem is the following.
$\RT^{2}+\WKLo$ is a $\Pi^{1}_{1}$-conservative extension of $\BN[3]$.
To show the main theorem, we will sharpen the argument from [@CJS], which is used for the proof of Theorem \[thm:CJS-RT2\].
Over $\RCAo$, $\RT^{2}$ is equivalent to $\D^{2}$ plus $\COH$.
Here, $\D^{2}$ and $\COH$ are the following statements.
- $\D^{2}$: for any $k\in \N$ and any $\Delta^{0}_{2}$-partition $\N=\bigsqcup_{i<k} \mc{A}_{i}$, there exists an infinite set $Z\subseteq \N$ such that $Z\subseteq \mc{A}_{i}$ for some $i<k$,
- $\COH$: for any infinite sequence of sets $\langle R_{i}: i\in\N \rangle$, there exists an infinite set $Z\subseteq \N$ such that $(Z\subseteq^{*} R_{i}\vee Z\subseteq^{*} \N\setminus R_{i})$ for any $i\in\N$.
(Note that $\N$ denotes the set of all natural numbers within $\RCAo$, *i.e.*, if $\mc M=(M,S)$ is a model of $\RCAo$, $\N^{\mc M}=M$.)
Since we already know that $\RCAo+\RT^{2}$ implies $\BN[3]$, we will consider the first-order strength of the above two statements over $\BN[3]$. Note that $\D^{2}$ and $\COH$ are both $\Pi^{1}_{2}$-statements, and $\Pi^{1}_{1}$-conservation results for $\Pi^{1}_{2}$-statements can be amalgamated, *i.e.*, if both of $\RCAo+\BN[3]+\D^{2}$ and $\RCAo+\BN[3]+\COH$ are $\Pi^{1}_{1}$-conservative over $\BN[3]$ then so is $\RCAo+\BN[3]+\D^{2}+\COH$, which is equivalent to $\RCAo+\RT^{2}$ (see [@Y2010]). The strength of $\COH$ (together with weak König’s lemma) over $\BN[3]$ is already known.
$\WKLo+\COH+\BN[3]$ is a $\Pi^{1}_{1}$-conservative extension of $\BN[3]$.
Thus, what we need is the following.
\[thm:main-conservation\] $\RCAo+\D^{2}+\BN[3]$ is a $\Pi^{1}_{1}$-conservative extension of $\BN[3]$.
In [@CJS], it is shown by a variant of Mathias forcing that a computable instance of $\D^{2}$ admits a $\low_{2}$-solution. On the other hand, $\low_{2}$-sets preserve $\BN[3]$ since they won’t add any new $\Sigma^{0}_{3}$-sets. Thus, the following theorem is essential for Theorem \[thm:main-conservation\].
\[thm:main-construction\] Let $(M,\{B\})$ be a countable model of $\BN[3]$, and let $M=\bigsqcup_{i<k} \mc{A}_{i}$ be a $\Delta^{B}_{2}$-partition of $M$ for some $k\in M$. Then there exists an unbounded $\Delta^{B}_{3}$-set $G\subseteq M$ such that $G\subseteq \mc{A}_{i}$ for some $i<k$, and any $\Sigma^{B\oplus G}_{3}$ subset of $M$ is already $\Sigma^{B}_{3}$ in $M$.
We will prove this theorem in the next section. Assuming this theorem, it is routine work to prove Theorem \[thm:main-conservation\].
Assume that $\BN[3]$ does not prove a $\Pi^{1}_{1}$-sentence $\A X\psi(X)$. Then there exists a countable model $(M,S)\models \BN[3]$ such that $(M,S)\models \neg\psi(B)$ for some $B\in S$. For $X,Y\subseteq M$, $X\le_{T} Y$ means that $X$ is $\Delta^{Y}_{1}$ in $M$. By using Theorem \[thm:main-construction\] repeatedly, one can construct an $\omega$-length sequence of subsets of $M$, $B=B_{0}\le_{T}B_{1}\le_{T} \dots$ so that
- for any $m\in\omega$ and $\Delta^{B_{m}}_{2}$-partition $M=\bigsqcup_{i<k} \mc{A}_{i}$, there exist $n\ge m$ and an unbouded set $G\le_{T} B_{n}$ such that $G\subseteq \mc{A}_{i}$ for some $i<k$, and,
- any $\Sigma^{B_{m}}_{3}$ subset of $M$ is already $\Sigma^{B}_{3}$ in $M$.
Put $\bar S=\{X\subseteq M: X\le_{T} B_{m}, m\in\omega\}$, then $(M,\bar S)\models \RCAo+\D^{2}+\BN[3]$ but $\neg\psi(B)$ is still true in $(M,\bar S)$. Hence $\RCAo+\D^{2}+\BN[3]$ does not prove $\A X\psi(X)$.
Construction
============
In this section, we will prove Theorem \[thm:main-construction\]. The main idea is formalizing a computability theoretic construction within a nonstandard model of arithmetic. The following theorem is a basic tool to formalize standard arguments for $\Pi^{0}_{1}$-classes, and we will use it freely throughout this section.
\[thm:Pi01-in-WKL\] Let $\varphi(X,A)$ be a $\Pi^{0}_{1}$-formula with exactly displayed the set variables.
1. There exists a $\Pi^{0}_{1}$-formula $\psi(A)$ such that $\WKLo$ proves $\E X\varphi(X,A)\leftrightarrow \psi(A)$.
2. $\WKLo$ proves that $\E X\varphi(X,A)$ is equivalent to the statement that there exists (a $\Delta^{A}_{2}$-code for) a low set $Y$ relative to $A$ such that $\varphi(Y,A)$.
3. For a given $\Delta^{0}_{2}$-definable set $\mc{A}$ (possibly not a second-order object), $\WKLo+\BII$ proves $\E X\varphi(X,\mc{A})\to \E X\E Y\varphi(X,Y)$. Thus, “there exists $\Delta^{0}_{2}$-definable set $\mc{A}$ such that $\E X\varphi(X,\mc A)$” can be described by a $\Pi^{0}_{1}$-formula.
1 is a well-known fact, see, e.g., [@SOSOA Lemma VIII.2.4]. 2 is a low basis theorem for $\Pi^{0}_{1}$-classes which is formalizable within $\II$ [@HK1989]. With $\BII$, one can mimic the proof of 1 for $\Delta^{0}_{2}$-sets, 3 easily follows from that.
As we mentioned in the previous section, we want to formalize the second $\low_{2}$-solution construction for $\D^{2}$ from [@CJS] within $\BN[3]$. However, that construction uses $\IN[3]$ in two parts, to find the right color for a solution, and to do $\mathbf{0}''$-primitive recursion. In the following construction, we need to avoid these. To overcome the first problem, we will construct solutions for all possible colors, and see that it works for at least one color in the end. For the second problem, we will still use $\mathbf{0}''$-primitive recursion. In a nonstandard model $(M,S)\models\BN[3]$, $\mathbf{0}''$-primitive recursion might end in nonstandard numbers of steps which form a proper cut of $M$. Thus, we will decide some finite collection of $\Sigma^{0}_{2}$-statements at each step, and finally decide all $\Sigma^{0}_{2}$-statements before $\mathbf{0}''$-primitive recursion ends, adapting Shore’s blocking argument.
Now we start the construction. Let $(M,\{B\})$ be a countable model of $\BN[3]$. By the following theorem, we will work within $(M,S)\models \WKLo+\BIII$ with $B\in S$.
Let $(M,\{B\})$ be a countable model of $\BN[3]$. Then there exists $S\subseteq \mc{P}(M)$ such that $B\in S$ and $(M,S)\models \WKLo+\BN[3]$.
In what follows, we will mimic the “double jump control” method in [@CJS]. Let $\bigsqcup_{i<k}\mc{A}_{i}=M$ be a $\Delta^{B}_{2}$-partition for some $k\in M$ and $B\in S$. A quintuple $p=(\bar{F}, X, \sigma, \ell_{0},\ell_{1})$ is said to be a *pre-condition* if
- $\ell_{0},\ell_{1}\in M$, $\sigma:\ell_{0}\times k\to 2$,
- $\bar{F}$ is a $k$-tuple of finite sets $\langle F_{i}: i<k \rangle$ such that $F_{i}\subseteq \mc{A}_{i}$,
- $X$ is coded by $\ell_{1}$ and (a $\Delta^{B}_{2}$ code for) an infinite $\low^{B}$ set $X_{0}$ as $X=X_{0}\cap(\ell_{1},\infty)$,
- $\max \bar{F}\cup \{\ell_{0}\}<\ell_{1}$, and a code for $X_{0}$ is bounded by $\ell_{1}$.
Here, we call a pair of $k$-tuple of finite sets and another set $(\bar{F},X)$ with $\min X>\max\bar{F}$ a Mathias pair. (In what follows, we will mainly deal with an infinite Mathias pair, *i.e.*, a Mathias pair with $X$ infinite, but quantification for Mathias pairs ranges over possibly finite Mathias pairs.) For finite sets $E,F$ and another set $X$, we write $E\in (F,X)\leftrightarrow F\subseteq E\subseteq F\cup X$. For two Mathias pairs $(\bar{F},X),(\bar{E}, Y)$, we say that $(\bar{E}, Y)$ *extends* $(\bar{F},X)$ (write $(\bar{F},X)\ge (\bar{E}, Y)$) if $E_{i}\in (F_{i},X)$ for every $i<k$, and $Y\subseteq X$.
Next, we define how Mathias pairs force $\Sigma^{0}_{1}$ and $\Sigma^{0}_{2}$-formulas at each color. To control the complexity of forcing formulas, we consider a triple of the form $(\bar{F},X,\ell)$, which is a Mathias pair $(\bar{F},X)$ with a bound $\ell\in M$. Let $\theta(n, G[n])$ be a $\Sigma^{0}_{0}$-formula with a new variable $G$. Then we define strong forcing $\Vdash^{+}$ for a pair of color $i$ and a $\Sigma^{0}_{1}$-formula $\E n\,\theta(n, G[n])$ as $$(\bar{F},X,\ell)\Vdash^{+}\langle i,\E n\,\theta(n, G[n]) \rangle\Leftrightarrow \E n\le \max F_{i}\,\theta (n, F_{i}[n]).$$ Similarly, let $\theta(m,n, G[n])$ be a $\Sigma^{0}_{0}$-formula with a new variable $G$. Then we define forcing $\Vdash$ for a pair of color $i$ and a $\Sigma^{0}_{2}$-formula $\E m\A n\,\theta(m,n, G[n])$ as $$(\bar{F},X,\ell)\Vdash\langle i,\E m\A n\,\theta(m,n,G[n]) \rangle\Leftrightarrow \E m\le \ell\,\A E\in (F_{i},X)\A n\le\max E\,\theta (m,n, E[n]).$$
Let $\pi(e, m, G)\equiv \A n\, \pi_{0}(e,m,n,G[n])$ be a universal $\Pi^{B,G}_{1}$-formula, *i.e.*, a universal $\Pi^{0}_{1}$-formulas with a new set variable $G$ (and a set parameter $B$). For a finite partial function $\sigma\subseteq M\times k\to 2$, we let $$\begin{aligned}
\sigma_{+}&:=\{\langle i, \E m\,\pi(e,m,G) \rangle: \sigma(e,i)=1\},\\
\sigma_{+,i,\le \ell}&:=\{\langle i, \E m(\pi(e,m,G)\wedge m\le \ell) \rangle: \sigma(e,i)=1\},\\
\sigma_{-}&:=\{\langle i, \E m\,\pi(e,m,G) \rangle: \sigma(e,i)=0\}.\end{aligned}$$
Let $\sigma$ be a finite partial function $\sigma\subseteq M\times k\to 2$.
1. A Mathias pair $(\bar{F},X)$ is said to be *$\sigma$-large* if for any finite sets of (possibly finite) Mathias pairs $\{(\bar{E}^{t}, Y^{t})\}_{t<s}$ and any bound $\ell'\in M$ such that for all $t<s$ and for all $i<k$, ${E}^{t}_{i}\subseteq \mc{A}_{i}$, $(\bar{E}^{t}, Y^{t})\le (\bar{F}, X)$, $\ell'\ge \max\bar{E}^{t}$, and $X\supseteq \bigsqcup_{t<s}Y^{t}\supseteq X\setminus \ell'$ (*i.e.*, $Y^{t}$’s partition a superset of $X\setminus \ell'$ which is included in $X$), there exists $t<s$ such that $(\bar{E}^{t}, Y^{t},\ell')\not\Vdash\langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \sigma_{+}$ and $Y^{t}$ is not bounded by $\ell'$.
2. Let $i<k$, $\ell\in M$. Then a Mathias pair $(\bar{F},X\cap \mc{A}_{i})$ is said to be *$\sigma$-large at $i$ up to $\ell$* if the largeness holds for $\sigma_{+,i,\le\ell}$ instead of $\sigma_{+}$ with considering all possible $\Delta^{0}_{2}$-definable sets for $Y^{t}$’s. Formally, $(\bar{F},X\cap \mc{A}_{i})$ is *$\sigma$-large at $i$ up to $\ell$* if for any $\Delta^{0}_{2}$-definable finite sets of Mathias pairs $\{(\bar{E}^{t}, Y^{t})\}_{t<s}$ and any bound $\ell'\in M$ such that for all $t<s$, ${E}^{t}_{i}\subseteq \mc{A}_{i}$, $(\bar{E}^{t}, Y^{t})\le (\bar{F}, X\cap\mc{A}_{i})$, $\ell'\ge \max\bar{E}^{t}$, and $X\cap \mc{A}_{i}\supseteq \bigsqcup_{t<s}Y^{t}\supseteq (X\cap \mc{A}_{i})\setminus \ell'$, there exists $t<s$ such that $(\bar{E}^{t}, Y^{t},\ell')\not\Vdash\langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \sigma_{+,i,\le\ell}$ and $Y^{t}$ is not bounded by $\ell'$. (Here, we consider all $\Delta^{0}_{2}$-definable sets in $(M,S)$ with any parameters from $S$. Be aware that we do not restrict to $\Delta^{B}_{2}$-sets.)
Roughly speaking, $\sigma$-largeness guarantees that one can find an extension without forcing any $\langle i,\psi \rangle\in \sigma_{+}$ in the future construction.
\[rem:largeness\]
1. The notion “$(\bar{F},X)$ is $\sigma$-large” won’t be changed whether we consider Mathias pairs $(\bar{E}^{t}, Y^{t})$ with $Y^{t}$ being a set in the structure or a $\Delta^{0}_{2}$-definable set by Theorem \[thm:Pi01-in-WKL\].3, and it is described by a $\Pi^{B}_{2}$-formula.
2. For the case “$(\bar{F},X\cap \mc{A}_{i})$ is $\sigma$-large at $i$ up to $\ell$”, it is essential to consider $\Delta^{0}_{2}$-definable sets, and thus the statement cannot be described by a $\Pi^{B}_{2}$-formula. In the following construction (which will be $B''$-primitive recursive), we will avoid checking this requirement directly.
A pre-condition $p=(\bar{F}^{p}, X^{p}, \sigma^{p}, \ell_{0}^{p},\ell_{1}^{p})$ is said to be a *condition* if
1. $(\bar{F}^{p}, X^{p})$ is $\sigma^{p}$-large,
2. $(\bar{F}^{p}, X^{p},\ell_{1}^{p})\Vdash \langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \sigma^{p}_{-}$,
3. if $(\bar{F}^{p}, X^{p}\cap \mc{A}_{i})$ is $\sigma^{p}$-large at $i$ up to $\ell_{0}^{p}$, then, $\A m\le \ell_{0}^{p}$, $(\bar{F}^{p}, X^{p},\ell_{1}^{p})\Vdash^{+} \langle i, \neg\pi(e,m,G) \rangle$ for any $e\le \ell_{0}^{p}$ with $\sigma^{p}(e,i)=1$.
Define $\mathbb{P}$ as the set of all conditions. For given two conditions $p,q\in \mathbb{P}$, $q$ properly extends $p$ ($p\succ q$) if $$(\bar{F}^{p}, X^{p})\ge(\bar{F}^{q}, X^{q})\wedge \ell_{1}^{p}\le \ell_{0}^{q}\wedge \sigma^{p}\subseteq\sigma^{q}.$$
For a given condition $p=(\bar{F}^{p}, X^{p}, \sigma^{p}, \ell_{0}^{p},\ell_{1}^{p})$, we want to find an extension of $p$. For this, we introduce a weaker version of the largeness notion.
Let $\sigma$ be a finite partial function $\sigma\subseteq M\times k\to 2$. A Mathias pair $(\bar{F},X)$ is said to be *$\sigma$-fair* if
- there exist a finite set of Mathias pairs $\{(\bar{E}^{t}, Y^{t})\}_{t<s}$ and a bound $\ell'\in M$ such that ${E}^{t}_{i}\subseteq \mc{A}_{i}$, $(\bar{E}^{t}, Y^{t})\le (\bar{F}, X)$, $\ell'\ge \max\bar{E}^{t}$, $X\supseteq \bigsqcup_{t<s}Y^{t}\supseteq X\setminus \ell'$ such that for any $t<s$,
- if $(\bar{E}^{t}, Y^{t},\ell')\not\Vdash\langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \sigma_{+}$, then $(\bar{E}^{t}, Y^{t},\ell')\Vdash\langle i,\psi \rangle$ for every $\langle i,\psi \rangle\in \sigma_{-}$, or,
- $Y^{t}$ is bounded by $\ell'$,
and,
- for any finite set of Mathias pairs $\{(\bar{E}^{t}, Y^{t})\}_{t<s}$ and a bound $\ell'\in M$ which witness the condition $(\dag)$, there exists $t<s$ such that $(\bar{E}^{t}, Y^{t},\ell')\not\Vdash\langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \sigma_{+}$ and $Y^{t}$ is not bounded by $\ell'$.
Note that “$(\bar{F},X)$ is $\sigma$-fair” can be described by a boolean combination of $\Sigma^{B}_{2}$ and $\Pi^{B}_{2}$ formulas.
\[lem:fair-extension\] Let $p=(\bar{F}^{p}, X^{p}, \sigma^{p}, \ell_{0}^{p},\ell_{1}^{p})$ be a condition, and let $\ell'\ge \ell_{1}^{p}$. Then $(\bar{F}^{p}, X^{p})$ is $\tau$-fair for some $\tau:\ell'\times k\to 2$ extending $\sigma^{p}$. Moreover, one can find a lexicographically maximal such $\tau$.
Since $p$ is a condition, $(\bar{F}^{p}, X^{p})$ is $\sigma^{p}$-fair. We will see by $\Sigma^{0}_{2}$-induction that for any finite set $H\subseteq M\times k$, there exists $\tau:\dom(\sigma^{p})\cup H\to 2$ such that $\tau\supseteq \sigma^{p}$ and $(\bar{F}^{p}, X^{p})$ is $\tau$-fair. For this, we only need to see that for any $\sigma'$ extending $\sigma^{p}$ such that $(\bar{F}^{p}, X^{p})$ is $\sigma'$-fair and $(e_{0},i_{0})\in M\times k\setminus \dom(\sigma')$, either $\sigma'\cup\{(e_{0},i_{0},0)\}$ or $\sigma'\cup\{(e_{0},i_{0},1)\}$ satisfies the fairness condition for $(\bar{F}^{p}, X^{p})$. Assume that $(\bar{F}^{p}, X^{p})$ is not $\sigma'\cup\{(e_{0},i_{0},1)\}$-fair. Since any finite set of Mathias pairs and a bound which witness the condition $(\dag)$ for $(\bar{F}^{p}, X^{p})$ to be $\sigma'$-fair actually witness $(\dag)$ for $(\bar{F}^{p}, X^{p})$ to be $\sigma'\cup\{(e,i,1)\}$-fair, the condition $(\dag\dag)$ for $(\bar{F}^{p}, X^{p})$ to be $\sigma'\cup\{(e_{0},i_{0},1)\}$-fair must fail. Thus, there exist a finite set of Mathias pairs $\{(\bar{E}^{t}, Y^{t})\}_{t<s}$ and a bound $\ell'\in M$ which witness the condition $(\dag)$ for $\sigma'\cup\{(e,i,1)\}$ such that for any $t<s$, $(\bar{E}^{t}, Y^{t},\ell')\Vdash\langle i,\psi \rangle$ for some $\langle i,\psi \rangle\in \sigma'_{+}\cup\{\langle i_{0}, \E m\,\pi(e_{0},m,G) \rangle\}$ or $Y^{t}$ is bounded by $\ell'$. Thus, for any $t<s$, if $Y^{t}$ is not bounded by $\ell'$, then $(\bar{E}^{t}, Y^{t},\ell')\not\Vdash\langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \sigma'_{+}$ implies $(\bar{E}^{t}, Y^{t},\ell')\Vdash\langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \sigma'_{-}\cup\{\langle i_{0}, \E m\,\pi(e_{0},m,G) \rangle\}$. This means $\{(\bar{E}^{t}, Y^{t})\}_{t<s}$ and $\ell'$ witness the condition $(\dag)$ for $(\bar{F}^{p}, X^{p})$ to be $\sigma'\cup\{(e_{0},i_{0},0)\}$-fair. The condition $(\dag\dag)$ for $\sigma'\cup\{(e_{0},i_{0},0)\}$ is automatically satisfied by the same condition for $\sigma'$.
\[lem:left-most-extension\] For any $p\in\mathbb{P}$, there exists $q\in \mathbb{P}$ such that $q\prec p$. Moreover, one can construct such an extension in a “left-most” way, *i.e.*, there is a canonical definable way to choose needed elements in the construction of an extension.
For a given condition $p=(\bar{F}^{p}, X^{p}, \sigma^{p}, \ell_{0}^{p},\ell_{1}^{p})\in \mathbb{P}$, put $\ell_{0}=\ell_{1}^{p}$. By Lemma \[lem:fair-extension\], there exists a lexicographically maximal $\tau:\ell_{0}\times k\to 2$ which extends $\sigma^{p}$ such that $(\bar{F}^{p}, X^{p})$ is $\tau$-fair. Then one can find a family of low Mathias pairs $\{(\bar{E}^{t}, Y^{t})\}_{t<s}$ (of smallest index) and a bound $\ell'\in M$ which witness $(\dag)$. By $(\dag\dag)$, pick the smallest $t<s$ such that $(\bar{E}^{t}, Y^{t})$ is $\tau$-large. Such a $t<s$ exists by $\BII$ since for any $\ell''\ge \ell'$ and for any $\{(\bar{D}^{t}, Z^{t})\}_{t<s''}$ which refines $\{(\bar{E}^{t}, Y^{t})\}_{t<s}$, one can apply $(\dag\dag)$ for $\{(\bar{D}^{t}, Z^{t})\}_{t<s''}$ and $\ell''$. Note that $\tau$-largeness implies that $(\bar{E}^{t}, Y^{t},\ell')\not\Vdash\langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \tau_{+}$ and $Y^{t}$ is infinite, thus, by $(\dag)$, $(\bar{E}^{t}, Y^{t},\ell')\Vdash\langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \tau_{-}$.
Now $(\bar{E}^{t}, Y^{t},\tau,\ell_{0},\ell')$ satisfies the first and second clauses to be a condition. For the third clause, we use the following claims. We say that $(\bar{D}', Z')$ is a finite extension of $(\bar{D}, Z)$ at $i$ if $(\bar{D}', Z')\le(\bar{D}, Z)$, $Z\setminus Z'$ is finite, and $D'_{i'}=D_{i'}$ for any $i'\neq i$. One can observe that finite extensions preserve $\tau$-largeness.
Let $(\bar{D}, Z)$ be a finite extension of $(\bar{E}^{t}, Y^{t})$ at $i$. If $(\bar{D}, Z\cap \mc{A}_{i})$ is $\tau$-large at $i$ up to $\ell_{0}$, then for any $e< \ell_{0}$ such that $\tau(e,i)=1$, there exists a finite extension $(\bar{D}', Z')\le(\bar{D}, Z)$ at $i$ such that $D'_{i}\in (D_{i},Z\cap\mc{A}_{i})$ and $(\bar{D}', Z',\max\bar{D}'\cup\{\ell'\})\Vdash^{+} \langle i, \A m\le \ell_{0}\,\neg\pi(e,m,G) \rangle$.
If $(\bar{E}^{t}, Y^{t}\cap\mc{A}_{i})$ is $\tau$-large at $i$ up to $\ell_{0}$, then there exists a finite extension $(\bar{D}', Z')\le(\bar{E}^{t}, Y^{t})$ at $i$ such that $D'_{i}\in ({E}^{t}_{i}, Y^{t}\cap\mc{A}_{i})$ and $(\bar{D}', Z',\max\bar{D}'\cup\{\ell'\})\Vdash^{+} \langle i, \A m\le \ell_{0}\,\neg\pi(e,m,G) \rangle$ for all $e<\ell_{0}$ such that $\tau(e,i)=1$.
One can easily check the first claim by unfolding the definition of $\tau$-largeness at $i$ up to $\ell_{0}$. Since finite extensions preserve $\tau$-largeness at $i$, the second claim is obtained by applying the first claim repeatedly. (This is possible within $\III$.)
Now we define $(\bar{D}^{*},Z^{*})\le (\bar{E}^{t}, Y^{t})$ as follows. For each $i<k$, check whether there exists a finite extension $(\bar{D}', Z')\le(\bar{E}^{t}, Y^{t})$ at $i$ such that $D'_{i}\in ({E}^{t}_{i}, Y^{t}\cap\mc{A}_{i})$ and $(\bar{D}', Z',\max\bar{D}'\cup\{\ell'\})\Vdash^{+} \langle i, \A m\le \ell_{0}\,\neg\pi(e,m,G) \rangle$ for all $e<\ell_{0}$ with $\tau(e,i)=1$. (Note that this condition can be expressed by a $\Sigma^{B}_{2}$-formula.) Put $D^{*}_{i}=D'_{i}$ if such $\bar{D}'$ exists, and put $D^{*}_{i}=E^{t}_{i}$ otherwise. (More precisely, one can pick minimal such $\bar{D}^{*}$ within $\III$.) Put $Z^{*}=Y^{t}\setminus[0,\max \bar{D}^{*}]$. Then, by the second claim, one can observe that for all $i<k$ and $e\le \ell_{0}$, $(\bar{D}^{*}, Z^{*}, \max\bar{D}^{*}\cup\{\ell'\})\Vdash^{+} \langle i, \A m\le \ell_{0}\,\neg\pi(e,m,G) \rangle$ if $(\bar{D}^{*}, Z^{*}\cap \mc{A}_{i})$ is $\tau$-large at $i$ up to $\ell_{0}$ and $\tau(e,i)=1$. Take the minimal $\ell_{1}$ so that $\ell_{1}$ bounds $\max\bar{D}^{*}\cup\{\ell'\}$ and a code for $Z^{*}$. Then $q=(\bar{D}^{*}, Z^{*}, \tau, \ell_{0},\ell_{1})$ is the desired extension.
For a given $p\in \mathbb{P}$, the extension constructed in the proof of Lemma \[lem:left-most-extension\] is said to be a *left-most successor of $p$.* Note that “$q$ is a left-most successor of $p$” can be described by a boolean combination of $\Sigma^{0}_{2}$ and $\Pi^{0}_{2}$ formulas.
Let $p_{0}\succ p_{1}\succ \dots$ be the left-most path of $\mathbb{P}$, *i.e.*, $p_{i+1}$ is a left-most successor of $p_{i}$. More formally, put $$\begin{aligned}
\mc{G}&=\{p_{n}: \E \langle p_{j}\mid j\le n \rangle (p_{0}=(\emptyset, \N, \emptyset, 0,1)\wedge \A j<n(p_{j+1}\mbox{ is a left-most successor of }p_{j}))\},\\
J&=\{n : \E \langle p_{j}\mid j\le n \rangle (p_{0}=(\emptyset, \N, \emptyset, 0,1)\wedge \A j<n(p_{j+1}\mbox{ is a left-most successor of }p_{j}))\}. \end{aligned}$$ Both of $J$ and $\mc{G}$ are $\Sigma^{B}_{3}$. Note that $J$ may form a proper cut of $M$.
\[lem:unbdd-G\] $\mc{G}$ is unbounded, *i.e.*, for any $x\in M$, there exists $p_{i}\in \mc{G}$ such that $\ell_{1}^{p_{i}}>x$.
Assume that $\mc{G}$ is bounded by some $\bar{\ell}\in M$. Then the first existential quantifier in the definition of $J$ is bounded. Thus it is defined by a boolean combination of $\Sigma^{B}_{2}$ and $\Pi^{B}_{2}$ formulas. Hence $J$ has a maximal element by $\III$, which contradicts Lemma \[lem:left-most-extension\].
Thus, $\mc{G}$ is cofinal in $M$. Our next task is to see that at some $i<k$, the construction of a solution works for any $j\in J$. If we can find such $i<k$, then $\bigcup_{j\in J} F_{i}^{p_{j}}$ is unbounded in $M$.
For each $j\in J$, put $$\begin{aligned}
\eta^{j}:=\{i<k: \A m\le \ell_{0}^{p_{j}} (\bar{F}^{p_{j}}, X^{p_{j}},\ell_{1}^{p_{j}})\Vdash^{+} \langle i, \neg\pi(e,m,G) \rangle\mbox{ for any }e\le \ell_{0}^{p_{j}}\mbox{ with }\sigma^{p_{j}}(e,i)=1\}. \end{aligned}$$ Here, $i\in\eta^{j}$ means that the construction for color $i$ is sill working at stage $j\in J$. Trivially, $\eta^{j}\supseteq\eta^{j'}$ if $j\le j'$.
\[lem:color-survive1\] $\eta^{j}\neq \emptyset$ for any $j\in J$.
By the definition of the condition, it is enough to show that $(\bar{F}^{p_{j}}, X^{p_{j}}\cap \mc{A}_{i})$ is $\sigma^{p_{j}}$-large at $i$ up to $\ell_{0}^{p_{j}}$ for some $i<k$. Assume not, then for each $i<k$ there exists a witness $\{(\bar{E}^{t,i}, Y^{t,i})\}_{t<s_{i}}$ so that $(\bar{F}^{p_{j}}, X^{p_{j}}\cap \mc{A}_{i})$ is not $\sigma^{p_{j}}$-large at $i$ up to $\ell_{0}^{p_{j}}$. Then the union $\{(\bar{E}^{t,i}, Y^{t,i})\}_{t<s_{i},i<k}$ indicates that $(\bar{F}^{p_{j}}, X^{p_{j}})$ is not $\sigma^{p_{j}}$-large by Remark \[rem:largeness\].1, which is a contradiction.
\[lem:color-survive2\] There exists $i<k$ such that $i\in \eta^{j}$ for any $j\in J$.
Assume that such $i<k$ does not exist. Then we have $\A i<k\,\E \bar{\ell}\,\E j\in J(i\notin \eta^{j}\wedge \ell_{1}^{p_{j}}<\bar{\ell})$. Thus, by $\BIII$, there exists $\ell\in\N$ such that $\A i<k\, \E j\in J(i\notin \eta^{j}\wedge \ell_{1}^{p_{j}}<\bar{\ell})$. By Lemma \[lem:unbdd-G\], there exists $p_{j'}\in\mc{G}$ such that $\ell_{1}^{p_{j'}}>\bar{\ell}$. Then $\eta^{j'}=\emptyset$ by the monotonicity of $\eta^{j}$, which contradicts Lemma \[lem:color-survive1\].
By Lemma \[lem:color-survive2\], pick a color $i<k$ such that $i\in \eta^{j}$ for every $j\in J$ and put $G:=\bigcup_{j\in J} F_{i}^{p_{j}}$. Then $G\subseteq \mc{A}_{i}$. Take $e_{\inf}\in \N$ so that $\A m\E n>m(n\in G)\leftrightarrow \A m\neg\pi(e_{\inf},m,G)$. Then, for large enough $j\in J$, $\sigma^{p_{j}}(e_{\inf},i)=1$ since “$G$ is finite” is never forced by an infinite Mathias pair. Thus, $G$ is infinite by the third clause of the definition of conditions. $G$ is $\Delta^{B}_{3}$ since $x\in G\leftrightarrow \E j\in J (\ell_{0}^{p_{j}}>x\wedge x\in F_{i}^{p_{j}})$ and $x\notin G\leftrightarrow \E j\in J (\ell_{0}^{p_{j}}>x\wedge x\notin F_{i}^{p_{j}})$. For any $e\in \N$, $\A m \neg\pi(e,m,G)$ holds if and only if $\E j\in J(\ell_{0}^{p_{j}}>e\wedge \sigma^{p_{j}}(e,i)=1)$. Thus, any $\Pi^{B\oplus G}_{2}$-formula is equivalent to a $\Sigma^{B}_{3}$-formula, and hence any $\Sigma^{B\oplus G}_{3}$-formula is equivalent to a $\Sigma^{B}_{3}$-formula.
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abstract: 'Motivated by lubrication problems, we consider a micropolar fluid flow in a 2D domain with a rough and free boundary. We assume that the thickness and the roughness are both of order $0<\e <<1$. We prove the existence and uniqueness of a solution of this problem for any value of $\e$ and we establish some a priori estimates. Then we use the two-scale convergence technique to derive the limit problem when $\e$ tends to zero. Moreover we show that the limit velocity and micro-rotation fields are uniquely determined via auxiliary well-posed problems and the limit pressure is given as the unique solution of a Reynolds equation.'
author:
- '[Mahdi Boukrouche Laetitia Paoli]{} [^1]'
title: Asymptotic analysis of a micropolar fluid flow in thin domain with a free and rough boundary
---
Lubrication, micropolar fluid, free and rough boundary, asymptotic analysis, two-scale convergence, Reynolds equation.
35Q35, 76A05, 76D08, 76M50.
Introduction
============
The theory of micropolar fluids, was introduced and formulated by A.C. Eringen in [@erin]. It aims to describe fluids containing suspensions of rigid particles in a viscous medium. Such fluids exhibit micro-rotational effects and micro-rotational inertia. Therefore they can support couple stress and distributed body couples. They form a class of fluids with nonsymmetric stress tensor for which the classical Navier-Stokes theory is inadequate since it does not take into account the effects of the micro-rotation. Experimental studies have showned that the micropolar model better represents the behavior of numerous fluids such as polymeric fluids, liquid crystals, paints, animal blood, colloidal fluids, ferro-liquids, etc., especially when the characteristic dimension of the flow becomes small (see for instance [@popel]). Extensive reviews of the theory and its applications can be found in [@ariman1; @ariman2] or in the books [@erin1999] and [@luk1] and also in more recent articles (see for example [@Bakr2011; @m2as05; @abdullallh2010; @geng2011]).
Motivated by lubrication theory where the domain of flow is usually very thin and the roughness of the boundary strongly affects the flow ([@bc]), we consider the motion of the micropolar fluid described by the equilibrium of momentum, mass and moment of momentum. More precisely, the velocity field of the fluid $u^{\e} = (u^{\e}_1,
u^{\e}_2 )$, the pressure $p^{\e}$ and the angular velocity of the micro-rotations of the particles $\om^{\e}$ satisfy the system
$$\begin{aligned}
&& u^{\e}_{t} - (\nu + \nu_r) \Delta u^{\e}
+ (u^{\e}\cdot\nabla) u^{\e}
+ \nabla p^{\e} = 2 \nu_r \; {\rm rot \,} \, \om^{\e} + f^{\e},
\label{2.1} \\
&& \qquad {\rm div } u^{\e} = 0,
\label{2.2} \\
&& \om^{\e}_{t} - \aal \Delta \om^{\e} + (u^{\e}\cdot\nabla) \; \om^{\e} +
4 \nu_{r} \om^{\e}
= 2 \nu_{r} \; {\rm rot \, } u^{\e} + g^{\e},\label{2.3}\end{aligned}$$
in the space-time domain $(0 , T)\times\Om^{\e}$ with $$\Omega^{\e} = \{z=(z_{1} , z_{2}) \in {{\mathbb R}}^{2}, \quad 0 < z_{1} < L, \quad
0 < z_{2} < \e h^{\e}(z_{1})\}, \quad h^{\e}(z_{1}) = h(z_{1} , {z_{1} \over \e})$$
where $h$ is a given smooth function, $f^{\e}$ and $g^{\e}$ are given external forces and moments, $\nu$ is the usual Newtonian viscosity, $\nu_r$ and $\aal$ are the micro-rotation viscosities, which are assumed to be positive constants ([@erin]).
The choice of the domain $\Omega^{\e}$ comes from one of the important fields of the theory of lubrication given by the study of self-lubricating bearings. These bearings are widely used in mechanical and electromechanical industry, to lubricate the main axis of rotation of a device, in order to prevent its endomagement.
Such bearings consist in an inner cylinder and a outer cylinder, and along a circumferencial section, one can see two non-concentric discs. The radii of the two cylinders are much smaller than their lengh and the gap between the two cylinders, which is fullfilled with a lubricant, is much smaller than their radii ([@Bayada-Chambat]). By assuming that the external fields and the flow do not depend on the coordinate along the longitudinal axis of the bearing, one can represent the fluid domain by $\Om^{\e}$ which is a 2D view of a cross section after a radial cut of the two circumferences. The boundary of $\Om^{\e}$ is $\partial\Om^{\e}=\bar{\Gamma}_{0}\cup\bar{\Gamma}_{L}^{\e}\cup\bar{\Gamma}_{1}^
{\e}$,
where $\Gamma_{0}=\{z\in \partial\Om^{\e} : z_{2}=0\}$ is the bottom, $\Gamma_{1}^{\e}=\{z\in \partial\Om^{\e} : z_{2}= \e h^{\e}(z_{1})\}$ is the upper strongly oscillating part, and $\Gamma_{L}^{\e}$ is the lateral part of the boundary. The surface of the inner cylinder, which corresponds to $\Gamma_0$, is in contact with the rotating axis of the device while the surface of the outer cylinder, which corresponds to $\Gamma_{1}^{\e}$, remains still.
Hence the boundary and initial conditions are given as follows $$\begin{aligned}
\om^{\e},\quad u^{\e} , \quad p^{\e}\quad \mbox{ are L-periodic with
respect to}~z_{1}
\label{eqn:er2.4a}\\
u^{\e}= U_0e_{1}=(U_0 , 0) , \quad \om^{\e} = W_0\quad \mbox{on} \quad (0 ,
T)\times\Gamma_{0}
\label{eqn:er2.5}\\
\om^{\e}=0, \quad u^{\e}\cdot n=0, \qquad {\partial u^{\e}\over \partial n} \cdot\tau =0
\qquad\mbox{on} \quad (0 , T)\times\Gamma_{1}^{\e}
\label{eqn:er2.4}\\
u^{\e} (0 , z) = u^{\e}_{0}(z) , \quad \om^{\e} (0 , z) =
\om^{\e}_{0}(z)\quad
{\rm for} \quad z\in\Om^{\e}
\label{eqn:er2.5a}
\end{aligned}$$
where $\tau$ and $n$ are respectively the tangent and normal unit vectors to the boundary of the domain $\Om^{\e}$. Let us observe that (\[eqn:er2.5\]) represents non-homogeneous Dirichlet conditions along $\Gamma_0$, which means adherence of the fluid to the boundary of the rotating inner cylinder, so $U_0$ and $W_0$ are two given functions of the time variable only. The second condition in (\[eqn:er2.4\]) is the nonpenetration boundary condition, while the last one is non-standard, and it means that the tangential component of the flux on $\Gamma^{\e}_{1}$ is equal zero ([@D.L]).
The choice of the particular scaling, with a roughness in inverse proportion to the thickness of the domain, is quite classical in lubrication theory. In [@bcc] and in [@bc] a Stokes flow is considered with adhering boundary conditions and Tresca boundary conditions at the fluid solid interface respectively. For other related works see also [@bucj1; @bucj2] or [@bacj] for instance.
We prove the existence and uniqueness of a weak solution $(u^{\e} ,
\om^{\e}, p^{\e})$ in adequate functional framework. Then we will establish some a priori estimates for the velocity, micro-rotation and pressure fields, independently of $\e$, and finally we will derive and study the limit problem when $\e$ tends to zero.
The paper is organized as follows. In Section \[secSP\] we give the variational formulation. Then, using an idea of J.L. Lions ([@JLLions78]), we consider the divergence free condition (\[2.2\]) as a constraint, which can be penalized, and we prove in Theorem \[th2.1\] the existence and uniqueness of a weak solution $(u^{\e} , \om^{\e}, p^{\e})$ for any value of $\e$. Let us emphasize that our proof ensures that the pressure (unique up to an additional function of time) belong to $H^{-1}(0 , T , L^{2}_{0}(\Om^{\e}))$. This result is more suitable for the next parts of our study, than $W^{-1, \infty}(0 , T ,
L^{2}_{0}(\Om^{\e}))$ obtained by J. Simon [@Simon1999] (see also Theorem 2.1 in [@galdi]).
In Section \[uniformEst\], we establish some a priori estimates for the velocity and micro-rotation fields in Proposition \[pro1\] and for the pressure in Proposition \[prop2\]. In Section \[twoscaleconv\], since we deal with an evolution problem, we extend first the classical two-scale convergence results ([@allaire; @nguetseng]) to a time-dependent setting and we use this technique to prove some convergence properties for the velocity in Proposition \[prop4.1\], the micro-rotation in Proposition \[prop4.2\], and the pressure in Proposition \[prop4.3\].
Then, in Section \[limitprobl\] we derive the limit problem when $\e$ tends to zero in Theorem \[limit\_pb\]. We notice that the trilinear and rotational terms, as well as the time derivative do not contribute when we pass to the limit. However the time variable remains in the limit problem as a parameter. We note also that the limit problem can be easily decoupled: we obtain a variational equality involving only the limit velocity and the limit pressure and another variational equality involving the limit micro-rotation. However, the micropolar nature of the fluid still appears in the limit problem for the velocity and pressure since we keep the viscosity $\nu + \nu_r$. Moreover we show in Proposition \[prop5.1\] that the limit velocity and micro-rotation fields are uniquely determined via auxiliary well-posed problems. In Proposition \[prop5.2\], we prove that the limit pressure is given as the unique solution of a Reynolds equation. Finally in Section \[open-Pbs\] we propose a generalization to the case where both the upper and the lower boundary of the fluid domain are oscillating.
Existence and uniqueness results {#secSP}
================================
We assume that $$\begin{aligned}
\label{Lper1}
{L\over \e}\in {\mathbb N}, \quad h : (z_{1} , \eta_1) \mapsto h(z_{1} , \eta_1)
\mbox{ is L-periodic in } z_{1}
\mbox{ and 1-periodic in } \eta_1,
\end{aligned}$$ so $h$ is L-periodic in $z_{1}$. We assume also that $$\begin{aligned}
\label{Lper2}
h \in {\cal C}^{\infty}([0 , L]\times {\mathbb R}), \quad {\partial h\over
\partial \eta_1} \quad \mbox{is 1-periodic in } \eta_1,
\end{aligned}$$ and there exist $h_{m}$ and $h_{M}$ such that $$\begin{aligned}
0< h_{m}=\min_{[0 , L]\times [0 , 1]}h(z_{1} , \eta_1),
\quad \mbox{and} \quad h_{M}=\max_{[0 , L]\times [0 , 1]}h(z_{1} ,
\eta_1).
\end{aligned}$$
\[lUW\] Let the functions ${\cal U}$, ${\cal W}$ be in ${\cal D}(-\infty , h_{m})$, and $U_{0}$, $W_{0}$ be in $H^{1}(0 , T)$, with ${\cal U}(0)=1$, ${\cal W}(0)=1$. We set $$U^{\e}(t ,z_{2})= {\cal U}^{\e}(z_{2})U_{0}(t)
= {\cal U}({z_{2}\over \e})U_{0}(t) ,
\qquad
W^{\e}(t ,z_{2})= {\cal W}^{\e}(z_{2})W_{0}(t)
= {\cal W}({z_{2}\over \e})W_{0}(t).$$ Then we have for all $ (t , z_1 )\in (0 , T)\times(0,L)$ $$\label{eqn:er2.6.1}
U^{\e} (t , 0)=U_0 (t), \, U^{\e} (t , \e h^{\e} (z_1 ))=0,
\, {\partial U^{\e}\over \partial z_{2}}(t , \e h^{\e} (z_1 ))=0,$$ $$\label{eqn:er2.7.1}
W^{\e} (t , 0)=W_0 (t), \quad W^{\e} (t , \e h^{\e} (z_1 ))=0.$$
Indeed, $U^{\e}(t , 0) = {\cal U}(0)U_{0}(t)=U_{0}(t)$, $U^{\e}(t , \e h^{\e}(z_{1})) = {\cal U}(h(z_{1} ,{z_{1}\over \e}
))U_{0}(t) =0$ and $${\partial U^{\e}\over \partial z_{2}}(t , \e h^{\e}(z_{1}))
= {1\over \e} {\cal U}' (h^{\e}(z_{1}))U_{0}(t)
= {1\over \e} {\cal U}' (h(z_{1} ,{z_{1}\over \e}))U_{0}(t)
=0,$$ thus (\[eqn:er2.6.1\]) follows. The proof is valid also for (\[eqn:er2.7.1\]).
We can now set $$\label{eqn:er2.6}
u^{\e} (t , z_{1} , z_{2}) = U^{\e} (t , z_{2})e_1 + v^{\e} (t , z_{1} , z_{2})$$ $$\label{eqn:er2.7}
\om^{\e} (t , z_{1} , z_{2}) = W^{\e} (t , z_{2} ) + Z^{\e} (t , z_{1} , z_{2})$$ with $U^{\e}$ and $W^{\e}$ satisfying (\[eqn:er2.6.1\]) (\[eqn:er2.7.1\]). Moreover $${\partial u^{\e}_{i}\over \partial z_{j}}= {\partial v^{\e}_{i}\over \partial
z_{j}} +
{\partial \over \partial z_{j}}(U^{\e} (\cdot , z_{2})e_1 )=
\left\{\begin{array}{ll}
{\partial v^{\e}_{i}\over \partial z_{j}} \quad {\rm if} \quad j=1,\\
{\partial v^{\e}_{i}\over \partial z_{j}} + {\partial U^{\e}\over
\partial z_{2}}(\cdot , z_{2})e_1 \quad {\rm if} \quad j=2
\end{array}\right.$$ and from (\[eqn:er2.6.1\]) $ {\partial U^{\e}\over\partial z_{2}}(t , z_{2})={\partial U^{\e}\over
\partial z_{2}}(t , \e h^{\e} (z_1))=0
\quad\mbox{ for } (t , z_{2})\in (0 , T)\times\Gamma_{1}^{\e}$ so $$\begin{aligned}
\label{e2.15}
{\partial u^{\e}_{i}\over \partial z_{j}}=
{\partial v^{\e}_{i}\over \partial z_{j}}\quad\mbox{ for} \quad j= 1, 2
\quad \mbox{ on } \quad (0 , T)\times\Gamma_{1}^{\e}.\end{aligned}$$ Recall also that $${\rm rot \, } u^{\e} = \frac{\partial u^{\e}_2}{\partial z_1}
- \frac{\partial u^{\e}_1}{\partial z_2}, \;\qquad \;
{\rm rot \, } \, \om^{\e} = \bigg(\frac{\partial \om^{\e}}{\partial z_2},
- \frac{\partial \om^{\e}}{\partial z_1}\bigg).$$ Then the problem (\[2.1\])-(\[eqn:er2.5a\]) becomes $$\begin{aligned}
\label{eqn:er2.8}
v^{\e}_{t} -(\nu +\nu_{r})\Delta v^{\e} + (v^{\e} \cdot \nabla) v^{\e}
+ U^{\e}{\partial v^{\e}\over \partial z_{1}}
+ (v^{\e})_2 {\partial U^{\e}\over \partial z_{2}} e_1 + \nabla p^{\e}
=
2 {\nu}_{r} {\rm rot \,}Z^{\e}
\\ + (\nu + \nu_r) {\partial^{2} U^{\e}\over \partial z_{2}^{2}}e_1
+ 2\nu_r {\partial W^{\e}\over \partial z_{2}}e_1
-{\partial U^{\e}\over \partial t}e_1 + f^{\e} \quad \mbox{in } (0 ,
T)\times\Om^{\e},\nonumber\end{aligned}$$ $$\begin{aligned}
{\rm div \,} v^{\e} = 0 \quad \mbox{in } \Om^{\e}, \quad t\in (0 , T),
\label{eqn:er2.9}\end{aligned}$$ $$\begin{aligned}
Z^{\e}_{t} - \aal\Delta Z^{\e}+ (v^{\e} \cdot \nabla)Z^{\e} + 4 \nu_{r}Z^{\e}
+ U^{\e}{\partial Z^{\e}\over \partial z_1 } +
(v^{\e})_2 {\partial W^{\e} \over \partial z_{2}}
=
2\nu_{r}{\rm rot \,} v^{\e}
\nonumber\\ + \aal {\partial^{2}
W^{\e}\over\partial
z_{2}^{2}}
- 2\nu_r {\partial U^{\e}\over\partial z_{2}} -
4\nu_rW^{\e} -{\partial W^{\e}\over \partial t} + g^{\e}\quad \mbox{in } (0 ,
T)\times\Om^{\e}, \label{eqn:er2.10}\end{aligned}$$ $$\label{eqn:er2.8a}
v^{\e}, Z^{\e} \ \mbox{\rm and } p^{\e}\ \mbox{\rm L-periodic in } z_{1},$$ $$\label{eqn:er2.11}
Z^{\e} = 0, \quad v^{\e} =0 \quad \mbox{\rm on} \quad (0 ,
T)\times\Gamma_{0},$$ $$\label{eqn:er2.11b}
Z^{\e}=0, \quad v^{\e}\cdot n=0, \qquad {\partial v^{\e}\over \partial n}
\cdot\tau =0
\quad \quad \mbox{\rm on} \quad (0 , T)\times\Gamma^{\e}_{1},$$ $$\begin{aligned}
v^{\e}(0, z) = v^{\e}_{0}(z) = u^{\e}_{0}(z) - U^{\e}(0 , z_2 )e_1 \quad
\mbox{\rm in } \Om^{\e},\label{eqn:er2.11a} \\
\quad Z^{\e}(0 , z) =
Z^{\e}_{0}(z) = \om^{\e}_{0}(z) - W^{\e}(0 , z_2 ) \quad \mbox{\rm in }
\Om^{\e}, \label{eqn:er2.11ab}\end{aligned}$$ where we have denoted by $(v^{\e})_2$ the second component of $v^{\e}$.
To define the weak formulation of the above problem (\[eqn:er2.8\])- (\[eqn:er2.11ab\]), we recall that $\Gamma^{\e}_{1}$ is defined by the equation $z_{2}= \e h^{\e}(z_{1})$, thus the unit outward normal vector to $\Gamma^{\e}_{1}$ is given by $$n=\frac{1}{\sqrt{ 1 + (\e (h^{\e})'(z_{1}))^2}} (-\e
(h^{\e})'(z_{1}) , 1)$$ and $v\cdot n= 0$ becomes $-\e (h^{\e})'(z_{1}) v_{1} + v_{2} = 0$ on $\Gamma^{\e}_{1}$. We consider now the following functional framework $$\tilde{V^{\e}}= \{ v \in {\cal C}^{\infty}(\overline{\Om^{\e}})^{2}:
\,\, v \,\,\mbox{\rm is L-periodic in }\,\, z_1,\,\,\, v_{|_{\Gamma_0}} =0,
\, -\e (h^{\e})'(z_{1}) v_{1} + v_{2} =0
\mbox{ on } \Gamma^{\e}_{1}\}$$ $$\tilde{H^{1}}^{\e}= \{Z \in {\cal C}^{\infty}(\overline{ \Om^{\e} }): \,\,\,Z
\,\,\,\mbox{\rm is L-periodic in }\,\,\, z_1,\,\,\,Z=0 \,\,\,{\rm
on }\,\,\,
\Gamma_0\cup\Gamma^{\e}_1\}$$ $$V^{\e} = {\rm closure \,\, of\,\, } \tilde{V^{\e}} \,\,
{\rm in } \,\, H^{1}(\Om^{\e})\times H^{1}(\Om^{\e}),\qquad
V^{\e}_{div} = \{v\in V^{\e} \, : \,
{\rm div \,} v = 0, \,\, \mbox{\rm in} \,\, \Om^{\e}\}$$ $$H^{\e} = {\rm closure \,\, of \, } \tilde{V^{\e}} \,\, {\rm in }\,\,
L^{2}(\Om^{\e}) \times L^{2}(\Om^{\e}), \qquad
{H^{1,}}^{\e} = {\rm closure \,\, of\,\, } \tilde{H^{1}}^{\e} \,\, {\rm in }
\,\, H^{1}(\Om^{\e}),$$ $${H^{0,}}^{\e} = {\rm closure \,\, of \, } \tilde{H^{1}}^{\e}\,\,
{\rm in }\,\, L^{2}(\Om^{\e}),
\quad L^{2}_{0}(\Om^{\e})=\{q\in L^{2}(\Om^{\e}) : \, \int_{\Om^{\e}} q(z)
dz=0\}.$$ We endowed these functional spaces with the inner products and norms defined by $$[\bar{v} ,\Theta] = (v , \varphi) + (Z , \psi) \mbox{ in }
H^{\e}\times {H^{0,}}^{\e} \mbox{ with the norm }
[\bar{v}] = [\bar{v}, \bar{v}]^{1\over 2}$$ $$[[\bar{v} , \Theta]] = (\nabla v , \nabla \varphi) + (\nabla Z ,
\nabla \psi)
\mbox{ in } V^{\e}\times {H^{1,}}^{\e} \mbox{ with the norm }
[[\bar{v}]] = [[\bar{v}, \bar{v}]]^{1\over 2}$$ for any pairs of functions $\bar{v}= (v , Z)$ and $\Theta =
(\varphi , \psi)$. The weak formulation of the problem (\[eqn:er2.8\])- (\[eqn:er2.11ab\]) is given by[**Problem $(P^{\e})$**]{} Find $$\bar{v}^{\e} = (v^{\e} , Z^{\e})\in
\Bigl( {\cal C}([0 , T]; H^{\e})\cap L^{2}(0 , T; V^{\e}_{div}) \Bigr) \times
\Bigr( {\cal C}([0 ,
T]; {H^{0,}}^{\e})\cap L^{2}(0 , T; {H^{1,}}^{\e}) \Bigr)$$ and $p^{\e}\in H^{-1}(0 , T ; L^{2}_{0}(\Om^{\e}))$, such that $$\begin{aligned}
\label{eqn:er2.14}
[{\partial\bar{v}^{\e} \over \partial t}(t) , \Theta^{\e}] +
a(\bar{v}^{\e}(t) , \Theta^{\e})
+B(\bar{v}^{\e}(t) ,\bar{v}^{\e}(t) , \Theta^{\e}) + {\cal
R}(\bar{v}^{\e}(t) ,
\Theta^{\e})=\nonumber\\
= (p^{\e}(t) \, , \, {\rm div }\varphi^{\e})
+ ({\cal F}(\bar{v}^{\e}(t)) , \Theta^{\e}) \qquad
\forall\Theta^{\e} = (\varphi^{\e} , \psi^{\e}) \in V^{\e}\times
{H^{1,}}^{\e},\end{aligned}$$ with the initial condition $$\label{eqn:er2.14a}
\bar{v}^{\e}(z , 0) = \bar{v^{\e}_{0}} (z) = ( v^{\e}_{0}(z) \, ,
\,Z^{\e}_{0}(z)),$$ where $$\begin{aligned}
({\cal F}(\bar{v}^{\e}(t)) , \Theta^{\e})=
-a(\bar{\xi^{\e}}(t),\Theta^{\e})
-B(\bar{\xi^{\e}}(t),\bar{v}^{\e}(t),\Theta^{\e})
- B(\bar{v}^{\e}(t),\bar{\xi^{\e}}(t), \Theta^{\e})
\nonumber\\
- {\cal
R}(\bar{\xi^{\e}}(t),\Theta^{\e})
- [{\partial \bar{\xi^{\e}} \over \partial t}(t) , \Theta^{\e}] +
[\bar{f}^{\e}(t) , \Theta^{\e}],
\quad \bar{\xi^{\e}} = (U^{\e}e_1 , W^{\e}),\end{aligned}$$ and for all $\bar{v}= (v , Z)$, $\bar{u}= (u , w)$, and $\Theta = (\varphi
, \psi)$ in $V^{\e}\times {H^{1,}}^{\e}$, $$\begin{aligned}
\left[\bar{f}^{\e} , \Theta\right] &=& (f^{\e} ,\varphi) +
(g^{\e} , \psi),\nonumber\\
a(\bar{v} , \Theta) &=& (\nu +\nu_{r})(\nabla v , \nabla \varphi)
+ \aal(\nabla Z, \nabla \psi),
\nonumber\\
{\cal R}(\bar{v} , \Theta) &=& - 2 \nu_{r}({\rm rot \,} Z , \varphi)
-
2 \nu_{r}({\rm rot \,} v , \psi) + 4\nu_{r} (Z , \psi) ,
\nonumber\\
B(\bar{v} ,\bar{u} , \Theta) &=& b(v , u , \varphi)
+ b_{1}(v , w , \psi)= \sum_{i, j=1}^{2}\int_{\Om^{\e}} v_{i}{\partial
u_{j}\over\partial z_{i}} \varphi_{j} dz
+\sum_{i=1}^{2}\int_{\Om^{\e}}v_{i} {\partial w\over\partial z_{i}} \psi dz.\end{aligned}$$
\[th2.1\] Let $T > 0$, $U^{\e}$ and $ W^{\e}$ be given as in [Lemma \[lUW\]]{}, $f^{\e}$ in $(L^{2}((0 , T)\times\Omega^{\e}))^{2}$, $g^{\e}$ in $L^{2}((0 , T)\times\Omega^{\e})$ and $(v^{\e}_{0} \, , \,Z^{\e}_{0})$ in $H^{\e}\times {H^{0,}}^{\e}$. Then problem $(P^{\e})$ admits a unique solution $(v^{\e} , Z^{\e}, p^{\e})$.
Following the techniques proposed by J.L.Lions in [@JLLions78], we construct a sequence of approximate solutions by relaxing the divergence free condition for the velocity field. More precisely we consider the following penalized problems $(P^{\e}_{\delta})$, with $\delta >0$:
[**Problem $(P^{\e}_{\delta})$**]{} Find $$\bar{v}^{\e}_{\delta} =(v^{\e}_{\delta}, Z^{\e}_{\delta} ) \in
\Bigl( {\cal C}([0 , T]; H^{\e})\cap L^{2}(0 , T; V^{\e}) \Bigr) \times \Bigl(
{\cal C}([0 ,
T]; {H^{0,}}^{\e})\cap L^{2}(0 , T; {H^{1,}}^{\e}) \Bigr)$$ such that $$\begin{aligned}
\label{cal(P)}
&&\qquad [{\partial \bar{v}^{\e}_{\delta}\over \partial t} , \Theta^{\e}] +
a(\bar{v}^{\e}_{\delta} , \Theta^{\e})
+B(\bar{v}^{\e}_{\delta} , \bar{v}^{\e}_{\delta} , \Theta^{\e})
+{1\over 2}\left\{
( v^{\e}_{\delta}{\rm div \,} v^{\e}_{\delta} \, , \, \varphi^{\e})
+(Z^{\e}_{\delta}{\rm div \,} v^{\e}_{\delta} \, , \, \psi^{\e}) \right\}
\nonumber\\
&&\qquad +{1\over \delta}({\rm div } v^{\e}_{\delta} \, , \, {\rm div
}\varphi^{\e})
= ({\cal F}(\bar{v}^{\e}_{\delta}) , \Theta^{\e})
- {\cal R}(\bar{v}^{\e}_{\delta} , \Theta^{\e}) \quad
\forall\Theta^{\e}
= (\varphi^{\e} , \psi^{\e}) \in V^{\e}\times {H^{1,}}^{\e},\end{aligned}$$ with the initial condition $$\begin{aligned}
\label{cal(Pc)}
\bar{v}^{\e}_{\delta}(0)= \bar{v^{\e}_{0}}.\end{aligned}$$ The first term on the right of the second line of (\[cal(P)\]) is the penalty term and the term $${1\over 2}\left\{
( v^{\e}_{\delta}{\rm div \,} v^{\e}_{\delta} \, , \, \varphi^{\e})
+(Z^{\e}_{\delta}{\rm div \,} v^{\e}_{\delta} \, , \, \psi^{\e}) \right\}$$ is added in order to vanish with $B ( \bar{v}^{\e}_{\delta} ,
\bar{v}^{\e}_{\delta} , \Theta^{\e})$ when $\Theta^{\e} = \bar v^{\e}_{\delta}$.
Hence the proof of Theorem \[th2.1\] is divided in two parts. First we prove the existence of a solution of $(P^{\e}_{\delta})$, for any $\delta >0$, by using a Galerkin method. Then we pass to the limit as $\delta$ tends to zero by applying compactness arguments and we prove that the limit solves problem $(P^{\e})$.
Since $V^{\e}$ and $ {H^{1,}}^{\e}$ are closed subspaces of $(H^1(\Om^{\e}))^2$ and $H^1(\Om^{\e})$, they admit Hilbertian bases, denoted as $(\Phi_{j})_{j \ge 1}$ and $ (\psi_{j})_{j\geq 1}$ respectively, which are orthonormal in $(H^1(\Om^{\e}))^2$ and $H^1(\Om^{\e})$ and are also orthogonal bases of $(L^2(\Om^{\e}))^2$ and $L^2(\Om^{\e})$. For all $m \ge 1$ we define $v^{\e}_{0 m}$ and $Z^{\e}_{0m}$ as the $L^2$-orthogonal projection of $v^{\e}_0$ and $Z^{\e}_0$ on the finite dimentional subspaces $\langle \Phi_1, \dots, \Phi_m \rangle$ and $\langle
\psi_1, \dots, \psi_m \rangle$ respectively and we let $\bar v^{\e}_{0 m} = (
v^{\e}_{0 m}, Z^{\e}_{0 m})$. Then we consider $\bar{v}^{\e}_{\delta m}=
(v^{\e}_{\delta m} , Z^{\e}_{\delta m})$, with $$\begin{aligned}
\label{jm}
v^{\e}_{\delta m} (t , x) = \sum_{j=1}^{m} v^{\e}_{\delta mj}(t)
\Phi_{j}(x),
\qquad Z^{\e}_{\delta m}(t , x) = \sum_{j=1}^{m} Z^{\e}_{\delta mj}(t)
\psi_{j}(x)\end{aligned}$$ such that $$\begin{aligned}
\label{cal(Pm)}
&&({\partial \bar{v}^{\e}_{\delta m}\over \partial t} , \Theta_{i}) +
a(\bar{v}^{\e}_{\delta m} , \Theta_{i})
+B(\bar{v}^{\e}_{\delta m} , \bar{v}^{\e}_{\delta m} , \Theta_{i})
+{1\over 2}
(v^{\e}_{\delta m}{\rm div \,} v^{\e}_{\delta m} \, , \, \Phi_{i})
\nonumber\\
&&+{1\over 2}(Z_{\delta m}{\rm div \,} v^{\e}_{\delta m} \, , \, \psi_{i})
+{1\over \delta}({\rm div } v^{\e}_{\delta m} \, , \, {\rm div
}\Phi_{i})
= ({\cal F}(\bar{v}^{\e}_{\delta m}) , \Theta_{i})
- {\cal R}(\bar{v}^{\e}_{\delta m} , \Theta_{i})
\nonumber\\ &&\forall \Theta_i =
(\Phi_i, \psi_i), \ 1\leq i\leq
m,
\\
&& {\bar{v}^{\e}_{\delta m}}(0)= {\bar{v}}^{\e}_{0m }.\label{cal(Pcm)}\end{aligned}$$ By taking $\psi_{i}= 0$ in (\[cal(Pm)\]) we deduce $$\begin{aligned}
\label{Pm1}
({\partial v^{\e}_{\delta m}\over \partial t} , \Phi_{i}) &+&
(\nu+\nu_{r}) (\nabla v^{\e}_{\delta m} , \nabla \Phi_{i})
+b(v^{\e}_{\delta m} , v^{\e}_{\delta m} , \Phi_{i})
+{1\over 2}(v^{\e}_{\delta m}{\rm div } v^{\e}_{\delta m} \, , \,
\Phi_{i})
\nonumber\\
&+&{1\over \delta}({\rm div \, } v^{\e}_{\delta m} \, , \, {\rm div }\Phi_{i})
= ({\cal F}_{1}(v^{\e}_{\delta m}) , \Phi_{i})
+ 2\nu_{r} ({\rm rot \,} Z^{\e}_{\delta m} , \Phi_{i}) \qquad 1\leq i\leq
m, \qquad
\\
&& v^{\e}_{\delta m} (0)= v^{\e}_{0m } ,\label{cal(Pcmv)}\end{aligned}$$ and by taking $\Phi_{i}= 0$ in (\[cal(Pm)\]) we deduce $$\begin{aligned}
\label{Pm2}
({\partial Z^{\e}_{\delta m} \over \partial t} , \psi_{i}) &+&
\alpha (\nabla Z^{\e}_{\delta m} , \nabla \psi_{i})
+b_{1}(v^{\e}_{\delta m} , Z^{\e}_{\delta m} , \psi_{i})
+{1\over 2}
(Z^{\e}_{\delta m}{\rm div \,} v^{\e}_{\delta m} \, , \psi_{i})
= ({\cal F}_{2}(v^{\e}_{\delta m}) , \psi_{i})
\nonumber\\
&&+2\nu_{r} ({\rm rot \,} v^{\e}_{\delta m} , \psi_{i})
- 4\nu_{r} (Z^{\e}_{\delta m} \, , \, \psi_{i}) \qquad
1\leq i\leq m,\qquad
\\
&& Z^{\e}_{\delta}(0)= Z^{\e}_{0 m} , \label{cal(Pcmz)}\end{aligned}$$ where $$\begin{aligned}
\label{F}
({\cal F}_{1}(v^{\e}_{\delta m}) , \Phi_{i})&=&
-(\nu+\nu_{r}) (\nabla U^{\e}e_{1} , \nabla \Phi_{i})
- b(U^{\e} e_{1}\, ,\, v^{\e}_{\delta m} \, ,\, \Phi_{i})
- b(v^{\e}_{\delta m} \, ,\, U^{\e}_{\delta}e_{1}\, ,\, \Phi_{i})
\nonumber\\
&&+ 2\nu_{r}({\partial W^{\e} \over \partial z_{2}}e_{1} \, ,\,
\Phi_{i})
- ({\partial U^{\e} \over \partial t}e_{1} \, ,\, \Phi_{i})
+ (f^{\e} \, ,\, \Phi_{i}),\end{aligned}$$ and $$\begin{aligned}
\label{FZ}
({\cal F}_{2}(v^{\e}_{\delta m}) , \psi_{i})&=&
-\alpha (\nabla W^{\e} , \nabla \psi_{i})
- b_{1}(U^{\e}e_{1} \, ,\, Z^{\e} \, ,\, \psi_{i})
- b_{1}(v^{\e}_{\delta m} \, ,\, W^{\e}\, ,\, \psi_{i})
\nonumber\\
&& - 2\nu_{r}( {\partial U^{\e} \over \partial z_{2}} \, ,\, \psi_{i}) -4
\nu_{r} (W^{\e} \, ,\, \psi_{i})
- ({\partial W^{\e} \over \partial t} \, ,\, \psi_{i})
+ (g^{\e} \, ,\, \psi_{i}).\end{aligned}$$
Taking (\[jm\]) into account, we deduce from (\[Pm1\])-(\[FZ\]) a system of (nonlinear) differential equations for the unknown scalar functions $( v^{\e}_{\delta m i}, Z^{\e}_{\delta m i})_{1 \le i \le m}$, which possesses an unique maximal solution in $(H^1(0, T_m))^m$ with $T_m \in (0, T]$.
In order to prove that this solution is defined on the whole time interval $[0,T]$, we will establish some a priori estimates for $v^{\e}_{\delta m}$ and $Z^{\e}_{\delta m}$, independently of $m$. More precisely, we multiply the two sides of (\[Pm1\]) by $v^{\e}_{\delta mi}(t)$ and the two sides of (\[Pm2\]) by $Z^{\e}_{\delta mi}(t)$, then we sum for $i$ from $1$ to $m$, to get, with $\|\cdot\|=\|\cdot\|_{L^{2}(\Omega^{\e})}$, the following equations $$\begin{aligned}
\label{Pm1ss}
{1\over 2}{\partial\over \partial t} (\|v^{\e}_{\delta m}\|^{2}) &+&
(\nu+\nu_{r}) \|\nabla v^{\e}_{\delta m}\|^{2}
+b(v^{\e}_{\delta m} , v^{\e}_{\delta m} , v^{\e}_{\delta m})
+{1\over 2}(v^{\e}_{\delta m}{\rm div } v^{\e}_{\delta m} \, , \,
v^{\e}_{\delta m})
+{1\over \delta}\|{\rm div } v^{\e}_{\delta m}\|^{2}
\nonumber\\
&&= ({\cal F}_{1}(v^{\e}_{\delta m}) , v^{\e}_{\delta m})
+ 2\nu_{r} ({\rm rot \,} Z^{\e}_{\delta m} , v^{\e}_{\delta m}),\qquad\end{aligned}$$ $$\begin{aligned}
\label{Pm2ss}
{1\over 2}{\partial \over \partial t} ( \||Z^{\e}_{\delta m}\|^{2}) &+&
\alpha \|\nabla Z^{\e}_{\delta m}\|^{2}
+b_{1}(v^{\e}_{\delta m} , Z^{\e}_{\delta m} , Z^{\e}_{\delta m})
+{1\over 2}
(Z^{\e}_{\delta m}{\rm div \,} v^{\e}_{\delta m} \, , Z^{\e}_{\delta m})
= ({\cal F}_{2}(v^{\e}_{\delta m}) , Z^{\e}_{\delta m})
\nonumber\\
&&+2\nu_{r} ({\rm rot \,} v^{\e}_{\delta m} , Z^{\e}_{\delta m})
- 4\nu_{r} (Z^{\e}_{\delta m} \, , \, Z^{\e}_{\delta m}) .\end{aligned}$$ By integration by parts and using the boundary conditions (\[eqn:er2.8a\])-(\[eqn:er2.11b\]), we obtain that $$\begin{aligned}
b(v^{\e}_{\delta m} , v^{\e}_{\delta m} , v^{\e}_{\delta m})
+b_{1}(v^{\e}_{\delta m} , Z^{\e}_{\delta m} , Z^{\e}_{\delta m})
+{1\over 2}(v^{\e}_{\delta m}{\rm div } v^{\e}_{\delta m} \, , \,
v^{\e}_{\delta m})
+{1\over 2}
(Z^{\e}_{\delta m}{\rm div \,} v^{\e}_{\delta m} \, , Z^{\e}_{\delta m})
= 0,\end{aligned}$$ and $$\begin{aligned}
b(U^{\e} e_{1}\, ,\, v^{\e}_{\delta m} \, ,\, v^{\e}_{\delta m})
+ b_{1}(U^{\e}e_{1} \, ,\, Z^{\e}_{\delta m} \, ,\, Z^{\e}_{\delta m})
= 0.\end{aligned}$$ Thus by the addition of (\[Pm1ss\]) and (\[Pm2ss\]) we obtain $$\begin{aligned}
\label{eq2.28E}
\qquad\qquad {1\over 2}{\partial\over \partial t} (\|v^{\e}_{\delta m}\|^{2}
+\||Z^{\e}_{\delta m}\|^{2})
+(\nu+\nu_{r}) \|\nabla v^{\e}_{\delta m}\|^{2} + \alpha \|\nabla
Z^{\e}_{\delta m}\|^{2}
+{1\over \delta}\|{\rm div } v^{\e}_{\delta m}\|^{2}
= \Xi\end{aligned}$$ with $$\begin{aligned}
\label{Xi}
\Xi&=&
2\nu_{r} ({\rm rot \,} Z^{\e}_{\delta m} , v^{\e}_{\delta m})
+2\nu_{r} ({\rm rot \,} v^{\e}_{\delta m} , Z^{\e}_{\delta m})
- 4\nu_{r} \|Z^{\e}_{\delta m}\|^{2}
-(\nu+\nu_{r}) (\nabla U^{\e}e_{1} , \nabla v^{\e}_{\delta m})
\nonumber\\
&&
-\alpha (\nabla W^{\e} , \nabla Z^{\e}_{\delta m})
- b(v^{\e}_{\delta m} \, ,\, U^{\e}e_{1}\, ,\, v^{\e}_{\delta m})
- b_{1}(v^{\e}_{\delta m} \, ,\, W^{\e}\, ,\, Z^{\e}_{\delta m})
+ 2\nu_{r}({\rm rot \,} W^{\e} \, ,\, v^{\e}_{\delta m})
\nonumber\\
&&
+ 2\nu_{r}({\rm rot \,} U^{\e}e_{1}\, ,\, Z^{\e}_{\delta m})
-4 \nu_{r} (W^{\e} \, ,\, Z^{\e}_{\delta m})
- ({\partial U^{\e} e_{1} \over \partial t} \, ,\, v^{\e}_{\delta m})
- ({\partial W^{\e} \over \partial t} \, ,\, Z^{\e}_{\delta m})
\nonumber\\
&&
+ (f^{\e} \, ,\, v^{\e}_{\delta m})
+ (g^{\e} \, ,\, Z^{\e}_{\delta m}).\end{aligned}$$ Using Young’s inequality we have $$\begin{aligned}
\label{esR}
2 \nu_{r}|({\rm rot \,} {Z^{\e}_{\delta m}} , {v^{\e}_{\delta m}})|
\leq
2 \nu_{r}\|{\rm rot \,} {Z^{\e}_{\delta m}}\| \|{v^{\e}_{\delta m}}\|
\leq {\aal\over 4}\|\nabla {Z^{\e}_{\delta m}}\|^{2} + {4\nu_{r}^{2}\over
\aal}\| {v^{\e}_{\delta m}}\|^{2},\end{aligned}$$ $$\begin{aligned}
2\nu_{r} | ( {\rm rot \,} v^{\e}_{\delta m} , Z^{\e}_{\delta m}) |
&& \leq
2 \nu_{r}\|{\rm rot \,} {v^{\e}_{\delta m}}\|\|{Z^{\e}_{\delta m}}\|
\leq {\nu_{r}\over 4} \| {\rm rot \,} {v^{\e}_{\delta m}}\|^{2}
+ 4\nu_{r}\|{Z^{\e}_{\delta m}}\|^{2} \\
&& \leq {\nu_{r}\over 2} \|\nabla {v^{\e}_{\delta m}}\|^{2}
+ 4\nu_{r}\|{Z^{\e}_{\delta m}}\|^{2},\end{aligned}$$ $$\begin{aligned}
\label{aa}
(\nabla U^{\e}e_{1} , \nabla v^{\e}_{\delta m})
\leq
{1\over 2}\|\nabla ({v^{\e}_{\delta m}})\|^{2}
+ {1\over 2}\|{\partial U^{\e}\over \partial z_{2}}\|^{2},\end{aligned}$$ $$\begin{aligned}
\label{al}
(\nabla W^{\e} , \nabla Z^{\e}_{\delta m})
\leq
{1 \over 4}\|\nabla {Z^{\e}_{\delta m}}\|^{2}
+ \|{\partial W^{\e}\over \partial z_{2}}\|^{2},\end{aligned}$$ $$\begin{aligned}
\label{bb}
b(v^{\e}_{\delta m} \, ,\, U^{\e} e_{1}\, ,\, v^{\e}_{\delta m})
\leq
\|({v^{\e}_{\delta m}})_{2}\|
\|{\partial U^{\e}\over \partial z_{2}}\|_{\infty}
\|({v^{\e}_{\delta m}})_{1}\|
\leq \|{\partial U^{\e}\over \partial z_{2}}\|_{\infty}\|{v^{\e}_{\delta
m}}\|^{2},\end{aligned}$$ $$\begin{aligned}
\label{bb1}
b_{1}(v^{\e}_{\delta m} \, ,\, W^{\e}\, ,\, Z^{\e}_{\delta m})
\leq \|({v^{\e}_{\delta m}})_{2}\|\|{\partial W^{\e} \over \partial
z_{2}}\|_{\infty}
\|Z^{\e}_{\delta m}\|
\leq \frac{1}{2} \|{\partial W^{\e} \over \partial z_{2}}\|_{\infty}\left(
\|{v^{\e}_{\delta m}}\|^{2} + \|Z^{\e}_{\delta m}\|^{2}\right),\end{aligned}$$ $$\begin{aligned}
\label{rr}
&& 2\nu_{r}( {\rm rot \,} W^{\e} \, ,\, v^{\e}_{\delta m})
+ 2\nu_{r}({\rm rot \,} U^{\e}e_{1}\, ,\, Z^{\e}_{\delta m})
-4 \nu_{r} (W^{\e} \, ,\, Z^{\e}_{\delta m})
\nonumber\\
&=& + 2 \nu_{r} \left({\partial W^{\e} \over \partial z_{2}} , {(v^{\e}_{\delta
m})}_{1}\right)
- 2 \nu_{r} \left({\partial U^{\e} \over \partial z_{2}} ,
Z^{\e}_{\delta
m}\right)
- 4\nu_{r} (W^{\e} , Z^{\e}_{\delta m})
\nonumber\\
&\leq& \nu_{r}\|{v^{\e}_{\delta m}}\|^{2} + 2 \nu_{r}\|Z^{\e}_{\delta
m}\|^{2} +
\nu_{r}\| {\partial W^{\e} \over \partial z_{2}}\|^{2} +
\nu_{r}\|{\partial U^{\e} \over \partial z_{2}}\|^{2} + 4\nu_{r}\|
W^{\e}\|^{2}.\end{aligned}$$ So we have $$\begin{aligned}
\label{XiMag}
\Xi&\leq&
({\nu\over 2}+\nu_{r})\|\nabla{v^{\e}_{\delta m}}\|^{2}
+ {\alpha\over 2}\|\nabla{Z^{\e}_{\delta m}}\|^{2}
+\left(2+ 4 \nu_{r} + \frac{1}{2} \|{\partial W^{\e}(t) \over \partial
z_{2}}\|_{\infty}
\right)
\|{Z^{\e}_{\delta m}}\|^{2}
\nonumber\\
&&+\left(2+ \nu_{r} + {4\nu_{r}^2\over \alpha} +\|{\partial U^{\e}(t)\over
\partial z_{2}}\|_{\infty}
+ \frac{1}{2} \|{\partial W^{\e}(t) \over \partial z_{2}}\|_{\infty}
\right)\| {v^{\e}_{\delta m}}\|^{2}
\nonumber\\
&&+{(\nu+\nu_{r})\over 2}\|{\partial U^{\e}(t)\over \partial z_{2}}\|^{2}
+ \alpha\|{\partial W^{\e}(t)\over \partial z_{2}}\|^{2}
+
\nu_{r}\| {\partial W^{\e}(t) \over \partial z_{2}}\|^{2}
+\nu_{r}\|{\partial U^{\e}(t) \over \partial z_{2}}\|^{2}
\nonumber\\ &&
+
4\nu_{r}\| W^{\e}(t)\|^{2}
+\left\|{\partial U^{\e} (t) \over \partial t}\right\|^{2}
+\left\|{\partial W^{\e}(t) \over \partial t}\right\|^{2}
+ \|f^{\e}(t)\|^{2} + \|g^{\e}(t)\|^{2}.\end{aligned}$$ From (\[eq2.28E\])-(\[XiMag\]), we get $$\begin{aligned}
\label{eq2.31}
{1\over 2}{\partial \over\partial t}(
[\bar{v}^{\e}_{\delta m}]^{2})
+
{k\over 2}[[\bar{v}^{\e}_{\delta m}]]^{2}
+{1\over \delta}\|{\rm div\, } v^{\e}_{\delta m}\|^{2}
\leq A(t) [\bar{v}^{\e}_{\delta m}]^{2} + B (t),\end{aligned}$$ where $k= \min \{\nu ,
\alpha\}$ and $A$ and $B$ belong to $L^1(0,T)$ such that $A(t) \ge 2$ and $B(t) \ge 0$ almost everywhere on $[0,T]$. Moreover $A$ and $B$ depend neither on $m$ nor on $\delta$.
For any $t \in (0,T_m)$ we can integrate the inequality (\[eq2.31\]) over $[0,t]$: we obtain $$\begin{aligned}
\label{eqint2.31}
[\bar{v}^{\e}_{\delta m}(t)]^{2} +
k \int_{0}^{t} [[\bar{v}^{\e}_{\delta m}(s)]]^{2} ds
+{2\over \delta}\int_{0}^{t} \|{\rm div\, } v^{\e}_{\delta m}(s)\|^{2}ds
\leq [\bar{v}^{\e}_{0}]^{2} \nonumber\\+ 2\int_{0}^{t} A(t)[\bar{v^{\e}_{\delta
m}}(s)]^{2}ds + 2{\cal B},
\end{aligned}$$ with ${\cal B}= \int_{0}^{T}B(t) dt$. So by Grönwall’s inequality, we deduce first that $$[\bar{v}^{\e}_{\delta m}(t)]^{2}
\leq ( [\bar{v}^{\e}_{0}]^{2} + 2 {\cal B}) e^{2 {\cal A}}
\quad \mbox{with}\quad {\cal A}= \int_{0}^{T}A(t) dt.$$ Thus $\bar{v}^{\e}_{\delta m}$ is defined on the whole interval $[0,T]$ and $$\begin{aligned}
\label{Maj1}
\sup_{t\in [0 , T]}[\bar{v}^{\e}_{\delta m}(t)]^{2} \leq C.\end{aligned}$$ Then from (\[eqint2.31\]) and (\[Maj1\]), we deduce $$\begin{aligned}
\label{Maj2}
{1\over \delta}\int_{0}^{T} \|{\rm div\, } v^{\e}_{\delta m}(t)\|^{2}dt \le C,
\quad \int_{0}^{T} [[\bar{v}^{\e}_{\delta m}(t)]]^{2} dt\leq C,
\end{aligned}$$ where here and in what follows $C's$ denotes various constants which depend neither on $m$ nor on $\delta$.
We need now to look at the time derivative of $v^{\e}_{\delta
m}$ and $Z^{\e}_{\delta m}$. Let $\Theta^{\e}=(\varphi^{\e} , \psi^{\e}) \in (H^1_0(\Om^{\e}))^2 \times
H^1_0(\Om^{\e}) \subset V^{\e}\times H^{1,
\e}$. There exists a sequence $(q^{\e}_{i} , k^{\e}_{i})_{i \ge 1}$ in ${\mathbb R}^{2}$ such that $$\Theta^{\e}_{p} = (\varphi^{\e}_{p}, \psi^{\e}_{p})
\to (\varphi^{\e} , \psi^{\e}) \quad \mbox{ strongly in } V^{\e}\times H^{1,
\e}$$ with $$\varphi^{\e}_{p} = \sum_{i=1}^{p}q^{\e}_{i}\Phi_{i},
\quad \psi^{\e}_{p} = \sum_{i=1}^{p}k^{\e}_{i}\psi_{i} \qquad \forall p \ge 1.$$ Let $p \ge m$. Reminding that $(\Phi_i)_{i \ge 1}$ and $(\psi_i)_{i \ge 1}$ are orthogonal bases of $(L^2(\Om^{\e}))^2$ and $L^2(\Om^{\e})$ respectively, we get $$\begin{aligned}
\left( \frac{\partial v^{\e}_{\delta m}}{\partial t}, \varphi^{\e}_p \right) =
\sum_{j=1}^m ( v^{\e}_{\delta m j})' (t) (\Phi_j, \varphi^{\e}_p) =
\sum_{j=1}^m ( v^{\e}_{\delta m j})' (t) (\Phi_j, \varphi^{\e}_m) = \left(
\frac{\partial v^{\e}_{\delta m}}{\partial t}, \varphi^{\e}_m \right),\end{aligned}$$ and $$\begin{aligned}
\left( \frac{\partial Z^{\e}_{\delta m}}{\partial t}, \psi^{\e}_p \right) =
\sum_{j=1}^m ( Z^{\e}_{\delta m j})' (t) (\Phi_j, \psi^{\e}_p) = \sum_{j=1}^m (
Z^{\e}_{\delta m j})' (t) (\Phi_j, \psi^{\e}_m) = \left( \frac{\partial
Z^{\e}_{\delta m}}{\partial t}, \psi^{\e}_m \right).\end{aligned}$$ Since $ \frac{\partial \bar v^{\e}_{\delta m}}{\partial t} \in L^2(0,T; V^{\e}
\times H^{1, \e})$, we can pass to the limit as $p$ tends to $+ \infty$ i.e $$\begin{aligned}
\left( \frac{\partial v^{\e}_{\delta m}}{\partial t}, \varphi^{\e} \right) =
\left( \frac{\partial v^{\e}_{\delta m}}{\partial t}, \varphi^{\e}_m \right),
\quad \left( \frac{\partial Z^{\e}_{\delta m}}{\partial t}, \psi^{\e} \right) =
\left( \frac{\partial Z^{\e}_{\delta m}}{\partial t}, \varphi^{\e}_m \right).\end{aligned}$$ Then, by using Green’s formula and (\[Pm1\]) $$\begin{aligned}
\label{dg1}
\left( {\partial { v^{\e}_{\delta m}} \over \partial t}, \varphi^{\e} \right)
=\Bigl( (\nu +\nu_{r})\Delta { v^{\e}_{\delta m}}
- ({ v^{\e}_{\delta m}} \cdot \nabla) { v^{\e}_{\delta m}}
-{1\over 2} { v^{\e}_{\delta m}}{\rm div\,}{ v^{\e}_{\delta m}}
\nonumber\\ + {\cal F}_{1}({ v^{\e}_{\delta m}})+ 2\nu_{r}{\rm
rot\,}{Z^{\e}_{\delta m}}
+ {1\over \delta} \nabla ({\rm div\,}{ v^{\e}_{\delta m}}), \varphi^{\e}_m
\Bigr),\end{aligned}$$ and from (\[Pm2\]) $$\begin{aligned}
\label{dg2}
\Bigl( {\partial Z^{\e}_{\delta m} \over \partial t}, \psi^{\e} \Bigr) =
\Bigl( \alpha\Delta {
Z^{\e}_{\delta m}}
- ({ v^{\e}_{\delta m}} \cdot \nabla) { Z^{\e}_{\delta m}}
-{1\over 2} { Z^{\e}_{\delta m}}{\rm div\,}{ v^{\e}_{\delta m}}
+ {\cal F}_{2}({ v^{\e}_{\delta m}})+ 2\nu_{r}{\rm rot\,}{v^{\e}_{\delta m}}
\nonumber \\
-
4\nu_{r} Z^{\e}_{\delta m},\psi^{\e}_m \Bigr)\end{aligned}$$ and from (\[F\]) $${\cal F}_{1}({ v^{\e}_{\delta m}})= (\nu+\nu_{r}) {\partial^{2} U^{\e}\over
\partial z^{2}_{2}}e_{1}
- U^{\e}{\partial { v^{\e}_{\delta m}} \over \partial z_{1}}
- ({ v^{\e}_{\delta m}})_{2} {\partial U^{\e}\over \partial z_{2}}e_{1}
+ 2\nu_{r} {\partial W^{\e}\over \partial z_{2}} e_{1}
- {\partial U^{\e}\over \partial t}e_{1}+ f^{\e},$$ and from (\[FZ\]) $${\cal F}_{2}({ v^{\e}_{\delta m}})= \alpha {\partial^{2} W^{\e}\over
\partial z^{2}_{2}}
- U^{\e}{\partial { Z^{\e}_{\delta m}} \over \partial z_{1}}
- ({ v^{\e}_{\delta m}})_{2} {\partial W^{\e}\over \partial z_{2}}
+ 2\nu_{r} {\partial U^{\e}\over \partial z_{2}} - 4 \nu_{r} W^{\e}-
{\partial W^{\e}\over \partial t} + g^{\e}.$$ As ${v^{\e}_{\delta m}}$ is bounded in $L^{2}(0 , T ; (H^{1}(\Omega^{\e})^{2}))$ independently of $m$ and $\delta$, then $\Delta { v^{\e}_{\delta m}}$ and $\nabla ({\rm div\,}{ v^{\e}_{\delta m}})$ are also bounded in $L^{2}(0 , T ;
(H^{-1}(\Omega^{\e}))^{2})$ independently of $m$ and $\delta$. Similarly, since ${Z^{\e}_{\delta m}}$ is bounded in $L^{2}(0 , T ;
H^{1}(\Omega^{\e}))$ independently of $m$ and $\delta$, then ${\rm rot\,}{Z^{\e}_{\delta m}}$ is also bounded in $L^{2}(0 , T ;
(L^{2}(\Omega^{\e}))^{2})$ independently of $m$ and $\delta$. By assumption, $f^{\e} \in (L^{2}((0 , T)\times\Om^{\e})^{2}$, $g^{\e} \in
L^{2}((0 ,
T)\times\Om^{\e})$, and from Lemma \[lUW\], $U^{\e}$ and $W^{\e}$ belong to $H^{1}(0 , T)\times {\cal
D}((-\infty , h_{m}))$. Thus we infer that ${\cal F}_{1}({ v^{\e}_{\delta m}})$ and ${\cal F}_{2}({ v^{\e}_{\delta m}})$ are bounded in $L^{2}(0 , T;(L^2(\Omega^{\e}))^{2})$ and $L^{2}(0 ,
T;L^2(\Omega^{\e})$, independently of $m$ and $\delta$. Moreover let $\varphi\in ( H^{1}(\Omega^{\e}))^2$, we have $$\begin{aligned}
| (({ v^{\e}_{\delta m}} \cdot \nabla){ v^{\e}_{\delta m}} \, ,\,
\varphi)|
\leq \|{ v^{\e}_{\delta m}}\|_{L^{3}(\Omega^{\e})}\|\nabla { v^{\e}_{\delta
m}}\|_{L^{2}(\Omega^{\e})}\|\varphi\|_{L^{6}(\Omega^{\e})}.
\end{aligned}$$ Using now the classical inequality $$\|u\|_{L^{3}(\Omega^{\e})}\leq
\|u\|^{1/2}_{L^{2}(\Omega^{\e})}\|u\|^{1/2}_{L^{6}(\Omega^{\e})}
\quad \forall u\in L^{6}(\Om^{\e}),$$ and the continuous injection of $H^{1}(\Omega^{\e})$ in $L^{6}(\Omega^{\e})$, there exists a constant $C$ such that $$\begin{aligned}
| (({ v^{\e}_{\delta m}} \cdot \nabla) { v^{\e}_{\delta m}} \, ,\,
\varphi)|
\leq \left( C\|{ v^{\e}_{\delta m}}\|^{1/2}_{L^{2}(\Omega^{\e})}\|\nabla {
v^{\e}_{\delta m}}\|^{3/2}_{L^{2}(\Omega^{\e})}\right)
\|\varphi\|_{H^{1}(\Omega^{\e})}.
\end{aligned}$$ So we get $$\begin{aligned}
\|({ v^{\e}_{\delta m}} \cdot \nabla) { v^{\e}_{\delta
m}}\|_{(H^{1}(\Omega^{\e}))'}
\leq C\|{ v^{\e}_{\delta m}}\|^{1/2}_{L^{2}(\Omega^{\e})}\|\nabla {
v^{\e}_{\delta m}}\|^{3/2}_{L^{2}(\Omega^{\e})}
\end{aligned}$$ then $$\begin{aligned}
\int_{0}^{T}\| ({ v^{\e}_{\delta m}} \cdot \nabla) { v^{\e}_{\delta
m}}\|^{4/3}_{(H^{1}(\Omega^{\e}))'} dt
&\leq& C^{4/3}\int_{0}^{T}\|{ v^{\e}_{\delta
m}}\|^{2/3}_{L^{2}(\Omega^{\e})}\|\nabla { v^{\e}_{\delta
m}}\|^{2}_{L^{2}(\Omega^{\e})}
\nonumber\\
&\leq& C^{4/3}\|{ v^{\e}_{\delta m}}\|^{2/3}_{L^{\infty}(0 , T ;
L^{2}(\Omega^{\e}))}\|\nabla { v^{\e}_{\delta m}}\|^{2}_{L^{2}((0 ,
T)\times\Omega^{\e})}.
\end{aligned}$$ With the same arguments, we deduce similar result for ${v^{\e}_{\delta m}}{\rm div\,}{ v^{\e}_{\delta m}}$, $({ v^{\e}_{\delta m}} \cdot \nabla) { Z^{\e}_{\delta m}}$ and ${ Z^{\e}_{\delta m}}{\rm div\,}{ v^{\e}_{\delta m}}$. Finally, recalling that $(\Phi_i)_{i \ge 1}$ and $(\psi_i)_{i \ge 1}$ are $H^1$-orthonormal, we get $$\begin{aligned}
\| \varphi^{\e}_m \|_{(H^1(\Om^{\e}))^2} \le \| \varphi^{\e}
\|_{(H^1(\Om^{\e}))^2}, \quad \| \psi^{\e}_m \|_{H^1(\Om^{\e})} \le \|
\psi^{\e} \|_{H^1(\Om^{\e})} \quad \forall m \ge 1.\end{aligned}$$ So from (\[dg1\]) and (\[dg2\]) we see that there exists a constant $C$ such that $$\begin{aligned}
\label{boudv}
\|{\partial {v^{\e}_{\delta m}} \over\partial t}
\|_{L^{4/3}(0 , T ;( H^{-1}(\Omega^{\e}))^{2})}\leq C,
\quad
\|{\partial {Z^{\e}_{\delta m}} \over \partial t}
\|_{L^{4/3}(0 , T ; H^{-1}(\Omega^{\e}))}\leq C.\end{aligned}$$ From the estimates (\[Maj1\])-(\[Maj2\]) we infer that there exists a subsequence (denoted also by) $\bar{v}^{\e}_{\delta m}$ such that $$\begin{aligned}
\label{lim1m}
\bar{v}^{\e}_{\delta m} \tow \bar{v}^{\e}_{\delta} \quad \mbox{ in }
L^{2}(0 , T; V^{\e})\times L^{2}(0 , T; H^{1 , \e}) \quad \mbox{ weakly for }
m \to +\infty,\end{aligned}$$ $$\begin{aligned}
\label{lim2m}
\bar{v}^{\e}_{\delta m} \tow \bar{v}^{\e}_{\delta} \quad \mbox{ in }
L^{\infty}(0 , T; H^{\e})\times L^{\infty}(0 , T; H^{0 , \e}) \quad \mbox{weak
star for }
m \to +\infty,\end{aligned}$$ and from (\[boudv\]), by Aubin’s compactness theorem A.11 in [@aubin], there are two subsequences (denoted also by) ${v^{\e}_{\delta m}}$, ${Z^{\e}_{\delta m}}$ satisfying for $m\to +\infty$ the following strong convergence $$\begin{aligned}
\label{cfVZm}
v^{\e}_{\delta m} \to v^{\e}_{\delta} \mbox{ in } L^{2}(0 , T;
(L^{4}(\Omega^{\e}))^{2}), \quad
Z^{\e}_{\delta m}\to Z^{\e}_{\delta} \mbox{ in }
L^{2}(0 , T;
L^{4}(\Omega^{\e})).\end{aligned}$$
In order to pass to the limit as $m \to +\infty$, we remind that for any $\Theta^{\e}=(\varphi^{\e} , \psi^{\e}) \in V^{\e}\times
H^{1,
\e}$, there exists a sequence $(q^{\e}_{i} , k^{\e}_{i})_{i \ge 1}$ in ${\mathbb R}^{2}$ such that $$\Theta^{\e}_{m} = (\varphi^{\e}_{m}, \psi^{\e}_{m})
\to (\varphi^{\e} , \psi^{\e}) \quad \mbox{ strongly in } V^{\e}\times H^{1,
\e}$$ with $$\varphi^{\e}_{m} = \sum_{i=1}^{m}q^{\e}_{i}\Phi_{i},
\quad \psi^{\e}_{m} = \sum_{i=1}^{m}k^{\e}_{i}\psi_{i} \qquad \forall m \ge 1.$$ We multiply first the two sides of (\[Pm1\]) by $q^{\e}_{i}$ then we sum for $i=1$ to $m$, and we multiply the two sides of (\[Pm2\]) by $k^{\e}_{i}$ then we sum also for $i=1$ to $m$, we obtain $$\begin{aligned}
\label{Pm1n}
({\partial v^{\e}_{\delta m}\over \partial t} , \varphi^{\e}_{m}) &+&
(\nu+\nu_{r}) (\nabla v^{\e}_{\delta m} , \nabla \varphi^{\e}_{m})
+b(v^{\e}_{\delta m} , v^{\e}_{\delta m} , \varphi^{\e}_{m})
+{1\over 2}(v^{\e}_{\delta m}{\rm div } v^{\e}_{\delta m} \, , \,
\varphi^{\e}_{m})
\nonumber\\
&&
+{1\over \delta}({\rm div } v^{\e}_{\delta m} \, , \, {\rm div
}\varphi^{\e}_{m})
= ({\cal F}_{1}(v^{\e}_{\delta m}) , \varphi^{\e}_{m})
+ 2\nu_{r} ({\rm rot \,} \, Z^{\e}_{\delta m} , \varphi^{\e}_{m}),
\\
&& v^{\e}_{\delta m} (0)= v^{\e}_{0 m} , \label{cal(Pcmvn)}\end{aligned}$$ and $$\begin{aligned}
\label{Pm2n}
({\partial Z^{\e}_{\delta m} \over \partial t} , \psi^{\e}_{m}) &+&
\alpha (\nabla Z^{\e}_{\delta m} , \nabla \psi^{\e}_{m})
+b_{1}(v^{\e}_{\delta m} , Z^{\e}_{\delta m} , \psi^{\e}_{m})
+{1\over 2}
(Z^{\e}_{\delta m}{\rm div \,} v^{\e}_{\delta m} \, , \psi^{\e}_{m})
\nonumber\\
&&
= ({\cal F}_{2}(v^{\e}_{\delta m}) , \psi^{\e}_{m})
+2\nu_{r} ({\rm rot \,} v^{\e}_{\delta m} , \psi^{\e}_{m})
- 4\nu_{r} (Z^{\e}_{\delta m} \, , \, \psi^{\e}_{m}),
\\
&& Z^{\e}_{\delta}(0)= Z^{\e}_{0 m}.\label{cal(Pcmzn)}\end{aligned}$$ Let $\theta\in {\cal D}(0 , T)$, we multiply (\[Pm1n\]) and (\[Pm2n\]) by $\theta(t)$ and we integrate over $[0,T]$. We get $$\begin{aligned}
\label{cal(PPPm)}
- \int_{0}^{T}(\bar{v}^{\e}_{\delta m} (t), \Theta^{\e}_{m})
\theta'(t) dt
+
\int_{0}^{T} \left\{
a(\bar{v}^{\e}_{\delta m} , \Theta^{\e}_{m})
+B(\bar{v}^{\e}_{\delta m} , \bar{v}^{\e}_{\delta m} ,
\Theta^{\e}_{m})\right\} \theta(t) dt
\nonumber\\
+{1\over\delta}\int_{0}^{T} ({\rm div\, } v^{\e}_{\delta m} \, , \, {\rm div
}\varphi^{\e}_{m}) \theta(t) dt
+{1\over 2}\int_{0}^{T}
\left\{v^{\e}_{\delta m} {\rm div \,} v^{\e}_{\delta m} \, , \,
\varphi^{\e}_{m})
+(Z^{\e}_{\delta}{\rm div \,} v^{\e}_{\delta m} \, , \, \psi^{\e}_{m})
\right\} \theta(t) dt
\nonumber\\
= \int_{0}^{T} \left\{ ({\cal F}(\bar{v}^{\e}_{\delta m}) ,
\Theta^{\e}_{m})
- {\cal R}(\bar{v}^{\e}_{\delta m} , \Theta^{\e}_{m}) \right\}
\theta(t) dt
.\quad\end{aligned}$$ Using the convergences (\[lim1m\])-(\[lim2m\]), we can now pass easily to the limit in all terms of (\[cal(PPPm)\]) except for the nonlinear terms $$\int_{0}^{T} B(\bar{v}^{\e}_{\delta m} , \bar{v}^{\e}_{\delta m} ,
\Theta^{\e}_{m}) \theta(t) dt
= \int_{0}^{T} b({ v^{\e}_{\delta m}} , { v^{\e}_{\delta m}} ,
\varphi^{\e}_{m}) \theta(t) dt
+ \int_{0}^{T} b_{1}({ v^{\e}_{\delta m}} , {Z^{\e}_{\delta}} ,
\psi^{\e}_{m}) \theta(t) dt$$ and $$\int_{0}^{T}
\left\{ v^{\e}_{\delta m} {\rm div \,} v^{\e}_{\delta m} \, , \,
\varphi^{\e}_{m})
+(Z^{\e}_{\delta}{\rm div \,} v^{\e}_{\delta m} \, , \, \psi^{\e}_{m})
\right\} \theta(t) dt.$$ We have first $$\begin{aligned}
\label{nonltv}
\int_{0}^{T} b(v^{\e}_{\delta m} , v^{\e}_{\delta m} , \varphi^{\e}_{m}) \theta
(t) dt
&=& - \int_{0}^{T} b(v^{\e}_{\delta m} , \varphi^{\e}_{m}, v^{\e}_{\delta
m} ) \theta(t) dt
-\int_{0}^{T} ( {\rm div\,} {v^{\e}_{\delta m}} , \varphi^{\e}_{m}\cdot
{v^{\e}_{\delta m}} ) \theta(t) dt
\nonumber\\
&&+ \int_{0}^{T} \int_{\partial\Om^{\e}}
(\varphi^{\e}_{m}\cdot {v^{\e}_{\delta m}})({v^{\e}_{\delta m}}\cdot n)
\theta(t)
d\sigma dt.\end{aligned}$$ Using the boundary conditions (\[eqn:er2.8a\])-(\[eqn:er2.11b\]), we obtain that the last integral is equal to zero, then for the first and the second integrals we use the strong convergence (\[cfVZm\]). So we get $$\begin{aligned}
\int_{0}^{T} b(v^{\e}_{\delta m} , v^{\e}_{\delta m} , \varphi^{\e}_{m}) \theta
(t) dt
\to
\int_{0}^{T} b(v^{\e}_{\delta} , v^{\e}_{\delta} , \varphi^{\e}) \theta (t) dt
\mbox{ for } m\to +\infty.\end{aligned}$$ Similarly $$\begin{aligned}
\label{nonltZ}
\int_{0}^{T} b(v^{\e}_{\delta m} , Z^{\e}_{\delta m} , \psi^{\e}_{m}) \theta (t)
dt
&=& - \int_{0}^{T} b(v^{\e}_{\delta m} , \psi^{\e}_{m} , Z^{\e}_{\delta m} )
\theta (t)
dt
-\int_{0}^{T} ( {\rm div\,} v^{\e}_{\delta m} , \psi^{\e}\cdot Z^{\e}_{\delta m}
) \theta (t) dt
\nonumber\\
&&+ \int_{0}^{T} \int_{\partial \Om^{\e}}
(\psi^{\e}_{m}\cdot Z^{\e}_{\delta m})(v^{\e}_{\delta m}\cdot n) \theta (t)
d\sigma dt.\end{aligned}$$ Using the boundary conditions (\[eqn:er2.8a\])-(\[eqn:er2.11b\]), we obtain that the last integral is equal to zero, then for the first and the second integrals we use the strong convergence (\[cfVZm\]). So we get $$\begin{aligned}
\int_{0}^{T} b(v^{\e}_{\delta m} , Z^{\e}_{\delta m} , \psi^{\e}_{m}) \theta (t)
dt
\to
\int_{0}^{T} b(v^{\e}_{\delta} , \psi^{\e}, Z^{\e}_{\delta}) \theta (t) dt
\mbox{ for } m\to +\infty.\end{aligned}$$ We can now pass to the limit ($m\to +\infty$) in all terms of (\[cal(PPPm)\]) to get $$\begin{aligned}
\label{cal(PPP1)}
\int_{0}^{T} (v^{\e}_{\delta} , \Theta^{\e}) \theta'(t) dt =
\int_{0}^{T} \left\{
a( \bar{v}^{\e}_{\delta} , \Theta^{\e})
+ B(\bar{v}^{\e}_{\delta} , \bar{v}^{\e}_{\delta} , \Theta^{\e})
+{1\over\delta} ({\rm div\,} v^{\e}_{\delta} \, , \, {\rm div
}\varphi^{\e})
\right\} \theta(t) dt
\nonumber\\
+\int_{0}^{T}\left\{
{1\over 2}( v^{\e}_{\delta}{\rm div\,} v^{\e}_{\delta} \, , \,
\varphi^{\e})
+ {1\over 2} ( Z^{\e}_{\delta} {\rm div\,} v^{\e}_{\delta} \, , \, \psi^{\e})
-
({\cal F}({\bar{v}^{\e}}_{\delta} ) ,\Theta^{\e})- {\cal
R}(\bar{v}^{\e}_{\delta} ,\Theta^{\e})
\right\} \theta(t) dt
\nonumber\\
\quad \forall\Theta^{\e}= (\varphi^{\e} , \psi^{\e}) \in V^{\e}\times
{H^{1,}}^{\e}, \qquad\end{aligned}$$ that is $\bar{v}^{\e}_{\delta}$ satisfy (\[cal(P)\]) in ${\cal D}'(0 , T)$ (distribution sense). Moreover as the two items between the brackets $\{ \}$, in the right hand side of (\[cal(PPP1)\]), are in $L^{4/3}(0 , T)$, we deduce that (\[cal(P)\]) holds for almost every $t \in (0 , T)$.
In the following we set $$\begin{aligned}
p^{\e}_{\delta}= -{1\over \delta} {\rm div \, } v^{\e}_{\delta},\end{aligned}$$ then, rewrite (\[cal(P)\]) as follows $$\begin{aligned}
\label{cal(PP)}
[{\partial \bar{v}^{\e}_{\delta}\over \partial t} , \Theta^{\e}] +
a(\bar{v}^{\e}_{\delta} , \Theta^{\e})
+B(\bar{v}^{\e}_{\delta} , \bar{v}^{\e}_{\delta} , \Theta^{\e})
+{1\over 2}\left\{
( v^{\e}_{\delta}{\rm div \,} v^{\e}_{\delta} \, , \, \varphi^{\e})
+(Z^{\e}_{\delta}{\rm div \,} v^{\e}_{\delta} \, , \, \psi^{\e}) \right\}
\nonumber\\
-(p^{\e}_{\delta} \, , \, {\rm div }\varphi^{\e})
= ({\cal F}(\bar{v}^{\e}_{\delta}) , \Theta^{\e})
- {\cal R}(\bar{v}^{\e}_{\delta} , \Theta^{\e}) \qquad
\forall\Theta^{\e}= (\varphi^{\e} , \psi^{\e})
\in V^{\e}\times {H^{1,}}^{\e}.\end{aligned}$$ The aim now is to pass to the limit for $\delta\to 0$ in (\[cal(PP)\]). Reminding that the different constants $C$ in (\[Maj1\])-(\[Maj2\]) and (\[boudv\]) are independent of $\delta$, the same estimates hold for $\bar{v}^{\e}_{\delta}$ i.e. $$\begin{aligned}
\label{Maj1d}
\sup_{t\in [0 , T]}[\bar{v}^{\e}_{\delta}(t)]^{2} \leq C,\end{aligned}$$ $$\begin{aligned}
\label{Maj2d}
\int_{0}^{T}\|{\rm div\,} v^{\e}_{\delta}\|^{2} dt \leq C \delta, \quad
\int_{0}^{T} [[\bar{v}^{\e}_{\delta}(t)]]^{2}
dt\leq C,
\end{aligned}$$ and $$\begin{aligned}
\label{boudvd}
\|{\partial {v^{\e}_{\delta }} \over\partial t}
\|_{L^{4/3}(0 , T ; H^{-1}(\Omega^{\e})^{2})}\leq C,
\quad
\|{\partial {Z^{\e}_{\delta }} \over \partial t}
\|_{L^{4/3}(0 , T ; H^{-1}(\Omega^{\e}))}\leq C.\end{aligned}$$ Hence, there exists $\bar{v}^{\e}$ such that, possibly extracting a subsequence still denoted by $\bar{v}^{\e}_{\delta}$: $$\begin{aligned}
\label{lim1}
\bar{v}^{\e}_{\delta} \tow \bar{v}^{\e} \quad \mbox{ in }
L^{2}(0 , T; V^{\e})\times L^{2}(0 , T; H^{1 , \e}) \quad \mbox{ weakly for }
\delta \to 0,\end{aligned}$$ $$\begin{aligned}
\label{lim2}
\bar{v}^{\e}_{\delta} \tow \bar{v}^{\e} \quad \mbox{ in }
L^{\infty}(0 , T; H^{\e})\times L^{\infty}(0 , T; H^{0 , \e}) \quad \mbox{weak
star for } \delta \to 0,\end{aligned}$$ $$\begin{aligned}
\label{lim3}
{\rm div\, } v^{\e}_{\delta} \to 0
\quad \mbox{ in } L^{2}(0 , T; L^{2}(\Om^{\e})) \quad \mbox{strongly for }
\delta \to 0,\end{aligned}$$ and $$\begin{aligned}
\label{cfVZ}
\bar v^{\e}_{\delta} \to \bar v^{\e} \mbox{ strongly in } L^{2}(0 , T;
(L^{4}(\Omega^{\e})^{2}))
\times
L^{2}(0 , T ; L^4(\Omega^{\e})).\end{aligned}$$ So from (\[lim1\]) and (\[lim3\]) we deduce $$\begin{aligned}
\label{div}
{\rm div \, } v^{\e} = 0 \quad \mbox{\rm in } \Om^{\e}, \quad \mbox{\rm a.e.
in } (0 , T).\end{aligned}$$ We check now that $p^{\e}_{\delta}$ remains in a bounded subset of $H^{-1}(0 , T ; L^{2}_0(\Om^{\e}))$. Reminding that $p^{\e}_{\delta} = -
\frac{1}{\delta} {\rm div \,} v^{\e}_{\delta}$, we have $p^{\e}_{\delta} \in L^2
(0,T; L^2_0 (\Om^{\e}))$. Now let us consider $\omega\in H^{1}_{0}(0 , T ; L^{2}_{0}(\Om^{\e}))$, then (see [@JLLions78] page 13-15) there exists $\varphi\in H^1_0 (0 , T ; H^{1}_{0}(\Om^{\e})^{2})$ such that $$\begin{aligned}
\label{Phi}
&&{\rm div \, }\varphi(t)= \omega(t), \mbox{ and } \varphi(t)=
P\omega(t), \nonumber\\
&& P \mbox{ is a linear continuous operator from } L^{2}_{0}(\Om^{\e})
\mbox{ to } H^{1}_{0}(\Om^{\e})^{2}.\end{aligned}$$ The choice of $\Theta= (\varphi(t) , 0)$ in (\[cal(PP)\]), gives $$\begin{aligned}
\label{equ2.36}
\int_{0}^{T}(p^{\e}_{\delta} \, , \, \omega)dt =
\int_{0}^{T}\left(- (v^{\e}_{\delta} , {\partial\varphi\over \partial t})
+ (\nu+\nu_{r})(\nabla v^{\e}_{\delta} , \nabla\varphi)\right)dt
+\int_{0}^{T}b(v^{\e}_{\delta} , v^{\e}_{\delta} , \varphi)dt
\nonumber\\
+{1\over 2}\int_{0}^{T}
(v^{\e}_{\delta}{\rm div \,} v^{\e}_{\delta} \, , \, \varphi) dt
-2\nu_{r}\int_{0}^{T} ({\rm rot \,} Z^{\e}_{\delta} , \varphi) dt
- \int_{0}^{T}({\cal F}_{1}(v^{\e}_{\delta}) , \varphi) dt,\end{aligned}$$ with $$\begin{aligned}
({\cal F}_{1}(v^{\e}_{\delta}) , \varphi ) &=&
-(\nu +\nu_{r}) ({\partial U^{\e}\over \partial z_{2}},
{\partial\varphi_{1}\over \partial z_{2}})
- b(U^{\e}e_{1} , v^{\e}_{\delta} , \varphi) - b(v^{\e}_{\delta} ,
U^{\e}e_{1} ,\varphi )
\nonumber\\&&
+2\nu_{r} ( {\partial W^{\e}\over \partial z_{2}} , \varphi_{1})
-({\partial U^{\e} \over \partial t}, \varphi_{1}) + (f^{\e} ,
\varphi).\end{aligned}$$ Since $\omega\in H^{1}_{0}(0 , T ; L^{2}_{0}(\Om^{\e}))\subset L^{\infty}(0 , T
; L^{2}_{0}(\Om^{\e}))$, with continuous injection, it follows that $\varphi$ in $L^{\infty}(0 , T ;
H^{1}_{0}(\Om^{\e})^{2})$, and ${\partial \varphi\over\partial t} \in L^{2}(0 , T ;
H^{1}_{0}(\Om^{\e})^{2})$, then also by the continuous injection of $H^{1}(\Om^{\e})$ in $L^{4}(\Om^{\e})$ we have $$\begin{aligned}
|\int_{0}^{T}b(v^{\e}_{\delta}, v^{\e}_{\delta}, \varphi) dt|&\leq&
\|v^{\e}_{\delta}\|_{L^{2}(0 , T; (L^{4}(\Om^{\e}))^{2})}
\|v^{\e}_{\delta}\|_{L^{2}(0 , T; H^{1} (\Om^{\e})^{2})}
\|\varphi\|_{L^{\infty}(0 , T; (L^{4}(\Om^{\e}))^{2})}
\nonumber\\
&\leq&
C^{2}\|v^{\e}_{\delta}\|^{2}_{L^{2}(0 , T
;H^{1}(\Om^{\e})^{2})}\|\varphi\|_{H^{1}(0 , T ;
H^{1}(\Om^{\e})^{2})}.\end{aligned}$$ Similarly for the first term in the second line of (\[equ2.36\]). Therefore using (\[Maj1d\])-(\[Maj2d\]) we get $$\begin{aligned}
\label{equ2.37}
|\int_{0}^{T}(p^{\e}_{\delta} \, , \, \omega)dt | \leq
C\|\varphi\|_{H^{1}(0 , T ; H^{1}(\Om^{\e})^{2})}
\quad \forall \varphi\in H^{1}_{0}(0 , T ; H^{1}_{0}(\Om^{\e})^{2}).\end{aligned}$$ As $P: \omega(t) \mapsto \varphi(t)$ is a linear continuous operator from $L^{2}_{0}(\Om^{\e})$ to $H^{1}_{0}(\Om^{\e})^{2}$, there exists another constant $C$, independent of $\delta$, such that $$\begin{aligned}
\label{equ2.39}
|\int_{0}^{T}(p^{\e}_{\delta} \, , \, \omega)dt | \leq
C\|\omega\|_{H^{1}_{0}(0 , T ; L^{2}_{0}(\Om^{\e}))}
\quad \forall \omega\in H^{1}(0 , T ; L^{2}(\Om^{\e})).\end{aligned}$$ Let us take now $\omega\in H^{1}_{0}(0 , T ; L^{2}(\Om^{\e}))$ arbitrary, we can apply (\[equ2.39\]) to $$\tilde{\omega}=\omega- {1\over meas(\Om^{\e})}\int_{\Om^{\e}}\omega dz$$ which is in $H^{1}_{0}(0 , T ; L^{2}_{0}(\Om^{\e}))$. But $p^{\e}_{\delta} \in
L^2(0,T; L^2_0 (\Om^{\e}))$, so $$\begin{aligned}
\int_{0}^{T}(p^{\e}_{\delta} \, , \, \tilde \omega)dt =
\int_{0}^{T}(p^{\e}_{\delta} \, , \, \omega)dt\end{aligned}$$ and (\[equ2.39\]) remains valid for all $\omega\in H^{1}_{0}(0 , T ; L^{2}(\Om^{\e})$. Thus $p^{\e}_{\delta}$ remains in a bounded subset of $H^{-1}(0 , T ;
L^{2}_0(\Om^{\e}))$. It follows that there exists $p^{\e}\in H^{-1}(0 , T ; L^{2}_0(\Om^{\e}))$ such that $$\begin{aligned}
\label{lim5}
p^{\e}_{\delta} \tow p^{\e} \quad \mbox{ in } H^{-1}(0 , T ; L^{2}(\Om^{\e}))
\mbox{ weak}.
\end{aligned}$$
In order to pass to the limit as $\delta\to 0$, let $\theta\in {\cal D}(0 , T)$, multiply (\[cal(PP)\]) by $\theta(t)$ and integrate over $[0,T]$. We get $$\begin{aligned}
\label{cal(PPPs)}
- \int_{0}^{T}(\bar{v}^{\e}_{\delta} (t), \Theta^{\e}) \theta'(t) dt
+
\int_{0}^{T} \left(
a(\bar{v}^{\e}_{\delta} , \Theta^{\e})
+B(\bar{v}^{\e}_{\delta} , \bar{v}^{\e}_{\delta} , \Theta^{\e})\right)
\theta(t) dt
-\int_{0}^{T} (p^{\e}_{\delta} \, , \, {\rm div }\varphi^{\e})
\theta(t) dt
\nonumber\\
+{1\over 2}\int_{0}^{T}
\left\{
(v^{\e}_{\delta}{\rm div \,} v^{\e}_{\delta} \, , \, \varphi^{\e})
+(Z^{\e}_{\delta}{\rm div \,} v^{\e}_{\delta} \, , \, \psi^{\e}) \right\}
\theta(t) dt
= \int_{0}^{T} \left\{ ({\cal F}(\bar{v}^{\e}_{\delta}) , \Theta^{\e})
- {\cal R}(\bar{v}^{\e}_{\delta} , \Theta^{\e}) \right\}
\theta(t) dt
\nonumber\\
\qquad \forall\Theta^{\e}= (\varphi^{\e} , \psi^{\e})
\in V^{\e}\times {H^{1,}}^{\e}. \\end{aligned}$$ Using (\[lim1\]), (\[lim3\]), (\[cfVZ\]) and (\[lim5\]), then taking into account (\[nonltv\])-(\[nonltZ\]) for $v^{\e}_{\delta}$ and $Z^{\e}_{\delta}$ instead of $v^{\e}_{\delta m}$ and $Z^{\e}_{\delta m}$ for the nonlinear terms, we can now pass to the limit in all the terms of (\[cal(PPPs)\]) to get $$\begin{aligned}
\int_{0}^{T}(v^{\e}, \Theta^{\e}) \theta'(t) dt =
\int_{0}^{T} \left\{
a(\bar{v}^{\e} , \Theta^{\e}) +B(\bar{v}^{\e} , \bar{v}^{\e} ,
\Theta^{\e})
-(p^{\e} \, , \, {\rm div }\varphi^{\e})
\right\} \theta(t) dt
\nonumber\\
-\int_{0}^{T} \left\{ ({\cal F}(\bar{v}^{\e}),\Theta^{\e})- {\cal
R}(\bar{v}^{\e} ,\Theta^{\e})\right\} \theta(t) dt
\qquad \forall\Theta^{\e}= (\varphi^{\e} , \psi^{\e}) \in
V^{\e}\times {H^{1,}}^{\e} ,\end{aligned}$$ that is ($\bar{v}^{\e}, p^{\e}$) satisfy (\[eqn:er2.14\]) in ${\cal D}'(0 , T)$ (distribution sense). Moreover we can see also that (\[eqn:er2.14\]) is satisfied for almost every $t \in (0 , T)$.
Finally, by considering test-functions $\Theta^{\e}\in V_{div}\times H^{1,
\e}$, we can prove the uniqueness of $(v^{\e}, Z^{\e})$ and its continuity in time as in Theorem 2.2 [@gl]. Thus the proof of the existence and uniqueness of a solution of Problem $(P^{\e})$ is complete.
A priori uniform estimates of $\bar{v}^{\e}$ and $p^{\e}$ {#uniformEst}
=========================================================
The aim in this section is to establish uniform estimates with respect to $\e$ for $\bar{v}^{\e}$ and $p^{\e}$, which will allow us to derive in the next sections the limit problem as $\e$ tends to zero by using the two-scale convergence technique. More precisely we consider first the following scaling $$\begin{aligned}
\label{2.2a}
x_{1} = z_{1}, \quad\mbox{ and }\quad x_{2} ={z_{2}\over \e},
\end{aligned}$$ which transforms the domain $\Om^{\e}$ into the domain $$\Om_{\e}=\bigl\{x=(x_{1} , x_{2})\in {\mathbb R}^{2} :\quad 0< x_{1}< L
\qquad 0< x_{2}< h^{\e}(x_{1})= h(x_{1}, {x_{1}\over \e}) \bigr\},$$ then we introduce a second scaling $$\begin{aligned}
\label{2.2b}
y_{1} = x_{1}, \quad\mbox{ and }\quad y_{2} ={x_{2}\over h^{\e}(x_{1})}
= {z_{2}\over \e h^{\e}(x_{1})}
\end{aligned}$$ which transforms the domain $\Om_{\e}$ into $\Om=\{y=(y_{1} , y_{2}) \in
\Gamma_{0} \times (0 , 1)\}$. With the chain rule, we get easily the following relations $$\begin{aligned}
\label{not}
{\partial \over \partial z_{2}}= {1\over \e h^{\e}(y_{1})}{\partial \over
\partial y_{2}},
\quad
{\partial \over \partial z_{1}}
&=&{\partial \over \partial y_{1}}{\partial y_{1}\over \partial z_{1}} +
{\partial \over \partial y_{2}}{\partial y_{2}\over \partial z_{1}}
= {\partial \over \partial y_{1}} + \left(-{y_{2}\over h^{\e}(y_{1})}
{\partial h^{\e}\over\partial y_{1}}\right)
{\partial \over \partial y_{2}}
\nonumber\\
&=&
\left(1 , -{y_{2}\over h^{\e}(y_{1})}
{\partial h^{\e}\over\partial y_{1}}\right)
\left(\begin{array}[c]{c}
{\partial \over\partial y_{1}} \\ \\
{\partial \over\partial y_{2}}
\end{array}
\right)
=b_{\e}\cdot\nabla.\end{aligned}$$
Now we define the functional setting in $\Om$: let $\Gamma_{1}=
\{(y_{1} ,y_{2})\in \overline{\Om} : \quad y_{2}=1\}$ and $$\tilde{V}= \{ v \in {\cal C}^{\infty}({\overline \Om})^{2}:
\,\, v \,\,\mbox{\rm is L-periodic in }\,\, y_1,\,\,\, v_{|_{\Gamma_0}} =0,
\,
-\e (h^{\e})'(y_{1}) v_{1} + v_{2} =0
\mbox{ on } \Gamma_{1}\}$$ $$V = {\rm closure \,\, of\,\, } \tilde{V} \,\,
{\rm in } \,\, H^{1}(\Om)\times H^{1}(\Om)$$ $$\tilde{H^{1}}= \{Z \in {\cal C}^{\infty}(\overline{\Om}): \,\,\,Z
\,\,\,\mbox{\rm is L-periodic in }\,\,\, y_1,\,\,\,Z=0 \,\,\,{\rm
on}\,\,\,
\Gamma_0\cup\Gamma_1\}$$ $$H= {\rm closure \,\, of \, } \tilde{V} \,\, {\rm in }\,\,
L^{2}(\Om) \times L^{2}(\Om), \qquad
H^{1} = {\rm closure \,\, of\,\, } \tilde{H^{1}} \,\, {\rm in }
\,\, H^{1}(\Om),$$ $${H^{0}} = {\rm closure \,\, of \, } \tilde{H^{1}}\,\,
{\rm in }\,\, L^{2}(\Om).$$
In order to avoid new notations, we have still denoted by $ v^{\e}$, $Z^{\e}$ and $p^{\e}$ the unknown velocity, micro-rotation and pressure fields as functions of the rescaled variables $(y_1, y_2)$ instead of $(z_1, z_2)$. Similarly, we still denote the data by $\bar f^{\e}$ and $\bar \xi^{\e}$ considered now as functions of $(y_1, y_2)$.
Let $\Theta = (\varphi, \psi) \in V \times H^1$ and let $\Theta^{\e} =
(\varphi^{\e}, \psi^{\e}) \in V^{\e} \times H^{1,\e} $ be given by $$\begin{aligned}
\varphi^{\e} (z_1, z_2) = \varphi \left( z_1, \frac{z_2}{\e h^{\e} (z_1)}
\right), \quad \psi^{\e} (z_1, z_2) = \psi \left( z_1, \frac{z_2}{\e h^{\e}
(z_1)} \right) \quad \forall (z_1, z_2) \in \Om^{\e}.\end{aligned}$$ Using (\[not\]) we obtain that $$\begin{aligned}
\label{fla}
a(\bar{v}^{\e}(t),\Theta^{\e}) &=&
(\nu + \nu_{r})
\sum_{i, j=1}^{2}\int_{\Om^{\e}}
{\partial v_{i}^{\e}(t)\over \partial z_{j}}
{\partial \varphi_{i}^{\e} (t)\over \partial z_{j}}dz
+\aal\sum_{i=1}^{2} \int_{\Om^{\e}}
{\partial Z^{\e}(t)\over \partial z_{i}}
{\partial \psi^{\e} \over \partial z_{i}}dz
\nonumber\\
&=& (\nu + \nu_{r})
\int_{\Om}
\sum_{i=1}^{2} \left((b_{\e}\cdot \nabla v_{i}^{\e}(t))(b_{\e}\cdot \nabla
\varphi_{i})
+ {1\over (\e h^{\e})^{2}}{\partial v^{\e}_{i}(t)\over \partial y_{2}}
{\partial \varphi_{i}\over \partial y_{2}}\right) \e h^{\e} dy
\nonumber\\
&&+\aal\int_{\Om}
\left((b_{\e}\cdot \nabla Z^{\e}(t))(b_{\e}\cdot \nabla \psi) +
{1\over (\e h^{\e})^{2}}{\partial Z^{\e}(t)\over \partial y_{2}}
{\partial \psi\over \partial y_{2}}\right) \e h^{\e} dy
\nonumber\\
&=& {(\nu + \nu_{r})\over \e}
\int_{\Om}
\sum_{i=1}^{2} \left((\e b_{\e}\cdot \nabla v_{i}^{\e}(t))(\e b_{\e}\cdot \nabla
\varphi_{i})
+ {1\over (h^{\e})^{2}}{\partial v^{\e}_{i}(t)\over \partial y_{2}}
{\partial \varphi_{i}\over
\partial y_{2}}\right) h^{\e} dy
\nonumber\\
&&+{\aal\over \e}\int_{\Om}
\left((\e b_{\e}\cdot \nabla Z^{\e}(t))(\e b_{\e}\cdot \nabla \psi) +
{1\over (h^{\e})^{2}}{\partial Z^{\e}(t)\over \partial y_{2}}
{\partial\psi\over \partial y_{2}}\right) h^{\e} dy
={1\over \e}\hat{a}(\bar{v}^{\e}(t),\Theta),\end{aligned}$$ $$\begin{aligned}
\label{trifb}
B(\bar{v}^{\e}(t),\bar{v}^{\e}(t),\Theta^{\e}) &=& b(v^{\e}(t), v^{\e}(t),
\varphi^{\e}) +
b_{1}(v^{\e}(t), Z^{\e}(t), \psi^{\e}) \nonumber\\
&=&\int_{\Om^{\e}} \sum_{i , j =1}^{2} v^{\e}_{i}(t){\partial v^{\e}_{j}(t)\over
\partial z_{i}} \varphi_{j}^{\e} dz
+ \sum_{i =1}^{2}\int_{\Om^{\e}} v^{\e}_{i}(t){\partial Z^{\e}(t)\over \partial
z_{i}} \psi^{\e} dz
\nonumber\\
&=& \int_{\Om}\left(\sum_{j =1}^{2} v^{\e}_{1}(t)
\left(\e b_{\e}\cdot \nabla v^{\e}_{j}(t) \right)\varphi_{j} +
{v^{\e}_{2}(t)\over h^{\e}}{\partial v^{\e}_{j}(t) \over \partial y_{2}}
\varphi_{j}\right) h^{\e} dy
\nonumber\\
&+&
\int_{\Om} \left(\sum_{j =1}^{2}v^{\e}_{1}(t)(\e b_{\e}\cdot \nabla Z^{\e}(t))
\psi +
{v^{\e}_{2}(t)\over h^{\e}} {\partial Z^{\e}(t)\over \partial y_{2}} \psi
\right) h^{\e} dy
\nonumber\\
&=&
\hat{B}(\bar{v}^{\e}(t),\bar{v}^{\e}(t),\Theta),\end{aligned}$$ $$\begin{aligned}
\label{Rot}
{\cal R}(\bar{v}^{\e}(t) , \Theta^{\e})
&=& -2\nu_{r}
\int_{\Om^{\e}}\left( {\partial Z^{\e}(t)\over\partial z_{2}}\varphi_{1}^{\e}
- {\partial Z^{\e}(t)\over\partial z_{1}}\varphi_{2}^{\e}\right)
+\left({\partial v_{2}^{\e}(t)\over\partial z_{1}}- {\partial
v_{1}^{\e}(t)\over\partial z_{2}}\right)\psi^{\e} dz
\nonumber\\
&& +4\nu_{r} \int_{\Om^{\e}} Z^{\e}(t)\psi^{\e} dz
= -2\nu_{r} \int_{\Om} \left({1\over h^{\e} }
{\partial Z^{\e}(t)\over \partial y_{2}}\varphi_{1} -
(\e b_{\e}\cdot \nabla Z^{\e}(t)) \varphi_{2}\right) h^{\e} dy
\nonumber\\
&&-2\nu_{r} \int_{\Om} \left(
(\e b_{\e}\cdot \nabla v^{\e}_{2}(t))
- {1\over h^{\e}} {\partial v_{1}^{\e}(t)\over \partial y_{2}}\right)\psi h^{\e} dy
+ 4\nu_{r}\e \int_{\Om} Z^{\e}(t)\psi \, h^{\e} dy
\nonumber\\
&=& \hat{{\cal R}}(\bar{v}^{\e}(t) , \Theta).\end{aligned}$$ Using Lemma \[lUW\] we have $U^{\e}(t , z_{2})= {\cal U}({z_{2}\over \e}) U_{0}(t)
= {\cal U}(y_{2}h^{\e}(y_{1})) U_{0}(t)$, so $$\begin{aligned}
\label{For3.8}
b_{\e}\cdot \nabla {\cal U}(y_{2}h^{\e}(y_{1}))&=&
\left( {\partial \over \partial y_{1}}
-{y_{2}\over h^{\e}} {\partial h^{\e}\over \partial y_{1}} {\partial
\over \partial y_{2}} \right)
{\cal U}(y_{2}h^{\e}(y_{1}))
= {\cal U}'(y_{2}h^{\e}(y_{1})) \left(
y_{2} {\partial h^{\e}\over \partial y_{1}}
- {y_{2}\over h^{\e} } {\partial h^{\e}\over \partial y_{1}}
h^{\e}
\right)
\nonumber\\
&=& 0,
\end{aligned}$$ and similarly for $W^{\e}(t , z_{2})= {\cal W}({z_{2}\over \e}) W_{0}(t)
= {\cal W}(y_{2}h^{\e}(y_{1}))W_{0}(t)$, so $$\begin{aligned}
\label{For3.9}
b_{\e}\cdot \nabla {\cal W}(y_{2}h^{\e}(y_{1}))= 0.
\end{aligned}$$ Then $$\begin{aligned}
\label{For3.10}
a(\bar{\xi^{\e}},\Theta^{\e}) &=& (\nu + \nu_{r}) \int_{\Om^{\e}} \nabla
U^{\e}(z_{2},t)e_{1}\nabla\varphi^{\e} dz + \aal \int_{\Om^{\e}} \nabla
W^{\e}(z_{2},t) \nabla \psi^{\e} dz \nonumber\\
&=& (\nu + \nu_{r}) \int_{\Om^{\e}} \sum_{i=1}^{2}{\partial U^{\e}\over
\partial z_{i}} {\partial \varphi_{1}^{\e}\over \partial z_{i}} dz
+ \aal \int_{\Om^{\e}} \sum_{i=1}^{2}{\partial W^{\e}\over \partial z_{i}}
{\partial \psi^{\e}\over \partial z_{i}} dz
\nonumber\\
&=&(\nu + \nu_{r})U_{0}(t) \int_{\Om} \left( (b_{\e}\cdot \nabla {\cal U})
(b_{\e}\cdot \nabla \varphi_{1}) +
{1\over (\e h^{\e})^{2}} {\cal U}'(
y_{2}h^{\e}) h^{\e} {\partial \varphi_{1}\over \partial y_{2}}
\right) \e h^{\e} dy
\nonumber\\
&&+ \aal W_{0}(t)\int_{\Om} \left( (b_{\e}\cdot \nabla {\cal W}) (b_{\e}\cdot
\nabla \psi) +
{1\over (\e h^{\e})^{2}} {\cal W}'( y_{2}h^{\e}) h^{\e} {\partial
\psi\over \partial y_{2}}
\right) \e h^{\e} dy
\nonumber\\
&=&
{(\nu + \nu_{r})\over \e} U_{0}(t) \int_{\Om}
{\cal U}'( y_{2}h^{\e}) {\partial \varphi_{1}\over
\partial y_{2}} dy
+ {\aal \over \e} W_{0}(t)\int_{\Om}
{\cal W}'( y_{2}h^{\e}) {\partial \psi\over \partial
y_{2}} dy
= {1\over \e}\hat{a}(\bar{\xi^{\e}},\Theta).\end{aligned}$$ We have also $$\begin{aligned}
\label{Rot1}
B(\bar{\xi^{\e}},\bar{v}^{\e},\Theta^{\e})
&=& b(U^{\e}e_{1}, v^{\e}, \varphi^{\e}) + b_{1}(U^{\e}e_{1} , Z^{\e},
\psi^{\e})
\nonumber\\
&=&\int_{\Om^{\e}} \sum_{j =1}^{2} U^{\e}{\partial v^{\e}_{j}\over \partial
z_{1}} \varphi_{j}^{\e} dz
+ \int_{\Om^{\e}} U^{\e}{\partial Z^{\e}\over \partial z_{1}} \psi^{\e} dz
\nonumber\\
&=& U_{0}(t) \int_{\Om} {\cal U}( y_{2}h^{\e}) \left(
\sum_{j =1}^{2}
(\e b_{\e}\cdot \nabla v^{\e}_{j})\varphi_{j}
+
(\e b_{\e}\cdot \nabla Z^{\e}) \psi\right) h^{\e} dy
=\hat{B}(\bar{\xi^{\e}},\bar{v}^{\e},\Theta),\end{aligned}$$ $$\begin{aligned}
\label{Rott}
B(\bar{v}^{\e},\bar{\xi^{\e}}, \Theta^{\e})&=& -B(\bar{v}^{\e},
\Theta^{\e},\bar{\xi^{\e}})
= -b(v^{\e}, \varphi^{\e}, U^{\e}e_{1}) - b_{1}(v^{\e} , \psi^{\e}, W^{\e})
\nonumber\\
&=&-\sum_{i=1}^{2} \int_{\Om^{\e}}
v^{\e}_{i}{\partial\varphi_{1}^{\e}\over \partial z_{i}} U^{\e} dz -
\sum_{i=1}^{2} \int_{\Om^{\e}}
v^{\e}_{i}{\partial\psi^{\e}\over \partial z_{i}} W^{\e} dz
\nonumber\\
&=& - U_{0}(t) \int_{\Om} {\cal U}( y_{2}h^{\e})\left(
v^{\e}_{1} (\e b_{\e}\cdot \nabla \varphi_{1}) h^{\e} +
v^{\e}_{2} {\partial \varphi_{1}\over \partial y_{2}} \right) dy
\nonumber\\
&&- W_{0}(t) \int_{\Om} {\cal W}( y_{2}h^{\e})\left(
v^{\e}_{1} (\e b_{\e}\cdot \nabla \psi) h^{\e} +
v^{\e}_{2} {\partial \psi\over \partial y_{2}} \right)
dy
=\hat{B}(\bar{v}^{\e},\bar{\xi^{\e}}, \Theta),\end{aligned}$$ $$\begin{aligned}
\label{Rotxi}
{\cal R}(\bar{\xi^{\e}} , \Theta^{\e})
&=& -2\nu_{r} W_{0}(t) \int_{\Om} \left(\varphi_{1} {1\over \e h^{\e} }
{\partial \over \partial y_{2}} {\cal W}(y_{2}h^{\e}) - (b_{\e}\cdot \nabla
{\cal W}) \varphi_{2}\right)\e h^{\e} dy
\nonumber\\
&&-2\nu_{r} U_{0}(t)\int_{\Om} \left(- {1\over \e h^{\e} }{\partial\over
\partial y_{2}}{\cal U}(y_{2}h^{\e})\right)\psi \e h^{\e} dy
+ 4\nu_{r} W_{0}(t) \int_{\Om} {\cal W}(y_{2}h^{\e}) \psi \, \e
h^{\e} dy\nonumber\\
&=& -2\nu_{r} W_{0}(t)
\int_{\Om} {\cal W}'(y_{2}h^{\e})\varphi_{1} h^{\e} dy+
2 \nu_r U_{0}(t)\int_{\Om} {\cal U}'(y_{2}h^{\e}) \psi h^{\e} dy
\nonumber\\
&&+
4\nu_{r} W_{0}(t) \int_{\Om} {\cal W}(y_{2}h^{\e}) \psi \, \e
h^{\e} dy
=\hat{{\cal R}}(\bar{\xi^{\e}} , \Theta),\end{aligned}$$ and $$\begin{aligned}
\label{pression}
\qquad (p^{\e}(t) , {\rm div\,} \varphi^{\e} )= \int_{\Om^{\e}} p^{\e}(t){\rm div\,}
\varphi^{\e} dz
= \int_{\Om} p^{\e}(t) \left(
(\e b_{\e}\cdot \nabla \varphi_{1}) + {1\over h^{\e}}{\partial
\varphi_{2} \over \partial y_{2}}
\right) h^{\e} dy.\end{aligned}$$
\[lemma3.1\] Using [(\[2.2a\])-(\[2.2b\])]{}, the variational identity [(\[eqn:er2.14\])]{} written in $\Om^{\e}$ leads to the following one in $\Om$: $$\begin{aligned}
\label{eqvar}
\e
\int_{\Om} {d\bar{v}^{\e}\over dt}(t)\Theta^{\e} h^{\e} dy + {1\over
\e}\hat{a}(\bar{v}^{\e}(t), \Theta^{\e})
+ \hat{B}(\bar{v}^{\e}(t), \bar{v}^{\e}(t), \Theta^{\e}) +
\hat{{\cal R}}(\bar{v}^{\e}(t), \Theta^{\e})
= -\e
\int_{\Om} {d\bar{\xi}^{\e}\over dt}(t)\Theta^{\e} h^{\e} dy
\nonumber\\
-{1\over
\e}\hat{a}(\bar{\xi}^{\e}(t) , \Theta^{\e}) -\hat{B}(\bar{\xi}^{\e}(t)
, \bar{v}^{\e}(t), \Theta^{\e})
-\hat{B}(\bar{v}^{\e}(t), \bar{\xi}^{\e}(t) , \Theta^{\e}) -
\hat{{\cal R}}(\bar{\xi}^{\e}(t) ,\Theta^{\e})
+ \e \int_{\Om} \bar{f}^{\e}(t) \Theta^{\e} h^{\e} dy
\nonumber\\
+
\int_{\Om} p^{\e}(t) \left( (\e b_{\e}\cdot \nabla \varphi^{\e}_{1})
+ {1\over h^{\e}}{\partial \varphi^{\e}_{2} \over \partial
y_{2}} \right) h^{\e} dy,
\quad \forall \Theta^{\e}=(\varphi^{\e} , \psi^{\e})\in V^{\e}\times H^{1 , \e},\end{aligned}$$ where $\hat{a}$, $\hat{B}$, and $\hat{{\cal R}}$, are defined by (\[fla\]), (\[trifb\]), and (\[Rot\]) respectively.
Indeed, from (\[fla\])-(\[pression\]), the variational identity (\[eqvar\]) follows.
We prove now the following uniform estimates, with respect to $\e$:
\[pro1\] Assume that $\e^{2}\bar{f}^{\e}$ and $\e \bar v^{\e}_0$ are bounded independently of $\e$ in $(L^2( (0,T) \times \Om))^3$ and in $(L^2(\Om))^3$ respectively and $U_{0} \in H^{1}(0, T)$, $W_{0} \in H^{1}(0, T)$. There exists a constant $C>0$ which does not depends on $\e$, such that, for $i=1,2$, we have the following estimates: $$\begin{aligned}
\label{E3.13}
\|(\e b_{\e}\cdot \nabla v^{\e}_{i})\|_{L^{2}((0 , T)\times \Om)}\leq C,\qquad
\|(\e b_{\e}\cdot \nabla Z^{\e})\|_{L^{2}((0 , T)\times \Om)}\leq C,\end{aligned}$$ $$\begin{aligned}
\label{E3.14}
\|{\partial v_{i}^{\e}\over \partial y_{2}}\|_{L^{2}((0 , T)\times \Om)}\leq
C, \qquad
\|{\partial Z^{\e}\over \partial y_{2}}\|_{L^{2}((0 , T)\times \Om)}\leq C,
\end{aligned}$$ $$\begin{aligned}
\label{E3.15}
\|{\partial v_{i}^{\e}\over \partial y_{1}}\|_{L^{2}((0 , T)\times \Om)}\leq
{C\over \e}, \qquad
\|{\partial Z^{\e}\over \partial y_{1}}\|_{L^{2}((0 , T)\times \Om)}\leq
{C\over \e},
\end{aligned}$$ $$\begin{aligned}
\label{E3.16}
\|v_{i}^{\e}\|_{L^{2}((0 , T)\times \Om)}\leq C,
\qquad \|Z^{\e}\|_{L^{2}((0 , T)\times \Om)}\leq C.\end{aligned}$$
Taking $\Theta^{\e}= \bar{v}^{\e}(t)$ in (\[eqvar\]), and observing that $B (\bar{v}^{\e}(t), \bar{v}^{\e}(t), \bar{v}^{\e}(t))=
B (\bar{\xi^{\e}}(t), \bar{v}^{\e}(t), \bar{v}^{\e}(t))=0$, we obtain $$\begin{aligned}
\label{eq117}
\e{d\over 2dt} \int_{\Om}(\bar{v}^{\e}(t))^{2} h^{\e} dy
+ {(\nu + \nu_{r})\over \e} \int_{\Om}(\e\, b_{\e}\cdot \nabla
v_{1}^{\e}(t))^{2} h^{\e} dy
+ {(\nu + \nu_{r})\over \e}\int_{\Om}(\e\, b_{\e}\cdot \nabla v_{2}^{\e}(t))^{2}
h^{\e} dy
+
\nonumber\\
+ {(\nu + \nu_{r})\over \e}\left(
\int_{\Om}\left({1\over h^{\e}}{\partial v^{\e}_{1}(t)\over \partial
y_{2}}\right)^{2} h^{\e} dy
+
\int_{\Om}\left({1\over h^{\e}}{\partial v^{\e}_{2}(t)\over \partial
y_{2}}\right)^{2} h^{\e} dy\right)
+{\aal \over \e} \int_{\Om}(\e\, b_{\e}\cdot \nabla Z^{\e}(t))^{2} h^{\e} dy+
\nonumber\\
+ {\aal \over \e} \int_{\Om}
\left({1\over h^{\e}}{\partial Z^{\e}(t)\over \partial y_{2}}\right)^{2}
h^{\e} dy
+ 4\nu_{r} \e \int_{\Om}(Z^{\e}(t))^{2} h^{\e} dy
= 2\nu_{r} \int_{\Om} \left( {\partial Z^{\e}(t)\over \partial
y_{2}}v^{\e}_{1}(t)
-(\e b_{\e}\cdot \nabla Z^{\e}(t)) v^{\e}_{2}(t) h^{\e} \right) dy
\nonumber\\
+2\nu_{r} \int_{\Om} \left( (\e b_{\e}\cdot \nabla v^{\e}_{2}(t))Z^{\e}(t)
h^{\e}
- {\partial v_{1}^{\e}(t)\over \partial y_{2}} Z^{\e}(t) \right) dy
-{(\nu + \nu_{r})\over \e} U_{0} (t) \int_{\Om} {\cal U}'(y_{2}h^{\e})
{\partial v_{1}^{\e}(t)\over \partial y_{2}} dy
\nonumber\\
-{\aal\over \e} W_{0}(t)\int_{\Om} {\cal W}'(y_{2}h^{\e}) {\partial
Z^{\e}(t)\over\partial y_{2}} dy
+2\nu_{r}W_{0}(t) \int_{\Om} {\cal W}'(y_{2} h^{\e}) v_{1}^{\e}(t)h^{\e} dy
- 2\nu_{r} U_{0}(t) \int_{\Om}{\cal U}'(y_{2} h^{\e}) Z^{\e}(t) h^{\e} dy
\nonumber\\
- 4\nu_{r}\e W_{0}(t) \int_{\Om} {\cal W}(y_{2} h^{\e}) Z^{\e}(t) h^{\e} dy
+ U_{0}(t)\int_{\Om} {\cal U}(y_{2} h^{\e}) \left(
v^{\e}_{1}(t) (\e b_{\e}\cdot \nabla v^{\e}_{1}(t)) h^{\e}
+ v^{\e}_{2}(t) {\partial v^{\e}_{1}(t)\over \partial y_{2}}\right) dy
\nonumber\\
+ W_{0}(t) \int_{\Om} \left( v^{\e}_{1}(t) (\e b_{\e}\cdot \nabla Z^{\e}(t))
{\cal
W}(y_{2} h^{\e}) h^{\e}
+ v^{\e}_{2}(t) {\partial Z^{\e}(t)\over \partial y_{2}} {\cal W}(y_{2}
h^{\e}) \right) dy
\nonumber\\
-\e U'_{0}(t)\int_{\Om}{\cal U}(y_{2} h^{\e}) v^{\e}_{1}(t) h^{\e} dy
-\e W'_{0}(t)\int_{\Om}{\cal W}(y_{2} h^{\e}) Z^{\e}(t) h^{\e} dy
\nonumber\\
+ \e \int_{\Om}g^{\e}(t)Z^{\e}(t) h^{\e} dy
+ \e \int_{\Om} (f^{\e}_{1}(t)v^{\e}_{1}(t)+ f^{\e}_{2}(t)v^{\e}_{2}(t)) h^{\e}dy.\end{aligned}$$ Now we estimate the right hand side of the above inequality (\[eq117\]). Let $\lambda_{j}$ for $1\leq j\leq 16$, which must be some strictly positive constants, such that $$\begin{aligned}
\label{es1}
2\nu_{r} \left|\int_{\Om} {\partial Z^{\e}(t)\over \partial y_{2}} v^{\e}_{1}(t)
dy\right|
\leq {\nu_{r} \over \e \lambda_{1}}\|{1\over h^{\e} }
{\partial Z^{\e}(t)\over \partial y_{2}}\|^{2}_{L^{2}(\Om)}
+ \e \nu_{r}\lambda_{1}h^{2}_{M}\|v^{\e}_{1}(t)\|^{2}_{L^{2}(\Om)},\end{aligned}$$ $$\begin{aligned}
\label{es2}
2\nu_{r} \left|\int_{\Om} (\e b_{\e}\cdot \nabla Z^{\e}(t)) \, v^{\e}_{2}(t)
h^{\e} dy\right|
\leq {\nu_{r}\over \e \lambda_{2}}\|(\e b_{\e}\cdot \nabla
Z^{\e}(t)\|^{2}_{L^{2}(\Om)}
\nonumber\\
+ \nu_{r}\e \lambda_{2}h^{2}_{M}\|v^{\e}_{2}(t)\|^{2}_{L^{2}(\Om)},\end{aligned}$$ $$\begin{aligned}
\label{es3}
2\nu_{r}\left| \int_{\Om} (\e b_{\e}\cdot \nabla v^{\e}_{2}(t))\,
Z^{\e}(t)h^{\e} dy\right|
\leq {\nu_{r}\over \e \lambda_{3}}
\|(\e b_{\e}\cdot \nabla v^{\e}_{2}(t))\|^{2}_{L^{2}(\Om)}
\nonumber\\
+\e \lambda_{3}\nu_{r}h^{2}_{M} \|Z^{\e}(t)\|^{2}_{L^{2}(\Om)},\end{aligned}$$ $$\begin{aligned}
\label{es4}
2\nu_{r} \left|\int_{\Om} {\partial v_{1}^{\e}(t)\over \partial y_{2}}
\, Z^{\e}(t) dy \right|
\leq {\nu_{r}\over \e \lambda_{4}}
\|{1\over h^{\e}}{\partial v_{1}^{\e}(t)\over \partial y_{2}}\|^{2}_{L^{2}(\Om)}
+ \nu_{r}\e \lambda_{4}h^{2}_{M}\|Z^{\e}(t)\|^{2}_{L^{2}(\Om)}\end{aligned}$$ $$\begin{aligned}
\label{es5}
{1\over \e} \left|\int_{\Om} U_{0}(t){\cal U}'(y_{2}h^{\e})
{\partial v_{1}^{\e}(t)\over \partial y_{2}} dy
\right|
\leq {U_{0}(t)^{2}\|{\cal U}'\|^{2}_{L^{2}(0, h_m )}h_{M} L \over 2\e
\lambda_{5}(\nu +\nu_{r})}
\nonumber\\
+ {\lambda_{5}(\nu +\nu_{r})\over 2\e}
\|{1\over h^{\e}} {\partial v_{1}^{\e}(t)\over \partial
y_{2}}|^{2}_{L^{2}(\Om)},\end{aligned}$$ $$\begin{aligned}
\label{es6}
{\aal\over \e} \left|
\int_{\Om}W_{0}(t){\cal W}'(t , y_{2}h^{\e}) {\partial Z^{\e}(t)\over\partial
y_{2}} h^{\e} dy\right|
\leq
{\aal W_{0}(t)^{2}\|{\cal W}'\|^{2}_{L^{2}(0, h_m )}h_{M} L\over
2\lambda_{6}\e}
\nonumber\\+ {\lambda_{6}\aal\over 2\e}
\|{1\over h^{\e}}{\partial Z^{\e}(t)\over \partial y_{2}}\|^{2}_{L^{2}(\Om)}
\end{aligned}$$ $$\begin{aligned}
\label{es11}
2\nu_{r}\left| W_{0}(t) \int_{\Om}{\cal W}'(y_{2} h^{\e}) v_{1}^{\e}(t) h^{\e}
dy\right|
\leq {\nu_{r}W_{0}(t)^{2}\|{\cal W}'\|^{2}_{L^{2}(0, h_m )}L\over \e \lambda_{7}
h_M}
\nonumber\\+\nu_{r} \lambda_{7} \e h^{2}_{M}\|v_{1}^{\e}(t)\|^{2}_{L^{2}(\Om)},
\end{aligned}$$ $$\begin{aligned}
\label{es12}
2\nu_{r}\left|U_{0}(t) \int_{\Om} {\cal U}'(y_{2} h^{\e}) Z^{\e}(t) h^{\e}
dy\right|
\leq {\nu_{r}U_{0}(t)^{2} \|{\cal U}'\|^{2}_{L^{2}(0, h_m)} h_{M} L
\over \e \lambda_{8}}
\nonumber\\
+\nu_{r} \lambda_{8} \e \|Z^{\e}(t))^{2}\|^{2}_{L^{2}(\Om)},
\end{aligned}$$ $$\begin{aligned}
\label{es14}
\left|U_{0}(t)\int_{\Om} v^{\e}_{1}(t) (\e b_{\e}\cdot \nabla v^{\e}_{1}(t))
{\cal U}(y_{2}h^{\e}) h^{\e} dy\right|
&\leq& {\e U^{2}_{0}(t) \|{\cal U}\|^{2}_{\infty} h^{2}_{M}
\lambda_{9}\over 2 }
\|v^{\e}_{1}(t)\|^{2}_{L^{2}(\Om)}
\nonumber\\
&&
+ {1\over 2\e\lambda_{9}} \|(\e b_{\e}\cdot \nabla
v^{\e}_{1})\|^{2}_{L^{2}(\Om)},\end{aligned}$$ $$\begin{aligned}
\label{es15}
\left|U_{0}(t)\int_{\Om} v^{\e}_{2}(t) {\partial v^{\e}_{1}\over \partial y_{2}}
{\cal U}(y_{2}h^{\e}) dy\right|
&\leq& {1\over 2\e \lambda_{10}}
\|{1 \over h^{\e}}{\partial v^{\e}_{1}\over \partial y_{2}}\|^{2}_{L^{2}(\Om)}
\nonumber\\
&&
+
{\e\lambda_{10}U^{2}_{0}(t)\|{\cal U}\|^{2}_{\infty}h^{2}_{M} \over 2}
\|v^{\e}_{2}(t)\|^{2}_{L^{2}(\Om)},\end{aligned}$$ $$\begin{aligned}
\label{es16}
\left|W_{0}(t)\int_{\Om} v^{\e}_{1}(t) (\e b_{\e}\cdot \nabla Z^{\e}) {\cal
W}(y_{2}h^{\e})
h^{\e} dy\right|
&\leq& {1\over 2\e \lambda_{11}} \|(\e b_{\e}\cdot \nabla
Z^{\e})\|^{2}_{L^{2}(\Om)}
\nonumber\\
&&
+{\e \lambda_{11}W^{2}_{0}(t)\|{\cal W}\|^{2}_{\infty}h^{2}_{M} \over 2}
\|v^{\e}_{1}(t)\|^{2}_{L^{2}(\Om)},\end{aligned}$$ $$\begin{aligned}
\label{es17}
\left|W_{0}(t)\int_{\Om} v^{\e}_{2}(t) {\partial Z^{\e}\over \partial y_{2}}
{\cal W}(y_{2}h^{\e}) dy
\right|
&\leq& {1\over 2\e \lambda_{12}}
\|{1\over h^{\e}} {\partial Z^{\e}\over \partial y_{2}} \|^{2}_{L^{2}(\Om)}
\nonumber\\
&&+{\e \lambda_{12}W^{2}_{0}(t)\|{\cal W}\|^{2}_{\infty}h^{2}_{M}
\over 2}\|v^{\e}_{2}(t)\|^{2}_{L^{2}(\Om)},\end{aligned}$$ $$\begin{aligned}
\label{es131}
4\nu_{r}\e\left|W_{0}(t) \int_{\Om} {\cal W}(y_{2}h^{\e}) Z^{\e}(t) h^{\e}
dy\right|
\leq {2\nu_{r}\e W_{0}(t)^{2} \|{\cal W}\|^{2}_{L^{2}(0,h_m)}h_{M} L
\over \lambda_{13}}
\nonumber \\
+ 2 \e \lambda_{13}\nu_{r} \|Z^{\e}(t)\|^{2}_{L^{2}(\Om)},
\end{aligned}$$ $$\begin{aligned}
\label{es13a}
\e \left| U'_{0}(t)\int_{\Om} {\cal U}(y_{2}h^{\e})v^{\e}_{1}(t)h^{\e}
dy\right|
\leq \frac{\e}{2} h^{2}_{M}\|v^{\e}_{1}(t)\|^{2}_{L^{2}(\Om)}
\nonumber \\
+ \frac{\e L}{2
h_M} |U'_{0}(t)|^{2}
\|{\cal U}\|^{2}_{L^{2}(0, h_m)},
\end{aligned}$$ $$\begin{aligned}
\label{es13aa}
\e |W'_{0}(t)\int_{\Om} {\cal W}(y_{2}h^{\e}) Z^{\e}(t) h^{\e} dy|
\leq \frac{\e}{2} h^{2}_{M}\|Z^{\e}(t)\|^{2}_{L^{2}(\Om)} \nonumber \\+ \frac{\e L}{2 h_M}
|W'_{0}(t)|^{2}\|
\|{\cal W}\|^{2} _{L^{2}(0, h_m)}.
\end{aligned}$$ Finally $$\begin{aligned}
\e \left|\int_{\Om}g^{\e}(t)Z^{\e}(t) h^{\e} dy\right|
\le \frac{h_M}{\e} \|\e^2 g^{\e} \|_{L^2(\Om)} \|Z^{\e}\|_{L^2(\Om)}.\end{aligned}$$ By using Poincaré’s inequality and the boundary conditions (\[eqn:er2.8a\])-(\[eqn:er2.11b\]), we get $$\begin{aligned}
\|Z^{\e}\|_{L^2(\Om)} \le \| \frac{\partial Z^{\e}}{\partial y_2} \|_{L^2(\Om)}
\quad \mbox{\rm a.e. in} \ (0,T),\end{aligned}$$ and $$\begin{aligned}
\label{es13}
&& \e \left|\int_{\Om}g^{\e}(t)Z^{\e}(t) h^{\e} dy\right|
\le \frac{h_M}{\e} \|\e^2 g^{\e} \|_{L^2(\Om)} \|\frac{\partial
Z^{\e}}{\partial y_2}\|_{L^2(\Om)}
\nonumber \\
&&
\le \frac{h_M^2}{\e } \|\e^2 g^{\e} \|_{L^2(\Om)} \|\frac{1}{h^{\e}}
\frac{\partial Z^{\e}}{\partial y_2}\|_{L^2(\Om)}
\nonumber \\
&& \le \frac{h_M^4}{2 \lambda_{14} \e } \|\e^2 g^{\e} \|_{L^2(\Om)}^2 +
\frac{\lambda_{14}}{2 \e} \|\frac{1}{h^{\e}} \frac{\partial Z^{\e}}{\partial
y_2}\|_{L^2(\Om)}^2.\end{aligned}$$ Similarly $$\begin{aligned}
\label{ess14}
\e \left|\int_{\Om}f_1^{\e}(t)v_1^{\e}(t) h^{\e} dy\right| \le
\frac{h_M^4}{2 \lambda_{15} \e} \|\e^2 f_1^{\e} \|_{L^2(\Om)}^2 +
\frac{\lambda_{15}}{2 \e} \|\frac{1}{h^{\e}} \frac{\partial v_1^{\e}}{\partial
y_2}\|_{L^2(\Om)}^2,\end{aligned}$$ and $$\begin{aligned}
\label{ess15}
\e \left|\int_{\Om}f_2^{\e}(t)v_2^{\e}(t) h^{\e} dy\right| \le
\frac{h_M^4}{2 \lambda_{16} \e } \|\e^2 f_2^{\e} \|_{L^2(\Om)}^2 +
\frac{\lambda_{16}}{2 \e} \|\frac{1}{h^{\e}} \frac{\partial v_2^{\e}}{\partial
y_2}\|_{L^2(\Om)}^2.\end{aligned}$$ So from (\[eq117\]) and (\[es1\])-(\[es13\]) we get $$\begin{aligned}
\label{eq3.33}
\frac{\e}{2} {d\over dt}([{\bar{v}}^{\e}(t)]^{2})
+ {c_{1}\over \e}\|(\e\, b_{\e}\cdot \nabla v_{1}^{\e}(t)\|^{2}_{L^{2}(\Om)}
+ {c_{2}\over \e}\|(\e\, b_{\e}\cdot \nabla v_{2}^{\e}(t)\|^{2}_{L^{2}(\Om)}
\nonumber\\
+ {c_{3}\over \e} \|{1\over h^{\e}}{\partial Z^{\e}(t)\over \partial
y_{2}}\|^{2}_{L^{2}(\Om)}
+ {c_{4}\over \e}\|{1\over h^{\e}}{\partial v^{\e}_{1}(t)\over \partial
y_{2}}\|^{2}_{L^{2}(\Om)}
+ {c_{5}\over \e} h_m \|{1\over h^{\e}}{\partial v^{\e}_{2}(t)\over
\partial y_{2}}\|^{2}_{L^{2}(\Om)}
\nonumber\\
+ {c_{6}\over \e}\|(\e\, b_{\e}\cdot \nabla Z^{\e}(t)\|^{2}_{L^{2}(\Om)}
+ \nu_{r}\e c_{7} \|Z^{\e}(t)\|^{2}_{L^{2}(\Om)} \leq
\e c_{8}(t)[{\bar{v}^{\e}}(t)]^{2} + {c_{9}(t)\over\e}\end{aligned}$$ where $$\begin{aligned}
&&c_{1}=(\nu+ \nu_{r}) h_m - {1\over 2\lambda_{9}}, \qquad
c_{2}= (\nu + \nu_{r}) h_m - {\nu_{r}\over\lambda_{3}},
\nonumber\\
&&c_{3}=\aal h_m - {\nu_{r}\over \lambda_{1}}- {\lambda_{6}\aal \over 2}
- {1\over 2 \lambda_{12}} - \frac{\lambda_{14}}{2}, \quad
c_{4}=(\nu + \nu_{r}) h_m - {\lambda_{5}(\nu + \nu_{r})^2 \over 2}
-{\nu_{r}\over \lambda_{4}} -{1 \over 2\lambda_{10}} -
\frac{\lambda_{15}}{2},
\nonumber\\
&& c_{5} = (\nu + \nu_r) h_m - \frac{\lambda_{16}}{2} , \quad
c_{6}=\aal h_m - {1\over 2 \lambda_{11}} -{\nu_{r}\over \lambda_{2}},\quad
c_{7}=4 h_m - \lambda_{8} - 2 \lambda_{13}, \nonumber \\
&&
c_{8}(t)=\max\{A(t) , B(t), h_M^2 (1 + \lambda_3 + \lambda_4) \}\end{aligned}$$ with $$A(t)=h_{M}^{2}
\left(1+\nu_{r}\lambda_{1}+ \nu_{r}\lambda_{7} +
{\lambda_{9} U_{0}^{2}(t) \|{\cal U}\|^{2}_{\infty}
+ \lambda_{11}W^{2}_{0}(t)\|{\cal W}\|^{2}_{\infty} \over 2}\right)$$ $$B(t)=h_{M}^{2}\left( \frac{1}{2} + \nu_{r}\lambda_{2} +
{\lambda_{10}U^{2}_{0}(t)\|{\cal U}\|^{2}_{\infty}
\over 2}
+ {\lambda_{12}W^{2}_{0}(t) \|{\cal W}\|^{2}_{\infty}\over 2} \right)$$ and $$\begin{aligned}
{c_{9}(t)\over\e}&=& {U^{2}_{0}(t) \|{\cal U}'\|^{2}_{L^{2}(0, h_m)}
h_{M} L
\over 2\e\lambda_{5} }
+ {\nu_{r} U^{2}_{0}(t) \|{\cal U}\|^{2}_{L^{2}(0, h_m)} h_{M} L
\over \e\lambda_{8} }
+{\aal W^{2}_{0}(t) \|{\cal W}'\|^{2}_{L^{2}(0, h_m)} h_{M} L \over 2
\e \lambda_{6} }
\nonumber\\
&&
+ {\nu_{r} W^{2}_{0}(t) \|{\cal W}'\|^{2}_{L^{2}(0, h_m )} L \over \e
\lambda_{7} h_M}
+ {2\nu_{r} \e W_{0}(t)^{2} \|{\cal W}\|^{2}_{L^{2}(0, h_m )} h_{M} L \over
\lambda_{13} }
+ \frac{\e L}{2 h_M} |U'_{0}(t)|^{2}\|{\cal U}\|^{2}_{L^{2}(0, h_m)}
\nonumber\\
&&
+ \frac{\e L}{2 h_M} |W'_{0}(t)|^{2} \|{\cal W}\|^{2}_{L^{2}(0, h_m)}
+ {h_M^4\over 2 \e}\left(\frac{1}{\lambda_{15}} \|\e^2 f_{1}^{\e}
(t)\|^{2}_{L^{2}(\Omega)}
+ \frac{1}{\lambda_{16}} \|\e^2 f_{2}^{\e} (t)\|^{2}_{L^{2}(\Omega)}
+ \frac{1}{\lambda_{14}} \|\e^2 g^{\e} (t)\|^{2}_{L^{2}(\Omega)}\right).\end{aligned}$$ Each $c_{i}$ for $i=1,\cdots, 6$ must be strictly positive, which is possible for example with $$\begin{aligned}
\lambda_{1}= \lambda_{2}= {4\nu_{r}\over \aal h_m },\qquad \qquad \lambda_{3}=
\lambda_{4}= \frac{1}{h_m} ,\qquad
\qquad \lambda_{5}={\nu h_m \over 2(\nu + \nu_{r})^2},
\nonumber\\
\qquad \lambda_{6}= \lambda_{8}=\lambda_{13}={h_m \over 2},\qquad
\lambda_{9}={4\over (\nu + \nu_{r}) h_m }, \qquad \lambda_{10}={2\over \nu h_m
}, \qquad
\lambda_{11}=\lambda_{12}={2\over \aal h_m }, \nonumber \\
\qquad \lambda_{14} = \frac{ \alpha h_m}{8} , \qquad \qquad \lambda_{15} =
\frac{ \nu h_m}{4}, \qquad \qquad \lambda_{16} = \frac{ (\nu + \nu_r) h_m}{2} .\end{aligned}$$ Note that $\lambda_{7}$ remains arbitrary and can be taken as $\lambda_7=1$. So from (\[eq3.33\]) we get $$\begin{aligned}
\frac{\e^{2}}{2} {d\over dt} ([\bar{v}^{\e}(t)]^{2})
\leq
\e^{2} c_{7}(t) [\bar{v}^{\e}(t)]^{2}
+ c_{8}(t).\end{aligned}$$ As $U_{0}$ and $W_{0}$ belong to $H^{1}(0 , T)$, then $c_8$ is bounded in $L^1(0,T)$ independently of $\e$ and by Grönwall’s lemma we deduce that there exists a constant $C$ independent of $\e$ such that $$\begin{aligned}
\label{eq3.34}
\e^2 [\bar{v}^{\e}(t)]^{2} \leq C
\quad \forall t\in [0 , T].\end{aligned}$$ Now we integrate the inequality (\[eq3.33\]) over the time interval $(0 , s)$ for $0< s\leq T$, we deduce $$\begin{aligned}
\label{eq3.35}
\frac{\e^{2}}{2} [{\bar{v}}^{\e}(s)]^{2}
+
C_{1}\int_{0}^{s}\|(\e\, b_{\e}\cdot \nabla v_{1}^{\e}(t)\|^{2} _{L^{2}(\Om)}
+\|(\e\, b_{\e}\cdot \nabla v_{2}^{\e}(t)\|^{2}_{L^{2}(\Om)}+
+ \|(\e\, b_{\e}\cdot \nabla Z^{\e}(t)\|^{2}_{L^{2}(\Om)} dt
\nonumber\\
+ C_{2}\int_{0}^{s}
\|{1\over h^{\e}}{\partial Z^{\e}(t)\over \partial y_{2}}\|^{2}_{L^{2}(\Om)}
+
\|{1\over h^{\e}}{\partial v^{\e}_{1}(t)\over \partial y_{2}}\|^{2}_{L^{2}(\Om)}
+ \|{1\over h^{\e}}{\partial v^{\e}_{2}(t)\over \partial
y_{2}}\|^{2}_{L^{2}(\Om)} dt
\nonumber\\
+ \e^{2} \nu_r c_6 \int_{0}^{s} \|(Z^{\e}(t)\|^{2}_{L^{2}(\Om)} dt
\leq
\e^{2} \int_{0}^{s} c_{7}(t)[{\bar{v}}^{\e}(t)]^{2}dt + \int_{0}^{s}
c_{8}(t) dt
+ \frac{\e^{2}}{2} [{\bar{v}}^{\e}(0)]^{2},\end{aligned}$$
where $C_{1}= \min\{c_{1}, c_{2}, c_{6}\}$, $C_{2}= \min\{c_{3},c_{4}, c_{5} \}$, are two constants independent of $\e$. Observing that $c_7 \in L^{\infty}(0,T)$ and $$\begin{aligned}
\int_{0}^{T}\int_{\Om} \left({1\over h^{\e}}{\partial v^{\e}_{i}(t)\over
\partial y_{2}}\right)^{2} \, dy dt
\geq {1\over h_{M}^{2}}\|{\partial v^{\e}_{i}\over \partial
y_{2}}\|_{L^{2}((0,T) \times\Om)}^{2}\end{aligned}$$ we deduce (\[E3.13\]) and (\[E3.14\]) from (\[eq3.34\]). Moreover, from (\[not\]) $$\begin{aligned}
\label{not2}
b_{\e}\cdot\nabla= {\partial \over \partial y_{1}} - {y_{2}\over
h^{\e}}{\partial h^{\e}\over \partial y_{1}} {\partial \over \partial
y_{2}}
\quad \mbox{ with }\quad
|{\partial h^{\e}\over \partial y_{1}}| =|{\partial h\over \partial
y_{1}} + {1\over \e} {\partial h\over \partial \eta_{1}}|\leq {C\over \e}.\end{aligned}$$ Thus we have $$\begin{aligned}
\int_{0}^{T}\int_{\Om} \left(\e{\partial v^{\e}_{i}(t)\over \partial
y_{1}}\right)^{2} \, dy dt
=\int_{0}^{T}\int_{\Om} \left((\e b_{\e}\cdot\nabla v^{\e}_{i}(t))
+ {y_{2}\e \over h^{\e}}{\partial h^{\e}\over \partial y_{1}} {\partial
v^{\e}_{i}(t) \over \partial y_{2}} \right)^{2} \, dy dt
\nonumber\\
\leq 2 \|(\e b_{\e}\cdot\nabla v^{\e}_{i})\|^{2}_{L^{2}((0 ,
T)\times\Om)}
+ 2 {C\over h_{m}}\| {\partial v^{\e}_{i} \over \partial
y_{2}}\|^{2}_{L^{2}((0
, T)\times\Om)}\end{aligned}$$ and a similar estimate holds for $Z^{\e}$. Finally with (\[E3.13\]) and (\[E3.14\]) we deduce (\[E3.15\]). Next, using again the boundary conditions (\[eqn:er2.8a\])-(\[eqn:er2.11b\]) and Poincaré’s inequality, we get $$\begin{aligned}
\|v^{\e}_{i}\|^{2}_{L^{2}((0 , T)\times\Om)}\leq \int_{0}^{T}\int_{\Om}
\| {\partial v^{\e}_{i} \over \partial y_{2}}\|^{2} dy dt=
\| {\partial v^{\e}_{i} \over \partial y_{2}}\|^{2}_{L^{2}((0 , T)\times\Om)}\end{aligned}$$ and we deduce (\[E3.16\]) from (\[E3.14\]).
\[prop2\] Assume that the proposition [\[pro1\]]{} holds. Then there exists a constant $C>0$ which does not depends on $\e$, such that we have $$\begin{aligned}
\label{E3.18}
\e^{2} \|p^{\e}\|_{H^{-1}(0 , T; L^{2}(\Om))} \leq C.\end{aligned}$$
Let $\varphi\in {\mathcal D}(0,T) \times {\mathcal D}(\Om)$, then choose $\Theta= ( 0, \varphi(t), 0)$ as a test-function in (\[eqvar\]) and multiply the two sides by $\e$: we obtain $$\begin{aligned}
\label{eq3.43}
\e \int_{0}^{T}\int_{\Om} p^{\e}
{\partial \varphi\over \partial y_{2}} dy dt
=
-\e^{2}\int_{0}^{T} \int_{\Om} v^{\e}_{2} {\partial\varphi\over \partial t}
h^{\e}
dy dt
\nonumber\\
+ (\nu+\nu_{r})\sum_{i=1}^{2}
\int_{0}^{T} \int_{\Om}\left(
h^{\e}(\e b_{\e}\cdot \nabla v^{\e}_{2}) (\e b_{\e}\cdot \nabla\varphi)
+{1\over h^{\e}}{\partial v^{\e}_{2}\over \partial y_{2}}
{\partial\varphi\over \partial y_{2}}\right) dy dt
\nonumber\\
+\int_{0}^{T}\int_{\Om} \left(\sum_{i=1}^{2}
\e v^{\e}_{1}(\e b_{\e}\cdot \nabla v^{\e}_{2})\varphi h^{\e}
+ \e v^{\e}_{2}{\partial v^{\e}_{2}\over \partial
y_{2}}\varphi\right) dy
dt
+ 2\nu_{r}
\int_{0}^{T} \int_{\Om}
(\e b_{\e}\cdot \nabla Z^{\e})\varphi \e h^{\e} dy dt
\nonumber\\
+ \int_{0}^{T}\int_{\Om}
U_{\e}(\e b_{\e}\cdot \nabla v^{\e}_{2})\varphi\e h^{\e} dy dt
- \int_{0}^{T} \int_{\Om}
\e^{2} f^{\e}_{2}\varphi h^{\e} dy dt,\end{aligned}$$ with $U_{\e} (t, y) = U_0(t) {\mathcal U}(y_2 h^{\e} (y_1))$ for all $(t, y_1, y_2) \in [0,T] \times \Om$. Using (\[E3.13\])-(\[E3.16\]), we get $$\begin{aligned}
\label{equPy1}
|\int_{0}^{T} \int_{\Om}
p^{\e} {\partial \varphi\over \partial y_{2}} dy dt
|\leq {C\over \e} \|\varphi\|_{H^{1}(0 , T ,
H^{1}_{0}(\Om))} \quad \forall \varphi\in {\mathcal D}(0,T) \times {\mathcal
D}(\Om) .
\end{aligned}$$ Now let $\phi \in {\mathcal D}(0,T) \times {\mathcal D}(\Om)$ and choose $\Theta
= ( \frac{\phi}{h^{\e}}, 0,0)$ as a test-function in (\[eqvar\]), then multiply the two sides by $\e$: we obtain $$\begin{aligned}
\label{eq3.45} \e^{2} \int_{0}^{T}\int_{\Om} p^{\e}
\left( {\partial\phi \over \partial y_{1}}
- \frac{\partial }{\partial y_2} \left( y_2 \frac{1}{h^{\e}} \frac{\partial
h^{\e}}{\partial y_1} \phi \right) \right)
dy dt
=
-\e^{2}\int_{0}^{T} \int_{\Om} v^{\e}_{1} {\partial\phi\over \partial t}
dy dt
\nonumber\\
+ (\nu+\nu_{r})
\int_{0}^{T} \int_{\Om} \e
(\e b_{\e}\cdot \nabla v^{\e}_{1})
\left(
\frac{\partial \phi}{\partial y_1} - \frac{\partial h^{\e}}{\partial y_1}
\frac{1}{h^{\e}} \phi - y_2 \frac{1}{h^{\e}} \frac{\partial h^{\e}}{\partial
y_1} \frac{\partial \phi}{\partial y_2}
\right) dy dt \nonumber\\
+
(\nu+\nu_{r})
\int_{0}^{T} \int_{\Om}
{1\over ( h^{\e} )^2}{\partial v^{\e}_{1}\over \partial y_{2}}
{\partial\phi\over \partial y_{2}} dy dt
+\int_{0}^{T}\int_{\Om} \left(
\e v^{\e}_{1}(\e b_{\e}\cdot \nabla v^{\e}_{1})\phi
+ \frac{\e}{h^{\e}} v^{\e}_{2}{\partial v^{\e}_{1}\over \partial
y_{2}}\phi\right) dy
dt
\nonumber\\
+ (\nu+\nu_{r})\int_{0}^{T} \int_{\Om} \frac{1}{(h^{\e})^2}
{\partial U_{\e}\over \partial y_{2}}{\partial\phi\over \partial y_{2}}
dy
dt
-2\nu_{r}
\int_{0}^{T} \int_{\Om} \frac{\e}{h^{\e}}
{\partial Z^{\e}\over \partial y_{2}}\phi dy dt
+ \int_{0}^{T} \int_{\Om} \e U_{\e}
(\e b_{\e}\cdot \nabla v^{\e}_{1})\phi dy dt
\nonumber\\
- \int_{0}^{T} \int_{\Om}
\e^2 v^{\e}_{1} \left( \frac{\partial \phi}{\partial y_1} - \frac{\partial
h^{\e}}{\partial y_1} \frac{1}{h^{\e}} \phi - y_2 \frac{\partial
h^{\e}}{\partial y_1}\frac{1}{h^{\e}} \frac{\partial \phi}{\partial y_2} \right)
U_{\e} dy dt
- \int_{0}^{T} \int_{\Om}
\frac{\e}{h^{\e}} v^{\e}_{2}{\partial \phi \over \partial y_{2}}U_{\e} dy dt
\nonumber\\
-2\e \nu_{r}\int_{0}^{T} \int_{\Om}
\frac{1}{h^{\e}} {\partial W_{\e}\over \partial y_{2}}\phi dy dt
- \int_{0}^{T} \int_{\Om}
\left( f^{\e}_{1}\phi -{\partial U^{\e} \over \partial
t}\phi\right) \e^{2} dy dt,\end{aligned}$$ where $W_{\e} (t, y) = W_0(t) {\mathcal W}(y_2 h^{\e} (y_1))$ for all $(t, y_1,
y_2) \in [0,T] \times \Om$.
Using the estimates (\[E3.13\])-(\[E3.16\]) and (\[not2\]), we infer that $$\begin{aligned}
|\int_{0}^{T}\int_{\Om} p^{\e}
\left( {\partial\phi \over \partial y_{1}}
- \frac{\partial }{\partial y_2} \left( y_2 \frac{1}{h^{\e}} \frac{\partial
h^{\e}}{\partial y_1} \phi \right) \right)
dy dt | \leq {C \over \e^{2}}\|\phi\|_{H^{1}(0 , T ; H^{1} (\Om))}.
\end{aligned}$$ By choosing now $\varphi= y_{2} \frac{1}{h^{\e}} {\partial h^{\e}\over\partial y_{1}}\phi$ in (\[eq3.43\]), we get $$\begin{aligned}
\frac{\partial \varphi}{\partial t} = y_{2} \frac{1}{h^{\e}} {\partial
h^{\e}\over\partial y_{1}} \frac{\partial \phi}{\partial t} , \quad
\frac{\partial \varphi}{\partial y_2} = \frac{1}{h^{\e}} {\partial
h^{\e}\over\partial y_{1}} \left( \phi + y_2 \frac{\partial \phi}{\partial y_2}
\right)\end{aligned}$$ and $$\begin{aligned}
b_{\e} \cdot \nabla \varphi =
- y_2 \frac{1}{(h^{\e})^2}
\left( \frac{\partial h^{\e}}{\partial y_1} \right)^2 \left( 2 \phi + y_2
\frac{\partial \phi}{\partial y_2} \right)
+ y_2 \frac{1}{h^{\e}} \left( \frac{\partial^2 h^{\e}}{\partial y_1^2} \phi +
\frac{\partial h^{\e}}{\partial y_1} \frac{\partial \phi}{\partial y_1} \right)
.\end{aligned}$$ Hence $$\begin{aligned}
&& \| \varphi \|_{L^{\infty}(0,T; L^4(\Om))} \le \frac{C}{\e} \| \phi\|_{H^1
(0,T; H^1(\Om))}, \quad
\| \frac{\partial \varphi}{\partial t} \|_{L^2 ((0,T) \times \Om)} \le
\frac{C}{\e} \| \phi\|_{H^1 (0,T; H^1(\Om))}, \\
&& \| \frac{\partial \varphi}{\partial y_2} \|_{L^2 ((0,T) \times \Om)} \le
\frac{C}{\e} \| \phi\|_{H^1 (0,T; H^1(\Om))}, \quad
\| \e b_{\e} \cdot \nabla \varphi \|_{L^2 ((0,T) \times \Om)} \le \frac{C}{\e}
\| \phi\|_{H^1 (0,T; H^1(\Om))}\end{aligned}$$ and with (\[eq3.43\]) $$\begin{aligned}
|\int_{0}^{T}\int_{\Om} p^{\e}
\frac{\partial }{\partial y_2} \left( y_2 \frac{1}{h^{\e}} \frac{\partial
h^{\e}}{\partial y_1} \phi \right)
dy dt | \leq {C \over \e^{2}}\|\phi\|_{H^{1}(0 , T ; H^{1}(\Om))}.
\end{aligned}$$ It follows that $$\begin{aligned}
\label{equPy2}
|\int_{0}^{T}\int_{\Om} p^{\e}
{\partial\phi \over \partial y_{1}}
dy dt | \leq {C \over \e^{2}}\|\phi\|_{H^{1}(0 , T ; H^{1} (\Om))} \quad
\forall \phi \in {\mathcal D}(0,T) \times {\mathcal D}(\Om).
\end{aligned}$$ By density of ${\mathcal D}(0,T) \times {\mathcal D}(\Om)$ into $H^1_0(0,T;
H^1_0(\Om))$ we get from (\[equPy1\])-(\[equPy2\]) $$\begin{aligned}
\label{equPy3}
\| \frac{\partial p^{\e}}{\partial y_2} \|_{H^{-1} (0,T; H^{-1} (\Om))} \le
\frac{C}{\e}, \quad \| \frac{\partial p^{\e}}{\partial y_1} \|_{H^{-1} (0,T;
H^{-1} (\Om))} \le \frac{C}{\e^2}.\end{aligned}$$ Finally we can deduce [@Temam1979] that $\e^{2}p^{\e}$ remains in a bounded subset of $H^{-1}(0 , T ; L^{2}(\Om))$.
Two-scale convergence properties {#twoscaleconv}
================================
Since our unknown functions depend on the time variable, we are not in the classical framework of two-scale convergence as it has been introduced by G. Allaire in [@allaire] or G. Nguetseng in [@nguetseng]. Nevertheless this technique can be easily adpated to a time-dependent framework (see for instance [@miller; @holm97; @gilbert-Mik2000; @wright00]). For the convenience of the reader we will provide a complete proof a the generalization of [@allaire] that will be used later for the study of the sequences $(v^{\e})_{\e >0}$, $(Z^{\e})_{\e >0}$ and $(p^{\e})_{\e >0}$.
Let us recall the following usual notations: $Y=[0,1]^2$, $ {\mathcal C}^{\infty}_{\sharp} (Y) $ is the space of infinitely differentiable functions in ${\mathbb R}^2$ that are $Y$-periodic and $$\begin{aligned}
L^2_{\sharp} (Y) = \overline{{\mathcal C}^{\infty}_{\sharp} (Y)}^{L^2(Y)}, \quad
H^1_{\sharp} (Y) = \overline{{\mathcal C}^{\infty}_{\sharp} (Y)}^{H^1(Y)}.\end{aligned}$$
The space $L^2_{\sharp} (Y)$ coincides with the space of functions of $L^2(Y) $ extended by $Y$-periodicity to ${\mathbb R}^2$.
We extend the definition of the two-scale convergence as follows
A sequence $(w^{\e})_{\e >0}$ of $L^2 \bigl( (0,T) \times
\Omega \bigr)$ (resp. in $H^{-1} \bigl( 0,T; L^2( \Omega) \bigr)$) two-scale converges to $w^0 \in L^2 \bigl( 0,T; L^2 (\Omega \times Y) \bigr)$ (resp. $w^0
\in H^{-1} \bigl( 0,T; L^2 (\Omega \times Y) \bigr)$) if and only if $$\begin{aligned}
\lim_{\e \to 0} \int_0^T \int_{\Omega} w^{\e} (t,y) \varphi \left( y,
\frac{y}{\e} \right) \theta (t) \, dy dt = \int_0^T \int_{\Omega \times Y} w^0 (
t, y, \eta) \varphi (y, \eta) \theta (t) \, d\eta dy dt\end{aligned}$$ for all $\theta \in {\mathcal D}(0,T)$, for all $\varphi \in {\mathcal D}
\bigl(\Omega; {\mathcal C}^{\infty}_{\sharp} (Y) \bigr)$. In such a case we will denote $w^{\e} {\twoheadrightarrow}w^0$.
Then we obtain
Let $(w^{\e})_{\e >0}$ be a bounded sequence of $L^2 \bigl( (0,T) \times \Omega
\bigr)$ (resp. in $H^{-1} \bigl( 0,T ; L^2( \Omega ) \bigr)$). There exists $w^0 \in L^2 \bigl( 0,T; L^2 (\Omega \times Y) \bigr)$ (resp. $w^0 \in H^{-1}
\bigl( 0,T; L^2 (\Omega \times Y) \bigr)$) such that, possibly extracting a subsequence still denoted $(w^{\e})_{\e >0}$, we have $$\begin{aligned}
w^{\e} {\twoheadrightarrow}w^0.\end{aligned}$$
The proof is similar to the proof of Theorem 1.2 in [@allaire]. Let us assume first that $(w^{\e})_{\e >0}$ is a bounded sequence of $L^2 \bigl( (0,T)\times \Omega \bigr)$. In our time-dependent framework we consider test-functions $\psi \in {\mathcal C} \bigl( [0,T]; {\mathcal C}
(\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) \bigr)$. Furthermore, For any $\psi \in {\mathcal C} \bigl( [0,T]; {\mathcal C} (\overline{\Omega};
{\mathcal C}_{\sharp} (Y) ) \bigr)$ and for any fixed $\e >0$, the mapping $
\displaystyle (t,y) \mapsto \psi^{\e} (t,y)= \psi \left( t, y, \frac{y}{\e}
\right)$ is mesurable on $(0,T) \times \Omega$ and satisfies $$\begin{aligned}
\| \psi^{\e} \|_{L^2 ( (0,T) \times \Omega)} = \left( \int_0^T \int_{\Omega}
\left( \psi \left( t, y, \frac{y}{\e} \right) \right)^2 \, dy dt \right)^{1/2}
\le \sqrt{ T | \Omega|} \| \psi\|_{{\mathcal C} ( [0,T]; {\mathcal C}
(\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) )}.\end{aligned}$$ Hence we can define $\Lambda_{\e} \in \Bigl( {\mathcal C} \bigl( [0,T];
{\mathcal C} (\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) \bigr) \Bigr)'$ by $$\begin{aligned}
\Lambda_{\e} (\psi) = \int_0^T \int_{\Omega} w^{\e} (t,y) \psi \left( t, y,
\frac{y}{\e} \right) \, dy dt
\quad \forall \psi \in {\mathcal C} \bigl( [0,T]; {\mathcal C}
(\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) \bigr).\end{aligned}$$ Since $(w^{\e})_{\e >0}$ is a bounded sequence of $L^2 \bigl( (0,T) \times
\Omega \bigr)$, we infer with Cauchy-Schwarz’s inequality that there exists a real number $C>0$, independent of $\e$, such that $$\begin{aligned}
\label{sec4.1}
\bigl| \Lambda_{\e} (\psi) \bigr| &\le& \|w^{\e} \|_{L^2 ( (0,T) \times \Omega)}
\| \psi^{\e} \|_{L^2 ( (0,T) \times \Omega)} \le C \| \psi^{\e} \|_{L^2 (
(0,T) \times \Omega)} \nonumber\\
&\le& C \sqrt{ T | \Omega|} \| \psi\|_{{\mathcal C} (
[0,T]; {\mathcal C} (\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) )}\end{aligned}$$ for all $ \psi \in {\mathcal C} \bigl( [0,T]; {\mathcal C} (\overline{\Omega};
{\mathcal C}_{\sharp} (Y) ) \bigr)$ and the sequence $(\Lambda_{\e})_{{\e} >0}$ is bounded in $\Bigl( {\mathcal C} \bigl( [0,T]; {\mathcal C}
(\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) \bigr) \Bigr)'$. Reminding that $ {\mathcal C} \bigl( [0,T]; {\mathcal C} (\overline{\Omega}; {\mathcal
C}_{\sharp} (Y) ) \bigr) $ is a separable Banach space, we infer that there exists $\Lambda_0 \in \Bigl( {\mathcal C} \bigl( [0,T]; {\mathcal C}
(\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) \bigr) \Bigr)'$ such that, possibly extracting a subsequence still denoted $(\Lambda_{\e})_{{\e} >0}$, $$\begin{aligned}
(\Lambda_{\e}) \rightharpoonup \Lambda_{0} \quad \hbox{\rm weak * in }
\Bigl( {\mathcal C} \bigl( [0,T]; {\mathcal C} (\overline{\Omega}; {\mathcal
C}_{\sharp} (Y) ) \bigr) \Bigr)'\end{aligned}$$ i.e. $$\begin{aligned}
\lim_{{\e} \to 0} \int_0^T \int_{\Omega} w^{\e} (t,y) \psi \left( t, y,
\frac{y}{\e} \right) \, dy dt = \Lambda_0 (\psi)
\quad \forall \psi \in {\mathcal C} \bigl( [0,T]; {\mathcal C}
(\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) \bigr).\end{aligned}$$ Observing that, for all $t \in [0,T]$, $\psi^2 ( t, \cdot, \cdot) \in L^1
\bigl(\Omega; {\mathcal C}_{\sharp} (Y) ) \bigr)$, we can also use Lemma 5.2 of [@allaire], which yields $$\begin{aligned}
\lim_{{\e} \to 0} \int_{\Omega} \left( \psi \left( t, y, \frac{y}{\e} \right)
\right)^2 \, dy = \int_{\Omega \times Y} \bigl( \psi( t, y, \eta) \bigr)^2 \, d
\eta dy \quad \forall t \in [0,T].\end{aligned}$$ Then, using Lebesgue’s convergence theorem, we obtain $$\begin{aligned}
\label{sec4.2}
\lim_{{\e} \to 0} \int_0^T \int_{\Omega} \left( \psi \left( t, y, \frac{y}{\e}
\right) \right)^2 \, dy dt = \int_0^T \int_{\Omega \times Y} \bigl( \psi( t, y,
\eta) \bigr)^2 \, d \eta dy dt .\end{aligned}$$ With (\[sec4.1\]) and (\[sec4.2\]) we get $$\begin{aligned}
\bigl| \Lambda_0 ( \psi) \bigr| \le C \| \psi\|_{L^2 (0,T; L^2 (\Omega \times
Y))} \quad \forall \psi \in {\mathcal C} \bigl( [0,T]; {\mathcal C}
(\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) \bigr).\end{aligned}$$ It follows that $\Lambda_0$ can be extended to $ \Bigl( L^2 \bigl(0,T; L^2
(\Omega \times Y) \bigr) \Bigr)'$ and with Riesz’s representation theorem we infer that there exists $w^0 \in L^2 \bigl(0,T; L^2 (\Omega \times Y) \bigr)$ such that $$\begin{aligned}
\Lambda_0 ( \psi) = \int_0^T \int_{\Omega \times Y} w^0 (t, y, \eta) \psi (t, y,
\eta) \, d\eta dy dt \quad \forall \psi \in L^2 \bigl(0,T; L^2 (\Omega \times Y)
\bigr)\end{aligned}$$ which allows us to conclude for the first part of the theorem.
Let us assume now that $(w^{\e})_{\e >0}$ is a bounded sequence of $H^{-1}
\bigl( 0,T; L^2( \Omega) \bigr)$ and let $$\begin{aligned}
{\mathcal C}^1_0 \bigl( [0,T]; {\mathcal C} (\overline{\Omega}; {\mathcal
C}_{\sharp} (Y) ) \bigr) = \bigl\{ \psi \in {\mathcal C}^1 \bigl( [0,T];
{\mathcal C} (\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) \bigr); \psi(0, y,
\eta) = \psi(T, y, \eta) =0 \ \forall (y, \eta) \in \overline{\Omega} \times Y
\bigr\}.\end{aligned}$$ It is a closed subspace of ${\mathcal C}^1 \bigl( [0,T]; {\mathcal C}
(\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) \bigr) $ for the usual norm of ${\mathcal C}^1 \bigl( [0,T]; {\mathcal C} (\overline{\Omega}; {\mathcal
C}_{\sharp} (Y) ) \bigr) $ and for any $\psi \in {\mathcal C}^1_0 \bigl( [0,T]; {\mathcal C}
(\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) \bigr)$, we have $$\begin{aligned}
&& \displaystyle \| \psi^{\e} \|_{H^1 ( 0,T ; L^2( \Omega) )} = \left(
\int_0^T \int_{\Omega} \left( \psi \left( t, y, \frac{y}{\e} \right) \right)^2
\, dy dt
+ \int_0^T \int_{\Omega} \left( \frac{\partial \psi}{\partial t} \left( t, y,
\frac{y}{\e} \right) \right)^2 \, dy dt \right)^{1/2} \\
&& \displaystyle \le \sqrt{ T | \Omega|} \| \psi\|_{{\mathcal C}^1 ( [0,T];
{\mathcal C} (\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) )}.\end{aligned}$$ Furthermore, we may now define $\Lambda_{\e} \in \Bigl( {\mathcal C}^1_0 \bigl(
[0,T]; {\mathcal C} (\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) \bigr)
\Bigr)'$ by $$\begin{aligned}
\Lambda_{\e} (\psi) = \int_0^T \int_{\Omega} w^{\e} (t,y) \psi \left( t, y,
\frac{y}{\e} \right) \, dy dt
\quad \forall \psi \in {\mathcal C}^1_0 \bigl( [0,T]; {\mathcal C}
(\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) \bigr).\end{aligned}$$ Since $(w^{\e})_{\e >0}$ is a bounded sequence of $H^{-1} \bigl( 0,T;
L^2(\Omega) \bigr)$, we infer that there exists a real number $C'>0$, independent of $\e$, such that $$\begin{aligned}
\bigl| \Lambda_{\e} (\psi) \bigr| \le \|w^{\e} \|_{H^{-1} ( 0,T ; L^2( \Omega))}
\| \psi^{\e} \|_{H^1 ( 0,T; L^2( \Omega))} \le C ' \| \psi^{\e} \|_{H^1 ( 0,T;
L^2( \Omega))} \le C ' \sqrt{ T | \Omega|} \| \psi\|_{{\mathcal C}^1 ( [0,T];
{\mathcal C} (\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) )}\end{aligned}$$ for all $ \psi \in {\mathcal C}^1_0 \bigl( [0,T]; {\mathcal C}
(\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) \bigr)$ and the sequence $(\Lambda_{\e})_{{\e} >0}$ is bounded in $\Bigl( {\mathcal C}^1_0 \bigl( [0,T];
{\mathcal C} (\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) \bigr) \Bigr)'$. Since $ {\mathcal C}^1 \bigl( [0,T]; {\mathcal C} (\overline{\Omega};
{\mathcal C}_{\sharp} (Y) ) \bigr) $ is a separable Banach space, we can conclude in the same way as previously.
We may observe that this proof shows that we can choose test-functions in $ {\mathcal C} \bigl( [0,T]; {\mathcal C} (\overline{\Omega}; {\mathcal
C}_{\sharp} (Y) ) \bigr)$ (resp. in $ {\mathcal C}^1_0 \bigl( [0,T]; {\mathcal
C} (\overline{\Omega}; {\mathcal C}_{\sharp} (Y) ) \bigr)$) instead of ${\mathcal D} (0,T) \times {\mathcal D} \bigl( \Omega ; {\mathcal
C}^{\infty}_{\sharp} (Y) \bigr)$.
Then the convergence results for the velocity, the micro-rotation and the pressure are given in the following three propositions.
\[prop4.1\] [**(Two-scale limit of the velocity)**]{} Under the assumptions of Proposition \[pro1\], there exist $v^0 \in \Bigl( L^2
\bigl( 0,T; L^2 (\Omega; H^1_{\sharp}(Y) ) \bigr) \Bigr)^2$ such that $\displaystyle \frac{\partial v^0}{\partial y_2} \in \Bigl( L^2 \bigl(0,T; L^2
(\Omega \times Y) \bigr) \Bigr)^2$ and $v^1 \in \Bigl( L^2 \bigl( 0,T; L^2
\bigl(\Omega \times (0,1) ; H^1_{\sharp}(0,1)_{/ {\mathbb R}} ) \bigr) \Bigr)^2$ such that, possibly extracting a subsequence still denoted $(v^{\e})_{\e>0}$, we have for $i=1,2$: $$\begin{aligned}
\label{eq:sec4-1}
v_i^{\varepsilon} {\twoheadrightarrow}v^0_i, \quad \frac{\partial v_i^{\varepsilon}}{\partial
y_2} {\twoheadrightarrow}\frac{\partial v^0_i}{\partial y_2} + \frac{\partial v^1_i}{\partial
\eta_2},\end{aligned}$$ and $$\begin{aligned}
\label{eq:sec4-2}
\varepsilon \frac{\partial v^{\e}_i}{\partial y_1} {\twoheadrightarrow}\frac{\partial
v^0_i}{\partial \eta_1}.\end{aligned}$$ Furthermore $v^0$ does not depend on $\eta_2$, $v^0$ is divergence free in the following sense $$\begin{aligned}
\label{eq:sec4-3}
h(y_1, \eta_1) \frac{\partial v^0_1}{\partial \eta_1} - y_2 \frac{\partial
h}{\partial \eta_1} (y_1, \eta_1) \frac{\partial v^0_1}{\partial y_2} +
\frac{\partial v^0_2}{\partial y_2} = 0 \quad \hbox{\rm in } (0,T) \times \Omega
\times (0,1),\end{aligned}$$ and $$\begin{aligned}
\label{eq:sec4-4}
&& \displaystyle v^0 = 0 \quad \hbox{\rm on } (0,T) \times \Gamma_0 \times (0,1)
, \Gamma_0 = (0,L) \times \{ 0 \}, \\
&& \displaystyle - v_1^0 \frac{\partial h}{\partial \eta_1} ( y_1, \eta_1) +
v_2^0 =0 \quad \hbox{\rm on } (0,T) \times \Gamma_1 \times (0,1), \Gamma_1 =
(0,L) \times \{1\}.\end{aligned}$$
The first part of the result is a direct consequence of the previous theorem and is obtained by using the same techniques as in Proposition 1.14 in [@allaire].
Indeed, from Proposition \[pro1\] we know that $(v^{\e}_i)_{{\e} >0}$, $
\displaystyle \left( \frac{\partial v^{\e}_i}{\partial y_2} \right)_{{\e} >0}$ and $ \displaystyle \left( \e \frac{\partial v^{\e}_i}{\partial y_1}
\right)_{{\e} >0}$ are bounded in $L^2 \bigl( (0,T) \times \Omega \bigr)$. It follows that, possibly extracting a subsequence, they two-scale converge to $v^0_i$, $\xi^0_i$ and $\xi^1_i$ respectively, i.e. $$\begin{aligned}
\label{sec4.3}
\lim_{\e \to 0} \int_0^T \int_{\Omega} v^{\e}_i (t,y) \varphi \left( y,
\frac{y}{\e} \right) \theta (t) \, dy dt = \int_0^T \int_{\Omega \times Y} v^0_i
( t, y, \eta) \varphi (y, \eta) \theta (t) \, d\eta dy dt\end{aligned}$$ $$\begin{aligned}
\label{sec4.4}
\begin{array}{l}
\displaystyle \lim_{\e \to 0} \int_0^T \int_{\Omega} \frac{\partial
v^{\e}_i}{\partial y_2} (t,y) \varphi \left( y, \frac{y}{\e} \right) \theta (t)
\, dy dt \\
\displaystyle = - \lim_{\e \to 0} \int_0^T \int_{\Omega} v^{\e}_i (t,y) \left(
\frac{\partial \varphi}{\partial y_2} \left( y, \frac{y}{\e} \right) +
\frac{1}{\e} \frac{\partial \varphi}{\partial \eta_2} \left( y, \frac{y}{\e}
\right) \right) \theta (t) \, dy dt \\
\displaystyle = \int_0^T \int_{\Omega \times Y} \xi^0_i ( t, y, \eta) \varphi
(y, \eta) \theta (t) \, d\eta dy dt
\end{array}\end{aligned}$$ and $$\begin{aligned}
\label{sec4.5}
\begin{array}{l}
\displaystyle \lim_{\e \to 0} \int_0^T \int_{\Omega} \e \frac{\partial
v^{\e}_i}{\partial y_1} (t,y) \varphi \left( y, \frac{y}{\e} \right) \theta (t)
\, dy dt \\
\displaystyle = - \lim_{\e \to 0} \int_0^T \int_{\Omega} v^{\e}_i (t,y) \left(
\e \frac{\partial \varphi}{\partial y_1} \left( y, \frac{y}{\e} \right) +
\frac{\partial \varphi}{\partial \eta_1} \left( y, \frac{y}{\e} \right)
\right) \theta (t) \, dy dt \\
\displaystyle = \int_0^T \int_{\Omega \times Y} \xi^1_i ( t, y, \eta) \varphi
(y, \eta) \theta (t) \, d\eta dy dt
\end{array}\end{aligned}$$ for all $\theta \in {\mathcal D}(0,T)$, $\varphi \in {\mathcal D} \bigl(\Omega;
{\mathcal C}^{\infty}_{\sharp} (Y) \bigr)$. From (\[sec4.5\]) and (\[sec4.3\]) we obtain $$\begin{aligned}
\label{sec4.5bis}
\begin{array}{l}
\displaystyle - \lim_{\e \to 0} \int_0^T \int_{\Omega} v^{\e}_i (t,y) \left( \e
\frac{\partial \varphi}{\partial y_1} \left( y, \frac{y}{\e} \right) +
\frac{\partial \varphi}{\partial \eta_1} \left( y, \frac{y}{\e} \right)
\right) \theta (t) \, dy dt \\
\displaystyle = - \int_0^T \int_{\Omega \times Y} v^0_i ( t, y, \eta)
\frac{\partial \varphi}{\partial \eta_1} (y, \eta) \theta (t) \, d\eta dy dt
\\
\displaystyle
= \int_0^T \int_{\Omega \times Y} \xi^1_i ( t, y, \eta) \varphi (y, \eta) \theta
(t) \, d\eta dy dt
\end{array}\end{aligned}$$ for all $\theta \in {\mathcal D}(0,T)$, $\varphi \in {\mathcal D} \bigl(\Omega;
{\mathcal C}^{\infty}_{\sharp} (Y) \bigr)$, which implies that $\displaystyle
\xi^1_i = \frac{\partial v^0_i}{\partial \eta_1} \in L^2 \bigl( 0,T; L^2 (\Omega
\times Y) \bigr)$. Thus (\[eq:sec4-2\]) holds.
Similarly, by multiplying (\[sec4.4\]) by $\e$ and taking into account (\[sec4.3\]) we get $$\begin{aligned}
\lim_{\e \to 0} \int_0^T \int_{\Omega} v^{\e}_i (t,y) \frac{\partial
\varphi}{\partial \eta_2} \left( y, \frac{y}{\e} \right) \theta (t) \, dy dt
= 0 = \int_0^T \int_{\Omega \times Y} v^0_i ( t, y, \eta) \frac{\partial
\varphi}{\partial \eta_2} (y, \eta) \theta (t) \, d\eta dy dt\end{aligned}$$ and thus $v_i^0$ does not depend on $\eta_2$. Moreover, by choosing $\varphi$ independent of $\eta$ (i.e $\varphi \in {\mathcal D} (\Omega)$) in (\[sec4.5bis\]), we get $$\begin{aligned}
\int_0^T \int_{\Omega \times (0,1)} \frac{\partial v^0_i}{\partial \eta_1} ( t,
y, \eta_1) \varphi (y) \theta (t) \, d\eta_1 dy dt =0 = \int_0^T \int_{\Omega}
\bigl( v^0_i (t, y, 1) - v^0_i (t,y, 0) \bigr) \varphi (y) \theta (t) \, dy dt\end{aligned}$$ and $v^0_i \in L^2 \bigl( 0,T; L^2 (\Omega ; H^1_{\sharp} (Y) ) \bigr)$.
Next, by choosing $\varphi \in {\mathcal D} \bigl(\Omega \times (0,1) \bigr)$ (i.e. $\varphi$ does not depend on $\eta_2$), we obtain now $$\begin{aligned}
&& \displaystyle \lim_{\e \to 0} \int_0^T \int_{\Omega} v^{\e}_i (t,y)
\frac{\partial \varphi}{\partial y_2} \left( y, \frac{y_1}{\e} \right) \theta
(t) \, dy dt = \int_0^T \int_{\Omega \times Y} v^0_i ( t, y, \eta_1)
\frac{\partial \varphi}{\partial y_2} (y, \eta_1) \theta (t) \, d\eta dy dt \\
&& \displaystyle = - \int_0^T \int_{\Omega \times Y} \xi^0_i ( t, y, \eta)
\varphi (y, \eta_1) \theta (t) \, d\eta dy dt.\end{aligned}$$ Hence $$\begin{aligned}
\int_0^T \int_{\Omega \times Y} \left( - \frac{\partial v^0_i}{\partial y_2}
(t,y, \eta_1) + \xi^0_i ( t, y, \eta) \right) \varphi (y_1, y_2, \eta_1) \theta
(t) \, d\eta dy dt =0.\end{aligned}$$ It follows that there exists $v^1_i \in L^2 \bigl( 0,T; L^2 \bigl(\Omega \times
(0,1); H^1_{\sharp} (0,1)_{| {{\mathbb R}}} \bigr) \bigr)$ such that $$\begin{aligned}
\frac{\partial v^{\e}_i}{\partial y_2} {\twoheadrightarrow}\xi^0_i = \frac{\partial
v^0_i}{\partial y_2} + \frac{\partial v^1_i}{\partial \eta_2},
\end{aligned}$$ and the second part of (\[eq:sec4-1\]) holds.
Now, let $\displaystyle \varphi^{\e} (z)= \varphi \left( z_1, \frac{z_2}{\e
h^{\e} ( z_1)}, \frac{z_1}{\e} \right)$ for all $(z_1, z_2 ) \in \Omega^{\e}$. Recalling that ${\rm div}_z v^{\e} =0$ in $\Omega^{\e}$, we get $$\begin{aligned}
&& \displaystyle 0 = \int_0^T \int_{\Omega^{\e}} \left( \frac{\partial
v_1^{\e}}{\partial z_1} (t,z) + \frac{\partial v_2^{\e}}{\partial z_2} (t,z)
\right)
\varphi^{\e} (z) \theta (t) \, dz dt \\
&& \displaystyle = - \int_0^T \int_{\Omega^{\e}} \left( v_1^{\e} (t,z)
\frac{\partial \varphi^{\e}}{\partial z_1}
(z) + v_2^{\e} (t,z) \frac{\partial \varphi^{\e}}{\partial z_2}
(z) \right) \theta (t) \, dz dt \\
&& = \displaystyle - \int_0^T \int_{\Omega} \left(
v_1^{\e} (t, y) \bigl(b_{\e} \cdot \nabla \varphi^{\e} \bigr) (y) + v_2^{\e} (t,
y) \frac{1}{\e h^{\e} (y_1)}
\frac{\partial \varphi^{\e}}{\partial y_2} (y) \right)\e h^{\e}(y_{1})
\theta(t) \, dy dt \\
&& = \displaystyle - \int_0^T \int_{\Omega}
v_1^{\e} (t,y) \left(
\e h \left( y_1, \frac{y_1}{\e} \right) \frac{\partial \varphi}{\partial y_1}
\left( y, \frac{y}{\e} \right)
+ h \left( y_1, \frac{y_1}{\e} \right) \frac{\partial \varphi}{\partial \eta_1}
\left( y, \frac{y}{\e} \right) \right.\\
&& \displaystyle \left. - y_2 \left(
\e \frac{\partial h}{\partial y_1} \left( y_1, \frac{y_1}{\e} \right) +
\frac{\partial h}{\partial \eta_1} \left( y_1, \frac{y_1}{\e} \right)
\right)
\frac{\partial \varphi}{\partial y_2} \left( y, \frac{y}{\e} \right)
\right) \theta (t) \, dy dt \\
&& \displaystyle - \int_0^T \int_{\Omega} v_2^{\e} (t,y)
\frac{\partial \varphi}{\partial y_2} \left( y, \frac{y}{\e} \right)
\theta (t) \, dy dt.\end{aligned}$$ We pass to the limit as $\e $ tends to zero: $$\begin{aligned}
&& \displaystyle 0 = - \int_0^T \int_{\Omega \times (0,1) } v_1^{0} (t, y,
\eta_1) \left(
h ( y_1, \eta_1) \frac{\partial \varphi}{\partial \eta_1} (y, \eta_1)
- y_2
\frac{\partial h}{\partial \eta_1} (y_1, \eta_1) \frac{\partial
\varphi}{\partial y_2} ( y, \eta_1) \right) \theta (t) \, d \eta_1 dy dt \\
&& \displaystyle - \int_0^T \int_{\Omega \times (0,1) } v_2^{0} (t, y,\eta_1)
\frac{\partial \varphi}{\partial y_2} (y, \eta_1) \theta (t) \, d \eta_1 dy dt
\\
&& \displaystyle = \int_0^T \int_{\Omega \times (0,1) } \left(
h(y_1, \eta_1) \frac{\partial v^0_1}{\partial \eta_1} (t, y, \eta_1)
- y_2 \frac{\partial h}{\partial \eta_1} (y_1, \eta_1) \frac{\partial
v^0_1}{\partial y_2} (t, y, \eta_1)
+ \frac{\partial v^0_2}{\partial y_2} (t, y, \eta_1) \right) \varphi (y, \eta_1)
\theta (t) \, d\eta_1 dy dt\end{aligned}$$ which gives (\[eq:sec4-3\]). But, taking into account the boundary conditions on $v^{\e}$, we may reproduce the same computation with $\varphi \in {\mathcal C}^{\infty} \bigl(\overline{
\Omega}; {\mathcal C}^{\infty}_{\sharp} (0,1) \bigr)$ such that $\varphi$ is $L$-periodic in $y_1$, so with (\[eq:sec4-3\]) it remains $$\begin{aligned}
&& \displaystyle \int_0^T \int_{\Gamma_1 \times (0,1)} \left( - v_1^0 (t, y,
\eta_1) \frac{\partial h}{\partial \eta_1} (y_1, \eta_1) + v_2^0 (t, y, \eta_1)
\right) \varphi (y, \eta_1) \theta (t) \, d \eta_1 dy_1 dt \\
&& \displaystyle - \int_0^T \int_{\Gamma_0 \times (0,1)} v_2^0 (t, y, \eta_1)
\varphi (y, \eta_1) \theta(t) \, d \eta_1 dy_1 dt = 0.\end{aligned}$$ We choose more precisely $\varphi (y_1, y_2, \eta_1) = \hat \varphi (y_2) \tilde
\varphi (y_1, \eta_1)$ with $\hat \varphi \in {\mathcal C}^{\infty} \bigl(
[0,1])$ and $\tilde \varphi \in {\mathcal C}^{\infty}_{\sharp} \bigl( [0,L];
{\mathcal C}^{\infty}_{\sharp} (0,1) \bigr)$. With $\hat \varphi (1)=0$ and $\hat \varphi (0) = 1$ we get first $$\begin{aligned}
v_2^0 = 0 \quad \hbox{\rm on } (0,T) \times \Gamma_0 \times (0,1).\end{aligned}$$ Next with $\hat \varphi(1)=1$ and $\hat \varphi (0)=0$ we get $$\begin{aligned}
- v_1^0 \frac{\partial h}{\partial \eta_1} + v_2^0 = 0 \quad \hbox{\rm on }
(0,T) \times \Gamma_1 \times (0,1).\end{aligned}$$
Finally, let $\varphi \in {\mathcal D} \bigl( 0,L; {\mathcal
C}^{\infty}_{\sharp} (0,1) \bigr)$ and $\displaystyle \varphi^{\e} (y_1, y_2) =
\varphi \left( y_1, \frac{y_1}{\e} \right) (1-y_2)$ for all $(y_1, y_2) \in
\Omega$. Taking into account the boundary conditions for $v_1^{\e}$ (see (\[eqn:er2.8a\])-(\[eqn:er2.11b\])) we have $$\begin{aligned}
\int_0^T \int_{\Omega} \frac{\partial v_1^{\e}}{\partial y_2} (t,y) \varphi^{\e}
(y) \theta (t) \, dy dt = \int_0^T \int_{\Omega} v_1^{\e} (t,y) \varphi \left(
y_1, \frac{y_1}{\e} \right) \theta (t) \, dy dt.\end{aligned}$$ By passing to the limit as $\e$ tends to zero we obtain $$\begin{aligned}
&& \displaystyle \int_0^T \int_{\Omega \times (0,1)} \frac{\partial
v_1^{0}}{\partial y_2} (t,y, \eta_1) \varphi (y_1, \eta_1) (1-y_2) \theta (t) \,
d \eta_1 dy dt \\
&& \displaystyle = \int_0^T \int_{\Omega \times (0,1)} v_1^{0} (t,y, \eta_1)
\varphi ( y_1, \eta_1) \theta (t) \, d \eta_1 dy dt.\end{aligned}$$ It follows that $$\begin{aligned}
\int_0^T \int_{\Gamma_0 \times (0,1)} v_1^{0} (t,y, \eta_1) \varphi ( y_1,
\eta_1) \theta (t) \, d \eta_1 dy dt =0\end{aligned}$$ which implies that $ v_1^0 = 0$ on $(0,T)\times \Gamma_0 \times (0,1)$.
Similarly we can define the two-scale limit of $Z^{\e}$.
\[prop4.2\] [**(Two-scale limit of the micro-rotation field)**]{} Under the assumptions of Proposition \[pro1\], there exist $Z^0 \in L^2 \bigl( 0,T; L^2 (\Omega;
H^1_{\sharp}(Y) ) \bigr)$ such that $\displaystyle \frac{\partial Z^0}{\partial
y_2} \in L^2 \bigl( 0,T; L^2 (\Omega \times Y ) \bigr)$ and $Z^1 \in L^2 \bigl(
0,T; L^2 \bigl(\Omega \times (0,1) ; H^1_{\sharp}(0,1)_{/ {\mathbb R}} \bigr) \bigr)$ such that, possibly extracting a subsequence still denoted $(Z^{\e})_{\e >0}$, we have $$\begin{aligned}
\label{eq:sec4-5}
Z^{\varepsilon} {\twoheadrightarrow}Z^0, \quad \frac{\partial Z^{\varepsilon}}{\partial y_2}
{\twoheadrightarrow}\frac{\partial Z^0}{\partial y_2} + \frac{\partial Z^1}{\partial \eta_2},\end{aligned}$$ and $$\begin{aligned}
\label{eq:sec4-6}
\varepsilon \frac{\partial Z^{\e}}{\partial y_1} {\twoheadrightarrow}\frac{\partial
Z^0}{\partial \eta_1}.\end{aligned}$$ Furthermore $Z^0$ does not depend on $\eta_2$, and $Z^0 \equiv 0$ on $(\Gamma_0
\cup \Gamma_1) \times (0,1) \times (0,T)$.
The first part of the proof is identical to the proof of the previous proposition. Let us establish now the boundary conditions for the limit $Z^0$. Let $\theta \in {\mathcal D}(0,T)$, $\varphi \in {\mathcal C}^{\infty}
\bigl(\overline{ \Omega}; {\mathcal C}^{\infty}_{\sharp} (0,1) \bigr)$ such that $\varphi$ is $L$-periodic in $y_1$ and we define $\displaystyle \varphi^{\e} (z)= \varphi \left( z_1, \frac{z_2}{\e h^{\e} (
z_1)}, \frac{z_1}{\e} \right)$ for all $(z_1, z_2) \in \Omega^{\e}$. With the boundary conditions (\[eqn:er2.8a\])-(\[eqn:er2.11b\]) for $Z^{\e}$ we get $$\begin{aligned}
&& \displaystyle \int_0^T \int_{\Omega^{\e}} \frac{\partial Z^{\e}}{\partial
z_2} (t,z) \varphi^{\e} (z) \theta (t) \, dz dt =
- \int_0^T \int_{\Omega^{\e}} Z^{\e} (t,z) \frac{\partial \varphi^{\e}
}{\partial z_2} (z) \theta (t) \, dz dt \\
&& \displaystyle = - \int_0^T \int_{\Omega} Z^{\e} (t,y) \frac{\partial
\varphi}{\partial y_2} \left( y, \frac{y_1}{\e} \right) \theta (t) \, dy dt =
\int_0^T \int_{\Omega} \frac{\partial Z^{\e}}{\partial y_2} (t,y) \varphi \left(
y, \frac{y_1}{\e} \right) \theta (t) \, dy dt\end{aligned}$$ and taking $\e \to 0^+$ we obtain $$\begin{aligned}
&& \displaystyle \int_0^T \int_{\Omega \times Y} \left( \frac{\partial
Z^0}{\partial y_2} (t, y, \eta_1) + \frac{\partial Z^1}{\partial \eta_2} (t, y,
\eta) \right) \varphi (y, \eta_1) \theta (t) \, d\eta_1 dy dt\\
&& \displaystyle = - \int_0^T \int_{\Omega \times (0,1)} Z^0 (t, y, \eta_1)
\frac{\partial \varphi}{\partial y_2} (y, \eta_1) \theta (t) \, d \eta_1 dy dt.\end{aligned}$$ But the periodicity properties of $Z^1$ with respect to $\eta_2$ imply that $$\begin{aligned}
\int_0^T \int_{\Omega \times Y} \frac{\partial Z^1}{\partial \eta_2} (t, y,
\eta) \varphi (y, \eta_1) \theta (t) \, d\eta_1 dy dt =0.\end{aligned}$$ Hence $$\begin{aligned}
\int_0^T \int_{\Omega \times (0,1)} \frac{\partial Z^0}{\partial y_2} (t, y,
\eta_1) \varphi (y, \eta_1) \theta (t) \, d\eta_1 dy dt = - \int_0^T
\int_{\Omega \times (0,1)} Z^0 (t, y, \eta_1) \frac{\partial \varphi}{\partial
y_2} (y, \eta_1) \theta (t) \, d \eta_1 dy dt.\end{aligned}$$ By Green’s formula we infer that $$\begin{aligned}
0 = - \int_0^T \int_{\Gamma_0 \times (0,1)} Z^0 (t, y, \eta_1) \varphi (y,
\eta_1) \theta (t) \, d \eta_1 dy dt + \int_0^T \int_{\Gamma_1 \times (0,1)} Z^0
(t, y, \eta_1) \varphi (y, \eta_1) \theta (t) \, d \eta_1 dy dt.\end{aligned}$$
Now we choose $\varphi (y_1, y_2, \eta_1) = \hat \varphi (y_2) \tilde \varphi
(y_1, \eta_1)$ with $\hat \varphi \in {\mathcal C}^{\infty} \bigl( [0,1] \bigr)$ and $\tilde \varphi \in {\mathcal C}^{\infty}_{\sharp}( [0,L]; {\mathcal
C}^{\infty}_{\sharp} (0,1) \bigr)$, with $\hat \varphi (1)=0$, $\hat \varphi (0)
= 1$ then $\hat \varphi (1)=1$, $\hat \varphi (0) = 0$, we get $$\begin{aligned}
0= \int_0^T \int_{\Gamma_0 \times (0,1)} Z^0 (t, y, \eta_1) \tilde \varphi (y_1,
\eta_1) \theta (t) \, d \eta_1 dy dt = \int_0^T \int_{\Gamma_1 \times (0,1)}
Z^0 (t, y, \eta_1) \tilde \varphi (y_1, \eta_1) \theta (t) \, d \eta_1 dy dt\end{aligned}$$ which allows us to conclude the proof of Proposition \[prop4.2\].
Finally we can define the two-scale limit of $p^{\e}$.
\[prop4.3\] [**(Two-scale limit of the pressure field)**]{} Under the assumptions of Proposition \[prop2\], there exists $p^0 \in H^{-1} \bigl( 0,T; L^2 (\Omega \times Y) \bigr)$ such that, possibly extracting a subsequence still denoted $(p^{\e})_{\e >0}$, we have $$\begin{aligned}
\e^2 p^{\e} {\twoheadrightarrow}p^0.\end{aligned}$$ Moreover $p^0$ depends only $t$ and $y_1$, $p^0 \in H^{-1} \bigl( 0,T;
H^1_{\sharp} (0,1) \bigr)$ and satisfies $ \displaystyle \int_0^L p^0 (t,y_1) \left( \int_0^1 h(y_1, \eta_1) \, d
\eta_1 \right) \, dy_1 =0$ almost everywhere in $(0,T)$.
The first part of the result is an immediate consequence of the estimate (\[E3.18\]) (see Proposition \[prop2\]). From proposition \[pro1\] and (\[eq3.43\]) we know that there exists a constant $C>0$, independent of $\e$, such that for all $\varphi^{\e} \in H^1_0
\bigl(0,T; H^1_0(\Omega) \bigr)$ we have $$\begin{aligned}
&& \displaystyle \left| \int_0^T \int_{\Omega} p^{\e} (t,y) \frac{\partial
\varphi^{\e}}{\partial y_2} (t,y) \, dy dt \right|
\le C \left( \| \varphi^{\e} \|_{L^{2} (0,T; L^2(\Omega))} + \e \left\|
\frac{\partial \varphi^{\e}}{\partial t} \right\|_{L^2(0,T; L^2(\Omega))}
\right) \\
&& \displaystyle + \frac{C}{\e} \left( \| \varphi^{\e} \|_{L^{\infty} (0,T;
L^4(\Omega))} + \| \e b_{\e} \cdot \nabla \varphi^{\e} \|_{L^{2} (0,T;
L^2(\Omega))} + \left\| \frac{\partial \varphi^{\e}}{\partial y_2}
\right\|_{L^2(0,T; L^2(\Omega))} \right) .\end{aligned}$$ Now let $\varphi \in {\mathcal D}\bigl( \Omega; {\mathcal C}^{\infty}_{\sharp}
(Y) \bigr)$ and $\theta \in {\mathcal D}(0,T)$. We define $\varphi^{\e}(t,y) =
\theta (t) \varphi \left( y, \frac{y}{\e} \right)$ for all $(t,y) \in (0,T)
\times \Omega$ and we get $$\begin{aligned}
\label{eq:4.3.1}
\left| \int_0^T \int_{\Omega} \e^2 p^{\e} (t, y) \left( \frac{\partial
\varphi}{\partial y_2} \left(t, y, \frac{y}{\e} \right) + \frac{1}{\e}
\frac{\partial \varphi}{\partial \eta_2} \left(t, y, \frac{y}{\e} \right)
\right) \theta (t) \, dy dt \right|
\nonumber\\
\le {\mathcal O}(\e) + C \|\theta\|_{C^0
([0,T])} \left\| \frac{\partial \varphi}{\partial \eta_2} \right\|_{C^0
(\overline{\Omega}; C_{\sharp}(Y))}.\end{aligned}$$ We multiply the two members of this inequality by $\e$ and we pass to the limit as $\e$ tends to zero. We obtain $$\begin{aligned}
\int_0^T \int_{\Omega \times Y} p^0 (t, y, \eta) \frac{\partial
\varphi}{\partial \eta_2} (y, \eta) \theta (t) \, d \eta dy dt =0.\end{aligned}$$ Hence $p^0$ does not depend on $\eta_2$. Now we consider $\varphi \in {\mathcal
D}\bigl( \Omega; {\mathcal C}^{\infty}_{\sharp} (0,1) \bigr)$ (i.e $\varphi$ is independent of $\eta_2$) and we pass to the limit in (\[eq:4.3.1\]) as $\e$ tends to zero. We get $$\begin{aligned}
\int_0^T \int_{\Omega \times (0,1)} p^0 (t, y, \eta) \frac{\partial
\varphi}{\partial y_2} (y, \eta_1) \theta (t) \, d \eta dy dt =0\end{aligned}$$ which implies that $p^0$ does not depend on $y_2$.
Now we take $\Theta = (\varphi^{\e}, 0, 0)$ in (\[eqvar\]) and we multiply by $\e$: we get $$\begin{aligned}
\label{e3.43}
\e^{2} \int_{0}^{T}\int_{\Om} p^{\e}
\left( {\partial\varphi^{\e} \over \partial y_{1}}
- {y_{2}\over h^{\e}} {\partial h^{\e}\over \partial y_{1}}
{\partial\varphi^{\e} \over \partial y_{2}}
\right) h^{\e} dy dt
=
-\e^{2}\int_{0}^{T} \int_{\Om} v^{\e}_{1} {\partial\varphi^{\e}\over \partial t}
h^{\e}
dy dt
\nonumber\\
+ (\nu+\nu_{r})
\int_{0}^{T} \int_{\Om}\left(
h^{\e}(\e b_{\e}\cdot \nabla v^{\e}_{1}) (\e b_{\e}\cdot\nabla\varphi^{\e})
+{1\over h^{\e}}{\partial v^{\e}_{1}\over \partial y_{2}}
{\partial\varphi^{\e}\over \partial y_{2}}\right) dy dt
\nonumber\\
+\int_{0}^{T}\int_{\Om} \left(
\e v^{\e}_{1}(\e b_{\e}\cdot \nabla v^{\e}_{1})\varphi^{\e} h^{\e}
+ \e v^{\e}_{2}{\partial v^{\e}_{1}\over \partial
y_{2}}\varphi^{\e} \right) dy
dt
+ (\nu+\nu_{r})\int_{0}^{T} \int_{\Om} \frac{1}{h^{\e}}
{\partial U_{\e}\over \partial y_{2}}{\partial\varphi^{\e} \over \partial y_{2}}
dy
dt
\nonumber\\
-2\nu_{r}
\int_{0}^{T} \int_{\Om} \e
{\partial Z^{\e}\over \partial y_{2}}\varphi^{\e} dy dt
+ \int_{0}^{T} \int_{\Om}U_{\e}
(\e b_{\e}\cdot \nabla v^{\e}_{1})\varphi^{\e} \e h^{\e} dy dt
\nonumber\\
- \int_{0}^{T} \int_{\Om}
\e v^{\e}_{1}(\e b_{\e}\cdot \nabla\varphi^{\e})U_{\e} h^{\e} dy dt
- \int_{0}^{T} \int_{\Om}
\e v^{\e}_{2}{\partial \varphi^{\e} \over \partial y_{2}}U_{\e} dy dt
-2\e \nu_{r}\int_{0}^{T} \int_{\Om}
{\partial W_{\e}\over \partial y_{2}}\varphi^{\e} dy dt
\nonumber\\
- \int_{0}^{T} \int_{\Om}
\left( f^{\e}_{1}\varphi^{\e} -{\partial U^{\e} \over \partial
t}\varphi^{\e} \right) \e^{2} h^{\e} dy dt,\end{aligned}$$ where we recall that $U_{\e} (t, y) = U_0(t) {\mathcal U}(y_2 h^{\e} (y_1))$ and $W_{\e} (t, y) = W_0(t) {\mathcal W}(y_2 h^{\e} (y_1))$ for all $(t, y_1, y_2)
\in [0,T] \times \Om$. With the results of Proposition \[pro1\], we infer that there exists a constant $C>0$, independent of $\e$, such that $$\begin{aligned}
&& \displaystyle \left| \int_0^T \int_{\Omega} \e^2 p^{\e} (t,y) ( b_{\e} \cdot
\nabla \varphi^{\e}) (t,y) h^{\e} (y) \, dy dt \right| \\
&& \displaystyle
\le C \left( \| \varphi^{\e} \|_{L^{2}(0,T; L^2(\Omega))} + \| \e b_{\e} \cdot
\nabla \varphi^{\e} \|_{L^{2}(0,T; L^2(\Omega))} + \left\| \frac{\partial
\varphi^{\e}}{\partial y_2} \right\|_{L^{2}(0,T; L^2(\Omega))}
+ \| \varphi^{\e} \|_{L^{\infty}(0,T; L^4(\Omega))}\right) \\
&& \displaystyle + C \e^2 \left\| \frac{\partial \varphi^{\e}}{\partial t}
\right\|_{L^{2}(0,T; L^2(\Omega))} .\end{aligned}$$ We multiply the two members of this inequality by $\e$ and we obtain $$\begin{aligned}
&& \displaystyle \left| \int_0^T \int_{\Omega} \e^2 p^{\e} (t,y) \left( \e
\frac{\partial \varphi}{\partial y_1}
\left(y_1, y_2, \frac{y_1}{\e} \right) + \frac{\partial \varphi}{\partial
\eta_1}
\left(y_1, y_2, \frac{y_1}{\e} \right)
\right) h \left(y_1, \frac{y_1}{\e} \right) \theta (t) \, dy dt \right. \\
&& \displaystyle \left.
- \int_0^T \int_{\Omega} \e^2 p^{\e} (t,y) y_2 \left( \e \frac{\partial
h}{\partial y_1}
\left(y_1, \frac{y_1}{\e} \right) + \frac{\partial h}{\partial \eta_1}
\left(y_1, \frac{y_1}{\e} \right)
\right) \frac{\partial \varphi}{\partial y_2} \left(y_1, y_2, \frac{y_1}{\e}
\right) \theta (t) \, dy dt
\right| \le {\mathcal O}(\e)\end{aligned}$$ By taking the limit as $\e$ tends to zero, we have $$\begin{aligned}
\int_0^T \int_{\Om \times (0,1)} p^0 (t, y_1, \eta_1)\left( \frac{\partial
\varphi}{\partial \eta_1} (y_1, y_2, \eta_1) h( y_1, \eta_1) - y_2
\frac{\partial h}{\partial \eta_1}
(y_1, \eta_1) \frac{\partial \varphi}{\partial y_2} (y_1, y_2, \eta_1) \right)
\theta(t) \, d \eta_1 d y dt =0.\end{aligned}$$ Reminding that $p^0$ is independent of $y_2$ and $\varphi \in {\mathcal D}\bigl(
\Omega; {\mathcal C}^{\infty}_{\sharp} (0,1) \bigr)$, we get $$\begin{aligned}
&& \displaystyle \int_0^T \int_{\Om \times (0,1)} p^0 (t, y_1, \eta_1)\left(
\frac{\partial \varphi}{\partial \eta_1} (y_1, \eta_1) h( y_1, y_2, \eta_1) -
y_2 \frac{\partial h}{\partial \eta_1}
(y_1, \eta_1) \frac{\partial \varphi}{\partial y_2} (y_1, y_2, \eta_1) \right)
\theta(t) \, d \eta_1 d y dt \\
&& \displaystyle = \int_0^T \int_{ \Om \times (0,1)} p^0 (t, y_1, \eta_1)\left(
\frac{\partial \varphi}{\partial \eta_1} (y_1, y_2, \eta_1) h( y_1, \eta_1)
+ \frac{\partial h}{\partial \eta_1}
(y_1, \eta_1) \varphi (y_1, y_2, \eta_1) \right)
\theta(t) \, d \eta_1 d y dt \\
&& \displaystyle = \int_0^T \int_{\Om \times (0,1)} p^0 (t, y_1, \eta_1) \frac{
\partial ( h \varphi) }{\partial \eta_1} (y_1, y_2, \eta_1) \, d \eta_1 dy dt
=0.\end{aligned}$$ Then for any $\phi \in {\mathcal D}\bigl( \Omega; {\mathcal C}^{\infty}_{\sharp}
(0,1) \bigr)$, we may define $\displaystyle \varphi = \frac{\phi}{h} \in
{\mathcal D}\bigl( \Omega; {\mathcal C}^{\infty}_{\sharp} (0,1) \bigr)$ and we obtain $$\begin{aligned}
\int_0^T \int_{\Om \times (0,1)} p^0 (t, y_1, \eta_1) \frac{ \partial
\phi}{\partial \eta_1} (y_1, y_2, \eta_1) \, d \eta_1 dy dt
=0.\end{aligned}$$ Thus we can conclude that $p^0 $ is independent of $\eta_1$.
Now let $\varphi \in {\mathcal C}^{\infty}_{\sharp} (0,L) $ and $\theta \in
{\mathcal D}(0,T)$. We define $\varphi^{\e}$ by $$\begin{aligned}
\label{eq:rajout1}
\varphi^{\e} (y) = \frac{\varphi(y_1)}{h\left(y_1, \frac{y_1}{\e} \right)}
\left( y_2 e_1 + \e y_2^2 \left( \frac{\partial h}{\partial y_1} \left(y_1,
\frac{y_1}{\e} \right) + \frac{1}{\e} \frac{\partial h}{\partial \eta_1}
\left(y_1, \frac{y_1}{\e} \right) \right) e_2 \right)\end{aligned}$$ for all $(y_1, y_2) \in \Om$. We can check that $\varphi^{\e} \in \tilde V$ and with Lemma \[lemma3.1\], Proposition \[pro1\] and (\[eq3.43\])-(\[e3.43\]), we obtain $$\begin{aligned}
\displaystyle \left| \int_0^T \int_{\Om} \e^2 p^{\e} \left( (b_{\e} \cdot \nabla
\varphi^{\e}_1) + \frac{1}{\e h^{\e}} \frac{\partial \varphi^{\e}_2}{\partial
y_2} \right) h^{\e} \theta (t) \, dy dt \right|
\displaystyle \le {\mathcal O}( \e) + C \| \varphi \theta \|_{L^2 ((0,T)
\times (0,L))}\end{aligned}$$ with a constant $C>0$ independent of $\e$. Hence $$\begin{aligned}
\left| \int_0^T \int_{\Om} \e^2 p^{\e} (t,y) y_2 \frac{\partial
\varphi}{\partial y_1} (y_1) \theta (t) \, dy dt \right|
\le {\mathcal O}( \e) + C \| \varphi \theta \|_{L^2 ((0,T) \times (0,L))} .\end{aligned}$$ We pass to the limit as $\e$ tends to zero: $$\begin{aligned}
\left| \int_0^T \int_{\Om} p^{0} (t,y_1) y_2 \frac{\partial
\varphi}{\partial y_1} (y_1) \theta (t) \, dy dt \right| = \frac{1}{2}
\left| \int_0^T \int_0^L p^{0} (t,y_1) \frac{\partial \varphi}{\partial y_1}
(y_1) \theta (t) \, dy dt \right|
\le C \| \varphi \theta \|_{L^2 ((0,T) \times (0,L))}\end{aligned}$$ and we infer that $p^0 \in H^{-1} \bigl( 0,T; H^1_{\sharp} (0,L) \bigr)$.
Finally, recalling that $\displaystyle \int_{\Omega^{\e}} p^{\e} (t,z) \, dz =0$ almost everywhere in $(0,T)$, we have $$\begin{aligned}
\int_0^T \int_{\Omega} {\e}^2 p^{\e} (t,y) h^{\e} (y) \theta (t) \, dy dt =0
\quad \forall \theta \in {\mathcal D} (0,T)\end{aligned}$$ and by passing to the limit as $\e$ tends to zero, we get $$\begin{aligned}
\int_0^T \int_{\Omega \times (0,1)} p^{0} (t,y_1, \eta_1) h (y_1, \eta_1)
\theta (t) \, d \eta_1 dy dt =0 \quad \forall \theta \in {\mathcal D} (0,T)\end{aligned}$$ which allows us to conclude the proof of Proposition \[prop4.3\].
The limit problem {#limitprobl}
=================
Now let us pass to the limit in equation (\[eqn:er2.14\]). It is convenient to introduce the following functional spaces: $$\begin{aligned}
&& \displaystyle \tilde V = \left\{ \varphi \in \bigl( {\mathcal C}^{\infty} (
\overline{ \Omega} ; {\mathcal C}^{\infty}_{\sharp} (0,1) \bigr) \bigr)^2; \,
\varphi \, \hbox{\rm is L-periodic in } y_{1}, \ \varphi = 0 \ \hbox{\rm on }
\Gamma_0 \times (0,1), \right. \\
&& \displaystyle \left. \qquad \ \ \ - \varphi_1 \frac{\partial h}{\partial
\eta_1}+ \varphi_2 =0 \ \hbox{\rm on } \Gamma_1 \times (0,1) \right\}\end{aligned}$$ $$\begin{aligned}
\tilde V_{div} = \left\{ \varphi \in \tilde V; \ h \frac{\partial
\varphi_1}{\partial \eta_1} - y_2 \frac{\partial h}{\partial \eta_1}
\frac{\partial \varphi_1}{\partial y_2} + \frac{\partial \varphi_2}{\partial
y_2} =0 \ \hbox{\rm in } \Omega \times (0,1) \right\}\end{aligned}$$ $$\begin{aligned}
\tilde H^1 = \left\{ \psi \in {\mathcal C}^{\infty} (\overline{ \Omega} ;
{\mathcal C}^{\infty}_{\sharp} (0,1) \bigr) ; \, \psi \mbox{ is
L-periodic in } y_{1}, \, \psi=0 \ \mbox{ on } (\Gamma_{0} \cup \Gamma_{1}
\times (0,1) \right\} ,\end{aligned}$$ $$\begin{aligned}
\begin{array}{l}
\displaystyle V_{div} = \mbox{ closure of } \tilde V_{div} \mbox{ in }
L^2_{\sharp}
\bigl(0,L; {\mathcal F} \bigr)^{2},\\
\displaystyle H^1_{0 , \sharp} = \hbox{\rm closure of } \tilde H^1 \mbox{ in }
L^2_{\sharp} \bigl( 0,L; {\mathcal F} \bigr)
\end{array}\end{aligned}$$ with $$\begin{aligned}
{\mathcal F} = \left\{ v \in L^2 \bigl( (0,1); H^1_{\sharp} (0,1) \bigr) ,
\frac{\partial v}{\partial y_2} \in L^2 \bigl((0,1) \times (0,1) \bigr)
\right\}.\end{aligned}$$
\[limit\_pb\] Assume that there exist $f \in {\mathcal C}\bigl( [0,T] ; {\mathcal C} \bigl(
\overline{\Om}; {\mathcal C}_{\sharp}(0,1) \bigr) \bigr)^2$ and $g \in {\mathcal
C}\bigl( [0,T] ; {\mathcal C} \bigl( \overline{\Om}; {\mathcal C}_{\sharp}(0,1)
\bigr) \bigr)$ such that $f$ and $g$ are $L$-periodic in $y_1$ and $$\begin{aligned}
\e^2 f^{\e} (t,y) = f \left( t, y, \frac{y_1}{\e} \right), \quad \e^2 g^{\e}
(t,y) = g \left( t, y, \frac{y_1}{\e} \right) \quad \forall (t,y) \in [0,T]
\times \overline{\Om}.\end{aligned}$$ Then the functions $v^0$, $Z^0$ and $p^0$ satisfy the following limit problem: $$\begin{aligned}
&& \displaystyle ( \nu + \nu_r) \int_0^T \int_{\Omega \times (0,1) }
\sum_{i=1}^2 \left(
h (\bar b \cdot \nabla v_i^0) (\bar b \cdot \nabla \varphi_i) + \frac{1}{h}
\frac{\partial v_i^0}{\partial y_2} \frac{\partial \varphi_i}{\partial y_2}
\right) \theta \, d\eta_1 d y dt \\
&& \displaystyle + \alpha \int_0^T \int_{\Omega \times (0,1)} \left(
h (\bar b \cdot \nabla Z^0) (\bar b \cdot \nabla \psi) + \frac{1}{h}
\frac{\partial Z^0}{\partial y_2}
\frac{\partial \psi}{\partial y_2} \right) \theta \, d\eta_1 d y dt \\
&& \displaystyle - \int_0^T \int_{\Omega \times (0,1)} \frac{\partial
p^0}{\partial y_1} h \varphi_1 \theta \, d \eta_1 dy dt \\
&& \displaystyle = - ( \nu + \nu_r) \int_0^T \int_{\Omega \times (0,1) }
\left(
h (\bar b \cdot \nabla \bar U) (\bar b \cdot \nabla \varphi_1) + \frac{1}{h}
\frac{\partial \bar U}{\partial y_2} \frac{\partial \varphi_1}{\partial y_2}
\right) \theta \, d\eta_1 d y dt \\
&& \displaystyle - \alpha \int_0^T \int_{\Omega \times (0,1)} \left(
h (\bar b \cdot \nabla \bar W) (\bar b \cdot \nabla \psi) + \frac{1}{h}
\frac{\partial \bar W}{\partial y_2}
\frac{\partial \psi}{\partial y_2} \right) \theta \, d\eta_1 d y dt \\
&& \displaystyle + \int_0^T \int_{\Omega \times (0,1)} f \varphi h \theta \, d
\eta_1 d y dt
+ \int_0^T \int_{\Omega \times (0,1)} g \psi h \theta \, d \eta_1 d y dt\end{aligned}$$ for all $\Theta = (\varphi, \psi) \in V_{div} \times H^1_{0 {\sharp}}$ and $\theta \in {\mathcal D} (0,T)$, where $\bar b \cdot \nabla$ is the differential operator defined by $$\begin{aligned}
\bar b \cdot \nabla = \left( 1, - \frac{y_2}{h (y_1, \eta_1)} \frac{\partial
h}{\partial \eta_1} (y_1, \eta_1) \right)
\left(\begin{array}[c]{c}
\displaystyle {\partial \over\partial \eta_{1}} \\ \\
\displaystyle {\partial \over\partial y_{2}}
\end{array}
\right)\end{aligned}$$ and $$\begin{aligned}
&& \displaystyle \bar U ( t, y_1, y_2, \eta_1) = U_0 (t) {\mathcal U} \left(
h(y_1, \eta_1) y_2 \right), \\
&& \displaystyle \bar W (t, y_1, y_2, \eta_1) = W_0 (t) {\mathcal W} \left(
h(y_1, \eta_1) y_2 \right)
\end{aligned}$$ for all $(t, y_1, y_2, \eta_1, t) \in [0,T] \times \overline{\Omega} \times
[0,1] $.
With the above assumptions for $f^{\e}$ and $g^{\e}$ we can check immediately that $$\begin{aligned}
\| \e^2 f^{\e} \|_{L^2 ((0,T) \times \Om)} \le \sqrt{ T | \Om | } \|
f\|_{{\mathcal C}\bigl( [0,T] ; {\mathcal C} \bigl( \overline{\Om}; {\mathcal
C}_{\sharp}(0,1) \bigr) \bigr)}, \quad
\| \e^2 g^{\e} \|_{L^2 ((0,T) \times \Om )} \le \sqrt{ T | \Om | } \| g
\|_{{\mathcal C}\bigl( [0,T] ; {\mathcal C} \bigl( \overline{\Om}; {\mathcal
C}_{\sharp}(0,1) \bigr) \bigr)}\end{aligned}$$ and $$\begin{aligned}
\e^2 f^{\e} {\twoheadrightarrow}f, \quad \e^2 g^{\e} {\twoheadrightarrow}g.\end{aligned}$$ Let us recall that $$\begin{aligned}
b_{\e} \cdot \nabla = \left( 1, - \frac{y_2}{h^{\e} (y_1)} \frac{\partial
h^{\e}}{\partial y_1} (y_1) \right)
\left(\begin{array}[c]{c}
\displaystyle {\partial \over\partial y_{1}} \\ \\
\displaystyle {\partial \over\partial y_{2}}
\end{array}
\right)\end{aligned}$$ Taking into account the convergence results of Proposition \[prop4.1\] and Proposition \[prop4.2\], we get $$\begin{aligned}
&& \displaystyle \e b_{\e} \cdot \nabla v_i^{\e} = \e \frac{\partial
v_i^{\e}}{\partial y_1} (y)
- \frac{y_2}{h \left( y_1, \frac{y_1}{\e} \right)} \left( \e \frac{\partial
h}{\partial y_1} \left( y_1, \frac{y_1}{\e} \right) + \frac{\partial
h}{\partial \eta_1} \left( y_1, \frac{y_1}{\e} \right) \right)
\frac{\partial v_i^{\e} }{\partial y_2} (y) \\
&& \displaystyle {\twoheadrightarrow}\frac{\partial v^0_i}{\partial \eta_1} (y, \eta_1)
- \frac{y_2}{h (y_1, \eta_1) } \frac{\partial h}{\partial \eta_1} (y_1,
\eta_1)
\left( \frac{\partial v^0_i }{\partial y_2} ( y, \eta_1 ) + \frac{\partial
v^1_i}{\partial \eta_2} (y, \eta) \right) = \bar b \cdot \nabla v^0_i -
\frac{y_2}{h } \frac{\partial h}{\partial \eta_1}
\frac{\partial v^1_i}{\partial \eta_2}
\end{aligned}$$ for $i=1,2$ and $$\begin{aligned}
\displaystyle b_{\e} \cdot \nabla Z^{\e} {\twoheadrightarrow}\bar b \cdot \nabla Z^0 -
\frac{y_2}{h } \frac{\partial h}{\partial \eta_1}
\frac{\partial Z^1}{\partial \eta_2} .
\end{aligned}$$
Similarly, let $\phi \in {\mathcal C}^{\infty} \bigl( \overline{ \Omega};
{\mathcal C}^{\infty}_{\sharp} (0,1) \bigr)$ and $\displaystyle \phi^{\e}
(y_1,y_2) = \phi \left( y_1, y_2, \frac{y_1}{\e} \right)$ for all $(y_1, y_2)
\in \Omega$. We have $$\begin{aligned}
&& \displaystyle b_{\e} \cdot \nabla \phi^{\e} = \frac{\partial
\phi^{\e}}{\partial y_1} (y)
- \frac{y_2}{h \left( y_1, \frac{y_1}{\e} \right)} \left( \frac{\partial
h}{\partial y_1} \left( y_1, \frac{y_1}{\e} \right) + \frac{1}{\e}
\frac{\partial h}{\partial \eta_1} \left( y_1, \frac{y_1}{\e} \right) \right)
\frac{\partial \phi^{\e} }{\partial y_2} (y) \\
&& \displaystyle = \frac{\partial \phi}{\partial y_1} \left( y, \frac{y_1}{\e}
\right) + \frac{1}{\e} \frac{\partial \phi}{\partial \eta_1} \left( y,
\frac{y_1}{\e} \right) - \frac{y_2}{h \left( y_1, \frac{y_1}{\e} \right)}
\left( \frac{\partial h}{\partial y_1} \left( y_1, \frac{y_1}{\e} \right) +
\frac{1}{\e} \frac{\partial h}{\partial \eta_1} \left( y_1, \frac{y_1}{\e}
\right) \right)
\frac{\partial \phi }{\partial y_2} \left( y, \frac{y_1}{\e} \right) .
\end{aligned}$$ Now let $\theta \in {\mathcal D}( 0,T)$, $\Theta = (\varphi, \psi) \in \tilde
V_{div} \times \tilde H^1$ and let $\Theta^{\e} = (\varphi^{\e}, \psi^{\e})$ with $$\begin{aligned}
\varphi^{\e} (z) = \varphi \left( z_1, \frac{ z_2}{\e h^{\e} (z_1)} ,
\frac{z_1}{\e} \right) + \frac{ z_2}{ h^{\e} (z_1)} \frac{\partial h}{\partial
y_1} \left( z_1, \frac{z_1}{\e} \right) \varphi_1 \left( z_1, \frac{z_2}{\e
h^{\e}(z_1)}, \frac{z_1}{\e} \right) e_2\end{aligned}$$ and $\displaystyle \psi^{\e} (z) = \psi \left( z_1, \frac{z_2}{\e h^{\e} (z_1)}
, \frac{z_1}{\e} \right)$ for all $(z_1, z_2) \in \Omega^{\e}$. We have $\Theta^{\e} \in \tilde V^{\e} \times \tilde H^{1, \e}$ and from (\[fla\]) $$\begin{aligned}
&& \displaystyle \e \int_0^T a \bigl( \bar v^{\e} (t) , \Theta^{\e} \bigr)
\theta (t) \, dt \to
( \nu + \nu_r) \int_0^T \int_{\Omega \times Y} \sum_{i=1}^2 \left(
h (\bar b \cdot \nabla v_i^0) (\bar b \cdot \nabla \varphi_i) + \frac{1}{h}
\frac{\partial v_i^0}{\partial y_2} \frac{\partial \varphi_i}{\partial y_2}
\right) \theta \, d\eta d y dt \\
&& \displaystyle + \alpha \int_0^T \int_{\Omega \times Y} \left(
h (\bar b \cdot \nabla Z^0) (\bar b \cdot \nabla \psi) + \frac{1}{h}
\frac{\partial Z^0}{\partial y_2}
\frac{\partial \psi}{\partial y_2} \right) \theta \, d\eta d y dt \\
&& \displaystyle + (\nu + \nu_r) \int_0^T \int_{\Omega \times Y} \sum_{i=1}^2
\left(
\left( - y_2 \frac{\partial h}{\partial \eta_1} \frac{\partial v_i^1}{\partial
\eta_2} \right)
(\bar b \cdot \nabla \varphi_i) + \frac{1}{h} \frac{ \partial v_i^1}{\partial
\eta_2} \frac{\partial \varphi_i}{\partial y_2} \right) \theta \, d \eta dy dt
\\
&& \displaystyle + \alpha \int_0^T \int_{\Omega \times Y} \left( \left( - y_2
\frac{\partial h}{\partial \eta_1} \frac{\partial Z^1}{\partial \eta_2} \right)
(\bar b \cdot \nabla \psi) + \frac{1}{h} \frac{ \partial Z^1}{\partial \eta_2}
\frac{\partial \psi}{\partial y_2} \right) \theta \, d \eta dy dt .\end{aligned}$$ But these last two integral terms vanish since $\varphi$, $\psi$ and $h$ do not depend on $\eta_2$ and $v^1$ and $Z^1$ are $\eta_2$-periodic. Hence we obtain $$\begin{aligned}
\e \int_0^T a \bigl( \bar v^{\e} (t) , \Theta^{\e} \bigr) \theta (t) \, dt
\to \int_0^T \bar a \bigl( \bar v^0 (t), \Theta \bigr) \theta (t) \, d t\end{aligned}$$ with $\bar v^0 = (v^0, Z^0)$ and $$\begin{aligned}
&& \displaystyle \bar a ( \bar v, \Theta) = ( \nu + \nu_r) \int_{\Omega \times
(0,1) } \sum_{i=1}^2 \left(
h (\bar b \cdot \nabla v_i) (\bar b \cdot \nabla \varphi_i) + \frac{1}{h}
\frac{\partial v_i}{\partial y_2} \frac{\partial \varphi_i}{\partial y_2}
\right) \, d\eta_1 d y \\
&& \displaystyle + \alpha \int_{\Omega \times (0,1)} \left(
h (\bar b \cdot \nabla Z) (\bar b \cdot \nabla \psi) + \frac{1}{h}
\frac{\partial Z}{\partial y_2}
\frac{\partial \psi}{\partial y_2} \right) \, d\eta_1 d y
\end{aligned}$$ for all $\bar v = (v,Z) \in V_{div} \times H^1_{0 {\sharp}}$, for all $\Theta =
(\varphi, \psi) \in V_{div} \times H^1_{0 {\sharp}}$.
From (\[trifb\]) and the estimates (\[E3.13\])-(\[E3.14\])-(\[E3.16\]) obtained at Proposition \[pro1\] we get $$\begin{aligned}
\e \int_0^T B \bigl(\bar v^{\e} (t), \bar v^{\e} (t) , \Theta^{\e} \bigr) \theta
(t) \, dt = {\mathcal O}(\e) \to 0
\end{aligned}$$ and similarly, from (\[Rot\]) and (\[E3.13\])-(\[E3.14\])-(\[E3.16\]) $$\begin{aligned}
\e \displaystyle \int_0^T {\mathcal R} \bigl(\bar v^{\e} (t), \bar v^{\e} (t) ,
\Theta^{\e} \bigr) \theta (t) \, dt
= {\mathcal O}(\e) \to 0.
\end{aligned}$$
Let us consider now the right hand side of equation (\[eqn:er2.14a\]). We recall that $\bar \xi^{\e} = (U^{\e} e_1, W^{\e} )$ with $$\begin{aligned}
&& \displaystyle U^{\e} (t,z) = U_0 (t) {\mathcal U} \left( h^{\e}(y_1) y_2
\right) = \bar U \left( t, y_1, y_2, \frac{y_1}{\e} \right) \\
&& \displaystyle W^{\e}(t,z) = W_0 (t) {\mathcal W} \left( h^{\e}(y_1) y_2
\right) = \bar W \left( t, y_1, y_2, \frac{y_1}{\e} \right)\end{aligned}$$ and $U_0$, $W_0$ belong to $H^1 (0,T)$, ${\mathcal U}$, ${\mathcal W}$ belong to ${\mathcal D} \bigl( (-\infty, h_{max}) \bigr)$. Hence $\bar U$ and $\bar W$ belong to ${\mathcal C} \bigl( [0,T]; {\mathcal
C}^{1} (\overline{ \Omega}; {\mathcal C}^{1}_{\sharp} (0,1) \bigr) \bigr)$ and with (\[For3.10\])-(\[Rot1\])-(\[Rott\])-(\[Rotxi\]) $$\begin{aligned}
&& \displaystyle \e \int_0^T a \bigl( \bar \xi^{\e} (t), \Theta^{\e} \bigr) \theta
(t) \, dt \to \int_0^T \bar a \bigl( \bar \xi (t), \Theta) \theta (t) \, dt \\
&& \displaystyle \e \int_0^T B \bigl( \bar \xi^{\e} (t), \bar v^{\e} (t) ,
\Theta^{\e} \bigr) \theta (t) \, dt = {\mathcal O} (\e) \to 0 \\
&& \displaystyle \e \int_0^T B \bigl( \bar v^{\e} (t) , \bar \xi^{\e} (t),
\Theta^{\e} \bigr) \theta (t) \, dt = {\mathcal O} (\e) \to 0 \\
&& \displaystyle \e \int_0^T {\mathcal R} \bigl( \bar \xi^{\e} (t), \Theta^{\e}
\bigr) \theta (t) \, dt = {\mathcal O} (\e) \to 0\end{aligned}$$ with $\bar \xi = ( \bar U e_1, \bar W)$.
Next, using (\[pression\]) and reminding that $\varphi^{\e} \in \tilde
V^{\e}$: $$\begin{aligned}
&& \displaystyle \e \int_0^T \int_{\Omega^{\e}} p^{\e} (t,z) {\rm div}_z
\varphi^{\e} (z) \theta (t) \, dz dt
= \int_0^T \int_{\Omega} \e p^{\e} \left( (\e b_{\e} \cdot \nabla
\varphi_1^{\e}) + \frac{1}{h^{\e}} \frac{\partial \varphi_2^{\e}}{\partial y_2}
\right) h^{\e} \theta \, dy dt \\
&& \displaystyle = \int_0^T \int_{\Omega} \e p^{\e}
\left(
\e h \left( y_1, \frac{y_1}{\e} \right) \frac{\partial \varphi_1}{\partial y_1}
\left( y, \frac{y}{\e} \right)
+ h \left( y_1, \frac{y_1}{\e} \right) \frac{\partial \varphi_1}{\partial
\eta_1} \left( y, \frac{y}{\e} \right) \right.\\
&& \displaystyle - y_2 \left(
\e \frac{\partial h}{\partial y_1} \left( y_1, \frac{y_1}{\e} \right) +
\frac{\partial h}{\partial \eta_1} \left( y_1, \frac{y_1}{\e} \right)
\right)
\frac{\partial \varphi_1}{\partial y_2} \left( y, \frac{y}{\e} \right)
+ \frac{\partial \varphi_2}{\partial y_2} \left( y, \frac{y}{\e} \right) \\
&& \displaystyle \left. + \e y_2 \frac{\partial h}{\partial y_1} \left( y_1,
\frac{y_1}{\e} \right) \frac{\partial \varphi_1}{\partial y_2} \left( y,
\frac{y}{\e} \right) + \e \frac{\partial h}{\partial y_1} \left( y_1,
\frac{y_1}{\e} \right) \varphi_1 \left( y, \frac{y}{\e} \right)
\right) \theta (t) \, dy dt \\
&& \displaystyle = \int_0^T \int_{\Omega} \e^2 p^{\e}
\left(
h \left( y_1, \frac{y_1}{\e} \right) \frac{\partial \varphi_1}{\partial y_1}
\left( y, \frac{y}{\e} \right)
+ \frac{\partial h}{\partial y_1} \left( y_1, \frac{y_1}{\e} \right) \varphi_1
\left( y, \frac{y}{\e} \right)
\right) \theta (t) \, dy dt \\
&& \displaystyle \to \int_0^T \int_{\Omega \times (0,1)} p^0 \frac{ \partial ( h
\varphi_1)}{\partial y_1} \theta \, d \eta_1 dy dt
= - \int_0^T \int_{\Omega \times (0,1)} \frac{\partial p^0}{\partial y_1} h
\varphi_1 \theta \, d \eta_1 dy dt.
\end{aligned}$$ Finally $$\begin{aligned}
&& \displaystyle \e^2 \int_0^T \frac{d}{dt} \bigl[ \bar v^{\e} , \Theta^{\e}
\bigr] (t) \theta (t) \, dt =
- \e^2 \int_0^T \bigl[ \bar v^{\e} , \Theta^{\e} \bigr] (t) \theta' (t) \, dt =
{\mathcal O} (\e^2) \to 0 \\
&&\displaystyle - \e^2 \int_0^T \left[ \frac{\partial \bar \xi^{\e}}{\partial t},
\Theta^{\e} \right] (t) \theta (t) \, dt
= {\mathcal O} (\e^2) \to 0.\end{aligned}$$
By multiplying equation (\[eqn:er2.14a\]) by $\e \theta (t)$, integrating over $[0,T]$ and passing to the limit as $\e $ tends to zero we obtain $$\begin{aligned}
&& \displaystyle
\int_0^T \bar a \bigl( \bar v^0 (t), \Theta) \theta (t) \, dt - \int_0^T
\int_{\Omega \times (0,1)} \frac{\partial p^0}{\partial y_1} h \varphi_1 \theta
\, d \eta_1 dy dt \\
&& \displaystyle
= - \int_0^T \bar a \bigl( \bar \xi (t), \Theta) \theta (t) \, dt + \int_0^T
\int_{\Om \times (0,1)} (f \varphi + g \psi) h \theta \, d \eta_1 dy dt\end{aligned}$$ for all $\Theta = (\varphi, \psi) \in \tilde V_{div} \times \tilde H^1$ and $\theta \in {\mathcal D} (0,T)$. By density of $\tilde V_{div} \times \tilde
H^1$ into $V_{div} \times H^1_{0 {\sharp}}$ we get the announced result.
We may observe that the limit problem is totally decoupled with respect to the velocity and micro-rotation fields. Furthermore the time variable appears as a parameter in the limit problem. More precisely, for all $y_1 \in [0,L] $, let $a_{y_1}$ be the bilinear symmetric form defined on ${\mathcal F}$ by $$\begin{aligned}
a_{y_1} (w, \psi) = \int_Y \left( h(y_1, \eta_1) ( \bar b \cdot \nabla w)(
y_2, \eta_1) (\bar b \cdot \nabla \psi)(y_2, \eta_1) + \frac{1}{h(y_1, \eta_1)}
\frac{\partial w}{\partial y_2} (y_2, \eta_1) \frac{\partial \psi}{\partial
y_2} (y_2, \eta_1) \right) \, d \eta_1 dy_2
\end{aligned}$$ for all $(w, \psi) \in {\mathcal F}$. The limit velocity, pressure and micro-rotation fields are solution of the problems $(P_{v^0,p^0})$ and $(P_{Z^0})$ given respectively by $$\begin{aligned}
&& \displaystyle \hbox{\rm Find } v^0 \in L^2( 0,T; V_{div}) \mbox{ and } p^0 \in
H^{-1} (0,T; H^1_{\sharp} (0,L) ) \mbox{ such that } \\
&& \displaystyle \displaystyle \int_0^L p^0 (t, y_1) \left( \int_0^1
h ( y_1, \eta_1) \, d \eta_1 \right) \, dy_1 =0 \mbox{ a.e. } t \in [0,T] \mbox{ and } \\
&& \displaystyle ( \nu + \nu_r) \int_0^L \sum_{i=1}^2 a_{y_1} (v^0_i,
\varphi_i) \, dy_1
- \int_0^L \frac{\partial p^0}{\partial y_1} \left( \int_0^1 h(y_1, \cdot )
\varphi_1 \, d \eta_1 \right) \, d y_1 \\
&& \displaystyle =- (\nu + \nu_r) \int_0^L a_{y_1} \bigl(\bar U (t) ,
\varphi_1\bigr) \, dy_1
+ \int_0^L \left( \int_Y f(t, y_1, \cdot, \cdot) h( y_1, \cdot) \varphi \, d
\eta_1 d y_2 \right) d y_1 \\
&& \displaystyle \forall \varphi \in V_{div} , \ {\rm a.e.} \ t \in [0,T]\end{aligned}$$ and $$\begin{aligned}
&& \displaystyle \hbox{\rm Find } Z^0 \in L^2 (0,T; H^1_{0, {\sharp}} ) \mbox{ such
that } \\
&& \displaystyle \alpha \int_0^L a_{y_1} (Z^0, \psi) \, dy_1
=- \alpha \int_0^L a_{y_1} \bigl(\bar W (t) , \psi \bigr) \, dy_1
+ \int_0^L \left( \int_Y g(t, y_1, \cdot, \cdot) h( y_1, \cdot) \psi \, d \eta_1
d y_2 \right) d y_1 \\
&& \displaystyle \forall \psi \in H^1_{0, {\sharp}}, \ {\rm a.e.} \ t \in
[0,T].\end{aligned}$$
\[prop5.1\] Under the assumptions of theorem \[limit\_pb\], the limit micro-rotation field $Z^0$ is uniquely given by $$\begin{aligned}
Z^0 (t, y_1, y_2, \eta_1) = W_0 (t) z^1_{ y_1} (y_2, \eta_1) + z^2_{ t, y_1}
(y_2, \eta_1) \quad \hbox{\rm a.e. in } (0,T) \times \Omega \times (0,1)\end{aligned}$$ where $z^1_{y_1} \in H^1_{0, \sharp} $ and $z^2_{t, y_1} \in H^1_{0, \sharp} $ are the unique solutions of the following auxiliary problems: $$\begin{aligned}
a_{y_1} (z^1_{ y_1}, \psi) = - a_{y_1} \bigl( {\mathcal W} (y_1, \cdot), \psi
\bigr)
\quad \forall \psi \in H^1_{0, \sharp}\end{aligned}$$ and $$\begin{aligned}
\alpha a_{y_1} (z^2_{t, y_1}, \psi) =
\int_Y g_{t, y_1} h( y_1, \cdot) \psi \, d \eta_1 d y_2
\quad \forall \psi \in H^1_{0, \sharp} .\end{aligned}$$
It is clear that, for all $y_1 \in [0,L] $, the mapping $a_{y_1}$ is continuous on ${\mathcal F}$. Moreover $$\begin{aligned}
a_{y_1} (w, w) \ge h_{min} \| \bar b \cdot \nabla w \|^2_{L^2(Y)} +
\frac{1}{h_{max}} \left\| \frac{\partial w}{\partial z_2} \right\|_{L^2(Y)}
\end{aligned}$$ and $$\begin{aligned}
&& \displaystyle \| \bar b \cdot \nabla w \|^2_{L^2(Y)} = \left\|
\frac{\partial w}{\partial \eta_1} \right\|^2_{L^2(Y)} + \int_Y
\frac{y_2^2}{h^2(y_1, \eta_1)} \left( \frac{\partial h}{\partial \eta_1} (y_1,
\eta_1, t) \right)^2 \left( \frac{\partial w}{\partial y_2} \right)^2 \, d
\eta_1 dy_2 \\
&& \displaystyle - 2 \int_Y \frac{y_2}{h(y_1, \eta_1)} \frac{\partial
h}{\partial \eta_1} (y_1, \eta_1) \frac{\partial w}{\partial y_2}
\frac{\partial w}{\partial \eta_1} \, d \eta_1 dy_2 \\
&& \displaystyle \ge (1 - \lambda) \left\| \frac{\partial w}{\partial \eta_1}
\right\|^2_{L^2(Y)} + \left( 1- \frac{1}{\lambda} \right)
\int_Y \frac{y_2^2}{h^2(y_1, \eta_1)} \left( \frac{\partial h}{\partial \eta_1}
(y_1, \eta_1) \right)^2 \left( \frac{\partial w}{\partial y_2} \right)^2 \, d
\eta_1 dy_2 \quad \forall \lambda >0.\end{aligned}$$ But, recalling that $h \in {\mathcal C} \bigl( [0,L] \times [0,1] \bigr)$, there exists $C>0$, independent of $y_1$, such that $$\begin{aligned}
\left| \frac{y_2}{h(y_1, \eta_1)} \frac{\partial h}{\partial \eta_1} (y_1,
\eta_1) \right| \le C \quad \forall (y_1, y_2, \eta_1) \in [0,L] \times Y\end{aligned}$$ and, for all $\lambda \in (0,1)$ $$\begin{aligned}
\label{coer_1}
a_{y_1} (w,w) \ge C_{1}(\lambda)\left\| \frac{\partial
w}{\partial
\eta_1} \right\|^2_{L^2(Y)} + C_{2}(\lambda)
\left\| \frac{\partial w}{\partial y_2} \right\|^2_{L^2(Y)},
\end{aligned}$$ where $C_{1}(\lambda) = h_{min} (1 - \lambda)$ and $C_{2}(\lambda)
=\left(\left(1-\frac{1}{\lambda}\right) C^2 h_{min} + \frac{1}{h_{max}}
\right)$. Then we may choose $\lambda$ such that $$\begin{aligned}
\label{coer_2}
\lambda \in \left( \frac{C^2 h_{max} h_{min}}{1+ C^2 h_{max} h_{min}}, 1
\right)
\end{aligned}$$ which shows that $a_{y_1}$ is coercive on $H^1_{0, \sharp}$, uniformly with respect to $y_1$. Since $g \in {\mathcal C} \bigl( [0,T];
{\mathcal C} \bigl( \overline{\Om}; {\mathcal C}_{\sharp} (0,1) \bigr) \bigr)$ the mapping $g_{t,y_1} = g(t, y_1, \cdot, \cdot) $ belongs to $L^2(Y)$ for all $(t, y_1) \in [0,T] \times [0,L]$. Then Lax-Milgram’s theorem implies that, for all $(t, y_1) \in [0,T] \times
[0,L] $ the problems $$\begin{aligned}
&& \displaystyle \hbox{\rm Find } z^1_{y_1} \in H^1_{0, \sharp} \mbox{ such that}
\\
&& \displaystyle a_{y_1} (z^1_{ y_1}, \psi) = - a_{y_1} \bigl( {\mathcal W}
(y_1, \cdot) , \psi \bigr)
\quad \forall \psi \in H^1_{0, \sharp}\end{aligned}$$ and $$\begin{aligned}
&& \displaystyle \hbox{\rm Find } z^2_{t, y_1} \in H^1_{0, \sharp} \mbox{ such that}
\\
&& \displaystyle \alpha a_{y_1} (z^2_{t, y_1}, \psi) =
\int_Y g_{t, y_1} h( y_1, \cdot) \psi \, d \eta_1 d y_2
\quad \forall \psi \in H^1_{0, \sharp}\end{aligned}$$ admit a unique solution. Furthermore, recalling that $W_0 \in H^1(0,T) \subset
{\mathcal C} \bigl( [0,T] \bigr)$ and $h \in {\mathcal C}^1 \bigl( [0,L] \times
[0,1] ; {\mathbb R}\bigr) $ with values in $[h_{min}, h_{max} ] \subset {\mathbb R}^+_*$, we infer that the mapping $(t, y_1 ) \mapsto Z^0_{t, y_1} = W_0 z^1_{ y_1} + z^2_{
t, y_1}$ is continuous on $[0,T] \times [0,L]$ with values in $H^1_{0, \sharp}$ and is $L$-periodic in $y_1$.
Thus the mapping $Z^0: (t, y_1, y_2, \eta_1) \mapsto Z^0_{t, y_1} (y_2,
\eta_1)$ belongs to $L^2 (0,T ; H^1_{ 0, {\sharp}} \bigr)$ and solves the problem $(P_{Z_0})$. Indeed, let $\psi \in \tilde H^1$. Then $\psi (
y_1, \cdot, \cdot) \in H^1_{\sharp} $ and we get $$\begin{aligned}
&& \displaystyle \alpha a_{y_1} \bigl(Z^0_{t, y_1}, \psi ( y_1, \cdot, \cdot)
\bigr) = - \alpha a_{y_1} \bigl( \bar W(t, y_1, \cdot, \cdot) , \psi ( y_1,
\cdot, \cdot) \bigr) \\
&& \displaystyle + \int_Y g (t, y_1, \cdot, \cdot) h( y_1, \cdot) \psi \, d
\eta_1 d y_2
\quad \forall y_1 \in [0,L] .\end{aligned}$$ Both sides of this equality are continuous on $[0,L]$, hence we may integrate with respect to $y_1$ and $$\begin{aligned}
\int_0^L a_{y_1} (Z^0_{t, y_1}, \psi) \, dy_1 = - \int_0^L a_{y_1} \bigl( \bar
W , \psi \bigr) \, dy_1
+ \int_0^L \left( \int_Y g_{t, y_1} h( y_1, \cdot) \psi \, d \eta_1 d y_2
\right) d y_1
\quad \forall \psi \in \tilde H^1.\end{aligned}$$ It follows that $$\begin{aligned}
&& \displaystyle \int_0^L a_{y_1} (Z^0 (t), \psi) \, dy_1 = - \int_0^L a_{y_1}
\bigl( \bar W (t) , \psi \bigr) \, dy_1
+ \int_0^L \left( \int_Y g (t) h( y_1, \cdot) \psi \, d \eta_1 d y_2 \right) d
y_1 \\
&& \displaystyle \forall \psi \in \tilde H^1, \ \hbox{\rm a.e.} \ t \in [0,T]\end{aligned}$$ and the density of $\tilde H^1$ into $H^1_{ 0, {\sharp}} $ allows us to conclude the existence part of the proof. Then we observe that the uniqueness is a immediate consequence of the uniform coercivity of $a_{y_1}$ with respect to $y_1$.
Now, for all $y_1 \in [0,L] $, let $$\begin{aligned}
\tilde V_{y_1} = \left\{ \varphi \in \bigl( {\mathcal C}^{\infty} \bigl([0,1];
{\mathcal C}^{\infty}_{\sharp} (0,1) \bigr) \bigr)^2; \ \varphi (0, \cdot)= 0 \
\hbox{\rm on } (0,1), \ - \varphi_1 (1, \cdot) \frac{\partial h}{\partial y_1}
(y_1, \cdot) + \varphi_2 (1, \cdot) =0 \ \hbox{\rm on } (0,1) \right\},\end{aligned}$$ $$\begin{aligned}
\tilde V_{y_1, div} = \left\{ \varphi \in \tilde V_{y_1}; \ h(y_1, \cdot)
\frac{\partial \varphi_1}{\partial \eta_1} - y_2 \frac{\partial h }{\partial
\eta_1} (y_1, \cdot) \frac{\partial \varphi_1}{\partial y_2} + \frac{\partial
\varphi_2}{\partial y_2} =0 \ \hbox{\rm in } Y \right\}\end{aligned}$$ and $$\begin{aligned}
V_{y_1, div} = \hbox{\rm closure of } \tilde V_{y_1, div} \mbox{ in } {\mathcal F}^{2}.\end{aligned}$$ Let $\displaystyle \bar a_{y_1} (w, \varphi) = (\nu + \nu_r) \sum_{i=1}^2
a_{y_1} (w_i, \varphi_i)$ for all $(w, \varphi) \in V_{y_1, div}^2$. With Poincaré’s inequality we know that $w \mapsto \| \nabla w\|_{L^2 (Y)}$ defines a norm on $V_{y_1, div}$ which is equivalent to the $H^1$-norm. Furthermore, with (\[coer\_1\])-(\[coer\_2\]), we may infer that $\bar
a_{y_1}$ is coercive on $V_{y_1, div}$ for all $y_1 \in [0,L] $, uniformly with respect to $y_1$. It follows that we can define $w_{y_1}^1 \in V_{y_1, div}$, $w_{y_1}^2 \in V_{y_1, div}$ and $w_{t,y_1}^3 \in V_{y_1, div}$ as the unique solutions of $$\begin{aligned}
\bar a_{y_1} (w^1_{y_1}, \varphi) = - \int_Y h (y_1, \cdot ) \varphi_1 \, d
\eta_1 d y_2 \quad \forall \varphi \in V_{y_1, div},\end{aligned}$$ $$\begin{aligned}
\bar a_{y_1} (w^2_{y_1}, \varphi) = - (\nu + \nu_r) a_{y_1} \bigl({\mathcal U}
( y_1, \cdot ) , \varphi_1 \bigr) \quad \forall \varphi \in V_{y_1, div}\end{aligned}$$ and $$\begin{aligned}
\bar a_{y_1} (w^3_{t,y_1}, \varphi) = \int_Y f_{t,y_1} h (y_1, \cdot )
\varphi_1 \, d \eta_1 d y_2 \quad \forall \varphi \in V_{y_1, div}\end{aligned}$$ with $f_{t,y_1} = f(t, y_1, \cdot, \cdot)$ for all $(t,y_1) \in [0,T] \times
[0,L]$.
Then we have
\[prop5.2\] Under the assumptions of theorem \[limit\_pb\], the limit velocity $v^0$ is uniquely given by $$\begin{aligned}
v^0 (t, y_1, y_2, \eta_1) = \frac{\partial p^0}{\partial y_1} (t, y_1) w^1_{y_1}
(y_2, \eta_1) + U_0 (t) w^2_{ y_1} (y_2, \eta_1) + w^3_{t,y_1} (y_2, \eta_1)
\quad \hbox{\rm a.e. in } (0,T) \times \Omega\times (0,1).\end{aligned}$$ Furthermore, for almost every $t \in [0,T]$, the limit pressure $p^0 (t, \cdot)
$ is the unique solution in $H^1_{\sharp} (0,L)_{| {\mathbb R}}$ of the following homogenized Reynolds equation $$\begin{aligned}
\int_0^L \frac{\partial p^0}{\partial y_1} \frac{\partial \psi}{\partial y_1}
\bar a (w^1_{y_1} , w^1_{y_1} ) \, dy_1 = - \int_0^L U_0 (t) \frac{\partial
\psi}{\partial y_1} \bar a \bigl(w^1_{y_1} , w^2_{ y_1} \bigr) \, dy_1
- \int_0^L \frac{\partial \psi}{\partial y_1} \bar a \bigl(w^1_{y_1} , w^3_{
t, y_1} \bigr) \, dy_1
\quad \forall \psi \in H^1_{\sharp} (0,L)\end{aligned}$$ satisfying $\displaystyle \int_0^L p^0 \left( \int_0^1 h( \cdot , \eta_1) \, d
\eta_1 \right) \, dy_1 =0$.
The first part of the result is obtained by using the same kind of arguments as in Proposition \[prop5.1\].
Let $\theta \in {\mathcal D} (0,T)$, $\psi \in {\mathcal C}^{\infty}_{\sharp}
\bigl( [0,L] \bigr)$ and $\psi^{\e} (z) = \psi(z_1)$ for all $z = (z_1, z_2) \in
\Omega^{\e}$. Recalling that ${\rm div}_z v^{\e} =0$ in $\Omega^{\e}$ and using the boundary conditions (\[eqn:er2.8a\])-(\[eqn:er2.11\])-(\[eqn:er2.11b\]) we get $$\begin{aligned}
&& \displaystyle 0 = \frac{1}{\e} \int_0^T \int_{\Omega^{\e}} \left(
\frac{\partial v^{\e}_1 }{\partial z_1} (t,z)+ \frac{\partial v^{\e}_2
}{\partial z_2} (t,z) \right) \psi^{\e} (z) \theta (t) \, dz dt \\
&& \displaystyle 0 = - \frac{1}{\e} \int_0^T \int_{\Omega^{\e}} v_1^{\e} (t,z)
\frac{\partial \psi^{\e} }{\partial z_1} (z) \theta (t) \, dz dt = -
\int_0^T \int_{\Omega} v_1^{\e} (t,y) (b_{\e} \cdot \nabla \psi^{\e} ) (y)
h^{\e} (y) \theta (t) \, dy dt \\
&& \displaystyle = - \int_0^T \int_{\Omega} v_1^{\e} (t,y) \frac{\partial \psi
}{\partial y_1} (y_1) h \left( y_1, \frac{y_1}{\e} \right) \theta (t) \, dy dt.\end{aligned}$$ By passing to the limit as $\e$ tends to zero we get $$\begin{aligned}
0 = \int_0^T \int_{\Omega \times (0,1)} v_1^0 (t, y, \eta_1) \frac{\partial
\psi }{\partial y_1} (y_1) h (y_1, \eta_1) \theta (t) \, d \eta_1 dy dt.\end{aligned}$$ It follows that $$\begin{aligned}
&& \displaystyle \int_0^L \frac{\partial p^0}{\partial y_1} \frac{\partial
\psi}{\partial y_1}\left( \int_Y w_{y_1, 1}^1 h (y_1, \cdot) \, d \eta_1 d y_2
\right) \, dy_1 + \int_0^L U_0 (t) \frac{\partial \psi}{\partial y_1} \left(
\int_Y w_{y_1, 1}^2 h (y_1, \cdot) \, d \eta_1 d y_2 \right) \, dy_1 \\
&& \displaystyle + \int_0^L \frac{\partial \psi}{\partial y_1} \left( \int_Y
w_{t, y_1, 1}^3 h (y_1, \cdot) \, d \eta_1 d y_2 \right) \, dy_1
=0 \quad {\rm a.e.} \ t \in [0,T].\end{aligned}$$ But $$\begin{aligned}
&& \displaystyle \int_Y w_{y_1, 1}^1 h (y_1, \cdot) \, d \eta_1 d y_2 = - \bar
a_{y_1} (w_{y_1}^1, w_{y_1}^1) , \\
&& \displaystyle \int_Y w_{y_1, 1}^2 h (y_1, \cdot) \, d \eta_1 d y_2 = - \bar
a_{y_1} (w_{y_1}^1, w_{y_1}^2), \\
&& \displaystyle \int_Y w_{t,y_1, 1}^3 h (y_1, \cdot) \, d \eta_1 d y_2 = -
\bar a_{y_1} (w_{y_1}^1, w_{y_1}^3)\end{aligned}$$ and by density of $ {\mathcal C}^{\infty}_{\sharp} \bigl( [0,L] \bigr)$ in $H^1_{\sharp} (0,L)$ we get $$\begin{aligned}
&& \displaystyle \int_0^L \frac{\partial p^0}{\partial y_1} \frac{\partial
\psi}{\partial y_1} \bar a (w^1_{y_1} , w^1_{y_1} ) \, dy_1 = - \int_0^L U_0 (t)
\frac{\partial \psi}{\partial y_1} \bar a \bigl(w^1_{y_1} , w^2_{ y_1} \bigr)
\, dy_1 \\
&& \displaystyle - \int_0^L \frac{\partial \psi}{\partial y_1} \bar a
\bigl(w^1_{y_1} , w^3_{ t, y_1} \bigr) \, dy_1 \quad \forall \psi \in
H^1_{\sharp} (0,L) , \ {\rm a.e.} \ t \in [0,T].\end{aligned}$$ We can check that this Reynolds problem admits a unique solution in $H^1_{\sharp} (0,L)_{| {\mathbb R}}$. Indeed, let $\displaystyle \varphi_{y_1} (y_2, \eta_1) = \left( \frac{ -y_2 +
y_2^2}{h(y_1, \eta_1)}, \frac{\partial h}{\partial \eta_1} (y_1, \eta_1) \frac{
y_2^2 (y_2 -1)}{h(y_1, \eta_1)} \right)$ for all $(y_2, \eta_1) \in Y$, for all $y_1 \in [0,L]$. Then we obtain $ \varphi_{y_1} \in V_{y_1, div}$ and $$\begin{aligned}
\bar a_{y_1} (w^1_{y_1}, \varphi_{y_1}) = - \int_Y h (y_1, \eta_1)
\varphi_{y_1, 1} (y_2 , \eta_1) \, d \eta_1 d y_2 = \frac{1}{6}.\end{aligned}$$ Since $\bar a_{y_1}$ defines an inner product on $V_{y_1, div}$, we have $$\begin{aligned}
\frac{1}{6 } = \bar a_{y_1} (w^1_{y_1}, \varphi_{y_1}) \le \bar a_{y_1}
(w^1_{y_1}, w^1_{y_1} )^{1/2} \bar a_{y_1} ( \varphi_{y_1},
\varphi_{y_1})^{1/2} .\end{aligned}$$ But the mapping $y_1 \mapsto \bar a_{y_1} ( \varphi_{y_1}, \varphi_{y_1})$ is continuous on $[0,L]$ and does not vanish since $ \varphi_{y_1} \not \equiv 0$. It follows that there exists $\alpha >0$ such that $ \bar a_{y_1} (
\varphi_{y_1}, \varphi_{y_1}) \ge \alpha$ for all $y_1 \in [0,L]$ and $ \bar a
(w^1_{y_1} , w^1_{y_1} ) \ge \frac{1}{ 36 \alpha}$ for all $y_1 \in [0,L]$.
We may observe also that the mapping $\displaystyle \psi \mapsto - \int_0^L U_0 (t) \frac{\partial \psi}{\partial
y_1} \bar a \bigl(w^1_{y_1} , w^2_{ y_1} \bigr) \, dy_1 - \int_0^L
\frac{\partial \psi}{\partial y_1} \bar a \bigl(w^1_{y_1} , w^3_{ t, y_1}
\bigr) \, dy_1 $ is linear and continuous on $H^1_{\sharp} (0,L)$ for every $t
\in [0,T]$ and the mapping $\displaystyle (p, \psi) \mapsto \int_0^L \frac{\partial p}{\partial y_1}
\frac{\partial \psi}{\partial y_1} \bar a (w^1_{y_1} , w^1_{y_1} ) \, dy_1$ is bilinear, symmetric, continuous and coercive on $H^1_{\sharp} (0,L)_{|{\mathbb R}}$. We can apply Lax-Milgram’s theorem to conclude the proof of Proposition \[prop5.2\].
As a consequence of the uniqueness of $p^0$, we can state the next result:
The whole sequences $( \e^2 p^{\e})_{\e >0}$, $( v^{\e})_{\e >0}$ and $(Z^{\e})_{\e >0}$ satisfy the following convergence: $$\begin{aligned}
&& \e p^{\e} {\twoheadrightarrow}p^0 \\
&& v^{\e} {\twoheadrightarrow}v^0 \\
&& Z^{\e} {\twoheadrightarrow}Z^0.\end{aligned}$$
Concluding remarks {#open-Pbs}
==================
A possible generalization of this study consists in considering a domain $\Om^{\e}$ where both the upper and lower boundary are oscillating. More precisely, let us assume that $$\begin{aligned}
\Om^{\e} =\bigl\{ (z_{1} , z_{2}) \in {\mathbb R}^{2}; \quad 0 < z_{1}< L, \quad
-\e \beta(z_{1}) h^{\e}(z_{1})< z_{2} < \e h^{\e}(z_{1}) \bigr\}\end{aligned}$$ where $\beta$ belongs to $ {\cal C}^{\infty}([0 , L]; {\mathbb R}^+)$ and is $L$-periodic in $z_1$ (with $\beta \equiv 0$ we recognize the case presented in the previous sections). Now we should denote by $\Gamma_0^{\e}$ the lower boundary of $\Om^{\e}$ and we can choose the functions ${\cal U}$ and ${\cal W}$ (see Lemma \[lUW\]) such that ${\cal U}$ and ${\cal W}$ belong to ${\cal C}^{\infty}({\mathbb R}, {\mathbb R})$ with ${\cal U}(\sigma) = {\cal W} (\sigma) = 1$ for all $\sigma \le 0$ and ${\rm Supp} ({\cal U}) \subset (- \infty, h_m)$, ${\rm Supp} ({\cal W}) \subset (- \infty, h_m)$. Then we define again $$U^{\e}(t ,z_{2})= {\cal U}^{\e}(z_{2})U_{0}(t)
= {\cal U}({z_{2}\over \e})U_{0}(t) ,
\quad
W^{\e}(t ,z_{2})= {\cal W}^{\e}(z_{2})W_{0}(t)
= {\cal W}({z_{2}\over \e})W_{0}(t)$$ and we get the same variational problem $(P^{\e})$. It follows that the existence and uniqueness result given at Theorem \[th2.1\] is still valid. Furthermore, we can use the same scalings (see (\[2.2a\]) and (\[2.2b\])) which transforms the domain $\Om^{\e}$ into $$\begin{aligned}
\Om= \bigl\{ (y_1, y_2) \in {\mathbb R}^2; \quad 0 < y_1 <L, \quad - \beta (y_1) < y_2 <1 \bigr\}\end{aligned}$$ and by reproducing the same computations, we obtain the same a priori estimates as in Proposition \[pro1\] and Proposition \[prop2\].
Finally we may apply once again the two-scale convergence technique to pass to the limit as $\e$ tends to zero. We obtain the same convergence properties for the velocity and the micro-rotaion field as in Proposition \[prop4.1\] and Proposition \[prop4.2\] with $\Gamma_0 = \bigl\{ \bigl(y_1, - \beta(y_1) \bigr); \ 0< y_1 < L \bigr\}$. For the convergence of the pressure, we follow the same arguments as in Proposition \[prop4.3\] with a natural modification of the test-function $\varphi^{\e}$ introduced at formula (\[eq:rajout1\]) which may be chosen now as $$\begin{aligned}
\varphi^{\e} (y) = \frac{\varphi(y_1)}{h\left(y_1, \frac{y_1}{\e} \right)} \left( \bigl(y_2 + \beta(y_1) \bigr) e_1 + \e y_2 \bigl(y_2 + \beta(y_1) \bigr) \left( \frac{\partial h}{\partial y_1} \left(y_1, \frac{y_1}{\e} \right) + \frac{1}{\e} \frac{\partial h}{\partial \eta_1} \left(y_1, \frac{y_1}{\e} \right) \right) e_2 \right)\end{aligned}$$ for all $(y_1, y_2) \in \Om$, which leads to $$\begin{aligned}
\left| \int_0^T \int_0^L p^{0} (t,y_1) \frac{\partial }{\partial y_1}\bigl( \frac{1}{2} (1 + \beta)^2 \varphi \bigr) (y_1) \theta (t) \, dy dt \right|
\le C \| \varphi \theta \|_{L^2 ((0,T) \times (0,L))}.\end{aligned}$$ Then we may conclude by considering any $\phi \in {\mathcal C}^{\infty}_{\sharp} (0,L) $ and letting $\varphi = \frac{2 \phi}{(1+ \beta)^2}$.
Hence the limit problem remains the same as in Theorem \[limit\_pb\]: $Z^0$ and $v^0$ can be decomposed by using the same auxiliary problems and $p^0$ is the unique solution of the same Reynolds equation, with obvious adaptations in the definition of $a_{y_1}$ and $\bar a_{y_1}$, i.e. for all $y_1 \in [0,L]$: $$\begin{aligned}
a_{y_1} (w, \psi) = \int_{ - \beta(y_1)}^1 \int_0^1 \Bigl( && h(y_1, \eta_1) ( \bar b \cdot \nabla w)( y_2, \eta_1) (\bar b \cdot \nabla \psi)(y_2, \eta_1) \\
&& + \frac{1}{h(y_1, \eta_1)} \frac{\partial w}{\partial y_2} (y_2, \eta_1) \frac{\partial \psi}{\partial y_2} (y_2, \eta_1) \Bigr) \, d \eta_1 dy_2
\end{aligned}$$ for all $\displaystyle (w, \psi) \in {\mathcal F}_{y_1} = \Bigl\{ v \in L^2 \bigl( (- \beta(y_1), 1 ); H^1_{\sharp} (0,1) \bigr) ; \frac{\partial v}{\partial y_2} \in L^2 \bigl( (- \beta(y_1), 1 ) \times (0,1) \bigr) \Bigr\}$ and $\displaystyle \bar a_{y_1} (w, \varphi) = (\nu + \nu_r) \sum_{i=1}^2 a_{y_1} (w_i, \varphi_i)$ for all $(w, \varphi) \in V_{y_1, div}^2$ where $V_{y_1, div}$ is the closure of $\tilde V_{y_1, div}$ in ${\mathcal F}_{y_1}^2$ and $$\begin{aligned}
\tilde V_{y_1, div} = \Bigl\{&& \varphi \in \bigl( {\mathcal C}^{\infty} \bigl([- \beta(y_1),1]; {\mathcal C}^{\infty}_{\sharp} (0,1) \bigr) \bigr)^2; \ \varphi (- \beta(y_1), \cdot)= 0 \ \hbox{\rm on $(0,1)$,} \\
&& - \varphi_1 (1, \cdot) \frac{\partial h}{\partial y_1} (y_1, \cdot) + \varphi_2 (1, \cdot) =0 \ \hbox{\rm on $ (0,1)$,} \\
&&
h(y_1, \cdot) \frac{\partial \varphi_1}{\partial \eta_1} - y_2 \frac{\partial h }{\partial \eta_1} (y_1, \cdot) \frac{\partial \varphi_1}{\partial y_2} + \frac{\partial \varphi_2}{\partial y_2} =0 \ \hbox{\rm in $(- \beta(y_1), 1) \times (0,1)$}
\Bigr\}.\end{aligned}$$
[**Acknowledgements:**]{} We would like to thank the anonymous referees for their constructive comments.
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, [*Homogenized elliptic equations and variational inequallities with oscillating parameters. Application to the study of thin flow behavior with rough surfaces.*]{} Nonlinear Anal.: Real World Appli. 7, no. 5, 2006, 950-966.
, [*Existence and uniqueness for several non-linear elliptic problems arising in lubrication theory.*]{} J. Diff. Equations 218, no. 1, 2005, 187-215.
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, [*Asymptotic behaviour of pressure and stresses in a thin flow with a rough boundary.*]{} Quart. Appl. Math. 63, 2005, 369-400.
, [*Attractor dimension estimate for plane shear flow of micropolar fluid with free boundary*]{} Math. Meth. Appl. Sci. 28, 2005, 1673 -1694.
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[^1]: PRES Lyon University, UJM F-42023 Saint-Etienne, CNRS UMR 5208 Institut Camille Jordan, 23 Docteur Paul Michelon 42023 Saint-Etienne Cedex 2, France. Fax: +33 4 77 48 51 53, Phone +33 4 77 48 15 00, Mahdi.Boukrouche@univ-st-etienne.fr, laetitia.paoli@univ-st-etienne.fr
|
---
abstract: 'The [Mutation$^{++}$]{} library provides accurate and efficient computation of physicochemical properties associated with partially ionized gases in various degrees of thermal nonequilibrium. With v1.0.0, users can compute thermodynamic and transport properties, multiphase linearly-constrained equilibria, chemical production rates, energy transfer rates, and gas-surface interactions. The framework is based on an object-oriented design in C++, allowing users to plug-and-play various models, algorithms, and data as necessary. [Mutation$^{++}$]{} is available open-source under the GNU Lesser General Public License v3.0.'
address: 'von Karman Institute for Fluid Dynamics, B-1640 Rhode-St-Genèse, Belgium'
author:
- 'James B. Scoggins'
- Vincent Leroy
- 'Georgios Bellas-Chatzigeorgis'
- Bruno Dias
- 'Thierry E. Magin'
bibliography:
- 'biblio.bib'
title: '[Mutation$^{++}$]{}: MUlticomponent Thermodynamic And Transport properties for IONized gases in C++'
---
0[0]{}
partially ionized gases ,thermochemical nonequilibrium ,multiphase equilibrium ,gas-surface interaction
Motivation and Significance {#s:introduction}
===========================
The evaluation of thermochemical nonequilibrium, partially ionized gas properties is essential for a wide range of applications, including hypersonic flows, solar physics and space weather, ion thrusters, medical plasmas, combustion processes, meteor phenomena, and biomass pyrolysis. For example, the prediction of hypersonic flow plays an important role in the development of thermal protection systems for atmospheric entry vehicles. Such flows span a broad range of temporal scales, from local thermodynamic equilibrium to thermo-chemical nonequilibrium. As such, myriads of physicochemical models, data, and algorithms are used in today’s hypersonic Computational Fluid Dynamics (CFD) codes and represent a significant body of work in the scientific literature. The thermochemical models employed in these codes directly affect the evaluation of gas properties necessary to close the conservation laws governing the fluid. These include mixture thermodynamic and transport properties, species chemical production rates, and energy transfer rates. Each of these properties further depends on the selection of a variety of specialized algorithms and data, such as species partition functions, transport collision integrals, and reaction rate coefficients.
The implementation, testing, and maintenance of the models, algorithms, and data required to simulate thermal nonequilibrium flows represent a significant cost, in terms of human resources and time necessary to develop a simulation tool. As new models, algorithms, or data become available, additional effort is required to update existing codes, especially when models are “hard-coded.” A number of commercial and academic software packages are available which provide gaseous properties, including CEA [@Gordon1994], EGlib [@Ern1996a], PEGASE [@Bottin1999], Chemkin [@Kee2000], MUTATION [@MaginThesis], Cantera [@Goodwin2017], and KAPPA [@Campoli2019], however, these libraries tend to focus on a specific application, a narrow range of collisional time-scales, or are specialized in providing only certain types of properties.
These observations have led to the desire to reduce the work necessary to implement new models and algorithms and centralize their development into a single software library which may be used by multiple CFD codes to maximize code reuse, testing, and open collaboration. This paper presents the [Mutation$^{++}$]{} library, which has been developed to meet this objective. The library is designed with several goals in mind, including
1. provide accurate thermodynamic, transport, and chemical kinetic properties for multicomponent, partially ionized gases,
2. ensure the efficient evaluation of these properties using state-of-the-art, object-oriented algorithms and data structures in C++,
3. be easily extendable to incorporate new data or algorithms as they become available,
4. interface to any simulation tool based on the solution of conservation laws through a consistent and logical interface,
5. use self-documenting database formats to decrease data transcription errors and increase readability, and
6. be open source to promote code and data sharing among different research communities.
The latest version of [Mutation$^{++}$]{} (v1.0.0) has recently been released open-source under the Lesser GNU Public License (LGPL v3) and is freely available on Github[^1]. In the remainder of the paper, we present an overview of the library and its impact on the research community to-date. In particular, the four main modules of the library — thermodynamics, transport, chemical kinetics, and gas-surface interaction — are presented, with a few examples to illustrate the library’s use.
Software Description {#s:software}
====================
Generalized Conservation Equations
----------------------------------
While it is beyond the scope of this article to describe in detail all the various physicochemical models that are present in the literature, it is useful to briefly present a generalized model which has been used in the design of the library. For a more complete discussion, see the work of Scoggins [@ScogginsThesis-Chap2]. We consider a generalized conservation law of the form, $$\partial_t \bm{U} + \nabla_{\bm{x}} \bm{F} = \bm{S},
\label{e:general-conservation}$$ where $\bm{U} = \begin{bmatrix}
\tilde{\rho}_i & \rho \bm{u} & \rho E & \rho \tilde{e}^m
\end{bmatrix}^T$, is a vector of species mass, momentum, and total and internal energy densities, $\bm{F}(\bm{U}, \nabla_{\bm{x}}\bm{U})$ represents their flux, and $\bm{S}(\bm{U})$ is a source function. The tilde over the indexed variables in the density vector denotes that these quantities must be expanded over their indices. The exact forms of $\bm{U}$, $\bm{F}$, and $\bm{S}$ depend on 1) the coordinate system, 2) the physical model (i.e.: Euler, Navier-Stokes), and 3) the thermochemical model of the gas (i.e.: equilibrium, reacting, multi-temperature, state-to-state).
![Overview of the [Mutation$^{++}$]{} library and its coupling to CFD.[]{data-label="f:overview"}](figures/overview.pdf){width="48.00000%"}
We define the thermochemical state-vector as $\hat{\bm{U}} = \begin{bmatrix}
\tilde{\rho}_i & \rho e & \rho \tilde{e}^m
\end{bmatrix}^T$ where $\rho e = \rho E - \rho \bm{u}\cdot\bm{u} / 2$ is the static energy density of the gas. The flux and source functions are closed by constitutive relations for thermodynamic, transport, and chemical properties of the gas. These include quantities such as pressure, enthalpy, viscosity, thermal conductivity, diffusion coefficients, chemical production rates, and energy transfer source terms. In general, these properties are only functions of the local state-vector $\hat{\bm{U}}$ and possibly its gradient. This fact allows us to separate the solution of [Eq. \[e:general-conservation\]]{} into two separate domains with limited coupling controlled by the CFD solver and [Mutation$^{++}$]{}, as shown in [Fig. \[f:overview\]]{}.
Software Architecture
---------------------
[Mutation$^{++}$]{} is designed with a strong focus on Object-Oriented Programming (OOP) patterns in C++. The library’s Application Programming Interface (API) is thoroughly documented using the Doxygen format. A continuous integration strategy has been employed. Regression and black box testing are performed through a combination of the Catch2 header-only testing framework and CTest. The primary access to the library is through a [`Mixture`]{} object, which is implemented as a set of submodules encapsulating clearly separated physical quantities as depicted in the simplified Unified Modeling Language (UML) diagram in [Fig. \[f:uml\]]{}.
{width="95.00000%"}
Software Functionalities
------------------------
Each module in [Fig. \[f:uml\]]{} is described in the following subsections. Specific examples of some of the outputs that the library can provide are given in the [Sec. \[s:examples\]]{}.
\
\
### Thermodynamics {#s:thermo}
The thermodynamics module provides pure species and mixture thermodynamic quantities, such as enthalpy, entropy, specific heats, or Gibbs free energies. Mixture thermodynamic quantities are derived as sums of pure species properties, weighted by the composition of the mixture. Thermodynamic data for pure species can be found in several references [@Ruscic2004; @Ruscic2005; @Burcat2005; @Ruscic2013; @Blanquart2007; @Blanquart2009; @Narayanaswamy2010; @Blanquart2015; @Goldsmith2012]. Differences exist between each database, such as their format, temperature range of applicability, or degree of nonequilibrium supported. Such differences often sway simulation tool designers to select a single database format to support, or hard-code thermodynamic data directly into their models. This approach makes it difficult to update data as needed, or compare with other tools using different databases.
The [Mutation$^{++}$]{} framework provides an abstraction layer which enforces a weak coupling between the concrete thermodynamic database, used for any given set of species, and the computation of mixture thermodynamic quantities. Such a design provides the flexibility to swap out different databases as needed, with minimal effort. The NASA 7- and 9-coefficient polynomial databases [@McBride1993; @McBride1993a; @Gordon1999; @McBride2002] and a custom XML format which implements a Rigid-Rotor/Harmonic-Oscillator (RRHO) model are currently implemented. The NASA format is widely used and provides thermodynamic properties for pure species in thermal equilibrium. The RRHO model is suitable for thermal nonequilibrium calculations. In addition, a new database including more than 1200 neutral and ionized species containing , , , and is shipped with the library in the NASA 9-coefficient format. The details of this database have been published in [@Scoggins2017]. The user can specify the concrete thermodynamic model when creating a mixture ([Fig. \[f:mixture-file\]]{}).
A closely related task to the calculation of thermodynamic properties is the solution of chemical equilibrium compositions. The efficient and robust computation of multiphase, constrained equilibrium compositions is an important topic in several fields, including combustion, aerospace and (bio)chemical engineering, metallurgy, paper processes, and the design of thermal protection systems for atmospheric entry vehicles (e.g., [@Chan1992; @Pajarre2008; @Koukkari2011; @Milos1997; @Howard2012; @Rabinovitch2014]). Several challenges associated with computing chemical equilibria make conventional methods hard to converge under some conditions [@Reynolds1986; @Gordon1994; @McBride1996; @Bishnu1997; @Bishnu2001]. A new multiphase equilibrium solver, based on the single-phase Gibbs function continuation method [@Pope2003; @Pope2004], has been developed specifically for [Mutation$^{++}$]{}. The Multiphase Gibbs Function Continuation (MPGFC) solver is robust for all well-posed constraints. More details about the solver can be found in [@Scoggins2015b].
### Transport
Closure of transport fluxes is achieved through a multiscale Chapman-Enskog perturbative solution of the Boltzmann equation, yielding asymptotic expressions for the necessary transport coefficients, such as thermal conductivity, viscosity, and diffusion coefficients [@Ferziger1972; @Mitchner1973; @Giovangigli1999; @Graille2009; @Nagnibeda2009]. Explicit expressions for these coefficients are derived in terms of linear transport systems through a Laguerre-Sonine polynomial approximation of the Enskog expansion at increasing orders of accuracy [@Magin2004; @Magin2004a; @Scoggins2016]. These linear systems are functions of transport collision integrals and the local state-vector, and may be solved through a variety of methods.
Collision integrals represent Maxwellian averages of collision cross-sections for each pair of species considered in a given mixture [@Hirschfelder1954; @Ferziger1972], weighted depending on the Laguerre-Sonine polynomial order used [@Chapman1998]. The preferred method to compute collision integrals is to numerically integrate from accurate *ab initio* potential energy surfaces. Such data is available for several important collision systems [@Wright2005; @Wright2007; @Bruno2010]. When potential energy surfaces are not available, collision integrals are integrated from model interaction potentials. The evaluation of these model integrals can be partitioned in several ways — neutral-neutral, ion-neutral, electron-neutral, and charged interactions, heavy, electron-heavy, and electron interactions — each with different functional dependencies on temperature and degree of ionization. [Mutation$^{++}$]{} introduces a custom XML format for storing collision integral data ([Fig. \[f:collision-file\]]{}) which
1. is self-documenting,
2. is easily extensible in data and model type,
3. provides customizable default behavior for missing data, and
4. enforces consistency of standard ratios.
Just-in-time loading and efficient evaluation of this data is handled by a [`CollisionDB`]{} object, as shown in [Fig. \[f:uml\]]{}.
The solution of the linear transport systems represents a significant CPU time for some CFD applications. Several algorithms have been proposed in the literature for reducing this cost [@Giovangigli1991; @Ern1995; @Ern1997; @Magin2004a; @Giovangigli2010]. [Mutation$^{++}$]{} provides plug-and-play transport algorithms through the use of self-registering algorithm classes. For example, the abstract class [`ThermalConductivityAlgorithm`]{}, shown in [Fig. \[f:uml\]]{}, provides the necessary interface that all thermal conductivity algorithms must include, namely functions for computing the thermal conductivity and thermal diffusion ratios. Specific algorithms are then implemented by creating a concrete class which implements the interface. This pattern has been used for the calculation of the multicomponent diffusion matrix and shear viscosity as well.
### Kinetics
The goal of the chemical kinetics module is the efficient and robust computation of species production rates due to finite-rate chemical reactions. For reaction set ${\ensuremath{\mathcal{R}}}$ involving species in ${\ensuremath{\mathcal{S}}}$, we consider production rates of the form $$\frac{\dot{\omega}_k}{M_k}= \sum_{r\in{\ensuremath{\mathcal{R}}}} \Delta \nu_{kr} \bigg[k_{fr}\prod_{j\in{\ensuremath{\mathcal{S}}}}\tilde{\rho}_j^{\nu^{'}_{jr}}-k_{br}\prod_{j\in{\ensuremath{\mathcal{S}}}}\tilde{\rho}_j^{\nu^{''}_{jr}}\bigg] \Theta_r,
\label{e:species-prodrate}$$ where a full description of each term is given in [@ScogginsThesis-Chap2]. The forward reaction rate is assumed to be a function of a single, reaction-dependent temperature $k_{fr} = k_{fr}(T_{fr})$ and the backward rate is determined from equilibrium as $\smash{k_{br}(T_{br}) = k_{fr}(T_{br}) / K_{\text{eq},r}(T_{br})}$ where $T_{br}$ is a reaction-dependent temperature for the backward rate.
Apart from the reaction rate temperatures, knowledge of reaction types is essential in some energy exchange mechanisms. Manually inputting the type of every reaction in a mechanism of hundreds or thousands of reactions can be a tedious and error-prone process. Therefore, [Mutation$^{++}$]{} provides a unique feature which determines the type of reaction automatically when a mechanism is loaded. The problem is formulated as classification tree [@Loh2011], which can be constructed automatically using simple characteristics of each reaction. An example of such a classification tree is provided in [Fig. \[f:reaction-classification\]]{}.
[m[9.5cm]{}m[3.5cm]{}]{}  & \
In principle, the evaluation of [Eq. \[e:species-prodrate\]]{} is straight-forward, though great care is required to do it robustly and efficiently. A simplified class diagram of the kinetics module is presented in [Fig. \[f:uml\]]{}. The module contains a list of [`Reaction`]{} objects provided by the user through an XML reaction mechanism file (see [Fig. \[f:mechanism-file\]]{}). The rest of the module is comprised of a set of computational “managers”, which are responsible for the efficient evaluation of individual parts of [Eq. \[e:species-prodrate\]]{}. These include the evaluation of reaction rates, operations associated with the reaction stoichiometry (the sum and products in [Eq. \[e:species-prodrate\]]{}), and the evaluation of the third-body term, $\Theta_r$. An additional manager class is responsible for evaluating the Jacobian of species production rates, necessary for implicit time-stepping CFD algorithms. Finally, the [`Kinetics`]{} class orchestrates the use of each of these managers to evaluate [Eq. \[e:species-prodrate\]]{} and its Jacobian with respect to species densities and temperatures.
### Gas-Surface Interactions
The Gas-Surface Interaction (GSI) module provides surface boundary conditions for [Eq. \[e:general-conservation\]]{}. They are obtained by applying the conservation of mass, momentum, and energy in a thin control-volume on a surface at steady-state in the form $$\left(\bm{F}_{\text{g}} - \bm{F}_{\text{b}} \right) \cdot \bm{n} = \bm{S}_{\text{s}},
\label{e:gsi-balances}$$ where $\bm{F}_{\text{g}}$ and $\bm{F}_{\text{b}}$ are gas and bulk phase fluxes, $\bm{n}$ is the surface normal, and $\bm{S}_{\text{s}}$ is the source term associated with surface processes.
[Mutation$^{++}$]{} provides several built-in terms that can be mixed to create a model through a custom XML format, as shown in [Fig. \[f:gsi-file\]]{}. Once a model is specified, balance equations are created dynamically through an object-oriented approach forming a [`Surface`]{} object ([Fig. \[f:uml\]]{}). The resulting nonlinear equations are functions of the surface thermodynamic state (stored as $\hat{\bm{U}} = \begin{bmatrix} \tilde{\rho}_i & \rho e & \rho \tilde{e}^m \end{bmatrix}^T$ in [`SurfaceState`]{}), thermochemical properties of the interface, and the two connecting phases (e.g. [`SurfaceProperties`]{} and [`SolidProperties`]{}). This framework provides flexibility for the description of a variety of surfaces (e.g. chemically active, impermeable, porous with fixed outgassing, etc.).
For each type of surface, [Mutation$^{++}$]{} provides the fluxes and source terms expressed in [Eq. \[e:gsi-balances\]]{} to the client code. A very unique feature of the library is that it, on demand, solves the steady-state balances in a robust and efficient way, to obtain the boundary condition necessary for the CFD or material solvers. More information about the GSI models available can be found in [@BellasChatzigeorgis2018].
Illustrative examples {#s:examples}
=====================
Thermodynamic and Transport Properties
--------------------------------------
[Fig. \[f:air-properties\]]{} presents a selection of thermodynamic and transport properties computed by [Mutation$^{++}$]{} for an 11-species, isobaric air mixture in thermochemical equilibrium. The equilibrium mole fractions and thermodynamic properties are provided using both the NASA-9 and RRHO thermodynamic databases. Where possible, comparisons with equilibrium air curve-fits of D’Angola [et al.]{} [@DAngola2008] and thermal conductivity data from Murphy [@Murphy1995] and Azinovsky [et al.]{} [@Asinovsky1971] are also shown. Both “frozen” and “equilibrium” curves are shown for the specific heat at constant pressure and specific heat ratio. The frozen curves neglect the dependence of the species composition on the temperature through the equilibrium reactions, while equilibrium curves do not. This distinction is important, as it shows that these properties can vary substantially, depending on the thermochemical model employed by the user. For more information regarding the models, data, algorithms, and interpretation of these figures, please see the discussion in [@ScogginsThesis-Chap4].
{width="95.00000%"}
Equilibrium Ablation Rates
--------------------------
An important problem in the prediction of material response for thermal protection systems (TPS) of atmospheric entry vehicles is the solution of so-called “B-prime” tables. B-prime tables describe the equilibrium gas composition at the surface of an ablating TPS as well as its mass loss rate due to reactions at the surface, such as oxidation or nitridation. Assuming a thin control volume over an ablating TPS with equal diffusion coefficients for each species, conservation of elements inside the control volume yields $$y_{w} = \frac{B'_c\;y_{c} + B'_g\;y_{g} + y_{e}}{B'_c + B'_g + 1},
\label{e:bprime}$$ where $y$ is the elemental mass fraction for any element in the mixture, the subscripts $w$, $c$, $g$, and $e$ refer to wall, char, pyrolysis gas, and boundary layer edge properties respectively, $B' \equiv \dot{m}/(\rho_e u_e C_M)$ is a mass blowing rate, nondimensionalized by the boundary layer edge mass flux, and $C_M$ is the local Stanton number for mass transfer. When coupled with the minimization of Gibbs energy at a known surface condition, [Eq. \[e:bprime\]]{} can be solved to obtain species composition and char mass blowing rate $B'_c$. [Fig. \[f:bprime\]]{} shows such a calculation performed using [Mutation$^{++}$]{} with the custom thermodynamic database discussed in [Sec. \[s:thermo\]]{}. The $B'_c$ results are compared with those obtained with the thermodynamic database from the NASA Chemical Equilibrium with Applications (CEA) code. More discussion on the differences in these databases can be found in [@Scoggins2017].
![Equilibrium char mass blowing rate and wall mole fractions for a carbon-phenolic ablator in air computed with [Mutation$^{++}$]{} [@Scoggins2017].[]{data-label="f:bprime"}](figures/bprime.pdf){width="48.00000%"}
Impact {#s:impact}
======
The main goal of [Mutation$^{++}$]{} is to promote open collaboration between different research groups and communities working in the broad areas of hypersonics, combustion, and plasma physics, by lowering the cost of developing new physicochemical models, data, and algorithms, for modeling gas and gas-surface phenomena. With an efficient and extensible framework, [Mutation$^{++}$]{} can be easily coupled with existing CFD tools, allowing researchers to test effectively new thermodynamic, transport, or chemical models, physicochemical data, or numerical algorithms. In addition, users of the library can benefit from the work of others through collaborative testing, bug fixing, and maintenance which is supported by a continuous development and integration strategy.
Since its creation, [Mutation$^{++}$]{} has been used in many diverse applications, branching out from the original motivation of hypersonic flows for atmospheric entry [@delValBenitez2015; @Bellas-Chatzigeorgis2015; @Bellemans2015; @Fossati2019]. These include the study of biomass pyrolysis [@Lachaud2017], solar physics, magnetized transport [@Scoggins2016], and meteor phenomena [@Dias2015; @Dias2016]. Application of the library to fields other than originally intended serves to highlight the extensibility of the framework and the impact it can have on basic research. The library has also been used in limited commercial settings. Recently, [Mutation$^{++}$]{} has been coupled to the material response codes Amaryllis of LMS Samcef (Siemens) and the Porous material Analysis Toolbox (PATO), developed jointly between C la Vie and NASA Ames Research Center.
Conclusions
===========
The [Mutation$^{++}$]{} library provides an OOP framework for computing thermodynamic, transport, and kinetic properties of non to fully ionized gas mixtures at all points on thermochemical nonequilibrium spectrum. The library leverages the simple dependence of thermochemical properties on the local thermodynamic state of the gas to implement a weak coupling between the computation of those properties and the simulation tools which need them, through a clean and consistent API.
The code is freely available on GitHub with an LGPL v3.0 license, following a continuous integration development strategy with periodic versioning to alleviate backward compatibility concerns for its users. This paper marks version v1.0.0 for the library. Future versions will aim to provide additional features and greater flexibility for the end-user.
Acknowledgements {#acknowledgements .unnumbered}
================
The development of [Mutation$^{++}$]{} v1.0.0 was partially supported under the European Research Council Starting Grant \#259354 and Proof of Concept Grant \#713726. The authors would like to thank Drs. Nagi N. Mansour, Jean Lachaud, and Alessandro Turchi for invaluable discussions and guidance during development of the library. Other contributions, including development and testing of the library, are listed on the project’s GitHub site.
Required Metadata {#required-metadata .unnumbered}
=================
Current code version {#current-code-version .unnumbered}
====================
**Nr.** **Code metadata description** **Please fill in this column**
--------- ----------------------------------------------------------------- -----------------------------------------------------
C1 Current code version v1.0.0
C2 Permanent link to code/repository used for this code version https://github.com/mutationpp/Mutationpp
C3 Legal Code License LGPL-3.0
C4 Code versioning system used git
C5 Software code languages, tools, and services used C++, Python, Fortran, CMake, Eigen, Catch2, Doxygen
C6 Compilation requirements, operating environments & dependencies Linux, Mac OS X
C7 If available Link to developer documentation/manual See Github project for documentation.
C8 Support email for questions scoggins@vki.ac.be
: Code metadata (mandatory)
[^1]: <https://github.com/mutationpp/Mutationpp>
|
---
author:
- Claire Levaillant
date: '*Work with the Microsoft Research Station Q team*'
title: 'Topological quantum computation within the anyonic system the Kauffman-Jones version of $SU(2)$ Chern-Simons theory at level $4$'
---
Introduction {#introduction .unnumbered}
============
We present protocols allowing to perform any unitary operation on $n$-qubits on a topological quantum computer within the anyonic system the Kauffman-Jones version of $SU(2)_4$. The minor modifications resulting from studying $SU(2)_4$ instead of the Kauffman-Jones version of $SU(2)_4$ get explained in [@Qet]. The quantum gates are made by braiding and measuring quasiparticles called anyons.\
The first two chapters deal with $1$-qudit gates for $d=2,3$, and the last chapter deals with $2$-qubit gates. In $2001$, Ranee and Jean-Luc Brylinski showed as part of their work that if we can approximate any $1$-qubit gate and if we have only one $2$-qubit entangling gate, then we can approximate any $n$-qubit gate.\
The goal of our paper is to tell how to braid and to measure the particles in order to make a few gates (a few $1$-qubit gates and at least one $2$-qubit entangling gate) from which one can approximate any other gate.\
By braiding only, we obtain only finitely many $1$-qubit gates. But, by adding measurement based operations, we are able to make an irrational phase gate, which makes the group of gates issued from braiding become infinite and dense in $SU(2)$ (overall phases do not affect quantum computation). The gate is made from an ancilla which we prepare so that it has an irrational relative phase. Making ancillas is simpler than making gates and is quite comfortable because any measurement operation is allowed. Indeed, in case of a bad measurement outcome, we can simply dispose of the ancilla. Also, when making quantum gates, a big challenge of measurement based operations is that the effect of the measurement on the system should be independent from the input. This is a difficult condition to have. In order to bypass this difficulty, we prepare an ancilla by braiding and measurement and then fuse the ancilla into the input in order to form the gate. The difficulty has however not completely vanished. In the process we must do measurements. And so we must have adequate ancillas so that no quantum information gets lost, nor distorted during the fusion of the ancilla with the input. Making such ancillas is a priori a challenge but is still a more reasonable challenge than making a gate directly, due to the large number of possibilities offered in terms of braiding, measurements, number of particles used.\
The third and last chapter deals with realizing all the $2$-qubit permutation gates, some of which are entangling gates (like the CNOT gate). The chapter also relies on other published protocols which are not part of this paper.
From ancilla to quantum gate
============================
Note
This chapter grew out of many inspiring discussions with Michael Freedman and is the first stone in a series of two chapters leading to the production of irrational phase gates in the Kauffman-Jones version of $SU(2)$ Chern-Simons theory at level $4$.
Abstract
We provide a way to turn an ancilla into a gate for specific ancillas in the Kauffman-Jones version of $SU(2)$ Chern-Simons theory at level $4$. We deal with both the qubit $1221$ and the qutrit $2222$. Together with ancilla preparations, our protocols are later used to make irrational phase gates, leading to universal single qubit and qutrit gates.
Motivation of the work
----------------------
The framework of this work is the Kauffman-Jones version of $SU(2)$ Chern-Simons theory at level $4$. In this theory, there are five particle types, called anyons, with respective topological charges $0$, $1$, $2$, $3$ and $4$. The anyons obey to fusion rules that are governed by $$\left\lbrace\begin{array}{l}
a\leq b+c,\,b\leq c+a,\,c\leq a+b\\
a+b+c\leq 8\\
a+b+c\,\;\text{is even}\end{array}\right.,$$ where $a$, $b$ and $c$ denote the topological charges of the particles. The operations we do consist of braiding the anyons, fusing them and measuring them. Some measurements are fusion measurements and other measurements are done by interferometry, see [@BO], [@BO2], [@BO3]. For basic facts about recoupling theory, we refer the reader to [@KL], except the theory which we use is unitary. In particular we deal with unitary theta symbols and unitary $6j$-symbols (see [@ZW] and Appendix of [@CL]). The value of the Kauffman constant is, using the same notations as in [@KL], $$A=i\,e^{-i\frac{\pi}{12}}$$ The main four moves which we use throughout the paper are summarized below.
- The “$F$-move”
The brackets are called unitary $6j$-symbols.
- The “$R$-move”
- The “theta move”
The delta’s are the quantum dimensions and the $\theta^u$ are the unitary theta symbols.
- The “bubble removal”
The coefficient is obtained by doing an F-move, followed by a Theta move. Note that $i$, $j$ and $k$ must obey to the fusion rules, else the evaluation is zero.
We study here two anyonic systems: the quantum bit formed by four anyons of topological charges $1221$ and the quantum trit formed by four anyons of topological charges $2222$.
We show in each case how, starting with adequate ancillas, we are able to turn them into quantum gates. We got inspired by [@BK]. The main idea is to fuse the ancilla into the input and avoid leakage of quantum information during the process by imposing some conditions on the ancillas we use. Inventing protocols that produce interesting ancillas is much easier than inventing protocols that directly produce interesting quantum gates. Indeed, when doing measurements during the ancilla production, if we don’t measure the precise outcome we wish to measure, then we can just dispose of the ancilla. This allows to not worry about recoveries for bad measurement outcomes. And these recoveries can be challenging. By the fact that the range of allowed possibilities is considerably enhanced since any measurement is permitted, it is possible to make ancillas with interesting amplitude ratios and interesting relative phases.\
\
**In what follows, the ancillas and the gates are all defined up to some overall complex scalars.**\
\
The method which we expose in this paper is the following.
- For the qubit $1221$, given an ancilla $$a\,|1> + b\,|3>$$ where $a$ and $b$ are two complex numbers of equal norms, how to make the gate $$\begin{pmatrix}
a&0\\0&b
\end{pmatrix}$$
- For the qutrit $2222$, given an ancilla $$|0>+\sqrt{2}\,e^{i\,{\alpha}}\,|2>+|4>$$ where ${\alpha}$ is any phase, how to make the gate $$\begin{pmatrix}
1&0&0\\
0&e^{i\,{\alpha}}&0\\
0&0&1\end{pmatrix}$$
The protocols above are later used in Chapter $2$ to make infinite order phase gates for the qubit $1221$ and the qutrit $2222$, after we took care of producing adequate ancillas. Thus leading to universal single qubit and qutrit gates by only three operations consisting of braiding, measuring and fusing particles.
The qubit $1221$
----------------
The anyons will be numbered from left to right. We bring the input and the ancilla close to each other and do an F-move like on the picture below.
We measure anyons $4$ and $5$. If the measurement outcome is $0$, then we have fused anyons $4$ and $5$ and we have a pair of $1$’s which we dispose of.\
If we rather measure $2$, then we do an F-move on the edge labeled $2$ of the measurement (no physical action here, only for the clarity of the presentation) and we measure the last four anyons by interferometry like on the figure below.
If we measure $0$, then we start all over again. If we rather measure $2$, then we braid anyons $4$ and $5$ and we remeasure, a process we do until we measure a $0$ and are back to the initial configuration of input and ancilla. Alternatively, as suggested by Parsa Bonderson, we measure anyons $4$ and $5$ and hope to measure $0$ which would achieve the desired fusion. If we measure $2$ instead, we are simply back to the initial wrong outcome and proceed just like before. In summary, we could see the algorithm as an alternation of measuring anyons $4$ and $5$ on one hand and measuring anyons $5$, $6$, $7$, $8$ on the other hand, until the expected fusion is achieved. Anyhow, after possibly several tries, we have now fused anyons $4$ and $5$. We will continue fusing the ancilla into the input one step further by now measuring anyons $3$ and $4$, like below.
The measurement outcome is either $0$, $2$ or $4$.
- If the measurement outcome is $0$, then we obtain the proposed gate.
- If the measurement outcome is $4$, we do two additional steps: we fuse anyons $3$ and $4$, and we then fuse anyons $3$ and $4$ again where it is understood that the anyons get renumbered from left to right after each action. For both fusions, the outcome is unique (4 for the first fusion, 2 for the second fusion) by the theta rule and by the fusion rules respectively. It yields the gate $$\begin{array}{ccc}\begin{pmatrix}
b&0\\
0&a
\end{pmatrix}&\text{instead of the gate}&\begin{pmatrix} a&0\\0&b\end{pmatrix}\end{array}$$ Since all our gates are defined up to some overall complex scalar, this is simply the inverse gate. Then, by doing a random walk, we are able to eventually produce the gate we want. No need of recovery here: only an iteration of the overall process.
- If the measurement outcome is $2$, suppressing the bubble with three external edges introduces some minus signs when the input edge and the ancilla edge carry a distinct label. However, these minus signs get exactly compensated after doing an F-move on the edge of the measurement. Then, by running an interferometric measurement on the last three anyons, we get back to the initial configuration after the first fusion of the ancilla into the input with the only difference that the middle horizontal edge could now carry the label $3$ instead of $1$, depending on the outcome of the interferometric measurement. However, whatever be the label, the recovery is complete and we start all over again by measuring anyons $3$ and $4$.
Below are some figures which illustrate the different measurement outcomes and their treatment in the same order in which they were discussed above.\
\
\
In the second series of unitary $6j$-symbols given above, if the upper right corner number is $3$ instead of $1$, get the same values with opposite signs.
The qutrit $2222$
-----------------
Before we state the theorem, we will outline how we fuse the “last” anyon from the input and the “first” anyon from the ancilla with a “forced” measurement to $0$. We measure the collective charge of both anyons. If the outcome is $0$, success. If the outcome is $2$, by forthcoming Eq. $(1.1)$ we see that after doing an F-move, we have the following configuration
Like on the figure above, run an interferometric measurement. If we measure $0$, we are back to the initial configuration and try again. If we measure a $4$, it will suffice to fuse a pair of $4$’s like on the figure in order to cancel the two $4$ charge lines. This is described as a Freedman fusion operation in [@CL], but it can also be interpreted as
after doing an F-move on the $0$ labeled edge between the two $4$ charge lines. Finally if the measurement outcome is $4$, then use the trick from before and fuse a pair of $4$’s like on the figure.
By a careful inspection at the unitary $6j$-symbols, $$\begin{array}{ccc}\left\lbrace\begin{array}{ccc}2&2&2\\2&2&4\end{array}\right\rbrace^u=\,-\,\frac{1}{\sqrt{2}}&\&&
\left\lbrace\begin{array}{ccc}2&2&0\\2&2&4\end{array}\right\rbrace^u=\left\lbrace\begin{array}{ccc}2&2&4\\2&2&4\end{array}\right\rbrace^u=\frac{1}{2}\end{array},$$ we have accomplished the fusion we wanted, except the $|2>$ trit from the ancilla has been switched to $-|2>$. It then suffices to fuse another pair of $4$’s like on the figure in order to switch the $-|2>$ trit of the ancilla back to $|2>$ by the following Lemma.
\
We now state the Theorem.
Given an ancilla $|0> + \sqrt{2}\,e^{i{\alpha}_2}|2>+|4>$, the following protocol
There are three cases.\
*Case $(i)$*. The outcome of the measurement is $0$. Then, we have formed the gate $$\begin{pmatrix}
1&0&0\\
0&e^{i\,{\alpha}}&0\\
0&0&1\end{pmatrix}$$
*Case $(ii)$*. The outcome of the measurement is $4$. Then, fuse anyons $3$ and $4$ necessarily to $4$. Then fuse anyons $3$ and $4$ necessarily to $2$. We have formed the same gate as in $(i)$.\
*Case $(iii)$*. The outcome of the measurement is $2$. Then, fuse anyons $3$ and $4$ necessarily to $2$. Then fuse anyons $3$ and $4$ necessarily to $2$. We have then formed the inverse gate $$\begin{pmatrix}
e^{i\,{\alpha}}&0&0\\
0&1&0\\
0&0&e^{i\,{\alpha}}\end{pmatrix}$$ To finish do a random walk in order to eventually produce the gate of $(i)$.
<span style="font-variant:small-caps;">Proof.</span> When the outcome is zero, the bubble removal introduces a factor $\frac{1}{\sqrt{2}}$ in front of the $|2>$ trit, which compensates exactly the $\sqrt{2}$ from the ancilla. We get the phase gate announced.\
We next deal with Case $(ii)$. On the figure below, all the unitary $6j$-symbols that are useful to the computations and got omitted take the value $1$. This case illustrates why it is important to have an ancilla that has identical phases in $|0>$ and $|4>$. We obtain the same gate as in $(i)$.
Finally, when the measurement outcome is $2$, the $(|2>,|2>)$ contribution vanishes as specific to this theory, we have $$\left\lbrace\begin{array}{ccc}2&2&2\\2&2&2\end{array}\right\rbrace^u=0,$$ We obtain the $2$-qutrit vector $$|2>\otimes(|0>-|4>)+e^{i{\alpha}}\,(|0>-|4>)\otimes |2>$$ Next we fuse anyons $3$ and $4$ and after that we fuse anyons $3$ and $4$ again (same protocol as when the measurement outcome was $4$).\
We claim that the outcome from the second fusion must be $2$ and not $0$ or $4$. This is partly because the $(|0>-|4>)\otimes |2>$ contribution of the superposition would vanish if the fusion outcome were $0$ or $4$ by the fusion rules and partly because the $|2>\otimes\, (|0>-|4>)$ contribution of the superposition would also vanish if the fusion outcome were $0$ or $4$ since all the $6j$-symbols that are involved are equal. Thus, there wouldn’t be any contribution left. And so, we will never measure $0$ or $4$ as illustrated on the figure below.
The straightforward computations lead to producing the gate $$\begin{pmatrix}
e^{i\,{\alpha}}&0&0\\
0&1&0\\
0&0&e^{i\,{\alpha}}\end{pmatrix}$$ which, up to overall phase $e^{i{\alpha}}$ is the inverse gate of the one from the first two cases, that is $$\begin{pmatrix}
1&0&0\\
0&e^{-i{\alpha}}&0\\
0&0&1\end{pmatrix}$$
In order to always produce the gate of $(i)$ or $(ii)$, it will suffice to iterate the process, that is do a random walk until we obtain the desired gate.
**Acknowledgements.** The author thanks Michael Freedman for quite helpful and inspiring discussions and Parsa Bonderson for many nice comments. She thanks Matt Hastings for enlightening discussions.
Irrational phase gates
======================
Note
The author thanks Michael Freedman for providing these thrilling problems, for guiding her with endless generosity in time and encouragements and for communicating his own excitement during the pursuit of this research.
Abstract
In Chapter $1$, it is shown with respect to two different anyonic systems, the same as those considered in this paper, how to fuse an ancilla into the input in order to form a gate. In each case, the ancilla must satisfy to some adequate amplitude ratios properties. In the present chapter, we show how to prepare such ancillas, leading to infinite order phase gates. We obtain universal $1$-qubit and $1$-qutrit gates. Together with [@CL2] and [@CL4] in which are respectively shown how to make a $2$-qubit and a $2$-qutrit entangling gate, we get qubit and qutrit gate sets that are universal for quantum computation.
Introduction
------------
The framework of this work is the Kauffman-Jones version of $SU(2)$ Chern-Simons theory at level $4$. By braiding only of $4$ anyons of topological charges $1221$ or $2222$, we obtain gates that generate a finite subgroup of $SU(2)$ or $SU(3)$ respectively. By adding some fusion operations, it is possible to enlarge the size of the respective groups but it is still not possible to make them infinite and dense. The case of the qutrit was extensively studied in [@BL] and [@CL]. Measurements assisted gates appear to become a necessity for universality. In [@F], a general protocol for $1$-qubit and $1$-qutrit “gate hunting” is provided which uses braids and interferometric measurement. It is composed of a few main steps which are summarized on the figure below. The idea is to fuse the right most anyon from the input and the left most anyon from the ancilla by doing a measurement with recovery which leaves a pair of anyons in position $4-5$, then do braids involving all the other anyons, then do an interferometric measurement forced to $0$ on anyons $5,6,7,8$ to separate the input and the ancilla. The latter measurement having an easy recovery if the outcome is not $0$. There are three major difficulties in this protocol. First, we must check for no-leakage of quantum information when we operate the interferometric measurement. This leads to conditions on the ancilla we use. These conditions depend heavily on the braids we perform before running the interferometric measurement. Next, we must be able to prepare an ancilla which pleases these conditions. This is not always easy. For the qubit $1221$, some ancillas not issued from braids are provided in [@QIP] which are useful in this general protocol. Third, after the interferometric measurement, the input and the ancilla will be topologically separated but could be entangled. It can be very hard to find a disentangler which uses braids and perhaps also measurements, though it was shown possible in some cases. For instance the $2$-qutrit entangler of [@CL2] is used in [@F] as a disentangler. We may have to run more interferometric measurements and have to check for no-leakage again. Last, after overcoming all these difficulties, we must hope for an interesting gate, that is a gate which is not already in the image of the braid group.
On the figure above, the letter $I$ stands for “input”, the letter $A$ for “ancilla”, the letter $b$ for “braid”, the letter $d$ for “disentangler”.\
This theoretically interesting protocol did not so far yield any interesting results. In the present paper, we show how to produce infinite order phase gates for both the qubit and the qutrit by using a different protocol.\
The paper is divided into two parts, one that deals with the qubit $1221$ and one that deals with the qutrit $2222$. The scheme is the same for the $1$-qubit gate or the $1$-qutrit gate. First, we do an ancilla preparation. Second we fuse the ancilla into the input following Chapter $1$ in order to form the gate. In Chapter $1$ it is shown that an ancilla $$x|1>+y|3>\;\text{with $|x|=|y|$}$$ for the qubit $1221$ can be turned into a gate $$\begin{pmatrix}x&0\\0&y\end{pmatrix}$$ And an ancilla $$|0>+\,\sqrt{2}\,e^{i{\alpha}}|2>+|4>$$ for the qutrit $2222$ can be turned into a gate $$\begin{pmatrix}1&&\\&e^{i{\alpha}}&\\&&1\end{pmatrix}$$ The present paper focuses on ancilla preparations.\
The ancilla preparation for the qubit can be divided into two parts. First, we must find a way to create an interesting relative phase, say $\theta$. Second, we must have equal complex norms in $|1>$ and $|3>$ in order to avoid leakage during the fusion of the ancilla into the input. An idea to get an interesting phase is simply to add two complex numbers. An idea to get equal norms is to notice that for two given complex numbers $a$ and $b$, we have $$|a\bar{b}|=|b\bar{a}|$$ Thus, suppose we can prepare an ancilla $\left(\begin{array}{l}a\\b\end{array}\right)$ with interesting relative phase and suppose we are able to prepare the conjugate ancilla $\left(\begin{array}{l}\bar{a}\\\bar{b}\end{array}\right)$. By the following Freedman operation
we also know how to prepare the ancilla $\left(\begin{array}{l}\bar{b}\\\bar{a}\end{array}\right)$. Then, we can fuse the two ancillas together in order to obtain a third ancilla which is $$\left(\begin{array}{l}a\,\bar{b}\\b\,\bar{a}\end{array}\right)$$ As a matter of fact, the complex norms in $|1>$ and $|3>$ are equal and the relative phase is now $2\theta$ instead of $\theta$, which is still an “interesting” relative phase. The goal is then reached.\
For the qutrit $2222$, it is slightly more difficult to prepare an adequate ancilla. While the idea to get an interesting phase between the $|0>$ trit and the $|2>$ trit or between the $|2>$ trit and the $|4>$ trit is rather similar as for the qubit - that is use any braids, any measurements and any anyons to make such a qutrit - there are two main difficulties once this is achieved.
First, we must have equal phases in $|0>$ and $|4>$, else the “ancilla turning into gate” protocol described in lengthy details in Chapter $1$ will not work.
Second, we must have the correct norm ratios like provided in the discussion above. Again, the strategy we used for the qubit won’t apply here and we must come up with new ways.
In order to solve the problem of “identical phases” in $|0>$ and $|4>$, it will suffice to use a process of “pair fusion” which allows by fusing a pair of $2$’s into anyons $2$ and $3$ of the qutrit to transfer and clone the contribution from the $|2>$ trit onto the $|0>$ and $|4>$ trits (then get identical phases, like aimed at), while the two contributions in $|0>$ and $|4>$ get added to yield the contribution in $|2>$.
The most tricky part remains to have the correct norm ratios. In order to achieve this, we use a subtle combination of three protocols, namely the pair fusion that was just mentioned, together with what we called the “qutrit fusion” and the “qutrit projection”. The qutrit projection allows, given a qutrit ancilla, to eliminate exactly one of the two contributions arising from the $|0>$ trit or the $|4>$ trit. What is important is that we are left with two trits which carry the interesting relative phase. If furthermore, the ratio of the norms in $|0>$ (or $|4>$) and $|2>$ equals $\sqrt{2}$, then by applying a pair fusion, we will get the three desired things: adequate norms ratios, the same interesting relative phase between the $|0>$ and $|2>$ trits and identical phases for the $|0>$ and $|4>$ trits. Note where the importance of the trit elimination precisely is. If you don’t do it, you can still get the same phases in $|0>$ and $|4>$ by pair fusion and perhaps still have an interesting relative phase, but you will loose the adequate norms ratios because you will have added two distinct complex numbers during the pair fusion process, which can create a new uncontrolled amplitude. Let us conclude by saying a word about the qutrit fusion. Suppose we are able to make an ancilla $a|0>+b|2>$ with an interesting relative phase and its conjugate ancilla. Given these two ancillas, the qutrit fusion allows by fusing them to get a third ancilla like below. $$\begin{array}{cc}\begin{array}{l}|0>\\|2>\\|4>\end{array}&\begin{pmatrix}a\\b\\0\end{pmatrix}\end{array}, \;
\begin{array}{cc}\begin{array}{l}|0>\\|2>\\|4>\end{array}&\begin{pmatrix}\bar{a}\\\bar{b}\\0\end{pmatrix}\end{array}
\longrightarrow\begin{array}{cc}\begin{array}{l}|0>\\|2>\\|4>\end{array}&\begin{pmatrix}a\bar{b}\\\frac{1}{\sqrt{2}}b\bar{a}\\0\end{pmatrix}\end{array}$$ The latter resulting ancilla has the appropriate norms ratio in order to successfully benefit from the pair fusion operation and twice the interesting relative phase we started from. It is an ancilla which is ready to be turned into a good final ancilla by pair fusion like in our discussion above. While, as we just saw, the qutrit pair fusion’s beneficial outcome relies on the nice shape of the ancilla issued from the qutrit fusion protocol, it is very important to note that the qutrit fusion’s beneficial outcome is itself made possible by the prior qutrit projection. This fact will be enlightened later on when we get into the details of the qutrit fusion. Thus, in this protocol, all the different parts fit together and intertwine with each other beautifully.\
\
We illustrate the latter comment by outlining the different steps. Given an ancilla with interesting relative phase in $|0>$ (or $|4>$) and $|2>$ and its conjugate ancilla,
- The **qutrit projections** allow a successful qutrit fusion of both ancillas.\
- The **qutrit fusion** allows adequate norms ratios for a successful qutrit pair fusion.\
- The **qutrit pair fusion** allows identical phases in $|0>$ and $|4>$ and correct norms ratios in preparation for the ancilla fusion into the input of Chapter $1$.
In light of the outlines provided above, we are ready to now describe the ancilla preparations themselves. First for the qubit $1221$, then for the qutrit $2222$.\
The protocols which produce an ancilla ready to be turned into an irrational phase gate appear on pages $31$ and $39$ for the qubit and qutrit respectively.
Irrational qubit phase gate
---------------------------
### First ancilla $A_1$
The starting point is four anyons of topological charge $1$ forming the qubit $|2>$, together with a pair of $2$’s out of the vacuum. We do two braids, followed by two fusion measurements and one braid. The protocol is represented on the figure below.
Below, we will explain the ideas that are hidden behind the three steps of the figure.\
\
*Step $1$*. The left braid creates a superposition of $|0>$ and $|2>$ for the qubit $1111$. The second braid creates a superposition of $|1>$ and $|3>$ when braiding $(1122)_0$. The latter charge line will in the end carry the quantum information for the ancilla.
*Step $2$*. This fusion and the subsequent one are really at the origin of the interesting relative phase. The idea is based on the fact that whether the measurement outcome of the fusion is $1$ or $3$, for exactly one of the qubits $|1>$ or $|3>$ from the right horizontal charge line, both contributions of $|0>$ and $|2>$ remain on the left horizontal charge line. This is a consequence of the fusion rules. Two anyons of respective topological charges $3$ and $0$ cannot fuse into an anyon of topological charge $1$ and two anyons of respective topological charges $1$ and $0$ cannot fuse into an anyon of topological charge $3$. From now on, suppose the fusion measurement outcome is $1$, like illustrated below.
*Step $3$*. When the second fusion occurs (and we force the outcome from this fusion to be $2$), for the $1$ horizontal charge line, the bubble suppression results in summing two contributions: one from the $0$ horizontal charge line and one from the $2$ horizontal charge line. This sum yields a quite interesting phase. Whereas for the $3$ horizontal charge line, there is only one term, namely the one from the $2$ horizontal charge line. Now we get a quite interesting ***relative*** phase. It remains to braid the last two anyons in order to get to the qubit shape $1221$.
We now go over the computations themselves. First, recall the matrix for a $\sigma_2$-braid on four anyons of topological charge $1$.\
\
$$G_2[1,1,1,1]=\begin{pmatrix} \frac{e^{\frac{i\pi}{4}}}{\sqrt{3}}&&\sqrt{\frac{2}{3}}\,e^{-i\frac{5\pi}{12}}\\&&\\
\sqrt{\frac{2}{3}}\,e^{-i\frac{5\pi}{12}}&&\frac{e^{-i\frac{\pi}{12}}}{\sqrt{3}}\end{pmatrix}$$
We also recall from [@QIP] the matrix for the action by a $\sigma_2$-braid on four anyons $1122$.
$$\begin{array}{l}\qquad\qquad\qquad\qquad\qquad\qquad |0>\qquad\qquad |2>\\\\G_2[1,1,2,2]=\begin{array}{cc}\begin{array}{l}|1>\\\\|3>\end{array}&\begin{pmatrix} \frac{1}{\sqrt{2}}\,e^{i\frac{2\pi}{3}}&&\frac{1}{\sqrt{2}}\\&&\\\frac{1}{\sqrt{2}}\,e^{-i\frac{5\pi}{6}}&&-\frac{i}
{\sqrt{2}}
\end{pmatrix}\end{array}\end{array}$$ $$\begin{array}{l}\\\end{array}$$ So, after the two braids of Step $1$, we read from the first column of $G_2[1,1,2,2]$ and the second column of $G_2[1,1,1,1]$ that we have the superposition\
$\begin{array}{l}\\\end{array}$$
$$\begin{array}{cc}\begin{array}{l}|01>\\\\|03>\\\\|21>\\\\|23>\end{array}&\begin{pmatrix}\frac{1}{\sqrt{3}}
\,e^{i\frac{\pi}{4}}\\
\\\star\\\\\frac{1}{\sqrt{6}}\,e^{i\frac{7\pi}{12}}\\\\\frac{1}{\sqrt{6}}
\,e^{-i\frac{11\pi}{12}}
\end{pmatrix}\end{array}
$ $\begin{array}{l}\\\epsfig{file=two_qubit.jpg, height=4cm}\end{array}$ $$\begin{array}{l}\\\end{array}$$ Note, we only provided the useful contributions. Indeed, a triangle inside a net can be removed at the cost of a multiplication by a scalar, like on the figure below.\
\
When we fuse anyons $3$ and $4$ and measure a $1$, we see with the fusion rules that only the three $2$-qubits $|01>$, $|21>$, $|23>$ contribute to the measurement. After the triangle collapse, we get a superposition\
$$\begin{array}{cc}\begin{array}{l}|01>\\\\|21>\\\\|23>\end{array}&\begin{pmatrix}
2\sqrt{2}\,e^{i\frac{\pi}{4}}\\\\e^{-i\frac{5\pi}{12}}\\\\\sqrt{3}\,e^{-i\frac{11\pi}{12}}
\end{pmatrix}\end{array}$$ $\begin{array}{l}\\\end{array}$\
After the second fusion, we get an ancilla qubit $1212$. The contributions are as follows, where the brackets denote the unitary $6j$-symbols.
- For the $|1>$ qubit: $$\left\lbrace\begin{array}{ccc} 1&1&2\\1&1&0\end{array}\right\rbrace^{u}\;2\;\sqrt{2}\;e^{\frac{i\pi}{4}}\;+\;
\left\lbrace\begin{array}{ccc} 1&1&2\\1&1&2\end{array}\right\rbrace^{u}\,e^{-\frac{i5\pi}{12}}=\sqrt{7}\,ArcTan\left(\frac{14+3\sqrt{3}}{13}\right)$$
- For the $|3>$ qubit: $$\left\lbrace\begin{array}{ccc} 1&1&2\\1&3&2\end{array}\right\rbrace^{u}\,\sqrt{3}e^{-\frac{i11\pi}{12}}=\sqrt{3}e^{-\frac{i11\pi}{12}}$$
Then, after R-move, we finally get an ancilla qubit $1221$, which we name $A_1$. We have $$A_1=\frac{1}{\sqrt{10}}(\sqrt{7}e^{i\,arctan(\frac{-14-5\sqrt{3}}{11})}|1>+\sqrt{3}\,e^{-i\frac{\pi}{12}}|3>)$$ Note, if we apply a $\sigma_2$-full twist on this ancilla, we get another ancilla: $$A^{'}_1=\frac{1}{\sqrt{10}}(|1> + 3\,|3>)$$ More generally, by doing more braids, we can make many interesting ancillas for the qubit $1221$ which we would not be able to make by braiding only.
If the first fusion outcome is $3$ instead of $1$, it can be read from the diagrams and from the values of the unitary $6j$-symbols that we get an ancilla where the contributions in $|1>$ and $|3>$ are swapped. Hence, it does not matter whether we measure a $1$ or a $3$ after the first fusion. Note also that one ancilla can be obtained from the other one by a Freedman type fusion operation mentioned in the introduction. After the second fusion however, we want to measure a $2$. If we were to measure a $0$ instead, we would start the ancilla preparation all over again.
### Conjugate ancilla $\overline{A_1}$
This section enlightens quite particularly the necessity of the protocol that produced the first ancilla. Indeed, a quite specific and interesting fact about the matrix $G_2[1,1,1,1]$ is that its two anti-diagonal coefficients are identical. So that the conjugate transpose of this matrix equals its conjugate, that is the inverse matrix of $G_2[1,1,1,1]$ is the conjugate matrix of $G_2[1,1,1,1]$. Thus, if $$\left(\begin{array}{l}x_0\\x_2\end{array}\right)$$ denotes the second column of the matrix $G_2[1,1,1,1]$, the second column of $G_2[1,1,1,1]^{-1}$ is simply $$\left(\begin{array}{l}\overline{x_0}\\\overline{x_2}\end{array}\right)$$ After a quick glance at the first column of the matrix $G_2[1,1,2,2]$ and at the R-matrix $$R[1,2]=\begin{pmatrix} -e^{i\frac{\pi}{3}}&0\\0&-e^{-i\frac{\pi}{6}}\end{pmatrix},$$ we see that the two right most braids of the protocol of $\S\,2.1$ cancel exactly. This could also be seen as a Reidemeister’s move. Since removing bubbles from the fusion measurements only introduces real numbers in the computations, it appears clearly that if we do the exact same protocol as before with the exception of an inverse $\sigma_2$-braid on $(1111)_2$ instead of a $\sigma_2$-braid, we will produce the conjugate ancilla $\overline{A_1}$. For clarity, we provide below the protocol which produces $\overline{A_1}$ and which simply differs from the previous protocol by one braid being replaced by its inverse braid.
In the next section, we explain how by fusing $A_1$ and $\overline{A_1}$ together, we succeed to make an ancilla which has equal norms of $|1>$ and $|3>$ and where the interesting relative phase from $A_1$ is doubled.
### Final ancilla $A_f$
The general protocol described here applies to the qubit $1221$. It takes two ancillas $$\begin{array}{ccc}\left(\begin{array}{l}a\\b\end{array}\right)&\text{and}&\left(\begin{array}{l}x\\y\end{array}\right)
\end{array}$$ and returns a third ancilla which is $$\begin{array}{cccc}\text{either}&\left(\begin{array}{l}ax\\by\end{array}\right)&\text{or}&\left(\begin{array}{l}ay\\bx\end{array}\right)
\end{array}$$
We take two such ancillas. Following ideas from [@F]Chapter $1$, we measure anyons $4$ and $5$. If the outcome is $0$, we have fused the two anyons. If the outcome is $2$, we run an interferometry on anyons $5,6,7,8$ with a desired outcome of $0$. Then we measure anyons $4$ and $5$ again and iterate the process. If the interferometric measurement outcome is $2$, we measure again anyons $4$ and $5$ and hope to measure $0$. If we measure $2$ , we run an interferometric measurement on anyons $5,6,7,8$ again and proceed like before depending on the outcome. We remove the pair of $1$’s.\
, since we are dealing with ancilla preparations, we fuse anyons $4$ and $5$ together. If the outcome of the fusion is $0$, fine. If the oucome of the fusion is $2$, we start all over again with two fresh ancillas. In this alternate procedure, there is no pair of $1$’s to remove at the end.\
We get the well-known picture of [@F]Chapter $1$.
The idea next is to fuse anyons $3$ and $4$. We somehow got inspired by [@BK].
Our initial state is the tensor product $(a|1> +b|3>)\otimes (x|1>+y|3>)$.
- If we measure a $0$, since the quantum dimensions of $1$ and $3$ are equal, we get the ancilla $$a\,x\,|1>\, + \,b\,y\,|3>$$ where the contribution in $|1>$ comes from the tensor $|1>\otimes |1>$ and the contribution in $|3>$ comes from the tensor $|3>\otimes |3>$.
- If we measure a $4$, then do a second fusion, namely of anyons $3$ and $4$, whose outcome is necessarily a $2$. We get the ancilla $$a\,y\,|1>\,+\,b\,x\,|3>,$$ where the contribution in $|1>$ comes from the tensor $|1>\otimes |3>$ and the one in $|3>$ from the tensor $|3>\otimes\, |1>$.
- If we measure a $2$, we fail, but we may be able to retrieve one of the two ancillas by doing an extra fusion measurement. Namely, we will retrieve the ancilla to the left $a|1> + b|3>$ if anyons $3$ and $4$ fuse into $2$ and we will retrieve the ancilla to the right $x|1> + y|3>$ if anyons $2$ and $3$ fuse into $2$. Of course in either case, there is no insurance that the fusion outcome will be $2$.
, we could measure anyons $3$ and $4$ instead of fusing them. If the outcome is $0$, no change. If the outcome is $4$, we can fuse anyons $3$ and $4$ and then conclude like before. If the outcome is $2$, the nice thing is that we are able to retrieve both ancillas instead of at most one. The process is the following. Run an interferometric measurement on the last three anyons whose oucome can be either $1$ or $3$. If the outcome is $3$, then fuse a pair of $4$’s into anyons $3$ and $4$ in order to simulate having measured a $1$. Next, bring a pair of $1$’s in between anyons $3$ and $4$ and measure the last four anyons by interferometry. If measure $0$, end of the process. If rather measure $2$, then braid anyons $4$ and $5$ and remeasure by interferometry anyons $5,6,7,8$. Iterate in the case when the outcome of the latter measurement is $2$.\
By $\S\,2.1$, $\S\,2.2$ and $\S\,2.3$, we are able to make the ancilla $$\boxed{\begin{array}{l}A_f=\,|1>\,+\,e^{i(2\theta^{'}\,+\,\frac{\pi}{6})}\,|3>\\\\\text{with}\;\;
\theta^{'}=\,ArcTan(\frac{-14-5\sqrt{3}}{11})\end{array}}$$ Set $$\theta=2\theta^{'}\,+\,\frac{\pi}{6}$$ Then, by the protocol of Chapter $1$, we are able to make the gate $$\Lambda(e^{i\theta})=\begin{pmatrix}1&0\\0&e^{i\theta}\end{pmatrix}$$ or the inverse gate $$\Lambda(e^{-i\theta})=\begin{pmatrix}1&0\\0&e^{-i\theta}\end{pmatrix}$$ By doing a random walk, we are able to make the gate $\Lambda(e^{i\theta})$, see for instance [@BK]. We conclude by using a statement of [@OL] which we recall below.
(by Olmsted) If $x$ is rational in degrees, then the only possible rational values of the tangent are: $$tan\,x=0,\,\pm\,1$$
Olmsted’s theorem is also proven independently by Jack S. Calcut in a more recent American Mathematical Monthly [@CAL] using Gaussian integers.
We also recall the following classical result.
If $\frac{\theta}{\pi}$ is irrational, then the sequence $(e^{in\theta})_{n\geq 0}$ is dense in the unit circle.
We will use Olmsted’s Theorem to show that our phase $\theta$ is irrational in degrees. We show a more general result.
Let $p$ be a prime number and $a$ and $b$ be any given two non-zero rational numbers such that $$\left\lbrace\begin{array}{l}a^2\neq 1+pb^2\hfill\qquad (\star)\\a\neq \pm 1\pm\sqrt{2+pb^2}\hfill \qquad(\star\star)\end{array}\right.$$ Then, $ArcTan(a+b\sqrt{p})$ is irrational in degrees.
<span style="font-variant:small-caps;">Proof.</span> First we show that at least one of $$ArcTan(a+b\sqrt{p})\;\text{ and }\;ArcTan(a-b\sqrt{p})$$ must be irrational in degrees. We then deduce that both numbers are in fact irrational in degrees.\
Our argument is based on three ingredients: the contrapositive of Olmsted’s theorem, an elementary equality giving the tangent of a sum and elementary facts about Galois conjugates.\
\
**Notation**. If $x=a+b\sqrt{p}$, denote by $\bar{x}$ the conjugate of $x$ that is, $\bar{x}=a-b\sqrt{p}$.\
\
We will use that $$\begin{array}{l}
\text{(i) $x+\bar{x}$ and $x\bar{x}$ are both rationals. }\\
\text{(ii) The sum (resp product) of two conjugates is the conjugate of the sum} \\\text{(resp product). }
\end{array}$$ A starting point is the classical congruence $$ArcTan(x)+ArcTan(y)\equiv ArcTan\bigg(\frac{x+y}{1-xy}\bigg)\qquad\qquad\textit{Modulo $\pi$}$$ Use congruence $(2.2)$ with $x=a+b\sqrt{p}$ and $y=\bar{x}$. By the contrapositive of Olmsted’s Theorem, if $r$ is a rational number distinct from $0,-1,1$, then $ArcTan(r)$ is irrational in degrees. The assumption $(\star\star)$ on $a$, $b$ and $p$ guarantees that $$\frac{2a}{1-a^2+pb^2}\not\in\lbrace 0,-1,1\rbrace$$ Then the sum $$ArcTan(x)+ArcTan(\bar{x})$$ is irrational in degrees. And so at least one of $ArcTan(x)$, $ArcTan(\bar{x})$ must be irrational in degrees. Without any loss of generality, suppose $ArcTan(x)$ is irrational in degrees. If $ArcTan(\bar{x})$ were rational in degrees, then we could write $$ArcTan(\bar{x})=\frac{p\pi}{q},$$ some $p\in\mathbb{Z}$, $q\in\mathbb{N}-\lbrace 0\rbrace$ with $p\wedge q=1$. Then, $$Tan(q\,ArcTan(\bar{x}))=0$$ Now use $$Tan(x+y)=\frac{Tan(x)+Tan(y)}{1-Tan(x)Tan(y)}$$ in order to prove by induction on the integer $q\geq 2$ the following claim.
$Tan(q\,ArcTan(x))\in\mathbb{Q}(\sqrt{p})$ and $$\overline{Tan(q\,ArcTan(x))}=Tan(q\,ArcTan(\bar{x}))$$
<span style="font-variant:small-caps;">Proof.</span> We have $$Tan(2ArcTan(x))=\frac{2x}{1-x^2}\in\mathbb{Q}(\sqrt{p})$$ and $$\begin{aligned}
\overline{Tan(2ArcTan(x))}&=&\frac{2\bar{x}}{1-\bar{x}^2}\\
&&\notag\\
&=&Tan(2ArcTan(\bar{x}))\end{aligned}$$ where Eq. $(2.5)$ follows from point $(ii)$ above. So the claim holds for $q=2$. Let $q\geq 3$ and suppose the claim holds for $q-1$. Write $$Tan(q\,ArcTan(x))=\frac{Tan((q-1)ArcTan(x))+x}{1-x\,Tan((q-1)ArcTan(x))}$$ By induction hypothesis and point $(ii)$, the claim obviously holds for $q$ and so it holds for all $q\geq 2$.\
\
Eq. (2.3) and Claim $1$ imply that $$Tan(q\,ArcTan(x))=0$$ And so we have $$q\,ArcTan(x)\in\mathbb{Z}\pi$$ In other words, $ArcTan(x)$ is rational in degrees, a contradiction.\
Below, we draw a summary of one possible complete protocol which realizes the ancilla $A_f$.
Irrational qutrit phase gate
----------------------------
Again, the phase gate is built by ancilla preparation, then fusion of the ancilla into the input to form the gate. The second part appears in Chapter $1$. Only the ancilla preparation is detailed here.\
Like announced and explained in the introduction of the paper, the ancilla preparation relies crucially on three different lemmas whose statements and proofs appear below.
### Qutrit projection
The goal here is the following. Given any qutrit $a_0|0>+a_2|2>+a_4|4>$, find a protocol which projects this qutrit onto either the first two trits or the last two trits, up to some changes in the amplitudes.
The respective protocols below project the qutrit onto either $|0>$ and $|2>$ (protocol to the left) or $|4>$ and $|2>$ (protocol to the right).
$$\begin{array}{cc}a_0|0>+a_2|2>+a_4|4>\;\longrightarrow\;a_0|0>+\frac{1}{2}\,a_2|2>
&a_0|0>+a_2|2>+a_4|4>\;\longrightarrow\;a_4|4>+\frac{1}{2}\,a_2|2>\end{array}$$
The idea is to get rid of one of the $|0>$ or $|4>$ contributions by fusion. Recall that $$\left\lbrace\begin{array}{l}\text{$1$ and $0$ can fuse into $1$}\\
\text{$1$ and $4$ cannot fuse into $1$}\end{array}\right.$$ Also, $$\left\lbrace\begin{array}{l}\text{$1$ and $4$ can fuse into $3$}\\
\text{$1$ and $0$ cannot fuse into $3$}\end{array}\right.$$ So bring a pair $1$’s out of the vacuum and do two fusions like on the figures.
The second fusion outcome determines which contribution $|0>$ or $|4>$ vanishes. By the “triangle collapse” move (cf Fig. $1$), the fusion rules must indeed be respected between $1$, the qutrit and $1$ (or $3$) depending on the outcome. In order to get back to a qutrit $2222$, it will suffice to bring a pair of $2$’s and do additional fusions.
Regarding the coefficients, the only relevant symbols are
- For the first part, $$\left\lbrace\begin{array}{ccc}2&2&1\\1&1&2\end{array}\right\rbrace^u=\frac{1}{\sqrt{2}},\;
\left\lbrace\begin{array}{ccc}2&2&3\\1&1&2\end{array}\right\rbrace^u=-\frac{1}{\sqrt{2}},\;\left\lbrace\begin{array}{ccc}4&2&3\\1&1&2\end{array}
\right\rbrace^u=1$$
- For the second part, $$\left\lbrace\begin{array}{ccc}2&1&2\\1&2&1\end{array}\right\rbrace^u=\frac{1}{\sqrt{2}},\; \left\lbrace\begin{array}{ccc}2&3&2\\1&2&1\end{array}\right\rbrace^u=-\frac{1}{\sqrt{2}},\;\left\lbrace\begin{array}{ccc}4&3&2\\1&2&1\end{array}\right\rbrace^u=1$$
The next step allows to fuse together two qutrits that have previously been projected to $|0>$ and $|2>$ (or to $|4>$ and $|2>$).
### Qutrit fusion
The idea underlying this fusion is to keep the $(|0>,|2>)$ contribution and the $(|2>,|0>)$ contribution, having in mind that if one ancilla is the conjugate of the other one, then we will get equal norms in $|0>$ and $|2>$ and twice the interesting relative phase we had. Thus, we want to fuse both ancillas into a $2$, that is fuse anyons $4$ and $5$ to $0$, then fuse anyons $3$ and $4$ to $2$. The issue of the $(|2>,|2>)$ contribution gets resolved by the fact that, specific to this theory, the $6j$-symbols with only $2$’s in it is zero, hence this contribution vanishes during the process. The second magic thing about this fusion is that when we fuse again in order to retrieve a qutrit $2222$, we create the desired factor of $\frac{1}{\sqrt{2}}$.
The following protocol realizes the fusion of two “projected” ancillas into a third ancilla, where the amplitudes have been multiplied term by term like indicated below, and where a factor $\frac{1}{\sqrt{2}}$ has been generated in front of the $|2>$ trit.
$$\begin{array}{cccccccc}
\begin{array}{l}|0>\\|2>\\|4>\end{array}&{\negthickspace}{\negthickspace}{\negthickspace}\begin{pmatrix}a_0\\a_2\\0\end{pmatrix}&,&
\begin{array}{l}|0>\\|2>\\|4>\end{array}&{\negthickspace}{\negthickspace}{\negthickspace}\begin{pmatrix}b_0\\b_2\\0\end{pmatrix}&\overset{Fusion}{\longrightarrow}&
\begin{array}{l}|0>\\|2>\\|4>\end{array}&{\negthickspace}{\negthickspace}{\negthickspace}\begin{pmatrix}a_0b_2\\\frac{1}{\sqrt{2}}a_2b_0\\0\end{pmatrix}
\end{array}$$
<span style="font-variant:small-caps;">Proof.</span> The proof is straigtforward. The factor $\frac{1}{\sqrt{2}}$ arises from the $6j$-symbol $$\left\lbrace\begin{array}{ccc}2&2&2\\2&2&0\end{array}\right\rbrace^u=\frac{1}{\sqrt{2}}$$$\square$\
Here is how we apply the lemma. Suppose we know how to prepare two ancillas $$\begin{array}{ccc}\begin{array}{cc}\begin{array}{l}|0>\\|2>\\|4>\end{array}&{\negthickspace}{\negthickspace}{\negthickspace}\begin{pmatrix}a\\\frac{1}{2}b\\0\end{pmatrix}\end{array}&\text{and}&
\begin{array}{cc}\begin{array}{l}|0>\\|2>\\|4>\end{array}&{\negthickspace}{\negthickspace}{\negthickspace}\begin{pmatrix}\bar{a}\\\frac{1}{2}\bar{b}\\0\end{pmatrix}\end{array}\end{array}$$
with relative phase $\theta$ between complex numbers $a$ and $b$. Then, by fusion, we are able to make a third ancilla $$\begin{array}{cc}\begin{array}{l}|0>\\|2>\\|4>\end{array}&{\negthickspace}{\negthickspace}{\negthickspace}\begin{pmatrix}a\bar{b}\\\frac{1}{\sqrt{2}}b\bar{a}\\0\end{pmatrix}\end{array}$$ which has the norms we want and an interesting relative phase of $2\theta$.
### Qutrit pair fusion
The next protocol allows, given a qutrit ancilla, to get another ancilla which is obtained from the first one by shifting and cloning its $|2>$ contribution to the $|0>$ and $|4>$ trits and by adding its $|0>$ and $|4>$ contributions to yield the $|2>$ contribution of the new ancilla.
Pair of $2$’s fusion into the qutrit $2222$.
<span style="font-variant:small-caps;">Proof.</span> The proof can be read out of the following figure.\
We do an F-move on the diagram to the left on the zero labeled edge. Then, on the first diagram of the sum, the $|2>$ trit contribution vanishes. Indeed, if we attempt to remove the bubble, the $6j$-symbol arising during the removal process is $$\left\lbrace\begin{array}{ccc}2&2&2\\2&2&2\end{array}\right\rbrace$$ which is zero in this theory. Up to an overall scalar, we get $${\negthickspace}{\negthickspace}{\negthickspace}{\negthickspace}{\negthickspace}{\negthickspace}{\negthickspace}{\negthickspace}{\negthickspace}{\negthickspace}{\negthickspace}\begin{array}{cc}\begin{array}{l}|0>\\\\\\|2>\\\\\\|4>\end{array}&{\negthickspace}{\negthickspace}{\negthickspace}\begin{pmatrix}a_2\bigg(\left\lbrace\begin{array}{ccc}2&2&2\\2&0&2\end{array}
\right\rbrace^u\bigg)^2\left\lbrace\begin{array}{ccc}2&2&0\\2&2&0\end{array}
\right\rbrace^u\\\\a_0\bigg(\left\lbrace\begin{array}{ccc}2&2&2\\2&2&0\end{array}
\right\rbrace^u\bigg)^2\left\lbrace\begin{array}{ccc}2&2&2\\0&0&0\end{array}
\right\rbrace^u+a_4\bigg(\left\lbrace\begin{array}{ccc}2&2&2\\2&2&4\end{array}
\right\rbrace^u\bigg)^2\left\lbrace\begin{array}{ccc}2&2&2\\4&4&0\end{array}
\right\rbrace^u\\\\a_2\bigg(\left\lbrace\begin{array}{ccc}2&2&2\\2&4&2\end{array}
\right\rbrace^u\bigg)^2\left\lbrace\begin{array}{ccc}2&2&4\\2&2&0\end{array}
\right\rbrace^u\end{pmatrix}\end{array}$$
The squared values or values of the $6j$-symbols are in the same order as in which they appear,
$1,\frac{1}{2}$\
$\;$\
$\frac{1}{2},1,\frac{1}{2},1$\
$\;$\
$1,\frac{1}{2}$
Hence we get an overall $\frac{1}{2}$ and the coefficients announced. $\square$\
\
If we apply this protocol of pair fusion to the ancilla of last section, here is what we get $$\begin{array}{cc}\begin{array}{l}|0>\\|2>\\|4>\end{array}&{\negthickspace}{\negthickspace}{\negthickspace}\begin{pmatrix}a\bar{b}\\\frac{1}{\sqrt{2}}b\bar{a}\\0\end{pmatrix}\end{array}
\overset{\text{After pair fusion}}{\longrightarrow} \begin{array}{cc}\begin{array}{l}|0>\\|2>\\|4>\end{array}&{\negthickspace}{\negthickspace}{\negthickspace}\begin{pmatrix}\frac{1}{\sqrt{2}}b\bar{a}\\a\bar{b}\\\frac{1}{\sqrt{2}}b\bar{a}
\end{pmatrix}\end{array}$$ It remains to construct an ancilla $$\begin{array}{cc}\begin{array}{l}|0>\\|2>\\|4>\end{array}&{\negthickspace}{\negthickspace}{\negthickspace}\begin{pmatrix}a\\b\\0\end{pmatrix}\end{array}$$ and its conjugate ancilla with interesting relative phase between the two complex numbers $a$ and $b$. This is the object of the next part.
### Final ancilla preparation
We start with the same configuration as for the qubit, that is four anyons $(1111)_2$ and a pair of $2$’s. We do the following braids and fusions.
The full twist on $1122$ swaps the $|0>$ and $|2>$ bits, see [@QIP], hence creates a $2$ charge line between the qubit $1111$ and the pair of $2$’s. The single braid on $(1111)_2$ creates a superposition of $|0>$ and $|2>$. After the three fusions like on the picture, we get a qutrit $2222$, which is up to overall complex scalar $$\sqrt{2}e^{-i\frac{\pi}{3}}|0>+\frac{1}{2}|2>$$ Of course the relative phase is not yet satisfactory, but our preparatory efforts will be rewarded by a $\sigma_2$-braid on that vector. It yields, still up to overall complex scalar, $$\begin{array}{cc}\begin{array}{l}|0>\\\\|2>\\\\|4>\end{array}&{\negthickspace}{\negthickspace}{\negthickspace}\begin{pmatrix}\sqrt{7}e^{i\gamma^{'}}\\\\e^{i\frac{\pi}{4}}\\\\\sqrt{3}e^{i\frac{\pi}{12}}
\end{pmatrix}\end{array}\text{with $\gamma^{'}=\pi+ArcTan\bigg(\frac{-14+3\sqrt{3}}{13}\bigg)$}$$ Our Lemma $3$ does not apply here because condition $(\star)$ is not satisfied.\
However, we can still conclude by [@CAL2]. In [@CAL2], J.S. Calcut determines all of the numbers $Tan(\frac{p\pi}{q})$ with degree two or less over $\mathbb{Q}$. He shows that for such numbers, the only possible quadratic irrational values are $$\left\lbrace\begin{array}{l}\pm\sqrt{3}\\\\\pm\frac{\sqrt{3}}{3}\\\\\pm 1\pm\sqrt{2}\\\\\pm 2\pm\sqrt{3}\end{array}\right.$$
Thus, the relative phase of our newly produced qutrit ancilla is irrational in degrees. Moreover, we claim that the same protocol where the two single braids of the picture have been replaced with their inverse braids as on the new picture below produces the conjugate ancilla. This is simply because each matrix corresponding to a $\sigma_2$ action on $1111$ and $2222$ is self-transpose, so that the inverse matrices are the conjugate matrices. We obtain the conjugate ancilla $$\begin{array}{cc}\begin{array}{l}|0>\\\\|2>\\\\|4>\end{array}&{\negthickspace}{\negthickspace}{\negthickspace}\begin{pmatrix}\sqrt{7}e^{-i\gamma^{'}}\\\\e^{-i\frac{\pi}{4}}\\\\\sqrt{3}e^{-i\frac{\pi}{12}}
\end{pmatrix}\end{array}$$
By applying a qutrit projection on both ancillas, then a qutrit fusion of the two ancillas, followed by one qutrit pair fusion, we are able to make the final ancilla, say $B_f$, which is up to overall complex scalar, $$\boxed{B_f=\begin{array}{ccc}\begin{array}{cc}\begin{array}{l}|0>\\|2>\\|4>\end{array}&{\negthickspace}{\negthickspace}{\negthickspace}\begin{pmatrix}1\\\sqrt{2}\,e^{i\gamma}\\1\end{pmatrix}\end{array}
&\text{with}&\gamma=2\,ArcTan\bigg(\frac{3\sqrt{3}-14}{13}\bigg)-\frac{\pi}{2}\end{array}}$$ By Chapter $1$, we are able to turn this ancilla into an irrational phase gate $$\begin{pmatrix} 1&&\\&e^{i\gamma}&\\&&1\end{pmatrix}$$ Below, we drew a picture which assembles all the different parts of the protocol together to produce the ancilla $B_f$.
To conclude, we comment on the similarities and differences between this ancilla preparation and the one we did for the quantum bit. We only focus on the first step of each protocol, namely the step that yields the “interesting relative phase” and whose starting point is in both cases the qubit $1111$ and a pair of $2$’s.
For the qubit, the information is carried ultimately by the edge between the qubit $1111$ and the ancilla pair $22$, whereas for the qutrit, the information is carried by the edge from the qubit $1111$.
Another aspect is that for the qubit the interesting phase is really arising from one bit and one bit only (either $|1>$ or $|3>$) as a sum of two contributions carried by the edge of the qubit $1111$. Similarly for the qutrit, the interesting phase is carried by one trit only, namely the $|0>$ trit, after the braiding of the superposition of $|0>$ and $|2>$ has occurred. There would have been a priori two candidates, namely $|0>$ or $|4>$ (it would not be $|2>$ by the shape of the $\sigma_2$ matrix possessing a zero in its central position), but by the game of the coefficients and phases, it is the $|0>$ trit which will carry the irrational phase.
Finally, in the protocol for the qutrit, the pair fusion is crucial, whereas in the protocol for the qubit, we only do simple fusions (this is allowed by adjusting the outcome from the first fusion, thus avoiding the Freedman pair of $4$’s fusion).
**Acknowledgements**
We are pleased to thank Bela Bauer, Stephen Bigelow, Parsa Bonderson, Meng Cheng, Spiros Murakalis, Chetan Nayak and Zhenghan Wang for helpful discussions.
Realizing the $2$-qubit permutation gates
=========================================
**Abstract**
We give a protocol which generates one or the other of two $2$-qubit entangling gates. By acting on these two gates with one of the $2$-qubit entangling gates that was recently made in [@QIP] and by additional braiding, we are able to make the gate $$CNOT.\,SWAP=\begin{pmatrix}1&0&0&0\\0&0&1&0\\0&0&0&1\\0&1&0&0\end{pmatrix}$$ Since the SWAP gate is easily realized by braiding only, we are thus able to make the CNOT gate. Further, with additional pair fusion operations, we are able to generate all the permutation gates of $Sym(4)$.
Introduction
------------
The Brylinski couple, in a work [@BRY] dating from $2001$, proves that universal single qudit gates together with any $2$-qudit entangling gate are sufficient for universal quantum computation. For qubits, there is an independent proof of this fact by [@BCD]. In [@CL2], we give a protocol which produces a $2$-qubit entangling gate whose definition appears in Theorem $1$ of the paper. With additional braiding, we also make a circulant $2$-qubit entangling gate. The goal of the present paper is to make all the $2$-qubit quantum gates from $Sym(4)$ in the Kauffman-Jones version of $SU(2)$ Chern-Simons theory at level $4$. Our qubit is formed by four anyons of topological charges $1221$, just like in [@CL2]. The difficult part is to make a $3$-cycle. For instance the products of two transpositions with disjoint support are easily obtained by some Freedman type [@CL] fusion operations. The main protocol developed in our paper relies on two novel ideas, which we name “qubit transfer” and “qubit demolition”. First we move the information carried by the right qubit, second we destroy the right qubit without leaking any quantum information. During the second phase, namely the demolition phase, we use a process that was described in [@CL2] as “Process (P)” which operates a projection for a qutrit $0/2/4$ onto either $2$ or $0/4$. For each of both outcomes of the projection, we get a distinct gate. By acting on each of these two gates to the left by $$\begin{array}{cc}&\begin{array}{cccc}|11>&|13>&|31>&|33>\end{array}\\\begin{array}{l}|11>\\\\|13>\\\\|31>\\\\|33>\end{array}&\begin{pmatrix}
-\frac{1}{2}&0&0&-i\frac{\sqrt{3}}{2}\\\\0&-\frac{1}{2}&-i\frac{\sqrt{3}}{2}&0\\\\
0&-i\frac{\sqrt{3}}{2}&-\frac{1}{2}&0\\\\-i\frac{\sqrt{3}}{2}&0&0&-\frac{1}{2}\end{pmatrix}\end{array}=AUX$$
a $2$-qubit entangling gate that was produced in [@CL2], and together with some additional braiding, we are able to always make the CNOT.SWAP gate independently from the resulting outcome of Process (P).
While by the Brylinski result, only one $2$-qubit entangling gate like built in [@CL2], together with a universal set of $1$-qubit gates like resulting from Chapter $2$ would be sufficient in order to approximate any permutation gate of $Sym(4)$, one of the highlights of our paper is to give a way to physically make these permutation gates accurately instead of simply by approximation.
Throughout the paper, we assume that the reader is familiar with the Kauffman-Jones theory and a nice reference for it is [@KL]. For the unitary version of this theory, a good reference is [@ZW][@CL]. We also assume that the reader is familiar with interferometric measurements and good references are [@BO][@BO2][@BO3][@FL].
Our main protocol uses braids, interferometric measurements, fusions and unfusions of anyons, vacuum pair creation and recovery procedures, ideas which originated in [@MO].
Main protocol
-------------
The protocol is based on qubit transfer, that is the qubit outputs can be different from the qubit inputs. Which means a second part of the protocol must deal with qubit demolition: after the information has been transferred onto another qubit, the old qubit must be destroyed without leaking any quantum information. In the protocol which we present, the left qubit input is also the left qubit output, but the right qubit input is not the right qubit output and must be destroyed. The ancillas are simply pairs created out of the vacuum which help with the destruction process as well as with the final separation process. Whereas, in the protocol of [@CL2], there are two qubit ancillas which become the qubit outputs. We provide the protocol, then explain it. As part of the protocol, some recovery procedures are needed and get provided in the discussion which follows the theorem. In order to give a figure that is readable, the recovery procedures don’t appear in the figure, but are nevertheless a crucial part of the protocol. Thus, the figure only provides a winning protocol, that is one with favorable measurement outcomes. The current section deals with the statement of Theorem $3$ and its proof.
The following protocol
produces the entangling gate $$EG^{(1)}=\begin{array}{cc}\begin{array}{l}\\\\|11>\\\\|13>\\\\|31>\\\\|33>\end{array}&{\negthickspace}{\negthickspace}\begin{array}{l}\begin{array}{cccc}\;\;\;|11>&\;|13>&\;\;\;|31>&\;\;\;{\negthickspace}|33>\end{array}\\\\
\begin{pmatrix}1/4&i\sqrt{3}/4&3/4&-i\sqrt{3}/4\\&&&\\3/4&-i\sqrt{3}/4&1/4&i\sqrt{3}/4\\&&&\\
i\sqrt{3}/4&-3/4&-i\sqrt{3}/4&-1/4\\&&&\\-i\sqrt{3}/4&-1/4&i\sqrt{3}/4&-3/4\end{pmatrix}\end{array}\end{array}$$ The next protocol
produces the entangling gate $$EG^{(2)}=\begin{array}{cc}\begin{array}{l}\\\\|11>\\\\|13>\\\\|31>\\\\|33>\end{array}
&{\negthickspace}{\negthickspace}\begin{array}{l}\begin{array}{cccc}\;\;\;|11>&\;|13>&\;\;\;|31>&\;\;\;{\negthickspace}|33>\end{array}\\\\
\begin{pmatrix}1/4&-i\sqrt{3}/4&3/4&i\sqrt{3}/4\\&&&\\3/4&i\sqrt{3}/4&1/4&-i\sqrt{3}/4\\&&&\\
i\sqrt{3}/4&3/4&-i\sqrt{3}/4&1/4\\&&&\\-i\sqrt{3}/4&1/4&i\sqrt{3}/4&3/4\end{pmatrix}\end{array}\end{array}$$
**Recovery:** if at point $(t^{'})$, the measurement rather produces the outcome $0$ (resp $4$) instead of $2$, then remove the pair of $2$’s composed by the last two anyons (resp fuse the last three anyons two by two from the right, then fuse a pair of $4$’s in the last two anyons). By doing so, get back to a previous stage in time, namely to point $(t)$. Proceed again from there and generate either the first gate of the Theorem or the second gate depending on the subsequent measurement outcomes.
<span style="font-variant:small-caps;">Proof of Theorem $3$.</span> **Unless otherwise mentioned in the discussion, it is understood that we deal with the first protocol.**\
The protocol can be divided into a few main steps.\
*Step $1$*. Measure anyons $4$ and $5$ to zero in order to fuse them (we refer the reader to Chapter $1$ for the recovery to follow when the outcome of this first measurement is not zero).\
*Step $2$*. The first two braids of the picture are the entangling braids. The $2$-qubit output is carried by the edge from the left qubit input and the edge from the fusion of Step $1$. It will remain to destroy the right qubit input independently from what the $2$-qubit input was, then get back to a $2$-qubit shape.\
*Step $3$*. The qubit demolition is made possible by a series of two braids, followed by a measurement assisted by a pair of $2$’s out of the vacuum.\
*Step $4$*. All the steps which follow Step $3$ deal with the “back into shape” process, that is get back to a $2$-qubit. This step is made possible by three fusions, one braid, one unfusion, an interferometric measurement, adding a pair out of the vacuum and running a second interferometric measurement. Though this seems like a long and complicated process, all these steps arise “naturally” from reading the figure. Thus, the key ideas of this protocol remain the qubit transfer and the qubit demolition.\
We will go over each step, except Step $1$ which is extensively detailed in Chapter $1$. In order to understand the entangling braids, we will need to know the $\sigma_2$-actions on the qubits $1221$, $3223$, $1223$ and $3221$. The first two actions and the last two are identical, hence there are only two matrices of interest here. We have $$\begin{array}{l}G_2(1,2,2,1)=G_2(3,2,2,3)=\begin{pmatrix}-\frac{1}{2}&i\frac{\sqrt{3}}{2}\\&\\i\frac{\sqrt{3}}{2}&-\frac{1}{2}
\end{pmatrix}\\\\G_2(1,2,2,3)=G_2(3,2,2,1)=\begin{pmatrix}-i\frac{\sqrt{3}}{2}&\frac{1}{2}\\&\\\frac{1}{2}&-i\frac{\sqrt{3}}{2}\end{pmatrix}
\end{array}$$ And so by reading the first columns of the matrices above, the matrix representing the action of the first braid
is the following
Each row of the matrix is indexed in the same order as the horizontal edges appearing from left to right on the figure of the protocol. So that the first and last bits correspond to the input bits and the middle bit corresponds to the label carried by the middle edge after the braid has occurred. If the first and last bits are identical, then we read the superposition of the middle edge on the first column of the first two by two matrix above. When the first and last bits are distinct, the superposition is given by the first column of the second two by two matrix above.
Now the second braid. We braid $1221$ or $1223$ on the left input depending on whether the middle horizontal edge carries the label $1$ or the label $3$. We get the following superposition. For clarity we ordered the $2$-qubit input states differently, namely we permuted $|13>$ and $|31>$. We kept the same ordering as before on the rows. Since the right qubit input never gets braided, its label remains what it is as an input. Thus, if the right qubit input is $1$, the last four rows of the matrix are filled with zeroes and if the right qubit input is $3$, the first four rows of the matrix are filled with zeroes. This motivated our choice of reordering the input basis. The reader gets invited to compute some of the coefficients provided in the matrix below in order to grasp the entangling action.
The goal now is to get to a four by four matrix with the $2$-qubit output read out of the first two bits, that is find a way to suppress the right qubit. The situation is complex since we have a superposition of two qubits, namely $1221$ and $3221$, but we developed a technique that deals with it. We will need the respective matrices from the following actions
These are $$\begin{array}{cc}\begin{array}{cc}&\begin{array}{cc}|1>&|3>\end{array}\\&\\\begin{array}{l}|0>\\\\|2>\end{array}&\begin{pmatrix}
\frac{e^{i\frac{2\pi}{3}}}{\sqrt{2}}&\frac{e^{-i\frac{5\pi}{6}}}{\sqrt{2}}\\&\\
\frac{1}{\sqrt{2}}&-\frac{i}{\sqrt{2}}\end{pmatrix}\end{array}
&\begin{array}{cc}&\begin{array}{cc}|1>&|3>\end{array}\\&\\\begin{array}{l}|2>\\\\|4>\end{array}&\begin{pmatrix}
\frac{i}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\&\\
\frac{e^{-i\frac{5\pi}{6}}}{\sqrt{2}}&\frac{e^{-i\frac{\pi}{3}}}{\sqrt{2}}
\end{pmatrix}\end{array}
\end{array}$$ $$\begin{array}{l}\end{array}$$ Suppose we have put the right qubit under a superposition of shapes $1212$ and $3212$ by a single $\sigma_1$-braid exchanging the right most anyons of topological charges $1$ and $2$. The matrix of this exchange is $$R(1,2)=\begin{pmatrix}
-e^{i\frac{\pi}{3}}&\\
&-e^{-i\frac{\pi}{6}}
\end{pmatrix}$$ The matrices gathering the two actions above read $$\begin{array}{ccc}\begin{array}{ccc}&\begin{array}{cc}|1>&|3>\end{array}&\\&&\\\frac{1}{\sqrt{2}}&\begin{pmatrix}
1&1\\
-e^{i\frac{\pi}{3}}&e^{i\frac{\pi}{3}}\end{pmatrix}&\begin{array}{l}|0>\\|2>\end{array}\end{array}
&\begin{array}{ccc}&\begin{array}{cc}|1>&|3>\end{array}\\&&\\\frac{1}{\sqrt{2}}&\begin{pmatrix}
e^{-i\frac{\pi}{6}}&-e^{-i\frac{\pi}{6}}\\
i&i
\end{pmatrix}&\begin{array}{l}|2>\\|4>\end{array}\end{array}&(\text{\textbf{M}})
\end{array}$$ $$\begin{array}{l}\end{array}$$
We then proceed after making a few observations. Those observations that got omitted are those which appear even more clearly on the matrices (M).\
- For the $|2>$ outcome of the braid on $1212$ (second row of the left hand side matrix), the phases arising from both right input bits $|1>$ and $|3>$ are opposite and are, up to overall phase $-e^{i\frac{\pi}{3}}$, respectively $1$ and $-1$.\
- For the $|2>$ outcome of the braid on $3212$ (first row of the right hand side matrix), the phases arising from both right input bits $|1>$ and $|3>$ are opposite and are, up to overall phase $e^{-i\frac{\pi}{6}}$, respectively $1$ and $-1$.\
- Doing a full $\sigma_2$-twist on $1122$ swaps the bits $|0>$ and $|2>$ with overall phase $e^{i\frac{2\pi}{3}}$, see [@QIP].\
- Likewise, doing a full $\sigma_2$-twist on $3122$ swaps the bits $|2>$ and $|4>$ with overall phase $-e^{i\frac{2\pi}{3}}$.
Suppose we do a series of operations consisting of doing the $\sigma_1$-braid, then doing the $\sigma_2$-braid on the superposition $1212$ and $3212$, then doing the full twist exchanging the bits $|0>$ and $|2>$ and $|2>$ and $|4>$ of the superposition, followed by bringing a pair of $2$’s placed like on the main figure and measuring either $0$ or $4$ during the interferometric measurement of the main figure. These outcomes have the effect of projecting the right qubit onto $2$ since $2$ and ($0$ or $4$) can only fuse into $2$. Next, we fuse the anyons two by two from the right hand side until getting to a single anyon of topological charge $2$. Its left neighbor carries the topological charge $1$. Now interchange the relative positions of these two anyons by doing a $\sigma_1$-braid on them. By doing so, introduce a relative phase of $-i$ between the $|1>$ and the $|3>$ bits carried by the central horizontal edge. This relative phase of $-i$ compensates exactly the $i$ relative phase observed before between the two matrices $(M)$ after the projection onto $|2>$.\
\
**Thus, the only relative phase to keep track of during the process is the minus sign arising from the qubit swap**.\
\
In what follows, we deal with both protocols of Theorem $3$ in a subtle way. This will appear clear later in the discussion. Suppose the interferometric measurement outcome is $2$ instead of $0$ or $4$. Then, follow process (P) of [@CL2], that is do the following braid on the last three anyons
We recall from [@BL] or [@CL] the matrix of this action on the qutrit $2222$. Up to overall phase, it is $$\begin{array}{cc}&\begin{array}{ccc}|0>&|2>&|4>\end{array}\\&\\\begin{array}{l}|0>\\\\|2>\\\\|4>\end{array}&\begin{pmatrix} \frac{1}{2}&\frac{1}{\sqrt{2}}&\frac{1}{2}\\&&\\\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}\\&&\\\frac{1}{2}&-\frac{1}{\sqrt{2}}&\frac{1}{2}\end{pmatrix}
\end{array}$$ Then measure the last two anyons.\
If the outcome is $0$, after removing the pair of $2$’s to the right, we are back to the configuration we had before the interferometric measurement. Then, we bring another pair of $2$’s and run the interferometric measurement again. We are back to a previous time in history, namely to point $(t)$ like indicated on the two protocols. Up to point $(t)$ the two protocols are identical, hence we may still produce one gate or the other.\
If the outcome is $4$, we fuse the last three anyons two by two, then fuse a pair of $4$’s in the last two anyons, using Lemma $1$ of Chapter $1$. Again, we are back to the configuration $(t)$ before the interferometric measurement and proceed from there. Again, we may still produce one gate or the other.\
Finally, if the outcome is $2$, we have projected onto $|0>$ and $|4>$. In that case, we will produce the second gate of the Theorem. The protocol to finish is the following. Fuse the last three anyons two by two from the right hand side. We must keep track of a relative phase of $\pi$ between the $|0>$ and the $|4>$ as a result of the braids and the projection. Now do the qubit swap in order to switch the $|0>$ into a $|2>$ and switch the $|4>$ into a $|2>$. The two minus signs, the one arising from the projection and the one arising from the qubit swap compensate beautifully. We next finish like before by fusion followed by a $\sigma_1$-braid on the last two anyons. Again the relative phase of $-i$ introduced during the last braiding operation compensates exactly the relative phase of $i$ from the matrices (M).\
\
**Thus, this time, the relative phases to keep track are:**\
\
**1) the minus sign when the right input bits are distinct**\
**2) the minus sign from the qubit swap, the one occurring right before point $(t)$**.\
Work is not yet over as in both cases, we must still get back to a $2$-qubit shape now that the right qubit input has vanished. Below, we drew the configuration we are at.
The two edges carrying the $1/3$ superpositions will be the two edges carrying the $2$-qubit output. There will be two additional steps in order to reach the goal. First, we obviously don’t have the right number of anyons. By unfusing the third anyon into two anyons of topological charge $2$, we increase this number and allow to get to the same topological shape as after the two inputs were initially fused. Except the central edge is a superposition of $1$ and $3$ from the F-move.\
\
**The F-symbols involved are opposite for distinct output bits.**
We force a label of either $1$ or $3$ of the central horizontal edge by running an interferometric measurement on the last three anyons. If the resulting projection is onto $1$ (resp $3$), we next bring a pair of $1$’s (resp $3$’s) out of the vacuum and place it on top of the central horizontal edge. Then by running an interferometric measurement forced to $0$ of the last four anyons, we form a final “pair of $2$-qubit output”. The goal is reached. If the outcome of the interferometric measurement is $2$ rather than $0$, we simply braid and remeasure, process we iterate until we have a successful outcome of $0$. Another way is to measure alternatively the last four anyons and the central two anyons until the separation occurs.\
We are now able to conclude. In order to get the two entangling gates of the Theorem, we look back at the second entangling braid matrix and make the following changes.\
- The last qubit digit of the rows which used to correspond to the right qubit input is no longer there. Therefore, we collapse the last four rows and the lower right quadrant of the matrix has to be moved up.\
- We take care of interchanging the two middle columns so that our input and output bases are ordered in the same and usual way.\
- Once this is done, the middle rows get a minus sign in order to take into account the third remark in bold.\
- Rows $2$ and $4$ get a minus sign because of the qubit swap occurring right before point $(t)$.\
- **For the second entangling gate only**, Columns $2$ and $4$ get a minus sign when the right qubit input is $3$ in order to take into account the second remark in bold.
$$\begin{array}{l}\end{array}$$ All together, we get the two entangling gates announced in Theorem $3$.
In the discussion above, we assumed that the outcome of the interferometric measurement of the three anyons at the top of the main figure of the Theorem is $1$ (just like on the figure of the Theorem). If the outcome is $3$ instead, the first and last rows of the gate get an overall minus sign (instead of the middle two). Anyhow, the two gates we get are the same since they simply differ by an overall minus sign. Also, it will suffice to fuse a pair of $4$’s into anyons $3$ and $4$ on one hand and $5$ and $6$ on the other hand in order to get back to a $2$-qubit $1221$. These two pair fusions operations get achieved at no cost.
The reader might wonder why we did the first full twist qubit swap. This is for good reasons which will be enlightened later. The way we operated all the different braids is the only way which makes the protocol of next section optimal in a sense we will define later.
The CNOT. SWAP gate
-------------------
This section enlightens the fact that the $2$-qubit entangling gate which we made in [@CL2] is not only useful for universal quantum computation with the qubit $1221$ but is also a nice gate to have in its own. Together with our main protocol from previous section, we will use it to produce the product gate CNOT.SWAP; then we will deduce how to make the CNOT gate. In what follows, the auxiliary gate AUX is the one defined in $\S\,1$. The protocol producing it is the main protocol of [@CL2]. It is also the auxiliary gate used to make the circulant $2$-qubit entangling gate of [@CL2].
By the main protocol of Theorem $3$, we are able to make one of the two gates $EG_1$ or $EG_2$. Then the product gate $CNOT.SWAP$ is given by
<span style="font-variant:small-caps;">Proof of Theorem $2$.</span> We have $$AUX.EG_1=\begin{array}{cc}&\begin{array}{cccc}|11>\qquad&|13>\;&|31>\;&\;\;\;\;|33>\end{array}\\&\\\begin{array}{l}|11>\\\\|13>\\\\|31>\\\\|33>\end{array}&\begin{pmatrix}
-\frac{1}{2}\qquad&0\qquad&0\qquad&\frac{i\sqrt{3}}{2}\\&&&\\
0\qquad&\frac{i\sqrt{3}}{2}\qquad&-\frac{1}{2}\qquad&0\\&&&\\
-\frac{i\sqrt{3}}{2}\qquad&0\qquad&0\qquad&\frac{1}{2}\\&&&\\
0\qquad&\frac{1}{2}\qquad&-\frac{i\sqrt{3}}{2}\qquad&0
\end{pmatrix} \end{array}$$ and $$AUX.EG_2=\begin{array}{cc}&\begin{array}{cccc}|11>\qquad&|13>\;&|31>\;&\;\;\;\;|33>\end{array}\\&\\\begin{array}{l}|11>\\\\|13>\\\\|31>\\\\|33>\end{array}&\begin{pmatrix}
-\frac{1}{2}\qquad&0\qquad&0\qquad&-\frac{i\sqrt{3}}{2}\\&&&\\
0\qquad&-\frac{i\sqrt{3}}{2}\qquad&-\frac{1}{2}\qquad&0\\&&&\\
-\frac{i\sqrt{3}}{2}\qquad&0\qquad&0\qquad&-\frac{1}{2}\\&&&\\
0\qquad&-\frac{1}{2}\qquad&-\frac{i\sqrt{3}}{2}\qquad&0
\end{pmatrix} \end{array}$$ Doing a $\sigma_2$-braid on the left input side yield the respective two matrices\
$$\begin{array}{cc}&\begin{array}{cccc}|11>&|13>\;&\;|31>\;&\;|33>\end{array}\\&\\\begin{array}{l}|11>\\\\|13>\\\\|31>\\\\|33>\end{array}&\begin{pmatrix}
1\qquad&0\qquad&0\qquad&0\\&&&\\
0\qquad&0\qquad&1\qquad&0\\&&&\\
0\qquad& 0\qquad&0\qquad&-1\\&&&\\
0\qquad&-1\qquad&0\qquad&0
\end{pmatrix} \end{array}$$ $$\begin{array}{l}\end{array}$$ and $$\begin{array}{cc}&\begin{array}{cccc}|11>&|13>\;&\;|31>\;&\;|33>\end{array}\\&\\\begin{array}{l}|11>\\\\|13>\\\\|31>\\\\|33>\end{array}&\begin{pmatrix}
1\qquad&0\qquad&0\qquad&0\\&&&\\
0\qquad&0\qquad&1\qquad&0\\&&&\\
0\qquad& 0\qquad&0\qquad&1\\&&&\\
0\qquad&1\qquad&0\qquad&0
\end{pmatrix} \end{array}$$ $$\begin{array}{l}\end{array}$$
It will now suffice to do a $\sigma_1$-full twist in order to swap the two $(-1)$’s of these matrices to $1$’s. Indeed, the matrix for a $\sigma_1$-full twist is up to overall phase $$\begin{array}{cc}&\begin{array}{cc}|1>&|3>\end{array}\\&\\\begin{array}{l}|1>\\\\|3>\end{array}&\begin{pmatrix}
1&&0\\&&\\0&&-1\end{pmatrix}\end{array}$$ After following the procedure of the figure, we get in each case $$CNOT.SWAP=\begin{pmatrix}1&0&0&0\\0&0&1&0\\0&0&0&1\\0&1&0&0\end{pmatrix}$$
To finish the section, we comment on Very important remark $1$. Let us introduce some notations. By operation $R_{ij}$ (resp $C_{ij}$), we mean that the rows (resp columns) $i$ and $j$ of a matrix get an overall minus sign. Once the second entangling braid matrix is put under the format of collapsing the rows to make the third bit vanish and exchanging the relative positions of the two middle columns, we note that the two operations we do are:\
$R_{24}$ for the first entangling gate $EG_1$.\
$R_{24}$ and $C_{24}$ for the second entangling gate $EG_2$.\
If we did other kinds of operations guided by different braiding schemes, we may need to act on an entangling gate twice by the auxiliary matrix instead of only once. This would be less efficient. For instance, if we did $C_{24}$ **only** (that would happen if we did not do the full twist prior to point $(t)$), that would be the case.
All the $2$-qubit permutation gates
-----------------------------------
In what follows, we will sometimes refer to “permutation gates” by the permutation that they encode. So, for instance the freshly made $CNOT.SWAP$ gate will be referred to as $(243)$. Because the SWAP gate is available to us by
we have the transposition $(23)$. And by Theorem $4$, we now have the controlled NOT gate which is the transposition $(34)$. Further, by simple pair fusion operations due to Mike Freedman [@CL], thus called Freedman fusion operation (FFO), we also have all the products of two transpositions with disjoint supports, like follows.
The matrix product is the anti-identity matrix $(14)(23)$ and is obtained physically by realizing both fusions (on the right qubit and on the left qubit).\
Multiplying a gate to the left by $FFO_r$ has the effect of swapping the first two rows and swapping the last two rows.\
Multiplying a gate to the left by $FFO_l$ has the effect of swapping the quadrants up and the quadrants down.\
Any $3$-cycle and product of two transpositions with disjoint supports suffice to generate $A_4$. Adding a transposition generator is enough to then generate the whole symmetric group $Sym(4)$. We thus obtain all the $2$-qubit permutation gates.\
**Acknowledgements.** We thank Michael Freedman and Stephen Bigelow for helpful discussions.
[ll]{} R. Ainsworth and J.K. Slingerland, Topological Qubit Design and Leakage New J. Phys. $13$, $065030$ $(2011)$ B. Bauer, P. Bonderson, M.H. Freedman, M. Hastings, C. Levaillant, Z. Wang, J. Yard, Anyonic gates beyond braiding, Unpublished B. Bauer and C. Levaillant, A new set of generators and a physical interpretation for the $SU(3)$ finite subgroup $D(9,1,1;2,1,1)$, Quantum Information Processing Vol. $12$, Issue $7$ $(2013)$ $2509-2521$ P.H. Bonderson, Non-Abelian anyons and interferometry, Ph.D. thesis California Institute of Technology $(2007)$ P. Bonderson, M.H. Freedman, C. Nayak, Measurement only topological quantum computation via anyonic interferometry, Annals Phys. $324$, $787-826$ $(2009)$ P. Bonderson, K. Shtengel, J.K. Slingerland, Interferometry of non-Abelian anyons, Annals Phys. $323$, $2709-2755$ $(2008)$ Michael J. Bremner, Christopher M. Dawson, Jennifer L. Dodd, Alexei Gilchrist, Aram W. Harrow, Duncan Mortimer, Michael A. Nielsen and Tobias J. Osborne, A practical scheme for quantum computation with any two-qubit entangling gate, Phys. Rev. Lett. $89$, $247902$ $(2002)$ S. Bravyi and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, Phys. Rev. A $71$, $022316$ $(2005)$ J-L. Brylinski and R. Brylinski, Universal quantum gates, arXiv:quant-ph/0108062 $(2001)$ J. Calcut, Gaussian integers and arctangent identities for $\pi$, The American Mathematical Monthly $116$ $(2009)$ $515-530$ J. Calcut, Rationality and the tangent function,
http://www.oberlin.edu/faculty/jcalcut/tanpap.pdf
M.H. Freedman and C. Levaillant, Interferometry versus projective measurement of anyons, arXiv:1501.01339 L. Kauffmann and S. Lins, *Temperley-Lieb recoupling theory and invariants of $3$-manifolds* Ann. Math. Studies, Vol $134$, Princeton, NJ:Princeton Univ. Press $1994$ C. Levaillant, The Freedman group: a physical interpretation for the $SU(3)$-subgroup $D(18,1,1;2,1,1)$ of order $648$, Journal of Physics A: Mathematical and Theoretical $47$, $285203$ $(2014)$ C. Levaillant, On some projective unitary qutrit gates, arXiv:1401.0506 C. Levaillant, Making a circulant $2$-qubit entangling gate, arXiv:$1501.01013$ C. Levaillant, Protocol for making a $2$-qutrit entangling gate in the Kauffman-Jones version of $SU(2)_4$, arXiv:$1501.01019$ C. Levaillant, B. Bauer, M. Freedman, Z. Wang and P. Bonderson, Fusion and measurement operations for $SU(2)_4$ anyons, to appear. C. Mochon, Anyons from non-solvable finite groups are sufficient for universal quantum computation Phys. Rev. A $67$, $022315$ $(2003)$ J.M.H. Olmsted, Rational values of trigonometric functions, The American Mathematical Monthly, Vol. $52$ No. $9$ $507-508$ $(1945)$ Z. Wang, *Topological quantum computation*, CBMS monograph, Vol $112$, American Mathematical Society $2010$
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---
abstract: 'Motivated by the discovery of young, massive stars in the nuclei of some LINER/ “transition” nuclei such as NGC 4569, we have computed photoionization models to determine whether some of these objects may be powered solely by young star clusters rather than by accretion-powered active nuclei. The models were calculated with the photoionization code CLOUDY, using evolving starburst continua generated by the the [<span style="font-variant:small-caps;">STARBURST99</span>]{} code of @lei99. We find that the models are able to reproduce the emission-line spectra of transition nuclei, but only for instantaneous bursts of solar or higher metallicity, and only for ages of $\sim3-5$ Myr, the period when the extreme-ultraviolet continuum is dominated by emission from Wolf-Rayet stars. For clusters younger than 3 Myr or older than 6 Myr, and for models with a constant star-formation rate, the softer ionizing continuum results in an emission spectrum more typical of regions. This model predicts that Wolf-Rayet emission features should appear in the spectra of transition nuclei. While such features have not generally been detected to date, they could be revealed in observations having higher spatial resolution. Demographic arguments suggest that this starburst model may not apply to the majority of transition nuclei, particularly those in early-type host galaxies, but it could account for some members of the transition class in hosts of type Sa and later. The starburst models during the Wolf-Rayet-dominated phase can also reproduce the narrow-line spectra of some LINERs, but only under conditions of above-solar metallicity and only if high-density gas is present ($n_e \gtrsim 10^5$ [cm$^{-3}$]{}). This scenario could be applicable to some “Type 2” LINERs which do not show any clear signs of nonstellar activity.'
author:
- 'Aaron J. Barth'
- 'Joseph C. Shields'
title: 'LINER/ “Transition” Nuclei and the Nature of NGC 4569'
---
Introduction
============
Emission-line nebulae in galactic nuclei are generally considered to fall into three major categories: star-forming or nuclei, Seyfert nuclei, and low-ionization nuclear emission-line regions, or LINERs. The formal divisions between these classes are somewhat arbitrary, as the observed emission-line ratios of nearby galactic nuclei fall in a continuous distribution between LINERs and Seyfert nuclei and between LINERs and nuclei [e.g., @hfs93]. Traditionally, LINERs have been defined as those nuclei having emission-line flux ratios which satisfy the relations \[\] [$\lambda$]{}3727/\[\] [$\lambda$]{}5007 $> 1$ and \[\] [$\lambda$]{}6300/\[\] [$\lambda$]{}5007 $> 1/3$ [@h80]. It is possible to construct alternative but practically equivalent definitions, based on other line ratios, which can be applied to datasets that do not include the wavelengths of the \[\], \[\], or \[\] lines [e.g., @hfs97a].
A sizeable minority of galactic nuclei has emission-line ratios which are intermediate between those of “pure” LINERs and those of typical regions powered by hot stars; these galaxies would be classified as LINERs except that their \[\] [$\lambda$]{}6300 line strengths are too small in comparison with other lines to meet the formal LINER criteria. Objects falling into this category have been dubbed “transition” galaxies by @hfs93, and although this nomenclature is somewhat ambiguous we adopt it here for consistency with the spectroscopic survey of @hfs97a. That survey defined the transition class in terms of the following flux ratios:
[\[\]]{} [$\lambda$]{}5007/[H$\beta$]{} $<$ 3,\
0.08 $\leq$ \[\] [$\lambda$]{}6300/[H$\alpha$]{} $<$ 0.17,\
[\[\]]{} [$\lambda$]{}6583/[H$\alpha$]{} $\geq$ 0.6,\
[\[\]]{} [$\lambda\lambda$]{}6716, 6731 /[H$\alpha$]{} $\geq$ 0.4.
@ft92 have used the term “weak-\[\] LINERs” to refer to galaxies having \[\] [$\lambda$]{}6583/[H$\alpha$]{} $\gtrsim 0.6$ (typical of LINERs) but which have \[\] [$\lambda$]{}6300/[H$\alpha$]{} $< 1/6$. This category is essentially identical to the transition class of @hfs93 [@hfs97a], and we will refer to these galaxies as transition objects in this paper.
According to the survey results of @hfs97b, this transition class accounts for 13% of all nearby galaxies, making them about as numerous as Seyfert nuclei. The Hubble type distribution of transition galaxies is intermediate between that of LINERs, which are most common in E/S0/Sa galaxies, and that of nuclei, which occur most often in Hubble types later than Sb [@hfs97b]. Roughly $20\%$ of galaxies with Hubble types ranging from S0 to Sbc belong to the transition class. There is not a consensus, however, as to whether these transition objects should be regarded as star-forming nuclei, as accretion-powered active nuclei, or as composite objects powered by an AGN and by hot stars in roughly equal proportion.
There is a large body of literature on the subject of the excitation mechanism of LINERs which is relevant to the similar transition class. A variety of physical mechanisms has been proposed to explain the emission spectra of LINERs, including shocks, photoionization by a nonstellar ultraviolet (UV) and X-ray continuum, and photoionization by hot stars. (See Filippenko 1996 for a review.) The possibility that LINERs (and Seyfert nuclei as well) might be photoionized by starlight was raised by @tm85, who suggested that very hot (${\ensuremath{T_{\rm eff}}}\sim 10^5$ K) Wolf-Rayet (W-R) stars in a metal-rich starburst could give rise to an ionizing continuum with a nearly power-law shape in the extreme-UV. More recent atmosphere models have indicated substantially lower temperatures for W-R stars, however, casting doubt on the Warmer hypothesis [@lgs92]. Subsequent photoionization models have attempted to explain LINER and transition-type spectra as resulting from massive main-sequence stars. @ft92 found that the spectra of weak-\[\] LINERs could be explained in terms of photoionization by O3–O4 stars having effective temperatures of $\gtrsim45,000$ K, at ionization parameters of $U \approx 10^{-3.7}$ to $10^{-3.3}$. @shi92 carried this line of argument farther, proposing that genuine LINER spectra could be generated by early O stars with ${\ensuremath{T_{\rm eff}}}\approx 50,000$ K, provided that a high-density component ($n_e \approx 10^{5.5}$ [cm$^{-3}$]{}) is present in the NLR; the high densities are needed to boost the strengths of high critical-density emission lines, most notably \[\] [$\lambda$]{}6300. Similar conclusions were reached by @sf94, who explored the effects of absorption by ionized gas as a means to harden the effective ionizing spectrum. Recent observations, particularly in the UV and X-ray bands, have provided convincing evidence that many LINERs are in fact AGNs, particularly the “Type 1” LINERs which have a broad component to the [H$\alpha$]{} emission line [for a recent review see @ho99]. The possibility has remained, however, that some LINERs and transition nuclei are powered entirely by bursts of star formation.
An important shortcoming of the model calculations performed by @ft92 and @shi92 is that the ionizing continua used as input were those of single O-type stars; these studies did not address the question of whether a LINER or transition-type spectrum could result from the the *integrated* ionizing continuum of a young stellar cluster. Compared with these single-star models, the contribution of late-O and B stars will soften the ionizing spectrum, making the emission-line ratios tend toward those of normal regions. W-R stars, on the other hand, will harden the ionizing spectrum during the period when these stars are present, roughly $3-6$ Myr after the burst. Another drawback of the O-star models is that they require the presence of stars with effective temperatures higher than are thought to occur in regions of solar or above-solar metallicity, in order to produce a LINER or transition-type spectrum rather than an region spectrum. Their applicability to galactic nuclei is therefore somewhat unclear.
Other mechanisms have been proposed for generating LINER or transition-type spectra. Shock excitation by supernova remnants in an aging starburst may give rise to some transition objects; the nucleus of NGC 253 is a likely candidate for such an object [@eng98]. Also, post-AGB stars and planetary nebula nuclei will produce a diffuse ionizing radiation field which could be responsible for the very faint LINER emission (with [H$\alpha$]{} equivalent widths of $\sim1$ Å) observed in some ellipticals and spiral bulges [@bin94].
An alternate possibility is that the transition galaxies may simply be composite systems consisting of an active nucleus surrounded by star-forming regions. For a galaxy at a distance of 10 Mpc, for example, a 2-wide spectroscopic aperture will include regions within 50 pc of the nucleus. Galaxies having emission lines both from a LINER nucleus and from surrounding star-forming regions, in roughly equal proportions, will appear to have a transition-type spectrum. This interpretation was advocated by @hfs93 as the most likely explanation for the majority of transition galaxies, and is consistent with the observed Hubble type distribution for the transition class. Other authors have similarly contended that transition galaxies are AGN/ region composites, based on optical line-profile decompositions [@vgv97; @gvv99] and near-infrared spectra [@hil99]. Two of the 65 transition nuclei observed in the @hfs97b survey have a broad component to the [H$\alpha$]{} emission line, indicating the likely presence of an AGN, and it is probable that many more transition nuclei contain obscured AGNs which were not detected in the optical spectra. On the other hand, radio observations do not appear to support the composite AGN/starburst interpretation. In a VLA survey of nearby galactic nuclei, Nagar [et al.]{} (1999) find compact, flat-spectrum radio cores in more than 50% of LINER nuclei, but in only 6% (1 of 18) of transition objects. This discrepancy suggests that the simple picture of an ordinary LINER surrounded by star-forming regions may not apply to the majority of transition objects.
Recent results from the *Hubble Space Telescope* ([*HST*]{}) have shed new light on the question of the excitation mechanism of transition nuclei. As shown by @mao98, the UV spectrum of the well-known transition nucleus in NGC 4569 over 1200-1600 Å is virtually identical to that of a W-R knot in the starburst galaxy NGC 1741, indicating that O stars with ages of a few Myr dominate the UV continuum. @mao98 find that the nuclear star cluster in NGC 4569 is producing sufficient UV photons to ionize the surrounding narrow-line region, a key conclusion which provides fresh motivation to study stellar photoionization models. The brightness of the NGC 4569 nucleus, and the consequently high S/N observations that have been obtained, make it one of the best objects with which to study the transition phenomenon.
The recent availability of the [<span style="font-variant:small-caps;">STARBURST99</span>]{} model set [@lei99] has prompted us to reexamine the issue of ionization by hot stars in LINERs and transition nuclei. These models give predictions for the spectrum and luminosity of a young star cluster, for a range of values of cluster age, metal abundance, and stellar initial mass function (IMF) properties. Using the photoionization code CLOUDY [@fer98] in combination with the [<span style="font-variant:small-caps;">STARBURST99</span>]{} model continua, we have calculated the expected emission-line spectrum of an region illuminated by a young star cluster, to test the hypothesis that some LINERs and transition nuclei may be powered by starlight. Similar calculations have been performed by @sl96, but for the physically distinct case of metal-poor objects representing galaxies. Other examples of photoionization calculations for regions using evolving starburst continua are presented by @gd94, @gbd95, and [@bkg99].
The Nucleus of NGC 4569 {#section4569}
=======================
Before describing the photoionization modeling, we review the properties of NGC 4569, as it is among the best-known examples of the transition class. NGC 4569 is a Virgo cluster spiral of type Sab, with a heliocentric velocity of $-235$ [km s$^{-1}$]{}, and we assume a distance of 16.8 Mpc for consistency with the catalog of @hfs97a. Its nucleus is remarkably bright for a non-Seyfert, and so compact in the optical that @hum36 suspected it to be a foreground Galactic star. It is also an unusually bright UV source, with the highest 2200 Å luminosity of the LINERs and transition objects observed by @mao95 and @bar98. [*HST*]{} Faint Object Spectrograph (FOS) spectra show that the UV continuum is dominated by massive stars, with prominent P Cygni profiles of [$\lambda$]{}1549, [$\lambda$]{}1400, and [$\lambda$]{}1240 [@mao98]. The UV spectrum is nearly an exact match to the spectrum of one of the starburst knots in the W-R galaxy NGC 1741, an object with a likely age in the range 3–6 Myr [@clv96]. The optical spectrum of the NGC 4569 nucleus is dominated by the light of A-type supergiants, providing additional evidence for recent star formation [@kee96].
One key result of the @mao98 study was the conclusion that the nuclear starburst in NGC 4569 is producing sufficient numbers of ionizing photons to power the narrow-line region, assuming that the surrounding nebula is ionization-bounded, *even without correcting for the effects of internal extinction on the UV continuum flux*. In fact, there appears to be substantial extinction within NGC 4569, as demonstrated by the UV continuum slope as well as the presence of deep interstellar absorption features [@mao98]. @hfs97a derive an internal reddening of $E(B-V)
= 0.46$ mag from the [H$\alpha$]{}/[H$\beta$]{} ratio, while @mao98 estimate a UV extinction of $A \approx 4.8$ mag at 1300 Å by comparison of the observed UV slope with the expected spectral shape of an unreddened starburst.
Despite the fact that NGC 4569 is often referred to as a LINER, and in some cases presumed to contain an AGN on the basis of that classification, there is no single piece of evidence which conclusively demonstrates that an AGN is in fact present at all. The [*HST*]{} images and spectra are all consistent with the nucleus being a young, luminous, and compact starburst region. No broad-line component is detected on the [H$\alpha$]{} emission line [@hfs97a], and no narrow or broad emission lines are visible at all in the UV spectrum other than the P Cygni features that are generated in O-star winds [@mao98]. Only the optical emission-line ratios point to a possible AGN classification. In a thorough study of optical and *IUE* UV spectra, @kee96 concluded that there was at best weak evidence for the presence of an AGN in NGC 4569, and that any AGN continuum component, if present, must have an unusually steep spectrum.
Furthermore, while the nucleus of NGC 4569 is certainly extremely compact, the UV and optical [*HST*]{} images show that the nucleus is not dominated by a central point source. At 2200 Å, the nucleus appears extended in WFPC2 images with FWHM sizes of 13 and 9 pc along its major and minor axes [@bar98]. Optical WFPC2 images have been discussed recently by @pog99, who state that the nucleus is unresolved by [*HST*]{}. We have obtained these same images from the [*HST*]{}archive. While the nucleus is certainly compact, we find that it is clearly extended even at the smallest radii. A 12-second, CR-SPLIT exposure in the F547M ($V$-band) filter is unsaturated and allows a radial profile measurement. We find a FWHM size of 14 pc by 8 pc along the major and minor axes, consistent with the size of the nuclear cluster measured at 2200 Å. @bar98 estimated that at most 23% of the nuclear UV flux could come from a central point source. From the equivalent widths of stellar-wind features in the UV spectrum, @mao98 give a similar upper limit of $\sim20\%$ to the possible contribution of a truly featureless continuum to the observed UV flux.
X-ray observations of NGC 4569 with *ROSAT* have revealed a source coincident with the nucleus which is unresolved at the 2 resolution of the HRI camera [@cm99]. This does not necessarily indicate that an AGN is present, however, as the optical/UV size of the starburst core is an order of magnitude smaller than the HRI resolution. *ASCA* observations show that the X-ray emission is extended over arcminute scales in both the hard (2–7 keV) and soft (0.5–2 keV) bands [@ter99]. Interestingly, the compact source seen in the *ROSAT* image is detected only in the soft *ASCA* band, while there is no detectable contribution from a compact, hard X-ray source. The spectral shape of the compact soft X-ray component is consistent with an origin either in an AGN or in X-ray binaries [@th99], but the lack of a compact hard X-ray source argues against the AGN interpretation. If an AGN is present, it must be highly obscured even at hard X-ray energies, with an obscuring column of $N_H > 10^{23}$ cm[$^{-2}$]{} [@ter99]. In radio emission, VLA observations show that the NGC 4569 nucleus is an extended source with a size of 4 and no apparent core [@nh92], in contrast with the compact, AGN-like cores found in some LINERs [@fal98].
Shock excitation has often been considered as a mechanism to power the narrow emission lines in LINERs. However, the lack of narrow emission features in the UV spectrum of NGC 4569 argues against shock-heating models for this object, as existing shock models generally predict strong UV line emission [e.g., @dop96]. Shock-excited filaments in supernova remnants show strong emission in high-excitation UV lines such as [$\lambda$]{}1549 and [$\lambda$]{}1640 [e.g., @bla91; @bla95] which are altogether absent from the NGC 4569 spectrum. Similarly, the shock-excited nuclear disk of M87 [@dop97] has a high-excitation UV line spectrum which bears no resemblance to the NGC 4569 spectrum. From an analysis of infrared spectra, @ah99 proposed that the NGC 4569 nucleus is powered by an 8–11 Myr-old starburst, by a combination of stellar photoionization and shock heating from supernova remnants. While this hypothesis may be applicable to some LINERs and transition galaxies, the UV spectrum of NGC 4569 shown by @mao98 is inconsistent with a burst of such advanced age, as the P Cygni features of , , and would have disappeared from a single-burst population after about 6 Myr.
The overall picture emerging from these observations is that the NGC 4569 nucleus is a compact, luminous, and young starburst. The only reason to invoke the presence of an AGN at all would be to explain the higher strengths of the low-ionization forbidden lines in comparison with values observed in normal nuclei. If it were indeed possible for a young starburst to produce transition or LINER-type emission lines in the surrounding gas, then there would be no reason to consider AGN models for NGC 4569.
Photoionization Calculations {#sectioncalc}
============================
The Ionizing Continuum
----------------------
As discussed by @ft92 and @shi92, the key ingredient necessary for generating a LINER or transition-type emission-line spectrum is an ionizing continuum which is harder than that produced by typical clusters of OB stars. A harder continuum will produce a more extended partially-ionized zone in the surrounding region, boosting the strength of the low-ionization lines which are typical of LINER spectra: \[\] [$\lambda$]{}6300, \[\] [$\lambda$]{}3727, \[\] [$\lambda$]{}[$\lambda$]{}6548,6583, and \[\] [$\lambda$]{}[$\lambda$]{}6716,6731.
To represent the ionizing continuum of a young starburst, we have chosen the [<span style="font-variant:small-caps;">STARBURST99</span>]{} model set; we refer the reader to @lei99 for the details of the methods used to construct these models. Briefly, the [<span style="font-variant:small-caps;">STARBURST99</span>]{} code employs the Geneva stellar evolution models of @mey94, with enhanced mass-loss rates, for high-mass stars. Atmospheres are represented by the models compiled by @lcb97 and @sch92. Figures 1–12 of @lei99 display the spectral energy distributions of the [<span style="font-variant:small-caps;">STARBURST99</span>]{} model clusters for a range of burst ages and for a variety of initial conditions. From the figures, some important trends are readily apparent. During the first 2 Myr after an instantaneous burst, the continuum is dominated by the hottest O stars, and there is essentially no emission below 228 Å, corresponding to the ionization energy of He$^{+}$. The appearance of W-R stars during the period 3–5 Myr after the burst results in a dramatic change in the UV continuum, as these stars emit strongly in the He$^{++}$ continuum below 228 Å. From 6 Myr onwards, the W-R stars disappear and the UV continuum rapidly fades and softens as the burst ages. Only the models with an upper mass limit of [$M_{\rm up}$]{} = 100[$M_{\odot}$]{} generate the hard, W-R-dominated UV continuum; the model sequences with [$M_{\rm up}$]{} = 30[$M_{\odot}$]{} do not generate significant numbers of photons below 228 Å for any ages because the progenitors of W-R stars are not present in the initial burst. Constant star-formation rate models with [$M_{\rm up}$]{} = 100[$M_{\odot}$]{} form W-R stars continuously after 3 Myr, but the overall shape of the UV continuum is softer than in the instantaneous burst models, because of the continuous formation of luminous O stars.
These results provide a useful starting point for the photoionization calculations. If it is possible for the region surrounding a young cluster to resemble a LINER or transition object, then this is most likely to occur when the ionizing continuum is hardest, when W-R stars are present during $t \approx 3-5$ Myr after a burst. Very massive stars (in the range 30–100 [$M_{\odot}$]{} or greater) must be present in the burst or else the requisite W-R stars will not appear. The formation of W-R stars is enhanced at high metallicity, so the ability to generate a LINER or transition-type spectrum may be a strong function of metal abundance as well as age.
Model Grid
----------
To create grids of photoionization models, we fed the UV continua generated by the [<span style="font-variant:small-caps;">STARBURST99</span>]{} models into the photoionization code CLOUDY [version 90.04; @fer98]. For each time step, a grid of models was calculated by varying the nebular density and the ionization parameter, which is defined as the ratio of ionizing photon density to the gas density at the ionized face of a cloud. Real LINERs and transition nuclei are likely to contain clouds with a range of values of density and ionization parameter, and more general models incorporating density and ionization stratification can be constructed as linear combinations of these simple single-zone models.
All models were run with the following range of parameters: burst age from 1 to 10 Myr at increments of 1 Myr, with log $U$ ranging from $-2$ to $-4$ at increments of 0.5, and a constant density ranging from log ([$n_{\rm H}$]{}/[cm$^{-3}$]{}) = 2 to 6 at increments of 1. As a starting point, we computed a grid for an instantaneous burst with an IMF having a power-law slope of $-2.35$, [$M_{\rm up}$]{} = 100 [$M_{\odot}$]{}, solar metallicity in stars and gas, and a single plane-parallel slab of gas with no dust; we will refer to this as model grid A. The solar abundance set was taken from @ga89 and @gn93. Other grids were computed as variations on this basic parameter set, with the following modifications made in different model runs: a constant star-formation rate; metallicity 0.2, 0.4, or 2[$Z_{\odot}$]{} in both stars and gas; and spherical geometry for the nebula. To assess the effects of the highest-mass stars, we also ran custom model grids, via the [<span style="font-variant:small-caps;">STARBURST99</span>]{}web site, with ${\ensuremath{M_{\rm up}}}= 70$ and 120 [$M_{\odot}$]{}.
The depletion of heavy elements onto grains can result in marked changes to the emergent emission-line spectrum of an region, both by removing gas-phase coolants from the nebula and by grain absorption of ionizing photons, which will modify the effective shape of the ionizing continuum. In metal-rich regions, these effects will tend to boost the strengths of the low-ionization emission lines relative to the dust-free case [@sk95]. To assess the effects of dust in transition nuclei, we calculated additional model grids which included dust grains with a Galactic ISM dust-to-gas ratio along with the corresponding gas-phase depletions. The dusty models were all calculated using the solar abundance set for the undepleted gas. Dust grains were assumed to have the optical properties of Galactic ISM grains, as described by @mrn77, @dl84, and @mr91. From the CLOUDY output files, we tabulated the strengths relative to [H$\beta$]{} of the major emission lines which are prominent in LINERs.
The calculations were performed under the assumption that the region is ionization bounded. For this case, the outer extension of the cloud was set to be the radius at which $T_e$ falls to 4000 K, beyond which essentially no emission is generated in the optical or UV lines. As a test, we ran a grid of models with the stopping temperature set to 1000 K, and we verified that the emission-line ratios were essentially identical to the default case of 4000 K stopping temperature. We also verified that the important diagnostic line ratios differed by $\lesssim0.1$ dex between the spherical and plane-parallel cases when all other input parameters were unmodified, and all results discussed in this paper refer to the plane-parallel models. In the calculations, the longest timescales for atomic species to reach equilibrium were of order $10^3$ years, much shorter than the evolution timescale of the stellar cluster, justifying the assumption that each time step of the cluster evolution could be used independently to calculate the nebular conditions. Table \[table1\] gives a summary of the model parameters, for the model grids which appear in the following discussion.
Discussion {#sectiondiscuss}
==========
Model Results
-------------
The model results are displayed in Figures \[oratio\]–\[ohimetal\]. To compare the model outputs with the observed properties of a variety of galaxy types, we have used the emission-line data compiled by @hfs97a. This catalog has the advantages of a homogeneous classification system, small measurement aperture ($2\arcsec \times 4\arcsec$), and careful starlight subtraction to ensure accurate emission-line data. In order to reduce confusion and to keep the sample of comparison objects to a reasonable number, we included only objects with unambiguous classifications as , LINER, transition, or Seyfert. Objects with borderline or ambiguous classifications, such as “LINER/Seyfert,” were excluded for clarity. The comparison sample was further reduced by excluding galaxies in which any of the emission lines [H$\alpha$]{}, [H$\beta$]{}, \[\] [$\lambda$]{}5007, \[\] [$\lambda$]{}6300, \[\] [$\lambda$]{}6583, or \[\] [$\lambda$]{}[$\lambda$]{}6716,6731 was undetected or was flagged as having a large uncertainty in flux (“b” or “c” quality flags). The measured line ratios are corrected for both Galactic and internal reddening.
Figure \[oratio\] plots the ratio \[\] [$\lambda$]{}5007/[H$\beta$]{}against \[\] [$\lambda$]{}6300/[H$\alpha$]{} at a burst age of 4 Myr, for the solar-metallicity model grids A, B, C, and D. The model results are plotted for a density of ${\ensuremath{n_{\rm H}}}= 10^3$ [cm$^{-3}$]{}, as an approximate match to the density of $n_e = 600$ [cm$^{-3}$]{} measured for NGC 4569 [@hfs97a]. The diagram shows that the instantaneous burst models (A and B) are a good match to the line ratios of the transition nuclei, for log $U \approx -3.5$. The dusty models from grid B fall more centrally within the region defined to contain transition objects, but the models without dust still closely match transition nuclei having lower \[\] [$\lambda$]{}6300/[H$\alpha$]{} ratios. Figures \[nratio\] and \[sratio\] show the corresponding diagrams for \[\] [$\lambda$]{}6583/[H$\alpha$]{} and for \[\] [$\lambda\lambda$]{}6716, 6731/[H$\alpha$]{}, respectively. In both cases we find that the single-burst models span the region occupied by transition nuclei in the diagnostic diagrams.
In the constant star-formation rate models (C and D), the UV continuum remains softer than in the W-R-dominated phase of the instantaneous burst models, because of the ongoing formation of luminous O stars. As a result, the low-ionization emission lines are significantly weaker than in the instantaneous burst models at 4 Myr. These constant star-formation rate sequences are a reasonable match to the region of nuclei in the diagram, and for \[\]/[H$\alpha$]{}and \[\]/[H$\alpha$]{} the agreement with nuclei is improved at lower densities of $n_e = 10^2$ [cm$^{-3}$]{}, a value more typical of nuclei. These constant star-formation rate models are probably appropriate for galaxies having spatially extended, ongoing star formation in their nuclei.
We note that the model curves shown in the figures should not be expected to follow the locus of nuclei in each plot, despite the fact that the models are generated with a starburst continuum. The range of line ratios observed in regions is primarily a sequence in metal abundance [@mrs85], while our models are shown as sequences in $U$ for a given metallicity and density. Another point to note about the diagrams is that some of the transition galaxies fall outside the region nominally defined for transition objects, particularly in the \[\]/[H$\alpha$]{} ratio (Figure \[oratio\]). These galaxies were classified as transition objects by @hfs97a on the basis of meeting the majority of the classification criteria. Similarly, some overlap can be seen in the diagrams between the regions occupied by LINERs and Seyfert nuclei; this again reflects the fact that galaxies span a continuous range in the values of these emission-line ratios.
The variation of \[\] line strength as a function of density is shown in Figure \[odensity\] for model grid A at $t = 4$ Myr. For the range of densities considered (${\ensuremath{n_{\rm H}}}= 10^2$ to $10^6$ [cm$^{-3}$]{}), the models closely overlap the transition region in the diagram at log $U \approx -3.5$. Introducing ISM depletion and dust grains to the nebula primarily increases the \[\]/[H$\alpha$]{} ratio at low density. As expected, the \[\]/[H$\alpha$]{} ratio increases with [$n_{\rm H}$]{} up to densities of $10^5$ [cm$^{-3}$]{}, while at densities approaching the critical density of the [$\lambda$]{}6300 transition ($1.6
\times 10^6$ [cm$^{-3}$]{}) this ratio saturates and begins to turn over.
Figure \[oage\] shows the \[\]/[H$\alpha$]{} ratio as a function of burst age (at $n_e = 10^3$ [cm$^{-3}$]{}) for model grid A, for ages of 2 to 6 Myr. This diagram highlights the dramatic changes that W-R stars generate in the surrounding nebula. From 3 to 5 Myr after the burst, when W-R stars dominate the UV continuum, the harder ionizing continuum boosts the strength of \[\] by an order of magnitude and the model sequences appear adjacent to the transition region, with relatively little evolution in the line ratios during this period. As the burst ages beyond 6 Myr, the \[\]/[H$\alpha$]{} ratio continues to fall, and the emission-line strengths drop rapidly as the ionizing continuum softens and its luminosity decreases. At $Z = 2{\ensuremath{Z_{\odot}}}$, the WR-dominated phase occurs slightly later, during time steps 4, 5, and 6 Myr. The \[\]/[H$\alpha$]{} and \[\]/[H$\alpha$]{} ratios have a similar dependence on burst age, and are displayed in Figures \[nage\] and \[sage\], respectively; these results are quite similar to the calculations presented by @lgs92 to illustrate the effects of the W-R continuum on the \[\]/[H$\alpha$]{} ratio. While the \[\]/[H$\alpha$]{} ratios in the models at $\log U = -3.5$ are too low by $\sim0.1-0.2$ dex to fit within the nominal transition region, they still closely match those transition nuclei having relatively low values for \[\]/[H$\alpha$]{}, and the low-ionization line strengths can be further enhanced by the inclusion of dust and depletion (as in Figure \[oratio\]).
One puzzling aspect of Figure \[oage\] is that for ages outside the range 3–5 Myr, the models predict \[\]/[H$\alpha$]{} ratios too low to match the majority of nuclei. @sl96 and @m97 discuss this same problem in the context of low-metallicity starburst galaxies. They propose that shocks generated by supernovae and stellar winds provide the additional \[\] emission, without making a significant contribution to the \[\] or \[\] line strengths. Shocks could play a similar role in transition galaxies, as in the model of [@ah99], although the lack of high-excitation UV line emission in transition nuclei is problematic for the shock hypothesis. A higher upper mass cutoff alleviates this problem to some extent, at least for very young bursts. Increasing [$M_{\rm up}$]{} to 120 [$M_{\odot}$]{} boosts \[\]/[H$\alpha$]{} by $\sim0.2$ dex for ages of $\lesssim3$ Myr. Due to the very short lifetimes of the highest-mass stars, however, the [$M_{\rm up}$]{} = 120, 100, and 70 [$M_{\odot}$]{} model grids result in identical emission-line spectra from $\sim4$ Myr onward.
Metal abundance is an additional parameter which must be considered. The models displayed up to this point were all calculated for a solar abundance set, while the nuclei of early-type spirals are likely to have enhanced heavy-element abundances. As discussed by @lei99, the continuum shortwards of 228 Å is strongest in the high-metallicity models, because the increased mass-loss rates lead preferentially to the formation of W-R stars at high metal abundance. Figure \[olometal\] plots the \[\]/[H$\alpha$]{} ratio for abundances of $Z$ = 0.2, 0.4, 1, and 2 [$Z_{\odot}$]{}, at a density of ${\ensuremath{n_{\rm H}}}= 10^3$ [cm$^{-3}$]{} and $t = 4$ Myr. From this diagram, it is clear that solar or higher abundances are necessary to match the \[\] strengths of the transition nuclei; at lower abundances the line ratios are a better match to those of the high-excitation (low-metallicity) nuclei. Figure \[ohimetal\] displays the density dependence of the \[\]/[H$\alpha$]{} ratio for the $Z =
2{\ensuremath{Z_{\odot}}}$ model grid; by comparison with the solar-metallicity model grid in Figure \[odensity\], the higher abundances result in a lower-excitation spectrum with enhanced \[\] emission, due to the harder extreme-UV continuum.
It would be advantageous to compare the model results with a wider variety of emission lines. Unfortunately, measurements of other optical emission lines are scarce for transition nuclei. The Ho [et al.]{} survey did not include the \[\] [$\lambda$]{}3727 line, and there is no other homogeneous catalog of \[\] measurements for transition galaxies. To be consistent with a LINER or transition-type classification, a model calculation must result in the flux ratio \[\] [$\lambda$]{}3727 / \[\] [$\lambda$]{}5007 $>1$. In fact, all of the models with log $U \leq -3$ do satisfy this criterion. Thus, any of our models which is consistent with the Ho [et al.]{} LINER or transition classification criteria is also consistent with the original @h80 criterion for the \[\]/\[\] ratio in LINERs. The relative strengths of UV lines such as \] [$\lambda$]{}2326, \] [$\lambda$]{}1909, and [$\lambda$]{}1549 can provide further diagnostics, but none of these lines is detected in NGC 4569 [@mao98]. The only other transition nucleus having [*HST*]{} UV spectra available is NGC 5055, and its spectrum appears to be devoid of UV emission lines as well [@mao98].
We ran one additional model grid to test whether different model atmospheres for O stars would lead to different results. The [<span style="font-variant:small-caps;">STARBURST99</span>]{}continua were calculated using stellar atmosphere models compiled by @lcb97, which are based on the @kur92 model set for the massive stellar component. The recent CoStar model grid of @sdk97, which includes non-LTE effects, stellar winds, and line blanketing for O stars, makes dramatically different predictions for the ionizing spectra. As shown by @sv98, the CoStar models yield a luminosity in the He$^{++}$ continuum which is four orders of magnitude greater than that predicted by the Kurucz models, for the most massive O stars which dominate the UV luminosity at burst ages of $<3$ Myr. In the CoStar-based models the photon output of the cluster below 228 Å is essentially constant from 0 to 5 Myr. To investigate the effects of this harder O-star continuum on the emission-line spectra, we ran a grid of models using the evolving starburst continua computed by @sv98 with the CoStar atmospheres. Model parameters were the same as for model grid A except that an upper mass limit of 120 [$M_{\odot}$]{} was used. We find that using the CoStar atmospheres has a relatively minor effect on our results. In comparison with the [<span style="font-variant:small-caps;">STARBURST99</span>]{}-based models having [$M_{\rm up}$]{} = 120[$M_{\odot}$]{}, the CoStar model grid yields an increase in the \[\]/[H$\alpha$]{} ratio of $\sim0.1-0.15$ dex for $t < 6$ Myr, while \[\]/[H$\alpha$]{} and \[\]/[H$\alpha$]{} are essentially unaffected. During the period $t < 3$ Myr, the CoStar-based models result in an region spectrum, demonstrating that W-R stars are still required in order to generate LINER or transition-type line ratios.
The strength of the \[\] emission lines at 7291 and 7324 Åis often used as a diagnostic of dust and depletion, because in the absence of depletion these lines are predicted to be strong in photoionized gas [e.g., @kff95; @vmb96]. (The [$\lambda$]{}7291 line is a cleaner diagnostic since [$\lambda$]{}7324 is blended with \[\] [$\lambda$]{}7325.) However, for a 4 Myr-old burst with nebular conditions of $n_e = 10^3$ [cm$^{-3}$]{}, $\log U = -3.5$, and an undepleted solar abundance set, our calculations yield a maximum prediction of only 0.2 for the ratio of \[\] [$\lambda$]{}[$\lambda$]{}7291, 7324 to [H$\beta$]{}. Only at very low ionization parameters ($\lesssim 10^{-4.5}$) does the \[\] emission become stronger than [H$\beta$]{}. Since [H$\beta$]{} is only barely visible in the spectra of many transition objects (prior to careful starlight subtraction, at least), typical observations may not have sufficient sensitivity to detect faint \[\] lines in these objects. High-quality spectra of LINERs do not show \[\] emission [@hfs93], indicating that Ca is likely to be depleted onto grains in these objects, but similar data are not generally available for transition nuclei. If the \[\] lines are found to indicate a high level of depletion onto dust grains in transition nuclei, this would also provide a further argument against shock-heating models, as shocks will tend to destroy grains [e.g., @mrw96].
The Nature of Transition Nuclei {#sectiontransition}
-------------------------------
The results shown in the preceding figures demonstrate that the starburst models are in fact able to reproduce the major diagnostic emission-line ratios of transition nuclei with reasonable accuracy, during the period $t$ = 3–5 Myr when W-R stars are present. For a density of $10^3$ [cm$^{-3}$]{} and an age of 4 Myr, the solar-metallicity models with and without depletion bracket the range of values observed in real transition nuclei for the line ratios \[\]/[H$\alpha$]{}, \[\]/[H$\alpha$]{}, and \[\]/[H$\alpha$]{}. We do not attempt to fine-tune a model to produce an exact match with the spectrum of NGC 4569, but the basic solar-metallicity dust-free model at $t = 4$ Myr with $n_{\rm H} = 10^3$ [cm$^{-3}$]{} and log $U = -3.5$ closely fits the observed \[\]/[H$\alpha$]{} ratio, while overpredicting \[\]/[H$\alpha$]{} by $\sim0.2$ dex and underpredicting \[\]/[H$\alpha$]{} by $\sim0.1$ dex.
We emphasize that the starburst models are only able to produce transition-type spectra for the case of an instantaneous burst; that is, when the burst duration is shorter than the timescale for evolution of the most massive stars. Multiple-burst populations can only yield a transition spectrum if the dominant population is $\sim3-5$ Myr old and the older or younger bursts do not contribute significantly to the ionizing photon budget. Models with a constant star-formation rate produce region spectra at all ages, as the softer ionizing continua do not produce sufficient \[\] [$\lambda$]{}6300 emission in the surrounding region to match transition-type spectra. The parameter which is most important for determining the hardness of the ionizing continuum is the number ratio of W-R stars to O stars, which exceeds $\sim0.15$ during the W-R-dominated phase in the [<span style="font-variant:small-caps;">STARBURST99</span>]{} models at solar metallicity, and approaches or exceeds unity at $Z = 2{\ensuremath{Z_{\odot}}}$. In the constant star-formation rate models at solar metallicity, the W-R/O ratio levels off at $\sim0.06$ after about 4 Myr. The compact size of the NGC 4569 nucleus is consistent with the requirement that the burst duration must be brief ($\lesssim1$ Myr) in order to generate a transition-type spectrum. The FWHM size of the starburst core in NGC 4569 is only $\sim10$ pc. For such a burst to occur in $\lesssim1$ Myr would require a propagation speed for star formation of only $\sim10$ [km s$^{-1}$]{}. In fact, the typical velocities in the NGC 4569 nucleus are much greater than 10 [km s$^{-1}$]{}: the \[\] [$\lambda$]{}6583 line has a velocity width of 340 [km s$^{-1}$]{} [@hfs97a]. Thus, the NGC 4569 nucleus could represent the result of a single, rapid burst of star formation.
Although our results suggest that transition galaxy spectra may be attributed to a starburst with a high W-R/O-star ratio, the demographics of transition nuclei and nuclei indicate that many transition galaxies are probably not formed by this mechanism. In the [<span style="font-variant:small-caps;">STARBURST99</span>]{} models, the W-R-dominated phase in an instantaneous burst lasts for $\sim3$ Myr (i.e., 3 time steps in the calculations). An region surrounding an instantaneous burst will be visible for $\sim6$ Myr, after which the emission lines will fade rapidly [e.g., @gd94]. Thus, for an instantaneous burst population, the transition phase and the nucleus phase will have approximately equal lifetimes. If all nuclei consisted of instantaneous burst stellar populations with nebular conditions conducive to the formation of transition-type spectra, then nuclei and transition nuclei should be roughly equal in number. In reality, it is likely that a large fraction of star-forming nuclei contain multiple bursts of star formation and/or conditions of low density or low metallicity, so all star-forming nuclei should not be expected to evolve through a transition-type phase. Although it is difficult to make specific predictions, it is probably safe to conclude that for a given Hubble type, transition nuclei generated solely by starbursts should be considerably less numerous than ordinary nuclei.
The statistics compiled by @hfs97b provide a basis for comparison. In early-type galaxies (E and S0), transition nuclei outnumber nuclei by a 3-to-1 margin. Only for Hubble types Sb and later do nuclei begin to outnumber transition nuclei by a factor of 2 or more. The most straightforward interpretation of this trend is that in early-type host galaxies, the majority of transition nuclei are actually AGN/ region composites, as proposed by @hfs93 and others. At intermediate and late Hubble types, the population of transition nuclei may consist of both composite objects and “pure” starbursts evolving through the W-R-dominated phase.
The presence of transition nuclei in a small fraction ($\sim10\%$) of elliptical galaxies [@hfs97b] presents a particularly intriguing problem. The Ho [et al.]{} survey detected five transition nuclei in ellipticals but not a single case of an elliptical galaxy hosting an nucleus. Given that the models which have been considered for transition nuclei involve star formation, either alone or in combination with an AGN, this observation is rather puzzling. Perhaps faint AGNs in elliptical nuclei can produce transition-type spectra without substantial star formation activity. Four of the five transition nuclei found in ellipticals by [@hfs97a] have borderline or ambiguous spectroscopic classifications, however, so “pure” transition nuclei in ellipticals are evidently quite rare.
Given these results, one might expect to see transition-type emission spectra in some fraction of disk regions in spiral galaxies, but in fact such spectra are never found. Single-burst models for disk region spectra are only compatible with observed line ratios for model ages of $t < 3$ Myr [@bkg99], as the harder ionizing spectrum after 3 Myr makes the models overpredict the strengths of the low-ionization lines. Bresolin [et al.]{} suggest that either current stellar evolution models are at fault, or that disk regions are disrupted before reaching an age of 3 Myr, in which case the W-R phase would not be observed in the nebular gas. An alternate (and perhaps more attractive) possibility is that the majority of nuclei, as well as disk regions, are better described by the models with constant star-formation rate, or contain multiple bursts of star formation with an age spread of a few Myr, which would result in a spectrum similar to the constant star-formation rate models.
For understanding the physical nature of transition objects, the observational challenge is to search for any unambiguous signs of nonstellar activity. Detection of broad [H$\alpha$]{} emission, or a compact source of hard X-ray emission with a power-law spectrum, would provide evidence for an AGN component. High-resolution optical spectra (from [*HST*]{}) could provide a means to spatially resolve a central AGN-dominated narrow-line region from the surrounding starburst-dominated component. Since direct evidence for accretion-powered nuclear activity in transition nuclei is generally lacking, it should not be assumed that any given transition object actually contains an AGN unless observations specifically support that interpretation.
One further effect that should be considered in starburst models in the future is photoionization by the X-rays generated by the starburst. X-ray binaries and supernova remnants will provide high-energy ionizing photons, resulting in a spatially extended source of soft X-ray emission as observed in the nucleus of NGC 4569 [@ter99], for example. (The massive main sequence stars will contribute only a negligible amount to the total X-ray luminosity of a starburst; see Helfand & Moran 1999.) Photoionization by X-rays will naturally lead to an enhancement of the low-ionization forbidden lines, and this could contribute to the excitation of some transition galaxies.
LINERs {#sectionliner}
------
The strength of \[\] [$\lambda$]{}6300 is the key distinguishing factor between LINERs and transition nuclei, and matching the observed strength of this line is the major challenge of starburst models for LINERs. Our calculations show that LINER spectra can only be generated by the [<span style="font-variant:small-caps;">STARBURST99</span>]{} clusters under a very specific and limited range of circumstances. Model grids A and B, while matching the \[\] / [H$\alpha$]{} ratio of transition nuclei quite well, do not overlap at all with the main cluster of LINERs in Figure \[oratio\], even at high densities and even when depletion and dust grains are included. Only grid G with $Z = 2{\ensuremath{Z_{\odot}}}$ is able to replicate the high \[\] / [H$\alpha$]{} ratios of most LINERs, and only during $t
\approx$ 4–6 Myr and at densities of [$n_{\rm H}$]{} $\gtrsim 10^5$ [cm$^{-3}$]{}. In agreement with previous models, we find that values of log $U
\approx$ $-3.5$ to $-3.8$ reproduce the observed \[\] / [H$\alpha$]{}ratios of LINERs. However, at such high densities the models underpredict the strengths of \[\] and \[\] relative to [H$\alpha$]{}. Single-zone models require $n_e \lesssim 10^5$ [cm$^{-3}$]{} to match the \[\]/[H$\alpha$]{} ratios of LINERs and $n_e \lesssim 10^4$ [cm$^{-3}$]{} for \[\]/[H$\alpha$]{}.
Agreement with LINER spectra can be achieved with a simple two-zone model, in which high-density and low-density components are present, similar to the scenario proposed by @shi92. As an example, a two-component model constructed from grid A containing gas at ($n_e =
10^3$ [cm$^{-3}$]{}, $U = 10^{-3.5}$) and at ($n_e = 10^5$ [cm$^{-3}$]{}, $U =
10^{-4}$) produces emission-line ratios which are consistent with all the LINER classification criteria of both the @h80 and @hfs97a systems, if the two density componenets are scaled so as to contribute equally to the total [H$\beta$]{} luminosity. As a local comparison, observations of near-infrared emission indicate the presence of clouds having $n_e > 10^5$ [cm$^{-3}$]{} in the Galactic center region [@dep92], so it is plausible that other galactic nuclei may contain ionized gas at similarly high densities even in the absence of an observable AGN. A starburst origin for some LINER 2 nuclei would provide a natural explanation for the lower values of the X-ray/[H$\alpha$]{} flux ratio seen in these objects, in comparison with AGN-like LINER 1 nuclei [@ter99].
It seems unlikely, however, that many LINERs are generated by this starburst mechanism. About 15% of LINERs are known to have a broad component of the [H$\alpha$]{} emission line, indicating a probable AGN [@hfs97b]. By analogy with the Seyfert population, a much larger fraction of LINERs is likely to have broad-line regions which are either obscured along our line of sight, or are simply too faint to be detected in ground-based spectra against a bright background of starlight. Many LINERs show signs of nuclear activity that cannot be explained by stellar processes: compact flat-spectrum radio sources or jets, compact X-ray sources with hard power-law spectra, or double-peaked broad Balmer-line emission, for example. As an increasing body of observational work supports the idea that many LINERs are in fact AGNs, there is less incentive to consider purely stellar models for their excitation.
Demographic arguments, similar to those given above for transition nuclei, can be applied for the LINER population. Since the LINER phase only occurs for instantaneous bursts at high density and high metallicity, the starburst scenario implies that nuclei should be considerably more numerous than starburst-generated LINERs for a given Hubble type. While LINERs are common in early-type hosts, nuclei are not found in elliptical hosts and are seen in fewer than 10% of S0 galaxies [@hfs97b]. This disparity is a strong argument against a starburst origin for those LINERs in early-type galaxies. In later Hubble types the situation is less clear, however. nuclei occur in $\sim80\%$ of spirals of type Sc and later, while LINERs occur in just 5% of these galaxy types [@hfs97b]. It is conceivable that some of the LINERs in intermediate to late-type hosts could have a starburst origin, and this issue could be resolved by further UV and X-ray observations in the future. Interestingly, the Ho [et al.]{} survey did not find any examples of broad [H$\alpha$]{} emission in LINERs or transition objects with hosts of type Sc or later; perhaps star formation plays a more prominent role than accretion-powered activity in these objects.
While a few LINERs show spectral features of young stars in the UV [@mao98], the quality of the observational data is poor in comparison with the NGC 4569 UV spectrum, and it is difficult to set meaningful constraints on the age of the young stellar population. NGC 404 is a possible candidate for a starburst-generated LINER, but in its UV spectrum the P Cygni features are weak in comparison with NGC 4569, indicating either an older burst population or dilution by a featureless AGN continuum [@mao98]. The LINERs having UV spectral features from massive stars may also host obscured AGNs which can be detected in other wavebands. For example, the UV continuum of the LINER NGC 6500 appears to have its origin in hot stars [@bar97; @mao98], but observations of a parsec-scale radio jet unambiguously demonstrate that nonstellar activity is occurring as well [@fal98].
W-R Galaxies with LINER or Transition-Type Spectra
--------------------------------------------------
The starburst models presented here run into two obvious problems. First, W-R galaxies are almost never known to have LINER or transition-type spectra. Second, LINERs and transition nuclei almost never show W-R features in their spectra. Is there any way to reconcile the starburst models with these facts?
W-R galaxies are identified by the appearance of the 4650 Å blend in their spectra [e.g., @ks81]. Since the formation of W-R stars is enhanced at high $Z$, the strength of this feature relative to [H$\beta$]{} increases dramatically with metallicity, from $\lesssim0.1$ at $Z < 0.4{\ensuremath{Z_{\odot}}}$ to $\sim0.5 - 4$ at $Z \geq {\ensuremath{Z_{\odot}}}$ [@sv98]. However, in the nuclei of early-type spirals where high metallicities are expected, a nuclear starburst will be surrounded by the old stellar population of the galactic bulge, making the detection of the W-R bump extremely difficult [@mkc99]. Most of the currently known W-R galaxies are late-type spirals or irregular galaxies [@scp99] in which the W-R bump is visible against the nearly featureless starburst continuum. When the W-R bump is detected in galaxies, its amplitude above the continuum level is generally far smaller than that of [H$\beta$]{} or even [H$\gamma$]{}[e.g., @ks81]. In most of the LINER and transition galaxy spectra in the catalog of @hfs95, however, [H$\beta$]{} barely appears and [H$\gamma$]{} is too weak to be visible at all prior to continuum subtraction. Even in a high-metallicity environment, where the total intensity of the W-R bump can be comparable to that of [H$\beta$]{}, the amplitude of the W-R bump above the continuum will be much lower than that of [H$\beta$]{} because the flux in the W-R feature is spread over $\sim70-100$ Å. Thus, the detection of W-R emission in galactic nuclei is strongly biased toward late-type, bulgeless galaxies. In late-type or dwarf irregular galaxies where the W-R bump is visible, the W-R/O-star ratio is expected to be much smaller owing to the lower metallicity, and the resulting softer ionizing spectrum will tend to produce an region spectrum rather than a transition object. The gas density as a function of Hubble type may play a role as well; in a study of nuclei, @hfs97c find a weak trend toward lower nebular densities in later-type host galaxies.
Observational detection of W-R features in transition nuclei is perhaps the clearest test of the starburst models, if sufficiently sensitive observations can be obtained. The UV spectrum of NGC 4569 is consistent with an age of $\sim3-6$ Myr, an age at which W-R stars are expected to be present. Previous optical spectra have not revealed the 4650 Å W-R bump in NGC 4569, but further observations with high S/N and small apertures would be worthwhile. The lack of [$\lambda$]{}1640 emission in the UV spectrum of NGC 4569 is potentially a more serious problem since the burst population should dominate at short wavelengths. The models of @sv98 predict an equivalent width of at least 2 Å in the W-R-generated [$\lambda$]{}1640 line during the period 3–6 Myr for an instantaneous burst of solar or higher metallicity, while the observed upper limit of $f(1640) < 2.0
\times 10^{-15}$ erg s[$^{-1}$]{} cm[$^{-2}$]{} [@mao98] corresponds to an equivalent width limit of $\lesssim 0.3$ Å. It should be noted, however, that the [$\lambda$]{}1640 line lies at the extremely noisy blue end of the FOS G190H grating setting, in a region where detection of emission or absorption features is difficult.
Two W-R galaxies may provide useful points of reference for the starburst models. NGC 3367 is classified by @hfs97a as an nucleus on the basis of its \[\]/[H$\alpha$]{} and \[\]/[H$\alpha$]{} ratios, although its \[\]/[H$\alpha$]{} ratio of 0.83 is more consistent with a LINER or transition-type classification and its emission lines are markedly broader than those of typical nuclei. @ah99 describe NGC 3367 as a starburst-dominated transition object [see also @dda88]. The 4650 Å W-R bump was noted by @hfs95, who also suggested a LINER/ classification and a composite source of ionization. As a borderline nucleus/transition object with clear evidence for W-R stars, this object deserves further study, to determine whether there is indeed an AGN or whether the enhanced low-ionization emission may be the result of ionization by the W-R population. The electron density of 835 [cm$^{-3}$]{} measured from the \[\] doublet [@hfs97a] is also noteworthy, as this is among the highest densities found for an nucleus in the Ho [et al.]{} survey.
Another intriguing object is the nucleus of NGC 6764, which has been classified variously as a Seyfert, a LINER, and a starburst by different authors [see @gvv99]. This galaxy exhibits prominent emission in the 4650 Å W-R blend [@oc82]. A recent study by @eck96 demonstrates that the narrow emission lines are consistent with a LINER classification, but there are no unambiguous signs of nonstellar activity in the nucleus. @eck96 find that the nucleus contains $\sim3600$ W-R stars, and that the overall properties of the object are consistent with ionization by the starburst alone, rather than by a starburst/AGN composite. If this conclusion is confirmed by further observations, NGC 6764 could be considered the best candidate for a LINER photoionized by a starburst during its W-R-dominated phase. @ah99 derived an age of 9–10 Myr for the starburst in NGC 6764 based on near-infrared emission-line diagnostics, but this age is inconsistent with the presence of W-R stars, at least for the case of an instantaneous burst.
Compared with typical LINERs and transition nuclei, conditions for detection of W-R spectral features in these two objects are perhaps more favorable. Both host galaxies are of late Hubble types (SBc for NGC 3367 and SBbc for NGC 6764) in comparison with the majority of LINERs and transition nuclei, so the level of contamination by the surrounding old stellar population is relatively low. Furthermore (and partly as a result of this), the emission-line equivalent widths in these two nuclei are relatively high for LINERs or transition nuclei. If the emission-line spectra of NGC 3367 or NGC 6764 had been superposed on a luminous early-type spiral bulge, the W-R emission might never have been noticed.
The detection of W-R emission in a LINER does not automatically imply a purely starburst origin for the emission lines, of course, since starbursts and AGNs are often known to coexist. Mrk 477 is a well-known example of a Seyfert 2 galaxy having a large population of W-R stars in its nucleus [@h97]. In the context of the starburst model, however, the crucial test is to search for additional examples of LINERs or transition nuclei which exhibit a high ratio of W-R to O stars but no signs of accretion-powered activity.
Caveats and Limitations
-----------------------
The most important limitation of these calculations comes from the accuracy of the input continua. The conclusion that the starburst models are able to reproduce transition or LINER spectra under some circumstances depends crucially on the presence of W-R stars to provide a hard and luminous ionizing continuum. Unfortunately, the continuum shape and luminosity of W-R stars in the extreme-UV band are quite uncertain, particularly in the He$^{++}$ continuum where stellar winds have a dramatic effect. Several up-to-date reviews of the numerous difficulties involved in modeling W-R spectra can be found in the volume edited by [@vdh99]. The [<span style="font-variant:small-caps;">STARBURST99</span>]{} model grid uses the W-R atmospheres of @sch92, which are calculated for a pure helium composition, but more recent atmosphere models are beginning to include the effects of line-blanketing, as well as clumping and departures from spherical symmetry. As discussed by @lei99, the W-R/O-star ratio is also extremely model dependent, and may be revised in future generations of models. This would have a direct effect on the strength of the extreme-UV continuum and consequences for the nebular emission lines. Furthermore, the [<span style="font-variant:small-caps;">STARBURST99</span>]{} models neglect binary evolution, although this is more likely to affect the W-R/O ratio at low metallicity. Since the results of the transition-object models are highly dependent on the most uncertain portion of the W-R spectrum, new photoionization calculations should be computed to assess the impact of different W-R evolution and atmosphere models in the future.
The shape of the IMF in starburst regions is a subject of some debate, and the possible variation of the IMF with metallicity is of particular importance for galactic nuclei, which are likely to have $Z
> {\ensuremath{Z_{\odot}}}$. @k74 and @st76 suggested that [$M_{\rm up}$]{} should be lower in regions of higher metal abundance, but this issue has not been settled definitively. Star-count observations demonstrate that the IMF slope and [$M_{\rm up}$]{} do not appear to vary with metallicity [@mjd95], at least for $Z \leq {\ensuremath{Z_{\odot}}}$. While nebular diagnostics in galaxies are generally consistent with a Salpeter IMF with ${\ensuremath{M_{\rm up}}}\approx 100$ [$M_{\odot}$]{} at subsolar metallicity [e.g., @sl96], at high metallicity the observational situation is somewhat ambiguous. @bkg99 find that the mean stellar temperature in regions decreases significantly with increasing $Z$, and that the [$\lambda$]{}5876 / [H$\beta$]{} ratios of regions at $Z \approx 2{\ensuremath{Z_{\odot}}}$ are more consistent with [$M_{\rm up}$]{} = 30 [$M_{\odot}$]{} than with [$M_{\rm up}$]{} = 100 [$M_{\odot}$]{}. Such a low value for [$M_{\rm up}$]{} would pose serious difficulties for any starburst models of LINERs and transition nuclei, as the massive progenitors of W-R stars would not be present. Counterbalancing this trend, the strong tidal forces, turbulence, and magnetic field strengths in galactic nuclei may act to raise the Jeans mass and favor the formation of more massive stars [@mor93]. In the Galactic center, there are stars with initial masses of $\sim100$ [$M_{\odot}$]{} [@kra95], and one Galactic center object (the Pistol star) may have $M_{\rm initial}$ as high as 200–250 [$M_{\odot}$]{} [@fig98]. Thus, the proposed trend toward lower values of [$M_{\rm up}$]{} at high metallicity in disk regions may not apply to galactic nuclei. Detailed comparison of the UV spectra of galaxies such as NGC 4569 with starburst population synthesis models can provide useful constraints on the population of high-mass stars in nuclear starbursts.
Despite these uncertainties, these photoionization models have a major advantage compared with previous generations of W-R or O-star models for LINERs and transition nuclei, in that the [<span style="font-variant:small-caps;">STARBURST99</span>]{} models with standard parameters are constructed to represent the actual stellar populations in starbursts, to the best of current knowledge. Previous O-star models [@ft92; @shi92] required the presence of hypothetical, unusually hot stars in order to explain LINER or transition spectra, and they did not address the evolution of the young stellar population at all. The starburst models presented here provide a more plausible mechanism to generate a transition-type spectrum, even if this model may apply only to a relatively small fraction of the population of transition galaxies.
Conclusions {#sectionconclusions}
===========
Our primary conclusion is that for standard starburst parameters and for nebular conditions which may be typical of galactic nuclei, the starburst models are able to reproduce the important diagnostic emission-line ratios for LINER/ transition galaxies, otherwise known as weak-\[\] LINERs. The key ingredient needed to generate a transition-type spectrum is a UV continuum dominated by W-R stars, a condition which occurs during $t =$ 3–5 Myr after an instantaneous burst. A transition-type emission spectrum may thus be a phase in the evolution of some nuclear regions in which the ionizing continuum is generated by a single-burst stellar population. The models are also able to produce an \[\] / [H$\alpha$]{} ratio high enough to match LINER spectra, but only for conditions of above-solar metallicity combined with the presence of high-density ($\gtrsim10^5$ [cm$^{-3}$]{}) clouds. A sensitive search for W-R spectral features in transition nuclei would provide a test of this starburst scenario. This model may apply only to a small fraction of LINERs and transition nuclei; many LINERs and some transition objects show clear signs of nonstellar activity, and the starburst models may not apply at all to objects in early-type host galaxies. Further multiwavelength observations of transition nuclei will be of great utility for determining what fraction of them contain genuine active nuclei, and what fraction appear to be purely the result of stellar phenomena.
Research by A.J.B. is supported by a postdoctoral fellowship from the Harvard-Smithsonian Center for Astrophysics. This research was also supported financially by grant AR-07988.02-96A, awarded to J.C.S. by STScI, which is operated by AURA for NASA under contract NAS5-26555. This work would not have been possible without the excellent software created and distributed by Gary Ferland and the Cloudy team, and by Claus Leitherer and the [<span style="font-variant:small-caps;">STARBURST99</span>]{} team. We also thank Gary Ferland for providing a helpful referee’s report, Claus Leitherer for additional helpful comments on the manuscript, and Daniel Schaerer for supplying model starburst spectra in electronic form.
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[lccc]{} A & I & 1.0 & N\
B & I & 1.0 & Y\
C & C & 1.0 & N\
D & C & 1.0 & Y\
E & I & 0.2 & N\
F & I & 0.4 & N\
G & I & 2.0 & N\
**Figure Captions**
Line-ratio diagram of \[\] [$\lambda$]{}5007 / [H$\beta$]{} against \[\] [$\lambda$]{}6300 / [H$\alpha$]{}, for model grids A (solid line), B (long-dashed line), C (short-dashed line), and D (dot-dashed line), at a burst age of 4 Myr and $n_e = 10^3$ [cm$^{-3}$]{}. The input continuum has solar metallicity, IMF power-law slope $-2.35$, and [$M_{\rm up}$]{} = 100 [$M_{\odot}$]{}. The following description applies to this and all subsequent plots: the small squares along each model line correspond to the model grid points at log $U$ = $-4, -3.5, -3, -2.5$, and $-2$, with $U$ increasing upward along the line. The points plotted represent galaxies from the Ho [et al.]{} catalog, as follows: *Small circles*: nuclei. *Squares:* LINER/ transition objects. NGC 4569 is represented by an open square. *Triangles:* “pure” LINERs. *Crosses:* Seyfert nuclei. The dotted line encloses the region defined for transition nuclei according to the criteria of @hfs97a.
|
---
abstract: 'The electronic structure of the perovskite LaCoO$_3$ for different spin states of Co ions was calculated in the LDA+U approach. The ground state was found to be a nonmagnetic insulator with Co ions in a low-spin state. Somewhat higher in energy we found two intermediate-spin states followed by a high-spin state at significantly higher energy. The calculation results show that Co 3$d$ states of $t_{2g}$ symmetry form narrow bands which could easily localize whilst $e_g$ orbitals, due to their strong hybridization with the oxygen 2$p$ states, form a broad $\sigma^*$ band. With the increase of temperature which is simulated by the corresponding increase of the lattice parameter, the transition from the low- to intermediate-spin states occurs. This intermediate-spin (occupation $t_{2g}^5e_g^1$) can develop an orbital ordering which can account for the nonmetallic nature of LaCoO$_3$ at 90 K$<$T$<$500 K. Possible explanations of the magnetic behavior and gradual insulating-metal transition are suggested.'
author:
- |
M.A.Korotin, S.Yu.Ezhov, I.V.Solovyev, V.I.Anisimov\
*Institute of Metal Physics, Russian Academy of Sciences,\
*620219 Yekaterinburg GSP-170, Russia\
\
D.I.Khomskii[^1], G.A.Sawatzky\
*Solid State Physics Laboratory, University of Groningen,\
*Nijenborg 4, 9747 AG Groningen,\
The Netherlands****
title: 'Intermediate-spin state and properties of LaCoO$_3$'
---
INTRODUCTION
============
Among the systems showing semiconductor to metal transition LaCoO$_3$ is especially interesting due to the fact that is also displays very unusual magnetic behavior, often associated with low-spin (LS) – high-spin (HS) transition [@Good]. Although a large number of investigations have been carried out since the early 1960’s the character of the transition and the nature of the temperature dependence of the spin state is still unclear. For example the temperature dependence of the magnetic susceptibility shows a strong maximum at around 90 K followed by a Curie-Weiss-like decrease at higher temperatures [@Asai] which was interpreted by the authors as a LS to a HS transition. The semiconductor to metal-like transition occurs in a range of 400 – 600 K, well above this transition. The insulating nature of LaCoO$_3$ below 400 – 600 K was attributed by Raccah and Goodenough to an ordering of LS and HS Co$^{3+}$ ions in a NaCl like structure [@Good], with the itinerant electrons in a broad band formed by the transition metal $e_g$ orbitals. This structural change however has not been observed in crystallographic studies. Recent neutron scattering experiments [@Neutron] suggest that the semiconductor – metal transition is not predominantly magnetic in origin but a clear picture of what its nature is absent.
Photoemission (XPS) and X-ray absorption (XAS) studies on the other hand have been interpreted in terms of a spin change at the semiconductor – metal transition [@Abbspec]. There are however several discrepancies in the interpretation. The Co 2$p$ XAS line shape at room temperature looks like LS Co$^{3+}$ which is not consistent with a spin state transition at 90 K. However, the valence band XPS spectrum at 300 K is quite different from that of a HS compound. So the interpretation of the two transitions is still uncertain.
In most of the previous studies one has assumed a rather ionic and ligand field-like starting point. In this picture the possibility of intermediate spin (IS) states as well as local symmetry lowering, orbital ordering and the large bandwidth of the $e_g$ states were not considered. It has recently been pointed out by several authors that the oxides corresponding to high formal oxidation states may be negative charge transfer systems in the Zaanen-Sawatzky-Allen scheme [@ZSA] resulting in an essential modification of the electronic structure, in particular in a possible stabilization of an IS states [@PotzeSCO].
A first physical model to explain the transitions in LaCoO$_3$ was proposed and recently revised by Goodenough in [@Good]. It was suggested that for trivalent cobalt in LaCoO$_3$ the crystal field energy $\Delta_{cf}$ is only slightly larger than the intraatomic (Hund) exchange energy $\Delta_{ex}$. Thus the ground state of the Co ion is LS $^1A_1$ (S=0) and the excited HS state $^5T_2$ (S=2), according to [@Good], is only 0.08 eV higher in energy. The corresponding one-electron configurations are $t_{2g}^6e_g^0$ for the LS state and $t_{2g}^4e_g^2$ for the HS state. The increase in temperature leads to the population of the HS states which is reflected in the magnetic susceptibility measurements. The semiconductor-metal transition was interpreted in a rather vague way as a formation of a $\sigma^*$ band from the localized ionic $e_g$ states. Long-range order of the LS and HS Co ions was assumed based on the results of the X-ray diffraction measurements. The recent neutron diffraction experiments [@Neutron] did not confirm it, but did not exclude a short-range order.
The theoretical description of LaCoO$_3$ is a difficult task, because this system exhibits transition from the localized to the delocalized behavior, and the two main existing methods: one-electron band structure calculations based on the Local Density Approximation (LDA) and “model hamiltonian configuration interaction” approach (which were both used for the compound under consideration [@Abbspec; @Abbband]) were constructed for the, respectively, completely itinerant and fully localized cases.
Recently a paper of Sarma [*et al*]{} was published [@sarma] where the results of the LSDA calculations for La$M$O$_3$ ($M$=Mn, Fe, Co, Ni) were presented. It was claimed there that LSDA results gave good agreement with the X-ray photoemission spectra. Such an agreement is not necessary a proof of the adequate description of the electronic structure. A well known example is NiO, where LSDA gives a sharp peak at the top of the valence band in agreement with XPS, but the nature of this peak is Ni3d $t_{2g}$ minority-spin character, while it is generally accepted that this peak is derived from $d^8\underline{L}$ final state. The real proof of the correctness of the one-electron approximation would be an absence of the satellites in the photoemission spectra. Also the XPS spectra, with which a comparison was made in [@sarma], were measured at the room temperature, where LaCoO$_3$ is already not in the low-spin state. So XPS must be compared with LSDA result corresponding to the magnetic solution and not a non-magnetic one as in [@sarma].
Anisimov [*et al*]{} proposed a so-called LDA+U method [@Anis] which combines in one calculation scheme LDA and Hubbard model approaches and is able to treat on a “first-principle” basis the systems with strong Coulomb correlation. It was demonstrated that this method (in contrast to standard LDA) could describe the existence of the different states of the system under consideration which are close in total energy (holes in doped copper oxides [@PRL] and transition metal impurities in insulators [@mgo]).
In this paper we present a study of an electronic structure of LaCoO$_3$ in the LDA+U approach. In contrast to the standard LDA there are several stable solutions corresponding to different local minima of the LDA+U functional. We have found a nonmagnetic insulating ground state in agreement with the experiment. We have also found two orbitally-polarized magnetic solutions corresponding to IS states (one of them is gapless semiconductor and the other is metal) and semiconducting magnetic solution corresponding to a HS state of the system, which lies much higher in energy. In addition we found that the $e_g$ states form broad bands whereas the $t_{2g}$ states exhibit narrow bands, split by Coulomb interactions into the lower (occupied) and upper (unoccupied) Hubbard bands. Using the results of these calculations, we propose an interpretation of the behavior of LaCoO$_3$. According to our scheme, there first occurs with increasing temperature a transition from a LS (nonmagnetic) insulating ground state to a state with an IS (configuration $t_{2g}^5e_g^1$). Due to strong Jahn-Teller nature of this configuration, this state may develop orbital ordering. The orbitally-ordered state turns out to be nonmetallic (actually nearly zero-gap semiconductor) in our calculations. With the further increase of the temperature the orbital ordering may be gradually destroyed which can explain the transition to a metallic state observed in LaCoO$_3$ at 400 – 600 K. We hope that this new information will eventually lead to a better understanding of LaCoO$_3$ and other similar materials.
CALCULATION METHOD
==================
The LDA+U method [@Anis] is based on the assumption that it is possible to separate all electrons in the system in two subsets: the localized states (for LaCoO$_3$ these are the Co 3$d$-orbitals), for which Coulomb intrashell interactions are described by the Hubbard-like term, and the itinerant states, where the averaged LDA energy and potentials are good approximations.
It is known that LDA calculations can provide all the necessary model parameters (such as the Coulomb parameter $U$ [@agun], the exchange $J$, hopping parameters describing hybridization [*etc.*]{}) on a first-principle basis, but the one-electron structure of the LDA equations with the orbital-independent potential does not allow to use these parameters in the full variational space. LDA+U overcomes this problem by using the framework of the degenerate Anderson model in the mean-field approximation. In this method, the trial function is chosen as a single Slater determinant, so it is still a one-electron method, but the potential becomes orbital-dependent, and that allows one to reproduce the main features of strongly correlated systems: the splitting of the $d$-band to the occupied lower Hubbard band and unoccupied upper Hubbard band.
The main idea of our LDA+U method is that the LDA gives a good approximation for the average Coulomb energy of $d$-$d$ interactions $E_{av}$ as a function of the total number of $d$-electrons $N=\sum_{m\sigma} n_{m\sigma}$, where $n_{m\sigma}$ is the occupancy of a particular $d_{m\sigma}$-orbital:
$$E_{av}=U\frac{N(N-1)}{2}-J\frac{N(N-2)}{4}.$$
But LDA does not properly describe the full Coulomb and exchange interactions between $d$-electrons in the same $d$-shell. So we suggested to subtract $E_{av}$ from the LDA total energy functional and to add orbital- and spin-dependent contributions to obtain the exact (in the mean-field approximation) formula:
$$E=E_{LDA}-\bigl(U\frac{N(N-1)}{2}-J\frac{N(N-2)}{4}\bigr)
+\frac{1}{2}\sum_{m,m^\prime,\sigma}U_{mm^\prime}
n_{m\sigma}n_{m^\prime -\sigma}
+\frac{1}{2}\sum_{m\neq m^\prime,m^\prime,\sigma}
(U_{mm^\prime}-J_{mm^\prime})n_{m\sigma}n_{m^\prime \sigma} \; .$$
The derivative of Eq.(2) over orbital occupancy $n_{m\sigma}$ gives the expression for the orbital-dependent one-electron potential:
$$V_{m\sigma}(\vec{r})=V_{LDA}(\vec{r})+
\sum_{m^\prime}(U_{mm^\prime}-U_{eff})n_{m^\prime -\sigma}
+\sum_{m^\prime \neq m}(U_{mm^\prime}-J_{mm^\prime}
-U_{eff})n_{m\sigma}
+U_{eff}(\frac{1}{2}-n_{m\sigma})-\frac{1}{4}J \;.$$
The Coulomb and exchange matrices $U_{mm^\prime}$ and $J_{mm^\prime}$ are:
$$\begin{aligned}
U_{mm^\prime} & = & \sum_{k}a_{k}F^{k} , \\
J_{mm^\prime} & = & \sum_{k}b_{k}F^{k} , \\
a_{k} & = & \frac{4\pi}{2k+1}\sum_{q=-k}^{k} \langle lm|Y_{kq}|lm\rangle
\langle m^\prime|Y_{kq}^\ast|l m^\prime\rangle , \\
b_{k} & = & \frac{4 \pi}{2k+1}\sum_{q=-k}^{k}
|\langle lm|Y_{kq}|l m^\prime\rangle |^2 \; ,\end{aligned}$$
where the $F^{k}$ are Slater integrals and $\langle lm|Y_{kq}|l
m^\prime\rangle $ are integrals over products of three spherical harmonics $Y_{lm}$.
For $d$-electrons, one needs $F^0, F^2$ and $F^4$ and these can be linked to the parameters $U$ (direct Coulomb interaction) and $J$ (intraatomic exchange) obtained from the LSDA-supercell procedures [@agun] via $U=F^0$ and $J=(F^2+F^4)/14$, while the ratio $F^2/F^4$ is to a good accuracy constant $\sim 0.625$ for 3$d$ elements [@groot]. For LaCoO$_3$, the Coulomb parameter U was found to be 7.8 eV and the exchange parameter J=0.92 eV.
The LDA+U approximation was applied to the full potential linearized muffin-tin orbitals (FP-LMTO) calculation scheme [@FPLMTO]. Crystallographic data being used in the calculations were taken from [@Neutron]. According to them, LaCoO$_3$ has a pseudocubic perovskite structure with a rhombohedral distortion along the (111) direction. The unit cell contains two formula units. Since this rhombohedral distortion is small (the largest rhombohedral angle is 60.990$^\circ$ at 4 K), we use the conception of $t_{2g}$ and $e_g$ orbitals, as referred to in the cubic setting in the following discussion. Temperature was introduced in our calculations only by the change of lattice parameter and atomic positions according to the data of Ref.[@Neutron]. The most detailed description of the technical aspects of FP LMTO calculations for the perovskite-type complex oxides can be found in [@Andrei]. The optimal choice of the basis set for describing the valence band and the bottom of conduction band of the LaCoO$_3$ is presented in Table \[basisset\]. Since U-correction is applied for the Co $d$-orbitals, the value of the [*muffin-tin*]{} (MT) radius for Co was chosen close to its value in metallic Co in order to get the full 3$d$-density inside the sphere. For the correct description of the wave functions in the interstitial region, we expanded the spherical harmonics up to the value of l$_{max}$=5, 4, 3 for La, Co and O MT-spheres correspondingly. The Brillouin zone (BZ) integration in the course of the self-consistency iterations was performed over a mesh of 65 [**k**]{}-points in the irreducible part of the BZ. Densities of states (DOS) were calculated by the tetrahedron method with 729 [**k**]{}-points in the whole BZ.
[llllc]{} & & & &\
La &6$s$6$p$5$d$4$f$ &6$s$6$p$5$d$4$f$ &6$s$6$p$5$d$ &1.77\
Co &4$s$4$p$3$d$ &4$s$4$p$3$d$ &4$s$4$p$ &1.26\
O &2$s$2$p$ &2$s$2$p$ &2$s$ &0.66\
As the potentials for the various $d$-orbitals of Co are different in LDA+U, it is not obvious [*a priori*]{} what will be the final symmetry of the system under consideration. In preliminary calculations we assumed for the simplicity that LaCoO$_3$ has a perfect perovskite-type cubic lattice. To allow the system to choose the appropriate symmetry by itself, we perform an integration not over 1/48 part of BZ as for cubic symmetry but over 1/8 (D$_{2h}$ symmetry group). The resulting symmetries were found to be cubic O$_{h}$ for the ground (LS) state and tetragonal D$_{4h}$ for the excited states. Practically the same situation occurs in real R$\bar{3}$c symmetry: the occupancies of $xy,
yz, zx$ orbitals are about the same for the low-spin configuration as well as that of $3z^2-r^2, x^2-y^2$ orbitals. The degeneracy of $xy$ and $yz, zx$ orbitals is broken in the other spin states.
RESULTS AND DISCUSSION
======================
Homogeneous Solutions
---------------------
We start by considering the results of the calculations for homogeneous regimes, without extra superstructure. The possibility to get a solution with an orbital ordering is considered in section \[sec:orbord\].
The detailed results of the calculations with crystallographic data corresponding to 4 K are presented in Table \[tab:res\]. One must start with the equal occupancies of all three $t_{2g}$ orbitals and also of two $e_g$ orbitals (e.g. $t_{2g}^6e_g^0$) to obtain the LS configuration. The initial spin polarization vanished during self-consistency iterations resulting in a nonmagnetic final solution. The charge excitation spectrum has a semiconducting character in accordance with experimental data [@Gap] (and in contrast to the LDA result [@Abbband]) with the energy gap equal to 2.06 eV. The top of the valence band (Fig.1) is formed by the mixture of oxygen 2$p$ states with Co $t_{2g}$ orbitals and the bottom of conduction band predominantly by the $e_g$ orbitals.
[cllclllcccc]{} & & & &\
& && & & & & & & &\
$t_{2g}^6e_g^0$& –& 2.06& $\uparrow$,$\downarrow$ & 0.98& 0.98& 0.99& 0.32& 0.33& 7.20& 0\
low-spin\
$t_{2g}^5\sigma^*$ & 0.24& 0 $^b$ & $\uparrow$ & 0.98& 0.98& 0.98& 0.84& 0.84& 7.13& 2.11\
intermediate-spin& & & $\downarrow$ & 0.07& 0.98& 0.99& 0.23& 0.24\
$t_{2g}^4e_g^2$& 0.65& 0.13& $\uparrow$ & 0.99& 0.99& 0.99& 1.00& 1.00& 6.78& 3.16\
high-spin&&& $\downarrow$& 0.99& 0.08& 0.11& 0.33& 0.30\
\
\
![The total and partial densities of states obtained in LDA+U calculations for LaCoO$_3$ with Co ions in low-spin state ($t_{2g}^6e_g^0$ configuration). For $d$-Co partial density of states solid line denotes $t_{2g}$ orbitals and dashed line – $e_g$ orbital.[]{data-label="Fig.1"}](fig1_Co.eps){width="60.00000%"}
From the X-ray photoemission and absorption spectra [@Abbspec] the gap was estimated to be 0.9 $\pm$ 0.3 eV. The optical measurements [@arima] gave $\approx$ 0.5 eV. The larger value of the calculated gap can be explained by the well known fact that a mean-field approximation, which is the basis of our LDA+U approach, usually overestimates the tendency to localization and hence the values of the gap.
The aim of our work was to find solutions corresponding to the higher-spin states of Co ions. Usually the HS state is described as a $t_{2g}^4e_g^2$ configuration with the maximum value of spin S=2 (magnetic moment $\mu_{Co}$=4 $\mu_B$). This corresponds to the purely ionic model, and hybridization of the Co 3$d$-orbitals with the oxygen 2$p$-orbitals and band formation in a solid can renormalize significantly this ionic value. Such kind of renormalization was obtained in our calculations (see Table \[tab:res\]). The initial $t_{2g}^4e_g^2$ configuration (two holes on $d_{xz}^\downarrow$, $d_{yz}^\downarrow$ orbitals of the $t_{2g}$ set and two on $d_{3z^2-r^2}^\downarrow$, $d_{x^2-y^2}^\downarrow$ of the $e_g$ set with spin-down (minority) spin projection) results in the self-consistent solution with a magnetic moment of $\mu_{d-Co}$=3.16 $\mu_B$. The total energy of this HS solution is 0.65 eV per formula unit higher than the ground state LS $t_{2g}^6e_g^0$ configuration. The unexpected result is that there exists an [*intermediate-spin*]{} solution (second line in Table \[tab:res\], magnetic moment value $\mu_{d-Co}$=2.11 $\mu_B$) which is lower in total energy than the HS solution (0.24 eV per formula unit relative to the LS ground state). This solution was obtained when we started from the initial configuration $t_{2g}^5e_g^1$, where only one electron was transferred from the $t_{2g}$ to $e_g$ states. The final self-consistent configuration is better described as a $t_{2g}^5$ state with a partially filed $\sigma^*$-band ($t_{2g}^5\sigma^*$) with the occupancies of the $d_{3z^2-r^2}^\uparrow$ and $d_{x^2-y^2}^\uparrow$ orbitals equal to 0.84. In a configuration interaction language used in the cluster calculations this may be compared to a $d^6$+$d^7\underline{L}$ state where $\underline{L}$ denotes a hole on the oxygen.
The IS state in our calculations turns out to be metallic (see, however, section \[sec:orbord\]), and HS state is semiconducting (Fig.2,3). The reason for this is that the antibonding $\sigma^*$ band (formed by $e_g$ orbitals) is very broad and the band splitting is not strong enough to create a gap in the case of IS state. When the $e_g^\uparrow$ band becomes completely filled (HS state), a small gap between $e_g^\uparrow$ and e$_g^\downarrow$ bands appears.
![The total and partial densities of states obtained in LDA+U calculations for LaCoO$_3$ with Co ions in intermediate-spin state ($t_{2g}^5\sigma^*$ configuration). Fermi level is denoted by vertical dashed line. Arrows corresponds to spin-up and spin-down spin projection. The other denotations are the same as on Fig.1.[]{data-label="Fig.2"}](fig2_Co.eps){width="70.00000%"}
In Table \[tab:res\] the occupation of different orbitals in different configurations is presented. One notices that the actual occupation is different from the formal “chemical” one and corresponds on the average not to 6 but rather to 7 electrons in $d$-shell. The reason for this is the strong hybridization of the empty $e_g$ orbitals with oxygen 2$p$-orbitals. In the LS ground state every $e_g$ orbital, which is formally empty, has actually an occupancy of about 0.33 resulting in 1.3 additional $d$-electrons above the formal six-electron configuration. In the case of excited IS state we have partially filled broad $\sigma^*$-band with the total number of $e_g$ electrons increased by 0.85 as compared to the LS state and the number of $t_{2g}$-electrons 0.92 less than in LS state with as a result practically unchanged total number of $d$-electrons. Therefore, we used the notation $t_{2g}^5\sigma^*$ in the Table \[tab:res\] to prevent misunderstanding and in accordance with notation proposed in [@Good]. One may also say that despite the formal oxidation state Co$^{3+}$ the real configuration e.g. in the IS state is a mixture of $t_{2g}^5e_g^1$ and $t_{2g}^5e_g^2\underline{L}$ configurations.
![The total and partial densities of states obtained in LDA+U calculations for LaCoO$_3$ with Co ions in high-spin state ($t_{2g}^4e_g^2$ configuration). Denotations are the same as on Fig.1 and Fig.2.[]{data-label="Fig.3"}](fig3_Co.eps){width="70.00000%"}
Some other points concerning the results obtained are worth mentioning. As one can see from Fig.1, the photoemission holes in a LS state are mainly formed by oxygen states (note that in Figs.1-3 the partial densities of states of constituent atoms are given per one atom, and there are 3 of oxygen atoms per formula unit). Consequently the final state of the XPS for LS state formally corresponds to d$^6\underline{L}$ configuration. At the same time the electronic excitations mostly reside on Co $e_g$ orbitals hybridized with $p$-states of oxygen. However, in the IS case which in our calculations is metallic (Fig.2), the states at the Fermi level contain comparable weight of both Co $e_g$ and O $p$ orbitals.
The IS state obtained is fully polarized which is due to the half-metallic ferromagnetic nature of the solution: the magnetic moment per formula unit is 2 $\mu_B$, i.e. it corresponds to the spin S=1. In that sense the calculation agrees with the ionic picture, in which the IS configuration of Co is d$^6$ ($t_{2g}^5e_g^1$), with S=1.
The IS state of LaCoO$_3$ in our results turned out to be half-metal [@hmgroot] ferromagnet, in which only the electrons of one spin projection are at the Fermi level (the other spin subbands are fully occupied). Ferromagnetic arrangement of the magnetic moments may be of course an artifact of the scheme used. In principle, LDA+U method is intended to describe the local correlation effects such as formation of the Hubbard gaps and local magnetic moments, which should still be present in the paramagnetic phase. However technically in the calculations we have to assume certain long-range magnetic order in accordance with translation symmetry of the crystal. What the results obtained in such a way do really tell you, is whether there is a gap or not and what is the value of the [*local*]{} magnetic moment.
Although the IS state in the calculations turns out to be metallic, the density of states at the Fermi level is very low, n(E$_F$)=0.36 states per eV per one formula unit. This probably indicates that it would not be so difficult to make this state semiconducting, see the discussion below (section \[sec:orbord\]).
The results presented above show that the first excited configuration – that with the IS – lies only 0.24 eV higher than the ground state with the LS. Experimentally it is established that there is a transition from LS nonmagnetic state to the magnetic one with the increase of temperature. According to the Ref.[@Asai] this transition occurs in the vicinity of 90 K. As the closest magnetic state is that with the IS (HS state lies, according to our results, much higher at 0.65 eV), we ascribe nonmagnetic – magnetic transition in LaCoO$_3$ to the LS – IS transition.
It is well known (see, e.g., [@Good]) that Co in the HS state has larger ionic radii than in LS one, and LS – HS transitions are accompanied by the increase of the volume (and vice versa). Keeping that in mind we carried out the calculations of the electronic structure of LaCoO$_3$ at different lattice parameters which can imitate the influence of the temperature (via thermal expansion). Fig.4 demonstrates the values of total energies for various spin states of Co relative to the energy of $t_{2g}^6e_g^0$ state at the lattice parameter corresponding to 4 K. In this figure actual lattice parameters used in calculations are shown on the horizontal axis together with the temperature scale (we used the lattice parameters as a function of the temperature measured in Ref.[@Neutron]). One sees that with the increase of the specific volume (or with the increase of temperature) the energy of the IS state crosses that of the LS state, which corresponds to the LS – IS transition. In our calculation this crossover occurs at about 150 K – somewhat larger that the experimental value about 90 K. Nevertheless this transition temperature is correct by order of magnitude which for such [*ab initio*]{} calculations is rather satisfactory. Note that the HS state still lies high enough even at the temperatures about 600 K although we can not exclude that it could have become the ground state at still larger specific volumes or temperatures.
![The total energies for various spin states of LaCoO$_3$ relative to the energy of $t_{2g}^6e_g^0$ state at 4 K versus R$\bar{3}$c lattice constant. The correspondent temperatures are marked also.[]{data-label="Fig.4"}](fig4_Co.eps){width="\textwidth"}
The total energy for all three solutions (LS, IS, and HS) decreases with the increasing volume and minima are achieved only for lattice parameters corresponding to high temperature. At this value of the volume the total energy of the IS state is lightly lower than LS. It is well known that the equilibrium values of volume calculated with LSDA are always a few percent less than the experimental volume, because LSDA overestimates cohesion. LSDA+U, on the other hand, underestimates cohesion due to its treating the $d$ states as localized. In both cases the calculated lattice parameters significantly deviate from the experimental values. Goodenough suggested that the magnetic transition in LaCoO$_3$ is caused by the fact that the crystal-field energy $\delta{_cf}$ is only slightly larger than the intra-atomic (Hund) exchange energy $\delta{_ex}$. The crystal-field energy is determined by the 3$d$-2$p$ hopping parameters, which strongly depend on crystal volume. Hence if one will perform calculations with lattice parameters corresponding to the equilibrium volume, then the delicate balance between the $\delta{_cf}$ and $\delta{_ex}$. Our results show that for low-temperature lattice parameters is larger $\delta{_cf}$ than the $\delta{_ex}$, but already a small increase on volume (corresponding to increasing the temperature to 150 K) would reverse this ratio.
![Scheme representation of various Co $d^6$+$d^7\underline{L}$ configurations in low- ([**a**]{}), intermediate- ([**b**]{}) and high- ([**c**]{}) spin states. Open circle denotes a hole in oxygen $p$-shell.[]{data-label="Fig.5"}](fig5_Co.eps){width="\textwidth"}
Thus from the results obtained we conclude that the nonmagnetic – magnetic transition in LaCoO$_3$ most probably occurs not between LS and HS states, but between [*low*]{}- and [*intermediate*]{}-spin states. The reason for the stabilization of the IS state may be understood from the following consideration. Different configurations of Co$^{3+}$ ($d^6$) are illustrated in Fig.5. Here $t_{2g}$ and $e_g$ are atomic $d$-states splitted by the crystal field (the splitting $\Delta$=10 Dq). In purely ionic picture one should expect that, depending on the ratio between $\Delta$ and intraatomic exchange interaction J either LS or HS states should be stable (if the energy of the LS state E$_{LS}$ is taken to be zero, then E$_{IS}$=$\Delta$-J and E$_{HS}$=2$\Delta$-6J, so that LS state is the ground state for $\Delta>$3J and HS state – if $\Delta<$3J; IS state in this scheme always would lie higher). However, as we mentioned in the Introduction, the oxides with unusually high valence of transition metals have a tendency to have the $d$-shell occupancy corresponding to a lower valence, with the extra hole being predominantly located on oxygen (one gets a configuration $d^7\underline{L}$ instead of $d^6$). In this situation the $d$-$p$ hybridization is especially strong and plays a crucial role. These considerations are also supported by our calculations, see Table \[tab:res\].
Especially important is the hybridization with the $e_g$ orbitals. One can show that it is stronger for the IS state than for the HS one: there are more channels of $p$-$e_g$ hybridization in this case (the most favorable configurations of the type $d^7\underline{L}$, admixed to $d^6$, are also shown in Fig.5). In other words, if these configurations ($d^7\underline{L}$) would be dominant, then the state corresponding to the ground state of the $d^7$ configuration would have lowest energy – and it is known that for $d^7$ (Co$^{2+}$) it is just the configuration of Fig.5b – i.e., the one giving a total state of IS. That is the physical reason why the IS configuration lies below the HS one and may become the ground state for an expanded lattice (higher temperatures).
The nonmagnetic – magnetic transition occurring at $\sim$90 K which we now associate with the LS – IS transition, is described quite reasonably by our model. The transition temperature obtained in our calculations ($\sim$150 K) lies not so far from that experimentally observed. In Ref.[@Asai] the relative change of the lattice parameter ($\Delta a$) at this transition was fitted by the relation (see [@Ches])
$$\Delta a = \frac{\nu*exp(-E_q/kT)}{1+\nu*exp(-E_q/kT)},$$
where $E_q$ is the energy gap (it was taken in [@Asai] as a fitting parameter), and $\nu$ is degeneracy of the magnetic state. As the high-temperature magnetic ground state was believed to be the HS state in Ref.[@Asai], the degeneracy was taken $\nu$=15 (triply degenerated $t_{2g}$ orbital times the spin degeneracy (2S+1) with S=2, see Fig.5c). In our new interpretation the degeneracy will be $\nu$=18 (triply degenerated $t_{2g}$ orbital times double degeneracy of $e_g$ orbital times (2S+1) with S=1, see Fig.5b). Thus the fit of $\Delta a$ would be as successful with this interpretation as with the one given in [@Asai], with actually nearly the same value of the energy gap $E_q$.
The picture of an IS state can also help to resolve one more problem mentioned in Ref.[@Asai]: that the correlations between magnetic sites in LaCoO$_3$ are not antiferromagnetic but (weakly) ferromagnetic. One should expect only antiferromagnetic correlations between [*high-spin*]{} Co$^{3+}$ ions in which the $e_g$-shell is half-filled ($e_g^2$). However in IS Co ions have nominally $e_g^1$ configuration, and especially if these $e_g$ orbitals are ordered (see discussion below) one can have a ferromagnetic exchange interaction (the well-known example is e.g. ferromagnetic K$_2$CuF$_4$ [@KKh]).
Possible Orbital Ordering in LaCoO$_3$ {#sec:orbord}
--------------------------------------
The conception of LS – IS transition describes quite reasonably the nonmagnetic – magnetic transition observed in LaCoO$_3$. At the same time it was established that the first transition at $\sim$90 K leaves this compound semiconducting whereas our calculated IS has a metallic character of the energy spectrum. Is it possible to overcome this disagreement?
As is clear from Fig.5b, the IS state has a strong double-degeneracy ($e_g^1$ configuration). For localized electrons it would be the typical Jahn-Teller situation. In particular for KCuF$_3$ it was proposed in Ref.[@KKh] and it was confirmed by calculations in Ref.[@Licht] that there exist the special kind of orbital and magnetic ordering of $d$-ions in simple cubic lattice with one electron or hole on doubly-degenerate $e_g$ level. The situation in IS LaCoO$_3$ is very similar to the case of KCuF$_3$ (keeping in mind that there exist permanent rhombohedral distortion and there are already two Co ions in an elementary cell in LaCoO$_3$, the “antiferro” orbital ordering may be consistent with the existing structural data).
We checked this possibility by repeating the calculations, assuming now that there exist an orbital ordering. The structure of the corresponding ordered IS state was taken as consisting of ferromagnetic planes (001) whereas the direction of spins is opposite in the nearest planes. The occupied $e_g$ orbitals were assumed to alternate; as a starting point the orbitals in two sublattices were taken as $d_{x^2-z^2}$ and $d_{y^2-z^2}$, see Fig.6. To supplement this Figure with the information about the other $d$-orbitals, let’s consider the Co ion in $t_{2g}^5e_g^1$ configuration which has $d_{y^2-z^2}^\uparrow$ occupied $e_g$ orbital (corresponding site is marked as [**1**]{} on Fig.6). For this state the other three $e_g$ orbitals are empty and all the $t_{2g}$ orbitals are occupied, with the exclusion of $d_{yz}^\downarrow$ one. This choice minimizes the Coulomb interaction energy of $t_{2g}$ and $e_g$ electrons. For the neighboring Co ion (site [**2**]{} in Fig.6) the $e_g$ electron is placed in $d_{x^2-z^2}^\uparrow$ orbital and $t_{2g}$ hole is placed in $d_{xz}^\downarrow$ orbital. Calculations were made for the real rhombohedral crystal structure with lattice parameters corresponding to 71 K. With this state as a starting point the self-consistent calculation was now carried out. The resulting densities of states corresponding to such orbital ordered IS state are presented in Fig.7.
![Spin and orbital ordering in orbital ordered intermediate spin state for occupied $e_g$ orbitals. For the simplicity the perfect cubic structure for Co ions is shown.[]{data-label="Fig.6"}](fig6_Co.eps){width="\textwidth"}
First of all it is necessary to point out that such kind of spin and orbital ordering is stable with respect to the self-consistency iterations and hence there exists the solution with such symmetry. The character of an energy spectrum turned out to be gapless semiconducting-like. The wide $e_g$ band which crossed the Fermi level in the case of IS state without orbital order (see Fig.2) is now splitted. The top of the valence band is formed (for one sublattice, e.g. for Co site [**1**]{}) by $d_{y^2-z^2}$ states and the bottom of conduction band – by $d_{3x^2-r^2}$ states (Fig.7b,c) hybridized with the oxygen states. As to the $t_{2g}$ densities of states, the character of their energy distribution is practically not changed in comparison with the case of IS state without orbital ordering. The value of spin magnetic moment of Co is 1.87 $\mu_B$ in this orbitally ordered IS state and it is less than for one without orbital order (2.11 $\mu_B$).
![The total and partial densities of states obtained in LDA+U calculations for LaCoO$_3$ with Co ions in ordered intermediate-spin state. [**a**]{}: Total density of states (per 4 formula units and both spins); [**b**]{}: partial Co $e_g$ densities of states e.g. for Co site [**1**]{} in Fig.6: solid line - for $d_{y^2-z^2}$ states, dashed line - for $d_{3x^2-r^2}$ states; [**c**]{}: the same as in [**b**]{} in the vicinity of the Fermi level. Denotations are the same as on Fig.2.[]{data-label="Fig.7"}](fig7_Co.eps){width="60.00000%"}
When we compare the two IS states with the fixed spin magnetic moment 1.87 $\mu_B$, the total energy of the orbitally ordered IS state is 0.11 eV lower. The necessity of using a fixed magnetic moment is due to the limitation of the band structure scheme. The magnetic state of LaCoO$_3$ is known to be paramagnetic (with the possible short-range spin or orbital ordering). At the same time as we already mentioned above, it is necessary to set a long-range magnetic order in calculations (antiferromagnetic for orbitally ordered IS state and ferromagnetic for one without orbital order). This long-range magnetic ordering imposed (in addition to the intraatomic exchange) influences somehow both the local magnetic moment and the total energy. In the case of long-range antiferromagnetic order the value of the magnetic moment is apparently underestimated in comparison with the real paramagnetic state, for the ferromagnetic one – overestimated. Thus to estimate the relative stability of different phases we compare the energies calculated for the same value of the magnetic moment. Our calculations show that in the case of equal spin values the orbitally ordered IS state is more preferable than the one without orbital order.
One can interpret now the first nonmagnetic – magnetic transition (at about 90 K) as a transition of Co ions from LS to IS state with the specific orbital ordering of occupied $e_g$ orbitals. Our results show that this transition occurs at temperatures lower than 150 K. Under this transition a magnetic moment on Co sites appears whereas the material is still nonmetallic. The spin-only value of the [*effective*]{} magnetic moment, in the case of IS state (S=1), is $\mu_{eff}$=2.83 $\mu_B$ – somewhat below experimental value of 3.1–3.4 $\mu_B$. It will be increased by the orbital contribution, because there is in principle an unquenched moment of $t_{2g}$ electrons for the configuration $t_{2g}^5e_g^1$ (which e.g. for Co$^{2+}$ ions the increases the value of $\mu_{eff}$ typically by $\sim$0.3 $\mu_B$). We cannot treat this effect numerically because our codes do not account for the spin-orbit interaction; however we expect on physical grounds that this effect should be present (which, by the way, would change somewhat the occupation of $t_{2g}$ orbitals). An extra change of $\mu_{eff}$ may be due to the very strong $d$-$p$ hybridization which leads to a large contribution to the total wave function of a state $d^7\underline{L}$ ($t_{2g}^5e_g^2\underline{L}$). In this state the effective moment of Co ion is that of the high-spin Co$^{2+}$ (S=3/2) compensated by the opposite polarization of the oxygen $p$-shell (S=-1/2) (note that this picture is supported by the calculations (see Table \[tab:res\]) where local Co spin moment is 2.11 $\mu_B$ and the total spin moment per unit cell is 2 $\mu_B$). This fact may significantly change the value of $\mu_{eff}$ measured in the susceptibility experiments.
The IS state with the nominal configuration $t_{2g}^5e_g^1$ may have an orbital ordering because of the doubly degenerate $e_g$ orbitals (strong Jahn-Teller nature of this configuration). This turned out to be the case in our calculations: the orbitally ordered state may be crudely described as an alternation of the occupied $d_{z^2-x^2}$ and $d_{z^2-y^2}$ orbitals.
According to our calculations, this ordering leads to the splitting of the $e_g$ ($\sigma^*$)-band, leading practically to the zero-gap situation, with the Fermi level lying in the gap. The second gradual transition to the metallic state ($\sim$600 K) with the increase of the effective magnetic moment value up to $\mu_{eff}$=4.0 $\mu_B$ may then be associated with the disappearance of the orbital ordering with increased temperature still within the IS state. This IS state without orbital order is calculated to be a metal with a larger magnetic moment. Our calculations show that the second transition has no relations to the HS state of Co ions because the HS solution lies still high in energy and if realized would have been semiconductor.
CONCLUSION
==========
The calculations of the electronic structure in LDA+U approximation for LaCoO$_3$ were performed. At 4 K the lowest total energy has the nonmagnetic insulating solution with Co ions in LS state. Three excited state configurations (two IS and one HS) are also found to be stable, with local magnetic moments on Co sites. The $t_{2g}$ states exhibit narrow bands whilst the $e_g$ states form broad bands. With the increase of lattice parameter corresponding to the thermal lattice expansion two transitions can be expected: the first occurs from LS state to IS state with the orbital ordering which in our calculation is a zero-gap semiconductor. The second transition occurs within the IS state and is connected with gradual disorder of occupied $e_g$ orbitals. We believe that although many details are not clear yet, the concept of the nonmagnetic – magnetic and semiconductor – metal transitions in LaCoO$_3$ as connected mainly with IS states is a good candidate to explain the properties of this interesting material which remains a puzzle for so long.
We thank Dr.A.Postnikov and R.Potze for helpful discussions. This work was supported by the Netherlands Foundation for Fundamental Research on Matter (FOM), the Netherlands Foundation for Chemical Research (SON), the Netherlands Organization for the advancement of Pure Research (NWO), the Committee for the European Development of Science and Technology (CODEST) program and the Netherlands NWO special fund for scientists from the former Soviet Union.
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[^1]: Also at the P.N.Lebedev Physical Institute, Moscow, Russia
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---
abstract: 'Spin-orbit (SO) coupling – the interaction between a quantum particle’s spin and its momentum – is ubiquitous in nature, from atoms to solids. In condensed matter systems, SO coupling is crucial for the spin-Hall effect[@Kato2004; @Konig2007] and topological insulators[@Kane2005; @Bernevig2006; @Hsieh2008], which are of extensive interest; it contributes to the electronic properties of materials such as GaAs, and is important for spintronic devices[@Koralek2009]. Ultracold atoms, quantum many-body systems under precise experimental control, would seem to be an ideal platform to study these fascinating SO coupled systems. While an atom’s intrinsic SO coupling affects its electronic structure, it does not lead to coupling between the spin and the center-of-mass motion of the atom. Here, we engineer SO coupling (with equal Rashba[@Bychkov1984] and Dresselhaus[@Dresselhaus1955] strengths) in a neutral atomic Bose-Einstein condensate by dressing two atomic spin states with a pair of lasers[@Liu2009]. Not only is this the first SO coupling realized in ultracold atomic gases, it is also the first ever for bosons. Furthermore, in the presence of the laser coupling, the interactions between the two dressed atomic spin states are modified, driving a quantum phase transition from a spatially spin-mixed state (lasers off) to a phase separated state (above a critical laser intensity). The location of this transition is in quantitative agreement with our theory. This SO coupling – equally applicable for bosons and fermions – sets the stage to realize topological insulators in fermionic neutral atom systems.'
author:
- 'Y.-J. Lin$^1$, K. Jim[é]{}nez-Garc[í]{}a$^{1,2}$ & I. B. Spielman$^1$'
title: 'Spin-orbit-coupled Bose-Einstein condensates'
---
Joint Quantum Institute, National Institute of Standards and Technology, and University of Maryland, Gaithersburg, Maryland, 20899, USA
Departamento de Física, Centro de Investigación y Estudios Avanzados del Instituto Politécnico Nacional, México D.F., 07360, México
Quantum particles have an internal “spin” angular momentum; this can be intrinsic for fundamental particles like electrons, or a combination of intrinsic (from nucleons and electrons) and orbital for composite particles like atoms. Spin-Orbit (SO) coupling links a particle’s spin to its motion, and generally appears for particles moving in static electric fields, such as the nuclear field of an atom or the crystal field in a material. The coupling results from the Zeeman interaction $-\vec{\mu}\cdot\vec{B}$ between a particle’s magnetic moment $\vec{\mu}$, parallel to the spin $\vec{\sigma}$, and a magnetic field $\vec{B}$ present in the frame moving with the particle. For example, Maxwell’s equations dictate that a static electric field $\vec{E}=E_0\hat z$ in the lab frame (at rest) gives a magnetic field $\vec{B}_{\rm SO}=E_0(\hbar/m c^2)
(-k_y,k_x,0)$ in the frame of an object moving with momentum $\hbar\vec{k}=\hbar(k_x,k_y,k_z)$, where $c$ is the speed of light in vacuum and $m$ is the particle’s mass. The resulting momentum dependent Zeeman interaction $-\vec{\mu}\cdot\vec{B}_{\rm
SO}({\mathbf k})\propto \sigma_x k_y -\sigma_y k_x$ is known as the Rashba[@Bychkov1984] SO coupling. In combination with the Dresselhaus[@Dresselhaus1955] coupling $\propto -\sigma_x k_y -
\sigma_y k_x$, these describe two dimensional (2D) SO coupling in solids to first order.
In materials, the SO coupling strengths are generally intrinsic properties, which are largely determined by the specific material and the details of its growth, and thus only slightly adjustable in the laboratory. We demonstrate SO coupling in a $\Rb87$ Bose-Einstein condensate (BEC) where a pair of Raman lasers create a momentum-sensitive coupling between two internal atomic states. This SO coupling is equivalent to that of an electronic system with equal contributions of Rashba and Dresselhaus[@Liu2009] couplings, and with a uniform magnetic field $\vec{B}$ in the $\hat
y$-$\hat z$ plane, which is described by the single particle Hamiltonian $$\label{H_SO_extended}
\hat H\!=\!\frac{\hbar^2\hat{\mathbf k}^2}{2m}\check{1}
-\left[\vec{B}\!+\!\vec{B}_{\rm SO}(\hat {\mathbf k})\right]\cdot\vec\mu
= \frac{\hbar^2\hat{\mathbf k}^2}{2m}\check{1}+\frac{\Omega}{2}\check{\sigma}_z+\frac{\delta}{2}\check{\sigma}_y+2 \alpha \hat{k}_x\check{\sigma}_y.$$ $\alpha$ parametrizes the SO coupling strength; $\Omega=-g\mu_{\rm
B} B_z$ and $\delta=-g\mu_{\rm B} B_y$ result from the Zeeman fields along $\hat z$ and $\hat y$, respectively; and $\check\sigma_{x,y,z}$ are the 2$\times$2 Pauli matrices. Absent SO coupling, electrons have group velocity $v_x=\hbar k_x/m$, independent of their spin. With SO coupling, their velocity becomes spin-dependent, $v_x=\hbar(k_x\pm 2\alpha m/\hbar^2)/m$ for spin $\ket{\uparrow}$ and $\ket{\downarrow}$ electrons (quantized along $\hat y$). In two recent experiments, this form of SO coupling was engineered in GaAs heterostructures where confinement into 2D planes linearized GaAs’s native cubic SO coupling to produce a Dresselhaus term, and asymmetries in the confining potential gave rise to Rashba coupling. In one experiment a persistent spin helix was found[@Koralek2009], and in another the SO coupling was only revealed by adding a Zeeman field[@Quay2010].
SO coupling for neutral atoms enables a range of exciting experiments, and importantly, it is a key ingredient to realize neutral atom topological insulators. Topological insulators are novel fermionic band insulators including integer-, and now spin-quantum Hall states that insulate in the bulk, but conduct in topologically protected quantized edge channels. The first known topological insulators – integer quantum Hall states[@Klitzing1980] – require large magnetic fields that explicitly break time-reversal symmetry. In a seminal paper[@Kane2005], Kane and Mele showed that in some cases SO coupling leads to zero magnetic field topological insulators preserving time-reversal symmetry. Absent the bulk conductance that plagues current materials, cold atoms can potentially realize these insulators in their most pristine form, perhaps revealing their quantized edge (in 2D) or surface (in 3D) states. To go beyond the form of SO coupling we created, virtually any SO coupling, including that needed for topological insulators, is possible with additional lasers[@Ruseckas2005; @Stanescu2007; @Dalibard2010].
To create SO coupling, we select two internal “spin” states from within the $\Rb87$ $5{\rm S}_{1/2}$, $F=1$ ground electronic manifold, and label them pseudo-spin up and down in analogy with an electron’s two spin states: $\ket{\uparrow}=\ket{F=1,m_F=0}$ and $\ket{\downarrow}=\ket{F=1,m_F=-1}$. A pair of $\lambda=804.1\nm$ Raman lasers, intersecting at $\theta=90^\circ$ and detuned by $\delta$ from Raman resonance (Fig. \[setup\]a), couple these states with strength $\Omega$; here $\hbar\kl=\sqrt{2}\pi\hbar/\lambda$ and $E_L=\hbar^2 \kl^2/2m$ are the natural units of momentum and energy. In this configuration, the atomic Hamiltonian is given by Eq. \[H\_SO\_extended\], with $k_x$ replaced by a quasimomentum $q$ and an overall $E_L$ energy offset. $\Omega$ and $\delta$ give rise to effective Zeeman fields along $\hat z$ and $\hat y$, respectively. The SO coupling term $2E_L q
\check\sigma_{y}/k_L$ results from the laser geometry, and $\alpha=E_L/k_L$ is set by $\lambda$ and $\theta$, independent of $\Omega$ (see Methods). In contrast with the electronic case, the atomic Hamiltonian couples bare atomic states $\ket{\uparrow,{\mathpzc k}_x=q+\kl}$ and $\ket{\downarrow,{\mathpzc
k}_x=q-\kl}$ with different velocities, $\hbar {\mathpzc
k}_x/m=\hbar(q\pm\kl)/m$.
The spectrum, a new energy-quasimomentum dispersion of the SO coupled Hamiltonian, is displayed in Fig. \[setup\]b at $\delta=0$ and for a range of couplings $\Omega$. The dispersion is divided into upper and lower branches $E_{\pm}(q)$, and we focus on $E_-(q)$. For $\Omega<4 E_L$ and small $\delta$ (see Fig. \[PhaseDiagram\]a), $E_-(q)$ consists of a double-well in quasi-momentum[@Higbie2004], where the group velocity $\partial E_-(q)/\partial\hbar q$ is zero. States near the two minima are dressed spin states, labeled as $\ket{\uparrow'}$ and $\ket{\downarrow'}$. As $\Omega$ increases, the two dressed spin states merge into a single minimum and the simple picture of two dressed spins is inapplicable. Instead, that strong coupling limit effectively describes spinless bosons with a tunable dispersion relation[@Lin2009] with which we engineered synthetic electric[@Lin2010] and magnetic fields[@Lin2009b] for neutral atoms.
Absent Raman coupling, atoms with spins $\ket{\uparrow}$ and $\ket{\downarrow}$ spatially mixed perfectly in a BEC. By increasing $\Omega$ we observed an abrupt quantum phase transition to a new state where the two dressed spins spatially separated, resulting from a modified effective interaction between the dressed spins.
We studied SO coupling in oblate $\Rb87$ BECs with $\approx 1.8
\times 10^5$ atoms in a $\lambda=1064$ nm crossed dipole trap with frequencies $(f_x,f_y,f_z)\approx(50,50,140)$ Hz. The bias magnetic field $B_0 \hat y$ generated a $\omega_Z/2\pi\approx~4.81\MHz$ Zeeman shift between $\ket{\uparrow}$ and $\ket{\downarrow}$. The Raman beams propagated along $\hat y \pm \hat x$ and had a constant frequency difference $\Delta \omega_L/2\pi\approx 4.81$ MHz. The small detuning from Raman resonance $\delta=\hbar(\Delta
\omega_L-\omega_Z)$ was set by $B_0$, and $\ket{m_F=+1}$ was decoupled due to the quadratic Zeeman effect (see Methods).
We prepared BECs with an equal population of $\ket{\uparrow}$ and $\ket{\downarrow}$ at $\Omega,\delta=0$, adiabatically increased $\Omega$ to a final value up to $7E_L$ in $70\ms$, and then allowed the system to equilibrate for $t_h=70\ms$. We abruptly ($t_{\rm off}< 1\us$) turned off the Raman lasers and the dipole trap–thus projecting the dressed state onto their constituent bare spin and momentum states–and absorption-imaged them after a $30.1\ms$ time-of-flight (TOF). For $\Omega>4 E_L$ (Fig. \[setup\]d), the BEC was located at the single minimum $q_{\rm 0}$ of $E_-(q)$ with a single momentum component in each spin state corresponding to the pair $\left\{\ket{\uparrow,q_{\rm 0}+\kl},\ket{\downarrow,q_{\rm
0}-\kl}\right\}$. However, for $\Omega<4 E_L$ we observed two momentum components in each spin state, corresponding to the two minima of $E_-(q)$ at $q_{\uparrow}$ and $q_{\downarrow}$. The agreement between the data (symbols), and the expected minima-locations (curves), demonstrates the existence of the SO coupling associated with the Raman dressing. We maintained $\delta\approx 0$ when turning on $\Omega$ by making equal populations in bare spins $\ket{\uparrow},\ket{\downarrow}$ (see Fig. \[setup\]d).
We experimentally studied the low temperature phases of these interacting SO coupled bosons as a function of $\Omega$ and $\delta$. The zero-temperature mean-field phase diagram (Fig. \[PhaseDiagram\]a,b) includes phases composed of: a single dressed spin state, a spatial mixture of both dressed spin states, and coexisting but spatially phase-separated dressed spins.
This phase diagram can be largely understood from non-interacting bosons condensing into the lowest energy single particle state, and can be divided into three regimes (Fig. \[PhaseDiagram\]a). In the region of positive detuning marked $\ket{\downarrow'}$, there are double minima at $q=q_{\uparrow},q_{\downarrow}$ in $E_{-}(q)$ with $E_{-}(q_{\downarrow})<E_{-}(q_{\uparrow})$ and the bosons condense at $q_{\downarrow}$. In the region marked $\ket{\uparrow'}$ the reverse holds. The energy difference between the two minima is $\Delta(\Omega,\delta)=E_-(q_{\uparrow})-E_-(q_{\downarrow})\approx
\delta$ for small $\delta$ (see Methods). In the third “single minimum” regime, the atoms condense at the single minimum $q_0$. These dressed spins act as free particles with group velocity $\hbar
K_x/m$ (with an effective mass $m^*\approx m$, for small $\Omega$), where $K_x=q-q_{\uparrow,\downarrow,0}$ for the different minima.
We investigated the phase diagram using BECs with initially equal spin populations prepared as described previously, but with $\delta\neq 0$ and $t_h$ up to $3$ s. We probed the atoms after abruptly removing the dipole trap, and then ramping $\Omega\rightarrow0$ in $1.5\ms$. This approximately mapped $\ket{\uparrow'}$ and $\ket{\downarrow'}$ back to their undressed counterparts $\ket{\uparrow}$ and $\ket{\downarrow}$ (see Methods). We absorption-imaged the atoms after a $30\ms$ TOF, during the last $20\ms$ of which a Stern-Gerlach magnetic field gradient along $\hat y$ separated the spin components.
Figure \[dynamics\]a shows the condensate fraction $f_{\downarrow'} =N_{\downarrow'} / (N_{\downarrow'} +
N_{\uparrow'})$ in $\ket{\downarrow'}$ at $\Omega=0.6E_L$ as a function of $\delta$, at $t_h=0.1$, $1$ and $3$ s, where $N_{\uparrow'}$ and $N_{\downarrow'}$ denote the number of condensed atoms in $\ket{\uparrow'}$ and $\ket{\downarrow'}$, respectively. The BEC is all $\ket{\uparrow'}$ for $\delta\lesssim0$ and all $\ket{\downarrow'}$ for $\delta\gtrsim0$, but both dressed spin populations substantially coexisted for detunings within $\pm w_{\delta}$ (obtained by fitting $f_{\downarrow'}$ to the error function where $\delta=\pm w_{\delta}$ corresponds to $f_{\downarrow'}=0.50 \pm
0.16$). Figure \[dynamics\]b shows $w_{\delta}$ versus $\Omega$ for hold times $t_h$. $w_{\delta}$ decreases with $t_h$; even by our longest $t_h=3$ s it has not reached equilibrium.
Conventional $F=1$ spinor BECs have been studied in $\Na23$ and $\Rb87$ without Raman coupling[@Stenger1998; @Chang2004; @Ho1998]. For our $\ket{\uparrow}$ and $\ket{\downarrow}$ states, the interaction energy depends on the local density in each spin state, and is described by $$\begin{aligned}
\hat H_{\rm I}\! =& \frac{1}{2}\!\int\!d^3r \bigg[\!
\left(c_0\!+\!\frac{c_2}{2}\right)\!\left(\hat \rho_{\uparrow}\!+\!\hat \rho_{\downarrow} \right)^2
+ \frac{c_2}{2}\!\left(\hat \rho^2_{\downarrow}\!-\!\hat
\rho^2_{\uparrow} \right) + (c_2\!+\!c^\prime_{\uparrow \downarrow})
\hat \rho_{\uparrow} \hat \rho_{\downarrow} \bigg],\end{aligned}$$ where $\hat \rho_{\uparrow}$ and $\hat \rho_{\downarrow}$ are density operators for $\ket{\uparrow}$ and $\ket{\downarrow}$. In the $\Rb87$ $F\!=\!1$ manifold, the spin independent interaction is $c_0 = 7.79\times10^{-12}\Hz\cm^3$, the spin dependent interaction[@Widera2006] is $c_2 = -3.61\times10^{-14}\Hz\cm^3$, and $c^\prime_{\uparrow \downarrow}=0$. Since $|c_0|\gg |c_2|$ the interaction is almost spin independent, but because $c_2<0$, the two-component mixture of $\ket{\uparrow}$ and $\ket{\downarrow}$ has a spatially mixed ground state (is miscible). When $\hat H_{\rm I}$ is re-expressed in terms of the dressed spin states, $c^\prime_{\uparrow \downarrow}\approx c_0
\Omega^2/(8E_L^2)$ is nonzero and corresponds to an effective interaction between $\ket{\uparrow'}$ and $\ket{\downarrow'}$. This modifies the ground state of our SO coupled BEC (mixtures of $\ket{\uparrow'}$ and $\ket{\downarrow'}$) from phase-mixed to phase-separated above a critical Raman coupling strength $\Omega_c$. This transition lies outside the common single-mode approximation[@Chang2004].
The effective interaction between $\ket{\uparrow'}$ and $\ket{\downarrow'}$ is an exchange energy resulting from the non-orthogonal spin part of $\ket{\uparrow'}$ and $\ket{\downarrow'}$ (see Methods): a spatial mixture produces total density modulations[@Higbie2004] with wavevector $2\kl$ in analogy with the spin-textures of the electronic case[@Koralek2009]. These increase the state-independent interaction energy in $\hat
H_{\rm I}$ wherever the two dressed spins spatially overlap, contributing to the $c^\prime_{\uparrow \downarrow}$ term. (Such a term does not appear for rf-dressed states, which are always spin-orthogonal.) Because $c^\prime_{\uparrow\downarrow}$ and $c_2$ have opposite sign here, the dressed BEC can go from miscible to immiscible, at the miscibility threshold[@Stenger1998] for a two-component BEC $c_0 + c_2 + c^\prime_{\uparrow\downarrow}/2 = \sqrt{c_0
(c_0+c_2)}$, when $\Omega=\Omega_c$ (this result is in agreement with an independent theory presented in Ref. [@Ho2010]).
Figure \[PhaseDiagram\]b depicts the mean field phase diagram *including* interactions, computed by minimizing the interaction energy $H_{\rm I}$ plus the single particle detuning $\Delta(\Omega,\delta)\approx\delta$. This phase diagram adds to the non-interacting picture both mixed (hashed) and phase-separated (bold line) regimes. The $c_2\!\left(\hat \rho^2_{\downarrow}\!-\!\hat
\rho^2_{\uparrow} \right)/2$ term in $\hat
H_{\rm I}$ implies that the energy difference between a $\ket{\uparrow}$ BEC and a $\ket{\downarrow}$ BEC is proportional $N^2 c_2$. The detuning required to compensate for this difference slightly displaces the symmetry point of the phase diagram downwards. As evidenced by the width of the metastable window $2w_{\delta}$ in Fig. \[PhaseDiagram\]b, for $|\delta|<w_{\delta}$ the spin-population does not have time to relax to equilibrium. Since the miscibility condition does not depend on atom number, the phase line in Fig. \[PhaseDiagram\]c shows the system’s phases for $\left|\delta\right|<w_\delta$: phase-mixed for $\Omega<\Omega_c$ and phase-separated for $\Omega>\Omega_c$ where $\Omega_c\approx\sqrt{-8c_2/c_0}E_L\approx0.19 E_L$.
We measured the miscibility of the dressed spin components from their spatial profiles after TOF, for $\Omega=0$ to $2 E_L$ and $\delta\approx 0$ such that $N_{\rm T\uparrow'}\approx N_{\rm
T\downarrow'}$, where $N_{\rm T\uparrow',\downarrow'}$ is the total atom number including both the condensed and thermal components in $\ket{\uparrow'},\ket{\downarrow'}$. For each TOF image, we numerically recentered the Stern-Gerlach-separated spin distributions (Fig. \[PhaseDiagram\]c, and see Methods), giving condensate densities $n_{\uparrow'}(x,y)$ and $n_{\downarrow'}(x,y)$. Since the self-similar expansion of BECs released from harmonic traps essentially magnifies the in-situ spatial spin distribution, these reflect the in-situ densities[@Hall1998].
A dimensionless metric $s=1-\langle n_{\uparrow'}
n_{\downarrow'}\rangle/\left(\langle n_{\uparrow'}^2\rangle\langle
n_{\downarrow'}^2\rangle\right)^{1/2}$ quantifies the degree of phase separation ($\langle\ldots \rangle$ is the spatial average over a single image): $s=0$ for any perfect mixture $n_{\uparrow'}(x,y) \propto
n_{\downarrow'}(x,y)$, and $s=1$ for complete phase separation. Figure \[transition\] displays $s$ versus Raman coupling $\Omega$ with a hold time $t_h=3$ s, showing that $s\approx 0$ for small $\Omega$ (as expected given our miscible bare spins) and $s$ abruptly increases above a critical $\Omega_c$. The inset to Fig. \[transition\] plots $s$ as a function of time, showing that $s$ reaches steady-state in $0.14(3)\second\ll t_h$. To obtain $\Omega_c$, we fit the data in Fig. \[transition\] to a slowly increasing function below $\Omega_c$ and the power-law $1-(\Omega/\Omega_c)^{-a}$ above $\Omega_c$. The resulting $\Omega_c=0.20(2) E_L $ is in agreement with the mean field prediction $\Omega_c=0.19 E_L$. This demonstrates a quantum phase transition for a two-component SO coupled BEC, from miscible when $\Omega<\Omega_c$ to immiscible when $\Omega > \Omega_c$.
Even below $\Omega_c$, $s$ slowly increased with increasing $\Omega$. To understand this effect, we numerically solved the 2D spinor Gross-Pitaevskii equation in the presence of a trapping potential. This demonstrated that the differential interaction term $c_2\!\left(\hat \rho^2_{\downarrow}\!-\!\hat \rho^2_{\uparrow} \right)/2$ in $\hat H_I$ favors slightly different density profiles for each spin component, while the $(c_2+c^\prime_{\uparrow\downarrow}) \hat\rho_{\uparrow} \hat\rho_{\downarrow}$ term favors matched profiles. Thus, as $c_2+c^\prime_{\uparrow\downarrow}$ approached zero from below this balancing effect decreased, leading $s$ to increase.
An infinite system should fully phase separate ($s=1$) for all $\Omega>\Omega_c$. In our finite system, the boundary between the phase separated spins, set by the spin-healing length ($\xi_s=\sqrt{\hbar^2/2m\left|c_2+c^\prime_{\uparrow\downarrow}\right|n}$, where $n$ is the local density), can be comparable to the system size. We interpret the increase of $s$ above $\Omega_c$ as resulting from the decrease of $\xi_s$ with increasing $\Omega$.
We realized SO coupling in a $\Rb87$ BEC, and observed a quantum phase transition from spatially mixed to spatially separated. By operating at lower magnetic field (with a smaller quadratic Zeeman shift), our method extends to the full $F=1$ or $F=2$ manifold of $\Rb87$ or $\Na23$, enabling a new kind of tuning for spinor BECs, without the losses associated with Feshbach tuning[@Erhard2004]. Such modifications may allow access to the expected non-abelian vortices in some $F=2$ condensates[@Kobayashi2009]. Since our SO coupling is in the small $\Omega$ limit, this technique is practical for fermionic $^{40}{\rm K}$, with its smaller fine-structure splitting and thus larger spontaneous emission rate[@Goldman2010a]. When the Fermi energy lies in the gap between the lower and upper bands (e.g., Fig. \[setup\]b) there will be a single Fermi surface; this situation can induce $p$-wave coupling between fermions[@Zhang2008] and more recent work anticipates the appearance of Majorana fermions[@Sau2010].
System preparation
------------------
Our experiments began with nearly pure $\approx 1.8\times 10^5$ atom $\Rb87$ BECs in the $\ket{F=1,m_F=-1}$ state[@Lin2009a] confined in a crossed optical dipole trap. The trap consisted of a pair of $1064\nm$ laser beams propagating along $\hat{x}-\hat{y}$ ($1/e^2$ radii of $w_{\hat{x}+\hat{y}}\approx 120\micron$ and $w_{\hat z}\approx 50\micron$) and $-\hat{x}-\hat{y}$ ($1/e^2$ radii of $w_{\hat{x}-\hat{y}}\approx w_{\hat z}\approx 65\micron$).
We prepared equal mixtures of $\ket{F=1,m_F=-1}$ and $\ket{1,0}$ using an initially off resonant rf magnetic field $B_{\text{rf}}(t)
\hat{x}$. We adiabatically ramped $\delta$ to $\delta \approx0$ in $15\ms$, decreased the rf coupling strength $\Omega_{\text{rf}}$ to about $150\Hz \ll \hbar\omega_q$ in $6\ms$, and suddenly turned off $\Omega_{\text{rf}}$, projecting the BEC into an equal superposition of $\ket{m_F=-1}$ and $\ket{m_F=0}$. We subsequently ramped $\delta$ to its desired value in $6\ms$ and then linearly increased the intensity of the Raman lasers from zero to the final coupling $\Omega$ in $70\ms$.
Magnetic fields
---------------
Three pairs of Helmholtz coils, orthogonally aligned along $\hat{x}+\hat{y}$, $\hat{x}-\hat{y}$ and $\hat{z}$, provided bias fields $(B_{x+y},B_{x-y},\text{and } B_{z})$. By monitoring the $\ket{F=1,m_F=-1}$ and $\ket{1,0}$ populations in a nominally resonant rf dressed state, prepared as above, we observed a short-time (below $\approx10$ minutes) RMS field stability $g
\mu_{\rm B} B_{\rm RMS}/h\lesssim 80\Hz$. The field drifted slowly on longer time scales (but changed abruptly when unwary colleagues entered through our laboratory’s ferromagnetic doors). We compensated for the drift by tracking the rf and Raman resonance conditions.
Due to the small energy scales involved in the experiment, it was crucial to minimize magnetic field gradients. We detected stray gradients by monitoring the spatial distribution of $\ket{m_F=-1}$-$\ket{m_F=0}$ spin mixtures after TOF. Small magnetic field gradients caused this otherwise miscible mixture to phase separate along the direction of the gradient. We canceled the gradients in the $\hat x\!-\!\hat y$ plane with two pairs of anti-Helmholtz coils, aligned along $\hat x\!+\!\hat y$ and $\hat x\!-\!\hat
y$, to $g\mu_B B'/h\lesssim 0.7\Hz/\mu\text{m}$.
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, , & . ** ****, ().
, , , & . ** ****, ().
, , , & . ** ****, ().
We thank E. Demler, T.-L. Ho, and H. Zhai for conceptual input; and we appreciate conversations with J. V. Porto, and W. D. Phillips. This work was partially supported by ONR, ARO with funds from the DARPA OLE program, and the NSF through the Physics Frontier Center at JQI. K.J.-G. acknowledges CONACYT.
The authors declare that they have no competing financial interests.
All authors contributed to writing of the manuscript. Y.-J. L. lead the data taking effort in which K. J.-G. participated. I. B. S. conceived the experiment; performed numerical and analytic calculations; and supervised this work.
Correspondence and requests for materials should be addressed to\
I. B. S. (ian.spielman@nist.gov).
SO coupled Hamiltonian
----------------------
Our system consisted of a $F=1$ BEC with a bias magnetic field along $\hat y$ at the intersection of two Raman laser beams propagating along $\hat{x}+\hat{y}$ and $-\hat{x}+\hat{y}$ with angular frequencies $\omega_L$ and $\omega_L+\Delta\omega_L$, respectively. The rank-1 tensor light shift of these beams produced an effective Zeeman magnetic field along the $z$ direction with Hamiltonian $\hat H_R=\Omega_R\check\sigma_{3,z} \cos(2\kl \hat x + \Delta\omega_L t)$, where $\check\sigma_{3,x,y,z}$ are the $3\times3$ Pauli matrices and we define $\check 1_3$ as the $3\times3$ identity matrix. If we take $\hat y$ as the natural quantization axis (by expressing the Pauli matrices in a rotated basis $\check\sigma_{3,y}\rightarrow\check\sigma_{3,z}$, $\check\sigma_{3,x}\rightarrow\check\sigma_{3,y}$, and $\check\sigma_{3,z}\rightarrow\check\sigma_{3,x}$) and make the rotating wave approximation, the Hamiltonian for spin states $\left\{\ket{m_F=+1},\ket{0},\ket{-1} \right\}$ in the frame rotating at $\Delta \omega_L$ is $$\begin{aligned}
\hat H_3 & = & \frac{\hbar^2\hat{\mathbf{k}}^2}{2m}\check 1_3 +
\left(
\begin{array}{ccc}
3\delta/2+\hbar\omega_q & 0 & 0 \\0 & \delta/2 & 0 \\0 & 0 &
-\delta/2
\end{array}
\right) + \\
& & \frac{\Omega_R}{2}\check\sigma_{3,x}\cos(2 \kl \hat x) - \frac{\Omega_R}{2}\check\sigma_{3,y}\sin(2 \kl \hat x).\nonumber\end{aligned}$$ As we justify below, $\ket{m_F=+1}$ can be neglected for large enough $\hbar\omega_q$, which gives the effective two-level Hamiltonian $$\begin{aligned}
\hat H_2 & = & \frac{\hbar^2\hat{\mathbf{k}}^2}{2m}\check 1 +
\frac{\delta}{2}\check\sigma_z +
\frac{\Omega}{2}\check\sigma_{x}\cos(2 \kl \hat x) -
\frac{\Omega}{2}\check\sigma_{y}\sin(2 \kl \hat x)\nonumber\end{aligned}$$ for the pseudo-spin $\ket{\uparrow}=\ket{m_F=0}$ and $\ket{\downarrow}=\ket{-1}$ where $\Omega = \Omega_R/\sqrt{2}$. After a *local* pseudo-spin rotation by $\theta(\hat x) = 2 \kl \hat x$ about the pseudo-spin $\hat z$ axis followed by a global pseudo-spin rotation $\check\sigma_z\rightarrow\check\sigma_y$, $\check\sigma_y\rightarrow\check\sigma_x$, and $\check\sigma_x\rightarrow\check\sigma_z$, the $2\times2$ Hamiltonian takes the SO coupled form $$\label{H_SO_extended2}
\hat H_{2} = \frac{\hbar^2\hat{\mathbf k}^2}{2m}\check{1}+\frac{\Omega}{2}\check{\sigma}_z+\frac{\delta}{2}\check{\sigma}_y+2 \frac{\hbar^2\kl\hat{k}_x}{2m}\check{\sigma}_y + E_L\check{1}.\nonumber$$ The SO term linear in $\hat k_x$ results from the non-commutation of the spatially-dependent rotation about the pseudo-spin $z$ axis and the kinetic energy.
Effective two-level system
--------------------------
For atoms in $\ket{m_F=-1}$ and $\ket{m_F=0}$ with velocities $\hbar
{\mathpzc k}_x/m\approx0$ and Raman-coupled near resonance, $\delta
\approx0$, the $\ket{m_F=+1}$ state is detuned from resonance owing to the $\hbar \omega_q=3.8E_L$ quadratic Zeeman shift. For $\delta/4E_L\ll1$ and $\Omega <4E_L$, $\Delta(\Omega,\delta)\approx
\delta[1-(\Omega/4E_L)^2]^{1/2}$.
Effect of the neglected state
-----------------------------
In our experiment, we focused on the two level system formed by the $\ket{m_F=-1}$ and $\ket{m_F=0}$ states. We verified the validity of this assumption by adiabatically eliminating the $\ket{m_F=+1}$ state from the full three level problem. To second order in $\Omega$, this procedure modifies the detuning $\delta$ and SO coupling strength $\alpha$ in Eq. \[H\_SO\_extended\] by $$\begin{aligned}
\delta^{(2)}&=&\left(\frac{\Omega}{2}\right)^2\frac{1}{4 E_L+\hbar\omega_q}\approx\frac{1}{32}\frac{\Omega^2}{E_L}\nonumber \\
\alpha^{(2)}&=&\left(\frac{\Omega}{2}\right)^2\frac{\alpha}{(4E_L+\hbar\omega_q)^2}\approx\frac{\alpha}{256}\left(\frac{\Omega}{E_L}\right)^2.\nonumber\end{aligned}$$ In these expressions, we have retained only largest term in a $1/\omega_q$ expansion. In our experiment, where $\hbar\omega_q=3.8E_L$, $\delta$ is substantially changed at our largest coupling $\Omega=7 E_L$. To maintain the desired detuning $\delta$ in the simple 2-level model (i.e., $\Delta\approx \delta+\delta^{(2)}=0$ in Fig. \[setup\]c), we changed $g\mu_{\rm B} B_0$ by as much as $3E_L$ to compensate for $\delta^{(2)}$. We did not correct for the always small change to $\alpha$.
Although both terms are small at the $\Omega=0.2 E_L$ transition from miscible to immiscible, slow drifts in $B_0$ prompted us to locate $\Delta=0$ empirically from the equal population condition, $N_{\rm T\uparrow'}=N_{\rm T\downarrow'}$. As a result, $\delta$ in Eq. \[H\_SO\_extended\] implicitly includes the perturbative correction $\delta^{(2)}$.
Origin of the effective interaction term
----------------------------------------
The additional $c^\prime_{\uparrow\downarrow}$ term in the interaction Hamiltonian for dressed spins directly results from transforming into the basis of dressed spins, which are $$\begin{aligned}
\ket{\uparrow',K_x} \approx& \ket{\uparrow,{\mathpzc k}_x=K_x+q_{\uparrow}+\kl}-\epsilon\ket{\downarrow,{\mathpzc k}_x=K_x+q_{\uparrow}-\kl},\ {\rm and} \nonumber\\
\ket{\downarrow',K_x}\approx&\ket{\downarrow,{\mathpzc
k}_x=K_x+q_{\downarrow}-\kl}-\epsilon\ket{\uparrow,{\mathpzc
k}_x=K_x+q_{\downarrow}+\kl},\label{DressedSpins}\end{aligned}$$ where $\hbar K_x/m$ is the group velocity, $K_x=q-q_{\uparrow}$ for $\ket{\uparrow'}$ and $K_x=q-q_{\downarrow}$ for $\ket{\downarrow'}$, and $\epsilon = \Omega/8E_L\ll1$. Thus, in second quantized notation, the dressed field operators transform according to $$\begin{aligned}
\hat \psi_\uparrow(r) &= \hat \psi_{\uparrow'}(r) + \epsilon
e^{2i\kl x}\hat \psi_{\downarrow'}(r)\end{aligned}$$ and $$\begin{aligned}
\hat \psi_\downarrow(r) &= \hat \psi_{\downarrow'}(r) + \epsilon
e^{-2i\kl x}\hat \psi_{\uparrow'}(r),\end{aligned}$$ where $q_\uparrow\approx-\sqrt{1-4\epsilon ^2}\kl\approx-\kl$ and $q_\downarrow\approx\sqrt{1-4\epsilon ^2}\kl\approx\kl$. Inserting the transformed operators into $$\begin{aligned}
\hat H_{\rm I} = \frac{1}{2}\int d^3r \bigg[ \left(c_0 +
\frac{c_2}{2}\right)\left(\hat \rho_{\downarrow} + \hat
\rho_{\uparrow} \right)^2 + \frac{c_2}{2} \left(\hat
\rho^2_{\downarrow} - \hat \rho^2_{\uparrow} \right) + c_2 \hat
\rho_{\downarrow} \hat \rho_{\uparrow} \bigg]\nonumber\end{aligned}$$ gives the interaction Hamiltonian for dressed spins which can be understood order-by-order (both $c_2/c_0$ and $\epsilon$ are treated as small parameters). In this analysis, the terms proportional to $c_2$ are unchanged to order $c_2/c_0$, and we only need to evaluate the transformation of the spin-independent term (proportional to $c_0$). At $O(\epsilon)$ and $O(\epsilon^3)$ all the terms in the expansion include high spatial frequency $e^{\pm2i\kl x}$ or $e^{\pm4i\kl x}$ prefactors. For density distributions that vary slowly on the $\lambda/2$ length scale these average to zero. The $O(\epsilon^2)$ term, however, has terms without these modulations, and is $$\begin{aligned}
\hat H_{\rm I}^{(\epsilon^2)} =& \frac{1}{2}\int d^3r\left( 8c_0\epsilon^2
\hat\psi^\dagger_{\downarrow'}\hat\psi^{\dagger}_{\uparrow'}
\hat\psi_{\downarrow'} \hat\psi_{\uparrow'} \right),\end{aligned}$$ giving rise to $c^\prime_{\uparrow\downarrow} =
c_0\Omega^2/(8E_L^2)$.
Mean field phase diagram
------------------------
We compute the mean-field phase diagram for a ground state BEC composed of a mixture of dressed spins in an infinite homogeneous system. This applies to our atoms in a harmonic trap in the limit of $R\gg\xi_s$, where $R$ is the system size, $\xi_s=\sqrt{\hbar^2/2m|c_2+c^\prime_{\uparrow\downarrow}|n}$ is the spin healing length and $n$ is the density. We first minimize the interaction energy $\hat H_{\rm I}$ at fixed $N_{\uparrow',\downarrow'}$, with an effective interaction $c^{\prime}_{\uparrow \downarrow}$ as a function of $\Omega$. The two dressed spins are either phase-mixed, both fully occupying the system’s volume $V$, or phase-separated with a fixed total volume constraint $V=V_{\uparrow'} + V_{\downarrow'}$. For the phase-separated case, minimizing the free energy gives the volumes $V_{\uparrow'}$ and $V_{\downarrow'}$, determined by $N_{\uparrow',\downarrow'}$ and $V$. The interaction energy of a phase-mixed state is smaller than that of a phase-separated state for the miscibility condition $c_0 +
c_2 + c^\prime_{\uparrow\downarrow}/2 < \sqrt{c_0 (c_0+c_2)}$, corresponding to $\Omega<\Omega_c$. This condition is independent of $N_{\uparrow',\downarrow'}$: for any $N_{\uparrow',\downarrow'}$ the system is miscible at $\Omega<\Omega_c$. Then, at a given $\Omega$, we minimize the sum of the interaction energy and the single-particle energy from the Raman detuning, $(N_{\uparrow'}-N_{\downarrow'})\delta/2$, allowing $N_{\uparrow',\downarrow'}$ to vary. For the miscible case $(\Omega<\Omega_c)$, the BEC is a mixture with fraction $N_{\downarrow'}/(N_{\uparrow'}+N_{\downarrow'})\in(0,1)$ only in the range of detuning $\delta\in(\delta_0-W_{\delta},\delta_0+W_{\delta})$, where $\delta_0=c_2n/2$, $W_{\delta}=|\delta_0|(1-\Omega/\Omega_c)^{1/2}$ and $n=(N_{\uparrow'}+N_{\downarrow'})/V$. For the immiscible case ($\Omega>\Omega_c$), $W_{\delta}=(c_2/8c_0)c_2n$ is negligibly small compared to $c_2n$.
Figure \[PhaseDiagram\]b shows the mean field phase diagram as a function of $(\Omega,\delta)$, where $\delta/E_L$ is displayed with a quasi-logarithmic scaling, ${\rm sgn}(\delta/E_L)\left[\log_{10}\left(|\delta/E_L|+|\delta_{\rm min}/E_L|\right)-\log_{10}|\delta_{\rm{
min}}/E_L|\right]$, in order to display $\delta$ within the range of interest. This scaling function smoothly evolves from logarithmic for $|\delta|\gg \delta_{\rm{min}}$, $\approx {\rm
sgn}(\delta/E_L)\log_{10}|\delta/E_L|$, to linear for $|\delta|\ll
\delta_{\rm min}$, $\approx \delta$, where $\delta_{\rm
min}/E_L=0.001 E_L=1.5~$Hz.
In our measurement of the dressed spin fraction $f_{\downarrow'}$ (see Fig. \[dynamics\]a), $\delta=0$ is determined from the $N_{\rm
T\uparrow'}=N_{\rm T\downarrow'}$ condition. We identify this condition as $\delta=\delta_0$ and apply it for all hold time $t_h$. Because $|\delta_0|\approx 3~$Hz is below our $\approx 80~$Hz RMS field noise, we are unable to distinguish $\delta_0$ from 0.
Recombining TOF images of dressed spins
---------------------------------------
To probe the dressed spin states (Eq. \[DressedSpins\]), each of which is a spin and momentum superposition, we adiabatically mapped them into bare spins, $\ket{\uparrow,{\mathpzc k}_x=q_{\uparrow}+\kl}$ and $\ket{\downarrow,{\mathpzc k}_x=q_{\downarrow}-\kl}$, respectively. Then, in each image outside a $\approx90\micron$ radius disk containing the condensate for each spin distribution, we fit $n_{\rm
T{\uparrow'},\rm T{\downarrow'}}(x,y)$ to a gaussian modeling the thermal background and subtracted that fit from $n_{\rm
T{\uparrow'},\rm T{\downarrow'}}(x,y)$ to obtain the condensate 2D density $n_{\uparrow',\downarrow'}(x,y)$. Thus, for each dressed spin we readily obtained the temperature, total number $N_{\rm
T\uparrow',\rm T\downarrow'}$, and condensate densities $n_{\uparrow',\downarrow'}(x,y)$.
To analyze the miscibility from the TOF images where a Stern-Gerlach gradient separated individual spin states, we recentered the distributions to obtain $n_{\uparrow'}(x,y)$ and $n_{\downarrow'}(x,y)$. This took into account the displacement due to the Stern-Gerlach gradient and the nonzero velocities $\hbar{\mathpzc k}_x/m$ of each spin state (after the adiabatic mapping). The two origins were determined by the following: we loaded the dressed states at a desired coupling $\Omega$ but with detuning $\delta$ chosen to put all atoms in either $\ket{\downarrow'}$ or $\ket{\uparrow'}$. Since $q_{\uparrow,\downarrow}=\mp (1-\Omega^2/32E_L^2)\kl$ (see Fig. \[setup\]c), these velocities $\hbar{\mathpzc
k}_x/m=\hbar(q_{\uparrow}+k_L)/m,\hbar(q_{\downarrow}-k_L)/m$ depend slightly on $\Omega$, and our technique to determine the distributions’ origin accounts for this effect.
Calibration of Raman Coupling
-----------------------------
Both Raman lasers were derived from the same Ti:Sapphire laser at $\lambda\approx804.1$ nm, and were offset from each other by a pair of AOMs driven by two phase locked frequency synthesizers near $80\MHz$. We calibrated the Raman coupling strength $\Omega$ by fitting the three-level Rabi oscillations between the $m_F=-1,0,\
{\rm and}\ +1$ states driven by the Raman coupling to the expected behavior.
|
---
abstract: 'Despite outstanding contribution to the significant progress of Artificial Intelligence (AI), deep learning models remain mostly black boxes, which are extremely weak in explainability of the reasoning process and prediction results. Explainability is not only a gateway between AI and society but also a powerful tool to detect flaws in the model and biases in the data. Local Interpretable Model-agnostic Explanation (LIME) is a recent approach that uses a linear regression model to form a local explanation for the individual prediction result. However, being so restricted and usually over-simplifying the relationships, linear models fail in situations where nonlinear associations and interactions exist among features and prediction results. This paper proposes an extended Decision Tree-based LIME (TLIME) approach, which uses a decision tree model to form an interpretable representation that is locally faithful to the original model. The new approach can capture nonlinear interactions among features in the data and creates plausible explanations. Various experiments show that the TLIME explanation of multiple black-box models can achieve more reliable performance in terms of understandability, fidelity, and efficiency.'
address: |
$^{\dagger}$ AI Laboratory, Lenovo Research, Beijing 100094, China\
$^{\star}$ University of Chinese Academy of Sciences, Beijing 100049, China
bibliography:
- 'strings.bib'
- 'refs.bib'
title: Explaining the Predictions of Any Image Classifier via Decision Trees
---
Explainable AI, Interpretable Model, Decision Tree, Local Fidelity, Model Agnostic
Introduction {#sec:intro}
============
In recent years, the fast-growing computing power, enormous consumer and commercial data, and emerging advanced machine learning algorithms jointly stimulate the prosperous of AI [@AI01][@AI02], which has gone from a science-fiction dream to a critical part of our daily life. Compared to traditional machine learning methods, deep learning has achieved superior performance in perception tasks such as object detection and classification. However, because of the nested non-linear structure, deep learning models usually remain black boxes that are particularly weak in the explainability of the reasoning process and prediction results. In many real-world mission-critical applications, transparency of deep learning models and explainability of the model outputs are essential and necessary in their real deployment process.
Explanable AI is not only a gateway between AI and society but also a powerful tool to detect flaws in the model and biases in the data. The development of techniques on explainability and transparency of deep learning models has recently received much attention in the research community [@One01][@One02][@One03][@One04]. The relevant research roughly falls into two categories: global explainability and local explainability. Global explainability aims at making the reasoning process wholly transparent and comprehensive [@global03][@global04], while local explainability focuses on extracting input regions that are highly sensitive to the network output to provide explanations for each decision [@local01][@local02][@local03][@local04].
An effective way to achieve explainability is to use a light-weight function family to create interpretable models. Local interpretable model-agnostic explanations (LIME) identify an interpretable model over the human-interpretable representation that is locally faithful to the original model [@local01]. LIME adopts the linear regression as its interpretable function, which represents the prediction as a linear combination of a few selected features to make the prediction process transparent. However, being so restricted and usually over-simplifying the relationships, linear regression models fail in some situations where non-linear associations and interactions exist among features and prediction results In this paper, we propose a Decision Tree-based Local Interpretable Model-agnostic Explanation (TLIME). The decision tree structure creates good explanations as the data ends up in distinct groups that are often easy to understand. Moreover, the tree structure can capture interactions between features in the data. We perform various experiments on explaining two black-box models, the random-forest classifier and Google’s pre-trained Inception neural network[@Inception]. The results show that decision tree explanations achieve more reliable performance than original LIME in terms of understandability, fidelity, and efficiency.
Interpretable models {#sec:format}
====================
Using a subset of algorithms from a light-weight function family to create interpretable models is an effective way to achieve interpretability. In this section, we analyze two representative interpretable models - the linear regression model and the decision tree model. Table \[tab:1\] shows the properties of two interpretable models. The linear regression displays the prediction as a linear combination of features, while the decision tree represents the reasoning process in a hierarchical structure, which is suitable for capturing the nonlinear association between features and predictions. The monotonicity constraint shown in both models is necessary to ensure the consistency between a feature and the target outcome. Moreover, the decision tree model can automatically capture the diverse interactions between features to predict the target outcome, applicable to both classification and regression tasks.
Models Linearity Monotonicity Feature Interaction Task
----------------- ----------- -------------- --------------------- ----------------------------
Line regression Yes Yes No Regression
Decision trees No Some Yes Classification, Regression
\[tab:1\]
Depending on the different criteria, various algorithms are capable of constructing a decision tree. The CART [@ref02] is the most popular algorithm which can handle both classification and regression tasks. In this paper, we mainly construct regression decision trees to explain the prediction probability of the image classifier. Figure \[fig:1\] illustrates a simple regression tree to explain image classification prediction made by Google’s Inception neural network. The predicted top $1$ class label is $African$ $chanmeleon$ $(p=0.9935)$. The highlighted superpixels give intuition as to why the model would choose that class. The decision tree shows that if feature $28$, $22$, and $30$ exist, then the prediction probability is $0.991$, which is the mean value of the instances $y$ in this node. Moreover, The importance of the three features is $0.7164$, $0.0709$, and $0.0259$, showing the contribution of the three features in improving the variance.
{width="100.00000%"}
\[fig:1\]
The TLIME Approach {#sec:pagestyle}
==================
Characteristics of TLIME
------------------------
Despite the fact that the amount of research in explainable AI is growing actively, there is no universal consensus on the exact definition of interpretability and its measurement criterion [@ref01]. Ruping first noted that interpretability is composed of three goals - accuracy, understandability, and efficiency [@ref03]. We argue that fidelity is a better description than accuracy since accuracy is easily confused with the performance evaluation criteria of the original black box model. These three goals are inextricably intertwined and competing with each other, as shown in Figure \[fig:2\]. An explainable model with good interpretability should be faithful to the data and the original model, understandable to the observer and graspable in a short time so that the end-users can make wise decisions.
![[]{data-label="fig:2"}](figures/process/characteristics.pdf){width="50.00000%"}
TLIME has many appealing characteristics, such as interpretable, local fidelity, and model-agnostic. It provides a qualitative understanding of features and predictions. It is challenging, if not impossible, to be utterly faithful to the black box model on a global scale. TLIME takes a feasible approach by approximating it in the vicinity of an instance being predicted. Besides, TLIME, as a model-agnostic interpretation, shows excellent flexibility and capability of explaining any underlying machine learning model.
Explanation System of TLIME
---------------------------
Considering the poor interpretability and high computational complexity of the pixel-based image representation, we adopt a superpixel based explanation system. Each superpixel, as the basic processing unit, is a group of connected pixels with similar colors or gray levels. Figure \[fig:3\] shows the pixel-based image, superpixel image, and superpixel-based explanation. The interpretable representation of an image $x\in{\mathbb{R}^d} $ consisting of $d$ pixels and $d'$ superpixels is a binary vector $x'\in{\{{0,1}\}^{d'}}$ where $1$ indicates the presence of original superpixel and $0$ indicates absence of original superpixel.
![[]{data-label="fig:3"}](figures/process/chanmeleon){width="100.00000%"}
![[]{data-label="fig:3"}](figures/process/chanmeleon_p){width="100.00000%"}
![[]{data-label="fig:3"}](figures/process/chanmeleon_p1){width="100.00000%"}
We denote the original image classification model being explained as $f$, the interpretable decision tree model as $g$, and the locality fidelity loss as $L(f,g,\pi_x)$, which is calculated by the locally weighted square loss:
$$\L(f,g,\pi_x)=\sum_{z,z'\in Z}e^{(-D(x,z)^2/{\sigma}^2)}(f(z)-g (z'))^2.\\$$
The database $Z$ is composed of perturbed samples $z'\in{\{{0,1}\}^{d'}}$ which are sampled around $x'$ by drawing nonzero elements at random. Given a perturbed sample $z'$, we recover the sample in the original representation $z\in {\mathbb{R}^d}$ and get $f(z)$. Moreover, $\pi_x(z)=exp(-D(x,z)^2/{\sigma}^2)$ where distance function $D$ is the $L_2$ distance of image $x$ and $z$ is used to capture locality.
We denote the decision tree explanation produced by TLIME as below: $$\xi(x)=argmin\quad{L(f,g,\pi_x)+ dep(g)}.$$ The depth of decision tree $dep(g)$ is a measure of model complexity. A smaller depth indicates a stronger understandability of model $g$. In order to ensure both local fidelity and understandability, formula (2) minimizes locality-fidelity loss $L(f,g,\pi_x)$ while holding $dep(g)$ low enough. Algorithm 1 shows a simplified workflow diagram of TLIME. Firstly, TLIME gets the superpixel image by using a standard segmentation method. Then the database $Z$ is constructed by running multiple iterations of the perturbed sampling operation. Finally, within the allowable range of prediction error, TLIME gets the minimum depth decision tree by using the CART method.
Classifier $f$; Number of samples $N$; Instance $x$; Max depth of tree $d$; time and prediction error of TLIME; get superpixel image $x'$ by segment method; initial $Z=\{\}$; get $z'$ by sampling around $x'$; get $f(x')$ by classifier $f$; get $z$ by recovering $z'$; $Z=Z+(z'_i,f(z_i),\pi_x(z_i))$ get decision tree $g=CART(Z, maxdepth=j)$ $error=\|f(x)-g(x')\|$; output decision tree $g$, time and prediction error;
Experimental Results {#sec:typestyle}
====================
In this section, TLIME and LIME explain the predictions of RandomForeset Classifier and Google’s pre-trained Inception neural network. We compare the experimental results of the two algorithms in terms of understandability, fidelity, and efficiency.
RandomForeset Classifier on MNIST database
------------------------------------------
The MNIST database is one of the most common databases used for image classification. It consists of $7\times{10^5}$ small $28\times28$ grayscale images of handwritten digits. In this experiment, the image data is split into $70\%$ as the training set and $30\%$ as the test set. Table \[tab:2\] shows the performance of the random forest classifier. For instance $x$, the predicted top $1$ classe is $Seven$ (p=1.0). Figure \[fig:6\] shows the decision tree explanations by TLIME.
precision recall f1-score support
-------------- ----------- -------- ---------- ---------
weighted avg 0.95 0.95 0.95 21000
\[tab:2\]
Comparing with LIME, which can only provide a one-shot explanation, the decision tree structure by TLIME provides a more intuitive explanation. Figure \[fig:6\] shows that if feature $0$ and feature $3$ exist, then the prediction probability is $1.0$. Moreover, the tree structure can capture the interaction between features in the data. The importance of feature $0$ and $3$ is $0.9416$ and $0.0402$, respectively, which tells us the feature $0$ makes a significant contribution to predicting the outcome. The prediction error is calculated to measure local fidelity. The prediction error of TLIME is $0.0$, showing better fidelity than LIME with an error of $0.0529$.
Efficiency is highly related to the time necessary for a user to grasp the explanation. The runtime of TLIME is $0.0020s$, which is faster than that of LIME - $0.0080s$. Note that the runtime does not include perturbed sampling operation, which takes the same time for LIME and TLIME.
{width="100.00000%"}
{width="100.00000%"}
{width="100.00000%"}
{width="100.00000%"}
\
{width="90.00000%"}
{width="90.00000%"}
{width="90.00000%"}
{width="90.00000%"}
\[fig:7\]
![[]{data-label="fig:6"}](figures/process/mnist_visio_tree0700){width="90.00000%"}
Inception prob pred prob pred error
------- ---------------- ----------- ------------ --
TLIME 0.4309 0.0476
LIME 0.6264 0.1479
TLIME 0.4682 0.0479
LIME 0.6814 0.2611
TLIME 0.0191 0.0036
LIME 0.0168 0.0059
\[tab:3\]
$table$ $lamp$ $studio$ $couch$ $pillow$
------- ---------------- ------------------ ----------
TLIME 0.0060s 0.0030s 0.0090s
LIME 0.0289s 0.0150s 0.0120s
\[tab:4\]
Google’s Inception neural network on Image-net database
-------------------------------------------------------
We explain the prediction of Google’s pre-trained Inception neural network on the image shown in Figure \[fig:7\]a. The top $5$ predicted classes are listed. Figures \[fig:7\]b, \[fig:7\]c, \[fig:7\]d show the superpixels explainations for the top $3$ predicted classes: $table$ $lamp (p=0.4785)$, $studio$ $couch (p=0.4203)$ and $pillow (p=0.0227)$ respectively. The prediction provides reasonable insight into what the neural network picks upon for each of the classes. This kind of explanation enhances trust in the classifier. Moreover, Table \[tab:3\] lists the prediction errors of TLIME and LIME. Table \[tab:4\] lists the runtime of TLIME and LIME. We can conclude from the above results that under less time, TLIME not only has a better understandability but also has a higher fidelity than LIME.
Conclusion
==========
We propose a decision tree-based local interpretable model-agnostic explanation (TLIME) for improving explainable AI. The goal of TLIME is to construct an interpretable decision tree model over the interpretable representation that is locally faithful to the oringal classifier. We compare TLIME and LIME in explaining the predictions of RandomForeset Classifier and Google’s pre-trained Inception neural network. Experimental results have shown that TLIME exhibits a better understandability and higher fidelity than LIME using less process time, which covers the ingredients of an ideal explainable AI model - understandability, fidelity, and efficiency.
|
---
abstract: 'What are the implications if the total ’information’ in the universe is conserved? Black holes might be ’logic gates’ recomputing the ’lost information’ from incoming ’signals’ from outside their event horizons into outgoing ’signals’ representing evaporative or radiative decay ’products’ of the reconfiguration process of the black hole quantum logic ’gate’. Apparent local imbalances in the information flow can be corrected by including the effects of the coupling of the vacuum ’reservoir’ of information as part of the total information involved in any evolutionary process. In this way perhaps the ’vacuum’ computes the future of the observable universe.'
author:
- |
**Scott M. Hitchcock**\
National Superconducting Cyclotron Laboratory (NSCL)\
Michigan State University, East Lansing, MI 48824-1321\
E-mail: hitchcock@nscl.msu.edu
date: 'August 1, 2001'
title: '**Is There a ’Conservation of Information Law’ for the Universe?**'
---
Introduction
============
Let us begin with the assumption that there may exist a Conservation of Total Information ’law’ for the entire universe. This means that the total information content in the current epoch is the same as that in the early universe [@mmc2] regardless of the limitations on what we observe as the ’visible’ forms it takes. The motivation for this is based in the idea of conservation of total mass-energy for the universe regardless of the forms matter takes during the reconfiguration processes of matter within the framework of an expanding vacuum filled with growing quantum networks [@qgn].
If all current visible structures floating on the sea of the vacuum constitute a very small percent of the total ’information’ in the entire universe and the ’expansion’ of ’space’ combined with local gravitationally driven aggregation of mass into ’information’ sources and sinks (such as stars and planets for instance) provide a means for ’computing’ new configurations of matter (biological systems for instance), then perhaps the remainder of the ’invisible information’ is in the vacuum ’reservoir’. All unstable ’visible’ physical systems such as atoms and molecules represent the building blocks for complex hierarchical systems.
If one were to take this view then the apparent information ’loss’ by ’trapping’ in black holes could be recast as the ’computation’ of new forms of information (Hawking radiation) by the black hole ’logic gate’ in a quantum computer space (vacuum). The signals emanating from the black hole carry information content about the logical operations performed on the incoming mass ’signals’ contributing to the process of black hole formation. The black hole recomputed its unstable state (coupled to the vacuum) into a more stable one in which outgoing signals are ’emitted’. The analogy is similar to the processing of incoming photons by the electrons around an atom into emission spectra.
Since new structures clearly emerge from previous ’unstable’ configurations of matter in the universe, new information also emerges as a function of the entanglement of new combinations of quantum systems into a single system with collective behaviors that are more than a linear sum (actually a ’direct product’) of the component systems by themselves. New information can be created locally then but overall information may be conserved for the universe as a whole due to the conversion of information from the expanding vacuum reservoir into novel configuration information defining the islands of matter throughout the universe.
Black Hole Quantum Logic ’Gates’?
=================================
Let us begin by looking at black holes as quantum computer logic gates in which incoming matter (information) signals are recomputed into outgoing decay product ’signals’. The ’difference’ in the form and content of the ’information’ between incoming and outgoing signals is the result of the coupling of the black hole to the expanding space (vacuum) information reservoir. In this sense the composite system of space and matter forms at least a network of local quantum computers in the neighborhood of black holes.
The universe may then be a form of ’quantum computer’ [@lloyd] *network* of black hole logic gates or other complex matter ’computers’ whose power to recompute the evolutionary progression of global states of the expanding universe is due their use of the vacuum reservoir of information created by the gravitational interaction of matter bound to the sea of vacuum energy upon which they are small components.
Milan M. Cirkovic’s idea ([@mmc]) that there may be no primordial black holes (certainly believable in light of the absence of appropriate gamma ray spectra) combined with the possibility that any black holes formed during the evolution of the universe might not trap or evaporate enough ’information’ at a rate significant enough to create an imbalance in the total information of the universe might be seen as a ’hint’ that total information content in the universe may to first order be conserved. Black holes are generally regarded as information ’sinks’ but they are still within the universe. While the information is usually thought of as being ’lost’ inside a BH, if we take the view that the BH is a sort of ’memory’ or logic gate connected with the rest of the universe through its coupling to the vacuum (expanding space) then the expanding universe is engaged in a ’race’ between the ’creation’, ’processing’, ’transformation’ and ’destruction’ of ’information’ encoded in complex physical system ’islands’ in the vacuum info-space. The rate of expansion (possibly accelerating according to recent supernovae observations) can be used to test whether a conservation of information law is valid.
The information processing time (lifetime?) of a black hole seems to be dependent on the information (e.g. consumed ’signals’ plus remnant mass of its stellar progenitor) that it holds. Even if this information is an entangled ’collective excitation’ mess, it can decohere (by a vacuum-BH interaction that triggers the ’output’ of decay products?) via the vacuum induced evaporation of the BH at a critical transition ’mass’. The key to accounting for all the information is to remember that the black hole exists only because there is an environment (the vacuum) that defines it.
Black holes represent one of the fundamental testing grounds for a conservation of information law for the entire universe. Any unstable system presents the opportunity to test conservation of information. Black holes and the interest in irreversible loss of information seems to have caught the imagination of cosmologists and those interested in quantum processes in which information is encoded in the form of ’observable’ properties of ’physical’ systems. There is a possibility that the ’entropy’ issues can be clarified when considering the ’remainder’ of information shifted into the vacuum ’information reservoir’ after processing of the ’lost’ signals (e.g. signal trapping and mass accretion resulting in creation of at least one ’signal’ in the form of the ’inflation of event horizon’, etc.) into outgoing signals during the evaporation or decay process. Gravity is a sort of adaptive ’wiring’ between BHs and matter outside the event horizon.
Black holes viewed as computational logic gates that recompute gravitationally wired signals into new forms of information provide a logical source for Hawking radiation and other possible ’evaporation’ signals. This perspective might illuminate the features of a general information conservation law that can be used to establish how complex systems can arise as the result of the computational effects of expanding vacuum upon the matter within it.
Info-flow through BH Gates
--------------------------
The state of the entire universe, $\left| U\right\rangle $, is the direct product its entangled component sub-states including the vacuum $\left|
V_{U}\right\rangle $, the physical (observable) ’matter’ sub-systems $\left|
S_{U}\right\rangle $ (such as particles, nuclei, atoms, molecules and gravitational aggregations of these in planets, stars, galaxies and life-forms), ’signals’ $\left| \lambda_{U}\right\rangle $(e.g. photons)and the expansion ’boundary condition’ or ’surface area’ of the ’event horizon’ of the expanding vacuum energy density characterized by the expansion front bubble surface area, $\left| A_{U}\right\rangle $: $$\left| U\right\rangle {\Large =}\left| V_{U}\right\rangle \bigotimes\left|
S_{U}\right\rangle \bigotimes\left| \lambda_{U}\right\rangle \bigotimes
\left| A_{U}\right\rangle$$
Which becomes: $$\left| U\right\rangle {\Large =}\left| V_{U}\right\rangle \bigotimes\left[
{\displaystyle\bigotimes\limits_{i=1}^{\infty}}
{\LARGE \left| S_{i}\right\rangle }\right] \bigotimes\left[
{\displaystyle\bigotimes\limits_{j=1}^{\infty}}
{\LARGE \left| \lambda_{j}\right\rangle }\right] \bigotimes\left|
A_{U}\right\rangle$$
Where causal networks can be formed by destabilized physical sub-systems and the signals between them in a general environment of an expanding vacuum ’information’ reservoir bounded by a computational enclosure of the expansion front metricized by the Hubble flow and large scale dynamics of galaxies. We note here that the ’vacuum’ is not empty, but to the contrary, contains most of the information in the universe. The observable matter we consider to be the sources of information about the large scale structure of the universe are relic computational perturbations from the anisotropic phonon-like collective excitation decay modes of the inflationary epoch. The Planck scale relics were ’computed’ into the particles we are built by the conversion of the localized energy of the Planck epoch universe into ’decay products’ like matter and the vacuum.
If we look at information flow through a single black hole logic gate, $\left| BH\right\rangle $, we see that an incoming signal, $\left|
\lambda_{IN}\right\rangle $, ’lost’ by entering a BH gate is computed into to new signals such as the quantized increase in the area of the event horizon, and a quantized increase in the total mass of the BH. If the BH mass is below the critical excited state at which quantum evaporation processes result in signal emissions, then the BH act like a long term RAM storage device or memory for the incoming signals delivered by the gravitational ’wiring’ of the vacuum to external sub-systems of the universe. At this point total information is still conserved and the ’lost’ information embedded in the BH has been computed into observable changes in the collective excitation states such as total BH mass and event area.
The state of the BH subsystem, $\left| S_{BH}\right\rangle $, encompassing the space in which incoming signals make a transition from ’classical’ separation (i.e. superposition) to quantum coupling (detection, absorption, or entanglement) before gravitational detection of a signal, ignoring for the moment the expansion area boundary conditions, $\left| A_{U}\right\rangle $, for the rest of the universe, is a ’classical’ quantum system: $$\left| S_{BH}\right\rangle _{C}{\Large =}\left| BH_{0}\right\rangle
{\Large +}\left| \lambda_{IN}\right\rangle$$
Upon detection of the signal by gravitational trapping of the ’quantum’ state of the BH system, $\left| S_{BH}\right\rangle _{Q}$, and the ’excited’ state, $\left| BH^{\ast}\right\rangle $, of the BH information processing system is a composite system of the entangled signal, $\left| \lambda_{IN}\right\rangle
$, with the BH ’mass’ inside its now expanded event horizon. The increase in the event horizon area is $\left| \delta A_{BH}\right\rangle $, and therefore information content of the BH is characterized by $\left| BH^{\ast
}\right\rangle $: $$\left| S_{BH}\right\rangle _{Q}{\Large =}\left| M_{BH}\right\rangle
\bigotimes\left| \lambda_{IN}\right\rangle =\left| M_{BH}\right\rangle
\bigotimes\left| \delta A_{BH}\right\rangle =\left| BH^{\ast}\right\rangle$$
The decay of this state occurs only if it is the critical evaporation threshold state, $\left| S_{BH}\right\rangle _{Critical}$. The system can gravitationally detect a ’less than infinite’ number of signals in a series of hierarchical excited states corresponding to the increased mass and computational (event horizon) surface area before it reaches this state in which the gate converts the event horizon and accretion mass information as Hawking evaporation signals. If the BH system is at a point where the emission of signals by ’evaporation’ occurs then the coupling of the vacuum to the remnant links the differences in the form and content of the pre-BH processed signals to the emitted post computation signals.
Since all forms of information during the process of ’loss’ and delayed ’evaporation’ in black holes (including any entropy terms that are in fact communicated to the vacuum information reservoir) can be accounted for, we see that black holes obey conservation of total information when all system components and environment involved in the reconfiguration of information from input to output are taken into account.
This limiting case of conservation of information for a black hole illustrates that a conservation of information law for the entire universe may provide a convenient tool for understanding fundamental processes, the evolution of complex systems and cosmological effects in local ’matter’ islands.
Note that this process pertains also to non-BH physical systems that detect and process information such as Feynman Cocks (FCs), Collective Excitation Networks (CENs) and Sequential Excitation Networks (SENs) [@hitchcock], [@hitchcock2], [@hitchcock3].
I would like to thank Paola A. Zizzi and Milan M. Cirkovic for inspiration to pursue this speculative direction. They are both blameless for my errors. Special thanks to poolside Toni. For further information and papers see: http://www.nscl.msu.edu/departments/facilities/hitchcock/index.htm
[9]{} ”**Is the Universe Really so Simple?”** by *Milan M. Cirkovic*, LANL e-print archives quant-ph/0107070, 13 Jul 2001. http://arxiv.org/ftp/quant-ph/papers/0107/0107070.pdf
**”The Early Universe as a Quantum Growing Network”** by Paola A. Zizzi, LANL e-print archives, gr-qc/0103002 v3, 6 Apr 2001, http://arxiv.org/ftp/gr-qc/papers/0103/0103002.pdf
**”Universe as quantum computer”** by *Seth Lloyd*, LANL e-print archives, quant-ph/9912088, 17 Dec 1999.
Private communication with Milan M. Cirkovic, Astronomical Observatory Belgrade, Yugoslavia dated August 1, 2001.
”**Quantum Clocks and the Origin of Time in Complex Systems**” by *Scott Hitchcock*, LANL e-print archives; gr-qc/9902046 v2, 20 Feb 1999, also NSCL Publication: MSUCL-1123, 1999. Available at http://www.nscl.msu.edu/news/nscl\_library/nscl\_preprint/MSUCL1123.pdf
”**Feynman Clocks, Causal Networks, and the Origin of Hierarchical ’Arrows of Time’ in Complex Systems from the Big Bang to the Brain. Part I ’Conjectures”’**by *Scott Hitchcock*, LANL e-print archives; gr-qc/0005074, 16 May 2000, also NSCL Publication: MSUCL-1135, 2000. Available at http://www.nscl.msu.edu/news/nscl\_library/nscl\_preprint/MSUCL1135.pdf
**”Feynman Clocks, Causal Networks, and Hierarchical Arrows of Time in Complex Systems From the Big Bang to the Brain”** An invited talk and paper given by *Scott Hitchcock* at the ’XXIII International Workshop on the Fundamental Problems of High Energy Physics and Field Theory’ at the Institute for High Energy Physics (IHEP), Protvino, Russia, June 21-23, 2000. To be published in the Proceedings of the Workshop and as NSCL as pre-print (MSUCL -1172). It is also at the Los Alamos National Labs, LANL e-Print Archives report number quant-ph/00100014. Available at http://www.nscl.msu.edu/news/nscl\_library/nscl\_preprint/MSUCL1172.pdf
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abstract: 'Recently a model, which is equivalent to the scalar form of G" ursey model, is shown to be a nontrivial field theoretical model when it is gauged with a $SU(N)$ field. In this paper we study another model that is equivalent to the vector form of the Gürsey model. We get a trivial theory when it is coupled with a scalar field. This result changes drastically when it is coupled with an additional $SU(N)$ field. We find a nontrivial field theoretical model under certain conditions.'
author:
- 'B.C. Lütfüoglu$^1$'
- 'F. Taşkin$^{1,2}$'
title: 'Renormalization Group Analysis of a G" ursey Model Inspired Field Theory II'
---
Introduction
============
Historically, there has always been a continuing interest in building nontrivial field theoretical models. A while ago it was shown that perturbative expansions are not adequate in deciding whether a model is nontrivial or not. Baker et al. showed that the $\phi^{4}$ theory, although perturbatively nontrivial, went to a free theory as the cutoff was lifted in four dimensions [@ba_ki_79; @ba_ki_81]. Continuing research is going on this subject [@kl_06]. Alternative methods become popular. Renormalization group (RG) methods are the most commonly used one. They were first introduced by Wilson et al. [@wi_ko_74]. Another method is using exact RG algorithm which were proposed by Polchinski [@po_84]. Recent studies gave important insights on both methods [@so_07_40_5733; @so_07_40_9675; @ig_it_so_07].
Another endeavor is building a model of nature using only fermions. Here all the observed bosons are constructed as composites of these ingredient spinors. In solid state physics, electrons come together to form bosonic particles [@mi_93; @ba_co_sc_57]. Historically, the first work on models with only spinors goes back to the work of Heisenberg [@he_54]. Two years later Gürsey proposed his model as a substitute for the Heisenberg model [@gu_56]. This Gürsey’s spinor model is important since it is conformally invariant classically and has classical solutions [@ko_56] which may be interpreted as instantons and merons [@ak_82], similar to the solutions of pure Yang-Mills theories in four dimensions [@fu_sh_90]. This original model can be generalized to include vector, pseudovector and pseudoscalar interactions.
We have worked on different forms of the G" ursey model [@ho_lu_06; @ho_ta_07; @ho_lu_ta_07] using the earlier works [@ak_ar_du_ho_ka_pa_82-34; @ak_ar_du_ho_ka_pa_82-41; @ak_ar_ho_pa_83; @ar_ho_83; @ar_ho_ka_85] as a starting point. In those references it was claimed that a polynomial lagrangian could be written equivalently to G" ursey’s non-polynomial lagrangian. Recently it is shown that they are equivalent only in a naive sense [@ho_lu_06; @ho_ta_07]. In [@ho_lu_06], using perturbative methods, we showed that only composite particles took part in physical processes whereas the constituent fields did not interact with each other. Recently in [@ho_lu_ta_07], we showed that, when this model is coupled to a constituent $U(1)$ gauge field, we were mimicking a gauge Higgs-Yukawa (gHY) system, which had the known problems of the Landau pole, with all of its connotations of triviality. There, our motivation was the famous Nambu-Jona-Lasinio model [@na_jo_61], which was written only in terms of spinor fields. This model was shown to be trivial [@ko_ko_94; @zi_89]. Recent attempts to gauge this model to obtain a nontrivial theory are given in references [@ha_ki_ku_na_94; @ao_mo_su_te_to_99; @ao_mo_su_te_to_00; @ku_te_99; @ko_ta_ya_93].
The essential point in our analysis is the factor of $\epsilon$ in the composite propagator [@ho_lu_06; @ho_ta_07]. This main difference makes many of the diagrams convergent when the cutoff is removed. Consequently, we find that we can construct a nontrivial model from the scalar G" ursey model when a non-Abelian gauge field is coupled to the fermions [@ho_lu_07]. In this paper we will investigate the vector form of the G" ursey model. Here we will closely follow the line of discussion followed in the references [@ha_ki_ku_na_94; @ho_lu_07].
This article is organized as follows. In the next section we describe the vector form of the G" ursey like model. There we derive the composite vector field propagator. In section 3, we couple a constituent scalar field to our model and discuss the new results. Then we solve the renormalization group equations (RGE’s) and find a Landau pole in the solution. In section 4, we introduce another field, a non-Abelian gauge field to the model. In the subsections we write the new RGE’s and derive the solutions by using some RG invariants. We discuss some limiting cases of the coupling constant solutions before giving the criteria’s of the nontriviality condition in section 5. Then we find the fixed point solutions. In the following subsections we analyze the solutions of the coupled equations and find their asymptotic behaviors. The final section is devoted to conclusions.
The Model
=========
The vector form of the pure spinor G" ursey model [@ak_ar_ho_pa_83] is given as
L=(i/ -ig /g\^[-1]{}-m)+\^[2/3]{}. \[gursey lagrangian\] Here only the spinor fields have kinetic part. The $g$ field is a pure gauge term to restore the local gauge symmetry, when the spinor field is transformed. This non-polynomial Lagrangian has been converted to an equivalent polynomial form by introducing auxiliary fields $\lambda_{\mu}$ and $G_{\mu}$ in [@ak_ar_ho_pa_83]. The constrained Lagrangian in the polynomial form is given as
L\_[c]{}= -e\^4\_G\^G\^[2]{} +. \[s yenilagran\] Recently it was shown that this equivalence should be taken only “naively” [@ho_ta_07]. This expression contains two constraint equations, obtained from writing the Euler-Lagrange equations for the auxiliary fields. Hence it should be quantized by using the constraint analysis $\grave{a}$ *la* Dirac [@di_64]. This calculation is performed using the path integral method. We find out that one can write the effective Lagrangian as L\_[eff]{}=-e\^4\_G\^G\^[2]{}+ \^(g\_G\^2 + 2G\_G\_)w\^. \[etkinlagran\] Here $\overline{w}^\mu$ and $w^{\nu}$ are the ghost fields. With a suitable redefinition of the fields the effective action can be given as S\_[eff]{}=Tr(i/+eJ/+m) + dx\^4 , \[eff\_action\]where $ J_{\mu}=-ig\partial_{\mu} g^{-1}+ G_{\mu}+ \lambda_{\mu}$. The second derivative of the effective action with respect to the $J_{\mu}$ field gives us the induced inverse propagator as . |\_[J\_=0]{}= -(q\_q\_-g\_q\^2) . Here dimensional regularization is used for the momentum integral and $\epsilon = 4-n$. All the other fields not shown in this expression, including ghost fields arising from the constrained equations, decouple from the model. The only remaining fields are the spinors and the $J_{\mu}$ field. This procedure is explicitly carried out in [@ho_ta_07]. In the Feynman gauge the propagator of the composite vector field can be written as $\epsilon\frac{g^{\mu\nu}}{p^2}$ where the spinor propagator is the usual Dirac propagator in the lowest order.
Although the original Lagrangian does not have a kinetic term for the vector field, one loop corrections generate this term and make this composite field as a dynamical entity like it is done in [@ho_lu_06], where the composite vector field is replaced by composite scalar field. In the literature there are also other similar models with differential operators in the interaction Lagrangian [@am_ba_da_ve_81].
In reference [@ho_ta_07], the contributions to the fermion propagator at higher orders were investigated by studying the Dyson-Schwinger equations for the two point function. We found that there is a phase which has no additions to the existing fermion mass.
Coupling with A Scalar Field
============================
We may add a constituent complex scalar field to the model and investigate the consequences of this addition. Our motivation is the work of Bardeen et al. [@ba_le_lo_86; @le_lo_ba_86]. When they added a vector field to the Nambu-Jona-Lasinio model, a complementary procedure to our work, they got interesting results. Since we already have a composite vector field, we can couple a massless scalar field which has its kinetic term, a self interacting term with coupling constant $a$ and an interaction term with new coupling constant $y$ in the Lagrangian. Then the effective Lagrangian becomes L\_[eff]{}=-e\^4\_G\^G\^[2]{}+ \^(g\_G\^2 + 2G\_G\_)w\^+ - \^[4]{}-y. \[etkinlagran+skalar1\] Since the $G_{\mu}$, $\lambda_{\mu}$ and ghost fields decouple, this Lagrangian reduces to the effective expression given below. L\_[eff]{}=(i/ +eJ/- y-m)-e\^4J\^[4]{}+ - \^[4]{}. \[etkinlagran+skalar2\] If our fermion field had a color index $i$ where $i=1...N$, we could perform an 1/N expansion to justify the use of only ladder diagrams for higher orders for the scattering processes. Although in our model the spinor has only one color, we still consider only ladder diagrams anticipating that one can construct a variation of the model with N colors. In the following subsection we summarize the changes in our results for this new model.
New Results and Higher Orders
-----------------------------
In the model described in reference [@ho_ta_07], it is shown that only composites can scatter from each other with a finite expression, due to the presence of $\epsilon$ in the composite vector propagator. There is also a tree-diagram process where the spinor scatters from a composite particle, a Compton-like scattering, with a finite cross-section. This diagram can be written in the other channel, which can be interpreted as spinor production out of vector particles. Note that in the original model the four spinor kernel was of order $\epsilon $. The lowest order diagram, vanishes due to the presence of the composite vector propagator. In higher orders this expression can be written in the quenched ladder approximation [@mi_93], where the kernel is separated into a vector propagator with two spinor legs joining the proper kernel. If the proper kernel is of order $\epsilon$, the loop involving two spinors and a vector propagator can be at most finite that makes the whole diagram in first order in $\epsilon$. This fact shows that there is no nontrivial spinor-spinor scattering in the original model.
These results changes drastically with scalar field coupling. Two fermion scattering is now possible due to the presence of the scalar field instead of vector field channel. In lowest order this process goes through the tree diagram given in Figure \[fig789\].a. At the next higher order the box diagram with two spinors and two scalar particles, Figure \[fig789\].b, is finite from dimensional analysis. If the scalar particles are used as intermediaries, the spinor production from scattering of composite vector particles becomes possible as shown in Figure \[fig789\].c where the dotted, straight and wiggly lines represent scalar, spinor and composite vector particles, respectively.
$\begin{array}{c@{\hspace{1cm}}c@{\hspace{5mm}}c}
\multicolumn{1}{l}{\mbox{\bf }}&
\multicolumn{1}{l}{\mbox{\bf }}&
\multicolumn{1}{l}{\mbox{\bf }}\\
[-0.53cm]
\epsfxsize=20mm \epsffile{51-1}&
\epsfxsize=18mm \epsffile{52-0}&
\epsfxsize=22mm \epsffile{41-1} \\
[0.4cm]
\mbox{\bf (a)} &
\mbox{\bf (b)} &
\mbox{\bf (c)}
\end{array}$
Renormalization Group Equations and Solutions
---------------------------------------------
In reference [@ho_ta_07], it is widely discussed that the $<{\overline{\psi}}\psi J_\mu>$ vertex and the spinor box diagram give finite results. The higher diagrams do not change this result, since each momentum integration is accompanied by an $\epsilon$ term in the composite vector propagator. Therefore, there is no need for infinite coupling constant renormalization.
In the new model where a massless scalar field is added, all the three coupling constants are renormalized. One can write the first order RGE’s for these coupling constants, similar to the analysis in [@ha_ki_ku_na_94]. We take $\mu_0$ as a reference scale at low energies, $t=ln (\mu/\mu_0)$, where $\mu$ is the renormalization point. 16\^2y(t) &=& A y \^3(t),\[y3\]\
16\^2e(t) &=& B e(t) y\^2(t),\[eg2\]\
16\^2a(t) &=& C a\^2(t)-D y\^4(t) \[g4\]. Here $A$, $B$, $C$ and $D$ are positive numerical constants. We find out that Yukawa and $<{\overline{\psi}}\psi J_\mu>$ vertices have only scalar correction. The composite vector correction to these vertices are finite due to the $\epsilon$ in the propagator. Therefore, our equations differ from those in reference [@ho_lu_07; @ha_ki_ku_na_94]. These processes are illustrated in diagrams shown in Figure \[fig456\].
$\begin{array}{c@{\hspace{1cm}}c@{\hspace{5mm}}c@{\hspace{5mm}}c}
\multicolumn{1}{l}{\mbox{\bf }}&
\multicolumn{1}{l}{\mbox{\bf }}&
\multicolumn{1}{l}{\mbox{\bf }}&
\multicolumn{1}{l}{\mbox{\bf }}\\
[-0.53cm]
\epsfxsize=13mm \epsffile{01-1} &
\epsfxsize=13mm \epsffile{01-2} &
\epsfxsize=20mm \epsfysize=8mm \epsffile{01-3} &
\epsfxsize=16mm \epsffile{01-4} \\
[3mm]
\mbox{\bf (a)} &
\mbox{\bf (b)} &
\mbox{\bf (c)} &
\mbox{\bf (d)}
\end{array}$
The RGE’s have the immediate solutions y\^2(t)&=&,\
e(t)&=&e\_0Z(t)\^[-B/2A]{},\
a(t)&=& , where $Z(t)=1-\frac{Ay_0^2}{8\pi^2}t$.
The main problem of models with U(1) coupling, namely the Landau pole, is expected to make our new model a trivial one. We expect that coupling to a non-Abelian gauge theory will remedy this defect by new contributions to the RGE’s. Thus, obtaining a nontrivial model will be possible. Coupling to a non-Abelian gauge field will also give us more degrees of freedom in studying the behavior of the beta function. This may allow us to find the critical number of gauge and fermion fields to obtain a zero of this function at nontrivial values of the coupling constants of the model.
Coupling with a Non-Abelian Field
=================================
In this section we consider our model with $SU(N_{C})$ gauge field interaction, where the spinors have $N_{f}$ different flavors. Although we study in the leading order of $\frac{1}{N_{C}}$ expansion, where all the planar diagrams contribute to the RGE’s, we are interested in the high-energy asymptotic region where the gauge coupling is perturbatively small; $\frac{g^{2}N_{C}}{4\pi}\ll 1$. However, the number of fermions is in the same order as $N_{C}$. Only $n_{f}$ fermions have a degenerate large Yukawa coupling. We start with the effective Lagrangian L\_[eff]{}=\_[i=1]{}\^[N\_[f]{}]{}\_[i]{}(iD / +eJ/ -m)\_[i]{}-e\^4J\^[4]{}+ - \^[4]{}-\_[i=1]{}\^[n\_[f]{}]{} \_[i]{}y\_[i]{}- Tr \[F\_ F\^\] + L\_+ L\_. \[etkinlagran+skalar+gauge\] The gauge field belongs to the adjoint representation of the color group $SU(N_C) $ where $D_{\mu}$ is the color covariant derivative. $y$, $a$, $e$ and $g$ are the Yukawa, quartic scalar, composite vector and gauge coupling constants, respectively.
There are two kind of ghost fields in the model. The first one, which comes from the composite constraints, decouples from our model [@ak_ar_ho_pa_83; @ak_ar_du_ho_ka_pa_82-41]. The second one, coming from the gauge condition on the vector field, do not decouple and contribute to the RGE’s in the usual way.
Renormalization Group Equations and Solutions
---------------------------------------------
In this subsection we will analysis the RGE’s in the leading order of the approximation given above. In the one loop approximation the RGE’s are 16\^2g(t) &=& -A g\^3(t),\[5g3\]\
16\^2y(t) &=& B y\^3(t)-Cy(t)g\^2(t),\[5y3\]\
16\^2e(t) &=& D e(t)y\^2(t)-Ee(t)g\^2(t),\[5eg2\]\
16\^2a(t) &=& F a(t)y\^2(t)-Gy\^4(t) \[5L4\]. Here $A$, $B$, $C$, $D$, $E$, $F$ and $G$ are positive constants.
In the RGE’s we see that the diagrams, where the composite vector field takes part, are down by order of $\epsilon$. Therefore we do not have contributions proportional to $e^2(t)$, $e^3(t)$, $y(t)e^2(t)$ and $g(t)e^2(t)$. Also we neglect the scalar loop contribution to the gauge coupling $g(t)$, similar to the work of [@ha_ki_ku_na_94].
The solutions of the first RG equation (\[5g3\]) can be given as g\^[2]{}(t)=g\_[0]{}\^[2]{}(1+t)\^[-1]{}, \[g\_nin\_cozumu\]where $\alpha_0=\frac{g_{0}^2}{4\pi}$. We define (t), where $g_0=g(t=0)$ which is the initial value at the reference scale $\mu_0$. For the solution of the second RG equation (\[5y3\]), we can propose a RG invariant $H(t)$ as H(t)=-\^[-1+C/A]{}(t). Since $H(t)$ is a constant, we call it $H_0$. Then, the solution of the Yukawa coupling constant can be written as y\^2(t)=g\^2(t)\^[-1]{}. \[y\_nin\_cozumu\] The solution of the third RG equation (\[5eg2\]) can be defined by another RG invariant $P(t)$ if and only if the constants $B$ equals to $D$ and $C$ equals to $E$. Then the invariant becomes P(t)=-\^[-1+C/A]{}(t). The solution of the composite vector coupling $e(t)$ can be written as e\^2(t)=-()\^[2]{}g\^2(t)\^[-1]{}. \[e\_nin\_cozumu\] where $P_{0}$ denotes the value of the invariant $P(t)$. The solution of the last RG equation (\[5L4\]) can be defined by another RG invariant $K(t)$, given as K(t)=-\^[-1+]{}(t). We can rewrite the solution with a value of the invariant $K(t)$ as $K_{0}$ a(t)=g\^[2]{}(t) \[a\_nin\_cozumu\].Here we notice that the RG constants $H_{0}$, $P_{0}$ and $K_{0}$ play important roles on the behavior of the solutions of the coupling equations (\[g\_nin\_cozumu\]), (\[y\_nin\_cozumu\]), (\[e\_nin\_cozumu\]), (\[a\_nin\_cozumu\]). Similar works have been studied in [@ha_ki_ku_na_94; @ho_lu_07]. The values of the constants are given in these equations as A=, B=D==2n\_fN\_C, C=E=6C\_2(R), F=G. Here $C_2(R)$ is a second Casimir, $C_2(R)=\frac{(N_{C}^{2}-1)}{2N_{C}}$, $R$ is the fundamental representation with $T(R)=\frac{1}{2}$.
Before entering the analysis of the fixed point, we briefly investigate the results of some limits.
### The limiting case A$\rightarrow$$+0$ for finite $t$
In this case the coupling constants solutions can be written as g\^2(t)&=&g\^2\_0,\
y\^2(t)&=& \^[-1]{},\
e\^2(t)&=&- \^[-1]{},\
a(t)&=& . Here $\alpha_{0}=\alpha$ and $\frac{C}{2\pi}=\frac{1}{\alpha_c}$.
### The limiting case A$\rightarrow$$C$ for finite $t$
In this limit case the solutions of the couplings (\[y\_nin\_cozumu\]), (\[e\_nin\_cozumu\]) and (\[a\_nin\_cozumu\]) seem to vanish. If we suggest new RG invariant $H_{1}$, instead of $H_{0}$, as $H_{0}=-1+\frac{C-A}{A}H_{1}$ , we find that two of the coupling solutions do not vanish, whereas composite vector coupling goes to zero. These behaviors are given below y\^[2]{}(t)&=&g\^[2]{}(t)\^[-1]{},\
e\^[2]{}(t)&=&P\_[0]{}()( )g\^[2]{}(t)\^[-1]{},\
a(t)&=& g\^[2]{}(t) . It is amusing to see that the added interactions nullify the original vector-spinor coupling.
### The limiting case A$\rightarrow$$2C$ for finite $t$
In this limit case only the quartic coupling constant solution (\[a\_nin\_cozumu\]) behaves critically. Similarly we can redefine RG invariant $K_{1}$ instead of $K_{0}$ as $K_{0}=-1+\frac{2C-A}{A}K_{1}$, then the quartic coupling solution takes the form a(t)=g\^[2]{}(t). This limit is not allowed because it does not give asymptotic freedom.
In the next section we will mention which criteria are needed to define a nontrivial theory.
Nontriviality of the system {#nontriviality}
===========================
To have a nontrivial theory all the running coupling constants should not diverge at any finite energy, which means the absence of Landau poles of the system. For a consistent theory these solutions should not vanish identically and must have real and positive values. These conditions make the model unitary and satisfy the vacuum stability criterion. Note that if we decouple the scalar and composite vector field from the system, we have a nontrivial theory, similar to QCD. Therefore, $e(t)\equiv
g(t)\equiv a(t)\equiv 0$ solution will not be named as the nontriviality of our composite model. The mass parameter can be renormalized in the $\overline{MS}$ scheme and the mass can be chosen as zero.
Remember that we are restricted by neglecting the scalar loop contributions to the gauge coupling where the composite vector contributions are not neglected but down due to the presence of $\epsilon$ in its propagator. If the Yukawa and/or quartic scalar couplings become so large and break the $1/N_{C}$ expansion then the behavior of the gauge coupling might be affected.
These restriction conditions are the same as the ones in the gHY system which was discussed widely in [@ha_ki_ku_na_94]. A while ago, one of us, B.C.L., with a collaborator, studied the scalar form of the Gürsey model in this fashion [@ho_lu_07]. In that model, there is a composite scalar field with a propagator completely different from a constituent scalar field used in reference [@ha_ki_ku_na_94]. There, we showed that a restriction is not needed between the scalar and the gauge field coupling since the contribution of the scalar field to the gauge field is down by the factor of $\epsilon$ in the scalar propagator. In this work, the vector form of the Gürsey Model, we have constituent scalar field and composite vector field which is missing in gHY system. This composite field adds a new RGE to the system but does not contribute to the former ones in gHY system with a totally different reason.
After these remarks we will discuss the nontriviality conditions of our model in the following subsections.
Fixed Point Solution
--------------------
The RGE’s can be rewritten as 8\^[2]{}&=& Bg\^[2]{}(t) ,\
8\^[2]{}&=&(C-A)g\^[2]{}(t) ,\
8\^[2]{}&=&(2C-A)g\^[2]{}(t) . The fixed point solutions can be given as &=&,\
&=&,\
&=&. These are also the solutions of the equations (\[y\_nin\_cozumu\]), (\[e\_nin\_cozumu\]) and (\[a\_nin\_cozumu\]) where the RG invariants are $P_{0}=H_{0}=K_{0}=0$ as $P_{0}=\zeta H_{0}$. Here $\zeta$ is a constant. It is clear that the behavior of all the coupling constants are determined by the gauge coupling which means the Kubo, Sibold and Zimmermann’s “coupling constant reduction” [@ku_si_zi_89]. This corresponds to the Pendleton-Ross fixed point [@pe_ro_81] in the context of the RGE. Remark that only the case, $C>A$, prevents the violation of the unitarity and keeps the stability of the vacuum. This gives rise to nontriviality of the model when the RG invariants are set to zero. In the following subsections we will analysis the coupling constant solutions only in this case with non zero RG invariants.
Yukawa Coupling
---------------
The Yukawa coupling solution is given in equation (\[y\_nin\_cozumu\]). It is obvious that the sign of the RG invariant, $H_{0}$, plays an important role in the behavior of the solution where B is positive. The ultraviolet (UV) limit of $\eta(t)$ is needed before continuing the analysis in $C>A$ case. \^[1-]{}(t )
[ll]{} +.\
The UV behavior of Yukawa coupling with a non zero RG invariant $H_{0}$ is y\^2(t) {
[lrl]{} +0, &&\
, & &\
-0, &&\
. For $-1<H_{0}<0$ case, there exists a finite value of $t$ before it goes to infinity 1+t=()\^[A/(C-A)]{}. In this $t$ value Yukawa coupling diverges and changes its sign. These asymptotic behaviors show that the theory is nontrivial if and only if the RG invariant $H_{0}$ is positive.
The RG flows in the $(g^{2}(t),y^{2}(t))$ plane are shown in Figure \[y\_kare\_g\_kare\]. The upper bound of the figure denotes the “Landau Pole”.
Composite Vector Field Coupling
-------------------------------
The composite vector coupling solution is given in equation (\[e\_nin\_cozumu\]). In this case not only the sign of $H_{0}$ but also the sign of $P_{0}$ is crucial for nontriviality. Since $H_{0}$ is positive, $P_{0}$ must be negative. The composite vector field coupling behaves similarly to the Yukawa coupling up to a constant multiplier. In figure \[e\_kare\_y\_kare\] we plot $e^2(t)$ vs. $y^2(t)$ where $P_{0}<0$, $H_{0}>0$. Both coupling constants approach the origin as $t$ goes to infinity. Thus, our model fulfills the condition required by the asymptotic freedom criterion.
Quartic Scalar Field Coupling
-----------------------------
Finally quartic scalar coupling solution given in equation (\[a\_nin\_cozumu\]) can be analyzed. We have already restricted ourselves with $C>A$, $H_{0}>0$ and $P_{0}<0$ for nontriviality. In the limit where $t\gg 1$ , the $\eta$ terms in the last fraction of equation (\[a\_nin\_cozumu\]) become dominant therefore $1$ can be neglected. Hence we can express the solution as a(t)g\_[0]{}\^[2]{}(t) , which is equal to a(t)=g\_[0]{}\^[2]{} . This asymptotic behavior shows that to have a nontrivial model the RG invariant $K_{0}$ should be equal to zero. The other possibilities for a non zero solution for $K_{0}$ is been widely discussed in the reference [@ha_ki_ku_na_94]. In Figure \[a\_y\_kare\], we plot the RG flows in $(a(t),y^2(t))$ plane for different values of $H_0$ higher than zero while the gauge coupling $\alpha(t=0)$ is fixed to one. The origin is the limit where $t$ goes to infinity, there both coupling constants approach zero when $K_{0}=0$.
Conclusion
==========
A while ago, one of us, F.T., with a collaborator, showed that the scattering of composite vector particles gives nontrivial results while the constituent spinors do not. In that work [@ho_ta_07], a polynomial Lagrangian model inspired by the vector form of G" ursey model was used. Here we couple a constituent massless scalar field to our previous model. We find out that many of the features, related to the creating and scattering of the spinor particles of the original model, are not true anymore. In the one loop approximation we find the RGE’s whose solutions have all the problems associated with the Landau pole, like the case in reference [@ho_lu_ta_07]. To remedy this defect we couple a $SU(N_{C})$ non-Abelian gauge field to the new model. We solve the new RGE’s and conclude that if the conditions $C>A$, $H_{0}>0$, $P_{0}\leq 0$ and $K_{0}=0$ are satisfied, the model gives a result which can be interpreted as a nontrivial field theoretical model. We find fixed point solutions where the coupling constants are not equal to zero. In section \[nontriviality\] we plot the UV region behavior of the coupling constants. There, they all go to zero asymptotically which means asymptotic freedom, which is another feature of a nontrivial model.
Our calculation shows that one can construct nontrivial field theory starting from constrained Lagrangians.
**Acknowledgement**: We thank to Mahmut Hortaçsu for discussions and both scientific and technical assistance while preparing this manuscript. We also thank Nazmi Postacioglu for technical discussions. This work is supported by the ITU BAP project no: 31595. This work is also supported by TUBITAK, the Scientific and Technological Council of Turkey.
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|
---
abstract: 'In this work we address the problem of distributed optimization of the sum of convex cost functions in the context of multi-agent systems over lossy communication networks. Building upon operator theory, first, we derive an ADMM-like algorithm that we refer to as relaxed ADMM (R-ADMM) via a generalized *Peaceman-Rachford Splitting* operator on the Lagrange dual formulation of the original optimization problem. This specific algorithm depends on two parameters, namely the averaging coefficient $\alpha$ and the augmented Lagrangian coefficient $\rho$. We show that by setting $\alpha=1/2$ we recover the standard ADMM algorithm as a special case of our algorithm. Moreover, by properly manipulating the proposed R-ADMM, we are able to provide two alternative ADMM-like algorithms that present easier implementation and reduced complexity in terms of memory, communication and computational requirements. Most importantly the latter of these two algorithms provides the first ADMM-like algorithm which has guaranteed convergence even in the presence of lossy communication under the same assumption of standard ADMM with lossless communication. Finally, this work is complemented with a set of compelling numerical simulations of the proposed algorithms over cycle graphs and random geometric graphs subject to i.i.d. random packet losses.'
author:
- 'N. Bastianello$^\dagger$, M. Todescato$^\ddagger$, R. Carli$^\dagger$, L. Schenato$^\dagger$ [^1] [^2]'
bibliography:
- './IEEEabrv.bib'
- './references.bib'
title: '**Distributed Optimization over Lossy Networks via Relaxed Peaceman-Rachford Splitting: a Robust ADMM Approach** '
---
distributed optimization, ADMM, operator theory, splitting methods, Peaceman-Rachford operator
Introduction {#sec:intro}
============
From classical control theory to more recent Machine Learning applications, many problems can be cast as optimization problems [@Slavakis:14] and, in particular, as large-scale optimization problems given the advent of Internet-of-Things we are witnessing with its ever-increasing growth of large-scale cyber-physical systems. Hence, stemming from classical optimization theory, in order to break down the computational complexity, parallel and distributed optimization methods have been the focus of a wide branch of research [@BD:TP:1989]. Within this vast topic, typical applications, going under the name of *distributed consensus optimization*, foresee distributed computing nodes to communicate in order to achieve a desired common goal. More formally, the distributed nodes seek to $$\min_{x}\sum_{i=1}^Nf_i(x)$$ where, usually, each $f_i$ is owned by one node only. Toward this application among very many different optimization algorithms explored in past as well as in current literature, e.g. subgradient methods [@BJ:MR:MJ:2010], the well known Alternating Direction Method of Multipliers (ADMM), first introduced in [@glowinski1975approximation] and [@gabay1976dual], is recently receiving an ever-increasing interest because of its numerical efficiency and its natural structure which makes it well-suited for distributed and parallel computing. In particular, the relatively recent monograph [@boyd2011distributed] reveals the ADMM in detail presenting a broad set of selected applications to which ADMM is suitably applied. For a wider set of applications together with some convergence results we refer the interested reader to [@fukushima1992application; @eckstein1994some; @eckstein1992douglas; @EG:AT:ES:MJ:2015].\
While ADMM can be proficiently applied to distributed setups, rigorous convergence results are usually provided only in scenarios characterized by synchronous updates and lossless communications.However, practically it is rarely possible and often difficult to ensure synchronization and communication reliability among computing nodes. And even when this is possible via specific communication protocols, it is clear how the impossibility to deal with asynchronous and lossy updates would majorly compromise the algorithm applicability.\
Hence, an extensive body of work has been devoted to overcoming this limitation by adapting the ADMM to operate in an asynchronous fashion. Among the first steps in this direction, [@iutzeler2013asynchronous] proves convergence when only one randomly selected coordinate is updated at each iteration. Similarly, [@wei20131] suggests to update only the variables related to a subset of constraints randomly selected at each iteration, showing convergence of the algorithm with a rate of $O(1/k)$. To deal with asynchronous updates, a master-slave architecture is proposed in [@zhang2014asynchronous]. The more recent [@bianchi2016coordinate] extends the formulation introduced in [@iutzeler2013asynchronous] to allow the update of a subset of coordinates at each instant. In [@chang2016asynchronous], in view of large-scale optimization, the convergence rate of a partially asynchronous ADMM – i.e., subject to a maximum allowed delay – is studied. Finally [@peng2016arock] defines a framework for asynchronous operations used to solve a broad class of optimization problems and showing how to derive an asynchronous ADMM formulation.\
Conversely to the above works that deal with asynchronous updates, to the best of our knowledge, no work explicitly focuses on the robustness of ADMM to packet losses. Yet, to set the stage for the analysis of robustness of the ADMM algorithm to losses in the communication, we resort to a different body of literature on operator theory. Here, the underlying idea is to convert optimization problems into the problem of finding the fixed points of suitable nonexpansive operators [@bauschke2011convex]. However, the mere application of the so-called *proximal point algorithm* (PPA) – introduced in [@rockafellar1976monotone] and the later [@parikh2014proximal] – to look for the fixed points can be unwieldy in complex optimization problems. Hence, particular credits have been given to *splitting methods* which exploit the problem’s structure to break it in smaller and more manageable pieces. It is in the framework of splitting operators, and in particular the well recognized Peaceman-Rachford (PRS) [@peaceman1955numerical] and Douglas-Rachford (DRS) [@douglas1956numerical; @lions1979splitting] splitting, that the ADMM comes into place. Indeed, the classical formulation of the ADMM naturally arises as application of the DRS to the Lagrange dual problem of the original optimization problem [@eckstein2012augmented]. For further details on a variety of splitting operators and their application in asynchronous setups we refer to [@davis2016convergence] and [@hannah2016unbounded], respectively.\
In this paper we present and analyze different formulation for the ADMM algorithm. We are particularly interested to the broad class of distributed consensus optimization problems. Our final goal is to present a novel robustness result in scenarios characterized by synchronous but possibly lossy updates among distributed nodes. To achieve our result we start by considering a prototypical optimization problem assuming reliable loss-free communication. In this case, by leveraging the general framework arising from the Krasnosel’skii-Mann (KM) iteration for averaging operators [@krasnosel1955two; @mann1953mean], we derive a relaxed version of the ADMM (R-ADMM). Next, we draw the attention to the problem of interest, i.e., distributed consensus optimization. We first present the natural algorithmic implementation of the R-ADMM tailored for the problem. Then we propose two different implementations which are particularly favorable for storage and communication purposes. Moreover, the latter turns out extremely advantageous and yet robust in the presence of lossy communication. As natural byproduct we obtain a comprehensive and self-contained overview on the algorithm and a plethora of possible practical implementations.\
The remainder of the paper is organized as follows. Section \[sec:operators-background\] presents the necessary background on splitting operators. Section \[sec:ADMMandRADMM\] reviews the classical ADMM algorithm and its generalized version. Section \[sec:distributed\_consensus\] focuses on the analysis of distributed consensus optimization. Section \[sec:simulation\] collects some numerical simulations and Section \[sec:conclusions\] concludes the paper. The technical proofs can be found in the Appendices.
Background on Splitting Operators {#sec:operators-background}
=================================
This Section introduces some background on operator theory on Hilbert spaces and, in particular, on nonexpansive operators. The interest for operator theory stems from the fact that a convex optimization problem can be cast into the problem of finding the fixed point(s) of a suitable nonexpansive operator $T$ [@davis2016convergence; @bauschke2011convex], that is the points $x^*$ such that $Tx^*=x^*$.
Definitions and Properties {#subsec:definitions}
--------------------------
Let $\mathcal{X}$ be a Hilbert space, an operator $T:\mathcal{X}\rightarrow\mathcal{X}$ is said to be *nonexpansive* if it has unitary Lipschitz constant, *i.e.* it verifies $\|Tx-Ty\|\leq\|x-y\|$ for any two $x,y\in\mathcal{X}$.
Let $\mathcal{X}$ be a Hilbert space, $T:\mathcal{X}\rightarrow\mathcal{X}$ a nonexpansive operator and $\alpha\in(0,1)$. We define the *$\alpha$-averaged operator* $T_\alpha$ as $T_\alpha=(1-\alpha)I+\alpha T$, where $I$ is the identity operator on $\mathcal{X}$.
Notice that $\alpha$-averaged operators are also nonexpansive, indeed nonexpansive operators are $1$-averaged. Moreover, the $\alpha$-averaged operator $T_\alpha$ has the same fixed points of $T$ [@bauschke2011convex].
Let $\mathcal{X}$ be a Hilbert space and $f:\mathcal{X}\rightarrow\mathbb{R}\cup\{+\infty\}$ be a closed, proper and convex function. We define the *proximal operator* of $f$ with penalty $\rho>0$, $\operatorname{prox}_{\rho f}:\mathcal{X}\rightarrow\mathcal{X}$, as $$\operatorname{prox}_{\rho f}(y)=\operatorname*{arg\,min}_{x\in\mathcal{X}}\left\{f(x)+\frac{1}{2\rho}\|x-y\|^2\right\}.$$ Moreover, we define the relative *reflective operator* as $\operatorname{refl}_{\rho f}=2\operatorname{prox}_{\rho f}-I$.
It can be seen that the proximal operator is $1/2$-averaged and the reflective operator is nonexpansive [@bauschke2011convex].
Finding the Fixed Points of Nonexpansive Operators {#subsec:fixpoints}
--------------------------------------------------
One of the prototypical algorithm for finding the fixed points of $T$ is the Krasnosel’skii-Mann (KM) iteration [@bauschke2011convex] $$\label{eq:km-iteration}
x(k+1)=T_\alpha x(k)=(1-\alpha)x(k)+\alpha Tx(k)$$ where in general the step size $\alpha$ can be time-varying. Notice that the KM iteration is equivalent to $x(k+1)=x(k)-\alpha Sx(k)$, where $S=I-T$, that is, finding the fixed points of $T$ coincides with finding the zeros of $S$.\
Now, consider the general unconstrained problem $$\label{eq:MinProblem}
\min_{x\in\mathcal{X}} \{f(x)+g(x)\},$$ where $f,g$ are closed proper and convex not necessarily smooth functions. Further, assume that simultaneous minimization of $f+g$ is unwieldy while minimizing $f$ and $g$ separately is manageable. In this case, to compute the solution of we can apply the KM iteration to the *Peaceman-Rachford Splitting operator*, defined as (see [@davis2016convergence; @peng2016arock]), $$T_{PRS}=\operatorname{refl}_{\rho f}\circ\operatorname{refl}_{\rho g}.$$ As show in [@bauschke2011convex], the iteration $$\label{eq:KM_T_PRS}
x(k+1)=(1-\alpha)x(k)+\alpha T_{PRS}x(k)$$ can be conveniently implemented by the following updates $$\begin{aligned}
\psi(k)&=\operatorname{prox}_{\rho g}(z(k))\label{eq:prs-1}\\
\xi(k)&=\operatorname{prox}_{\rho f}(2\psi(k)-z(k))\label{eq:prs-2}\\
z(k+1)&=z(k)+2\alpha(\xi(k)-\psi(k))\label{eq:prs-3}\end{aligned}$$ where $\psi, \xi, z$ are suitable auxiliary variables while the optimal solution $x^*$ to is recovered from the limit $z^*$ of the iterate $z(k)$ by computing $x^*=\operatorname{prox}_{\rho g}(z^*)$. This algorithm goes under the name of *relaxed Peaceman-Rachford splitting* (R-PRS), where “relaxed” denotes the fact that the KM iteration is $\alpha$-averaged. In case $\alpha=1$ we recover the classic *Peaceman-Rachford splitting* introduced in [@peaceman1955numerical], and in case $\alpha=1/2$ we recover the *Douglas-Rachford splitting* [@douglas1956numerical].\
The important feature of splitting schemes such as the R-PRS is that they divide the computational load of iterate into smaller subproblems that can be solved more efficiently.
From the ADMM to the Relaxed ADMM {#sec:ADMMandRADMM}
=================================
In this Section, we first review the popular ADMM algorithm [@gabay1976dual; @boyd2011distributed], then we introduce the more general *relaxed* ADMM (R-ADMM) algorithm, and compare the two methods.
The ADMM Algorithm {#subsec:ADMM}
------------------
Consider the following optimization problem $$\begin{aligned}
\label{eq:primal-problem}
\begin{split}
&\min_{x\in\mathcal{X},y\in\mathcal{Y}} \{f(x)+g(y)\}\\
&\text{s.t.}\ Ax+By=c
\end{split}\end{aligned}$$ where $\mathcal{X}$ and $\mathcal{Y}$ are Hilbert spaces, $f:\mathcal{X}\rightarrow\mathbb{R}\cup\{+\infty\}$ and $g:\mathcal{Y}\rightarrow\mathbb{R}\cup\{+\infty\}$ are closed, proper and convex functions[^3].\
To solve problem via the ADMM algorithm, we first define the *augmented Lagrangian* as $$\begin{aligned}
\label{eq:augmented-lagr}
\begin{split}
\mathcal{L}_{\rho}(x,y;w)&=f(x)+g(y)-w^\top\left(Ax+By-c\right)\\&+\frac{\rho}{2}\|Ax+By-c\|^2
\end{split}\end{aligned}$$ where $\rho>0$ and $w$ is the vector of Lagrange multipliers. The ADMM algorithm consists in keeping alternating the following update equations $$\begin{aligned}
y(k+1)&=\operatorname{arg\,min}_y \mathcal{L}_\rho(x(k),y;w(k))\label{eq:admm-1}\\
w(k+1)&=w(k)-\rho(Ax(k)+By(k+1)-c)\label{eq:admm-2}\\
x(k+1)&=\operatorname{arg\,min}_x \mathcal{L}_\rho(x,y(k+1);w(k+1))\label{eq:admm-3}.\end{aligned}$$ Notice that the above formulation is equivalent to the one proposed in [@boyd2011distributed] except for a change in the order of the updates which however does not affect the convergence properties of the algorithm. Moreover, the ADMM algorithm is provably shown to converge to the optimal solution of for any $\rho>0$ assuming that $\mathcal{L}_0$ has a saddle point [@boyd2011distributed].\
We conclude this section by stressing the following fact. While the ADMM in its classical form – is typically presented as an augmented Lagrangian method computed with respect to the primal problem , the algorithm naturally arises from the application of the DRS to the Lagrange dual of problem . This will be made clear in the next section.
The Relaxed ADMM {#subsec:R-ADMM}
----------------
The Relaxed ADMM algorithm can be derived applying the R-PRS method described in Section \[sec:operators-background\] to the Lagrange dual of problem , that is to $$\label{eq:dual-problem}
\min_{w\in\mathcal{W}}\left\{d_f(w)+d_g(w)\right\}$$ where $$\begin{aligned}
d_f(w)&=f^*(A^\top w)\\
d_g(w)&=g^*(B^\top w)-w^\top c,\end{aligned}$$ and $f^*$, $g^*$ are the convex conjugates of $f$ and $g$[^4]. The derivation of problem can be found in [@davis2016convergence; @peng2016arock].\
Observe that, given the structure of problem (i.e., proper closed and convex functions and linear constraints) there is no duality gap and, in turn, the optimal solutions of and of attain the same optimal value.\
The motivation for dealing with the Lagrange dual problem relies on the fact that the minimization in is performed over a single variable, thus allowing for the use of the R-PRS algorithm described in , and .\
Lemma $11$ in [@davis2016convergence] shows that the update and the update , applied to the dual problem, can be conveniently computed by, respectively, $$\begin{aligned}
y(k)&=\operatorname*{arg\,min}_y\left\{g(y)-z^\top(k)(By-c)+\frac{\rho}{2}\|By-c\|^2\right\}\nonumber\\
\psi(k)&=z(k)-\rho(By(k)-c)\label{eq:psi-update}
\end{aligned}$$ and $$\begin{aligned}
x(k)&=\operatorname*{arg\,min}_x\left\{f(x)-(2\psi(k)-z(k))^\top Ax+\frac{\rho}{2}\|Ax\|^2\right\}\nonumber\\
\xi(k)&=2\psi(k)-z(k)-\rho Ax(k)\label{eq:xi-update}\end{aligned}$$ The so called *relaxed ADMM* algorithm (in short R-ADMM) consists in applying iteratively the set of five equations given by the two equations in , the two equations in and equation .\
It is worth stressing a fundamental difference regarding the auxiliary variables $z$ in – and those used in ,. Indeed, when implementing the R-PRS –, the KM iteration is applied directly to the primal problem . Hence $z$ has the same dimension of the primal variable $x$. Conversely, the R-ADMM as in , and , is derived on the dual problem . Hence, in this case $z$ has dimension of the constraints. This will become more clear later when dealing with distributed consensus optimization problems.\
Next, we derive a more compact formulation of the R-ADMM, that shows clearly the relation with the popular ADMM algorithm described in , and .\
First of all, by adding $\psi(k)$ on both sides of the second equation in , we obtain $$2\psi(k)-z(k)=\psi(k)-\rho(By(k)-c)\label{eq:equality-1}$$ and substituting this equation in we get $$\xi(k)=\psi(k)-\rho(Ax(k)+By(k)-c)\label{eq:equality-2}.$$ Getting $z(k)$ from and, substituting back into the first equation in we obtain $$\begin{aligned}
\begin{split}
x(k)&=\operatorname*{arg\,min}_x\{f(x)-\psi^\top(k)(Ax+By(k)-c)\\&+\frac{\rho}{2}\|Ax+By(k)-c\|^2\}.
\end{split}\end{aligned}$$ where terms independent of $x$ were added. By substituting and in we get $$z(k+1)=\psi(k)-\rho Ax(k)-\rho(2\alpha-1)(Ax(k)+By(k)-c)\label{eq:z-update}$$ and by plugging into the first equation in and adding some terms that do not depend on $y$, we get $$\begin{aligned}
\begin{split}
y&(k+1)=\operatorname*{arg\,min}_y\{g(y)\\
&-\psi^\top(k)(Ax(k)+By-c)+\rho\|By-c\|^2\\
&+\rho\left[Ax(k)+(2\alpha-1)(Ax(k)+By(k)-c)\right]^\top(By-c)\}.
\end{split}\end{aligned}$$ Finally, recalling the defintion of augmented Lagrangian and renaming $\psi$ as $w$, we arrive to the three updates that represent the R-ADMM algorithm $$\begin{aligned}
\begin{split}
y(k+1)&=\operatorname*{arg\,min}_y\{\mathcal{L}_\rho(x(k),y;w(k))\\&+\rho(2\alpha-1)\langle By,(Ax(k)+By(k)-c)\rangle\}\label{eq:r-admm-1}
\end{split}\\
\begin{split}
w(k+1)&=w(k)-\rho(Ax(k)+By(k+1)-c)\\&-\rho(2\alpha-1)(Ax(k)+By(k)-c)\label{eq:r-admm-2}
\end{split}\\
x(k+1)&=\operatorname*{arg\,min}_x\mathcal{L}_\rho(x,y(k+1);w(k+1)).\label{eq:r-admm-3}\end{aligned}$$ The ADMM algorithm in – can be recovered from this formulation of the R-ADMM by setting $\alpha=1/2$, which cancels the additional terms weighted by $2\alpha-1$.\
It is of notice that the R-ADMM has two tunable parameters, $\rho$ and $\alpha$, against the only one of the ADMM, $\rho$, which is the cause of the greater reliability of the R-ADMM.
Figure \[fig:relationships-algorithms\] depicts the relationships between the splitting operators derived from the relaxed Peaceman-Rachford, and the classic and relaxed ADMM.
(r-prs) at (0,0) [R-PRS]{}; (prs) at (2.5,2.5) [PRS]{}; (drs) at (2.5,-2.5) [DRS]{}; (r-admm) at (-2.5,2.5) [R-ADMM]{}; (admm) at (-2.5,-2.5) [ADMM]{};
(r-prs) edge node\[right\] [$\alpha=1/2$]{} (drs) (r-prs) edge node\[right\] [$\alpha=1$]{} (prs) (r-prs) edge node\[align=center\] [applied to\
Lagrange\
dual of ]{} (r-admm) (drs) edge node\[align=center\] [applied to Lagrange\
dual of ]{} (admm) (r-admm) edge node\[left\] [$\alpha=1/2$]{} (admm);
Distributed Consensus Optimization {#sec:distributed_consensus}
==================================
This Section introduces the distributed consensus convex optimization problem that we are interested in, and the solutions obtained by applying the R-ADMM algorithm.
Problem Formulation {#subsec:distributed_problem}
-------------------
Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be a graph, with $\mathcal{V}$ the set of $N$ vertices, labeled $1$ through $N$, and $\mathcal{E}$ the set of undirected edges. For $i \in \mathcal{V}$, by $\mathcal{N}_i$ we denote the set of neighbors of node $i$ in $\mathcal{G}$, namely, $$\mathcal{N}_i =\left\{j \in V \,:\, (i,j) \in \mathcal{E} \right\}.$$ We are interested in solving the following optimization problem $$\begin{aligned}
\label{eq:opt_problem}
\begin{split}
&\min_{x}\sum_{i=1}^Nf_i(x)
\end{split}\end{aligned}$$ where $f_i:\mathbb{R}^n \rightarrow\mathbb{R}\cup\{+\infty\}$ are closed, proper and convex functions and where $f_i$ is known only to node $i$. In the following we denote by $x^*$ the optimal solution of .\
Observe that can be equivalently formulated as $$\begin{aligned}
\label{eq:distributed-primal}
\begin{split}
&\min_{x_i,\forall i}\sum_{i=1}^Nf_i(x_i)\\
&\text{s.t.}\ x_i=x_j,\ \forall (i,j)\in\mathcal{E}
\end{split}\end{aligned}$$ By introducing for each edge $(i,j)\in\mathcal{E}$ the two *bridge variables* $y_{ij}$ and $y_{ji}$, the constraints in can be rewritten as $$\begin{aligned}
\begin{split}
& x_i=y_{ij}\\
& x_j=y_{ji}\\
& y_{ij}=y_{ji}
\end{split}\ \ \ \forall (i,j)\in\mathcal{E}.\end{aligned}$$ Defining $\mathbf{x}=[x_1^\top,\ldots,x_N^\top]^\top$, $f(\mathbf{x})=\sum_if_i(x_i)$, and stacking all bridge variables in $\mathbf{y} \in \mathbb{R}^{n |\mathcal{E}|}$, we can reformulate the problem as $$\begin{aligned}
& \min_{\mathbf{x}} f(\mathbf{x})\\
& \text{s.t.}\ \ A\mathbf{x}+\mathbf{y}=0\\
& \mathbf{y}=P\mathbf{y}\end{aligned}$$ for a suitable $A$ matrix and with $P$ being a permutation matrix that swaps $y_{ij}$ with $y_{ji}$. Making use of the indicator function $\iota_{(I-P)}(\mathbf{y})$ which is equal to 0 if $(I-P)\mathbf{y}=0$, and $+\infty$ otherwise, we can finally rewrite problem as $$\begin{aligned}
\label{eq:primal-indicator-f}
\begin{split}
& \min_{\mathbf{x},\mathbf{y}}\left\{f(\mathbf{x})+\iota_{(I-P)}(\mathbf{y})\right\}\\
& \text{s.t.}\ \ A\mathbf{x}+\mathbf{y}=0.
\end{split}\end{aligned}$$ In next Section we apply the R-ADMM algorithm described in Section \[subsec:R-ADMM\] to the above problem.
R-ADMM for Convex Distributed Optimization {#subsec:distributed_R-ADMM}
------------------------------------------
In this section we employ , and to solve problem . To do so we introduce the dual variables $w_{ij}$ and $w_{ji}$ which are associated to the constraints $x_i=y_{ij}$ and $x_j=y_{ji}$, respectively. The resulting algorithm is described in Algorithm \[alg:r-admm-three-eqs\]. Observe that R-ADMM applied to is amenable of a *distributed* implementation, in the sense that node $i$ stores in memory only the variables $x_i$, $y_{ij}$, $w_{ij}, j\in\mathcal{N}_i$, and updates these variables exchanging information only with its neighbors, i.e, with nodes in $\mathcal{N}_i$.
$k\leftarrow0$
Notice that, in the update of $x_i$, the term $w_{ij}(k+1)-\rho y_{ij}(k+1)$ can be rewritten, using the previous updates, as a function of the variables computed at time $k$ only. Therefore, only one round of transmissions is necessary.\
The above implementation of the R-ADMM is quite straightforward and popular but very unwieldy due to the fact that, depending on the number of neighbors, there might be nodes which need to store, update and transmit a large number of variables. The derivation of Algorithm \[alg:r-admm-three-eqs\] is reported in Appendix \[app:derivation-alg-3eqs\].
In the following we provide an alternative algorithm which is derived directly from the application of the set of five equations in , and to the dual of problem . Notice, that since the vector $z$ has the same dimension of the vector $w$, this implies the presence of also the variables $z_{ij}$ and $z_{ji}$ for any $(i,j) \in \mathcal{E}$.\
We have the following Proposition, which is proved in Appendix \[app:proof-prop-1\].
\[pr:r-admm-five-eqs\] The implementation of the R-ADMM algorithm described in the set of five equations given in , and applied to the dual of problem , reduces to alternating between the following two updates $$\begin{aligned}
\label{eq:x-update-distributed}
& x_i(k)=\operatorname*{arg\,min}_{x_i}\left\{f_i(x_i)-\left(\sum_{j\in\mathcal{N}_i}z_{ji}^\top(k)\right) x_i\right.\\
&\,\,\,\qquad\qquad\qquad \qquad \qquad \qquad\qquad \Biggl.+\,\frac{\rho}{2}|\mathcal{N}_i|\|x_i\|^2 \Biggr\},\nonumber
$$ for all $i \in V$, and $$\begin{aligned}
\label{eq:z-update-distributed}
\begin{split}
& z_{ij}(k+1)=(1-\alpha)z_{ij}(k)-\alpha z_{ji}(k)+2\alpha\rho x_i(k)\\
& z_{ji}(k+1)=(1-\alpha)z_{ji}(k)-\alpha z_{ij}(k)+2\alpha\rho x_j(k)
\end{split}\end{aligned}$$ for all $(i,j) \in \mathcal{E}$.
Observe that the reformulation of , and as in Proposition \[pr:r-admm-five-eqs\] is possible for the particular structure of Problem and, in particular, for the structure of the constraints $Ax+y=0$. In general, given a set of constraints $Ax+By=c$ being $A$, $B$ and $c$ generic matrices and vector, such reformulation might not be possible.
The previous proposition naturally suggests an alternative distributed implementation of the R-ADMM Algorithm \[alg:r-admm-three-eqs\], in which each node $i$ stores in its local memory the variables $x_i$ and $z_{ij},j\in\mathcal{N}_i$. Then, at each iteration of the algorithm, each node $i$ first collects the variables $z_{ji},j\in\mathcal{N}_i$; second, updates $x_i$ and $z_{ij}$ according to and the first of , respectively; finally, it sends $z_{ij}$ to $j\in\mathcal{N}_i$.\
Differently to the natural implementation just briefly described, we present a slightly different implementation building upon the observation that each node $i$, to update $x_i$ as in requires the variables $z_{ji}$ rather than $z_{ij}$ for $j\in\mathcal{N}_i$. Consequently, we assume node $i$ stores in its memory and is in charge for the update of $z_{ji},j\in\mathcal{N}_i$. The implementation is described in Algorithm \[alg:smart-distributed-r-admm\].
$k\leftarrow0$
As we can see, both Algorithms \[alg:r-admm-three-eqs\] and \[alg:smart-distributed-r-admm\] need a single round of transmissions at each time $k$. However, they differ for the number of packets that each node has to transmit and for the number of variables that a node has to update. Table \[tab:variables-counts\] reports the comparison between the two algorithms.
Alg. \[alg:r-admm-three-eqs\] Alg. \[alg:smart-distributed-r-admm\]
----------------- ------------------------------- ---------------------------------------
Update and Send $2|\mathcal{N}_i|+1$ $|\mathcal{N}_i|+1$
Store $3|\mathcal{N}_i|$ $|\mathcal{N}_i|$
: Comparison of R-ADMM implementations.[]{data-label="tab:variables-counts"}
Therefore, exploiting the auxiliary $z$ variables we have obtained an algorithm with smaller memory and computational requirements.\
We conclude this section by stating the convergence properties of Algorithms \[alg:r-admm-three-eqs\], \[alg:smart-distributed-r-admm\]. The proof can be found in Appendix \[app:proof-convergence\].
\[prop:convergence\] Consider Algorithm \[alg:smart-distributed-r-admm\]. Let $(\alpha, \rho)$ be such that $0<\alpha <1$ and $\rho >0$. Then, for any initial conditions, the trajectories $k \to x_i(k)$, $i \in V$, generated by Algorithm \[alg:smart-distributed-r-admm\], converge to the optimal solution of , i.e., $$\lim_{k \to \infty} x_i(k) = x^*, \qquad \forall i \in \mathcal{V},$$ for any $x_i(0)$ and $z_{ji}(0)$, $j \in \mathcal{N}_i$. The same result holds true also for Algorithms \[alg:r-admm-three-eqs\].
Distributed R-ADMM over lossy networks {#sec:robustADMM}
======================================
The distributed algorithms illustrated in the previous section work under the standing assumption that the communication channels are reliable, that is, no packet losses occur. The goal of this section is to relax this communication requirement and, in particular, to show that Algorithm \[alg:smart-distributed-r-admm\] still converges, under a probabilistic assumption on communication failures which is next stated.
\[ass:lossy\] During any iteration of Algorithm \[alg:smart-distributed-r-admm\], the communication from node $i$ to node $j$ can be lost with some probability $p$.
In order to describe the communication failure more precisely, we introduce the family of independent binary random variables $L_{ij}(k)$, $k=0,1,2,\ldots$, $i \in \mathcal{V}$, $j \in \mathcal{N}_i$, such that[^5] $$\mathbb{P}\left[L_{ij}=1\right]=p, \qquad \mathbb{P}\left[L_{ij}=0\right]=1-p.$$ We emphasize the fact that independence is assumed among all $L_{ij}(k)$ as $i, j$ and $k$ vary. If the packet transmitted, during the $k$-th iteration by node $i$ to node $j$ is lost, then $L_{ij}(k)=1$, otherwise $L_{ij}(k)=0$.\
In this lossy scenario, Algorithm \[alg:smart-distributed-r-admm\] is modified as shown in Algorithm \[alg:robust-smart-distributed-r-admm\].
$k\leftarrow0$
In this case, at $k$-th iteration node $i$ updates $x_i$ as in . Then, for $j \in \mathcal{N}_i$, it computes $q_{i \to j}$ as in and transmits it to node $j$. If node $j$ receives $q_{i \to j}$, then it updates $z_{ij}$ as $z_{ij}(k+1)=(1-\alpha)z_{ij}(k)+ \alpha q_{i \to j}$, otherwise $z_{ij}$ remains unchanged, i.e., $z_{ij}(k+1)=z_{ij}(k)$. This last step can be compactly describes as $$\begin{aligned}
z_{ij}(k+1)&=L_{ij}(k)z_{ij}(k) + \\
&\qquad +\left(1-L_{ij}(k)\right) \,\left( (1-\alpha)z_{ij}(k)+ \alpha q_{i \to j}\right)\end{aligned}$$ We have the following Proposition, whose proof is reported in Appendix \[app:convergence-lossy\].
\[prop:convergence\_lossy\] Consider Algorithm \[alg:robust-smart-distributed-r-admm\] working under the scenario described in Assumption \[ass:lossy\]. Let $(\alpha, \rho)$ be such that $0<\alpha <1$ and $\rho >0$. Then, for any initial conditions, the trajectories $k \to x_i(k)$, $i \in \mathcal{V}$, generated by Algorithm \[alg:robust-smart-distributed-r-admm\], converge almost surely to the optimal solution of , i.e., $$\lim_{k \to \infty} x_i(k) = x^*, \qquad \forall i \in \mathcal{V},$$ with probability one, for all $i \in \mathcal{V}$, for any $x_i(0)$ and $z_{ji}(0)$, $j \in \mathcal{N}_i$.
We stress that the underling idea behind the result of Proposition \[prop:convergence\_lossy\] relies on rewriting Algorithm \[alg:robust-smart-distributed-r-admm\] as a stochastic KM iteration and then to resort to a different set of methodological tools from probabilistic analysis [@bianchi2016coordinate].
We have restricted the analysis to the case of synchronous communication since we were mainly interested in investigating the algorithm performance in the presence of packet losses. The practically more appealing asynchronous scenario will be the focus of future research.
Interestingly, while the robustness result that we provide in the lossy scenario holds true for Algorithm \[alg:robust-smart-distributed-r-admm\], we cannot prove the same for Algorithms \[alg:r-admm-three-eqs\] which, in the case of synchronous and reliable communications, is instead characterized by the same convergent behavior despite of the different communication and memory requirements.
Observe that Proposition \[prop:convergence\], for the case of reliable communications, and Proposition \[prop:convergence\_lossy\], regarding the lossy scenario, share exactly the same region of convergence in the space of the parameters. This means that Algorithm \[alg:smart-distributed-r-admm\] remains provably convergent if $0<\alpha<1$ and $\rho>0$ in both cases. However, observe that the result is not *necessary and sufficient* and, in particular, the convergence might hold also for value of $\alpha\geq 1$. Indeed, in the simulation Section \[sec:simulation\] we show that, for the case of quadratic functions $f_i,\ i\in\mathcal{V}$, the region of attraction in parameter space is larger. Moreover, despite what suggested by the intuition, the larger the packet loss probability $p$, the larger the region of convergence. However, this increased region of stability is counterbalanced by a slower convergence rate of the algorithm.
Simulations {#sec:simulation}
===========
In this section we provide some experimental simulations to test the proposed R-ADMM Algorithm \[alg:robust-smart-distributed-r-admm\] to solve distributed consensus optimization problems . We are particularly interested in showing the algorithm performances in the presence of packet losses in the communication among neighboring nodes. To simplify the numerical analysis we restrict to the case of quadratic cost functions of the form $$f_i(x_i)=a_ix_i^2 + b_ix_i + c_i$$ where, in general, the quantities $a_i,b_i,c_i\in\mathbb{R}$ are different for each node $i$. In this case the update of the primal variables becomes linear and, in particular, Eq. reduces to $$\begin{aligned}
&x_i(k)=\frac{\sum_{j\in\mathcal{N}_i}z_{ji}(k)-b_i}{2a_i+\rho|\mathcal{N}_i|}\, .
$$ We consider the family of random geometric graphs with $N=10$ and communication radius $r=0.1$\[p.u.\] in which two nodes are connected if and only if their relative distance is less that $r$. We perform a set of 100 Monte Carlo runs for different values of packet losses probability $p$, step size $\alpha$ and penalty parameters $\rho$.\
First of all, for different values of packet loss probability $p$ and for fixed values of step size $\alpha=1$ and penalty $\rho=1$, Figure \[fig:evolution\_different\_losses\] shows the evolution of the relative error $$\log\frac{\|x(k)-x^*\|}{\|x^*\|}$$ computed with respect to the unique minimizer $x^*$ and averaged over 100 Monte Carlo runs. As expected, the higher the packet loss probability, the smaller the rate of convergence. Indeed, failures in the communication among neighboring nodes negatively affect the computations.\
![Evolution, in log-scale, of the relative error of Alg. \[alg:robust-smart-distributed-r-admm\] computed w.r.t. the unique optimal solution $x^*$ as function of different values of packet loss probability $p$ for step size $\alpha=1$ and penalty $\rho=1$. Average over 100 Monte Carlo runs.[]{data-label="fig:evolution_different_losses"}](ErrorEvolution_DifferentPacketLoss_RandGeomGraph){width="\columnwidth"}
Figure \[fig:randgeom\_stability\_boundaries\] plots the stability boundaries of the R-ADMM Algorithm \[alg:robust-smart-distributed-r-admm\] as function of step size $\alpha$ and penalty $\rho$ for different packet loss probabilities $p$. More specifically, each curve in Figure \[fig:randgeom\_stability\_boundaries\] represents the numerical boundary below which the algorithm is found to be convergent and above which, conversely, the algorithm diverges. In this case the results turn out extremely interesting. Indeed, given $\alpha$ and $\rho$, for increasing packet loss probability $p$, the stability region enlarges. This means that the higher the loss probability is, the more robust the algorithm is. The numerical findings are perfectly in line with the result of Proposition \[prop:convergence\_lossy\], telling us that for $\alpha\in (0,1)$ the algorithm converges for any value of $\rho$. However, it suggests the additional interesting fact that the theory misses to capture a larger area – in parameters space and depending on $p$ – for which the algorithm still converges. This will certainly be a direction of future investigation.
![Stability boundaries of Alg. \[alg:robust-smart-distributed-r-admm\] as function of the step size $\alpha$ and the penalty $\rho$ for different values of loss probability $p$ for the family of random geometric graphs. Average over 100 Monte Carlo runs.[]{data-label="fig:randgeom_stability_boundaries"}](StabilityBoundariesDifferentLosses_RandGeomGraph){width="\columnwidth"}
Finally, Figure \[fig:randgeom\_different\_stepsizes\] reports the evolution of the error as a function of different values of the step-size $\alpha$. Notice that to values of $\alpha$ that are larger than $1/2$ correspond faster convergences. Recalling that setting $\alpha=1/2$ yields the standard ADMM, then it is clear that the use of the R-ADMM can speed up the convergence, which motivates its use against the use of the classic ADMM.
![Evolution, in log-scale, of the relative error of Alg. \[alg:robust-smart-distributed-r-admm\] computed w.r.t. the unique optimal solution $x^*$ as function of different values of the step size $\alpha$, with fixed packet loss probability $p=0.6$ and penalty $\rho=1$. Average over 100 Monte Carlo runs.[]{data-label="fig:randgeom_different_stepsizes"}](ErrorEvolution_DifferentStepSize_RandGeomGraph){width="\columnwidth"}
Conclusions and Future Directions {#sec:conclusions}
=================================
In this paper we addressed the problem of distributed consensus optimization in the presence of synchronous but unreliable communications. Building upon results in operator theory on Hilbert spaces, we leveraged the relaxed Peaceman-Rachford Splitting operator to introduced what is referred to R-ADMM, a generalization of the well known ADMM algorithm. We started by drawing some interesting connections with the classical formulation as typically presented. Then, we introduced several algorithmic reformulations of the R-ADMM which differs in terms of computational, memory and communication requirements. Interestingly the last implementation, besides being extremely light from both the communication and memory point of views, turns out the be provably robust to random communication failures. Indeed, we rigorously proved how, in the lossy scenario, the region of convergence in parameters space remains unchanged compared to the case of reliable communication; yet, we numerically showed that the region of convergence is positively affected by a larger packet loss probability. The drawback lies in a slower convergence rate of the algorithm.\
There remain many open questions paving the paths to future research directions such as analysis of the asynchronous case and generalization of the results to more general distributed optimization problems.
Derivation of Algorithm \[alg:r-admm-three-eqs\] {#app:derivation-alg-3eqs}
================================================
First of all we derive the augmented Lagrangian for problem , and obtain $$\begin{aligned}
\label{eq:augmented-lagr-distributed}
\begin{split}
\mathcal{L}_\rho(x,y;w)=\sum_{i=1}^Nf_i(x_i)&+\iota_{(I-P)}(y)+\\&-w^\top(Ax+y)+\frac{\rho}{2}\|Ax+y\|^2,
\end{split}\end{aligned}$$ where $\|Ax+y\|^2=\|Ax\|^2+\|y\|^2+2\langle Ax,y\rangle$. We can now proceed to derive equations – for the problem at hand.
### Equation
By and discarding the terms that do not depend on $y$ we get $$\begin{aligned}
y(k+1)=\operatorname*{arg\,min}_y&\Big\{\iota_{(I-P)}(y)-w^\top(k)y+\frac{\rho}{2}\|y\|^2\\&+2\alpha\rho\langle Ax(k),y\rangle+\rho(2\alpha-1)\langle y,y(k)\rangle\Big\}\end{aligned}$$ where we summed the terms with the inner product $\langle Ax(k),y\rangle$. Therefore we need to solve the problem $$\begin{aligned}
y(k+1)=\operatorname*{arg\,min}_{y=Py}&\Big\{-w^\top(k)y+\frac{\rho}{2}\|y\|^2 \\
&+2\alpha\rho\langle Ax(k),y\rangle+\rho(2\alpha-1)\langle y,y(k)\rangle\Big\}\end{aligned}$$ that for simplicity we can write as $$\label{eq:problem-y}
y(k+1)=\operatorname*{arg\,min}_{y=Py}\{h_{\alpha,\rho}(y;x(k),w(k))\}.$$\
We apply now the Karush-Kuhn-Tucker (KKT) conditions [@boyd2004convex] to problem and obtain the system $$\begin{aligned}
&\nabla\Big[h_{\alpha,\rho}(y;x(k),w(k))-\nu^\top(I-P)y\Big|_{y(k+1),\nu^*}=0\label{eq:kkt-1}\\
&y(k+1)=Py(k+1)\label{eq:kkt-2}\end{aligned}$$ where $\nu^*$ is the optimal value of the Lagrange multipliers of the problem.\
By computing the gradient in we obtain $$\begin{aligned}
\label{eq:kkt-1-bis}
\begin{split}
y(k+1)=\frac{1}{\rho}\big[w(k)&-2\alpha\rho Ax(k)\\&-\rho(2\alpha-1)y(k)+(I-P)\nu^*\big].
\end{split}\end{aligned}$$ We substitute this formula for $y(k+1)$ in the right-hand side of which results in $$\begin{aligned}
\label{eq:kkt-2-bis}
\begin{split}
y(k+1)=\frac{1}{\rho}\big[P&w(k)-2\alpha\rho PAx(k)\\&-\rho(2\alpha-1)Py(k)-(I-P)\nu^*\big]
\end{split}\end{aligned}$$ for the fact that $P^2=I$ and hence $P(I-P)=-(I-P)$.\
We sum now equations and and obtain $$\begin{aligned}
\label{eq:y-update-final}
\begin{split}
y(k+1)=\frac{1}{2\rho}(I+P)\big[w(k)-2\alpha\rho Ax(k)-\rho(2\alpha-1)y(k)\big].
\end{split}\end{aligned}$$ Finally noting that, given a vector $t$ of dimension equal to that of $y$, the $ij$-th element of $(I+P)t$ is equal to $t_{ij}+t_{ji}$, then the update for $y_{ij}(k+1)$ follows.
### Equation
By equation and we can write $$\begin{aligned}
w(k+1)=&w(k)-2\alpha\rho Ax(k)-\rho(2\alpha-1)y(k)+\\
&-\frac{1}{2}(I+P)[w(k)-2\alpha\rho Ax(k)-\rho(2\alpha-1)y(k)]\\
=&\frac{1}{2}(I-P)[w(k)-2\alpha\rho Ax(k)-\rho(2\alpha-1)y(k)]\end{aligned}$$ and by the definition of $I-P$ we get the update equation for $w_{ij}(k+1)$ stated in Algorithm \[alg:r-admm-three-eqs\].
### Equation
Finally we apply equation to the problem at hand, which means that we need to solve $$\begin{aligned}
x(k+1)=&\operatorname*{arg\,min}_x\Bigg\{\sum_{i=1}^Nf_i(x_i)+\\&-\Big(w(k+1)-\rho y(k+1)\Big)^\top Ax+\frac{\rho}{2}\|Ax\|^2\Bigg\}.\end{aligned}$$ We know that each variable $x_i$ appears in $|\mathcal{N}_i|$ constraints and therefore $\|Ax\|^2=\sum_{i=1}^N|\mathcal{N}_i|\|x_i\|^2$. Moreover, given a vector $t$ with the same size as $y$, we have $$\begin{aligned}
t^\top Ax&=
\begin{bmatrix}
\cdots & t_{ji}^\top & \cdots & t_{ji}^\top & \cdots
\end{bmatrix}
\begin{bmatrix}
\vdots\\-x_i\\ \vdots\\-x_j\\ \vdots
\end{bmatrix}\\
&=\sum_{(i,j)\in\mathcal{E}}\left(t_{ji}^\top x_i+t_{ij}^\top x_j\right)\\
&=\sum_{i=1}^N\left(\sum_{j\in\mathcal{N}_i}t_{ji}^\top\right)x_i.\end{aligned}$$ and we get the update equation for $x_i(k+1)$ substituting $\Big(w(k+1)-\rho y(k+1)\Big)$ to $t$. Notice that by the results obtained above we have $$\begin{aligned}
\Big(w(k+1)-&\rho y(k+1)\Big)=\\&=-P[w(k)-2\alpha\rho Ax(k)-\rho(2\alpha-1)y(k)]\end{aligned}$$ which means that $x(k+1)$ can be computed as a function of the $x$, $y$ and $w$ variables at time $k$ only.
Proof of Proposition \[pr:r-admm-five-eqs\] {#app:proof-prop-1}
===========================================
### Equations
The following derivation shares some points with the derivation described in the section above. Indeed, applying the first equation of to the problem at hand requires that we solve $$y(k)=\operatorname*{arg\,min}_{y=Py}\left\{-z^\top(k)y+\frac{\rho}{2}\|y\|^2\right\},$$ which can be done by solving the system of KKT conditions of the problem as performed above. The result is $$\label{eq:y-update-proof}
y(k)=\frac{1}{2\rho}(I+P)z(k).$$ It easily follows from that $\psi(k)=\frac{1}{2}(I-P)z(k)$.
### Equations
First of all we have $(2\psi(k)-z(k))=-Pz(k)$, hence according to the same reasoning employed above to derive the expression for $x(k+1)$ we find . Moreover, we have $\xi(k)=-Pz(k)-\rho Ax(k)$.
### Equation
By the results derived above we can easily compute $$z(k+1)=(1-\alpha)z(k)-\alpha Pz(k)-2\alpha\rho Ax(k)$$ which gives equations .
Notice that to compute the variables $y(k)$, $\psi(k)$, $x(k)$ and $\xi(k)$ we need only the variables $z(k)$. Moreover, to update $z$ we require only $z(k)$ and $x(k)$. Hence the five update equations reduce to the updates for $x$ and $z$ only.
Proof of Proposition \[prop:convergence\] {#app:proof-convergence}
=========================================
To prove convergence of the R-ADMM in the two implementations of Algorithms \[alg:r-admm-three-eqs\] and \[alg:smart-distributed-r-admm\], we resort to the following result, adapted from [@bauschke2011convex Corollary 27.4].
\[pr:convergence-deterministic\] Consider problem and assume that it has solution; let $\alpha\in(0,1)$, $\rho>0$, and $x(0)\in\mathcal{X}$. Assume to apply equations – to the problem. Then there exists $z^*$ such that
- $x^*=\operatorname{prox}_{\rho g}(z^*)\in\operatorname*{arg\,min}_x\{f(x)+g(x)\}$, and
- $\{z(k)\}_{k\in\mathbb{N}}$ converges weakly to $z^*$.
We need to show now that this result applies to the dual problem of problem . First of all, by formulation of the problem we have that $f$ is convex and proper (and also closed). Moreover, by [@bauschke2011convex Example 8.3] we know that the indicator function of a convex set is convex (and, by definition, proper). But the set of vectors $y$ that satisfy $(I-P)y=0$ is indeed convex, hence also $g$ is convex and proper.\
Now [@rockafellar2015convex Theorem 12.2] states that the convex conjugate of a convex and proper function is closed, convex and proper. Therefore both $d_f$ and $d_g$ are closed, convex and proper, which means that we can apply the convergence result in Proposition \[pr:convergence-deterministic\] to the dual problem of .\
Therefore we have that $w^*=\operatorname{prox}_{\rho d_g}(z^*)$ is indeed a solution of the dual problem and $\{z(k)\}_{k\in\mathbb{N}}$ converges to $z^*$. But since the duality gap is zero, then when we attain the optimum of the dual problem we have obtained that of the primal as well.
Proof of Proposition \[prop:convergence\_lossy\] {#app:convergence-lossy}
================================================
In order to prove the convergence of Algorithm \[alg:robust-smart-distributed-r-admm\] we need to introduce a probabilistic framework in which to reformulate the KM update. For this stochastic version of the KM iteration we can state a convergence result adapted from [@bianchi2016coordinate Theorem 3] and show that indeed Algorithm \[alg:robust-smart-distributed-r-admm\] is represented by this formulation.
We are therefore interested in altering the standard KM iteration in order to include a stochastic selection of which coordinates in $\mathcal{I}=\{1,\ldots,M\}$ to update at each instant. To do so we introduce the operator $\hat{T}^{(\xi)}:\mathcal{X}\rightarrow\mathcal{X}$ whose $i$-th coordinate is given by $\hat{T}^{(\xi)}_ix=T_ix$ if the coordinate is to be updated ($i\in \xi$), $\hat{T}^{(\xi)}_ix=x_i$ otherwise ($i\not\in \xi$). In general the subset of coordinates to be updated changes from one instant to the next. Therefore, on a probability space $(\Omega,\mathcal{F},\mathbb{P})$, we define the random i.i.d. sequence $\{\xi_k\}_{k\in\mathbb{N}}$, with $\xi_k:\Omega\rightarrow 2^\mathcal{I}$, to keep track of which coordinates are updated at each instant. The stochastic KM iteration is finally defined as $$\label{eq:stochastic-km}
x(k+1)=(1-\alpha)x(k)+\alpha\hat{T}^{(\xi_{k+1})}x(k)$$ and consists of the $\alpha$-averaging of a stochastic operator.
The stochastic iteration satisfies the following convergence result, which is particularized from [@bianchi2016coordinate] using the fact that a nonexpansive operator is $1$-averaged, and a constant step size.
Let $T$ be a nonexpansive operator with at least a fixed point, and let the step size be $\alpha\in(0,1)$. Let $\{\xi_k\}_{k\in\mathbb{N}}$ be a random i.i.d. sequence on $2^\mathcal{I}$ such that $$\forall i\in\mathcal{I},\ \exists I\in2^\mathcal{I}\ \text{s.t.}\ i\in I\ \text{and}\ \mathbb{P}[\xi_1=I]>0.$$ Then for any deterministic initial condition $x(0)$ the stochastic KM iteration converges almost surely to a random variable with support in the set of fixed points of $T$.
We turn now to the distributed optimization problem, in which the stochastic KM iteration is performed on the auxiliary variables $z$. In particular we assume that the packet loss occurs with probability $p$, and that in the case of packet loss the relative variable is not updated. As shown in the main paper, this update rule can be compactly written as $$\label{eq:operator-packet-loss}
\hat{T}^{(\xi_{k+1})}z(k)=L_kz(k)+(I-L_k)Tz(k)$$ where $L_k$ is the diagonal matrix with elements the realizations of the binary random variables that model the packet loss at time $k$. Recall that these variables take value 1 if the packet is lost.\
Substituting now the operator into we get the update equation $$\label{eq:stochastic-km-order-1}
z(k+1)=(1-\alpha)z(k)+\alpha\left[L_kz(k)+(I-L_k)Tz(k)\right]$$ which conforms to the stochastic KM iteration for which the convergence result is stated.\
Finally, notice that in the main article the $\alpha$-averaging is applied before the stochastic coordinate selection, that is the update is given by $$\label{eq:stochastic-km-order-2}
z(k+1)=L_kz(k)+(I-L_k)\left[(1-\alpha)z(k)+\alpha Tz(k)\right].$$ However it can be easily shown that and do indeed coincide, hence proving the convergence of our update scheme.
[^1]: $^\dagger$ Department of Information Engineering (DEI), University of Padova, Italy. [nicola.bastianello.3@studenti.unipd.it, \[carlirug|schenato\]@dei.unipd.it]{}.
[^2]: $^\ddagger$ Bosch Center for Artificial Intelligence. Renningen, Germany. [mrc.todescato@gmail.com]{}. The work was carried out during the author’s postdoctoral fellowship at DEI.
[^3]: A function $f:\mathcal{X}\rightarrow\mathbb{R}\cup\{+\infty\}$ is said to be *closed* if $\forall a \in\mathbb{R}$ the set $\{x\in\operatorname{dom}(f)\ |\ f(x)\leq a\}$ is closed. Moreover, $f$ is said to be *proper* if it does not attain $-\infty$ [@boyd2011distributed].
[^4]: The *convex conjugate* of a function $f$ is defined as $f^*(y)=\sup_{x\in\mathcal{X}}\{\langle y,x\rangle-f(x)\}$.
[^5]: We highlight that the results of this section can be extended to the case where the loss probability is different for edge.
|
---
address:
- University of Illinois at Chicago
- 'University of Illinois at Urbana-Champaign'
author:
- Henri Gillet
- 'Daniel R. Grayson'
bibliography:
- 'papers.bib'
date: 'JanuaryFebruaryMarchAprilMay JuneJulyAugustSeptemberOctoberNovemberDecember , '
title: Volumes of symmetric spaces via lattice points
---
[^1]
Introduction {#introduction .unnumbered}
============
In this paper we show how to use elementary methods to prove that the volume of $\Sl_k\R/\Sl_k\Z$ is $\zeta(2) \zeta(3) \cdots \zeta(k) / k$; see Corollary \[volzeta\]. Using a version of reduction theory presented in this paper, we can compute the volumes of certain unbounded regions in Euclidean space by counting lattice points and then appeal to the machinery of Dirichlet series to get estimates of the growth rate of the number of lattice points appearing in the region as the lattice spacing decreases.
In section \[padicvol\] we present a proof of the closely related result that the Tamagawa number of $\Sl_{k,\Q}$ is $1$ that is somewhat simpler and more arithmetic than Weil’s in [@MR83m:10032]. His proof proceeds by induction on $k$ and appeals to the Poisson summation formula, whereas the proof here brings to the forefront local versions (\[localTam\]) of the formula, one for each prime $p$, which help to illuminate the appearance of values of zeta functions in formulas for volumes.
The volume computation above is known; see, for example, [@Siegel36], and formula (24) in [@siegel45]. The methods used in the computation of the volume of $\Sl_k\R/\Sl_k\Z$ in the book [@MR91d:11070 Lecture XV] have a different flavor from ours and do not involve counting lattice points. One positive point about the proof there is that it proceeds by induction on $k$, making clear how the factor $\zeta(k)$ enters in at $k$-th stage. See also [@MR99g:20090 §14.12, formula (2)]. The proof offered there seems to have a gap which consists of assuming that a certain region (denoted by $T$ there) is bounded, thereby allowing the application of [@MR99g:20090 §14.4, Theorem 3][^2]. The region in Example \[unboundedexample\] below shows that filling the gap is not easy, hence if we want to compute the volume by counting lattice points, something like our use of reduction theory in Section \[sec:redthy\] is needed.
An almost equivalent result was proved by Minkowski — he computed the volume of $SO(k) {\setminus} \Sl_k\R/\Sl_k\Z$. The relationship between the two volume computations is made clear in the proof of [@MR99g:20090 §14.12, Theorem 2].
Some of the techniques we use were known to Siegel, who used similar methods in his investigation of representability of integers by quadratic forms in [@siegel37; @siegel38; @siegel39]. See especially [@siegel38 Hilfssatz 6, p. 242], which is analogous to our Lemma \[cusps\] and the reduction theory of Section \[sec:redthy\], where we show how to compute the volume of certain unbounded domains in Euclidean space by counting lattice points; see also the computations in [@siegel37 §9], which have the same general flavor as ours. See also [@MR6:38b p. 581] where Siegel omits the laborious study, using reduction theory, of points at infinity; it is those details that concern us here.
We thank Harold Diamond for useful information about Dirichlet series and Ulf Rehmann for useful suggestions, advice related to Tamagawa numbers, and clarifications of Siegel’s work.
Counting with zeta functions
============================
As in [@MR89m:11084] we define the zeta function of a group $G$ by summing over the subgroups $H$ in $G$ of finite index. $$\label{zetadef}
\zeta(G,s) = \sum_{H \subseteq G} [G:H]^{-s}$$ Evidently, $\zeta(\Z,s) = \zeta(s)$ and the series converges for $s > 1$. For good groups $G$ the number of subgroups of index at most $T$ grows slowly enough as a function of $T$ that $\zeta(G,s)$ will converge for $s$ sufficiently large.
Let’s pick $k \ge 0$ and compute $\zeta(\Z^k,s)$. Any subgroup $H$ of $\Z^k$ of finite index is isomorphic to $\Z^k$; choosing such an isomorphism amounts to finding a matrix $A : \Z^k \to \Z^k$ whose determinant is nonzero and whose image is $H$. Any two matrices $A$, $A'$ with the same image $H$ are related by an equation $A' = A S$ where $S \in \Gl_k\Z$.
Thus the terms in the sum defining $\zeta(\Z^k,s)$ correspond to the orbits for the action of $\Gl_k\Z$ via column operations on the set of $k\times
k$-matrices with integer entries and nonzero determinant. A unique representative from each orbit is provided by the matrices $A$ that are in [*Hermite normal form*]{} (see [@MR94i:11105 p. 66] or [@MR49:5038 II.6]), i.e., those matrices $A$ with $A_{ij} = 0$ for $i >
j$, $A_{ii} > 0$ for all $i$, and $0 \le A_{ij} < A_{ii}$ for $ i < j $.
Let $\HNF$ be the set of integer $k\times k$ matrices in Hermite normal form. Given positive integers $n_1, \dots, n_k$, consider the set of matrices $A$ in $\HNF$ with $A_{ii} = n_i$ for all $i$. The number of matrices in it is $n_1^{k-1} n_2^{k-2} \cdots n_{k-1}^1 n_k^0$. Using that, we compute formally as follows. $$\label{zetalatt}
\begin{split}
\zeta(\Z^k,s)
& = \sum_{H \subseteq \Z^k} [\Z^k:H]^{-s} \\
& = \sum_{A \in \HNF} (\det A)^{-s} \\
& = \sum_{n_1>0, \dots, n_k>0}
(n_1^{k-1} n_2^{k-2} \cdots n_{k-1}^1 n_k^0)
(n_1 \cdots n_k)^{-s} \\
& = \sum_{n_1>0, \dots, n_k>0}
n_1^{k-1-s}
n_2^{k-2-s} \cdots
n_{k-1}^{1-s}
n_k^{-s} \\
& = \sum_{n_1>0} n_1^{k-1-s}
\sum_{n_2>0} n_2^{k-2-s} \cdots
\sum_{n_{k-1}>0} n_{k-1}^{1-s}
\sum_{n_{k}>0} n_{k}^{-s} \\
& = \zeta(s-k+1) \zeta(s-k+2)
\cdots \zeta(s-1) \zeta(s)
\end{split}$$ The result $\zeta(s-k+1) \zeta(s-k+2) \cdots \zeta(s-1) \zeta(s)$ is a product of Dirichlet series with positive coefficients that converge for $s>k$, and thus $\zeta(\Z^k,s)$ also converges for $s>k$. This computation is old, and appears in various guises. See, for example: proof 2 of Proposition 1.1 in [@MR89m:11084]; Lemma 10 in [@MR21:1966]; formula (1.1) in [@MR57:286]; page 64 in [@MR47:3318]; formula (5) and the lines following it in [@Siegel36], where the counting argument is attributed to Eisenstein, and its generalization to number rings is attributed to Hurwitz; and pages 37–38 in [@MR83m:10032].
\[HNFcount\] $
\# \{ H \subseteq \Z^k \mid [\Z^k:H] \le T \}
\sim
\zeta(2) \zeta(3) \cdots \zeta(k) T^k / k
$ for $k \ge 1$.
The right hand side is interpreted as $T$ when $k = 1$. The notation $f(T) \sim g(T)$ means that $\lim_{T \to \infty}
f(T)/g(T) = 1$.
We give two proofs.
The first one is more elementary, and was told to us by Harold Diamond. Writing $\zeta(s-k+1) =
\sum n^{k-1} n^{-s}$ and letting $B(T) = \sum_{n \le T} n^{k-1}$ be the corresponding coefficient summatory function we see that $B(T) = T^k/k + O(T^{k-1})$. If $k \ge 3$ we may apply Theorem \[product\] to show that the coefficient summatory function for the Dirichlet series $\zeta(s) \zeta(s-k+1)$ behaves as $\zeta(k) T^k/k + O(T^{k-1})$. Applying it several more times shows that the coefficient summatory function for the Dirichlet series $\zeta(s) \zeta(s-1)
\cdots \zeta(s-k+3) \zeta(s-k+1)$ behaves as $\zeta(k) \zeta(k-1) \cdots \zeta(3) T^k/k +
O(T^{k-1})$. Applying it one more time we see that the coefficient summatory function for $\zeta(\Z^k,s) = \zeta(s) \cdots \zeta(s-k+1)$ behaves as $\zeta(k) \zeta(k-1) \cdots \zeta(2)
T^k/k + O(T^{k-1} \log T)$, which in turn implies the result.
The second proof is less elementary, since it uses a Tauberian theorem. From (\[zetalatt\]) we know that the rightmost (simple) pole of $\zeta(\Z^k,s)$ occurs at $s = k$, that the residue there is the product $ \zeta(2) \zeta(3) \cdots \zeta(k)$, and that Theorem \[IkeTau\] can be applied to get the result.
Now we point out a weaker version of lemma \[HNFcount\] whose proof is even more elementary.
\[easybound\] If $T>0$ then $
\# \{ H \subseteq \Z^k \mid [\Z^k:H] \le T \} \le T^k.
$
As above, we obtain the following formula. $$\begin{split}
\# \{ H \subseteq \Z^k \mid [\Z^k:H] \le T \}
& = \# \{ A \in \HNF \mid \det A \le T \} \\
& = \sum_{{n_1>0, \dots, n_k>0} \atop {n_1 \cdot \dots \cdot n_k \le T}}
n_1^{k-1} n_2^{k-2} \cdots n_{k-1}^1 n_k^0
\end{split}$$ We use it to prove the desired inequality by induction on $k$, the case $k=0$ being clear. $$\begin{split}
\# \{ H \subseteq \Z^k \mid [\Z^k:H] \le T \}
& = \sum_{n_1 = 1}^{\lfloor T \rfloor} n_1^{k-1}
\sum_{{n_2>0, \dots, n_k>0} \atop {n_2 \cdots n_k \le T/n_1}}
n_2^{k-2} \cdots n_{k-1}^1 n_k^0 \\
& = \sum_{n_1 = 1}^{\lfloor T \rfloor} n_1^{k-1}
\cdot \# \{ H \subseteq \Z^{k-1} \mid [\Z^{k-1}:H] \le T/n_1 \} \\
& \le \sum_{n_1 = 1}^{\lfloor T \rfloor} n_1^{k-1} (T / n_1)^{k-1}
\qquad \text{[by induction on $k$]} \\
& = \sum_{n_1 = 1}^{\lfloor T \rfloor} T^{k-1}
= {\lfloor T \rfloor} \cdot T^{k-1} \le T^k
\end{split}$$
Volumes
=======
Recall that a bounded subset $U$ of Euclidean space $\R^k$ is said to have [*Jordan content*]{} if its volume can be approximated arbitrarily well by unions of boxes contained in it or by unions of boxes containing it, or in other words, that the the characteristic function $\chi_U$ is Riemann integrable. Equivalently, the boundary $\partial U$ of $U$ has (Lebesgue) measure zero (see [@MR53:8338 Theorem 105.2, Lemma 105.2, and the discussion above it]). If $U$ is a possibly unbounded subset of $\R^k$ whose boundary has measure zero, its intersection with any ball will have Jordan content.
Now let’s consider the Lie group $G = \Sl_k\R$ as a subspace of the Euclidean space $M_k\R$ of $k\times k$ matrices. Siegel defines a Haar measure on $G$ as follows (see page 341 of [@siegel45]). Let $E$ be a subset of $G$. Letting $I = [0,1]$ be the unit interval and considering a number $T > 0$, we may consider the following cones. $$\begin{split}
I \cdot E &= \{ t \cdot B \mid B \in E, 0 \le t \le 1\} \\
T \cdot I \cdot E &= \{ t \cdot B \mid B \in E, 0 \le t \le T\} \\
\R^+ \cdot E &= \{ t \cdot B \mid B \in E, 0 \le t\}
\end{split}$$ Observe that if $B \in T \cdot I \cdot E$, then $0 \le \det B \le T^k$.
\[muinfdef\] We say that $E$ is measurable if $I \cdot E$ is, and in that case we define $\mu_\infty(E) = \vol(I \cdot E) \in [0,\infty]$.
The Jacobian of left or right multiplication by a matrix $\gamma$ on $M_k\R$ is $(\det B)^k$, so for $\gamma \in \Sl_k\R$ volume is preserved. Thus the measure is invariant under $G$, by multiplication on either side. According to Siegel, the introduction of such invariant measures on Lie groups goes back to Hurwitz (see [@MR27:4723b p. 546] or [@Hurwitz97]).
Let $F \subseteq G$ be the fundamental domain for the action of $\Gamma = \Sl_k\Z$ on the right of $G$ presented in [@MR21:1966 section 7]; it’s an elementary construction of a fundamental domain which is a Borel set without resorting to Minkowski’s reduction theory. In each orbit they choose the element which is closest to the identity matrix in the standard Euclidean norm on $M_k\R
\cong \R^{k^2}$, and ties are broken by ordering $M_k\R$ lexicographically. This set $F$ is the union of an open subset of $G$ (consisting of those matrices with no ties) and a countable number of sets of measure zero.
The intersection of $T \cdot I \cdot F$ with a ball has Jordan content. To establish that, it is enough to show that the measure of the boundary $\partial F$ in $G$ is zero. Suppose $g \in
\partial F$. Then it is a limit of points $g_i \not\in F$, each of which has another point $g_i
h_i$ in its orbit which is at least as close to $1$. Here $h_i$ is in $\Sl_k (\Z)$ and is not $1$. The sequence $i \mapsto g_i h_i$ is bounded, and thus so is the sequence $h_i$; since $\Sl_k (\Z)$ is discrete, that implies that $h_i$ takes only a finite number of values. So we may assume $h_i =
h$ is independent of $i$, and is not $1$. By continuity, $gh$ is at least as close to $1$ as $g$ is. Now $g$ is also a limit of points $f_i$ in $F$, each of which has $f_i h$ not closer to $1$ than $f_i$ is. Hence $gh$ is not closer to $1$ than $g$ is, by continuity. Combining, we see that $gh$ and $g$ are equidistant from $1$. The locus of points $g$ in $\Sl_k (\R)$ such that $gh$ and $g$ are equidistant from $1$ is given by the vanishing of a nonzero quadratic polynomial, hence has measure zero. The boundary $\partial F$ is contained in a countable number of such sets, because $\Sl_k (\Z)$ is countable, hence has measure zero, too.
We remark that $\HNF$ contains a unique representative for each orbit of the action of $\Sl_k\Z$ on $\{ A \in M_k\Z \mid \det A > 0 \}$. The same is true for $\R^+ \cdot F$. Restricting our attention to matrices $B$ with $\det B \le
T^k$ we see that $ \# ( T \cdot I \cdot F \cap M_k \Z ) = \# \{ A \in \HNF \mid \det A \le T^k \}$.
Warning: $\HNF$ is not contained in $\R^+ \cdot F$. To convince yourself of this, consider the matrix $A = \begin{pmatrix}5&-8\\3&5 \end{pmatrix}$ of determinant $49$. Column operations with integer coefficients reduce it to $B = \begin{pmatrix}49&18\\0&1\end{pmatrix}$, but $(1/7)A$ is closer to the identity matrix than $(1/7)B$ is, so $B \in HNF$, but $B \not\in \R^+ \cdot
F$.
We want to approximate the volume of $T \cdot I \cdot F$ by counting the lattice points it contains, i.e., by using the number $\# ( T \cdot I \cdot F
\cap M_k \Z )$, at least when $T$ is large. Alternatively, we may use $\# (
I \cdot F \cap r \cdot M_k \Z )$, when $r$ is small.
\[muZdef\] Suppose $U$ is a subset of $\R^n$. Let $$N_r(U) = r^n \cdot \# \{ U \cap r
\cdot \Z^n \}$$ and let $$\mu_\Z(U) = \lim_{r \to 0} N_r(U),$$ if the limit exists, possibly equal to $+\infty$. An equation involving $\mu_\Z(U)$ is to be regarded as true only if the limit exists.
\[muIF\] $\mu_\Z(I \cdot F) = \zeta(2) \zeta(3) \cdots \zeta(k) / k$
We replace $r$ above with $1/T$: $$\begin{split}
\mu_\Z(I \cdot F)
& = \lim_{T \to \infty} T^{-k^2} \cdot \# ( T \cdot I \cdot F \cap M_k \Z ) \\
& = \lim_{T \to \infty} T^{-k^2} \cdot \# \{ A \in \HNF \mid \det A \le T^k \} \\
& = \lim_{T \to \infty} T^{-k^2} \cdot \# \{ H \subseteq \Z^k \mid [\Z^k:H] \le T \} \\
& = \zeta(2) \zeta(3) \cdots \zeta(k) / k
\qquad \text{[using lemma \ref{HNFcount}]}
\end{split}$$
\[jcon\] If $U$ is a bounded subset of $\R^n$ with Jordan content, then $\mu_\Z(U) =
\vol U $.
Subdivide $\R^n$ into cubes of width $r$ (and of volume $r^n$) centered at the points of $r\Z^n$. The number $\# \{ U \cap r \cdot \Z^n \}$ lies between the number of cubes contained in $U$ and the number of cubes meeting $U$, so $r^n \cdot \# \{ U \cap r \cdot \Z^n \}$ is captured between the total volume of the cubes contained in $U$ and the total volume of the cubes meeting $U$, hence approaches the same limit those two quantities do, namely $\vol U$.
\[cusps\] Let $B_R$ be the ball of radius $R>0$ centered at the origin, and let $U$ be a subset of $\R^n$ whose boundary has measure zero.
1. \[two\] For all $R$, the quantity $\mu_\Z(U)$ exists if and only if $\mu_\Z(U - B_R)$ exists, and in that case, $\mu_\Z(U) = \vol(U \cap
B_R) + \mu_\Z(U-B_R)$.
2. \[twoo\] If $\mu_\Z(U)$ exists then $\mu_\Z(U) = \vol(U) + \lim_{R
\to \infty} \mu_\Z(U-B_R)$.
3. \[one\] If $\vol(U) = +\infty$, then $\mu_\Z(U) = +\infty$.
4. \[four\] If $\lim_{R \to \infty} \limsup_{r \to 0} N_r(U-B_R) =
0$, then $\mu_\Z(U) = \vol(U)$.
Writing $U = (U \cap B_R) \cup (U - B_R)$ we have $$N_r(U) = N_r(U \cap B_R) + N_r(U - B_R).$$ For each $R>0$, the set $U \cap B_R$ is a bounded set with Jordan content, and thus lemma \[jcon\] applies to it. We deduce that $$\liminf_{r \to 0} N_r(U) = \vol(U \cap B_R) + \liminf_{r \to 0} N_r(U - B_R)$$ and $$\limsup_{r \to 0} N_r(U) = \vol(U \cap B_R) + \limsup_{r \to 0} N_r(U - B_R),$$ from which we can deduce (\[two\]), because $\vol(U \cap B_R) < \infty$. We deduce (\[twoo\]) from (\[two\]) by taking limits. Letting $R \to \infty$ in the equalities above we see that $$\liminf_{r \to 0} N_r(U) = \vol(U) + \lim_{R \to \infty} \liminf_{r \to 0} N_r(U - B_R)$$ and $$\limsup_{r \to 0} N_r(U) = \vol(U) + \lim_{R \to \infty} \limsup_{r \to 0} N_r(U - B_R),$$ in which some of the terms might be $+\infty$. Now (\[one\]) follows from $\liminf_{r \to 0} N_r(U) \ge \vol(U)$, and (\[four\]) follows because if $$\lim_{R \to \infty} \limsup_{r \to 0}
N_r(U-B_R) = 0,$$ then $$\lim_{R \to \infty} \liminf_{r \to 0} N_r(U-B_R) =
0$$ also.
If $U$ is a subset of $\R^n$ whose boundary has measure zero, and $ \mu_\Z(U) = \vol(U)$, then $\vol(T \cdot U) \sim \# ( T \cdot U \cap
\Z^n )$ as $T \to \infty$.
The statement follows immediately from the definitions.
Care is required in trying to compute the volume of $I \cdot F$ by counting lattice points in it, for it is not a bounded set (even for $k=2$, because $\begin{pmatrix}a&0\\0&1/a\end{pmatrix} \in
F$).
\[unboundedexample\] It’s easy to construct an unbounded region where counting lattice points does not determine the volume, by concentrating infinitely many very thin spikes along rays of rational slope with small numerator and denominator. Consider, for example, a bounded region $B$ in $\R^2$ with Jordan content and nonzero area $v = \vol B$, for which (by Lemma \[jcon\]) $\mu_\Z B =
\vol B$. Start by replacing $B$ by its intersection $B'$ with the lines through the origin of rational (or infinite) slope – this doesn’t change the value of $\mu_\Z$, because every lattice point is contained in a line of rational slope, but now the boundary $\partial B'$ does not have measure zero. To repair that, we enumerate the lines $M_1, M_2, \dots$ through the origin of rational slope, and for each $i = 1,2,3,\dots$ we replace $R_i =
B \cap M_i$ by a suitably scaled and rotated version $L_i$ of it contained in the line $N_i$ of slope $i$ through the origin, with scaling factor chosen precisely so $L_i$ intersects each $r \cdot \Z^2$ in the same number of points as does $R_i$, for every $r>0$. The scaling factor is the ratio of the lengths of the shortest lattice points in the lines $M_i$ and $N_i$. The union $L = \bigcup L_i$ has $\mu_\Z L = \mu_\Z B = v \ne 0$, but it and its boundary have measure zero.
Reduction Theory {#sec:redthy}
================
In this section we apply reduction theory to show that the volume of $I \cdot
F$ can be computed by counting lattice points.
We introduce a few basic notions about lattices. For a more leisurely introduction see [@MR86h:22018].
A [*lattice*]{} is a free abelian group $L$ of finite rank equipped with an inner product on the vector space $L \otimes \R$.
We will regard $\Z^k$ or one of its subgroups as a lattice by endowing it with the standard inner product on $\R^k$.
If $L$ is a lattice, then a sublattice $L' \subseteq L$ is a subgroup with the induced inner product. The quotient $L/L'$, if it’s torsion free, is made into a lattice by equipping it with the inner product on the orthogonal complement of $L'$.
There’s a way to handle lattices with torsion, but we won’t need them.
If $L$ is a lattice, then $\covol L$ denotes the volume of a fundamental domain for $L$ acting on $L \otimes \R$.
The covolume can be computed as $|\det(\theta v_1,\cdots,\theta v_k)|$, where $\theta : L \otimes \R \to \R^k$ is an isometry, $\{v_1,\dots,v_k\}$ is a basis of $L$, and $(\theta v_1,\dots,\theta v_k)$ denotes the matrix whose $i$-th column is $\theta v_i$. We have the identity $\covol(L) = \covol(L') \cdot
\covol(L/L')$ when $L/L'$ is torsion free.
If $L$ is a subgroup of $\Z^k$ of finite index, then $\covol L = [\Z^k : L]$.
If $L$ is a nonzero lattice, then $\min L$ denotes the smallest length of a nonzero vector in $L$.
If $L$ is a lattice of rank 1, then $\min L = \covol L$.
\[inequ\] For any natural number $k > 0$, there is a constant $c$ such that for any $S \ge 1$ and for any $T > 0$ the following inequality holds. $$c S^{-k} T^{k^2}
\ge
\# \{ L \subseteq \Z^k \mid [\Z^k: L] \le T^k {\rm \ and\ } \min L \le T/S \}.$$
For $k=1$ we may take $c=2$, so assume $k \ge 2$. Letting $N$ be the number of these lattices $L$, we bound $N$ by picking within each $L$ a nonzero vector $v$ of minimal length, and counting the pairs $(v,L)$ instead. For each $v$ occurring in such pair we write $v$ in the form $v = n_1 v_1$ where $n_1 \in \N$ and $v_1$ is a primitive vector of $\Z^k$, and then we extend $\{v_1\}$ to a basis $B = \{v_1,\dots,v_k\}$ of $\Z^k$. We count the lattices $L$ occurring in such pairs with $v$ by putting a basis $C$ for $L$ into Hermite normal form with respect to $B$, i.e., it will have the form $C = \{n_1 v_1, A_{12} v_1 + n_2 v_2, \dots
, A_{1k} v_1 + \dots + A_{k-1,k} v_{k-1} + n_k v_k\}$, with $n_i > 0$ and $0 \le A_{ij} < n_i$. Notice that $n_1$ has been determined in the previous step by the choice of $v$. The number of vectors $v \in \Z^k$ satisfying $\| v \| \le T/S$ is bounded by a number of the form $c(T/S)^k$; for $c$ we may take a large enough multiple of the volume of the unit ball. With notation as above, and counting the bases for $C$ in Hermite normal form as before, we see that $$\begin{split}
N & \le \sum_{\| v \| \le T/S} \sum_{{n_2>0, \dots, n_k>0}\atop{n_1 \cdots n_k \le T^k}}
n_1^{k-1}
n_2^{k-2} \cdots
n_{k-1}^{1}
n_k^{0} \\
& = \sum_{\| v \| \le T/S} n_1^{k-1} \sum_{{n_2>0, \dots, n_k>0}\atop{n_2 \cdots n_k \le T^k/n_1}}
n_2^{k-2} \cdots n_{k-1}^{1} n_k^{0} \\
& = \sum_{\| v \| \le T/S} n_1^{k-1} \cdot
\# \{ H \subseteq \Z^{k-1} \mid [\Z^{k-1}:H] \le T^k/n_1 \} \\
& \le \sum_{\| v \| \le T/S} n_1^{k-1} (T^k / n_1)^{k-1}
\qquad \text{[by Lemma \ref{easybound}]} \\
& = \sum_{\| v \| \le T/S} T^{ k (k-1) } \\
& \le c(T/S)^k T^{ k (k-1) } \\
& = c S^{-k} T^{k^2}.
\end{split}$$
\[limz\] The following equality holds. $$0 = \lim_{S \to \infty}
\limsup_{T \to \infty}
T^{-k^2} \cdot \# \{ L \subseteq \Z^k \mid [\Z^k: L] \le T^k {\rm \ and\ } \min L \le T/S \}$$
The following two lemmas are standard facts. Compare them, for example, with [@MR39:5577 1.4 and 1.5].
\[lifting\] Let $L$ be a lattice and let $v \in L$ be a primitive vector. Let $\bar L = L / \Z v$, let $\bar w \in \bar L$ be any vector, and let $w \in L$ be a vector of minimal length among all those that project to $\bar w$. Then $\| w \|^2 \le \| \bar w \| ^2 + (1/4) \| v \| ^2$.
The vectors $w$ and $w \pm v$ project to $\bar w$, so $\| w \| ^2 \le \|w\pm v\|^2 =
\|w\|^2+\|v\|^2\pm 2 {\langle}w,v{\rangle}$, and thus $| \langle w,v \rangle | \le (1/2) \|v\|^2$. We see then that $$\begin{split}
\|\bar w\|^2 & = \| w - \frac{\langle w,v \rangle}{\|v\|^2} v \|^2 \\
& = \|w \|^2 - \frac{\langle w,v\rangle^2}{\|v\|^2} \\
& \ge \|w\|^2 - \frac14 \|v\|^2.
\end{split}$$
\[plane\] Let $L$ be a lattice of rank $2$ with a nonzero vector $v \in L$ of minimal length. Let $L' =
\Z v$ and $L'' = L/L'$. Then $\covol L'' \ge (\sqrt 3 / 2) \covol L'$.
Let $\bar w \in L''$ be a nonzero vector of minimal length, and lift it to a vector $w \in L$ of minimal length among possible liftings. By lemma \[lifting\] $\| w \|^2 \le \| \bar w \| ^2 +
(1/4) \| v \| ^2$. Combining that with $\|v\|^2 \le \|w\|^2$ we deduce that $\covol L'' = \| \bar w \|
\ge (\sqrt 3 / 2) \| v \| = (\sqrt 3 / 2) \covol L'$.
If $L$ is a lattice, then $\minbasis L$ denotes the smallest value possible for $(\| v_1 \|^2 + \dots + \| v_k \|^2)^{1/2}$, where $\{v_1,\dots,v_k\}$ is a basis of $L$.
\[minbound\] Given $k \in \N$ and $S \ge 1$, for all $R \gg 0$, for all $T > 0$, and for all lattices $L$ of rank $k$ with $\covol L \le T^k$, if $\minbasis L \ge RT$ then $\min L \le T/S$.
We show instead the contrapositive: provided $\covol L \le T^k$, if $\min L > T/S$ then $\minbasis L < RT$. There is an obvious procedure for producing an economical basis of a lattice $L$, namely: we let $v_1$ be a nonzero vector in $L$ of minimal length; we let $v_2$ be a vector in $L$ of minimal length among those projecting onto a nonzero vector in $L/(\Z v_1)$ of minimal length; we let $v_3$ be a vector in $L$ of minimal length among those projecting onto a vector in $L/(\Z v_1)$ of minimal length among those projecting onto a nonzero vector in $L/(\Z v_1 + \Z
v_2)$ of minimal length; and so on. A vector of minimal length is primitive, so one can show by induction that the quotient group $L/(\Z v_1 + \dots + \Z v_i)$ is torsion free; the case where $i=k$ tells us that $L = \Z v_1 + \dots + \Z v_k$. Let $L_i = \Z v_1 + \dots + \Z v_i$, and let $\alpha_i = \covol(L_i/L_{i-1})$, so that $\alpha_1 = \| v_1 \| = \min L > T/S$.
Applying Lemma \[plane\] to the rank $2$ lattice $L_i / L_{i-2}$ shows that $\alpha_{i} \ge A
\alpha_{i-1}$, where $A = \sqrt 3/2$, and repeated application of Lemma \[lifting\] shows that $\| v_i \|^2 \le \alpha_i^2 + (1/4)( \alpha_{i-1}^2 + \dots + \alpha_{1}^2)$, so of course $
\| v_i \|^2 \le (1/4)( \alpha_{k}^2 + \dots + \alpha_{i+1}^2) + \alpha_i^2 + (1/4)( \alpha_{i-1}^2
+ \dots + \alpha_{1}^2)$. We deduce that $$\label{foo}
\minbasis L
\le ( \sum_{i=1}^k \| v_i \|^2 )^{1/2}
\le \biggl(\frac{k+3}{4} \sum \alpha_i^2\biggr)^{1/2}.$$ Going a bit further, we see that $$\begin{split}
T^k & \ge \covol L \\
& = \alpha_1 \cdots \alpha_k \\
& \ge A^{0+1+2+\dots+(i-2)} \alpha_1^{i-1} \cdot A^{0+1+2+\dots+(k-i)} \alpha_{i}^{k-i+1} \\
& > c_1 (T/S)^{i-1} \alpha_{i}^{k-i+1}
\end{split}$$ where $c_1$ is some constant depending on $S$ which we may take to be independent of $i$. Dividing through by $T^{i-1}$ we get $T^{k-i+1} > c_2 \alpha_{i}^{k-i+1}$, from which we deduce that $T > c_3
\alpha_{i}$, where $c_2$ and $c_3$ are new constants (depending only on $S$). Combining these latter inequalities for each $i$, we find that $(((k+3)/4) \sum \alpha_i^2)^{1/2} < R T$, where $R$ is a new constant (depending only on $S$); combining that with (\[foo\]) yields the result.
\[minbb\] The following equality holds. $$0 = \lim_{R \to \infty} \limsup_{T \to \infty} T^{-k^2} \cdot
\# \{ L \subseteq \Z^k \mid [\Z^k : L] \le T^k {\rm \ and\ } \minbasis L \ge
RT \}$$
Combine (\[limz\]) and (\[minbound\]).
If in the definition of our fundamental domain $F$ we had taken the smallest element of each orbit, rather than the one nearest to $1$, we would have been almost done now. The next lemma takes care of that discrepancy.
If $L$ is a (discrete) lattice of rank $k$ in $\R^k$, then $\size L$ denotes the value of $(\| w_1 \|^2 + \dots + \| w_k \|^2)^{1/2}$, where $\{w_1,\dots,w_k\}$ is the (unique) basis of $L$ satisfying $(w_1,\dots,w_k) \in \R^+ \cdot F$.
\[minbsize\] For any (discrete) lattice $L \subseteq \R^k$ of rank $k$ the inequalities $$\minbasis L \le \size L \le \minbasis L + 2 \sqrt k (\covol L)^{1/k}$$ hold.
Let $\{v_1,\dots,v_k\}$ be the basis envisaged in the definition of $\minbasis
L$, let $\{w_1,\dots,w_k\}$ be the basis of $L$ envisaged the definition of $\size L$, and let $U = (\covol L)^{1/k} = (\det (v_1,\dots,v_k))^{1/k} = (\det
(w_1,\dots,w_k))^{1/k}$. The following chain of inequalities gives the result. $$\begin{aligned}
\minbasis L & = \|(v_1 , \dots , v_k ) \| \le \size L \notag\\
& = \|(w_1 , \dots , w_k )\| \le \|(w_1 , \dots , w_k ) - U \cdot 1_k \| + U \sqrt k \notag\\
& \le \|(v_1 , \dots , v_k ) - U \cdot 1_k \| + U \sqrt k \notag\\
& \le \|(v_1 , \dots , v_k ) \| + 2 U \sqrt k = \minbasis L + 2 U \sqrt k \notag\end{aligned}$$
\[sizeb\] The following equality holds. $$0 = \lim_{Q \to \infty} \limsup_{T \to \infty} T^{-k^2}
\cdot \# \{ L \subseteq \Z^k \mid [\Z^k : L] \le T^k {\rm \ and\ } \size L
\ge Q T \}$$
It follows from (\[minbsize\]) that given $R > 0$, for all $Q \gg 0$ (namely $Q \ge R + 2 \sqrt
k$) if $\covol L \le T^k$ and $\size L \ge QT$ then $\minbasis L \ge RT$. Now apply (\[minbb\]).
\[red\] $ \vol(I \cdot F) = \mu_\Z(I \cdot F) $.
Observe that $\# \{ L \subseteq \Z^k \mid \covol L \le T^k {\rm \ and\ } \size L \ge Q T \} = \#
((T \cdot I \cdot F - B_{QT}) \cap M_k \Z) = \# ((I \cdot F - B_{Q}) \cap T^{-1} M_k \Z) $, so replacing $1/T$ by $r$, Corollary \[sizeb\] implies that $\lim_{Q \to \infty} \limsup_{r \to
0} N_r(I \cdot F - B_Q) = 0$, which allows us to apply Lemma \[cusps\] (\[four\]).
The theorem allows us to compute the volume of $F$ arithmetically, simultaneously showing it’s finite.
\[volzeta\] $ \mu_\infty(G/\Gamma) = \zeta(2) \zeta(3) \cdots \zeta(k) / k $
Combine the theorem with lemma \[muIF\] as follows. $$\mu_\infty(G/\Gamma) = \mu_\infty(F) = \vol(I \cdot F) =
\mu_\Z(I \cdot F) = \zeta(2) \zeta(3) \cdots \zeta(k) / k$$
$p$-adic volumes {#padicvol}
================
In this section we reformulate the computation of the volume of $G/\Gamma$ to yield a natural and informative computation of the Tamagawa number of $\Sl_k$. We are interested in the form of the proof, not its length, so we incorporate the proofs of (\[volzeta\]) and (\[zetalatt\]) rather than their statements. The standard source for information about $p$-adic measures and Tamagawa measures is Chapter II of [@MR83m:10032], and the proof we simplify occurs there in sections 3.1 through 3.4. See also [@MR36:171] and [@MR31:2249].
We let $\mu_p$ denote the standard translation invariant measure on $\Q_p$ normalized so that $\mu_p(\Z_p) = 1$. Let $\mu_p$ also denote the product measure on the ring of $k$ by $k$ matrices, $M_k(\Q_p)$. Observe that $\mu_p(M_k(\Z_p)) =
1$.
For $x \in \Q_p$, let $|x|_p$ denote the standard valuation normalized so that $|p|_p= 1/p$
If $A \in M_k(\Q_p)$ and $U \subseteq \Q_p^k$, then $\mu_p(A\cdot U) = |\det
A|_p \cdot \mu_p(U)$. (To prove this, first diagonalize $A$ using row and column operations, and then assume that $U$ is a cube.) It follows that if $V \subseteq M_k(\Q_p)$, then $\mu_p(A\cdot V) = |\det A|_p^k \cdot \mu_p(V)$.
Consider $\Gl_k(\Z_p)$ as an open subset of $M_k(\Z_p)$. The following computation occurs on page 31 of [@MR83m:10032]. $$\label{measGL}
\begin{split}
\mu_p(\Gl_k(\Z_p) )
& = \# (\Gl_k(\F_p)) / p^{k^2} \\
& = (p^k-1)(p^k-p)\cdots (p^k-p^{k-1}) / p^{k^2} \\
& = (1 - p^{-k}) (1 - p^{-k+1}) \cdots (1 - p^{-1})
\end{split}$$
Weil considers the open set $M_k(\Z_p)^* = \{A \in M_k(\Z_p) \mid \det A
\ne 0 \}$.
\[Mstar\] $ \mu_p(M_k(\Z_p)^*) = 1 $
Let $Z = M_k(\Z_p) \setminus M_k(\Z_p)^*$ be the set of singular matrices. If $A \in Z$, then one of the columns of $A$ is a linear combination of the others. (This depends on $\Z_p$ being a discrete valuation ring – take any linear dependency with coefficients in $\Q_p$ and multiply the coefficients by a suitable power of $p$ to put all of them in $\Z_p$, with at least one of them being invertible.) For each $n\ge 0$ we can get an upper bound for the number of equivalence classes of elements of $Z$ modulo $p^n$ by enumerating the possibly dependent columns, the possible vectors in the other columns, and the possible coefficients in the linear combination: $\mu_p(Z) \le
\lim_{n \to \infty} k \cdot (p^{nk})^{k-1} \cdot (p^n)^{k-1} / (p^n)^{k^2}
= \lim_{n \to \infty} k \cdot p^{-n}
= 0$.
We call rank $k$ submodules $J$ of $\Z_p^k$ [*lattices*]{}. To each $A \in
M_k(\Z_p)^*$ we associate the lattice $J = A \Z_p^k \subseteq \Z_p^k$. This sets up a bijection between the lattices $J$ and the orbits of $\Gl_k(\Z_p)$ acting on $M_k(\Z_p)^*$. The measure of the orbit corresponding to $J$ is $\mu_p(A \cdot \Gl_k(\Z_p)) = |{\det A}|_p^k \cdot \mu_p(\Gl_k(\Z_p)) =
[\Z_p^k:J]^{-k} \cdot \mu_p(\Gl_k(\Z_p))$. Now we sum over the orbits. $$\label{localTam}
\begin{split}
1
& = \mu_p (M_k(\Z_p)^*) \\
& = \sum_J \Bigl([\Z_p^k:J]^{-k} \cdot \mu_p(\Gl_k(\Z_p))\Bigr) \\
& = \Bigl(\sum_J [\Z_p^k:J]^{-k}\Bigr) \cdot \mu_p(\Gl_k(\Z_p))
\end{split}$$ An alternative way to prove (\[localTam\]) would be to use the local analogue of (\[zetalatt\]), which holds and asserts that $\sum_J
[\Z_p^k:J]^{-s} = (1-p^{k-1-s})^{-1} (1-p^{k-2-s})^{-1} \cdots
(1-p^{-s})^{-1}$; we could substitute $k$ for $s$ and compare with the number in (\[measGL\]). The approach via lemma \[Mstar\] and (\[localTam\]) is preferable because $M_k(\Z_p)^*$ provides natural glue that makes the computation seem more natural.
The product $\prod_p \mu_p(\Gl_k(\Z_p))$ doesn’t converge because $\prod_p
(1-p^{-1})$ doesn’t converge, so consider the following formula instead. $$1 = \Bigl((1-p^{-1}) \sum_J [\Z_p^k:J]^{-k}\Bigr) \cdot
\Bigl( (1-p^{-1})^{-1} \mu_p(\Gl_k(\Z_p)) \Bigr)$$ Now we can multiply these formulas together. $$\label{globalTam}
1 = \Bigl(\prod_p (1-p^{-1}) \sum_J [\Z_p^k:J]^{-k}\Bigr) \cdot
\prod_p \Bigl((1-p^{-1})^{-1} \mu_p(\Gl_k(\Z_p)) \Bigr)$$ We’ve parenthesized the formula above so it has one factor for each place of $\Q$, and now we connect each of them with a volume involving $\Sl_k$ at that place.
We use the Haar measure on $\Sl_k(\Z_p)$ normalized to have total volume $$\#
\Sl_k(\F_p) / p^{\dim \Sl_k} .$$ The normalization anticipates (\[meas\]), which shows how a gauge form could be used to construct the measure, or alternatively, it ensures that the exact sequence $1 \to \Sl_k(\Z_p) \to
\Gl_k(\Z_p) \to \Z_p^\times \to 1$ of groups leads to the desired assertion $ \mu_p(\Gl_k(\Z_p)) = \mu_p(\Z_p^\times) \cdot \mu_p(\Sl_k(\Z_p))$ about multiplicativity of measures. We rewrite the factor of the right hand side of (\[globalTam\]) corresponding to the prime $p$ as follows. $$\label{fubini}
\begin{split}
(1-p^{-1})^{-1} \mu_p(\Gl_k(\Z_p))
& = \mu_p(\Z_p^\times)^{-1} \cdot \mu_p(\Gl_k(\Z_p)) \\
& = \mu_p(\Sl_k(\Z_p)).
\end{split}$$
To evaluate the left hand factor of the right hand side of (\[globalTam\]), we insert the complex variable $s$. Because the ring $\Z$ is a principal ideal domain, any finitely generated sub-$\Z$-module $H \subseteq \Z^k$ is free. Hence a lattice $H
\subseteq \Z^k$ is determined freely by its localizations $H_p = H \otimes_\Z
\Z_p \subseteq \Z_p^k$ (where $H_p = \Z_p^k$ for all but finitely many $p$), and its index is given by the formula $$\label{localglobal}
[\Z^k:H] = \prod_p [\Z_p^k:H_p],$$ in which only a finite number of terms are not equal to $1$. $$\label{work1}
\begin{split}
\res_{s = k} & \zeta(\Z^k,s) \\
& = \res_{s = k} \sum_{H} [\Z^k:H]^{-s} \qquad \text{[by (\ref{zetadef})]} \\
& = \lim_{s \to k+} \zeta(s-k+1)^{-1} \cdot \sum_{H} [\Z^k:H]^{-s} \\
& = \lim_{s \to k+} \biggl(
\zeta(s-k+1)^{-1} \bigl( \sum_{H \subseteq \Z^k} \prod_p [\Z_p^k:H_p]^{-s} \bigr)
\biggr)
\qquad \text{[by (\ref{localglobal})]} \\
& = \lim_{s \to k+} \biggl(
\zeta(s-k+1)^{-1} \bigl( \prod_p \sum_{J \subseteq \Z_p^k} [\Z_p^k:J]^{-s} \bigr)
\biggr)
\qquad \text{[positive terms]}
\\
& = \lim_{s \to k+} \prod_p \bigl( (1-p^{-s+k-1}) \sum_J [\Z_p^k:J]^{-s} \bigr)
\\
& = \prod_p (1-p^{-1}) \sum_J [\Z_p^k:J]^{-k}
\\
\end{split}$$ Starting again we get the following chain of equalities. $$\label{work2}
\begin{split}
\res_{s = k} \zeta(\Z^k,s)
& = \res_{s = k} \zeta(s-k+1) \zeta(s-k+2) \cdots \zeta(s-1) \zeta(s) \\
& = \zeta(2) \cdots \zeta(k-1) \zeta(k) \\
& = k \cdot \lim_{T \to \infty} T^{-k} \# \{ H \subseteq \Z^k \mid [\Z^k:H] \le T \}
\qquad \text{[by \ref{HNFcount}]} \\
& = k \cdot \lim_{T \to \infty} T^{-k^2} \# \{ H \subseteq \Z^k \mid [\Z^k:H] \le T^k \} \\
& = k \cdot \lim_{T \to \infty} T^{-k^2} \# \{ A \in \HNF \mid \det A \le T^k \} \\
& = k \cdot \mu_\Z (I \cdot F ) \qquad \text{[by definition \ref{muZdef}]} \\
& = k \cdot \mu_\infty ( \Sl_k(\R) / \Sl_k(\Z) ) \qquad \text{[by \ref{red} and \ref{muinfdef}]}
\end{split}$$
Combining (\[work1\]) and (\[work2\]) we get the following equation. $$\label{work3}
\prod_p (1-p^{-1}) \sum_J [\Z_p^k:J]^{-k}
=
k \cdot \mu_\infty ( \Sl_k(\R) / \Sl_k(\Z) )$$
We combine (\[globalTam\]), (\[fubini\]) and (\[work3\]) to obtain the following equation. $$\label{almost}
1 = k \cdot \mu_\infty ( \Sl_k(\R) / \Sl_k(\Z) ) \cdot \prod_p \mu_p(\Sl_k(\Z_p))$$
To relate this to the Tamagawa number we have to introduce a gauge form $\omega$ on the algebraic group $\Sl_k$ over $\Q$, invariant by left translations, as in sections 2.2.2 and 2.4 of [@MR83m:10032]. We can even get gauge forms over $\Z$. Let $X$ be a generic element of $\Gl_k$. The entries of the matrix $X^{-1} dX$ provide a basis for the 1-forms invariant by left translation on $\Gl_k$. On $\Sl_k$ we see that $\tr(X^{-1}
dX) = d(\det X) = 0$, so omitting the element in the $(n,n)$ spot will provide a basis of the invariant forms on $\Sl_k$. We let $\omega$ be the exterior product of these forms. Just as in the proof of Theorem 2.2.5 in [@MR83m:10032] we obtain the following equality. $$\label{meas}
\int_{\Sl_k(\Z_p)} \omega_{p} = \mu_p(\Sl_k(\Z_p))$$ The measure $\omega_p$ is defined in [@MR83m:10032 2.2.1] in a neighborhood of a point $P$ by writing $\omega = f \, dx_1 \wedge \dots \wedge dx_n$ and setting $\omega_p = |f(P)|_p (dx_1)_p
\dots (dx_n)_p$, where $(dx_i)_p$ is the Haar measure on $\Q_p$ normalized so that $\int_{\Z_p}
(dx_i)_p = 1$, and $|c|_p$ is the $p$-adic valuation normalized so that $d(cx)_p = |c|_p (dx)_p$.
Now we want to determine the constant that relates our original Haar measure $\mu_\infty$ on $\Sl_k(\R)$ to the one determined by $\omega_\infty$. For this purpose, it will suffice to evaluate both measures on the infinitesimal parallelepiped $B$ in $\Sl_k(\R)$ centered at the identity matrix and spanned by the tangent vectors $\varepsilon e_{ij}$ for $i
\ne j$ and $\varepsilon ( e_{ii} - e_{kk} )$ for $i < k$. Here $\varepsilon$ is an infinitesimal number, and $e_{ij}$ is the matrix with a $1$ in position $(i,j)$ and zeroes elsewhere. For the purpose of this computation, we may even take $\varepsilon = 1$. We compute easily that $\int_B \omega_\infty = 1$ and $$\label{normaliz}
\begin{split}
\mu_\infty(B) &= \vol(I \cdot B) \\
&= (1/k^2) \cdot |\det(e_{11}-e_{kk},\cdots,e_{k-1,k-1}-e_{kk},\sum e_{ii})| \\
&= (1/k^2) \cdot |\det(e_{11}-e_{kk},\cdots,e_{k-1,k-1}-e_{kk},k e_{kk})| \\
&= (1/k^2) \cdot |\det(e_{11},\cdots,e_{k-1,k-1},k e_{kk})| \\
&= 1/k
\end{split}$$ We obtain the following equation. $$\label{relate}
\mu_\infty ( \Sl_k(\R) / \Sl_k(\Z) ) = \frac 1 k \int_{ \Sl_k(\R) / \Sl_k(\Z) } \omega_\infty$$ See [@MR99g:20090 §14.12, (3)] for an essentially equivalent proof of this equation. We may now rewrite (\[almost\]) as follows. $$\label{final}
1 = \int_{\Sl_k(\R) / \Sl_k(\Z)} \omega_\infty \cdot \prod_p \int_{\Sl_k(\Z_p)} \omega_{p}$$ (If done earlier, this computation would have justified normalizing $\mu_\infty$ differently.)
The Tamagawa number $\tau(\Sl_{k,\Q}) = \int_{\Sl_k(\A_\Q)/\Sl_k(\Q)} \omega$ is the same as the right hand side of (\[final\]) because $F \times \prod_p
\Sl_k(\Z_p)$ is a fundamental domain for the action of $\Sl_k(\Q)$ on $\Sl_k(\A_\Q)$. Thus $\tau(\Sl_{k,\Q}) = 1$. This was originally proved by Weil in Theorem 3.3.1 of [@MR83m:10032]. See also [@MR35:4226], [@MR90e:11075], and [@MR99g:20090 §14.11, Corollary to Langlands’ Theorem].
See also [@MR35:2900 §8] for an explanation that Siegel’s measure formula amounts to the first determination that $\tau(SO)=2$.
Dirichlet series
================
\[convergence\] Suppose we are given a Dirichlet series $f(s) := \sum_{n=1}^\infty a_n n^{-s}$ with nonnegative coefficients. Let $A(T) := \sum_{n \le T} a_n$. If $ A(T) = O(T^k)$ as $T \to \infty$, then $\sum_{n=T}^\infty a_n n^{-s} = O(T^{k-s})$ as $T \to \infty$, and thus $f(s)$ converges for all complex numbers $s$ with $\re s > k$.
Write $\sigma = \re s$ and assume $\sigma > k$. We estimate the tail of the series as follows. $$\begin{split}
\sum_{n=T}^\infty a_n n^{-s}
& = \int_{T}^\infty x^{-s} \, d A(x) \\
& = x^{-s} A(x) \big ]_{T}^\infty - \int_{T}^\infty A(x) \, d(x^{-s}) \\
& = x^{-s} A(x) \big ]_{T}^\infty + s \int_{T}^\infty x^{-s-1} A(x) \, dx \\
& = O(x^{k-\sigma}) \big ]_T ^\infty + s \int_{T}^\infty x^{-s-1} O(x^k) \, dx \\
& = O(T^{k-\sigma}) + s \int_{T}^\infty O(x^{k-\sigma-1}) \, dx \\
& = O(T^{k-\sigma})
\end{split}$$
\[product\] Suppose we are given two Dirichlet series $$f(s) := \sum_{n=1}^\infty a_n n^{-s} \qquad \qquad
g(s) := \sum_{n=1}^\infty b_n n^{-s}$$ with nonnegative coefficients and corresponding coefficient summatory functions $$A(T) := \sum_{n \le T} a_n
\qquad \qquad
B(T) := \sum_{n \le T} b_n$$ Assume that $A(T) = O(T^i)$ and $B(T) = cT^k + O(T^j)$, where $i \le j < k$. Let $h(s) := f(s) g(s) = \sum_{n=1}^\infty c_n n^{-s}$, and let $C(T) := \sum_{n \le T} c_n$. Then $C(T) = c f(k) T^k + O(T^j \log T)$ if $i=j$, and $C(T) = c f(k) T^k + O(T^j)$ if $i<j$.
The basic idea for this proof was told to us by Harold Diamond.
Observe that Theorem \[convergence\] ensures that $f(k)$ converges. Let’s fix the notation $\beta(T) = O(\gamma(T))$ to mean that there is a constant $C$ so that $|\beta(T)| \le C
\gamma(T)$ for all $T \in [1,\infty)$, and simultaneously replace $O(T^j \log T)$ in the statement by $O(T^j (1 + \log T))$ in order to avoid the zero of $\log T$ at $T = 1$. We will use the notation in an infinite sum only with a uniform value of the implicit constant $C$.
We examine $C(T)$ as follows. $$\begin{split}
C(T) & = \sum_{n \le T} c_n
= \sum_{n \le T} \sum_{pq=n} a_p b_q
= \sum_{pq \le n} a_p b_q \\
& = \sum_{p \le T} a_p \sum_{q \le T/p} b_q
= \sum_{p \le T} a_p B(T/p) \\
& = \sum_{p \le T} a_p \{ c (T/p)^k + O((T/p)^j) \} \\
& = c T^k \sum_{p \le T} a_p p^{-k} + O(T^j) \sum_{p \le T} a_p p^{-j} \\
& = c T^k \{ f(k) + O(T^{i-k}) \} + O(T^j) \sum_{p \le T} a_p p^{-j} \\
& = c f(k) T^k + O(T^{i}) + O(T^j) \sum_{p \le T} a_p p^{-j}
\end{split}$$
If $i < j$ then $\sum_{p \le T} a_p p^{-j} \le f(j) = O(1)$. Alternatively, if $i = j$, then $$\begin{split}
\sum_{p \le T} a_p p^{-j}
& = \sum_{p \le T} a_p p^{-i}
= \int_{1-}^T p^{-i} \, d(A(p)) \\
& = p^{-i} A(p) \bigr]_{1-}^T - \int_{1-}^T A(p) \, d(p^{-i}) \\
& = T^{-i} A(T) + i \int_{1-}^T A(p) p^{-i-1} \, dp \\
& = O(1) + O(\int_{1-}^T p^{-1} \, dp)
= O(1 + \log T)
\end{split}$$ In both cases the result follows.
The proof of the following “Abelian” theorem for generalized Dirichlet series is elementary.
\[elementary\] Suppose we are given numbers $R$, $k \ge 1$, and $1 \le \lambda_1 \le \lambda_2 \le \dots \to \infty $. Suppose that $$N(T) := \sum_{\lambda_n \le T} 1 = (R + o(1)) \frac{T^k}{k} \qquad ( T \to \infty )$$ for some number $R$. Then the generalized Dirichlet series $\psi(s) := \sum \lambda_n^{-s}$ converges for all real numbers $s > k$, and $\lim_{s \to k+}(s-k)\psi(s) = R$.
In the case $R \ne 0$, the proof can be obtained by adapting the argument in the last part of the proof of [@MR33:4001 Chapter 5, Section 1, Theorem 3]: roughly, one reduces to the case where $k=1$ by a simple change of variables, shows $\lambda_n \sim n/R $, uses that to compare a tail of $\sum \lambda_n^{-s}$ to a tail of $\zeta(s) = \sum n^{-s}$, and then uses $\lim_{s \to 1+}(s-1)\zeta(s) = 1$.
Alternatively, one can refer to [@MR97e:11005b Theorem 10, p. 114] for the statement about convergence, and then to [@MR97e:11005b Theorem 2, p. 219] for the statement about the limit. Actually, those two theorems are concerned with Dirichlet series of the form $F(s) = \sum a_n
n^{-s}$, but the first step there is to consider the growth rate of $\sum_{n \le x} a_n$ as $x \to \infty$. Essentially the same proof works for $F(s) = \psi(s)$ by considering the growth rate of $N(x)$ instead.
The result also follows from the following estimate, provided to us by Harold Diamond. Assume $s
> k$. $$\begin{split}
\psi(s)
& := \sum \lambda_n^{-s} \\
& = \int_{1-}^\infty x^{-s} \, dN(x) \\
& = x^{-s} N(x) \big ]_{1-}^\infty + s \int_{1}^\infty x^{-s-1} N(x) \, dx \\
& = O(x^{k-s}) \big ] ^\infty + s \int_{1}^\infty x^{-s-1} (R + o(1))
\frac{x^k}{k} \, dx \qquad ( x \to \infty ) \\
& = \frac{s ( R + o(1) )}{k}
\int_{1}^\infty x^{-s-1+k} \, dx \qquad ( s \to k+ ) \\
& = \frac{s ( R + o(1) )}{k(s-k)} \qquad ( s \to k+ )
\end{split}$$ Notice the shift in the meaning of $o(1)$ from one line to the next, verified by writing $\int_{1}^\infty = \int_{1}^b + \int_{b}^\infty$ and letting $b$ go to $\infty$; it turns out that for sufficiently small $\epsilon$ the major contribution to $\int_1^\infty x^{-1-\epsilon} \, dx$ comes from $\int_b^\infty x^{-1-\epsilon} \, dx$.
The following Wiener-Ikehara “Tauberian” theorem is a converse to the previous theorem, but the proof is much harder.
\[IkeTau\] Suppose we are given numbers $R>0$, $k>0$, $1 \le \lambda_1 \le \lambda_2 \le \dots \to \infty $, and nonnegative numbers $a_1, a_2, \dots$. Suppose that the Dirichlet series $\psi(s) = \sum a_n \lambda_n^{-s}$ converges for all complex numbers with $\Re s > k$, and that the function $\psi(s) - R/(s-k)$ can be extended to a function defined and continuous for $\Re s \ge k$. Then $$\sum_{\lambda_n \le T} a_n \sim R T^k / k.$$
Replacing $s$ by $k s$ allows us to reduce to the case where $k=1$, which can be deduced directly from the Landau-Ikehara Theorem in [@Bochner], from Theorem 2.2 on p. 93 of [@MR88j:40011], from Theorem 1 on p. 464 of [@MR50:268], or from Theorem 1 on p. 534 of [@MR91h:11107]. See also Theorem 17 on p. 130 of [@Wiener33] for the case where $\lambda_n = n$, which suffices for our purposes. A weaker prototype of this theorem was first proved by Landau in 1909 [@MR16:904d §241]. Other relevant papers include [@Wiener32], [@MR16:921e], and [@MR12:405a]. See also Bateman’s discussion in [@MR16:904d Appendix, page 931] and the good exposition of Abelian and Tauberian theorems in chapter 5 of [@MR3:232d].
[^1]: Supported by NSF grants DMS 01-00587 and 99-70085.
[^2]: called Dirichlet’s Principle in [@MR33:4001 §5.1, Theorem 3]
|
---
abstract: 'New shell model Hamiltonians are derived for the $T=1$ part of the residual interaction in the f$_{5/2}$ p$_{3/2}$ p$_{1/2}$ g$_{9/2}$ model space based on the analysis and fit of the available experimental data for $^{57}_{28}$Ni$_{29}$–$^{78}_{28}$Ni$_{50}$ isotopes and $^{77}_{29}$Cu$_{50}$– $^{100}_{50}$Sn$_{50}$ isotones. The fit procedure, properties of the determined effective interaction as well as new results for valence-mirror symmetry and seniority isomers for nuclei near $^{78}$Ni and $^{100}$Sn are discussed.'
author:
- 'A. F. Lisetskiy$^{1}$, B. A. Brown$^1$, M. Horoi$^{2}$, and H. Grawe$^3$'
title: |
New $T=1$ effective interactions for the f$_{5/2}$ p$_{3/2}$ p$_{1/2} $ g$_{9/2}$ model space;\
Implications for valence-mirror symmetry and seniority isomers
---
Neutron-rich nickel isotopes in the vicinity of $^{78}_{28}$Ni$_{50}$ are currently in the focus of modern nuclear physics and astrophysics studies [@Bro95; @Grz98; @Graw02; @Saw03; @Ish00; @Sor02; @Lang03]. The enormous interest in this region is motivated by several factors. The primary issue concerns the doubly magic nature of $^{78}_{28}$Ni and understanding the way in which the neutron excess will affect the properties of nearby nuclei and the $^{78}$Ni core itself. The shell-model orbitals for neutrons in nuclei with $Z=28$ and N=28-50 ($^{56}$Ni-$^{78}$Ni) are the same as those for protons in nuclei with N=50 and Z=28-50 ($^{78}$Ni-$^{100}$Sn). Thus it is of interest to understand the similarities and differences in the properties of these nuclei with valence-mirror symmetry (VMS) [@Wir88]. The astrophysical importance is related to the understanding of the nuclear mechanism of the rapid capture of neutrons by seed nuclei through the r-process. The path of this reaction network is expected in neutron-rich nuclei for which there is little experimental data, and the precise trajectory is dictated by the details of the shell structure far from stability.
Experimental investigations of neutron-rich nuclei have greatly advanced the last decade providing access to many new regions of the nuclear chart. Nuclear structure theory in the framework of shell-model configuration mixing has also advanced from, for example, the elucidation of the properties of the sd-shell nuclei ($A=16-40$) in the 1980’s [@sd] to those of the pf-shell ($A=40-60$) in current investigations [@pf1; @pf2; @pf3]. Full configuration mixing in the next oscillator shell (sdg) is presently at the edge of computational feasibility. For heavy nuclei the spin-orbit interaction pushes the $g_{9/2}$ and $f_{7/2}$ orbits down relative to the lower-$l$ orbits. Thus the most important orbitals for neutrons in the region of $^{68}$Ni to $^{78}$Ni are $p_{3/2}$, $ f_{5/2}$, $p_{1/2}$ and $ g_{9/2}$ (referred to from now on as the $pf_{5/2}g_{9/2}$ model space). It is noteworthy that this model space is not affected by center-of-mass spurious components. Full configuration mixing calculations for neutrons or protons in this model space are relatively easy. The work we describe here on the $T=1$ effective interactions will provide a part of the input for the larger model space of both protons and neutrons in these orbits where the maximum m-scheme dimension is 13,143,642,988. This proton-neutron model space is computationally feasible with conventional matrix-diagonalization techniques for many nuclei in the mass region A=56-100, and Quantum Monte Carlo Diagonalization techniques [@Monte] or Exponential Convergence Methods [@Hor03] can be used for all nuclei.
The present paper reports on new effective interactions for the $pf_{5/2}g_{9/2}$ model space derived from a fit to experimental data for Ni isotopes from $A=57$ to $A=78$ and $N=50$ isotones from $^{79}$Cu to $^{100}$Sn for neutrons and protons, respectively. Predictions for the $^{72-76}$Ni isotopes are made using the new effective interaction. For the first time the calculated structures of the $^{68,70,72,74,76}$Ni isotopes and the $^{90}$Zr, $^{92}$Mo, $^{94}$Ru, $^{96}$Pd, $^{98}$Cd are compared and analyzed with respect to the VMS concept [@Wir88]. Our work provides a much improved Hamiltonian for $Z=28$ over those considered in smaller model spaces [@Bro95; @Grz98; @Graw02; @Graw97], and also provides a new Hamiltonian for $N=50$ that is similar to those obtained previously [@Ji88_effint; @Sin92].
The effective interaction is specified uniquely in terms of interaction parameters consisting of four single-particle energies and 65 $T=1$ two-body matrix elements (TBME). The starting point for the fitting procedure was a realistic G-matrix interaction based on the Bonn-C $NN$ potential together with core-polarization corrections based on a $^{56}$Ni core [@HJen_private]. The low-energy levels known experimentally are not sensitive to all of these parameters, and thus not all of them can be well determined by the selected set of the energy levels. Instead, they are sensitive to certain linear combinations of the parameters. The weights and the number of the most important combinations can be found with the Linear Combination Method (LCD) [@Honma_LCD]. Applying LCD for our fit we found that convergence of the $\chi^2$ in the first iteration is achieved already at 20 linear combinations and we have chosen this as a reasonable number for all following iterations. We performed iterations (about six) until the eigen-energies converged.
![Calculated and experimental excitation energies of the $2^+_1$ and $4^+_1$ states in $A=58-76$ even-even nickel isotopes. Calculated levels are given by circles connected by the dashed line. Experimental data are depicted by squares connected by the solid line. []{data-label="2and4n"}](fig1)
The values of the neutron interaction parameters are adjusted to fit 15 experimental binding energies for $^{57-78}$Ni and 91 energy levels for $^{60-72}$Ni. The nuclei below $^{60}$Ni were not emphasized in the fit due to the increased role of excitations from the $f_{7/2}$ orbit as $^{56}$Ni is approached [@pf3]. In the absence of experimental data on the binding energy of nuclei near $^{78}$Ni we include the SKX [@skx] Hartree-Fock value of -542.32 MeV for the binding energy of $^{78}$Ni as a “data” for the fit. Our calculated binding energies for $^{73-77}$Ni isotopes agree well with the recent corresponding extrapolations from Ref. [@Aud03]. For protons 19 binding energies and 113 energy levels were used in a fit (this data set is similar to that used in Ref. [@Ji88_effint]). The average deviation in binding and excitation energies between experiment and theory is 241 keV and 124 keV for neutrons and protons, respectively. A detailed report of the new interactions will be given elsewhere [@Lis04_int].
In this letter we emphasize some of the interesting results for known nuclei and the extrapolation to properties of unknown nuclei. To illustrate some general properties of the new interactions we plot the excitation energies of the $2^+_1$ and the $4^+_1$ states for neutrons and protons in Figs. 1 and 2, respectively. The systematics shows good agreement between shell-model calculation and experiment. There is some similarity in the trends for the nuclei with $A=68-76$ and $A=90-98$, that is referred to as the VMS [@Wir88]. However, the left part of Figs. 1 and 2 ($A=58-66$ for nickel isotopes and $A=80-88$ for $N=50$ isotones) are drastically different. Two nuclei, $^{66}$Ni and $^{88}$Sr, show the most profound differences in the location of the $4^+_1$ state. The energy gaps between the $4^+_1$ and the $2^+_1$ states in the $^{62}$Ni and the $^{84}$Se are also obviously distinct.
![The same as in the Fig. \[2and4n\] for $A=80-98$ even-even $N=50$ isotones. []{data-label="2and4p"}](fig2)
![The neutron ($^{57}$Ni) and proton ($^{79}$Cu) single-particle energies (SPE) relative to the $^{56}$Ni and the $^{78}$Ni cores, respectively. The SPE for neutron holes in $^{78}$Ni and proton holes in $^{99}$In are also shown. The SPE values relative to the corresponding cores ($p_{3/2}$ orbital) are given below (above) the plotted lines. The relative SPE for $^{57}$Cu and $^{99}$Sn are similar to those of the mirror nuclei $^{57}$Ni and $^{99}$In, respectively. []{data-label="spe"}](fig3)
These differences may be qualitatively understood from the ordering of the single-particle energies (SPE) for both cases (see Fig. \[spe\]). For neutrons the lowest orbital is $p_{3/2}$, which is followed by the $f_{5/2}$, $p_{1/2}$ and $g_{9/2}$ orbitals. This ordering is similar to the familiar cases of interactions in the $pf$-shell. For the protons we obtain the $f_{5/2}$ orbital as the lowest similar to the previous Ji and Wildenthal interaction [@Ji88_effint]. One notes that the spacing between $p_{3/2}$, $p_{1/2}$ and $g_{9/2}$ is rather similar in both cases. The fact that the $f_{5/2}$ orbital is pushed down in energy in $^{79}$Cu relative to $^{57}$Cu may be attributed to the neutron mean field of the $^{78}$Ni core. The strongly binding monopole interaction in the proton-neutron ($\pi \nu$) spin-flip configuration $\pi f_{5/2} \nu g_{9/2}$ as compared to $\pi p_{3/2} \nu g_{9/2}$ causes a dramatic down sloping of the $\pi f_{5/2}$ level in $Z=29$ (Cu) isotopes upon filling of the $\nu g_{9/2}$ orbit [@20a; @20b].
The difference in the ordering of the proton orbitals as compared to neutrons is the main reason for the differences (e.g. for the $4^+$ states) observed in Figs. 1 and 2. The SPE’s impact ground states as well: the $f_{5/2}^6p_{3/2}^4$ component ($f_{5/2}$ and $p_{3/2}$ are filled) constitutes 59.8$\%$ for the $^{88}$Sr and only 21.4$\%$ for the $^{66}$Ni (the VMS partner of the $^{88}$Sr). This difference determines what happens beyond the $^{66}$Ni ( e.g. $^{68}$Ni, see also Refs. [@Sor02; @Lang03]) or $^{88}$Sr upon filling the $p_{1/2}$ and the $g_{9/2}$ orbitals.
To compare the details of the low energy spectra of the even $^{68-76}$Ni isotopes and even $A=90-98$ $N=50$ isotones we show the calculated and experimental energies for some levels of interest in Figs. \[fig3\] and \[fig4\], respectively.
![Yrast level schemes for even-even Ni isotopes with $A=68-76$. Calculated levels on left side and experimental levels on the right side. []{data-label="fig3"}](fig4)
![ The same as in the Fig. \[fig3\] for even-even $N=50$ isotones with $A=90-98$. []{data-label="fig4"}](fig5)
The energies of the $2^+_1$, $4^+_1$, $6^+_1$ and $8^+_1$ states are, approximately, the same in all four $^{70-76}$Ni nuclei. A similar situation holds for the four isotones $^{92}$Mo-$^{98}$Cd, where the validity of the generalized seniority approximation is well established [@Ji88_trans]. Indeed, the calculated structure of the wave functions indicate a large contribution of the $g_{9/2}$ orbital for these nuclei. However, this contribution is not so large to conclude the dominance for all four nuclei. This is especially well illustrated by the structure of the ground states: the $[g_{9/2}]^{A-68}_{0^+}$ component in the $0^+_1$ wave functions for $^{70,72,74,76}$Ni is 44 $\%$, 53$\%$, 67$\%$ and 83$\%$, respectively. It is obvious that, in contrast to the single $g_{9/2}$ orbital approximation, the structures of $^{70}$Ni and $^{76}$Ni are significantly different. The contributions of the $[g_{9/2}]^{A-68}_{2^+}$ component to the $2^+_1$ states have approximately the same weight. The other components of the wave functions play a very important role. For instance, the difference between the effective neutron $g^2_{9/2};J=0$ and $g^2_{9/2};J=2$ TBME’s is 0.373 MeV, however the $2^+-0^+$ energy gap in $^{70}$Ni or in $^{76}$Ni is 1.2-1.1 MeV. Thus the largest contribution to the gap (0.8-0.7 MeV) is mixing with other configurations. The $g_{9/2}$ wave function content of the corresponding valence mirror partners $^{92}$Mo-$^{98}$Cd is slightly larger : 51 $\%$, 60$\%$, 71$\%$ and 84$\%$ of $[g_{9/2}]^{A-90}_{0^+}$ in the ground states of each A-isotone, respectively, but the overall situation is rather similar to $^{70,72,74,76}$Ni.
The energies of the $2^+_1$ states for the nickel isotopes are systematically lower ($\sim 0.3$ MeV) than for the $N=50$ isotones. This effect is due to the properties of effective $g^2_{9/2}$ TBME’s, since the energy gap between the effective $g^2_{9/2};J=0$ and $g^2_{9/2};J=2$ TBME’s is 0.727 MeV and 0.373 MeV for protons and neutrons, respectively ( i.e. they differ by approximately the same $\sim 0.3$ MeV) and the calculated structure of the $^{70-76}$Ni and $^{92}$Mo-$^{98}$Cd is similar (the energy gap in the starting renormalized Bonn-C Hamiltonian is 0.538 MeV). We interpret this as an indication that the $Z=28$ proton shell gap near $^{68}$Ni is relatively weak compared to the $N=50$ neutron shell gap near $^{88}$Sr. Thus, the nuclei $^{56-78}$Ni have substantial amounts of proton core excitations that are not included explicitly in the model space, but are implicitly taken into account by the effective TBME. One would like to treat the nickel isotopes in a model space which explicitly includes the proton excitations, but this is presently not feasible in terms of computational power.
The lowering of the effective $J=2$-$J=0$ gap for the nickel isotopes compared to $N=50$ has important consequences for non-yrast states. This is illustrated by the properties of the $6^+$ states. For $^{94}$Ru and $^{96}$Pd the second $6^+$ state is dominantly seniority $\nu=4$ and lies above the $8^+_1$ state, while in $^{72}$Ni and $^{74}$Ni it is well below the $8^+_1$ state and is almost degenerate with the $6^+_1$ seniority $\nu=2$ state. Despite the very small splitting between two $6^+$ states with dominant seniority $\nu=2$ and $\nu=4$ they are only slightly mixed. The structure and location of the $6^+$ states has important implications for the isomeric properties of the $8^+$ states. The B(E2; $8^+_1 \rightarrow 6^+_{1,2}$) values are given in Table \[be2n\]. The E2 transition between the $8^+$ and $6^+$ states of the same seniority is forbidden in the middle of the shell.
$J_i^\pi$ $J_f^\pi$ $^{70}$Ni $^{72}$Ni $^{74}$Ni $^{76}$Ni
--------------- ----------- --------------- -------------- -------------- ------------
$2^+_1$ $0^+_1$ 64 84 76 46
$4^+_1$ $2^+_1$ 51 94 85 54
$6^+_1$ $4^+_1$ 31 29 34 37
$8^+_1$ $6^+_1$ 12 1.9 9.2 15
$6^+_2$ 3.3 52 47 -
$\tau(8^+_1)$ Th. 326.0 ns 6.1 ns 5.1 ns 1.2 $\mu$s
Expt. 335(4)[^1] ns $<$26$^b$ ns $<$87$^b$ ns
: Calculated B(E2; $J^\pi_i \rightarrow J^\pi_f)$ values for the $A=70-76$ Ni isotopes. A reasonable value of 1.0 is assigned, tentatively, to an effective quadrupole charge $e_n$ . B(E2) values are given in units of e$^2$fm$^4$. Calculated and available experimental lifetimes $\tau$ for the $8^+_1$ state are given in last two rows. Experimental excitation energies were used in lifetime calculations for $^{70}$Ni . For $^{72,74,76}$Ni isotopes theoretical excitation energies were used. []{data-label="be2n"}
It is well known that this seniority selection rule leads to the isomerism of $8^+_1$ states in $^{94}$Ru and $^{96}$Pd [@nndc], with measured lifetimes $\tau$ of 102(6) $\mu$s and 3.2(4) $\mu$s, respectively. Our calculations result in lifetimes of the order of $\mu$s as well. The discrepancy between theoretical and experimental B(E2) values for the $^{94}$Ru is relatively large, and this is common to previous calculations [@Sin92; @Ji88_trans].
$J_i^\pi$
----------- ----- ----------- ----- ----------- ----- ----------- ----- -----------
Th. Expt.[^2] Th. Expt.$^a$ Th. Expt.$^a$ Th. Expt.$^b$
$2^+_1$ 235 207(12) 304 - 283 - 181 -
$4^+_1$ 164 $<$605 9.2 - 40 - 214 -
$6^+_1$ 110 81(2) 8.2 2.9(1) 20 20(3) 149 -
$8^+_1$ 42 32.4(5) 2.7 0.09(1) 7.1 8.9(12) 60 35(11)
: Calculated and experimental B(E2;$J^\pi_i \rightarrow J^\pi_i-2)$ values for $A=92-98$, $N=50$ isotones. A reasonable value of 2.0 is assigned, tentatively, to an effective quadrupole charge $e_p$ [@Ji88_trans]. B(E2) values are given in e$^2$fm$^4$ units. []{data-label="be2p"}
Turning back to Ni-isotopes, one notes, that pushing down of the $6^+_2$ $\nu=4$ states opens up a new channel for the fast E2 decay of the $8^+$ states that results in the disappearance of isomeric states in $^{72,74}$Ni –the corresponding lifetimes move down to the ns region (see Table \[be2n\]). Our results with the newly derived interactions fully support the above explanation of the absence of an isomeric $8^+$ state in $^{72}$Ni proposed in [@Graw02; @Saw03] on the basis of single $g_{9/2}$ orbital shell-model calculations since our calculated wave functions of the $8^+,6^+$ states in $^{72,74}$Ni are dominated by the $g^4_{9/2}$ (70$\%$) and $g^6_{9/2}$ (85$\%$) configurations, respectively. However, it is shown above that this is not the case for the $0^+_1$ and $2^+_1$ states.
Clearly more detailed experimental studies of the nuclei in the vicinity of the $^{78}$Ni are required to verify the shell-model predictions. The next step will be to combine these new neutron-neutron and proton-proton effective Hamiltonians with a proton-neutron Hamiltonian to describe the wide variety of spherical, collective and co-existing features for the $A=56-100$ mass region as well as to apply the wavefunctions to the calculations of weak-interaction and astrophysical phenomena.
We acknowledge support from NSF grant PHY-0244453.
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|
---
abstract: 'Let $\Gamma$ be a group of type rotating automorphisms of a building ${{\mathcal B}}$ of type $\widetilde A_2$, and suppose that $\Gamma$ acts freely and transitively on the vertex set of ${{\mathcal B}}$. The apartments of ${{\mathcal B}}$ are tiled by triangles, labelled according to $\Gamma$-orbits. Associated with these tilings there is a natural subshift of finite type, which is shown to be irreducible. The key element in the proof is a combinatorial result about finite projective planes.'
address:
- 'Mathematics Department, University of Newcastle, Callaghan, NSW 2308, Australia'
- 'Istituto Di Matematica e Fisica, Università degli Studi di Sassari, Via Vienna 2, 07100 Sassari, Italia'
author:
- Guyan Robertson
- Tim Steger
date: 'March 4, 2002'
title: 'Irreducible subshifts associated with $\tilde A_2$ buildings.'
---
Introduction
============
Let ${{\mathcal B}}$ be a locally finite thick affine building of type ${\widetilde}A_2$ [@gar]. Such a building ${{\mathcal B}}$ is a two dimensional simplicial complex which is a union of two dimensional subcomplexes, called [*apartments*]{}. The apartments are Coxeter complexes of type $\widetilde A_2$, which may be realized as a tilings of the Euclidean plane by equilateral triangles. Buildings of type $\widetilde A_2$ ``are contractible as topological spaces and are natural two dimensional analogues of homogeneous trees. (A homogeneous tree is a building of type $\widetilde A_1$.) Each vertex $v$ of ${{\mathcal B}}$ is labeled with a [*type*]{} $\tau (v) \in {{\mathbb Z}}/3{{\mathbb Z}}$, and each chamber has exactly one vertex of each type. An automorphism $\alpha$ of ${{\mathcal B}}$ is said to be [*type rotating*]{} if there exists $i \in
{{\mathbb Z}}/3{{\mathbb Z}}$ such that $\tau(\alpha(v)) = \tau(v)+i$ for all vertices $v \in {{\mathcal B}}$.
\[fig5\]
units <0.5cm, 0.866cm> x from -2.5 to 5, y from -1 to 2.5 \[l\] at -1.6 2.2 \[l\] at 0.4 2.2 \[l\] at 2.4 2.2 \[l\] at 3.4 -0.8 \[l\] at 1.4 -0.8 \[l\] at -0.6 -0.8 \[l\] at -2.6 -0.8 \[l\] at -3.6 0.2 \[l\] at -1.6 0.2 \[l\] at 4.4 0.2 \[l\] at 2.4 0.2 \[l\] at 0.4 0.2 \[l\] at -2.4 1.2 \[l\] at -0.4 1.2 \[l\] at 1.6 1.2 \[l\] at 3.6 1.2 from -2.5 2 to 2.5 2 from -3.5 1 to 3.5 1 from -4.5 0 to 4.5 0 from -3.5 -1 to 3.5 -1 -4.3 -0.3 -1.7 2.3 / -3.3 -1.3 0.3 2.3 / -1.3 -1.3 2.3 2.3 / 0.7 -1.3 3.3 1.3 / 2.7 -1.3 4.3 0.3 / -4.3 0.3 -2.7 -1.3 / -3.3 1.3 -0.7 -1.3 / -2.3 2.3 1.3 -1.3 / -0.3 2.3 3.3 -1.3 / 1.7 2.3 4.3 -0.3 /
If ${{\mathcal B}}$ is a building of type $\widetilde A_2$ then the set $S_v$ of vertices of ${{\mathcal B}}$ adjacent to any vertex $v$ may be given the structure of a finite projective plane. The projective planes corresponding to different vertices $v$ may be nonisomorphic [@rt], but they all have the same order $q$. If a vertex $v$ of ${{\mathcal B}}$ has type $i$ then the set $P$ of vertices of type $i+1$ in $S_v$ correspond to the $q^2+q+1$ points of the projective plane. The set $L$ of vertices of type $i+2$ in $S_v$ correspond to the $q^2+q+1$ lines of the projective plane. A point $p\in P$ and a line $l\in L$ are incident in the projective plane if and only if there is an edge connecting them in the building. The integer $q$ is called the order of the building and each edge in ${{\mathcal B}}$ lies on $q+1$ triangles. The reason for this is that every line in the projective plane is incident with $q+1$ points and every point is incident with $q+1$ lines. These facts will be used repeatedly below.
Suppose that ${{\mathcal B}}$ is a building of type $\widetilde A_2$ and that ${{\Gamma}}$ is a group of type rotating automorphisms of ${{\mathcal B}}$ which acts freely and transitively on the vertex set of ${{\mathcal B}}$. Such groups ${{\Gamma}}$ are called $\widetilde A_2$ groups. In some ways, $\widetilde A_2$ groups are rank two analogues of finitely generated free groups, which act in a similar way on buildings of type $\widetilde A_1$ (trees). The theory of $\widetilde A_2$ groups has been developed in detail in [@cmsz]. The $\widetilde A_2$ groups have a detailed combinatorial structure which makes them an ideal place to attack problems involving higher rank groups.
An $\widetilde A_2$ group can be described as follows [@cmsz I,§3]. Let $(P,L)$ be a projective plane of order $q$. Let $\lambda : P \rightarrow L$ be a bijection (a [*point–line correspondence*]{}). Let ${\mathcal T}$ be a set of triples $(x, y, z)$ where $x, y, z \in P$, with the following properties.
\(i) Given $x, y \in P$, then $(x, y, z) \in {\mathcal T}$ for some $z \in P$ if and only if $y$ and $\lambda(x)$ are incident (i.e. $y \in
\lambda(x)$).
\(ii) $(x, y, z) \in {\mathcal T} \Rightarrow (y, z, x) \in {\mathcal T}$.
\(iii) Given $x, y \in P$, then $(x, y, z) \in {\mathcal T}$ for at most one $z \in P$.
${\mathcal T}$ is called a [*triangle presentation*]{} compatible with $\lambda$. A complete list is given in [@cmsz] of all triangle presentations for $q = 2$ and $q = 3$.
Let $\{a_x : x \in P\}$ be $q^2 + q + 1$ distinct letters and form the group $$\Gamma = \big\langle a_x, x \in P \ |\ a_x a_y a_z = 1 \hbox { for } (x,
y, z) \in {\mathcal T}
\big \rangle$$
The Cayley graph of $\Gamma$ with respect to the generators $a_x, x \in P$ is the $1$-skeleton of an affine building of type $\widetilde A_2$. It is convenient to identify the point $x \in P$ with the generator $a_x \in \Gamma$. If $x\in P$ then the line $\lambda(x)$ corresponds to the inverse $a_x^{-1}$ [@cmsz]. We therefore write $x^{-1}$ for $a_x^{-1}$ and identify $x^{-1}$ with $\lambda(x)$. From now on the notation $x$ and $\lambda(x)$ is used to represent $a_x$ and $a_{\lambda (x)}$ respectively. Note that, with this notation, $${\mathcal T} = \{ (x,y,z) : x,y,z \in P \hbox { and } \ xyz = 1 \}.$$ This means that if $x,y \in P$ then $y \in \lambda(x)$ if and only if $xyz = 1$ for some $z \in P$.
The Cayley graph of $\Gamma$ will be regarded as a directed graph. Vertices are identified with elements of ${{\Gamma}}$ and a directed edge of the form $(a,as)$ with $a\in{{\Gamma}}$ is labeled by a generator $s\in P$. Figure \[A4\] illustrates a typical triangle based at a vertex $a\in{{\mathcal B}}$.
\[A4\]
units <1cm, 1.732cm> x from -2 to 2, y from -0.1 to 1 at 0 0 at -1 1 \*1 2 0 / \[t\] at 0 -0.1 \[r\] at -1.1 1 \[l\] at 1.2 1 \[.2, .67\] from 0.2 1 to 0 1 \[.2, .67\] from -0.7 0.7 to -0.5 0.5 \[.2, .67\] from 0.3 0.3 to 0.5 0.5 \[r \] at -0.7 0.5 \[ l\] at 0.7 0.5 \[ b\] at 0 1.1 from -1 1 to 1 1 -1 1 0 0 1 1 / .
If $q=2$ there are eight $\widetilde A_2$ groups ${{\Gamma}}$, all of which embed as lattices in the linear group ${{\text{\rm{PGL}}}}(3,{{\mathbb F}})$ over a local field ${{\mathbb F}}$. If $q=3$ there are 89 possible $\widetilde A_2$ groups, of which 65 have buildings which are not associated with linear groups [@cmsz].
\[C1\] The group C.1 of [@cmsz] has presentation $$\langle
x_i, 0\le i \le 6\,
|\,
x_0x_0x_6,
x_0x_2x_3,
x_1x_2x_6,
x_1x_3x_5,
x_1x_5x_4,
x_2x_4x_5,
x_3x_4x_6
\rangle.$$ For this group, $q=2$, and there are $q^2+q+1=7$ generators. Thus $P=\{x_0, \dots , x_6\}$ and $L=\{x_0^{-1}, \dots , x_6^{-1}\}$.
Two triangles lie in the same $\Gamma$-orbit if and only if they have the same edge labels, where each edge label is a generator of ${{\Gamma}}$. The combinatorics of the finite projective plane $(P,L)$ shows that there are precisely $(q+1)(q^2+q+1)$ such labellings, which we refer to as [*$\widetilde A_2$ triangle labellings*]{}. Triangle labellings are in bijective correspondence with the elements of the triangle presentation ${{\mathcal T}}$. In Figure \[A6\] we illustrate a triangle labelling (one of three) corresponding to the second relation in Example \[C1\].
\[A6\]
units <1cm, 1.732cm> x from -2 to 2, y from 0 to 1.2 \[b \] at 0 1.1 \[l \] at 0.6 0.5 \[ r\] at -0.6 0.5 \[.2, .67\] from 0.2 1 to 0 1 \[.2, .67\] from -0.7 0.7 to -0.5 0.5 \[.2, .67\] from 0.3 0.3 to 0.5 0.5 from -1 1 to 1 1 -1 1 0 0 1 1 /
The edge labels (or equivalently the tiles) induce a tiling of the apartments in ${{\mathcal B}}$, as illustrated in Figure \[tiling\].
\[tiling\]
units <1cm, 1.732cm> x from -2 to 2, y from -1 to 1.5 \[r \] at -0.6 -0.5 \[ l\] at 0.6 -0.5 \[r \] at 0.4 0.5 \[ l\] at 1.6 0.5 \[r \] at -1.5 0.4 \[ l\] at -0.4 0.5 \[t\] at 0.0 -0.1 \[b\] at -1.0 0.8 \[b\] at 1.0 0.8 at 0 0 1 1 -1 1 /
There is a natural ${{\mathbb Z}}^2$ action on the space of tiled apartments, which gives rise to a so called 2-dimensional subshift of finite type.
Consider the set of all apartments of ${{\mathcal B}}$, with each triangle labelled as above. Two matrices $M_1$, $M_2$ with entries in $\{0, 1\}$ are defined as follows. If $\alpha, \beta \in {{\mathcal T}}$, say that $M_1(\alpha, \beta)=1$ if and only if the triangle labellings $\alpha=(a_1,a_2,a_3)$ and $\beta=(b_1,b_2,b_3)$ lie as shown on the right of Figure \[XX1\]. A similar definition applies for $M_2(\alpha,\gamma)=1$, as on the left of Figure \[XX1\].
\[XX1\]
units <0.5cm,0.866cm> point at -6 0 x from -5 to 5, y from -1 to 2 at 1 1.6 at 0 0.7 at 0 -1 -1 1 0 0 1 1 / -1 1 0 2 1 1 -1 1 / 1 1 2 2 0 2 / units <0.5cm,0.866cm> point at 6 0 x from -5 to 5, y from -1 to 2 at -1 1.6 at 0 0.7 at 0 -1 -1 1 0 0 1 1 / -1 1 0 2 1 1 -1 1 / -1 1 -2 2 0 2 /
The commuting matrices $M_1$, $M_2$ are the transition matrices associated with a 2-dimensional subshift, with alphabet ${{\mathcal T}}$. This subshift is said to be irreducible if for all $\alpha, \beta \in {{\mathcal T}}$, there exist integers $r, s > 0$ such that the $(\alpha, \beta)$ component of the matrix $M_1^rM_2^s$ satisfies $$(M_1^rM_2^s)(\alpha,\beta)>0.$$ A geometric interpretation of this condition is that any two triangle labellings $\alpha,\beta\in {{\mathcal T}}$ can be realized so that $\beta$ lies in some sector with base labelled triangle $\alpha$, as in Figure \[XX2\].
\[XX2\]
units <0.5cm,0.866cm> x from -6 to 6, y from 0.5 to 6 at 0 0.6 at 1 4.6 -6 6 0 0 6 6 / -1 1 1 1 / 1 4 2 5 0 5 1 4 /
It is important for the simplicity of the $C^*$-algebras considered in [@rs] that this subshift is irreducible. In this article we prove irreducibility by showing that we can actually choose $r>0$ such that $M_1^r(\alpha,\beta)>0.$ Thus $\beta$ lies on the wall of a sector as in Figure \[XX3\]. A similar statement is true for the matrix $M_2$.
\[XX3\]
units <0.5cm,0.866cm> x from -6 to 6, y from 0.5 to 6 at 0 0.6 at 4 4.6 -6 6 0 0 6 6 / -1 1 1 1 / 4 4 3 5 5 5 /
Another way of viewing this is to say that irreducibility is proved for the one dimensional subshift associated with tilings of strips between parallel walls in apartments, as illustrated in Figure \[strip\]. This is considerably stronger than irreducibility of the 2-dimensional subshift.
\[strip\]
units <0.5cm, 0.866cm> x from -3.5 to 5, y from -1.3 to 2.3 from -2.5 2 to 2.5 2 from -3.5 1 to 3.5 1 from -4.5 0 to 4.5 0 from -3.5 -1 to 3.5 -1 -4.3 -0.3 -1.7 2.3 / -3.3 -1.3 0.3 2.3 / -1.3 -1.3 2.3 2.3 / 0.7 -1.3 3.3 1.3 / 2.7 -1.3 4.3 0.3 / -4.3 0.3 -2.7 -1.3 / -3.3 1.3 -0.7 -1.3 / -2.3 2.3 1.3 -1.3 / -0.3 2.3 3.3 -1.3 / 1.7 2.3 4.3 -0.3 / span <1.5pt> -3.3 -1.3 -1.3 <,z,,> -1.3 -1.3 0.7 <z,,,> 0.3 0.3 2.3 <z,,,> 2.3 2.3 2.3 /
Let $\Gamma$ be an $\widetilde A_2$ group. If $\Gamma$ has the property that the 2-dimensional subshift described above is irreducible, then the theory developed in [@rs Section 7] applies. This means that one may construct an associated simple $C^*$-algebra whose structure was analyzed in [@rs]. The required irreducibility result was proved in [@rs Theorem 7.10] only for the case where $\Gamma$ is a lattice in ${{\text{\rm{PGL}}}}_3({{\mathbb K}})$, where ${{\mathbb K}}$ is a local field of characteristic zero. The argument of [@rs Theorem 7.10] does not apply if ${{\mathcal B}}$ is the building of ${{\text{\rm{PGL}}}}_3({{\mathbb K}})$, where ${{\mathbb K}}$ is a local field of positive characteristic, which is the case for the group C.1 of Example \[C1\]. Neither does it apply to many examples constructed in [@cmsz], for which ${{\mathcal B}}$ is not the Bruhat-Tits building of a linear group. The purpose of the present article is to show that irreducibility holds for all $\widetilde A_2$ groups. This means that the theory of [@rs] now applies to any such group.
The subshift studied in [@rs] was defined in terms of labelled parallelograms formed by a union of two labelled triangles of the following form.
\[modeltile\]
units <0.5cm,0.866cm> x from -5 to 5, y from 0.5 to 2 from -1 1 to 1 1 -1 1 0 0 1 1 / -1 1 0 2 1 1 /
However, irreducibility of that subshift is an easy consequence of the result presented here.
We now state our main result.
\[main\] Given any two $\widetilde A_2$ triangle labellings, these labellings can be realized as the initial and final triangles of a sequence of triangles arranged along some wall in ${{\mathcal B}}$ as follows:
\[fig1\]
units <0.5cm, 0.866cm > x from -6 to 4, y from -1 to 0 at -7 -0.7 at 5 -0.7 from -8 -1 to 6 -1 -8 -1 -7 0 -6 -1 / 4 -1 5 0 6 -1 / from -7 0 to 5 0 -6 -1 -5 0 -4 -1 -3 0 -2 -1 -1 0 0 -1 1 0 2 -1 3 0 4 -1 /
The rest of the article is devoted to the proof of Theorem \[main\].
Proof of Irreducibility of the 1-dimensional subshift
=====================================================
Fix once and for all the triangle labellings $I$ and $F$. Consider a triangle labelling of the form below (which we refer to as $\underset{b}{\Delta}$).
units <1cm, 1.732cm> x from -1 to 1, y from -2 to 0.1 \[.2, .67\] from 0 -1 to 0.2 -1 \[.2, .67\] from -0.5 -0.5 to -0.7 -0.7 \[.2, .67\] from 0.5 -0.5 to 0.3 -0.3 \[r,b\] at -0.7 -0.5 \[l,b\] at 0.7 -0.5 \[ t\] at 0 -1.1 at 0 -1.6 from -1 -1 to 1 -1 -1 -1 0 0 1 -1 /
Call such a labelling $\underset{b}{\Delta}$ [*reachable*]{} from the [*left*]{} if it is the final triangle labelling in some sequence with initial triangle $I$.
units <0.5cm, 0.866cm > x from -8 to 1, y from -2 to 0 from -8 -1 to 1 -1 -8 -1 -7 0 -6 -1 / \[.2, .67\] from 0 -1 to 0.2 -1 \[.2, .67\] from -0.5 -0.5 to -0.7 -0.7 \[.2, .67\] from 0.5 -0.5 to 0.3 -0.3 \[r,b\] at -0.7 -0.5 \[l,b\] at 0.7 -0.5 \[ t\] at 0 -1.1 at -7 -0.7 -1 -1 0 0 1 -1 /
Similarly define [*reachable*]{} from the [*right*]{}.
\[fig3r\]
units <0.5cm, 0.866cm > x from -1 to 8, y from -1 to 0 from -1 -1 to 8 -1 6 -1 7 0 8 -1 / \[.2, .67\] from 0 -1 to 0.2 -1 \[.2, .67\] from -0.5 -0.5 to -0.7 -0.7 \[.2, .67\] from 0.5 -0.5 to 0.3 -0.3 \[r,b\] at -0.7 -0.5 \[l,b\] at 0.7 -0.5 \[ t\] at 0 -1.1 at 7 -0.7 -1 -1 0 0 1 -1 /
Note that for each edge labelling $b$ there are $q+1$ triangles of the form $\underset{b}{\Delta}$. Therefore if we can show that there exists $b$ such that $\underset{b}{\Delta}$ is reachable from the left for more than $(q+1)/2$ values of the pair $(b_2,b_3)$ and reachable from the right for more that $(q+1)/2$ values of $(b_2,b_3)$, then there exists a labelling $(b,b_2,b_3)$ which is reachable both ways. This will prove Theorem \[main\].
\[fig3\*\]
units <0.5cm, 0.866cm > x from -8 to 8, y from -1 to 0 from -8 -1 to 8 -1 -8 -1 -7 0 -6 -1 / 6 -1 7 0 8 -1 / \[.2, .67\] from 0 -1 to 0.2 -1 \[.2, .67\] from -0.5 -0.5 to -0.7 -0.7 \[.2, .67\] from 0.5 -0.5 to 0.3 -0.3 \[r,b\] at -0.7 -0.5 \[l,b\] at 0.7 -0.5 \[ t\] at 0 -1.1 at -7 -0.7 at 7 -0.7 -1 -1 0 0 1 -1 /
In subsequent arguments, we will need to use a criterion for a triangle labelling of the form $\underset{c}{\Delta}$ to be reachable in one step from a triangle labelling of the form $\underset{b}{\Delta}$, as in Figure \[newfig\].
\[newfig\]
units <1cm, 1.732cm> x from 1 to 7, y from -1.2 to 0.1 \[.2, .67\] from 2 -1 to 2.2 -1 \[.2, .67\] from 1.5 -0.5 to 1.3 -0.7 \[.2, .67\] from 2.5 -0.5 to 2.3 -0.3 from 1 -1 to 3 -1 1 -1 2 0 3 -1 / x from -1 to 1, y from -1.2 to 0.1 \[.2, .67\] from 4 -1 to 4.2 -1 \[.2, .67\] from 3.5 -0.5 to 3.3 -0.7 \[.2, .67\] from 4.5 -0.5 to 4.3 -0.3 \[t\] at 2 -1.1 \[t\] at 4 -1.1 at 3 -1 at 1 -1 at 5 -1 at 2 0 at 4 0 from 3 -1 to 5 -1 3 -1 4 0 5 -1 / from 2 0 to 4 0
\[added\] Figure \[newfig\] is possible in an apartment of ${{\mathcal B}}$ if and only if $c\not\in \lambda(b)$.
Fix a vertex $v\in {{\mathcal B}}$. Since the 1-skeleton of ${{\mathcal B}}$ is the Cayley graph of $(\Gamma, P)$, the vertex $v$ may be considered as an element of $\Gamma$. The choice of $v$ is irrelevant, by transitivity of the action of $\Gamma$.
As explained in the introduction, the set $S_v$ of vertices adjacent to $v$ has the structure of a finite projective plane. The points of this projective plane are $\{ vx \, ; \, x\in P \}$ and the lines are $\{ v\lambda(x) \, ; \, x\in P \}$. Recall that $\lambda(x)=x^{-1}$ in the group $\Gamma$. Figure \[newfig\] is therefore equivalent to Figure \[newfigA\].
\[newfigA\]
units <1cm, 1.732cm> x from 1 to 7, y from -1.2 to 0.1 \[.2, .67\] from 2 -1 to 2.2 -1 \[.2, .67\] from 1.5 -0.5 to 1.3 -0.7 \[.2, .67\] from 2.5 -0.5 to 2.3 -0.3 from 1 -1 to 3 -1 1 -1 2 0 3 -1 / x from -1 to 1, y from -1.2 to 0.1 \[.2, .67\] from 4 -1 to 4.2 -1 \[.2, .67\] from 3.5 -0.5 to 3.3 -0.7 \[.2, .67\] from 4.5 -0.5 to 4.3 -0.3 \[r\] at 0.9 -1.1 \[l\] at 5.1 -1.1 \[t\] at 3 -1.1 at 3 -1 at 1 -1 at 5 -1 at 2 0 at 4 0 from 3 -1 to 5 -1 3 -1 4 0 5 -1 / from 2 0 to 4 0
If $c\in \lambda(b)$, then there is an edge in ${{\mathcal B}}$ between $v\lambda(b)$ and $vc$. Figure \[newfigA\] is therefore impossible, by contractibility of the building ${{\mathcal B}}$.
On the other hand, if $c\not\in \lambda(b)$ then $v\lambda(b)$ and $vc$ are not adjacent in $S_v$. Now $v\lambda(b)$ and $vc$ lie in a hexagon $H$ whose vertices belong to $S_v$. This is because the projective plane $S_v$ has the structure of a spherical building, whose apartments are hexagons. The vertices of the hexagon $H$ are alternately points and lines of the projective plane $S_v$. The only way in which the line $v\lambda(b)$ and the point $vc$ can fail to be adjacent in the hexagon $H$ is if they are opposite vertices of the hexagon, as shown in Figure \[hexagon\].
\[hexagon\]
units <0.5cm, 0.866cm> x from -2 to 3, y from -1 to 1 1 1 -1 1 -2 0 -1 -1 1 -1 2 0 1 1 / at -2 0 at 2 0 \[r\] at -2.4 0 \[l\] at 2.3 0
This means that Figure \[newfigA\] is possible in ${{\mathcal B}}$, where each labelled triangle has one edge on the hexagon $H$.
\[red\] If $b\in P$ then the numbers $${\mathcal L}(b) = \#\{(b_2,b_3): (b,b_2,b_3)\text{ is reachable from the left}\},$$ $${\mathcal R}(b) = \#\{(b_2,b_3): (b,b_2,b_3)\text{ is reachable from the right}\}$$ are [*independent of $b$*]{}.
It is clearly enough to prove the assertion for ${\mathcal L}(b)$. Given $b'\in P$, we must show that ${\mathcal L}(b)= {\mathcal L}(b')$. Now the diagram in Figure \[fig4\] can be completed by choosing $c$ such that $c\not\in\lambda(b)$ and $b'\not\in\lambda(c)$. This is possible, since there exist $q+1$ elements $c\in\lambda(b)$, there exist $q+1$ elements $c$ such that $b'\in\lambda(c)$, and $2(q+1)<q^2+q+1=\#(P)$.
\[fig4\]
units <0.5cm, 0.866cm > x from -8 to 8, y from -1 to 0 from -8 -1 to 8 -1 -8 -1 -7 0 -6 -1 / 6 -1 7 0 8 -1 / \[ t\] at 3 -1.1 \[ t\] at 5 -1.1 \[ t\] at 7 -1.1 at -7 -0.7 2 -1 3 0 4 -1 5 0 6 -1 7 0 8 -1 /
Choose and fix such an element $c\in P$. Then each labelling of $\underset{b}{\Delta}$ uniquely determines the labelling of $\underset{b'}{\Delta}$, and vice versa. That is, for fixed $b,c,b'$, the number of labellings of $\underset{b}{\Delta}$ is the same as the number of labellings of $\underset{b'}{\Delta}$. It follows that ${\mathcal L}(b)\le {\mathcal L}(b')$. By symmetry, ${\mathcal L}(b)= {\mathcal L}(b')$.
It follows from Lemma \[red\] that, in order to prove Theorem \[main\], it is enough to find an elements $b_1, b_2\in P$ such that
\[enough\] $$\begin{aligned}
{\mathcal L}(b_1) > (q+1)/2\, , \label{enough1a}\\
{\mathcal R}(b_2) > (q+1)/2. \label{enough1b}\end{aligned}$$
It is clearly enough to verify (\[enough1a\]).
Given the initial triangle labelling $I$, denote by $D$ the set of all $d\in P$ for which Figure \[fig55\] is possible. Thus $D$ contains precisely $q$ elements. For each $d\in D$ let $S_d$ denote the set of $f\in P$ such that Figure \[fig55\] is possible. Therefore $\#(S_d)=q$.
\[fig55\]
units <1cm, 1.732cm> x from -1 to 1, y from -1.2 to 0.1 \[.2, .67\] from 0 -1 to 0.2 -1 \[.2, .67\] from -0.5 -0.5 to -0.7 -0.7 \[.2, .67\] from 0.5 -0.5 to 0.3 -0.3 from -1 -1 to 1 -1 -1 -1 0 0 1 -1 / \[.2, .67\] from 2 -1 to 2.2 -1 \[.2, .67\] from 1.5 -0.5 to 1.3 -0.7 \[.2, .67\] from 2.5 -0.5 to 2.3 -0.3 at 0 -0.7 \[b,r\] at 1.5 -0.4 \[l,b\] at 2.7 -0.5 from 1 -1 to 3 -1 from 0 0 to 2 0 1 -1 2 0 3 -1 /
\[technical\] If $d_1, d_2\in D$ and $d_1\neq d_2$, then $S_{d_1}\cap S_{d_2}$ contains at most one element.
If $f\in S_{d_1}\cap S_{d_2}$ then $d_1$, $d_2\in\lambda(f)$. The two points $d_1$, $d_2$ in the projective plane determine the line $\lambda(f)$ uniquely. That is, $f$ is uniquely determined.
Let $S=\displaystyle \bigcup_{d\in D} S_d$. Then $S$ is the set of elements $f\in P$ such that a diagram like Figure \[fig55\] is possible, for the given initial triangle $I$. There are $q(q-1)/2$ sets of the form $S_{d_i}\cap S_{d_j}$, each of which contains at most one element. It follows from the exclusion-inclusion principle that $$\label{recall}
\#(S)\geq q.q-\frac{q(q-1)}{2}=\frac{q^2+q}{2}\,.$$ This gives a lower bound on the number of possible edge labels $f$ in Figure \[fig55\]. Now let $f\in S$ be such an edge label. Then $f\in S_d$ for some $d\in P$. Consider diagrams of the form illustrated in Figure \[fig66\].
\[fig66\]
units <1cm, 1.732cm> x from -1 to 5, y from -1.2 to 0.1 \[.2, .67\] from 0 -1 to 0.2 -1 \[.2, .67\] from -0.5 -0.5 to -0.7 -0.7 \[.2, .67\] from 0.5 -0.5 to 0.3 -0.3 from -1 -1 to 1 -1 -1 -1 0 0 1 -1 / \[.2, .67\] from 2 -1 to 2.2 -1 \[.2, .67\] from 1.5 -0.5 to 1.3 -0.7 \[.2, .67\] from 2.5 -0.5 to 2.3 -0.3 at 0 -0.7 \[r,b\] at 1.3 -0.5 \[l,b\] at 2.6 -0.6 from 1 -1 to 3 -1 from 0 0 to 2 0 1 -1 2 0 3 -1 / x from -1 to 1, y from -1.2 to 0.1 \[.2, .67\] from 4 -1 to 4.2 -1 \[.2, .67\] from 3.5 -0.5 to 3.3 -0.7 \[.2, .67\] from 4.5 -0.5 to 4.3 -0.3 \[l\] at 3.6 -0.6 \[l,b\] at 4.7 -0.5 \[t\] at 4 -1.1 \[t\] at 3 -1.1 at 3 -1 from 3 -1 to 5 -1 3 -1 4 0 5 -1 / from 2 0 to 4 0
In the projective plane of nearest neighbours of $x$ label the points $p_f$, $p_g$ and lines $l_f$, $l_h$ as in Figure \[pp\]. (By duality, the words ‘point’ and ‘line’ could be interchanged here. The specified choice makes the wording of a later argument easier.)
\[pp\]
units <1cm, 1.732cm> x from 1 to 7, y from -1.2 to 0.1 \[.2, .67\] from 2 -1 to 2.2 -1 \[.2, .67\] from 1.5 -0.5 to 1.3 -0.7 \[.2, .67\] from 2.5 -0.5 to 2.3 -0.3 \[r,b\] at 1.3 -0.5 \[l,b\] at 2.6 -0.6 from 1 -1 to 3 -1 1 -1 2 0 3 -1 / x from -1 to 1, y from -1.2 to 0.1 \[.2, .67\] from 4 -1 to 4.2 -1 \[.2, .67\] from 3.5 -0.5 to 3.3 -0.7 \[.2, .67\] from 4.5 -0.5 to 4.3 -0.3 \[l\] at 3.6 -0.6 \[l,b\] at 4.7 -0.5 \[t\] at 4 -1.1 \[t\] at 3 -1.1 \[r\] at 0.9 -1.1 \[l\] at 5.1 -1.1 \[r\] at 1.9 0.1 \[l\] at 4.1 0.1 at 3 -1 at 1 -1 at 5 -1 at 2 0 at 4 0 from 3 -1 to 5 -1 3 -1 4 0 5 -1 / from 2 0 to 4 0
Then $(g,h,k)$ is reachable from $f$, (i. e. the diagram is possible), if and only if $l_h\neq l_f$, $p_g\neq p_f$ and $p_g\in l_h\cap l_f$. That is $p_g = (l_f-\{p_f\})\cap l_h$, where $l_h\neq l_f$.
For $h\in P$ the set of possible $g$ is in bijective correspondence with the set $$\begin{aligned}
T_h&=&\bigcup_{f\in S-\{h\}}\{(l_f-\{p_f\})\cap l_h\}\\
&=& l_h\cap\bigcup_{f\in S-\{h\}}l_f-\{p_f\}\,.\end{aligned}$$
If we can show that $\#(T_h)>\frac{q+1}{2}$ for some $h$, then (\[enough1a\]) is satisfied with $b_1=h$.
The proof of Theorem \[main\] therefore reduces to the following combinatorial result about projective planes. Recall from (\[recall\]) that $\#(S)\geq\frac{q^2+q}{2}$.
In a projective plane of order $q$, let $\left\{l_j:1\leq j\leq \frac{q^2+q}{2}\right\}$ be a family of distinct lines. For each $j$, let $p_j$ be a point on $l_j$ and let $l_j'=l_j-\{p_j\}$. Then there exists a line $m$ such that $$\#\left(m\cap \bigcup\{l_j':l_j\neq m\}\right)>\frac{q+1}{2}\,.$$
This divides into 3 separate cases, which are dealt with in increasing order of difficulty.\
[**Case 1:**]{} $q=2$. Here $\frac{q^2+q}{2}=3$, and so there are three distinct lines $l_1$, $l_2$, $l_3$, each containing three points. Each set $l_j'$ therefore contains exactly two points.
Choose a line $m$ which meets a point of $l_1'-l_2$ and a point of $l_2'-l_1$. Then $m\cap(l_1'\cup l_2')$ contains $2>\frac{3}{2}$ elements.
\[spp\]
units <1cm, 0.5cm > x from -4 to 4, y from -3 to 3 from -2.5 1 to 2.5 1 -1 -3 2 3 / 1 -3 -2 3 / at 0 -1 at 0.5 -2 at -0.5 -2 at 1 1 at -1 1 \[b\] at 0 1.3 \[r\] at -2 2.8 \[l\] at 2.1 2.8 \[l\] at 0.7 -2 \[r\] at -0.7 -2
[**Case 2:**]{} $q\geq 4$.\
Each line contains $q+1$ points, so $\#(l_j')=q$. Two distinct lines meet in exactly one point. Hence $$\label{ineq1}
\#(l_1'\cup l_2'\cup l_3')\geq 3q-3\,.$$ Assume that the conclusion of the Lemma is false. Then we [*claim*]{} that for $3\leq k\leq\frac{q^2+q}{2}$, $$\label{ineq2}
\#(l_1'\cup l_2'\cup\cdots\cup l_k')\geq(3q-3)+(k-3)\lceil\frac{q-1}{2}
\rceil\,,$$ where $\lceil t \rceil$ denotes the ceiling of $t$, the least integer not less than $t$.
We prove the claim by induction. If $k=3$ then it is true, by (\[ineq1\]). Assume that (\[ineq2\]) holds for a given value of $k$. Since we are assuming that the conclusion of the Lemma fails, $$\#(l_{k+1}'\cap(l_1'\cup\cdots \cup l_k'))\leq\#(l_{k+1}\cap(l_1'\cup\cdots\cup l_k'))\leq\frac{q+1}{2}\,.$$ Hence, $$\#(l'_{k+1}-(l'_1\cup\cdots\cup l'_k))\geq q-\frac{q+1}{2}=\frac{q-1}{2}\,.$$ Therefore $$\begin{aligned}
\#(l_1'\cup l_2'\cup\cdots\cup l_k'\cup l_{k+1}')
&\geq (3q-3)+(k-3)\lceil\frac{q-1}{2}\rceil+\lceil\frac{q-1}{2}\rceil \\
&=(3q-3)+((k+1)-3)\lceil\frac{q-1}{2}\rceil\,.\end{aligned}$$ Thus we have established (\[ineq2\]).
In particular, since (\[ineq2\]) holds for $k=(q^2+q)/2$, and there are $q^2+q+1$ points in the projective plane, we have $$\label{ineq3}
q^2+q+1\geq(3q-3)+\left(\frac{q^2+q}{2}-3\right)\lceil\frac{q-1}{2}\rceil\,.$$
Now (\[ineq3\]) has been derived from the assumption that the conclusion of the Lemma was false. Therefore all that is required now is to show that (\[ineq3\]) is false. Now (\[ineq3\]) fails when $q=4$, since in that case $$q^2+q+1=21\not\geq 23=(3q-3)+\left(\frac{q^2+q}{2}-3\right)\lceil\frac{q-1}{2}\rceil\,.$$ On the other hand, if $q\geq 5$, write $q=r+5$, $r\geq 0$. Then $$\begin{aligned}
4\left(3q-3+\left(\frac{q^2+q}{2}-3\right)\left(\frac{q-1}{2}\right)-(q^2+q+1)\right)
&=&q^3-4q^2+q-10\\
&=&r^3+11r^2+36r+20\\
&\geq& 20\,.\end{aligned}$$ Therefore (\[ineq3\]) also fails when $q\geq 5$. This proves Case 2.
[**Case 3:**]{} $q=3$. This requires separate treatment. Here $\frac{q^2+q}{2}=6$, $\frac{q+1}{2}=2$.
Given distinct lines $l_1, l_2 \ldots, l_6$ we delete a point from each to obtain sets $l_1',\ldots,l_6'$. We must find a line $m$ such that $$\label{three}
\#\left(m\cap\bigcup\{l_j':l_j\neq m\}\right)>2\,.$$ It is known that there is a unique projective plane of order 3, namely the Desarguesian plane arising from a 3-dimensional vector space over ${{\mathbb F}}_3$ [@bl Theorem 2.3.1].
There are thirteen points and thirteen lines in the projective plane. Label the points $0,1,2,\ldots,12$ and label the lines $(0),(1),(2),\ldots,(12)$, as indicated in the table below [@bl Section 1.4]. For example, line $(8)$ contains the points $5,6,8,1$.
(12) (11) (10) (9) (8) (7) (6) (5) (4) (3) (2) (1) (0)
------ ------ ------ ----- ----- ----- ----- ----- ----- ----- ----- ----- -----
1 2 3 4 5 6 7 8 9 10 11 12 0
2 3 4 5 6 7 8 9 10 11 12 0 1
4 5 6 7 8 9 10 11 12 0 1 2 3
12 0 1 2 3 4 5 6 7 8 9
By permuting the lines $l_1, l_2 \ldots, l_6$, if necessary, we may suppose that $l_1\cap l_2$ is not equal to either of the excluded points $p_1$ or $p_2$.
To check this assertion, suppose that it does not already hold for the given choice of $l_1, l_2$. Since each point is incident at most four of the lines $l_1, l_2 \ldots, l_6$, we may assume that $l_1\cap l_2=p_1$ but that $l_1\cap l_5\ne p_1$ and $l_1\cap l_6\ne p_1$. If $l_1\cap l_5\ne p_5$ or $l_1\cap l_6\ne p_6$ we are done. On the other hand, if $l_1\cap l_5 = p_5$, $l_1\cap l_6 = p_6$ and $p_5\ne p_6$ then $l_5 \cap l_6$ is not equal to either $p_5$ or $p_6$. It remains to consider the case $l_1\cap l_5 = p_5$, $l_1\cap l_6 = p_6$ with $p_5=p_6$. In that case, $l_2\cap l_5\ne p_5$, $l_2\cap l_6\ne p_6$ (since $l_1\cap l_2 =p_1$) and either $l_2\cap l_5\ne p_2$ or $l_2\cap l_6\ne p_2$.
Having verified this assertion, we can assume that $l_1\cap l_2$, $p_1$ and $p_2$ are three noncollinear points. Now the automorphism group ${{\text{\rm{PGL}}}}_3({{\mathbb F}}_3)$ acts transitively on triples of noncollinear points. Map these three points to the points $2, 10, 11$ respectively We may therefore suppose that $l_1$, $l_2$ are lines $(12), (11)$ respectively with excluded points $10, 11$ (underlined in the table). Thus $$l_1'=\{1,2,4\}\qquad l_2'=\{2,3,5\}\,.$$ Now for $j=3,4,5,6$, the set $l_j'$ contains a point not in $l_1$ or $l_2$, namely one of the points 0,6,7,8,9,12. Let $j\in\{3,4,5,6\}$.
- If $0\in l_j'$ then $$(0)\cap(l_1'\cup l_2'\cup l_j')=\{1,3,0\}\,,$$ and $$(9)\cap(l_1'\cup l_2'\cup l_j')=\{4,5,0\}\,.$$ One can choose as line $m$ to satisfy inequality (\[three\]) whichever of $(0)$, $(9)$ is not equal to $l_j$. Both choices of $m$ may be possible.\
- If $6\in l_j'$ then $$(8)\cap(l_1'\cup l_2'\cup l_j')=\{1,5,6\}\,,$$ and $$(10)\cap(l_1'\cup l_2'\cup l_j')=\{4,3,6\}\,.$$ One can choose as line $m$ whichever of $(8)$, $(10)$ is not equal to $l_j$.\
- If $7\in l_j'$ then $$(9)\cap(l_1'\cup l_2'\cup l_j')=\{4,5,7\}\,,$$ so if $l_j\neq(9)$ we can choose $m=(9)$.
If $8\in l_j'$ then $$(8)\cap(l_1'\cup l_2'\cup l_j')=\{1,5,8\}\,,$$ so if $l_j\neq(8)$ we can choose $m=(8)$.
If $9\in l_j'$ then $$(0)\cap(l_1'\cup l_2'\cup l_j')=\{1,3,9\}\,,$$ so if $l_j\neq(0)$ we can choose $m=(0)$.
If $12\in l_j'$ then $$(10)\cap(l_1'\cup l_2'\cup l_j')=\{4,3,12\}\,,$$ so if $l_j\neq(10)$ we can choose $m=(10)$.\
- By choosing $j=3,4,5,6$ in parts (a), (b) and (c) above that, we see that we can choose $m$ to satisfy inequality (\[three\]) except in one case. Up to a permutation of the set $\{3, 4, 5, 6\}$, this is the case where $$l_3,\,l_4,\,l_5,\,l_6=(9),\,(8),\,(0),\,(10)$$ respectively with $$7\in(9)',\, 8\in(8)',\, 9\in(0)',\, 12\in(10)'\,.$$
We work with the three lines $l_1=(12)$, $l_3=(9)$, $l_6=(10)$. There are two possibilities to consider:
If $6\in (10)'$ then $(7)\cap(l_1'\cup l_3'\cup l_6')=\{2,7,6\}$; so take $m=(7)$.
If $6\not\in(10)'$ then $(10)'=\{3,4,12\}$, $(2)\cap(l_1'\cup l_3'\cup l_6')=\{1,7,12\}$; so take $m=(2)$.
Careful examination of the proof of Theorem \[main\] shows that six steps are enough to get from initial to final triangle, exactly as indicated in Figure \[fig1\].
[CMSZ]{}
J. W. Blattner, [*Projective Plane Geometry*]{}, Holden-Day, San Francisco, 1968.
D. I. Cartwright, A. M. Mantero, T. Steger and A. Zappa, Groups acting simply transitively on the vertices of a building of type $\widetilde A_2$, I,II, [*Geom. Ded.*]{} [**47**]{} (1993), 143–166 and 167–223.
P. Garrett, [*Buildings and Classical Groups*]{}, Chapman and Hall, London, 1997.
G. Robertson and T. Steger, Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras, [*J. reine angew. Math.*]{} [**513**]{} (1999), 115–144.
M. A. Ronan and J. Tits, [Building buildings]{}, [*Math. Ann.*]{} [**278**]{} (1987), 291–306.
|
---
author:
- 'Tomoya <span style="font-variant:small-caps;">Murata</span>$^{1}$, Wataru <span style="font-variant:small-caps;">Horiuchi</span>$^{2}$, Toru <span style="font-variant:small-caps;">Sato</span>$^{1}$ and Satoshi X. <span style="font-variant:small-caps;">Nakamura</span>$^{1}$'
title: |
Neutrino Induced $^4$He Break-up Reaction\
[-Application of the Maximum Entropy Method in Calculating Nuclear Strength Function-]{}
---
Introduction
============
The neutrino-nucleus reactions play an important role for the heating and cooling mechanisms in the core collapse supernova explosion. Particularly the neutrino-$^4$He reactions have been of interest, and their effects on the accelerating shock wave [@haxton1988] and nucleosynthesis [@fuller1995] have been studied. The typical neutrino energy inside the supernova is tens of MeV, and the dominant reaction channel is $^4$He breakup. Therefore an accurate theoretical treatment of the four-body scattering state is essential, which, however, is a hard task. The neutrino-$^4$He inclusive cross sections have been evaluated in a shell-model approach in ref.[@suzuki2006], while an [*ab-initio*]{} calculation has been done by making use of the Lorentz Integral Transformation (LIT) method[@gazit2007]. Also, a recent work of strength function (SF) based on the correlated Gaussians and the complex scaling method[@horiuchi2013] has been applied to the neutrino reaction[@Murata].
Among those methods, the LIT method has been widely applied to the break-up reactions of few-nucleon systems, where function $L(\sigma_R,\sigma_I)$ defined with the SF $R(\omega)$ ($\omega$: excitation energy) by an integral transformation, $$\begin{aligned}
L(\sigma_R,\sigma_I) = \int d\omega
\frac{R(\omega)}{(\omega-\sigma_R)^2+\sigma_I^2},\end{aligned}$$ plays a central role. The function $L$ can be calculated by using bound state like wave functions of the many body system. Then the SF $R(\omega)$ is obtained by the inverse transformation of the above integral. Though the inversion of function expressed by the convolution is in principle possible using Fourier transformation, the method used in the literature is $\chi^2$-fitting by using assumed functional form of SF such as a sum of exponentials [@Efros2007; @Leidemann2008]. However, the $\omega$-dependence of SF is not known a priori and the assumed functional form might lead to a false SF even the $\chi^2$ minimum is achieved. In this report, we examine the maximum entropy method (MEM) as a new tool for the inversion of the LIT. In the MEM, we do not need to assume any functional form of the SF. The MEM is widely used in the fields of the condensed-matter physics, the Lattice QCD [@R.N.SilverD.S.Sivia1990; @Gubernatis1991; @Asakawa2001], and the Green’s functions Monte Carlo method [@Lovato2015].
In section 2 and 3, we briefly explain the LIT method and the MEM. In section 4, we apply the MEM to the inversion of LIT. The results are reported for spin-dipole SF of $^4$He as an example.
Lorentz Integral Transformation Method
======================================
We briefly explain the LIT method for calculating nuclear SFs. The SF of transition operator $O$ is given as $$\begin{aligned}
R(\omega) = \Sigma_f \left|\bra{\psi_f}O
\ket{\psi_0}\right|^2 \delta(E_f-E_0-\omega),\end{aligned}$$ where $\psi_0$ and $\psi_f$ are the initial and final states, respectively. Both the initial and final states are eigenstates of a Hamiltonian $H$ with the energies $E_0$ and $E_f$. Here we define the LIT of SF as $$\begin{aligned}
L(\sigma_R,\sigma_I) = \int d\omega
\frac{R(\omega)}{(\omega+E_0-\sigma_R)^2+\sigma_I^2}.
\label{lit-def}\end{aligned}$$ From the definition of SF, the LIT can be written as $$\begin{aligned}
L(\sigma_R, \sigma_I)
&=& \bra{\psi_0}O^\dagger
(H+E_0-\sigma_R+i\sigma_I)^{-1}(H+E_0-\sigma_R-i\sigma_I)^{-1}O\ket{\psi_0}\\
&=& \braket{\tilde{\psi}|\tilde{\psi}},\end{aligned}$$ where $$\begin{aligned}
\ket{\tilde{\psi}} = (H+E_0-\sigma_R- i \sigma_I)^{-1}O\ket{\psi_0}.
\label{eq-psi}\end{aligned}$$ We see that the norm of $\ket{\tilde{\psi}}$ is finite and thus $\ket{\tilde{\psi}}$ can be treated like a bound state, which is a great advantage of this method, otherwise one has to construct the scattering state of a few-nucleon system. In the LIT method, one at first calculates $L(\sigma_R,\sigma_I)$ and then SF $R(\omega)$ is obtained by the inverse transformation of Eq. (\[lit-def\]).
Maximum Entropy Method
======================
We briefly explain how the MEM [@MEM] can be used to extract the SF $R(\omega)$, knowing only a set of $L(\sigma_R,\sigma_I)$ obtained from the many-body calculation. According to the Bayes’ theorem [@bayes], the most plausible SF ($R^{\rm MEM}(\omega)$) is given by the functional integral of the SF $R(\omega)$: $$\begin{aligned}
R^{\rm MEM} & = & \int [dR] R P[R|\bar{L},m],\end{aligned}$$ where $P[R|\bar{L},m]$ is a conditional probability of SF $R$ for a given LIT denoted as $\bar{L}$ and prior information, denoted by $m$, for the SF. $m$ is called the default model of the SF. Introducing an auxiliary variable $\alpha$, we can rewrite the above formula as $$\begin{aligned}
R^{\rm MEM} & = & \int d\alpha R^\alpha P[\alpha|\bar{L},m], \label{eq-smem}\end{aligned}$$ where $R^\alpha$ depends on $\alpha$ in addition to our inputs $\bar{L}$ and $m$, and is given by $$\begin{aligned}
R^\alpha & = & \int [dR] R P[R|\alpha,\bar{L},m]
\propto \int [dR] R P[\bar{L}|R,\alpha,m] P[R|\alpha,m].\end{aligned}$$ Here $P[\bar{L}|R,\alpha,m]$ and $P[R|\alpha,m]$ are called the likelihood function and prior probability, respectively and are given by $$\begin{aligned}
P[\bar{L}|R,\alpha,m] = \frac{1}{Z_\chi}\exp\left(-\frac{1}{2}\chi^2\right),\ \ P[R|\alpha,m] = \frac{1}{Z_S} \exp\left(\alpha S\right).\end{aligned}$$ with $$\begin{aligned}
\chi^2 & =& \Sigma_{l=1}^{N_{\sigma_R}}\frac{(\bar{L_l}-L_l)^2}{\delta_l^2},\ \
S = \Sigma_i^{N_\omega} \left(R_i - m_i - R_i \ln\frac{R_i}{m_i} \right)\\
Z_\chi &=& \Pi_{l=1} \sqrt{2\pi\delta_l^2},\ \
Z_S = \left(\frac{2\pi}{\alpha}\right)^{N_\omega/2}. \end{aligned}$$ Here $S$ is called the Shannon-Jaynes entropy. The excitation energy $\omega$ is discretized as $\omega_i$ ($i=1,2,...,N_\omega$) with equal spacing $\Delta_\omega$. $R_i$ and $m_i$ are given by $R_i=R(\omega_i)\Delta_\omega$ and $m_i = m(\omega_i)\Delta_\omega$. $\bar{L}_l$ is the given $L$ at $\sigma_R = \sigma_R^l$ ($l=1,2,...,N_{\sigma_R}$) with the error $\delta_l$. $L_l$ is calculated from $R_i$ using Eq. (3) at $\sigma_R = \sigma_R^l$. Combining the likelihood function and the prior probability, the SF $R^\alpha$ is chosen to maximize the probability of $\displaystyle P[R|\alpha,\bar{L},m] \propto e^{Q(R)}$ with $Q(R) = \alpha S - \frac{1}{2}\chi^2$.
Finally, as in Eq. (8), $R^\alpha$ is convoluted with the probability $P[\alpha|\bar{L},m]$ that has a sharp peak as a function of $\alpha$ and is written as $$\begin{aligned}
P[\alpha|\bar{L},m] & = & \int [dR] P[R, \alpha|\bar{L},m]
\propto P[\alpha|m] \int [dR] \frac{1}{Z_S Z_\chi}e^{Q(R)}.\end{aligned}$$ Assuming that $P[\alpha|m]$ is constant, we obtain the SF $R^{\rm MEM}$ by integrating with respect to $\alpha$ around the sharp peak of $P[\alpha|\bar{L},m]$ as in Eq. (\[eq-smem\]).
Application of MEM to Inversion of LIT
======================================
Our question is whether the MEM described in the previous section is useful to invert the integral transformation in Eq. (3). For this purpose, we start from pseudo data of LIT, $L(\sigma_R, \sigma_I)$, that are generated from a ’known’ SF $R^{\rm orig}(\omega)$ by using Eq. (3). We then use the MEM to obtain SF from the pseudo LIT data without assuming any functional form for the ’reconstructed’ $R(\omega)$. We use the SF of $^4$He for the spin-dipole operator as $R^{\rm orig}$; this SF has been calculated in Ref. [@horiuchi2013]. The SF for the spin-dipole operator that can induce neutrino and anti-neutrino reactions ($\pm$) is given by summing the final scattering states $\ket{\psi_f}$ as $$\begin{aligned}
R^{J\pm}(\omega) &=& \Sigma_f \left| \bra{\psi_f}\Sigma_j
\left[{\mib \rho}_j\otimes{\mib \sigma}_j\right]_{(J)}\tau^\pm_j\ket{\psi_0}\right|^2
\delta(E_f-E_0-\omega)\end{aligned}$$ where $\Sigma_j$ denotes sum of the nucleons, ${\mib \rho}_j$, ${\mib \sigma}_j$ and $\tau_j$ are the internal coordinate, spin, and isospin of the [*j*]{}-th nucleon, respectively. $\psi_0$ is the ground state of $^4$He.
The pseudo LIT data $L(\sigma_R, \sigma_I)$ as a function of $\sigma_R$ are shown in Fig. \[fig:lit-data\] for $\sigma_I=3, 5, 10$ MeV. The peak of $R^{\rm orig}$ becomes broader in $L(\sigma_R, \sigma_I)$ as $\sigma_I$ is increased.
![Pseudo LIT data of $^4$He spin-dipole SF. The green dot-dashed, the blue solid and the red dotted curves show $L(\sigma_R, \sigma_I)$ for $\sigma_I$=3, 5, 10 MeV, respectively.[]{data-label="fig:lit-data"}](lit-data.eps){width="10cm"}
We then apply the MEM to those pseudo data $L(\sigma_R, \sigma_I)$ to obtain the $R^{\rm MEM}$. Here we adopt a constant SF as the default model that is $m_i$ in Eq. (11). The obtained $R^{\rm MEM}$ in comparison with the original SF $R^{\rm orig}$ are shown in Fig. \[fig:sf-mem\]. The left panel shows the $R^{\rm MEM}$ reconstructed from $L(\sigma_R, \sigma_I)$ with $\sigma_I$=3, 5, 10 MeV. The right panel shows the ratio $R^{\rm MEM}/R^{\rm orig}$, showing deviation of the reconstructed SF from the original one. For $\sigma_I=3, 5$ MeV, the reconstructed SF $R^{\rm MEM}(\omega)$ agree well with $R^{\rm orig}(\omega)$ except near the threshold. The deviation of the ratio $R^{\rm MEM}/R^{\rm orig}$ from one is within 1% in the region $25$ MeV $\leq \omega \leq$ $120$MeV. On the other hand, for $\sigma_I=$10 MeV, the peak of $R^{\rm MEM}(\omega)$ shifts by 1 MeV, and $R^{\rm MEM}$ shows an oscillatory behavior for higher $\omega$ region. The deviation from $R^{\rm orig}$ is more than 5%, and is nonnegligible for $\omega <$ 30 MeV.
![The reconstructed SF $R^{\rm MEM}$(Left) and the ratio $R^{\rm MEM}/R^{\rm orig}$(Right).[]{data-label="fig:sf-mem"}](sf.eps){width="8cm"}
![The reconstructed SF $R^{\rm MEM}$(Left) and the ratio $R^{\rm MEM}/R^{\rm orig}$(Right).[]{data-label="fig:sf-mem"}](ratio-sf.eps){width="8cm"}
We have shown that the MEM can be successfully applied to the inversion of LIT without assuming any functional form for SF $R(\omega)$, provided we use LIT data with $\sigma_I$ sufficiently smaller than 10 MeV. In general, the structures of $R(\omega)$ narrower than $\sigma_I$ are smeared out in $L(\sigma_R, \sigma_I)$, and it is very hard to reconstruct the narrow structure of $R(\omega)$. In our example, the width of the peak structure of $R(\omega)$ is about 10 MeV.
In this work, we used the known SF, $R^{\rm orig}(\omega)$, to generate the pseudo data, $L(\sigma_R,\sigma_I)$. Thus, the pseudo data are very accurate at any $\sigma_R$ and $\sigma_I$. In practice, however, we need obtain $L(\sigma_R,\sigma_I)$ from a discrete spectrum that is calculated with the eigenvalue method or the Lanczos algorithm [@Efros2007]. Therefore, more study is needed to examine if the MEM method applied to the LIT inversion works in the practical situations.
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|
---
abstract: 'In this paper, we describe our experience implementing some of classic software engineering metrics using Boa—a large-scale software repository mining platform—and its dedicated language. We also aim to take an advantage of the Boa infrastructure to propose new software metrics and to characterize open source projects by software metrics to provide reference values of software metrics based on large number of open source projects. Presented software metrics, well known and proposed in this paper, can be used to build large-scale software defect prediction models. Additionally, we present the obstacles we met while developing metrics, and our analysis can be used to improve Boa in its future releases. The implemented metrics can also be used as a foundation for more complex explorations of open source projects and serve as a guide how to implement software metrics using Boa as the source code of the metrics is freely available to support reproducible research.'
address: 'Wrocław University of Science and Technology, Faculty of Computer Science and Management, Wrocław, Poland'
author:
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-
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-
-
-
title: 'Software Metrics in Boa Large-Scale Software Mining Infrastructure: Challenges and Solutions'
---
,
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,
and
boa, large-scale software mining, software metrics, prediction models
Introduction
============
Boa is a tool that can be used for data mining repositories of open-source projects. It contains the full history of a repository—from every revision’s date and author, data on added, deleted and modified files to the complete state of the repository at the moment of commit. All data can be obtained by using the dedicated language. Boa provides a set of functions, which can be used for advanced data filtering [@boaDocumentation][@dyer2013].
Boa has already been used for a variety of studies, including developers’ willingness to adapt new Java features [@dyer2014] or the licenses used in open-source projects [@vendome2015license]. So far they have not been metrics-oriented, even though the tool is intended to be used this way, as implicated by the inclusion of appropriate examples in the documentation of Boa [@boaExamples] (e.g., [*What are the number of attributes (NOA), per-project and per-type?*]{}, [*What are the number of public methods (NPM), per-project and per-type?*]{}).
In this paper we focus on using Boa infrastructure to answer three research questions:
1. Which of the classic, widely known, software engineering metrics can be implemented in Boa?
The implementation of classic software engineering metrics in Boa and publication of calculation scripts will make it easier to extend existing small-scale empirical software engineering research using software metrics, performed usually on a small number of projects, to a large-scale research.
2. What new metrics, that take advantage of the Boa’s unique infrastructure, can be proposed?
This paper will serve as a guide, for other researchers and practitioners, which shows how to implement new software metrics taking into account the unique features, as well as limitations, of the Boa large-scale software repository mining platform.
3. What is the feasibility of defect prediction models based on large number of projects data obtained from Boa data sets?
According to our knowledge, this is one of the first attempts (if not the first) to build large-scale software defect prediction models based on a very large number of projects. Existing software defect prediction models usually base on a very limited number of projects.
Presented study refers to state of Boa framework during October 2015 - January 2016 period – when the source material was gathered.
Research methodology {#sec:Methodology}
====================
In this section we introduce briefly into the following topics: how we selected projects for further investigation (see Section \[sec:ProjectsSelection\]), how we implemented software metric scripts using the Boa language (see Section \[sec:ImplementationOfSEMetrics\]), and how we built software defect prediction models using software metrics from Boa (see Section \[sec:DefectPredictionModel\]), including also how we obtained data from the Boa output files (see Section \[sec:weka\]).
Projects selection {#sec:ProjectsSelection}
------------------
Boa source code described in this paper has been developed and tested on two Boa data sets: September 2015 GitHub, and September 2013 SourceForge. A special filtering has been applied to select projects passing some entry criteria. The software projects explored in our study had to pass the following criteria:
1. **They have to have a code repository with revisions.** The *2013 September/SourceForge* data set consists of 700k projects. Our analysis with Boa queries has shown that 30% of them have no code repository [@onlineAppendix Section 2.1]. Out of remaining 489k (amount close to this stated by Boa developers - 494,158 [@boaStatistics]) 4,767 projects have two repositories. Repositories in those projects have common history of revisions [@onlineAppendix Section 1.1]. In case of projects with multiple repositories, only the first of them is considered during study to avoid data duplication. Out of 489k projects with one code repository, 423k of them had no code revisions (commits) [@onlineAppendix Section 1.2]. It is difficult to determine whatever or not Boa is missing some data—the data sets have been defined for a given month in a given year, and current state of the repository might be different.
The *2015 September/GitHub* data set has 7.83 million projects. 95% of them have no code repository in the Boa framework, even though the majority of them is available from the GitHub website. They are active and public, but most of them have had no commits since 2013 [@onlineAppendix Section 1.3]. From 380k projects with repository, only 2486 of them had commits in 2015 [@onlineAppendix Section 1.4]. Out of the entire GitHub dataset, 4% of projects have code repositories with revisions [@onlineAppendix Section 1.5].
2. **They have to have over 100 commits.** The projects picked should be mature enough for metrics calculation. A larger number of commits usually means a larger number of *fixing revisions*, which are in turn used for development of software defect prediction models.
3. **They have to be written in Java.** Java has been picked for this research due to being a mature, object-oriented language, popular among developers. It is also worth mentioning that Boa is written in Java, as well as provides extra Java-specific options, such as recognizing Java source files with and without parsing errors.
The Boa language implementation of filters to select projects fulfilling the above mentioned criteria is presented in Listing \[lst:filtersBoaImpl\].
The final number of projects that passed our entry criteria is presented in Table \[table:datasets\].
**Dataset** **All projects** **Accepted projects**
------------- ------------------ -----------------------
GH small 7,988 29
GH medium 783,982 2485
GH large 7,830,023 25307
SF small 7,029 50
SF medium 69,735 666
SF large 699,331 7407
: Data sets
\[table:datasets\]
Implementation of SE metrics {#sec:ImplementationOfSEMetrics}
----------------------------
All of the metrics are calculated for classes. Each of the metric is implemented as a different Boa query, and is run on all Boa data sets mentioned in Section \[sec:Methodology\]. Due to long execution time, only data from GH small and SF small data sets are used for creating prediction models later on.
The output file of a query has to have the following data:
- [the ID of the project]{}
- [the ID of the class]{}
- [the value of the calculated metric or the expected value]{}
This approach makes it possible to effortlessly merge all values gathered as the outputs of Boa queries, so they can be used as an input data set for a prediction model.
Defect prediction model {#sec:DefectPredictionModel}
-----------------------
Software defect prediction model is aiming to find the classes that cause the most defects. A simple strategy to find them is searching for the classes that had been fixed most frequently.
### Expected value - NCFIX {#sssec:expValue}
The expected value in our defect prediction model is Number of Class Fixes. Based on Boa’s abilities, it is assumed the class has been fixed, if the two following conditions have been met:
- the file containing the class has been modified in a revision;
- the revision is marked as a fixing revision by the Boa’s function\
`isfixingrevision` [@boaDocumentation].
The list of classes and their fixes is obtained by the following algorithm:
1. Create an empty key-value collection for storing respectively: files in projects, number of fixing revisions for each file.
2. Visit a project’s repository revision.
3. Check if it’s a fixing revision.
4. Investigate the files changed in this revision.
1. If a file is marked as deleted, remove it from the collection.
2. If a file is added to the project in the current revision, add it to the collection:
1. with a value of 1 if the revision is a fixing one;
2. with a value of 0 otherwise.
3. If a file is modified in the current revision, update it in the collection
1. increment the number of fixes by one, if the revision is a fixing one;
2. leave it otherwise.
5. Repeat steps 2-4 until you reach the most recent revision and there is no more revisions to check.
6. For all files stored in the collection, select only the ones that declare classes. Return the identifiers of the classes, and numbers of fixes corresponding to their files as the output.
The algorithm is inspired by the `getsnapshot` function implemented by Boa [@boaDocumentation], which returns the state of the repository at given time stamp.
The use of Boa API and Weka {#sec:weka}
---------------------------
To allow easy management of Boa jobs and connecting job outputs with development of defect prediction models, a simple Java program [@sourceCodeGithub] has been written. The software uses Boa Java API [@boaAPI] release 0.1.0 to run jobs. Data from Boa is transformed into `.arff` file of following format:
``` {basicstyle="\footnotesize"}
@RELATION classes
@ATTRIBUTE classID string
@ATTRIBUTE M_1 NUMERIC
...
@ATTRIBUTE M_N NUMERIC
@ATTRIBUTE fixingRevisions NUMERIC
```
where `classID` is an identifier of a studied class; `M_1 ... M_N` is a vector of calculated metrics for a class from latest repository SNAPSHOT; `fixingRevisions` attribute is the expected value described in Section \[sssec:expValue\].
Results
=======
In this section three kinds of contribution are discussed, related to implementation of classic and new software metrics in Boa, as well as development of software defect prediction models on a basis of very large number of software projects. The latter can be seen as a way to address external validity threats common for most of the the empirical studies focused on software defect prediction. All metrics’ implementations are available to download via links provided in appendix [@onlineAppendix Section 3].
Implementation of classic software engineering metrics
------------------------------------------------------
This section presents how to implement scripts to collect some of the well-known, classic software metrics [@chidamber1994metrics] in Boa. The metrics were chosen based on their popularity and Boa’s limitations.
### Obtaining classes
Using `getsnapshot` function implemented in Boa, all files available in the most recent revision of the project are gathered. Then, they are filtered so that only the files containing classes are taken into consideration. The data stored in the `Declaration` [@boaDocumentation] and its attributes are used for calculating the value of a metric.
### Inheritance issue
Each declaration (class or interface) node in Boa has its array of parents [@boaDocumentation]. However, those parents are presented only as *Types*, meaning, they only have *TypeKind* (determining if it’s a class, interface, or something else) and name, without its full package path or any other identifier. If two classes or interfaces in a project have the same name, but they are in different packages, it is impossible to determine which one is the ancestor of a given declaration. Therefore, all metrics using inheritance (such as all of the MOOD metrics [@moodMetrics], Depth of Inheritance Tree, Number of Children and Coupling between Object Classes [@chidamber1994metrics]) had to be, unfortunately, excluded from the study.
### Metrics obtained directly from the `Declaration` node
Weighted Methods per Class (WMC) in its base version—the sum of methods in a class, Number of Fields (NoF) and Number of Nested Declarations (NoND), presented in Table \[table:declarationmetrics\], have been successfully implemented using the structure of the `Declaration` node alone.
**Attribute** **Metric**
---------------------- ------------
methods WMC
fields NoF
nested\_declarations NoND
: `Declaration` attributes and associated metrics
\[table:declarationmetrics\]
For each of those metrics, the value is a length of the attribute array. The execution time for those metrics is relatively small, up to 10 minutes for the biggest data sets, which clearly shows the advantages of using Boa and the approach to calculate metrics using the structure of the `Declaration` node, presented in this paper.
### Response For a Class (RFC)
The RFC metric was implemented as a number of methods in the class, added to number of remote methods directly called by methods of the class. The issue with the implementation of this metric is that Boa makes it difficult to recognize the difference between class’ inner method and method of the external classes of the same identifier. For example: the method `getId()` of `class A`, called in `class B`, is seen as the same as method `getId()` in `class B`. If `class A` called two methods of the same name from different classes (`class B` and `class C`), those would be indistinguishable as well. There is no direct method that would allow to instantly determine the types of called methods’ arguments [@boaDocumentation] as well as the type of instance of variable from which the method was called [@onlineAppendix Section 1.6]. Such information can be obtained only by deeper analysis of Boa’s AST tree, to the level of single `Statement`s.
The simplified version of the metric, that ignores this nuance, has been successfully implemented and ran for both Boa’s data sets.
Implementation of new software metrics
--------------------------------------
The metrics presented below have been developed by us upon learning more about the Boa architecture and its tree structure.
### Number of Statements in Methods
The NoSiM metric is calculated as a sum of all statements in class methods. The nodes calculated are of the Boa type `Statement`. For studied Java classes, those nodes are either blocks of code marked by `{}` or single code expressions. The implementation of this metric is a starting point for implementation of a Lines of Code (LoC) metric. To achieve the LoC metric, all class’ fields, number of methods, and such, would have to be added.
### Maximum Depth of Declaration Nesting {#sssec:MDoDN}
MDoDN is the maximum level of class nesting in a class. For the following code:
``` {basicstyle="\footnotesize"}
class A {
class B {
class C {}
}
class D {}
}
```
the result for `class A` would be 3 (the depth of `C` class). The metric is not calculated for nested classes (in the example: `B, C,` and `D`). For implementation of this metric, Boa’s stack functions are used. Every time the node of a nested `Declaration` is entered, it is pushed onto the stack. The metric value is the stack’s element count.
### Number of Anonymous Declarations
NoAD for Java is a sum of all anonymous children classes in the parent class. To calculate this metric, the `Expression` Boa node is tested for having a `Declaration` with a parameter of `ANONYMOUS` type.
### Cumulative metrics {#sssec:CumulativeMetrics}
Metrics NoM, NoF, NoSiM, NoAD and NoND have been also successfully implemented in cumulative versions (CNoM [@onlineAppendix Section 1.7], CNoF [@onlineAppendix Section 1.8], CNoSiM [@onlineAppendix Section 1.9], CNoAD [@onlineAppendix Section 1.10], CNoND [@onlineAppendix Section 1.11]), where calculated value is a sum of metric for not only a class, but also all its nested and local classes.
Defect prediction model {#subsec:predictionModel}
-----------------------
The defect prediction model presented below is a single defect prediction model calculated for a high number of Boa projects. This is different from a traditional approach, with a single, or several projects used to develop defect prediction models.
Data obtained from the Boa output files (described in Section \[sec:weka\]) is randomly separated into training set and testing set (in 9:1 proportion). The `fixingRevisions` attribute in the testing set is nulled out, so it can be calculated using prediction model.
We used Random Forest to build defect prediction model. Random Forest generates a lot of random samples which are the subsets of training data set. A decision tree is generated for each of the samples [@Breiman2001]. The parameters listed below have been determined experimentally:
- [number of trees : 200]{},
- [max depth : 12]{},
- [number of features : 12]{},
- [cross-validation folds: 10]{},
- [random seed: 1]{}
The results of 10-fold cross-validation are presented in Table \[table:evalModelResults\]. Pearson product-moment correlation coefficient $r$ shows a low correlation between the results from defect prediction model and real values, with high error ratio. Those results are further analyzed in Section \[sec:dicussion\].
**Evaluation attribute**
------------------------------------ ------- -------
Correlation coefficient (R) 0.215 0.244
Mean absolute error (MAE) 2.16 0.603
Root mean squared error (RMSE) 9.96 1.32
Relative absolute error (RAE) 102% 93.3%
Root relative squared error (RRSE) 100% 97.8%
: Results of evaluation of the prediction model
\[table:evalModelResults\]
Reference values of software metrics
------------------------------------
The subsequent goal was to characterize a large number of open source projects available from Boa by means of software metrics in order to create reference values of software metrics. Table \[table:stats\] presents descriptive statistics for each of calculated metrics among the data sets.
\[table:stats\]
Discussion {#sec:dicussion}
==========
The presented prediction model was tested on small data sets, but with correct resources it can be easily scaled to use full data sets with up to 25k subjects. This use case would be, to the best of our knowledge, the first attempt to create a large-scale defect prediction model, as other examples from literature show prediction models developed using less than 200 projects [@JureczkoMadeyski10; @JureczkoMadeyski15; @Madeyski15SQJ].
The performance of the prediction model is poor due to the fact that a majority of classes studied has zero fixing revisions and therefore input data is highly unbalanced, see Table \[table:classFixes\]. However, the quality of prediction model and employing methods to deal with the class imbalance problem are not the main objectives of the study. Our aim was to show that it is possible to collect all the data necessary to build a large-scale software defect prediction model using the Boa platform.
Results from Table \[table:stats\] show that not for all metrics standard deviation is lower for filtered datasets. This can be caused by the nature of metrics (such as NoND, NoAD, MDoDN), which are unlikely to have a high mean value in majority of projects.
------- --------------- ---------------
0 13296 (58.9%) 30244 (80.1%)
>0 9260 (41.1%) 7504 (19.9%)
------- --------------- ---------------
: Number of classes with zero and more than zero fixes in datasets
\[table:classFixes\]
Further research
----------------
It is worth to look at the way the fix in the revision is identified. Boa-provided function `isfixingrevision` is based only on the commit message text analysis. We assume this function is not ideal and integrating Boa API with outside software, such as bug tracking systems, can be a better solution to determine existing bugs in code revisions.
The data used for building prediction models in our study has big disproportions. Applying different filters and criteria (more mature projects, different languages and so on) could provide better data set for analysis, with more fixing revisions.
An interesting path of further research are process metrics [@Madeyski15SQJ; @Madeyski11], which reflect changes over time and are becoming the crucial ingredients of software defect prediction models.
Conclusions
===========
Overall, the goal of the research, as described with research questions — implementation of software metrics in Boa and collecting data sets from a large number of projects, e.g., for the sake of prediction models —has been achieved. We were able to implement some of the classic software engineering metrics using Boa, we presented some Boa-specific metrics, and we made an attempt to create a defect prediction model with the data we gathered. This proves that Boa can be a useful tool for data mining analysis in this particular field, as well as for creating sophisticated queries regarding its data sets. However, Boa is still a new framework that comes with a few disadvantages, and some of the metrics and operations were impossible to implement at the moment. In the following sections, the challenges met and our solutions are presented.
Challenges {#subsection:challenges}
----------
Boa uses visitor pattern—one of Boa’s greatest strengths—which sometimes might provide unexpected results if queries are not written properly.
### Local and nested classes
One of the first issues we encountered creating Boa queries was a different size of output jobs. For our metrics, we gathered all classes from all projects. Therefore, for the same data set, all queries should return the same number of rows. As it turned out, the difference was caused by the behaviour of the visitor pattern, used by Boa. When source code contains a local class (class defined inside one of the methods) or a nested class (a class declared inside of another class), this class is visited by the visitor pattern before the analysis of the class containing it ends. Upon returning to the class-container, some of it’s metrics and calculations had been assigned to the local or nested class.
**Solution:** Boa offers implementation of stacks, which we started using while visiting local and nested classes. We took advantage of this solution implementing the Maximum Depth of Declaration Nesting metric described in Section \[sssec:MDoDN\].
### Boa code compilers
Boa uses two different code compilers for SourceForge and GitHub data sets. As the framework is still in early development, sometimes the same query acts differently depending on the data set used.
**Example:** One of Boa sample queries “How many committers are there for each project?” [@boaExampleCommiters] works fine in SF [@onlineAppendix Section 1.12], but causes compilation error in GH [@onlineAppendix Section 1.13]. In that case, a small change in the code notation solved the issue [@onlineAppendix Section 1.14]:
- Code resulting with error:
``` {basicstyle="\footnotesize"}
committers[p.code_repositories[i].revisions[j].committer.username] = true;
```
- Code resulting with success:
``` {basicstyle="\footnotesize"}
username : string = p.code_repositories[i].revisions[j].committer.username;
committers[username] = true;
```
This example shows that a person creating queries with Boa might run into different issues depending on the data set picked.
During our research, we often used Boa dictionaries. Dictionaries are defined by Boa as `map[key_type] of [value_type]`. Boa returns an error, if `int` is used as a `value_type`. We must have stored our integer values as strings, which resulted in converting value to integer each time it was used in calculations, and then back to string to update the map.
### Debugging process
The errors reported by Boa are often lacking any sort of description. The debugging process comes down to commenting out parts of queries to check which fragments are causing errors. Each code test takes about a minute (and then some follow-up time to check if the output data is correct), and sometimes multiple tests are required to find the source of an error. There is no way of tracking the execution of the queries.
**Solution:** All variables used during the debugging process have to be initiated, by defining its type and aggregation method, and then returned in the output file.
Contribution
------------
The paper describes our experience with using Boa platform for implementing software engineering metrics and defect prediction models. Our findings can be useful for both researchers—with solutions presented in Section \[subsection:challenges\] and provided source codes for metrics we implemented—as well as developer teams and project managers, providing an example for obtaining large-scale SE metrics for projects of particular profile (i.e. number of commits, used programming language and so on). The metric implementations proposed by us are scalable—calculated for classes, but could be as well implemented for packages or projects.
Based on our findings, we confirm that Boa can be a powerful data mining tool, which can be used for a variety of research, alone and with usage of other software, like Weka, as demonstrated in Section \[sec:weka\].
[99]{}
Iowa State University of Science and Technology: The Boa Programming Guide. <http://boa.cs.iastate.edu/docs/> (October 2015), accessed: 2015-11-18 Dyer, R., Nguyen, H.A., Rajan, H., Nguyen, T.N.: Boa: A language and infrastructure for analyzing ultra-large-scale software repositories. In: Proceedings of the 2013 International Conference on Software Engineering. pp. 422–431. IEEE Press (2013) Dyer, R., Rajan, H., Nguyen, H.A., Nguyen, T.N.: Mining billions of ast nodes to study actual and potential usage of java language features. In: Proceedings of the 36th International Conference on Software Engineering. pp. 779–790. ACM (2014) Vendome, C., Linares-Vásquez, M., Bavota, G., Di Penta, M., German, D., Poshyvanyk, D.: License usage and changes: A largescale study of java projects on github. In: The 23rd IEEE International Conference on Program Comprehension, ICPC (2015) Iowa State University of Science and Technology: Example Boa Programs, <http://boa.cs.iastate.edu/examples/> (October 2015), accessed: 2015-11-18 Patalas, A., Cichowski, W., Malinka, M., Stępniak, W., Maćkowiak, P., Madeyski, L.: Appendix to “Software Metrics in Boa Large-Scale Software Mining Infrastructure: Challenges and Solutions” (2016), <http://madeyski.e-informatyka.pl/download/PatalasEtAl16Appendix.pdf> Iowa State University of Science and Technology: Boa. Mining Ultra-Large-Scale Software Repositories. Dataset Statistics, <http://boa.cs.iastate.edu/stats/> (October 2015), accessed: 2015-11-18 Java research software, source code for metrics and statistical tests, <https://github.com/Aknilam/metrics-research-software> Iowa State University of Science and Technology: Boa. Mining Ultra-Large-Scale Software Repositories. Client API, <http://boa.cs.iastate.edu/api/> (October 2015), accessed: 2015-11-18 Chidamber, S.R., Kemerer, C.F.: A metrics suite for object oriented design. IEEE Transactions on Software Engineering 20(6), 476–493 (1994) e Abreu, F.B.: Design quality metrics for object-oriented software systems. ERCIM News 23 (1995) Breiman, L.: Random forests. Machine Learning 45(1), 5–32 (2001) Jureczko, M., Madeyski, L.: Towards Identifying Software Project Clusters with Regard to Defect Prediction. In: Proceedings of the 6th International Conference on Predictive Models in Software Engineering. pp. 9:1–9:10. PROMISE ’10, ACM, New York, USA (2010) Jureczko, M., Madeyski, L.: Cross–project defect prediction with respect to code ownership model: An empirical study. e-Informatica Software Engineer-ing Journal 9(1), 21–35 (2015) Madeyski, L., Jureczko, M.: Which Process Metrics Can Significantly Improve Defect Prediction Models? An Empirical Study. Software Quality Journal 23(3), 393–422 (2015) Jureczko, M., Madeyski, L.: A review of process metrics in defect prediction studies. Metody Informatyki Stosowanej 30(5), 133–145 (2011) Iowa State University of Science and Technology: Example Boa Programs, <http://boa.cs.iastate.edu/examples/> (October 2015), accessed: 2015-11-18
|
---
author:
- 'Arjun Bagchi,'
- 'Amartya Saha,'
- 'and Zodinmawia.'
- '\'
title: BMS Characters and Modular Invariance
---
Introduction
============
Conformal field theories (CFTs) [@DiFrancesco:1997nk] are a high energy theorist’s dream. Symmetries of a relativistic conformal theory are constraining enough to determine many quantities of interest, e.g. the form of the two and three point functions in arbitrary dimensions, even without requiring the details of the underlying Lagrangian description. Powerful methods of the conformal bootstrap [@Ferrara:1973yt; @Polyakov:1974gs; @Rattazzi:2008pe] (for a modern introduction see e.g. [@Simmons-Duffin:2016gjk]), which relies on the crossing symmetry of 4-point functions, constrain the system further and quite severely, and these are being currently utilised to chart out the allowed parameter space of all relativistic CFTs. The spectrum of primary operators and the coefficients of the three point functions (and the central charge) is all the information that is required to specify a CFT. The bootstrap equation tells us which of these sets of data constitute CFT consistent with fundamental requirements like crossing symmetry. As is well known, this programme of constraining CFTs has far reaching consequences. The modern way of understanding all relativistically invariant quantum field theories (QFTs) is by renormalisation group flows away from fixed points governed by CFTs. So a classification of all CFTs in a sense would lead to the classification of all QFTs.
[*[Conformal field theories and two dimensions]{}*]{}
In two spacetime dimensions, CFTs become even more special [@Belavin:1984vu]. The underlying symmetry algebra is enhanced to two copies of the infinite dimensional Virasoro algebra. When defined on the 2d plane, methods of complex variables allow us to have tremendous analytic control on the theory and the 2d theory holomorphically factorises into a chiral and an anti-chiral sector, which are treated separately. There seems to be no prior restriction to how one should put the chiral and anti-chiral sectors together. 2d CFTs arise in the context of string theory, as the residual symmetries on the string worldsheet after the fixing of conformal gauge, and the formulation on the complex plane, or equivalently the Riemann sphere, corresponds to tree-level string scattering. Loop diagrams are, of course, an integral part of any scattering computation and the higher loop contributions arise in string theory by placing 2d CFT on higher genus surfaces. Consistency on higher genus Riemann surfaces impose further conditions on 2d CFTs. For one loop diagrams, the surface of interest is the torus and these consistency requirements lead to modular invariance of 2d CFTs. Of late, these consistency requirements have led to what is called the modular bootstrap and has contributed in further constraining 2d CFTs.
Modular invariance of 2d CFTs not only constrains possible CFT data, but has led to the famous Cardy formula which computes the entropy of the theory by relating the high energy regime to the low energy by specifically the modular S-transformation [@Cardy:1986ie]. In the context of holography, the Cardy formula has been used to great effect in matching up with the entropy of the BTZ black holes in the bulk dual AdS$_3$ theory [@Strominger:1997eq; @Carlip:1998qw]. Recently, these ideas have been generalised to obtain averaged three-point function coefficients of two heavy and one light operator based on modular properties of 1-point torus amplitudes [@Kraus:2016nwo; @Romero-Bermudez:2018dim; @Hikida:2018khg; @Brehm:2018ipf; @Das:2017vej]. It has also been shown that crossing symmetry can be translated to modular properties, making this another avenue of implementing bootstrap-like techniques [@Maldacena:2015iua; @Das:2017cnv].
[*[Holography and flatspace]{}*]{}
Holography in AdS spacetime suggests that the asymptotic symmetry group of the gravitation theory dictates the symmetries of the putative dual boundary theory. A natural way to extend the notion of holography to non-AdS spacetimes is thus the following: one should use canonical methods to calculate the asymptotic symmetry group of a bulk theory. This is the group of allowed diffeomorphisms given a particular set of boundary conditions, modded out by the trivial diffeos (the ones which lead to zero charge). One should then attempt to realise this as the symmetry group of a field theory that lives on the asymptotic boundary of that spacetime.
In particular, we are interested in formulating the dual theory of asymptotically flat spacetimes. The asymptotic symmetry group of asymptotically Minkowski spacetime at its null boundary is given by the infinite dimensional Bondi-Metzner-Sachs (BMS) group [@Bondi:1962px; @Sachs]. This flies in the face on conventional wisdom, which would have suggested that the ASG would be the Poincare group. It has been shown that even though in higher dimensions ($D>4$) using stringent boundary conditions one can restrict to the Poincare group, in $D=3, 4$, the infinite dimensional group is unavoidable. While taken to be mostly a curiosity after its initial discovery in the 1960’s, it has been released in the recent past that the BMS symmetries have a fundamental role to play in the physics of the infra-red and link soft graviton theorems and memory effects in a triangle of relations (for a review of recent developments, see e.g. [@Strominger:2017zoo]).
In this paper, we will be interested in BMS$_3$ [@Barnich:2006av] and its realisation in field theories in $D=2$, which according to the formulation above, will lead to a dual theory of 3d flat space, living on its null boundary [@Bagchi:2010zz] [[^1]]{}. Specifically, we would be interested in the construction of characters of the theory and the notion of modularity in these field theories. The calculation of characters help us construct the partition function of the BMS field theory. In a manner similar to 2d CFTs, the partition functions of 2d theories also admit a deformed version of modular invariance, which leads to a BMS-Cardy formula [@Bagchi:2012xr]. There are solutions in 3D flatspace which are obtained by quotienting Minkowski spacetime by a boost and a translation [@Cornalba:2003kd]. These are cosmological solutions called Flat Space Cosmologies, which have a cosmological horizon. The Bekenstein-Hawking entropy of these FSCs are reproduced by the BMS-Cardy formula [@Bagchi:2012xr; @Barnich:2012xq].
[*[BMS, strings and other things]{}*]{}
In connection with the earlier motivation of string theory, it is of interest to note that the same algebra arises as residual symmetries on the worldsheet of the tensionless bosonic string [@Isberg:1993av; @Bagchi:2013bga; @Bagchi:2015nca]. So this notion of modularity would be of interest when one considers scattering of strings in the tensionless limit. In a series of famous papers Gross and Mende [@Gross:1987kza; @Gross:1987ar], and later Gross [@Gross:1988ue] found that there was a very large simplification in the behaviour of string scattering amplitudes in this limit, leading to an infinite number of linear relations between scattering amplitudes of different string states valid order by order in perturbation theory. This pointed to some higher symmetry structure in this extreme stringy limit. Modular invariance of BMS would be central to understanding this Gross-Mende regime from the point of view of the symmetries on the worldsheet.
Intriguingly, it has been recently found that these same BMS$_3$ symmetries arise in the formulation of the ambi-twistor string [@Casali:2016atr; @Casali:2017zkz]. Ambitwistor strings [@Mason:2013sva] are a variant of the original twistor string theory that provide a basis for understanding of the Cachazo-He-Yuan (CHY) formula for scattering of massless fields in quantum field theories [@Cachazo:2013hca]. Modular invariance of BMS constructed earlier in [@Bagchi:2012xr; @Bagchi:2013qva] would find its uses here as well, as has been noted in [@Casali:2017zkz].
Finally, BMS$_3$ algebras are isomorphic to 2d Galilean CFTs [@Bagchi:2010zz]. In a manner similar to relativistic CFTs, Galilean CFTs [@Bagchi:2009my; @Bagchi:2009pe] would govern renormalization group flow fixed points for Galilean invariant quantum field theories. So the bootstrap programme we have initiated in [@Bagchi:2016geg], augmented by constraints arising out of modularity, would help in charting out the allowed space of GCFTs and thereby Galilean field theories.
[*[A short summary of the paper]{}*]{}
In this paper, we formulate the construction of characters of the BMS$_3$ algebra in the highest weight representations. We show two different ways of arriving at the formula for the character in these highest weight representation, one of which follows from observations in the algebra and the other relies on the construction of the Gram matrix of inner products. As a robust cross-check of our answers, we reproduce the same characters in singular limits of 2d CFT answers. We then comment at length on the surprising relation of the characters in the highest weight representations of the BMS$_3$ with those of the very different induced representations computed earlier in [@Oblak:2015sea] and reproduced by a calculation of 1-loop determinants in 3d flat space [@Barnich:2015mui]. Through our analysis, we find that the character makes sense as a quantity that counts the number of possible states at a given level in the 2d field theory. This is a strong suggestion that the BMS-Cardy formula which accounts for the entropy of cosmological horizons in 3d flat spacetimes, could have a microscopic origin along the lines of the Strominger-Vafa construction [@Strominger:1996sh].
[*[Outline of the paper]{}*]{}
We start in Sec 2 with a brief review of the building blocks of the holographic correspondence in 3d asymptotically flat space. In Sec 2.1, we discuss the properties of the putative 2d field theory invariant under the BMS$_3$ algebra. The representation of interest, the highest weight representation, is introduced and elaborated on. We also touch upon correlation functions, the construction of the partition function and the BMS-Cardy formula for the counting of states. In Sec 2.2, we move on to aspects of bulk 3d physics and introduce Flat Space Cosmologies (FSC) of asymptotically flat spacetime and comment on how the BMS-Cardy formula captures the entropy of the cosmological horizon of the FSCs.
In Sec 3, we derive the character formula for these highest weight representations, first from the commutation relations of the underlying algebra alone and then by looking at the Gram matrix of inner products and constructing the trace of the operator from there. The Gram matrix for the BMS throws up some interesting structure, the details of which we delve into in the two appendices at the end of the paper.
In Sec 4, we reproduce the character formula earlier obtained by intrinsically BMS methods, by singular limits from 2d CFT. We remind the reader of the two limits, the non-relativistic and the ultra-relativistic, that take one from the two copies of the Virasoro algebra to the BMS$_3$ algebra. The character formula is reproduced by both these limits. We point out why this is a big surprise and then go on to offer an explanation based on a novel automorphism in the parent 2d CFT.
In Sec 5, as an application of the characters obtained, we derive a formula for the density of primary states, based on the deformed modular properties of the BMS-invariant field theory. We construct the partition function based on the characters of the highest weight representations and then impose modularity on this to obtain a BMS-Cardy formula for just the primaries of the field theory. In the holographic limit, this matches with the usual BMS-Cardy formula and thus we can say that the majority of the states contributing to the entropy of FSCs are BMS primaries. We conclude in Sec 6 with a summary of the results and a discussion of future directions of work.
BMS$_3$ holography: a lightning review
======================================
In this section, we briefly review some important aspects of Minkowskian holography in 3d bulk and 2d boundary case.
Aspects of dual 2d field theory
-------------------------------
The asymptotic symmetry group for Einstein gravity in 3d asymptotically flat spacetimes at its null boundary is given by the BMS group, the associated algebra of which is given by [@Barnich:2006av] && \[L\_n, L\_m\] = (n-m) L\_[n+m]{} + [c\_L]{}\_[n+m, 0]{} (n\^3 - n) && \[L\_n, M\_m\] = (n-m) M\_[n+m]{} + [c\_M]{}\_[n+m, 0]{} (n\^3 - n), && \[M\_n, M\_m\] = 0. \[bms3\] As motivated in the introduction, we would like to construct the notion of 3D flat holography by demanding that there exists a putative dual 2D field theory living on the null boundary that inherits this asymptotic BMS$_3$ algebra as its underlying symmetry [@Bagchi:2010zz]. This 2d field theory is then used to reproduce gravitational physics in the 3D asymptotically flat spacetime. See [@Bagchi:2016bcd; @Riegler:2016hah] for a review of important work in this direction.
Highest weight representations {#highest-weight-representations .unnumbered}
------------------------------
The states of the gravitational theory form representations of the underlying symmetry algebra. Again taking a cue out of holography in AdS and also its extensions to higher spins and dS spacetimes, we will consider the highest weight representation of BMS$_3$. We label the states of the dual 2d theory with the centre of the algebra which turns out to be $L_0$ and $M_0$. L\_0 |, = |, , M\_0 |, = |, . We wish to build on our CFT intuition and thus define BMS primary states as the states $|{\Delta}, \xi{\rangle}$ which have the lowest value of ${\Delta}$ for a given $\xi$. Since acting $L_n$ and $M_{n}$ on $|{\Delta}, \xi{\rangle}$ lowers the eigenvalue of $L_0$ by $n$ L\_0 L\_[n]{} |, = (-n) L\_[n]{} |, ,L\_0 M\_[n]{} |, = (-n) M\_[n]{} |, , we would impose that L\_n |, = M\_n |, = 0 n>0. It will be particularly convenient (but not essential) to have the BMS analog of the state-operator correspondence and hence we will demand that the states $|{\Delta}, \xi{\rangle}$ are created by acting the primary field $\phi_{{\Delta},\xi}$ on the vacuum \_[,]{}(0,0)|0= |,. Here the vacuum is the one which is annihilated by the global sub-algebra $\{L_{0, \pm1}, M_{0, \pm1} \}$, which is the Poincare sub-algebra $iso(2,1)$ of BMS$_3$.
We can increase the eigenvalue of $L_0$ by acting the raising operator $L_{-n}$ and $M_{-n}$ on the BMS primary states. The set of all states obtained from $|{\Delta},
\xi \rangle$ and their linear combination is called the BMS module for $|{\Delta}, \xi \rangle$. We will denote this module by $\mathcal{B}(c_L,c_M,\Delta,\xi)$. Then the Hilbert space of the BMS theory is the direct sum of the BMS module of all primaries present in the theory \_[BMS]{}(c\_L,c\_M)= (c\_L,c\_M,,). States in the module has the general form L\_[-1]{}\^[k\_1]{}L\_[-2]{}\^[k\_2]{}....L\_[-l]{}\^[k\_l]{} M\_[-1]{}\^[q\_1]{}M\_[-2]{}\^[q\_2]{}....M\_[-r]{}\^[q\_r]{}|, L\_M\_|, , where $\vec{k}=(k_1,k_2,....,k_l)$ and $\vec{q}=(q_1,q_2,....,q_r)$ and its $L_0$ eigenvalue is given by L\_0 L\_M\_|, = (N + ) L\_M\_|,, N=\_[i]{}ik\_i + \_j jq\_j. $N$ is called the level of the state and states in the BMS module are grouped according to their level. For example, level 0 consists of the BMS primary $|\Delta,\xi\rangle$ and level 1 consists of $L_{-1}|\Delta,\xi\rangle$ and $M_{-1}|\Delta,\xi\rangle$. We have given the states upto level 3 in table below.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Level States
------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
N=0 $|\Psi_{1}\rangle=|\Delta,\xi\rangle$
N=1 $|\Psi_{1}\rangle=L_{-1}|\Delta,\xi\rangle,\,\,\,|\Psi_{2}\rangle=M_{-1}|\Delta,\xi\rangle$
N=2 $\begin{array}{l}
|\Psi_{1}\rangle=L_{-1}^{2}|\Delta,\xi\rangle,\,\,\,|\Psi_{2}\rangle=L_{-2}|\Delta,\xi\rangle,\,\,\,|\Psi_{3}\rangle=L_{-1}M_{-1}|\Delta,\xi\rangle,\,\,\, |\Psi_{4}\rangle = M_{-2}|\Delta,\xi\rangle\\
|\Psi_{5}\rangle=M_{-1}^{2}|\Delta,\xi\rangle
\end{array}$
N=3 $\begin{array}{l}
|\Psi_{1}\rangle=L_{-1}^3|\Delta,\xi\rangle,\,\,\,|\Psi_{2}\rangle=L_{-1}L_{-2}|\Delta,\xi\rangle,\,\,\,|\Psi_{3}\rangle=L_{-1}^2M_{-1}|\Delta,\xi\rangle,\\
|\Psi_{4}\rangle=L_{-3}|\Delta,\xi\rangle,\,\,\,|\Psi_{5}\rangle=L_{-2}M_{-1}|\Delta,\xi\rangle,\,\,\,|\Psi_{6}\rangle=L_{-1}M_{-2}|\Delta,\xi\rangle,\\
|\Psi_{7}\rangle=M_{-3}|\Delta,\xi\rangle,\,\,\,|\Psi_{8}\rangle=L_{-1}M_{-1}^2|\Delta,\xi\rangle,\,\,\,|\Psi_{9}\rangle=M_{-1}M_{-2}|\Delta,\xi\rangle,|\Psi_{10}\rangle=M_{-1}^3|\Delta,\xi\rangle
\end{array}$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: States in the BMS module upto level 3[]{data-label="BMS_states"}
We will denote the number of states at level $N$ by $\widetilde{\dim}_{N}$. This will be equal to the number of partitioning the integer $N$ using two color (corresponding to the $L$’s and $M$’s). Correspondingly, given a state at level $N$, we can separate out the contribution of the $L$’s and $M$’s to the level as N=n\_L+n\_M,n\_L = k\_1+2k\_2+....+lk\_l,n\_M=q\_1+2q\_2+...sq\_s. Then it can be seen that the number of states at level $N$ is \_[N]{}=\_[n\_L,n\_M,n\_L+n\_M=N]{} p(n\_L)p(n\_M) = \_[m=0]{}\^[N]{} p(N-m)p(m), \[dimN\] where $p(n)$ is the number of ways to partition an integer $n$. As $L_{-1}$ and $M_{-1}$ annihilate the vacuum, we will not have any states containing these generator in the BMS module of the vacuum. In this case, the number of states is given by \_[N]{}([vac]{})&=&\_[n\_L,n\_M,n\_L+n\_M=N]{} (p(n\_L)-p(n\_L-1))(p(n\_M)-p(n\_M-1)) &= &\_[m=0]{}\^N (p(N-m)-p(N-m-1))(p(m)-p(m-1)). \[dimN\_vac\]
Gram matrix of BMS module {#gram-matrix-of-bms-module .unnumbered}
-------------------------
We can construct a matrix by taking inner products of the states in the BMS module K\_[ij]{} = \_i |\_j . These are the Gram matrices of the BMS module. Here $|\Psi_i{\rangle}$ are the states of the BMS module, the first few of which are listed in Table \[BMS\_states\]. The inner product is defined through the Hermiticity properties of the generators L\_n\^= L\_[-n]{}, M\_n\^= M\_[-n]{} Since states belonging to different levels are orthogonal to each other, this matrix is block diagonal. So, we can study them for each level separately and we use the notation $K^{(N)}$ for the matrix at level $N$. For level 1 we have two states and the Gram matrix $K^{(1)}$ is a $2\times2$ matrix given by $$\begin{aligned}
K^{(1)} = \left[ \begin{array}{cc}
\langle\Delta,\xi|L_{1}L_{-1}|\Delta,\xi\rangle & \langle\Delta,\xi|L_{1}M_{-1}|\Delta,\xi\rangle\\
\langle\Delta,\xi|M_{1}L_{-1}|\Delta,\xi\rangle & \langle\Delta,\xi|M_{1}M_{-1}|\Delta,\xi\rangle
\end{array} \right]
&= \left [ \begin{array}{cc}
2\Delta & 2\xi\\
2\xi & 0
\end{array} \right].
\label{GM_level1}\end{aligned}$$ For level 2 the Gram matrix $K^{(2)}$ is a $5\times 5$ matrix given by K\^[(2)]{} = . \[GM\_level2\] Note that the matrix $K^{(1)}$ and $K^{(2)}$ has a particular triangular structure i.e., non-zero anti-diagonal elements and all matrix entries on the right hand side of the anti-diagonal line are zero. This is due to the particular way we order the basis states in Table \[BMS\_states\]. It is helpful to know the structure of the Gram matrix. For example, we can see that due to the triangular structure we have just mentioned, the determinant of $K^{(1)}$ and $K^{(2)}$ are simply given by the product of their anti-diagonal elements. We explore in detail the structure of Gram matrix for general levels in Appendix \[appendix\_1\]. This will in turn be used to find the character of the BMS module in Sec \[innerprod\].
Correlation functions, partition function and modular invariance {#correlation-functions-partition-function-and-modular-invariance .unnumbered}
----------------------------------------------------------------
The BMS$_3$ invariant field theories that we propose as dual field theories to asymptotically 3d flat spacetimes, live on its null boundary. These theories are thus defined on a spacetime metric that is degenerate: ds\^2\_[\^+]{} = 0 du\^2 + d\^2, where $u$ is the null-direction and $\theta$ is the angular co-ordinate at null infinity. The topology of the boundary is ${\rm I\!R}_u \times \mathbb{S}^1$. The vector fields that represent the BMS$_3$ algebra on this boundary are given by: L\_n = e\^[in ]{} ( \_- i n u \_u ), M\_n = e\^[in ]{} \_u We can fix the 2 and 3 point correlation functions of BMS primary operators upto constants by considering the action of only the Poincare sub-algebra $\{L_{0, \pm1}, M_{0, \pm 1} \}$ [@Bagchi:2013qva]. The expression for the 2-point function for primary operators with weights $(\Delta, \xi)$ turns out to be [@Bagchi:2013qva] G\^[(2)]{}\_(\_1, u\_1, \_2, u\_2) = C\_1 ( 2 )\^[-2]{} ([- u\_[12]{} ]{}). One can glue together the two ends of the boundary cylinder to define the theory on a torus and here is where the notion of modular invariance comes in. We will define the partition function of the BMS field theory by \_ (, ) = e\^[2 i L\_0 ]{} e\^[2 i M\_0]{} = \_[, ]{} d(, ) e\^[2 i ]{} e\^[2 i ]{}, where $d(\Delta, \xi)$ denotes the density of states. The BMS version of modular invariance [@Bagchi:2013qva] reads , . This can be derived as a limit from a parent CFT. We will have more to say about this later in the paper.
The invariance of the partition function under BMS modular transformations, specifically the S-transformation, leads to the BMS-Cardy formula by relating the low-energy spectrum to the high energy spectrum. The BMS-Cardy formula is given by [@Bagchi:2012xr]: \[bmscardy\] S\_\^[(0)]{} = d(, ) = 2( c\_L + ). This is the leading contribution to the entropy in the saddle-point approximation of the integral which gives the density of states. One can also readily compute the next-to-leading piece, which are logarithmically suppressed. This gives the total entropy to be[[^2]]{}: S = S\_\^[(0)]{} + S\_\^[(1)]{} = 2 (c\_L + ) - () + Further corrections can be found systematically. Actually, the exact integral giving the density of states should be computable analytically following [@Loran:2010bd] [[^3]]{}.
3d bulk physics
---------------
We have already stated that a canonical analysis of 3d Einstein gravity with asymptotically flat boundary conditions leads to the BMS$_3$ algebra [(\[bms3\])]{} on its null boundary. These asymptotically flat 3d spacetimes are characterised by metrics of the form [@Barnich:2010eb; @Barnich:2012rz] ds\^2 = () du\^2 - 2 du dr + du d+ r\^2 d\^2 where $u=t-r$ is the retarded time. $\Theta(\phi)$ and $\Xi(\phi)$ are arbitrary functions labelling the solutions called the mass aspect and angular momentum aspect respectively.
Flat Space Cosmologies {#flat-space-cosmologies .unnumbered}
----------------------
For the zero mode solutions, the above arbitrary functions just become the mass $(M)$ and the angular momentum $(J)$ upto constants: $\Theta(\phi) = M$ and $\Xi(\phi)=J/2$ and hence the metric takes the form: \[g1\] ds\^2 = M du\^2 - 2 du dr + J du d+ r\^2 d\^2 For non-zero $J$, these solutions are obtained as orbifolds of 3d Minkowski spacetime quotiented by a boost and a translation. The solutions are called shifted-boost orbifolds [@Cornalba:2003kd] and also flat space cosmologies (FSC) [@Bagchi:2013lma]. These are the analogues of non-extremal BTZ black holes in AdS$_3$. Together with their $J=0$ cousins, the boost orbifold, and the $M=0, J=0$ member, the null orbifold, these form the $M\geq0$ zero mode sector of 3d asymptotically flat gravity. The solution space is depicted in Fig 1(b). For convenience, we also give the zero mode solutions in AdS$_3$ in Fig 1(a)[[^4]]{}. In this case, angular deficit is the region bounded by the two lines $\ell M = -|J|$ and the parabola $\ell M=-\frac{l}{8G}-\frac{2G}{l}J^2$, and the range of $J$ given by $|J|\leq \frac{\ell}{4G}$.
From the figure above, it seems that FSCs can be understood as a singular limit of the non-extremal BTZ black hole. To understand this, we start off with the global AdS$_3$ metric: ds\^2 = - (1+ ) dt\^2 + [(1+ )]{}\^[-1]{} [dr\^2]{}+ r\^2 d\^2 Here $\ell$ is the AdS radius. To obtain global flat space from AdS, we need to take $\ell \to \infty$. Now consider doing the same on the non-extremal BTZ metric: ds\^2 = - dt\^2 + dr\^2 + r\^2(d- dt)\^2. Here $r_\pm$ are the outer and inner horizons related to the mass and angular momentum of the BTZ black hole by r\_=
\[fig\]
When we consider the singular flat space limit on the BTZ, we are faced with an apparent conundrum. There are no black holes in 3d flatspace. But following our nose, the solution becomes immediate. When we take $\ell \to \infty$, the outer horizon goes to infinity, while the inner horizon stays put: r\_+ = \_+ , r\_[-]{} J = r\_0 The resulting metric becomes \[g2\] ds\_\^2 = \_+\^2 dt\^2 - dr\^2 + 2 \_+ r\_0 ddt + r\^2 d\^2.
The entire spacetime becomes the inside of the outer horizon of the original BTZ black hole. The radial and temporal directions switch roles and hence this is a time dependent cosmological solution. The Penrose diagram of this spacetime is given in Fig. 2. With appropriate rescaling of coordinates, it is easy to see that [(\[g2\])]{} reduces to [(\[g1\])]{}.
These FSCs have many interesting properties, the most striking of which is the creation of an FSC by a Hawking-Page like phase transition from empty flat spacetime [@Bagchi:2013lma]. Another interesting property, viz. relating FSC on the two separate null boundaries of flat spacetime, $\mathcal{I}^+$ and $\mathcal{I}^-$, is explored in [@Prohazka:2017equ].
Entropy and bulk-boundary matching {#entropy-and-bulk-boundary-matching .unnumbered}
----------------------------------
FSCs are time-dependent solutions with cosmological horizons. $r=r_0$ is the location of the cosmological horizon and one can associate a Bekenstein-Hawking entropy to it: S\_ = = For the excited state corresponding to the BTZ blackhole in a 2d CFT, the weights are given by [@Kraus:2006wn] h = (M + J) + , [[|h]{}]{}= (M - J) + , where $c=\bar{c}=\frac{3\ell}{2G}$ is the Brown-Henneaux central charge for the 2d CFT [@Brown:1986nw]. Similarly, the BMS weights of the FSC are = M + , = J + For 3d Einstein gravity, we have $c_L=0, c_M = 1/4G$. In the limit of large weights, we get \_ = M, \_ = J. We plug these values into the BMS-Cardy formula [(\[bmscardy\])]{}, and we obtain a perfect matching: S\^[(0)]{} = 2\_ = = S\_ The logarithmic corrections to this also turn out to be of a form expected from the gravitational analysis. For more details, the reader is referred to [@Bagchi:2013qva]. We should also point out here that the BMS-Cardy formula has also been derived as a limit from the inner horizon Cardy formula [@Castro:2012av] in [@Riegler:2014bia; @Fareghbal:2014qga]. This has been further used to compute entanglement entropy [@Bagchi:2014iea; @Basu:2015evh] in BMS-invariant 2d field theories via the so-called Rindler method in [@Jiang:2017ecm].
Highest Weight Characters for BMS
=================================
A torus can be obtained by gluing two ends of a cylinder. We can also twist the cylinder by an angle and then glue the two ends. Then the partition function on a torus twisted by an angle $\theta$ is given by e\^[-H+iP]{} , where H is the Hamiltonian which generate transformation along the length of the torus and P generate transformation along the circumference of the torus. If we mapped the cylinder to the plane, we can rewrite $H$ and $P$ in terms of the generator $L_0$ and $M_0$ on the plane e\^[-H+iP]{} = [Tr]{}e\^[2i(L\_[0]{}-c\_[L]{}/2)]{}e\^[2i(M\_[0]{}-c\_[M]{}/2)]{}Z\_(,).
Formula for character and partition function
--------------------------------------------
Let us first briefly recall the definition of trace of a linear operator $\hat{O}$ acting on a vector space $V$. If we choose the basis of $V$ to be $|\Psi_i\rangle$, the action of $\hat{O}$ on any basis is given by |\_i= \_j O\_[ij]{}|\_j. Trace of the operator $\hat{O}$ over $V$ is defined as sum of the diagonal elements \_i O\_[ii]{}. \[trace\_O\] For the BMS partition function, $\hat{O}=e^{2\pi i\sigma (L_{0}-c_L/2)}e^{2\pi i\rho (M_{0}-c_M/2)}$, and $V$ is the Hilbert space $\mathcal{H}_{BMS}(c_L,c_M)$ of the BMS theory. As the operator $\hat{O}$ does not mix states belonging to different BMS modules, we can take trace over each module separately. This give us the character of the module \_[(c\_[L]{},c\_[M]{},,)]{}(,) & = & [Tr]{}\_[,]{}e\^[2i(L\_[0]{}-c\_[L]{}/2)]{}e\^[2i(M\_[0]{}-c\_[M]{}/2)]{}, where ${\rm Tr}_{\Delta,\xi}$ means trace over the states belonging to the module $ \mathcal{B}(c_L,c_M,\Delta,\xi)$. The partition function is then given by summing over the characters of the primary fields in the theory Z\_(,) = \_[, ]{} D(,) \_[(c\_[L]{},c\_[M]{},,)]{}(,), where $D(\Delta,\xi)$ is multiplicity or density of the primaries with weight $(\Delta,\xi)$.
[*[Trace at level 1]{}*]{}
Now, let us try to find an expression for the character $\chi_{(c_{L},c_{M},\Delta,\xi)}(\sigma,\rho)$. As we just stated above, $\hat{O}$ does not mix states with different levels, and hence we will consider the trace for each level separately. Let us first look at level 1 which consists of the states $L_{-1}|\Delta,\xi\rangle$ and $M_{-1}|\Delta,\xi\rangle$. First of all, since all the states in the BMS modules are eigenstates of $L_0$, we have && L\_[0]{}L\_M\_|,=(+N)L\_M\_|,\
&& e\^[2iL\_[0]{}]{}L\_M\_|,=e\^[2i(N+)]{}L\_M\_|,. \[act\_L0\] On the other hand, for $M_0$, we have && e\^[2iM\_[0]{}]{}M\_[-1]{}|,= e\^[2i]{}M\_[-1]{}|,,\
&& e\^[2iM\_[0]{}]{}L\_[-1]{}|,= (2i)e\^[2i]{}M\_[-1]{}|,+e\^[2i]{}L\_[-1]{}|,. Combining all these we have (
[c]{} L\_[-1]{}|,\
M\_[-1]{}|,
)=e\^[-2i(+)]{}e\^[2i(1+)]{}e\^[2i]{}(
[cc]{} 1 & 2i0 & 1
)(
[c]{} L\_[-1]{}|,\
M\_[-1]{}|,
). So, the trace over level 1 is given by \_[,]{}\_[1]{} = 2 e\^[-2i(+)]{}e\^[2i(1+)]{}e\^[2i]{}. The factor of two is the number of states that we have in the BMS module at the first level.
[*[Trace at general level]{}*]{}
It is straightforward to find the diagonal elements of the operator $\hat{O}$ for general level. If we take any state $L_{\vec{k}}M_{\vec{q}}|\Delta,\xi\rangle$ at level $N$, the action of $\hat{O}$ on this state is given by L\_M\_|,&=& e\^[-2i(+)]{} e\^[2iL\_[0]{}]{} e\^[2iM\_[0]{}]{} L\_M\_|,\
&& = e\^[-2i(+)]{}e\^[2i(N+)]{}(\[e\^[2iM\_[0]{}]{},L\_M\_\]|,+ L\_M\_e\^[2iM\_[0]{}]{}|,)\
&&= e\^[-2i(+)]{}e\^[2i(N+)]{}(\[e\^[2iM\_[0]{}]{},L\_M\_\]|,+ e\^[2i]{}L\_M\_|,) \[act\_O\] Since commutator of $M_0$ with $L_n$ changes $L_n$ to $M_n$, i.e. $[M_0,L_n]=-nM_n$, the states $[e^{2\pi i\rho M_{0}},L_{\vec{k}}M_{\vec{q}}]|\Delta,\xi\rangle$ will not contain $L_{\vec{k}}M_{\vec{q}}|\Delta,\xi\rangle$. Therefore, from we have L\_M\_|,&=& e\^[-2i(+)]{}e\^[2i(N+)]{}() && + e\^[-2i(+)]{}e\^[2i(N+)]{}e\^[2i]{}L\_M\_|,. So, the diagonal elements of $\hat{O}$ are all the same and given by $e^{-2\pi i(\sigma \frac{c_{L}}{2}+\rho \frac{c_{M}}{2})}e^{2\pi i\sigma(N+\Delta)}e^{2\pi i\rho \xi}$. Then the trace at level $N$ is simply given by \_[,]{}\_[N]{} = \_[i=1]{}\^[\_N]{} O\_[ii]{} &=& \_[i=1]{}\^[\_N]{} e\^[-2i(+)]{} e\^[2i(N+)]{}e\^[2i]{} &=& \_N e\^[-2i(+)]{} e\^[2i(N+)]{}e\^[2i]{}, \[trace\_level\_N\] where $\widetilde{\dim}_N$ is the number of linearly independent descendant states at level $N$. Hence \_[(c\_[L]{},c\_[M]{},,)]{}(,) &=& \_N [Tr]{}\_[,]{}\_N =e\^[-2i(+)]{}e\^[2i(+)]{}\_[N]{}\_[N]{}e\^[2iN]{}. \[BMS\_character\] Substituting the expression for $\widetilde{\dim}_{N}$ given by in the above formula, we have \_[(c\_[L]{},c\_[M]{},,)]{}(,) &=& e\^[-2i(+)]{}e\^[2i(+)]{}\_[N]{}\_[n\_L+n\_M=N]{} p(n\_L) p(n\_M)e\^[2i(n\_L+n\_M)]{}\_[(c\_[L]{},c\_[M]{},,)]{} &=& e\^[-2i(+)]{}e\^[2i(+)]{} \_[n\_L]{} p(n\_L) e\^[2in\_L]{} \_[n\_M]{} p(n\_M) e\^[2in\_M]{}. \[BMS\_character2\] Using the generating function of partition numbers and the definition of the Dedekind eta function \_[n=1]{}\^=\_[n=0]{}\^ p(n) x\^n, ()=e\^\_[n=1]{}\^(1-e\^[2i n]{}), we can rewrite as \_[(c\_[L]{},c\_[M]{},,)]{}(,) = . \[BMS\_character3\] The above formula is for non-vacuum states. For the vacuum $(\Delta=0,\xi=0)$, we have to use for number of states giving us \_[(c\_[L]{},c\_[M]{},0,0)]{}(,) &=& (1-e\^[2i ]{})\^2. Finally, the partition function is given by Z\_(,) &&= \_[,]{} D(,) \_[(c\_[L]{},c\_[M]{},,)]{}(,)\
&& = ((0,0)(1-e\^[2i ]{})\^2+\_[, 0]{} D(,) e\^[2i(+)]{})\[partition\_function\] where we use the notation $\tilde{D}(0,0)$ for the density of vacuum state. If the vacuum is non-degenerate we can simply take this to be 1. If we introduce $D(0,0)=\tilde{D}(0,0)(1-e^{2\pi i \sigma})^2$, the above formula can be rewritten as Z\_(,) &=& \_[, ]{} D(,) e\^[2i(+)]{}. \[full\_partition\]
Character in terms of inner product {#innerprod}
-----------------------------------
If a vector space is equipped with an inner product, we can write the trace of an operator using this inner product. Let $K_{ij}$ be the Gram matrix formed from the inner product of the basis states $|\Psi_i\rangle$’s K\_[ij]{}=\_i|\_j, then trace of an operator $\hat{O}$ is given by = \_[i,j]{} K\^[ij]{}\_i||\_j, \[tr\_2\] where $K^{ij}$ is the matrix inverse of $K_{ij}$. It is easy to check that this is independent of the basis we use. Now, let us use to express the character of the BMS module $\mathcal{B}(c_L,c_M,\Delta,\xi)$. For the character, the operator $\hat{O}$ is once again $e^{2\pi i\sigma (L_{0}-c_L/2)}e^{2\pi i\rho (M_{0}-c_M/2)}$. Since states in different levels are orthogonal to each other we may write \_[(c\_[L]{},c\_[M]{},,)]{}(,) = \_[i,j,N]{} K\^[ij]{}\_[(N)]{}\_i\^[(N)]{}||\_j\^[(N)]{}\_[i,j,N]{} K\^[ij]{}\_[(N)]{}\_[ij]{}\^[(N)]{}, where $K_{(N)}$ is the inverse of the level $N$ Gram matrix $K^{(N)}$ and the $|\Psi_i^{(N)}\rangle$’s are basis states of the BMS module at level $N$. So, we can calculate the trace for each level separately and then add everything at the end. We will calculate this for level 1 and level 2 below and give a general proof for arbitrary level in Appendix \[appendix\_2\].
For level 1 we have two basis states (see Table ) with the Gram matrix $K^{(1)}$ given in . The inverse matrix $K_{(1)}$ is given by $$\begin{aligned}
K_{(1)} = \left[
\begin{array}{cc}
0 & \frac{1}{2 \xi } \\
\frac{1}{2 \xi } & -\frac{\Delta }{2 \xi ^2} \\
\end{array}
\right].\end{aligned}$$ For $\tilde{O}^{(1)}$ we have \^[(1)]{} = e\^[-2i(+)]{}\
where $\langle \ldots \rangle = \langle\Delta,\xi| \ldots |\Delta,\xi\rangle$. Hence we have \^[(1)]{} = e\^[-2i(+)]{} e\^[2i(+1)]{}e\^[2i]{}. Then the trace over level 1 is \_[i,j]{}K\^[ij]{}\_[(1)]{}\_[ij]{}\^[(1)]{} = 2 e\^[-2i(+)]{} e\^[2i(+1)]{}e\^[2i]{}. Here the pre-factor 2 is the number of states at level 1.
Now let us calculate the trace for level 2. Here we have five descendant states (see Table ) and from we can see that $K^{(2)}$ has a triangular structure with non-zero anti-diagonal elements and all the matrix elements on the right hand side of the anti-diagonal line being zero. Then we can see that $K_{(2)}$ will have opposite structure $$\begin{aligned}
K_{(2)} = \left[
\begin{array}{ccccc}
0 & 0 & 0 & 0 & \frac{1}{8 \xi ^2} \\
0 & 0 & 0 & \frac{1}{4 \xi +6 c_M} & K^{25} \\
0 & 0 & \frac{1}{4 \xi ^2} & K^{34} & K^{35} \\
0 & \frac{1}{4 \xi +6 c_M} & K^{43} & K^{44} & K^{45} \\
\frac{1}{8 \xi ^2} & K^{52} & K^{53} & K^{54} & K^{55} \\
\end{array}
\right],\end{aligned}$$ with all matrix elements on the left hand side of the anti-diagonal line being zero and the anti-diagonal elements being the inverse of that of $K^{(2)}$ K\^[ii]{}\_[(2)]{} = . \[rel\_1\] As for the matrix $\tilde{O}^{(2)}$ we have \^[(2)]{}= e\^[-2i(+)]{} e\^[2i(+2)]{}e\^[2i]{} . Note that this matrix is triangular in the same way as the Gram matrix $K^{(2)}$ and its anti-diagonal elements are proportional to that of $K^{(2)}$ by a common factor \_[ii]{}\^[(2)]{} = e\^[-2i(+)]{} e\^[2i(+2)]{}e\^[2i]{} K\_[ii]{}\^[(2)]{}. \[rel\_2\] Since the matrix $\tilde{O}^{(2)}$ and $K_{(2)}$ are triangular in the opposite way, only the anti-diagonal elements of both matrix will contribute to the trace $\sum_{i,j}K^{ij}_{(2)}\tilde{O}_{ij}^{(2)}$. Using this information along with and we have \_[i,j]{}K\^[ij]{}\_[(2)]{}\_[ij]{}\^[(2)]{} &=& \_[i=1]{}\^[5]{} K\^[ii]{}\_[(2)]{}\_[ii]{}\^[(2)]{} = e\^[-2i(+)]{} e\^[2i(+2)]{}e\^[2i]{} \_[d=1]{}\^[5]{} 1&=& 5 e\^[-2i(+)]{} e\^[2i(+2)]{}e\^[2i]{} , where the sum, $\sum_d 1$, just gave us number of states at level 2 which is 5. So, we can already guess that for arbitrary level $N$ \_[i,j]{}K\^[ij]{}\_[(N)]{}\_[ij]{}\^[(N)]{} = \_N e\^[-2i(+)]{} e\^[2i(+N)]{}e\^[2i]{}, \[trace\_level\_N\_2\] in agreement with . The upper/lower triangular structure of the above matrices don’t survive to higher order. But one can systematically organise them to arrive at the above expression for the trace at a general level. We will prove this in Appendix \[appendix\_2\]. Using the above formula, the character is the same as \_[(c\_[L]{},c\_[M]{},,)]{}(,) = \_[i,j,N]{}K\^[ij]{}\_[(N)]{}\_[ij]{}\^[(N)]{} = e\^[-2i(+)]{} e\^[2i(+ )]{} \_N \_N e\^[2iN]{}.
Limiting analysis from character of 2d CFT
==========================================
In this section we will re-derive our formula for BMS character by taking limits on the 2d CFT character.
The Virasoro character
----------------------
The holomorphic part of the character for the Virasoro module generated by the primary field with conformal weight $(h,\bar{h})$ is given by $$\begin{aligned}
\chi_{(c,h)}(\tau)& ={\rm Tr}_h\,q^{\mathcal{L}_{0}-c/24}.\end{aligned}$$ where $q=e^{2\pi i\tau}$. This is easy to calculate as all the states of the Virasoro module \_|h(\_[-1]{})\^[k\_[1]{}]{}(\_[-2]{})\^[k\_[2]{}]{}...(\_[-r]{})\^[k\_[r]{}]{} |hare eigenstates of $\mathcal{L}_0$ \_0 \_|h= (n+h) \_|h,n=\_l lk\_l. The number of states with eigenvalue $(h+n)$ (except for $h=0$) is given by the partition number $p(n)$. So we have \_[(c,h)]{}()=\_n p(n)q\^[(n+h)-c/24]{}. We have the same structure for the anti-hlomorphic part. The character for $(h,\bar{h})$ is then given by $$\begin{aligned}
\chi_{(c,h)}(\tau)\chi_{(\bar{c},\bar{h})}(\bar{\tau})
& = & q^{-c/24}\bar{q}^{-\bar{c}/24}\sum_{n}p(n)q^{n+h}\sum_{\bar{n}}p(\bar{n})\bar{q}^{\bar{n}+\bar{h}}.
\label{eq:cft_character}\end{aligned}$$ For the vacuum $(h=0,\bar{h}=0)$, we will not have the states containing $\mathcal{L}_{-1}(\bar{\mathcal{L}}_{-1})$ as these generators annihilate the vacuum state. So, the character for this case will be given by $$\begin{aligned}
\chi_{(c,0)}(\tau)\chi_{(\bar{c},0)}(\bar{\tau}) = q^{-c/24}\bar{q}^{-\bar{c}/24}\sum_{n}(p(n)-p(n-1))q^{n}\sum_{\bar{n}}(p(\bar{n})-p(\bar{n}-1))\bar{q}^{\bar{n}}.\end{aligned}$$
The two limits and the limiting characters
------------------------------------------
We expect to get the character of the BMS module by taking limit on the Virasoro character. We will do this analysis for non-vacuum primaries for both the non-relativistic (NR) and the ultra-relativistic (UR) limits. We first remind the reader of the two different contraction that gets one from the two copies of the Virasoro algebra to the BMS$_3$. The first one is the non-relativistic contraction: L\_n = Ł\_n + [|]{}\_n, M\_n = -(Ł\_n - [|]{}\_n) \[NR\] The name is derived from the fact that if one looks at the generators of the Virasoro algebra on the cylinder and takes a spacetime contraction where the speed of light is taken to infinity, these are the linear combination of generators required to give finite answers. One can also take the Carrollian ($c\to 0$) or the ultra-relativistic limit instead of the NR limit. In this case, the linear combination of generators are: L\_n = Ł\_n - [|]{}\_[-n]{}, M\_n = (Ł\_n + [|]{}\_[-n]{}) \[UR\] The mappings of the central terms, the weights and the modular parameters from the relativistic to the NR/UR theory are given in the equations below:
$$\begin{aligned}
&{\mbox{NR limit:}} \quad (c, \bar{c}) = 6 \left(c_{L}\mp\frac{c_{M}}{\epsilon}\right); \ (h,\bar{h}) = \frac{1}{2}\left(\Delta \mp \frac{\xi}{\epsilon}\right); \
(\tau, \bar{\tau}) = \pm \sigma-\epsilon\rho; \\
&{\mbox{UR limit:}} \quad (c, \bar{c}) = 6 \left(\pm c_{L} + \frac{c_{M}}{\epsilon}\right); \ (h,\bar{h}) = \frac{1}{2}\left(\pm \Delta + \frac{\xi}{\epsilon}\right); \ (\tau, \bar{\tau}) = \sigma \pm \epsilon\rho.\end{aligned}$$
In the NR limit, the CFT character reduces to $$\begin{aligned}
\chi^{NR}(\sigma,\rho)& = & \lim_{{\epsilon}\to0} \ q^{-c/24}\bar{q}^{-\bar{c}/24}\sum_{n}p(n)q^{n+h}\sum_{\bar{n}}p(\bar{n})\bar{q}^{\bar{n}+\bar{h}}\cr
& = & \lim_{{\epsilon}\to0} \ e^{-2\pi i(\sigma-\epsilon\rho)\frac{6}{24}(c_L-\frac{c_M}{\epsilon})}e^{-2\pi i(\sigma+\epsilon\rho)\frac{6}{24}(c_L+\frac{c_M}{\epsilon})}\cr
&&\times \sum_{n}p(n)e^{2\pi i(\sigma-\epsilon\rho)(n+\frac{1}{2}(\Delta-\frac{\xi}{\epsilon}))}\sum_{\bar{n}}p(\bar{n})e^{2\pi i(\sigma+\epsilon\rho)(\bar{n}+\frac{1}{2}(\Delta+\frac{\xi}{\epsilon}))}\cr
\implies \chi^{NR}(\sigma,\rho) &= & e^{-2\pi i(\sigma \frac{c_L}{2}+\rho \frac{c_M}{2})}e^{2\pi i(\sigma\Delta+\xi\rho)} \frac{1}{|\phi(\sigma)|^2}
\label{NRchar}.\end{aligned}$$ where $\phi(\sigma)=\prod_{n=1}^{\infty}(1-e^{2\pi i \sigma})$. Similarly, for the UR limit, we have $$\begin{aligned}
\chi^{UR}(\sigma,\rho)& = & \lim_{{\epsilon}\to0} \ q^{-c/24}\bar{q}^{-\bar{c}/24}\sum_{n}p(n)q^{n+h}\sum_{\bar{n}}p(\bar{n})\bar{q}^{\bar{n}+\bar{h}}\cr
& = & \lim_{{\epsilon}\to0} \ e^{-2\pi i(\sigma+\epsilon\rho)\frac{6}{24}(c_L+\frac{c_M}{\epsilon})}e^{-2\pi i(\sigma-\epsilon\rho)\frac{6}{24}(-c_L+\frac{c_M}{\epsilon})}\cr
&&\times \sum_{n}p(n)e^{2\pi i(\sigma+\epsilon\rho)(n+\frac{1}{2}(\Delta+\frac{\xi}{\epsilon}))}\sum_{\bar{n}}p(\bar{n})e^{2\pi i(\sigma-\epsilon\rho)(-\bar{n}+\frac{1}{2}(\Delta-\frac{\xi}{\epsilon}))}\cr
\implies \chi^{UR}(\sigma,\rho) & = & e^{-2\pi i(\sigma \frac{c_L}{2}+\rho \frac{c_M}{2})}e^{2\pi i(\sigma\Delta+\xi\rho)}\frac{1}{|\phi(\sigma)|^2}.\end{aligned}$$ The characters in the limit, viz. $\chi^{NR}(\sigma,\rho), \chi^{UR}(\sigma,\rho)$ are the same and are identical to what we have obtained using intrinsic methods in .
Equal limiting characters: the problem
--------------------------------------
At the outset, the fact that the characters in the two limits are the same and reproduce the answer in the intrnisic analysis is not surprising given that both limits brought us to the same algebra from the two copies of the Virasoro. But there is something deeply profound about the above statements. We shall try to address why the matching of answers in the two limits is extremely surprising and then go on to give some partial answer to the puzzle.
Characters are properties of the representations of a particular algebra and not the algebra per se. For finite groups, the characters of a certain representation are the trace of representative matrix. If two representations have equal characters, we know that these representations are linked by similarity transformations. The Virasoro characters [(\[eq:cft\_character\])]{} are constructed for the highest weight representations that are characterised by weights $(h, \bar{h})$. As a reminder, highest weight representations of the Virasoro are built on primary states that defined as \[vhw\] Ł\_0 | h, |[h]{}= h | h, |[h]{}, [|]{}\_0 | h, |[h]{}= |[h]{} | h, |[h]{}; Ł\_n | h, |[h]{}= 0, [|]{}\_n | h, |[h]{}= 0 n >0. The Virasoro modules are built by acting raising operators on these states. What about the BMS characters? In the first part of the paper, we constructed these in terms of highest weight representations as well.
So far, there seems to be no trouble. But let’s look at the two limits. The NR limit [(\[NR\])]{} maps Virasoro highest weight states to BMS highest weight states. [(\[vhw\])]{} L\_0 |, = |, , M\_0 |, = |, ; L\_n |, = 0 = M\_n |, n >0. Hence the fact that we reproduced the BMS highest weight characters as a NR limit of the Virasoro characters in [(\[NRchar\])]{} is very natural, and is a robust check of our previous analysis.
On the other hand, due to the mixing of positive and negative modes in the linear combination, the UR limit [(\[UR\])]{} definitely does not take the Virasoro highest weights to BMS highest weights. This in fact maps to something call the BMS induced representations [@Barnich:2014kra; @Barnich:2015uva; @Campoleoni:2016vsh; @Oblak:2016eij], which are defined by: [(\[vhw\])]{} L\_0 |, = |, , M\_0 |, = |, ; M\_n |, = 0 n 0. These representations are clearly different from the highest weight representations that we have talked about so far. In fact, the characters for these representations have been computed by group theoretic methods in [@Oblak:2015sea] (see also [@Garbarz:2015lua]). The UR limit thus gives us the character of the induced representation.
The disturbing aspect of the statements made above is the fact that the characters in the NR and UR limits are one and the same. This means that for the BMS$_3$ algebra, the highest weight representation and the induced representation have identical characters. What makes things even more disturbing is the question of unitarity of the representations. The major advantage of the induced representations is the fact that these are manifestly unitary [@Barnich:2014kra; @Barnich:2015uva; @Campoleoni:2016vsh; @Oblak:2016eij]. The highest weight representations, on the other hand, although extremely useful in various holographic applications, are explicitly non-unitary. A quick look at the equivalent of the Kac determinant at level one is enough to convince one of this. The Gram matrix at the first level is given by [(\[GM\_level1\])]{}. The determinant is K\^[(1)]{} = - 4 \^2. This is negative for all real values of $\xi$ and hence the representation is non-unitary for all $\xi>0$. The only chance of unitarity is $\xi=0$. It can be shown that this also needs to be combined with $c_M=0$ [@Grumiller:2014lna]. But one can also show that when one considers the sub-sector with $(\xi=0, c_M=0)$, there is a truncation of the algebra from BMS$_3$ to a single copy of the Virasoro algebra [@Bagchi:2009pe]. One of the principle reasons we are interested in the BMS$_3$ because of the potential connection to holographic physics in flatspace. For that $c_M\neq0$. Hence the $(\xi=0, c_M=0)$ is not a very interesting sub-sector[[^5]]{}.
So, we have seen that in a generic BMS-invariant theory, the highest weight representation is non-unitary whereas the induced representations are constructed to be unitary. Their characters are however identical. This is a source of great intrigue.
Hints of a solution
-------------------
One might assume that the very unexpected mapping between the two apparently inequivalent representations is inherently a property of the BMS$_3$ algebra, and this happens because of the vagaries of the limiting procedure from the 2d CFT. It has been argued that the reason that the NR and the UR limits give the same algebra starting out from two copies of the Virasoro is because there are only two directions in the 2d field theory and the process of contraction is blind to this.
But there is something more fundamental about this weird NR $\leftrightarrow$ UR mapping. This actually is not a property of the limit, but a strange intrinsic property of 2d CFT itself. Notice that the following operation: Ł\_n - Ł\_[-n]{}, c-c is an automorphism of the Virasoro algebra[[^6]]{}. Hence for two copies of the Virasoro algebra, if we perform the above operation on the anti-holomorphic sector, we get the following automorphism Ł\_n + [|]{}\_n Ł\_n - [|]{}\_[-n]{}, Ł\_n - [|]{}\_n Ł\_n + [|]{}\_[-n]{}, c|[c]{} c|[c]{} Without the factors of ${\epsilon}$, this is precisely the NR $\leftrightarrow$ UR swapping. This exchanges Ł\_0 + [|]{}\_0 Ł\_0 - [|]{}\_0 Hence the usual highest weight representation theory of the 2d CFT gets mapped to one where the $\bar{h}$ eigenvalue is not bounded from below, but bounded from above. The characters of these very different representations are the same because of this automorphism. We believe that the route to understanding the conflicting issues in the BMS representation theory is to carefully study this particular automorphism in the parent 2d CFT.
BMS-Cardy formula for primary states
====================================
We have obtained the characters for the BMS$_3$ algebra in the previous sections by a variety of methods. As we emphasised in the introduction and later in Sec 2, the BMS$_3$ algebra is central to understanding holography in 3d asymptotically flat spacetimes. Using our expressions for the characters found in the earlier sections, in this section we will obtain a Cardy like formula for density of primaries $D(\Delta,\xi)$ with large $\Delta$ and $\xi$.
We have in Sec 3 used the characters to write down the partition function of the BMS invariant field theory. In this section, we will use this and the modular invariance of the partition function $Z_{\mbox{\tiny{BMS}}}(\sigma,\rho)$ to arrive at an expression for the entropy of primary states. From , we have Z\_(,) = (,) \[partition\_tilde\] where $\tilde{Z}(\sigma,\rho) = \sum_{\Delta,\xi} D(\Delta,\xi) e^{2\pi i(\sigma\Delta+\rho\xi)}$. We can invert the above formula to give us the density of the primary states D(,) = dde\^[-2i(+)]{} (,). \[density\_ps\] Using the modular properties of the partition function and Dedekind eta function Z\_(,)=Z\_(-,), () = (-), we can deduce that (-,) &=& Z\_(-,)(-)\^2e\^[2i(-+ )]{}\
(,) &=& (-,)()e\^[2i(- ++)]{}. Substituting this in , we have D(,) &=& dd (-,)()e\^[f(,)]{}, \[density\_int\] where f(,)= 2i(- ++--). For large $(\Delta, \xi)$, we employ the saddle-point approximation to calculate the integral : D(,) (-,)()e\^[f(\_c,\_c)]{}, \[density\_saddle\] where $(\sigma_c,\rho_c)$ is the critical point of the function $f(\sigma,\rho)$: $\partial_{\sigma}f(\sigma_c,\rho_c)=0, \, \partial_{\rho}f(\sigma_c,\rho_c)=0.$ These are given by -+ (6 c\_L-1)+-= 0, -+-= 0 . This has two solutions which we denote by $(\sigma_0,\rho_0)$ and $(\tilde{\sigma}_0,\tilde{\rho}_0)$. For large $\Delta$ and $\xi$, they are given by \_[0]{} i,\_0i, \_0 -i,\_0-i, and we have to choose as $(\sigma_c,\rho_c)$ whichever solution maximizes $f(\sigma,\rho)$. We can see that f(\_0,\_0) && 2(c\_L-)+2, f(\_0,\_0) && -2(c\_L-)-2. When taking $\Delta$ and $\xi$ to large values, we assume that, we do it in such a way that \_[,]{} = , where $\gamma$ is a finite number. Then we have
f(\_0,\_0) && 2 (c\_L- + c\_M),\
f(,\_0) && -2 (c\_L- + c\_M).
$(\sigma_0,\rho_0)$ will be the maxima if $f(\sigma_0,\rho_0)>f(\tilde{\sigma_0},\tilde{\rho}_0)$. From the above equation, we can see that this will be the case when c\_L- + c\_M > 0. Likewise, $(\tilde{\sigma_0},\tilde{\rho}_0)$ is the maxima when -c\_L + - c\_M > 0 . Now, the saddle point approximation is valid only when $\tilde{Z}\left(-\frac{1}{\sigma_c},\frac{\rho_c}{\sigma_c^2}\right)$ is dominated by the vacuum. We will separately analyse what conditions we have to impose for this to happen for the two cases mentioned above.
Case 1: $(\sigma_0,\rho_0)$ is the maximum {#case-1-sigma_0rho_0-is-the-maximum .unnumbered}
------------------------------------------
The arguments of the function $Z(-\frac{1}{\sigma_0},\frac{\rho_0}{\sigma_0^2})$ are - = i, = i i . From and , we have (-,) = (0,0)(1-e\^[-2]{})\^2 + \_[(\^,\^)(0,0)]{} D(\^,\^)e\^[-2\^]{}e\^[-2 \^]{}. We assume that in our theory $L_0$ is bounded from below and hence $\Delta$’s takes only positive values. So, for large $\xi$ e\^[-2\^]{} 0, and we don’t want $e^{-2\pi \frac{1}{c_M}\sqrt{\frac{\xi}{2}} \lambda \xi^{\prime}}$ to diverge in this limit. This means that $\lambda$ has to be either a finite positive number or zero. In other words = -c\_L + c\_M 0. If this is satisfied, then $\tilde{Z}(-\frac{1}{\sigma_0},\frac{\rho_0}{\sigma_0^2})$ is dominated by the contribution from the vacuum (-,) (0,0).
Case 2: $(\tilde{\sigma}_0,\tilde{\rho}_0)$ is the maximum {#case-2-tildesigma_0tilderho_0-is-the-maximum .unnumbered}
----------------------------------------------------------
Now let us look at the conditions we have to impose so that $Z(-\frac{1}{\tilde{\sigma}_0},\frac{\tilde{\rho}_0}{\sigma_0^2})$ is dominated by the vacuum. The arguments of the function are - = -i, = i i . And from and , we have (-,) = (0,0)(1-e\^[2]{})\^2 + \_[(\^,\^)(0,0)]{} D(\^,\^)e\^[2\^]{}e\^[-2 \^]{}. Let us look at the second term. We can see that for large $\xi$, the factor $e^{2\pi \sqrt{\frac{2\xi}{c_M}}\Delta^{\prime}}$ diverges. So, we need at least the other factor, $e^{-2\pi \frac{1}{c_M}\sqrt{\frac{\xi}{2}} \lambda \xi^{\prime}}$, to vanish in this limit. In other words, $\tilde{\lambda}$ has to be a positive number -+c\_L - c\_M >0. We also have -c\_L +- c\_M >0. The above two inequalities does not have any common overlap. So, we can conclude that $Z(-\frac{1}{\tilde{\sigma}_0},\frac{\tilde{\rho}_0}{\sigma_0^2})$ cannot be dominated by the vacuum and thus the saddle point analysis is not useful for this case.
We conclude that the density of primaries with large $\Delta$ and $\xi$ satisfying $c_L-\frac{1}{6} + \gamma c_M > 0$ and $\frac{1}{6} - c_L + \gamma c_M > 0$, using , is given by D(,) (0,0)(2(c\_L-)+2+()).\[Dprim\] Notice that the leading piece of the entropy, which is obtained by taking a logarithm of the density of states, obtained from [(\[Dprim\])]{} equals S\_ = 2((c\_L-)+). This is the same as [(\[bmscardy\])]{} with a replacement $c_L \to c_L - 1/6$. The entropy obtained from primaries is clearly the principle part of the whole entropy and it is clear that these contribute the largest when one is looking at holographic applications.
To make contact with gravity in asymptotically flatspace, we recall that in 3d flat Einstein gravity $c_L=0$. The overlap of the inequalities thus reads $$\gamma c_M > \frac{1}{6}.$$ This is a perfectly acceptable range of values and the saddle-point analysis works well in this regime. Also, the holographic regime is given by the limit of large central charge, which in this context means large $c_M$. The contribution to the correction piece $$\tilde{S}= S_{\mbox{\tiny{total}}} - S_{\mbox{\tiny{Primary}}}= - \frac{\pi}{3} \sqrt{\frac{\xi}{2c_M}}$$ is clearly subleading in this limit. So the density of these BMS primaries captures the entropy of the flat space cosmologies rather well.
Conclusions
===========
Summary {#summary .unnumbered}
-------
To summarise, we constructed the characters for the highest weight representations of the BMS$_3$ algebra in two different ways at the beginning of this paper. Some of the details of the proof using the BMS Gram matrix are detailed in Appendices \[appendix\_1\] and \[appendix\_2\].
We then used two singular limits from 2d CFTs to reproduce our character formula from the well-known Virasoro characters. This led to a conundrum, since one of the limits (the non-relativistic one) took Virasoro highest representations to BMS highest weight representations, and the other (the ultra-relativistic one) took the Virasoro highest representations to the very different BMS induced representation. We attempted to explain this by alluding to a novel automorphism in the parent 2d CFT.
Finally, we used the form of the characters to construct the partition function and from there derived the density of BMS primary states in the spectrum. We were able to see, interestingly, the BMS primaries capture the principle part of the BMS-Cardy entropy, of e.g. the FSC solutions.
Future directions {#future-directions .unnumbered}
-----------------
As we have already emphasised, we wish to investigate this novel automorphism in the context of 2d CFTs. This, to the best of our knowledge, is the first time this rather peculiar automorphism has been noticed in literature [[^7]]{}. This should lead to a deeper understanding of the rather bizarre duality between non-relativistic and ultra-relativistic physics, and also should go a long way to understand the representation theory of the BMS$_3$ algebra. This should also lead to a clarification of which particular representation to use while constructing the field theory dual to 3d asymptotically flat space. The highest weight representations are very clearly more useful for computational purposes. It seems that because of this isomorphism, some answers we get are independent of the underlying representation. We wish to understand in detail to what extent this feature is true.
We investigate various structure related to the BMS Gram matrix in the two appendices. A very natural question is to investigate the equivalent of the Kac determinant for the BMS$_3$ algebra. We should be able to relate this to the null vectors of the BMS module investigated e.g. in [@Bagchi:2009pe]. This, unfortunately, does not seem as straightforward as it may seem. Our constructions so far only reproduce the most “singular" null vector (constructed out of $M_n$’s alone) for an arbitrary level. We are looking to solve this issue and understand whether there is a minimal series for the BMS$_3$ algebra like the Virasoro. In this context, it is interesting to point out that in [@Hijano:2018nhq] while generalising the monodromy method of calculating BMS blocks (beyond the LLLL or Poincare blocks as computed in [@Bagchi:2016geg; @Bagchi:2017cpu] and holographically in [@Hijano:2017eii]), some rather strange representations, which are reducible but indecomposable like logarithmic CFTs, were used. It may be of interest to reconsider these in the context of the BMS Kac determinant and matching of null states.
There are of course rather straight-forward generalisations of this work that we are interested in pursuing. We would like to find the characters of higher-spin versions of BMS [@Afshar:2013vka; @Gonzalez:2013oaa] (called BMW algebras!), and different supersymmetric versions of BMS$_3$ (e.g. the “homogeneous" [@Barnich:2014cwa; @Barnich:2015sca; @Bagchi:2016yyf; @Lodato:2016alv; @Lodato:2018gyp] and the “inhomogeneous" [@Lodato:2016alv; @Bagchi:2016yyf; @Bagchi:2017cte; @Lodato:2018gyp] algebras). Some of the above (the spin-3 case and the “homogeneous" superalgebra) have been computed for the induced representations in [@Campoleoni:2015qrh]. It is of interest to see how much of their analysis is reproduced in the highest weight analysis. It would be even more interesting to see if there are any departures from the answers of the induced representations. One expects to see some differences in the higher spin versions.
On a gravitational side, the one loop determinant of the bulk theory equals the character of the boundary symmetry [@Maloney:2007ud; @Giombi:2008vd; @David:2009xg]. This has been computed also for flat-spacetime in [@Barnich:2015mui], where the answer is exactly the character formula we have computed here. A generalisation of this analysis is the computation of one-loop determinants for higher spin theories in the asymptotically flat 3d bulk and also the 3d supergravity theory which has the homogeneous super-BMS algebras as its asymptotic symmetries [@Barnich:2014cwa; @Barnich:2015sca]. This has been performed in [@Campoleoni:2015qrh]. We would like to investigate the inhomogeneous super-algebra [@Lodato:2016alv; @Bagchi:2016yyf; @Bagchi:2017cte] arising out of the “twisted" supergravity construction of [@Lodato:2016alv].
One of the more ambitious programmes is to attempt a microscopic counting of states of the Flatspace Cosmologies, [*[[á]{} la]{}*]{} Strominger-Vafa [@Strominger:1996sh]. The fact that the character of the BMS module has a positive integral expansion in terms of the levels and clearly counts the number of states of the putative dual field theory, gives us hope of a more fundamental string theoretic understanding of the underlying degrees of freedom. A string/M theory embedding of the FSC was initially discussed in [@Cornalba:2002fi]. We wish to construct the equivalent of the D1-D5 CFT on the field theory side, and one of the first things to attempt is a contraction of the superalgebra and this should lead to a better understanding of the symmetries on the field theoretic side. It should turn out to be one of the different $\mathcal{N}=4$ Super-BMS algebras that can be constructed in close analogy with the homogeneous and inhomogeneous algebras discussed above. The systematic limit could also be a way to understand the brane construction analogue to the D1-D5 system.
Acknowledgements {#acknowledgements .unnumbered}
================
Discussions and correspondence with Rudranil Basu, Diptarka Das, Abhijit Gadde, Daniel Grumiller, and Gautam Mandal are gratefully acknowledged. This work was presented in TIFR Mumbai, SINP Kolkata and Durham University prior to publication. AB thanks the string theory groups at these places for interesting discussions.
AB also acknowledges the warm hospitality of the Department of Mathematics and Grey College at Durham University and the financial support of the Institute of Advanced Study, Durham through a Senior Research Fellowship (COFUNDed between Durham University and the European Union under grant agreement number 609412), during the final stages of this work.
This work is also partially supported by the following grants: DST-INSPIRE faculty award, SERB Early Career Research Award (ECR/2017/000873), DST-Max Planck mobility award, DST-BMWF India-Austria bilateral grant, Royal Society International Exchange grant, SERB extra-mural grant (EMR/2016/008037), SERB National Post Doctoral Fellowship PDF/2016/002166.
APPENDICES {#appendices .unnumbered}
==========
Structure of the BMS Gram matrix \[appendix\_1\]
================================================
In this appendix we will show that the Gram matrix can always be put in the following form if we order the basis states according to some rules which we will lay out. K\^[(N=)]{} = (
[ccccc]{} K\_[1,1]{} & & & & A\_[\_N]{}\
& & & & 0\
& & & &\
& A\_[2]{}\
A\_[1]{} & 0 & & & 0
) , \[oddmatrix\] K\^[(N=)]{}=(
[cccccccccc]{} K\_[1,1]{} & & & & & & & & & A\_[\_[N]{}]{}\
& & & & & & & & & 0\
& & & & & & & A\_[k+l]{} & &\
& & & D\_[1]{} & 0 & & 0\
& & & 0 & D\_[2]{} & & 0 & & &\
& & & & & & & & &\
& & & 0 & 0 & & D\_[l]{}\
& & A\_[k]{} & & & & & 0\
& & & & & & & & &\
A\_[1]{} & 0 & & & & & & & & 0
). \[evenmatrix\] We can see from the above figure that the difference between the Gram matrix for odd and even level is that for even level we have a diagonal matrix at the matrix. The dimension of this diagonal matrix is $p(N/2)$. With these structure for $K^{(N)}$, the inverse matrix $K_{(N)}$ will have the form K\_[(N=)]{} = (
[ccccc]{} 0 & & 0 & &\
0 & & & &\
& & & &\
&\
& & & & K\^[\_N,\_N]{}
) , \[oddmatrix\_in\] K\_[(N=)]{}=(
[cccccccccc]{} 0 & & & & & & 0 & & &\
& & & & & & & & &\
& & & & & & & & &\
& & & & 0 & & 0\
& & & 0 & & & 0 & & &\
0 & & & & & & & & &\
& & & 0 & 0 & &\
& & & & & & &\
& & & & & & & & &\
& & & & & & & & & K\^[\_N,\_N]{}
). \[evenmatrix\_in\]
Conditions for Non-zero BMS Inner Products
------------------------------------------
In order to find the ordering rules, we first have to find the conditions for inner products of states in the BMS module to be non-zero. In the usual eigenbasis of $L_0$, the linearly independent descendant states of a primary state $|\Delta, \xi \rangle$ are chosen to be $$L_{-i_1}^{l_1}L_{-i_2}^{l_2} \ldots L_{-i_r}^{l_r}M_{-j_1}^{m_1}M_{-j_2}^{m_2} \ldots M_{-j_s}^{m_s}|\Delta, \xi \rangle,$$ where $1\leqslant i_1 < i_2 <...< i_r$ and $1\leqslant j_1 < j_2 <...< j_s$. This is a descendant state at level $N= \sum_{p=1}^r i_p l_p + \sum_{q=1}^s j_q m_q $. In this basis, we can readily check the conditions for non-vanishing inner product between the basis vectors. Let us consider two generic states: $$L_{-i_1}^{l_1}L_{-i_2}^{l_2}...L_{-i_r}^{l_r}M_{-j_1}^{m_1}M_{-j_2}^{m_2}...M_{-j_s}^{m_s}|\Delta, \xi \rangle \quad \& \quad L_{-i'_1}^{l'_1}L_{-i'_2}^{l'_2}...L_{-i'_{r'}}^{l'_{r'}}M_{-j'_1}^{m'_1}M_{-j'_2}^{m'_2}...M_{-j'_{s'}}^{m'_{s'}}|\Delta, \xi \rangle.$$ There are two basic conditions that are required to be satisfied for non-zero inner products between these two states:
1. These two states must be at the same level \_[p=1]{}\^r i\_p l\_p + \_[q=1]{}\^s j\_q m\_q = \_[p=1]{}\^[r’]{} i’\_p l’\_p + \_[q=1]{}\^[s’]{} j’\_q m’\_q.
2. Inside the bra-ket of inner product, i.e., in the expression , |M\_[j’\_[s’]{}]{}\^[m’\_[s’]{}]{}...M\_[j’\_2]{}\^[m’\_2]{}M\_[j’\_1]{}\^[m’\_1]{}L\_[i’\_[r’]{}]{}\^[l’\_[r’]{}]{}...L\_[i’\_2]{}\^[l’\_2]{}L\_[i’\_1]{}\^[l’\_1]{}L\_[-i\_1]{}\^[l\_1]{}L\_[-i\_2]{}\^[l\_2]{}...L\_[-i\_r]{}\^[l\_r]{}M\_[-j\_1]{}\^[m\_1]{}M\_[-j\_2]{}\^[m\_2]{}...M\_[-j\_s]{}\^[m\_s]{}|, , for every $M_{j'_{q}}$ ($1\leqslant q \leqslant s'$) we must have combinations of L-operators that give (after using the commutation relations or just simply summing the L-indices) $L_{-j'_{q}}$. Similarly, for every $M_{-j_{q}}$ we must have combinations of L-operators that gives $L_{j_{q}}$ . Note that if we have more than one $M_{j'_{q}}$ we need separate $L_{-j'_{q}}$ for each of them. The same thing applies for $M_{-j_{q}}$. The combinations (or sets) of L-operators once used, are not further considered (all the elements of those sets) to find suitable combinations for remaining M-operators.
From the above two basic rules and working out some examples, we can derive some easy-to-work-with conditions for non-zero inner product. Before we give these rules, let us first introduce some notations and definitions which will makes things simpler. We associate two numbers $\alpha$ and $\beta$ for each basis states $L_{-i_1}^{l_1}L_{-i_2}^{l_2}...L_{-i_r}^{l_r}M_{-j_1}^{m_1}M_{-j_2}^{m_2}...M_{-j_s}^{m_s}|\Delta, \xi \rangle$ which are given by &=& = \_[p=1]{}\^r l\_p -\_[q=1]{}\^s m\_q, &=& = \_[p=1]{}\^r i\_pl\_p -\_[q=1]{}\^s j\_p m\_q. In particular, at level $N$, the value of $\alpha$ range from $N$ (associated with the basis state $L_{-1}^N|\Delta,\xi\rangle$) to $-N$ (associated with the state $M_{-1}^N|\Delta,\xi\rangle$). There will be symmetry in the sense that the number of states sharing the same value of $(\alpha,\beta)$ will be equal to the number of states with same value of $(-\alpha,-\beta)$. For example at level 5, there are two states $(L_{-2}^2M_{-1}|\Delta,\xi\rangle,\,L_{-1}L_{-3}M_{-1}|\Delta,\xi\rangle)$ with $(\alpha=1,\beta=3)$. Similarly there are two states $(L_{-1}M_{-2}^2|\Delta,\xi\rangle,\,L_{-1}M_{-1}M_{-3}M_{-1}|\Delta,\xi\rangle)$ with $(\alpha=-1,\beta=-3)$. It can be seen that these two sets of states are related by swapping $L$ and $M$. We will call this operation conjugation ,L M, and pairs like $L_{-2}^2M_{-1}|\Delta,\xi\rangle$ and $L_{-1}M_{-2}^2|\Delta,\xi\rangle$ which are related through conjugation to be a conjugate pair. So, conjugation flips the sign of $\alpha$ and $\beta$. We could also have self conjugate states i.e., states which remain the same under conjugation. The simplest example is $L_{-1}M_{-1}|\Delta,\xi\rangle$. These kind of states only exists for even level and have $(\alpha=0,\beta=0)$. Using these notations, the non-zero conditions for the inner product of basis states $|\Psi\rangle$ and $|\Psi^{\prime}\rangle$ are:
1. Inside the bra-ket of inner product, the total number of L-operators must not be less than the total number of M-operators. Otherwise, the inner product is zero |\^= 0, +\^<0. \[rule1\]
2. If the total number of L-operators is greater than that of M, the sum of L-indices must not be less than the sum of M-indices && +\^>0, +\^>0 .
3. The next rule is for the case when the total number of L-operators and M-operators are the same i.e., when $\alpha+\alpha^{\prime}=0$ + \^=0,[werequire]{}&& i)r=s’,r’=s, && ii) i\_p=j’\_p, l\_p=m’\_p,[for]{} 1p r, && iii)i’\_q=j\_q,l’\_q=m\_q,[for]{} 1q s . \[rule3\] This means that whenever the total number of L-operators and M-operators are the same, the two states must be a conjugate pair.
All other pairs of states not obeying the above rules give vanishing inner product. Using the above rules, we can readily deduce the conditions (not sufficient but necessary) for a generic state to have non-zero norms. These are && 0 ,\
&& >0,>0,\
&& = 0,. We have already mentioned that at odd level we could not have self conjugate states. So, at odd levels, all the states with equal length of L-string and M-string i.e., $\alpha=0$ have vanishing norm.
The ordering rules
------------------
At a particular level of a generic BMS module, we can now find the pairs of basis states that have vanishing inner product using the above rules. From this we can guess a method of arranging the basis states such that the resulting Gram matrix has a structure mentioned in the beginning of this appendix. These arrangement rules are given by:
1. Arrange the basis states in decreasing order of $\alpha$, keeping $L_{-1}^n|\Delta, \xi \rangle $ at the beginning and $M_{-1}^n|\Delta, \xi \rangle $ at the ending of the queue. Keep the states having the same $\alpha$ one after another as a bunch.
2. For states with the same value of $\alpha$, ordering is done in decreasing value of $\beta$.
3. If we have $r$ states with the same value of $(\alpha,\beta)\neq (0,0)$ and arrange them in the order $(|\Phi_1\rangle,\,\,|\Phi_2\rangle,....,|\Phi_{r-1}\rangle,\,\,|\Phi_r\rangle)$, then the $r$ conjugate of these states with $(-\alpha,-\beta)$ must be arranged in the order $(|\Phi_r^c\rangle,\,\,|\Phi_{r-1}^c\rangle,....,|\Phi_2^c\rangle,\,\,|\Phi_1^c\rangle)$, where $c$ means a conjugation. In other words, if a state is at position $n$, then its conjugate state should be at the position $\widetilde{\dim}_N-n+1$, where $\widetilde{\dim}_N$ is the number of states at level N. So, every state will pair up with its conjugate state at the anti-diagonal line of the Gram matrix.
4. For even level $N$, we could have $p(N/2)^2$ states with $(\alpha=0,\beta=0)$ and $p(N/2)$ of these are self-conjugate states. We first put the self-conjugate states (in any order) at the centre and arrange the other states in such way that if a states is at position $s$, somewhere on the left side of the centre, its conjugate should be on the right side of the centre at position $p(N/2)^2-s+1$. This arrangement is similar to the preceding rule where the aim is again to let conjugate pairs meet at the anti-diagonal line in the Gram matrix.
We have already seen that the above ordering rules for level 1 and level 2 && |\_1= L\_[-1]{}|,, |\_2= M\_[-1]{}|, && |\_1= L\_[-1]{}\^2|,, |\_2= L\_[-2]{}|,, |\_3= L\_[-1]{}M\_[-1]{}|,&& |\_4=M\_[-2]{}|,, |\_5=M\_[-1]{}\^2 |,, give us a triangular Gram matrix in and . For level 3, the ordering of the states are given in Table \[BMS\_states\]. We shown the states again below along with their values of $\alpha$ and $\beta$ && (
[l|llll]{} & =3 & =1 & =-1 & =-3\
=3 & |\_[1]{}=L\_[-1]{}\^[3]{}|,\
=2 & |\_[2]{}=L\_[-1]{}L\_[-2]{}|,& & &\
=1 & |\_[3]{}=L\_[-3]{}|,& |\_[4]{}=L\_[-1]{}\^[2]{}M\_[-1]{}|,\
=0 & & |\_[5]{}=L\_[-2]{}M\_[-1]{}|,& |\_[6]{}=L\_[-1]{}M\_[-2]{}|,\
=-1 & & & |\_[7]{}=L\_[-1]{}M\_[-1]{}\^[2]{}|,& |\_[8]{}=M\_[-3]{}|,\
=-2 & & & & |\_[9]{}=M\_[-1]{}M\_[-2]{}|,\
=-3 & & & & |\_[10]{}=M\_[-1]{}\^[3]{}|,
).&& The form of the Gram matrix with this ordering ordering of basis is indeed the one we expected (
[c|cccccccccc]{} & |\_[1]{}& |\_[2]{}& |\_[3]{}& |\_[4]{}& |\_[5]{}& |\_[6]{}& |\_[7]{}& |\_[8]{}& |\_[9]{}& |\_[10]{}\
\_[1]{}| & K\_[1,1]{} & K\_[1,2]{} & K\_[1,3]{} & K\_[1,4]{} & K\_[1,5]{} & K\_[1,6]{} & K\_[1,7]{} & K\_[1,8]{} & K\_[1,9]{} & K\_[1,10]{}\
\_[2]{}| & K\_[2,1]{} & K\_[2,2]{} & K\_[2,3]{} & K\_[2,4]{} & K\_[2,5]{} & K\_[2,6]{} & K\_[2,7]{} & K\_[2,8]{} & K\_[2,9]{} & 0\
\_[3]{}| & K\_[3,1]{} & K\_[3,2]{} & K\_[3,3]{} & K\_[3,4]{} & K\_[3,5]{} & K\_[3,6]{} & 0 & K\_[3,8]{} & 0 & 0\
\_[4]{}| & K\_[4,1]{} & K\_[4,2]{} & K\_[4,3]{} & K\_[4,4]{} & K\_[4,5]{} & K\_[4,6]{} & K\_[4,7]{} & 0 & 0 & 0\
\_[5]{}| & K\_[5,1]{} & K\_[5,2]{} & K\_[5,3]{} & K\_[5,4]{} & 0 & K\_[5,6]{} & 0 & 0 & 0 & 0\
\_[6]{}| & K\_[6,1]{} & K\_[6,2]{} & K\_[6,3]{} & K\_[6,4]{} & K\_[6,5]{} & 0 & 0 & 0 & 0 & 0\
\_[7]{}| & K\_[7,1]{} & K\_[7,2]{} & 0 & K\_[7,4]{} & 0 & 0 & 0 & 0 & 0 & 0\
\_[8]{}| & K\_[8,1]{} & K\_[8,2]{} & K\_[8,3]{} & 0 & 0 & 0 & 0 & 0 & 0 & 0\
\_[9]{}| & K\_[9,1]{} & K\_[9,2]{} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\
\_[10]{}| & K\_[10,1]{} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
). For level 4, a possible ordering of the basis states using our arrangement rules is given by && |\_1= L\_[-1]{}\^4|,,|\_2= L\_[-1]{}\^2L\_[-2]{}|,,|\_3= L\_[-2]{}\^2|,, |\_4= L\_[-1]{}L\_[-3]{}|,, && |\_5= L\_[-1]{}\^3M\_[-1]{}|,, |\_6= L\_[-4]{}|,,|\_7= L\_[-1]{}L\_[-2]{}M\_[-1]{}|,,|\_8= L\_[-1]{}\^2M\_[-2]{}|,,&& |\_9= L\_[-3]{}M\_[-1]{}|,,|\_[10]{}= L\_[-2]{}M\_[-2]{}|,,|\_[11]{}= L\_[-1]{}\^2M\_[-1]{}\^2|,,|\_[12]{}= L\_[-1]{}M\_[-3]{}|,&& |\_[13]{}= L\_[-2]{}M\_[-1]{}\^2|,, |\_[14]{}= L\_[-1]{}M\_[-1]{}M\_[-2]{}|,, |\_[15]{}= M\_[-4]{}|,,|\_[16]{}= L\_[-1]{}M\_[-1]{}\^3|,&& |\_[17]{}= M\_[-1]{}M\_[-3]{}|,,|\_[18]{}= M\_[-2]{}\^2|,, |\_[19]{}= M\_[-1]{}\^2M\_[-2]{}|,, |\_[20]{}= M\_[-1]{}\^4|,.&&
Now let us explain why the above rules give us a triangular structure for the Gram matrix. We first consider the case of odd level. Here we don’t have to deal with self-conjugate states and our arrangement rules is such that the matrix entries of the anti-diagonal line are the inner products of a state and its conjugate and is thus non-zero due to . Let us look at a particular anti-diagonal element pairing up a state $|\Psi_a\rangle$ with its conjugate $|\Psi_b\rangle$ K\_[a,b]{}=\_a | \_b 0, \_a+\_b=0, \_a+\_b=0. Any matrix elements on its right side is given by K\_[a,b+c]{} = \_a | \_[b+c]{} . Since we are arranging the states in such a that way $\alpha_b\geq\alpha_{b+1}$, we could have $\alpha_b=\alpha_{b+c}$ or $\alpha_b>\alpha_{b+c}$. For the first case we have \_a + \_[b+c]{} = 0. From , for the inner product $\langle \Psi_a|\Psi_{b+c}\rangle$ to be non-zero, $|\Psi_{b+c}\rangle$ should be a conjugate of $|\Psi_a\rangle$. But this is not possible as $|\Psi_b\rangle$ is the conjugate of $|\Psi_a\rangle$. Therefore \_a | \_[b+c]{} =0. For the next case, from \_[a]{}|\_[b+c]{}= 0, \_a+\_[b+c]{}<0. So all matrix entries on the right side of $\langle \Psi_a|\Psi_b\rangle$ are zero. In other words, the Gram matrix is triangular.
For even level $N$, we have $p(N/2)$ self-conjugate states. These have non-zero norms and are orthogonal to each other due to . So, the Gram matrix constructed from these sub-set of states will be diagonal and this diagonal matrix will sit at the centre of the Gram matrix due to our arrangement rules of putting self-conjugate states at the middle while ordering the basis states. Similar arguments made for odd level can be used to conclude that all matrix elements on the right hand side of this diagonal matrix will be zero. Just like odd level, all the non self-conjugate states pair up with their conjugate at the anti-diagonal line outside the diagonal matrix, with all the entries on the right side of this anti-diagonal line vanishing again due to the same argument. Thus we have proven that with the use of our arrangement rules the Gram matrix will have the form for odd level and the form for even level.
Form of $\sum_{i,j}K^{ij}_{(N)}\tilde{O}_{ij}^{(N)}$ for general level $N$ \[appendix\_2\]
==========================================================================================
In this appendix we will prove the formula for $\sum_{i,j}K^{ij}_{(N)}\tilde{O}_{ij}^{(N)}$ given in . For this, we have to prove that, using our ordering rules for the basis states, the matrix $\tilde{O}^{(N)}$ have the following structure: \^[(N=)]{} = e\^[-2i(+)]{}e\^[2i(+N)]{}e\^[2i]{}(
[ccccc]{} \_[1,1]{}\^ & & & & A\_[\_N]{}\
& & & & 0\
& & & &\
& A\_[2]{}\
A\_[1]{} & 0 & & & 0
) , \^[(N=)]{} = e\^[-2i(+)]{}e\^[2i(+N)]{}e\^[2i]{}(
[cccccccccc]{} \_[1,1]{}\^ & & & & & & & & & A\_[\_[N]{}]{}\
& & & & & & & & & 0\
& & & & & & & A\_[k+l]{} & &\
& & & D\_[1]{} & 0 & & 0\
& & & 0 & D\_[2]{} & & 0 & & &\
& & & & & & & & &\
& & & 0 & 0 & & D\_[l]{}\
& & A\_[k]{} & & & & & 0\
& & & & & & & & &\
A\_[1]{} & 0 & & & & & & & & 0
). In the above equation, the $A$’s and $D$’s are the same as those appearing in and for $K^{(N)}$ and $K_{(N)}$. Before we prove the structural property of $\tilde{O}^{(N)}$ given above, let us first see how it lead to the formula . For odd level the reasoning is exactly same as what we did for level 2. More precisely, $K_{(N)}$ and $\tilde{O}^{(N)}$ have the opposite triangular structure so that when taking the trace, only the anti-diagonal elements will contribute. The anti-diagonal elements of these matrix are also inverse of each other apart from a factor of $e^{-2\pi i(\sigma\frac{c_L}{2}+\rho\frac{c_M}{2})}e^{2\pi i\sigma(\Delta+N)}e^{2\pi i\rho\xi}$. So we have \_[i,j]{}K\^[ij]{}\_[(N)]{}\_[ij]{}\^[(N)]{} &=& \_[i=1]{}\^[\_N]{} K\^[ii]{}\_[(N)]{}\_[ii]{}\^[(N)]{} = e\^[-2i(+)]{} e\^[2i(+N)]{}e\^[2i]{} \_[d=1]{}\^[\_N]{} 1&=& \_N e\^[-2i(+)]{} e\^[2i(+N)]{}e\^[2i]{}. This reasoning can be easily extended for even level.
Now let us give a proof why $\tilde{O}^{(N)}$ have the structure shown above if we use our ordering rules for the basis states. We know from that the factor $e^{-2\pi i(\sigma\frac{c_L}{2}+\rho\frac{c_M}{2}}e^{2\pi i\sigma(\Delta+N)}$ is due to $e^{2\pi i\sigma (L_{0}-c_L/2)}e^{-2\pi i\rho c_M/2)}$. So, what is left is to see the matrix element of $e^{2\pi i\rho M_{0}}$. First let us note that using the commutator e\^[2iM\_[0]{}]{} L\_M\_|,&=& L\_M\_ e\^[2iM\_[0]{}]{} |,+ \[e\^[2iM\_[0]{}]{}, L\_M\_\] |,& = & e\^[2i]{} L\_M\_|,+ \_[n=1]{}\^ (2i )\^n \[M\_0\^n, L\_M\_\] |,. \[act\_M0\_gen\] As an example, we have |,&=& e\^[2i]{}(4iL\_[-1]{}M\_[-1]{} + M\_[-1]{}M\_[-1]{})|,. The point is that since the commutator of $M_0$ with $L$’s changes $L$’s to $M$’s, all the states in $[e^{2\pi i\rho M_{0}}, L_{\vec{k}}M_{\vec{q}}] |\Delta,\xi\rangle$ will have less number of $L$’s, i.e., less value of $\alpha$, than that of $L_{\vec{k}}M_{\vec{q}}|\Delta,\xi\rangle$. The exception is for states of the form $M_{\vec{q}}|\Delta,\xi\rangle$ which will not be relevant for our discussion below. Now let us look at a particular anti-diagonal element of $O^{(N)}$ for odd level which pair up a state $|\Psi_a\rangle$ with its conjugate $|\Psi_b\rangle=L_{\vec{q}}M_{\vec{k}}|\Delta,\xi\rangle$. Using , we have \^[(N)]{}\_[ab]{} &=& e\^[-2i(+)]{}e\^[2i(+N)]{} \_a |e\^[2iM\_[0]{}]{} L\_M\_|,&=& e\^[-2i(+)]{}e\^[2i(+N)]{} (e\^[2i]{} \_a |\_b + \_a|\[e\^[2iM\_[0]{}]{}, L\_M\_\] |,).&& Now since $|\Psi_b\rangle$ and $|\Psi_b\rangle$ are conjugate pair we have \_a + \_b = 0. As we have argued before, all the states in $[e^{2\pi i\rho M_{0}}, L_{\vec{q}}M_{\vec{k}}] |\Delta,\xi\rangle$ will have less value of $\alpha$ than that of $\alpha_b$. So, for these states we will have \_a + < 0. Therefore the inner products of these states with $|\Psi_a\rangle$ is zero due to . In other words, \_a|\[e\^[2iM\_[0]{}]{}, L\_M\_\] |,= 0. Thus we have shown that \^[(N)]{}\_[ab]{} = e\^[-2i(+)]{}e\^[2i(+N)]{}e\^[2i]{} K\^[(N)]{}\_[ab]{}. for anti-diagonal elements. Now let us look at matrix entries on the right side of the anti-diagonal element $\tilde{O}_{ab}^{(N)}$. Particularly, let us look at $\tilde{O}_{a(b+c)}^{(N)}$ with $|\Psi_{b+c}\rangle = L_{\vec{q^{\prime}}}M_{\vec{k^{\prime}}}$. Using $K_{a(b+c)}=\langle \Psi_a | \Psi_{b+c}\rangle = 0$, we have \^[(N)]{}\_[a(b+c)]{} &=& e\^[-2i(+)]{}e\^[2i(+N)]{} \_a|\[e\^[2iM\_[0]{}]{}, L\_M\_\] |,. Due to our ordering rule we have $\alpha_{b+c}\leq\alpha_b$. We can again use the previous argument to conclude that all the states in $[e^{2\pi i\rho M_{0}}, L_{\vec{q^{\prime}}}M_{\vec{k^{\prime}}}] |\Delta,\xi\rangle$ will have $\alpha$ such that $\alpha<\alpha_{b+c}$. Thus \_a + < 0, which imply that \^[(N)]{}\_[a(b+c)]{} = 0. We can use similar steps for even level to prove our claim.
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[^1]: For higher dimensional putative dual theories to flat space, see discussions in e.g. [@Bagchi:2016bcd; @Bagchi:2019xfx].
[^2]: The computations are done in the microcanonical ensemble
[^3]: This observation has been made by Navya Gupta.
[^4]: We thank an anonymous referee for pointing out errors in a previous version of this figure.
[^5]: If, however, we venture beyond Einstein gravity, there are interesting holographic duals to this theory. See [@Bagchi:2012yk] for the bosonic theory and [@Bagchi:2018ryy] for a recent fermionic generalisation.
[^6]: The automorphism in the Witt algebra (i.e. without the central term) has been noticed before in [@Oblak:2016eij].
[^7]: A supersymmetric version of this automorphism has been reported in [@Bagchi:2018wsn] in relation to tensionless strings, but observation was made in the context of this work earlier.
|
---
abstract: 'We calculate the effect of electron-electron interactions involving vanishing momentum transfer (forward scattering) on the orbital linear magnetic response of disordered metal rings pierced by a magnetic flux $\phi$. Using the bulk value of the Landau parameter $F_0$ for copper, we find that in the experiment by Lévy [*[et al.]{}*]{} \[Phys. Rev. Lett. [**[64]{}**]{}, 2074 (1990)\] the forward scattering contribution to the [*[linear]{}*]{} magnetic response is larger than the corresponding contribution from large momentum transfers considered by Ambegaokar and Eckern \[Phys. Rev. Lett. [**[65]{}**]{}, 381 (1990)\]. However, outside the regime of validity of linear response and to first order in the effective screened interaction the persistent current is dominated by scattering processes involving large momentum transfers.'
author:
- Long Phi Chau and Peter Kopietz
date: 'February 12, 2004'
title: 'Linear Magnetic Response of Disordered Metallic Rings: Large Contribution from Forward Scattering Interactions'
---
= 10000
Introduction
============
More than a decade ago the measurement by Lévy [*[et al.]{}*]{} [@Levy90] of persistent currents in mesoscopic normal metal rings pierced by an Aharonov-Bohm flux $\phi$ has triggered a lot of theoretical activity [@Imry97; @Schwab02]. Yet, up until now a truely convincing and generally accepted theoretical explanation of the surprisingly large persistent currents observed in Ref. [@Levy90] and in subsequent experiments [@Mohanty99; @Jariwala01] has not been found. It has become clear, however, that this effect cannot be explained within a model of non-interacting electrons. Ambegaokar and Eckern (AE) [@Ambegaokar90] were the first to examine the effect of electron-electron interactions on mesoscopic persistent currents: they realized that, to first order in the screened Coulomb-interaction, the dominant contribution to the disorder averaged persistent current can be obtained from the two diagrams shown in Fig. \[fig:AE\], representing a special correction $ \overline{\Omega}_{\rm AE} ( \phi )$ to the disorder averaged thermodynamic potential which depends strongly on the Aharonov-Bohm flux $\phi$.
Here the overline denotes averaging over the disorder. Given the grand canonical potential $\Omega ( \phi )$, the corresponding persistent current $I ( \phi )$ can be obtained from the thermodynamic relation $$I ( \phi ) = - c \frac{\partial \Omega ( \phi ) }{ \partial \phi }
\; .$$
In a bulk metal at high densities the bare Coulomb-interaction $V_{0} ( {\bf{q}} ) = 4 \pi e^2 / {\bf{q}}^2$ is strongly screened. A simple way to take the screening into account diagrammatically is the random-phase approximation (RPA). Following this procedure, AE approximated the effective interaction (in the imaginary frequency formalism) as follows $$\overline{V}_{\rm RPA} ( {\bf{q}} , i \omega ) = \frac{ V_0 ( {\bf{q}} ) }{
1 + \overline{ \Pi}_0 ( {\bf{q}} , i \omega ) V_0 ( {\bf{q}} ) }
\; .
\label{eq:RPA}$$ For momentum transfers $| {\bf{q}}|$ small compared with the inverse elastic mean free path $\ell^{-1}$, and for frequency transfers $| \omega |$ small compared with the inverse elastic lifetime $\tau^{-1}$ the disorder averaged polarization is given by $$\overline{ \Pi}_0 ( {\bf{q}} , i \omega )
\approx 2 \nu_0 \frac{ D_0 {\bf{q}}^2}{
D_0 {\bf{q}}^2 + | \omega | }
\; ,
\label{eq:pol}$$ where $D_0$ is the diffusion coefficient and $\nu_0$ is the average density of states at the Fermi energy (per spin) in the absence of interactions. Note that $\nu_0 = ( \Delta_0 {\cal{V}} )^{-1}$, where ${\cal{V}}$ is the volume of the system and $\Delta_0$ is the average level spacing (per spin) at the Fermi energy. It turns out that both diagrams in Fig. \[fig:AE\] are dominated by momentum transfers of the order of the Fermi momentum $k_F$, which for a metallic system is large compared with $\ell^{-1}$. Eqs. (\[eq:RPA\]) and (\[eq:pol\]) are therefore not suitable for a quantitatively accurate calculation of persistent currents. To make some progress analytically, AE estimated the contribution from the diagrams in Fig. \[fig:AE\] by replacing the effective interaction by a constant $$\overline{V}_{\rm RPA} (
{\bf{k}} - {\bf{k}}^{\prime} , i \omega )
\rightarrow
\langle \overline{V}_{\rm RPA} (
{\bf{k}}_F - {\bf{k}}_F^{\prime} , i 0 ) \rangle
\equiv \overline{V}
\; ,
\label{eq:Vbar}$$ where $\langle \ldots \rangle $ denotes the Fermi surface average over ${\bf{k}}_F$ and ${\bf{k}}_F^{\prime}$. For simplicity, it is assumed that the ring is quasi one-dimensional, with transverse thickness $L_{\bot}$ in the range $ k_F^{-1} \ll L_{\bot} \ll \ell \ll L$, where $L$ is the circumference of the ring. Then diffusive motion is only possible along the circumference. At temperature $T=0$ the resulting average persistent current can be written as [@Ambegaokar90] $$\overline{I}^{\rm AE} ( \phi ) = \sum_{k=1}^{\infty}
I_k^{\rm AE} \sin ( 4 \pi k \phi / \phi_0 )
\; ,
\label{eq:IAE}$$ where $\phi_0 = hc / e$ is the flux quantum [@weprefer] and the Fourier coefficients of the current are $$I_k^{\rm AE}
= \frac{c}{\phi_0} \frac{16 \lambda_c }{k^2} E_c
e^{ - k \sqrt{\gamma}} [ 1 + k \sqrt{{\gamma}} ]
\; .
\label{eq:ImAE}$$ Here $E_c = \hbar D_0 / L^2$ is the Thouless energy and $\gamma = \Gamma / E_c \ll 1$, where at zero temperature $\Gamma = \Delta_0 / \pi$ is the cutoff energy that regularizes the singularity in the Cooperon in a finite system [@Voelker96], see Eqs. (\[eq:gwl\]) and (\[eq:Pdef\]) below. The coupling constant ${\lambda}_c = \nu_0 \bar{V} $ can be identified with the dimensionless effective interaction in the Cooper channel to first order in perturbation theory. AE estimated ${\lambda}_{c} \approx 0.3 $, assuming that the validity of the RPA can be extended to momentum transfers of the order of $k_F$. However, higher order ladder diagrams in the Cooper channel strongly reduce the effective interaction, so that $\lambda_c \approx 0.06$ is a more realistic estimate [@Eckern91] for the Cu-rings in the experiment [@Levy90].
In real space Eq. (\[eq:Vbar\]) amounts to replacing the electron-electron interaction by a local effective density-density interaction, $$\overline{V}_{\rm eff} ( {\bf{r}} - {\bf{r}}^{\prime} ) \rightarrow
\overline{V} \delta ( {\bf{r}} - {\bf{r}}^{\prime} )
\; .
\label{eq:Veff}$$ More precisely, this replacement means that for distances $| {\bf{r}} - {\bf{r}}^{\prime} |$ larger than $ \ell$, the interaction is effectively local. In a recent letter Schechter, Oreg, Imry, and Levinson [@Schechter03] pointed out that a different type of effective interaction can possibly lead to a much larger persistent current. Specifically, they used the BCS model to calculate the leading interaction correction to the orbital linear magnetic response and found [@Schechter03; @weprefer] $$\left. \frac{\partial \overline{ I }^{\rm BCS} }{\partial \phi }
\right|_{\phi =0}
= \frac{c}{\phi_0^2} 32 \pi \lambda_{\rm BCS} E_c \ln \left( \frac{ E_{\rm co}}{
\Delta_0 } \right)
\; ,
\label{eq:chiBCS}$$ where $\lambda_{\rm BCS}<0 $ is the attractive dimensionless interaction in the BCS model, and the coherence energy $E_{\rm co}$ is the smaller energy of $\hbar / \tau$ and the Debye energy $ \hbar \omega_D$. Eq. (\[eq:chiBCS\]) should be compared with the corresponding result for the local interaction model used by AE, which implies according to Eqs. (\[eq:IAE\]) and (\[eq:ImAE\]), $$\left. \frac{\partial \overline{ I }^{\rm AE} }{\partial \phi }
\right|_{\phi =0}
= \frac{c}{\phi_0^2} 32 \pi \lambda_{c} E_c \ln \left( \frac{ E_{c}}{
\Delta_0 } \right)
\; ,
\label{eq:chiAE}$$ where we have used $\Gamma = \Delta_0 / \pi $ and retained only the leading logarithmic order. Note that the logarithm is due to the slow decay ($ \propto k^{-1} $) of the Fourier coefficients $4 \pi k I_k^{\rm AE} / \phi_0$ of $ \partial \overline{ I }^{\rm AE} / {\partial \phi }$, so that all coefficients with $ k \lesssim 1/\sqrt{\gamma}$ contribute to the linear response. For $E_{ \rm co} \gg E_c$ the linear magnetic response in the BCS model is parametrically larger than the linear response in the local interaction model. Whether or not this remains true beyond the linear response has not been clarified. Note also that in the BCS model the linear magnetic response is diamagnetic because the effective interaction is attractive ($\lambda_{\rm BCS} < 0$), whereas the linear response in the local interaction model is paramagnetic, corresponding to a repulsive effective interaction ($\lambda_c > 0)$.
Magnetic response due to forward scattering
===========================================
An interesting observation made by the authors of Ref. [@Schechter03] is that an effective interaction different from the local interaction used by AE can lead to a much larger persistent current, at least for sufficiently small flux $\phi$, where it is allowed to calculate the current from the linear response. Given the rather crude approximations in the microscopic derivation of the local interaction model, it seems worth while to explore the magnetic response for other types of effective interactions. A possibility which so far has not been thoroughly analyzed is an interaction which is dominated by small momentum transfers. Note that the assumption that only forward scattering processes (corresponding to vanishing momentum transfer) have to be taken into account for a consistent description of the low-energy and long-wavelength properties of normal metals lies at the heart of the Landau’s Fermi liquid theory. The Landau model is in a sense the opposite extreme of the local interaction model, because the effective interaction in the Landau model is proportional to a Kronecker-delta in momentum space, $$\overline{V}_{\rm eff} ( {\bf{q}} , i \omega )
\rightarrow \delta_{ {\bf{q}} , 0 } f_0
\; ,
\label{eq:f0def}$$ where the Landau parameter $f_0$ can be determined from experiments. In fact, the dimensionless Landau parameter [@notethat] $ F_0 \equiv 2 \nu_0 f_0 $ can be written as $F_0 = \frac{B}{B_0} \frac{ m_{\ast}}{m_0} -1$, where $B$ is the bulk modulus, $m_{\ast}$ is the effective mass, and $B_0$ and $m_0$ are the corresponding quantities in the absence of interactions. Inserting the known bulk values for Cu [@Ashcroft76], $m_{\ast}/ m \approx 1.3$ and $B/B_0 \approx 2.1$, we find $F_0 \approx 1.7$, which is a factor of $30$ larger than the corresponding estimate $\lambda_c \approx 0.06 $ in the local interaction model. Note that in real space Eq. (\[eq:f0def\]) corresponds to a constant effective interaction, proportional to the inverse volume of the system $$\overline{V}_{\rm eff} ( {\bf{r}} - {\bf{r}}^{\prime} ) \rightarrow
\frac{f_0}{\cal{V}}
\; .
\label{eq:Veff2}$$ Given an effective interaction of the form (\[eq:f0def\]), the dominant flux-dependent contributions to the average potential $\overline{\Omega} ( \phi )$ to first order in the interaction are shown in Fig. \[fig:AS\].
The Fock diagrams (b)–(d) have been discussed previously in Refs. [@BealMonod92; @Kopietz98]; as first pointed out by Béal-Monod and Montambaux [@BealMonod92], to leading order in the small parameter $ (k_F \ell )^{-1}$, the three Fock diagrams in Fig. \[fig:AS\] (b)–(d) cancel, so that a direct evaluation of the sum of these diagrams is rather difficult. To calculate the leading contribution of these diagrams, we note that the fermion loops in Fig. \[fig:AS\] (b)–(d) can be identified with contributions to the disorder averaged polarization, which for general frequencies and small wavevectors can be written as [@Vollhardt80] $$\overline{ \Pi}_0 ( {\bf{q}} , i \omega )
= 2 \nu_0 \frac{ D ( i \omega ) {\bf{q}}^2}{
D ( i \omega ) {\bf{q}}^2 + | \omega | }
\; ,
\label{eq:pol2}$$ where $D ( i \omega )$ is a generalized frequency-dependent Diffusion coefficient. The crucial observation is now that the sum of the three Fock diagrams in Fig. \[fig:AS\] (b)–(d) corresponds to the usual weak localization correction to the average conductance [@Kopietz98], $$D ( i \omega ) \approx D_0 [ 1 + g_{\rm WL} ( i \omega ) ]
\; ,$$ where $$g_{\rm WL} ( i \omega ) = - \frac{2 \Delta_0}{\pi} \sum_{ {\bf{q}}}
\frac{1}{ \hbar D_0 {\bf{q}}^2 + | \omega | + \Gamma }
\; .
\label{eq:gwl}$$ Essentially we have used the equation of continuity to replace the charge vertices in Fig. \[fig:AS\] by current vertices, which cannot be renormalized by singular diffusion corrections. The fact that a gauge transformation replacing charge vertices by current vertices can be used to avoid the explicit calculation of vertex corrections has also been employed in Ref.[@Kopietz98PRL] to calculate the zero bias anomaly in the tunneling density of states of two-dimensional disordered electrons interacting with Coulomb forces.
The evaluation of the contribution of the three Fock diagrams in Fig. \[fig:AS\] to the persistent current is now straightforward. Note that for a thin ring with $ L_{\bot} \ll \ell \ll L$ the ${\bf{q}}$-summation is one-dimensional, with quantized wave-vectors $ 2 \pi ( n + 2 \phi/ \phi_0 ) /L$, $n = 0, \pm 1 , \pm 2 , \ldots$. Then we obtain for the $k$-th Fourier component of the average current due to the Fock diagrams (b)–(d) in Fig. \[fig:AS\] for the Landau model [@Kopietz98], $$I_k^{\rm L, Fock} \propto k^{-1} \frac{f_0 }{ {\cal{V}}}
\; .
\label{eq:ILFock}$$ Due to the extra factor of inverse volume, this contribution is, for experimentally relevant parameters [@Levy90], negligible compared with corresponding result in the local interaction model given in Eq. (\[eq:ImAE\]).
The Hartree diagram in the Landau model is more interesting. The fact that the diagram with two Cooperons shown in Fig. \[fig:AS\] (a) dominates the persistent current due to electron-electron interactions with momentum transfers $| {\bf{q}} | \lesssim \ell^{-1}$ has already been pointed out in Ref. [@Kopietz93]. A similar diagram with two Cooperons (but without interaction line) dominates the fluctuations of the number of energy levels in a fixed energy window centered at the Fermi energy [@Altshuler86]. Using the approximate relation $$I_{N} ( \phi ) = - \frac{c}{2} {\Delta}_0
\frac{
\partial (\delta N )^2 }{ \partial \phi}$$ between the persistent current $I_N ( \phi )$ at constant particle number and the fluctuation $ ( \delta N )^2$ of the particle number at constant chemical potential $\mu$, several authors have realized [@Schmid91; @Altshuler91; @Oppen91] that without interactions the two-Cooperon diagram determines the average persistent current in a canonical ensemble. Note that the Hartree diagram in Fig. \[fig:AS\] (a) does not contain any vertex corrections analogous to the diffusion corrections of the vertices in the Fock diagams (b)–(d). This is due to the fact that the interaction line in the Hartree process does not transfer any energy. Hence, the two Green functions attached to the vertex of a Hartree interaction are either both retarded or both advanced, so that it is impossible to attach a singular Diffuson to the vertex.
For the Landau model the Hartree diagram in Fig. \[fig:AS\] (a) yields at finite temperature $T$ the following correction to the disorder averaged grand canonical potential, $$\begin{aligned}
\overline{\Omega}^{\rm L, Hartree} ( \phi ) & = & \frac{f_0}{2 \cal{V}} 4 \sum_{ {\bf{q}}}
T^2 \sum_{ \tilde{\omega}_n , \tilde{\omega}_{n^{\prime}} }
\theta ( - \tilde{\omega}_n \tilde{\omega}_{n^{\prime}} )
\nonumber
\\
& & \hspace{-15mm} \times \left ( \frac{ \Delta_0 }{ 2 \pi } \frac{\hbar}{\tau} \right)^2
\left[ \frac{ \hbar / \tau }{
\hbar D_0 {\bf{q}}^2 + | \tilde{ \omega}_n - \tilde{\omega}_{n^{\prime}} | + \Gamma }
\right]^2
\nonumber
\\
& & \hspace{-15mm} \times
\sum_{ {\bf{k}}} [ \overline{G}_0 ( {\bf{k}} , i \tilde{\omega}_n ) ]^2 \overline{G}_0 ( - {\bf{k}} + {\bf{q}} ,
i \tilde{\omega}_{n^{\prime}} )
\nonumber
\\
& & \hspace{-15mm} \times
\sum_{ {\bf{k}}^{\prime} } [ \overline{G}_0 ( {\bf{k}}^{\prime} ,
i \tilde{\omega}_{n^{\prime}} ) ]^2 \overline{G}_0 ( - {\bf{k}}^{\prime} + {\bf{q}} ,
i \tilde{\omega}_{n} )
\; .
\label{eq:OmegaH0}
\end{aligned}$$ Here $ \tau = \ell / v_F$ is the elastic lifetime, $\tilde{\omega}_n = 2 \pi ( n + \frac{1}{2} ) T$ are fermionic Matsubara frequencies, and $$\overline{G}_0 ( {\bf{k}} , i \tilde{\omega}_n )
= \frac{1}{ i \tilde{\omega}_n - \frac{ \hbar^2 {\bf{k}}^2}{2m} +
\mu + i \frac{\hbar}{2 \tau} {\rm sign} \tilde{\omega}_n }$$ is the disorder averaged non-interacting Matsubara Green function. Since the Cooperons (i.e. the second line) in Eq. (\[eq:OmegaH0\]) are only singular for $ | {\bf{q}} | \lesssim \ell^{-1}$, and because the ${\bf{k}}$- and ${\bf{k}}^{\prime}$- sums are dominated by momenta of the order of the Fermi momentum, we may approximate $\overline{G}_0 ( - {\bf{k}} + {\bf{q}} ,
i \tilde{\omega}_{n^{\prime}} )
\approx
\overline{G}_0 ( - {\bf{k}} ,
i \tilde{\omega}_{n^{\prime}} )$ and $
\overline{G}_0 ( - {\bf{k}}^{\prime} + {\bf{q}} ,
i \tilde{\omega}_{n} )
\approx
\overline{G}_0 ( - {\bf{k}}^{\prime} ,
i \tilde{\omega}_{n} )$ in Eq. (\[eq:OmegaH0\]). The product of the last two lines of Eq. (\[eq:OmegaH0\]) gives then rise to a factor of $ [ (2 \pi / \Delta_0 ) ( \tau / \hbar )^2 ]^2$, so that we obtain $$\overline{\Omega}^{\rm L, Hartree} ( \phi ) = \frac{f_0}{2 \cal{V}} P ( \phi )
\; ,
\label{eq:OmegaH}$$ with the dimensionless coefficient $$P ( \phi ) = \frac{4 T}{\pi}
\sum_{ 0 < \omega_m < \hbar / \tau } \sum_{ {\bf{q}} }
\frac{ \omega_m}{ [ \hbar D_0 {\bf{q}}^2 + \omega_m + \Gamma ]^2 }
\label{eq:Pdef}
\; ,$$ where $\omega_m = 2 \pi m T$ are bosonic Matsubara frequencies. Assuming again a thin ring with $ L_{\bot} \ll \ell \ll L$, we find in the limit $T \rightarrow 0$ for the Fourier components of the persistent current, $$I_k^{\rm L, Hartree}
= \frac{16 }{\pi} \frac{c}{\phi_0} \frac{f_0}{ {2 \cal{V}}}
e^{ - k \sqrt{\gamma}}
= \frac{c}{\phi_0} 8 F_0 \frac{ \Delta_0}{\pi}
e^{ - k \sqrt{\gamma}}
\; .
\label{eq:IH}$$ Comparing this expression with the corresponding result (\[eq:ImAE\]) of the local interaction model, we see that in the Landau model the Fourier components $I_k^{\rm L,Hartree}$ are independent of $k$ as long as $k \lesssim 1/ \sqrt{\gamma}$. Therefore the linear magnetic response is determined by all Fourier components up to $k \lesssim \sqrt{ E_c/ \Gamma}$, $$\begin{aligned}
\left. \frac{\partial \overline{ I }^{\rm L, Hartree} }{\partial \phi }
\right|_{\phi =0}
& = & \frac{ 4 \pi}{\phi_0} \sum_{k=1}^{\infty} k I_k^{\rm L, Hartree}
\nonumber
\\
& \approx & \frac{c}{\phi_0^2} 16 \pi F_0 E_c
\label{eq:chiL}
\; ,
\end{aligned}$$ where we have assumed that $\gamma = \Gamma / E_c \ll 1$, so that $$\sum_{k=1}^{\infty} k e^{- k \sqrt{\gamma }}
= \frac{e^{\sqrt{\gamma}}}{[ 1 - e^{ - \sqrt{\gamma} }]^2}
\approx \frac{1}{\gamma}
\approx \frac{ \pi E_c}{\Delta_0}
\; .$$ Note that the small energy scale $\Delta_0$ of Eq. (\[eq:IH\]) has disappeared on the right-hand side of Eq. (\[eq:chiL\]), and is replaced by the much larger Thouless energy $E_c$. Due to the faster decay of the Fourier components (\[eq:ILFock\]) of the Fock contribution in the Landau model, the linear response due to the Fock diagrams shown in Fig. \[fig:AS\] (b)–(d) is a factor of $\sqrt{\Gamma / E_c }$ smaller than the corresponding Hartree contribution. Interestingly, the anomalously large linear magnetic response in the BCS model given in Eq. (\[eq:chiBCS\]) is also dominated by the Hartree process [@Schechter03]. Thus, the importance of Hartree interactions for persistent currents is some extent independent of a specific model for the interaction. For the Cu-rings used in the experiment [@Levy90] we estimate $F_0 \approx 1.7$, $\lambda_c \approx 0.06$ and $E_c / \Delta_0 \approx 25$; with these values the linear magnetic response due to forward scattering is more than four times larger than the linear response in the local interaction model considered by AE [@Ambegaokar90]. To take both contributions into account one should parameterize the total effective interaction as $$\overline{V}_{\rm eff} ( {\bf{r}} - {\bf{r}}^{\prime} ) =
\overline{V} \delta ( {\bf{r}} - {\bf{r}}^{\prime} ) +
\frac{f_0 - \overline{V}}{\cal{V}}
\; ,
\label{eq:Veff3}$$ which in momentum space amounts to $$\overline{V}_{\rm eff} ( {\bf{q}} ) = \left\{
\begin{array}{cl}
\overline{V} & \mbox{for $ {\bf{q}} = 0 $} \\
f_0 & \mbox{for $ {\bf{q}} \neq 0 $}
\end{array}
\right.
\; .$$ Using the estimate for bulk Cu given above, $F_0 = 2 \nu_0 f_0 \approx 1.7$ and $ \lambda_c = \nu_0 \overline{V} \approx 0.06$, we find $f_0 / \overline{V} \approx 14$, which supports our assumption that there is indeed a strong enhancement of the effective interaction in the forward scattering channel.
Conclusions
===========
In summary, we have shown that a forward scattering excess interaction, involving a vanishing momentum transfer, yields the dominant contribution to the [*[linear]{}*]{} magnetic response of mesoscopic metal rings for experimentally relevant parameters [@Levy90]. On the other hand, outside the linear response regime the persistent current is dominated by the term $\overline{V}$ involving large momentum transfers, at least if we use the bulk estimates for $\overline{V}$ and $F_0$ for Cu. However, for a mesoscopic disordered Cu-ring it is not obvious that the bulk estimates are reliable. Note also that in the bulk the normal Fermi liquid is stable as long as $F_0 > -1 $, so that for $0> F_0 > -1 $ the linear magnetic response in the normal state can be diamagnetic, in spite of the fact that the effective coupling $\lambda_c$ in the Cooper channel is positive. Moreover, in the vicinity of an $s$-wave Pomeranchuk instability [@Pomeranchuk58; @Murthy03], where $F_0 < 0$ and $ | 1 + F_0 | \ll 1$, one should replace $F_0$ by $ F_0 / (1 + F_0)$. In this case we predict a strongly enhanced diamagnetic linear response. In fact, the forward scattering channel might then dominate the persistent current even beyond the linear order in the flux $\phi$. To clarify this point, a better microscopic theory of the effective electron-electron interaction in mesoscopic disordered metals is necessary. In particular, a microscopic theory should properly treat the problem of screening in a finite system and incorporate the breakdown of Fermi liquid theory in quasi one-dimensional disordered metals at sufficiently low temperatures [@Altshuler85].
We thank M. Schechter for his clarifying remarks concerning Ref. [@Schechter03] and for his comments on this manuscript.
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|
---
abstract: |
We study homomorphisms between quantized generalized Verma modules $M(V_{\Lambda})\stackrel{\phi_{\Lambda,\Lambda_1}}{\rightarrow}M(V_{\Lambda_1})$ for ${\mathcal U}_q(su(n,n))$. There is a natural notion of degree for such maps, and if the map is of degree $k$, we write $\phi^k_{\Lambda,\Lambda_1}$. We examine when one can have a series of such homomorphisms $\phi^1_{\Lambda_{n-1},\Lambda_{n}}\circ
\phi^1_{\Lambda_{n-2},\Lambda_{n-1}}\circ\cdots\circ\phi^1_{\Lambda,\Lambda_1}
=\textrm{Det}_q$, where $\textrm{Det}_q$ denotes the map $M(V_{\Lambda})\ni p\rightarrow \textrm{Det}_q\cdot p\in M(V_{\Lambda_n})$. If, classically, $su(n,n)^{\mathbb
C}={\mathfrak p}^-\oplus(su(n)\oplus su(n)\oplus {\mathbb
C})\oplus {\mathfrak p}^+$, then $\Lambda=(\Lambda_L,\Lambda_R,\lambda)$ and $\Lambda_n=(\Lambda_L,\Lambda_R,\lambda+2)$. The answer is then that $\Lambda$ must be one-sided in the sense that either $\Lambda_L=0$ or $\Lambda_R=0$ (non-exclusively). There are further demands on $\lambda$ if we insist on ${\mathcal
U}_q({\mathfrak g}^{\mathbb C})$ homomorphisms. However, it is also interesting to loosen this to considering only ${\mathcal
U}^-_q({\mathfrak g}^{\mathbb C})$ homomorphisms, in which case the conditions on $\lambda$ disappear. By duality, there result have implications on covariant quantized differential operators. We finish by giving an explicit, though sketched, determination of the full set of ${\mathcal
U}_q({\mathfrak g}^{\mathbb C})$ homomorphisms $\phi^1_{\Lambda,\Lambda_1}$.
address: 'Department of Mathematical Sciences, University of Copenhagen, Denmark'
author:
- Hans Plesner Jakobsen
title: 'Special classes of homomorphisms between generalized Verma modules for ${\mathcal U}_q(su(n,n))$'
---
Dedicated to I.E. Segal (1918-1998) in commemoration of the centenary of his birth.
Introduction
============
Generalized and quantized Verma modules have physically attractive properties similar to the Fock space. There is a “vacuum vector”, here called a highest weight vector, which is annihilated by the “upper diagonal” operators, is an eigenvector for the “diagonal operators”, and which generate the whole space when acted upon by the algebra of “lower diagonal operators”. Since it may happen that there is a second vacuum vector, it is of interest to determine cases in which this may happen. This is further interesting because by duality, such cases correspond to quantized covariant differential operators such as the Maxwell equations. We give here a complete proof of the one-sidedness and we give a sketch of the case of an arbitrary first order. Further details as well as the dual picture will appear in a forthcoming article. For the “classical” analogue, see e.g. [@jak]. On a personal note: The explicitness presented here is in line with how mathematical physics was taught to me by Segal, my Ph.D. advisor.
Set-up
======
$$\begin{array}{ccccccccc}{\mathfrak g}^{\mathbb
C}&=&su(n,n)^{\mathbb C}\qquad&=&{\mathfrak k}^{\mathbb C}\oplus
{\mathfrak p}&=&{\mathfrak p}^-\oplus {\mathfrak k}^{\mathbb
C}\oplus {\mathfrak p}^+&=&{\mathfrak p}^-\oplus {\mathfrak
k}^{\mathbb C}\oplus {\mathfrak p}^+, \\{\mathfrak k}^{\mathbb
C}&=&su(n)^{\mathbb C}\oplus {\mathbb C}\oplus su(n)^{\mathbb
C}&=&{\mathfrak k}_L^{\mathbb C}\oplus {\zeta} \oplus {\mathfrak
k}_R^{\mathbb C}\ .&&&& \end{array}$$
$$\begin{array}{ccccccccc} \zeta \textrm{ is the
center, }&{\mathfrak p}^{\pm}\textrm{ are abelian ${\mathcal
U}({\mathfrak k}^{\mathbb C})$ modules,}&\textrm{ and }&
{\mathcal U}({\mathfrak g}^{\mathbb C})&={\mathcal P}({\mathfrak
p}^-)\cdot {\mathcal U}({\mathfrak k}^{\mathbb C})\cdot
{\mathcal P}({\mathfrak p}^+) \end{array} .$$
We let the simple roots be denoted $\Pi=\{\mu_1,\dots,\mu_{n-1}\}\cup\{\beta\}\cup
\{\mu_1,\dots,\mu_{n-1}\}$, where $\beta$ is the unique non-compact roots and where the decomposition of simple roots corresponds to the decomposition of ${\mathfrak k}^{\mathbb C}$ above. In the quantum group ${\mathcal
U}_q({\mathfrak g}^{\mathbb C})$, we denote the generators by $E_\alpha,F_\alpha,K_\alpha^{\pm1}$ for $\alpha\in\Pi$. There are also decompositions $$\begin{aligned}
{\mathcal
U}_q({\mathfrak g}^{\mathbb C})&=&{\mathcal A}_q^-\cdot
{\mathcal U}_q({\mathfrak k}^{\mathbb C})\cdot {\mathcal
A_q}^+,\\ {\mathcal U}_q({\mathfrak k}^{\mathbb C})&=&{\mathcal
U}_q({\mathfrak k}_L^{\mathbb C})\cdot {\mathbb
C}[K_\beta^{\pm1}] \cdot {\mathcal U}_q({\mathfrak k}_R^{\mathbb
C}). \end{aligned}$$ Here, ${\mathcal A}_q^\pm$ are quadratic algebras which are furthermore ${\mathcal U}_q({\mathfrak
k}^{\mathbb C})$. Specifically, $$\begin{aligned}
{\mathcal
A}_q^-&=&{\mathbb C}[W_{i,j}\mid i,j=1,\dots,n\},\\ {\mathcal
A}_q^+&=&{\mathbb C}[Z_{i,j}\mid i,j=1,\dots,n\}, \end{aligned}$$ with relations $$\begin{aligned}
\label{a}Z_{ij}Z_{ik} &=&
q^{-1}Z_{ik}Z_{ij} \textrm{ if }j < k;\\ \label{b}Z_{ij}Z_{kj}
&=& q^{-1}Z_{kj}Z_{ij}\textrm{ if }i< k;\\\label{c} Z_{ij}Z_{st}
&=& Z_{st}Z_{ij} \textrm{ if }i < s\textrm{ and }t <
j;\\\label{cross} Z_{ij}Z_{st} &=&
Z_{st}Z_{ij}-(q-q^{-1})Z_{it}Z_{sj} = \textrm{ if }i < s\textrm{
and } j < t. \end{aligned}$$ The algebra ${\mathcal A}_q^-$ have the same relations, but the algebras ${\mathcal A}_q^\pm$ are different as ${\mathcal U}_q({\mathfrak k}^{\mathbb C})$ modules. The elements $Z_{ij}$ and $W_{ij}$ are constructed by means of the Lusztig operators. References [@l] and [@jan] are general references of much of this. Using the Serre relations one gets, setting $\mu_0=Id$,
\[2.1\]$$\begin{aligned}
Z_{i,j}&=&T_{\nu_{j-1}}
T_{\nu_{j-2}}\dots T_{\nu_{0}}\cdot T_{\mu_{i-1}}
T_{\mu_{i-2}}\dots T_{\mu_{0}}(E_\beta),\\
W_{i,j}&=&T_{\nu_{j-1}} T_{\nu_{j-2}}\dots T_{\nu_{0}}\cdot
T_{\mu_{i-1}} T_{\mu_{i-2}}\dots
T_{\mu_{0}}(F_\beta).\label{26}\end{aligned}$$
For later use, we give the relations in the full algebra:
$$\begin{aligned}
E_{\mu_k} W_{i,j}&=&W_{i,j}E_{\mu_k}\textrm{ if
}k\neq i-1,\\
E_{\mu_k}W_{i,j}^a&=&(-q)[a]W_{i-1,j}W_{i,j}^{a-1}K_{\mu_k}+W^a_{i,j}E_{\mu_k}
\textrm{ if }k=i-1,\\ F_{\mu_k}
W_{i,j}&=&W_{i,j}F_{\mu_k}\textrm{ if }k\neq i,i-1,\\
F_{\mu_k}W_{i,j}^a&=&-q^{-1}[a]W_{i,j}^{a-1}W_{i+1,j}+q^{-a}W^a_{i,j}F_{\mu_k}
\textrm{ if }k=i,\\ F_{\mu_k}
W_{i,j}&=&qW_{i,j}F_{\mu_k}\textrm{ if }k= i-1,\\ F_{\mu_k}
Z_{i,j}&=&Z_{i,j}F_{\mu_k}\textrm{ if }k\neq i-1,\\
F_{\mu_k}Z_{i,j}^a&=&[a]Z_{i-1,j}Z_{i,j}^{a-1}K^{-1}_{\mu_k}+W^a_{i,j}E_{\mu_k}
\textrm{ if }k=i-1,\\ E_{\mu_k}
Z_{i,j}&=&Z_{i,j}E_{\mu_k}\textrm{ if }k\neq i,i-1,\\
E_{\mu_k}Z_{i,j}^a&=&[a]Z_{i,j}^{a-1}Z_{i+1,j}+q^{-a}Z^a_{i,j}E_{\mu_k}
\textrm{ if }k=i,\\ E_{\mu_k}
Z_{i,j}&=&qZ_{i,j}E_{\mu_k}\textrm{ if }k= i-1. \end{aligned}$$
There are similar formulas for the commutators involving $E_{\nu_k}$ and $F_{\nu_k}$. If e.g. $S$ denotes the obvious automorphism defined on generators by $W_{ij}\rightarrow
W_{j,i}$, and similarly, $Z_{ij}\rightarrow Z_{j,i}$ then $E_{\nu_k}=SE_{\mu_k}S$ and $F_{\nu_k}=SF_{\mu_k}S$.
Finite dimensional ${\mathcal U}_q({\mathfrak
k}^{\mathbb C})$ modules
=============================================
A non-zero vector $v_\Lambda$ of a finite dimensional module $V_\Lambda$ of ${\mathcal
U}_q({\mathfrak k}^{\mathbb C})$ is a highest weight vector of highest weight $\Lambda$, and $V_\Lambda$ is a highest weight module of highest weight $\lambda$, if$$\begin{array}{lccccccc} \forall i=1,\dots,n-1:
K_{\mu_i}^{\pm1}=q^{\pm\lambda^\mu_i}v_\Lambda,&
K_{\nu_i}^{\pm1}= q^{\pm \lambda^\nu_i}v_\Lambda, &\textrm{ and
} K_{\beta}^{\pm1}=q^{\pm\lambda}v_\Lambda.\\ \textrm{Finally,
}{\mathcal U}^+_q({\mathfrak k}^{\mathbb C})v_\Lambda=0,
&\textrm{and } {\mathcal U}^-_q({\mathfrak k}^{\mathbb
C})v_\Lambda=V . \end{array}$$ We set $\Lambda=((\lambda^\mu_1,\dots,\lambda^\mu_{n-1}),(\lambda^\nu_1,\dots,
\lambda^\nu_{n-1});\lambda)=(\Lambda_L, \Lambda_R,\lambda)$.
As a vector space, $V_\Lambda=V_{\Lambda_L}\otimes
V_{\Lambda_R}$ where $V_{\Lambda_L}$ and $V_{\Lambda_R}$ are highest weight representations of ${\mathcal U}_q({\mathfrak
k}_L^{\mathbb C})$ and ${\mathcal U}_q({\mathfrak k}_R^{\mathbb
C})$, respectively, of highest weights $\Lambda_L=(\lambda^\mu_1,\dots,\lambda^\mu_{n-1})$ and $\Lambda_R=(\lambda^\nu_1,\dots,\lambda^\nu_{n-1})$, respectively. The highest weight vector can then be written as $v_\Lambda=v_{\Lambda_L}\otimes v_{\Lambda_R}$ with the stipulation that $K_\beta^{\pm1}v_{\Lambda_L}\otimes
v_{\Lambda_R}=q^{\pm\lambda}v_{\Lambda_L} \otimes
v_{\Lambda_R}$.
Generalized quantized Verma modules and their homomorphisms
===========================================================
Consider a finite dimensional module $V_\Lambda=V_{\Lambda_L,\Lambda_R,\lambda}$ over ${\mathcal
U}_q({\mathfrak k}^{\mathbb C})$ with highest weight is defined by $\Lambda=(\Lambda_L,\Lambda_R,\lambda)$ where $\Lambda_L=(\lambda^\mu_1,\lambda^\mu_2,\dots,\lambda^\mu_{n-1},0)$, $\Lambda_R=(\lambda^\nu_1,\lambda^\nu_2,\dots,\lambda^\nu_{n-1},0)$, and $\lambda\in{\mathbb C}$.
We extend such a module to a ${\mathcal U}_q({\mathfrak
k}^{\mathbb C}){\mathcal A}_q^+$ module, by the same name, by letting ${\mathcal A}_q^+$ act trivially.
The quantized generalized Verma module $M(V_\Lambda)$ is given by $$M(V_\Lambda)={\mathcal U}_q({\mathfrak
g}^{\mathbb C})\bigotimes_{{\mathcal U}_q({\mathfrak k}^{\mathbb
C}){\mathcal A}_q^+}V_\Lambda$$ with the natural action from the left.
As a vector space, $$M(V_\Lambda)={\mathcal
A}_q^-\otimes V_\Lambda.$$ We are interested in structure preserving homomorphisms between quantized generalized Verma modules. We call such maps intertwiners, covariants, or equivariants, indiscriminately. Dually, they will be quantized covariant differential operators. In abstract notation, the structure under investigation is$$Hom_{{\mathcal
U}_q({\mathfrak g}^{\mathbb
C})}(M(V_{\Lambda}),M(V_{\Lambda_1})).$$However, for the time being we will consider$$\label{former}Hom_{{\mathcal
A}_q^-{\mathcal U}_q({\mathfrak k}^{\mathbb
C})}(M(V_{\Lambda}),M(V_{\Lambda_1})).$$ An element $\phi_{\Lambda,\Lambda_1}$ in the latter space is completely determined by the ${\mathcal U}_q({\mathfrak k}^{\mathbb C})$ equivariant map, denoted by the same symbol: $$V_{\Lambda}\stackrel
{\phi_{\Lambda,\Lambda_1}}{\rightarrow} {\mathcal A}_q^-\otimes
V_{\Lambda_1}\textrm{ leads to } {\mathcal A}_q^-\otimes
V_{\Lambda}\stackrel{\phi_{\Lambda,\Lambda_1}}{\rightarrow}
{\mathcal A}_q^-\otimes V_{\Lambda_1}.$$Specifically, ${\phi_{\Lambda,\Lambda_1}}$ does not depend on $\lambda$ and is completely given by the condition that the image of the highest weight vector ${\phi_{\Lambda,\Lambda_1}}(v_{\Lambda})$ is a highest weight vector for ${\mathcal U}_q({\mathfrak k}^{\mathbb C})$. For the map $\phi_{\Lambda,\Lambda_1}$ to belong to the former space (\[former\]) it is necessary, and sufficient that, additionally, ($Z_\beta$ acting in $M(V_{\Lambda_1}$)) $$\label{zbeta} Z_\beta\left(
{\phi_{\Lambda,\Lambda_1}}(v_{\Lambda})\right)=0.$$ This equation depends heavily on $\lambda$. It is clear that such maps, whether of the first or second kind, can be combined: $$\phi_{\Lambda_1,\Lambda_2}\circ
\phi_{\Lambda,\Lambda_1}=\phi_{\Lambda,\Lambda_2}$$ though it may happen that the composite is zero.
We use the terminology of degree of elements of ${\mathcal
A}_q^-$ in the obvious way, and we let, for $k=1,\dots$, $
{\mathcal A}_q^-(k)$ denote the ${\mathcal U}_q({\mathfrak
k}^{\mathbb C})$ module spanned by homogeneous elements of degree $k$. If the elements $p_{ij}$ all belong to ${\mathcal
A}_q^-(k)$, we write $\phi^k_{\Lambda,\Lambda_1}$.
[**General Problem:**]{} When is it possible to have $\phi^1_{\Lambda_{n-1},\Lambda_{n}}\circ
\phi^1_{\Lambda_{n-2},\Lambda_{n-1}}\circ\cdots\circ\phi^1_{\Lambda,\Lambda_1}
={{\operatorname{Det}}_q}$? In this case, if $\Lambda=(\lambda_L,\Lambda_R,\lambda)$ , then $\Lambda_{n}=(\Lambda_L,\Lambda_R,\lambda+2)$.
Laplace expansion
=================
If $m=n$, one may define the quantum determinant $det_q$ in ${\mathcal A}_q-$ as follows: $$\begin{aligned}
\label{2b}
{{\operatorname{det}}_q}(n)={{\operatorname{det}}_q}&=&\Sigma_{\sigma\in
S_n}(-q^{-1})^{\ell(\sigma)}W_{1,\sigma(1)}W_{2,\sigma(2)}
\cdots W_{n,\sigma(n)}\\\label{3b}&=&\Sigma_{\delta\in
S_n}(-q^{-1})^{\ell(\delta)}W_{\delta(1),1}W_{\delta(2),2}
\cdots W_{\delta(n),n}. \end{aligned}$$
If $m=n$ and $I=\{i_1<1_2< \dots <
i_{n-1}\}=\{1,2,\cdots,n\}\setminus\{i\},J=\{j_1<j_2< \dots <
j_{n-1}\}=\{1,2,\cdots,n\}\setminus\{j\}$, we set $$\begin{aligned}
\label{2} A(i,j)&=&\Sigma_{\sigma\in
S_{n-1}}(-q^{-1})^{\ell(\sigma)}W_{i_1,j_{\sigma(1)}}W_{i_2,j_{\sigma(2)}}
\cdots
W_{i_{n-1},j_{\sigma({n-1})}}\\\label{3}&=&\Sigma_{\tau\in
S_{n-1}}(-q^{-1})^{\ell(\tau)}
W_{i_{\tau(1)},j_1}W_{i_{\tau(2)},j_2} \cdots
W_{i_{\tau({n-1})},j_{n-1}}.\end{aligned}$$ These elements are quantum $(n-1)\times(n-1)$ minors. The following was proved by Parshall and Wang [@pw]:
${{\operatorname{det}}_q}$ is central. Furthermore, let $i,k\le n$ be fixed integers. Then
$$\begin{aligned}
\label{325}\delta_{i,k}{{\operatorname{det}}_q}&=&\sum_{j=1}^n(-q^{-1})^{j-k}W_{i,j}
A(k, j)=\sum_j(-q^{-1})^{i-j} A(i,j)W_{k,j}\\\label{326}
&=&\sum_j(-q^{-1})^{j-k}W_{j,i}A(j,k) =
\sum_j(-q^{-1})^{i-j}A(j,i)W_{j,k}.\end{aligned}$$
1. order
========
Any finite dimensional highest weight representation of ${\mathcal U}_q({\mathfrak k}^\mathbb C)$ of the form $\Lambda=(\Lambda_L,\Lambda_R,\lambda)$ in which either $\Lambda_L=0$ or $\Lambda_R=0$ will be called one-sided. We will now give an explicit form for a highest weight vector $v_1$ of an irreducible sub-representation of ${\mathcal A}^-(1)\otimes
V_{\Lambda=(\Lambda_L,0,\lambda)}$. Specifically, consider$$\label{hwv}
v_1=W_{N+1,1}v_0+W_{N,1}u_Nv_0
+W_{N-1,1}u_{N-1,1}v_0+W_{N-2,1}u_{N-2}v_0
+\dots+W_{1,1}u_{1}v_0,$$ where $\forall
i=1,\dots,N: u_i\in {\mathcal
U}^{-\mu_N-\mu_{N-1}-\dots-\mu_{i}}_q({\mathfrak k}^\mathbb C)$.
Because of this, we first want to consider a basis of ${\mathcal
U}_q^{-\mu_N+\cdots- \mu_\ell}({\mathfrak k}^{\mathbb C}_L)$.
Set ${\mathcal E}_{\ell,N}=\{\ell,\ell+1,\cdots,N\}\subseteq
\{1,2,\dots,n-1\}$. Any sequence $I_{\ell,N}=(i_\ell,i_{\ell+1},\cdots,i_N)$ made up of pairwise different elements of ${\mathcal E}^\mu_{\ell,N}$ defines a non-zero element $$F_{\mu_{i_\ell}}F_{\mu_{i_{\ell+1}}}\cdots
F_{\mu_{i_N}}=F^\mu(I_{\ell,N})\in {\mathcal
U}_q^{-\mu_\ell+\cdots- \mu_N}({\mathfrak k}^{\mathbb C}_L).$$ We will call such a sequence [**allowed**]{}. We reserve the name $E_{\ell,N}$ for the special sequence $(\ell,\ell+1,\cdots,N)$.
We will say that a transposition $(i_\ell,i_{\ell+1},\dots,
i_k,i_{k+1},\dots,i_N)\rightarrow (i_\ell,i_{\ell+1},\dots,
i_{k+1},i_k,\dots,i_N)$ is legal if $\mid i_{k+1}-i_k\mid>1$.
Recall that $F_{\mu_i}F_{\mu_j}=F_{\mu_j}F_{\mu_i}$ if $\vert
i-j\vert>1$. We will say that two allowed sequences $I^{(1)}$ and $I^{(2)}$ are equivalent if one can be obtained from the other by a series of legal transpositions. It is clear that any allowed sequence $I$ can be brought, uniquely, and by legal transpositions, into the form $J_1J_2\cdots J_r$ which is the concatenation of sequences $J_t$ that are either descending or ascending, and such that the following are satisfied: Firstly, the elements of $J_s$ are smaller than the elements of $J_t$ if $s<t$, and $\cup_s J_s=\{\ell,\ell+1,\dots,N\}$. Secondly, two neighboring sequences cannot both be ascending (maximality), and thirdly, singletons are ascending.
We denote by ${\mathcal J}_{\ell,N}$ the set of such sequences. The following is then obvious:
$$\{F^\mu(I_{\ell,N})\mid
I_{\ell,N}\in {\mathcal J}_{\ell,N}\}$$
is a basis of $ {\mathcal U}_q^{-\mu_\ell+\cdots- \mu_N}({\mathfrak
k}^{\mathbb C}_L)$.
We furthermore have from e.g. [@dpw lemma 6.27]:
\[basis\] Let $V=V(\Lambda_L)$ be a finite dimensional highest weight representation of ${\mathcal
U}_q({\mathfrak k}^{\mathbb C}_L)$ with $\Lambda_L=(\lambda^\mu_1,\lambda^\mu_2,\cdots,
\lambda^\mu_{n-1})$ satisfying: $\lambda^\mu_\ell>0,
\lambda^\mu_{\ell+1}>0,\dots, \lambda^\mu_N>0$. Let $v_0$ denote a highest weight vector (unique up to a non zero constant). Then $$\{F^\mu(I_{\ell,N})v_0\mid I_{\ell,N}\in
{\mathcal J}_{\ell,N}\}$$ is a basis of $V^{\Lambda_L-\mu_\ell+\cdots- \mu_N}$.
If $I_{\ell,N}=J_1J_2\cdots J_s\in {\mathcal J}_{\ell,N}$ as above, we attach to it a sequence $C^\mu(I_{\ell,N})=(c_{i_\ell},c_{i_{ell+1}},\cdots c_{i_N})$ where $c_k=a_k$ if either $i_k$ belongs to an ascending sub-sequence $J_x$ of $I_{\ell,N}$ or if $i_k$ is the biggest element in a descending sub-sequence $J_y$ of $I_{\ell,N}$. Here, $x,y\in\{1, 2,\dots,s\}$. In the remaining cases, $c_{i_k}=b_{i_k}$. We furthermore set $f^\mu(C^\mu(I_{\ell,N}))=\prod_{t=\ell}^Nc_{i_t}$.
We can then state, maintaining the assumptions from Lemma \[basis\]:
\[Prop-basis\] If the vector $v_1$ in (\[hwv\]) is a highest weight vector in ${\mathcal
A}_1^-\otimes V(\Lambda_L)$ then $$\forall
\ell=1,\dots,N:u_\ell=\sum_{I_{\ell,N}\in{\mathcal
J}_{\ell,N}}f^\mu(C^\mu(I_{\ell,N}))F^\mu(I_{\ell,N})v_0.$$
Later, we shall find it convenient to set ${\mathcal J}_{N+1,N}=\emptyset$ and $f^\mu(C^\mu(\emptyset))=1 =F^\mu(\emptyset)$. Likewise, $E_{N+1,N}=\emptyset$.
Our general case of interest is where we only assume $\lambda^\mu_N\neq0$. Bear in mind that in the sequence $C(I_{\ell,N})$, $c_0=b_i$ signals that the corresponding $\mu_i$, taking part in $F(I_{\ell,N})$, can be moved all the way to the right without changing $F(I_{\ell,N})$. If we allow $\lambda^\mu_i=0$ this means that such elements, when applied to $v_0$, give zero. Hence if we let ${\mathcal
Z}_{\ell,N}=\{i={\ell,\cdots,N}\mid\lambda_i^\mu=0\}$ and if we let ${\mathcal J}_{\ell,N}^{\mathcal Z}$ denote those sequences $I$ in ${\mathcal J}_{\ell,N}$ for which any index $i$ from ${\mathcal Z}_{\ell,N}$ either belongs to an increasing sequence or is the biggest index in a decreasing sequence, then we have:
\[6.4\]$$\{F^\mu(I_{\ell,N})v_0\mid
I_{\ell,N}\in {\mathcal J}^{\mathcal Z}_{\ell,N}\}$$ is a basis of $V^{\Lambda_L-\mu_\ell+\cdots-
\mu_N}$.
Clearly there is an analogue to Proposition \[Prop-basis\] for this general case (just as long as $\lambda^\mu_N>0$).
There is yet another helpful way to view the various sets ${\mathcal J}_{\ell,N}$, $\ell=N,N-1,\dots,1$, namely as a labeled, directed rooted tree with root at $F_{\mu_N}$: $$\begin{array}{rcl} &F^\mu(I_{\ell,N})&\\
\stackrel{L_{\ell-1}}{\swarrow}&&\stackrel{R_{\ell-1}}{\searrow}\\
F_{\mu_{\ell-1}}F^\mu(I_{\ell,N})
&&F^\mu(I_{\ell,N})F_{\mu_{\ell-1}} \end{array}.$$ Here, it is really only the relative positions of $F_{\mu_\ell}$ and $F_{\mu_{\ell-1}}$ that matter.
If we have $\lambda_i^\mu=0$ we just modify the tree by removing all branches labeled by $R_i$ - as well as everything above these branches - from the tree. (In this picture, the root is lowest.)
In this way, there is an obvious bijection between the paths in the modified tree and the basis.
We now return to (\[hwv\]). To obtain the following equations, it is used that $E_{\mu_{i-1}}(W_{i,j})=-qW_{i-1,j}K_{\mu_{i-1}}
+(W_{i,j})E_{\mu_{i-1}}$, which follows from Lemma \[2.1\]. Furthermore, for the vector in (\[hwv\]) to be a ${\mathcal
U}_q({\mathfrak k}^\mathbb C)$ highest weight vector we clearly only need to look at ${\mathcal U}_q({\mathfrak k}_L^\mathbb
C)$. Here we must have: $$\begin{aligned}
\forall i=1,\dots,N:
(-q)W_{i,1}K_{\mu_i}u_{i+1}v_0
+W_{i,1}E_{\mu_i}u_{i}v_0&=&0\\\forall i,j=1,\dots,N:
E_{\mu_j}u_{i}v_0&=&0\textrm{ if }i\neq j. \end{aligned}$$ We assume throughout that $\lambda^\mu_N\neq0$.
Using Proposition \[Prop-basis\], we set $u_{N+1}=1$ and $$\forall i=1,\dots,N, u_{i}:=a_{i}
F_{\mu_{i}}u_{i+1}+b_{i}u_{i+1}F_{\mu_{i}} \textrm{ (except
$b_{N}:=0$)}.$$
The vector $v_1$ in (\[hwv\]) is a highest weight vector if and only if $$\begin{aligned}
a_N=\frac{q^{1+\lambda^\mu_N}}{[\lambda^\mu_N]},&\\
(a_{N-1}[\lambda^\mu_{N-1}+1]+b_{N-1}[\lambda^\mu_{N-1}])u_{N}v_0=q^{\lambda^\mu_{N-1}+2}
u_{N}v_0,&\\
(a_{N-1}[\lambda^\mu_{N}]+b_{N-1}[\lambda^\mu_{N}+1])F_{\mu_{N-1}}u_{N+1}v_0=0.&\\
\textrm{ For $i<N-1$: }\qquad\qquad&\\
(a_{i}[\lambda^\mu_{i}+1]+b_{i}[\lambda^\mu_{i}])u_{i+1}v_0=q^{\lambda^\mu_{i}+2}u_{i+1}
v_0,&\\
\left(a_{i}(a_{i+1}[\lambda^\mu_{i+1}+1]+b_{i+1}[\lambda^\mu_{i+1}])\right)F_{\mu_i}u_{i+2}
v_0\ +&\\\left(b_{i}(a_{i+1}[
\lambda^\mu_{i+1}+2]+b_{i+1}[\lambda^\mu_{i+1}+1])\right)F_{\mu_i}u_{i+2}
v_0=0.\label{lambda-1}&\nonumber \end{aligned}$$
In continuation of the discussion following Proposition \[6.4\], notice that if $\lambda^\mu_i=0$ then equation (\[lambda-1\]) should be stricken, $b_i=0$, and $a_i=q^2$.
Returning to the general case: If all $\lambda^\mu_i\neq0$:
$$\begin{aligned}
a_N&=&\frac{q^{\lambda^\mu_N+1}}{[\lambda^\mu_N]}.\\\forall
k=1,\dots,N-1:\\ \label{a-n-k}
a_{N-k}&=&q^{\lambda^\mu_{N-k}+2}\frac{[\lambda^\mu_N+\dots+\lambda^\mu_{N-k+1}+k]}{[
\lambda^\mu_N+\dots+\lambda^\mu_{N-k+1}+\lambda^\mu_{N-k}+k]},\\
b_{N-k}&=&-q^{\lambda^\mu_{N-k}+2}\frac{[\lambda^\mu_N+\lambda^\mu_{N-k+1}+k-1]}{[
\lambda^\mu_N+\dots+\lambda^\mu_{N-k+1}+\lambda^\mu_{N-k}+k]}.\end{aligned}$$
If $\lambda^\mu_{N-1}=0=\dots=\lambda^\mu_{N-R}$ the $a_{N-k}$ just become $q^2$ for $k=1,\dots,R$. This is just the limit of the equations (\[a-n-k\]). The corresponding $b_{N-k}=0$ seemingly do not have a nice limit, but recall that instead, we just cut all branches of the tree marked by $R_{N-i}$, $i=1.\dots,R$. Actually, in this sense there is a nice limit for any case in which $\lambda^\mu_i=0$ for some values of $i=1,\dots, N-1$.
One-sidedness
=============
Recall that ${{\operatorname{det}}_q}$ is central in ${\mathcal A}^-$.
If $\Lambda=(\Lambda_L,\Lambda_R,\lambda)$ and if $$\label{60}
\phi^1_{\Lambda_{n-1},\Lambda_{n}}\circ
\phi^1_{\Lambda_{n-2},\Lambda_{n-1}}\circ\cdots\circ\phi^1_{\Lambda,\Lambda_1}
={{\operatorname{Det}}_q},$$ where ${{\operatorname{Det}}_q}$ denotes the operator $M(V_{\Lambda})\ni p\rightarrow {{\operatorname{det}}_q}\cdot p \in
M(V_{\Lambda_n})$, then at least one of the pair $\Lambda_L,\Lambda_R$ is 0.
We call such a representation [*one-sided*]{}. We shall see later that there is a converse to this.
The proof (sketched) is obtained in 10 installments:
[**1.**]{} We shall need the following elementary result:
Let $a,b\in{\mathbb N}$ with $b\leq a$. Then $$[a]_q[b]_q=[a+b-1]_q+[a+b-3]_q+\dots +[a-b+1]_q.$$
[*Proof of Lemma:*]{} Using that $[a+1]_q=q^{-a}+q^{-a+2}+\dots+q^a$, this follows easily by counting $q$ exponents.
[**2.**]{} We have that ${{\operatorname{det}}_q}\otimes V\subseteq
{\mathcal A}_1^-\otimes {\mathcal A}_{n-1}^-\otimes V={\mathcal
A}_1^-\otimes ( {\mathcal A}_{n-1}^-\otimes V)=({\mathcal
A}_1^-\otimes {\mathcal A}_{n-1}^-)\otimes V$ and ${\mathcal
A}_{n-1}^-$ is a sum of double tableaux of box size $(n-1)\times(n-1)$ and similarly ${\mathcal A}_{n}^-$ is a sum of double tableaux of box size $(n)\times(n)$. By the Littlewood-Richardson rule, to get ${{\operatorname{det}}_q}$ we need to use the invariant subspace ${\mathcal A}_{n-1}^-(n-1)$ of $(n-1)\times
(n-1)$ minors in ${\mathcal A}_{n-1}^-\otimes V$. We can ignore contributions from other minors.
[**3.**]{} We now extend the notation used in Proposition \[Prop-basis\] to also cover the cases of representations of ${\mathcal U}_q({\mathfrak k}^{\mathbb C}_R)$ in the obvious way. We then have the following extension of said proposition:
$$\label{1}\textrm{If }\quad
v_1=\sum_{k=1,\ell=1}^{i+1,j+1}W_{k,\ell}u_{k,\ell}$$
is a highest weight vector and $u_{i+1,j+1}=1$, then
$$\begin{aligned}
\label{2}\forall k=1,\dots, i+1,
\ell=1,\dots,j+1:\\u_{k\ell}=\sum_{I_{k,i}\in{\mathcal
J}_{k,i},I_{\ell,j}\in{\mathcal
J}_{\ell,j}}f^\mu(C^\mu(I_{k,i}))f^\nu(C^\nu(I_{\ell,j}))
F^\mu(I_{k,i})F^\nu(I_{\ell,j})v_0.\nonumber \end{aligned}$$
[**4.**]{} Let ${\mathcal A}_{n-1}^-(n-1)$ denote the space generated by the $(n-1)\times (n-1)$ minors in ${\mathcal
A}^-$. This is a ${\mathcal U}_q({\mathfrak k}^{\mathbb C})$ module of highest weight $\Lambda^\mu=(0,0,\dots,0,1)=\Lambda^\nu$. The same kind of reasoning can be applied to ${\mathcal A}_{n-1}^-(n-1)\otimes
V$. (Notice that ${\mathcal A}_{n-1}^-(n-1)$ is the dual to ${\mathcal A}_{1}^-$.) A highest weight vector $v_0$ in an irreducible submodule $V_0\subseteq {\mathcal
A}_{n-1}^-(n-1)\otimes \tilde V$ has the form $$v_0= \sum_{k,\ell} A(a+k,b+\ell) \tilde
u_{a+k,b+\ell}\tilde v_0,\label{6}$$ where the vectors $\tilde u_{a+k,b+\ell}\tilde v_0$, if $k+\ell>0$, have weights strictly smaller that $\tilde v_0$.
[**5.**]{} If we insert (\[6\]) into (\[1\]) and isolate the $\tilde v_0$ terms, we get in particular, using (\[60\], (\[2\]), and since clearly here $(a,b)=(i+1,j+1)$ that $$\sum_{k=1}^{i+1}\sum_{\ell=1}^{j+1}
W_{k,\ell}\sum_{I_{k,i}\in{\mathcal
J}_{k,i},I_{\ell,j}\in{\mathcal
J}_{\ell,j}}f^\mu(C^\mu(I_{k,i}))f^\nu(C^\nu(I_{\ell,j}))
F^\mu(I_{k,i})F^\nu(I_{\ell,j})A(i+1,j+1)=\kappa\cdot{{\operatorname{det}}_q}$$ for some constant $\kappa\neq0$. It is easy to see that $F^\mu(I_{k,i})F^\nu(I_{\ell,j}) A(i+1,j+1))=0$ unless $(I_{k,i}),I_{\ell,j})=(E_{k,i},E_{\ell,j})$. In the latter case we get, by (74) in Chapter 1, $(-q^{-1})^{i+j-k-\ell
}A(k,\ell)$.
So $$\sum_{k=1}^{i+1}\sum_{\ell=1}^{j+1}
W_{k,\ell}(-q^{-1})^{i+j-k-\ell
}f^\mu(C^\mu(E_{k,i}))f^\nu(C^\nu(E_{\ell,j})A(k,\ell)=\kappa\cdot{{\operatorname{det}}_q}.\label{9}$$
[**6.**]{} If both $i+1<n$ and $j-1<n$ we can apply $F_{\nu_{n-1}}\dots F_{\nu_{j+1}}F_{\mu_{n-1}}\dots
F_{\mu_{i+1}}$ to both sides of (\[9\]) and get that $W_{n,n}A(i+1,j+1)=0$; a contradiction.
[**7.**]{} Let us first assume that $i=j=n$. If we set $d_{k,\ell}=f^\mu(C^\mu(E_{k,i}))f^\nu(C^\nu(E_{\ell,j})$, (\[9\]) becomes
$$\sum_{k=1}^{n}\sum_{\ell=1}^{n}
W_{k,\ell}d_{k,\ell}(-q^{-1})^{2n-k-\ell
}A(k,\ell)=\kappa\cdot{{\operatorname{det}}_q}.\label{99}$$
Using (\[325\]) we can subtract a certain multiple of ${{\operatorname{det}}_q}$ in each row such that in the resulting equations
$$\sum_{k=0}^{n-1}\sum_{\ell=0}^{n-1}
W_{n-k,n-\ell}b_{k,\ell}A(n-k,n-\ell)=\tilde
\kappa\cdot{{\operatorname{det}}_q},\label{999}$$
we may assume: $\forall k:b_{k,n}=0$. Of course, this may change the constant into $\tilde\kappa$. A) If all the remaining $b_{k,\ell}$s are zero then, naturally, the resulting $\tilde\kappa$ is zero but that will also imply that each row of the original system satisfies, up to a constant non-zero multiple, equation (\[325\]). In particular, $$\sum_{k=1}^{n}
W_{k,n}d_{k,n}(-q^{-1})^{n-k}A(k,n)=\tilde\kappa\cdot{{\operatorname{det}}_q}.\label{10}$$ B) If a non-zero system remains, we can subtract using column equations (\[326\]) to remove the terms $W_{nj}A(n,j)$; $j=1,\dots,n-1$ (the term with $j=n$ has already been removed. If there still remains an equation $$\sum_{k=0}^{i-1}\sum_{\ell=0}^{j-1}
W_{i-k,j-\ell}b_{k,\ell}A(i-k,j-\ell)=\kappa\cdot{{\operatorname{det}}_q},\label{9a}$$ we reach a contradiction as in [**6**]{}.
In conclusion:
There is either a column equation
$$\sum_{k=1}^{n}
W_{k,n}d_{k,n}(-q^{-1})^{n-k}A(k,n)=\tilde\kappa\cdot{{\operatorname{det}}_q},\label{100}$$
or an analogous row equation
$$\sum_{\ell=1}^{n}
W_{n,\ell}d_{n,\ell}(-q^{-1})^{n-\ell}A(n,\ell)=\tilde\kappa\cdot{{\operatorname{det}}_q}.\label{101}$$
[**8.**]{} Suppose that we have a row equation
$$\sum_{\ell=1}^{n}
W_{n,\ell}d_{n,\ell}(-q^{-1})^{n-\ell}A(n,\ell)=\tilde\kappa\cdot{{\operatorname{det}}_q}.\label{101}$$
Then $\lambda^\mu_{n-1}=1$ and $\forall
i=1,\dots,n-2:\lambda^\mu_i=0$.
We have a PBW basis made up of monomials $W_{n,j_n,
W_{n,j_{n-1}}}, \dots, W_{1,j_1}$. It follows that $\kappa=1$ and it follows from (\[101\]) and (\[325\]) that $\forall
k=d_{k,n}=q^{2(n-k)}$. It is easy to see (see [**7.**]{}) that $d_{k,n}=a_{n-1}a_{n-2}\cdots a_k$. This clearly implies that $a_{k}=q^2$ for all $k=1,\dots. n-1$.
In particular, $a_{n-1}=q^2$, hence $$q^2=\frac{q^{1+\lambda^\mu_{n-1}}}{[\lambda^\mu_{n-1}]_q}\Rightarrow
q^{2\lambda^\mu_{n-1}-2}=1\Rightarrow \lambda^\mu_{n-1}=1.$$
Inductively, it follows from (\[a-n-k\]) that $$q^{\lambda^\mu_{N-k}}\frac{[1+k]}{[\lambda^\mu_{N-k}+1+k]}=1\Rightarrow
\lambda^\mu_{N-k}=0.$$
[**9.**]{} If there is a column equation, it follows in the same way that $\Lambda_R=(0,0,\dots,0,1)$.
[**10.**]{} By [**6, 7**]{} what remains are the cases $i<n, j=n$ and $i=n, j<n$. However, it is clear that they, by inspection, are covered by the arguments of the case $i=j=n$ simply by eliminating one possibility, so that if $j=n$, we must have $\Lambda_R=0$ and if $i=n$ we must have $\lambda_L=0$.
We have the following converse which is quite straightforward:
Let $V_\Lambda=V(\Lambda_L,0,\lambda)$. Set $\Lambda_0=\Lambda$. Then there exist ${{\mathcal
A}_q^-{\mathcal U}_q({\mathfrak k}^{\mathbb C})}$ intertwining maps maps $\psi^1_{\Lambda_i,\Lambda_{i+1}}:
V_{\Lambda_{i}}\rightarrow V_{\Lambda_{i+1}} \subset
V_{\Lambda_{i}}\otimes {\mathcal A}^-_q(1)$, for $i=0,1\dots,n-1$, independent of $\lambda$, such that, with $\Lambda_n=(\Lambda^0_\mu,0,\lambda+2)$, $$\psi^1_{\Lambda_{n-1},\Lambda_n}\circ\psi^1_{\Lambda_{n-2},\Lambda_{n-1}}
\circ\cdots\circ\psi^1_{\Lambda,\Lambda_1}={{\operatorname{Det}}_q}.$$ This decomposition is not unique. Furthermore the maps may be grouped together to form maps of higher degrees, defined by means of minors of the given degree.
First order intertwiners
========================
It is clear that any submodule $V_{\Lambda_1}$ of ${\mathcal
A}_q^-(1)\otimes V_\Lambda$ defines a ${{\mathcal
A}_q^-{\mathcal U}_q({\mathfrak k}^{\mathbb C})}$ equivariant map $M(V_{\Lambda_1})\rightarrow M(V_\Lambda)$. We shall now see that there is a unique $\lambda=\lambda(\Lambda_L,\Lambda_R)$ for which this becomes a ${\mathcal U}_q({\mathfrak g}^{\mathbb
C})$ equivariant map. See our forthcoming article for details. Notice also that the integrality assumption on $(\Lambda_L,\Lambda_R)$ is not used.
We need the following extra information. Modulo ${\mathcal
A}_q^-E_\beta$ it holds:$$\begin{aligned}
Z_\beta(W_{i,1})&=&
T_{\mu_{i-1}}T_{\mu_{i-2}}\dots
T_{\mu_{2}}(F_{\mu_1})K_\beta^{-1},\\Z_\beta(W_{i,j})&=&-(q-q^{-1})T_{\nu_{j-1}}T_{\nu_{j-2}}\dots
T_{\nu_{2}}(F_{\nu_1}) T_{\mu_{i-1}}T_{\mu_{i-2}}\dots
T_{\mu_{2}}(F_{\mu_1})K_\beta^{-1} \textrm{ if
}i,j\geq2.\end{aligned}$$
To any ${\mathcal U}_q({\mathfrak k}^{\mathbb C})$ homomorphism $V_{\Lambda_1}\rightarrow {\mathcal
A}_q^-(1)\otimes V_\Lambda$ there corresponds a unique $\lambda$ such that $\psi_{\Lambda_1,\Lambda}\in Hom_{{\mathcal
U}_q({\mathfrak g}^{\mathbb
C})}(M(V_{\Lambda_1}),M(V_\Lambda))$.
We focus on the case where $\Lambda=(\Lambda_L,0,\lambda)$. Recall (\[zbeta\]) and consider $$\begin{aligned}
Z_\beta(W_{N+1}v_0+W_{N}u_Nv_0
+W_{N-1}u_{N-1}v_0+W_{N-2}u_{N-2}v_0 +\dots+W_{1}u_{1}v_0)=\\
q^{-\lambda}\sum_{k=0}^{N-1}T_{\mu_{N-k}}\cdots
T_{\mu_2}T_{\mu_1}(F_{\mu_1})u_{N-k+1}v_0+[\lambda_1+1]u_1v_0=0.\label{sum}\end{aligned}$$
We may expand the equation into equations for each vector $F^\mu(I_{1,N})$ in the basis. We claim that the general case can be reduced by contraction of trees to just the equation for $F^\mu(E_{1,N})=F_{\mu_1}F_{\mu_2}\cdots F_{\mu_N}v_0$. Here we get$$q^{-\lambda}(1-a_N+a_Na_{N-1}+
a_Na_{N-1}a_{N-2}+\cdots+a_Na_{N-1}a_{N-2}\cdots
a_2+[\lambda+1]a_Na_{N-1}a_{N-2}\cdots a_2 a_1=0.$$ $$\begin{aligned}
1+a_N&=&q\frac{[\lambda^\mu_N+1]}{[\lambda^\mu_N]},\\
1+a_N+a_Na_{N-1}&=&q^2\frac{[\lambda^\mu_N+1]}{[\lambda^\mu_N]}\frac{[\lambda^\mu_N+\lambda^\mu_
{N-1}+2]}{[\lambda^\mu_N+\lambda^\mu_{N-1}+1]},\\
1+a_N+a_Na_{N-1}+a_Na_{N-1}a_{N-2}&=&q^3
\frac{[\lambda^\mu_N+1]}{[\lambda^\mu_N]}\frac{[
\lambda^\mu_N+\lambda^\mu_{N-1}+2]}{[\lambda^\mu_N+\lambda^\mu_{N-1}+1]}
\frac{[\lambda^\mu_N+\lambda^\mu_{N-1}+\lambda^\mu_{N-2}+
3]}{[\lambda^\mu_N+\lambda^\mu_{N-1}
+\lambda^\mu_{N-2}+2]}.\nonumber\\\end{aligned}$$ $$\begin{aligned}
\label{S}S:=1+a_N+a_Na_{N-1}+a_Na_{N-1}a_{N-2}+\cdots+a_Na_{N-1}a_{N-2}\cdots
a_2&=&\\q^{N-1}\frac{[\lambda^\mu_N+1]}{[\lambda^\mu_N]}\cdots
\frac{[\lambda^\mu_N+\cdots+\lambda^\mu_{N-k}+k+1]}{[\lambda^\mu_N+\cdots+\lambda^\mu_{N-k}+k]}
\cdots
\frac{[\lambda^\mu_N+\lambda^\mu_{N-1}+\dots+\lambda^\mu_{2}+
N-1]} {[\lambda^\mu_N+\lambda^\mu_{N-1}+\dots+
\lambda^\mu_{2}+N-2]}. \end{aligned}$$
Comparing to $$\label{T}T:=a_Na_{N-1}a_{N-2}\cdots
a_2a_1,$$ one easily obtains $$\label{lambdaeq}
q^{-\lambda}S+[\lambda+1]T=0,$$ which upon divison becomes $$q^{-\lambda}+[\lambda+1]q^{\lambda^\mu_1+\lambda^\mu_2+\cdots+
\lambda^\mu_N+N}
\frac1{[\lambda^\mu_1+\lambda^\mu_2+\cdots+\lambda^\mu_N+N-1]}=0.$$ Using the equation $[a+b]=q^{-a}[b]+q^b[a]$, one easily concludes: $$=0.$$ This result can easily be generalized to the general first order case. It is related to the $q$-Shapovalov form [@mu].
References {#references .unnumbered}
==========
[9]{} Deng B, Du J, Parshall B and Wang J 2008 [*Finite Dimensional Algebras and Quantum Groups*]{} Mathematical Surveys and Monographs [**150**]{} (Providence RI: American Math. Soc.)
Jakobsen HP 1985 [*Basic covariant differential operators on hermitian symmetric spaces*]{} Ann. scient. Éc. Norm. Sup. [**18**]{} 421-436
Jantzen J 1996 [*Lectures on Quantum Groups*]{} Graduate Studies in Mathematics Vol. 6 (Providence RI: American Math. Soc.)
Lusztig G 1993 [*Introduction to Quantum Groups*]{} Progress In Mathematics [**11**]{} (Boston: Birkh[ä]{}user)
Mudrov A 2015 [*Orthogonal basis for the Shapovalov form on $U_q((n + 1)$*]{} Reviews in Mathematical Physics [**27**]{} Issue 2 1550004
Parshall P and Wang J 1991 [*Quantum Linear Groups*]{} Memoirs of the A.M.S. [**89**]{} Nr. [**439**]{} (Providence RI: American Math. Soc.)
|
---
abstract: 'P. Albano and P. Cannarsa proved in 1999 that, under some applicable conditions, singularities of semiconcave functions in ${{\mathbb R}}^n$ propagate along Lipschitz arcs. Further regularity properties of these arcs were proved by P. Cannarsa and Y. Yu in 2009. We prove that, for $n=2$, these arcs are very regular: they can be found in the form (in a suitable Cartesian coordinate system) $\psi(x) = (x, y_1(x)-y_2(x)), \ x \in [0,\alpha]$, where $y_1$, $y_2$ are convex and Lipschitz on $[0,\alpha]$. In other words: singularities propagate along arcs with finite turn.'
address: 'Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Praha 8-Karlín, Czech Republic'
author:
- Luděk Zajíček
title: A note on propagation of singularities of semiconcave functions of two variables
---
[^1]
Introduction
============
Let $u$ be a function defined on an open set $\Omega \subset {{\mathbb R}}^n$ which is locally (linearly) semiconcave; i.e., $f$ is locally representable in the form $u(x)= g(x) + K \|x\|^2$, where $g$ is concave (cf. [@CaSi]).
Let $\Sigma(u)$ be the singular set of $u$, i.e. $$\Sigma(u) = \{x \in \Omega:\ u \ \ \text{is not differentiable at}\ \ x\}.$$ It is clear that in many questions concerning $\Sigma(u)$ we can suppose that $u$ is concave (or convex), since the results for semiconcave functions then easily follow. But it is reasonable to formulate theorems for semiconcave functions, since these functions are important in a number of applications (see [@CaSi]).
It is well-known that $\Sigma(u)$ is a rather small set: it can be covered by countably many Lipschitz DC hypersurfaces ([@zajkonv]). (Note that for $A \subset {{\mathbb R}}^n$ there exists a convex (resp. semiconcave) function $u$ on ${{\mathbb R}}^n$ such that $A = \Sigma(u)$, if and only if $A$ is an $F_{\sigma}$ set which can be covered by countably many Lipschitz DC hypersurfaces, see [@Pa].)
The set $\Sigma(u)$ can have isolated points, but P. Albano and P. Cannarsa [@AC] found applicable conditions ensuring that $\Sigma(u)$ is in a sense big in each neigbourhood of a given $x_0 \in \Sigma(u)$. (The results of [@AC] can be found also in the book [@CaSi].) In particular, they proved that if $\partial D^+ u(x_0) \setminus D^* u(x_0) \neq \emptyset$ (see Preliminaries for the definitions), then a Lipschitz arc $\xi: [0,\tau] \to \Omega$ emanating from $x_0$ is a subset of the singular set $\Sigma(u)$. The results of [@AC] were refined in [@CY]; in particular it is proved in [@CY Corollary 4.3] that $\xi$ has nonzero (right continuous) right derivative at all points.
The purpose of the present note is to show that in ${{\mathbb R}}^2$ the results of [@CY] and methods from [@zajkonv] and [@VeZa] easily imply that the restriction of $\xi$ to an interval $[0,\tau']$ has an equivalent parametrization of the form (in a suitable Cartesian coordinate system) $\psi(x) = (x, y_1(x)-y_2(x)), \ x \in [0,\alpha]$, where $y_1$, $y_2$ are convex and Lipschitz on $[0,\alpha]$. (This result is equivalent to the assertion that the restriction of $\xi$ to an interval $[0,\tau^*]$ has finite turn, cf. Remark \[2\]). In particular, $\xi$ has (left continuous) left halftangents at all points.
The question whether the results can be generalized to the case $n >2$ remains open.
Preliminaries
=============
By $B(x,r)$ we denote the open ball with center $x$ and radius $r$. The scalar product of $v,w \in {{\mathbb R}}^n$ is denoted by $\langle v,w\rangle$. If $A \subset {{\mathbb R}}^n$, $c\in {{\mathbb R}}$ and $v \in {{\mathbb R}}^n$, then we define the sets $A+v$ and $c A$ by the usual way and similarly set $\langle v,A\rangle := \{\langle v,a\rangle:\ a \in A\}$. The boundary and the convex hull of a set $A \subset {{\mathbb R}}^n$ are denoted by $\partial A$ and ${\operatorname{conv}}A$, respectively. The (Fr' echet) derivative $Df(a)$ of a function $f$ on ${{\mathbb R}}^n$ at $a \in {{\mathbb R}}^n$ is considered as an element of ${{\mathbb R}}^n$. The one-sided derivatives of a real or vector function $\xi$ of one variable at $x \in {{\mathbb R}}$ are denoted by $\xi'_+(x)$ and $\xi'_-(x)$.
If $f$ is a function defined on a subset of ${{\mathbb R}}^n$, $x \in {{\mathbb R}}^n$ and $v\in {{\mathbb R}}^n$, then we define the one-sided directional derivative as $$f'_+(x,v):=\lim_{h\to 0+}\frac{f(x+hv)-f(x)}{h}.$$
Let $\Omega \subset {{\mathbb R}}^n$ be an open set and $u$ a locally semiconcave function on $\Omega$ (see Introduction). Then $u$ is locally Lipschitz and so differentiable a.e. in $\Omega$. For $x \in \Omega$, we define (see [@AC] or [@CaSi p. 54]) the set $$D^*u(x) = \{p \in {{\mathbb R}}^n:\ \Omega \ni x_i \to x,\, Du(x_i) \to p\}$$ of all [*reachable gradients*]{} of $u$ at $x$ (note that $D^*u(x) $ is also called limiting subdifferential, cf. [@AC p. 725]).
The [*superdifferential*]{} $D^+u(x)$ of $u$ at $x$ can be defined as the convex hull of $ D^*u(x)$ (see [@AC p. 723], cf. [@CaSi Theorem 3.3.6]).
Always $D^*u(x) \subset \partial D^+u(x)$ (see [@CaSi Proposition 3.3.4]). Note that the superdifferential $D^+u(x)= {\operatorname{conv}}D^*u(x)$ coincides with the Clarke’s subdifferential $\partial^C u(x)$ (since $\partial^C u(x)= {\operatorname{conv}}D^*u(x)$, see, e.g., [@Cl]).
Let $u(x)= g(x) + K \|x\|^2$, where $g$ is concave, on a ball $B(x_0,\delta) \subset \Omega$. Set $f:=-g$. Since $D(K\|x\|^2) = 2Kx$, we easily obtain that $ D^*u(x_0)= -D^*f(x_0)+ 2Kx_0$, and therefore $$\label{redconv}
D^+u(x_0) = -\partial f(x_0) + 2Kx_0,$$ where $\partial f$ is the classical subdifferential of the convex function $f$.
Recall that a function defined on an open convex subset of ${{\mathbb R}}^n$ is a [*DC function*]{} if it is a difference of two convex functions. We will need the following simple lemma which is a special case of the “mixing lemma” [@VeZa Lemma 4.8].
\[mix\] Let ${\varphi}_1,\dots,{\varphi}_p$ be DC functions on ${{\mathbb R}}$, and let $h$ be a continuous function on ${{\mathbb R}}$ such that $$h(x) \in \{{\varphi}_1(x),\dots,{\varphi}_p(x)\}\ \ \ \text{for each}\ \ \ x \in {{\mathbb R}}.$$ Then $h$ is DC on ${{\mathbb R}}$.
We will need also the well-known fact that convex functions are semismooth (see [@Mi Proposition 3], cf. also [@Sp Proposition 2.3]). In other words:
\[semismooth\] Let $f$ be a convex function on an open convex set $C \subset {{\mathbb R}}^n$ and $x_0 \in C$. Let $0 \neq q \in {{\mathbb R}}^n$, $q_n \to q$, $t_n \searrow 0$, and $z_n \in \partial f(x_n)$, where $x_n:= x_0 +t_nq_n$, be given. Then $\langle q,z_n\rangle \to f_+'(x_0,q)$. In particular, $$\label{diamknule}
{\operatorname{diam}}\langle q, \partial f(x_n)\rangle \to 0.$$
The result and its proof
========================
The following result is an immediate consequence of [@CY Corollary 4.3].
\[ACY\] Let $u$ be a semiconcave function on an open set $\Omega \subset {{\mathbb R}}^n$, $x_0 \in \Sigma(u)$ be a singular point of $u$ and $$\partial D^+ u(x_0) \setminus D^* u(x_0) \neq \emptyset.$$ Then there exists $q \in {{\mathbb R}}^n$ with $\|q\|=1$, $\tau>0$, and a Lipschitz curve $\xi: [0,\tau] \to \Sigma(u)$ such that
1. $\xi'_+(0) =q$,
2. $\lim_{s \to 0+} \xi'_+(s) = q$, and
3. $\inf_{s \in [0,\tau]}\ {\operatorname{diam}}\,D^+ u(\xi(s)) \, > 0$.
Note that it is proved in [@CY] also that $\xi'_+(s)$ exists for each $s \in [0,\tau)$ and $\xi'_+$ is right continuous on $[0,\tau)$. Further note that the result without (ii) was proved already in [@AC].
Using Theorem CY and the method of the proof of the implicit function theorem for DC functions [@VeZa Theorem 4.4], we easily prove the following result.
\[main\] Let $u$ be a semiconcave function on an open set $\Omega \subset {{\mathbb R}}^2$, $x_0 \in \Sigma(u)$ be a singular point of $u$ and $$\partial D^+ u(x_0) \setminus D^* u(x_0) \neq \emptyset.$$ Then there exist a Cartesian coordinate system in ${{\mathbb R}}^2$ given by a map $A:{{\mathbb R}}^2\to {{\mathbb R}}^2$ such that $A(x_0)=(0,0)$, and convex Lipschitz functions $y_1, y_2$ on some $[0,\alpha]$ ($\alpha>0$) such that, denoting $\psi(x):= (x, y_1(x)-y_2(x))$, $x \in [0,\alpha]$, we have $\psi(0) = (0,0)$ and $A^{-1}(\psi([0,\alpha])) \subset \Sigma(u)$.
Let $\xi: [0,\tau] \to \Sigma(u)$ and $q \in {{\mathbb R}}^2$ have properties from Theorem CY. We will proceed in four steps. In steps 1-3 we will suppose that $$\label{spec}
x_0 = (0,0)\ \ \text{and}
\ \ \ q= (1,0).$$
[*Step 1*]{} Set $e_2:= (0,1)$. Let $u(x)= g(x) + K \|x\|^2$ for $x \in B(x_0,\delta) \subset \Omega$, where $g$ is concave and Lipschitz with a constant $L>0$ on $B(x_0,\delta)$. Set $f:=-g$. Applying to any point $x \in B(x_0,\delta)$, we obtain $D^+u(x) = -\partial f(x) + 2Kx, \ x \in B(x_0,\delta)$. So (iii) (of Theorem CY) easily implies that, for some $0 <\tau_1< \tau$, we have that $f(\xi(s)) \in B(x_0,\delta)$ and $\partial f(\xi(s)) \subset B(0,L)$ for each $s \in [0,\tau_1]$, and $$\label{pdvel}
\inf_{s \in [0,\tau_1]}\ {\operatorname{diam}}\, \partial f(\xi(s)) \, > 0.$$
We will show that there exists $0< \tau_2 < \tau_1$ such that $$\label{diam1}
\delta := \inf_{s \in (0,\tau_2]}\ {\operatorname{diam}}\,\langle e_2, \partial f(\xi(s))\rangle \, > 0$$ Suppose on the contrary that there exits a sequence $(t_n)$ such that $t_n\searrow 0$ and $$\label{diam2}
\lim_{n \to \infty}
{\operatorname{diam}}\,\langle e_2, \partial f(\xi(t_n))\rangle = 0.$$ Set $q_n := \xi(t_n)/t_n$ and $x_n := \xi(t_n) = t_nq_n$. Since $q_n \to q$ by (i), Lemma \[semismooth\] gives that $$\label{diamx}
\lim_{n \to \infty}
{\operatorname{diam}}\,\langle q, \partial f(\xi(t_n))\rangle = 0.$$ Since and clearly imply $\lim_{n \to \infty}
{\operatorname{diam}}\, \partial f(\xi(t_n)) = 0$, we obtain a contradiction with .
[*Step 2*]{} Let $\xi = (\xi_1,\xi_2)$. By (ii), we have $\lim_{s\to 0+} (\xi_1)_+'(s) = 1$ and therefore there exits $0 < \tau_3 <\tau_2$ such that $1/2 \leq (\xi_1)'(s)$ for a.e. $s \in (0,\tau_3)$. So $\xi_1$ is Lipschitz strictly increasing on $[0,\tau_3]$ and $(\xi_1)^{-1}$ is Lipschitz on $[0, \alpha]$, where $\alpha:=\xi_1(\tau_3)$. Set $g(x):= \xi_2 \circ (\xi_1)^{-1}(x),\ x \in [0,\alpha]$. Then $g$ is Lipschitz and $\psi(x): = (x,g(x)),\ x \in [0,\alpha]$, is an equivalent parametrization of $\xi|_{[0, \tau_3]}$.
[*Step 3*]{} Choose a partition $-L=y_0 <y_1 <\dots<y_p=L\}$ of the interval $[-L,L]$ such that $\max\{y_{i} - y_{i-1},\ i=1,\dots,p\} < \delta/2$. For each $x \in (0,\alpha)$, the set $\langle e_2, \partial f(\psi(x))\rangle \subset [-L,L]$ is a closed interval of length at least $\delta$ and so we can choose $i_x \in \{1,\dots,p\}$ such that $$\label{oba}
y_{i_x} \in \langle e_2, \partial f(\psi(x))\rangle\ \ \ \text{and}\ \ \ y_{i_x-1} \in \langle e_2, \partial f(\psi(x))\rangle.$$ For $i\in \{1,\dots,p\}$, set $A_i := \{x\in (0,\alpha):\ i_x = i\}$. We will show that, for each $i \in \{1,\dots,p\}$ with $A_i \neq \emptyset$, the function $g|_{A_i}$ can be extended to a Lipschitz DC function ${\varphi}_i$ on ${{\mathbb R}}$.
To this end, fix a such $i$ and set $$\omega_1(x):= f(x,g(x))-y_i g(x)\ \ \text{and}\ \ \omega_2(x):= f(x,g(x))-y_{i-1} g(x)\ \ \text{for}\ x \in
A_i.$$ Since $\omega_1(x) - \omega_2(x) = (y_{i-1}- y_i) g(x),\ x \in A_i$, it is sufficient to prove that $\omega_i$ ($i=1,2$) can be extended to a Lipschitz convex function $c_i$ defined on ${{\mathbb R}}$.
For each $x\in A_i$, choose $p_x \in {{\mathbb R}}$ such that $(p_x,y_i) \in \partial f (x,g(x))$ and consider the affine function $$a_x(t) := \omega_1(x) + p_x(t-x),\ \ t \in {{\mathbb R}}.$$ Set $$c_1(t) := \sup \{a_x(t): \ x \in A_i\},\ \ t \in {{\mathbb R}}.$$ Since $\omega_1$ is clearly bounded on $A_i$ and $|p_x| \leq L$ for $x\in A_i$, it is easy to see that $c_1$ is a Lipschitz convex function on ${{\mathbb R}}$.
Now consider arbitrary $x, t \in A_i,\ x\neq t$. Since $(p_x,y_i) \in \partial f (x,g(x))$, we have $$f(t,g(t))-f(x,g(x)) \geq p_x(t-x) + y_i(g(t)-g(x)),$$ and therefore $$\omega_1(t) = f(t,g(t))- y_i g(t) \geq f(x,g(x)) - y_i g(x) + p_x(t-x) = a_x(t).$$ Since $a_t(t) = \omega_1(t),\ t \in A_i$, we obtain that $c_1$ extends $\omega_1$. Quite similarly we can find a convex Lipschitz extension $c_2$ of $\omega_2$.
Since $g(x) \in \{{\varphi}_1(x),\dots,{\varphi}_p(x)\}$ for each $x \in (0,\alpha)$, and $g,\ {\varphi}_1,\dots,{\varphi}_p$ are continuous on $[0,\alpha]$, we can clearly find $i_0, i_{\alpha} \in \{1,\dots,p\}$ such that $g(0) = {\varphi}_{i_0}(0)$ and $g(\alpha) = {\varphi}_{i_{\alpha}}(\alpha)$.
Let $h$ be the extension of $g$ with $h(x) = {\varphi}_{i_0}(x),\ x<0$ and $h(x) ={\varphi}_{i_{\alpha}}(x), x>\alpha$. Then $h$ is continuous on ${{\mathbb R}}$ and $h(x) \in \{{\varphi}_1(x),\dots,{\varphi}_p(x)\}$ for each $x \in {{\mathbb R}}$. Thus Lemma \[mix\] implies that $h$ is DC on ${{\mathbb R}}$, i.e., $h= \gamma_1-\gamma_2$, where $\gamma_1$ and $\gamma_2$ are convex on ${{\mathbb R}}$. Then $y_i := \gamma_i|_{[0,\alpha]}$, $i=1,2$, are clearly convex Lipschitz functions, and $\psi(x)= (x,y_1(x)-y_2(x)),\ x \in [0,\alpha]$.
[*Step 4*]{} If does not hold, we can choose a Cartesian system of coordinates given by a map $A: {{\mathbb R}}^2 \to {{\mathbb R}}^2$ such that $A(x_0) = (0,0)$ and $A(q)= (1,0)$. Applying steps 1-3 to $u^* := u \circ A^{-1}$ and $\xi^*:= A \circ \xi$, we obtain $\psi$ of the demanded form with $\psi([0,\alpha]) \subset \Sigma(u^*) = A(\Sigma(u))$.
\[1\] Well-known elementary properties of convex functions on ${{\mathbb R}}$ easily imply that the one-sided derivative $\psi_+'$ ($\psi_-'$) exists and is right (left) continuous on $[0,\alpha)$ ($(0,\alpha])$ and has finite variation on this interval. In other words, $\psi$ has [*bounded convexity*]{} (see [@VZ Theorem 3.1] or [@Du Lemma 5.5]). Further, since clearly $|\psi'_+|\geq 1,\ |\psi'_-|\geq 1$ we obtain that the curve $\psi$ has [*finite turn*]{} (see [@AR Theorem 5.4.2] or [@Du Theorem 5.11]). So the curve $\psi^*: = A^{-1} \circ \psi$, for which $\psi^*([0,\alpha]) \subset \Sigma(u)$, has also bounded convexity and finite turn.
\[2\] The proof of Theorem \[main\] and Remark \[1\] show that, for the curve $\xi: [0,\tau] \to \Sigma(u)$ from Theorem CY, there exists $0<\tau^*<\tau$ such that $\xi|_{[0,\tau^*]}$ has finite turn. In fact, this assertion “is not weaker” than Theorem \[main\], since it implies quickly by standard methods Theorem \[main\].
\[3\] We [*did not shown*]{} that the curve $\xi$ from Theorem CY has near $0$ (left-continuous) left derivative $\xi'_-$ at all points. However, the proof of Theorem \[main\] clearly implies that $\xi$ has (left-continuous) left half-tangent on $(0,\tau^*]$ for some $0<\tau^*<\tau$.
We will not give detailed proofs of facts from Remarks \[1\]-\[3\], since they would be inadequately long, and these facts are not essential for the present short note.
[WWW]{}
P. Albano, P. Cannarsa, [*Structural properties of singularities of semiconcave functions*]{}, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 719–740.
A.D. Alexandrov, Yu. G. Reshetnyak, [*General theory of irregular curves*]{}, Mathematics and its Applications (Soviet Series) Vol. 29., Kluwer Academic Publishers, Dordrecht, 1989.
P. Cannarsa, C. Sinestrari: [*Semiconcave functions, Hamilton-Jacobi equations, and optimal control*]{}, Progress in Nonlinear Differential Equations and their Applications 58, Birkh" auser, Boston, 2004.
F.H. Clarke, [*Optimization and nonsmooth analysis*]{}, 2nd edition, Classics in Applied Mathematics 5, SIAM, Philadelphia, 1990.
P. Cannarsa, Y. Yu, [*Singular dynamics for semiconcave functions*]{}, J. Eur. Math. Soc. 11 (2009), 999–1024.
J. Duda, [*Curves with finite turn*]{}, Czechoslovak Math. J. 58 (133) (2008), 23–49.
R. Mifflin, [*Semismooth and semiconvex functions in constrained optimization*]{}, SIAM J. Control Optimization 15 (1977), 959–972.
D. Pavlica, [*On the points of non-differentiability of convex functions*]{}, Comment. Math. Univ. Carolin. 45 (2004), 727–734.
J.E. Spingarn, [*Submonotone subdifferentials of Lipschitz functions*]{}, Trans. Amer. Math. Soc. 264 (1981), 77–89.
L. Veselý, L. Zajíček, [*Delta-convex mappings between Banach spaces and applications,*]{} Dissertationes Math. (Rozprawy Mat.) [289]{} (1989).
L. Veselý, L. Zajíček, [*On vector functions of bouned convexity*]{}, Math. Bohemica 133 (2008), 321–335.
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[^1]: The research was supported by the grant MSM 0021620839 from the Czech Ministry of Education and by the grant GAČR 201/09/0067.
|
---
abstract: 'We report low-dimensional tunneling in an independently contacted vertically coupled quantum wire system. This nanostructure is fabricated in a high quality GaAs/AlGaAs parallel double quantum well heterostructure. Using a novel flip chip technique to align top and bottom split gates to form low-dimensional constrictions in each of the independently contacted quantum wells we explicitly control the subband occupation of the individual wires. In addition to the expected 2D-2D tunneling results, we have found additional tunneling features that are related to the 1D quantum wires.'
author:
- 'E. Bielejec'
- 'J. A. Seamons'
- 'J. L. Reno'
- 'M. P. Lilly'
title: 'Tunneling and nonlinear transport in a vertically coupled GaAs/AlGaAs double quantum wire system'
---
Coupled nanostructures show promise in leading to new understanding of non-fermi liquid physics, many-body effects and electron-electron interactions [@Zulicke2002; @Eugster1991; @Eugster1994; @Smoliner1996; @Ploner2000; @Auslaender2002; @Tserkovnyak2002; @Bird2003]. The vertically coupled double quantum wire system described in this letter is a simple realization of such a coupled nanostructure. Proposals of coupled quantum wire devices that utilize tunneling and spin-orbit coupling to make a spin filter [@Governale2002] and coherent tunneling oscillations to form a qubit [@Bertoni2000] rely on exquisite control of the wire density, number of subbands occupied and coupling both between the wires and to the macroscopic environment.
In this letter we report tunneling measurements between vertically coupled double quantum wires fabricated using split gates on both sides of a double quantum well GaAs/AlGaAs heterostructure. While semiconductor based nanoelectronics promise a flexible platform, care must be taken that implementation of the quantum wires and the control over their subband occupation does not also alter the properties of the wires under investigation. With this issue in mind, we devise a tunneling geometry where the quantum wires are separated from the gates that are necessary for independent contacts. The price for this isolation is a 2D-2D tunneling component. We present tunneling results where 2D-2D resonances and additional structures are visible. The origin of the additional features and their relationship to 1D physics is investigated.
For the samples reported here, 18 nm wide GaAs quantum wells are separated with an AlGaAs barrier. For sample A the barrier is 7.5 nm and for sample B the barrier is 10 nm. The top and bottom quantum wells in sample B have individual layer densities of $1.16$ and $1.96 \times 10^{11}$ cm$^{-2}$ respectively and a combined mobility of $0.91 \times 10^{6}$ cm$^{2}$/Vs; sample A is similar. The heterostructure is thinned to approximately 0.4 $\mu$m using an epoxy-bond-and-stop-etch (EBASE) process [@EBASE1996]. This allows for top and bottom split gates $\sim$ 150 nm from the top and bottom quantum wells, aligned laterally with sub-0.1 $\mu$m resolution. The advantages of this device structure are three-fold. First, by making use of molecular beam epitaxy (MBE) growth of the tunneling barrier we have a rigid potential barrier between the layers. Second, the close proximity of the top and bottom split gates to the electron layers leads to a well-defined confinement potential for the 1D wires. Third, independent contact to individual electron layers in combination with the top and bottom split gates allows for the independent formation and control of the number of occupied subbands in both the top and bottom quantum wires.
In Fig. 1(a), a top view scanning electron micrograph of a coupled quantum wire device is shown. The dark areas are the GaAs/AlGaAs heterostructure described above. The active region of the device consists of six TiAu gates. The four gates in the center form pairs of split gates defining the quantum wires. Only the two gates on the top surface are visible due to the accurate alignment of the top and bottom gates. The quantum wires are formed by electrostatic confinement in the 0.5 $\mu$m wide and 1.0 $\mu$m long gap between the split gates. The remaining two gates, spanning the entire field of view, are used to independently contact the individual electron layers. The lateral separation between the quantum wire and these gates is chosen to be $\sim$ 10 times greater than the depth of the quantum well to ensure a uniform density of the quantum wires. Low frequency measurements of the parallel transport and tunneling conductance are made using a standard ac method with a constant excitation voltage of 100 $\mu$V at 13 to 143 Hz; dc IV’s are used as a consistency check. All measurements were taken at base temperature of a dilution refrigerator with an approximate electron temperature of 50 mK.
In Fig. 2 we plot the two-terminal linear conductance of the two layers in parallel as a function of the top and bottom split gate voltage for sample A. This figure shows the transition from 2D to 1D and finally to zero conductance as negative top and bottom split gate voltage is applied to the system. Consider the case where the bottom split gate voltage is fixed at zero and we vary the top split gate voltage. For top split gate voltages up to point A (Fig. 2 bold line) we have two 2D electron layers present. As we apply a larger negative top split gate voltage (-0.33 to -0.81 V, A to B) a wide 1D channel is formed in the top electron layer in parallel with the lower 2D layer. For larger negative top split gate voltage (-0.81 to -1.76 V, B to C) 1D constrictions form in each of the layers, although the top constriction is much narrower than the bottom constriction. By a top split gate voltage of -1.76 V (C) the top layer is completely completely depleted and conductance steps are observed in the bottom wire. The presence of quantized conductance in region III is an indication of the occupation of the bottom wire only. Similarly, in region I only the top wire is occupied. In the center region (II) of the figure we observe a more complicated pattern when both of the wires are occupied and contribute to the conductance. The complicated crossing pattern provides a means to map out the subband occupation and illustrates the control we have over the states of the individual wires.
A schematic view of the resonant tunneling geometry is shown in Fig. 1(b). Using this tunneling geometry we expect a combined 1D-1D and 2D-2D tunneling signal. The 1D-1D results from the overlap of the quantum wires themselves and the 2D-2D results from the small 2D areas on either side of the quantum wires. Resonant tunneling occurs when an electron in one wire tunnels to the other wire conserving both energy and momentum [@Zulicke2002].
In Fig. 3(a) we show the 2D-2D ac tunneling conductance for several values of magnetic field, $H_{||}$, applied parallel to the plane of the two-dimensional electron system and perpendicular to the tunneling current (see Fig. 1(b)). 2D-2D tunneling occurs when the top and bottom quantum wire split gates are grounded, while maintaining independent contact to the individual layers (see Fig, 1(a)). For $H_{||} =$ 0 a sharp resonance occurs at $\sim$ 2.8 mV. We observe a decrease in the tunneling conductance and a splitting of the resonance with increasing magnetic field. This decrease and splitting has been previously observed [@Eisenstein1991; @Smoliner1989; @Smoliner1995] and is attributed to the magnetic field shifting the dispersion curves in k-space which reduces the phase space for tunneling and leads to a complex dependence on both the magnetic field and applied V$_{Source-Drain}$ (V$_{SD}$). The observed 2D-2D tunneling at $\sim$ 2.8 mV is in good agreement with the measured densities of the two layers via Shubnikov-deHaas oscillations and gate depletion studies. Due to the presence of small 2D areas on either side of the quantum wires, the 2D-2D tunneling features described here are visible to some extent in all our tunneling results.
Fig. 3(b) shows the tunneling conductance in regions I, II and III for sample B with the top and bottom quantum wire split gates set to a parallel conductance of $\sim 4e^{2}/h$. The resonance at $\sim$ 2.8 mV is 2D-2D tunneling from the pair of 1 $\mu$m $\times$ 2 $\mu$m 2D areas shown in Fig. 1. As we move from region I to region II we observe a doubling of the peak tunneling conductance. This is attributed to a doubling of the 2D-2D tunneling area as we move from only one wire occupied (region I) to both wires occupied (region II). The peak tunneling conductance is halved as we move from region II to region III for the same reason. An additional peak is observed at V$_{SD}
\approx$ 13 mV in region I and at V$_{SD} \approx$ - 7 mV in region III enlarged view in Fig. 3(b). The position of these additional peaks are sensitively dependent on the top and bottom split gate voltage and are only visible when one of the wires is depleted. It should be noted that these additional peaks are [*not*]{} due to second 2D subband tunneling which occurs at $\sim$ 40 mV independent of the top and bottom split gate voltages. As we move towards region II the additional peaks move towards and collapse into the 2D-2D tunneling resonance. For a fixed top and bottom split gate voltage, the peak position is independent of magnetic field, but the amplitude can have a strong dependence as shown in Fig. 3(c).
In region II of Fig. 2 we have observed a 2D-2D tunneling signature as shown in Fig. 3(b). However, we have not observed a clear 1D-1D signal as expected. From previous work on vertically coupled quantum wires [@Tserkovnyak2002] we expect a 1D-1D conductance signal on the order of $\sim$ 0.01 $\mu$S, which should be clearly observable in this experiment. Comparing 2D-2D tunneling results to the tunneling signal from region II we estimate that any 1D-1D signal buried in the 2D-2D background to be no larger than $\sim$ 0.001 $\mu$S.
Finally we turn our attention to the side peaks in the tunneling when only one wire is occupied (regions I and III). From the strong magnetic field dependence of the amplitude (Fig. 3c) and the fact that the peak position in $V_{SD}$ depends on the exact combination of top and bottom split gate voltages, it is tempting to conclude that the peak arises from 1D-2D tunneling. This tunneling could be from either the occupied or unoccupied wire to the 2D region in the other layer. If these features are indeed 1D-2D tunneling signatures, the tunneling process is quite different from the resonant 2D-2D tunneling. First, the insensitivity of peak position with $H_{||}$ suggests an inelastic origin. Second, we might expect each of the multiple 1D subbands to contribute additional tunneling channels; however, no other tunneling peaks are observed until the second 2D subband tunneling at $\sim$ 40 mV.
It is important to note the large $V_{SD}$ values of the additional tunneling structures. Clearly nonlinear processes can be important. One example is lateral transport through the unoccupied wire at high bias. This could allow the floating 2D paddle (see diagrams in Fig. 3b) to contribute to tunneling. Finally, the effect of a depleted quantum wire at high bias is unknown in this system. While the origin of the additional tunneling peaks remains unknown, possibilities of 1D-2D tunneling and nonlinear transport are being explored to solve this important problem.
In conclusion, we have fabricated a nanoelectronic structure consisting of a pair of vertically coupled quantum wires and measured tunneling between the layers. The key to fabricating this device is the combination of EBASE thinning, electron beam lithography and depletion gates for separate contact. Transport measurements demonstrate external control over the number of occupied subbands in each 1D wire. With independent contacts, tunneling is measured between the quantum wire systems. An easily identified 2D-2D component is visible, and additional structure is present at high bias voltages when only one wire is occupied. The additional peak in the tunneling measurement is possibly related to 1D-2D tunneling or nonlinear transport of the unoccupied 1D wire. The successful combination of semiconductor heterostructures and dual-side electron beam lithography can be used to create a wide variety of electrically coupled nanostuctures for both fundamental and applied studies.
We thank S. K. Lyo for useful discussions and we acknowledge the outstanding technical assistance of R. Dunn and D. Tibbets. This work has been supported by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, U.S. Department of Energy. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000.
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|
-0.6cm PAR–LPTHE 02–01
hep-th/0201110
1.4truecm
[Quantum Field Theory with Extra Dimensions]{} Laurent Baulieu$^\dagger$ [*$^\dagger$ LPTHE, Universités Pierre et Marie Curie Paris 6 et Denis-Diderot Paris 7*]{}
[**Abstract**]{}: We explain that a bulk with arbitrary dimensions can be added to the space over which a quantum field theory is defined. This gives a TQFT such that its correlation functions in a slice are the same as those of the original quantum field theory. This generalizes the stochastic quantization scheme, where the bulk is one dimensional.
------------------------------------------------------------------------
[*Postal address: Laboratoire de Physique Théorique et des Hautes Energies, UMR CNRS 7589, Université Pierre et Marie Curie, boîte postale 126, 4, place Jussieu, F–75252 PARIS Cedex 05.*]{}
Introduction
============
In recent works [@BaZw][@BaGrZw], the basic ideas of stochastic quantization [@pawu][@huffel] have been elaborated in a systematic approach, called bulk quantization. These papers give a central role to the introduction of a symmetry of a topological character. The correlation functions for equal values of the bulk time define the correlations of the physical theory. Pertubatively, bulk quantization and the usual quantization method are equivalent because the observables satisfy the same Schwinger-Dyson equations in both approaches; basic concepts such as the definition of the $S$-matrix (in the LSZ sense) and the Cutkowski rules can be also directly addressed in 5 dimensions [@BaZw]. We believe that the difficulty of giving a consistent stochastic interpretation to all details of the formalism, especially in the case of gauge theories, justifies to directly postulate that bulk quantization is a particular type of a topological field theory. Moreover, there is an interesting geometrical interpretation for many of the ingredients that are needed in bulk quantization. The idea of a topological field theory is actually relevant, since one wishes to define observable that are independent of most of the details of the bulk, such as the metrics components $g_{tt}$ and $g_{\mu
t}$. Quite interestingly, the interpretation of anomalies in gauge theories is that, the limit of taking the limit of an equal bulk time can be ambiguous.
As shown in [@BaZw], the additional dimension $t$ does not take part in the Poincaré group of symmetries for the case of an additional non compact dimension. The homogeneity of the Lagrangian requires that $t$ has the dimension of the square of the ordinary coordinates. We investigated how this method can be applied to gravity and supersymmetric theories in [@gravity][@susy]. Here, we will show that bulk quantization can be generalized to the case where the bulk has an arbitrary dimension $n\geq
1$, according to the following picture: .8cm
.7cm
The interest of this generalization, when $n$ is larger than one, is yet to be discovered, but we find that the aesthetics of the whole construction makes it worth being presented.
The fields
==========
We consider for simplicity the case of a commuting scalar field $\phi (x)$ in 4 dimensions, with a Lagrangian $L_0(\phi)$ and action $I_0[\phi ]=\int d^4x L_0(\phi)$. We want to find another formulation of this theory with an action in a space with dimensions $4+n$, where $n$ denotes generically the dimension of the bulk, with observables that are defined in a slice of dimension 4. The number of bosonic fields that are needed depends on $n$, and increases as $2^n$, according to: n=0, \_0 &=& \_1(x) = (x)\
[level]{} n=1, \_1 &=& \_1(x,t\^1) ,\_2(x,t\^1)\
.... & &\
[level]{} n, \_n&=& \_1(x,t\^1,…,t\^n ) ,\_2(x,t\^1,…,t\^n), …\_[2\^n]{}(x,t\^1,…,t\^n)\
....& & Indeed, if we apply from level $n$ to $n+1$ the process explained in [@BaZw], the number of field degrees of freedom that are needed doubles. Thus, we have a multiplet with $2^n$ components when the bulk has dimension $n$. For $n=1$, $\phi_2(x,t^1)$ can be heuristically interpreted as the Gaussian noise of a stochastic process. A more fundamental interpretation is that $\phi_2(x,t^1)$ is the canonical moment with respect to the bulk time of the field $\phi_1=\phi$, and so on.
At every level $n$, there is a hidden BRST symmetry, defined by the graded differential operator $\Q$: where, $1\leq p \leq 2^{n-1}$. The $\Psi $’s and $\bar \Psi $’s are ghosts and antighosts, with the opposite statistics to the $\phi$’s. The set of fields $\phi$, $\Psi$ and $\bar \Psi$ determine a BRST topological quartet. We can define an anti-BRST operator, which merely interchanges the ghosts and antighosts: The action that describes the formulation of the theory with a bulk of dimension $n$ must be $\Q$-exact, that is, it must be of the form: d\^4x dt\^1…dt\^n (\_[p=1]{}\^[2\^[n-1]{}]{} |\_[p+2\^[n-1]{}]{} Z\_p ), that is, d\^4x dt\^1…dt\^n ( \_[p=1]{}\^[2\^[n-1]{}]{} \_[p+2\^[n-1]{}]{} Z\_p - \_[p=1]{}\^[2\^[n-1]{}]{} |\_[p+2\^[n-1]{}]{}Z\_p ) To determine the action, it is sufficient to determine the expression of the functionals $ Z_p[\Phi_n]$, for every level $n$. At level $n$, the ghosts have parabolic propagators along $t_n$. Thus, they can be integrated out exactly when one computes correlations functions of the $\Phi$’s. Indeed, the latter cannot contain closed loops of the ghosts. It follows that it is sufficient to determine the following part of the action: I\_n= d\^4x dt\^1…dt\^n \_[p=1]{}\^[2\^[n-1]{}]{} \_[p+2\^[n-1]{}]{} Z\_p. The rest of the action will be determined by the requirement of BRST symmetry. The determination of the factors $Z_p$ is restricted by power counting and by symmetries. The latter include the parity in the bulk, that is, the invariance under $t_p
\to - t_p$, for $1 \leq p
\leq n$, and by translation invariance. The covariance of the fields under this parity will be shortly determined.
We define the following self-consistent assignments for the dimensions of the bulk coordinates and of the fields: \[powerc\]\[t\^n\]\^[-1]{}= 2\^n=2p-1 With this assignments, the dimension of the Lagrangian at level $n$ must be $2+2^{n+1}$. This ensures that the theory generated by the action $I_n$ is renormalizable by power counting, as a generalization of [@BaZw]. It is of relevance to note that: +\[\_[2\^[n-1]{}]{}\]+\[t\^n\]\^[-1]{}= 2+2\^[n+1]{} This will imply that, in the formulation at level $n$, $\phi_{2^{n-1}}$ is the conjugate momentum of $\phi_1$ with respect to $t^n$. This turns out to be one of the key facts for proving by induction that the physical content of the theory at level $n$ is the same physics as for the theory at level $n-1$, and so on, down to the ordinary formulation with the action $I_0$.
The action at level $n$
=======================
At level zero, the theory is defined by the standard action $I_0[\phi]=\int d^4x L_0(\phi(x))$. At level $n=1$, it is defined by: I\_1\[\_1,\_2\]=d\^4xdt\^1 ( \_2\_1 \_1 +\_2 (\_2+ [[ I\_0]{}]{} ) ) The exponential of this action must be inserted in the path integral with measure $[d\phi_{1 }]_{ x,t}[d\phi_{ 2}]_{x,t}$. $I_1$ satisfies power counting according to equation (\[powerc\]) (here $[t^1]^{-1}=2$, $[\phi_1]=1, [\phi_2]=3$) and the bulk-parity symmetry $P_1$ is: t\^1 && -t\^1\
\_1& & \_1\
\_2 && -\_2 - [[ I\_0]{}]{} The action $I_1$ and its symmetry $P_1$ have been discussed in details in [@BaZw], where we have also shown that it describes the same physics as the action $I_0$. Notice that the existence of the symmetry $P_1$ is obvious after the elimination of $ \phi_2$ by its equation of motion.
At level $n= 2$, the action is: \[actiondeux\] I\_2\[\_1,\_2,\_3,\_4\]=d\^4xdt\^1dt\^2 ( \_3\_2 \_1 +\_3 (\_3+ [[ I\_1]{}]{} )+\_4 ( \_2+[[ I\_0]{}]{}+\_1\_1 ) ) $I_2$ is invariant under the bulk-parity transformations $P_2$ and $P_1$. $P_2$ is defined as: t\_2 && -t\_2\
\_1, \_2 && \_1, \_2\
\_3 && - \_3-[[ I\_1]{}]{}\
\_4 && \_4 +\_2 \_2\
The action of the symmetry $P_2$ on $\phi_4$ is such that $\delta I_2=-\int \partial _2( I_1)$. $P_1$ is defined as: \[paritedeux\] t\_1 && -t\_1\
\_1, \_3 && \_1, \_3\
\_2 && - \_2-[[ I\_0]{}]{}\
\_4 && -\_4 +\_3 [[ \^2 I\_0]{}]{} $P_1$ transforms $\phi_4$ in such a way that the variation of the term $\phi_4( \delta I_2/ \delta \phi _2)$ compensates that of $\phi_3 ( \delta I_1 / \delta \phi _1) $.
The parity symmetry under $P_1$ and $P_2$ implies that a $I_2$ has the form displayed in eq. (\[actiondeux\]). In this action, power conting implies that $I_0[\phi]$ is a local functional, which can be identified as an action that is renormalizable in 4 dimensions. Thus, no new parameter of physical relevance can be introduced when one switches from the ordinary formulation to the formulation with a bulk. By a straightforward generalization of [@BaZw], one can than prove that the correlations functions, computed from $I_2$ at a given point of the two-dimensional bulk, satisfy the same Dyson–Schwinger equations as those computed from $I_1$, at a given point of the one-dimensional bulk. In turn, there is the equivalence of the physics computed either from $I_1$ from $I_0$, which gives the desired result that we can use a two-dimensional bulk to compute physical quantities with the same result as in the ordinary formulation. .
We can now give the general expression of the action at level $n$, which satisfies power counting and is invariant under all parity transformations in the bulk, $t_p\to -t_p$, for $1\leq p\leq n$. It reads: \[actionN\] I\_n=d\^4xdt\^1dt\^2…dt\^n ( & \_[1+2\^[n-1]{}]{}\_n \_1 +\_[1+2\^[n-1]{}]{} (\_[1+2\^[n-1]{}]{} + [[ I\_[n-1]{}]{}]{})\
&+ \_[p=2+2\^[n-1]{}]{} \^[2\^n]{} \_p [[ I\_[n-1]{}\[\_1 …,\_[2\^[n-1]{}]{}\]]{}]{} ) ) $I_n$ is invariant under the parity transformation $P_n$, with: t\_n && -t\_n\
\_[1+2\^[n-1]{}]{} && - \_[1+2\^[n-1]{}]{} - [[ I\_[n-1]{}]{}]{}\
\_p && \_p 2.05cm [for]{} p < 1+2\^[n-1]{}\
\_p && \_p +\_n \_[p-2\^[n-1]{}]{} p > 1+2\^[n-1]{}\
Under the symmetry $P_n$, the Lagrangian density varies by a pure derivative, Ł[[L]{}]{} ${\L}_n\to {\L}_n -\pa_n({\L}_{n-1})$.
As for the rest of the parity transformations $P_p$ of the fields in the bulk, with $1\leq p\leq n-1$, their existence can be proven by induction.
Assume that the full bulk parity symmetry exists at level $n-1$, that is, field transformations exist that leave invariant $I_{n-1}[\phi_1,\ldots,\phi_{2^{n-1}}]$ for all transformations $t_p\to
-t_p$, $1\leq p\leq n-1$. Then, the triangular nature of the Jacobian of the transformation $(\phi_1,\ldots,\phi_{2^{n-1}}) \to
P_p(\phi_1,\ldots,\phi_{2^{n-1}})$ at level $n-1$ implies that one can extend this transformation law for the new fields that occur at level $n$, $(\phi_1,\ldots,\phi_{2^{n}}) \to P_p(\phi_{1},\ldots,\phi_{2^{n }})$ and that $I_n$, as given in eq. (\[actionN\]), is invariant under $P_p$, in a way that generalizes eq.(\[paritedeux\]).
Conversely, the parity symmetry and power counting imply that $I_n$ must be of the form (\[actionN\]). This shows that the number of parameters of the theory is the same in bulk quantization, with any given choice of the the bulk dimension n, as in the standard formulation. These parameters are just those of an action that is renormalizable by power counting in 4 dimensions.
We can now write the action in the following form: \[QactionN\] d\^4xdt\^1dt\^2…dt\^n ( &|\_[1+2\^[n-1]{}]{} (\_n \_1 +\_[1+2\^[n-1]{}]{} [[ I\_[n-1]{}\[\_1…,\_[2\^[n-1]{}]{}\]]{}]{} )\
& + \_[p=2 ]{} \^[2\^[n-1]{}]{} |\_p [[ I\_[n-1]{}\[\_1…,\_[2\^[n-1]{}]{}\]]{}]{} )
A propagation occurs in the new direction $t^n$, while the equation of motions of the formulation at degree $n-1$ are enforced in a BRST invariant way. Because the action is $\Q$ exact, and $\phi_{1+2^{n-1}}$ is the momentum with respect to $t_n$ of $\phi_1$, the correlation functions that one can compute in the ($4+n$)-dimensional theory, at an equal bulk component $t_n$, are identical to those computed in the theory defined by $I_{n-1}$. The proof is just as in the case of a one-dimensional bulk, and uses the BRST invariance and the translation and parity symmetries in the bulk. Finally, the correlation functions computed in the ($4+n$)-dimensional theory, where all argument only involve a single point $T$ in the bulk, are identical to those computed from the basic four dimensional $I_0$, that is, G \_N\^[I\_0]{}(x\_1,…,x\_N) = G \_N\^[I\_n]{}( (x\_1,[ t\^[p\_1]{}]{})…,(x\_N, [ t\^[p\_N]{}]{})) \_[[ t\^[p\_1]{}]{}=…=[ t\^[p\_N]{}]{}=T\^[p]{}]{} Due to translation invariance, such a correlation function is independent on the choice of $T$. An another interesting expression of the action at level $n$ is: \[QactionNb\] d\^4xdt\^1dt\^2&…& dt\^n ( ( |\_[1+2\^[n-1]{}]{} \_n \_1 ) && + ( |\_[1+2\^[n-1]{}]{}\_1 + I\_[n-1]{}\[\_1,…,\_[2\^[n-1]{}]{}\] ) )
It shows that the Hamiltonian at level $n$ is a supersymmetric term, $H~=~\demi\{Q_n,\bar Q_n\}$, which involves in the very simple way the action at level $n-1$.
[**Acknowledgments**]{}: This is a pleasure to thank Daniel Zwanziger for discussions related to this work. The research of Laurent Baulieu was supported in part by DOE grant DE-FG02-96ER40959.
[10]{}
Laurent Baulieu and Daniel Zwanziger, [*[QCD]{}$_4$ From a Five-Dimensional Point of View*]{}, hep-th/9909006, Nucl.Phys. B581 (2000) 604; [*From stochastic quantization to bulk quantization: Schwinger-Dyson equations and S-matrix*]{} JHEP 0108 (2001) 016, hep-th/0012103; [*Bulk Quantization of Gauge Theories: Confined and Higgs Phases*]{}, JHEP 0108 (2001) 015, hep-th/0107074.
Laurent Baulieu, Pietro Antonio Grassi, and Daniel Zwanziger. , Nucl.Phys. B597 (2001) 583, hep-th/0006036.
G. Parisi and Y.S. Wu. Sci. Sinica [**24**]{} (1981), 484.
P.H. Daamgard and H. Huffel. Phys. Rep. [**152**]{} (1983), 227.
Laurent Baulieu, , Talk given in the memory of E.S. Fradkin, hep-th/0007027.
Laurent Baulieu and Marc Bellon, [*Bulk quantization of supersymmetric gauge theories*]{}, hep-th/0102075, Phys.Lett. B507 (2001) 265-269.
|
---
author:
- Arkadiusz O l e c h
date: |
Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland\
e-mail: olech@sirius.astrouw.edu.pl
title: '[**V485 Centauri: the Shortest Period SU UMa Star**]{}[^1]'
---
[**Key words:**]{} [binaries: close – novae, cataclysmic variables – Stars: individual: V485 Centauri]{}
Introduction
============
Cataclysmic variable stars are binary systems containing white dwarf primary and late-type main sequence, low mass secondary. The secondary fills its Roche lobe and loses material through the inner Lagrangian point toward the white dwarf primary. In case of non-magnetic systems falling material forms a disc around the white dwarf.
Dwarf novae are a subclass of cataclysmic variable systems (for recent reviews see Warner 1995 and Osaki 1996). Usually they are divided into three additional classes. The first one is called U Geminorum (or SS Cygni) type stars. Objects belonging to this type are characterized by orbital periods in the range 3–10 hours and by having outbursts with an amplitude from 2 to about 8 mag which last a few days, separated by a few weeks period of quiescence. The mechanism of such outbursts is a thermal instability in the disc which causes episodes of enhanced mass transfer from the disc into the primary. The second subgroup are Z Camelopardalis stars. The periods of these objects are in the same range as U Gem-type variables but additionally during the outbursts one can observe so-called “standstills”. It is believed that Z Cam stars lay on the border line between stars with thermally stable and unstable discs.
The stars belonging the the third group are called SU Ursa Majoris systems. They have orbital periods in range 80–120 minutes and from time to time they show additional, slightly brighter and longer lasting outbursts called superoutbursts or super-maxima. A characteristic feature of the superoutbursts is presence of “tooth-shape” periodic light oscillations called superhumps. The periods of superhumps are 1%–9% longer than the binary orbital periods. The amplitude of light modulations is typically in the range 0.1–0.4 mag.
In the Catalog and Atlas of Cataclysmic Variables (Downes and Shara 1993) variable star V485 Centauri is classified as U Geminorum type nova. The magnitude range of variability is given as 12.9–17.9 mag. The first photometric and spectroscopic study of this star made by Augusteijn et al. (1993) revealed clear photometric oscillations with the period 0.$^d$041096 (59.0 min) and amplitude about 0.3 mag. The spectra presented in the same paper were fairly typical for cataclysmic variables and showed clear ${\rm H\alpha}$, HeI and CaII emission lines. Due to the presence of hydrogen lines Augusteijn et al. (1993) excluded the hypothesis that V485 Cen is a double-degenerate AM CVn system. Also the value of the period was too long for the rotational period of the white dwarf primary which is observed in the intermediate polars. The main conclusion of that paper was that V485 Cen contains hydrogen-deficient main-sequence star and the observed brightness oscillations reflect the orbital period of the system.
The second paper with more extensive quiescent photometry and spectroscopy of V485 Cen was published by Augusteijn et al. (1996). They confirmed the 59 min periodicity is the orbital period of the system and gave its more exact value equal to 0.$^d$040995001. They also gave an estimate of a few parameters of the system. The mass ratio $q$ defined as ${M_{\rm WD}}/{M_{\rm
sec}}$ was estimated to be about 2.6 (lower limit for a mass of the white dwarf was ${M_{\rm WD}\approx0.7M_\odot}$ and a lower limit of the mass of the secondary ${M_{\rm sec}\approx0.14M_\odot}$), the inclination was equal to $i=20-30^\circ$ and mass transfer from the secondary ${\dot{M}}$ was estimated between ${1\times 10^{-10}}$ and ${1\times 10^{-9}~M_\odot/{\rm year}}$.
In the middle of May 1997 we were notified by Rod Stubbings ([*VSNET-alert*]{} no. 908) that a new outburst of V485 Cen had just begun. In the present paper we report on results of CCD photometry of V485 Cen performed during that outburst.
Observations and Data Reduction
===============================
The entire set of observations presented in this paper was carried out at Las Campanas Observatory in Chile, which is operated by Carnegie Institution of Washington. Data were collected with the 1.3 m Warsaw telescope equipped with a 2048$\times$2049 SITe thinned CCD with a scale 0.417$"/$pixel. For the purpose of observing of V485 Cen we used only part of CCD chip trimming it to 512$\times$512 pixels. The detailed description of the system used is given by Udalski (1997).
Observations of V485 Cen were collected as a subproject of the Optical Gravitational Lensing Experiment (OGLE-2). The main goal of the OGLE-2 project is a search for dark matter in our Galaxy using microlensing phenomena (Paczyński 1986, Udalski et al. 1992). When atmospheric conditions are poor (seeing $>1.6"$, cirrus clouds) and photometry of dense stellar regions is not reliable some sub-projects like described in this paper are conducted.
We have monitored V485 Cen in $B$, $V$ and $I$ filters on 14 nights from May 16 through June 2, 1997. The exposure times varied between 60 and 300 seconds, depending on atmospheric conditions and the brightness of the star. Dead time between the consecutive frames was about 20 seconds. Journal of observations with duration of each run, filters used and exposure times is given in Table 1.
Table 1\
\
[|l|c|c|c|c|]{} Date & Time of start & Length of & Filter & Exp. Time\
1997 & HJD 2450000. + & run (h) & & (sec)\
May 16/17 & 585.5286 & 3.2 & $B$,$V$,$I$ & 60\
May 18/19 & 587.7642 & 1.1 & $I$ & 60\
May 19/20 & 588.4735 & 4.0 & $V$,$I$ & 60\
May 20/21 & 589.4684 & 5.7 & $V$,$I$ & 60\
May 21/22 & 590.4893 & 4.3 & $V$,$I$ & 60\
May 22/23 & 591.4569 & 5.8 & $I$ & 60,90\
May 23/24 & 592.4646 & 5.0 & $V$,$I$ & 60\
May 24/25 & 593.4539 & 1.9 & $V$,$I$ & 90,120\
May 25/26 & 594.5252 & -$^*$ & $I$ & 180,300\
May 27/28 & 596.5169 & 1.2 & $I$ & 180\
May 27/28 & 596.6380 & 0.8 & $I$ & 180\
May 30/31 & 599.5640 & 2.1 & $V$,$I$ & 90\
May 31/01 & 600.4396 & 1.3 & $I$ & 90\
June 01/02 & 601.4681 & 1.0 & $I$ & 180\
June 02/03 & 602.4490 & 1.0 & $I$ & 180\
\
The data reduction (debiasing and flatfielding) was performed using IRAF[^2] software. The profile photometry was done with the DAOphotII package. All data were differentially reduced using a comparison star located ${\sim50"}$ W and ${\sim10"}$ S with respect to the variable star. According to Augusteijn et al. (1993) the $V$ brightness of the comparison star is ${15.04\pm0.02}$ mag and its color ${B-V=0.64\pm0.04}$. Because our $V$-band is very close to the standard Johnson’s band we simply added Augusteijn et al. (1993) value to our differential measurement to have absolute scale.
Mean errors during the first stage of outburst were between 0.005 and 0.01 mag depending mainly on atmospheric conditions. During a brightness dip and the second stage of outburst errors were in the range 0.007–0.030 mag except for the night May 25/26 when only six exposures through thick cirrus clouds were made and errors were about 0.15 mag. Seeing varied from $1.1"$ on the best night to $2.1"$ during the worse run.
Long-term Behavior of V485 Cen
==============================
Based on observations made by members of the Variable Star section of the Royal Astronomical Society of New Zealand, Augusteijn et al. (1993) mentioned that about twenty outbursts of V485 Cen were observed with a duration between 1 and 7 days.
Long-term behavior of V485 Cen during May 1997 outburst is presented in Fig. 1. Rough estimate of the zero point on the magnitude axis is 15.0 mag. The first observations were made on May 17.024 UT. Certainly the outburst began a little earlier because the first positive detection of this star in outburst was made by Rod Stubbings at 9:16 UT on May 15 ([*VSNET-alert*]{} no. 908).
During the period May 16/17 – May 23/24 we observed linear decline of the brightness of the star with the rate 0.11 mag/day. Very rapid drop of brightness occurred on May 24/25 when the star faded by more than 0.5 mag in comparison to previous night. During two further nights brightness reached level by more than 3 mag below the maximum. Unfortunately due to bad weather conditions we do not have any observations from nights May 28/29 and 29/30. After this break we observed the star again on May 30/31 and it was brighter over 1.5 mag in comparison to measurements made on May 27/28. It is clear that V485 Cen showed about 2 mag brightness dip – a characteristic feature seen in light curves of some SU UMa stars.
Since May 30/31 to June 1/2 the decline of brightness was much steeper than during the first stage of superoutburst and its rate was equal to ${\sim1.1}$ mag/day.
It is clearly visible from Fig. 1 that May 1997 outburst lasted at least 16 days. In comparison with durations of other outbursts reported by observers from New Zealand this time is relatively long.
All above facts, i.e. long lasting outburst with plateau and the $\sim$2 mag brightness dip are the evidences for calling this eruption superoutburst. As we will see in the next section another property characteristic for superoutburst – superhumps, was also detected.
Superhumps
==========
Fig. 2 presents nightly $I$-band light curves of V485 Cen for eight first nights. The first run from May 16/17 is separated from other nights by one night and next seven runs from May 18/19 to 24/25 are consecutive. The superhumps with their characteristic shape of steeper increase to the maximum and slower decrease are clearly visible on each night. Their amplitude is about 0.25 mag on May 16/17 and decreases slowly to 0.1 mag on May 23/24. During the last night presented in Fig. 2 the amplitude increases to about 0.15 mag.
To obtain the value of superhump period we have used the Lomb–Scargle (Lomb 1976, Scargle 1982) method of Fourier analysis for unevenly spaced data. Before the calculation of power spectra we have removed the nightly mean and a longer-scale change trend from each individual run. The resulting periodogram is shown in Fig. 3. The highest peak is detected at frequency 23.7254 cycles/day which corresponds to a period ${0.^d04215\pm0.^d00009}$. Additionally at frequency 47.443 cycles/day the first harmonic of main periodicity is clearly visible.
We also determined 27 times of maxima of superhumps. They are listed in Table 2. The best linear fit to these maxima computed by least squares method is given below: $$
----------------- --- ---------------- ------- ---------- -----
HJD$_{\rm Max}$ = 2450585.5522 + 0.042156 $E$
${\pm}$ 0.0005 $\pm$ 0.000004
----------------- --- ---------------- ------- ---------- -----
(1)$$
This value is in very good agreement with the period obtained from Fourier transform but its accuracy is much better, so we may conclude that the superhump period of V485 Cen is equal to ${0.^d042156\pm0.^d000004}$ (60.$^m$7). Preliminary value of superhump period of V485 Cen, equal to 57.7 min, reported by Olech (1997) was based on one observing run only, and therefore turned out to be slightly incorrect. Thus, V485 Cen has the shortest known period among SU UMa-type stars. The shortest orbital and superhump periods were previously observed in WZ Sge and AL Com and were 81.$^m$6 and 82.$^m$3 for WZ Sge and 81.$^m$6 and 82.$^m$6 for AL Com (Patterson et al. 1981, Patterson et al. 1996, Howell et al. 1996), respectively.
Knowing the value of the superhump period we used our $V$-band measurements to plot phased $V$-band light curves of V485 Cen for five nights of superoutburst. Result is shown in Fig. 4. Phase 0.0 corresponds to HJD=2450585.5553. Two cycles are shown for clarity.
Table 2\
\
------------ ----- ---------- ------------ ----- ----------
HJD $E$ ${O-C}$ HJD $E$ ${O-C}$
2450000. + cycles 2450000. + cycles
585.5553 0 0.0738 590.5234 118 –0.0767
585.5973 1 0.0695 590.5667 119 –0.0478
585.6386 2 0.0507 590.6096 120 –0.0304
587.7878 53 0.0319 590.6512 121 –0.0449
588.5021 70 –0.0241 591.5398 142 0.0035
588.5444 71 –0.0217 591.5800 143 –0.0116
588.5870 72 –0.0101 591.6214 144 –0.0290
588.6296 73 0.0000 592.4685 164 0.0652
589.4707 93 –0.0478 592.5102 165 0.0535
589.5138 94 –0.0246 592.5519 166 0.0420
589.5542 95 –0.0666 592.5939 167 0.0391
589.5970 96 –0.0521 592.6356 168 0.0290
589.6393 97 –0.0478 593.4812 188 0.0869
589.6810 98 –0.0579
------------ ----- ---------- ------------ ----- ----------
The superhump period given in ephemeris (1) is a mean value averaged from eight nights of superoutburst. In Table 2 we also list the ${O-C}$ values calculated with the ephemeris (1). These residuals evidently indicate an increase of the period. The quadratic ephemeris obtained as the best least squares fit to the same 27 maxima listed in Table 2 is the following: $$
----------------- --- ---------------- ------- ---------- ----- ------- ------------------------
HJD$_{\rm Max}$ = 2450585.5555 + 0.042047 $E$ $+$ 5.96$\times10^{-6}E^2$
${\pm}$ 0.0006 $\pm$ 0.000012 $\pm$ 0.63
----------------- --- ---------------- ------- ---------- ----- ------- ------------------------
(2)$$
The increase of the superhump period is shown in Fig. 5, where we have plotted the ${O-C}$ residuals taken from Table 2. The solid line in Fig. 5 presents the fit corresponding to the quadratic ephemeris (2).
As it was already pointed out by Semeniuk et al. (1997) it might suggest that the SU UMa stars with the shortest orbital periods exhibit increasing superhump periods contrary to the other objects from this group which derivative of superhump period is negative (see Patterson et al. 1993).
Interpulses
===========
Beginning from night of May 23/24 we observed clear secondary superhumps (sometimes called interpulses) located on the light curve of the star between maxima of ordinary superhumps. In the upper panel of Fig. 6 we plotted again light curve form May 23/24 and marked interpulses by arrows. Additionally in the lower panel we plotted the light curve from May 23/24 phased with the period 0.$^d$042156. In both panels interpulses are clearly visible.
One additional interpulse was also detected on May 24/25, but this was the last night before the brightness dip occurred and after that moment interpulses were not observed. The moments of maxima of detected interpulses are listed in Table 3.
Table 3\
\
------------ -------- ------------ --------
HJD $E$ HJD $E$
2450000. + cycles 2450000. + cycles
592.4777 0 592.6067 3
592.5227 1 592.6480 4
592.5633 2 593.4905 24
------------ -------- ------------ --------
Similar interpulses were also observed in other SU UMa stars like SW UMa (Semeniuk et al. 1997) or SU UMa itself (Udalski 1990). It was suggested (Schoembs and Vogt 1980, Warner 1995) that late superhumps in VW Hyi may develop out of such interpulses. However, we did not have possibility to study this hypothesis because of the brightness dip, which started on the second night in which we observed interpulses.
Post–Dip Behavior
=================
We have collected 5 observing runs during and after the brightness dip in light curve of V485 Cen. The observations made during the first night, that is May 27/28, showed that star faded over 3 mag below its maximum brightness. After two nights we detected the star at brightness only 1.4 mag below the maximum. Apparently the brightness dip ended and the second stage of superoutburst began. This stage lasted only 2 or 3 days because on June 1 the star was again over 3 mag below the maximum brightness.
Fig. 7 presents nightly light curves of V485 Cen from May 27 to June 2, i.e. during and after the brightness dip. The periodic light oscillations with amplitude from 0.05 mag to 0.25 mag are clearly visible. It is obvious that at this time light variations showed double humped structure with two humps (one with higher amplitude than the other) present at each cycle.
Again to obtain the value of the period of these variations we have used the Lomb–Scargle (Lomb 1976, Scargle 1982) method of Fourier analysis. Before the calculation of power spectra we have also removed the nightly mean and a longer-scale change trend from each individual run. The resulting periodogram is shown in Fig. 8. The highest peak in the power spectrum corresponds to frequency 48.782 cycles/day, i.e. ${0.^d020499\pm0.^d00003}$. Due to the double structure of light modulations the real value of the period should be twice of that. We also detect a peak at 24.352 cycle/day which corresponds to ${0.^d04106\pm0.^d00009}$ and which is within errors twice the value of 0.$^d$020499.
From the light curves presented in Fig. 7 we determined 8 times of maxima of higher peaks. They are marked by arrows and additionally listed in Table 4.
Table 4\
\
------------ -------- ------------ --------
HJD $E$ HJD $E$
2450000. + cycles 2450000. + cycles
596.5194 0 599.6336 76
596.5601 1 600.4565 96
596.6425 3 601.4783 121
599.5937 75 602.4644 145
------------ -------- ------------ --------
The best linear fit to these maxima calculated by the least squares method gives the following ephemeris: $$
----------------- --- ---------------- ------- ---------- -----
HJD$_{\rm Max}$ = 2450596.5192 + 0.040996 $E$
${\pm}$ 0.0006 $\pm$ 0.000007
----------------- --- ---------------- ------- ---------- -----
(3)$$
which is in very good agreement with the value of period obtained from Fourier spectrum.
We can summarize that during the five nights from May 27 to June 2 the light curve was dominated by double humped structure with a period $0.^d0409961\pm0.^d0000066$ (${59.^m0}$). This is in excellent agreement with the value of orbital period of V485 Cen given by Augusteijn et al. (1996) and equal to $0.^d040995001$.
Discussion
==========
Recently a few groups published their results concerning 1995 superoutburst of AL Comae Berenices (Pych and Olech 1995, Kato et al. 1996, Howell et al. 1996, and Patterson et al. 1996). The conclusions were in all cases very similar – AL Com is a twin star of previously well studied dwarf nova WZ Sge. Both objects have one of the shortest known orbital periods among SU UMa stars, both show very rare superoutburst with amplitude about 8–9 mag and almost no ordinary outburst. Both also show a 2–3 mag brightness dip in the light curve, and in both stars dominant feature during quiescence is very symmetrical double hump with the period equal to the orbital period of the system and full amplitude of 0.12 mag. Up to now there are only two dwarf novae with orbital wave of that shape. Additionally AL Com during the main stage of superoutburst, when ordinary superhumps were clearly visible, showed increase of the period of superhumps. There is only one known SU UMa star, except for AL Com, with positive superhump period derivative. That object is SW UMa (Semeniuk et al. 1997) and it is also characterized by very short orbital and superhump periods and long time between superoutbursts.
Our CCD photometry of 1997 superoutburst of V485 Cen indicates that the star belongs to the SU UMa type variables and shows features very similar to those observed in the above mentioned stars. We found that present superoutburst of V485 Cen lasted at least 16 days and it is the only one superoutburst known for this object. Before, only ordinary outbursts with duration between 1 and 7 days were observed (Bateson 1979, 1982).
During the first stage of superoutburst we detected clear periodic light modulations called superhumps characteristic for SU UMa stars. Their period was equal to ${60.^m705\pm0.^m006}$. Thus, V485 Cen has the shortest known period among SU UMa type stars, as much as 25% shorter than previously known objects (superhump periods above mentioned stars are about 82 min). As it was shown by Paczyński (1981) and Paczyński and Sienkiewicz (1981) the theoretical value of minimal orbital period of system containing hydrogen-rich secondary is about 81 minutes. That was in very good agreement with observational results because the shortest orbital periods of SU UMa stars were about 81 min. Short values of superhump period equal to 60.$^m$705 and orbital period equal to 59.$^m$03 (Augusteijn et al. 1996, this work) observed in V485 Cen may suggest that this star belongs to a group of AM CVn systems containing degenerate helium secondary. However, the detection of hydrogen emission lines in spectrum of V485 Cen (Augusteijn et al. 1993, 1996) excludes this possibility. The only remaining possibility is that the secondary star in V485 Cen is not degenerate, but a hydrogen deficient main-sequence star. Theoretical calculations made by Sienkiewicz (1984), Nelson et al. (1986) and Tutukov and Yungelson (1996) imply that depending on opacities and hydrogen fraction in the secondary used in calculations the limiting value of minimal orbital period is between 60 and 80 min. According to Paczyński and Sienkiewicz (1981) absolute minimal orbital period for a non-degenerate secondary with critical mass equal to $0.084M_\odot$ is about 49 min.
The period excess defined as $(P_{\rm sh}-P_{\rm orb})/P_{\rm orb}$ is equal to 0.028 for V485 Cen. It is known that SU UMa stars show linear relation between the period excess and orbital period with the smallest excesses at short orbital periods (Stolz and Schoembs 1981). The periods excesses for the shortest period SU UMa stars like WZ Sge, AL Com, HV Vir and SW UMa are 0.008, 0.011, 0.011 and 0.024, respectively. It is clearly visible that V485 Cen does not follow the Stolz and Schoembs’ relation. According to Fig. 1 of Molnar and Kobulnicky (1992) there is another exception from this rule – T Leo, which period excess is considerably too large for its orbital period. On the other hand, calculations made by Whitehurst (1988) show that there is also a linear relation between period excess and mass ratio of the system defined as ${q={M_{\rm sec}/{M_{\rm WD}}}}$. Observational results seem to confirm these calculations without any exceptions as can be seen in Fig. 2 of Molnar and Kobulnicky (1992). Assuming that V485 Cen is similar to T Leo in the sense that it follows only the relation between period excess and $q$ we can roughly estimate its mass ratio. From linear relation obtained by Molnar and Kobulnicky (1992), for period excess equal to 0.028, $q$ should be around 0.17. This value is in disagreement with estimate made by Augusteijn et al. (1996). From spectroscopy they obtained $M_{\rm WD}/{M_{\rm sec}}$ about 2.6, that is ${q\approx0.38}$. According to Molnar and Kobulnicky (1992 and references therein) the absolute upper limit for $q$ for SU UMa stars is equal to 0.33 and is very likely less than 0.22. For higher values of the mass ratio $q$ it is hard to obtain tidal instability of accretion disc caused by effect of 1:3 resonance which is believed to be a cause of presence of superhumps. Detection of superhumps in V485 Cen implies that its mass ratio should be smaller than 0.33. Our rough estimate equal to 0.17 is very close to the optimal value ${q=0.16}$ for which there is the highest possibility of developement of superhumps (Molnar and Kobulnicky 1992).
We have also demonstrated that the value of the superhump period increases during the superoutburst. The period derivative ${\dot{P_{\rm sh}}}$, obtained from a parabolic fit to the ${O-C}$ diagram, is equal to ${28.3\times10^{-
5}}$. Such a rate of the superhump period change is more than two times greater than the typical value for SU UMa stars. Moreover the SU UMa stars generally show decreasing superhump periods during the plateau phase of superoutbursts. All SU UMa stars with measured superhump period changes listed by Patterson et al. (1993) have negative ${\dot{P}_{\rm sh}}$. The newly discovered exceptions from that rule are AL Com (Howell et al. 1996, Patterson et al. 1996) and SW UMa (Semeniuk et al. 1997) both having the shortest orbital periods among SU UMa stars and longest intervals between superoutbursts. The discovery of positive value of ${\dot{P}_{\rm sh}}$ in V485 Cen confirms hypothesis of Semeniuk et al. (1997) that the SU UMa stars with the shortest orbital periods and the longest superoutburst recurrence times exhibit increasing superhump periods contrary to the other SU UMa stars whose ${\dot{P_{\rm sh}}}$ have negative values.
The first stage of superoutburst ended with over 2 mag brightness dip which lasted 2–3 days. During the dip and later when star brightened for 2–3 days we observed double humped light variations with the period ${59.^m03\pm0.^m01}$, which is exactly equal to the orbital period of the system (Augusteijn et al. 1993, 1996). These modulations were also seen after the end of the superoutburst. There are only two other cataclysmic variable stars known with orbital waves similar to the observed. These are WZ Sge and AL Com – again both having the shortest orbital periods among SU UMa stars and the longest intervals between superoutbursts.
Kato et al. (1996), Howell et al. (1996), and Patterson et al. (1996) interpreted a brightness dip in the light curve of AL Com as an effect of propagating cooling wave in the disc. This cooling wave due to the large amount of material which still exists in the disc is not stable and is reflected by a heating wave which starts the next normal outburst which subsequently triggers the second fainter superoutburst. This hypothesis was partially confirmed by detection of superhumps in the post-dip light curve of AL Com. In case of V485 Cen we did not observe superhumps in the post-dip light curve, but we detected clear orbital humps. This may suggest that brightening of the star after the dip was a normal outburst. Situation is, however, unclear because observations of Augusteijn et al. (1993) did not reveal any modulations during regular normal outburst and their observations during quiescence also did not show any double humped structure.
Augusteijn et al. (1996) gave an estimate of the mass transfer from the secondary in V485 Cen between ${1\times10^{-10}}M_\odot$/y and ${1\times10^{-9}}M_\odot$/y. From theoretical calculations (see Osaki 1996 and references therein) we know that such a large values of ${\dot{M}}$ are characteristic for systems called “permanent superhumpers” – dwarf novae below the “period gap” which are in permanent superoutburst. The values of ${\dot{M}}$ obtained for systems such as WZ Sge, AL Com or SW UMa are around ${{\dot{M}}\approx2\times10^{-11}}M_\odot$/y, in other words a few times smaller than estimate of Augusteijn et al. (1996). According to the fact that V485 Cen often shows normal outbursts its mass transfer should be slightly higher than in systems like WZ Sge and AL Com. This is in good agreement with theoretical calculations made by Nelson et al. (1986) who, for system with secondary containing 50% of hydrogen, obtained minimal period equal to ${P_{\rm orb}=0.^d99}$, mass of the secondary ${M_{\rm sec}=0.057M_\odot}$ and mass transfer rate ${{\dot{M}}=0.58\times10^{-10}}M_\odot$/y.
[**Acknowledgments**]{} We would like to thank Prof. Andrzej Udalski for his helpful hints, reading and commenting on the manuscript. We are also grateful to Prof. Janusz Ka[ł]{}użny for his help with reduction of the raw data. This work was partly supported by the KBN grant BW to the Warsaw University Observatory.
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[^1]: Based on observations obtained with the 1.3 m Warsaw telescope at the Las Campanas Observatory of the Carnegie Institution of Washington
[^2]: IRAF is distributed by National Optical Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with National Science Foundation.
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One of the phenomena which is presently attracting interest both in physics and in biology is the so called ratchet effect [@Ajd92]. In simple terms, a ratchet system can be described as a periodically forced brownian particle moving in an asymmetric potential in presence of damping and periodic driving [@hangggi]. The periodic forcing keeps the system out of equilibrium so that the thermal energy can assist the conversion of the ac driver into effective work (direct motion of the particle) without any conflict with the second law of thermodynamics. This phenomenon has been found in several physical [@physical] and biological [@biological] systems and is presently considered as a possible mechanism by which biological motors perform their functions [@Ast94]. For ode systems with damping, additive forcing and noise, the ratchet effect can be viewed as a phase locking phenomenon between the motion of the particle in the periodic potential and the external driver [@Barbi]. Ratchet dynamics have been observed also in more complicated systems such as overdamped $\phi^4$ models [@marchesoni], chains of coupled particles with degenerate on-site potentials [@savin], long Josephson junctions with modulated widths [@goldobin], inhomogeneous parallel Josephson arrays [@trias], 3D helical models [@zolo], etc. These are infinite dimensional systems described by continuous or discrete equations of soliton type, with asymmetric potentials, damping, and periodic forcing, in which the ratchet phenomenon manifests as unidirectional motion of the soliton (soliton ratchet). For overdamped systems one can reduce the soliton ratchet to the usual single particle ratchet by using a collective coordinate approach for the center of mass of the soliton [@marchesoni]. For underdamped or moderately damped systems, however, this could be unappropriate, since the radiation field present in the system can play an important role for the generation of the phenomenon.
The aim of this Letter is to investigate the mechanism underlying soliton ratchets both in absence and in the presence of noise. To this end we use an asymmetric double sine-Gordon equation as a working model for studying the effect (the phenomena, however, will not depend on the particular model used). We show that the asymmetry of the potential induce a spatially asymmetric internal mode on the soliton profile which can be excited by the periodic force. In presence of damping, this mode can exchange energy with the translational mode so that the soliton can have a net motion under the action of the ac force. In this mechanism, the damping plays the role of coupling between the internal mode and the translational mode of the soliton. We find that, for fixed amplitude and frequency of the ac force, there is an optimal value of the damping for which the transport (i.e. the velocity achieved by the soliton) becomes maximal. In this case the frequency of the internal mode and the one of the external driver, perfectly match (phase locking). A similar resonant behavior is also observed by varying the frequency of the ac force, keeping fixed the other parameters of the system. At very high damping and fixed amplitude of the forcing, the internal mode oscillation becomes very small, and the transport due to soliton-ratchet is strongly reduced. At low damping and higher forcing we find, quite surprisingly, that current reversals can occur. Finally, we show that soliton ratchets can survive the presence of noise in the system.
We start by introducing the following generalized double sine-Gordon equation $$\phi _{tt}-\phi _{xx}=- \sin(\phi)- \lambda \sin (2\phi +\theta)
\, \equiv -\frac{dU(\phi )}{d\phi }, \label{nlkg}$$ with the potential $U(\phi )=C-\cos (\phi )-\frac{\lambda }{2}\cos
(2\phi +\theta )$. Here $\lambda $ is the asymmetry parameter, $\theta $ is a fixed phase and $C$ a constant which fix the zero of the potential. A discrete version of this equation was introduced in Ref. [@ms85] in terms of a chain of elastically coupled double pendula assembled by a gear of ratio $1/2$ with a phase angle $\theta$ between them. For $\lambda =0$ Eq. (\[nlkg\]) gives the well known sine-Gordon equation (SGE) with exact soliton solutions, while for $\lambda \neq 0$ and $\theta
=0$ (mod $\pi $) it reduces to the proper double sine-Gordon equation (note that in both cases the potential is periodic and symmetric in $\phi$). In the following we are interested in the case $\theta \neq 0$ (mod $\pi )$ for which the periodic potential becomes asymmetric. We shall refer to this case as the asymmetric double sine-Gordon equation (ADSGE). In particular, we fix $\theta
=\pi/2$ in Eq. (\[nlkg\]) in order to have maximal asymmetry, and choose $C=\cos \left(\phi_{0}\right) -{\lambda }/{2}\sin
\left( 2\phi_{0}\right)$, with $\phi _{0 }=\arcsin
[(1-A)/{4\lambda }]+2n\pi$ and $A=\sqrt{1+8\lambda ^{2}}$, to have the zero of the potential in correspondence with its absolute minima $\phi_{0 }$. We also assume, for simplicity, $\lambda \in
[-1,1]$ to avoid relative minima appearing in the potential. Besides the mentioned mechanical model, Eq. (\[nlkg\]) is also linked to another interesting physical system i.e. a one dimensional array of inductively coupled SQUIDs, each consisting of a loop of a Josephson junction in parallel with a serie of two identical Josephson junctions. The single element of this array was studied in Ref. [@hangii] in which it was shown that, due to the ratchet effect, the system can rectify periodic signals. In analogy with the single particle ratchet, it is reasonable to introduce in the distributed model, periodic forcing, damping, and noise, this leading to the following perturbed ADSGE $$\begin{aligned}
\phi _{tt}-\phi _{xx} &+&\sin (\phi )+\lambda \cos (2\phi )= \label{asydsg}
\\
&-&\alpha \phi _{t}+\epsilon \sin (\omega t+\theta_{0})+n(x,t).
\nonumber\end{aligned}$$ Here $\alpha $ denotes the damping constant, $n(x,t)$ is white noise with autocorrelation $$<n(x,t){n}(x^{\prime },t^{\prime })>=D\delta (x-x^{\prime })\delta
(t-t^{\prime }), \label{noise}$$ and $\epsilon $, $\omega $, $\theta _{0}$ are, respectively, the amplitude the frequency and the phase of the driver. Travelling wave solutions of the unperturbed ADSGE, i.e. solutions which depend on $\xi\equiv {(x-Vt)}/{\sqrt{1-V^{2}}}$, (note that Eq. (\[nlkg\]) is Lorentz invariant), can be found by substituting $\phi \equiv \phi(\xi)$ in the left hand side of Eq. (\[asydsg\]) and equating it to zero. After one integration in $\xi $ we obtain $$\int_{\phi_{0}}^{\phi }\frac{d\phi }{\sqrt{2(E+U)}}=\xi -\xi _{0},
\label{integral}$$ which gives, after the inversion of the integral at $E=0$ (top of the reversed potential), the $2\pi$-kink (antikink) solutions as $$\label{exact} \phi^{\pm}_{K}=\phi_{0}+2\tan^{-1}
\{\frac{sign(\lambda)\,A\,B}{A-1- \eta \sinh \left[\pm
\frac{\xi}{2}\sqrt{{AB}/{|\lambda |}}\right]}\}$$ where $\eta=2\lambda\sqrt{2(1+A)}$, and $B=\sqrt{2(4\lambda
^{2}-1+A)}$ (the plus and minus signs refer to the kink and antikink solutions, respectively). Note that in the limit $\lambda
\rightarrow 0$ Eq. (\[exact\]) reduces to the well known soliton solution of SGE. To investigate the existence of internal modes in the system, we linearize the ADSGE around the solution in Eq. (\[exact\]), i.e. we look for solutions of the form $\phi=
\phi^{\pm}_{K}+ \psi$ with $$\psi (x,t)=\exp (i\omega t)f(x), \quad f(x) << 1.$$ This leads to the following eigenvalue problem on the whole line $$\begin{aligned}
f_{xx}+\left( \omega ^{2}-\cos [\phi^{\pm}_{K}]+2\lambda \sin
(2\phi^{\pm}_{K})\right) f &=&0, \label{eq17}\end{aligned}$$ with $f_{x}(\pm \infty )=0$, which can be easily solved by numerical methods for any finite length of the system. In Fig. \[fig1\] we report the numerical spectrum of Eq. (\[eq17\]) as a function of $\lambda$. We see that except for the sine-Gordon limit ($\lambda = 0$), there is an internal mode frequency $\Omega_{i}$ below the spectrum of the phonon band (the zero mode, existing for all values of $\lambda$, is not plotted for graphical convenience). In the inset of the figure the shape of the internal mode is also reported, from which we see that the asymmetry of the potential induces a spatial asymmetry in $f(x)$. In the presence of a periodic force this internal mode can be easily excited.
To understand the role of the various elements of the problem, (i.e. asymmetry of the potential, internal mode, damping, forcing, and noise) it is better to consider first the zero noise case (deterministic soliton ratchet). By viewing the soliton as a string lying on the potential surface $\it{S}(U,\phi,x)$ and connecting adjacent minima, the following picture of the phenomenon can be given. If the potential is asymmetric (in $\phi
$) the transition from the top of the potential to one minimum and from the top of the potential to the other minimum, is also asymmetric (it will be more rapid for the part of the string lying on the region where the potential is more stiff). Thus, the potential asymmetry in $\phi$ induces an asymmetry in space which can be seen both in the $2\pi$-kink profile and in the internal mode. In presence of an ac force, but in absence of damping, this asymmetry will not produce transport, i.e. the string will slide, without any friction, back and forth on the potential profile along the $x-$direction. The presence of damping, however, introduces friction in this sliding, and the part of the string moving on the stiff part of the potential profile dissipate more that the other. This asymmetry in the dissipation produces net motion for the soliton (the string moves in the direction in which it approaches the potential minimum more smoothly). We can say that the effect of the damping is to couple the internal mode to the translational mode. The mechanism underlying deterministic soliton ratchet can then be described as follows: the ac force pumps energy in the internal mode which is converted into net dc motion by the coupling with the zero mode induced by the damping. >From this picture one can easily predict that in absence of the internal mode, or in absence of damping, no soliton ratchet can exist. Moreover, one expects that the maximal effect in transport, is observed when the internal oscillation and the external force are synchronized (phase-locked).
In order to confirm this picture, we have performed direct numerical integrations of the ADSGE for different values of the system parameters. First, we have checked that in the SGE limit, i.e. when $\lambda=0$, the ratchet dynamics does not exists. This agrees with the fact that in this case there is no asymmetry in the potential and no internal mode. At this point we remark that the dynamics of a SG kink subject to a periodic force was also investigated in Ref. [@olsa], in which it was shown that in absence of damping the kink can acquire a finite velocity depending on the initial phase of the ac force. Net motion of a SGE soliton was shown to be possible also in presence of a small damping, if the ac force excites a phonon mode which exchange energy with the soliton [@rc]. These cases, however, should not be confused with soliton ratchets since they strongly depend on initial conditions (if one average on initial conditions the transport disappears). Moreover, in contrast with soliton ratchets, these effects exist only at zero or at very low damping. Second, we have checked that for $\lambda\neq 0$ (asymmetric potential) but in absence of damping, soliton ratchets also do not exist (for brevity we will not expand on these cases here). From this analysis we conclude that, in analogy with the deterministic single particle ratchets, the asymmetry of the potential, the damping and, obviously the periodic forcing, are crucial ingredient for soliton ratchets. In Fig. \[fig2\] we show a prospectic view of the soliton ratchet dynamics as obtained from numerical integration of Eq. (\[asydsg\]). We remark that the direction of the motion is fixed by the asymmetry of the potential and can be inverted by changing the sign of $\lambda$. We see that, except for the soliton profile which is wobbling, no phonons are present in the system. This confirms the relevance of the internal mode in the phenomenon. We also find that for some parameter values, the net motion is more effective. To investigate the dependence of the phenomenon on parameters, we have performed numerical simulations of Eq. (\[asydsg\]) both by fixing all parameters and changing $\omega$, and by fixing all parameters and changing $\alpha$. In Fig.\[fig3\] we show the average velocity (computed by using an integration time $t=1000$) of the soliton center of mass versus the frequency of the external driver, for two different values of the amplitude of the ac force (the low damping part of the curves was not computed due to the longer integration times required in this case). From this figure we see that $\langle V \rangle$ has a maximum at $\omega \sim 1$, (i.e $\omega=1.04$ and $\omega=1$ for $\epsilon=0.4, \epsilon=0.6$, respectively), this being very close to the internal mode frequency $\Omega_{i} \approx 1.0562$ (the discrepancy is within the numerical accuracy of our numerical scheme). By increasing the amplitude of the forcing, the dynamics gets more complicated (breather-like excitations can appear) and the resonance peak in frequency more pronounced.
A similar resonant behavior is expected to exists also as function of the damping. When the damping is very high, indeed, the internal mode is almost suppressed by the damping, while when it is very low the coupling between the internal mode and the translational mode is very small, both cases giving minimal transport. In between these extremes, a value of the damping which allow the internal mode to synchronize with the external driver and optimize transport, should then exist. This is what we observe in Fig. \[fig4\] where the velocity vs damping is reported as a continuous curve. Although the ratchet velocity can be slightly increased by optimizing parameters, it remains small in comparison to the velocities achieved under the action of dc forces. This can be due to the fact that the internal and the translational mode seems to couple only to second order in a perturbation scheme. To increase the velocity one could increase the amplitude of the ac force, but this is limited by the threshold above which the system becomes chaotic. In practical applications, however, a small drift velocity is sufficient to manipulate solitons under the action of external ac forces.
To show that the internal mode is phase locked with of external driver we have plotted in the inset of this figure the $\phi_x$ profiles in the co-moving frame (i.e. the drift motion was subtracted) at two fixed times $t_{1}=211.6$ (solid line) and $t_{2}=227.3$ (circles) separated by one period $T=2\pi/\omega=15.7$ of the driver. We see that the profiles overlap each other, i.e. the oscillation on the kink profile is perfectly synchronized with the external driver (phase locking). >From Fig. \[fig4\] we see, quite surprisingly, that at low damping current reversals can occur (note that the average velocity becomes negative for $\alpha$ less than $\alpha_{cr}\sim
0.24$). We find that the value of $\alpha_{cr}$ increases as the amplitude of the driver is increased. The occurrence of this phenomenon, which resemble the one observed in single particle ratchets at low damping [@Barbi; @cha95], seems to be related more to the phonon-soliton interaction than to the internal mode mechanism described above (at low dampings a complicate transferral of energy between phonons, internal mode, and translational mode, can arise). To understand this phenomenon, however, a more detailed study is required.
Finally, we have investigated the effect of the noise on deterministic soliton ratchets. A preliminary analysis shows that for low noise amplitudes the effect of the noise on the phenomenon is minimal, in the sense that the dynamics gets dressed by the noise, but after averaging on the noise, almost the same soliton mean velocity $\langle V\rangle$ is obtained. This is shown by the stars in Fig. \[fig4\] which represent the numerical values of $\langle V\rangle$ calculated in presence of a noise of amplitude $D=0.01$. The fact that soliton ratchets can survive the presence of weak amplitude noise can be understood as consequence of the structural stability of phase locking phenomena against “small” fluctuations. This indicates that the phenomenon can exist also in real systems such as one dimensional arrays of inductively coupled SQUIDs. The mechanism of soliton ratchets discussed in this Letter is expected to be valid also for other soliton systems.
.2cm We thank M.Barbi and M.R. Samuelsen for interesting discussions. Financial support from the European grant LOCNET n.o HPRN-CT-1999-00163 is acknowledged.
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---
abstract: 'We are interested in reconstructing the initial condition of a non-linear partial differential equation (PDE), namely the Fokker-Planck equation, from the observation of a Dyson Brownian motion at a given time $t>0$. The Fokker-Planck equation describes the evolution of electrostatic repulsive particle systems, and can be seen as the large particle limit of correctly renormalized Dyson Brownian motions. The solution of the Fokker-Planck equation can be written as the free convolution of the initial condition and the semi-circular distribution. We propose a nonparametric estimator for the initial condition obtained by performing the free deconvolution via the subordination functions method. This statistical estimator is original as it involves the resolution of a fixed point equation, and a classical deconvolution by a Cauchy distribution. This is due to the fact that, [in free probability,]{} the analogue of the Fourier transform is the R-transform, related to the Cauchy transform. In past literature, there has been a focus on the estimation of the initial conditions of linear PDEs such as the heat equation, but to the best of our knowledge, this is the first time that the problem is tackled for a non-linear PDE. The convergence of the estimator is proved and the integrated mean square error is computed, providing rates of convergence similar to the ones known for non-parametric deconvolution methods. Finally, a simulation study illustrates the good performances of our estimator.'
author:
- |
Mylène Maïda[^1], Tien Dat Nguyen[^2], Thanh Mai Pham Ngoc[^3],\
Vincent Rivoirard[^4],Viet Chi Tran[^5]
title: 'Statistical deconvolution of the free Fokker-Planck equation at fixed time'
---
Keywords: PDE with random initial condition; free deconvolution; inverse problem; kernel estimation; Fourier transform; mean integrated square error; Dyson Brownian motion\
AMS 2000: 35Q62; 65M32; 62G05; 46L53; 35R30; 60B20; 46L54
Introduction
============
Motivations
-----------
Letting the initial condition of a partial differential equation (PDE) be random is interesting for considering complex phenomena or for introducing uncertainty and irregularity in the initial state. There is a large literature on the subject, and we can mention that this has been studied for the Navier-Stokes equation, to account for the turbulence arising in fluids with high velocities and low viscosities (see [@vishikfursikov; @constantinwu]), for the Burgers equation that is used in astrophysics (see [@burgers; @bertoingiraudisozaki; @giraud2003; @flandoli2018] or also the survey by [@vergassollaetal]), for the wave equations, to study the solutions with low-regularity initial data (see [@burqtzevtkovI; @burqtzevtkovII; @tzvetkov]) or for the Schrödinger PDE (see [@bourgain]). The Burgers PDE or the vortex equation, associated to the Navier-Stokes PDE by considering the curl of the velocity, are of the McKean-Vlasov type as introduced and studied in [@sznitman; @meleardcime]. Numerical approximations of such PDEs with random initial conditions have been considered in [@talayvaillant; @transolstat]. In this paper, we are interested in the Fokker-Planck PDE which is another case of McKean-Vlasov PDE [@carrillomccannvillani]. This equation models the motion of particles with electrostatic repulsion and a probabilistic interpretation that we will adopt has been considered in [@bianespeicher].\
A question naturally raised in this context is to estimate the random initial condition, given the observation of the PDE solution at a given fixed time $t>0$. For linear PDEs, this inverse problem is solved by deconvolution techniques, and this has been explored for PDEs such as the heat equation or the wave equation by Pensky and Sapatinas [@penskysapatinas09; @penskysapatinas10]. For the 1d-heat equation, it is known that the solution at time $t$, say $\nu_t(dx)$, is the convolution of the initial condition $\nu_0(dx)$ with Green function $G_t$, which is a Gaussian transition function associated with the standard Brownian motion $(B_t)_{t\geq 0}$. The probabilistic interpretation of the heat equation is built on this observation, and $\nu_t$ can be viewed as the distribution of $X_t=X_0+B_t$ where $X_0$ is distributed as $\nu_0$. Taking the Fourier transforms changes the convolution problem into a multiplication, which paves the way to reconstruct the initial condition.\
Here, we are interested in estimating the initial condition of a non-linear PDE, namely the Fokker-Planck equation, from the observation of its solution at time $t$. Recall that the Fokker-Planck equation is: $$\label{eq:fokkerplanck}
\partial_t p(t,x)=-\partial_x \int_{\R^2} Hp(t,x) p(t,x){\mathrm{d}}x,$$with $$Hp(t,x)= \lim_{\varepsilon \rightarrow 0} \int_{\R \setminus [x-\varepsilon,x+\varepsilon]} \frac{1}{x-y}p(t,y){\mathrm{d}}y,$$ and for $t\in\R_+$, $x\in \R$, and initial condition $p_0(x) \in L^1(\R)$. Contrarily to the examples considered in [@penskysapatinas09; @penskysapatinas10], this PDE is non-linear of the McKean-Vlasov type with logarithmic interactions. To the best of our knowledge, this is the first work devoted to the deconvolution of a non-linear PDE to recover the initial condition. The choice of this equation is motivated by its strong similarities with the heat equation: the standard Brownian motion of the probabilistic interpretation is replaced here by the free Brownian motion $(\mathbf{h}_t)_{t\geq 0}$ (operator-valued), and the usual convolution by a Gaussian distribution is replaced by the free convolution by a semi-circular distribution $\sigma_t$ characterized by its density with respect to the Lebesgue measure: $$\label{def:semicircle}
\sigma_t(dx)=\frac{1}{2\pi t}\sqrt{4t-x^2} \ind_{[-2\sqrt{t},2\sqrt{t}]}(x) \ dx.$$ If $\mathbf{x}_0$ admits the spectral measure $\mu_0$, then $\mathbf{x}_t=\mathbf{x}_0+\mathbf{h}_t$ admits $$\label{eq:freeconvolution}
\mu_t= \mu_0 \boxplus \sigma_t,$$as spectral measure, where the operation $\boxplus$ is the free convolution and has been introduced by Voiculescu in [@voiculescu]. It can be proved that [the density $p(t,\dot )$ of]{} $\mu_t$ solves .
For the Fokker-Planck equation, the inverse problem boils down to a free deconvolution, where it was a usual deconvolution for the heat equation. Recently, the problem of free deconvolution has been studied by Arizmendi, Tarrago and Vargas [@Tarrago1]. To solve in a general setting, subordination functions are used. Here, if the Cauchy transform of a measure $\mu$ is defined as $G_{\mu}(z)=\int_{\R} (z-x)^{-1}{\mathrm{d}}\mu(x)$ for $z\in {\mathbb{C}}^+$, where ${\mathbb{C}}^+$ is the set of complex numbers with positive imaginary part, the subordination function $w_{fp}(z)$ at time $t$ is related to $G_{\mu_t}$ by the functional equation $$\label{eq:intro}
w_{fp}(z)=z+t G_{\mu_t}(w_{fp}(z)).$$From this, we can recover $G_{\mu_0}$ with the formula $G_{\mu_0}(z)=G_{\mu_t}(w_{fp}(z))$ and thus $p_0$ (see Lemma \[lem:Gmu0\] and in the paper). More precisely, we prove in Section \[sec:defest\] that for any $\gamma>2\sqrt{t}$, $f_{\mu_0 * \mathcal C_{\gamma}}$ the density of the classical convolution of $\mu_0$ with the Cauchy distribution of parameter $\gamma$, defined by its density $$f_\gamma(x):= \frac{\gamma}{\pi(x^2+\gamma^2)},$$ satisfies $$\label{deconv-intro}
f_{\mu_0 * \mathcal C_{\gamma}}(x)= \frac{1}{\pi t} \left[ \gamma - \Im w_{fp}(x+ i \gamma)\right], \quad x \in \R.$$ Then, estimating $p_0$, the density of $\mu_0$, requires an estimation of the subordination function $w_{fp}$ combined with a classical deconvolution step from a Cauchy distribution.
Observations
------------
Additionally to the free deconvolution problem, our observation does not consist in the operator-valued random variable $\mathbf{x}_t$ but in its matricial counterpart. More precisely, we observe a matrix $X^n(t)$ for a given $t>0$, assumed to be fixed in the sequel, where $$\label{eq:Xn}
X^n(t)=X^n(0)+H^n(t),\qquad t\geq 0$$with $X^n(0)$ a diagonal matrix whose entries are the ordered statistic $\lambda^n_1(0)<\cdots <\lambda_n^n(0)$ of a vector $(d_i^n)_{i\in \{1,\dots n\}}$ of $n$ independent and identically distributed (i.i.d.) random variables distributed as $\mu_0(dx)=p_0(x)\ dx$, absolutely continuous with respect to the Lebesgue measure on $\R$, and $H^n(t)$ a standard Hermitian Brownian motion, as defined in Definition \[def:Hermit.BrowMotion\]. The purpose is to estimate $p_0$ without observing directly the initial condition $X^n(0)$. As the distribution of $H^n(t)$ is invariant by conjugation, choosing $X^n(0)$ to be a diagonal matrix is not restrictive. It is known, see [@AGZ page 249, section 4.3.1], that the eigenvalues $(\lambda_1^n(t),\cdots,\lambda_n^n(t))$ of $X^n(t)$ solve the following [system of]{} stochastic differential equations (SDE): $$\label{eq:SDE.lambda}
{\mathrm{d}}\lambda^{n}_i(t)= \frac{1}{\sqrt{n}}{\mathrm{d}}\beta_i(t)+ \frac{1}{n}\sum_{j\not= i}\frac{{\mathrm{d}}t}{\lambda^{n}_i(t)-\lambda^{n}_j(t)},\quad 1\leq i\leq n,$$where $\beta_i$ are i.i.d. standard [real]{} Brownian motions. If we denote by $$\label{def:empiricalmeasure}
\mu^n_t=\frac{1}{n}\sum_{i=1}^n \delta_{\lambda_i^n(t)}$$ the empirical measure of these eigenvalues at time $t$, then the process $(\mu^n_t)_{t \ge 0}$ converges weakly almost surely as $n$ goes to infinity to the process [$(\mu_t)_{t \ge 0}$]{} with density $(p(t,\cdot))_{t \ge 0}$ solution of . For $n=1$, we recover the classical heat equation as the Dyson Brownian motion boils down to a standard Brownian motion.\
Contributions
-------------
Relying on the analysis of Arizmendi et al. [@Tarrago1], we provide, in Theorem-Definition \[th:defestimator\], a statistical estimator $\widehat{w}^n_{fp}(z)$ for the subordination function. As the Cauchy transform $G_{\mu_t}$ in is not invertible on the whole domain ${\mathbb{C}}^+$, the subordination function $w_{fp}(z)$ will be defined only for $z\in {\mathbb{C}}_{2\sqrt{t}}$ where ${\mathbb{C}}_\gamma:=\{z\in {\mathbb{C}}^+,\ \Im(z)>\gamma\}$. We shall prove the following result.
\[propsition.consistency.estimate.w\_fp\]Let $\gamma>2\sqrt{t}$. Suppose $\lambda^{n}(0)$ satisfies the condition $$\label{C0}
\sup_{n \geq 1} \dfrac{1}{n} \sum_{i=1}^{n} \log \left( \lambda^{n}_{i}(0)^2 + 1 \right)<\infty \textrm{ almost surely (a.s.)} $$ Then, we have:\
(i) For any $z \in \mathbb{C}_{2\sqrt{t} }$, the estimator $\widehat{w}^{n}_{fp}(z)$ converges almost surely to $w_{fp}(z)$ as $n \rightarrow \infty$.\
(ii) The convergence is uniform on ${\mathbb{C}}_\gamma$.\
(iii) We have the following convergence rate on ${\mathbb{C}}_\gamma$: $$\sup_{n \in \N} \sup_{z \in {\mathbb{C}}_{\gamma}} \mathbb{E} \left[ n\big| \widehat{w}^{n}_{fp}(z) - w_{fp}(z) \big|^{2} \right] <+\infty.$$
To obtain uniform convergence and fluctuations ((ii) and (iii)), we will need to restrict to strict subdomains of ${\mathbb{C}}_{2\sqrt{t}}$. The fluctuations (iii) are established in the line of the work of Dallaporta and Février [@Fevrier1].\
Proposition \[propsition.consistency.estimate.w\_fp\] is the crucial tool to reach the main goal of this paper, namely providing an estimator of $p_0$. As explained previously, we estimate $p_0$ by combining a free deconvolution step via the use of $\widehat{w}^{n}_{fp}$ with then a classical deconvolution step. We define our final estimator $\widehat{p}_{0,h}$ via its Fourier transform, denoted $\widehat{p}_{0,h}^{\star}$; from Equation , it is natural to define it as follows: $$\widehat{p}_{0,h}^{\star}(\xi) = e^{\gamma |\xi| }. K_{h}^{\star}(\xi). \dfrac{1}{\pi t} \left[ \gamma - \Im \hspace{0.05cm} \widehat{w}^n_{fp}(\cdot+ i \ \gamma )^{\star}(\xi) \right],\quad \xi\in\R.$$ Note that, as usual in nonparametric statistics, the last expression depends on $K_{h}^{\star}$, a regularization term defined through the Fourier transform of a kernel function $K_{h}$ depending on a bandwidth parameter $h$. See Equation in Definition \[def:est\] for more details.\
We study theoretical properties of $\widehat{p}_{0,h}$ by deriving asymptotic rates of the mean integrated square error of $\widehat{p}_{0,h}$ decomposed as the sum of bias and variance terms. The study of the variance term is intricate and is based on the sharp controls of the difference $\widehat{w}^{n}_{fp}(z) - w_{fp}(z)$ provided by Proposition \[propsition.consistency.estimate.w\_fp\]. We show in Theorem \[variance\] that the variance term is of order $\frac{e^{\frac{2\gamma}{h}}}{n}$ as desired for deconvolution with the Cauchy distribution with parameter $\gamma$. The bias term is driven by the smoothness properties of the function $p_0$. In particular, when we assume that $p_0$ belongs to a space of supersmooth densities (see ), we can establish convergence rates, after an appropriate (non-adaptive) choice of the bandwidth parameter $h$. For instance, if $$\int_{\R} |p_0^\star(\xi)|^2 e^{2a |\xi|}{\mathrm{d}}\xi \leq L$$ for $0<L<\infty$, then $$\E\Big[\|\widehat{p}_{0,h}-p_0\|^2\big] = O\big(n^{-\frac{a}{a+\gamma}}\big).$$ The previous rate is optimal when we address the statistical deconvolution problem involving the Cauchy distribution with parameter $\gamma$. See Corollary \[thm:MISE\] for more details and more general results that establish the optimality of our procedure. Note that the exponent in the previous bound reflects the difficulty of our statistical problem: the larger $\gamma$, the smaller the rate. Remembering that $\gamma$ is connected to the observational time $t$ through the condition $\gamma>2\sqrt{t}$, it means that for the previous example, our estimate can achieve the polynomial rate $n^{-\frac{a}{a+2\sqrt{t}+\epsilon}}$ for any $\epsilon>0$. The question of whether it is possible to consider smaller values for $\gamma$ constitutes a challenging problem. Adaptive choices for $h$ are also a very interesting issue. These problems will be investigated in another work.
Overview of the paper
---------------------
In Section \[section:freedeconv\], we study the free deconvolution and explain the construction of the estimator $\widehat{p}_{0,h}$ of $p_0$. Existence results and properties of the subordination functions are precisely stated and proved. Then, in Section \[section:estimationw\], we prove Proposition \[propsition.consistency.estimate.w\_fp\]. In Section \[section:MISE\], rates of convergence of $\widehat{p}_{0,h}$ are established. Numerical simulations are provided in Section \[section:numerical\].
#### Notations: {#notations .unnumbered}
For any $z=u+iv\in{\mathbb{C}}+$, we denote $\sqrt{z}:=a+ib\in{\mathbb{C}}$ with $$a=\sqrt{\frac{\sqrt{u^2+v^2}+u}{2}},\quad b=\sqrt{\frac{\sqrt{u^2+v^2}-u}{2}}.$$ We denote the Fourier transform of a function $g \in {\mathbb L}^{1}(\R)$ by $g^{\star}$: $$\label{Fourier}
g^\star(\xi):=\int_\R g(x)e^{ix\xi}dx,\quad \xi\in\R.$$
Free deconvolution of the Fokker-Planck equation {#section:freedeconv}
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Dyson Brownian motions
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Let us denote by $\mathcal{H}_{n}({\mathbb{C}})$ the space of $n$-dimensional matrices $H_n$ such that $\left(H_{n}\right)^{*} = H_{n}$.
\[def:Hermit.BrowMotion\] Let $\big( B_{i,j}, \tilde{B}_{i,j}, 1 \leq i \leq j \leq n \big)$ be a collection of i.i.d. real valued standard Brownian motions, the Hermitian Brownian motion, denoted $H^{n} \in \mathcal{H}_{n}({\mathbb{C}})$, is the random process with entries $\left\{ (H^{n}(t))_{k,l}, t \geq 0,1 \le k, l \leq n\right\}$ equal to $$\label{eq:Hermit.BrowMotion}
(H^{n})_{k,l} =
\begin{cases}
\dfrac{1}{\sqrt{2n}} \left( B_{k,l} + i\ \tilde{B}_{k,l} \right), &\textrm{if } k < l \\[0.5cm]
\dfrac{1}{\sqrt{n}} B_{k,k}, &\textrm{if } k = l
\end{cases}$$
Let us now define the initial condition, that we will choose independent of the Hermitian Brownian motion $H^n$. Recall that $\mu_0$ is a probability measure with density $p_0(x)$ with respect to the Lebesgue measure on $\R$. Without loss of generality, we can choose the initial condition $X^n(0)$ to be a diagonal matrix, with entries $(\lambda^n_1(0), \ldots, \lambda^n_n(0))$ the ordered statistics of i.i.d. random variables [$(d_i^n)_{1\le i \le n}$]{} with distribution $\mu_0$.
For $t \geq 0$, let $\lambda^{n}(t) = \big( \lambda^{n}_{1}(t), \dots, \lambda^{n}_{n}(t) \big)$ denote the ordered collection of eigenvalues of $$\label{eq:Dyson.BrowMotion}
X^{n}(t) = X^{n}(0) + H^{n}(t).$$
\[thm.Dyson\] The process $\big( \lambda^{n}(t) \big)_{t \geq 0}$ is the unique solution in $C \left( \mathbb{R}_{+}, \mathbb{R}^{n} \right)$ of the system $$d\lambda^{n}_{i}(t) = \dfrac{1}{\sqrt{n}}d\beta_{i}(t) + \dfrac{1}{n} \sum_{j\not= i} \dfrac{dt}{\lambda^{n}_{i}(t) - \lambda^{n}_{j}(t)}, \quad 1 \leq i \leq n, \tag{\ref{eq:SDE.lambda}}$$ with initial condition $\lambda^{n}_{i}(0)$ and where $\beta_{i}$ are i.i.d. real valued standard Brownian motions. With probability one and for all $t> 0$, $ \lambda^{n}_{1} (t)<\ldots <\lambda^{n}_{n} (t) $.
Moreover, if for fixed $T > 0$, we denote by $\mathcal{C} \big(\left[0,T\right], \mathcal{M}_{1}\left(\mathbb{R}\right) \big)$ the space of continuous processes from $\left[0 , T\right]$ into $\mathcal{M}_{1}\left(\mathbb{R}\right),$ the space of probability measure on $\mathbb{R}$, equipped with its weak topology, we now prove the convergence of the process of empirical measures $\mu^{n}$ as defined in , viewed as an element of $C \big(\left[0,T\right], \mathcal{M}_{1}\left(\mathbb{R}\right) \big)$.
\[prop:LGN.mu.t\] Under Assumption , for any fixed time $T < \infty$, $\big( \mu^{n}_{t} \big)_{t \in [0,T]}$ converges almost surely in $\mathcal{C} \big( [0,T] , \mathcal{M}_{1}\left(\mathbb{R}\right) \big)$. Moreover, its limit is the unique measure-valued process $\left( \mu_{t} \right)_{t \in [0,T]}$ whose densities satisfy with initial condition [$p_0.$]{}
For deterministic initial conditions, Theorem \[thm.Dyson\] and Proposition \[prop:LGN.mu.t\] are classical results and we refer to [@AGZ Section 4.3] for a proof. Both results can be easily extended to random initial conditions, independent of the Hermitian Brownian motion itself. For details, we refer to [@datthesis].
Free deconvolution by subordination method
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Our starting point is , for a fixed time $t >0.$ Recovering $\mu_0$ knowing $\mu_t$ is a free deconvolution problem. The generic problem of free deconvolution has been introduced and studied by Arizmendi et al. [@Tarrago1] with the use of the Cauchy transform instead of the Fourier transform. Before stating their result, we need to introduce a few notations and definitions.
\[def:Cauchy.transform\] Let $\mu$ be a probability measure on $\R$, the Cauchy transform of $\mu$ is defined by: $$G_\mu(z)=\int_\R \frac{d\mu(x)}{z-x}, \quad z \in \mathbb{C} \setminus \mathbb{R}.$$
The fact is that $G_{\mu}\left( \overline{z} \right) = \overline{G_{\mu}(z)}$, so the behavior of the Cauchy transform in the lower half-plan $\mathbb{C}^{-} = \left\{ z \in {\mathbb{C}}| \Im(z) < 0 \right\}$ can be determined by its behavior in the upper half-plan $\mathbb{C}^{+} = \left\{ z \in {\mathbb{C}}| \Im(z) > 0 \right\}$. The function $G_\mu$ is a bijection from a neighbourhood of infinity to a neighbourhood of zero (see [@Biane1] for example) and we can define the $R$-transform of $\mu$ by: $$R_\mu(z)=G_\mu^{<-1>}(z)-\frac{1}{z},$$where $G_\mu^{<-1>}(z)$ is the inverse function of $G_\mu$ on a proper neighbourhood of zero. This $R$-transform plays the role of the logarithm of the Fourier transform for the free convolution in the sense that for any probability measures $\mu_1$ and $\mu_2$, $$\label{eq:R-transform}
R_{\mu_1\boxplus \mu_2}=R_{\mu_1}+R_{\mu_2}.$$ Using this formula for statistical deconvolution requires the computation of two inverse functions, and Arizmendi et al. [@Tarrago1] propose to use subordination functions [which also characterize the free convolution as in]{} .\
Let us recall the definition of subordination functions due to Voiculescu [@voiculescu93]. We first introduce $F_\mu(z)=1/G_\mu(z)$. As $G_\mu$ does not vanish on ${\mathbb{C}}^+$, $F_\mu$ is well defined on ${\mathbb{C}}^+$. Then:
\[def:subordinationfunction\]There exist unique subordination functions $\alpha_{1}$ and $\alpha_{2}$ from ${\mathbb{C}}^+$ onto ${\mathbb{C}}^+$ such that:\
(i) for $z \in {\mathbb{C}}^+$, $\Im\big(\alpha_{1}(z)\big) \geq \Im(z)$ and $\Im\big(\alpha_{2}(z)\big) \geq \Im(z)$, with $$\lim_{y\rightarrow +\infty} \frac{\alpha_{1}(iy)}{iy}=\lim_{y\rightarrow +\infty} \frac{\alpha_{2}(iy)}{iy}=1.$$ (ii) for $z \in {\mathbb{C}}^+$, $F_{\mu_1\boxplus \mu_2}(z)=F_{\mu_1}(\alpha_{1}(z))=F_{\mu_2}(\alpha_{2}(z))$ and $\alpha_1(z)+\alpha_2(z)=F_{\mu_1\boxplus \mu_2}(z)+z$.
Using this result, Belinschi and Bercovici [@belinschibercovici Theorem 3.2] introduce a fixed-point construction of the subordination functions, which Arizmendi et al. [@Tarrago1] adapt for the deconvolution problem. We state their result in the special case of the deconvolution by a semi-circular distribution defined in . In this case, we have an explicit formula for its Cauchy transform $G_{\sigma_t}(z)$ and its reciprocal function $F_{\sigma_t}(z):$ $$\label{eq.relation.FGSigmat}
G_{\sigma_t}(z)=\frac{z-\sqrt{z^2-4t}}{2t},\qquad \mbox{ and }\qquad
z - F_{\sigma_t}(z) = t \ G_{\sigma_t}(z).$$ Before stating the result, let us define, for any $\gamma > 0,$ $$\mathbb{C}_{\gamma} = \left\{ z \in \mathbb{C}^{+} \big| \textrm{Im}(z) > \gamma \right\}.$$ These domains will appear since $G_\mu$ is not invertible on the whole plane $\mathbb{C}$.
\[thm.Fixpoint\] There exist unique subordination functions $w_{1}$ and $w_{fp}$ from ${\mathbb{C}}_{2\sqrt{t}}$ onto ${\mathbb{C}}^+$ such that following properties are satisfied.\
(i) For $z \in {\mathbb{C}}_{ 2\sqrt{t} }$, $\Im\big(w_{1}(z)\big) \geq \dfrac{1}{2} \Im(z)$ and $\Im\big(w_{fp}(z)\big) \geq \dfrac{1}{2} \Im(z)$, and also $$\lim_{y\rightarrow +\infty} \frac{w_{1}(iy)}{iy}=\lim_{y\rightarrow +\infty} \frac{w_{fp}(iy)}{iy}=1.$$ (ii) For $z \in {\mathbb{C}}_{2\sqrt{t}}$ : $$\label{eq.Subordi.ii} F_{\mu_0}(z)=F_{\sigma_t}(w_{1}(z))=F_{\mu_t}(w_{fp}(z)).$$ (iii) For $z\in {\mathbb{C}}_{2\sqrt{t}}$ : $$w_{fp}(z)=z+w_{1}(z)-F_{\mu_0}(z). \label{eq.Subordi.func}$$ (iv) Denote $h_{\sigma_t}(w) = w - F_{\sigma_t}(w) = t\ G_{\sigma_t}(w)$ and $\widetilde{h}_{\mu_t}(w)=w+F_{\mu_t}(w)$ on $\mathbb{C}^{+}$. We can define the function ${L}_z$ as $$\begin{aligned}
L_z(w) :&= h_{\sigma_t} \big( \widetilde{h}_{\mu_t}(w) - z \big) + z \nonumber\\
&= t. G_{\sigma_t} \big( \widetilde{h}_{\mu_t}(w) - z \big) + z . \label{fixpoint.func.form}
\end{aligned}$$ For any $z\in {\mathbb{C}}_{2\sqrt{t}}$, we have $$\label{eq:fix-point}L_z\big(w_{fp}(z)\big)=w_{fp}(z),$$ and for all $w$ such that $\Im(w) > \frac{1}{2} \Im(z)$, the iterated function $L_z^{\circ m}(w)$ converges to $w_{fp}(z) \in \mathbb{C}^{+}$ when $m\rightarrow +\infty$.
One difference between Theorem \[thm.Fixpoint\] and Theorem-definition \[def:subordinationfunction\] lies in the fact that the subordination functions are expressed in terms of $F_{\mu_0\boxplus \sigma_t}$ and $F_{\sigma_t}$ whereas in Theorem-definition \[def:subordinationfunction\] it would have been $F_{\mu_0}$ and $F_{\sigma_t}$. Here the restriction to the domain ${\mathbb{C}}_{2\sqrt{t}}$ comes from the fact that $\Im(\widetilde{h}_{\mu_t}(w) - z )$ appearing in the definition of $L_z$ has to be positive.\
The proof of Theorem \[thm.Fixpoint\] is postponed to the last subsection of this section, Section \[sec:proofTheoremFixPoint\]. We now explain how the subordination functions allow us to construct the estimator of $p_0$.
Construction of the estimator of $p_0$ {#sec:defest}
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#### Overview of the estimation strategy
Based on Theorem \[thm.Fixpoint\], we devise the estimation strategy of the paper. The theorem allows us to get the subordination function $w_{fp}$ as a fixed point of $L_z$. From there, we will be able to recover the Cauchy transform of the initial condition $\mu_0$ from $w_{fp}$, as stated in the following lemma proved at the end of the section:
\[lem:Gmu0\] For any $z \in {\mathbb{C}}_{ 2\sqrt{t} },$ $$\label{eq:lemGmu0} G_{\mu_0}(z) = \frac{1}{t}(w_{fp}(z)-z)=G_{\mu_t}\big(w_{fp}(z)\big).$$ Consequently, $$\label{estimee:wfp}
\left| w_{fp}(z) - z \right| \leq \sqrt{t}.$$
Moreover, for any $\gamma >0,$ if we denote by $\mathcal C_\gamma$ the centered Cauchy distribution with density: $$f_\gamma(x):= \frac{\gamma}{\pi(x^2+\gamma^2)},$$ one can check that, for any probability measure $\mu$ on $\R,$ the density $f_{\mu * \mathcal C_\gamma}$ of the classical convolution of $\mu$ by $\mathcal C_\gamma$ is given, for $x \in \R,$ by $$\label{eq:lien_f_G} f_{\mu * \mathcal C_\gamma}(x)= - \frac{1}{\pi}\Im G_{\mu}(x+ i \gamma).$$ Using the expression of $G_{\mu_0}$ given by Lemma \[lem:Gmu0\] with $\gamma>2\sqrt{t}$, we get that for any $x \in \R,$ $$\label{eq:densiteconvolee}
f_{\mu_0 * \mathcal C_{\gamma}}(x)= \frac{1}{\pi t} \left[ \gamma - \Im w_{fp}(x+ i \gamma)\right].$$ From this, we can recover the density $p_0$ of $\mu_0$ by a classical deconvolution of by $f_{\gamma}$. The subordination function $w_{fp}$ in is estimated using the second equality of Lemma \[lem:Gmu0\]. In parallel with our work, Tarrago [@tarrago] has used the formula to perform spectral deconvolution in a more general setting (including the multiplicative free convolution), but neither the approximation of $w_{fp}$ by its estimator $\widehat{w}^n_{fp}$ defined Theorem-Definition \[th:defestimator\] below nor the (classical) deconvolution of the Cauchy distribution are treated, which are key difficulties encountered in our paper. Tarrago uses a different approach based on concentration inequalities when we use fluctuations in view of the work of Février and Dallaporta [@Fevrier1]. To prove the rates announced in the introduction, we need to establish very precise estimates of the error terms (see Section \[section:MISE\]).
Now, from (\[eq.Subordi.func\]), and , we write for $z\in {\mathbb{C}}_{2\sqrt{t}}$, $$w_{fp}(z) = z + w_{1}(z) - F_{\mu_0}(z) = z + w_{1}(z) - F_{\sigma_t}\big( w_{1}(z) \big) = z + t. G_{\sigma_t} \big( w_{1}(z) \big).$$ So, we obtain $$G_{\sigma_t}\big( w_{1}(z) \big) = \dfrac{1}{t} \big( w_{fp}(z) - z \big).$$Using again , we obtain both equalities of . From there $$\left| w_{fp}(z) - z \right| = t.\left| G_{\sigma_t}(w_{1}(z))\right| = t.\left| G_{\mu_{t}}(w_{fp}(z)) \right| \leq \dfrac{t}{| \Im(w_{fp}(z)) |} \leq \sqrt{ t},$$using Theorem \[thm.Fixpoint\] (i).
#### Estimator of $p_0$
We do not observe directly the measure $\mu_t$. The observation is the matrix $X^n(t)$ at time $t>0$ for a given $n$. From this observation, we can construct the empirical spectral measure as defined in . Then, for $z \in {\mathbb{C}}^{+}$, a natural estimator of $G_{\mu_t}(z)$ is obtained as follows: $$\label{def:Ghat}
\widehat{G}_{\mu^{n}_t}(z):=\int_{\R} \frac{d\mu^n_t(\lambda)}{z-\lambda}=\frac{1}{n}\sum_{j=1}^n \frac{1}{z-\lambda^{n}_j(t)}=\frac{1}{n}\mbox{tr}\Big(\big(zI_n-X_n(t) \big)^{-1}\Big).$$
\[th:defestimator\] There exists a unique fixed-point to the following functional equation in $w(z)$:$$\label{eq:fixed-point.empirical}
\dfrac{1}{t} (w(z)-z)= \widehat G_{\mu^{n}_t}(w(z)), \quad \mbox{ for } z \in {\mathbb{C}}_{ 2\sqrt{t} }$$ This fixed-point is denoted by $\widehat{w}^n_{fp}(z)$. We have $\Im(\widehat{w}^{n}_{fp}(z))>\Im(z)/2$ and $\left| \widehat{w}^{n}_{fp}(z) - z \right| \leq \sqrt{t}$.
The theorem is proved at the end of this section. We shall prove in Section \[section:estimationw\] that $\widehat{w}^n_{fp}(z)$ is a convergent estimator of $w_{fp}(z)$ and establish a fluctuation result associated with this convergence. This is the result announced in Proposition \[propsition.consistency.estimate.w\_fp\]. Let us now explain how the estimator of $p_0$ can be obtained from $\widehat{w}^n_{fp}(z)$.\
Recall that the Fourier transform of the Cauchy distribution $\mathcal{C}_{\alpha}$ with $\alpha>0$ is $f^{\star}_{\alpha} (\xi) = e^{-\alpha \left|\xi\right|}$ for $\xi \in \R$. Performing the deconvolution from , the Fourier transform of $p_0$ is the division of the Fourier transform of the right-hand side of by $f^{\star}_{\gamma} (\xi)$ with $\gamma>2\sqrt{t}$. It is now classical to define our ultimate estimator for the density function $p_0$ from its Fourier transform:
\[def:est\] Let us consider a bandwidth $h>0$ and a regularizing kernel $K$. We assume that the kernel $K$ is such that its Fourier transform $K^\star$ is bounded by a positive constant $C_K<+\infty$ and has a compact support, say $[-1,1]$. We define the estimator $\widehat{p}_{0,h}$ of $p_0$ by its Fourier transform: $$\label{Fourier.f_mu0.hat}
\widehat{p}_{0,h}^{\star}(\xi) = e^{\gamma |\xi| }. K_{h}^{\star}(\xi). \dfrac{1}{\pi t} \left[ \gamma - \Im \hspace{0.05cm} \widehat{w}^n_{fp}(\cdot+ i \ \gamma )^{\star}(\xi) \right],$$ where we have defined $K_h(\cdot)=\frac{1}{h}K\big(\frac{\cdot}{h}\big)$.
Note that the assumption on $K$ ensures the finiteness of the estimator. These assumptions are for instance satisfied for $K(x)={\mbox{sinc}}(x)=\sin(x)/(\pi x)$ whose Fourier transform is $K^{\star}(\xi)=\mathds{1}_{[-1, 1]}(\xi)$, and in this case $C_K=1$.
Proof of Theorem \[thm.Fixpoint\] {#sec:proofTheoremFixPoint}
---------------------------------
The constants of Theorem \[thm.Fixpoint\] are better than the ones of Arizmendi et al. [@Tarrago1] who work in full generality. We sketch here the main steps of the proof in our context, using the explicit formula for the semi-circular distribution.\
In the whole proof, we consider $z \in \mathbb{C}_{2\sqrt{t}}$.\
**Step 1:** We first prove that the function ${L}_z(w) = h_{\sigma_t}\big( \widetilde{h}_{\mu_t}(w) - z \big) + z$ is well-defined and analytic on $\mathbb{C}_{\frac{1}{2} \Im(z)}$. Since $h_{\sigma_t}$ is defined on $\mathbb{C}^{+}$, we need to check that $\widetilde{h}_{\mu_t}(w) - z \in \mathbb{C}^{+}$ for $w \in \mathbb{C}_{\frac{1}{2} \Im(z)}$. This is satisfied since for such $w$, $$\begin{aligned}
\label{etape2}
\Im\big( \widetilde{h}_{\mu_t}(w) - z \big) = \Im \big( w + F_{\mu_t}(w) - z \big) \geq 2 \Im(w) - \Im(z) > 0,
\end{aligned}$$ where we have used $\Im F_{\mu_t}(w) \geq \Im(w)$ for the first inequality. Indeed, if $w=w_1+iw_2$, we have $$(F_{\mu_t}(w))^{-1}=G_{\mu_t}(w)=\int\frac{d\mu_t(x)}{w_1+iw_2-x}=\int \frac{(w_1-x)d\mu_t(x)}{(w_1-x)^2+w_2^2}-iw_2\int \frac{d\mu_t(x)}{(w_1-x)^2+w_2^2}$$ and $$\begin{aligned}
\label{mino}
\Im(F_{\mu_t}(w))&=w_2\times\frac{\int \frac{d\mu_t(x)}{(w_1-x)^2+w_2^2}}{\left(\int \frac{(w_1-x)d\mu_t(x)}{(w_1-x)^2+w_2^2}\right)^2+w_2^2\left(\int \frac{d\mu_t(x)}{(w_1-x)^2+w_2^2}\right)^2}\nonumber\\
&\geq w_2\times\frac{\int \frac{d\mu_t(x)}{(w_1-x)^2+w_2^2}}{\int \frac{(w_1-x)^2d\mu_t(x)}{\left((w_1-x)^2+w_2^2\right)^2}+w_2^2\int \frac{d\mu_t(x)}{\left((w_1-x)^2+w_2^2\right)^2}}=w_2\end{aligned}$$
**Step 2:** We show that $L_z({\mathbb{C}}_{\frac{1}{2} \Im(z)})\subset \overline{{\mathbb{C}}_{\frac{1}{2} \Im(z)}}$ and that $L_z$ is not a conformal automorphism.\
First, let us show that ${L}_z\left( \mathbb{C}_{\frac{1}{2} \Im(z)} \right) \subset \overline{\mathbb{C}_{\frac{1}{2} \Im(z)}}$. Let $w \in \mathbb{C}_{\frac{1}{2} \Im(z)}$, we have: $$\begin{aligned}
\Im \left({L}_z(w)\right) &= \Im \left[t.G_{\sigma_t} \big( \widetilde{h}_{\mu_t}(w) - z \big) + z \right] \nonumber\\
&= \Im \left( \dfrac{\widetilde{h}_{\mu_t}(w) - z - \sqrt{\left(\widetilde{h}_{\mu_t}(w) - z\right)^2 - 4t} }{2} + z \right).\label{etape1}
\end{aligned}$$To lower bound the right hand side, note that for all $v\in {\mathbb{C}}^+$, one can check that: $$\begin{aligned}
\Im\big(\sqrt{v^2-4t}\big)\leq &\sqrt{\Im^2(v)+4t}.\end{aligned}$$Therefore, we have: $$\Im \left( \sqrt{\big( \widetilde{h}_{\mu_t}(w) - z \big)^2 - 4t} \right) \leq \sqrt{\big[\Im \big( \widetilde{h}_{\mu_t}(w) - z \big) \big]^2 + 4t} .$$ Hence, yields: $$\begin{aligned}
\Im \big( L_z(w)\big) \geq \dfrac{1}{2} \left[ \Im\big( \widetilde{h}_{\mu_t}(w) - z \big) - \sqrt{\big[\Im \big( \widetilde{h}_{\mu_t}(w) - z \big) \big]^2 + 4t} \right] + \Im(z) .
\end{aligned}$$The function $g(s) = s - \sqrt{s^2 + 4t}$ is non-decreasing on $\R_+$ and for all $s > 0$, $ g(s) \geq -2\sqrt{t}$. This implies that $$\begin{aligned}
\label{eq:sqrtt}
\Im\big( {L}_z(w) \big) \geq \Im(z) -\sqrt{t}> \frac{1}{2} \Im(z),
\end{aligned}$$since $z\in {\mathbb{C}}_{2\sqrt{t}}$. This guarantees that $L_z(w) \in \mathbb{C}_{\frac{1}{2} \Im(z)}$.\
Let us now prove that ${L}_z$ is not an automorphism of $\mathbb{C}_{\frac{1}{2} \Im(z)}$. Consider $$\begin{aligned}
\left| {L}_z(w) - z \right| = & \left| F_{\sigma_t}\big( \widetilde{h}_{\mu_t}(w) - z \big) - \big( \widetilde{h}_{\mu_t}(w) - z \big) \right| = \left|t G_{\sigma_t}\big(\widetilde{h}_{\mu_t}(w) - z\big)\right|.
\end{aligned}$$ For $v\in {\mathbb{C}}^+$, if $|v|>3\sqrt{t}$, since the support of $\sigma_t$ is $[-2\sqrt{t},2\sqrt{t}]$, $$\left|t G_{\sigma_t}(v)\right| = \left| \int_{-2\sqrt{t}}^{2\sqrt{t}} \frac{t}{v-x}{\mathrm{d}}\sigma_t(x) \right|\leq \sqrt{t}.$$ If $|v|\leq 3\sqrt{t}$, $$\left|t G_{\sigma_t}(v)\right| =\left| \frac{v-\sqrt{v^2-4t}}{2} \right|\leq \frac{2|v|+2\sqrt{t}}{2}\leq 4\sqrt{t}.$$ Hence, for all $w\in \mathbb{C}_{\frac{1}{2} \Im(z)}$, $$\label{eq:ball}
\left| {L}_z(w) - z \right|\leq 4\sqrt{t}.$$ This implies that ${L}_z\left( \mathbb{C}_{\frac{1}{2} \Im(z)} \right)$ is included in the ball centered at $z$ with radius $4\sqrt{t}$. As a result, ${L}_z$ is not surjective and hence is not an automorphism of $\mathbb{C}_{\frac{1}{2} \Im(z)}$.\
**Step 3:** Existence and uniqueness of $w_{fp}$, which is a fixed point of $L_z$.\
By Steps 1 and 2, $L_z$ satisfies the assumptions of Denjoy-Wolff’s fixed-point theorem (see e.g. [@belinschibercovici; @Tarrago1]). The theorem says that for all $w\in {\mathbb{C}}_{\frac{1}{2} \Im(z)}$ the iterated sequence $L_z^{\circ m}(w)=L_z\circ L_z^{\circ (m-1)}(w)$ converges to the unique Denjoy-Wolff point of $L_z$ which we define as $w_{fp}(z)$. The Denjoy-Wolff point is either a fixed-point of $L_z$ or a point of the boundary of the domain. Let us check that $w_{fp}$ is a fixed point of $L_z$. For any $z \in {\mathbb{C}}_{2\sqrt t},$ there exists $\gamma >2$ such that $z \in {\mathbb{C}}_{\gamma\sqrt t}$ and from , $L_z({\mathbb{C}}_{\frac{1}{2} \Im(z)}) \subset {\mathbb{C}}_{(1-\frac{1}{\gamma}) \Im(z)}.$ Moreover, from , $L_z({\mathbb{C}}_{\frac{1}{2} \Im(z)}) \subset B(z, 4\sqrt t).$ Therefore, $w_{fp}(z) \in \overline{{\mathbb{C}}_{(1-\frac{1}{\gamma}) \Im(z)} \cap B(z, 4\sqrt t) } \subset {\mathbb{C}}_{\frac{1}{2} \Im(z)},$ so that it is [necessarily]{} a fixed point.\
We now define $$w_1 (z) := F_{\mu_t}(w_{fp}(z) ) + w_{fp}(z) -z.$$
One can check that $$\begin{aligned}
F_{\sigma_t}(w_1(z)) &= &w_1(z) - h_{\sigma_t}(w_1(z)) \\
& = & F_{\mu_t}(w_{fp}(z) ) + w_{fp}(z) -z - h_{\sigma_t}(F_{\mu_t}(w_{fp}(z) ) + w_{fp}(z) -z)\\
& = & \tilde h_{\mu_t}(w_{fp}(z) ) - z - h_{\sigma_t}(\tilde h_{\mu_t}(w_{fp}(z) ) - z ) \\
& = & \tilde h_{\mu_t}(w_{fp}(z) ) - w_{fp}(z) = F_{\mu_t}(w_{fp}(z)).\end{aligned}$$ One can therefore rewrite $$\label{eq:w1}
w_1(z) = F_{\sigma_t}(w_1(z)) + w_{fp}(z)-z.$$ From and the fact that $w_{fp}(z)$ is a fixed point of $L_z,$ one easily gets that $$\lim_{y\rightarrow +\infty} \frac{w_{fp}(iy)}{iy}=1,$$ which implies that $$\lim_{y\rightarrow +\infty} \frac{F_{\mu_t}(w_{fp}(iy))}{iy}=1,\quad \mbox{ and }\quad \lim_{y\rightarrow +\infty} \frac{w_1(iy)}{iy}=1.$$
Now we connect $F_{\mu_0}$ to the previous quantities. For $z$ large enough, all the functions we consider are invertible and we have $$F_{\mu_t}(w_{fp}(z)) + w_{fp}(z) = z+ w_1 (z) = z + F_{\sigma_t}^{<-1>}(F_{\mu_t}(w_{fp}(z))).$$ On the other hand, for $z$ large enough, using Theorem-definition \[def:subordinationfunction\] for $\mu_1= \sigma_t$ and $\mu_2= \mu_0,$ we get $$F_{\mu_t}(w_{fp}(z)) + w_{fp}(z) = \alpha_1(w_{fp}(z)) + \alpha_2(w_{fp}(z)) = F_{\sigma_t}^{<-1>}(F_{\mu_t}(w_{fp}(z))) + F_{\mu_0}^{<-1>}(F_{\mu_t}(w_{fp}(z))).$$ Comparing the two equalities gives $$F_{\mu_0}^{<-1>}(F_{\mu_t}(w_{fp}(z))) = z,$$ so that, for $z$ large enough, $$F_{\mu_t}(w_{fp}(z)) = F_{\mu_0}(z).$$ The two functions being analytic on ${\mathbb{C}}_{2 \sqrt t},$ the equality can be extended to any $z \in {\mathbb{C}}_{2 \sqrt t}.$
Finally, since $$w_1(z) = F_{\mu_t}(w_{fp}(z) ) + w_{fp}(z) -z=F_{\mu_0}(z) + w_{fp}(z) -z,$$ we have, using with $\mu_0$ instead of $\mu_t$, $$\Im (w_1(z))=\Im(F_{\mu_0}(z)) + \Im(w_{fp}(z)) -\Im(z)\geq \Im(w_{fp}(z))\geq \frac{1}{2} \Im(z).$$
This ends the proof of Theorem \[thm.Fixpoint\].
Proof of Theorem-Definition \[th:defestimator\]
-----------------------------------------------
The proof of this theorem follows the steps of the proof of Theorem \[thm.Fixpoint\]. First, $\widehat{L}_z(w):=t \widehat{G}_{\mu^n_t}(w)+z$ is a well-defined and analytic function on ${\mathbb{C}}^+$. Let us check that $\widehat{L}_z\big({\mathbb{C}}_{\frac{1}{2}\Im(z)}\big)\subset {\mathbb{C}}_{\frac{1}{2}\Im(z)}$ for $z\in {\mathbb{C}}_{2\sqrt{t}}$. For $w=u+iv \in {\mathbb{C}}_{\frac{1}{2}\Im(z)}$, $$\Im\big(\widehat{G}_{\mu^n_t}(w)\big)=\frac{1}{n}\sum_{j=1}^n \Im\Big(\frac{u-\lambda_j^n(t) - i v}{(u-\lambda_j^n(t))^2+v^2}\Big)>-\frac{1}{v}=-\frac{1}{\Im(w)}.\label{eq:ImG}$$ Thus, $$\begin{aligned}
\Im\big(\widehat{L}_z(w)\big)= & t \ \Im\big(\widehat{G}_{\mu^n_t}(w)\big)+\Im(z)> -\frac{t}{\Im(w)}+\Im(z)
> -\frac{2t}{\Im(z)}+\Im(z)
> \frac{1}{2}\Im(z).\end{aligned}$$The second inequality comes from the choice of $w\in {\mathbb{C}}_{\frac{1}{2}\Im(z)}$, and the last inequality is a consequence of $\Im(z)>2\sqrt{t}$.\
Moreover, $\widehat{L}_z$ is not an automorphism since: $$\begin{aligned}
\label{Kcircle}\left|\widehat{L}_z(w)-z\right|= & \left|t \widehat{G}_{\mu^n_t}(w)\right|= \left| \frac{1}{n}\sum_{j=1}^n \frac{t}{w-\lambda_j^n(t)}\right| \leq \frac{t}{\Im(w)}\leq \sqrt{t}\end{aligned}$$ since $\Im(w)> \frac{1}{2}\Im(z)>\sqrt{t}$. We use again the Denjoy-Wolff fixed-point theorem. Because the inclusion of $\widehat{L}_z\big({\mathbb{C}}_{\frac{1}{2}\Im(z)}\big)$ into $ {\mathbb{C}}_{\frac{1}{2}\Im(z)}$ is strict, the unique Denjoy-Wolff point of $\widehat{L}_z$ is necessarily a fixed point that we denote $\widehat{w}_{fp}(z)$. From the construction, $\Im(\widehat{w}_{fp}(z))>\Im(z)/2$. Finally, the last announced estimate is a straightforward consequence of .
Study of the subordination function {#section:estimationw}
===================================
This section is devoted to the proof of Proposition \[propsition.consistency.estimate.w\_fp\]. We show that $\widehat{w}_{fp}^n(z)$ converges uniformly to $w_{fp}(z)$ on ${\mathbb{C}}_\gamma$ with $\gamma>2\sqrt{t}$. Next, we establish that the fluctuations are of order $1/\sqrt{n}$.
Proof of (i) and (ii) of Proposition \[propsition.consistency.estimate.w\_fp\]
------------------------------------------------------------------------------
We first state a useful lemma.
\[lem.Lipschitz.CauchyTransform\] For any probability measure $\mu$ on $\R$ and $\alpha>0$, the Cauchy transform $G_{\mu}$ is Lipschitz on ${\mathbb{C}}_{\alpha}$ with Lipschitz constant $ \dfrac{1}{\alpha^2}$, and one has for any $z\in {\mathbb{C}}_\alpha$, $\left|G_{\mu}(z)\right|\leq \dfrac{1}{\alpha}$.
For $z, z' \in {\mathbb{C}}_{\alpha}$, $$\begin{aligned}
\left| G_{\mu}(z) - G_{\mu}(z') \right| = & \left| \int_{\R} \dfrac{d\mu(x)}{z-x} - \int_{\R} \dfrac{d\mu(y)}{z'-y} \right| \leq \left|z-z'\right| \ \int_{\R} \dfrac{d\mu(x)}{\left|(z-x)(z'-x)\right|} \\
&\leq \frac{|z-z'|}{\Im(z) \Im(z')} \leq \dfrac{|z-z'|}{ \alpha^2 }.
\end{aligned}$$ This implies the Lipschitz property of $G_{\mu_t}$. Also, $$\begin{aligned}
\left| G_{\mu}(z) \right| = \left| \int_{\R} \dfrac{d\mu(x)}{z-x} \right| \leq \dfrac{1}{\Im(z)} \leq \dfrac{1 }{\alpha} .
\end{aligned}$$This finishes the proof.
We are now ready to prove the points (i) and (ii) of Proposition \[propsition.consistency.estimate.w\_fp\].
Consider $z\in {\mathbb{C}}_{\gamma}$ with $\gamma>2\sqrt{t}$. Using the equations and characterizing $w_{fp}(z)$ and $\widehat w_{fp}^{n}(z)$, we have $$\begin{aligned}
\left| \widehat w_{fp}^{n}(z) - w_{fp}(z) \right| &= t \left| \widehat G_{\mu^{n}_t} ( \widehat w^{n}_{fp}(z)) - G_{\mu_t} ( w_{fp}(z)) \right| \nonumber \\
&\leq t \left| \widehat G_{\mu^{n}_t} ( \widehat w^{n}_{fp}(z)) - \widehat{G}_{\mu^n_t} ( w_{fp}(z)) \right| + t \left| \widehat{G}_{\mu^n_t} ( w_{fp}(z)) - G_{\mu_t} ( w_{fp}(z)) \right| .
\end{aligned}$$ By Theorem \[thm.Fixpoint\], $\Im(w_{fp}(z)) \geq \dfrac{1}{2}\Im(z)$ and since $\widehat{G}_{\mu^n_{t}}$ is a Lipschitz function on $\mathbb{C}_{\frac{1}{2} \Im(z)}$ with Lipschitz constant $ \dfrac{4}{\Im^2(z)}\leq \frac{4}{\gamma^2}$, by Lemma \[lem.Lipschitz.CauchyTransform\], we have an upper bound for the first term $$\begin{aligned}
\left |\widehat{G}_{\mu^n_t} ( \widehat w^{n}_{fp}(z)) - \widehat{G}_{\mu^{n}_t} ( w_{f_p}(z)) \right| \leq \dfrac{4}{ \gamma^2} \times \left| \widehat{w}^{n}_{fp}(z) - w_{fp}(z) \right|.
\end{aligned}$$ Thus, $$\begin{aligned}
\left| \widehat{w}^{n}_{fp}(z) - w_{fp}(z) \right| \leq \frac{4t}{\gamma^2} \left| \widehat{w}^{n}_{fp}(z) - w_{fp}(z) \right| + t \left| \widehat{G}_{\mu^{n}_t} \big(w_{fp}(z)\big) - G_{\mu_t}\big(w_{fp}(z)\big) \right| ,
\end{aligned}$$ implying that $$\begin{aligned}
\label{proof.consistency.w^n_fp.upperbound.by.Cauchytransform}
\left| \widehat{w}^{n}_{fp}(z) - w_{fp}(z) \right| & \leq \left( \dfrac{ t \gamma^2}{\gamma^2 - 4t} \right) \times \left| \widehat{G}_{\mu^{n}_t} \big(w_{fp}(z)\big) - G_{\mu_t}\big(w_{fp}(z)\big) \right|.
\end{aligned}$$ By Proposition \[prop:LGN.mu.t\], since the function $x\mapsto \frac{1}{z-x}$ is continuous and bounded on $\R$ for any $z \in {\mathbb{C}}_{\sqrt{t}}$, $\widehat G_{\mu^n_t}(w_{fp}(z))= \int_\R \frac{1}{w_{fp}(z)-x}\mu^n_t({\mathrm{d}}x)$ converges almost surely to $G_{\mu_t}(w_{fp}(z))= \int_\R \frac{1}{w_{fp}(z)-x}\mu_t({\mathrm{d}}x)$. This concludes the proof of (i).\
To prove the uniform convergence (ii), we will need Vitali’s convergence theorem, see e.g. [@BaiSilver Lemma 2.14, p.37-38]: on any bounded compact set of ${\mathbb{C}}_{2\sqrt{t}}$, the simple convergence is in fact a uniform convergence. Moreover, the functions $G_{\mu_t}(z)$ and $\widehat G_{\mu^n_t}(z)$ decay as $1/|z|$ when $|z|\rightarrow +\infty$, implying the uniform convergence of the right-hand side of on ${\mathbb{C}}_\gamma$, for $\gamma>2\sqrt{t}$ and of $\widehat{w}^n_{fp}(z)$ to $w_{fp}(z)$.
Fluctuations of the Cauchy transform of the empirical measure {#section:Fluctuations of Cauchy transform}
-------------------------------------------------------------
We now prove point (iii) of Proposition \[propsition.consistency.estimate.w\_fp\]. For this purpose, we first decompose: $$\begin{aligned}
\lefteqn{\widehat{G}_{\mu^{n}_t}(z)-G_{\mu_t}(z)}\nonumber\\
= & \widehat{G}_{\mu^{n}_t}(z) - \mathbb{E} \big[\widehat{G}_{\mu^{n}_t}(z) | X^n(0)\big] + \mathbb{E} \big[\widehat{G}_{\mu^{n}_t}(z)| X^n(0)\big] - G_{\mu^{n}_{0} \boxplus \sigma_{t} }(z)+ G_{\mu^{n}_{0} \boxplus \sigma_{t} }(z) -G_{\mu_t}(z)\nonumber\\
=: & A_1^n(z)+A_2^n(z)+A_3^n(z).\label{decomp:G}\end{aligned}$$ The first term is related to the variance of $\widehat{G}_{\mu^{n}_t}(z)$ (conditional on $X^n(0)$). The second term heuristically compares the evolution with the Hermitian Brownian motion to its limit. The third term deals with the fluctuations of the empirical initial condition. A similar decomposition for the first two terms is done in Dallaporta and Février [@Fevrier1] (without the problem of the random initial condition) and we will adapt their results. We will show that the fluctuations of the first two terms are of order $1/n$, and this is treated in Propositions \[proposition.D&F19.1st.term\] and \[prop.b\_n(z).bound\] below. The third term, which is associated to a classical central limit theorem, is of order $1/\sqrt{n}$. This is proved in Proposition \[prop.fluctuG2\].\
For the term $A_1^n(z)$, the result is a direct consequence of Proposition 3 in [@Fevrier1] and we refer to the detailed computation in [@datthesis].
\[proposition.D&F19.1st.term\] For $z \in {\mathbb{C}}^+$ and $n \in \N$, $${\mbox{Var}}\big(n A^n_1(z)|X^n(0)\big) = {\mbox{Var}}\big(n \widehat{G}_{\mu^{n}_t}(z) |X^n(0)\big) \leq \dfrac{10 t}{\Im^4(z)}.$$
### Fluctuations of $A^n_2(z)$
We start with some additional notations. Let us denote the resolvent of $X^{n}(t)$ by $$\label{eq:Rnt}
R_{n,t}(z) := \left( z I_{n} - X^{n}(t) \right)^{-1}.$$ Then one can write $$\widehat{G}_{\mu^{n}_{t}}(z) = \dfrac{1}{n} \ \mbox{Tr}\left(R_{n,t}(z)\right).$$ Then, the bias term is: $$n A^n_2(z)
= \mathbb{E}\left[ {\mbox{Tr}}\left(R_{n,t}(z)\right) \ |\ X^n(0)\right] - n G_{\mu^{n}_{0} \boxplus \sigma_{t}}(z),$$ and it is given by an adaptation of [@Fevrier1 Proposition 4] to the case of a random initial condition:
\[prop.b\_n(z).bound\] For $z \in {\mathbb{C}}^+$ and $n \in \N$, $$\label{eq:but1}
\left| n A^n_2(z) \right| \leq \left( 1 + \frac{4t}{ \Im^2(z)}\right). \left(\dfrac{2t}{\Im^3(z)} + \dfrac{12 t^2}{\Im^5(z)} \right).$$
The term $A^n_2(z)$ compares $\mathbb{E} \big[\widehat{G}_{\mu^{n}_t}(z)| X^n(0)\big]$ with $G_{\mu_0^n\boxplus \sigma_t}(z)$. Proceeding as in Theorem-Definition \[def:subordinationfunction\], with $\mu^n_0$ and $\sigma_t$, we can define a subordination function $\overline{w}_{fp}(z)$ such that $$\label{def:woverline}
G_{\mu_0^n\boxplus \sigma_t}(z)=G_{\mu^n_0}\big(\overline{w}_{fp}(z)\big).$$In what follows, it will be natural to introduce and use this subordination function.
Note that by definition of the resolvent, we have for all $z\in {\mathbb{C}}^+$, $$\label{et2}|n A^n_2(z)|\leq 2n \Im^{-1}(z),$$ which is suboptimal due to the factor $n$.\
We follow the ideas of [@Fevrier1] for their ‘approximate subordination relations’. Since our initial condition is random, the strategy has to be adapted and we introduce the following analogues of $R_{n,t}(z)$ and $A^n_2(z)$, which differ from [@Fevrier1]: $$\begin{aligned}
\widetilde{R}_{n,t}(z) & := \Big( \big( z - \frac{t}{n}\mathbb{E} \big[ {\mbox{Tr}}\left(R_{n,t}(z)\right) \ |\ X^n(0)\big] \big).I_{n} - X^{n}(0) \Big)^{-1} \label{eq:Rtildent}\\
n \widetilde A^n_2(z) & := \mathbb{E} \big[ {\mbox{Tr}}\left(R_{n,t}(z)\right)\ |\ X^n(0)\big] - {\mbox{Tr}}\big( \widetilde{R}_{n,t}(z) \big).\nonumber\end{aligned}$$ We will bound $A^n_2(z)$ by using its approximation $\widetilde{A}^n_2(z)$.\
**Step 1:** First, we prove an upper bound for $\widetilde{A}^n_2(z)$:
\[lem:but1\] For $z \in {\mathbb{C}}^+$, $$\big| n \widetilde{A}^n_2(z) \big| \leq \dfrac{2t}{\Im^3(z)} + \dfrac{12 t^2}{\Im^5(z)} .$$
The proof of this lemma is postponed in Appendix.\
**Step 2:** If $| \widetilde{A}^n_2(z)|\geq \Im(z)/(2t)$ then, by $$|n A^n_2(z)| \leq \frac{ 4 t n |\widetilde{A}^n_2(z)|}{\Im^2(z)}.$$ And we conclude with Lemma \[lem:but1\].\
**Step 3:** We now consider the case where $|\widetilde{A}^n_2(z)|< \Im(z)/(2t)$. We have: $$\label{et13}
A^n_2(z)= \widetilde{A}^n_2(z)+ \big[A^n_2(z)-\widetilde{A}^n_2(z)\big]$$ We will control the difference $|A^n_2(z)-\widetilde{A}^n_2(z)|$ by $\widetilde{A}^n_2(z)$ and conclude with Lemma \[lem:but1\].\
By their definitions: $$\begin{aligned}
\label{et1}
n\big(A^n_2(z)-\widetilde{A}^n_2(z)\big)= & {\mbox{Tr}}\big( \widetilde{R}_{n,t}(z) \big)-n G_{\mu^n_0\boxplus \sigma_t}(z).\end{aligned}$$ We follow the trick in [@Fevrier1] which consists in going back to the fluctuations of the subordination functions. In view of , it is natural to express the first term ${\mbox{Tr}}\big( \widetilde{R}_{n,t}(z) \big)$ of similarly. As $\widetilde{R}_{n,t}(z)$ is a diagonal matrix, $$\label{et3}
{\mbox{Tr}}\big( \widetilde{R}_{n,t}(z) \big)=\sum_{j=1}^n \frac{1}{z- \frac{t}{n}\E\big[{\mbox{Tr}}\big( R_{n,t}(z) \big)\ |\ X^n(0)\big]-\lambda_j^n(0)}=n G_{\mu^n_0}(\widetilde{w}_{fp}(z)),$$where $$\label{def:wwildetilde}\widetilde{w}_{fp}(z):=z- \frac{t}{n}\E\big[{\mbox{Tr}}\big( R_{n,t}(z) \big)\ |\ X^n(0)\big]$$and where $\lambda_j^n(0)$ are the eigenvalues of $X^n(0)$. Thus: $$A^n_2(z)-\widetilde{A}^n_2(z)= G_{\mu^n_0}(\widetilde{w}_{fp}(z))-G_{\mu^n_0}(\overline{w}_{fp}(z)).\label{et4}$$
To continue, we first need the following result proved in Appendix.
\[lem:inverse\] (i) The function $\overline{w}_{fp}(z)$, defined in , solves $$\overline{w}_{fp}(z)=z-t G_{\mu^n_0\boxplus \sigma_t}(z).$$ (ii) The function $\zeta(z) =z+tG_{\mu_0^n}(z)$ is well-defined on ${\mathbb{C}}^+$ and is the inverse of $\overline{w}_{fp}(z)$ on $\overline{\Omega}=\{z\in {\mathbb{C}}^+,\ \Im(\zeta(z))>0\}$. For such $z\in \overline{\Omega}$, we denote this function $\overline{w}^{<-1>}_{fp}(z)$.
Let us prove that under the condition of Step 3, $\widetilde{w}_{fp}(z)\in \overline{\Omega}$ for all $z\in {\mathbb{C}}^+$. [ $$\begin{aligned}
\zeta(\widetilde{w}_{fp}(z)) - z &= \widetilde{w}_{fp}(z) + t G_{\mu^{n}_{0}} \big(\widetilde{w}_{fp}(z) \big) - z
\nonumber\\
&= z - \frac{t}{n} \mathbb{E} \big[ {\mbox{Tr}}\big( R_{n,t}(z) \big)\ |\ X^n(0) \big] + t G_{\mu^{n}_{0}} \big( \widetilde{w}_{fp}(z) \big) - z
\nonumber\\
&= -t \widetilde{A}^n_2(z),\label{et5}
\end{aligned}$$]{}by . Therefore, $$\begin{aligned}
\label{controlektilde}
\left|\Im\big(\zeta(\widetilde{w}_{fp}(z)) \big) - \Im(z)\right| \leq \left| \zeta(\widetilde{w}_{fp}(z)) -z \right| = t \big| \widetilde{A}^n_2(z) \big| \leq \frac{\Im(z)}{2}.
\end{aligned}$$ Thus, under the condition of Step 3, $\widetilde{w}_{fp}(z) \in \overline{\Omega}$. Denoting $\widetilde{z}=\overline{w}^{<-1>}_{fp}\big(\widetilde{w}_{fp}(z)\big)$, which is well-defined, we have $\overline{w}_{fp}(\widetilde{z})=\widetilde{w}_{fp}(z)$. Plugging this into , $$\begin{aligned}
A^n_2(z)-\widetilde{A}^n_2(z)
&= G_{\mu_{0}^{n} \boxplus \sigma_{t}} ( \widetilde{z} ) - G_{\mu_{0}^{n} \boxplus \sigma_{t}} (z)
\\
&= \big(z-\widetilde{z}\big) \int_{\R} \dfrac{ \mu_{0}^{n} \boxplus \sigma_{t}( dx) }{ \big(\widetilde{z} - x\big). \big( z - x \big) }
\\
&= t \widetilde{A}^n_2(z). \int_{\R} \dfrac{ \mu_{0}^{n} \boxplus \sigma_{t}(dx) }{ \big(\widetilde{z} - x\big). \big( z - x \big) }, \end{aligned}$$where we used for the last equality.\
From there, using , we get $$|A^n_2(z) | \le \Big| 1 + t.\int_{\R} \dfrac{ \mu_{0}^{n} \boxplus \sigma_{t}(dx) }{ \big(\widetilde{z} - x\big). \big( z - x \big) } \Big|.|\widetilde{A}^n_2(z)| \leq \left( 1 + \frac{2t}{\Im^2 (z)} \right) |\widetilde{A}^{n}_2(z)|.$$ This concludes the proof of Proposition \[prop.b\_n(z).bound\].
### Fluctuations of $A^n_3(z)$
Finally, the third step is to control $A^n_3(z)=G_{\mu^{n}_{0} \boxplus \sigma_{t} }(z) -G_{\mu_t}(z),$ with $\mu_t=\mu_0\boxplus \sigma_t$.
\[prop.fluctuG2\] For any $\gamma>2\sqrt{t}$ and for any $z$ such that $\Im (z) \ge \frac{\gamma}{2},$ we have: $$\left|A^n_3(z)\right|<\frac{\gamma^2}{\gamma^2-4t} \left|\int_{\R} \dfrac{1}{z - t.G_{\mu_{0} \boxplus \sigma_{t} }(z) - x} \big[ {\mathrm{d}}\mu^{n}_{0}(x) - {\mathrm{d}}\mu_{0}(x) \big] \right|$$ and $$\label{et8}
\sup_{n\in \N} \sup_{z\in {\mathbb{C}}_{\frac{\gamma}{2}}} \E\big[n\left|A^n_3(z)\right|^2\big]<\frac{8\gamma^2}{(\gamma^2-4t)^2}.$$
Using again the subordination function $\overline{w}_{fp}(z)$ defined in and Lemma \[lem:inverse\](i), we have $$\label{et6}
G_{\mu^{n}_{0} \boxplus \sigma_{t} }(z) = G_{\mu^{n}_{0}} \big( \overline{w}_{fp}(z) \big) = \int_{\R} \dfrac{{\mathrm{d}}\mu^{n}_{0}(x)}{\overline{w}_{fp}(z) - x} = \int_{\R} \dfrac{ {\mathrm{d}}\mu^{n}_{0}(x) }{ z - t G_{\mu^{n}_{0} \boxplus \sigma_{t} }(z) - x} .$$In this proof, $\Im(z)\geq \gamma/2\geq \sqrt{t}$. Note that $\Im(\overline{w}_{fp}(z))\geq\Im(z)$ (Theorem-Definition \[def:subordinationfunction\]) so that $$\label{et7}
| z - t G_{\mu^{n}_{0} \boxplus \sigma_{t} }(z) - x| \geq \frac{\gamma}{2}\geq \sqrt{t},$$ and the integrand in is well-defined and upper-bounded by $1/\sqrt{t}$. Similarly, we can establish that $$G_{\mu_0 \boxplus \sigma_{t}}(z) = \int_{\R} \dfrac{{\mathrm{d}}\mu_{0}(x)}{z - t G_{\mu_0 \boxplus \sigma_{t}}(z) - x}.$$ Then, we can write $$\begin{aligned}
G_{\mu^{n}_{0} \boxplus \sigma_{t} }(z) - G_{\mu_0 \boxplus \sigma_{t}}(z) =& \int_{\R} \dfrac{1}{z - t.G_{\mu^{n}_{0} \boxplus \sigma_{t} }(z) - x} {\mathrm{d}}\mu^{n}_{0}(x) - \int_{\R} \dfrac{1}{z - t.G_{\mu_{0} \boxplus \sigma_{t} }(z) - x} {\mathrm{d}}\mu^{n}_{0}(x)
\\
&+ \int_{\R} \dfrac{1}{z - t.G_{\mu_{0} \boxplus \sigma_{t} }(z) - x} {\mathrm{d}}\mu^{n}_{0}(x) - \int_{\R} \dfrac{1}{z - t.G_{\mu_0 \boxplus \sigma_{t}}(z) - x} {\mathrm{d}}\mu_{0}(x)
\\
=& \hspace{0.2cm} t. \int_{\R} \dfrac{G_{\mu^{n}_{0} \boxplus \sigma_{t} }(z)- G_{\mu_0 \boxplus \sigma_{t}}(z) }{\left(z - t. G_{\mu^{n}_{0} \boxplus \sigma_{t} }(z) - x \right) . \Big(z - t.G_{\mu_{0} \boxplus \sigma_{t} }(z) - x \Big)} {\mathrm{d}}\mu^{n}_{0}(x)
\\ &+ \int_{\R} \dfrac{1}{z - t.G_{\mu_{0} \boxplus \sigma_{t} }(z) - x} \left[ {\mathrm{d}}\mu^{n}_{0}(x) - {\mathrm{d}}\mu_{0}(x) \right].
\end{aligned}$$ Thus, $$\begin{gathered}
\left( G_{\mu^{n}_{0} \boxplus \sigma_{t} }(z) - G_{\mu_0 \boxplus \sigma_{t}}(z) \right).\left[ 1 - t. \int_{\R} \dfrac{ 1 }{\left(z - t.G_{\mu^{n}_{0} \boxplus \sigma_{t} }(z) - x \right) . \Big(z - t.G_{\mu_{0} \boxplus \sigma_{t} }(z) - x \Big)} {\mathrm{d}}\mu^{n}_{0}(x) \right]
\\
= \int_{\R} \dfrac{1}{z - t.G_{\mu_{0} \boxplus \sigma_{t} }(z) - x} \big[ {\mathrm{d}}\mu^{n}_{0}(x) - {\mathrm{d}}\mu_{0}(x) \big] .\end{gathered}$$ Similarly to , we can show that $|z - t.G_{\mu_{0} \boxplus \sigma_{t} }(z) - x| \geq \gamma/2.$ Thus $$\left| t. \int_{\R} \dfrac{ 1 }{\left(z - t.G_{\mu^{n}_{0} \boxplus \sigma_{t} }(z) - x \right) . \Big(z - t.G_{\mu_{0} \boxplus \sigma_{t} }(z) - x \Big)} {\mathrm{d}}\mu^{n}_{0}(x) \right| \le \frac{4t}{\gamma^2},$$ consequently, $$\left|A^n_3(z)\right|<\frac{\gamma^2}{\gamma^2 - 4t} \left|\int_{\R} \dfrac{1}{z - t.G_{\mu_{0} \boxplus \sigma_{t} }(z) - x} \big[ {\mathrm{d}}\mu^{n}_{0}(x) - {\mathrm{d}}\mu_{0}(x) \big] \right|,$$ which gives the first part of the proposition. For the second part ,
$$\begin{aligned}
\E\big[n\left|A^n_3(z)\right|^2\big]
\leq \left(\frac{\gamma^2}{\gamma^2 - 4t}\right)^2 n\mathbb E \Big[\left|\int_{\R} \dfrac{1}{z - t.G_{\mu_{0} \boxplus \sigma_{t} }(z) - x} \big[ {\mathrm{d}}\mu^{n}_{0}(x) - {\mathrm{d}}\mu_{0}(x) \big] \right|^2\Big].\end{aligned}$$
Now, for any $z$ such that $\Im( z) > \frac{\gamma}{2},$ the function $\varphi_z\ :\ x \mapsto \dfrac{1}{z - t.G_{\mu_{0} \boxplus \sigma_{t} }(z) - x}$ is bounded by $2/\gamma$. Then: $$\begin{aligned}
n\E\Big[\Big|\int_\R \varphi_z(x){\mathrm{d}}\mu^n_0(x) - \int_\R \varphi_z(x){\mathrm{d}}\mu_0(x)\Big|^2\Big]= & n \E\Big[\Big|\frac{1}{n} \sum_{j=1}^n \varphi_z\big(\lambda^n_j(0)\big)-\E\big[\varphi_z\big(\lambda^n_j(0)\big)\big]\Big|^2\Big]\\
= & n{\mbox{Var}}\Big(\frac{1}{n} \sum_{j=1}^n \varphi_z\big(d_j^n\big)\Big)\\
= & \int_{\R} |\varphi_z(x)|^2{\mathrm{d}}\mu_0(x)-\Big|\int_{\R} \varphi_z(x){\mathrm{d}}\mu_0(x)\Big|^2\leq \frac{8}{\gamma^2},\end{aligned}$$for any $z\in {\mathbb{C}}_{\frac{\gamma}{2}}$.
**Conclusion:** We can now conclude the proof of Proposition \[propsition.consistency.estimate.w\_fp\] (iii). From , Propositions \[proposition.D&F19.1st.term\], \[prop.b\_n(z).bound\] and the first part of Proposition \[prop.fluctuG2\], we obtain that for $z\in {\mathbb{C}}_{\gamma/2}$: $$\E\big[|\widehat{G}_{\mu^{n}_t}(z)-G_{\mu_t}(z)|^2 \ |\ X^n(0)\big] \leq
C(\gamma,t) \Big(\frac{1}{n^2}+ \left|\int_{\R} \dfrac{1}{z - t.G_{\mu_{0} \boxplus \sigma_{t} }(z) - x} \big[ {\mathrm{d}}\mu^{n}_{0}(x) - {\mathrm{d}}\mu_{0}(x) \big] \right|^2\Big),\label{eq:majoA123conditionnelle}$$ where $C(\gamma,t)$ depends only on $\gamma$ and $t$ (and converges to $+\infty$ when $\gamma\rightarrow 2\sqrt{t}$). Using the second part of Proposition \[prop.fluctuG2\], we get: $$\sup_{n\in \N}\sup_{z\in {\mathbb{C}}_{\gamma/2}} n\E\Big[ |\widehat{G}_{\mu^{n}_t}(z)-G_{\mu_t}(z)|^2 \Big]<+\infty.$$ Equation implies that for any $\gamma>2\sqrt{t}$, $$\sup_{n\in \N}\sup_{z\in {\mathbb{C}}_{\gamma} } \mathbb{E} \left[ n \big| \widehat{w}^{n}_{fp}(z) - w_{fp}(z) \big) \big|^{2} \right]\leq \big(\frac{t\gamma^2}{\gamma^2-4t}\big)^2 \sup_{n\in \N}\sup_{z\in {\mathbb{C}}_{\frac{\gamma}{2}}} n\E\Big[ |\widehat{G}_{\mu^{n}_t}(z)-G_{\mu_t}(z)|^2 \Big]< +\infty$$ since if $z\in {\mathbb{C}}_{\gamma}$ then $\Im\big(w_{fp}(z)\big) \geq \dfrac{1}{2} \Im(z)>\frac{\gamma}{2}$ (Theorem \[thm.Fixpoint\]) so that $w_{fp}(z)\in {\mathbb{C}}_{\frac{\gamma}{2}}$ and point (iii) of Proposition \[propsition.consistency.estimate.w\_fp\] is proved.
Study of the mean integrated squared error {#section:MISE}
==========================================
In Section \[sec:theo\], we state theoretical results associated with our nonparametric statistical problem. Section \[proof:variance\] is devoted to the proof of Theorem \[variance\].
Theoretical results {#sec:theo}
-------------------
The goal of this section is to study the rates of convergence of $\E\Big[\|\widehat{p}_{0,h}-p_0\|^2\big]$, the mean integrated squared error of $\widehat{p}_{0,h}$. To derive rates of convergence, we rely on the classical bias-variance decomposition of the quadratic risk. Using Parseval’s equality we obtain $$\begin{aligned}
\label{decomp:MISE}
{\left\lVert \widehat{p}_{0,h} - p_0 \right\rVert}^{2} = \dfrac{1}{2 \pi} {\left\lVert \widehat{p}_{0,h}^{\star} - p^{\star}_{0} \right\rVert}^{2}
\leq \dfrac{1}{\pi} {\left\lVert \widehat{p}_{0,h}^{\star} - K^{\star}_{h}.p_0^{\star} \right\rVert}^{2} + \dfrac{1}{\pi} {\left\lVert K^{\star}_{h}.p_0^{\star}- p_0^{\star} \right\rVert}^{2}.\end{aligned}$$
The expectation of the first term is a variance term whereas the second one is a bias term. To derive the order of the bias term, we assume that $p_0$ belongs to the space $\mathcal{S}(a,r,L)$ of supersmooth densities defined for $a>0$, $L>0$ and $r>0$ by: $$\label{def:Sobolev}
\mathcal{S}({a,r,L})= \left\{p_0 \; \textrm{density such that} \int_{\R} |p_0^\star(\xi)|^2 e^{2a |\xi|^r}{\mathrm{d}}\xi \leq L \right\}.$$ In the literature, this smoothness class of densities has often been considered (see [@lacourCRAS], [@ButuceaTsybakov], [@comtelacour]). Most famous examples of supersmooth densities are the Cauchy distribution belonging to $\mathcal{S}({a,r, L})$ with $r=1$ and the Gaussian distribution belonging to $\mathcal{S}({a,r, L})$ with $r=2$. To control the bias, we rely on Proposition 1 in [@ButuceaTsybakov] which states that:
\[Biais\] For $p_0 \in \mathcal{S}({a,r,L})$, we have $${\left\lVert K^{\star}_{h}.p_0^{\star}- p_0^{\star} \right\rVert} \leq C_B L^{1/2} e^{-ah^{-r}},$$ where $C_B$ is a constant.
Whereas the control of the bias term is very classical, the study of the variance term in is much more involved. The order of the variance term is provided by the following theorem.
\[variance\] Let $$\Sigma:={\left\lVert \widehat{p}_{0,h}^{\star} - K^{\star}_{h} p_0^{\star} \right\rVert}^{2}.$$ We assume that there exists a constant $C>0$ such that for sufficiently large $\kappa>0$, $$\label{H2}
\mu_0\big((\kappa,+\infty)\big)\leq \frac{C}{\kappa}.$$ Then, we have for any $h>0$, for any $\gamma > 2\sqrt t,$ $$\label{bound-variance}
\mathbb{E} (\Sigma) \leq \frac{C_{var}. e^{\frac{2\gamma}{h}}}{n},$$ for $C_{var}$ a constant.
In , the constant $C_{var}$ depends on all the parameters of the problem and may blow up when $\gamma$ tends to $2\sqrt{t}$. Theorem \[variance\] is proved in Section \[proof:variance\]. The main point will be to obtain the optimal $n$ factor appearing at the denominator. The term $e^{\frac{2\gamma}{h}}$ appearing at the numerator is classical in our setting. Note that Assumption is very mild and is satisfied by most classical distributions.\
Now, using similar computations to those in [@lacourCRAS], we can obtain from Proposition \[Biais\] and Theorem \[variance\] the rates of convergence of our estimator $\widehat p_{0,h}$. We indeed showed that: $$\label{bornerisque}
MISE:=\E\Big[\left\| \widehat{p}_{0,h} - p_0\right\|^{2}\Big] \leq C_{B}^2L e^{-2ah^{-r}} + \frac{C_{var}. e^{\frac{2\gamma}{h}}}{n}.$$ Minimizing in $h$ the right hand side of (\[bornerisque\]) provides the convergence rate of the estimator $\widehat{p}_{0,h}$. The rates of convergence are summed up in the following corollary, adapted from the computation of [@lacourCRAS]. One can see that there are three cases to consider to derive rates of convergence: $r=1$, $r<1$ and $r>1$, depending on which the bias or variance term dominates the other. For the sake of completeness Corollary \[thm:MISE\] is proved in Appendix.
\[thm:MISE\] Suppose that $\mu_0$ satisfies Assumption and the density $p_0$ belongs to the space $\mathcal{S}(a,r,L)$ for $a>0$, $r>0$ and $L>0$. Then, for any $\gamma > 2\sqrt t$ and by choosing the bandwidth $h$ according to equation , we have: $$\E\Big[\|\widehat{p}_{0,h}-p_0\|^2\big] = \begin{cases}
O\big(n^{-\frac{a}{a+\gamma}}\big) & \mbox{ if } r=1\\
O\Big(\exp\Big\{-\frac{2a}{(2\gamma)^{r}}\Big[\log n+(r-1)\log \log n+\sum_{i=0}^k b_i^*(\log n)^{r+i(r-1)}\Big]^r\Big\}\Big) & \mbox{ if } r<1\\
O\Big(\frac{1}{n}\exp\Big\{\frac{2\gamma}{(2a)^{1/r}} \Big[\log n+\frac{r-1}{r}\log \log n+\sum_{i=0}^k d_i^* (\log n)^{\frac{1}{r}-i\frac{r-1}{r}}\Big]^{1/r}\Big\}\Big) & \mbox{ if } r>1,
\end{cases}$$ where the integer $k$ is such that $$\frac{k}{k+1}<\min\big(r,\frac{1}{r}\big)\leq \frac{k+1}{k+2},$$ and where the constants $b_i^*$ and $d_i^*$ solve the following triangular system: $$\begin{aligned}
b^*_0 & =-\frac{2a}{(2\gamma)^{r}},\qquad \forall i>0,\ b^*_i= & -\frac{2a}{(2\gamma)^{r}}\sum_{j=0}^{i-1} \frac{r(r-1)\cdots (r-j)}{(j+1)!} \sum_{p_0+\cdots p_j=i-j-1} b^*_{p_0}\cdots b^*_{p_{j}},\\
d^*_0 & =-\frac{2\gamma}{(2a)^{1/r}},\qquad \forall i>0,\ d_i^*= & -\frac{2\gamma}{(2a)^{1/r}} \sum_{j =0}^{i-1} \frac{\frac{1}{r}\big(\frac{1}{r}-1\big)\cdots \big(\frac{1}{r}-j\big)}{(j+1)!} \sum_{p_0+\cdots p_j=i-j-1} d^*_{p_0}\cdots d^*_{p_j}\end{aligned}$$
\[remarque-fenetre\] For $r=1$, the choice $h=2(a+\gamma)/\log(n)$ yields the rate of convergence. The expressions of the optimal bandwidths for $r>1$ and $r<1$ are much more intricate (see and in Appendix, and also [@lacourCRAS]).
Recall that we have transformed the free deconvolution of the Fokker-Planck equation associated with observation of the matrix $X^n(t)$ into the deconvolution problem expressed in (\[eq:densiteconvolee\]). To solve the latter, we have then inverted the convolution operator characterized by the Fourier transform of the Cauchy distribution $\mathcal{C}_\gamma$. The parameter $\gamma$ represents the difficulty of our deconvolution problem and consequently, the rates of convergence heavily depend on $\gamma$. The larger $\gamma$ the harder the problem, as can be observed in rates of convergences of Corollary \[thm:MISE\]. This is not surprising: as $t$ grows, it becomes naturally harder to reconstruct the initial condition from the observations at time $t$ and as $\gamma$ has to be chosen larger than $2\sqrt t,$ $\gamma$ and therefore the difficulty of the deconvolution problem grows with $t$ accordingly. It remains an open question if we can take $\gamma$ smaller. For a given $\gamma$, the upper bound of the variance term given by Theorem \[variance\] is optimal. Analogously, the bound for the bias given by Proposition \[Biais\] is also optimal. In consequence, rates of convergence for $r=1$ and $r<1$ in Corollary \[thm:MISE\] are optimal (as proved by Tsybakov in [@TsyCRAS] for the case $r=1$ and by Butucea and Tsybakov in [@ButuceaTsybakov] for $r<1$). The optimality for $r>1$ remains an open problem.
Proof of Theorem \[variance\] {#proof:variance}
-----------------------------
By the definition of $\widehat{p}_{0,h}^\star$, we have: $$\begin{aligned}
\Sigma&:={\left\lVert \widehat{p}_{0,h}^{\star} - K^{\star}_{h} p_0^{\star} \right\rVert}^{2} \\
&= \int_{\R} \dfrac{1}{\pi^2 t^2} e^{2\gamma |\xi| } . \left| K_{h}^{\star}(\xi) \right|^{2} . \left| \left[\left(\Im \big( \widehat{w}^{n}_{fp} (\centerdot+i\gamma )\big) \right)^{\star} - \left( \Im \big(w_{fp} (\centerdot + i\gamma )\big) \right)^{\star} \right] (\xi) \right|^{2} {\mathrm{d}}\xi . \label{eq:MISE.mainTerm1.1}\end{aligned}$$ Recall that by Lemma \[lem:Gmu0\], we have $\Im \left( w_{fp}(z) \right) = t.\Im \big( G_{\mu_t}\big(w_{fp}(z)\big) \big)+ \Im (z)$, and similarly by Theorem-Definition \[th:defestimator\], $\Im \big( \widehat{w}^{n}_{fp}(z) \big) = t.\Im \big( \widehat{G}_{\mu^{n}_t}\big(\widehat{w}^{n}_{fp}(z)\big)\big) + \Im (z)$ for $z \in {\mathbb{C}}_{2 \sqrt{t}}$. Since $K^\star_h(\xi)=K^\star(h\xi)$, we have $$\begin{aligned}
\Sigma &= \int_{\R} e^{2\gamma |\xi| }. \left| K_{h}^{\star}(\xi) \right|^2. \dfrac{1}{\pi^{2}} \Big| \Big( \Im \widehat{G}_{\mu^{n}_{t}}\big(\widehat{w}^{n}_{fp}(\centerdot + i\gamma )\big) - \Im G_{\mu_{t}}\big( w_{fp}(\centerdot + i\gamma )\big) \Big)^{\star}(\xi) \Big|^{2} {\mathrm{d}}\xi
\nonumber
\\
& \leq e^{ \frac{2\gamma}{h}} . \dfrac{C_K^2}{\pi^{2}}. \left\| \Big( \Im \widehat{G}_{\mu^{n}_{t}}\big(\widehat{w}^{n}_{fp}(\centerdot + i \gamma )\big) - \Im G_{\mu_{t}}\big( w_{fp}(\centerdot + i \gamma )\big) \Big)^{\star} \right\|^{2}
\\
&= \frac{2C_K^2}{\pi}. e^{ \frac{2\gamma }{h}} . \left\| \Im \widehat{G}_{\mu^{n}_{t}}\big(\widehat{w}^{n}_{fp}(\centerdot + i\gamma )\big) - \Im G_{\mu_{t}}\big( w_{fp}(\centerdot + i\gamma )\big) \right\|^{2},\end{aligned}$$by Parseval’s equality. Taking the expectation, and introducing a constant $\kappa>0$ chosen later (depending on $n$), we have $$\label{eq:Sigma}
\mathbb{E}(\Sigma) \leq \frac{2C_K^2}{\pi}. e^{ \frac{2\gamma }{h}} . \big(I^\kappa + J^\kappa)$$ where $$\begin{aligned}
I^\kappa= & \int_{ \left\{ x \in \R : |x| \leq \kappa \right\} } \mathbb{E} \Big[ \Big| \Im \widehat{G}_{\mu^{n}_{t}}\big(\widehat{w}^{n}_{fp}(x + i\gamma )\big) - \Im G_{\mu_{t}}\big( w_{fp}(x + i\gamma )\big) \Big|^{2} \Big] {\mathrm{d}}x \\
J^\kappa= & \int_{ \left\{ x \in \R : |x| > \kappa \right\} } \mathbb{E} \Big[ \Big| \Im \widehat{G}_{\mu^{n}_{t}}\big(\widehat{w}^{n}_{fp}(x + i\gamma )\big) - \Im G_{\mu_{t}}\big( w_{fp}(x + i\gamma )\big) \Big|^{2} \Big] {\mathrm{d}}x.\end{aligned}$$ To obtain the announced rates of convergence for the MISE, we need to be very careful in establishing the upper bounds for $I^\kappa$ and $J^\kappa$. For this purpose, we recall Lemma 4.3.17 of [@AGZ], with a null initial condition, which will be useful in the sequel:
\[lem.lambda\_j.bounded\] Let $(\eta^n_1(t), \ldots, \eta^n_n(t))$ be the eigenvalues of $H^n(t).$ With large probability, all the eigenvalues $(\eta^n_j(t))$ of $H^n(t)$ belong to a ball of radius $M>0$ independent of $n$ and $t$. Introduce $$A_{M}^{n,t} := \left\{ \forall 1 \leq j \leq n : \left|\eta^n_{j}(t)\right| \leq M \right\} .$$ There exist two positive constants $C_{eig}$ and $D_{eig}$ depending on $t$ such that for any $M>D_{eig}$ and any $n\in{\mathbb N}^*$ $$\mathbb{P}\left((A_{M}^{n,t})^c\right) = \mathbb{P}\left(\left\{ \eta^n_*(t) > M\right\}\right) \le e^{- n.C_{eig}.M},$$ with $\eta_*^n(t):= \max_{i=1, \ldots,n} |\eta^n_i(t)|$.
Using this lemma, we can control the tail distribution of $\mathbb{E}[\mu^{n}_{t}]$, which is essential to establish very precise estimates without which the announced rate would not be derived. We recall that $\lambda_1^n(t) \le \ldots \le \lambda_n^n(t)$ are the eigenvalues of $X^n(t) = X^n(0) + H^n(t)$ in increasing order. By Weyl’s interlacing inequalities, we have that, for $1 \le j \le n,$ $$\label{interlacing}
\lambda_j^n(0) - \eta_*^n(t) \le \lambda_j^n(t) \le \lambda_j^n(0) + \eta_*^n(t).$$ Therefore, for $1 \le j \le n,$ $$\mathbb{E} \left[ \mu^{n}_{t}\left( \left\{|\lambda| > \frac{\kappa}{2}\right\}\right) \right]\le \mathbb{E} \left[ \mu^{n}_{0}\left( \left\{|\lambda| > \frac{\kappa}{4}\right\}\right) \right]
+ n \mathbb{P}\left(\left\{ \eta^n_*(t) > \frac{\kappa}{4}\right\}\right) \le \mathbb{E} \left[ \mu^{n}_{0}\left( \left\{|\lambda| > \frac{\kappa}{4}\right\}\right) \right]
+ n e^{- \frac{n.C_{eig}.\kappa}{4}}.$$ Recall that after Equation , we introduced the notation $d_1^n, \ldots, d_n^n$ for the i.i.d. random variables of distribution $\mu_0$ and whose order statistic constitutes the diagonal elements of $X_n(0)$, $\lambda_1^n(0) < \ldots < \lambda_n^n(0)$. We have $$\mathbb{E} \left[ \mu^{n}_{0}\left( \left\{|\lambda| > \frac{\kappa}{4}\right\}\right) \right] = \frac{1}{n} \sum_{i=1}^n \mathbb{P}\left( |d_i^n| > \frac{\kappa}{4}\right)
= \mu_{0}\left(\left\{|\lambda | > \frac{\kappa}{4}\right\}\right),$$ so that we finally get $$\label{eq:lemmekato}
\mathbb{E} \left[ \mu^{n}_{t}\left( \left\{|\lambda| > \frac{\kappa}{2}\right\}\right) \right]\le \mu_{0}\left(\left\{|\lambda | > \frac{\kappa}{4}\right\}\right)
+ n e^{- \frac{n.C_{eig}.\kappa}{4}}.$$ Now, we successively study $I^\kappa$ and $J^\kappa$.
### Upper bound for $I^\kappa$
\[lem:Ikappa\] There exist constants $C_I^2$, $C_I^2$ and $C_I^3$ (that can depend on $\gamma$ and $t$) such that: $$I^\kappa\leq \frac{C^1_I}{n}+\frac{\kappa C_I^2}{n^2} + C_I^3 \kappa e^{-n.C_{eig}.M}.$$
Before proving Lemma \[lem:Ikappa\], let us establish a result that will be useful in the sequel.
\[lem:xleqkappa\] Let us consider $\gamma>2\sqrt{t}$, $p>1$ and $M>0$. Then, we have $$\mathfrak{I}_{p,\gamma,M,t} := \int_0^{+\infty}\int\dfrac{ 1 }{\left[
\left\{\Big|\big| \lambda\big|- x\Big|-\sqrt{t}-M\right\}\vee\frac{\gamma}{2} \right]^p}{\mathrm{d}}\mu_0(\lambda){\mathrm{d}}x\leq C(p,\gamma,M,t),
\label{et11}$$ for $ C(p,\gamma,M,t)$ a finite constant only depending on $p,$ $\gamma,$ $M$ and $t$.
The supremum in the denominator equals to $\big| |\lambda|- x\big|-\sqrt{t}-M $ when $x<|\lambda|-\sqrt{t}-M-\gamma/2$ (which is possible only if $|\lambda|-\sqrt{t}-M-\gamma/2$ is positive) or $x>|\lambda|+\sqrt{t}+M+\gamma/2$. Otherwise the supremum is $\gamma/2$. Hence $$\begin{aligned}
\mathfrak{I}_{p,\gamma,M,t} \leq & \int_\R \left\{ \int_0^{\big(|\lambda|-\sqrt{t}-M-\frac{\gamma}{2}\big) \vee 0} \frac{1}{\Big(|\lambda|-x-\sqrt{t}-M\Big)^p} {\mathrm{d}}x + \int^{|\lambda|+\sqrt{t}+M+\frac{\gamma}{2}}_{\{|\lambda|-\sqrt{t}-M-\frac{\gamma}{2}\}\vee 0}\dfrac{2^p}{\gamma^p}{\mathrm{d}}x \right. \nonumber\\
& \hspace{4cm} \left. +\int_{|\lambda|+\sqrt{t}+M+\frac{\gamma}{2}}^{+\infty}\dfrac{ 1 }{\left[x-\big| \lambda\big|-\sqrt{t}-M \right]^p}{\mathrm{d}}x \right\} {\mathrm{d}}\mu_0(\lambda)\nonumber\\
\leq & \int_\R \left\{ \int^{(|\lambda|-\sqrt{t}-M)\vee \frac{\gamma}{2}}_{\frac{\gamma}{2}}\dfrac{ 1}{v^p}{\mathrm{d}}v
+\frac{2^{p}(2\sqrt{t}+2M+\gamma)}{\gamma^p}+ \int_{\frac{\gamma}{2}}^{+\infty} \dfrac{ 1}{v^p}{\mathrm{d}}v \right\} {\mathrm{d}}\mu_0(\lambda) \nonumber\\
\leq & C(p,\gamma,M,t)<+\infty.\end{aligned}$$This concludes the proof.
We decompose $I^\kappa$ into three parts, $I^\kappa \leq 2 \big(I^\kappa_1+I^\kappa_2+I^\kappa_3\big)$ where: $$\begin{aligned}
I^\kappa_{1} &:= \int_{ \left\{ |x| \leq \kappa \right\} } \mathbb{E} \Big[ \Big| \widehat{G}_{\mu^{n}_{t}}\big(\widehat{w}^{n}_{fp}(x + i\gamma )\big) - \widehat{G}_{\mu^{n}_{t}}\big( w_{fp}(x + i\gamma )\big) \Big|^{2} \Big] {\mathrm{d}}x ,\\
I^\kappa_{2} &:= \int_{ \left\{ |x| \leq \kappa \right\} } \mathbb{E} \Big[ \Big| \widehat{G}_{\mu^{n}_{t}}\big(w_{fp}(x + i\gamma )\big) - \mathbb{E} \Big[\widehat{G}_{\mu^{n}_{t}}\big( w_{fp}(x + i\gamma )\big) \ |\ X^n(0)\Big ] \Big|^{2} \Big] dx,
\\
I^\kappa_{3} &:= \int_{ \left\{ |x| \leq \kappa \right\} } \E\Big[ \Big| \mathbb{E} \Big[\widehat{G}_{\mu^{n}_{t}}\big( w_{fp}(x + i\gamma )\big) \ |\ X^n(0)\Big] - G_{\mu_{t}}\big( w_{fp}(x + i\gamma )\big) \Big|^{2} \Big] {\mathrm{d}}x .\end{aligned}$$
**Step 1:** Let us first upper bound $I^\kappa_{1}$. It is relatively easy to bound $I^\kappa_1$ by an upper bound in $C(\gamma,t) \kappa /n$, but this will not yield in the end the announced convergence rate. To establish more precise upper bounds, we use the event $A_M^{n,t}$ defined in Lemma \[lem.lambda\_j.bounded\]. We have $I^\kappa_{1}= I^\kappa_{11}+I^\kappa_{12}$ with $$\begin{aligned}
I^\kappa_{11} &:= \int_{ \left\{ |x| \leq \kappa \right\} } \mathbb{E} \bigg[ \Big| \widehat{G}_{\mu^{n}_{t}}\big(\widehat{w}^{n}_{fp}(x + i\gamma )\big) - \widehat{G}_{\mu^{n}_{t}}\big( w_{fp}(x + i\gamma )\big) \Big|^{2} 1_{A_{M}^{n,t}} \bigg] {\mathrm{d}}x,\\
I^\kappa_{12} &:= \int_{ \left\{ |x| \leq \kappa \right\} } \mathbb{E} \bigg[ \Big| \widehat{G}_{\mu^{n}_{t}}\big(\widehat{w}^{n}_{fp}(x + i\gamma )\big) - \widehat{G}_{\mu^{n}_{t}}\big( w_{fp}(x + i\gamma )\big) \Big|^{2} 1_{(A_{M}^{n,t})^c} \bigg] {\mathrm{d}}x.\end{aligned}$$
For the term $I^\kappa_{12}$, we have by Theorem \[thm.Fixpoint\](i) and Lemma \[lem.lambda\_j.bounded\]: $$\label{eq:majoI12}
I^\kappa_{12} \leq \dfrac{16}{\gamma^2}\kappa\mathbb P((A_{M}^{n,t})^c) \leq \dfrac{16}{\gamma^2}\kappa e^{- n.C_{eig}.M}.$$
Let us now consider the term $I^\kappa_{11}$: $$\begin{aligned}
I^\kappa_{11} &= \int_{ \left\{ |x| \leq \kappa \right\} } \mathbb{E} \bigg[ \bigg| \dfrac{1}{n} \sum_{j=1}^{n} \dfrac{w_{fp}(x + i \gamma) - \widehat{w}^{n}_{fp}(x + i \gamma ) }{ \big( \widehat{w}^{n}_{fp}(x + i \gamma ) - \lambda_{j}^{n}(t) \big) . \big( w_{fp}(x + i \gamma ) - \lambda_{j}^{n}(t) \big) } \bigg|^{2} 1_{A_{M}^{n,t}} \bigg] {\mathrm{d}}x
\\
&\leq \int_{ \left\{ |x| \leq \kappa \right\} } \mathbb{E} \Bigg[ \Big| \widehat{w}^{n}_{fp}(x + i \gamma) - w_{fp}(x + i \gamma ) \Big|^{2} . \dfrac{1}{n} \sum_{j=1}^{n} \dfrac{ 1_{A_{M}^{n,t}} }{ \big| \widehat{w}^{n}_{fp}(x + i \gamma ) - \lambda_{j}^{n}(t) \big|^{2} . \big| w_{fp}(x + i \gamma ) - \lambda_{j}^{n}(t) \big|^{2} } \Bigg] {\mathrm{d}}x
$$ by convexity. Using and , we have $$\begin{aligned}
\big| w_{fp}(x + i \gamma ) - \lambda_{j}^{n}(t) \big|&\geq\big| w_{fp}(x + i \gamma )\big|-\big| \lambda_{j}^{n}(t)\big|\\
&\geq \big|\Re(w_{fp}(x + i \gamma ))\big|-\big| \lambda_{j}^{n}(t)\big|\\
&\geq \big|x\big|-\sqrt{t}-\big| \lambda_{j}^{n}(0)\big|- \eta_*^n(t).\end{aligned}$$ Since $\lambda_{j}^{n}(t)$ is real, we also have: $$\begin{aligned}
\big| w_{fp}(x + i \gamma ) - \lambda_{j}^{n}(t) \big|&\geq \big| \Re(w_{fp}(x + i \gamma ) - \lambda_{j}^{n}(t)) \big|\\
&\geq\big|\lambda_{j}^{n}(t) \big|-\big| \Re(w_{fp}(x + i \gamma ))\big|\\
&\geq \big| \lambda_{j}^{n}(t)\big|- \big|x\big|-\sqrt{t}\\
&\geq \big| \lambda_{j}^{n}(0)\big|- \eta_*^n(t)- \big|x\big|-\sqrt{t}.\end{aligned}$$ Therefore, using Theorem \[thm.Fixpoint\], $$\big| w_{fp}(x + i \gamma ) - \lambda_{j}^{n}(t) \big|\geq\left\{\Big|\big| \lambda_{j}^{n}(0)\big|- \big|x\big|\Big|-\sqrt{t}- \eta_*^n(t)\right\}\vee\frac{\gamma}{2}.$$ In Theorem-Definition \[th:defestimator\], it is shown that $\widehat{w}^n_{fp}(z)$ satisfies a similar inequality as . Thus, we obtain with similar computations that: $$\big| \widehat{w}^{n}_{fp}(x + i \gamma ) - \lambda_{j}^{n}(t) \big|\big|\geq\left\{\Big|\big| \lambda_{j}^{n}(0)\big|- \big|x\big|\Big|-\sqrt{t}- \eta_*^n(t)\right\}\vee\frac{\gamma}{2}.$$ Then, using the definition of $A_M^{n,t}$, there exists a constant $C_{11}(\gamma,t)$ only depending on $\gamma$ and $t$ such that $$\begin{aligned}
I^\kappa_{11}
&\leq \int_{ \left\{ |x| \leq \kappa \right\} } \mathbb{E} \Bigg[ \Big| \widehat{w}^{n}_{fp}(x + i \gamma) - w_{fp}(x + i \gamma ) \Big|^{2} . \dfrac{1}{n} \sum_{j=1}^{n} \dfrac{ 1 }{\left[
\left\{\Big|\big| \lambda_{j}^{n}(0)\big|- \big|x\big|\Big|-\sqrt{t}-M\right\}\vee\frac{\gamma}{2} \right]^4} 1_{A_{M}^{n,t}} \Bigg] {\mathrm{d}}x\nonumber\\
&\leq \dfrac{1}{n}\sum_{j=1}^{n} \int_{ \left\{ |x| \leq \kappa \right\} } \mathbb{E} \Bigg[ \dfrac{ 1 }{\left[
\left\{\Big|\big| \lambda_{j}^{n}(0)\big|- \big|x\big|\Big|-\sqrt{t}-M\right\}\vee\frac{\gamma}{2} \right]^4} \mathbb{E} \big[ \big| \widehat{w}^{n}_{fp}(x + i \gamma) - w_{fp}(x + i \gamma ) \big|^{2} | X^n(0) \big] \Bigg] {\mathrm{d}}x\nonumber\\
& \leq \frac{C_{11}(\gamma,t)}{n} \sum_{j=1}^{n} \int_{ \left\{ |x| \leq \kappa \right\} } \mathbb{E} \Bigg[ \dfrac{ 1 }{\left[
\left\{\Big|\big| \lambda_{j}^{n}(0)\big|- \big|x\big|\Big|-\sqrt{t}-M\right\}\vee\frac{\gamma}{2} \right]^4} \left(\frac{1}{n^2} \right. \nonumber\\
& \hspace{5cm} \left.+ \left|\int_{\R} \dfrac{1}{x+i\gamma - t.G_{\mu_{0} \boxplus \sigma_{t} }(x+i\gamma) - \lambda} \big[ {\mathrm{d}}\mu^{n}_{0}(\lambda) - {\mathrm{d}}\mu_{0}(\lambda) \big] \right|^2\right) \Bigg] {\mathrm{d}}x\nonumber\\
& \leq \frac{C_{11}(\gamma,t)}{n} \big(I^\kappa_{111}+I^\kappa_{112}\big),\label{eq:majoI11}\end{aligned}$$ where the third inequality comes from , and where: $$\begin{aligned}
I^\kappa_{111}&:= \dfrac{1}{n^2}\sum_{j=1}^{n} \int_{ \left\{ |x| \leq \kappa \right\} }\mathbb{E} \Bigg[\dfrac{ 1 }{\left[
\left\{\Big|\big| \lambda_{j}^{n}(0)\big|- \big|x\big|\Big|-\sqrt{t}-M\right\}\vee\frac{\gamma}{2} \right]^4} \Bigg]{\mathrm{d}}x\\
I^\kappa_{112} & := \int_{ \left\{ |x| \leq \kappa \right\} }\mathbb{E} \Bigg[\int \dfrac{ 1 }{\left[
\left\{\Big|\big| \lambda\big|- \big|x\big|\Big|-\sqrt{t}-M\right\}\vee\frac{\gamma}{2} \right]^4}{\mathrm{d}}\mu^{n}_{0}(\lambda) \\
& \left| \sqrt{n}\int_{\R} \dfrac{1}{x+i\gamma - t.G_{\mu_{0} \boxplus \sigma_{t} }(x+i \gamma) - \lambda} \big[ {\mathrm{d}}\mu^{n}_{0}(\lambda) - {\mathrm{d}}\mu_{0}(\lambda) \big] \right|^2\Bigg]{\mathrm{d}}x.\end{aligned}$$ Now we wish to upper bound $I^\kappa_{111}$ and $I^\kappa_{112} $ independently of $\kappa.$ We first deal with $ I^\kappa_{111}.$ $$\begin{aligned}
I^\kappa_{111}
&= \frac 1 n \int_{ \left\{ |x| \leq \kappa \right\} }\int_\R \dfrac{ 1 }{\left[
\left\{\Big|\big| \lambda\big|- \big|x\big|\Big|-\sqrt{t}-M\right\}\vee\frac{\gamma}{2} \right]^4}{\mathrm{d}}\mu_0(\lambda){\mathrm{d}}x\nonumber\\
&\leq \frac 2 n \int_0^{+\infty}\int\dfrac{ 1 }{\left[
\left\{\Big|\big| \lambda\big|- x\Big|-\sqrt{t}-M\right\}\vee\frac{\gamma}{2} \right]^4}{\mathrm{d}}\mu_0(\lambda){\mathrm{d}}x. \label{eq:majoI111}\end{aligned}$$The double integral is upper bounded by a constant $C(\gamma,M,t)/n$ by Lemma \[lem:xleqkappa\].\
Let us now consider $I^\kappa_{112}$. Using Cauchy-Schwarz inequality, we have: $$\begin{aligned}
I^\kappa_{112}
& \leq \sqrt{ \mathbb{E} \Bigg[ \int_{ \left\{ |x| \leq \kappa \right\} }\Big(\int \dfrac{ 1 }{\left[
\left\{\Big|\big| \lambda\big|- \big|x\big|\Big|-\sqrt{t}-M\right\}\vee\frac{\gamma}{2} \right]^4}{\mathrm{d}}\mu^{n}_{0}(\lambda) \Big)^2 {\mathrm{d}}x \Bigg] }\nonumber\\
& \sqrt{ \mathbb{E} \Bigg[ \int_{ \left\{ |x| \leq \kappa \right\} }\left|\sqrt n \int_{\R} \dfrac{1}{x+i\gamma - t.G_{\mu_{0} \boxplus \sigma_{t} }(x+i \gamma) - \lambda} \big[ {\mathrm{d}}\mu^{n}_{0}(\lambda) - {\mathrm{d}}\mu_{0}(\lambda) \big] \right|^4{\mathrm{d}}x \Bigg]}\label{etape13}\end{aligned}$$ The first term can be treated exactly as $I^\kappa_{111}$ as: $$\begin{gathered}
\mathbb{E} \Bigg[ \int_{ \left\{ |x| \leq \kappa \right\} }\Big(\int \dfrac{ 1 }{\left[
\left\{\Big|\big| \lambda\big|- \big|x\big|\Big|-\sqrt{t}-M\right\}\vee\frac{\gamma}{2} \right]^4}{\mathrm{d}}\mu^{n}_{0}(\lambda) \Big)^2 {\mathrm{d}}x \Bigg] \\
\leq \mathbb{E} \Bigg[ \int_{ \left\{ |x| \leq \kappa \right\} } \frac{1}{\left(\frac{\gamma}{2}\right)^4}\Big(\int \dfrac{ 1 }{\left[
\left\{\Big|\big| \lambda\big|- \big|x\big|\Big|-\sqrt{t}-M\right\}\vee\frac{\gamma}{2} \right]^4}{\mathrm{d}}\mu^{n}_{0}(\lambda) \Big) {\mathrm{d}}x \Bigg]
=\frac{16n}{\gamma^4} I^\kappa_{111}.\label{etape14}\end{gathered}$$ We now focus on the second term of . As in the proof of Proposition \[prop.fluctuG2\], if we denote by $\phi_x:=\varphi_{x+i \gamma} : \lambda \mapsto (x+i\gamma - t.G_{\mu_{0} \boxplus \sigma_{t} }(x+i \gamma) - \lambda)^{-1},$ the last term can be rewritten as $$\begin{aligned}
I_{1121}^\kappa := & \mathbb{E} \Bigg[ \int_{ \left\{ |x| \leq \kappa \right\} }\left|\sqrt n \int_{\R} \phi_{x}(\lambda) \big[ {\mathrm{d}}\mu^{n}_{0}(\lambda) - {\mathrm{d}}\mu_{0}(\lambda) \big] \right|^4{\mathrm{d}}x \Bigg] \\
= & n^2 \int_{ \left\{ |x| \leq \kappa \right\} } \mathbb{E} \Bigg[ \left| \frac{1}{n} \sum_{j=1}^n \left(\phi_{x}(d_j^n(0)) - \mathbb E(\phi_{x}(d_j^n(0))\right) \right|^4\Bigg] {\mathrm{d}}x
\end{aligned}$$ where we used the notation $d_1^n, \ldots, d_n^n$ for the non-ordered diagonal elements of $X_n(0)$ (introduced after Equation ). Since the random variables $d_1^n, \ldots, d_n^n$ are i.i.d. with law $\mu_0,$ the random variables $(\phi_x(\lambda_j^n(0)) - \mathbb E(\phi_x(\lambda_j^n(0)) ))_{1 \le j \le n}$ are i.i.d. centered with finite fourth moment. By Rosenthal and then Cauchy-Schwarz inequality, we have $$\label{et12}
I_{1121}^\kappa \le \frac{C}{n^2}(n+n^2) \int_{ \left\{ |x| \leq \kappa \right\}} \int_\R \left|\phi_x(\lambda) - \int_\R \phi_x(\lambda) {\mathrm{d}}\mu_{0}(\lambda) \right|^4 {\mathrm{d}}\mu_{0}(\lambda){\mathrm{d}}x,$$ for $C$ a constant. We can conclude if the above double integral is bounded independently of $\kappa$. We would like to use Lemma \[lem:xleqkappa\] but the fact that we have a non-centered moment here implies that we should be careful, because a constant integrated with respect to ${\mathrm{d}}x$ on $\left\{ |x| \leq \kappa \right\}$ yields a term proportional to $\kappa$ that we should avoid.\
First let us recall some estimates for the functions $\phi_x$. As, we know that $\Im \big(G_{\mu_0 \boxplus \sigma_t }(x+i \gamma) \big)<0$, we have $$\label{et9}
| x+i\gamma - t G_{\mu_0 \boxplus \sigma_t }(x+i \gamma) - \lambda| \ge \Im ( x+i\gamma - t G_{\mu_0 \boxplus \sigma_t }(x+i \gamma) - \lambda) \ge \gamma \ge \frac{\gamma}{2}$$ and the functions $\phi_x$ are bounded by $2/\gamma$. This yields that $\big|\int_\R \phi_x(\lambda) {\mathrm{d}}\mu_{0}(\lambda) \big|\leq 2/\gamma$. By Lemma \[lem:Gmu0\], $|t G_{\mu_0 \boxplus \sigma_t }(x+i \gamma) | \le \frac{t}{\gamma} \le \frac{\sqrt t}{2} \le \sqrt t$ so that $$\label{et10} | x+i\gamma - t G_{\mu_0 \boxplus \sigma_t }(x+i \gamma) - \lambda| \ge (|x - \lambda|- \sqrt t) \geq (\big||x| - |\lambda|\big|- \sqrt t) .$$ As a consequence, $$| x+i\gamma - t G_{\mu_0 \boxplus \sigma_t }(x+i \gamma) - \lambda| \ge (\big||x| - |\lambda|\big|- \sqrt t) \vee \frac{\gamma}{2}.$$ Using that $d_1^n$ has distribution $\mu_0$, the double integral in the right hand side of can be rewritten as: $$\begin{gathered}
\int_{ \left\{ |x| \leq \kappa \right\}} \E\Big( \big|\phi_x(d_1^n) - \E\big[\phi_x(d_1^n)\big]\big|^4\Big){\mathrm{d}}x \\
\begin{aligned}
= & \int_{\left\{ |x| \leq \kappa \right\}} \Big\{\E\big[ |\phi_x(d_1^n)|^4 \big] - 2 \E\big[|\phi_x(d_1^n)|^2\phi_x(d_1^n)\big]\E\big[\overline{\phi_x}(d_1^n)\big]
+ \E\big[\phi^2_x(d_1^n)\big]\E\big[\overline{\phi_x}(d_1^n)\big]^2 \\
& - 2 \E\big[|\phi_x(d_1^n)|^2\overline{\phi_x}(d_1^n) \big]\E\big[\overline{\phi_x}(d_1^n)\big]
+ 4 \E\big[|\phi_x(d_1^n)|^2\big]\big|\E\big[\phi_x(d_1^n)\big]\big|^2 + \E\big[\overline{\phi_x}(d_1^n)^2\big]\big(\E\big[\phi_x(d_1^n)\big]\big)^2
- \big|\E\big[\phi_x(d_1^n)\big]\big|^4\Big\}\\
\leq & \mathfrak{I}_{4,\gamma,0,t}+ \frac{8}{\gamma}\mathfrak{I}_{3,\gamma,0,t}+ \frac{24}{\gamma^2} \mathfrak{I}_{2,\gamma,0,t}
\end{aligned}\end{gathered}$$ by using the notation of Lemma \[lem:xleqkappa\] and by neglecting the term $- \big|\E\big[\phi_x(d_1^n)\big]\big|^4<0$. The Lemma \[lem:xleqkappa\] allows us to conclude that $ I_{1121}^\kappa\leq C_{1121}(\gamma,t)<+\infty$.\
We can now conclude the Step 1. This last result, together with implies that $I_{112}^\kappa\leq C_{112}(\gamma,t)<+\infty$. From and , we have that $I_{11}^\kappa \leq C_1(\gamma,t) /n$ for $C_1(\gamma,t)$ a constant. Gathering this result with , we finally obtain that: $$\label{eq:majoI1}
I^\kappa_{1}\leq\dfrac{C_1(\gamma,t)}{n}+ \dfrac{16}{\gamma^2}\kappa e^{- n.C_{eig}.M}.$$
**Step 2:** Let us consider $I^\kappa_{2}$. Using Proposition \[proposition.D&F19.1st.term\], we have: $$\begin{aligned}
I^\kappa_2= & \int_{ \left\{ |x| \leq \kappa \right\} } \E\Big[{\mbox{Var}}\Big( \widehat{G}_{\mu^{n}_{t}}\big( w_{fp}(x + i\gamma)\big) \ |\ X^n(0)\Big) \Big] {\mathrm{d}}x
= \int_{ \left\{ |x| \leq \kappa \right\} } \E\Big[{\mbox{Var}}\Big( A^n_1\big( w_{fp}(x + i\gamma)\big) \ |\ X^n(0)\Big) \Big] {\mathrm{d}}x \nonumber\\
\leq & \int_{ \left\{ |x| \leq \kappa \right\} } \dfrac{10 \ t}{n^2 \Im^4\big(w_{fp}(x + i\gamma)\big) } dx
\leq \dfrac{10.2^{4}. t . 2\kappa}{n^{2} \gamma^{4}} \leq \dfrac{20 \kappa}{ n^{2} t}. \label{ett2}\end{aligned}$$
**Step 3:** Let us finally provide an upper bound for $I^\kappa_{3}$. Recall the definitions of $A^n_2(z)$ and $A^n_3(z)$ in : $$\begin{aligned}
I^\kappa_3 = & \int_{\left\{ |x| \leq \kappa \right\}} \E\Big[\big|A^n_2\big(w_{fp}(x+i\gamma)\big)+A^n_3\big(w_{fp}(x+i\gamma)\big)\big|^2 \Big] {\mathrm{d}}x\nonumber\\
\leq & 2 \int_{\left\{ |x| \leq \kappa \right\}} \E\Big[\big|A^n_2\big(w_{fp}(x+i\gamma)\big)\big|^2 \Big] {\mathrm{d}}x +2 \int_{\left\{ |x| \leq \kappa \right\}} \E\Big[\big|A^n_3\big(w_{fp}(x+i\gamma)\big)\big|^2\Big] {\mathrm{d}}x.\label{et14}\end{aligned}$$ By using Proposition \[prop.b\_n(z).bound\] together with Theorem \[thm.Fixpoint\] (i) and the fact that $\gamma>2\sqrt{t}$, we obtain that the first term in the right hand side is upper-bounded by $$2 \int_{\left\{ |x| \leq \kappa \right\}} \E\Big[\big|A^n_2\big(w_{fp}(x+i\gamma)\big)\big|^2 \Big]{\mathrm{d}}x \leq \frac{c \kappa }{n^2t},$$ where $c$ is an absolute constant. Let us now consider the second term in the right hand side of . Using the bound of Proposition \[prop.fluctuG2\], $$\begin{gathered}
2 \int_{\left\{ |x| \leq \kappa \right\}} \E\Big[\big|A^n_3\big(w_{fp}(x+i\gamma)\big)\big|^2 \Big] {\mathrm{d}}x \\
\leq 2 \frac{\gamma^4}{(\gamma^2-4t)^2}\int_\R \E\Big[\Big|\int_\R \frac{1}{w_{fp}(x+i\gamma) - t. G_{\mu_t}\big(w_{fp}(x+i\gamma)\big)-v}[{\mathrm{d}}\mu_0^n(v)-{\mathrm{d}}\mu_0(v)]\Big|^2\Big]{\mathrm{d}}x.\label{et15}\end{gathered}$$ Recall that $\mu_0^n$ is the empirical measure of independent random variables $(d_i^n)$ with distribution $\mu_0$ and whose order statistics are the $(\lambda_i^n(0))$. Recalling that $\big(w_{fp}(x+i\gamma)- t.G_{\mu_t}\big(w_{fp}(x+i\gamma)\big)-v\big)^{-1}=\varphi_{w_{fp}(x+i\gamma)}(v),$ we have that $$\begin{aligned}
\E\Big[\Big|\int_\R \varphi_{w_{fp}(x+i\gamma)}(v)[{\mathrm{d}}\mu_0^n(v)-{\mathrm{d}}\mu_0(v)]\Big|^2\Big]
&={\mbox{Var}}\Big[\frac{1}{n}\sum_{j=1}^n\varphi_{w_{fp}(x+i\gamma)}(\lambda_j^n(0))\Big] \nonumber \\
&\leq \frac{1}{n}\E\big[|\varphi_{w_{fp}(x+i\gamma)}(d_1^n)|^2\big]\nonumber\\
& = \frac{1}{n}\int_\R \frac{1}{|w_{fp}(x+i\gamma)- t.G_{\mu_t}\big(w_{fp}(x+i\gamma)\big)-v|^2}
{\mathrm{d}}\mu_0(v).\label{et16}\end{aligned}$$ Recall that from Lemma \[lem:Gmu0\] and Theorem \[thm.Fixpoint\] (i), $|w_{fp}(x+i\gamma)- t.G_{\mu_t}\big(w_{fp}(x+i\gamma)\big)-v| \geq |\Im\big(w_{fp}(x+i\gamma)\big)| \geq \gamma/2$, so that the integrand in the right hand side of is bounded. However, we have to work more to show that it is integrable with respect to $x$. We have: $$\begin{aligned}
|w_{fp}(x+i\gamma)- t.G_{\mu_t}\big(w_{fp}(x+i\gamma)\big)-v| &\geq \left|\Re\big(w_{fp}(x+i\gamma)- t.G_{\mu_t}\big(w_{fp}(x+i\gamma)\big)\big)-v\right|\\
&\geq \left|\Re\big(w_{fp}(x+i\gamma)\big)-v\right|-t\left|\Re\left(G_{\mu_t}\big(w_{fp}(x+i\gamma)\big)\right)\right|.\end{aligned}$$ By Theorem \[thm.Fixpoint\] (i), we obtain that: $$\left|\Re\left(G_{\mu_t}\big(w_{fp}(x+i\gamma)\big)\right)\right|\leq\left|\int_\R \frac{{\mathrm{d}}\mu_t(y)}{w_{fp}(x+i\gamma)-y}\right|\leq\frac{1}{|\Im(w_{fp}(x+i\gamma))|}\leq\frac{2}{\gamma}.$$ Also, by using , we get that $|\Re(w_{fp}(x+i\gamma))-x|\leq \sqrt{t}$. Therefore, $$|w_{fp}(x+i\gamma)- t.G_{\mu_t}\big(w_{fp}(x+i\gamma)\big)-v|\geq \big||x|-|v|\big|-\sqrt{t}-\frac{2t}{\gamma}.
\label{et17}$$ From , and , we have that: $$\begin{gathered}
2 \int_{\left\{ |x| \leq \kappa \right\}} \E\Big[\big|A^n_3\big(w_{fp}(x+i\gamma)\big)\big|^2 \Big] {\mathrm{d}}x \\
\leq \frac{2\gamma^4}{n(\gamma^2-4t)^2} \int_\R \int_\R \frac{1}{\left( \left\{\big||x|-|v|\big|-\sqrt{t}-\frac{2t}{\gamma}\right\}\vee \frac{\gamma}{2}\right)^2}{\mathrm{d}}\mu_0(v){\mathrm{d}}x = \frac{4\gamma^4}{n(\gamma^2-4t)^2} \mathfrak{I}_{2,\gamma,2t/\gamma,t},\end{gathered}$$ by Lemma \[lem:xleqkappa\]. We conclude as for $I^\kappa_{11}$ and we obtain $$\label{eq:majoI3}
I^\kappa_{3}\leq\dfrac{c \kappa}{n^2t}+ \frac{4\gamma^4}{n(\gamma^2-4t)^2} \mathfrak{I}_{2,\gamma,2t/\gamma,t}.$$ Gathering , and we obtain the result announced in Lemma \[lem:Ikappa\].
### Upper bound for $J^\kappa$
Recall the definition of $J^\kappa$ in . Our goal is to prove the following bound:
\[lem:Jkappa\] There exist constants $C_J^1$, $C^2_J$ and $C_J^3$ (that can depend on $\gamma$ and $t$) such that, for any $\kappa >\gamma,$ we have: $$J^\kappa\leq \frac{C_J^1}{ \kappa }+ C_J^2 n e^{- \frac{n.C_{eig}.\kappa}{4}} + C_J^3 \mu_{0}\left( \left\{|\lambda| > \frac{\kappa}{4}\right\}\right).$$
We decompose $J^\kappa\leq 2 (J^\kappa_1+J^\kappa_2)$ where $$\begin{aligned}
J^\kappa_1 &:= \int_{ \left\{ |x| > \kappa \right\} } \mathbb{E} \bigg( \Big| \int_{\R} \dfrac{ {\mathrm{d}}\mu^{n}_{t}(\lambda) }{ \widehat{w}^{n}_{fp}(x + i\gamma ) - \lambda } \Big|^{2} \bigg) {\mathrm{d}}x \\
J^\kappa_2 &:= \int_{ \left\{ |x| > \kappa \right\} } \Big| \int_{\R} \dfrac{ {\mathrm{d}}\mu_{t}(\lambda) }{ w_{fp}(x + i \gamma) - \lambda } \Big|^{2} {\mathrm{d}}x.\end{aligned}$$
Let us consider the first term $J^\kappa_{1}$. Using the estimate of Theorem-Definition \[th:defestimator\], we have for all $x\in \R$ that $\big| \Re \big(\widehat{w}^{n}_{fp}(x + i \gamma)\big) - x \big| \leq \sqrt{t}$ and $\Im\big( \widehat{w}^{n}_{fp}(x + i\gamma)\big)\geq \gamma/2.$ This allows us to prove that there exists a constant $C_{asymp}$ such that $$(x-\lambda)^2+\frac{\gamma^2}{4} \leq C_{asymp} \Big(\Re^2\big( \widehat{w}^{n}_{fp}(x + i\gamma) - \lambda \big)^{2} + \frac{\gamma^{2}}{4}\Big).\label{eq:Casympt}$$ Thus, $$\begin{aligned}
J^\kappa_1 &\leq \int_{ \left\{ |x| > \kappa \right\} } \mathbb{E} \int_{\R} \dfrac{ {\mathrm{d}}\mu^{n}_{t}(\lambda) }{ \Re^{2} \big( \widehat{w}^{n}_{fp}(x + i\gamma) - \lambda \big) + \Im^2\big(\widehat{w}^{n}_{fp}(x + i\gamma ) \big)^{2} } {\mathrm{d}}x
\\
& \leq C_{asymp} \int_{ \left\{ |x| > \kappa \right\} } \mathbb{E} \bigg[ \int_{\R} \dfrac{ {\mathrm{d}}\mu^{n}_{t}(\lambda) }{ (x -\lambda)^2 + \frac{\gamma^{2}}{4} } \bigg] {\mathrm{d}}x \\
& = C_{asymp} \, \mathbb{E} \bigg[ \int_{\R} {\mathrm{d}}\mu^{n}_{t}(\lambda) \int_{ \left\{ |x| > \kappa \right\} } \frac{1}{ (x -\lambda)^2 + \frac{\gamma^{2}}{4} } {\mathrm{d}}x\bigg] \\
& = \frac{2 C_{asymp} }{\gamma} \, \mathbb{E} \left[ \int_{\R} {\mathrm{d}}\mu^{n}_{t}( \lambda)\left(\pi - \arctan\left(\frac{2}{\gamma}(\kappa-\lambda)\right) -\arctan\left( \frac{2}{\gamma}(\kappa+\lambda)\right)\right)\right] \\
& = \frac{2 C_{asymp} }{\gamma} \, \mathbb{E} \left[ \int_{\R} {\mathrm{d}}\mu^{n}_{t}( \lambda)\left(\arctan\left(\frac{4\kappa \gamma}{4\kappa^2-4\lambda^2 -\gamma^2}\right)
+\pi \mathbf 1_{\left\{\lambda^2 > \kappa^2 - \frac{\gamma^2}{4}\right\}}\right) \right].\end{aligned}$$ We now use the simple bounds $|\arctan x| \le |x|$ and $|\arctan x| \le \frac{\pi}{2}$ for any $x \in \R.$ Moreover, one can easily check that, if $\lambda^2 \leq \frac{\kappa^2}{2}- \frac{\gamma^2}{4},$ then $$\frac{4\kappa \gamma}{4\kappa^2-4\lambda^2 -\gamma^2} \le \frac{2 \gamma}{\kappa}.$$ We therefore get $$\begin{aligned}
J^\kappa_1 &\leq \frac{2 C_{asymp} }{\gamma} \, \mathbb{E} \left[ \int_{\lambda} {\mathrm{d}}\mu^{n}_{t}( \lambda)\left( \frac{2 \gamma}{\kappa} + \frac{\pi}{2}
1_{\left\{\lambda^2 > \frac{\kappa^2}{2} - \frac{\gamma^2}{4}\right\}}
+\pi 1_{\left\{\lambda^2 > \kappa^2 - \frac{\gamma^2}{4}\right\}}\right)\right].\end{aligned}$$ If we assume moreover that $\kappa > \gamma,$ this can be simplified as follows: $$\begin{aligned}
J^\kappa_1 \leq & C_{asymp} \left(\frac{4 }{\kappa} + \frac{3\pi }{\gamma} \mathbb{E} \left[ \mu^{n}_{t}\left( \{|\lambda| > \frac{\kappa}{2}\}\right) \right]\right)\nonumber\\
\leq & C_{asymp} \left(\frac{4 }{\kappa} + \frac{3\pi }{\gamma}\mu_{0}\left(\left\{|\lambda | > \frac{\kappa}{4}\right\}\right) + \frac{3\pi }{\gamma}
n e^{- \frac{n.C_{eig}.\kappa}{4}}\right),\label{eq:Jkappa1}\end{aligned}$$ by using .
We now go to the second term $J^\kappa_2.$ The strategy will be very similar to what we did for $J^\kappa_1$ and we will give less details. Using the estimate , we have for all $x\in \R$ that $\big| \Re \big({w}_{fp}(x + i \gamma)\big) - x \big| \leq \sqrt{t}$, which allows us to get that $$(x-\lambda)^2+\frac{\gamma^2}{4} \leq C_{asymp} \Big(\Re^2\big( w_{fp}(x + i\gamma) - \lambda \big)^{2} + \frac{\gamma^{2}}{4}\Big),$$ with $C_{asymp}$ as above. Thus, $$\begin{aligned}
J^\kappa_2 &\leq C_{asymp} \int_{ \left\{ |x| > \kappa \right\} } \int_{\lambda} \dfrac{ {\mathrm{d}}\mu_{t}(\lambda) }{ (x -\lambda)^2 + \frac{\gamma^{2}}{4} } {\mathrm{d}}x \\
& \leq \frac{2 C_{asymp} }{\gamma} \,\int_{\lambda} {\mathrm{d}}\mu_{t}( \lambda)\left( \frac{2 \gamma}{\kappa} + \frac{\pi}{2}
1_{\left\{\lambda^2 > \frac{\kappa^2}{2} - \frac{\gamma^2}{4}\right\}}
+\pi 1_{\left\{\lambda^2 > \kappa^2 - \frac{\gamma^2}{4}\right\}}\right).\end{aligned}$$ Again, if we assume that $\kappa > \gamma,$ this can be simplified as follows: $$J^\kappa_2 \leq C_{asymp} \left(\frac{4 }{\kappa } + \frac{3\pi}{\gamma} \mu_{t}\left( \left\{|\lambda| > \frac{\kappa}{2}\right\}\right)\right).$$ Moreover, letting $n$ going to infinity in , by Proposition \[prop:LGN.mu.t\] and dominated convergence, we get that, for any $\kappa > \gamma,$ $$\mu_{t}\left( \left\{|\lambda| > \frac{\kappa}{2}\right\}\right) \le \mu_{0}\left( \left\{|\lambda| > \frac{\kappa}{4}\right\}\right),$$ so that $$\label{eq:Jkappa2}
J^\kappa_2
\le C_{asymp} \left(\frac{4 }{\kappa} + \frac{3\pi }{\gamma} \mu_{0}\left( \left\{|\lambda| > \frac{\kappa}{4}\right\}\right)\right).$$
Gathering the upper bounds and , we get that for any $\kappa >\gamma,$ $$J^\kappa\leq C_{asympt}\left(\frac{8}{\kappa}+ \frac{6\pi}{\gamma} \mu_{0}\left( \left\{|\lambda| > \frac{\kappa}{4}\right\} \right)+ \frac{3\pi}{\gamma}ne^{-\frac{n.C_{eig}.\kappa}{4}}\right).\label{eq:Jkappa}$$This ends the proof.
### Conclusion
As a result, combining Lemma \[lem:Ikappa\] and Lemma \[lem:Jkappa\], we have: $$\begin{aligned}
I^\kappa+J^\kappa \leq & \frac{C^1_I}{n}+ \frac{C_I^2 \kappa}{n^2}+ \frac{C_J^1}{\kappa }+ C_I^3\kappa e^{-n.C_{eig}.M} + C_J^2 n e^{- n C_{eig}.\frac{\kappa}{4}} + C_J^3 \mu_0\left(\left\{ |\lambda| > \frac{\kappa}{4}\right\} \right).\end{aligned}$$ We take $\kappa=n$. Using Assumption , we obtain $$\mu_0\left(\left\{ |\lambda| > n\right\} \right)\leq Cn^{-1},$$ for some absolute constant $C$. Then, from and previous computations, there exists a constant $C_{var}$ (that can depend on $\gamma$ and $t$) such that for $n$ sufficiently large: $$\mathbb{E} (\Sigma) \leq \frac{C_{var}. e^{\frac{2\gamma}{h}}}{n}$$ and Theorem \[variance\] is proved.
Numerical simulations {#section:numerical}
=====================
In this section, we conduct a simulation study to assess the performances of our estimator $\widehat {p}_{0,h}$ designed in Definition \[def:est\] based on the $n$-sample $\lambda^n(t):=\{ \lambda_1^n(t),\cdots,\lambda_n^n(t) \}$ of (non ordered) eigenvalues. We consider the sample size $n=4000$ and the time value $t = 1$. We focus on initial conditions following a Cauchy distribution with scale parameter $s_d= 5$: $$p_{0}(x) = \dfrac{1}{\pi}. \dfrac{s_d}{(s_d^2+x^2)},\quad x\in \mathbb{R}.$$ Expression is used with the kernel $K(x)={\mbox{sinc}}(x)=\sin(x)/(\pi x)$, and the value $\gamma = 2\sqrt{t}+0.01$ so that the condition $ \gamma > 2 \sqrt{t}$ is satisfied. To implement $\widehat {p}_{0,h}$, we approximate integrals involved in Fourier and inverse Fourier transforms by Riemann sums, so it may happen that $\widehat {p}_{0,h}(x)$ is not real. This is the reason why the density $p_0$ is estimated with $\Re (\widehat {p}_{0,h})$, the real part of $\widehat {p}_{0,h}$.
The theoretical bandwidth $h$ proposed in Section \[section:MISE\] cannot be used in practice and we suggest the following data-driven selection rule, inspired from the principle of cross-validation. We decompose the quadratic risk for $\Re(\widehat{p}_{0,h})$ as follows: $$\begin{aligned}
\left\| \Re(\widehat{p}_{0,h}) - p_{0}\right\|^{2} = \int_{\mathbb{R}} \left| \Re(\widehat{p}_{0,h}(x)) - p_{0}(x) \right|^{2} dx
= \left\| \Re(\widehat{p}_{0,h}) \right\|^{2} - 2\int_{\mathbb{R}} \Re( \widehat{p}_{0,h}(x)) p_{0}(x) dx + \left\| p_{0}\right\|^{2}.\end{aligned}$$ Then, an ideal bandwidth $h$ would minimize the criterion $J$ with $$J(h):=\left\| \Re(\widehat{p}_{0,h} )\right\|^{2} - 2\int_{\mathbb{R}} \Re(\widehat{p}_{0,h}(x))p_{0}(x) dx,\quad h\in\R_+^*.$$ Since $J$ depends on $p_{0}$ through the second term, we investigate a good estimate of this criterion. For this purpose, we divide the sample $\lambda^n(t)$ into two disjoints sets $$\mathbf{\lambda}^{n,E}(t) := (\lambda_i^n(t))_{i\in E}\quad\text{ and }\quad \mathbf{\lambda}^{n,E^c}(t) := (\lambda_i^n(t))_{i\in E^c}.$$ There are $V_{\max} := \binom{n}{n/2}$ possibilities to select the subsets $(E, E^c)$, which is huge. Hence, to reduce computational time, we draw randomly $V=10$ partitions denoted $(E_j, E_j^c)_{j=1,\ldots,V}$. Choosing the grid $\mathcal{H}$ of $50$ equispaced points lying between $h_{\min}=0.25$ and $h_{\max}=2.7$, our selected bandwidth is $$\begin{aligned}
\hat h = \underset{h \in \mathcal{H}}{\textrm{argmin }} \textrm{Crit}(h) \label{def:Crit(h)} $$ with $$\textrm{Crit}(h) := \min_{h' \in \mathcal{H}, h' \neq h} \dfrac{1}{V} \sum_{j=1}^{V} \left( \left\| \Re(\widehat{p}^{(E_{j})}_{0,h}) \right\|^{2} - 2\int_{\mathbb{R}} \Re( \widehat{p}^{(E_{j})}_{0,h}(x)) \Re( \widehat{p}_{0,h'}^{(E^{c}_{j})} (x)) dx \right)$$ and our final estimator is then $\Re(\widehat p_{0,\hat h})$. In the last expression, $ \widehat{p}^{(E_{j})}_{0,h}$ and $\widehat{p}_{0,h'}^{(E^{c}_{j})}$ are estimates based on the samples $E_j$ and $E_j^c$ respectively.
To evaluate our approach, Figure \[Crit\_vs\_MISE\] displays the plot of $h\in \mathcal{H}\mapsto\textrm{Crit}(h)$ and $h\in \mathcal{H}\mapsto J(h)$ for the Cauchy density $p_0$.
![Plots of $h\mapsto\textrm{Crit}(h)$ and $h\mapsto J(h)$ for the Cauchy density $p_0$[]{data-label="Crit_vs_MISE"}](Crit2_vs_J-h_V10.pdf)
A close inspection of the graphs shows that the first criterion is a good estimate of the second one. As expected, for both criterions, we observe a plateau containing minimizers of $J$ and $\textrm{Crit}$. Outside the plateau, both criterions take large values due to large variance when $h$ is too small and to large bias when $h$ is too large.
![Estimation of $p_{0}$[]{data-label="reconstruction-cauchy"}](cauchy_mean_reconstruction.pdf)
Figure \[reconstruction-cauchy\] gives the reconstruction provided by $\Re(\widehat p_{0,\hat h})$ for the Cauchy density $p_0$. The results are quite satisfying, meaning that our estimation procedure seems to perform well in practice for estimating initial conditions of the Fokker-Planck equation.
Proof of technical lemmas and Corollary \[thm:MISE\]
====================================================
Proof of Lemma \[lem:but1\]
---------------------------
Recall that $R_{n,t}(z)$ and $\widetilde{R}_{n,t}(z)$ are defined in and , and that $$n \widetilde{A}^n_2(z) = \sum_{k=1}^n \E\left[ \big(R_{n,t}(z)\big)_{kk}\ |\ X^n(0)\right]-\big(\widetilde{R}_{n,t}(z)\big)_{kk}.\label{etape9}$$ Proceeding as in Dallaporta and Février [@Fevrier1], we introduce some notations. Let $R^{(k)}_{n,t}(z)$ be the resolvent of the $(n-1) \times (n-1)$ obtained from $X^{n}(t)$ by removing the $k$-th row and column and $C^{(k)}_{k,t}$ be the $(n-1)$-dimensional vector obtained from the $k$-th column of $H^{n}(t)$ by removing its $k$-th component.\
Using Schur’s complement (see e.g. [@BaiSilver Appendix A.1]): $$\begin{aligned}
\Big(\big( R_{n,t}(z) \big)_{kk}\Big)^{-1} = z - \left(H^{n}(t)\right)_{kk} - \left(X^{n}(0)\right)_{kk} - C^{(k) \ast}_{k,t}. R^{(k)}_{n,t}(z). C^{(k)}_{k,t}.
\end{aligned}$$Because $\widetilde{R}_{n,t}(z)$ is a diagonal matrix, we have easily: $$\begin{aligned}
\big( R_{n,t}(z) \big)_{kk} =& \big( \widetilde{R}_{n,t}(z) \big)_{kk}
\\
&+ \big( \widetilde{R}_{n,t}(z) \big)_{kk}. \big( R_{n,t}(z) \big)_{kk}. \Big( \left(H^{n}(t)\right)_{kk} + C^{(k) \ast}_{k,t}. R^{(k)}_{n,t}(z). C^{(k)}_{k,t} - \frac{t}{n} \mathbb{E} \big[ {\mbox{Tr}}\left(R_{n,t}(z)\ |\ X^n(0)\right) \big] \Big) \quad .
\end{aligned}$$ Replacing $\big( R_{n,t}(z) \big)_{kk}$ in the right-hand side of the previous formula, we obtain: $$\begin{gathered}
\left( R_{n,t}(z) \right)_{kk} - \big( \widetilde{R}_{n,t}(z) \big)_{kk} \\
\begin{aligned}
=& \big( \widetilde{R}_{n,t}(z) \big)_{kk}^{2}. \Big( \left(H^{n}(t)\right)_{kk} + C_{k,t}^{(k) \ast}. R^{(k)}_{n,t}(z). C_{k,t}^{(k)} - \frac{t}{n} \mathbb{E}\left[ {\mbox{Tr}}\left(R_{n,t}(z)\Big)\ |\ X^n(0) \right] \right)
\\
+& \big( \widetilde{R}_{n,t}(z) \big)_{kk}^{2}. \big( R_{n,t}(z) \big)_{kk}. \Big( \left(H^{n}(t)\right)_{kk} + C_{k,t}^{(k) \ast}. R^{(k)}_{n,t}(z). C_{k,t}^{(k)} - \frac{t}{n} \mathbb{E}\left[ {\mbox{Tr}}\left(R_{n,t}(z)\right)\ |\ X^n(0) \right] \Big)^{2} .
\end{aligned}\label{etape7}\end{gathered}$$ Since $H^{n}(t)$ and $C_{k,t}^{(k)}$ are independent of $X_{n}(0)$, $$\begin{gathered}
\mathbb{E} \Big[ \Big| \left(H^{n}(t)\right)_{kk} + C_{k,t}^{(k) \ast}. R^{(k)}_{n,t}(z) . C_{k,t}^{(k)} - \frac{t}{n} \mathbb{E}\left[ {\mbox{Tr}}\left(R_{n,t}(z)\right) \ |\ X^n(0)\right] \Big|^{2} \, |\, X^n(0) \Big] \\
\begin{aligned}
= &
\mathbb{E} \Big[ \Big| \left(H^{n}(t)\right)_{kk} + C_{k,t}^{(k) \ast}. R^{(k)}_{n,t}(z) . C_{k,t}^{(k)} - \frac{t}{n} {\mbox{Tr}}\big( R^{(k)}_{n,t}(z) \big) + \frac{t}{n} {\mbox{Tr}}\big( R^{(k)}_{n,t}(z) \big) - \frac{t}{n} \mathbb{E} \big[ {\mbox{Tr}}\big( R^{(k)}_{n,t}(z) \big) \ |\ X^n(0) \big]
\\ & + \frac{t}{n}\mathbb{E} \big[ {\mbox{Tr}}\big( R^{(k)}_{n,t}(z) \big) \ |\ X^n(0)\big] - \frac{t}{n} \mathbb{E}\left[ {\mbox{Tr}}\left(R_{n,t}(z)\right) \ |\ X^n(0)\right] \Big|^{2} \big| X^n(0) \Big]
\\ = &
\mathbb{E} \left[ \left(H^{n}(t)\right)_{kk}^{2} \right] + \mathbb{E}\Big[ \Big| C_{k,t}^{(k) \ast}. R^{(k)}_{n,t}(z). C_{k,t}^{(k)} - \frac{t}{n} {\mbox{Tr}}\big( R^{(k)}_{n,t}(z) \big) \Big|^{2} \,|\, X^n(0) \Big]
\\
& + \frac{t^2}{n^2} \Big( \textrm{Var} \big[ {\mbox{Tr}}\big( R_{n,t}^{(k)}(z) \big) \big| X^n(0) \big] + \Big| \mathbb{E} \big[ {\mbox{Tr}}\big( R_{n,t}^{(k)}(z) \big) - {\mbox{Tr}}\left(R_{n,t}(z)\right)\, |\, X^n(0) \big] \Big|^{2} \Big) .
\end{aligned}\label{etape5}\end{gathered}$$ We now upper bound each of the term in the right-hand side of . The first term equals to $t/n$.\
**Step 1:** We upper bound the second term in . By Lemma 5 of [@Fevrier1], $$\label{eq:CRC=trR}
\E\Big[ C_{k,t}^{(k) \ast}. R^{(k)}_{n,t}(z). C_{k,t}^{(k)} \ | \ X^n(0)\Big]=\frac{t}{n} \E\Big[{\mbox{Tr}}\big( R^{(k)}_{n,t}(z) \big) \ |\ X^n(0)\Big].$$ Thus, the second term in equals to ${\mbox{Var}}\big(C_{k,t}^{(k) \ast}. R^{(k)}_{n,t}(z). C_{k,t}^{(k)} \ | \ X^n(0) \big)$ and we have:
$$\begin{aligned}
{\mbox{Var}}\left[ C_{k,t}^{(k) \ast}. R^{(k)}_{n,t}(z). C_{k,t}^{(k)} \ | \ X^n(0)\right] = & \frac{t^2}{n^2} \mathbb{E}\left[{\mbox{Tr}}\big(R^{(k),\ast}_{n,t}(z).R^{(k)}_{n,t}(z)\big) \ |\ X^n(0)\right]\\
\leq & \frac{t^2}{n^2} \mathbb{E}\left[ \sum_{j=1}^n \frac{1}{|z-\lambda_j^{(k)} |^2} \ |\ X^n(0)\right]\end{aligned}$$
where the $\lambda_j^{(k)}$’s are the eigenvalues of the matrix with resolvent $R^{(k)}_{n,t}(z)$. Hence, $${\mbox{Var}}\left[ C_{k,t}^{(k) \ast}. R^{(k)}_{n,t}(z). C_{k,t}^{(k)} \ | \ X^n(0)\right]\leq \frac{t^2}{n \Im^2(z)}.\label{ub2}$$
**Step 2:** We now upper bound the third and fourth terms of . Let us denote in the sequel by $\mathbb{E}_{k}$ the expectation with respect to $\left\{ \left(H^{n}(t)\right)_{jk} : 1 \leq j \leq n \right\}$, and by $\mathbb{E}_{\leq k}$ the conditional expectation on the sigma-field $\sigma \left( (\left(X^{n}(0)\right)_{ij} , 1 \leq i \leq j \leq n), (\left(H^{n}(t)\right)_{ij}, 1 \leq i \leq j \leq k) \right)$.\
We have: $$\label{etape8}
{\mbox{Var}}\big[ {\mbox{Tr}}\big( R_{n,t}^{(k)}(z) \big) \big| X^n(0) \big]\leq 2{\mbox{Var}}\big[ {\mbox{Tr}}\big( R_{n,t}(z) \big) \big| X^n(0) \big]+ 2 {\mbox{Var}}\big[{\mbox{Tr}}\big( R_{n,t}(z) \big) - {\mbox{Tr}}(R_{n,t}^{(k)}(z) \big| X^n(0) \big].$$For the first term, $$\begin{aligned}
\textrm{Var} \big[ {\mbox{Tr}}\big( R_{n,t}(z) \big) \big| X^n(0) \big] = & \sum_{k=1}^n \E\left[ \left| \big(\mathbb{E}_{\leq k} - \mathbb{E}_{\leq k-1} \big) {\mbox{Tr}}\left( R_{n,t}(z) \right) \right|^{2} \ |\ X^n(0)\right]\nonumber\\
= & \sum_{k=1}^n \E\left[ \left| \big(\mathbb{E}_{\leq k} - \mathbb{E}_{\leq k-1} \big) \big({\mbox{Tr}}( R_{n,t}(z) )-{\mbox{Tr}}( R_{n,t}^{(k)}(z) )\big) \right|^{2} \ |\ X^n(0)\right],\label{etape6}\end{aligned}$$as $\big(\mathbb{E}_{\leq k} - \mathbb{E}_{\leq k-1} \big){\mbox{Tr}}\left( R_{n,t}^{(k)}(z) \right)=0$. The Schur complement formula (see e.g. [@BaiSilver Appendix A.1]) gives that: $$\label{proof.Var.TrR_n.TrR^k_n}
{\mbox{Tr}}\big( R_{n,t}(z) \big) - {\mbox{Tr}}\big( R^{(k)}_{n,t}(z) \big) = \dfrac{1 + C^{(k) \ast}_{k,t}. R^{(k)}_{n,t}(z)^{2}. C^{(k)}_{k,t} }{ z - \left(H^{n}(t)\right)_{kk} - \left(X^{n}(0)\right)_{kk} - C^{(k) \ast}_{k,t}. R^{(k)}_{n,t}(z). C^{(k)}_{k,t} }.$$ Then, $$\begin{aligned}
\left| {\mbox{Tr}}\big( R_{n,t}(z) \big) - {\mbox{Tr}}\big( R^{(k)}_{n,t}(z) \big) \right| & \leq \dfrac{ \left| 1 + C^{(k) \ast}_{k,t}. R^{(k)}_{n,t}(z)^{2}. C^{(k)}_{k,t} \right| }{ \left| \Im \left( z - \left(H^{n}(t)\right)_{kk} - \left(X^{n}(0)\right)_{kk} - C^{(k) \ast}_{k,t}. R^{(k)}_{n,t}(z). C^{(k)}_{k,t} \right) \right| } \nonumber
\\
& \leq \dfrac{ 1 + \left| C^{(k) \ast}_{k,t}. R^{(k)}_{n,t}(z)^{2}. C^{(k)}_{k,t} \right| }{\left| \Im(z) - \Im\left( C^{(k) \ast}_{k,t}. R^{(k)}_{n,t}(z). C^{(k)}_{k,t} \right) \right|} \nonumber \\
&\leq \dfrac{ 1 + C^{(k) \ast}_{k,t}. R^{(k)}_{n,t}(z)^{\ast}. R^{(k)}(z). C^{(k)}_{k,t} }{ \left| \Im(z) + \Im\left(z\right) . C^{(k) \ast}_{k,t}. R^{(k)}_{n,t}(z)^{\ast}. R^{(k)}_{n,t}(z). C^{(k)}_{k,t} \right| } \nonumber \\
&= \dfrac{1}{\Im(z)} . \label{proof.R_n.rewrite.bound}
\end{aligned}$$ The second inequality it due to the fact that $\left(H^{n}(t)\right)_{kk}, \left(X^{n}(0)\right)_{kk} \in \R$ and the third inequality comes from the following equality: With $\Psi\ : \ M\in \mathcal{H}_n({\mathbb{C}})\mapsto C^* M C$ with $C\in {\mathbb{C}}^n$, then, for any $z\in{\mathbb{C}}$ and any resolvent matrix $R(z)$, we have (see [@Fevrier1 Lemma 1]) $$\Im\big(\Psi(R(z))\big)= -\Im (z) \Psi\big(R(z)^* R(z)\big).$$ The bound does not depend on $X^n(0)$. Plugging this bound into , we obtain: $$\textrm{Var} \big[ {\mbox{Tr}}\big( R_{n,t}(z) \big) \big| X^n(0) \big] \leq \frac{4n}{\Im^2(z)}.$$From there, using , $$\label{ub3}
\textrm{Var} \big[ {\mbox{Tr}}\big( R^{(k)}_{n,t}(z) \big) \big| X^n(0) \big] \leq \frac{8n+2}{\Im^2(z)}.$$
Similarly, also provides an upper bound for the fourth term of : $$\Big| \mathbb{E} \big[ {\mbox{Tr}}\big( R_{n,t}^{(k)}(z) \big) - {\mbox{Tr}}\left(R_{n,t}(z)\right) \ |\ X^n(0) \big] \Big|^{2} \leq \frac{1}{\Im^2(z)}.\label{ub4}$$
**Step 3:** In conclusion, using , , and , we obtain that: $$\begin{gathered}
\mathbb{E} \left[ \left| \left(H_{n}(t)\right)_{kk} + C_{k,t}^{(k) \ast}. R^{(k)}_{n,t}(z). C_{k,t}^{(k)} - \frac{t}{n} \mathbb{E}\left[ {\mbox{Tr}}\left(R_{n,t}(z)\right)\ |\ X^n(0) \right] \right|^{2} \big| X^n(0) \right]
\\
\leq \frac{t}{n} + \frac{t^2}{n \Im^2(z)} + \left( 8n + 3 \right)\frac{t^2}{n^2 \Im^2(z)}.\end{gathered}$$
Going back to and using to upper-bound the first term in the right-hand side: $$\begin{gathered}
\left| \mathbb{E} \left[ \left( R_{n,t}(z) \right)_{kk} - \big( \widetilde{R}_{n,t}(z) \big)_{kk} \ \big| \ X^{n}(0) \right] \right| \\
\begin{aligned}
& \leq \frac{t}{n} \big| \big(\widetilde{R}_{n,t}(z) \big)_{kk} \big|^{2}. \mathbb{E} \left[ \big| {\mbox{Tr}}\big(R^{(k)}_{n,t}(z)\big) - {\mbox{Tr}}\big(R_{n,t}(z)\big) \big| \hspace{0.2cm} \Big| \hspace{0.2cm} X^{n}(0) \right]
\\
& + \big| \big(\widetilde{R}_{n,t}(z) \big)_{kk} \big|^{2}. \mathbb{E} \left[ \left|\big(R_{n,t}(z) \big)_{kk}\right|. \big|\left(H^{n}(t)\right)_{kk} + C_{k,t}^{(k) \ast}. R^{(k)}_{n,t}(z). C_{k,t}^{(k)} \right.\nonumber\\
& - \left.\frac{t}{n} \mathbb{E}\left[ {\mbox{Tr}}\left(R_{n,t}(z)\right)\ |\ X^n(0) \right] \big|^{2} \hspace{0.2cm} \Big| \hspace{0.2cm} X^{n}(0) \right]
\\
& \leq \big| \big(\widetilde{R}_{n,t}(z) \big)_{kk} \big|^{2} . \left( \frac{t}{n \Im (z)} + \frac{t}{n \Im (z)} + \frac{t^2}{n \Im^3 (z)} + \frac{(8n+3)t^2}{n^2 \Im^3 (z)} \right) \\
& \leq \big| \big(\widetilde{R}_{n,t}(z) \big)_{kk} \big|^{2} . \frac{1}{n}\left( \frac{2t}{\Im (z)}+ \frac{12 t^2}{\Im^3 (z)} \right) .\end{aligned}\end{gathered}$$ Using this upper bound in , we obtain by summation the result and using that for any $k$, $$\big|\widetilde{R}_{n,t}(z) \big)_{kk} \big|^{2}\leq \frac{1}{\Im^2(z)}.$$
Proof of Lemma \[lem:inverse\]
------------------------------
From and introducing $\overline{w}_1(z)$ such that: $$G_{\mu_0^n\boxplus \sigma_t}(z)=G_{\mu^n_0}\big(\overline{w}_{fp}(z)\big)=G_{\sigma_t}(\overline{w}_1(z)).$$ We can derive from Theorem-Definition \[def:subordinationfunction\] that $\overline{w}_{fp}(z)$ solves the equation (i) of Lemma \[lem:inverse\] and that: $$z=\overline{w}_{fp}(z)+t G_{\mu_0^n}(\overline{w}_{fp}(z)),$$ for all $z\in {\mathbb{C}}^+$. The latter equation justifies (ii) of Lemma \[lem:inverse\].
Proof of Corollary \[thm:MISE\]
-------------------------------
Recall that from Proposition \[Biais\] and Theorem \[variance\], the mean integrated square error is $$MISE=\E\Big[\left\| \widehat{p}_{0,h} - p_0\right\|^{2}\Big] \leq C_{B}^2L e^{-2ah^{-r}} + \frac{C_{var}. e^{\frac{2\gamma}{h}}}{n}.$$ Minimizing in $h$ amounts to solving the following equation obtained by taking the derivative in the right hand side of (\[bornerisque\]): $$\label{tradeoff}
\psi(h):= \exp{(\frac{2\gamma}{h} +\frac{2a}{h^r}) } h^{r-1}= O(n).$$ Consequently for the minimizer $h_*$ of we get that $$\frac{e^{\frac{2\gamma}{h_*}}}{n}=Ch_*^{1-r} e^{-2ah_*^{-r}},$$ for some constant $C>0$. Hence, in view of , when $r<1$ the bias dominates the variance and the contrary occurs when $r>1$. Thus, there are three cases to consider to derive rates of convergence: $r=1$, $r<1$ and $r>1$. To solve the equation , we follow the steps of Lacour [@lacourCRAS].\
**Case $r=1$.**\
The case where $r=1$ provides a window $h_{*}=2(a+\gamma)/\log n $ and we get $$MISE = O \left (n^{-\frac{a}{a+\gamma}} \right ).$$
**Case $r<1$.**\
In this case, and in the case $r>1$, following the ideas in [@lacourCRAS], we will look for the bandwidth $h$ expressed as an expansion in $\log (n)$. In this expansion and when $r<1$, the integer $k$ such that $\frac{k}{k+1}<r\leq \frac{k+1}{k+2}$ will play a role. The optimal bandwidth is of the form: $$\label{eq:hstar1}h_*= 2\gamma \Big(\log(n) + (r-1) \log \log (n) + \sum_{i=0}^k b_i (\log n )^{r+i(r-1)}\Big)^{-1},$$ where the coefficients $b_i$’s are a sequence of real numbers chosen so that $\psi(h_*)=O(n)$. The heuristic of this expansion is as follows: the first term corresponds to the solution of $e^{2\gamma/h}=n$. The second term is added to compensate the factor $h^{r-1}$ in evaluated with the previous bandwidth, and the third term aims at compensating the factor $e^{2a/h^r}$. Notice that $r-1<0$ and that the definition of $k$ implies that $r>r+(r-1)>\dots >r+k(r-1)>0>r+(k+1)(r-1)$. This explains the range of the index $i$ in the sum of the right hand side of .\
Plugging into , $$\begin{aligned}
\psi(h_*)
= & n \big(\log n\big)^{r-1}\exp\Big( \sum_{i=0}^k b_i (\log n)^{r+i(r-1)}\Big)\\
& \times
\exp\Big(\frac{2a}{(2\gamma)^r} \big(\log n\big)^r \big(1 + \frac{(r-1) \log \log (n) + \sum_{i=0}^k b_i (\log n)^{r+i(r-1)}}{\log n}\big)^r \Big)\\
& \times (2\gamma)^{r-1} \big(\log n\big)^{-(r-1)} \Big(1 + \frac{(r-1) \log \log (n) + \sum_{i=0}^k b_i (\log n)^{r+i(r-1)}}{\log n}\Big)^{-(r-1)}\\
= & (2\gamma)^{r-1} n(1+v_n)^{1-r}\exp\Big( \sum_{i=0}^k b_i (\log n)^{r+i(r-1)}\Big)\\
& \times \exp\Big(\frac{2a}{(2\gamma)^r} \big(\log n\big)^r \Big[1+ \sum_{j=0}^{k} \frac{r(r-1)\cdots (r-j)}{(j+1)!}v_n^{j+1}+o(v_n^{k+1})\Big]\Big)\end{aligned}$$where $$v_n=\frac{(r-1) \log \log (n) + \sum_{i=0}^k b_i (\log n)^{r+i(r-1)}}{\log n}=(r-1)\frac{\log \log(n)}{\log n}+\sum_{i=0}^k b_i (\log n)^{(i+1)(r-1)}$$ converges to zero when $n\rightarrow +\infty$. We note that $$\begin{aligned}
v_n^{j+1}= & \sum_{i=0}^{k-j-1} \sum_{p_0+\cdots p_{j}=i}b_{p_0}\cdots b_{p_{j}} (\log n)^{(i+j+1)(r-1)}+O\Big(\big(\log n\big)^{(k+1)(r-1)}\Big)\\
= & \sum_{\ell=j+1}^{k} \sum_{p_0+\cdots p_j=\ell-j-1} b_{p_0}\cdots b_{p_{j}} (\log n)^{\ell(r-1)}+O\Big(\big(\log n\big)^{(k+1)(r-1)}\Big).\end{aligned}$$ So $$\begin{aligned}
\psi(h_*)= & (2\gamma)^{r-1} n (1+v_n)^{1-r}\exp\Big( \sum_{i=0}^k b_i (\log n)^{r+i(r-1)}\Big)\\
& \times \exp\Big\{\frac{2a}{(2\gamma)^r}(\log n\big)^r + \frac{2a}{(2\gamma)^r}\sum_{\ell=1}^k \sum_{j=0}^{\ell-1} \Big[\frac{r(r-1)\cdots (r-j)}{(j+1)!}\sum_{p_0+\cdots p_j=\ell-j-1} b_{p_0}\cdots b_{p_{j}} \Big](\log n)^{r+\ell(r-1)}\\
& +O\Big(\big(\log n\big)^{(k+1)(r-1)}\Big)\Big\} \\
= &
(2\gamma)^{r-1} n (1+v_n)^{1-r} \exp\Big(\sum_{i=0}^{k} M_i (\log n)^{i(r-1)+r} + o(1)\Big).\end{aligned}$$The condition $\psi(h_*)=O(n)$ implies the following choices of constants $M_i$’s: $$M_0= b_0+\frac{2a}{(2\gamma)^{r}},\qquad \forall i>0,\
M_i= b_i+\frac{2a}{(2\gamma)^{r}}\sum_{j=0}^{i-1} \frac{r(r-1)\cdots (r-j)}{(j+1)!} \sum_{p_0+\cdots p_j=i-j-1} b_{p_0}\cdots b_{p_{j}}.$$ Since $h_*$ solves if all the $M_i=0$ for $i\in \{0,\cdots k\}$, the above system provides equation by equation the proper coefficients $b_i^*$. $$\label{eq:bstar}
b^*_0=-\frac{2a}{(2\gamma)^{r}},\qquad b^*_i=-\frac{2a}{(2\gamma)^{r}}\sum_{j=0}^{i-1} \frac{r(r-1)\cdots (r-j)}{(j+1)!} \sum_{p_0+\cdots p_j=i-j-1} b^*_{p_0}\cdots b^*_{p_{j}}.$$ Replacing in , we get: $$MISE=O\Big(\exp\Big\{-\frac{2a}{(2\gamma)^{r}}\Big[\log n+(r-1)\log \log n+\sum_{i=0}^k b_i^*(\log n)^{r+i(r-1)}\Big]^r\Big\}\Big).$$ **Case $r>1$.**\
Here, let us denote by $k$ the integer such that $\frac{k}{k+1}<\frac{1}{r}\leq \frac{k+1}{k+2}$. We look here for a bandwidth of the form: $$\label{eq:hstar2}h_*^r= 2a \Big(\log n + \frac{r-1}{r} \log \log (n) + \sum_{i=0}^k d_i (\log n)^{\frac{1}{r}-i\frac{r-1}{r}}\Big)^{-1},$$ where the coefficients $d_i$’s will be chosen so that $\psi(h_*)=O(n)$.
Similar computations as for the case $r<1$ provide that: $$\begin{aligned}
\psi(h_*)= & (2a)^{\frac{r-1}{r}} n (1+v_n)^{-\frac{r-1}{r}}\times \exp\Big(\sum_{i=0}^k d_i (\log n)^{\frac{1}{r}-i \frac{r-1}{r}}\Big)\\
& \times \exp\Big(\frac{2\gamma}{(2a)^{1/r}} (\log n)^{1/r} \Big[1 +\\
& \quad \sum_{\ell=1}^{k} \sum_{j =0}^{\ell-1} \sum_{p_0+\cdots p_j=\ell-j-1} \frac{\frac{1}{r}\big(\frac{1}{r}-1\big)\cdots \big(\frac{1}{r}-j\big)}{(j+1)!} d_{p_0}\cdots d_{p_j} (\log n)^{\ell \frac{1-r}{r}} +O\big((\log n)^{k\frac{1-r}{r}}\big)\Big]\Big)\\
= & (2a)^{\frac{r-1}{r}} n (1+v_n)^{-\frac{r-1}{r}} \exp\Big(\sum_{i=0}^{k} M_i (\log n)^{\frac{1}{r}-i\frac{r-1}{r}}+o(1)\Big)\end{aligned}$$ where here $$v_n=\frac{\frac{r-1}{r} \log \log (n)+\sum_{i=0}^k d_i (\log n)^{\frac{1}{r}-i\frac{r-1}{r}}}{\log n},$$ and $$\label{eq:dstar}
M_0=d_0+\frac{2\gamma}{(2a)^{1/r}},\qquad \forall i>0,\ M_i=d_i+\frac{2\gamma}{(2a)^{1/r}} \sum_{j =0}^{i-1} \sum_{p_0+\cdots p_j=i-j-1} \frac{\frac{1}{r}\big(\frac{1}{r}-1\big)\cdots \big(\frac{1}{r}-j\big)}{(j+1)!} d_{p_0}\cdots d_{p_j}$$ Solving $M_0=\cdots =M_k=0$ provides the coefficients $d_i^*$ so that is satisfied.\
Plugging the bandwidth $h_*$ with the coefficients $d_i^*$ into , we obtain: $$MISE=O\Big(\frac{1}{n}\exp\Big\{\frac{2\gamma}{(2a)^{1/r}} \Big[\log n+\frac{r-1}{r}\log \log n+\sum_{i=0}^k d_i^* (\log n)^{\frac{1}{r}-i\frac{r-1}{r}}\Big]^{1/r}\Big\}\Big).$$ This concludes the proof of Corollary \[thm:MISE\].
### Acknowledgement {#acknowledgement .unnumbered}
The authors thank P. Tarrago for useful discussions. M.M. acknowledges support from the Labex CEMPI (ANR-11-LABX-0007-01). V.C.T. is partly supported by Labex Bézout (ANR-10-LABX-58) and by the Chair “Modélisation Mathématique et Biodiversité" of Veolia Environnement-Ecole Polytechnique-Museum National d’Histoire Naturelle-Fondation X.
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[^1]: Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France
[^2]: Laboratoire de Mathématiques, UMR 8628, Université Paris Sud, 91405 Orsay Cedex France
[^3]: Laboratoire de Mathématiques, UMR 8628, Université Paris Sud, 91405 Orsay Cedex France
[^4]: Ceremade, CNRS, UMR 7534, Université Paris-Dauphine, PSL Research University, 75016 Paris, France
[^5]: LAMA, Univ Gustave Eiffel, UPEM, Univ Paris Est Creteil, CNRS, F-77447, Marne-la-Vallée, France
|
---
abstract: 'We investigate the presence and importance of dark matter discs in a sample of 24 simulated Milky Way galaxies in the [<span style="font-variant:small-caps;">apostle</span>]{}project, part of the [<span style="font-variant:small-caps;">eagle</span>]{}programme of hydrodynamic simulations in Lambda-CDM cosmology. It has been suggested that a dark disc in the Milky Way may boost the dark matter density and modify the velocity modulus relative to a smooth halo at the position of the Sun, with ramifications for direct detection experiments. From a kinematic decomposition of the dark matter and a real space analysis of all 24 halos, we find that only one of the simulated Milky Way analogues has a detectable dark disc component. This unique event was caused by a merger at late time with an LMC-mass satellite at very low grazing angle. Considering that even this rare scenario only enhances the dark matter density at the solar radius by 35% and affects the high energy tail of the dark matter velocity distribution by less than 1%, we conclude that the presence of a dark disc in the Milky Way is unlikely, and is very unlikely to have a significant effect on direct detection experiments.'
author:
- |
Matthieu Schaller$^1$[^1], Carlos S. Frenk$^1$, Azadeh Fattahi$^2$, Julio F. Navarro$^{2}$[^2], Kyle A. Oman$^2$ & Till Sawala$^3$\
$^1$Institute for Computational Cosmology, Durham University, South Road, Durham, UK, DH1 3LE\
$^2$Department of Physics and Astronomy, University of Victoria, PO Box 1700 STN CSC, Victoria, BC, V8W 2Y2, Canada\
$^3$Department of Physics, University of Helsinki, Gustaf Hällströmin katu 2a, FI-00014 Helsinki, Finland
bibliography:
- 'bibliography.bib'
title: The low abundance and insignificance of dark discs in simulated Milky Way galaxies
---
\[firstpage\]
cosmology: theory, dark matter – Galaxy: structure, disc – methods: numerical
Introduction
============
The very successful [$\Lambda$CDM]{}cosmological model is based on the assumption that around 25% of the energy density of the Universe is in the form of as-yet undetected, weakly-interacting particles that make up the dark matter (DM). Validating this assumption requires detecting the particles either indirectly through their decay or annihilation products, or directly through interaction with the atoms of a detector, or by finding evidence of their existence in particle accelerators (see [@Bertone2005; @Bertone2010] for reviews). In the case of direct detection experiments knowledge of the local (solar neighbourhood) DM density and its velocity distribution is essential. This is particularly important for experiments sensitive to low mass particles for which the energy required to interact with an atom in the detector can only be reached by energetic particles in the high-velocity tail of the distribution.
Historically, the distribution of the DM velocity modulus has been characterised by a Maxwellian with a peak value of $220~\rm{km}/\rm{s}$; the local dark matter density is normally taken to be $0.3~\rm{GeV}/\rm{cm}^3$. High resolution N-body simulations of halos of Milky Way type galaxies [@Springel2008] have since confirmed this general picture except that the DM velocity distribution is anisotropic and better described by a multivariate Gaussian rather than a Maxwellian. These simulations have also revealed halo-to-halo variations in the velocity modulus typically of amplitude $30\%$ which are related to the assembly history of individual halos [@Vogelsberger2009].
While high resolution [$N$-body]{}simulations have fully characterized all the relevant properties for direct detection experiments, they ignore effects due to galaxy formation which could, in principle, modify this picture. For example, the contraction of the inner halo induced by the condensation of baryons towards the centre steepens the density profile, shifting the velocity distribution towards larger values. [@Barnes1984; @Blumenthal1986; @Gnedin2004]
Another potentially important baryon effect is the formation of a “dark disc” perhaps facilitated by the formation of the baryonic disc [@Read2008]. Such a dark disc could form by the accretion of a massive satellite in the plane of the galactic disc at late times [@Read2009]. The presence of the galactic disc modifies the potential well (relative to the DM-only case) and this may help confine dark matter stripped from the satellite onto co-planar orbits, boosting the creation of a dark disc co-rotating with the stars. If a large fraction of the DM at the position of the Sun were co-rotating with the stellar disc, fewer particles would have a large velocity in the detector frame, thus modifying the outcome of DM direct detection experiments, potentially to a very significant extent. It is therefore of central importance for these detection experiments to quantify how common and how massive such discs are in [$\Lambda$CDM]{}.
By comparing the morphological and kinematic properties of the Milky Way to idealized simulations of accretion events onto discs [@Purcell2009] concluded that the co-rotating dark matter fraction near the Sun is at most 30%. A hydrodynamical “zoom” simulation of a galaxy was found to have formed a dark disc contributing around $25\%$ of the DM at the solar radius co-rotating with the stars [@Ling2010]. On the other hand, using a chemodynamical template [@Ruchti2014] found no evidence for accreted stars near the Sun, leaving little room for a dark disc in our galaxy.
In this paper we search for dark discs in the twelve [<span style="font-variant:small-caps;">apostle</span>]{}(“A Project Of Simulating The Local Environment”) simulations [@Sawala2015; @Fattahi2015]) carried out as part of the “Evolution and Assembly of GaLaxies and their Environments” ([<span style="font-variant:small-caps;">eagle</span>]{}) programme [@Schaye2015; @Crain2015]. The [<span style="font-variant:small-caps;">apostle</span>]{}simulations are designed to reproduce the kinematic properties of the Local Group. They thus have merger histories resembling that of our own Galaxy and should provide informative predictions for the abundance of dark discs. These simulations have been shown to reproduce the satellite galaxy luminosity functions of the MW and Andromeda and do not suffer from the “too-big-to-fail” problem [@Boylan-Kolchin2011]. They were performed with the same code used in the [<span style="font-variant:small-caps;">eagle</span>]{}simulation, which provides an excellent match to many observed properties of the galaxy population as a whole, such as the stellar mass function and the size distribution at low and high redshift.
Of particular relevance for this study are the rotation curves of the [<span style="font-variant:small-caps;">eagle</span>]{}galaxies, which agree remarkably well with observations of field galaxies [@Schaller2015a]. This indicates that the matter distribution in the simulated galaxies is realistic and suggests that baryon effects on the DM are appropriately modelled. The same set of simulations have been used to make predictions for indirect detection experiments [@Schaller2015c]. One aim of this Letter is to help complete a consistent picture for both types of DM experiments from state-of-the-art simulations.
We assume the best-fitting flat [$\Lambda$CDM]{}cosmology to the [WMAP7]{} microwave background radiation data [@WMAP7] ($\Omega_b =
0.0455$, $\Omega_m = 0.272$, $h = 0.704$ and $\sigma_8= 0.81$), and express all quantities without $h$ factors. We assume a distance from the Galactic Centre to the Sun of $r_{\odot}=8~\rm{kpc}$ and a solar azimuthal velocity of $v_{\rm Sun}=220~\rm{km}/\rm{s}$.
Simulations and method {#sec:sim}
======================
Details of the [<span style="font-variant:small-caps;">eagle</span>]{}code used to carry out the [<span style="font-variant:small-caps;">apostle</span>]{}simulations used in this work may be found in [@Schaye2015; @Crain2015]. The [<span style="font-variant:small-caps;">apostle</span>]{}simulations are described by [@Sawala2015]. Here we briefly describe the parts of the model that are most relevant to dark discs; in this section we also describe our procedure for identifying dark discs.
Simulation setup and subgrid model
----------------------------------
The [<span style="font-variant:small-caps;">eagle</span>]{}simulation code is built upon the [<span style="font-variant:small-caps;">gadget</span>]{}code infrastructure [@Springel2005]. Gravitational interactions are computed using a Tree-PM scheme, and gas physics using a pressure-entropy formulation of smooth particle hydrodynamics (SPH) [@Hopkins2013], called [<span style="font-variant:small-caps;">anarchy</span>]{}(Dalla Vecchia ([*in prep.*]{}), see also [@Schaller2015b]). The astrophysical subgrid model includes the following processes: a star formation prescription that reproduces the Kennicutt-Schmidt relation [@Schaye2008], injection of thermal energy and metals in the ISM following [@Wiersma2009b], element-by-element radiative cooling [@Wiersma2009a], stellar feedback in the form of thermal energy injection [@DallaVecchia2012], supermassive black hole growth and mergers and corresponding AGN feedback [@Booth2009; @Schaye2015; @RosasGuevara2013]. Galactic winds develop naturally without imposing a preferred direction or a shutdown of cooling.
The free parameters of the model were calibrated (mainly by adjusting the efficiency of stellar feedback and the accretion rate onto black holes) so as to reproduce the observed present-day galaxy stellar mass function and observed relation between galaxy masses and sizes, as well as the correlation between stellar masses and central black hole masses [@Schaye2015; @Crain2015]. Galaxies are identified as the stellar and gaseous components of subhalos found using the [ subfind]{} algorithm [@Springel2001; @Dolag2009]. No changes to the original model used by [@Schaye2015] were made to match the Local Group properties [@Sawala2015].
Milky Way halo selection
------------------------
The 24 galactic halos analyzed in this study come from twelve zoom resimulations of regions extracted from a parent N-body simulation of a $100^3~\rm{Mpc}^3$ volume containing $1620^3$ particles. Halo pairs were selected to match the observed dynamical properties of the Local Group [@Fattahi2015]. In each volume a pair of halos with mass in the range $5\times10^{11} < M_{200}/ {{\rm{M}_\odot}}< 2.5\times10^{12}$ that will host analogues of the MW and Andromeda galaxies are found. We use the two halos in each volume to construct our sample. These selection criteria ensure that our sample is not biased towards particular halo assembly histories and consists of galaxies with a similar environment to that of the MW and thus, plausibly, with a relevant star formation history. This selection contrasts with that by [@Read2009] where three simulated galaxies “were chosen to span a range of interesting merger histories”.
The high-resolution region of our simulations always encloses a sphere larger than $2.5~\rm{Mpc}$ centred on the centre of mass of the Local Group at $z=0$. The primordial gas particle mass in the high resolution regions was set to $1.2\times10^5~{{\rm{M}_\odot}}$ and the DM particle mass to $5.7\times10^5~{{\rm{M}_\odot}}$; the Plummer-equivalent softening length for all particle types is $\epsilon=307~\rm{pc}$ (physical). DM-only simulations were run from the same initial conditions and are denoted as [<span style="font-variant:small-caps;">dmo</span>]{}in the remainder of this letter.
The galaxies that formed in our halos have a stellar mass in the range $1.3\times10^{10}{{\rm{M}_\odot}}$ to $4.6\times10^{10}{{\rm{M}_\odot}}$ and half-mass radii in the range $2.3$ to $6.9~\rm{kpc}$ in reasonable agreement with observational estimates for the MW [e.g. @Bovy2013]. The Bulge-over-Total ratios of our galaxies, obtained from a kinematical decomposition, vary from $0.1$ to $0.9$. The four haloes that formed an elliptical galaxies are kept in the sample for comparison purposes.
Stellar and dark matter velocity distributions
----------------------------------------------
For each galaxy in the sample, we define a North pole axis to be in the direction of the angular momentum vector of all the stars within a spherical aperture of $30~\rm{kpc}$ around the centre of potential of the halo. Note that, as in the lower-resolution study of [<span style="font-variant:small-caps;">eagle</span>]{}galaxies by [@Schaller2015d], there is no significant offset between the centre of the stellar and DM distributions. We then select matter in a torus, along the plane of the stellar disc, around the solar radius, $r_\odot=8~\rm{kpc}$, with both radial and vertical extents of $\pm1~\rm{kpc}$. These tori contain of the order of $4,500$ DM and $30,000$ star particles, allowing the density and velocity distributions to be well sampled. In the case of the [<span style="font-variant:small-caps;">dmo</span>]{}simulations, we place the torus in the same plane as in the corresponding full baryonic run in order to make sure that any bulk halo rotation is accounted for. We note that the alternative choice of aligning the planes with the inertial axes of the halos leads to qualitatively similar results. Similarly, shifting the radius of the torus to smaller or larger values does not change the results of our study. We also generated $10^4$ randomly oriented tori within which we compute the mean DM density.
Within each galactic plane torus we calculate the velocity distributions of the DM and stars in the radial, azimuthal and vertical directions in bins of width of $25~\rm{km}/\rm{s}$. The tori are split in a large number of angular segments to bootstrap-resample the distributions and construct an estimate of the statistical fluctuations induced by the finite particle sampling of our simulations. An example (halo $10$, see below) of the velocity distributions for the DM and stars is shown in Fig. \[fig:distribution\].
![The velocity distribution with respect to the galaxy’s frame in a torus at the radius of the Sun for the halo with the most prominent dark disc in our sample. The panels show the distributions of velocity modulus, $v$, and radial, azimuthal and vertical velocity, $v_r$, $v_\phi$ and $v_z$ respectively. The red lines show the DM distribution in the [<span style="font-variant:small-caps;">apostle</span>]{}simulation with the $1\sigma$ error shown as a shaded region; the blue lines correspond to the DM in the equivalent [<span style="font-variant:small-caps;">dmo</span>]{}halo. The stellar velocity distribution is shown by the yellow lines. The best-fitting Maxwellian (top left panel) or the Gaussian of Model 2 (the remaining three panels; see text) to the DM velocity distribution in the [<span style="font-variant:small-caps;">apostle</span>]{}simulation is shown as a black dashed line and the difference between this model and the actual simulation data is shown in the sub-panels at the top of each plot. For the azimuthal velocity distribution, a model with two Gaussians (Model 3, see text) is shown with green dots.[]{data-label="fig:distribution"}](Figures/distributions_MR_V5_2.pdf){width="\columnwidth"}
Results {#sec:results}
=======
In this section we formulate a criterion for identifying dark discs in the simulations using the velocity distribution (as in previous studies) and apply it to our sample of galaxies. We then perform an analysis of the spatial distribution to confirm our findings. Finally, we analyse the cases that present tentative evidence for a dark disc.
Azimuthal velocity distribution models {#ssec:models}
--------------------------------------
In order to quantify the prominence of a DM disc in velocity space, we fit three different models to the azimuthal DM velocity distribution in the simulations.
- **Model 1**: a single Gaussian, centred at $v_\phi=0$ with the root mean square value ([*rms*]{}) as the only free parameter;
- **Model 2**: A single Gaussian, with both the centre and [*rms*]{} as free parameters;
- **Model 3**: Two Gaussians, one centred at $v_\phi=0$ and the other one at location that is free to vary. The [*rms*]{} of both Gaussians, as well as their relative normalisation, are the other three free parameters.
The results of collsionless simulations are well described by the first model [@Vogelsberger2009], and this also applies to the azimuthal velocity distributions extracted from our [<span style="font-variant:small-caps;">dmo</span>]{}halos. The best-fitting Gaussian from Model 2 is shown as a dashed black line on the bottom left panel of Fig. \[fig:distribution\]. Finally, the third model, shown as two dotted green lines also in the bottom left panel of the figure, is the model used by [@Read2009], which includes a *halo component* (as in Model 1) and a *disc component* represented by the second Gaussian.
The azimuthal velocity distribution of a halo with a significant dark disc would either have a single-peaked distribution (Model 2) shifted to a mean velocity comparable to the typical stellar azimuthal velocity at the radius of the Sun, or require a clear second Gaussian in addition to the halo component (Model 3).
For each galaxy we find the best-fitting parameters for all three models of the DM azimuthal velocity distributions. We then use the Akaike Information Criterion [AIC: @Akaike], corrected for finite sample size [@Burnham2002], to select amongst the different models[^3]. The one with significantly lowest AIC, for a given halo, minimises information loss and should be favoured [^4].
Abundance of dark discs from velocity space analysis
----------------------------------------------------
The top panel of Fig. \[fig:abundance\] shows the AIC of all three models for all halos in our sample. For $23$ of our $24$ halos, either the model with two Gaussians (Model 3, red squares) is disfavoured or all three models are too close for a decision to be made. Only halo 10 (discussed further below) is clearly better modelled by two Gaussians.
For all other halos, the simple model consisting of a single Gaussian either centred at $v_\phi=0$ (Model 1, blue triangles) or at an adjustable value (Model 2, yellow circles) are either slightly or strongly favoured by the velocity distributions. For those cases where the preferred mean velocity is non-zero, to establish whether the off-centre Gaussians are located at large values of $v_\phi$, which would also indicate the presence of a dark disc, we show the position of the mean of the Gaussian in the central panel of Fig. \[fig:abundance\]. As can be seen, ignoring halo 10, the location of the centre of the shifted Gaussian (Model 2, yellow circles) is always below $45~\rm{km}/\rm{s}$. This implies that baryon effects have displaced the peak of the Gaussian by less than $\frac{1}{2}$ the typical [*rms*]{} value of Model 1 (top dashed blue line, located at $\sigma(v_\phi)=116~\rm{km}/\rm{s}$), thus rendering this shift inconsequential for DM direct detection experiments. This shifted Gaussian centre is also located at less than $\frac{1}{4}
v_{\rm{Sun}}$, implying that the rotation identified by the model is not commensurate with the rotation speed of the stars, one of the criteria signalling the presence of a dark disc. Note also that if these slightly shifted Gaussians were to represent genuine dark discs, we would infer the presence of a counter-rotating component in halo 2. We also don’t find any correlation between the dark disc AIC values and galaxy properties such as mass, size or Bulge-to-Total ratio. We conclude that in $23$ of our $24$ representative MW halos, no dark discs able to affect the tail of the velocity distribution are detected.
![*Top panel:* The @Akaike Information Criterion (AIC) for all three models of the azimuthal DM velocity distribution of each simulated galaxy. Blue arrows indicate values grater than $25$ for the AIC of Model 1. Only for halo $10$ (shaded region) should the model with two Gaussians be preferred. *Central panel:* The [*rms*]{}, $\sigma_1$, of the azimuthal velocity distribution of the centred Gaussian (Model 1, blue triangles) and the position of the centre, $v_2$, of the shifted Gaussian (Model 2, yellow circles) for all $24$ halos. The [*rms*]{} value, $\langle\sigma_1\rangle$, and $\frac{1}{2}\langle\sigma_1\rangle$ are both indicated with dashed blue lines. For all halos, the centre of the shifted Gaussian is at a position, $|v_2| < \frac{1}{2}\langle\sigma_1\rangle \approx
\frac{1}{4}v_{\rm Sun}$, indicating that the shifts in the velocity distribution caused by baryon effects is not significant. *Bottom panel:* The distributions of mean dark matter densities in $10^4$ randomly orientated torii for both the [<span style="font-variant:small-caps;">dmo</span>]{}and [<span style="font-variant:small-caps;">apostle</span>]{}halos. The $68^{\rm th}$, $95^{\rm th}$ and $99.7^{\rm th}$ percentiles are indicated by errorbars. Triangles give the mean density in the torus oriented in the plane of the stellar disc. Ignoring the overall shift in normalisation, halo $10$ is the only one for which the density in the plane of the disc has been significantly altered by baryon effects. The dotted line indicates the commonly adopted value of the local DM density.[]{data-label="fig:abundance"}](Figures/AIC.pdf){width="\columnwidth"}
Abundance of dark discs from real space analysis
------------------------------------------------
If a dark disc can be identified in velocity space, it should also be identifiable in real space. To verify this, we constructed $10^4$ randomly orientated tori of the same size as the original torus aligned with the stellar disc plane. We compute the DM density in each of them and construct a distribution of densities whose medians and percentiles are displayed on the bottom panel of Fig. \[fig:abundance\]. We then determine the location in this distribution of the DM density corresponding to the torus aligned with the plane of the stellar disc. The presence of a dark disc should manifest as an enhanced DM density in the aligned torus compared to the other tori. However, as galaxies (especially centrals and late types) are preferentially aligned with their halo [e.g. @vanDenBosch2002; @Bett2010; @Velliscig2015], the dark matter density in the plane of the stellar disc is likely to be enhanced as a result of the anisotropic dark matter distribution even in the absence of a dark disc.
For most halos in our sample, the DM density is indeed enhanced in the plane of the stellar disc (see bottom panel of Fig. \[fig:abundance\]), but this enhancement is also present in the [<span style="font-variant:small-caps;">dmo</span>]{}simulations. This implies that the stellar disc that formed in this halo is aligned with the halo and not that a dark disc has formed. Only halo $10$ shows signs of a dark disc: its [<span style="font-variant:small-caps;">dmo</span>]{}counterpart displays a DM density lower than the median in the plane of the stellar disc but the [<span style="font-variant:small-caps;">apostle</span>]{}simulation displays a vastly enhanced DM density (more than $2.5\sigma$ above the median density). This high density is not the result of an intrinsic alignment of baryons with the halo but rather evidence for the presence of a dark disc.
Examining the variation of density with torus orientation shows that only one out of our $24$ MW halos (halo 10 again) shows signs of the presence of a dark disc. This result confirms our earlier conclusion from the velocity analysis.
Detailed analysis of the galaxy with a dark disc
------------------------------------------------
We now carry out a more detailed analysis of halo 10, the only one[^5] for which a double Gaussian azimuthal velocity distribution (Model 3) is favoured and the only one for which the real space analysis also implies the presence of a significant overdensity in the plane of the stellar disc. The distributions of each velocity component for this halo are displayed in Fig. \[fig:distribution\]. For the azimuthal distribution, the best-fitting models 2 and 3 are plotted. The double Gaussian model is clearly favoured. The second Gaussian is centred on $v_\phi=117\pm19~\rm{km}/\rm{s}$, a displacement comparable to the [*rms*]{} of the best-fitting Gaussian distribution for the corresponding [<span style="font-variant:small-caps;">dmo</span>]{}halo torus velocity distribution ($\sigma_1 =
113\pm4~\rm{km}/\rm{s}$). Baryonic effects have induced a clear second peak in the distribution of $v_\phi$.
Apart from the position of the velocity peak, the other important property of a dark disc is the amount of dark matter that it contains. This can be characterized in different ways. Simply evaluating the integral of the two Gaussians required to fit the velocity distribution implies that $31.5\%$ of the total DM mass at the position of the Sun is in the component rotating with respect to the halo. Another way of measuring the mass of the rotating component is to apply a kinematical bulge/disc decomposition technique (e.g. [@Abadi2003; @Scannapieco2012]) whereby the orbital energies are compared to the energy of the equivalent circular orbit of the same radius. The bulge (in our case the halo component) will be distributed symmetrically around $0$ (if the halo is not rotating, or slightly displaced from $0$ if it is) and the disc will appear as a clear second peak. Comparing the mass in the halo component to the total shows that $65.3\%$ of the mass is non-rotating, in agreement with the simpler estimate given above.
Finally, we find that the DM density in the plane of the disc in halo 10 is $\rho_{\rm DM}=0.33~\rm{GeV}/\rm{cm}^3$. Compared to the median $\bar{\rho}_{\rm DM}=0.21~\rm{GeV}/\rm{cm}^3$ of all the tori, this again indicates an excess of $\approx35\%$, in agreement with the kinematically derived dark disc mass fraction.
It is interesting to note that despite containing roughly a third of the mass at the location of the Sun, the dark disc has only a minimal impact on the distribution of the velocity modulus (top left panel of Fig. \[fig:distribution\]), which remains essentially unchanged from the [<span style="font-variant:small-caps;">dmo</span>]{}case (blue line). The velocity distribution of this halo, as well as its expected signal in direct detection experiments, lie well within the halo-to-halo scatter measured by [@Bozorgnia2016] in their sample of simulated MW analogues which includes some of the [<span style="font-variant:small-caps;">apostle</span>]{}simulations. More quantitatively, the DM mass with a velocity magnitude larger than $350~\rm{km}/\rm{s}$ has increased by only $0.6\%$ in the baryonic simulation compared to its [<span style="font-variant:small-caps;">dmo</span>]{}counterpart. We conclude that the one dark disc that has formed in our 24 simulations would have a minimal impact on direct dark matter detection experiments.
Origin of the dark disc
-----------------------
To investigate the origin of the dark disc that formed in one of our simulations we trace back in time the particles that have the largest azimuthal velocities today. We find that these particles belonged to a subhalo that merged with the central galaxy at $z\approx0.4$. At the time of the merger, the subhalo had a mass of $3.8\times10^{10}{{\rm{M}_\odot}}$ and the galaxy in it a stellar mass of $3.3\times10^{9}{{\rm{M}_\odot}}$, comparable to the Large Magellanic Cloud. The satellite impacted the galaxy at an angle of only $9^\circ$ above the plane of the stellar disc. Most of its material is then tidally stripped in the plane of the disc forming of a stream. The formation of a dark disc in this galaxy is thus consistent with the formation mechanism proposed by [@Read2009].
Conclusion {#sec:conclusions}
==========
We searched for dark discs in a sample of 24 Milky Way analogues simulated as part of the [<span style="font-variant:small-caps;">apostle</span>]{}project of simulations of volumes selected to match the kinematical and dynamical properties of the Local Group. This environment is similar to that in which the Milky Way formed suggesting that the galaxies in our sample are representative of plausible formation paths for the Milky Way.
We find that only one out the 24 cases develops a dark disc aligned with the plane of the stellar disc. The dark disc was identified by fitting models with and without a disc to the azimuthal velocity distribution at the radial location of the Sun and selecting the best model according to the AIC. The identification was then confirmed by searching for unusual dark matter overdensities in the plane of the stellar disc (comparing to randomly oriented discs). None of the other 23 halos show any evidence for a dark disc. We conclude from our unbiased sample of MW halo analogues that dark discs are rare.
Our simulations reveal that rather unusual conditions are required for the formation of a dark disc. In our case, the disc resulted from a recent impact, at a very low grazing angle, of a satellite as large as the LMC. According to [@Purcell2009] and [@Ruchti2014], Milky Way kinematical data indicate that our galaxy could not have experienced an encounter of this kind.
For the dark disc that formed in our simulations we found that $\approx35\%$ of the mass in a torus at the location of the Sun is rotating at a mean velocity of $116~\rm{km}/\rm{s}$. However, this rotating dark matter component does not significantly modify the distribution of the velocity modulus measured in the counterpart of the halo in a dark matter only simulation. The azimuthal velocity distribution is still well fit by a Maxwellian, indicating that this disc would have a negligible impact on direct dark matter detection experiments.
We conclude that while evidence for recent satellite collisions with the Milky Way disc would be very interesting to find in, for example, the GAIA data, direct dark matter searches need not be concerned about the potentially confusing effects of a dark disc at the position of the Earth.
Acknowledgments
===============
This work would have not be possible without Lydia Heck and Peter Draper’s technical support and expertise. We thank Nassim Bozorgnia, Francesca Calore and Gianfranco Bertone for useful discussions on DM direct detection experiments.\
This work was supported by the Science and Technology Facilities Council (grant number ST/L00075X/1) and the European Research Council (grant numbers GA 267291 “Cosmiway” ).\
This work used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility ([www.dirac.ac.uk](www.dirac.ac.uk)). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, and STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure.
\[lastpage\]
[^1]: E-mail: matthieu.schaller@durham.ac.uk
[^2]: Senior CIfAR fellow
[^3]: For two models with $AIC_A$ and $AIC_B$, the relative likelihood of the two models given the data is given by $\exp\left[\frac{1}{2}(AIC_A-AIC_B)\right]$
[^4]: Using the Bayesian Information Criterion (e.g. [@BIC]) leads to similar results.
[^5]: This is the second halo in volume AP-5 of [@Sawala2015]. Detailed properties of this simulation volume and halo can be found in Table 2 of [@Fattahi2015]. This galaxy has a stellar mass of $2.3\times10^{10}{{\rm{M}_\odot}}$ and a half-mass radius of $5.5~\rm{kpc}$, well within our sample. With a Bulge-to-Total ratio of $0.15$, it is one of the most disk-dominated system of our sample.
|
---
author:
- Makoto Makita
- Takuya Matsuda
title: 3D Finite Volume Simulation of Accretion Discs with Spiral Shocks
---
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to\#2
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Introduction
============
One of the models of accretion discs in a close binary system is the spiral shock model. This model was first proposed by one of the present authors (Sawada, Matsuda & Hachisu 1986a, b, Sawada et al. 1987). A number of authors have confirmed since then that spiral shocks appear in two-dimensional discs. In the case of 3D, Sawada & Matsuda (1992) used a Roe scheme to calculate the case of $\gamma=1.2$ with mass ratio of unity. They found the existence of spiral shocks, but their calculation was done up to only half a revolution period. Yukawa, Boffin & Matsuda (1997), who made numerical simulations using SPH with mass ratio of unity, also demonstrated that spiral shocks existed in the case of $\gamma=1.2$.
The purpose of the present paper is to perform 2D and 3D finite volume numerical calculations with higher resolution up to enough time to confirm the existence of spiral shocks.
Calculations and Conclusions
============================
We considered a binary system composed of a mass-accreting primary star and a mass-losing secondary star. The mass ratio of the mass-losing star to the mass-accreting star was one. We set a numerical grid centered at the primary star and just touching the inner Lagrangian point L1. In addition, we used Cartesian coordinates having $200\times200$ grid points in 2D and $200\times200\times50$ grid points in 3D, respectively. Finally, we assumed a polytropic gas with constant specific heat ratio $\gamma$.
In this paper, we examined the cases of $\gamma=$1.2, 1.1, 1.05, and 1.01. Symmetry about the orbital plane was assumed in our 3D calculations. We used the Simplified Flux vector Splitting (SFS) scheme (Jyounouti et al. 1993; Shima & Jyounouti 1994). In 2D calculations, we found that the smaller $\gamma$ was, the more tightly the spiral wound. The density contours in the orbital plane in the cases of $\gamma=1.2$ and $\gamma=1.01$ of our 3D simulations are presented in Fig.\[fig:densex-y\]. These figures show the existence of spiral shocks in the cases of $\gamma=1.2$ as well as in all the other cases. However, they do not show as clear a difference in spiral structure as they do in our 2D calculations. (Other results can be seen in CD-ROM).
Steeghs, Harlaftis & Horne (1997) found the first convincing evidence of spiral structure in accretion discs observationally. We have now succeeded in reproducing their results using our numerical calculations. (Please see Matsuda et al. in this volume).
Acknowledgments {#acknowledgments .unnumbered}
===============
The 2D calculations were mainly performed on VX/1R at the NAOJ. The 3D calculations were performed on SX-4 at IPC at Kobe University.
Jyounouchi T., Kitagawa I., Sakashita, Yasuhara M., 1993, Proceedings of 7th CFD Symposium, in Japanese. Sawada K., Matsuda T., 1992, MNRAS, 255, s17. Sawada K., Matsuda T., Hachisu I., 1986a, MNRAS, 219, 75. Sawada K., Matsuda T., Hachisu I., 1986b, MNRAS, 221, 679. Sawada K., Matsuda T., Inoue M., Hachisu I., 1987, MNRAS, 224, 307. Shima E., Jyounouchi T., 1994, 25th Annual Meeting of Space and Aeronautical Society of Japan, pp.36-37. Steeghs D., Harlaftis E.T., Horne K., 1997, MNRAS, 290, L28. Yukawa H., Boffin H.M.J., Matsuda T., 1997, MNRAS, 292, 321.
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---
abstract: 'The legacy of solar neutrinos suggests that large neutrino detectors should be sited underground. However, to instead go underwater bypasses the need to move mountains, allowing much larger contained water Čerenkov detectors. Reaching a scale of $\sim 5$ Megatons, the size of the proposed Deep-TITAND, would permit observations of “mini-bursts” of neutrinos from supernovae in the nearby universe on a yearly basis. Importantly, these mini-bursts would be detected over backgrounds without the need for optical evidence of the supernova, guaranteeing the beginning of time-domain MeV neutrino astronomy. The ability to identify, to the second, every core collapse would allow a continuous “death watch” of all stars within $\sim 5$ Mpc, making previously-impossible tasks practical. These include the abilities to promptly detect otherwise-invisible prompt black hole formation, provide advance warning for supernova shock-breakout searches, define tight time windows for gravitational-wave searches, and identify “supernova impostors” by the non-detection of neutrinos.'
author:
- 'Matthew D. Kistler'
- 'Hasan Y[ü]{}ksel'
- 'Shin’ichiro Ando'
- 'John F. Beacom'
- Yoichiro Suzuki
date: 'October 10, 2008'
title: 'Core-Collapse Astrophysics with a Five-Megaton Neutrino Detector'
---
Introduction
============
Core-collapse supernovae have long been suspected to be the solution of many long-standing puzzles, including the production of neutron stars and black holes, radioactive isotopes and heavy elements, and cosmic rays [@Baade]. Understanding these issues, and the properties of neutrinos and hypothesized new particles, requires improving our knowledge of supernovae. It is not enough to record their spectacular visual displays, as these do not reveal the dynamics of the innermost regions of the exploding stars, with their extremes of mass and energy density. Moreover, sophisticated simulations of the core collapse of massive stars do not robustly lead to supernova explosions [@Buras:2003sn; @Burrows:2005dv; @Mezzacappa:2005ju], raising the suspicion that crucial physics is missing.
Neutrinos are the essential probe of these dynamics, as they are the only particle that escapes from the core to the observer (gravitational waves may be emitted, but they are energetically subdominant). There is an important corollary to this, namely [*until supernovae besides SN 1987A are detected by neutrinos, our fundamental questions about supernovae will never be decisively answered.*]{} In fact, the most interesting problems–associated with the presence, nature, variety, and frequency of core collapse in massive stars–can only be solved by detecting [*many*]{} supernova neutrino bursts.
![Probabilities to obtain the indicated numbers of $\bar{\nu}_e$ neutrino events (with $E_{e^+} > 18$ MeV) in a 5 Mton detector as a function of the supernova distance. We assume a Fermi-Dirac $\bar{\nu}_e$ spectrum with an average energy of 15 MeV and a total energy of $5\times 10^{52}$ erg. Optical supernovae observed in the last 10 years are noted at their distances; those in red indicate multiple supernovae in the same galaxy.[]{data-label="fig:yields"}](yields){width="3.25in"}
The challenges of supernova neutrino burst detection are that Milky Way sources are rare and that more common distant sources have little flux. The 32 kton Super-Kamiokande (SK) detector is large enough to detect with high statistics a burst from anywhere in the Milky Way or its dwarf companions, but the expected supernova rate is only 1–3 per century, and there is no remedy but patience. Proposed underground detectors [@Nakamura:2003hk; @Jung:1999jq; @deBellefon:2006vq; @Autiero:2007zj], like the $\sim 0.5$-Mton Hyper-Kamiokande (HK), could detect one or two neutrinos from supernovae in some nearby galaxies [@Ando:2005ka]. As shown in Fig. \[fig:yields\], to robustly detect all neutrino bursts within several Mpc, where recent observations show the supernova rate to be at least 1 (2) per year within $\sim 6$ $(10)$ Mpc, requires scaling up the detector mass of SK by about two orders of magnitude, to at least $\sim 5$ Mton.
A recent proposal for the Deep-TITAND detector shows in detail how it might be feasible to build such a large detector in a cost-effective way [@Suzuki:2001rb; @Suzuki]. To avoid the high costs and slow pace of excavating caverns underground, this proposal conceives of a modular 5 Mton undersea detector that could be constructed quickly. Key motivations for such a detector are superior exposure for studies of proton decay, long-baseline neutrinos, and atmospheric neutrinos. To reduce costs, the detector would be built with a shallower depth and lower photomultiplier coverage than SK; these decisions would sacrifice the low-energy capabilities for all but burst detection.
There is a compelling case for a 5 Mton detector based on supernova neutrino detection alone, and the science benefits that we discuss below will hold even if a Milky Way supernova is detected first. On an annual basis, one would expect a burst of $\gtrsim\,$3 events, and every several years, a burst comparable to the $\sim 10$ events from SN 1987A detected by each of Kamiokande-II [@Hirata:1987hu; @Hirata:1988ad] or IMB [@Bionta:1987qt; @Bratton:1988ww]. Indeed, a 5 Mton supernova neutrino detector is one of the most promising prospects for developing an observatory for non-photon time-domain astrophysics. There are no serious uncertainties in the number of sources or the strengths of their signals. The minimal size of the required detector is known now, and it is not out of reach, with costs comparable to those of existing or near-term high-energy neutrino and gravitational-wave observatories.
Before elaborating on details concerning detection rates, we will begin by exploring how the data obtained from multiple neutrino bursts would transform the way that we consider questions about supernovae; these considerations are a major part of our new results. We will then examine recent developments concerning the rate and properties of supernovae observed in the nearby universe. This will lead into our discussion of the detector properties required to measure neutrino bursts from these supernovae, focusing on the Deep-TITAND proposal [@Suzuki:2001rb; @Suzuki], and the quantitative neutrino yields expected.
Discovery Prospects {#prospects}
===================
Our primary interest is on the scientific impact of measuring neutrino “mini-bursts,” detectable signals of 3 or more events within 10 seconds (the observed duration of the SN 1987A neutrino burst), from many supernovae in the nearby universe. As we will show in Sections \[rate\] and \[detection\], the minimum detector size for achieving this purpose is about 5 Mton. We emphasize in advance that such signals can be separated from backgrounds even at shallow depth, so that the presence of a core collapse can be deduced independently of photon-based observations. Additionally, for nearby transients identified through photons, a non-detection in neutrinos means that a conventional supernova neutrino flux was not present. These facts have new and profound implications.
While our principal focus is thus on individual objects, the aggregate data would, of course, also be useful. For science goals that require a large number of accumulated events, the most certain signal is the Diffuse Supernova Neutrino Background (DSNB), which is a steady flux arising from all core-collapse supernovae in the universe (e.g., Ref. [@DSNB] and references therein). In the proposed $\sim 0.5$ Mton HK detector, with added gadolinium to reduce backgrounds by neutron tagging [@Beacom:2003nk], $\sim\,$50–100 DSNB signal events with little background could be collected per year. The ratio of DSNB signal to detector background in Deep-TITAND would be the same as in the background-dominated SK search of Ref. [@Malek:2002ns], which set an upper limit. To reach the smallest plausible DSNB signals, one needs an improvement of about a factor 3 in signal sensitivity and thus a factor of about 10 in exposure. After four years, as in the SK search, the Deep-TITAND exposure would be about 100 times larger than that of Ref. [@Malek:2002ns], thus allowing a robust detection of the DSNB flux. (To measure the spectrum well, HK with gadolinium would be needed.) The fortuitous occurrence of a supernova in the Milky Way, or even Andromeda (M31) or Triangulum (M33), would also give a very large number of neutrino events (see Table \[tab:detectors\]). The physics prospects associated with such yields from a single supernova have been discussed for underground detectors at the 0.5 Mton scale [@MTnu].
-------- ------------- ----------- ---------- ---------------
32 kton 0.5 Mton 5 Mton
(SK) (HK) (Deep-TITAND)
10 kpc (Milky Way) $10^4$ $10^5$ $10^6$
1 Mpc (M31, M33) $1$ $10$ $10^2$
3 Mpc (M81, M82) $10^{-1}$ $1$ $10$
-------- ------------- ----------- ---------- ---------------
: Approximate neutrino event yields for core-collapse supernovae from representative distances and galaxies, as seen in various detectors with assumed fiducial volumes. Super-Kamiokande is operating, and Hyper-Kamiokande and Deep-TITAND are proposed.[]{data-label="tab:detectors"}
Probing the core collapse mechanism
-----------------------------------
The optical signals of supposed core-collapse supernovae show great diversity [@Zwicky(1940); @Filippenko:1997ub], presumably reflecting the wide range of masses and other properties of the massive progenitor stars. In contrast, the neutrino signals, which depend on the formation of a $\sim 1.4 M_\odot$ neutron star, are presumed to be much more uniform. However, since we have only observed neutrinos from SN 1987A, it remains to be tested whether all core-collapse supernovae do indeed have comparable neutrino emission. The total energy emitted in neutrinos is $\simeq 3 G M^2 / 5 R$, and some variation is expected in the mass $M$ and radius $R$ of the neutron star that is formed, though proportionally much less than in the progenitor stars.
With at least $\sim 1$ nearby supernova per year, a wide variety of supernovae can be probed, including less common types. For example, the observational Types Ib and Ic are now believed to be powered by core collapse, despite their original spectroscopic classification that defined them as related to Type Ia supernovae, which are thought to be powered by a thermonuclear runaway without significant neutrino emission. While each of the Types Ib/Ic and Ia are only several times less frequent than Type II, some of each should occur nearby within a reasonable time, so that the commonality of the Type II/Ib/Ic explosion mechanism can be tested.
While the nature of the explosion in the above supernova types is very likely as expected, there are other bright transients observed for which the basic mechanism is much more controversial. For these events, the detection or non-detection of neutrinos could decisively settle debates that are hard to resolve with only optical data. One type of so-called “supernova impostor” is thought to be the outburst of a Luminous Blue Variable (LBV) [@Humphreys], which seem to require a stellar mass of $M_* \gtrsim 20\,M_\odot$. Since this type of outburst affects only the outer layers, with the star remaining afterward, there should be no detectable neutrino emission.
There are several recent examples in nearby galaxies where neutrino observations could have been conclusive, including the likely LBV outburst SN 2002kg in NGC 2403 [@VanDyk:2006ay]. SN 2008S in NGC 6946 [@Prieto:2008bw] and a mysterious optical transient in NGC 300 [@Thompson08] warrant further discussion for another reason. In neither case was a progenitor seen in deep, pre-explosion optical images; however, both were revealed as relatively low-mass stars ($M_* \sim 10\,M_\odot$) by mid-infrared observations made years before the explosions. This suggests that they were obscured by dust expelled from their envelopes, a possible signature of stars dying with cores composed of O-Ne-Mg instead of iron [@Prieto:2008bw; @Thompson08]. As we will address in detail later, these events were sufficiently near for a 5 Mton detector to have identified them as authentic supernovae or impostors.
Measuring the total core collapse rate
--------------------------------------
In the previous subsection, we implicitly considered supernovae for which the optical display was seen. However, as we will calculate, the detection of $\ge 3$ neutrinos is sufficient to establish that a core collapse occurred, including those events not later visible to telescopes. This provides a means of measuring the total rate of true core collapses in the nearby universe. A successful supernova may be invisible simply if it is in a very dusty galaxy, of which there are examples quite nearby, such as NGC 253 and M82. These are supposed to have very high supernova rates, perhaps as frequent as one per decade each, as deduced from radio observations of the number of young supernova remnants [@Muxlow]. However, only a very few supernovae have been seen [@CBAT].
More interestingly, it remains unknown if, as in numerical models of supernova explosions, some core collapses are simply not successful at producing optical supernovae. This can occur if the outgoing shock is not sufficiently energetic to eject the envelope of the progenitor star, in which case one expects the prompt formation of a black hole with very little optical emission [@Heger:2002by]. Indirect evidence for such events follows from a deficit of high-mass supernova progenitors compared to expectations from theory [@Kochanek:2008mp; @Smartt:2008zd], as well as from the existence of black holes recently discovered to have $M_{\rm BH}\gtrsim 15\,M_\odot$ [@Orosz:2007ng]. One way to probe this exotic outcome would be to simply watch the star disappear [@Kochanek:2008mp]. However, a detectable burst of neutrinos should be emitted before the black hole forms (and typically, if the duration of the emission is shorter, the luminosity is higher) [@Burrows:1988ba; @Beacom:2000qy; @Sumiyoshi:2008zw]. Taken together, these would be a dramatic and irrefutable signal of an otherwise invisible event. While the rate of prompt black hole formation probably cannot exceed the visible supernova rate without violating constraints on the DSNB, reasonable estimates indicate that up to $\gtrsim 20\%$ of core collapses may have this fate [@Kochanek:2008mp].
Testing the neutrino signal
---------------------------
By measuring neutrinos from many supernovae, the deduced energy spectra and time profiles could be compared to each other and to theory. In most cases, only several events would be detected, but this is enough to be useful. The highest neutrino energies range up to $\simeq 50$ MeV. The thermal nature of the neutrino spectrum makes it relatively narrow, and since it is falling exponentially at high energies, even a small number of events can help determine the temperature. Recall that for SN 1987A, the Kamiokande-II and IMB detectors collected only $\sim 10$ events each [@Hirata:1987hu; @Bionta:1987qt], but that this data strongly restricts the details of the collapsed core.
The time profile is thought to rise quickly, over perhaps at most 0.1 s, and then decline over several seconds, as seen for SN 1987A. The neutrino events collected would most likely be at the early peak of the emission, and hence the most relevant for the question of whether heating by the emergent neutrino flux is adequate for shock revival [@Bethe:1984ux; @Thompson:2002mw; @Murphy:2008dw] or whether $\nu$-$\nu$ many-body effects are important [@nu-nu].
Over time, as many supernovae are detected, the average energy spectrum and time profile will be built up. (For the time profile, there will be some uncertainties in the start times.) If there are large variations from one supernova to the next, then these average quantities will ultimately provide a more useful template for comparison than the theoretical results that must be used at present. If there is no evidence for significant variations between supernovae, then the accumulated data will be equivalent to having detected one supernova with many events. It is quite likely that such a detector would observe a supernova in one of the Milky Way, M31, or M33; the high-statistics yield from these would also provide a point of comparison. Taken together, all of these data will provide new and exacting tests of how supernovae work.
With enough accumulated events, it is expected that neutrino reactions besides the dominant inverse beta decay process will be present in the data. One oddity still remaining from SN 1987A is that the first event in Kamiokande-II seems to be due to $\nu_e + e^-\rightarrow \nu_e + e^-$ scattering and points back to the supernova [@Hirata:1988ad], which is improbable based upon standard expectations [@Beacom:2006]. This can be tested, however, and if it turns about to be ubiquitous, could be exploited in determining the directionality of the larger future bursts without optical signals, as the inverse beta decay signal is not directional [@Vogel:1999zy].
Since Earth is transparent to supernova neutrinos, the whole sky can be monitored at once. For neutrinos that pass through Earth, particularly those which cross the core, matter-enhanced neutrino mixing can significantly affect the spectrum relative to those which do not [@earth]. Dividing the accumulated spectra appropriately based on optical detections, this would allow a new test of neutrino mixing, sensitive to the sense of the neutrino mass hierarchy. Detecting neutrinos from distant sources would also allow tests of neutrino decay [@decay], the equivalence principle [@equivalence], and other exotic possibilities [@exotic].
Revealing other transient signals
---------------------------------
Detection of a neutrino burst means detection of the instant of core collapse, with a precision of $\sim 1$ second determined by the sampling of the peak of the $\simeq 10$ second time profile. This would provide a much smaller time window in which to search for gravitational wave signals [@LIGO; @GW] from core-collapse supernovae; otherwise, one must rely on the optical signal of the supernova, which might optimistically be determined to a day ($\sim 10^5$ seconds). This is important, since the gravitational-wave signal remains quite uncertain, making searches more difficult. Knowing the instant of core collapse would also be useful for searches for high-energy neutrinos from possible choked jets that do not reach the surface of the star [@Jets], where again the timing information can be used to reduce backgrounds and improve sensitivity.
Once core collapse occurs, the outward appearance of the star initially remains unchanged. Knowing that a signal was imminent would give unprecedented advanced warning that photons should soon be on the way, allowing searches to commence for the elusive UV/X-ray signal of supernova shock breakout [@SBO] and also the early supernova light curve. Those signals are expected to emerge within hours and days, respectively. While the neutrino signal is likely not directional, the number of events detected will provide constraints for triggered searches [@Kistler].
Finally, it is possible that such large detector would find not only core-collapse supernovae in nearby galaxies, but also other types of transients that are presently unknown. In the Milky Way, there would be sensitivity to any transient with a supernova-like neutrino signal, as long as its overall strength is at least $\sim 10^{-6}$ as large as that for a supernova. To be detectable, the key requirement is a $\gtrsim 15$ MeV $\bar{\nu}_e$ component.
![Estimates of the core-collapse supernova rate in the nearby universe, based on that expected from the optical luminosities of known galaxies (line) and supernovae observed within the last decade (bins). Note that SN 2002kg is a likely LBV outburst, while SN 2008S and the NGC 300 transient are of unusual origin. These estimates are all likely to be incomplete.[]{data-label="fig:snrates"}](snrates){width="3.25in"}
Nearby Supernova Rate {#rate}
=====================
Over the past decade, there has been rapid growth in the level of interest among astronomers in measuring the properties of core-collapse supernovae. There is also a renewed interest in completely characterizing the galaxies in the nearby universe, within 10 Mpc. In nearby galaxies, both amateurs and automated surveys (e.g., KAIT [@Li:1999sd]) are finding many supernovae. For these, archival searches have revealed pre-explosion images of about a dozen supernova progenitor stars, allowing a better understanding of which types of massive stars lead to which kinds of core-collapse supernovae (e.g., [@Prieto:2008bw; @Smartt:2008zd; @Li07; @GalYam:2006iy]).
Figure \[fig:snrates\] shows the expected rate of core-collapse supernovae in the nearby universe (dashed line) calculated using the galaxy catalog of Ref. [@Karachentsev:2004dx] (designed to be $\sim$70–80% complete up to 8 Mpc), with a conversion from $B$-band optical luminosity to supernova rate from Ref. [@Cappellaro:1999qy]. The effects of clustering and of incompleteness at large distances can clearly be seen, since the histogram would rise as the distance squared for a smooth universe of identical galaxies. Ultimately, a more accurate result could be obtained by combining the information from star-formation rate measurements in the ultraviolet [@Salim:2004dg], H$\alpha$ [@Halpha], and infrared [@Kennicutt:2003dc], likely leading to a larger prediction for the supernova rates.
Also displayed in Fig. \[fig:snrates\] is the rate deduced from supernovae discovered in this volume in the last 10 years [@CBAT], with distances primarily from Ref. [@Karachentsev:2004dx] (when available; otherwise from [@WEBdist]). While the observed rate is already $\sim 2$ times larger than the above calculation, even this estimate is likely incomplete, as supernova surveys under-sample small galaxies and the Southern hemisphere. As previously mentioned, supernovae with little or no optical signal, e.g., due to direct black hole formation or dust obscuration, would also have been missed. This is particularly important for nearby dusty starburst galaxies with large expected, but low observed, supernova rates, like NGC 253 and M82.
Distance measurements of nearby galaxies also stand to be improved. For example, at the largest distances, SN 1999em, SN 1999ev, SN 2002bu, and SN 2007gr are probably not all truly within 10 Mpc, as some distance estimates put them outside. We emphasize that their inclusion or not does not affect our approximate supernova rates, and barely matters for the neutrino bursts of sufficient multiplicity, which are dominantly from closer supernovae. It would be very helpful to refine distance measurements, not just for star formation/supernova rate estimates, but also to determine the absolute neutrino luminosities once a supernova has been detected.
Overall, there is a strong case that the core-collapse supernova rate within $\sim $ 6 (10) Mpc is at least 1 (2) per year. This can be compared to the estimated Milky Way rate of $2 \pm 1$ per century (see Ref. [@Diehl:2006cf] and references therein), with Poisson probabilities ultimately determining the odds of occurrence, as shown in Fig. \[fig:forecast\].
![Probabilities for one or more supernovae in the Milky Way over time spans relevant for the lifetimes of large neutrino detectors, depending on the assumed supernova rate.[]{data-label="fig:forecast"}](forecast){width="3.25in"}
Neutrino Burst Detection {#detection}
========================
A goal of measuring supernova neutrino “mini-bursts” from galaxies at a few Mpc necessitates a large detector, roughly $\sim$100 times the size of SK. We focus on the Deep-TITAND proposal for a 5 Mton (fiducial volume) enclosed water-Čerenkov detector [@Suzuki:2001rb; @Suzuki]. The detector would be constructed in modules sized by Čerenkov light transparency and engineering requirements. We assume a photomultiplier coverage of $20\%$, similar to that of SK-II (half that of the original SK-I and the rebuilt SK-III). As in SK, the detection efficiency at the energies considered here would be nearly unity.
The backgrounds present in deep detectors have been well-characterized by SK and other experiments. Deep-TITAND is proposed to be at a relatively shallow depth of 1000 meters of seawater, which would increase the downgoing cosmic ray muon rate per unit area by a factor $\simeq 100$ compared to SK, which is at a depth of 2700 meters water equivalent. A nearly perfect efficiency for identifying cosmic ray muons in the outer veto or the detector itself is required. This was achieved in SK, where the only untagged muons decaying in the detector were those produced inside by atmospheric neutrinos [@Malek:2002ns]. Simple cylinder cuts around cosmic ray muon tracks would veto all subsequent muon decays while introducing only a negligible detector deadtime fraction.
Low-energy backgrounds include natural radioactivities, solar neutrinos, photomultiplier noise, and beta decays from nuclei produced following spallation by cosmic ray muons. Of these, only the last is depth-dependent, and this would be much larger than in SK (a factor $\simeq 30$ for the higher muon rate per area but lower muon average energy, and a factor $\simeq 30$ for the larger detector area). The high muon rate means that it would not be possible to use the cylinder cuts employed in SK to reduce spallation beta decays without saturating the deadtime fraction (note that these beta decays have lifetimes more than $10^6$ times longer than the muon lifetime). At low energies, the above background rates are large, but the spectrum falls steeply with increasing energy, essentially truncating near 18 MeV [@Malek:2002ns; @Ikeda:2007sa].
This allows for a significant simplification and reduction in the background rate by considering only events with a reconstructed energy greater than 18 MeV (a neutrino energy of 19.3 MeV). Which events to reconstruct would be determined by a simple cut on the number of hit photomultipliers, just as in SK, but with a higher threshold. The backgrounds above this cut are due to atmospheric neutrinos, and thus the rates scale with the detector volume but are independent of depth. The dominant background contribution is from the decays of non-relativistic muons produced by atmospheric neutrinos in the detector, i.e., the so-called invisible muons. The background rate in 18–60 MeV in SK is about 0.2 events/day, of which the energy-resolution smeared tail of the low-energy background is only a minor component [@Malek:2002ns; @Ikeda:2007sa].
Scaling this rate to a 5 Mton detector mass ($\sim 5 \times 10^{-4}$ s$^{-1}$) and considering an analysis window of 10 sec duration (comparable to the SN 1987A neutrino signal) allows calculation of the rate of accidental coincidences [@Ikeda:2007sa]. For $N = 3$ events, this corresponds to about only once every five years, and when it does, examination of the energy and timing of the events will allow further discrimination between signal and background (a subsequent optical supernova would confirm a signal, of course). For $N \ge 4$, accidental coincidences are exceedingly rare ($\sim\,$1 per 3000 years), therefore we require at least $N = 3$ signal events to claim detection of a supernova (a somewhat greater requirement than in Ref. [@Ando:2005ka], where a smaller detector was assumed). Since the backgrounds observed by SK in this energy range are from atmospheric neutrinos, we expect no correlated clusters of background events.
To estimate detection prospects, for the $\bar{\nu}_e$ flavor we assume a Fermi-Dirac spectrum with an average energy of 15 MeV and a total energy of $5\times 10^{52}$ erg. The dominant interaction for the neutrino signal is inverse-beta decay, $\bar{\nu}_e + p\rightarrow n + e^+$, where $E_{e^+} \simeq E_{\bar{\nu}_e} - 1.3$ MeV and the positron direction is nearly isotropic [@Vogel:1999zy]. Combining the emission spectrum, cross section, and number of free target protons in 5 Mton of water, we find that the average number of neutrino events (for $E_{e^+} > 18$ MeV) from a burst at distance $D$ is $$\mu (D; E_{e^+} > 18 ~ \mathrm{MeV}) \simeq 5 \left(\frac{D}{3.9 ~ \mathrm{Mpc}}\right)^{-2}.$$ This is the key normalization for the supernova signal. In Table \[tab:yields\], we list recent nearby supernovae within 6 Mpc, with type, host galaxy name, distance, and the expected neutrino yields $\mu$ in a 5 Mton detector. As can be seen in Fig. 3 of Ref. [@Ando:2005ka], our $E_{e^+} > 18$ MeV threshold still allows us to detect $\sim 70\%$ of the total supernova signal.
The probability to detect $\geq N$ neutrino events from a given core collapse is then $$P( \geq N; D) = \sum_{n = N}^{\infty} P_n [\mu (D)] = \sum_{n=N}^\infty \frac{\mu^n(D)}{n!}e^{-\mu (D)},$$ where $P_n(\mu)$ represents the Poisson probability. $P(\geq N; D)$ is shown in Fig. \[fig:yields\] as a function of $D$ for several values of $N$. From this figure, we see, for example, that from a 4 Mpc supernova, we have an excellent chance ($\gtrsim 90\%$) to get more than 3 neutrino events. For 8 Mpc, like those shown in Fig. \[fig:snrates\], there is still a $\lesssim 10\%$ chance to get $\ge 3$ events.
SN Type Host D \[Mpc\] $\nu$ events
-------- --------- ---------------- ----------- --------------
2002hh II-P NGC 6946 5.6 2.4
2002kg IIn/LBV NGC 2403 3.3 6.8
2004am II-P NGC 3034 (M82) 3.53 5.9
2004dj II-P NGC 2403 3.3 6.8
2004et II-P NGC 6946 5.6 2.4
2005af II-P NGC 4945 3.6 5.7
2008S IIn NGC 6946 5.6 2.4
2008bk II-P NGC 7793 3.91 4.8
2008? II? NGC 300 2.15 16.0
: Recent core-collapse supernova candidates within 6 Mpc, with their expected neutrino event yields ($E_{e^+} > 18$ MeV) in a 5 Mton detector.[]{data-label="tab:yields"}
For a particular supernova rate, $R_{{\rm SN},i}$, we can get the expected total rate of $N$-tuplet detections from distances $D_i$ as $$R_{N,{\rm burst}} = \sum_i R_{{\rm SN},i} P_N [\mu (D_i)].$$ In Fig. \[fig:multiplicities\], we show this as an annual rate, $R_{N,{\rm burst}}$, plotted versus $N$. For the supernova rate $R_{{\rm SN},i}$, we have adopted three different models: (i) all supernova candidates shown in Fig. \[fig:snrates\] (20 in total); (ii) same as (i), except excluding SN 2002kg, SN 2008S, and the NGC 300 transient as exceptional events (17 in total); the catalog-based rate estimate (line in Fig. \[fig:snrates\]). As the detection criterion is $N \geq 3$, the annual rate of detectable mini-bursts is obtained by summing $R_{N,{\rm burst}}$ for $N \ge 3$, which yields 0.8, 0.6, and 0.4 supernovae per year, for supernova rate models (i), (ii), and (iii), respectively. Adding supernovae from beyond 10 Mpc would not change the rate of $N \ge 3$ multiplets, only increasing the number of unremarkable lower-$N$ multiplets (which, as shown, are already dominated by supernovae in the 8–10 Mpc range).
The total neutrino event counts, $N_{\rm total}$, can be obtained from $R_{N,{\rm burst}}$ by $$N_{\rm total} = \sum_{N = 3}^{\infty} N R_{N,{\rm burst}},$$ which are 48, 31, and 22 per decade, for rate estimates (i), (ii), and (iii), respectively. Since each burst is triggered with $E_{e^+} > 18$ MeV events, one would also look for somewhat lower-energy events in the same time window, potentially raising the total yield by $\simeq 20\%$.
![Frequency of neutrino mini-bursts expected with a 5 Mton detector. The bins with $N = 3$ or more can be used for burst detection because the background rate is small enough. Three different estimates of the supernova rate are shown, as labeled.[]{data-label="fig:multiplicities"}](multiplicities){width="3.25in"}
Conclusions
===========
The $\sim 10$ neutrino events associated with SN 1987A in each of the Kamiokande-II and IMB detectors [@Hirata:1987hu; @Bionta:1987qt] were the first and, thus far, only detection of neutrinos from a supernova. This detection showed that we can learn a great deal even from a small number of events, and revealed that an immense amount of energy is released in the form of neutrinos ($> 10^{53}$ erg) during a core collapse. Measuring “mini-bursts” of neutrino events from multiple supernovae would allow for the study of the core-collapse mechanism of a diverse range of stellar deaths, including optically-dark bursts that appear to be relatively common [@Kochanek:2008mp; @Smartt:2008zd].
This would be made possible by a $\sim\,$5 Mton scale water Čerenkov detector [@Suzuki:2001rb; @Suzuki], which has the special advantages of being able to trigger on supernovae using neutrinos alone, and to guarantee detection if neutrinos are produced with the expected flux. Moreover, for burst detection, a relatively-high low-energy background rate can be tolerated, significantly decreasing the required detector depth, so that construction could be relatively quick and inexpensive. Our conservative estimates shows that the occurrence rate of mini-bursts that give $\ge 3$ neutrino events is likely $\sim$1 yr$^{-1}$ or higher.
In conclusion, we wish to reiterate that, even if a supernova occurs in the Milky Way tomorrow, the important problems discussed in Section \[prospects\] will remain unresolved, and can only be addressed by a suitable “census” of core collapses in the nearby universe. The possibilities mentioned here almost certainly do not exhaust the scientific potential of such an instrument. As is now almost commonplace in the business of observing supernovae with photons, it would be surprising [*not*]{} to find new and unexpected phenomena.
[**Acknowledgments:**]{} We thank Shunsaku Horiuchi, Chris Kochanek, Y. Ohbayashi, José Prieto, Stephen Smartt, Michael Smy, Kris Stanek, Todd Thompson, and Mark Vagins for helpful discussions. This work was supported by Department of Energy grant DE-FG02-91ER40690 (MDK); National Science Foundation CAREER grant PHY-0547102 to JFB (HY and JFB); and by the Sherman Fairchild Foundation at Caltech (SA).
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|
---
author:
- 'Xue-Jian Jiang'
- 'Jun-Zhi Wang'
- Yu Gao
- 'Qiu-Sheng Gu'
bibliography:
- 'jiang\_hc3n.bib'
date: 'Received 2016 month day; accepted 2016 month day'
title: 'HC$_3$N Observations of Nearby Galaxies$^*$ [^1] '
---
[We aim to systematically study the properties of the different transitions of the dense molecular gas tracer [HC$_3$N]{} in galaxies.]{} [We have conducted single-dish observations of [HC$_3$N]{} emission lines towards a sample of nearby gas-rich galaxies. [HC$_3$N]{}($J$=2-1) was observed in 20 galaxies with Effelsberg 100-m telescope. [HC$_3$N]{}($J$=24-23) was observed in nine galaxies with the 10-m Submillimeter Telescope (SMT).]{} [[HC$_3$N]{} 2-1 is detected in three galaxies: , and ($> 3\,\sigma$). [HC$_3$N]{} 24-23 is detected in three galaxies: , and . This is the first measurements of [HC$_3$N]{} 2-1 in a relatively large sample of external galaxies, although the detection rate is low. For the [HC$_3$N]{} 2-1 non-detections, upper limits (2$\sigma$) are derived for each galaxy, and stacking the non-detections is attempted to recover the weak signal of [HC$_3$N]{}. But the stacked spectrum does not show any significant signs of [HC$_3$N]{} 2-1 emission. The results are also compared with other transitions of [HC$_3$N]{} observed in galaxies. ]{} [The low detection rate of both transitions suggests low abundance of [HC$_3$N]{} in galaxies, which is consistent with other observational studies. The comparison between [HC$_3$N]{} and HCN or [HCO$^+$]{}shows a large diversity in the ratios between [HC$_3$N]{} and HCN or [HCO$^+$]{}. More observations are needed to interpret the behavior of [HC$_3$N]{} in different types of galaxies.]{}
Introduction {#sect:intro}
============
Molecular lines play an essential role in our understanding of star formation activity and galaxy evolution. With molecular lines of different species and their different transitions, not only the chemical composition of the interstellar medium can be investigated, but other important physical parameters, such as temperature, pressure, density, and non-collisional pumping mechanism can be derived as well (e.g., @Henkel1991; @Evans:1999; @Fukui:2010; @Meier:2012; @Meier:2014). New facilities providing wide band and highly sensitive instruments are making weak line surveys and multi-species analysis feasible, and the detections and measurements of a variety of species, are helping us to reveal the gas components of galaxies, and how their abundances, densities, ratios reflect the radiative properties of galaxies. Multi-species, multi-transition molecular lines can be combined to diagnose the evolution stage of galaxies [@Baan:2014], because different species are sensitive to different physical environments, such as photo dissociation regions (PDRs) dominated by young massive stars, X-rays dominated regions (XDRs) induced by active galactic nucleus (AGNs), and shock waves by cloud-cloud collisions [@Aladro2011; @Greve2009; @Aladro2011; @Costagliola2011; @Viti2014].
One of the interstellar species that benefits from the upgraded facilities is cyanoacetylene ([HC$_3$N]{}). [HC$_3$N]{} was firstly detected in 1971 at 9.0977 GHz ($J$=2-1) in the Galactic star forming region Sgr B2 [@Turner1971]. The critical density of [HC$_3$N]{} is comparable to the widely-used dense gas tracer HCN and can also trace dense molecular gas around star forming sites. [HC$_3$N]{} has been detected in many star formation regions in the Milky Way with several transitions from centimeter to sub-millimeter (e.g., @Suzuki1992). Due to the small rotational constant ($\sim$ 1/13 of CO), there are many closely spaced rotational transitions of [HC$_3$N]{} (separated by only 9.1 GHz) at centimeter and millimeter wavelengths, and its levels are very sensitive to the changes in excitation [@Meier:2012]. This makes it easier to conduct multi-transition observations of [HC$_3$N]{} lines than other dense molecular gas tracers, and can help better understand the excitation conditions of star forming regions. In contrast, the high-$J$ lines of other dense molecular gas tracers such as HCN and HCO$^+$ are at very high frequencies, thus it is difficult to observe them with ground-based telescopes. Another advantage of using [HC$_3$N]{} lines is that [HC$_3$N]{} is very likely optical thin even in low-$J$ transitions, due to the relatively low abundance (@Irvine:1987; @Lindberg2011). And low opacity is important for accurate estimate of dense molecular gas mass for the study of relationship between dense molecular gas and star formation [@Gao2004a; @Gao2004b; @Wang2011; @Zhang:2014].
There have been efforts to detect [HC$_3$N]{} in nearby galaxies, mainly in millimeter band. Observations suggest that [HC$_3$N]{} is related to the warm, dense star forming gas, and is easily dissociated by UV radiation (@Henkel1988; @Costagliola2011; @Costagliola2011; @Lindberg2011; @Aladro2011, @Aladro2015). [HC$_3$N]{} was found to be unusually luminous in , and it is attributed to its high abundance (10$^{-7}$) as well as the intense radiation field in the dense and warm gas at the center of [@Aalto2007; @Costagliola2010]. @Meier2005 [@Meier:2012; @Meier:2011; @Meier:2014] presented high resolution observations of [HC$_3$N]{} ($J$=5-4, 10-9, 12-11 and 16-15) of a few very nearby galaxies, and gave detailed analysis of the galactic structures and morphology that traced by [HC$_3$N]{} and other dense gas tracers (HNC, HCN, CS, etc.). However, these results are still limited by their sample size, and the chemical process of [HC$_3$N]{} is still unclear. Larger samples are still necessary for analyzing the properties of [HC$_3$N]{} and how it relate to other galactic parameters. In this paper we present the first systematic survey of [HC$_3$N]{} ($J=2-1$) and [HC$_3$N]{} ($J=24-23$) in a relatively large sample of nearby galaxies. And the results are compared with the observations of [HC$_3$N]{} in other transitions. The critical densities of [HC$_3$N]{} $J=2-1$ and [HC$_3$N]{} $J=24-23$ are about $3\times 10^3\ \rm cm^{-3}$ and $4 \times 10^6\ \rm cm^{-3}$, respectively. And the upper state energies ($E_u$) of the two transitions are 1.3 K and 131 K, respectively [@Costagliola2010].
[lccrrrrl]{} Galaxy & & $V_{\rm Helio}$ & Distance & log $L_{\rm IR}$ & $\Delta V_{\rm CO}$ & Telescope\
& h m s & & (km s$^{-1}$) & (Mpc) & ($L_\odot$) & (km s$^{-1}$)\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8)\
NGC 520 & 01 24 34.9 & $+$03 47 30.0 & 2281 & 30.22 & 10.91 & 270 & Effelsberg\
NGC 660 & 01 43 02.4 & $+$13 38 42.0 & 850 & 12.33 & 10.49 & 280 & Effelsberg\
NGC 891 & 02 22 33.4 & $+$42 20 57.0 & 528 & 8.57 & 10.27 & 110 & Effelsberg\
NGC 972 & 02 34 13.4 & $+$29 18 41.0 & 1543 & 20.65 & 10.67 & 220 & Effelsberg\
NGC 1068 & 02 42 41.4 & $-$00 00 45.0 & 1137 & 13.7 & 11.27 & 280 & Effelsberg & SMT\
IC 342 & 03 46 48.5 & $+$68 05 46.0 & 31 & 4.60 & 10.17 & 72.8$^a$ & Effelsberg & SMT\
UGC 2855 & 03 48 20.7 & $+$70 07 58.0 & 1200 & 19.46 & 10.75 &…& Effelsberg\
UGC 2866 & 03 50 14.9 & $+$70 05 40.9 & 1232 & 20.06 & 10.68 &…& Effelsberg\
NGC 1569 & 04 30 49.0 & $+$64 50 53.0 &$-$104& 4.60 & 9.49 & 90 & Effelsberg\
NGC 2146 & 06 18 39.8 & $+$78 21 25.0 & 882 & 16.47 & 11.07 & 320 & Effelsberg & SMT\
NGC 2403 & 07 36 51.3 & $+$65 36 29.9 & 161 & 3.22 & 9.19 & 90 & Effelsberg\
M 82 & 09 55 53.1 & $+$69 40 41.0 & 187 & 3.63 & 10.77 & 150 & Effelsberg & SMT\
NGC 3079 & 10 01 57.8 & $+$55 40 47.0 & 1116 & 18.19 & 10.73 & 380 & Effelsberg\
NGC 3310 & 10 38 45.9 & $+$53 30 12.0 & 993 & 19.81 & 10.61 & 140 & Effelsberg\
M 66 & 11 20 15.0 & $+$12 59 30.0 & 727 & 10.04 & 10.38 & 180 & Effelsberg\
IC 694 & 11 28 33.8 & $+$58 33 45.0 & 3120 & 47.74 & 11.63 & 250 & Effelsberg$^b$ & SMT\
NGC 3690 & 11 28 30.8 & $+$58 33 43.0 & 3120 & 47.74 & 11.32 & 260 & Effelsberg$^b$ & SMT\
Mrk 231 & 12 56 14.2 & $+$56 52 25.0 &12600 & 171.84& 12.51 & 167 & Effelsberg\
Arp 220 & 15 34 57.1 & $+$23 30 10.0 & 5352 & 79.90 & 12.21 & 360 & Effelsberg & SMT\
NGC 6240 & 16 52 58.9 & $+$02 24 03.0 & 7160 & 103.86& 11.85 & 420 & SMT\
NGC 6946 & 20 34 52.6 & $+$60 09 12.0 & 53 & 5.32 & 10.16 & 130 & Effelsberg & SMT\
\[Tab:source\]
Observations & Data reduction {#sect:Obs}
=============================
We select nearby infrared bright galaxies [@Sanders:2003] with IRAS 60 $\mu$m flux greater than 30 Jy and declination greater than $-21\degr$ to do this survey. It is not a complete but representative sample of infrared bright galaxies. The sample consists of 21 galaxies. Note that due to the different beam size of the two telescopes we used, the merger Arp 299 ( and ) were observed as a single pointing by Effelsberg 100-m, while the two galaxies were observed separately by the SMT 10-m.
[HC$_3$N]{} 2-1 observations with the Effelsberg 100-m {#sect:obs_21}
------------------------------------------------------
[HC$_3$N]{} ($J=2-1$) ($\nu_{\rm rest} =$ 18.196 GHz) of 20 galaxies was observed with Effelsberg 100-m telescope in 2010. The Half Power Beam Width (HPBW) is 46.5$''$ at 18 GHz for the 100-m telescope. We used the 1.9 cm band receiver, 500 MHz bandwidth with 16384 channels correlator setup, which provided $\sim$ 8300 [km s$^{-1}$]{} velocity coverage and $\sim$ 0.5 [km s$^{-1}$]{} velocity resolution during the observations. Position-switching mode with beam-throws of about $\pm
2'$ was used. Pointing and focus were checked about every two hours. The typical system temperature of the Effelsberg observations was about 46 K. The on-source time for each galaxy is about 14–47 minutes. The weather during the observations is not ideal, and the baselines of many sources are affected and induced artificial features which are hard to remove.
[HC$_3$N]{} 24-23 observations with the SMT 10-m {#sect:obs_2423}
------------------------------------------------
[HC$_3$N]{} ($J=24-23$) ($\nu_{\rm rest} =$ 218.324 GHz) of nine galaxies was observed in 2009 with the SMT 10-m telescope. The HPBW is about 33$''$ at $\sim$218 GHz for SMT, and a single pointing was used for each galaxy toward their central positions. We used the ALMA Sideband Separating Receiver and the Acousto-Optical-Spectrometers (AOS), which have dual polarization, 970 MHz ($\sim$ 1300 [km s$^{-1}$]{}) bandwidth and 934kHz channel spacing. Observations were carried out with the beam-switching mode with a chop throw of $2'$ in azimuth (AZ) and a chopping frequency of 2.2 Hz. Pointing and focus were checked about every two hours by measuring nearby QSOs with strong millimeter continuum emission. The typical system temperature at 218 GHz was less than 300 K, and the on-source time for each galaxy was $\sim$ 60–168 minutes.
Data Reduction {#sect:reduction}
--------------
The basic parameters of our sample galaxies are listed in Table\[Tab:source\]. The data were reduced with the CLASS program of the GILDAS[^2] package. First, we checked each spectrum and discarded the spectra with unstable baseline. Most of the Effelsberg spectra do not have flat baselines, but over several hundred [km s$^{-1}$]{}near the line the baselines can still be fixed. In the SMT spectra the image signal of strong CO 2-1 in the upper sideband affect the baseline of the lower sideband and for and the [HC$_3$N]{} 24-23 is contaminated. But for other galaxies the image CO line does not affect the [HC$_3$N]{} line. Then we combined spectra with both polarizations of the same source into one spectrum. Depending on the quality of the spectral baselines, a first-order or second-order fitting was used to subtract baselines from all averaged spectra. The identifications of the transition frequencies of [HC$_3$N]{} have made use of the NIST database [*Recommended Rest Frequencies for Observed Interstellar Molecular Microwave Transitions*]{}[^3].
To reduce the noise level, the spectra are smoothed to velocity resolutions $\sim 20-40$ [km s$^{-1}$]{}. The velocity-integrated intensities of the [HC$_3$N]{} line are derived from the Gaussian fit of the spectra, or integrated over a defined window if the line profiles significantly deviate from a Gaussian. The intensities are calculated using $ I = \int T_{\rm mb}{\rm d}v$, where $T_{\rm mb}$ is the main beam brightness temperature. Molecular line intensity in antenna temperature ($T_A^{*}$) is converted to main beam temperature $T_{\rm mb}$ via $T_{\rm mb} = T_A^{*} /{\rm MBE}$, with the main beam efficiency MBE = 53% at 18 GHz for Effelsberg telescope, and 70% at 218 GHz for SMT during the observations. The flux density is then derived from $T_{\rm mb}$, using $S/T_{\rm mb}$ = 0.59 Jy/K for the Effelsberg telescope, and 24.6 Jy/K for the SMT.
Results & Discussion {#sect:results}
====================
The spectral measurements and estimated intensities of the [HC$_3$N]{} lines, including RMS noise and on-source time, are listed in Table\[Tab:results\] ([HC$_3$N]{}2-1) and Table\[Tab:results2\] ([HC$_3$N]{} 24-23).
[HC$_3$N]{} 2-1 {#sect:hcccn21}
---------------
Among the 20 galaxies observed by Effelsberg 100-m telescope, [HC$_3$N]{} 2-1 is detected in three galaxies: , and (See Figure\[Fig:2\_1\]). This is the first report of [HC$_3$N]{} 2-1 detections in external galaxies, although limited by the SNR (Signal to Noise Ratio) the detection rate is low.
#### **:
IC 342 has the strongest peak intensity ($T_{\rm mb}\sim$ 14 mK) of [HC$_3$N]{} 2-1 in the sample, which is about twice the strength as the [HC$_3$N]{}(9-8) line of IC 342 detected by IRAM 30-m telescope [@Aladro2011], while the line width (FWHM $\sim$ 60 [km s$^{-1}$]{}) is similar to their result.
#### **:
The detected [HC$_3$N]{} 2-1 in has a similar linewidth (FWHM $\sim$ 294.7 [km s$^{-1}$]{}) to CO 1-0 ($\sim$ 280 [km s$^{-1}$]{}). While the [HC$_3$N]{} survey by @Lindberg2011 did not observe , its 10-9 and 12-11 transitions were not detected by @Costagliola2011. This difference in the detection of [HC$_3$N]{} lines may imply that there is little warm and dense gas content in , thus the high-$J$ [HC$_3$N]{} lines can not be excited.
#### **:
In [HC$_3$N]{} 2-1 is only detected on about 2 $\sigma$ level, but this is the first tentative detection of [HC$_3$N]{} in . It was not observed by @Costagliola2011 nor @Lindberg2011.
#### *non-detections*:
Due to the poor quality (and probably insufficient integration time) of the [HC$_3$N]{} 2-1 data, 16 out of 19 galaxies were not detected. Assuming their linewidth is approximate to CO 1-0 linewidth (FWHM, from @Young1995), we derive upper limits of the integrated intensity for each galaxy (2 $\sigma$, where $\sigma =$ RMS $\sqrt{\delta V
\cdot \Delta V}$) and show them in Table\[Tab:results\]. Note that the linewidth of [HC$_3$N]{} is likely narrower than that of CO, and such assumption might overeistimate the upper limits of integrated intensity, thus is only a rough estimate. The upper limits are in the range of $\sim$ 0.3 – 1.2 K [km s$^{-1}$]{}. For those non-detection galaxies, We also stack their spectra together, weighted by the RMS level of each galaxy, to exam if a cumulated signal can be obtained (see Figurer \[Fig:2\_1\]). Although the RMS of the the stacking [HC$_3$N]{} 2-1 spectrum is reduced down to 0.66 mK, we do not see any signs of emission (at a resolution of 30 [km s$^{-1}$]{}). Since these galaxies have similar linewidth (100–400 [km s$^{-1}$]{}), we can estimate the stacked upper limit assuming a linewidth of 200 [km s$^{-1}$]{}, based on the RMS (0.66 mK) of the stacked spectrum. Thus the 2 $\sigma$ upper limit of these galaxies is about 0.26 K [km s$^{-1}$]{}. To eliminate the possible effect induced by different linewidths of galaxies, we also tried to group the non-detection galaxies based on their CO linewidth. Galaxies with CO FWHM (Table \[Tab:source\]) wider than 200 [km s$^{-1}$]{}are stacked as one group, and other galaxies are stacked as another group. Neither group shows any signs of emission.
[HC$_3$N]{} 24-23
-----------------
Among the nine galaxies observed by SMT, [HC$_3$N]{}($J$=24-23) is detected in three galaxies: , and (Figure\[Fig:24\_23\]).
#### ** :
[HC$_3$N]{} 24-23 of IC 342 was previously detected and measured by [@Aladro2011], and our observation obtains consistent result, although comparing to their observation we do not detect [H$_2$CO]{}simultaneously. In our observations, IC 342 is the only galaxy detected in both 2-1 and 24-23 transitions. The line center and width of the two transitions are similar, considering observational uncertainties. This might imply that the two transitions have similar emitting area. And the ratio between the integrated intensities of [HC$_3$N]{} 24-23/[HC$_3$N]{} 2-1 is about 0.6.
#### ** :
In NGC 1068, the integrated intensity of [HC$_3$N]{} 24-23 is about 2.0 K [km s$^{-1}$]{} (in $T_{\rm mb}$), which is stronger than that of [HC$_3$N]{} 10-9 ($\sim$1.1 K [km s$^{-1}$]{}) reported by [@Costagliola2011]. It may imply that there is sufficient warm and dense gas, which is able to excite the high transition [HC$_3$N]{} 24-23 line. Besides, it could also be affected by the strong AGN signature of this galaxy [@Wang:2014; @Tsai:2012].
#### ** :
Previous observation only obtained upper limits of [HC$_3$N]{} 12-11 For IC 694 [@Lindberg2011]. In our observations, a tentative detection in IC 694 ($> 2
\sigma$) is obtained. The line profile of IC 694 obviously deviates from a Gaussian, so we derive the [HC$_3$N]{} intensity by integrating the line within a window of 400 [km s$^{-1}$]{} width (Table \[Tab:results2\]).
We note that, in NGC 1068 and IC 694, [HC$_3$N]{} 24-23 is possibly blended with H$_2$CO 3(0,3) – 2(0,2) emission ($f_\nu=$218.22219 GHz). The upper state energy of this is para-H$_2$CO line is about 10.5 K, which is likely to be excited in these cases. The H$_2$CO line is shifted by 141.1 [km s$^{-1}$]{}or -102 MHz from the [HC$_3$N]{} 24-23 line, and it is unclear that how much intensity of [HC$_3$N]{} 24-23 in NGC 1068 and IC 694 is contributed by H$_2$CO (see Figure\[Fig:24\_23\]). We still lack enough data to disentangle this issue, and can only compare with other observations. For example, in the observation of M 82 by @Ginard2015, they showed that near the frequency of 145 GHz, H$_2$CO 2(0,1) – 1(0,1) is as strong as [HC$_3$N]{} 16-15. H$_2$CO is not detected in M 82 in 3mm band [@Aladro2015]. In the observations toward NGC 4418 by @Aalto2007, they showed that [HC$_3$N]{} 16-15 is blended with H$_2$CO ,and H$_2$CO may contribute 20% of the total integrated line intensity.
#### *non-detections*:
The spectra of M 82 and Arp 220 are seriously contaminated by the image signal of CO 2-1 from the upper side-band ($\nu$ = 230 GHz), which is strong and wide hence difficult to remove. As a consequence we could not extract the spectrum of [HC$_3$N]{} properly. We treat the [HC$_3$N]{} 24-23 in M 82 and Arp 220 as non-detections, and their 2$\sigma$ upper limits are also only indicative. Although not contaminated by adjacent CO image signal, [HC$_3$N]{} 24-23 was not detected in NGC 2146, NGC 6946, NGC 3690 and NGC 6240. For these non-detection we present 2$\sigma$ upper limit of the integrated intensity of [HC$_3$N]{} 24-23 in Table\[Tab:results2\]. Only four galaxies are not contaminated by CO image signal, thus no stacking is implemented for their [HC$_3$N]{} 24-23 spectra.
 \
 \
 \
 \
 
[ccccccccc]{} Source & On-time & RMS & $\delta V$ & $\Delta V$ & $V_0$ & $I(\rm HC_3N)$ & $S(\rm HC_3N)$ & $\frac{\rm HC_3N\ 2-1}{\rm HCN\ 1-0}$\
& (min) & (mK) & (km s$^{-1}$) &(km s$^{-1}$) & (km s$^{-1}$) & (K km s$^{-1}$) & (Jy km s$^{-1}$)\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9)\
NGC 660 & 28 & 4.2 & 32.2 & 260.7 (90.6) & 825 (36)& 1.47 (0.40) & 0.86 (0.24) & $\sim$ 0.034$^a$\
IC 342 & 20 & 6.5 & 20.1 & 52.0 (12.9) & 45 (6) & 0.77 (0.18) & 0.45 (0.11) & $\sim$ 0.005$^b$\
M 66 & 18 & 7.0 & 20.1 & 44.8 (17.7) & 775 (7) & 0.68 (0.21) & 0.4 (0.12) & $\sim$ 0.114$^c$\
NGC 520 & 14 & 5.7 & …& …& …& $<$ 1.02 & $<$ 0.6 & $<$ 0.087$^d$\
NGC 891 & 30 & 2.3 & …& …& …& $<$ 0.26 & $<$ 0.15 & $<$ 0.033$^e$\
NGC 972 & 33 & 3.8 & …& …& …& $<$ 0.61 & $<$ 0.36 & …\
NGC 1068 & 25 & 4.0 & …& …& …& $<$ 0.73 & $<$ 0.43 & $<$ 0.013$^f$\
UGC 2855 & 23 & 4.9 & …& …& …& $<$ 0.76$^*$ & $<$ 0.45 & …\
UGC 2866 & 22 & 3.6 & …& …& …& $<$ 0.56$^*$ & $<$ 0.33 & …\
NGC 1569 & 33 & 3.0 & …& …& …& $<$ 0.31 & $<$ 0.18 & …\
NGC 2146 & 24 & 3.4 & …& …& …& $<$ 0.67 & $<$ 0.39 & $<$ 0.021$^e$\
NGC 2403 & 30 & 3.2 & …& …& …& $<$ 0.33 & $<$ 0.19 & …\
M 82 & 26 & 5.5 & …& …& …& $<$ 0.73 & $<$ 0.43 & $<$ 0.011$^e$\
NGC 3079 & 19 & 5.5 & …& …& …& $<$ 1.17 & $<$ 0.69 & $<$ 0.225$^e$\
NGC 3310 & 36 & 2.3 & …& …& …& $<$ 0.29 & $<$ 0.17 & …\
IC 694+NGC3690 &47& 4.2 & …& …& …& $<$ 0.72 & $<$ 0.42 & $<$ 0.158$^g$\
Mrk 231 & 44 & 6.4 & …& …& …& $<$ 0.91 & $<$ 0.54 & $<$ 0.191$^g$\
Arp 220 & 26 & 4.5 & …& …& …& $<$ 0.94 & $<$ 0.55 & $<$ 0.069$^h$\
NGC 6946 & 28 & 5.1 & …& …& …& $<$ 0.64 & $<$ 0.38 & $<$ 0.018$^e$\
\
[ccccccccc]{} Source & On-time & RMS & $\delta V$& $\Delta V$ & $V_0$ & $I(\rm HC_3N)$ & $S(\rm HC_3N)$ & $\frac{\rm HC_3N\ 24-23}{\rm HCN\ 1-0}$\
& (min) & (mK) & (km s$^{-1}$) & (km s$^{-1}$) &(km s$^{-1}$) & (K km s$^{-1}$) & (Jy km s$^{-1}$)\
(1) & (2) & (3) & (4) & (5) & (6) &(7) &(8) &(9)\
NGC 1068& 121 & 1.2 & 39.1 & 257.3 (24.0)$^a$ & 1102 (13) & 2.03 (0.18) & 49.9 (4.4)& $\sim$ 0.59\
IC 342 & 132 & 2.7 & 20.9 & 100$^b$ & 43 & 0.47 (0.12) & 11.6 (3.0)& $\sim$ 0.12\
IC 694 & 127 & 0.87 & 41.7 & 400$^b$ & 3095 & 0.90 (0.11) & 22.1 (2.7)& $\sim$ 2.76\
NGC 2146& 115 & 1.01 & 20.9 & 320$^{c}$ & …& $<$ 0.17 & 4.2& $<$ 0.26\
M 82 & 60 & 3.0 & 20.9 & 150$^{c}$ & …& $<$ 0.34 & 8.4& $<$ 0.26\
NGC 3690& 139 & 0.73 & 42.0 & 260$^c$ & …& $<$ 0.15 & 3.7& $<$ 1.27\
ARP 220 & 162 & 0.75 & 39.1 & 420$^c$ & …& $<$ 0.19 & 4.7& $<$ 0.48\
NGC 6240& 97 & 0.92 & 20.1 & 420$^{c}$ & …& $<$ 0.17 & 4.2& $<$ 1.15\
NGC 6946& 168 & 1.76 & 21.0 & 130$^{c}$ & …& $<$ 0.18 & 4.4& $<$ 0.26\
Discussion: [HC$_3$N]{} in galaxies
-----------------------------------
The HPBW of SMT and Effelsberg observations are 33$''$ and 46$''$, respectively, which should be able to cover the bulk of the sample galaxies, especially the galaxy center. Thus our observations should be able to cover the region where the majority of dense gas resides. However, with single-dish observations we can not constrain the emission size of either [HC$_3$N]{} 2-1 or [HC$_3$N]{} 24-23, and can not easily estimate the filling factors. Along with the large uncertainty of the emission intensities measurements, it is difficult to estimate the brightness temperature of the sample.
To better understand the excitation environment of [HC$_3$N]{}, the effect of free-free and synchrotron emission near 18 GHz should be also taken into account, as they are more prominent than that in millimeter band that is dominated by dust thermal emission. We detect [HC$_3$N]{} 2-1 lines in emission and not in absorption, and this may be due to the fact that the beam filling factor of the [HC$_3$N]{} gas is higher than the radio continuum. In the high resolution radio observations towards a few nearby galaxies [@Tsai2006], it is found that compact radio sources contribute 20% – 30% of the total 2 cm (15 GHz) emission from the central kiloparsec of these galaxies. In contrast, the distribution of gas with moderate critical density such as [HC$_3$N]{} 2-1 is likely more diffuse.
Comparing to other dense molecular gas tracers such as the popular HCN and [HCO$^+$]{}, [HC$_3$N]{} is generally optically thin in galaxies owing to its relatively low abundance, which makes it an ideal dense gas tracer for calculating the column density and/or mass of molecular hydrogen content of galaxies. In the observations by @Lindberg2011 and @Costagliola2011 a low detection rate of [HC$_3$N]{} was reported and was explained as the intrinsically faint emission of [HC$_3$N]{} , and our stacked result also implies that the [HC$_3$N]{} is quite weak in the non-detected galaxies (2$\sigma$ upper limit=0.14 K [km s$^{-1}$]{}), which is also in favor of this explanation. The non-detection in M 82 is consistent with the low abundance of [HC$_3$N]{} in M 82 suggested by [@Aladro2011], that [HC$_3$N]{} traces a nascent starburst of galaxy, and it can be easily destroyed by the UV radiation in PDRs, which is ubiquitous in active galaxies.
In very recent line surveys of a few local active galaxies (AGN and/or Starbursts, @Aladro2015; @Costagliola2015), several [HC$_3$N]{}transitions in 3mm band ([HC$_3$N]{} $J$=10-9, $J$=11-10 and $J$=12-11) were detected. The ALMA observations by @Costagliola2015 even reported the [HC$_3$N]{} $J$=32-31 rotational transition, and some of the vibrationally excited [HC$_3$N]{} lines. The latest high resolution line surveys in a few very nearby galaxies [@Meier2005; @Meier:2012] and @Meier:2014 [@Meier:2015] show that, the derived [HC$_3$N]{} abundances (on $\sim$ 100 pc, roughly GMC scales) are about several 10$^{-10}$ (relative to H$_2$), which is about an order of magnitude lower than the abundance of HCN and some other molecules.
The results in @Aladro2015 show that, the [HC$_3$N]{} fractional abundance is generally several times lower than that of HCN, [HCO$^+$]{} and CS. And comparing to other AGN or starburst galaxies in their sample, [HC$_3$N]{} abundance is significantly higher in the two ULIRGs and , implying it is suited for studying the activity of ULIRGs. Besides, there was no obvious evidence of the affection by AGN on the intensity of [HC$_3$N]{}. Four galaxies in our sample (, , and ) were also studied in @Aladro2015. We compare our data with their results, and the [HC$_3$N]{} spectra of and from @Aladro2015 are shown in Figure \[Fig:2\_1\] and \[Fig:24\_23\], to be compared with the non-detection of [HC$_3$N]{} 2-1 in , and the detection of [HC$_3$N]{} 24-23 in , respectively. Their results show that, in 3mm band, the intensities between the three transitions of [HC$_3$N]{} (10-9, 11-10 and 12-11) differ not too much, and the peak temperature ($T_{\rm mb}$) of [HC$_3$N]{} are $\sim$ 4 mK for NGC 1068, $\sim$ 11 mK for M 82, and $\sim$ 1.1 – 1.7 mK for Mrk 231, and $\sim$ 10 mK for Arp 220. In our results, the detection of [HC$_3$N]{} 24-23 in NGC 1068 shows a peak $T_{\rm mb} \sim$ 7 mK, while the non-detection of [HC$_3$N]{} 2-1 in Mrk 231 and Arp 220 show that, the RMS we have ($\sim$ 4–6 mK) might not be low enough to detection the [HC$_3$N]{} lines. Here we conclude that, besides the low abundance of [HC$_3$N]{}, insufficient integration time and not ideal observing conditions are the main cause for the low detection rate of [HC$_3$N]{}.
It would be interesting to compare the intensity ratios between [HC$_3$N]{} and other dense gas tracers, such as HCN and [HCO$^+$]{}. We list the ratio between the flux density of [HC$_3$N]{} and HCN 1-0 in Table \[Tab:results\] and \[Tab:results2\], respectively. Because the data quaity of this work is not good enough for us to present an accurate estimate on the [HC$_3$N]{} flux density, the ratios are only tentative. We see a large variation in the ratios, which could be an evidence of the essentially large variation of [HC$_3$N]{} luminosities among galaxies. In the [HC$_3$N]{} survey by @Lindberg2011, ratios like [HC$_3$N]{}/HCN were used to compare [HC$_3$N]{} between galaxies. Based on that ratio, IC 342 and M 82 were classified as [HC$_3$N]{}-luminous galaxies. In our observation we detect both [HC$_3$N]{} 2-1 and [HC$_3$N]{} 24-23 in IC 342, but neither [HC$_3$N]{}transition is detected in M 82. On the other hand, we obtained [HC$_3$N]{} 24-23 detections in NGC 1068 which were classified as a [HC$_3$N]{}-poor galaxy in @Lindberg2011. In the sample of some nearby galaxies observed by @Aladro2015, the ratio between the peak temperature ($T_{\rm mb}$) of [HC$_3$N]{}/HCN or [HC$_3$N]{}/[HCO$^+$]{}also show large variance. In NGC 253 and M 82, [HC$_3$N]{} 10-9 is only $\sim$ 1/20 as strong as [HCO$^+$]{}1-0, while in Arp 220 [HC$_3$N]{} 10-9 is nearly as strong as [HCO$^+$]{}1-0. In our results such line ratios also show large diversity. It is not yet clear how to interpret the ratio between [HC$_3$N]{} and other molecular lines, and more data of [HC$_3$N]{} in different transitions would be helpful to disentangle its properties in different types of galaxies.
Our observations and other works have presented detection of [HC$_3$N]{}emission lines from near 18 GHz up to $\sim$ 292 GHz. The newly commissioned Tianma 65 m telescope in Shanghai, China, is able to observe low transition [HC$_3$N]{} emission, and has great potential for further [HC$_3$N]{} 2-1 surveys for large sample of galaxies.
Summary {#sect:summary}
=======
We carry out single-dish observations towards a sample of nearby gas-rich galaxies with the Effelsberg telescope and the Submillimeter Telescope. This is the first measurements of [HC$_3$N]{} 2-1 in a relatively large sample of external galaxies. Our results include:
1. [HC$_3$N]{}($J$=2-1) ($\nu$ = 18.196 GHz) was observed with the 100-m telescope in 20 galaxies and only three galaxies are detected ($> 3 \sigma$): , and . This is the first measurements of [HC$_3$N]{} 2-1 reported in external galaxies, and the first [HC$_3$N]{} detection in M 66. We stack the spectra of those non-detections yet there is still no sign of [HC$_3$N]{} emission. The $2 \sigma$ upper limit of [HC$_3$N]{} intensity from the stacked spectrum is about 0.12 K [km s$^{-1}$]{}.
2. [HC$_3$N]{}($J$=24-23) ($\nu$ = 218.324 GHz) was observed in nine galaxies with the SMT, and it is detected in three galaxies: , and .
3. IC 342 is the only galaxies detected in both [HC$_3$N]{} 2-1 and [HC$_3$N]{} 24-23 transitions in our observations, and the two transitions have similar line center and width, suggesting a similar emitting area. The ratio of integrated intensity of [HC$_3$N]{} 24-23/[HC$_3$N]{} 2-1 is about 0.82. Due to the contamination of CO 2-1 image signal in the upper sideband, M 82 and Arp 220 are treated as non-detection of [HC$_3$N]{} 24-23.
4. The ratios between [HC$_3$N]{} and HCN, HCO+ show a large variance among the galaxies with [HC$_3$N]{} detections, implying different behavior of the molecular lines in galaxies. More sample are needed to better understand the relationship between [HC$_3$N]{} and other molecules.
We thank the staff of the Effelsberg telescope and the SMT for their kind help and support during our observations. This project is funded by China Postdoctoral Science Foundation (grant 2015M580438), National Natural Science Foundation of China (grant 11420101002, 11311130491, 11590783 and 11603075), and the CAS Key Research Program of Frontier Sciences. This research has made use of NASA’s Astrophysics Data System, and the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
\[lastpage\]
[^1]: Based on observations with the 100-m telescope of the MPIfR (Max-Planck-Institut für Radioastronomie) at Effelsberg, and the Submillimeter Telescope (SMT). The SMT is operated by the Arizona Radio Observatory (ARO), Steward Observatory, University of Arizona.
[^2]: <http://iram.fr/IRAMFR/GILDAS/>
[^3]: <http://www.nist.gov/pml/data/micro/index.cfm>
|
---
abstract: 'Giant second-harmonic generation (SHG) in the terahertz (THz) frequency range is observed in a thin film of an s-wave superconductor NbN, where the time-reversal ($\mathcal{T}$-) and space-inversion ($\mathcal{P}$-) symmetries are simultaneously broken by supercurrent injection. We demonstrate that the phase of the second-harmonic (SH) signal flips when the direction of supercurrent is inverted, i.e., the signal is ascribed to the nonreciprocal response that occurs under broken $\mathcal{P}$- and $\mathcal{T}$-symmetries. The temperature dependence of the SH signal exhibits a sharp resonance, which is accounted for by the vortex motion driven by the THz electric field in an anharmonic pinning potential. The maximum conversion ratio $\eta_{\mathrm{SHG}}$ reaches $\approx10^{-2}$ in a thin film NbN with the thickness of 25nm after the field cooling with a very small magnetic field of $\approx1$Oe, for a relatively weak incident THz electric field of 2.8kV/cm at 0.48THz.'
author:
- Sachiko Nakamura
- Kota Katsumi
- Hirotaka Terai
- Ryo Shimano
title: Nonreciprocal Terahertz Second Harmonic Generation in Superconducting NbN under Supercurrent Injection
---
Nonreciprocal electric transport phenomena, where electrical conductivity depends on the direction of the current ($I$), play crucial roles in modern electronics. Most typical example is a diode consisted of a p-n junctions, where the space-inversion ($\mathcal{P}$-) symmetry is *macroscopically* broken. The nonreciprocal phenomena are found to emerge even in bulk materials where $\mathcal{P}$-symmetry is *microscopically* broken [@PhysRevLett.87.236602], and the nonreciprocity can become prominent especially in the diagonal components of the conductivity $\sigma$ (i.e. $\sigma_{xx}(I)-\sigma_{xx}(0) \propto I$). The extension of the phenomena to higher frequency ranges such as microwave, terahertz (THz), and optical frequencies is highly demanded for high-speed optoelectronic devices . For the linear and diagonal nonreciprocity, the time-reversal ($\mathcal{T}$-) symmetry should also be broken in addition to the $\mathcal{P}$-symmetry breaking according to the Onsager’s reciprocal theorem [@PhysRev.37.405]. Generally, the $\mathcal{T}$-symmetry breaking is induced by applying external magnetic fields or in the presence of magnetization [@PhysRevLett.94.016601; @10.1038/nphys3356; @PhysRevLett.117.127202; @giantSHG2019nature; @NPhys13.578; @PhysRevB.92.184419; @NComm5.3757; @doi:10.1063/1.1523895; @SciAdv3.e1602390; @NatComm8.14465; @NatComm10.2734]. In superconductors, it has been recently proposed that the nonreciprocal signal is gigantically enhanced [@PhysRevB.100.220501; @PhysRevLett.121.026601] and can be used as a new probe to detect small $\mathcal{P}$-symmetry breaking in superconductors [@SciAdv3.e1602390; @NatComm8.14465; @NatComm10.2734].
In this Letter, we illuminate the effect of supercurrent which breaks both of the $\mathcal{T}$- and $\mathcal{P}$-symmetries as depicted in Fig. \[fig1\](a). Notably, the supercurrent is non-dissipative, and thus can be treated as a sort of symmetry-breaking field in microscopic considerations, in contrast to normal currents where the dissipation plays a role of macroscopic irreversibility. We studied a thin film of s-wave superconductor NbN as a testing ground which can carry very large dc currents ($>1$MA/cm$^2$) without accompanying dc electric field due to zero resistance. In addition, the induced magnetic field is expelled from the sample as it is smaller than the lower critical field, and other properties such as optical conductivities and superfluid densities are almost unaffected as long as the current is kept under the critical value [@PhysRevLett.122.257001]. In the presence of a dc electric supercurrent $I_0$, we observed nonreciprocal second harmonic generation (SHG) in response to the oscillating current $I_\omega$ at THz frequencies. The SHG signal shows the phase inversion when the direction of dc current is reversed, which is the hallmark of the linear nonreciprocal response, e.g., $\sigma_{xx}(I_\omega,I_0)-\sigma_{xx}(0,0) \propto I_\omega I_0$, arising from the current-induced $\mathcal{T}$-symmetry breaking.
![(Color online) (a) The dc supercurrent $I_0$ indicated by the arrow changes the direction under each time-reversal and space-inversion operations. (b) Schematic view of the terahertz transmittance experiments under the dc supercurrent. (c) Configuration of the sample and currents. \[fig1\]](fig1){width="40.00000%"}
The SHG was measured using a THz time-domain spectroscopy (THz-TDS) technique in a transmission geometry as schematically illustrated in Fig. \[fig1\](b), to realize the phase-resolved detection. The intense monocycle THz pulse was generated by the tilted-pulse front method with a LiNbO$_3$ [@Hebling:02; @Watanabe:11] and spectrally narrowed by using band-pass filters with different center frequencies. Typical peak values of the electric field were 2.8, 2.7, 4.0, and 4.6kV/cm for 0.48, 0.6, 0.8, and 1THz sources, respectively. A regenerative amplified Ti:sapphire laser system with 800nm center wavelength, 100fs pulse duration, 1kHz repetition rate, and pulse energy of 1 or 4mJ was used as a light source. The THz intensity and polarization angle were controlled with wire grid polarizers. The transmitted THz pulse after the sample was detected by electro-optic (EO) sampling using a ZnTe crystal. The superconducting film was an epitaxial NbN films of $25$nm in thickness grown on a 400-$\mu$m-thick MgO (100) substrate. The dc current was injected into the 4-mm-square area and THz pulse was focused on its center. The critical temperature $T_\mathrm{c}$ is $14.5\pm0.2$K and the critical current $I_\mathrm{c}$ is 3.3A (3.3MA/cm$^2$) at $T=5$K. The critical current is defined as the current where the quench occurs by increasing the current by 0.01A with 5sec intervals.
![(Color online) THz transmission measurements on a NbN film at $T=11.6$K and $\omega/2\pi=0.48$THz (peak value: 5.4kV/cm). (a)–(c) Real-time waveform of the transmitted THz electric fields $E(\pm)$ and $E(0)$ measured with $I_0=\pm 2.1$ and $0$A, respectively, and current-induced changes $\delta E(\pm)\equiv E(\pm)-E(0)$, $\delta E\equiv \left[E(+)-E(-)\right]/2$, and $\Sigma E\equiv \left[\delta E(+)+\delta E(-)\right]/2$. The dashed line in (c) shows the expected change due to the current-induced optical conductivity change [@PhysRevLett.122.257001]. (d),(e) Power spectra of the transmitted THz electric fields and incident THz electric field $E_\mathrm{in}$. The FH, SH, and TH in (d) stand for the fundamental, second, and third harmonics. Power spectra of the current induced changes are shown in (e).\[fig2\]](fig2){width="48.00000%"}
Figure \[fig2\](a) shows waveforms of the transmitted THz pulse through the NbN film with a dc current in one direction $E(+)$, in the opposite direction $E(-)$, and without current $E(0)$ at $T=11.6$K. The polarization of the incident THz electric field is parallel to the dc current. The center frequency of the incident THz pulse $\omega/2\pi$ is 0.48THz, and the peak value of the incident THz field is 5.4kV/cm. Figure \[fig2\](d) shows the power spectra of the waveforms. Without the current injection, the power spectrum of the transmitted THz pulse \[$E(0)$\] has a peak at the fundamental harmonic (FH) frequency, and the small peak observed at the second harmonic (SH) frequency $2\omega$ is attributed to the transmission of the leakage of the band-pass filter which is inserted before the film to narrow the incident FH. A peak at the third harmonic (TH) frequency $3\omega$ originates from the Higgs mode [@Matsunaga1145]. By injecting a dc current of 2.1A, a clear peak appears at $2\omega$, whose intensity is $10^{-3}$ of the FH intensity. The SH intensity is larger than the tail component of the incident field \[$E_\mathrm{in}$ in Fig. \[fig2\](d)\] at the frequency $2\omega$ by a factor of eight, which confirms that the peak is attributed to SHG and not due to the current-induced transmittance change. The SH intensity is proportional to square of the FH intensity (shown in Fig. S2(a) in Supplemental Material), exhibiting the fundamental characteristics of second-order nonlinear phenomena. The SHG conversion ratio (up to $10^{-2}$ [^1]) is extraordinarily large for a 25-nm-thick film. The observed conversion efficiency corresponds to the effective nonlinear susceptibility, ${\chi^{(2)}_\mathrm{eff}}\approx 2.2\times 10^{8}$pm/V, if we dare to plug the value into the conventional expression for the second order nonlinear susceptibility of nonlinear optical crystals, $\chi^{(2)}_\mathrm{eff}=\sqrt{\dfrac{\mathrm{SH}}{\mathrm{FH}}\dfrac{n_\omega n_{2\omega} {\lambda_{2\omega}}^2}{d^2 \pi^2 |E_\omega|^2}}$ [@doi:10.1002/adom.201500723] where the refractive indices $n_\omega$ and $n_{2\omega}$ are assumed to be $\approx 1$, $d$ is the crystal thickness, and $\lambda_{2\omega}$ is the SH wavelength. This value is about $10^4$ times larger than that of LiTaO$_3$, a material known to possess a large $\chi^{(2)}$ in the THz frequency range [@PhysRevB.33.6954]. Figure \[fig2\](b) clearly manifests that the SH ($2\omega$) oscillation arises from the nonreciprocal part of the current-induced signal, $\delta E \equiv \left[E(+)-E(-)\right]/2$, whereas the reciprocal part $\Sigma E \equiv \left[\delta E(+)+\delta E(-)\right]/2$ oscillates in $\omega$ as shown in Fig. \[fig2\](c). The observed reciprocal part $\Sigma E$ shows a good accordance with the calculated one and thus is attributed to the current-induced conductivity change [@PhysRevLett.122.257001] as plotted by a dashed line in Fig. \[fig2\](c). The small peak at $2\omega$ appeared in the power spectrum of $\Sigma E$ originates from the leakage of the filter.
![(Color online) (a) Temperature ($T$) dependence of SH conversion factor $\eta$. (b) $T$-$I_0$ phase diagram. The color map shows SHG enhancement factor (defined in the text) as a function of the $T$ and $I_0$.\[fig3\]](fig3){width="45.00000%"}
The temperature dependence of the SH conversion efficiency $\eta$ (intensity ratio of SH and FH) is represented in Fig. \[fig3\](a). It is nearly constant in the low temperature range, and exhibits a prominent peak at high temperature which shifts to lower temperature side with increasing the dc current $I_0$. Regardless of the temperature, $\eta$ shows a quadratic dependence to the dc current $I_0$ [^2]. A similar peak is found for other incident fundamental frequencies $\omega$, while the peak height strongly depends on the incident frequency (see Fig. S1 in Supplementary Materials). When the probe THz polarization is perpendicular ($\perp$) to the dc current, the SH intensity is smaller than that with parallel configuration ($\parallel$) by more than two-orders of magnitude [^3]. This result is reasonable since there exists a mirror symmetry $m_y$ \[Fig. \[fig1\](c)\] where $I_0$ goes along the x-axis, with which the linear and diagonal nonreciprocity for the oscillating current along the y-axis ($I_{\omega}^{y}$) is forbidden. Figure \[fig3\](b) represents the color map of SHG enhancement factor as defined by SH/FH$^2/{I_0}^2$ for $\omega/2\pi=0.48$THz as a function of temperature $T$ and dc current $I_0$. The boundary of the superconducting phase and the normal metal phase is indicated by the black solid curve which identifies the critical current $I_\mathrm{c}$. Clearly, the SHG is enhanced in the vicinity of the $I_\mathrm{c}$ boundary. The resonant peak positions observed in Fig. \[fig3\](a) for $\omega/2\pi=0.8$THz source and for other frequencies are plotted by closed circles in the phase diagram of Fig. \[fig3\](b). The resonant points appear slightly below the phase boundary, and are located inward for higher $\omega$. The observed current-induced SHG accompanied by a characteristic resonance behavior can be phenomenologically described by an anharmonic oscillator model [@anharmonicmodel1965] where the $\mathcal{P}$-symmetry breaking is induced by the dc supercurrent as discussed in detail in the following.
![(Color online) (a) Schematic view of a vortex normal to the surface subjected to a force $F$ along Y-axis due to a current $I$ along X-axis. Corresponding pinning potentials without dc current ($I_0=0$), with current ($I_0>0$), and with the critical current ($I_0=I_\mathrm{c}$) are shown in (b), (c), and (d), respectively. (e) Trapped magnetization $M$ measured with increasing temperature at $H=0$Oe after cooling from $T=20$K to 5K across $T_\mathrm{c}=14.5$K in the indicated preparation field ($H_\mathrm{pre}=$0.5, 1, ..., 200Oe) applied normal to the film. (f, g) SHG intensity scaled by FH intensity for $\omega/2\pi$=1THz (f) after cooling with permanent magnets (FC, 1Oe) and (g) in the ambient field (AFC). \[fig4\]](fig4){width="48.00000%"}
As a microscopic origin of the anharmonic oscillator, here we consider the motion of pinned vortices induced by the remnant magnetic field. In type-II superconductors, the critical current $I_\mathrm{c}$ corresponds to the depinning current where the pinning force balances with the Lorentz force $F$ [@PhysRevLett.16.734] due to the dc current $I_0$. Figure \[fig4\](a) shows a typical configuration of a vortex, current, and the force in the film. Without the dc currents, the vortex is trapped in a symmetric pinning potential $V_\mathrm{pin}$ as schematically shown in Fig. \[fig4\](b). When the dc current ($I_0$) is applied, the pinning potential is tilted and becomes asymmetric as shown in Fig. \[fig4\](c), and the equilibrium point shifts to the left. When the current reaches $I_\mathrm{c}$, the vortex slips off the pinning potential as shown in Fig. \[fig4\](d). Then, the incident THz field induces an oscillating current $I_\omega$ along the X-axis which causes the oscillation of the vortex along the Y-axis. The equation of motion for a vortex along the Y-axis is given by [@jisokuSM] $$\label{eq:EOM1}
m\frac{\mathrm{d} v}{\mathrm{d} t}
=-\frac{\partial }{\partial y}\left(V_\mathrm{pin}+J_0 d_\mathrm{f} \Phi_0 y\right) - \eta d_\mathrm{f} v - d_\mathrm{f} \Phi_0 J_\omega$$ where $m$, $v$, and $y$ are the mass, velocity, and position of the vortex, respectively, $\Phi_0$ is the flux quantum, $d_\mathrm{f}$ is the thickness of the film, and $J_i$ ($i=0, \omega$) is the current density corresponding to $I_i$. The first term of eq. (\[eq:EOM1\]) is the restoring force due to the $V_\mathrm{pin}$ tilted by $J_0$, the second term is the viscous drag, and the last term is the driving force oscillating at $\omega$. If there are higher harmonic components in $V_\mathrm{pin}$ (i.e., $y^4, y^6, ...$), the Taylor series of the potential around the new equilibrium point has odd power terms which gives rise to even harmonics oscillation along the Y-axis. Because the moving vortex induces electric field perpendicular to the vortex motion $\vec{E}_\mathrm{ind}=\Phi_0 \vec{z} \times \vec{v}$ [@JOSEPHSON1965242; @doi:10.1143/JPSJ.57.3941], the even harmonics of electromagnetic field is expected to be emitted along the X-axis, which is parallel to the incident THz. According to this mechanism, the SH intensity should be proportional to $\mathrm{FH}^2$, $I_0^2$, and square of the vortex density. The first two features are consistent with our observations. To examine this picture further, we tried to change the vortex density.
In our configuration, the vortex density is determined by the applied magnetic field normal to the film when the sample is cooled across $T_\mathrm{c}$. The magnetic field induced by the dc current, which is parallel to the film, is expelled from the film [^4]. Figure \[fig4\](e) shows the temperature dependence of the magnetization $M$ of the NbN film which was first cooled under the presence of indicated magnetic fields $H_\mathrm{pre}$, and then warmed up without the field. The magnetization $M$ corresponds to the total magnetic flux of pinned vortices, i.e., the vortex density, which increases by increasing the $H_\mathrm{pre}$ and saturates at $H_\mathrm{pre}\approx5$Oe for $T=5$K and $H_\mathrm{pre}\approx 1$Oe for $T=13$K. We tried field cooling (FC) with applying an external magnetic field of $H_\mathrm{pre}\approx$1Oe using a pair of neodymium magnets placed outside the cryostat, and compare with ambient field cooling (AFC) where $H_\mathrm{pre}\approx 0.5$Oe. As the external magnetic field suppresses the $I_\mathrm{c}$, we removed the magnets before the measurements so that the $I_\mathrm{c}$ comes back to the original value. The SH intensity is clearly enhanced by FC compared to that for AFC without changing the resonance temperature, as shown in Figs. \[fig4\](f) and (g). Because the FC under the very weak field changes only the density of the frozen vortices, this observation indicates that the nonreciprocal SH originates from the pinned vortices in the sample. According to the mechanism proposed above, the observed decrease of the resonant frequency with increasing the $T$ or $I_0$ is accounted for by the shallowing of the effective pinning potential.
In the phase diagram shown in Fig. \[fig3\](b), the critical current $I_\mathrm{c}$ gradually decreases with increasing the temperature. Notably, one can see peaks at 8.5, 10.8, and 11.2K on the $I_\mathrm{c}$ curve marked by the arrows, which suggests that the vortices slowly creep to optimize the vortex lattice and deepen the effective pinning potential at substantially high temperature [@PhysRevLett.9.306; @PhysRevLett.9.309]. Under the THz irradiation, as plotted by broken lines in the same figure, the peaks are clearly suppressed, which suggests that the vortices are certainly oscillated by the THz irradiation and slip out of the pinning potential before optimizing their configuration. Having seen that the experimental results are consistently described by the vortex motion, based on eq. (\[eq:EOM1\]), we evaluate the parameters, i.e., vortex mass ($m\approx m_e$ where $m_e$ is the electron mass), viscosity \[$\eta\approx 10^{-10}$kg /(sec m)\], and eigenfrequency of the vortex motion (in the THz frequency range), from the current and temperature dependence of the resonant frequency [@jisokuSM]. The mass per unit length $m/d$ is estimated as $\approx 10^7 m_e$/meter where $d$ is the film thickness, 25nm. This value shows a reasonable agreement with that calculated for the vortex core in dirty limit superconductors [@PhysRevLett.14.226; @PhysRevB.57.575], whereas previous experiments have reported much heavier mass [@doi:10.1063/1.2747080; @PhysRevB.85.060504]. The difference suggests that the high frequency THz response in the present experiment is sensitive to the bare mass of vortex core and less affected by dissipative dynamics [@doi:10.1063/1.2747080; @PhysRevB.85.060504; @RevModPhys.66.1125]. The observed SH electric field is also in reasonable accordance with the calculated value of 170V/m, which is estimated, e.g. for $T=$10.4K, $\omega/2\pi=0.8$THz, and $I_0=2.6$A [@jisokuSM].
Finally, we address the contribution of Higgs mode in the SH signal. In the presence of dc current, the Higgs mode is directly excited by the fundamental wave ($\omega$) [@PhysRevLett.118.047001; @PhysRevLett.122.257001; @2020arXiv200108091P] which in turn should induce the SHG. The intensity of this Higgs-mediated SHG is expected to be comparable with that of the Higgs-mediated THG \[shown in Fig. \[fig2\](e)\] because $I_0$ is as large as $I_\omega$[^5]. In the present case, the Higgs-mediated SHG is overwhelmed by much stronger SHG induced by the vortex motion [^6]. Recently, SHG has been observed in a clean limit superconductor where the $\mathcal{P}$-symmetry breaking is caused by the effect of THz field-induced persistent photocurrent [@NatPhys13.707; @vaswani2019discovery]. The method is advantageous for extracting ultrafast $\mathcal{P}$-symmetry breaking phenomena which can follow the time scale of THz wave cycle. In contrast, the static bias of dc supercurrent enables the investigation of nonreciprocal responses in a $\mathcal{P}$- and $\mathcal{T}$-symmetry broken system under a nonequilibrium steady state that accompanies slow dynamics, e.g., vortex reconfiguration, lattice distortion, or thermal relaxation.
In summary, we have demonstrated the giant THz SHG in a type-II superconducting film of NbN under dc supercurrent injection. Based on the orientation dependence on the dc current and THz electric field polarization, we elucidate that the dc current breaks the time-reversal and space-inversion symmetries and gives rise to a colossal nonreciprocal SHG. This phenomenon would pave a new pathway for nonreciprocal THz electronics with superconductors. The observed maximum SH conversion efficiency reaches 1% for a few kV/cm of THz electric field input even for the film sample of 25nm in thickness. The SH is enhanced by the field cooling with the resonant peak always appearing slightly below the critical current line in the temperature-current phase diagram. A microscopic model taking into account the vortex dynamics is shown to describe well the observed experimental behaviors, indicating that the microscopic origin of the SHG is attributed to the motion of frozen vortices in anharmonic pinning potential that is tilted by the dc current. This method would also provide a new tool to access the detailed picture of the pinning potential in type-II superconductors.
This work was supported in part by JSPS KAKENHI (Grants No. 15H02102, No. 15H05452, and No. 18K13496), and by JST CREST Grant Number JPMJCR19T3, Japan.
[45]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****, ()](https://doi.org/10.1103/PhysRevLett.87.236602) [****, ()](https://www.nature.com/articles/s41467-018-05759-4) [****, ()](https://doi.org/10.1103/PhysRevApplied.10.047001) [****, ()](https://doi.org/10.1103/PhysRev.37.405) [****, ()](https://doi.org/10.1103/PhysRevLett.94.016601) @noop [****, ()]{} [****, ()](https://doi.org/10.1103/PhysRevLett.117.127202) [https://doi.org/10.1038/s41586-019-1445-3](https://doi.org/https://doi.org/10.1038/s41586-019-1445-3) () @noop [****, ()]{} [****, ()](https://doi.org/10.1103/PhysRevB.92.184419) @noop [****, ()]{} [****, ()](https://doi.org/10.1063/1.1523895), @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} [****, ()](https://doi.org/10.1103/PhysRevB.100.220501) [****, ()](https://doi.org/10.1103/PhysRevLett.121.026601) [****, ()](https://doi.org/10.1103/PhysRevLett.122.257001) [****, ()](https://doi.org/10.1364/OE.10.001161) [****, ()](https://doi.org/10.1364/OE.19.001528) [****, ()](https://doi.org/10.1126/science.1254697), [****, ()](https://doi.org/10.1002/adom.201500723), [****, ()](https://doi.org/10.1103/PhysRevB.33.6954) , ed., @noop [**]{} (, ) [****, ()](https://doi.org/10.1103/PhysRevLett.16.734) @noop [****, ()](https://doi.org/https://doi.org/10.1016/0031-9163(65)90826-7) [****, ()](https://doi.org/10.1143/JPSJ.57.3941), [****, ()](https://doi.org/10.1103/PhysRevLett.9.306) [****, ()](https://doi.org/10.1103/PhysRevLett.9.309) [****, ()](https://doi.org/10.1103/PhysRevLett.14.226) [****, ()](https://doi.org/10.1103/PhysRevB.57.575) [****, ()](https://doi.org/10.1063/1.2747080), [****, ()](https://doi.org/10.1103/PhysRevB.85.060504) [****, ()](https://doi.org/10.1103/RevModPhys.66.1125) [****, ()](https://doi.org/10.1103/PhysRevLett.118.047001) @noop (), @noop [****, ()]{} @noop (),
[^1]: Fig. S1(a) in Supplemental Material.
[^2]: Fig. S2(b) in Supplemental Material.
[^3]: Fig. S2(c) in Supplemental Material.
[^4]: Magnetization measured for another sample fabricated in very similar conditions is found in Supplemental Material.
[^5]: $I_\omega$ is calculated from the optical conductivity of the film and the internal electric field $E_0$ considering the multiple reflections.
[^6]: The $\eta$ for $\omega=0.48$THz shown in Fig. S1(a) of Supplemental Material shows a second peak for $I_0=0.5$ and 0.7A at $T=13.2$K where the resonance condition $\omega=2\Delta$ is fulfilled, which might be related to the Higgs mode.
|
---
abstract: 'An overview of the experimental results on high-[$p_T$]{} light hadron production and open charm production is presented. Data on particle production in elementary collisions are compared to next-to-leading order perturbative QCD calculations. Particle production in Au+Au collisions is then compared to this baseline.'
address: 'Lawrence Berkeley National Laboratory, Berkeley, California 94720'
author:
- M van Leeuwen
title: 'Overview of Hard processes at RHIC: high-[$p_T$]{} light hadron and charm production'
---
Introduction
============
The goal of research in high-energy heavy-ion collisions is to study the properties of strongly interacting matter at extreme energy density, including the possible phase transition to a colour-deconfined state: the Quark Gluon Plasma (QGP). The matter produced in these collisions can be probed using hadrons produced in partonic processes with a large momentum transfer (‘hard scatterings’). These processes take place early in the collision and are only sensitive to short distance scales. In the absence of nuclear effects the hard production yields in nucleus-nucleus collisions are therefore expected to scale as if the collision were an independent superposition of nucleon-nucleon collisions. Measurements at SPS have shown that dilepton production in the Drell-Yan process indeed follows this expectation [@Abreu:1997ji].
Among the first measurements at RHIC was the measurement of light hadron production at high transverse momentum [$p_T$]{} in Au+Au collisions, which shows a suppression with respect to the scaled p+p results. Since these first observations, measurements in d+Au collisions have confirmed that the observed suppression is a final state effect. Recently there has also been an increased activity to verify that high-[$p_T$]{} light-hadron production in proton-proton collisions can be understood in terms of perturbative QCD (pQCD) calculations, as expected for hard processes. Some of the relevant results will be reviewed in the next section.
While for light hadron production much of the groundwork has been done and analyses are clearly moving towards more advanced observables like identified hadron spectra and correlation measurements, first results on open charm production are becoming available. Like high-[$p_T$]{} light hadrons, open charm is expected to be dominantly produced in hard processes and can therefore serve as a calibrated probe of the medium. Due to their large mass, however, charm quarks and hadrons are expected to be affected differently by the medium than high-[$p_T$]{} light hadrons [@molnar_sqm04].
In the second part of this paper an overview will be presented of the existing results on open charm production at RHIC. The present results are based on run-2 and run-3 data. First results from the large statistics Au+Au data sample from run-4 are to be expected soon. These will greatly improve the precision and [$p_T$]{}-coverage of the open charm measurements in Au+Au collisions at RHIC.
High-[$p_T$]{} light hadron production
======================================
High-[$p_T$]{} hadron production at is the most readily accessible observable for hard processes at RHIC. At sufficiently high [$p_T$]{}, all hadrons are expected to be produced in jet fragmentation. The non-perturbative dynamics of jet-fragmentation can be characterised by a universal fragmentation function $D(z)$ which is parametrised using data from $e^+e^-$ collisions at different energies [@Kniehl:2000fe; @Kretzer:2000yf]. With these fragmentation functions and the parton densities from deep inelastic scattering experiments, the expected cross sections for high-[$p_T$]{} hadron production can be calculated in perturbative QCD (pQCD).
Neutral pions and charged hadrons in p+p collisions
---------------------------------------------------
shows [$p_T$]{}-spectra of [$\pi^0$]{} measured by PHENIX [@Adler:2003pb] (left panel) and charged hadrons from STAR [@Adams:2003kv] and BRAHMS [@Arsene:2004ux] (right panel) in p+p collisions at $\sqrt{s}=200$ GeV. The data are compared to a next-to-leading order (NLO) pQCD calculation [@Jager:2002xm]. The uncertainty in the theoretical calculation is estimated by varying the renormalisation and factorisation scales $\mu_R$ and $\mu_F$ to half and twice the nominal value of $\mu_R=\mu_F={\ensuremath{p_T}}$. These scale variations change the calculated cross-sections by about 20% for ${\ensuremath{p_T}}>5$ GeV. To illustrate the uncertainty in the fragmentation functions, the calculation was performed with two different sets of fragmentation functions, from Kniehl, Kramer and Potter (KKP) [@Kniehl:2000fe] and from Kretzer [@Kretzer:2000yf]. Both sets were independently determined from similar selections of $e^+e^-$ data using slightly different assumptions about relations between the fragmentation functions for different partons. This turns out to be the dominant source of uncertainty for the [$\pi^0$]{} spectrum: variations of the order of 50% are seen, mainly due to uncertainties in the gluon fragmentation function. Note, however, that these are partly normalisation uncertainties and do not change the shape of the spectra very much. The measurements have an overall normalisation uncertainty of about 10%, which is not indicated in the figures. The systematic offset between the STAR and BRAHMS results in can probably be attributed to this normalisation uncertainty.
Both for neutral pions and charged hadrons, data and theory agree over more than 5 orders of magnitude, which gives confidence that hadron production at high [$p_T$]{} ($>3$ GeV) is indeed governed by hard point-like processes.
Strange hadron production in p+p collisions
-------------------------------------------
The above comparisons can be extended to the strange hadrons $K^0_S$ and $\Lambda$. While the kaon fragmentation function is relatively well-constrained by the data from $e^+e^-$ collisions, data on $\Lambda$ production are scarce [@deFlorian:1997zj].
A first comparison of $K^0_S$ and $\Lambda$ spectra in $\sqrt{s}=200$ GeV p+p collisions to NLO calculations as presented at this conference is shown in [@heinz_sqm04]. The agreement between the measured $K^0_S$ spectrum and the expectation from NLO pQCD is reasonable, although the shape of the calculated spectrum is slightly more concave than the measured one. This difference is mainly apparent at relatively low [$p_T$]{} (1-2 GeV), where soft production processes may still contribute significantly.
For the $\Lambda$ on the other hand, the agreement between the data and the NLO calculations is far from satisfactory. This might be indicative of the breakdown of the massless formalism and the factorization ansatz for particles with mass that is significant compared to ${\ensuremath{p_T}}$ [@Kretzer:2004ie]. Before drawing this conclusion, however, the uncertainties in the $\Lambda$ fragmentation functions should be better quantified.
Suppression in Au+Au collisions
-------------------------------
To compare measured the particle spectra in Au+Au collisions to the expected $N_{coll}$ scaling from p+p, the nuclear modification factor $$R_{AA}= \frac{\left.dN/d{\ensuremath{p_T}}\right|_{Au+Au}}{N_{coll}
\left.dN/d{\ensuremath{p_T}}\right|_{p+p}}$$ is generally used. In these ratios are shown for peripheral and central Au+Au collisions at $\sqrt{s}=200$ GeV, both for [$\pi^0$]{} from PHENIX (left panel) [@Adler:2003qi] and charged hadrons from STAR (right panel) [@Adams:2003kv]. The suppression ratio $R_{AA}$ in peripheral collisions is close to unity, while for central collisions a suppression of up to a factor 5 is observed. This suppression was one of the first indications of a strong final state modification of particle production in Au+Au collisions that is now generally ascribed to energy loss of the fragmenting parton in the hot and dense medium.
Open charm production in d+Au and Au+Au collisions
==================================================
While light hadron production is only expected to be calculable in perturbative QCD at higher [$p_T$]{}, the charm quark mass ($m_c\approx 1.35$ GeV) is large enough to expect pQCD calculations to be valid for all [$p_T$]{}. Final state effects on charm quarks and hadrons are expected to be smaller than for the light hadrons due to the large charm mass [@molnar_sqm04].
Charmed meson spectra in d+Au collisions
----------------------------------------
None of the RHIC experiments currently has a vertex detector with sufficient resolution to reconstruct secondary vertices of charm decays. Even without secondary vertex reconstruction, STAR has been able to statistically reconstruct decays to charged hadrons in d+Au collisions at $\sqrt{s}=200$ GeV using an invariant mass method [@Tai:2004bf]. Different charmed mesons ($D^0$, $D^{\pm}$, and $D^{*\pm}$) can be reconstructed in different [$p_T$]{} ranges, thus providing a charm measurement up to ${\ensuremath{p_T}}{}=11$ GeV, as shown in the left hand panel of . The $D^\pm$ (triangles) and $D^{*\pm}$ (squares) spectra have been scaled to match the $D^0$ spectra (circles). Note that the $D^0$ has been measured at low [$p_T$]{} and at high [$p_T$]{} using different decay modes and thus provides a normalisation for the whole [$p_T$]{} range.
Also shown in (left) are NLO pQCD calculations of [*charm quark spectra*]{} [@Vogt:2001nh]. To illustrate the sensitivity of those calculations to the charm quark mass $m_c$ and the choice of renormalisation scale, curves are drawn for $m_c=1.2$ GeV and $m_c=1.5$ GeV and with two choices for the renormalisation and factorisation scales: $\mu_R=\mu_F={\ensuremath{m_T}}$ and $\mu_R=\mu_F=2{\ensuremath{m_T}}$. Note that the shape of the spectra at low [$p_T$]{} is most sensitive to both the charm quark mass and the choice of scales.
For a detailed comparison of the data to theory, the calculated charm quark spectrum should be convoluted with the charm fragmentation function. This would lead to a softening of the spectrum and a reduction of the yield, both of which may in principle depend on the meson species. Given the limited [$p_T$]{} range of the spectra for the separate species and the relatively large uncertainties in their fragmentation functions, we have chosen to compare the shape of the combined meson spectrum directly to the charm quark spectra. For this purpose, the calculated spectra were scaled up by a factor of 4 to approximately match the data. The shapes of the calculated charm quark spectra and the measured meson spectra are surprisingly similar, leaving little room for softening due to fragmentation.
All in all, it is far from clear that the present data can be matched with a NLO pQCD calculation. Before drawing any conclusions about charm production in elementary collisions at RHIC, though, we should wait until the present data are finalised. Although it is expected that $N_{coll}$ scaling is valid for charm production in d+Au, it would be good to confirm this by similar measurements in p+p. There are also some open questions for theory. For example, a matched next-to-leading logarithm calculation is needed to describe beauty production at the Tevatron [@Cacciari:2003uh]. In addition, a new way of extracting fragmentation functions, by fitting the Mellin moments instead of a direct $z$-space fit is found to lead to an effectively harder fragmentation function [@Cacciari:2002pa]. Similar considerations may also affect calculations of charm production at RHIC.
Total charm cross section
-------------------------
The right-hand panel of shows the measured energy dependence of the total charm quark cross section [@Adams:2004fc; @Kelly:2004qw], compared to NLO pQCD calculations [@Vogt:2001nh]. Estimating the total charm quark cross section from experimental data involves substantial extrapolations to the unmeasured regions of momentum space and corrections to include unmeasured charmed hadron species. These corrections lead to sizeable systematic uncertainties on the data, as can be seen from the figure. The uncertainties on the NLO pQCD calculations due to higher order corrections and the choice of the charm mass are also significant. It is therefore preferable to directly compare data and calculations in the measured regions (see also [@Frixione:2004md]).
Note also that it seems that the charm cross section per nucleon-nucleon collision as measured by STAR in d+Au collisions from a combination of the electron measurements and the invariant mass method is somewhat higher than expected from the trend observed by other experiments and the pQCD calculations. The deviations are within the present uncertainties on the measurements.
Centrality dependence of charm production in Au+Au collisions
-------------------------------------------------------------
A first indication of the centrality dependence of charm production in Au+Au collisions can be taken from electron spectra. After subtraction of the contributions from light hadrons (mainly through photon conversions, but also from Dalitz decays of [$\pi^0$]{}, $\eta$, $\eta'$, $\rho$, $\omega$ and $\phi$) the electrons from heavy flavour decays remain (‘non-photonic’ electrons). In the left hand panel of , the electron spectra from heavy flavour decays as measured by PHENIX in centrality-selected Au+Au collisions [@Adler:2004ta] are shown. The lines show reference spectra obtained from a fit to the measured spectrum in p+p, scaled by the number of collisions. At each centrality, the spectra agree with the expected $N_{coll}$ scaling from p+p, albeit within large errors.
In the right-hand panel of , the yields of non-photonic electrons with $0.8<{\ensuremath{p_T}}<4.0$ GeV per nucleon-nucleon collision are shown as a function of centrality. There is no indication of a suppression as seen for light hadrons (see ). One should keep in mind, however, that electrons with ${\ensuremath{p_T}}>0.8$ GeV have contributions from semi-leptonic charm decays at all [$p_T$]{}. The presented results are therefore not very sensitive to a possible suppression of charm production at moderate or high [$p_T$]{} ($>2$ GeV).
Charm flow
----------
A measurement of the elliptic flow $v_2$ of charmed mesons is an independent way of assessing the sensitivity of charmed mesons to final state interactions. Here again, we have to rely on measurements of decay electrons for the time being. In the elliptic flow of non-photonic electrons is shown [@laue_sqm04]. Both STAR and PHENIX observe non-zero electron flow, which is a strong indication that charmed mesons flow. This is an intriguing possibility, because it would show decisively that charm production is sensitive to the dense hadronic or even partonic environment in the collision. At the moment the statistical and systematic errors are still large, precluding a precise quantitative extraction of flow values. The situation is expected to dramatically improve with the larger Au+Au data samples which were recorded this year.
Summary and outlook
===================
A comparison of neutral pion and charged hadron [$p_T$]{}-spectra measured in p+p collisions at $\sqrt{s}=200$ GeV at RHIC to NLO pQCD calculations shows that high-[$p_T$]{} light hadron production is well described by perturbative QCD, albeit within relatively large uncertainties, mainly from the fragmentation functions. This gives confidence that high-[$p_T$]{} particle production is governed by hard, point-like processes, for which the cross section in Au+Au collisions is expected to scale with the number of nucleon-nucleon collisions. A suppression of light hadrons by approximately a factor of 5 compared to the expected scaling is observed in central Au+Au collisions, due to final state interactions of the fragmenting quarks and/or the produced hadrons.
For strange hadrons, $K_S^0$ and $\Lambda$, the agreement between data and NLO pQCD calculations is not as good, or even unsatisfactory (for $\Lambda$). In those cases, however, the data do not extend to very high [$p_T$]{} and the fragmentation functions are not as well known as for the light hadrons. This needs more investigation before conclusions can be drawn about the applicability of pQCD.
The same is true for open charm production in d+Au collisions, where shape of the measured $D$ meson spectra is similar to the calculated charm quark spectra, leaving little room for softening due to fragmentation.
First results on electron production from PHENIX indicate that there is no or very little suppression of charm production in Au+Au collisions. Measurements of electron flow, on the other hand, indicate significant flow of the charmed mesons, which can only be due to significant final state interactions.
In the near future, a measurement of the [$p_T$]{}-dependence of nuclear modification factors for non-photonic electrons or maybe even open charm can be expected from the large statistics Au+Au data samples collected in run-4 at RHIC. This, together with a more accurate measurement of charm flow, will map out the interactions of charm quarks and hadrons with the medium and, through comparison with the light hadron results, may eventually shed more light on the nature of these interactions.
{#section .unnumbered}
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abstract: 'The quantum admittance of an interacting/coupled mesoscopic system and its series expansion are obtained by using the refermionization method. With the help of these non-perturbative results, it is possible to study the dependencies of the admittance according to the applied dc voltage, temperature, and frequency without any restriction on the relative values of these variables. Explicit expressions of the admittance are derived both in the limits of weak and strong interactions/coupling strength, giving clear indication of the inductive or capacitive nature of the mesoscopic system. They help to determine the conditions under which the phase of the current with respect to the ac voltage is positive.'
author:
- Adeline Crépieux
title: Series expansion of the quantum admittance in mesoscopic systems
---
Introduction
============
There is a growing interest regarding the quantum admittance of mesoscopic systems from both experimental and theoretical points of view. The reason is that this quantity contains in itself information about the dynamics of these systems: its real part provides the value of low-frequency effective resistance and is related to the anti-symmetrized current fluctuations,[@bena07; @safi08] whereas its imaginary part gives information about the inductive or capacitive nature of the system. Moreover both its real and imaginary parts make it possible to determine the phase shift of the current response with respect to the applied voltage, and appear in the expression of non-symmetrized noise[@zamoum12] which is the relevant quantity in high frequency experiments.[@aguado00; @deblock03; @billangeon06; @onac06]
Measurement of quantum admittance has been achieved recently in two dimensional electron gas,[@gabelli06; @gabelli07; @gabelli12; @hashisaka12] carbon nanotube double quantum dot,[@chorley12] superconducting junction[@basset12] and quantum dot coupled to a two dimensional electron gas.[@frey12] A formalism to calculate the quantum admittance for non-interacting systems based on scattering theory is available in the literature.[@fu93; @christien96a; @christien96b; @pretre96; @buttiker96] Effect of strong electron interactions on the admittance of a quantum dot coupled to an edge state has also been considered,[@hamamoto10] but distinct approaches were used to describe the perturbative regime (weak barrier case) and the non-perturbative one. Admittance of a double quantum dot has been calculated with an emphasis on the effective capacitance obtained in the low frequency limit.[@cottet11]
The present work focuses on the calculation of the quantum admittance using a refermionization method which makes it possible to describe entire frequency, temperature, and voltage ranges for various coupled or interacting mesoscopic systems (see Fig. \[figure1\]) in an unified way. Indeed, our results apply to: (i) a one-channel conductor coupled to a measurement circuit the resistance of which equals the quantum of resistance, an experimental configuration which is realizable with a quantum point contact;[@parmentier11] (ii) a constriction in a two dimensional electron gas in the fractional quantum Hall regime with a specific filling factor; and (iii) a quantum dot coupled to two reservoirs.
![Pictures of (a) a one-channel conductor with transmission $\tau$ coupled to a resistance equals the quantum of resistance $R_q=h/e^2$, (b) a constriction in a fractional quantum Hall bar with backscattering amplitude $\Gamma_B$, and (c) a double barrier quantum dot whose coupling amplitudes with the left (L) and right (R) reservoirs are denoted $\Gamma_L$ and $\Gamma_R$, respectively. The associated characteristic energy, $\hbar\omega_M$, is indicated for each of these systems. The coefficient $\nu$ (see text) equals $1/2$ for systems (a) and (b), and $1$ for system (c). \[figure1\]](figure1.eps){width="9cm"}
The paper is organized as follows: In Sec. II, the procedure for calculating the admittance is formulated and the model used to describe the systems of Fig. \[figure1\] is presented. The results are reported in Sec. III. In Sec. IV are emphasized the different regimes that are reached when voltage, temperature and frequency vary with respect to each other. The conclusion is given in Sec. V. Technical details of calculations are described in the Appendixes.
Formulation
===========
When an ac voltage superposed to a dc voltage $V(t)=V_0+V_\omega\cos(\omega t)$ is applied to an electronic circuit, the current becomes time dependent and can be written as a series (photo-assisted current): $I(t)=\sum_{N=-\infty}^{\infty}I^{(N)}(\omega)e^{iN\omega t}$, where $I^{(N)}$ is the $N^{th}$ harmonic of the current. By using the fact that it is a real quantity, the current can be written equivalently under the form: $$\begin{aligned}
I(t)=I^{(0)}+2\sum_{N=1}^{\infty}\mathrm{Re}\{I^{(N)}(\omega)e^{iN\omega t}\}~.\end{aligned}$$ In the linear response regime with respect to the ac voltage, i.e. $e^*V_\omega\ll\{e^*V_0,\hbar\omega\}$ where $e^*=\nu e$ is the effective charge of the carrier that travels through the mesoscopic system, the current reduces to:[@note1] $$\begin{aligned}
\label{current}
I(t)\approx I^\mathrm{dc}(V_0)+2\nu V_\omega\mathrm{Re}\{Y(\omega)e^{i\omega t}\}~,\end{aligned}$$ where $I^\mathrm{dc}$ is the dc current response to a constant voltage $V_0$, and $Y$ is the admittance defined as the derivative of the first harmonic of the current with respect to the amplitude of the ac voltage: $Y(\omega)=\nu^{-1}\partial I^{(1)}(\omega)/\partial V_\omega$. The use of the coefficient $\nu$ makes it possible to treat the three systems depicted in Fig. \[figure1\] in an unified way. From Eq. (\[current\]), one can see that the real part of the admittance (the conductance) gives the instantaneous response to the ac voltage, whereas the imaginary part of the admittance (the susceptance) is responsible for the phase difference between the current response and the ac voltage. Indeed, introducing the phase $\varphi(\omega)=\arctan[\mathrm{Im}\{Y(\omega)\}/\mathrm{Re}\{Y(\omega)\}]$, the current reads as: $I(t)=I^\mathrm{dc}(V_0)+2\nu|Y(\omega)|V_\omega\cos(\omega t+\varphi(\omega))$. The method used here to get the admittance consists of calculating the photo-assisted current, then extracting its first harmonic and finally taking its derivative with respect to $V_\omega$. Studies of photo-assisted current and electrical response to time-dependent voltage in interacting mesoscopic systems have been achieved by several authors[@sharma01; @feldman03; @schmidt07; @ma11; @safi11; @dolcini12] but they did not notice the connections with the admittance.
The systems depicted in Fig. \[figure1\] are modeled in the framework of the Tomonaga-Luttinger theory,[@tomonaga50; @luttinger63] by the Hamiltonian:[@note2] $$\begin{aligned}
~\label{hamiltonian}
H&=&\frac{\hbar v_F}{4\pi}\int_{-\infty}^{\infty}dx[(\partial_x\phi_-(x))^2+(\partial_x\phi_+(x))^2]\nonumber\\
&&+\frac{\hbar\omega_M}{4\pi}e^{i[\phi_-(x)+\phi_+(x)]/\sqrt{2}-ie^*\chi(t)/(\hbar c)}+hc~,\end{aligned}$$ where $\phi_{-}$ and $\phi_{+}$ are the bosonic fields associated with the left ($-$) and right ($+$) moving electrons, and $v_F$ is the Fermi velocity. The function $\chi(t)=-c\int V(t)dt$ is included in order to treat the time-dependent applied voltage. The energy $\hbar\omega_M$ characterizes the mesoscopic system: (i) in the case where the mesoscopic system is coupled to a measurement circuit with a resistance equal to the quantum of resistance (see Fig. \[figure1\](a)), then $\hbar\omega_M\propto (1-\tau)\tau$, where $\tau$ is the transmission through the mesoscopic system, and $\nu=1/2$ because the bias voltage seen by the mesoscopic system is $V(t)/2$;[@zamoum12] (ii) in the case of a constriction in a two dimensional electron gas in the fractional quantum Hall regime with a filling factor $\nu=1/2$ (see Fig. \[figure1\](b)), then $\hbar\omega_M\propto\Gamma_B$ where $\Gamma_B$ is the backscattering amplitude;[@crepieux04] (iii) in the case of a quantum dot coupled to left (L) and right (R) reservoirs with amplitudes $\Gamma_L$ and $\Gamma_R$ (see Fig. \[figure1\](c)), then $\hbar\omega_M\propto\Gamma_L+\Gamma_R$,[@fu93] and $\nu=1$. In the first two cases, the calculated current is the backscattering current whereas in the latter case, it corresponds to the flux of charges across the barriers.
The justification of Eq. (\[hamiltonian\]) is obvious for system (b) of Fig. \[figure1\] since the transport of charge carriers in such a system takes place along the edge states of the Hall bar: due to their one-dimensional character the edge states are well described within the Tomonaga-Luttinger theory allowing one to treat the Coulomb interactions.[@note2] Concerning the system (a) of Fig. \[figure1\], the justification for using Eq. (\[hamiltonian\]) is based on the mapping that has been established between a one-channel conductor coupled to a measurement circuit and one impurity in a Luttinger liquid.[@safi04] Successful description of transport properties has been recently achieved by this means.[@zamoum12; @parmentier11; @jezouin13] Moreover, since the expression of the admittance obtained from Eq. (\[hamiltonian\]) is identical to the result obtained for system (c) of Fig. \[figure1\] (see Appendix \[appA\] for a detailed calculation), it is possible to include the double barrier quantum dot in the list of interacting/coupled systems described here.
Results
=======
From the Hamiltonian of Eq. (\[hamiltonian\]), the photo-assisted current can be calculated with the help of a refermionization procedure[@chamon96] which has the great advantage to be a non-perturbative method: it leads to exact results whatever the value of the energy $\hbar\omega_M$. The first order harmonic of the ac current in the linear response regime with $V_\omega$ has been derived in Ref. , it leads to the admittance: $$\begin{aligned}
\label{exp_admi}
Y(\omega)&=&\frac{e^2}{2h\omega}\int_{-\infty}^{\infty}\big[t(\Omega)-t(\Omega-\omega)\big] \nonumber\\
&&\times\big[f(\hbar\Omega+e^*V_0)-f(-\hbar\Omega+e^*V_0)\big]d\Omega~,\end{aligned}$$ where $f$ is the Fermi-Dirac distribution function, and $t(\Omega)=(i\omega_M/2)/(\Omega+i\omega_M/2)$ is the transmission amplitude through: the one-channel conductor of Fig. \[figure1\](a), the constriction of Fig. \[figure1\](b), or the barriers of Fig. \[figure1\](c). Note that the real part of the admittance obeys the relation[@tucker85; @safi10; @zamoum12] $\mathrm{Re}\{Y(\omega)\}=e\sum_\pm [\pm I^\mathrm{dc}(V_0\pm\hbar\omega/e^*)/(2\hbar\omega)]$.
To characterize its low frequency behavior, the admittance can be expanded in powers of $\omega$ according to $Y(\omega)=\sum_{n=0}^\infty Y^{(n)}\omega^n$, $Y^{(n)}$ is the $n$-order harmonics, with the help of the relation: $$\begin{aligned}
t(\Omega-\omega)=\sum_{n=0}^\infty \left(\frac{2\omega}{i\omega_M}\right)^nt^{n+1}(\Omega)~.\end{aligned}$$ The details of the calculation are given in Appendix \[appB\], we obtain: $$\begin{aligned}
\label{coeff_n}
Y^{(n)}&=&-\frac{2^{n}e^2}{h(i\omega_M)^{n+1}}\int_{-\infty}^{\infty}\left[t(\Omega)\right]^{n+2}\nonumber\\
&&\times\left[f(\hbar\Omega+e^*V_0)-f(-\hbar\Omega+e^*V_0)\right]d\Omega~,\end{aligned}$$ which is purely real for odd values of $n$ and purely imaginary for even values of $n$. Eqs. (\[exp\_admi\]) and (\[coeff\_n\]) are the central results of this work since they make it possible to calculate the admittance, and all the coefficients of its expansion in powers of $\omega$, whatever the dc voltage $V_0$, temperature $T$, and frequency $\omega_M$. In the next section, the behavior of the admittance in various regimes is discussed by examining the combined effects of the energies $\hbar\omega_M$, $e^*V_0$, $\hbar\omega$, and $k_BT$.
Discussion
==========
Zero temperature limit
----------------------
In this limit, the integration over frequency in Eq. (\[exp\_admi\]) can be performed explicitly: $$\begin{aligned}
\label{Y_T0}
Y_{T=0}(\omega)&=&\frac{e^2}{4i h}\frac{\omega_M}{\omega}\sum_\pm\mathrm{ln}\left(1+\frac{i\hbar\omega}{\hbar\omega_M/2\pm ie^*V_0}\right)~.\nonumber\\\end{aligned}$$ The real and imaginary parts of Eq. (\[Y\_T0\]), which are related by the Kramers-Kronig relation, coincide with the one obtained with the help of scattering theory in the case of a double barrier quantum dot.[@fu93; @buttiker96] However, the compact writing of the zero temperature admittance as formulated by Eq. (\[Y\_T0\]) is novel.
At zero temperature, the associated coefficients of the series expansion read as: $$\begin{aligned}
\label{series_T0}
Y_{T=0}^{(n)}&=&\frac{e^2}{4h}\sum_\pm\frac{(-i)^n\hbar^{n+1}\omega_M}{(n+1)(\hbar\omega_M/2\pm ie^*V_0)^{n+1}}~.\end{aligned}$$ In Fig. \[figure2\](a) is plotted the real part of the admittance at zero temperature as a function of voltage. Steps are observed at positions $e^*V_0=\pm\hbar\omega$ (see black solid and purple dotted lines) which disappear when $\hbar\omega_M$ increases. The profile of the imaginary part of the admittance shown in Fig. \[figure2\](b) is in agreement with recent experimental data[@basset12] confirming that the distance between the two peaks observed at weak $\hbar\omega_M$ is equal to $2\hbar\omega$. Fig. \[figure2\](c) reveals interesting features which highlight the fact that the regimes of weak and strong $\hbar\omega_M$ differ fundamentally. Indeed, in the limit $\hbar\omega_M\gg e^*V_0$, the Nyquist diagram is a quasi-circle (see the green long dashed line in Fig. \[figure2\](c)) which means that the admittance is a function of a complex variable $z$ with a single pole $z_p$ of order 1, i.e. of the form: $(z-z_p)^{-1}$. When $\hbar\omega_M$ decreases, singularities and loops appear (see red dashed, blue dash-dotted and purple dotted lines in Fig. \[figure2\](c)). This means that the admittance is a complex function with a single pole of higher order. As a consequence, the sign of the phase of the admittance will change when $\omega$ varies. It is precisely what is observed in Fig. \[figure2\](d) where the phase of the admittance is depicted as a function of frequency. The sign of the phase gives indication about the inductive (i.e. $\varphi(\omega)<0$) or capacitive (i.e. $\varphi(\omega)>0$) character of the mesoscopic system. For weak values of $\hbar\omega_M$ and for frequency within the interval $[-e^*V_0/\hbar,e^*V_0/\hbar]$, the current is in phase opposition with the ac voltage since $\varphi(\omega)\approx\pi/2$ (see the black solid line) and the mesoscopic system behaves as a capacitor. At high frequency, the system becomes inductive whatever the value of $\hbar\omega_M$ is. This dependence of the phase with the frequency is in qualitative agreement with experimental data obtained recently in carbon nanotube.[@chorley12]
![(a) Real part, (b) imaginary part, (c) Nyquist diagram, and (d) phase of the admittance at $T=0$, for $\hbar\omega_M=0.02e^*V_0$ (black solid line), $\hbar\omega_M=0.2e^*V_0$ (purple dotted line), $\hbar\omega_M=e^*V_0$ (blue dash-dotted line), $\hbar\omega_M=2e^*V_0$ (red dashed line) and $\hbar\omega_M=4e^*V_0$ (green long dashed line). The black solid line in graph (c) is not shown for visibility reasons. $\mathrm{Re}\{Y(\omega)\}$ and $\mathrm{Im}\{Y(\omega)\}$ are depicted in units of $G_q=e^2/h$, the quantum of conductance. \[figure2\]](figure2.eps "fig:"){width="4cm"} ![(a) Real part, (b) imaginary part, (c) Nyquist diagram, and (d) phase of the admittance at $T=0$, for $\hbar\omega_M=0.02e^*V_0$ (black solid line), $\hbar\omega_M=0.2e^*V_0$ (purple dotted line), $\hbar\omega_M=e^*V_0$ (blue dash-dotted line), $\hbar\omega_M=2e^*V_0$ (red dashed line) and $\hbar\omega_M=4e^*V_0$ (green long dashed line). The black solid line in graph (c) is not shown for visibility reasons. $\mathrm{Re}\{Y(\omega)\}$ and $\mathrm{Im}\{Y(\omega)\}$ are depicted in units of $G_q=e^2/h$, the quantum of conductance. \[figure2\]](figure3.eps "fig:"){width="4.1cm"} ![(a) Real part, (b) imaginary part, (c) Nyquist diagram, and (d) phase of the admittance at $T=0$, for $\hbar\omega_M=0.02e^*V_0$ (black solid line), $\hbar\omega_M=0.2e^*V_0$ (purple dotted line), $\hbar\omega_M=e^*V_0$ (blue dash-dotted line), $\hbar\omega_M=2e^*V_0$ (red dashed line) and $\hbar\omega_M=4e^*V_0$ (green long dashed line). The black solid line in graph (c) is not shown for visibility reasons. $\mathrm{Re}\{Y(\omega)\}$ and $\mathrm{Im}\{Y(\omega)\}$ are depicted in units of $G_q=e^2/h$, the quantum of conductance. \[figure2\]](figure4.eps "fig:"){width="3.8cm"} ![(a) Real part, (b) imaginary part, (c) Nyquist diagram, and (d) phase of the admittance at $T=0$, for $\hbar\omega_M=0.02e^*V_0$ (black solid line), $\hbar\omega_M=0.2e^*V_0$ (purple dotted line), $\hbar\omega_M=e^*V_0$ (blue dash-dotted line), $\hbar\omega_M=2e^*V_0$ (red dashed line) and $\hbar\omega_M=4e^*V_0$ (green long dashed line). The black solid line in graph (c) is not shown for visibility reasons. $\mathrm{Re}\{Y(\omega)\}$ and $\mathrm{Im}\{Y(\omega)\}$ are depicted in units of $G_q=e^2/h$, the quantum of conductance. \[figure2\]](figure5.eps "fig:"){width="4cm"}
Low frequency limit
-------------------
When the ac frequency $\omega$ is much smaller than all the other characteristic frequencies, i.e. $\omega_M$, $k_BT/\hbar$ and $e^*V_0/\hbar$, explicit expressions of the admittance up to the first order in $\omega$ can be obtained from Eq. (\[coeff\_n\]) by retaining only the terms $n=0$ and $n=1$ in the sum over $n$. The results are summarized in the two first lines of Table \[table1\]. Again, it is interesting to interpret them in terms of an effective RLC circuit. At strong $\hbar\omega_M$, since $\varphi(\omega)<0$, the mesoscopic system is inductive whatever the temperature is: the mesoscopic system behaves as a RL circuit in series, with an effective resistance independent of any characteristic energy: $R_{\mathrm{eff}}=G_q^{-1}$, and an effective inductance which depends on $\omega_M$: $L_{\mathrm{eff}}=(\omega_MG_q)^{-1}$. Note that the admittance converges to $G_q$ at very large $\hbar\omega_M$.
Other limits
------------
At weak $\hbar\omega_M$, the dynamics of the mesoscopic system is strongly temperature dependent: at low temperature (compared to the voltage), the system is capacitive ($\varphi(\omega)>0$) whereas it is inductive ($\varphi(\omega)<0$) at higher temperature. The fact that at zero temperature a change from capacitive to inductive circuit is observed when $\hbar\omega_M$ increases is in agreement with what was obtained in Ref. for a double barrier quantum dot. Indeed, at weak coupling $\hbar\omega_M\propto \Gamma_L+\Gamma_R$, this system is capacitive because of the small value of the transmission coefficient equal to $\mathcal{T}(\Omega)=t(\Omega)t^*(\Omega)$, with no possibility of activation by any energy since both temperature and frequencies are weak. It is interesting to notice that in the limit $\hbar\omega_M\approx 0$, the expression of the effective resistance, $R_\mathrm{eff}$, always contains the interactions/coupling energy, $\hbar\omega_M$, and the largest of the three other characteristic energies, i.e. either $k_BT$, $e^*V_0$, or $\hbar\omega$. Indeed, at high temperature: $G_qR_\mathrm{eff}=8k_BT/(\pi \hbar\omega_M)$, in agreement with Ref. ; at high voltage: $G_qR_\mathrm{eff}=(2e^*V_0)^2/(\hbar\omega_M)^2$; and at high frequency: $G_qR_\mathrm{eff}=4\omega/(\pi\omega_M)$. Furthermore, at high temperature and whatever the value of $\hbar\omega_M$ is, the effective resistivity reads as: $$\begin{aligned}
R_\mathrm{eff}^{-1}=\frac{\partial I^\mathrm{dc}(V_0)}{\partial V_0}=\frac{e^2\hbar\omega_M}{16\pi hk_BT}\Psi'\left(\frac{1}{2}+\frac{\hbar\omega_M}{8\pi k_BT}\right)~,\end{aligned}$$ in agreement with Ref. ($\Psi'$ is the derivative of the digamma function).
$\hbar\omega_M\approx 0$ $\hbar\omega_M\gg\{k_BT,|e^*V_0|,|\hbar\omega|\}$
------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------
$Y(\omega)=G_q\left(\frac{\hbar\omega_M}{2e^*V_0}\right)^2\left(1+i\frac{\omega}{\omega_M}\right)$
$\Rightarrow$ RC circuit in parallel
$R_\mathrm{eff}^{-1}=G_q\left(\frac{\hbar\omega_M}{2e^*V_0}\right)^2$, $C_\mathrm{eff}^{-1}=\omega_MR_\mathrm{eff}$
$Y(\omega)=G_q\frac{\pi \hbar\omega_M}{8k_BT}\left(1-i\frac{\hbar\omega}{4k_B T}\right)$ $Y(\omega)=G_q\left(1-i\frac{\omega}{\omega_M}\right)$
$\Rightarrow$ RL circuit in series $\Rightarrow$ RL circuit in series
$R_\mathrm{eff}^{-1}=G_q\frac{\pi \hbar\omega_M}{8k_BT}$, $L_\mathrm{eff}^{-1}=\pi\omega_MG_q/2$ $R_{\mathrm{eff}}^{-1}=G_q$, $L_{\mathrm{eff}}^{-1}=\omega_MG_q$
$Y(\omega)=G_q\frac{\omega_M}{4\omega}\left[\pi\mathrm{sign}(\omega)-i\ln\left(\frac{(\hbar\omega)^2}{(\hbar\omega_M/2)^2+(e^*V_0)^2}\right)\right]$
$k_BT\ll |e^*V_0|\ll|\hbar\omega|$ $\Rightarrow$ RL circuit in parallel
$R_\mathrm{eff}^{-1}=G_q\frac{\pi\omega_M}{4|\omega|}$,$L_\mathrm{eff}^{-1}=\frac{G_q\omega_M}{4}\ln\left(\frac{(\hbar\omega)^2}{(\hbar\omega_M/2)^2+(e^*V_0)^2}\right)$
: Admittance $Y(\omega)$ and its associated effective resistance and inductance/capacitance in various limits. $G_q=e^2/h$ refers to the quantum of conductance.[]{data-label="table1"}
In order to understand the relative effects of the various energies that characterize the system, the boundary between regions of positive and negative phases is drawn in Fig. \[figure3\] as a function of $\hbar\omega_M$ and $\hbar\omega$, both taken in units of $e^*V_0$. At zero temperature, the equation for the boundary line can be extracted from Eq. (\[Y\_T0\]), it reads as: $|\hbar\omega|=\sqrt{2[(e^*V_0)^2-(\hbar\omega_M/2)^2]}$. Below this line, the phase is positive whereas above it, the phase is negative (see Fig. \[figure3\](a)). The phase separation line moves to the origin when the temperature increases and so the region of positive phase collapses (see Fig. \[figure3\](b)). Accordingly, to get a positive value for the phase, i.e. a capacitive behavior of the mesoscopic system, both temperature and characteristic frequencies must be small enough in comparison to the dc voltage. More precisely, one must have: $\hbar\omega_M/2\lesssim|e^*V_0|$, $|\hbar\omega|\lesssim\sqrt{2[(e^*V_0)^2-(\hbar\omega_M/2)^2]}$, and $k_BT\lesssim|e^*V_0|$.
![(a) Sign of the phase of the admittance at zero temperature as a function of coupling/interaction $\hbar\omega_M$ and ac frequency $\hbar\omega$. (b) Evolution of the boundary between regions of positive and negative signs when temperature $T$ increases (in units of $e^*V_0/k_B$). \[figure3\]](figure6.eps "fig:"){width="4.2cm"} ![(a) Sign of the phase of the admittance at zero temperature as a function of coupling/interaction $\hbar\omega_M$ and ac frequency $\hbar\omega$. (b) Evolution of the boundary between regions of positive and negative signs when temperature $T$ increases (in units of $e^*V_0/k_B$). \[figure3\]](figure7.eps "fig:"){width="4.2cm"}
Conclusion
==========
The quantum admittance associated to one impurity in a Tomonaga-Luttinger liquid with $\nu=1/2$ has been calculated with the help of a refermionization method. Its expression is identical to the one of a double barrier quantum dot. Furthermore, in light of the mapping of Ref. , it is possible to consider in an unified view the three systems depicted in Fig. \[figure1\] and to give an unique and compact expression for their admittance. A detailed analysis of the various limits has shown that if at least one of the characteristic energies is larger than the dc applied voltage, the mesoscopic system behaves has an inductance since it is activated by either temperature, frequency or interactions/coupling frequency.
Acknowledgement
===============
The author thanks M. Lavagna, I. Safi and R. Zamoum for discussions.
Admittance calculation for system (c) depicted on Fig. \[figure1\] {#appA}
==================================================================
The time-dependent electric current which flows from the left (L) or the right (R) reservoir to the central region of a double barrier junction reads as:[@jauho94] $$\begin{aligned}
&&I_{L,R}(t)=\frac{2e\Gamma_{L,R}}{2\pi}\int_{-\infty}^{\infty} f(\hbar\Omega-eV_{L,R})\mathrm{Im}\{A(\Omega,t)\} d\Omega~.\nonumber\\\end{aligned}$$ In the case where an ac voltage superposed to a dc voltage $V(t)=V_0+V_\omega\cos(\omega t)$ is applied to the dot, the spectral function $A$ is given by: $$\begin{aligned}
A(\Omega,t)&=&\sum_{p=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}J_p\left(\frac{eV_\omega}{\hbar\omega}\right)J_m\left(\frac{eV_\omega}{\hbar\omega}\right)\nonumber\\
&&\times\frac{e^{i(p-m)\omega t}}{E_p(\Omega)+i\hbar\omega_M/2}~,\end{aligned}$$ where $\hbar\omega_M=\Gamma_L+\Gamma_R$, $E_p(\Omega)=\hbar\Omega-eV_0-p\hbar\omega$, and $J_p$ is the Bessel function of order $p$. From this expression, the harmonics of the photo-assisted current can be extracted: $$\begin{aligned}
I_{L,R}^{(N)}&=&\frac{ie\Gamma_{L,R}}{2\pi}\sum_{\pm}\Bigg\{\pm\sum_{p=-\infty}^{\infty}J_p\left(\frac{eV_\omega}{\hbar\omega}\right)J_{p\pm N}\left(\frac{eV_\omega}{\hbar\omega}\right)\nonumber\\
&&\times\int_{-\infty}^{\infty}\frac{f(\hbar\Omega-eV_{L,R})}{E_p(\Omega)\mp i\hbar\omega_M/2}d\Omega\Bigg\}~.\end{aligned}$$
By considering that the left and right reservoirs are set at the same voltage, i.e. $V_L=V_R=0$, the first harmonic of the current reduces to: $$\begin{aligned}
\label{first_harmonic}
I_{L,R}^{(1)}&=&\frac{ie\Gamma_{L,R}}{2\pi}\sum_{\pm}\Bigg\{\pm\sum_{p=-\infty}^{\infty}J_p\left(\frac{eV_\omega}{\hbar\omega}\right)J_{p\pm 1}\left(\frac{eV_\omega}{\hbar\omega}\right)\nonumber\\
&&\times\int_{-\infty}^{\infty}\frac{f(\hbar\Omega)}{E_p(\Omega)\mp i\hbar\omega_M/2}d\Omega\Bigg\}~.\end{aligned}$$
Using the fact that in the limit $x\rightarrow 0$, we have $J_0(x)\approx 1$ and $J_{\pm 1}(x)\approx \pm x/2$, the first harmonic of the current up to the first order in $V_\omega$ (linear response) reduces to: $$\begin{aligned}
I_{L,R}^{(1)}&=&\frac{ie\Gamma_{L,R}}{2\pi}\sum_{\pm}\left(\frac{eV_\omega}{2\hbar\omega}\right)\int_{-\infty}^{\infty}\frac{ f(\hbar\Omega)}{E_0(\Omega)\mp i\hbar\omega_M/2}d\Omega\nonumber\\
&&-\frac{ie\Gamma_{L,R}}{2\pi}\left(\frac{eV_\omega}{2\hbar\omega}\right)\int_{-\infty}^{\infty}\frac{ f(\hbar\Omega)}{E_1(\Omega)+ i\hbar\omega_M/2}d\Omega\nonumber\\
&&-\frac{ie\Gamma_{L,R}}{2\pi}\left(\frac{eV_\omega}{2\hbar\omega}\right)\int_{-\infty}^{\infty}\frac{ f(\hbar\Omega)}{E_{-1}(\Omega)- i\hbar\omega_M/2}d\Omega~.\nonumber\\\end{aligned}$$
The first, second and third terms in this expression correspond respectively to the terms $p=0$, $p=1$ and $p=-1$ in the sum over $p$ of Eq. (\[first\_harmonic\]). Finally: $$\begin{aligned}
I^{(1)}(\omega)&=&\frac{I_{L}^{(1)}+I_{R}^{(1)}}{2}=\frac{e^2V_\omega}{2h\omega}\int_{-\infty}^{\infty}f(\hbar\Omega)\nonumber\\
&&\times\Big[t(\Omega-eV_0/\hbar)-t(\Omega-\omega-eV_0/\hbar)\nonumber\\
&&-t^*(\Omega-eV_0/\hbar)+t^*(\Omega+\omega-eV_0/\hbar)\Big]d\Omega~,\nonumber\\\end{aligned}$$ which leads after a change of variables to the admittance: $$\begin{aligned}
\label{admittance_system_c}
Y(\omega)&=&\frac{\partial I^{(1)}(\omega)}{\partial V_\omega}=\frac{e^2}{2h\omega}\int_{-\infty}^{\infty}f(\hbar\Omega+eV_0)\nonumber\\
&&\times\Big[t(\Omega)-t(\Omega-\omega)-t^*(\Omega)+t^*(\Omega+\omega)\Big]d\Omega~.\nonumber\\\end{aligned}$$ This result is identical to Eq. (\[exp\_admi\]) since $e^*=e$ for the system (c) of Fig. 1.
Series expansion of the admittance {#appB}
==================================
Eq. (\[exp\_admi\]) can be expanded in powers of $\omega$. The starting point is the definition of the amplitude $t(\Omega-\omega)$ which appears in the expression of the admittance that can be transformed in this way: $$\begin{aligned}
t(\Omega-\omega)&=&\frac{i\omega_M/2}{\Omega-\omega+i\omega_M/2}\nonumber\\
&=&\frac{i\omega_M/2}{(\Omega+i\omega_M/2)\left(1-\frac{\omega}{\Omega+i\omega_M/2}\right)}~.\end{aligned}$$ It can be expressed in term of amplitude $t(\Omega)$ through: $$\begin{aligned}
t(\Omega-\omega)&=&t(\Omega)\sum_{n=0}^\infty\left(\frac{\omega}{\Omega+i\omega_M/2}\right)^n\nonumber\\
&=&t(\Omega)\sum_{n=0}^\infty\left(\frac{\omega}{i\omega_M/2}\right)^n\left(\frac{i\omega_M/2}{\Omega+i\omega_M/2}\right)^n\nonumber\\
&=&t(\Omega)\left(1+\sum_{n=1}^\infty\left(\frac{\omega}{i\omega_M/2}\right)^n\left[t(\Omega)\right]^n\right)~.\nonumber\\\end{aligned}$$
Reporting this result in the expression (\[exp\_admi\]) of the admittance, it leads to: $$\begin{aligned}
Y(\omega)&=&
-\frac{e^2}{2h}\sum_{n=0}^\infty\frac{\omega^{n}}{(i\omega_M/2)^{n+1}}\int_{-\infty}^{\infty}\left[t(\Omega)\right]^{n+2}\nonumber\\
&&\times\big[f(\hbar\Omega+e^*V_0)-f(-\hbar\Omega+e^*V_0)\big]d\Omega~,\nonumber\\\end{aligned}$$ which finally gives the expression (\[coeff\_n\]) of the harmonics of the admittance.
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abstract: 'We have entered the phase of extrasolar planets characterization, probing their atmospheres for molecules, constraining their horizontal and vertical temperature profiles and estimating the contribution of clouds and hazes. We report here a short review of the current situation using ground based and space based observations, and present the transmission spectra of HD189733b in the spectral range 0.5-24 microns.'
---
Introduction
============
Extrasolar Giant Planets (EGPs) are now being discovered at an ever increasing pace (Butler et al., 2007, Schneider, 2008). And as a result, planetary scientists and astronomers are increasingly called upon to make the transition from [*discovery*]{} to [*characterization*]{}, so that we can begin the long task of understanding these planets in the same way that we understand those in our own Solar System. For a growing sample of giant extrasolar planets orbiting very close to their parent star, we can already probe their atmospheric constituents using transit techniques. A stellar occultation (called primary transit) occurs when the light from a star is partially blocked by an intervening body, such as a planet, from reaching an observer. With this method, we can indirectly observe the thin atmospheric ring surrounding the optically thick disk of the planet -the limb- while the planet is transiting in front of its parent star. This method was traditionally used to probe the atmospheres of planets in our Solar System and most recently, thanks to the [*Hubble Space Telescope*]{} and [*Spitzer*]{}, was successfully applied to a growing sample of giant exoplanets orbiting very close to their parent star - the so called “Hot-Jupiters”-. The idea was first theoretically proposed by Seager and Sasselov in 2000, and confirmed experimentally by Charbonneau [*et al.*]{} in 2002, when he and his team first detected the presence of Sodium in the atmosphere of a hot-Jupiter.
In the secondary transit technique, we observe first the combined spectrum of the star and the planet. Then, we take a second measurement of the star alone when the planet disappears behind it: the difference between the two measurements consists of the planet’s contribution. This technique was pioneered by two different teams in 2005, using the Spitzer Space Telescope to probe two Hot-Jupiters in the Infrared (Deming et al., 2005; Charbonneau et al., 2005). A similar measurement was performed in 2006 by Harrington and collaborators on a non-transiting planet by monitoring through time the combined star-planet flux. The light-curve obtained in this way, allowed to understand some thermal properties of the planet Upsilon Andromedae b.
[**Primary transit.** ]{} The use of transmission spectroscopy to probe the outer layers of the transiting hot Jupiters has been particularly successful in the UV and visible spectral ranges (Charbonneau et al., 2002; Vidal-Madjar et al., 2003; Ballester, Sing & Herbert 2007; Ben Jaffel 2007, 2008; Knutson et al., 2007a, Pont et al., 2007, Redfield et al., 2008) and only more recently it was attempted in the Near and Middle Infrared spectral window, producing novel and extremely interesting results (Richardson et al., 2006, Deming et al. 2007, Knutson et al., 2007b, 2008a, Beaulieu et al., 2008, Tinetti et al., 2007, Swain et al., 2008a, Nutzman P., et al., 2008, Agol et al., Knutson et al., Harington et al. this volume). Transmission spectra are sensitive to atomic and molecular abundances and less to temperature variation. Temperature influences the transmission spectrum by way of its influence on the atmospheric scale height (Brown 2001) and the absorbtion coefficients.
[**Secondary transit.**]{} With this method we can probe the photons that are directly emitted (Charbonneau et al., 2005, Deming et al., 2005), or reflected by the planet (Cameron et al. 1999, Leigh et al., 2003, Rowe et al., 2006). So far, the focus has been on the brightest stars with transiting extrasolar planets, namely HD 209458b (Charbonneau et al., 2000), HD 189733b (Bouchy et al., 2005) and GJ436b (Butler et al., 2004), HD 149026b (Sato et al., 2005), TRES-1 (Alonso et al., 2004).
In the infrared spectral range, with this technique we can not only detect the molecular species showing a noticeable rotational/vibrational signature, but also constrain the bulk temperature and the vertical thermal gradient (Knuston et al., 2007b, 2008b; Burrows et al., 2007; Charbonneau 2008, Barman 2008, Harrington et al. 2006, 2007; Swain et al., 2008b). Compared to transmission spectroscopy, emission spectroscopy may scan different regions of the atmosphere for molecular signatures and cloud/hazes contributions (Brown, 2001; Richardson et al., 2007). Same considerations are valid in the UV-visible spectral range, except that the photons reflected by the planet do not bring any information about the planetary temperature and the thermal structure, but about the planetary albedo (Rowe et al., 2006) and the presence of atomic/ionic/molecular species having electronic transitions.
Finally, especially if the planet is tidally locked, with primary and secondary transit techniques we can observe different phases of the planet along its orbit. During the primary transit we can sound the terminator, whereas during secondary we can above all observe the day-side.
[**Light-curves.** ]{} Monitoring the light-curve of the combined star-planet spectrum, can be a useful approach both for transiting (Knutson et al., 2007, 2008) and non-transiting planets (Harrington et al., 2006). In the latter case the planetary radius can not be measured, but we can appreciate the temperature or albedo variations through time (depending if the observation is performed in the visible or infrared).
The problems that we can tackle with current telescopes are :
- Detection of the main molecular species in the hot transiting planets’ atmosphere.
- Constraint of the horizontal and vertical thermal gradients in the hot exoplanet atmospheres.
- Presence of clouds or hazes in the atmospheres.
Today we can use two approaches to reach these objectives:
1. Broad band or low resolution spectroscopy from a space based observatory. This can be accomplished by SPITZER or HST.
2. High resolution spectroscopy from ground based observatories in the optical and NIR.
The next steps with these indirect techniques will be:
- Detection of minor atmospheric species and constraint of their abundance.
- More accurate spectral retrieval to map thermal and chemistry gradients in the atmospheres.
- Cloud microphysics: understanding the composition, location and optical parameters of cloud/haze particles.
- Cooler and smaller planets, possibly in the habitable zone.
Futher into the future the James Webb Space Telescope or the JAXA/ESA SPICA mission concept (Nakagawa et al., 2003) will be the next generation of space telescopes to be online. They will guarantee high spectral resolution from space and the characterization of fainter targets, allowing us to expand the variety of “characterizable” extrasolar planets.
Temperature profiles
====================
With photometry, or low resolution spectroscopy in the Near and Mid Infrared we are today able to put some constraints on the thermal horizontal and vertical profiles of the planetary atmospheres. For instance, we are already in a position of appreciating the differences between HD189733b (Knutson et al., 2007b) and Upsilon Andromedae (Harrington, et al., 2006): at 8and 24 $\mu m$ HD189733b shows a well mixed temperature distribution between the day and the night side, while we have the opposite for Upsilon Andromedae at 24 $\mu $m. HD209458b shows clear signs of a thermal inversion at relatively low altitude (Burrows et al., 2007), the situation is different for HD189733b (Barman 2008, Swain et al., 2008b). However high resolution spectroscopy is needed to perform a more accurate spectral retrieval and better constrain the dynamics of planetary atmosphere.
{width="10cm"}
Molecules in the atmosphere of hot-Jupiters
===========================================
[**Water.**]{} In a star-planet system, a significant amount of water vapor can only exist in planetary atmospheres at orbital distances small enough (less than $\sim 1 $AU for a solar like star). This closeness requirement is well met by hot Jupiters. According to photochemical models, H$_2$O should be among the most abundant species (after H$_2$) in the lower atmosphere of giant planets orbiting close to their parent stars (Liang et al., 2003; 2004). Moreover, according to our calculations (Tinetti et al., 2007a), H$_2$O is the easiest of these species to detect in primary transit in the IR. A very accurate data list for water at hot Jupiter-like temperatures, has been calculated by Barber et al. (2006).
[**Carbon bearing molecules**]{}, such as $CO$, $CH_4$, $CO_2$, $C_2H_2$ are expected to be abundant as well, depending on the C/O ratio and the efficiency of the photochemistry in the upper atmosphere (Liang et al., 2004). $CO$, $CH_4$, $CO_2$ have already been detected (Barman, 2008, Charbonneau et al. 2008, Swain et al., 2008a, Swain et al., 2008b). For less abundant species, or with spectral signatures which are harder to detect, we need to make the leap to high resolution spectroscopy.
Also, improved line lists at high temperatures are needed to better interpret the measurements. From preliminary results and models, it is not excluded that the chemistry might vary substantially from the highly irradiated day-side to the non-illuminated night-side of these planets (Cooper and Showman, 2006; Swain et al., 2008a).
[**Nitrogen or sulfur-bearing molecules**]{} are also likely to be present in the atmospheres of hot-Jupiters, but their weaker signatures may be difficult to be caught with a low resolving power (Sharp and Burrows, 2007).
[**H$_3^+$**]{} – the simple molecular ion formed by the photo-ionisation of H$_2$ – could be a crucial indicator of the escape processes in the upper atmosphere (Yelle, 2004, 2006). Now that HST/STIS instrument is no longer operative to observe the Lyman alpha line in the UV (Vidal-Madjar et al. 2003, Ben-Jaffel, 2007), H$_3^+$ is the only molecular ion able to monitor the escape processes on hot Jupiters. So H$_3^+$ is a crucial detection target; even if detection is unsuccessful, such measurement provides at least an improved upper limit on its abundance (Shkolnik et al., 2006). Calculations of the H$_3^+$ abundance on a hot-Jupiter (Miller et al., 2000; Yelle, 2004) show that the contrast between H$_3^+$ emission and stellar brightness places it just on the current limit of detectability with a large ground-based telescope. We stress that even if it is a very challenging observation, it would be the best diagnostics to understand the properties of the upper atmospheres of Hot-Jupiters (Koskinen et al., 2007) whereas observations of the Lyman alpha line could be partially or totally contaminated by energetic neutral atoms from charge exchange between stellar wind protons and neutral hydrogen from the planetary exospheres (Holmstrom et al., 2008) or inadequately analyzed and understood as stressed by Ben-Jaffel (2008).
[**Clouds and hazes**]{}. At the spectral resolution we can obtain today from space, the best we can do is to assess their presence, as they are supposed to flatten the spectral signatures or modify the spectral shape. In the case of transmission spectroscopy, they cause the atmosphere to be opaque at higher altitudes (Brown, 2001). The HST observations from Pont et al., (2008) show an almost featureless transmission spectrum in the range $0.5-1 \mu $m which may suggest the presence of hazes.
{width="10cm"}
As an illustration we present the transmission spectrum of HD189733b in the wavelength range $0.5-25 \mu $m in Figure 1. The overall transmission spectrum is shaped by the water absorption in the infrared. H2-H2, methane and alkali metals absorptions are included, as well as a crude simulation of hazes opacity. Notice that the different data collected by instruments over a wide wavelength range are giving consistent results. Most probably additional molecules are present, but we are unable to appreciate their presence at this spectral resolution. For instance, in Figure 3., we show the additional contribution of a variety of plausible molecules as a function of wavelength: the contributions of Methane at 3.2 $\mu$m, CO at 4.5 $\mu$m and ammonia at 11 $\mu$m are quite noticeable. Although water can be detected with broad band photometry, it is clear that spectral resolution is needed in order to get the probe for the different species.
Towards smaller mass planets
============================
For both primary and secondary transit methods, smaller size/colder planets will increase the challenge. Transmission spectroscopy can benefit from very extended atmospheres : this scenario can occur if the main atmospheric component has a light molecular weight and a high temperature. The lighter, the hotter and the smaller the core, the easier is the observation in transmission spectroscopy (ie the more detectable are the spectral features). In the case of secondary transits, the parameters playing the major role are the size of the planet compared to its parent star and the planetary temperature for observations in the IR or the albedo in the visible.
With current telescopes we can already approach the case of hot Neptunes transiting later type stars, e.g. Gliese 436b (Deming et al., 2007). Figure 3 shows a simulated transmission spectrum of Gliese 436b. Note that even if the transit depth is smaller than a hot Jupiter one (0.7 %), molecules such as methane could leave signatures of similar order of magnitude ($\sim$0.05 %).
{width="10cm"}
For Earth-size planets and/or colder atmospheres, we need to wait for JWST (Cornia and Tinetti, 2007; Cavarroc et al., 2006, 2008).
Conclusion
==========
Probing the exoplanet atmospheres with transiting techniques has a bright and exciting future, both from space and from the ground. There are two main approaches: primary and secondary transit methods. In this proceeding we have reviewed how they can be complementary, and what are their inherent limitations. Taken together, and over a broad spectral range, these methods allow us to reach for exoplanet atmospheres similar level of knowledge that scientists had of the planets in the solar system at the time of Voyager 1.
[**Acknowledgements**]{}\
It is a great pleasure to thank our collaborators for many exciting and fruitful discussions, in particular Mark Swain, Gautam Vasisht, Ignasi Ribas, Sean Carey, Jeroen Bouwman, Danie Liang, Yuk Yung, Eric Agol, Jonathan Tennyson, Alan Aylward, Bob Barber, Steve Miller, Virginie Batista, Pieter Deroo, David Kipping and Tommi Koskinen. J.P.B. and G.T. acknowledge the financial support of the ANR HOLMES.
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abstract: 'Finite-difference Lattice Boltzmann (LB) models are proposed for simulating gas flows in devices with microscale geometries. The models employ the roots of half-range Gauss-Hermite polynomials as discrete velocities. Unlike the standard LB velocity-space discretizations based on the roots of full-range Hermite polynomials, using the nodes of a quadrature defined in the half-space permits a consistent treatment of kinetic boundary conditions. The possibilities of the proposed LB models are illustrated by studying the one-dimensional Couette flow and the two-dimensional driven cavity flow. Numerical and analytical results show an improved accuracy in finite Knudsen flows as compared with standard LB models.'
author:
- 'G. P. Ghiroldi'
- 'L. Gibelli'
title: 'A finite-difference lattice Boltzmann approach for gas microflows'
---
The outgrowing development of [microelectromechanical systems]{} (MEMS) has spurred interest in studying gas flows in devices with microscale geometries. These flows, which are referred to as gas microflows, are usually distinguished by relatively large Knudsen numbers and small Mach numbers. Because of the large Knudsen numbers, the conventional hydrodynamic approach breaks down and a description based on the Boltzmann equation is required. The low Mach numbers permit to replace the collision integral of the Boltzmann equation with a simpler kinetic model, such as the one proposed by Bathnagar, Gross and Krook and, independently, by Welander (BGKW), as well as to linearize the resulting kinetic equation around the equilibrium state [@c88]. Numerous studies on gas microflows based on the numerical solution of the linearized BGKW kinetic model equation [@lgffc07; @nv07; @aczgkp11] and/or the linearized Boltzmann equation [@cffggl08] have been reported over the years. Recently, the Lattice Boltzmann (LB) method has attracted considerable interest as an alternative tool for studying gas flows in microfluidic devices [@zgbe06; @iri11; @kpb08; @mz11a; @sbys11; @mzs11; @mz11b]. Although it was evolved from lattice-gas cellular automaton models for mimicking the Navier-Stokes hydrodynamics, the LB method can potentially describe gas microflows since it can be viewed as the discrete ordinate method to solve the linearized BGKW kinetic model equation [@hl87; @sh98]. By using the same notation as in Ref. [@mz11b], the non dimensional form of LB models in the absence of external force fields reads $$\label{eq:BGKW_lattice}
\frac{\partial f_\alpha}{\partial t} + {\bf {\boldsymbol{\xi}}_\alpha} \cdot \nabla{f_\alpha} = -\frac{1}{{\mbox{Kn}}} \left( f_\alpha -f_\alpha^{eq} \right),$$ where the Knudsen number is defined as ${\mbox{Kn}}=\mu_0 \sqrt{RT_0}/(p_0 l)$ with $\mu_0, T_0, p_0$ the reference gas viscosity, temperature and pressure, respectively, $l$ the characteristic length and $$\begin{aligned}
\label{eq:f_alpha}
f_\alpha ({\boldsymbol{r}},t)= w_\alpha f({\boldsymbol{r}},{\boldsymbol{\xi}}_\alpha,t)/\omega({\boldsymbol{\xi}}_\alpha), \\
\label{eq:feq_alpha}
f^{eq}_\alpha ({\boldsymbol{r}},t)= w_\alpha f^{eq}({\boldsymbol{r}},{\boldsymbol{\xi}}_\alpha,t)/\omega({\boldsymbol{\xi}}_\alpha),\end{aligned}$$ being $$\omega({\boldsymbol{\xi}}) = \frac{1}{\left(2\pi\right)^{d/2}} \exp{\left(-\frac{\xi^2}{2}\right)}.$$ In Eqs. and , $f$ is the distribution function, $f^{eq}$ is the $d$-dimensional equilibrium Maxwellian and $w_\alpha, {\boldsymbol{\xi}}_\alpha$ are the $n$ weights and nodes determined from a quadrature formula, respectively [@syc06]. Since we are considering isothermal flows, the terms related to the temperature in the second-order approximation of the equilibrium Maxwellian can be disregarded and $f^{eq}$ can thus be written as $$\label{eq:Maxwellian}
f^{eq} ({\boldsymbol{r}},t)=\omega({\boldsymbol{\xi}}) \rho({\boldsymbol{r}},t) \left[1+{\boldsymbol{\xi}}\cdot {\boldsymbol{u}}({\boldsymbol{r}},t) \right],$$ where $\rho$ and ${\boldsymbol{u}}$ are the density and macroscopic velocity, respectively, which can be computed by sums over the discrete velocity set
$$\begin{aligned}
\label{eq:density}
\rho ({\boldsymbol{r}},t) & = & \sum_{\alpha=1}^{n} f_\alpha ({\boldsymbol{r}},t), \\
\label{eq:velocity}
\rho ({\boldsymbol{r}},t) {\boldsymbol{u}}({\boldsymbol{r}},t) & = & \sum_{\alpha=1}^{n} {\boldsymbol{\xi}}_\alpha f_\alpha ({\boldsymbol{r}},t).\end{aligned}$$
Likewise, boundary conditions can be derived by a direct discretization of Maxwell’s diffuse-specular scattering kernel [@ak02]. For the sake of simplicity, let us consider a plane wall at $y=0$ with reference temperature and constant velocity $u_w$ in the positive $x$-direction. Let us further suppose that the gas fills the half space $y>0$ and molecules which strike the wall are re-emitted according to Maxwell’s scattering kernel with complete accommodation. The discrete form of the kinetic boundary condition at a point ${\boldsymbol{r}}_w$ of the solid surfaces reads
$$\label{eq:BC}
f_{\alpha}({\boldsymbol{r}}_w,t) = \omega({\boldsymbol{\xi}}_\alpha) \rho_{w} ({\boldsymbol{r}}_w,t) \left( 1+\xi_{x,\alpha} u_w \right),
\mbox{\hspace{0.2cm}} \xi_{y,\alpha}>0.$$
In Eq. (\[eq:BC\]), $\rho_{w}$ can be obtained through the [impermeability]{} condition, which states that the normal component of the gas velocity on the wall vanishes
$$\label{eq:rho}
\rho_{w} ({\boldsymbol{r}}_w,t) = -(2\pi)^{1/2} \sum_{\xi_{y,\alpha}<0} \xi_{y,\alpha} f_\alpha ({\boldsymbol{r}}_w,t).$$
For two-dimensional flows and very low Knudsen numbers, the $D_{2}Q_{9}$ model provides accurate results. We here use the standard terminology and denote by $D_mQ_n$ the $m$ dimensional LB models with $n$ discrete velocities. As ${\mbox{Kn}}$ increases, high-order LB models are needed to correctly reproduce non equilibrium effects, such as the velocity slip at the solid walls and the nonlinear stress-strain relationship within the Knudsen layer [@kpb08]. In the framework of single relaxation time modeling, several high-order LB models have been developed. Some have been derived by using a local mean free path in order to account for the presence of solid surfaces [@zgbe06]. Although these models have been shown to be effective in many applications, they are phenomenological in nature and, as such, not perfectly general. Composite models have also been developed which result from the superposition of discrete velocities determined from odd and even quadrature formula [@iri11]. However, the most common strategy is using a greater number of discrete velocities determined from a full-range Gauss-Hermite quadrature [@kpb08; @mz11a], possibly adopting a multiscale approach to contain the increase in the computational cost [@mzs11]. However, more discrete velocities do not guarantee an improved accuracy [@kpb08; @mz11a; @mz11b]. It has been demonstrated that this is due to the quadrature effect in dealing with the boundary conditions [@sbys11]. As a matter of fact, abscissae of the full-range Gauss-Hermite quadrature schemes are derived to obtain accurate evaluation of the moments of the distribution function defined over the entire velocity space. In contrast, they provide only an approximately estimate of the half-range integrals that enter in the formulation of kinetic boundary conditions [@ak02].
--------------------------------------------------------------------------------------------------------------
Quadrature $\xi_\alpha$ $w_\alpha$
------------------ ------------------------------------ ------------------------------------------------------
$D_{1}Q^{h}_{4}$ $\begin{array}[t]{c} $\begin{array}[t]{c}
\mp 0.4245383286 \\ 0.3613798911 \\
\mp 1.77119083 0.1386201089
\end{array}$ \end{array}$
$D_{1}Q^{h}_{6}$ $\begin{array}[t]{c} $\begin{array}[t]{c}
\mp 0.2694842630 \\ 0.2516453504 \\
\mp 1.199609295 \\ 0.2236832664 \\
\mp 2.545268446 0.0246713831 \\
\end{array}$ \end{array}$
$D_{1}Q^{h}_{8}$ $\begin{array}[t]{c} $\begin{array}[t]{c}
\mp 0.1891884657 \\ 0.1835325640 \\
\mp 0.8829284442 \\ 0.2375842404 \\
\mp 1.898635201 \\ 0.07528686870 \\
\mp 3.199890790 \\ 0.003596326913 \\
\end{array}$ \end{array}$
--------------------------------------------------------------------------------------------------------------
: \[tab:nodes\_weights\] Nodes, $\xi_\alpha$, and weights, $w_\alpha$, of the one-dimensional half-range Gauss-Hermite quadrature.
In this Rapid Communication, we want to show that non-equilibrium gas flows can be more accurately described by using a discrete velocity set different from the one employed by standard finite-difference LB models. More specifically, we propose finite-difference LB models which use the roots of half-range Hermite polynomials as quadrature nodes. Table \[tab:nodes\_weights\] gives nodes and weights of the one-dimensional half-range Gauss-Hermite quadrature. The two-dimensional quadrature can be obtained from the tensor product of the corresponding one-dimensional quadrature. Notice should be made that the roots of half-range Hermite polynomials are irrational and therefore the resulting fully-discrete numerical schemes are computationally more demanding than the simpler “stream-and-collide” algorithm. The exact space discretization of the advection step of on-lattice LB models is therefore no longer possible and the potential high efficiency of their parallel implementations is partially lost. However, off-lattice schemes may offer some advantages such as enhanced geometrical flexibility and, as shown in the present work, the capability of describing rarefaction effects. The use of high performance computing is still feasible but the porting of algorithms involving finite-difference approximations certainly requires some additional effort [@fgg11]. In comparison with alternative finite-difference LB models, the proposed velocity space discretization permits to explicitly account for the discontinuity of the distribution function and therefore leads to a faster convergence of the solution close to solid surfaces. In kinetic theory applications, the importance of a consistent treatment of boundary conditions has been recognized as early as the sixties of the past century [@gjz57] and the half-range discrete ordinate method has been widely used since then [@hg67; @bcrs01; @fgf09; @g12; @gg13]. By contrast, in LB simulations of gas flows in microchannels, most of high-order LB models use the roots of full-range Hermite polynomials as discrete velocities [@zgbe06; @iri11; @mzs11]. Although it is well known that finite-difference LB models can be developed from different quadrature formula [@syc06], to the authors’ knowledge, no previous works have pointed out that the quadrature based on half-range Hermite polynomials can easily address the issue of boundary conditions for the LB simulations of gas microflows. An important correspondence can be identified between the approach developed in the present work and the moment method presented in Ref. [@fgf09] for studying gas microflows. There, the isothermal linearized BGKW kinetic model equation has been solved by expanding the distribution function as a series of half-range Hermite polynomials. Expansion coefficients are the moments of the distribution function which, in turn, are strictly related to the macroscopic quantities. By using the half-range Gauss-Hermite quadrature formula for evaluating these integrals, a one-to-one correspondence can be identified between the expansion coefficients and the values of the distribution function at the roots of half-range Hermite polynomials. The two approaches are therefore equivalent even though, from the computational standpoint, the formulation which is here proposed is more efficient. It is worth noticing that already in Ref. [@ak02] it has been pointed out that the accuracy in dealing with boundary conditions can be improved by evaluating integrals which enter in their definition by means of a quadrature formula defined in the half-space. However, this possibility has not been further developed because of the mismatch between the nodes of the quadrature used at the boundary and those in the bulk. In the present work, however, the nodes of half-range Gauss-Hermite quadrature formula are used in the whole domain.
![Velocity slip at the upper plate versus the Knudsen number.[]{data-label="fig:slip"}](slip.eps)
![Reduced $xy$-component of the stress tensor versus the Knudsen number.[]{data-label="fig:pxy"}](pxy.eps)
We demonstrate, both analytically and numerically, that the proposed approach can be used to simulate gas microflows by studying two classical driven boundary value problems, i.e., the Couette and the cavity flows. Both problems are regarded as two-dimensional and the driven velocity is assumed sufficiently low so that the gas flow can be considered in the linearized regime [@ggdi12]. A fully discrete numerical scheme is derived by a first-order time splitting of the evolution operator and couples a first-order upwind scheme for the transport step with a first-order explicit Euler scheme for the relaxation step. More sophisticated methods should be used for both coupling and solving transport and relaxation steps if a more accurate and efficient numerical scheme is needed. However, the main aim of the work is to show that an improved accuracy in describing finite Knudsen flows can be achieved by using the roots of half-range Hermite polynomials as discrete velocities. This has been assessed by running the same numerical code twice, once with nodes and weights as given by full-range Gauss-Hermite quadrature and once with nodes and weights as given by half-range Gauss-Hermite quadrature. The validity of this comparison is therefore independent on the numerical scheme that has been used to solve the discrete kinetic equation.\
In the Couette flow problem, the upper plate moves in the $x$ direction with a velocity $u_w$ and the lower plate moves with a velocity $-u_w$. Diffuse boundary conditions have been implemented at the plates and periodic boundary conditions have been utilized at opposite ends of the channels. The two plates are separated by the distance $l$. It is worth noticing that the proposed models employ the quadrature nodes of the positive and negative half-range Gauss-Hermite quadrature formula as discrete velocities both in the $x$- and $y$-directions. However numerical computations showed that the same results are obtained if the $x$-direction of the velocity space is discretized by using the roots of full-range Hermite polynomials. By proceeding as in Ref. [@fgf09], it is not difficult to obtain a closed-form solution of the $D_{2}Q^{h}_{16}$ LB model for the stationary planar Couette flow
$$\begin{aligned}
\label{eq:velocity_exact}
u_x(y/l) & = & u_{w}
\frac{\left[\sinh{\left(\frac{0.9494}{{\mbox{Kn}}}\right)}+0.7978 \cosh{\left(\frac{0.9404}{{\mbox{Kn}}}\right)}\right]\frac{y}{l}+
0.5642 {\mbox{Kn}}\sinh{\left(\frac{1.881}{{\mbox{Kn}}}\frac{y}{l}\right)}}
{(0.7071 {\mbox{Kn}}+0.3989)\cosh{\left(\frac{0.9404}{{\mbox{Kn}}}\right)}+(1.128 {\mbox{Kn}}+0.5)\sinh{\left(\frac{0.9404}{{\mbox{Kn}}}\right)}}, \\
\label{eq:pressure_exact}
\frac{\sqrt{RT_0}p_{xy}}{u_w p_0} & = & -{\mbox{Kn}}\frac{\sinh{\left(\frac{0.9494}{{\mbox{Kn}}}\right)}+0.7979\cosh{\left(\frac{0.9404}{{\mbox{Kn}}}\right)}}
{(0.7071 {\mbox{Kn}}+0.3989)\cosh{\left(\frac{0.9404}{{\mbox{Kn}}}\right)}+(1.128 {\mbox{Kn}}+0.5)\sinh{\left(\frac{0.9404}{{\mbox{Kn}}}\right)}}.\end{aligned}$$
Beside their intrinsic interest, Equations and , as well as a similar closed-form solution obtained in Ref. [@akaap07] for the $D_{2}Q_{16}$ model, are of practical importance in that they permit to validate the numerical code. Figures \[fig:slip\] and \[fig:pxy\] show the velocity slip, $1-u_{x}/u_{w}$, at the upper plate and the ratio between the $xy$-component of the stress tensor, $p_{xy}$, and the value of the pressure in the free molecular regime, $p_{fm}=-2p_0 u_{w}/\sqrt{\pi R T_0}$, versus the Knudsen number, ${\mbox{Kn}}$, respectively. Both Figures use the same conventions. Solid line are the solutions of the linearized BGKW kinetic model equation obtained by means of the moment method described in Ref. [@fgf09]. Dashed lines are the closed-form solutions of the $D_{2}Q^{h}_{16}$ model, as given by Eqs. and evaluated at $y/l=1/2$, and of the $D_{2}Q_{16}$ model, which is reported in Ref. [@akaap07]. Solid circles are the numerical solutions of the $D_{2}Q^{h}_{16}$ and $D_{2}Q_{16}$ models. Only even-order LB models have been considered since they perform significantly better than those with an odd-order quadrature no matter how the order is [@kpb08; @mz11a]. As it is clearly shown, although the same number of discrete velocities is employed, the $D_{2}Q^{h}_{16}$ model gives much improved results as compared with standard $D_{2}Q_{16}$ model. The prediction of the velocity slip shows a very good match with the results obtained by solving the linearized BGKW kinetic model equation not only in the continuum and slip flow regimes but also in the early transition regime. The $xy$-component of the stress tensor shows a quite good agreement even in a wider range of Knudsen numbers.
![Nondimensional velocity profile through the half-channel. ${\mbox{Kn}}=1$. (a) $n=16$, (b) $n=144$.[]{data-label="fig:velocity"}](velocity.eps)
Convergence of the results at ${\mbox{Kn}}=1$ with increasing number of discrete velocities is reported in Fig. \[fig:velocity\] for the velocity profile. Solid line is the solution of the linearized BGKW kinetic model equation obtained by means of the moment method described in Ref. [@fgf09]. Dotted lines with solid circles are the numerical solutions obtained by using $n=16$ (left panel) and $n=144$ (right panel) discrete velocities, respectively. The $D_2Q_{16}$ model provides a slightly better description of the gas behavior in the bulk of the flow. A possible explanation is that the full-range Gauss-Hermite quadrature with $n=16$ discrete velocities integrates exactly continuous polynomials of order five in each velocity component whereas the half-range Gauss-Hermite quadrature with the same number of nodes integrates exactly linear polynomials with a possible discontinuity at the origin. It is thus reasonable that the former can be more accurate in the bulk of the flow where the distribution function is continuous whereas the latter can perform better close to the solid surface where the distribution function is expected to be discontinuous. Nevertheless, as results with $n=144$ clearly show, using the roots of half-range Hermite polynomials as discrete velocities greatly speed-up the convergence to the kinetic theory solution. A similar analysis for the $xy$-component of the stress tensor shows that the numerical prediction of the $D_{2}Q^{h}_{16}$ model at ${\mbox{Kn}}=1$ is affected by a relative error which is lower than $5\%$ and at least $12$ nodes per each velocity component should be used for the standard high-order LB models to achieve the same level of accuracy.
![Profile of the nondimensional horizontal component of the mean velocity crossing the center of the cavity. ${\mbox{Kn}}=0.05$. (a) $n=16$, (b) $n=64$.[]{data-label="fig:comparison_cavity"}](comparison_cavity.eps)
The possibilities of the proposed models are further illustrated by solving the square driven cavity flow problem [@nv05]. All the walls have a length $l$ and are fixed and isothermal. The flow is driven by the uniform translation of the top. Numerical simulations have been carried out for Knudsen numbers in the range $[0.05,0.4]$. This problem constitutes a severe test to assess the capability of the proposed velocity space discretization to describe the gas behavior in complex geometries. Indeed, the discontinuities which exist at the four corners, particularly the two at the top, propagate inside the computational domain and might cause the quadrature to fail. Since the discontinuities decay with distance owing to molecular collisions, it is expected that these numerical problems become more severe as the Knudsen number is increased. In Fig. \[fig:comparison\_cavity\], the profile of the nondimensional horizontal component of the macroscopic velocity, $u_{x}/u_w$, crossing the center of the cavity are thus show for the higher Knudsen number we considered, ${\mbox{Kn}}=0.4$. Solid line is the solution of the linearized BGKW equation obtained with the numerical method described in Ref. [@nv05]. Dotted lines with solid circles are the numerical solutions provided by full- and half-range LB models with $n=16$ (left panel) and $n=64$ (right panel) discrete velocities. As it was clearly shown, both methods suffer from an unphysical oscillatory behavior due to the difficulty of the quadratures to evaluate a region where discontinuities are present. Although more sophisticated methods can be adopted [@nv05], we here simply notice that the problem can be overcome by a reasonable increase in the number of discrete velocities. As for the case of the Couette flow, the proposed approach shows better convergence properties. For instance, the errors in the ${\cal L}^1$-norm of full- and half-range LB models using $n=16$ discrete velocities are $0.0445$ and $0.0280$, respectively, and reduce to $0.0218$ and $0.00354$ for $n=64$. Table \[tab:DeG\] reports a comparison of the values of the mean dimensionless shear stress along the moving plate, $D$, and the dimensionless flow rate of the main vortex, $G$, obtained by the different models [@nv05]. The predictions of the $D_{2}Q^{h}_{16}$ model show a better agreement with the kinetic theory results than those of the $D_{2}Q_{16}$ model. In spite of the unphysical oscillations which affect the macroscopic velocity, the drag coefficient converges up to two significant figures. Instead the error in the reduced flow rate is less than $1\%$ for ${\mbox{Kn}}=0.05$ but rapidly increases for greater Knuden numbers. However, as shown in Ref. [@fgf09], a quite good agreement can be found up to ${\mbox{Kn}}=10$ if the $D_{2}Q^{h}_{36}$ is used.
--------------- ------- ------------------- --------------- ------- ------------------- ---------------
${\mbox{Kn}}$ BGKW $D_{2}Q^{h}_{16}$ $D_{2}Q_{16}$ BGKW $D_{2}Q^{h}_{16}$ $D_{2}Q_{16}$
0.05 0.258 0.256 0.270 0.154 0.153 0.162
0.1 0.328 0.327 0.350 0.136 0.128 0.153
0.2 0.425 0.417 0.467 0.121 0.109 0.148
0.4 0.385 0.384 0.421 0.110 0.0961 0.145
--------------- ------- ------------------- --------------- ------- ------------------- ---------------
: \[tab:DeG\] Drag coefficient, $D$, and reduced flow rate, $G$, versus the Knudsen number, ${\mbox{Kn}}$.
To summarize, we have shown that, in comparison with full-range finite-difference high-order LB models, using the nodes of half-range Gauss-Hermite quadrature as discrete velocities permits to consistently deal with kinetic boundary conditions and thus to achieve a more accurate description of the gas behavior close to solid surfaces. Applications to one- and two-dimensional driven boundary value problems show that, even using a small number of discrete velocities, accurate results can be obtained in a wide range of Knudsen numbers which extends up to the early transition regime. The proposed approach is also of interest in that, as it can be deduced from Ref. [@fgf09], half-range LB models with $N$ discrete velocities simplify to full-range LB models with $N/2^d$ velocities far away from solid surfaces, being $d$ the dimension of the physical space. This suggest the development of a hybrid approach based on their coupling which combines the capability of half-range LB models to accurately describe the gas behavior in the Knudsen layers with the higher computational efficiency of full-range LB models when applied to the bulk of the gas.
The authors would like to thank Prof. A. [Frezzotti]{} for critically reading the paper and Prof. D. [Valougeorgis]{} for providing his numerical code. This work has been partly supported by “Progetto Giovani Ricercatori GNFM 2013”, Regione Lombardia and CILEA Consortium through a LISA Initiative (Laboratory for Interdisciplinary Advanced Simulation) 2011 grant \[link:http://lisa.cilea.it\].
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abstract: 'Analysing coronary artery plaque segments with respect to their functional significance and therefore their influence to patient management in a non-invasive setup is an important subject of current research. In this work we compare and improve three deep learning algorithms for this task: A 3D recurrent convolutional neural network (RCNN), a 2D multi-view ensemble approach based on texture analysis, and a newly proposed 2.5D approach. Current state of the art methods utilising fluid dynamics based fractional flow reserve (FFR) simulation reach an AUC of up to 0.93 for the task of predicting an abnormal invasive FFR value. For the comparable task of predicting revascularisation decision, we are able to improve the performance in terms of AUC of both existing approaches with the proposed modifications, specifically from 0.80 to 0.90 for the 3D-RCNN, and from 0.85 to 0.90 for the multi-view texture-based ensemble. The newly proposed 2.5D approach achieves comparable results with an AUC of 0.90.'
author:
- 'Felix Denzinger$^{1,2}$, Michael Wels$^2$, Katharina Breininger$^1$, Anika Reidelshöfer$^3$, Joachim Eckert$^4$, Michael Sühling$^2$, Axel Schmermund$^4$, Andreas Maier$^1$'
bibliography:
- '0000.bib'
title: Deep Learning Algorithms for Coronary Artery Plaque Characterisation from CCTA Scans
---
Introduction
============
Cardiovascular diseases (CVDs) remain the leading cause of natural death [@mendis15]. In diagnosis and treatment of CVDs, the identification of functionally significant atherosclerotic plaques that narrow the coronary vessels and cause malperfusion of the heart muscle plays an important role. In clinical practice, this is typically assessed using fractional flow reserve (FFR) measurements [@Cury16].
This measurement is performed minimally invasively and therefore induces a small but existing risk to the patient. A non-invasive modality capable of visualising and assessing coronary artery plaque segments is coronary computed tomography angiography (CCTA). Current research tries to simulate the FFR value from CCTA scans [@taylor13]. Approaches based on this mostly rely on a prior segmentation of the whole coronary tree which is computationally intensive, prone to errors and may need manual corrections [@wels16].
In this work, we investigate three lumen-extraction independent deep learning algorithms for the task of predicting the revascularisation decision and the significance of a stenosis on a lesion-level. We propose a multi-view 2.5D approach, which we compare with two previously published methods, a 3D-RCNN approach [@zreik18b] and a multi-view texture-based ensemble approach [@tejero19]. Additionally, we introduce adaptions to improve the performance of all approaches on our task. These include resizing lesions to an intermediate length instead of padding them and the usage of test-time augmentations. Also, we propose to use a different feature extraction backbone than described in [@tejero19] for the respective approach. Note that both reference approaches were originally used to detect lesions and characterise them. Contrary to this we characterise annotated lesions with a defined start and end point.
Material and Methods
====================
Data
----
The data collection used contains CCTA scans from 95 patients with suspected coronary artery disease taken within 2 years at the same clinical site. For each patient, the resulting clinical decision regarding revascularisation was made by trained cardiologists, based on different clinical indications. This decision was monitored on a branch level. Lesions were annotated using their start and end point on the centerline, which was extracted automatically using the method described in [@zheng13]. We binarise the stenosis grade, which is estimated based on the lumen segmentation and defined as the ratio between the actual lumen and an estimated healthy lumen, using a threshold of 50$\,$% according to [@Cury16]. The branch-wise revascularisation decision is propagated only to the lesion with the highest stenosis grade in branches known to be revascularised. Of the total of 345 lesions in our data set, 85 lesions exhibit a significant stenosis grade, and 93 require revascularisation.
Methods
-------
### 3D-RCNN
The first network we use is identical to the method described in [@zreik18b]. In this approach, after extracting the coronary centerlines, a multi-planar reformatted (MPR) image stack is created by interpolating an orthogonal plane for each centerline point. Next, the MPR image stack is cut into a sequence of 25 overlapping cubes with size 25x25x25 and a stride of 5. During training, data augmentation using random rotations around the centerline and random shifts in all directions is used. Moreover, the data set is resampled for batch creation to achieve class balance during training. Since detection instead of sole characterisation is performed in [@zreik18b], padding the inputs to the same length was not needed in their work.
### Texture-based Multi-view 2D-CNN
The second baseline approach is described in reference [@tejero19]. A VGG-M network backbone pretrained on the ImageNet challenge dataset is used as a texture-based feature extractor. The extracted features are encoded as Fisher vectors and used for classification using a linear support vector machine. As inputs for this classification pipeline, different 2D views of the MPR image stack are combined for a final vote.
### 2.5D-CNN
Both aforementioned methods utilize a sliced 3D representation of the lesion or a multitude of 2D representations, which is computationally expensive to obtain and to process by the subsequent machine learning pipeline. To mitigate this, we propose a 2.5D multi-view approach as shown in Figure \[fig1\].
![Algorithm overview: Extraction of two orthogonal views of the lesion of interest. These are concatenated and then used as an input for a 2D-CNN (conv = convolutional layer, bn = batch normalisation layer, dense = fully connected layer).[]{data-label="fig1"}](2andhalfDapproach.pdf){width="99.00000%"}
From the MPR image stack, only two orthogonal slices are selected, concatenated and forwarded to a 2D-CNN.
### Modifications
In this work, we examine the effect of three different padding strategies for all three approaches: zero-padding, stretching the volume stack to the longest lesion and resizing all lesions to an intermediate size. Stretching and squeezing of the image stacks along the centerline is performed with linear interpolation. Each MPR image stack for each lesion has a resolution of 64x32x32 and 170x32x32 after padding depending on the method used. For the 3D-RCNN approach, we downscale the y and x dimension further to 25x25 to match the original algorithm described in [@zreik18b]. For augmentation of the data set all single volumes are rotated around the centerline in steps of 20$^\circ$, which leads to an 18 times larger data collection. In order to create valid rotational augmentations of the image stack without cropping artefacts, we cut out a cylindrical ROI and set all values around it to zero. We confirmed in preliminary experiments that this computationally cheaper procedure does not to impact the results compared to cutting out a rotated view from the original data. In contrast to [@zreik18b; @tejero19], no class resampling was necessary during training, since the class imbalance is not as severe for classification given the start and end point of a lesion compared to detecting lesions as well. Instead of the originally proposed VGG-M backbone used in [@tejero19], we use the VGG-16 network architecture as a backbone since it was already shown to yield better performance in the original paper on texture-based filter banks [@fisher15]. The data set was normalised to fit ImageNet statistics. We also evaluate the performance of this approach using a pretrained Resnet50 architecture [@resnet16] as backbone.
### Evaluation
No hyperparameter optimisation is performed. Parameters are either taken from the references or default values are used. To reduce the influence of random weight initialisation and other random effects on the results, we repeat a 5-fold cross validation with five different initialisations, leaving a total of 25 splits. All splits are performed patient-wise. We also use the aforementioned rotational augmentation during test-time, and compare how the mean prediction over all rotations performs in comparison to a single input.
Results
=======
![Mean performance and standard deviation of all approaches for different padding strategies. These experiments are performed using only 8 instead of the 18 views (${\vert~~\vert}$ = resizing to intermediate size, ${\vert\to\vert}$ = resizing to the longest sequence, O = zero padding or no padding for the texture-based approach, MCC = Matthews correlation coefficient).[]{data-label="padding"}](Sten.pdf "fig:"){width="49.00000%"} ![Mean performance and standard deviation of all approaches for different padding strategies. These experiments are performed using only 8 instead of the 18 views (${\vert~~\vert}$ = resizing to intermediate size, ${\vert\to\vert}$ = resizing to the longest sequence, O = zero padding or no padding for the texture-based approach, MCC = Matthews correlation coefficient).[]{data-label="padding"}](Revasc.pdf "fig:"){width="49.00000%"}
Model/Metric AUC Accuracy F1-score Sensitivity Specificity MCC
------------------------------------------------ -------------- -------------- -------------- ------------- -------------- --------------
3D-RCNN[@zreik18b][@denzinger19] 0.89 0.85 0.67 **0.79** 0.86 0.59
3D-RCNN[@zreik18b]$^{\vert~~\vert}$ [**0.92**]{} [0.88]{} [0.69]{} [0.68]{} [0.93]{} [0.62]{}
2D[@tejero19]$^\ast$$^{\vert\to\vert}$$^{VGG}$ [0.85]{} [0.86]{} [0.62]{} [0.56]{} [0.94]{} [0.54]{}
2D[@tejero19]$^{+}$$^{\vert~~\vert}$$^{RES}$ [0.78]{} [0.82]{} [0.61]{} [0.70]{} [0.85]{} [0.50]{}
2D[@tejero19]$^\ast$$^{\vert~~\vert}$$^{RES}$ [0.90]{} [0.87]{} [0.68]{} [0.71]{} [0.91]{} [0.60]{}
2.5D$^{+}$$^{\vert~~\vert}$ [**0.92**]{} [0.89]{} [0.70]{} [0.64]{} [0.95]{} [0.64]{}
2.5D$^\ast$$^{\vert~~\vert}$ [**0.92**]{} [**0.90**]{} [**0.71**]{} [0.64]{} [**0.96**]{} [**0.66**]{}
: Results for predicting stenosis degree prediction on a lesion-level ($18$ and $8$ correspond to the amount of views considered for data augmentation during training and test time, $+$ = single view classification, $\ast$ = combined view classification, ${\vert~~\vert}$ = resizing to intermediate size, ${\vert\to\vert}$ = resizing to the longest sequence).[]{data-label="stenosisResults"}
Model/Metric AUC Accuracy F1-score Sensitivity Specificity MCC
------------------------------------------------ -------------- -------------- -------------- ------------- -------------- --------------
3D-RCNN[@zreik18b][@denzinger19] 0.80 0.76 0.55 **0.72** 0.77 0.42
3D-RCNN[@zreik18b]$^{\vert~~\vert}$ [0.90]{} [0.84]{} [0.63]{} [0.65]{} [0.90]{} [0.53]{}
2D[@tejero19]$^\ast$$^{\vert\to\vert}$$^{VGG}$ [0.86]{} [0.84]{} [0.56]{} [0.49]{} [0.93]{} [0.47]{}
2D[@tejero19]$^{+}$$^{\vert~~\vert}$$^{RES}$ [0.77]{} [0.81]{} [0.60]{} [0.68]{} [0.84]{} [0.48]{}
2D[@tejero19]$^\ast$$^{\vert~~\vert}$$^{RES}$ [0.90]{} [0.85]{} [0.66]{} [0.70]{} [0.89]{} [0.57]{}
2.5D$^{+}$$^{\vert~~\vert}$ [**0.90**]{} [0.87]{} [0.65]{} [0.60]{} [0.94]{} [0.58]{}
2.5D$^\ast$$^{\vert~~\vert}$ [**0.90**]{} [**0.88**]{} [**0.67**]{} [0.61]{} [**0.95**]{} [**0.60**]{}
: Results for predicting the revascularisation decision on a lesion-level (Abbreviations as in Table \[stenosisResults\]).[]{data-label="revascResults"}
The most important results are provided in Table \[stenosisResults\], Table \[revascResults\] and Figure \[padding\]. The results for the 3D-RCNN approach are also compared to the results of our previous work [@denzinger19], where similar experiments are performed on the same data set as here but with the workflow described in [@zreik18b], zero-padding and a different cross validation strategy. From the three padding methods examined, resizing all volume stacks of the data collection to one intermediate size yields the best results for most network approaches except for the texture-based approach with the VGG-16 backbone, where resizing all lesions to the size of the largest volume performs best. Interestingly, the same algorithm workflow with the Resnet50 backbone performs differently in that regard. A hypothesis that can be drawn from the intermediate padding performing best is that this scale provides on the one hand roughly the same amount of information per sample while on the other hand also keeping the input size in a range where it can be processed better. For the 3D-RCNN, we only look at classification in this work, in contrast to the task in [@zreik18b] which included the detection of lesions. For this target, the proposed adaptations to the workflow in terms of padding strategy and not resampling the data set during batch creation improves the performance of both predicting the stenosis degree and the revascularisation decision from an AUC of 0.89 to 0.92, and 0.80 to 0.90, respectively. Having a more powerful feature extractor network for the texture-based approach combined with slightly more data augmentation improves the AUC by 0.05 for classifying stenosis significance, and by 0.04 for classifying revascularisation decision. The method performs considerably better when using test augmentations than without. Our proposed approach performs similar to the other two approaches, outperforming them by a small margin with an AUC of 0.92/0.90 for predicting a significant stenosis/revascularisation decision. Interestingly, test augmentations only yield a small improvement. This suggests that the method already has all necessary information to predict the task at hand from two orthogonal slices.
Discussion
==========
In this paper, we compared and improved three segmentation independent deep learning-based algorithms for predicting both significant stenosis degree and clinical revascularisation decision for lesions annotated with a start and end point. We obtained comparable results for each method. Our proposed method – a 2.5D approach – slightly outperforms the other approaches and requires fewer views compared to the method previously described in [@tejero19]. Therefore, a faster training procedure and inference is possible. In future work, we will examine whether this method is also capable of detecting lesions instead of just classifying them, and whether it is able to predict an abnormal FFR value.
### Disclaimer {#disclaimer .unnumbered}
The methods and information here are based on research and are not commercially available.
|
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abstract: 'We show that large ($\gtrsim 100$ GeV) mass splittings between the charged Higgs boson ($H^\pm$) and the neutral Higgs bosons ($H^0$ and $A^0$) are possible in the Minimal Supersymmetric Standard Model (MSSM). Such splittings occur when the $\mu$ parameter is considerably larger than the common SUSY scale, $M_{SUSY}$, and have significant consequences for MSSM Higgs searches at future colliders.'
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0.26cm 0.26cm 0.5cm 16.cm
KIAS Preprint P01023\
May 2001\
Introduction
============
The Minimal Supersymmetric Standard Model (MSSM) [@prep] is currently the leading candidate for physics beyond the Standard Model (SM). At the tree–level the Higgs sector of the MSSM takes the form of a two Higgs doublet model (2HDM), where the coefficients of the quartic scalar terms are functions of the electroweak gauge couplings [@Gun]. Five physical Higgs bosons are predicted: a charged pair ($H^+$,$H^-$), two CP–even scalars ($h^0$,$H^0$), and a CP–odd $A^0$. In the tree–level approximation the lightest Higgs $h^0$ satisfies $M_{h^0} \le M_Z$, while $M_{H^\pm}\approx M_{H^0}\approx M_{A^0}$ for $M_{H^\pm}\gtrsim 200$ GeV. Only two parameters are needed to fully parametrize the tree–level potential, and these are usually taken to be $\tan\beta$ ($=v_2/v_1$, where $v_i$ is the vacuum expectation value of a Higgs doublet) and one of $M_{H^\pm},M_{A^0}$.
At the 1–loop level the coefficients in the Higgs potential receive contributions from virtual particle loops and thus become complicated functions of several (SUSY) parameters. In addition, new quartic scalar terms are generated. This changes the tree–level mass relationships, most notably weakening the above mass bound to $M_{h^0}\lesssim 130$ GeV [@1-loop]. The near degeneracy relationship, $M_{H^\pm}\approx M_{A^0}\approx M_{H^0}$ for the large ($\ge 200$ GeV) mass region, is only slightly affected, resulting in mass splittings of ${\cal O}(10)$ GeV. Recently there has been much interest in the 1–loop effective potential of the MSSM with unconstrained CP violating phases [@NPB553; @PilWag; @plb495; @CPH]. It has been shown that the mass splitting between the two heavier neutral scalars (now mixed states of CP) can be increased up to 30 GeV for large phases.
In this paper we will show that large mass splittings between the charged and neutral Higgs bosons, $|M_{H^{\pm}}-M_{A^0,H^0}|\gtrsim 100$ GeV, are possible in the CP conserving MSSM in a previously ignored parameter space. These large splittings occur when $\mu$ is larger than the supersymmetric (SUSY) scale (defined as the arithmetic mean of the stop masses) by a factor of 4 or more. Although sizeable corrections to the tree–level mass sum rules can be implicitly found in previous works e.g. [@Haber:1993an], we identify the parameter space for the largest splittings and discuss the phenomenological consequences. Importantly we show that this parameter space is consistent with current experiments.
Such splittings would break the commonly assumed degeneracy relation $M_{H^\pm}\approx M_{H^0}\approx M_{A^0}$ for $M_{H^\pm}\gtrsim 200$ GeV, which should no longer be taken as a prediction of the MSSM. The Higgs mass spectrum of the MSSM may resemble that of a general 2HDM or other extended (SUSY) Higgs sectors with scalar singlets etc. Knowledge of the maximum possible mass splittings among the Higgs bosons in the MSSM may be crucial in distinguishing different models at future colliders. In particular, measurements of the mass splittings provide important information on the structure of the underlying Higgs sector, especially since Higgs branching ratios (BR) and cross–sections are often very similar in many popular models. Moreover, new decays channels involving $H^\pm$,$H^0$ and $A^0$ would be open and may possess large branching ratios, thus affecting proposed MSSM Higgs search strategies at future colliders.
Our work is organized as follows. In Section 2 we outline our approach for evaluating the mass splittings. Section 3 presents our numerical results, while section 4 contains our conclusions.
Mass splittings in the MSSM
===========================
The most general CP violating Higgs potential of the MSSM may be described by the following effective Lagrangian:
$$\begin{aligned}
{\cal L}_V &=& \mu^2_{1}\Phi^{\dagger}_{1}\Phi_1+
\mu^2_{2}\Phi^{\dagger}_{2}\Phi_2+m^2_{12}\Phi^{\dagger}_{1}\Phi_2
+m^{*2}_{12}\Phi^{\dagger}_{2}\Phi_1+
\lambda_1(\Phi^{\dagger}_{1}\Phi_1)^2+
\lambda_2(\Phi^{\dagger}_{2}\Phi_2)^2 \nonumber \\
&&+\lambda_3(\Phi^{\dagger}_{1}\Phi_1)(\Phi^{\dagger}_{2}\Phi_2)
+\lambda_4(\Phi^{\dagger}_1\Phi_2)(\Phi^{\dagger}_2\Phi_1)
+\lambda_5(\Phi^{\dagger}_1\Phi_2)^2+\lambda^*_5(\Phi^{\dagger}_2\Phi_1)^2
\nonumber \\
&&+\lambda_6(\Phi^{\dagger}_1\Phi_1)(\Phi^{\dagger}_1\Phi_2)
+\lambda^*_6(\Phi^{\dagger}_1\Phi_1)(\Phi^{\dagger}_2\Phi_1)
+\lambda_7(\Phi^{\dagger}_2\Phi_2)(\Phi^{\dagger}_1\Phi_2)
+\lambda^*_7(\Phi^{\dagger}_2\Phi_2)(\Phi^{\dagger}_2\Phi_1)\end{aligned}$$
At tree–level $\lambda_1\to \lambda_4$ are functions of the $SU(2)$ and $U(1)$ gauge couplings while $\lambda_5=\lambda_6=\lambda_7=0$. In the 1-loop effective potential all $\lambda_i$ receive sizeable corrections from the enhanced Yukawa couplings of the third generation squarks. In particular $\lambda_5\ne \lambda_6\ne \lambda_7 \ne 0$ and are complex in the presence of SUSY phases in $A_t$,$A_b$ and $\mu$. Explicit formulae may be found in [@NPB553].
The restricted form of the tree–level potential limits the possible mass splittings between $M_{H^\pm}$,$M_{H^0}$ and $M_A$. It can be shown that $M_{H^\pm}$ and $M_A$ are related by: $$M^2_{H^\pm}-M^2_{A}=(\lambda_4/2-Re(\lambda_5))v^2 \label{split}$$ At tree–level $\lambda_4=g_W^2/2$ and $\lambda_5=0$, and the above relationship reduces to the familiar sum rule $$M^2_{H^\pm}-M^2_{A}=M^2_W$$ With the current LEP bound of $M_A\ge 90$ GeV one finds $M_{H^\pm}-M_A\le 30$ GeV, with approximate degeneracy for $M_A\ge 200$ GeV.
At the 1–loop level $\lambda_4$ and $\lambda_5$ may be written as (keeping the dominant terms): $$\begin{aligned}
\lambda_4 &\approx& {g_W^2\over 2}-{3\over 96\pi^2}\left[h_t^4\left(
{3|\mu|^2\over M^2_{SUSY}}-{3|\mu|^2|A_t|^2 \over M^4_{SUSY}}
\right)+h_b^4\left(
{3|\mu|^2\over M^2_{SUSY}}-{3|\mu|^2|A_b|^2 \over M^4_{SUSY}}
\right)\right] \nonumber\\
&&+{3\over {8\pi^2}}h_t^2h_b^2\left[{1\over 2}X_{tb}\right] \\
\lambda_5 &\approx& {3\over{192\pi^2}}\left[
h_t^4\left({\mu^2 A_t^2\over M^4_{SUSY}}\right)
+ h_b^4\left({\mu^2 A_b^2\over M^4_{SUSY}}\right)\right] \nonumber
\label{eq:lam45}\end{aligned}$$ where $X_{tb}$ is a function of $A_t$,$A_b$, $M_{SUSY}$ and $\mu$. From now on we will focus on the case of $\mu,A_{t,b}$ being real, with mention given to the case of complex phases where appropriate. It can be seen from eq. (4) that $\mu,A_t$ or $\mu,A_b \ge 4M_{SUSY}$ would overcome much of the suppression from the small coefficients, permitting $\lambda_4,\lambda_5=
{\cal O}(1)$. From eq. (2) this would give rise to large mass differences $M_{H^\pm}-M_A$.
One can see from eq. (2) that $M_{H^\pm} > M_A$ requires $\lambda_4 > \lambda_5$ while $M_{A} > M_{H^\pm}$ requires $\lambda_5 > \lambda_4$. We note that both terms in $\lambda_5$ are positive definite and the term proportional to $h_t^4$ will dominate unless $\tan\beta$ is large (which enhances $h_b$). In the expression for $\lambda_4$ there may be both constructive and destructive interference among the various terms and so the SUSY correction to the tree–level value may take either sign.
For the CP even Higgs bosons one has a mass matrix ${\cal M}_S$ which when diagonalized gives the mass eigenstates $h^0$ and $H^0$. Note that the 1–loop corrected ${\cal M}_S$ involves all $\lambda_i$, $i=1\to 7$, and its explicit form may be found in [@NPB553]. Of interest to us is the mass splitting between $M_{H^\pm}$ and $M_{H^0}$ which may be given approximately by: $$\begin{aligned}
M^2_{H^\pm} - M^2_{H^0} \sim v^2 \left(
{\lambda_4 \over 2} +{\rm Re} \lambda_5
+ { 2 {\rm Re } \lambda_6 \over \tan\beta}
\right).
\label{splita}\end{aligned}$$
Recently it has been shown that the SUSY radiative corrections to the effective Yukawa couplings [@Yuk] can be important (for a review see [@logan]). Such corrections comprise loops involving gluino–sbottom and chargino–stop. The modified $h_b$ has already been shown to sizeably affect $b \to s \gamma$ [@giudice],[@nierste], the effective $H^{\pm}tb$ coupling [@htb], and the Higgs decay $h^0\to b\overline b$ [@Borzumati; @loghab]. In our analysis we shall restrict ourselves to $\tan\beta\le 20$, which is the region where the above corrections to $h_b$ have minor impact.
Note that such large mass splittings occur more naturally in extended versions of the MSSM which include a singlet Higgs field in the superpotential e.g. [@Panagiotakopoulos:2001zy]. In such models eq. (2) is modified to include a term $\sim\lambda v^2$, where $\lambda=\mu/v_s$ ($v_s$ is the vacuum expectation value of the singlet Higgs field). Thus $\lambda={\cal O} (g_w)$ (i.e. gauge coupling strength) may be attained with $\mu= {\cal O} (100)$ GeV, giving rise to large mass splittings.
Numerical results
=================
We now present our numerical results for the mass splittings $M_{H^\pm}-M_{H^0}$ and $M_{A}-M_{H^\pm}$. We take two representative sets of parameters: set (A) gives large $M_{H^\pm}-M_{H^0}$, and set (B) gives large $M_{A}-M_{H^\pm}$. $$\begin{aligned}
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
& \tan\beta & M_{H^\pm} & M_{\tilde{Q_3}} & M_{\tilde{t}}
& M_{\tilde{b}} & \mu & A_t & A_b \\
\hline
\hline
(A) & 11 & 250 & 500 & 550 & 550 & 4000 & 1900 & 0 \\
\hline
(B) & 10 & 150 & 250 & 200 & 500 & 2800 & 0 & 0 \\
\hline
\end{array}
\label{sets}\end{aligned}$$ Here all masses are in GeV; $M_{\tilde{Q_3}},M_{\tilde{t}},M_{\tilde{b}}$ refer to third generation squark soft SUSY breaking masses; $A_t$ and $A_b$ are the analogous trilinear couplings. We take $\mu$ and $A_{t,b}$ to be real in the numerical analysis. For these values of $\tan\beta$ the terms $\sim h^4_t$ dominate the terms $\sim h^4_b$ in the expressions for the $\lambda_i$. We define $M_{SUSY}$ as $$M^2_{SUSY}=(m^2_{\tilde t_1}+m^2_{\tilde t_2})/2$$ where $\tilde t_1$ and $\tilde t_2$ refer to the lighter and heavier stop eigenstates respectively. For set(A) $M_{SUSY} \approx 450$ GeV and for set(B) $M_{SUSY} \approx 280$ GeV. For both parameter sets the mass splitting $m^2_{\tilde t_2}-m^2_{\tilde t_1}$ comfortably satisfies the RG analysis requirement [@NPB553] $${m^2_{\tilde t_2}-m^2_{\tilde t_1}\over
m^2_{\tilde t_2}+m^2_{\tilde t_1}} \lesssim 0.5$$ Note that the large value of $\mu$ taken in set (A) and (B) does not imply an unacceptably light $\tilde t_1$, since the corresponding entry in the stop mass matrix is $\mu\cot\beta$, which is comfortably suppressed for the assumed values of $\tan\beta$. Larger values of $\tan\beta$ together with large $\mu$ would invariably generate values of $m_{\tilde b_1}$ which are lighter than the current experimental bounds. We also require that all $\lambda_i$ remain in the perturbative region.
In Fig. 1 (a) and (b), we plot $M_{H^\pm}-M_{H^0}$ and $M_{A}-M_{H^\pm}$ as a function of $\tan\beta$ for $\mu$=$4500,4000,3500$ GeV (from above) and $\mu$=$3300,2800,2300$ GeV (from above), respectively. For the other parameters we take the choices in (\[sets\]).
In both Figs. 1 and 2, the line in thin typeset signifies that the corresponding mass for one of $\tilde t_1$, $\tilde b_1$ and $M_{h^0}$ becomes smaller than the present experimental bound.
In Fig. 1(a), we can understand the behaviour of the splittings $M_{H^\pm}-M_{H^0}$ from eqns.(4) and (\[splita\]). In eq. (4), increasing $\tan\beta$ enhances the term $\sim h_b^2h_t^2$ which is negative in this parameter space, and thus reduces the positive ${\lambda}_4/2$. ${\lambda}_5$ is also positive and is not so sensitive to $\tan\beta$ since the top quark Yukawa term dominates. However, $2 {\lambda}_6/\tan\beta$ is negative and its modulus becomes smaller as $\tan\beta$ increases. The combined effect is as follows: for small $\tan\beta$ the negative term in eq. (5) cancels the positive terms and the splitting is small. As $\tan\beta$ increases the positive terms dominate, giving rise to large splittings. Further increases in $\tan\beta$ reduce the still positive $\lambda_4$, while $\lambda_5$ remains approximately constant, which explains the descent of the curve for larger $\tan\beta$. We have also checked that the splittings are relatively insensitive to variations in $A_b$ since the relevant terms are suppressed by $h_b^4$. One might expect that the splittings could be further increased by changing the sign of $A_t$, which makes ${\lambda}_6$ positive. However it is not easy to satisfy the scalar quark mass bound with this choice in the CP conserving scenario. Larger values of $\mu$ give rise to larger splittings, which is explained in Fig. 2(a) below.
In Fig. 1(b) one can see that $M_{A}-M_{H^\pm}$ increases with $\tan\beta$, which is explained by the fact that $\lambda_4$ is decreasing (see above) and thus enhances the splitting from eq. (2). Large $\mu$ provides the largest splittings (see below).
In Fig. 2 (a) we plot $M_{H^\pm}-M_{H^0}$ as a function of $\mu/M_{SUSY}$ for $A_t =2000, 1900, 1800$ GeV (from above). For the other parameters we take the choices in (\[sets\]). In Fig. 2 (a), we can see that $\mu/M_{SUSY} \gtrsim 6$ allows splittings $M_{H^\pm}-M_{H^0} \gtrsim 80$ GeV. Increasing $A_t$ enhances the positive contributions to ${\lambda}_4$ and ${\lambda}_5$, which in turn enhances $M_{H^\pm}-M_{H^0}$ (from eq. (\[splita\])).
In Fig. 2 (b), we plot $M_{A}-M_{H^\pm}$ as a function of $\mu/M_{SUSY}$ for $M_{H^\pm} =150, 250, 350$ GeV (from above), since it turns out that the splittings are not very sensitive to $A_t$. As expected, we can see the decoupling behaviour, with the splitting decreasing for increasing $M_{H^\pm}$. For low values of $\mu/M_{SUSY}$ the 1-loop corrections are not so large and so one has approximately the tree–level result $M_{H^\pm}\ge M_A$. As $\mu/M_{SUSY}$ increases, $\lambda_5$ is enhanced to large positive values. In addition, $\lambda_4$ is decreased from its positive tree-level value. Hence the mass splitting $M_A-M_{H^\pm}$ increases and exceeds 80 GeV for $\mu/M_{SUSY}\gtrsim 10$.
If CP violating phases are allowed in $\mu$ and $A_{t,b}$ then ${\lambda}_{5,6}$ become complex numbers in general. The mass eigenstates $H_2^0$ and $H_3^0$ are now mixed states of CP, and the large splittings $M_{H^\pm}-M_{H_2^0}$ or $M_{H_3^0}-M_{H^\pm}$ are possible in a wider range of parameter space which satisfies the mass bounds on $H_1^0$ and $\widetilde{t}_1$ ($\widetilde{b}_1$).
The magnitude of the mass splittings has important consequences for the phenomenology of the MSSM. Our results show that $M_{H^\pm}$ need not be close in mass to $H^0$ and $A^0$, and the mass differences $|M_{H^\pm}-M_{H^0,A}|$ may be as large as the analogous values in a general 2HDM or in other extended Higgs sectors. The relation $M_{H^\pm}\approx M_{H^0,A}$ for $M_{H^\pm}\ge 200$ GeV, which is assumed in existing phenomenological analyses, may be broken in the region of large $\mu/M_{SUSY}$. We stress that the large values of $\mu$ considered here do not arise in popular SUSY models, such as minimal supergravity or gauge mediated models. As discussed in [@plb495], in order to be compatible with electroweak symmetry breaking, such large values of $\mu$ would require the soft SUSY breaking mass parameters $m^2_1$ and $m^2_2$ to be of the order of $|\mu|$ and negative, and considerable fine-tuning is necessary.
If $M_{H^\pm}-M_{H^0}\ge 80$ GeV then the 2 body decays $H^{\pm}\to H^0 W^\pm$ would be open. This would offer a new discovery channel for $H^\pm$ at the LHC, and one which is expected to offer a very promising signature. Ref. [@HAW] presented a signal–background analysis showing that the background is small, and any model which allows a large BR($H^{\pm}\to H^0W^\pm$) would provide a very clear signal in this channel. We shall show below that such large BRs are possible in the MSSM, and consequently would aid the search for $H^\pm$ at the LHC.
If $M_{A}-M_{H^\pm}\ge 80$ GeV then the 2 body decays $A^0\to H^\pm W^\mp$ would be open. This would have important consequences for the process $\mu^+\mu^-\to H^\pm W^\mp$ at a muon collider, which receives contributions from $A^0,H^0$ mediated s–channel diagrams, and was shown to be a promising production mechanism for $H^{\pm}$ in the MSSM with $M_{H^\pm}\approx M_{H^0,A^0}$ [@AAD]. Any splittings with $M_{A}\ge M_{H^\pm}$ would enhance the rate for $\sigma(\mu^+\mu^- \to H^{\pm}W^{\mp}$) compared to that in [@AAD]. For $M_{A}-M_{H^\pm}\ge 80$ GeV one could have resonant $H^\pm$ production via $\mu^+\mu^-\to A^0\to H^\pm W^\mp$, which was shown to allow very large cross–sections ($>1$ pb) in the case of the general 2HDM [@mumuHW]. These possibilities will be pursued in a future article [@progress].
In Fig. 3 we plot branching ratios (BRs) for the decay $H^{\pm}\to H^0 W^\pm$ as a function of $\tan\beta$. This two body decay is open if $M_{H^\pm}-M_{H^0}\ge 80$ GeV, and is proportional to $\sin^2(\beta-\alpha)$. Also plotted is BR($H^{\pm}\to h^0W^\pm$) which $\sim \cos^2(\beta-\alpha)$. We take the same parameters as Fig. 1 (a). The thick (thin) lines represent BR($H^{\pm}\to H^0 (h^0) W^\pm$) for $\mu=4500, 4000, 3500$ (from above), respectively. Note that we do not consider decays to SUSY particles $H^{\pm}\to \tilde t \tilde b$, $\chi^\pm\chi^0$ etc. The decays $H^\pm\to \tilde t_1\tilde b_1$ are not open for this choice of $M_{H^\pm}$ and $H^\pm \to \chi^\pm\chi^0$ can be closed by choosing suitable values for $M_1$,$M_2$ etc. It can be seen from Fig.3 that BR$\gtrsim 20\%$ is attainable for intermediate values of $\tan\beta$, clearly showing the importance of this new decay channel.
The behaviour of the BRs can be qualitatively understood as follows: for low $\tan\beta$ the splittings are not large enough to open the channel and for large $\tan\beta$ the newly opened channel $H^{\pm}\to H^0 W^\pm$ cannot compete with the fermionic decay modes. Thus the maximum inpact of $H^{\pm}\to H^0W^\pm$ decays is for intermediate values of $\tan\beta$, which is the most problematic region for the $H^{\pm}$ search at the LHC. The current search strategies utilize $H^{\pm}\to \tau\nu_{\tau}$ decays [@Roy], which is most effective for $\tan\beta\ge 15$, or $H^\pm\to tb$ decays which cover the regions $\tan\beta\le 3$ and $\tan\beta\ge 25$ [@Atlas9913]. We can also see that BR($H^{\pm}\to h^0 W^\pm$), although suppressed in the decoupling limit, can be significant. The sizeable BRs shown in Fig. 3 would offer very good detection prospects in these channels [@HAW].
In general, the mass splittings presented here may give rise to a large contribution to the precisely measured electroweak parameter $\rho$. For mass splittings $\ge 20$ GeV we find that the Higgs contribution is always positive and violates the $\delta\rho$ constraint at the $2(3)\sigma$ level for $|M_{H^\pm}-M_{H^0,A}|\ge 150(200)$ GeV. The dominant SUSY particle contribution ($\tilde t-\tilde b$ loops) to $\delta\rho$ has the same sign as the Higgs contribution [@hagiwara], thus further reducing the maximum allowed value of $|M_{H^\pm}-M_{H^0,A}|$. Therefore the mass splittings $|M_{H^\pm}-M_{H^0,A}|$ presented here are consistent with the $\delta\rho$ constraints.
Conclusions
===========
We have shown that very large mass splittings $M_{H^\pm}-M_{H^0,A}$ are possible in the MSSM. Such splittings occur in a previously ignored region of the MSSM parameter space and may exceed 100 GeV, thus strongly violating the commonly assumed degeneracy relation $M_{H^\pm}\approx M_{H^0}\approx M_{A}$. The largest splittings arise for relatively large $\mu\ge 6M_{SUSY}$. The previously neglected 2 body decays $H^{\pm}\to H^0W^\pm$ and $A^0\to H^+W^-,H^-W^+$ may become the dominant channels, which would have important consequences for Higgs searches at future colliders. If $M_{A}-M_{H^\pm} \gtrsim 80$ GeV, a muon collider could copiously produce $H^\pm$ at resonance.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank A. Arhrib, P. Ko and Y. Okada for useful comments. We are grateful to C. Wagner for confirming some of the results presented here.
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---
abstract: 'In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the Partial Differential Equation (PDE) and the goal functionals may be nonlinear. Our method is based on a posteriori error estimates in which the adjoint problem is used and a partition-of-unity is employed for the error localization that allows us to formulate the error estimator in the weak form. We provide a careful derivation of the primal and adjoint parts of the error estimator. The second objective is concerned with balancing the nonlinear iteration error with the discretization error yielding adaptive stopping rules for Newton’s method. Our techniques are substantiated with several numerical examples including scalar PDEs and PDE systems, geometric singularities, and both nonlinear PDEs and nonlinear goal functionals. In these tests, up to six goal functionals are simultaneously controlled.'
author:
- 'B. Endtmayer'
- 'U. Langer'
- 'T. Wick'
bibliography:
- './lit.bib'
title: 'Multigoal-Oriented Error Estimates for Non-linear Problems'
---
Introduction {#Introduction}
============
A posteriori error estimation and mesh adaptivity are well-developed methodologies for finite element computations, see, e.g., the monographs [@Verfuerth:1996a; @AinsworthOden:2000; @BaRa03; @NeittaanmaekiRepin:2004a; @Han:2005a; @RepinBook2008] and the references therein. Specifically, goal-oriented error estimation is a powerful method when the evaluation of certain functionals of interest (often these are technical quantities) is the main aim rather than the computation of global error norms. Here, the dual-weighted residual (DWR) method is often applied [@BeckerRannacher1995; @BeRa01].
Thanks to increasing computational resources, multiphysics applications such as multiphase flow, porous media applications, fluid-structure interaction and electromagnectics are currently one main focus in applied mathematics and engineering. Here, mesh adaptivity (ideally combined with parallel computing) can greatly reduce the computational cost while measuring functionals of interest with sufficient accuracy. Since in multiphysics, several physical phenomena interact, it might be desirable that more than one goal functional shall be controlled. However, only a few studies have appeared yet. A first methodology was proposed in [@HaHou03; @Ha08]. Other studies can be found in [@HouSenSue02; @PARDO20101953] and more recently in [@BruZhuZwie16; @EnWi17; @KerPruChaLaf2017].
Until a few years ago, one principle problem in using the DWR method was the fact that the error estimator was based on the strong form of the equations [@BeRa01] or the weak form using special interpolation operators working on patched meshes [@BraackErn02]. In [@RiWi15_dwr], the previous localization techniques were analyzed in more detail and additionally a novel localization strategy based on a partition-of-unity (PU) was proposed. The PU localization specifically allows for a much simpler application of the DWR method to multiphysics and nonlinear, vector-valued equations [@RiWi15_dwr; @Wi16_dwr_pff]. In addition, the PU-DWR method works well with other discretization techniques such as BEM-based FEM [@WeiWi18] or the finite cell method [@StolfoRademacherSchroeder2017]. On the other hand, the methodology of the PU-DWR method with multiple goal functionals has recently been worked out for linear, scalar-valued problems in [@EnWi17].
The first goal of this paper is to extend this work to nonlinear problems and PDE systems. Here, our focus is on a careful design of the error estimator that includes both the primal part and the adjoint part. The latter one is often neglected in the literature because the evaluation requires additional computational cost and renders the method even more expensive. It is clear and well-known (see e.g., [@BeRa01]) that, in the linear case, the primal and adjoint residuals yield the same error values, but possibly different locally refined meshes; see e.g. [@RiWi15_dwr]. In our current work, we will see that the adjoint estimator part is crucial to obtain good effectivity indices. Therefore, this term should not be neglected.
The second objective of this paper is concerned with balancing the discretization and the nonlinear iteration error. In recent years, there has been published some work on balancing the iteration error (of the linear or nonlinear solver) with the discretization error [@ErnVohral2013; @BeJohRa95; @MeiRaVih109; @RanVi2013; @RaWeWo10]. We base ourselves on [@RanVi2013], and we employ specifically the PU localization. Consequently, the DWR method is used to design an adaptive stopping criterion for Newton’s method that is in balance with the estimated discretization error. The main aspects comprise a careful choice of the weighting functions to design an appropriate joined goal functional. Moreover, we provide all details for the nonlinear solver, which is a Newton-type method with backtracking line-search. Since we know a solution on the previous mesh, we use this solution as initial guess for Newton’s method yielding a nested iteration. Specifically, nested solution methods or nonlinear nested iterations were developed, for instance, in [@BeRa01; @MultigridHackusch03a]. We refer to [@MultigridHackusch03a; @Reusken1988] for the analysis of nested iteration methods.
In summary, the goals of this work are two-fold:
- Design of the PU-DWR method for multigoal-oriented error estimation for nonlinear problems and PDE systems.
- Balancing iteration and discretization errors for nonlinear multigoal-oriented error estimation and mesh adaptivity. The nonlinearities may appear in the PDE itself as well as in the goal functionals.
The outline of this is paper is as follows: In Section \[Discretization\], our setting is described. Next, in Section \[PU-DWR-NONLINEAR-ONEFUNTIONAL\], we describe the methodology for one goal functional. This is followed by a detailed derivation of a multigoal-oriented approach presented in Section \[Multigoalfunctionals\]. The key algorithms are formulated in Section \[sec\_alg\]. In Section \[sec\_num\_tests\] several numerical tests substantiate our developments. We summarize our work in Section \[sec\_concl\].
An abstract setting {#Discretization}
===================
Let $U$ and $V$ be Banach spaces, and let $\mathcal{A}: U \mapsto V^*$ be a (possibly) nonlinear operator, where $V^*$ denotes the dual space of the Banach space $V$. We have in mind nonlinear differential operators $\mathcal{A}$ acting between Sobolev spaces. We now consider the following weak formulation of the operator equation $\mathcal{A}(u) = 0$ in $V^*$: Find $u \in U$ such that $$\label{Problem:primal}
\mathcal{A}(u)(v)=0 \quad \forall v \in V.$$ The discretization of the nonlinear variational problem (\[Problem:primal\]) can be performed by means of different methods. Our favored method is the Finite Element Method (FEM), see also Section \[subsec\_SpatialDiscretization\]. The corresponding discrete problem reads as follows: Find $u_h \in U_h$ such that $$\label{problem: discrete primal problem}
\mathcal{A}(u_h)(v_h)=0 \quad \forall v_h \in V_h,$$ where $U_h$ and $V_h$ are finite-dimensional subspaces of $U$ and $V$, respectively. For the time being, let us assume that both problems are solvable. Later we will specify our assumptions imposed on $\mathcal{A}$. We are primarily not interested in approximating a solution $u$ of (\[Problem:primal\]), but in the approximate computation of one or more possibly nonlinear functionals at a solution.
An example for such an operator $\mathcal{A}$ is given by the weak formulation of the regularized $p$-Laplace equation (see also [@DiRu07; @Hi2013; @ToWi2017]) that reads as follows: Find $u \in U := W^{1,p}_0(\Omega)$ such that $$\begin{aligned}
\label{pLaplace:weak}
\mathcal{A}(u)(v) :=& \langle{(\varepsilon^2+ |\nabla u|^2)^{\frac{p-2}{2}}\nabla u,\nabla v\rangle}_{(L^p(\Omega))^*\times L^p(\Omega)}-\langle{f,v\rangle}_{ (W^{1,p}_0(\Omega))^*\times W^{1,p}_0(\Omega)}
= 0
$$ for all $v \in V := W^{1,p}_0(\Omega)$, where $\varepsilon$ denotes a fixed positive regularization parameter, $f \in (W^{1,p}_0(\Omega))^*= W^{-1,q}(\Omega)$ is some given source, with $p^{-1} + q^{-1}=1$ and fixed $p>1$, and $\langle{\cdot,\cdot}\rangle$ denots the corresponding duality products. Here, $\Omega \subset \mathbb{R}^d$, $d=1,2,3$, is a bounded Lipschitz domain, and $W^{1,p}_0(\Omega)$ denotes the usual Sobolev space of all functions from the Lebesgue space $L^p(\Omega)$ with weak derivatives in $L^p(\Omega)$ and trace zero on the boundary $\partial \Omega$, see, e.g., [@Adams2003sobolev]. The notation $ | \cdot | $ is used for the Euclidean norm of some vector. The corresponding strong form is formally given by $$\begin{aligned}
-\text{div}((\varepsilon^2 + |\nabla u|^2)^{\frac{p-2}{2}}\nabla u) &=f \quad \mbox{in}\;\Omega, \\
u &= 0 \quad \mbox{on}\; \partial \Omega .\end{aligned}$$ In Subsection \[subsection: Example1\], the regularized $p$-Laplace (\[pLaplace:weak\]) serves as first example for our numerical experiments.
We refer the reader to [@Glowinski1975] for the investigation of the original $p$-Laplace problem.
The dual weighted residual method for nonlinear problems in the case of a single-goal functional {#PU-DWR-NONLINEAR-ONEFUNTIONAL}
================================================================================================
In this section, we apply the DWR method to nonlinear problems. The general method was developed in [@BeRa01]. The extension to balance discretization and iteration errors was undertaken in [@MeiRaVih109; @RaWeWo10; @RanVi2013]. We base ourselves on the latter study [@RanVi2013], in which algorithms for nonlinear problems have been worked out. This last paper, together with [@RiWi15_dwr; @EnWi17], form the basis of the current paper. We are interested in the goal functional evaluation $J:U\to\mathbb{R}$ with $u\mapsto J(u)$, where $u\in U$ is a solution of the primal problem (\[Problem:primal\]). Examples for such goal functionals are:
- point evaluation: $$J(u):= u(x_0),$$
- integral evaluation: $$J(u):= \int_{\Omega}^{}u(x)\xi(x)\,dx,$$
- nonlinear functional evaluation: $$J(u):= \int_{\Omega}^{}u(x)\xi(x)u(x_0)^2 \,dx\int_{\Omega}^{}u(y)\phi(y)\,dy,$$
where $\xi$ and $\phi$ are given functions from $L^2(\Omega)$ and $x_0$ a given point in $\Omega$. For the DWR approach we need to solve the adjoint problem: Find $z \in V$ corresponding to $u \in U$ such that $$\label{Problem: adjoint}
\mathcal{A}'(u)(v,z)=J'(u)(v) \quad \forall v \in U,$$ where $u$ denotes a (primal) solution of the primal problem (\[Problem:primal\]), and $\mathcal{A'}(u)$ and $J'(u)$ denote the Fréchet-derivatives of the nonlinear operator or functional, respectively, evaluated at $u$. Later we also need the corresponding discrete solution of the adjoint problem. This reads as follows: Find $z_h \in V_h$ corresponding to $u_h \in U_h$ such that $$\label{Problem: discrete adjoint on small space}
\mathcal{A}'(u_h)(v_h,z_h)=J'(u_h)(v_h) \quad \forall v_h \in U_h,$$ with $u_h$ as a solution of (\[problem: discrete primal problem\]).
Similarly to the findings in [@RanVi2013; @BeRa01; @RaWeWo10] for the Galerkin case ($U=V$), we derive an error representation in the following theorem:
\[Theorem: Error Representation\] Let us assume that $\mathcal{A} \in \mathcal{C}^3(U,V)$ and $J \in \mathcal{C}^3(U,\mathbb{R})$. If $u$ solves (\[Problem:primal\]) and $z$ solves (\[Problem: adjoint\]) for $u \in U$, then it holds for arbitrary fixed $\tilde{u} \in U$ and $ \tilde{z} \in V$ : $$\begin{aligned}
\label{Error Representation}
\begin{split}
J(u)-J(\tilde{u})&= \frac{1}{2}\rho(\tilde{u})(z-\tilde{z})+\frac{1}{2}\rho^*(\tilde{u},\tilde{z})(u-\tilde{u})
-\rho (\tilde{u})(\tilde{z}) + \mathcal{R}^{(3)},
\end{split}
\end{aligned}$$ where $$\begin{aligned}
\label{Error Estimator: primal}
\rho(\tilde{u})(\cdot) &:= -\mathcal{A}(\tilde{u})(\cdot), \\
\label{Error Estimator: adjoint}
\rho^*(\tilde{u},\tilde{z})(\cdot) &:= J'(u)-\mathcal{A}'(\tilde{u})(\cdot,\tilde{z}),
\end{aligned}$$ and the remainder term $$\begin{split} \label{Error Estimator: Remainderterm}
\mathcal{R}^{(3)}:=\frac{1}{2}\int_{0}^{1}[J'''(\tilde{u}+se)(e,e,e)
-\mathcal{A}'''(\tilde{u}+se)(e,e,e,\tilde{z}+se^*)
-3\mathcal{A}''(\tilde{u}+se)(e,e,e)]s(s-1)\,ds,
\end{split}$$ with $e=u-\tilde{u}$ and $e^* =z-\tilde{z}$.
For the completeness of the presentation we add the proof below, which is very similar to [@RanVi2013]. First we define $x := (u,z) \in X:=U \times V$ and $\tilde{x}:=(\tilde{u},\tilde{v}) \in X$. By assuming that $\mathcal{A} \in \mathcal{C}^3(U,V)$ and $J \in
\mathcal{C}^3(U,\mathbb{R})$ we know that the Lagrange function $$\mathcal{L}(\hat{x}):= J(\hat{u})-\mathcal{A}(\hat{u})(\hat{z}) \quad \forall (\hat{u},\hat{z})=:\hat{x} \in X,$$ is in $\mathcal{C}^3(X,\mathbb{R})$. Assuming this it holds $$\mathcal{L}(x)-\mathcal{L}(\tilde{x})=\int_{0}^{1} \mathcal{L}'(\tilde{x}+s(x-\tilde{x}))(x-\tilde{x})\,ds.$$ Using the trapezoidal rule [@RanVi2013], we obtain $$\int_{0}^{1}f(s)\,ds =\frac{1}{2}(f(0)+f(1))+ \frac{1}{2} \int_{0}^{1}f''(s)s(s-1)\,ds,$$ for $f(s):= \mathcal{L}'(\tilde{x}+s(x-\tilde{x}))(x-\tilde{x})$ we conclude $$\begin{aligned}
\mathcal{L}(x)-\mathcal{L}(\tilde{x}) =& \frac{1}{2}(\mathcal{L}'(x)(x-\tilde{x}) +\mathcal{L}'(\tilde{x})(x-\tilde{x})) + \mathcal{R}^{(3)}.
\end{aligned}$$ From the definition of $\mathcal{L}$ we observe that $$\begin{aligned}
J(u)-J(\tilde{u})=\mathcal{L}(x)-\mathcal{L}(\tilde{x}) +\underbrace{A(u)(z) }_{=0} + A(\tilde{u})(\tilde{z}) =& \frac{1}{2}(\mathcal{L}'(x)(x-\tilde{x}) +\mathcal{L}'(\tilde{x})(x-\tilde{x})) +A(\tilde{u})(\tilde{z})+ \mathcal{R}^{(3)}.
\end{aligned}$$ It remains to show that $\frac{1}{2}(\mathcal{L}'(x)(x-\tilde{x}) +\mathcal{L}'(\tilde{x})(x-\tilde{x})) = \frac{1}{2}\rho(\tilde{u})(z-\tilde{z})+\frac{1}{2}\rho^*(\tilde{u},\tilde{z})(u-\tilde{u}) $. But this is true since $$\begin{aligned}
\mathcal{L}'(x)(x-\tilde{x}) +\mathcal{L}'(\tilde{x})(x-\tilde{x}) = & \underbrace{J'(u)(e)-\mathcal{A}'(u)(e,z)}_{=0}-\underbrace{A(u)(e^*)}_{=0}+\underbrace{J'(\tilde{u})(e)-\mathcal{A}'(\tilde{u})(e,\tilde{z})}_{=\rho^*(\tilde{u},\tilde{z})(u-\tilde{u})}-\underbrace{A(\tilde{u})(e^*)}_{=-\rho(\tilde{u})(z-\tilde{z})}.
\end{aligned}$$
Instead of $\mathcal{A} \in \mathcal{C}^3(U,V)$ and $J \in \mathcal{C}^3(U,\mathbb{R})$ it is sufficient that $\mathcal{A} \in \mathcal{C}^2(U,V)$, $J \in \mathcal{C}^2(U,\mathbb{R})$ and $J''',$ $ \mathcal{A'''}$ exist and are bounded. Moreover one can further relax these assumptions. Indeed the boundedness of the derivatives is just needed in the set $\{ w \in U| w=(1-s)u+s\tilde{u} \}$ and just in direction $u-\tilde{u}$.
It might happen that $\mathcal{A} \in \mathcal{C}^3(U,V)$ and $J \in \mathcal{C}^3(U,\mathbb{R})$ do not hold for the continuous spaces. Since the result holds for general Banach spaces $U$ and $V$, it is sufficient to be shown for the discrete spaces $U_{h,u},V_{h,z}$, where $U_{h,u}:=\{w+cu| w \in U_h, c \in \mathbb{R}\},V_{h,z}:=\{v+cz|v\in V_h, c \in \mathbb{R}\}$.
In accordance with the literature, we denote the parts $\rho(\tilde{u})(z-\tilde{z})$ and $\rho^*(\tilde{u},\tilde{z})(u-\tilde{u})$ by *primal error estimator* and *adjoint error estimator*, respectively. The remainder term $\mathcal{R}^{(3)}$, as in (\[Error Estimator: Remainderterm\]), is of the order $\mathcal{O}(\Vert e \Vert_U^2
\text{max}(\Vert e \Vert_U, \Vert e^* \Vert_V))$. Therefore, it can be neglected if $\{\tilde{u},\tilde{z}\}$ are close enough to $\{u,z\}$.
As in [@RanVi2013] ,we can identify
$$\label{Error Estimator: discretization}
\eta_h:=|\frac{1}{2}\rho(\tilde{u})(z-\tilde{z})+\frac{1}{2}\rho^*(\tilde{u},\tilde{z})(u-\tilde{u})|,$$
as the *discretization error* and $$\eta_m:=|\rho(\tilde{u})(\tilde{z})|,$$ as the *linearization error* if we neglect the remainder term $\mathcal{R}^{(3)}$. Since Theorem \[Theorem: Error Representation\] is valid for arbitrary $\tilde{z}$ and $\tilde{u}$ it also holds for approximations $u_h$ and $z_h$, even if they are not computed exactly. Of course, formula (\[Error Estimator: discretization\]) still contains an exact solution $u$. Since $u$ is not known, we either use an approximation in an enriched discrete space (for example, in a finite element space, with higher polynomial degree), or we use an interpolant $I_h^{h_2}$, such as in [@BeRa01], to obtain a more accurate solution $u_h^{(2)}$. If not mentioned otherwise, we use the approximation in the enriched (finite element) space. An enriched discrete space is also used to compute an approximation $z_h^{(2)}$ of $z$. If one would use the same finite-dimensional space as for the test space used in the discrete primal problem (\[problem: discrete primal problem\]), then $\mathcal{A}(u_h)(z_h)=0 $ for our approximate solution $u_h$ of (\[problem: discrete primal problem\]) (if the nonlinear problem is solved exactly). Therefore, the discrete adjoint problem reads as follows: Find $z_h^{(2)}\in V_h^{(2)}$ such that $$\label{Problem: discrete adjoint}
\mathcal{A}'(u_h^{(2)})(v_h^{(2)},z_h^{(2)})=J'(u_h^{(2)})(v_h^{(2)})\quad \forall v_h^{(2)} \in U_h^{(2)},$$ where $U_h^{(2)}$ and $V_h^{(2)}$ denote the enriched finite dimensional spaces, and $u_h^{(2)}$ denotes the more accurate solution, obtained by solving (\[problem: discrete primal problem\]) with $U_h=U_h^{(2)}$ and $V_h=V_h^{(2)}$ or by interpolation $u_h^{(2)}=I_h^{h_2}u_h$. With these approximations, the practical error estimator reads: $$\label{Error Estimator: discretization inexact}
\eta_h:=|\frac{1}{2}\rho(u_h)(z_h^{(2)}-z_h)+\frac{1}{2}\rho^*(u_h,z_h)(u_h^{(2)}-u_h)|.$$ For localization of the error estimator, we use the partition of unity (PU) technique which is presented in [@RiWi15_dwr]. This means that we choose a set of functions $\{\psi_1, \psi_2, \cdots,\psi_N\}$ such that $ \sum_{i=1}^{N} \psi_i \equiv 1.$ Inserting this into (\[Error Estimator: discretization inexact\]) leads to $$\label{Local Error Estimator: discretization}
\eta_{h}:=|\sum_{i=1}^{N}\eta_i|,$$ with $$\label{eta_i_PU}
\eta_i:=\frac{1}{2}\rho(\tilde{u})((z_h^{(2)}-\tilde{z})\psi_i)+\frac{1}{2}\rho^*(\tilde{u},\tilde{z})((u_h^{(2)}-\tilde{u})\psi_i).$$ We notice that in the primal part of the error indicator $\tilde{z}$ is replaced by $i_h z_h^{(2)}$ as in [@BeRa01]. For instance, a typical partition of unity is given by the finite element basis. In this case, we distribute $|\eta_i|$ to the corresponding elements with a certain weight as for example illustrated in Figure \[fig: dist. of nodal error\].
Multiple-goal functionals {#Multigoalfunctionals}
=========================
Now let us assume that we are interested in the evaluation of $N$ functionals, which we denote by $J_1, J_2, \ldots,J_N$. From Section \[PU-DWR-NONLINEAR-ONEFUNTIONAL\], we know how to compute a local error estimator for one functional. We could compute the local error estimators separately. However, we would have to solve the adjoint problem (\[Problem: adjoint\]) $N$ times [@HaHou03; @Ha08]. Let us now assume that a solution $u$ of problem (\[Problem:primal\]) and the chosen $\tilde{u} \in U$ belong to $\bigcap_{i=1}^N \mathcal{D}(J_i)$, where $\mathcal{D}(J_i)$ describes the domain of $J_i$.
Let $ M \subseteq \mathbb{R}^N$. We say that $\mathfrak{E}: (\mathbb{R}^+_0)^N \times M \mapsto \mathbb{R}^+_0$ is an *error-weighting function* if $\mathfrak{E}(\cdot,m) \in
\mathcal{C}^1((\mathbb{R}^+_0)^N,\mathbb{R}^+_0)$ is strictly monotonically increasing in each component and $\mathfrak{E} (0,m)=0$ for all $m \in M$.
Let $\vec{J}: \bigcap_{i=1}^N \mathcal{D}(J_i) \subseteq U \mapsto \mathbb{R}^N$ be defined as $\vec{J}(v):=(J_1(v),J_2(v),\cdots, J_{N}(v) )$ for all $v \in \bigcap_{i=1}^N \mathcal{D}(J_i)$. Furthermore, we define the operation $|\cdot|_N:\mathbb{R}^N\mapsto (\mathbb{R}^+_0)^N$ as $|x|_N:= (|x_1|,|x_2|,\cdots,|x_N|)$ for $x \in \mathbb{R}^N $. This allows us to define the as follows $$\begin{aligned}
\label{ErrorrepresentationFunctional}
\tilde{J}_{\mathfrak{E}}(v):=\mathfrak{E}(|\vec{J}(u)-\vec{J}(v)|_N, \vec{J}(\tilde{u})) \qquad \forall v \in \bigcap_{i=1}^N \mathcal{D}(J_i).\end{aligned}$$ It is trivial to see from the definition of $\mathfrak{E}$ that $J_\mathfrak{E}(v) \in \mathbb{R}^+_0$ for all $v \in \bigcap_{i=1}^N \mathcal{D}(J_i)$.
The idea of the construction of $\tilde{J}_{\mathfrak{E}}(v)$ is that $\mathfrak{E}(|\vec{J}(u)-\vec{J}(v)|_N, \vec{J}(\tilde{u}))$ is a semi-metric (as in [@SierpinskiTopo52; @KhaKirMetric2001]) on the set of equivalence classes $(\vec{J})^{-1}(\mathcal{R}(\vec{J})):=\{(\vec{J})^{-1}(x): x \in
\mathcal{R}(\vec{J})\}$, where $(\vec{J})^{-1}(x):=\{v \in \bigcap_{i=1}^N
\mathcal{D}(J_i): \vec{J}(v)=x \}$, with $\mathcal{R}(\vec{J})$ denotes the range of $\vec{J}$, measuring the distance between the equivalence classes containing $u$ and $v$. Hence, $\tilde{J}_{\mathfrak{E}}(v)$ represents a semi-metric distance which ensures that $\tilde{J}_{\mathfrak{E}}$ is monotonically increasing if $|J_i(u)-J_i(\tilde{u})|$ is monotonically increasing.
If we drop the monotonicity condition in the definition of $\mathfrak{E}$, then, for example, $$\mathfrak{E}(|\vec{J}(u)-\vec{J}(v)|_N, \vec{J}(\tilde{u})):= \prod_{i=0}^{N} |J_i(u)-J_i(v)|,$$ would be an error-weighting function, resulting in $J_\mathfrak{E}(\tilde{u})=0$ iff $J_i(u)=J_i(\tilde{u})$ at least for one $i \in \{1,2, \cdots ,N\} $.
The derivation given in this section holds for a general $\tilde{u}$ such that $\vec{J}(\tilde{u}) \in M$. In particular, we are interested in $\tilde{u}$ to be an approximation to $u_h$ solving (\[problem: discrete primal problem\]).
The weak derivative of (\[ErrorrepresentationFunctional\]) in $U$ at $\tilde{u}$ is given by $$\begin{aligned}
\label{ErrorrepresentationFunctionalJ'}
\tilde{J}_{\mathfrak{E}}'(\tilde{u})(v):=-\sum_{i=1}^{N}\text{ sign}(J_i(u)-J_i(\tilde{u}))\frac{\partial \mathfrak{E}}{\partial x_i}(|\vec{J}(u)-\vec{J}(\tilde{u})|_N, \vec{J}(\tilde{u})) J_i'(\tilde{u})(v) \qquad \forall v \in \mathcal{D}(\tilde{J}_{\mathfrak{E}}'(\tilde{u})),\end{aligned}$$ with
$$\label{sign}
\text{sign}(x):=\begin{cases}
\frac{x}{|x|}, \quad \text{for }x\not =0 ,\\
0 \quad else
\end{cases}$$
In [@HaHou03; @Ha08; @EnWi17], the functionals where combined as follows $$\label{J_c}
\tilde{J}_c(v):=\sum_{i=1}^{N} \underbrace{\frac{\omega_i\text{ sign}(J_i(u)-J_i(\tilde{u}))}{|J_i(\tilde{u})|}}_{=:w_i }J_i(v) \quad \forall v\in \bigcap_{i=0}^N \mathcal{D}(J_i).$$
Carefully inspecting [@Ha08], we see that the following result can be established:
\[JeJc\] If $\tilde{J}_c$ is defined as in (\[J\_c\]) and $\tilde{J}_{\mathfrak{E}}$ as in (\[ErrorrepresentationFunctional\]), then we have $$\begin{aligned}
\tilde{J}_c(u)-\tilde{J}_c(\tilde{u})&= \tilde{J}_{\mathfrak{E}}(\tilde{u}), \label{Jcu-Jcuh=Jeuh}\\
-\tilde{J}_c'(\tilde{u})(v)&= \tilde{J}_{\mathfrak{E}}'(\tilde{u})(v), \qquad \forall v\in \mathcal{D}(\tilde{J}_{c}'(\tilde{u}))\cap\mathcal{D}(\tilde{J}_{\mathfrak{E}}'(\tilde{u})) ,\label{Jc'uh=Je'uh}\\
\mathcal{D}(\tilde{J}_{c}'(\tilde{u}))&=\mathcal{D}(\tilde{J}_{\mathfrak{E}}'(\tilde{u})) \label{DJe=DJc}
\end{aligned}$$ with $\mathfrak{E}(x,\vec{J}(\tilde{u})):= \sum_{i=1}^{N} \frac{\omega_i x_i}{|J_i(\tilde{u})|}$.
First we conclude that $$\begin{aligned}
\tilde{J}_c(u)-\tilde{J}_c(\tilde{u}) &= \sum_{i=1}^{N} \frac{\omega_i\text{ sign}(J_i(u)-J_i(\tilde{u}))}{|J_i(\tilde{u})|}(J_i(u)-J_i(\tilde{u}))\\
&= \sum_{i=1}^{N}
\frac{\omega_i|J_i(u)-J_i(\tilde{u})|}{|J_i(\tilde{u})|}\\
&=
\mathfrak{E}(|\vec{J}(u)-\vec{J}(\tilde{u})|_N,\vec{J}(\tilde{u}))
=\tilde{J}_{\mathfrak{E}}(\tilde{u}),
\end{aligned}$$ which already shows (\[Jcu-Jcuh=Jeuh\]). The weak derivative of $\tilde{J}_c$ is given by $$\label{J_c'}
\tilde{J}_c'(\tilde{u})(v)=\sum_{i=1}^{N} \frac{\omega_i\text{ sign}(J_i(u)-J_i(\tilde{u}))}{|J_i(\tilde{u})|}J_i'(\tilde{u})(v).$$ From $\frac{\partial \mathfrak{E}}{\partial x_i}(|\vec{J}(u)-\vec{J}(\tilde{u})|_N, \vec{J}(\tilde{u}))= \frac{\omega_i}{|J_i(\tilde{u})|}$ for all $i \in \{1,2,\cdots, N \}$, and because (\[J\_c’\]) and (\[ErrorrepresentationFunctionalJ’\]) coincide up to the sign, it holds that (\[Jc’uh=Je’uh\]) and (\[DJe=DJc\]) are valid.
$\mathfrak{E}(x,\vec{J}(\tilde{u})):= \sum_{i=1}^{N} \frac{\omega_i
x_i}{|J_i(\tilde{u})|}$ is an error-weighting function with $M:=\{x \in \mathbb{R}^N: \min(|x|)>0\}$ provided that $\omega_i > 0$ for all $i=1,2, \ldots, N$.
Proposition \[JeJc\] does not use the property that $u$ solves (\[Problem:primal\]). We just need that $u \in \bigcap_{i=0}^N \mathcal{D}(J_i)$. However, the goal is to measure the error to an exact solution.
Since an exact solution $u$ is not known, neither $\tilde{J}_c$ nor $\tilde{J}_{\mathfrak{E}}$ can be constructed. As in Section \[PU-DWR-NONLINEAR-ONEFUNTIONAL\], we use the approximation $u_h^{(2)}$ instead of an exact solution $u$ to approximate $\tilde{J}_c$ or $\tilde{J}_{\mathfrak{E}}$ by $J_c$ and $J_{\mathfrak{E}}$, respectively. This approximation reads as follows $$\begin{aligned}
\label{ErrorrepresentationFunctionalapprox}
J_{\mathfrak{E}}(v):=\mathfrak{E}(|\vec{J}(u_h^{(2)})-\vec{J}(v)|_N, \vec{J}(\tilde{u})) \qquad \forall v \in \bigcap_{i=1}^N \mathcal{D}(J_i),\end{aligned}$$ with the derivative $$\begin{aligned}
\label{ErrorrepresentationFunctionalapprox'}
J_{\mathfrak{E}}'(\tilde{u})(v):=-\sum_{i=1}^{N}\text{ sign}(J_i(u_h^{(2)})-J_i(\tilde{u}))\frac{\partial \mathfrak{E}}{\partial x_i}(|\vec{J}(u_h^{(2)})-\vec{J}(\tilde{u})|_N, \vec{J}(\tilde{u})) J_i'(\tilde{u})(v) \qquad \forall v \in \mathcal{D}(\tilde{J}_{\mathfrak{E}}'(\tilde{u})).\end{aligned}$$ Using this approximation of the error functional, we can apply the methods for the single-functional case in Section \[PU-DWR-NONLINEAR-ONEFUNTIONAL\] with $J=J_{\mathfrak{E}}$.
We notice that Theorem \[Theorem: Error Representation\] formally does not hold for $\tilde{J}_{\mathfrak{E}}$ since the sign-function enters. However, if $\mathfrak{E}(\cdot,m) \in \mathcal{C}^3((\mathbb{R}^+_0)^N,\mathbb{R}^+_0)$ and the functionals are sufficiently smooth, then the singularities (due to the signum function) in higher derivatives of $J_{\mathfrak{E}}$ just appear if $J_i(u)=J_i(u_h)$, or more precisely $J_i(u_h^{(2)})=J_i(u_h)$, since we use the better approximation $u_h^{(2)}$ instead of $u$. Alternatively, we can replace the signum function with a sufficiently smooth approximation.
Algorithms {#sec_alg}
==========
We now describe the algorithmic realization of the previous methods when we use the FEM as spatial discretization. To this end, we first introduce the finite element (FE) discretizations that we are going to use in our numerical experiments presented in Section \[sec\_num\_tests\]. Then we recapitulate the basic structure of Newton’s method including a line search procedure. Afterwards, we state the adaptive Newton algorithm for multiple-goal functionals followed by the structure of the final algorithm.
Spatial discretization {#subsec_SpatialDiscretization}
----------------------
For simplicity, we assume that $\Omega \subset \mathbb{R}^d$ is a polyhedral domain. Let $\mathcal{T}_h$ be a subdivision (trianglation) of $\Omega$ into quadrilateral elements such that $\bigcup_{K \in \mathcal{T}_h}\overline{K}=\overline{\Omega}$ and $K \cap K'= \emptyset$ for all $K,K' \in \mathcal{T}_h$ with $K \neq K'$. Furthermore, let $\psi_K$ be a multilinear mapping from the reference domain $\hat{K}=(0,1)^d$ to the element $K \in \mathcal{T}_h$. We now define the space $Q_c^r$ as $$\label{FE-space-part}
Q_c^{r}:=\{ v_h \in C(\overline{\Omega}): v_{h|K} \in Q_r(K), \, \forall K \in \mathcal{T}_h\},$$ with $Q_r(K):=\{v_{|\hat{K}}\circ\psi_K^{-1}:\, v(\hat{x})= \prod_{i=1}^{d} (\sum_{\beta=0}^{r} c_{\beta,i}\hat{x}_i^{\beta}),\, c_{\beta,i} \in \mathbb{R}\}$. Specifically, we use continuous tensor-product finite elements as described in [@Ciarlet:2002:FEM:581834] and [@Braess]. We also refer the reader to [@ArnBofFal2002] for the specific approximation properties of these finite element spaces. Let $\mathcal{T}_h^l$ be the triangulation of refinement level $l$. Then our finite element spaces are given by $U_h^l:= U \cap Q_c^r$ and $V_h^l:= V \cap Q_c^r$, whereas the enriched finite element spaces are defined by $U_h^{l,(2)}:= U \cap Q_c^{\tilde{r}}$ and $V_h^{l,(2)}:= V \cap Q_c^{\tilde{r}}$, where $Q_c^r$ and $Q_c^{\tilde{r}}$ are defined as in (\[FE-space-part\]) with $\mathcal{T}_h= \mathcal{T}_h^l$ and $\tilde{r} > r$. If $U$ and $V$ are spaces of vector-valued functions, then intersection has to be understood component-wise with possibly different $r$ in each component.
The algorithms presented in this section are formulated for FEM [@CaOd84; @Braess; @ArnBofFal2002; @Ciarlet:2002:FEM:581834]. However, we are not restricted to a particular discretization technique, but we must be able to realize the adaptivity in an appropriate way. For instance, in isogeometric analysis (IGA) that was originally introduced in [@HughesCottrellBazilevs:2005a] on tensor-product meshes, local mesh refinement is more challenging than in the FEM. Truncated hierarchical B-splines (THB-splines) are one possible choice to create localized basises which form a PU, see [@GiJuHeTHBsplines2012].
Higher-order B-splines of highest smoothness even on coaerser meshes can be used to construct enriched spaces $U_h^{l,(2)}$ and $V_h^{l,(2)}$ that lead to cheap problems on the enriched spaces, see [@KleissTomar:2015a; @LangerMatculevichRepin:2017a; @LangerMatculevichRepin:2017b] for the successful use of this technique in functional-type a posteriori error estimates.
Newton’s algorithm
------------------
Newton’s algorithm for solving the nonlinear variational problem (\[problem: discrete primal problem\]) belonging to refinement level $l$ is given by Algorithm \[newton\_algorithm\]. Below we identify $\mathcal{A}(u_h^{l,k})$ with the corresponding vector with respect to the chosen basis when we compute $\Vert \mathcal{A}(u_h^{l,k}) \Vert_{\ell_\infty}$.
Start with some initial guess $u^{l,0}_h \in
U_h^l$, set $k=0$, and set $TOL_{Newton}^l > 0$. Solve for $\delta u^{l,k}_h$, $$\mathcal{A}'(u^{l,k}_h)(\delta u^{l,k}_h,v_h)=-\mathcal{A}(u^{l,k}_h)(v_h) \quad \forall v_h \in
V_h^l.$$ Update : $u^{l,k+1}_h=u^{l,k}_h+\alpha \delta u^{l,k}_h$ for some good choice $\alpha \in (0,1]$. $k=k+1.$
In order to save computational cost we do not rebuild the matrices in every step. We rebuild the matrices if $\Vert \mathcal{A}(u_h^{l,k}) \Vert_{\ell_\infty}/\Vert \mathcal{A}(u_h^{l,k-1}) \Vert_{\ell_\infty} > 0.85 $ in Algorithm \[newton\_algorithm\].
Motivated by nested iterations, see, e.g., Section 6 in [@BeRa01], and the analysis for nonlinear nested iterations as given in Section 9.5 from [@MultigridHackusch03a], we use $TOL_{Newton}^1= 10^{-8}\Vert \mathcal{A}(u_h^{1,0}) \Vert_{\ell_\infty}$ and $TOL_{Newton}^l= 10^{-2}\Vert \mathcal{A}(u_h^{l,0}) \Vert_{\ell_\infty}$ for $l > 1$ as stopping criteria.
The parameter $\alpha$ can be obtained by means of a line search procedure. To obtain a good convergence, we used $\alpha=\gamma^L$ with $0<\gamma<1$, where the smallest $L$ that fulfills $$\Vert \mathcal{A}(u_h^{l,k}+\alpha \delta u^{l,k}_h ) \Vert_{\ell_\infty}<\text{c}(L,L_{max})\Vert \mathcal{A}(u_h^{l,k} ) \Vert_{\ell_\infty},$$ with $$c(L,L_{max}):=\begin{cases}
0.8\quad &L=0\\
0.888 \quad &L=1\\
(0.888+0.112\sqrt{\frac{L+1}{L_{max}}}) \quad &L>1
\end{cases},$$ $L=\{0, 1, 2 \cdots, L_{max}-1\}$ and $L_{max}=200$, is accepted. This choice of $\alpha$ was taken heuristically to obtain a better convergence of the Newton method in the numerical Example \[subsubsection Example 1c\]. In Algorithm \[newton\_algorithm\], we choose $\gamma=0.9$, and in Algorithm \[inexat\_newton\_algorithm\_for\_multiple\_goal\_functionals\], $\gamma=0.85$. We remark that a standard backtracking line search method also works, see, e.g., [@ToWi2017], but the previous exotic choice yields better iteration numbers.
Adaptive Newton algorithms for multiple-goal functionals
---------------------------------------------------------
In this section, we describe the key algorithm. The basic structure of the algorithm is similar to that presented in [@RanVi2013] and [@ErnVohral2013]. Our contribution is the extension to multiple-goal functionals.
Start with some initial guess $u^{l,0}_h \in U_h^l$ and $k=0$. For $z^{l,0}_h$, solve $$\mathcal{A}'(u^{l,0}_h)(v_h,z^{l,0}_h)=(J_{\mathfrak{E}}^{(0)})'(u^{l,0}_h)(v_h) \quad \forall v_h \in V_h^l,$$ with $(J_{\mathfrak{E}}^{(0)})'$ constructed with $u^{l,(2)}_h$ and $u^{l,0}_h$ as defined in (\[ErrorrepresentationFunctionalapprox’\]). For $\delta u^{l,k}_h$, solve $$\mathcal{A}'(u^{l,k}_h)(\delta u^{l,k}_h,v_h)=-\mathcal{A}(u^{l,k}_h)(v_h) \quad \forall v_h \in V_h^l.$$ Update : $u^{l,k+1}_h=u^{l,k}_h+\alpha \delta u^{l,k}_h$ for some good choice $\alpha \in (0,1]$. $k=k+1.$ For $z^{l,k}_h$, solve $$\mathcal{A}'(u^{l,k}_h)(v_h,z^{l,k}_h)=(J_{\mathfrak{E}}^{(k)})'(u^{l,k}_h)(v_h) \quad \forall v_h \in U_h^l,$$ $\quad$ with $(J_{\mathfrak{E}}^{(k)})'$ constructed with $u^{l,(2)}_h$ and $u^{l,k}_h$ as in (\[ErrorrepresentationFunctionalapprox’\]).
The final algorithm
-------------------
Now let us compose the final adaptive algorithm that starts from an initial mesh $\mathcal{T}_h^1$ and the corresponding finite element spaces $V_h^1$, $U_h^1$, $U_{h}^{1,(2)}$ and $V_{h}^{1,(2)}$, where $U_{h}^{1,(2)}$ and $V_{h}^{1,(2)}$ are the enriched finite element spaces as described in Section \[subsec\_SpatialDiscretization\]. The refinement procedure produces a sequence of finer and finer meshes $\mathcal{T}_h^l$ with the correponding FE spaces $V_h^l$, $U_h^l$, $U_{h}^{l,(2)}$ and $V_{h}^{l,(2)}$ for $l=2,3,\ldots$ .
Start with some initial guess $u_{h}^{0,(2)}$,$u_h^{0}$, set $l=1$ and set $TOL_{dis} > 0$. Solve (\[problem: discrete primal problem\]) for $u_h^{l,(2)}$ using Algorithm \[newton\_algorithm\] with the initial guess $u_h^{l-1,(2)}$ on the discrete space $U_{h}^{l,(2)}$. \[solve Uh2\] Solve (\[problem: discrete primal problem\]) and (\[Problem: discrete adjoint on small space\]) using Algorithm \[inexat\_newton\_algorithm\_for\_multiple\_goal\_functionals\] with the initial guess $u_h^{l-1}$ on the discrete spaces $U_{h}^l$ and $V_{h}^l$ . \[final algorithm: solveprimal\] Construct the combined functional $J_{\mathfrak{E}}$ as in (\[ErrorrepresentationFunctionalapprox\]). Solve the adjoint problem (\[Problem: adjoint\]) for $J_{\mathfrak{E}}$ on $V_h^{l,(2)}$. \[final algorithm: solveadjoint\] Construct the error estimator $\eta_K$ by distributing $\eta_i$ defined in (\[eta\_i\_PU\]) to the elements. Mark elements with some refinement strategy. \[final algorithm: refinement\] Refine marked elements: $\mathcal{T}_h^l \mapsto\mathcal{T }_h^{l+1}$ and $l=l+1$. If $|\eta_h| < TOL_{dis}$ stop, else go to \[solve Uh2\].
In step \[final algorithm: solveprimal\] of Algorithm \[final algorithm\], we replaced the estimated error $\eta_h^l$ by $\eta_h^{l-1}$ in Algorithm \[inexat\_newton\_algorithm\_for\_multiple\_goal\_functionals\], because we want to avoid the solution of the adjoint problem on the space $V_h^{l,(2)}$. Since the error in the previous estimate might be larger in general, we take $10^{-2} \eta_h^{l-1}$ instead of $10^{-1} \eta_h^{l}$, which was suggested in [@RanVi2013].
Thus, $\eta_h^{l-1}$ is not defined on the first level. Therefore, we set it to $\eta_h^{0}:=10^{-8}$. This means that we perform more iterations on the coarsest level. However, solving on this level is very cheap.
We notice that step \[solve Uh2\] in Algorithm \[final algorithm\] is costly, because we have to solve a problem corresponding to an enriched finite element space.
In step \[final algorithm: refinement\] of Algorithm \[final algorithm\], we mark all elements $K'$ where $\eta_{K'} \leq \frac{1}{|\mathcal{T}_h^l|}\sum_{K \in \mathcal{T}_h^l}^{}\eta_K$, where $|\mathcal{T}_h^l|$ denotes the number of elements.
Inspecting Algorithm \[final algorithm\], we need solve at each refinement level four problems: two are solved in step \[final algorithm: solveprimal\], and one in step \[solve Uh2\] and \[final algorithm: solveadjoint\], respectively. On the one hand, this is costly in comparison to other error estimators, e.g., residual-based, where only the primal problem needs to be solved. On the other hand, the adjoint solutions yield precise sensitivity measures for accurate measurements of the goal functionals. In addition, we control both the discretization and nonlinear iteration error for multiple goal functionals. Finally, the proposed approach is nonetheless much cheaper for many goal functionals. A naive approach (for a discussion in the linear case of multiple goal functionals or for using the primal part of the error estimator only, we refer the reader again to [@HaHou03; @Ha08]) would mean to solve $2N+2$ problems (i.e., $N+1$ for the primal part).
Numerical examples {#sec_num_tests}
==================
In this section, we perform numerical tests for two nonlinear problems, where the first problem contains two model parameters. We consider different choices of these parameters that lead to different levels of difficulty with respect to their numerical treatment.
- **Example 1** ($p$-Laplacian):
a) Smooth solution with homogeneous Dirichlet boundary conditions and right hand side on the unit square for $p=2$ and $p=4$ with $\varepsilon=1$ as regularization parameter, and an integral evaluation over the whole domain as functional of interest.
b) Smooth solution with inhomogeneous Dirichlet boundary conditions on the unit square with a disturbed grid and $p=5$ and $p=1.5$ with $\varepsilon=0.5$ and a point evaluation as functional of interest.
c) Solution with corner singularities and homogeneous Dirichlet boundary conditions on a cheese domain with $p=4$ and $p=1.33$ with a very small regularization parameter $\varepsilon=10^{-10}$, and two nonlinear and two linear functionals of interest.
- **Example 2** (a quasilinear PDE system): Solution with low regularity on a slit domain with mixed boundary conditions, and one linear and five nonlinear functionals of interest.
The implementation is based on the finite element library deal.II [@dealII84] and the extension of our previous work [@EnWi17].
Preliminaries
-------------
The following examples are discretized using globally continuous isoparametric quadrilateral elements as introduced in Section \[subsec\_SpatialDiscretization\]. If not mentioned otherwise, we use $U_h^{(2)}=Q^{r+1}_c \cap U$ and $V_h^{(2)}=Q^{r+1}_c \cap V$ for the enriched finite element spaces, if $U_h=Q^r_c \cap U$ and $V_h=Q^r_c \cap V$ is used for the original finite element spaces. In all numerical experiments we used $r=1$ except in Section \[subsubsection: Example1a\] Case 1, where the used discretization is given explicitly. To solve the arising linear systems, we used the sparse direct solver UMFPACK [@UMFPACK]. The error-weighting function $\mathfrak{E}(x,\vec{J}(u_h)):=\sum_{i=1}^{N}\frac{x_i}{|J_i(u_h)|}$ is used to construct $J_\mathfrak{E}$ as in (\[ErrorrepresentationFunctionalapprox\]). In our computations, we used the finite element function which is $1$ at the nodes which do not belong to the Dirichlet boundary and fulfills the boundary conditions at the nodes which belongs to the Dirichlet boundary as initial guess for $u_{h}^{0,(2)}$ and $u_h^{0}$.
To investigate how well our error estimator performs in estimating the error, we introduce the effectivity indices for the functional $J$ as follows: $$\begin{aligned}
I_{eff} &:= \frac{\eta_h}{|J(u)-J(u_h)|},\label{Ieff}\\
I_{effp} &:= \frac{|\rho(\tilde{u})(z_h^{(2)}-z_h)|}{|J(u)-J(u_h)|},\label{Ieffp}\\
I_{effa} &:= \frac{|\rho^*(\tilde{u},\tilde{z})(u_h^{(2)}-u_h)|}{|J(u)-J(u_h)|}, \label{Ieffa}\end{aligned}$$ where $\rho$ is defined by (\[Error Estimator: primal\]), $\rho^*$ as in (\[Error Estimator: adjoint\]), and $\eta_h$ as in (\[Error Estimator: discretization inexact\]). We call (\[Ieff\]) the effectivity index, (\[Ieffp\]) the primal effectivity index, and (\[Ieffa\]) the adjoint effectivity index. In the first part, we analyze the behavior of our algorithm for the regularized $p$-Laplace equation (\[p\_laplace\_problem\]). In Section \[subsubsection: Example1a\], Case 1, we apply our algorithm to the linear problem given in [@RiWi15_dwr], i.e., for $p=2$. For Section \[subsubsection: Example1a\], Case 2, we chose $p=4, \varepsilon=1$, and apply our algorithm to a nonlinear problem, and compare the refinement evolution for the different error estimators $|\rho(\tilde{u})(z_h^{(2)}-z_h)|,$ $|\rho^*(\tilde{u},\tilde{z})(u_h^{(2)}-u_h)|$ and $\eta_h$.
In Section \[subsubsection: Example 1b\], we solve the $p$-Laplace equation for $p=5$ and $p=1.5$ on a disturbed grid, aiming for a point evaluation. We compare the results of our algorithm with the results of global refinement and also to the different error estimators. The examples in Section \[subsubsection Example 1c\] consider several reentrant corners, several nonlinear functionals, and a very small regularization parameter $\varepsilon =10^{-10 }$. In Section \[QuasilinearvectorPDE\], we investigate the behavior of our algorithm for a quasilinear PDE system.
Example 1: $p$-Laplace {#subsection: Example1}
----------------------
Let $\varepsilon >0$ and $p \in \mathbb{R}$ with $p > 1$, and let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^2$. We again consider the Dirichlet problem for $p$-Laplace equation, cf. Section \[Discretization\], but now with inhomogeneous Dirichlet boundary conditions: Find $u$ such that: $$\begin{aligned}
\label{p_laplace_problem}
\begin{aligned}
-\text{div}((\varepsilon^2 + |{\nabla u}|^2)^{\frac{p-2}{2}}\nabla u) &=f \quad \forall \text{in }\Omega, \\
u &= g \quad \text{on } \partial \Omega .
\end{aligned}\end{aligned}$$ The Fréchet derivative $\mathcal{A}'(u)$ at $u$ of the nonlinear operator $\mathcal{A}$ corresponding to the $p$-Laplace problem problem \[p\_laplace\_problem\], cf. also Section \[Discretization\], is given by the variational identity
$$\begin{aligned}
\mathcal{A}'(u)(q,v) =& \langle{(\varepsilon^2+\Vert{\nabla u}\Vert_{\ell_2}^2)^{\frac{p-2}{2}}\nabla q,\nabla v}\rangle\\
+& \langle{(p-2)(\varepsilon^2+\Vert{\nabla u}\Vert_{\ell_2}^2)^{\frac{p-4}{2}}(\nabla u, \nabla q)_{\ell_2} \nabla u,\nabla v}\rangle \quad \forall q,v \in W^{1,p}_0(\Omega).
\end{aligned}$$
### Regular cases {#subsubsection: Example1a}
Here we consider a problem with a smooth solution and a smooth adjoint solution.
**Case 1 ($p=2$, i.e. Poisson problem):** This is the same example as Example 1 in [@RiWi15_dwr]. In this example, the data are given by $\Omega = (0,1) \times (0,1)$, $f=1$ and $g=0$. We are interested in the following functional evaluation: $$\begin{aligned}
J_1(u)&:=\int_{\Omega}u(x)\,d x \approx 0.03514425375\pm 10^{-10} .
$$ This reference value was taken from [@RiWi15_dwr]. If we compare our results in Table \[table: p=2 RiWiQ3Q6\] with the results in [@RiWi15_dwr], then we observe that they are quite similar. The estimated error $\eta_h$ is almost the same, and the DOFs exactly coincide with the DOFs in [@RiWi15_dwr]. However, using just one polynomial degree higher for $U_h^{(2)}$, we obtain similar results with less computational cost as is shown in Table \[table: p=2 RiWiQ3Q4\].
$l$ $|J(u)-J(u_h)|$ $\eta_h$ $I_{eff}$ $I_{effp}$ $I_{effa}$
----- ------- ----------------- ---------- ----------- ------------ ------------
1 169 8.51E-07 8.47E-07 1.00 1.00 1.00
2 317 1.12E-07 1.37E-07 1.23 1.23 1.23
3 937 5.57E-09 7.55E-09 1.35 1.35 1.36
4 1 813 1.15E-09 1.41E-09 1.22 1.23 1.22
5 3 877 6.48E-11 8.05E-11 1.24 1.24 1.24
6 7 057 2.81E-11 2.07E-11 0.74 0.74 0.74
\[table: p=2 RiWiQ3Q6\]
$l$ $|J(u)-J(u_h)|$ $\eta_h$ $I_{eff}$ $I_{effp}$ $I_{effa}$
----- ------- ----------------- ---------- ----------- ------------ ------------
1 169 8.51E-07 7.72E-07 0.91 0.91 0.91
2 317 1.12E-07 1.32E-07 1.18 1.18 1.18
3 789 5.12E-08 5.33E-08 1.04 1.04 1.04
4 1 301 4.11E-09 4.06E-09 0.99 0.99 0.99
5 1 977 1.06E-09 1.58E-09 1.49 1.49 1.5
6 4 149 6.56E-11 7.91E-11 1.2 1.2 1.21
7 7 273 2.65E-11 2.11E-11 0.8 0.8 0.8
\[table: p=2 RiWiQ3Q4\]
**Case 2 ($p=4,$ $\varepsilon =1$)**: We use the same setting as above, but with $p=4$ and $\varepsilon=1$. The finite element spaces are given by $U_h=Q^1_c$ and $U_h^{(2)}=Q^2_c$ . We are interested in the following functional evaluation $$\begin{aligned}
J_1(u)&:=\int_{\Omega}u(x) \,d x \approx 0.033553988572 \pm 10^{-6}.
$$ This reference value was computed on a fine grid with $263\,169$ $ \text{DOFs}$ ($9$ global refinement steps). In this example, we compare the refinements for different error estimators.
$l$ $|J(u)-J(u_h)|$ $I_{eff}$ $I_{effp}$ $I_{effa}$
----- -------- ----------------- ----------- ------------ ------------
1 9 1.08E-02 0.98 0.92 1.05
2 25 2.82E-03 0.99 0.92 1.07
3 81 7.11E-04 1.00 0.92 1.08
4 289 1.78E-04 1.00 0.92 1.08
5 1 089 4.44E-05 1.00 0.92 1.09
6 4 193 1.15E-05 1.07 0.98 1.15
7 6 545 9.45E-06 1.08 0.99 1.17
8 16 769 2.61E-06 1.07 0.98 1.16
9 36 009 1.75E-06 1.13 1.04 1.22
\[tableP4Primal\]
$l$ $|J(u)-J(u_h)|$ $I_{eff}$ $I_{effp}$ $I_{effa}$
----- -------- ----------------- ----------- ------------ ------------
1 9 1.08E-02 0.98 0.92 1.05
2 25 2.82E-03 0.99 0.92 1.07
3 81 7.11E-04 1.00 0.92 1.08
4 289 1.78E-04 1.00 0.92 1.08
5 913 7.54E-05 1.15 1.09 1.21
6 1 545 4.08E-05 1.09 1 .00 1.18
7 4 225 1.10E-05 1.02 0.93 1.10
8 10 513 6.56E-06 1.10 1.04 1.16
9 20 649 2.48E-06 1.12 1.03 1.22
\[tableP4Adjoint\]
$l$ $|J(u)-J(u_h)|$ $I_{eff}$ $I_{effp}$ $I_{effa}$
----- -------- ----------------- ----------- ------------ ------------
1 9 1.08E-02 0.98 0.92 1.05
2 25 2.82E-03 0.99 0.92 1.07
3 81 7.11E-04 1.00 0.92 1.08
4 289 1.78E-04 1.00 0.92 1.08
5 1 089 4.44E-05 1.00 0.92 1.09
6 3 137 2.26E-05 1.14 1.07 1.20
7 5 833 1.02E-05 1.10 1.01 1.19
8 16 641 2.61E-06 1.07 0.98 1.15
9 38 993 1.59E-06 1.15 1.08 1.21
\[tableP4PrimalAdjoint\]
In this example, we obtain quite good effectivity indices for the refinements based on the the primal part of the error estimator, cf. Table \[tableP4Primal\], the adjoint part of the error estimator, cf. Table \[tableP4Adjoint\], and the full error estimator $\eta_h$, cf. Table \[tableP4PrimalAdjoint\]. Furthermore, the convergence rates are also very similar. One might conclude that the adjoint error estimator is not required to obtain good effectivity indices. However, in the following examples, we observe that this is not the case for less regular solutions and adjoint solutions.
### Semiregular cases {#subsubsection: Example 1b}
As in the regular cases, we consider a smooth solution, but a low regular adjoint solution. This example is motivated by an example in [@ToWi2017]. We choose the right-hand side and the boundary conditions such that exact solution is given by $ u(x,y)=\text{sin}(6x+6y).$ The computation was done on the unit square $\Omega = (0,1) \times (0,1)$ on a slightly perturbed mesh (generated with the deal.II [@BangerthHartmannKanschat2007; @dealII84] command `distort_random` with $0.2$ on a $4$ times globally refined grid unit square). The resulting mesh is shown in Figure \[Example 1c: initial mesh\]. The functional of interest is $J(u)=u(0.6,0.6)$. We consider the following two cases:
- **Case 1 ($p=5,\varepsilon = 0.5$)**,
- **Case 2 ($p=1.5,\varepsilon =0.5$)**.
In both cases, the method also worked for the perturbed meshes. For the case $p=5$ and $\varepsilon=0.5$, we observe from Figure \[Example1b: adjoint solution\] that the adjoint solution almost vanishes in the set outside the domain which is covered by the condition $\nabla u =0$, and contains the point $(0.6,0.6)$. This was not observed in Case2. However, the condition $\nabla u=0$ seems to be important in both cases. The adaptively refined meshes shown in Figure \[Example1b: primal solution\] and Figure \[Example1b: 2:refined+marked\] have more refinement levels in these regions. In Figure \[Example1bCase1gnuplot\] and Figure\[Example1bCase2gnuplot\], we observe that we get the same convergence rate as in the case of uniform refinement. Since the solution is smooth, a global refinement already attains the optimal convergence rate. However, we get a reduction of the number of DOFs that are needed to obtain the same error. Furthermore, we monitor that the effectivity index is better on finer meshes. The reason might be the neglected remainder term from Theorem \[Theorem: Error Representation\]. From Table \[Example1bCase1Ieff\] and Table \[Example1bCase2Ieff\], we conclude that this does not necessarily hold for the primal and the adjoint error estimator separately.
set output “Figures/Example1bCase1gnuplottex” set datafile separator “|” set logscale set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set xlabel ’’ set ylabel ’$|u(0.6,0.6)-u_h(0.6,0.6)|$’ set format ’ plot \[180:2000000\] ’< sqlite3 Data/computationdata/disturbedunitsquare/p=5/datap=5.db “SELECT DISTINCT DOFS\_primal\_pf, Exact\_Error from data”’ u 1:2 w lp lw 2 title ’Error (adaptive)’, ’< sqlite3 Data/computationdata/disturbedunitsquare/p=5/datap=5.db “SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error from data”’ u 1:2 w lp lw 2 title ’$\eta_h$’, ’< sqlite3 Data/computationdata/disturbedunitsquare/p=5/datap=5l.db “SELECT DISTINCT DOFS\_primal\_pf, Exact\_Error from data\_global”’ u 1:2 w lp lw 2 title ’Error (uniform)’, 1/x lw 4 dashtype 2 title ’$\mathcal{O}(\text{DOFs}^{-1})$’ \#’< sqlite3 Data/computationdata/disturbedunitsquare/p=5/datap=5.db “SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error\_primal from data”’ u 1:2 w lp lw 2 title ’Estimated Error(primal)’, \#’< sqlite3 Data/computationdata/disturbedunitsquare/p=5/datap=5.db “SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error\_adjoint from data”’ u 1:2 w lp lw 2 title ’Estimated(adjoint)’,
set output “Figures/Example1bCase2gnuplottex” set datafile separator “|” set logscale set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set format ’ set xlabel ’’ set ylabel ’$|u(0.6,0.6)-u_h(0.6,0.6)|$’ plot \[180:2000000\] ’< sqlite3 Data/computationdata/disturbedunitsquare/p=1.5/datap15.db “SELECT DISTINCT DOFS\_primal\_pf, Exact\_Error from data”’ u 1:2 w lp lw 2 title ’Error (adaptive)’, ’< sqlite3 Data/computationdata/disturbedunitsquare/p=1.5/datap15.db “SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error from data”’ u 1:2 w lp lw 2 title ’$\eta_h$’, ’< sqlite3 Data/computationdata/disturbedunitsquare/p=1.5/datap15.db “SELECT DISTINCT DOFS\_primal\_pf, Exact\_Error from data\_global”’ u 1:2 w lp lw 2 title ’Error (uniform)’, 1/x lw 4 dashtype 2 title ’$\mathcal{O}(\text{DOFs}^{-1})$’ \#’< sqlite3 Data/computationdata/disturbedunitsquare/p=1.5/datap15.db “SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error\_primal from data”’ u 1:2 w lp lw 2 title ’Estimated Error(primal)’, \#’< sqlite3 Data/computationdata/disturbedunitsquare/p=5/datap=5.db “SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error\_adjoint from data”’ u 1:2 w lp lw 2 title ’Estimated(adjoint)’,
$l$ $|J(u)-J(u_h)|$ $I_{eff}$ $I_{effp}$ $I_{effa}$
----- ----------- ----------------- ----------- ------------ ------------
1 289 4.24E-03 0.48 0.60 1.56
2 599 5.23E-03 0.80 0.63 0.98
3 1 095 7.72E-04 0.16 0.02 0.34
4 2 418 8.52E-05 1.81 2.58 1.03
5 4 918 1.28E-05 4.92 3.35 6.49
6 10 112 2.64E-05 2.03 1.83 2.22
7 20 068 3.46E-06 5.59 10.33 0.86
8 40 302 1.02E-05 1.66 2.16 1.16
9 79 468 4.45E-06 1.51 1.60 1.43
10 157 272 2.68E-06 1.62 1.68 1.55
11 305 901 1.36E-06 1.32 1.62 1.01
12 602 720 8.52E-07 1.29 1.46 1.12
13 1 157 353 3.40E-07 1.28 1.55 1.01
\[Example1bCase1Ieff\]
$l$ $|J(u)-J(u_h)|$ $I_{eff}$ $I_{effp}$ $I_{effa}$
----- --------- ----------------- ----------- ------------ ------------
1 289 2.07E-02 0.61 0.47 0.75
2 503 5.55E-03 0.72 0.89 0.55
3 994 2.88E-03 0.89 1.30 0.49
4 2 090 8.55E-04 1.23 1.50 0.96
5 4 233 4.34E-04 1.45 1.90 1.00
6 8 667 1.42E-04 1.34 1.88 0.80
7 17 276 8.14E-05 1.40 2.71 0.09
8 34 846 3.54E-05 1.28 1.75 0.80
9 68 765 1.64E-05 1.36 2.58 0.14
10 136 267 8.59E-06 1.29 2.07 0.51
11 263 508 4.30E-06 1.19 2.20 0.18
12 514 223 2.18E-06 1.22 1.99 0.44
13 988 042 1.01E-06 1.22 2.20 0.24
\[Example1bCase2Ieff\]
### Low regularity cases {#subsubsection Example 1c}
As in Section \[subsubsection: Example1a\], we consider homogeneous Dirichlet conditions and $f=1$ as right-hand side for the $p$-Laplace equation (\[p\_laplace\_problem\]). However, here both the solution and adjoint solution have low regularity. The initial mesh is given as in Figure \[initcheese\], which was constructed using the deal.II [@BangerthHartmannKanschat2007; @dealII84] command cheese. With this data, we have singularities on each of the reentrant corners. Furthermore, in this example, we chose the regularization parameter $\varepsilon$ to be $10^{-10}$, which makes the problem very ill-conditioned (in fact it is practically the original $p$-Laplace problem) where $\nabla u=0$, but it is very close to the unregularized $p$-Laplace problem as in [@Lind2006] and[@Glowinski1975]. We are interested in the following four goal functionals: $$\begin{aligned}
J_1(u):= &(1+u(2.9,2.1))(1+u(2.1,2.9)),\\
J_2(u):= &\left(\int_{\Omega} u(x,y)-u(2.5,2.5)\,d(x,y)\right)^2,\\
J_3(u):= &\int_{(2,3)\times(2,3)} u(x,y)\,d(x,y),\\
J_4(u):= &u(0.6,0.6).\end{aligned}$$ These functionals will be combined to $J_{\mathfrak{E}}$ as formulated in (\[ErrorrepresentationFunctionalapprox\]).\
**Case 1 ($p=4$, $\varepsilon=10^{-10}$):** First we consider a case where $p>2$. The following values, which were computed on a fine grid (8 global refinements, $Q_c^2$ elements, 22038525 ) on the cluster RADON1[^1], are used to compute the reference values:\
$$\begin{aligned}
\int_{\Omega} u(x,y)\, d(x,y)\approx & 4.1285036414 \pm 4\times10^{-5},\\
\int_{(2,3)\times(2,3)} u(x,y)\,d(x,y)\approx & 0.31999986649\pm 10^{-5},\\
u(2.9,2.1)\approx & 0.16071095234\pm 10^{-5},
\end{aligned}$$
$$\begin{aligned}
u(2.9,2.1)\approx & 0.16071095234 \pm 10^{-5},\\
u(0.6,0.6)\approx & 0.35554352679\pm 2\times 10^{-6},\\
u(2.5,2.5)\approx & 0. 49244705234\pm 4\times10^{-6}.
\end{aligned}$$
Considering the accuracy of the functional evaluations above, we observe that the relative errors in the functionals $J_1$, $J_2$, $J_3$ and $J_4$ are less than $5\times 10^{-5}$. Our algorithm yields the results shown in Table \[Example 1b: errortable\]. In Figure \[Example1c: Case 1 gnuplot2\], we can see that the absolute error in the error functional $J_\mathfrak{E}$ bounds the relative errors of the functionals $J_1$, $J_2$, $J_3$ and $J_4$. Furthermore, we observe that $J_2$ is the dominating functional and $J_1$ is the one with the smallest error on most refinement levels. Therefore, we compare the convergence of this functionals in Figure \[Example1c: Case 1 gnuplot\]. For uniform refinement, we obtain an error behavior of approximately $\mathcal{O}(\text{DOFs}^{-\frac{3}{4}})$ for $J_2$ and $\mathcal{O}(\text{DOFs}^{-\frac{3}{5}})$ for $J_1$, whereas we obtain excellent convergence rates of $\mathcal{O}(\text{DOFs}^{-1})$ for both functionals using our refinement algorithm. We are not aware of a full convergence analysis on adaptive meshes for pointwise estimates for the $p$-Laplacian, but mention two related studies [@CaKlo03] for $p>2$ showing a posteriori estimates for the $W^{1,p}$ norm and [@Breit2017] with pointwise a priori estimates for the $p$-Laplacian. The bad convergence of $J_2$ might result from the fact that the point $(2.5,2.5)$ is the intersection of two lines, where the problem is ill-conditioned, and also leads to a kink in the solution at this point (see Figure \[Example 1c: p=4 solution\]). This kink is not visible in the case $p=1.33$ (see Figure \[Example 1c: p=1.33 solution\]). Comparing the number of Newton steps in Table \[Example1c\_inexact\_newton\] and Table \[Example1c\_exact\_newton\], we observe that the number of Newton steps is less than for Algorithm \[inexat\_newton\_algorithm\_for\_multiple\_goal\_functionals\]. However, the additional computational cost has to be considered, but we face a problem with nonlinear functionals, several reentrant corners and a very small regularization parameter $\varepsilon = 10^{-10}$. Furthermore, these tables also suggest that we should compute both the primal and the adjoint error estimator to obtain a better approximation of the error.
$l$ $I_{eff}$ $\left|\frac{J_1(u)-J_1(u_h)}{J_1(u)}\right|$ $\left|\frac{J_2(u)-J_2(u_h)}{J_2(u)}\right|$ $\left|\frac{J_3(u)-J_3(u_h)}{J_3(u)}\right|$ $\left|\frac{J_4(u)-J_4(u_h)}{J_4(u)}\right|$
----- --------- ----------- ----------------------------------------------- ----------------------------------------------- ----------------------------------------------- -----------------------------------------------
1 117 0.63 [ 5.05E-02]{} [ 3.02E-01 ]{} 1.10E-01 1.17E-01
2 161 0.53 [ 1.53E-02]{} 5.09E-02 4.94E-02 [ 1.17E-01]{}
3 290 0.84 [ 8.25E-03]{} 4.41E-02 2.14E-02 [ 1.09E-01]{}
4 447 0.81 [ 4.86E-03]{} [ 5.07E-02]{} 1.53E-02 1.51E-02
5 791 0.96 [ 2.09E-03]{} [ 3.26E-02]{} 1.12E-02 8.82E-03
6 1 331 1.14 [ 1.37E-03]{} [ 1.69E-02]{} 8.44E-03 2.40E-03
7 2 541 1.92 1.65E-03 3.37E-03 [ 4.38E-03]{} [ 1.21E-03 ]{}
8 4 582 1.38 [ 6.56E-04]{} [ 3.78E-03]{} 2.43E-03 8.57E-04
9 7 378 1.64 3.14E-04 [ 2.52E-03]{} 1.05E-03 [ 2.06E-04]{}
10 11 772 1.51 [ 2.72E-04]{} [ 1.73E-03]{} 8.83E-04 3.51E-04
11 20 443 1.87 9.65E-05 [ 5.65E-05]{} [ 5.24E-04]{} 5.84E-05
12 37 747 1.87 [ 6.09E-05]{} [ 3.05E-04]{} 2.17E-04 1.20E-04
13 64 316 1.63 [ 2.80E-05]{} 1.30E-04 [ 1.41E-04 ]{} 4.25E-05
14 104 832 1.44 [ 1.04E-05]{} [ 1.39E-04]{} 7.18E-05 2.02E-05
: Section \[subsubsection Example 1c\], Case 1. Relative errors for the goal functionals on several refinement levels ($l$) and effectivity index $I_{eff}$ that is computed for $J_{\mathfrak{E}}$ (\[ErrorrepresentationFunctionalapprox\]). []{data-label="Example 1b: errortable"}
$l$ Error in $J_{\mathfrak{E}}$ $I_{eff}$ $I_{effp}$ $I_{effa}$ Newton steps
----- --------- ----------------------------- ----------- ------------ ------------ --------------
1 117 7.43E-01 0.63 0.6 0.65 8
2 161 2.54E-01 0.53 0.27 0.79 2
3 290 1.94E-01 0.84 0.24 1.43 2
4 447 8.40E-02 0.81 0.28 1.34 5
5 791 5.39E-02 0.96 0.48 1.45 4
6 1 331 2.89E-02 1.14 0.24 2.05 1
7 2 541 1.06E-02 1.92 0.02 3.86 3
8 4 582 7.71E-03 1.38 0.41 2.36 4
9 7 378 4.09E-03 1.64 0.74 2.55 2
10 11 772 3.23E-03 1.51 0.7 2.32 4
11 20 443 6.23E-04 1.87 1.06 2.67 2
12 37 747 7.03E-04 1.87 0.66 3.07 6
13 64 316 3.41E-04 1.63 0.39 2.87 4
14 104 832 2.42E-04 1.44 0.5 2.38 3
\[Example1c\_inexact\_newton\]
$l$ Error in $J_{\mathfrak{E}}$ $I_{eff}$ $I_{effp}$ $I_{effa}$ Newton steps
----- --------- ----------------------------- ----------- ------------ ------------ --------------
1 117 7.43E-01 0.63 0.60 0.65 7
2 161 2.58E-01 0.52 0.26 0.79 4
3 290 1.94E-01 0.84 0.24 1.44 4
4 447 8.41E-02 0.81 0.28 1.34 6
5 791 5.40E-02 0.96 0.48 1.45 4
6 1 331 2.70E-02 1.38 0.38 2.39 6
7 2 198 2.02E-02 1.13 0.56 1.70 5
8 4 012 9.07E-03 1.43 0.70 2.16 6
9 6 879 4.02E-03 1.75 0.32 3.18 5
10 11 576 3.27E-03 1.40 0.62 2.19 6
11 20 187 8.20E-04 2.11 0.85 3.37 6
12 38 302 6.77E-04 1.78 0.54 3.02 7
13 64 740 3.12E-04 1.67 0.32 3.02 7
14 105 350 2.46E-04 1.35 0.47 2.22 5
\[Example1c\_exact\_newton\]
set output “Figures/Example1cCase1gnuplottex” set datafile separator “|” set logscale set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set xlabel ’’ set ylabel ’error’ set format ’ plot ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, relativeError0 from data WHERE DOFS\_primal\_pf <= 105000”’ u 1:2 w lp lw 2 title ’$J_1$(adaptive)’, ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, relativeError1 from data WHERE DOFS\_primal\_pf <= 105000”’ u 1:2 w lp lw 2 title ’$J_2$(adaptive)’, ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, relativeError2 from data WHERE DOFS\_primal\_pf <= 105000”’ u 1:2 w lp lw 2 title ’$J_3$(adaptive)’, ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, relativeError3 from data WHERE DOFS\_primal\_pf <= 105000”’ u 1:2 w lp lw 2 title ’$J_4$(adaptive)’, ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, Exact\_Error from data WHERE DOFS\_primal\_pf <= 105000”’ u 1:2 w lp lw 2 linecolor “red” title ’$J_\mathfrak{E}$(adaptive)’, 1.5\*x\*\*(-0.6) lw 4 dashtype 2 title ’$\mathcal{O}(\text{DOFs}^{-\frac{3}{5}})$’, 5/x lw 4 dashtype 2 title ’$\mathcal{O}(\text{DOFs}^{-1})$’ \# ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, Exact\_Error from data”’ u 1:2 w lp title ’Exact Error’, ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error from data”’ u 1:2 w lp title ’Estimated Error’, ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error\_primal from data”’ u 1:2 w lp title ’Estimated Error(primal)’, ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error\_adjoint from data”’ u 1:2 w lp title ’Estimated(adjoint)’,
\[Example1c: Case 1 gnuplot2\]
set output “Figures/Example1cCase2gnuplottex” set datafile separator “|” set logscale set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set xlabel ’’ set ylabel ’error’ set format ’ plot ’< sqlite3 Data/computation\_data/Cheese/p133epsE-10/p133computed.db “SELECT DISTINCT DOFS\_primal\_pf, relativeError0 from data WHERE DOFS\_primal\_pf <= 45000”’ u 1:2 w lp lw 2 title ’$J_1$(adaptive)’, ’< sqlite3 Data/computation\_data/Cheese/p133epsE-10/p133computed.db “SELECT DISTINCT DOFS\_primal\_pf, relativeError1 from data WHERE DOFS\_primal\_pf <= 45000”’ u 1:2 w lp lw 2 title ’$J_2$(adaptive)’, ’< sqlite3 Data/computation\_data/Cheese/p133epsE-10/p133computed.db “SELECT DISTINCT DOFS\_primal\_pf, relativeError2 from data WHERE DOFS\_primal\_pf <= 45000”’ u 1:2 w lp lw 2 title ’$J_3$(adaptive)’, ’< sqlite3 Data/computation\_data/Cheese/p133epsE-10/p133computed.db “SELECT DISTINCT DOFS\_primal\_pf, relativeError3 from data WHERE DOFS\_primal\_pf <= 45000”’ u 1:2 w lp lw 2 title ’$J_4$(adaptive)’, ’< sqlite3 Data/computation\_data/Cheese/p133epsE-10/p133computed.db “SELECT DISTINCT DOFS\_primal\_pf, Exact\_Error from data WHERE DOFS\_primal\_pf <= 45000”’ u 1:2 w lp lw 2 linecolor “red” title ’$J_\mathfrak{E}$(adaptive)’, 1.5\*x\*\*(-0.6) lw 4 dashtype 2 title ’$\mathcal{O}(\text{DOFs}^{-\frac{3}{5}})$’, 5/x lw 4 dashtype 2 title ’$\mathcal{O}(\text{DOFs}^{-1})$’
\[Example1c: Case 2 gnuplot2\]
set output “Figures/Example1cCase1gnuplot1tex” set datafile separator “|” set logscale set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set xlabel ’’ set ylabel ’relative error’ set format ’ plot ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, relativeError0 from data WHERE relativeError0 >= 1e-5”’ u 1:2 w lp lw 2 title ’$J_1$(adaptive)’, ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, relativeError0 from data\_global”’ u 1:2 w lp lw 2 title ’$J_1$(uniform)’, ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, relativeError1 from data WHERE DOFS\_primal\_pf <= 109000”’ u 1:2 w lp lw 2 title ’$J_2$(adaptive)’, ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, relativeError1 from data\_global”’ u 1:2 w lp lw 2 title ’$J_2$(uniform)’, 1.5\*x\*\*(-0.6) lw 4 dashtype 2 linecolor “red” title ’$\mathcal{O}(\text{DOFs}^{-\frac{3}{5}})$’, 5/x lw 4 dashtype 2 title ’$\mathcal{O}(\text{DOFs}^{-1})$’ \#5\*1e-5 lw 2 dashtype 1 linecolor “red” title ’$TOL$ \# ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, Exact\_Error from data”’ u 1:2 w lp title ’Exact Error’, ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error from data”’ u 1:2 w lp title ’Estimated Error’, ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error\_primal from data”’ u 1:2 w lp title ’Estimated Error(primal)’, ’< sqlite3 Data/computation\_data/Cheese/p=4eps=E-10/p4computed.db “SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error\_adjoint from data”’ u 1:2 w lp title ’Estimated(adjoint)’,
\[Example1c: Case 1 gnuplot\]
set output “Figures/Example1cCase2gnuplot1tex” set datafile separator “|” set logscale set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set xlabel ’’ set ylabel ’relative error’ set format ’ plot ’< sqlite3 Data/computation\_data/Cheese/p133epsE-10/p133computed.db “SELECT DISTINCT DOFS\_primal\_pf, relativeError0 from data WHERE DOFS\_primal\_pf <= 45000”’ u 1:2 w lp lw 2 title ’$J_1$(adaptive)’, ’< sqlite3 Data/computation\_data/Cheese/p133epsE-10/p133computed.db “SELECT DISTINCT DOFS\_primal\_pf, relativeError0 from data\_global”’ u 1:2 w lp lw 2 title ’$J_1$(uniform)’, ’< sqlite3 Data/computation\_data/Cheese/p133epsE-10/p133computed.db “SELECT DISTINCT DOFS\_primal\_pf, relativeError2 from data WHERE DOFS\_primal\_pf <= 45000”’ u 1:2 w lp lw 2 title ’$J_3$(adaptive)’, ’< sqlite3 Data/computation\_data/Cheese/p133epsE-10/p133computed.db “SELECT DISTINCT DOFS\_primal\_pf, relativeError2 from data\_global”’ u 1:2 w lp lw 2 title ’$J_3$(uniform)’, 4\*x\*\*(-0.75) lw 4 dashtype 2 linecolor “red” title ’$\mathcal{O}(\text{DOFs}^{-\frac{3}{4}})$’, 25/x lw 4 dashtype 2 title ’$\mathcal{O}(\text{DOFs}^{-1})$’ \#1e-5 lw 2 dashtype 1 linecolor “red” title ’$TOL$
\[Example1c: Case 2 gnuplot\]
**Case 2 ($p=1.33$, $\varepsilon=10^{-10}$):**\
We are interested in the same goal functional as in Case 1 but with $p=1.33$. The following values, which are computed on a fine grid (8 global refinements, $Q_c^2$ elements, $22\, 038\, 525$ ) on the cluster RADON1, are used to compute the reference values:
$$\begin{aligned}
\int_{\Omega} u(x,y)\,d(x,y)\approx & 0.48510099008 \pm 4\times10^{-5},\\
\int_{(2,3)\times(2,3)} u(x,y)\,d(x,y)\approx & 0.038058285978\pm 4\times 10^{-6},\\
u(2.9,2.1)\approx & 0.034930138311\pm 4\times 10^{-6},
\end{aligned}$$
$$\begin{aligned}
u(2.9,2.1)\approx & 0.034930138311\pm 4\times 10^{-6},\\
u(0.6,0.6)\approx & 0.024478640536\pm 2\times 10^{-6},\\
u(2.5,2.5)\approx & 0.039616834482\pm 4\times10^{-6}.\\
\end{aligned}$$
Considering again that the accuracy of the functional evaluations is valid, we observe that the relative error of $J_2$ is less than $8\times 10^{-4}$ and the relative error of $J_1,J_3,J_4$ is less than $10^{-4}$. As in Case 1, we compare the relative errors of the functionals in Figure \[Example1c: Case 2 gnuplot2\]. Here we see that the error in $J_\mathfrak{E}$ bounds the relative errors. However, we loose control of the single functionals as long as they do not dominate the error, as for $J_2$ in Figure \[Example1c: Case 2 gnuplot2\]. In Case 2, $J_3$ and $J_1$, are these functionals. In the error plot given in Figure \[Example1c: Case 2 gnuplot\], we observe that the error approximately behaves like $\mathcal{O}(\text{DOFs}^{-\frac{3}{4}})$ for a uniformly refined mesh, and $\mathcal{O}(\text{DOFs}^{-1})$ for adaptive refinement, as for $p=4$. It turns out that the regions of refinement (except for corner singularities and the point evaluations) have almost a complementary structure for $p=1.33$ and $p=4$ as we can conclude from Figure \[Example 1c: p=4 Mesh11\] and Figure \[Example 1c: p=1.33 Mesh11\].
Example 2: A quasilinear PDE system {#QuasilinearvectorPDE}
-----------------------------------
In this second numerical test, we further substantiate our approach for a nonlinear, coupled, PDE system. We consider the following nonlinear boundary value problem: Find $u=(u_1,u_2,u_3) $ such that $$\begin{aligned}
\begin{aligned}
-\Delta u_1 +u_2 +u_3& =1,\quad \text{in } \Omega, \\
-\Delta u_2 + g_1(1-u_2)-g_1(u_3)& =0,\quad \text{in } \Omega, \\
-\text{div}(g_2(u_1+u_2)\nabla u_3) + g_1(u_3)-g_1(u_1) &= 0,\quad \text{in } \Omega,
\end{aligned} $$ is fulfilled in a weak sense, where $$\begin{aligned}
u_1(x,y)=1-u_2(x,y)=u_3(x,y)& = \text{sign($y$)}\sqrt{\sqrt{x^2+y^2}-x} \qquad \text{on } \Gamma_D,\\
\nabla u_1.\vec{n} =\nabla u_2.\vec{n} = g_2(u_1+u_2) \nabla u_3 .\vec{n} & =0 \qquad \text{on } \Gamma_N.\end{aligned}$$ Here sign denotes the signum function as defined in (\[sign\]). The functions $g_1$ and $g_2$ are given by $ g_1(t):=e^t-\text{sin}(t-1) $ and $g_2(t):=e^{t^2-t}$, respectively. Obviously a solution is given by $u_1(x,y)=1-u_2(x,y)=u_3(x,y) = \text{sign(y)}\sqrt{\sqrt{x^2+y^2}-x}$ in $\Omega$. The computational domain is a slit domain as in [@EnWi17; @Wi16_dwr_pff; @AnMi15] and visualised in Figure \[Example 2: slit domain\]. The boundary conditions above introduces a discontinuity on the slit-boundary $(-1,0) \times \{0\}$ and consequently a discontinuity in the solution. The construction of this example was motivated by [@AnMi15; @BoDa01]. Let $J_A,J_B,J_C,J_D,J_E,J_F$ be defined as follows:\
$$\begin{aligned}
J_A(u):=& u_3(-0.5,0.01),\\
J_D(u):=&\int_{\Omega} \Phi_D(x,y) \cdot u(x,y) \,d(x,y),\end{aligned}$$
**** $$\begin{aligned}
J_B(u)&:=u_1(-0.01,0.01),\\
J_E(u)&:=u_1(-0.9,-0.9),
\end{aligned}$$
$$\begin{aligned}
J_C(u):=&\int_{\Omega} \Phi_C(x,y) \cdot u(x,y)\, d(x,y),\\
J_F(u):= &u_2(-0.9,-0.1),\\
\end{aligned}$$
where $\Phi_C(x,y) :=(0,0, \chi_C(x,y) )$ and $$\Phi_D(x,y) :=(-4 \chi_D(x,y),\frac{2\chi_D(x,y)}{1-\text{sign(y)}\sqrt{\sqrt{x^2+y^2}-x}},4\chi_D(x,y)),$$ with $$\chi_C(x,y):=\begin{cases}
y-x \quad &x<y \\
0 \quad &x \geq y
\end{cases}
\quad \mbox{and} \quad
\chi_D(x,y):=\begin{cases}
1 \quad &x,y>0 \\
0 \quad &\text{else}
\end{cases}
.$$
We are now interested in the six goal functionals
$$\begin{aligned}
J_{1}(u):=& J_B(u)J_D(u),\\
J_{4}(u):=& J_B(u)J_E(u),\\
\end{aligned}$$
$$\begin{aligned}
J_{2}(u):=& J_A(u)J_C(u),\\
J_{6}(u):=& J_B^3(u)J_E(u),\\
\end{aligned}$$
$$\begin{aligned}
J_{3}(u):=& J_A(u)J_C(u)J_F(u),\\
J_{6}(u)=& J_C(u).\\
\end{aligned}$$
For the functional $J_B$, we can not expect optimal convergence rates for uniform refinement due to the singularity at the slit tip. Consequently, the same is true for the functionals $J_1$, $ J_4$ and $ J_5$ as monitored in Figures \[Example\_2ErrorDOFsJ0J2\],\[Example\_2ErrorDOFsJ1J3\] and \[Example\_2ErrorDOFsJ4J5\]. For uniform refinement, we got a relative error in $J_1$ of about $1.409531 \times
10^{-2}$ with $3$ $153$ $411$ as visualized in Figure \[Example\_2ErrorDOFsJ0J2\]. To achieve a relative error less than $1.409531 \times 10^{-2}$ our adaptive algorithm just needs $2$ $538 $ DOFs ( $1.042219 \times 10^{-2}$ ). If we use a similar number of DOFs ($3$ $021$ $045$), then a relative error of $2.829422\times 10^{-6}$ is achieved. Figures \[Example\_2ErrorDOFsJ0J2\], \[Example\_2ErrorDOFsJ1J3\] and \[Example\_2ErrorDOFsJ4J5\] might also lead to the conclusion that we obtain a convergence rate $\mathcal{O}(\text{DOFs}^{-1})$ for all given functionals, where the functionals for uniform refinement just converge with approximately $\mathcal{O}(\text{DOFs}^{-\frac{1}{2}})$. This means, to obtain a relative error in $J_1$ of about $2.829422\times 10^{-6}$ for uniform refinement, we would need approximately $5 \times 10^{13}$ . This would mean just storing the solution would require approximately 400 Terabyte. Therefore, obtaining this accuracy by means of uniform refinement would even be a hard task on the supercomputer Sunway TaihuLight[^2], which is number one the of TOP500[^3] list from November 2017.
We remark that $I_{eff}$, illustrated in Figure \[Ieffs\], has no importance on course meshes since the approximations properties are bad anyway. On finer meshes, we see excellent behavior.
(-1.1859438838358034,-1.076009411190554) grid (1.1805682302792468,1.0396356943610634);(-1.1859438838358034,0) – (1.1805682302792468,0);in [-1,-0.5,0.5,1]{}(0pt,2pt) – (0pt,-2pt) node\[below\] [$\x$]{};(0,-1.076009411190554) – (0,1.0396356943610634);in [-1,-0.5,0.5,1]{}(2pt,0pt) – (-2pt,0pt) node\[left\] [$\y$]{};(0pt,-10pt) node\[right\] [$0$]{};(-1.1859438838358034,-1.076009411190554) rectangle (1.1805682302792468,1.0396356943610634); (-1,-1) – (1,-1) – (1,1) – (-1,1) – cycle;(-1,-1)– (1,-1);(1,-1)– (1,1);(1,1)– (-1,1);(-1,1)– (-1,-1);(-1,0)– (0,0);
(0.10811176867057129,-0.9227001718708244) node [$\Gamma_D$]{};(0,0) circle (2pt);(-0.3643544307905606,0.10857131538010847) node [$\Gamma_N$]{};
set output “Figures/Error\_nonlineartex” set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set datafile separator “|” set logscale set format “ \#set title ”Error in $J_n$“ set xlabel ”DOFs“ set ylabel ”relative error“ set key bottom left plot \[50:10000000\] ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, relativeError1 from data“’ u 1:2 w lp lw 2 title ’$J_2$(adaptive)’, ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, relativeError1 from data\_global“’ u 1:2 w lp lw 2 title ’$J_2$(uniform)’, ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, relativeError3 from data“’ u 1:2 w lp lw 2 title ’$J_4$(adaptive)’, ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, relativeError3 from data\_global“’ u 1:2 w lp lw 2 title ’$J_4$(uniform)’, 10/sqrt(x) lw 4 dashtype 2 linecolor ”red“ title ’$\mathcal{O}(\text{DOFs}^{-\frac{1}{2}})$’, 100/x lw 4 dashtype 2 title ’$\mathcal{O}(\text{DOFs}^{-1})$’ \# ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, Exact\_Error from data“’ u 1:2 w lp lw 2 title ’$J_{\mathfrak{E}}$(adaptive)’, \# ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, Exact\_Error from data\_global“’ u 1:2 w lp lw 2 title ’$J_{\mathfrak{E}}$(adaptive)’, \#’< sqlite3 Data/computationdata/disturbedunitsquare/p=5/datap=5.db ”SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error\_primal from data“’ u 1:2 w lp lw 2 title ’Estimated Error(primal)’, \#’< sqlite3 Data/computationdata/disturbedunitsquare/p=5/datap=5.db ”SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error\_adjoint from data"’ u 1:2 w lp lw 2 title ’Estimated(adjoint)’,
set output “Figures/Error\_nonlinear2tex” set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set datafile separator “|” set logscale set format “ \#set title ”Error in $J_n$“ set xlabel ”DOFs“ set ylabel ”relative error“ set key bottom left plot \[50:10000000\] ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, relativeError0 from data“’ u 1:2 w lp lw 2 title ’$J_1$(adaptive)’, ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, relativeError0 from data\_global“’ u 1:2 w lp lw 2 title ’$J_1$(uniform)’, ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, relativeError2 from data“’ u 1:2 w lp lw 2 title ’$J_3$(adaptive)’, ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, relativeError2 from data\_global“’ u 1:2 w lp lw 2 title ’$J_3$(uniform)’, 10/sqrt(x) lw 4 dashtype 2 linecolor ”red“ title ’$\mathcal{O}(\text{DOFs}^{-\frac{1}{2}})$’, 10/(x\*\*1.0) lw 4 dashtype 2 title ’$\mathcal{O}(\text{DOFs}^{-1})$’ \# ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, Exact\_Error from data“’ u 1:2 w lp lw 2 title ’$J_{\mathfrak{E}}$(adaptive)’, \# ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, Exact\_Error from data\_global“’ u 1:2 w lp lw 2 title ’$J_{\mathfrak{E}}$(adaptive)’, \#’< sqlite3 Data/computationdata/disturbedunitsquare/p=5/datap=5.db ”SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error\_primal from data“’ u 1:2 w lp lw 2 title ’Estimated Error(primal)’, \#’< sqlite3 Data/computationdata/disturbedunitsquare/p=5/datap=5.db ”SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error\_adjoint from data"’ u 1:2 w lp lw 2 title ’Estimated(adjoint)’,
set output “Figures/Error\_nonlinear3tex” set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set datafile separator “|” set logscale set format “ \#set title ”Error in $J_n$“ set xlabel ”DOFs“ set ylabel ”relative error“ set key bottom left plot \[50:10000000\] ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, relativeError4 from data“’ u 1:2 w lp lw 2 title ’$J_5$(adaptive)’, ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, relativeError4 from data\_global“’ u 1:2 w lp lw 2 title ’$J_5$(uniform)’, ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, relativeError5 from data“’ u 1:2 w lp lw 2 title ’$J_6$(adaptive)’, ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, relativeError5 from data\_global“’ u 1:2 w lp lw 2 title ’$J_6$(uniform)’, 10/sqrt(x) lw 4 dashtype 2 linecolor ”red“ title ’$\mathcal{O}(\text{DOFs}^{-\frac{1}{2}})$’, 100/x lw 4 dashtype 2 title ’$\mathcal{O}(\text{DOFs}^{-1})$’ \# ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, Exact\_Error from data“’ u 1:2 w lp lw 2 title ’$J_{\mathfrak{E}}$(adaptive)’, \# ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, Exact\_Error from data\_global“’ u 1:2 w lp lw 2 title ’$J_{\mathfrak{E}}$(adaptive)’, \#’< sqlite3 Data/computationdata/disturbedunitsquare/p=5/datap=5.db ”SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error\_primal from data“’ u 1:2 w lp lw 2 title ’Estimated Error(primal)’, \#’< sqlite3 Data/computationdata/disturbedunitsquare/p=5/datap=5.db ”SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error\_adjoint from data"’ u 1:2 w lp lw 2 title ’Estimated(adjoint)’,
set output “Figures/Error\_nonlinear4tex” set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set datafile separator “|” set format “ \#set title ”Error in $J_n$“ set xlabel ”$l$“ set key bottom plot ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT Refinementstep+1, Ieff from data“’ u 1:2 w lp lw 2 title ’$I_{eff}$ for $J_\mathfrak{E}$’, ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT Refinementstep+1, Ieff\_primal from data“’ u 1:2 w lp lw 2 title ’$I_{effp}$ for $J_\mathfrak{E}$’, ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT Refinementstep+1, Ieff\_adjoint from data“’ u 1:2 w lp lw 2 title ’$I_{effa}$ for $J_\mathfrak{E}$’, 1 lw 4 dashtype 2 \# ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, Exact\_Error from data“’ u 1:2 w lp lw 2 title ’$J_{\mathfrak{E}}$(adaptive)’, \# ’< sqlite3 Data/computationdata/nonlinear\_problem/Laplacetoy2.db ”SELECT DISTINCT DOFS\_primal, Exact\_Error from data\_global“’ u 1:2 w lp lw 2 title ’$J_{\mathfrak{E}}$(adaptive)’, \#’< sqlite3 Data/computationdata/disturbedunitsquare/p=5/datap=5.db ”SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error\_primal from data“’ u 1:2 w lp lw 2 title ’Estimated Error(primal)’, \#’< sqlite3 Data/computationdata/disturbedunitsquare/p=5/datap=5.db ”SELECT DISTINCT DOFS\_primal\_pf, Estimated\_Error\_adjoint from data"’ u 1:2 w lp lw 2 title ’Estimated(adjoint)’,
Conclusions {#sec_concl}
===========
In this work, we have further developed adaptive schemes for multigoal-oriented a posteriori error estimation and mesh adaptivity. First, we extended the existing methods to nonlinear problems. Second, we combined the estimation of the discretization error with an estimation of the nonlinear iteration error in order to obtain adaptive stopping rules for Newton’s method. In the key Sections \[Multigoalfunctionals\] and \[sec\_alg\], we formulated an abstract framework and its algorithmic realization. In Section \[sec\_num\_tests\], these developments were substantiated with several numerical tests. Here, we studied the regularized $p$-Laplace problem and a nonlinear, coupled PDE system. Our findings demonstrate the performance of the algorithms and specifically that the adjoint part of the error estimator, which is often neglected in the literature because of its higher computational cost, must be taken into account in order to achieve good effectivity indices. In view of the geometric singularities, nonlinearities in both the PDE and the goal functionals, our results show excellent performance of our algorithms.
Acknowledgments
===============
This work has been supported by the Austrian Science Fund (FWF) under the grant P 29181 ‘Goal-Oriented Error Control for Phase-Field Fracture Coupled to Multiphysics Problems’. The first author thanks the Doctoral Program on Computational Mathematics at JKU Linz the Upper Austrian Goverment for the support when starting the preparation of this work. The third author was supported by the Doctoral Program on Computational Mathematics during his visit at the Johannes Kepler University Linz in March 2018.
[^1]: <https://www.ricam.oeaw.ac.at/hpc/overview/>
[^2]: <http://www.nsccwx.cn/wxcyw/>
[^3]: <https://www.top500.org/lists/2017/11/>
|
---
author:
- 'A. Zanichelli'
- 'R. Scaramella'
- 'G. Vettolani'
- 'M. Vigotti'
- 'S. Bardelli'
- 'G. Zamorani'
date: 'Received ... / Accepted ...'
subtitle: 'II. The Cluster Sample [^1]'
title: 'Radio–Optically Selected Clusters of Galaxies'
---
Introduction {#sec:intr}
============
Groups and clusters of galaxies are the largest gravitationally bound, observable structures, and much can be understood about the global cosmological properties of the universe by studying their properties – such as their dynamical status and evolution, their morphological content and interactions with the environment. To this aim, it is of fundamental importance to gather cluster samples representative of different dynamical structures – from groups to rich clusters – in a wide range of redshift and covering large areas of the sky.
Existing wide-area cluster samples based on visual inspection of optical plates (Abell et al. [@Abell]) or obtained through objective algorithms (EDCC, Lumsden et al. [@Lumsden]; APM, Dalton et al. [@Dalton94]) are limited to redshift less than $0.2$, and suffer from the possibility of misclassification due to projection effects along the line of sight. Even more difficult is the detection of groups of galaxies, due to their low density contrast with respect to field galaxy distribution. In the optical band, cluster samples at higher $z$ have been built over selected areas of few square degrees (Postman et al. [@Postman]; Scodeggio et al. [@Scodeggio]). Alternatively, the X-ray emission of the hot intracluster medium has been widely used to build distant cluster samples, but this technique suffers from the limited sensitivity of wide-area X-ray surveys and from the possibility of evolutionary effects (Gioia et al. [@Gioia]; Henry et al. [@Henry]; RDCS, Rosati et al. [@Rosati]).
A different approach – complementary to purely optical or X-ray cluster selection methods – is the use of radiogalaxies as suitable tracers of dense environments. Faranoff–Riley I and II radio sources have been shown to inhabit different environments at different epochs and proved to be efficient tracers of galaxy groups and clusters (Prestage & Peacock [@Prestage]; Hill & Lilly [@Hill]; Allington–Smith et al. [@Allington--Smith]; Zirbel [@Zirbel97]). FRI sources are found on average in rich groups or clusters at any redshift, and are associated with elliptical galaxies, with the most powerful FRI often hosted by a cD or double nucleus galaxy. FRII radio sources are typically associated with disturbed ellipticals and they avoid rich clusters at low $z$ (Zirbel [@Zirbel96]).
Since there is no significant correlation between the radio properties of galaxies within a cluster with the cluster X-ray luminosity (Feigelson et al. [@Feigelson]; Burns et al. [@Burns]), or richness (Zhao et al. [@Zhao]; Ledlow & Owen [@Ledlow]), as well as between the properties of group members and the radio characteristics of the radiogalaxies (Zirbel [@Zirbel97]), radio selection should not impact on the X-ray or optical properties of the clusters found in this way.
Radiogalaxies can thus be used to study the global properties of galaxy groups and clusters, such as their morphological content, dynamical status and number density, as well as the effect of the environment on the radio emission phenomena.
We used the NRAO VLA Sky Survey (NVSS, Condon et al. [@Condon]) publicly available data to build a sample of radio-optically selected clusters through optical identifications of radio sources and search of excesses in the surface density of galaxies around these radiogalaxies. The NVSS survey offers indeed an unprecedented possibility to study a wide-area, homogeneous sample of radio sources down to relatively low flux levels, together with a positional accuracy suitable for optical identifications.
In a previous paper (Zanichelli et al. [@Zanichelli], hereafter Paper I) we described how we extracted a radio source catalogue from the NVSS maps and the optical identification procedure that led to the compilation of a radiogalaxy sample. In this paper we discuss the cluster finding method used for the compilation of a new sample of candidate groups and clusters of galaxies, and present the first observational results that spectroscopically confirmed the presence of a group or cluster for $9$ out of the $11$ successfully observed candidates.
This paper is structured as follows: in Sect. \[sec:data\] we give a summary of the properties of the radio and optical data samples used for the cluster search. In Sect. \[sec:clussel\] we describe the cluster finding method. The new sample of candidate clusters is presented in Sect. \[sec:clussample\]. Spectroscopic observations of a subsample of candidate clusters, aimed to obtain an observational confirmation of the presence of a cluster are presented and discussed in Sects. \[sec:observations\] and \[sec:results\].
The data {#sec:data}
========
In the following two Sects. we recall the properties of the radiogalaxy sample and of the optical galaxy catalogue that have been used during this search. For more details on the radio source catalogue and the definition of the radiogalaxy sample, refer to Paper I.
The radiogalaxy sample {#sec:rgsample}
----------------------
The radio source catalogue has been extracted from $31$ maps of the $1.4$ GHz NRAO VLA Sky Survey (Condon et al. [@Condon]) and consists of $13\,340$ pointlike and $2662$ double radio sources down to a flux limit of $2.5$ over an area of $\approx 550$ sq. degrees at the South Galactic Pole.
Optical identifications of NVSS radio sources have been made with galaxies brighter than $b_\mathrm{J} = 20.0$ in the EDSGC catalogue (Nichol et al. [@Nichol]) using a search radius of $15\arcsec$, i.e. $\approx 3\sigma$ positional accuracy for the faintest sources. The initial sample of optical counterparts consists of $1288$ radiogalaxies, $926$ of them having a pointlike radio morphology at the NVSS resolution of $45\arcsec$.
As shown in Table 2 of Paper I, the contamination level due to spurious identifications varies according to the radio morphological classification, ranging from about $16\%$ for the lists of optical counterparts of pointlike radio sources and “close” radio pairs (separation between components $D \le 50\arcsec$), to about $28\%$ for the list of optical counterparts of “wide” radio doubles ($50\arcsec < D < 100\arcsec$).
In order to obtain a more reliable sample, the radiogalaxy data set used in the search of candidate clusters has been selected among these optical identifications on the basis of radio-optical distance and galaxy magnitude. The uncertainty in the optical identification sample is indeed the only source of contamination that can be limited when selecting cluster candidates by looking for excesses in surface galaxy density near the identified radiogalaxies. Other contamination terms – like the probability of detecting a candidate by chance coincidence of the radiogalaxy position with an optical density excess, or the possibility that the optical excess itself is intrinsically spurious, i.e. due to chance superpositions of galaxies along the line of sight – cannot in fact be reduced unless one knows the redshift distribution of the galaxies.
From the initial sample of optical counterparts we thus selected those radiogalaxies having $d_\mathrm{r-o} \le 7\arcsec$. This constraint introduces a selection effect against faint sources in the radio sample, whose positional uncertainty is typically $\sim 5\arcsec$.
Furthermore, as our aim is to select candidate clusters at intermediate redshifts, we discarded those radiogalaxies brighter than magnitude $b_\mathrm{J} = 17.5$. In fact, considering the magnitude – redshift relation typical of radiogalaxies obtained in the R band by Grueff & Vigotti ([@Grueff]), and using color indexes for elliptical galaxies given in Frei & Gunn ([@Frei]), this cut in $b_\mathrm{J}$ magnitude corresponds to a redshift lower limit of $z \simgt 0.1$.
With these constraints, the final radiogalaxy sample that has been taken into account for the search of candidate clusters consists of $661$ radiogalaxies, and the mean, expected contamination level due to spurious optical identifications has been lowered to about $10\%$.
The galaxy catalogue {#sec:galcatalog}
--------------------
The Edinburgh–Durham Southern Galaxy Catalogue (EDSGC, Nichol et al. [@Nichol]) lists $\approx 1.5 \times 10^6$ galaxies over a contiguous area of $\sim 1200$ sq. degrees at the South Galactic Pole. About one half of this area is currently covered by our radiogalaxy sample and has been considered for the search of cluster candidates.
The EDSGC has been obtained from COSMOS scans of IIIa–J ESO/SERC plates at high galactic latitude ($\mid b_\mathrm{II}\mid \ge 20\degr$). The automated star-galaxy separation algorithm used for the EDSGC guarantees a completeness $> 95\%$ and a stellar contamination $< 12\%$ down to magnitudes $b_\mathrm{J}= 20.0$.
Magnitudes have been calibrated via CCD sequences, providing a plate-to-plate accuracy of $\Delta b_\mathrm{J}\simeq 0.1$ and an rms plate zero-point offset of $0.05$ magnitudes.
The EDSGC incompleteness starts to exceed the $5\%$ only above $b_\mathrm{J} =20.5$ (Collins et al. [@Collins92]). When looking for candidate clusters we thus decided to make optical galaxy counts down to the magnitude limit $b_\mathrm{J}=20.5$: as the radiogalaxy sample reaches $b_\mathrm{J}=20.0$, this choice makes it possible to point out also those regions of high galaxy surface density associated to the optically faintest radio sources in our sample.
Joint radio–optical cluster selection {#sec:clussel}
=====================================
The cluster finding method we adopted is based on optical counts of galaxies in cells, followed by a smoothing of these counts with a Gaussian function and by the definition of a detection threshold for the selection of significative excesses in the surface galaxy density. The density peaks that are found near an optically identified NVSS radio source are included in the cluster sample.
Both to keep in evidence any possible inhomogeneities in optical counts and to make the data handling easier, we divided the EDSGC galaxies brighter than $b_\mathrm{J} = 20.5$ in $21$ adjacent sky maps corresponding to the $5\degr \times 5\degr$ central regions of the UKST plates that cover the radio source catalogue area.
The detection threshold is built in terms of the mode and rms of galaxy counts over each of these sky regions. Despite the possibility of small intra-plate variations in the photometric accuracy of the optical data, we considered the choice of a “local” threshold for each plate preferable to a “global” one, over the whole sky region, as the latter choice would introduce in the cluster sample incompleteness effects that depend on the cluster location in the sky.
The optical count matrix for each $5\degr \times 5\degr$ sky region has been built by defining a regular grid of $600 \times 600$ cells and by counting galaxies in these cells. The size of each cell has been chosen to be $30\arcsec \times 30\arcsec$, to optimize the statistics of galaxy counts as well as to point out the presence of structure in the spatial distribution of galaxies.
Since radiogalaxies tend to reside in different environments – from groups of galaxies to rich clusters – depending on their Faranoff–Riley morphological classification (Zirbel [@Zirbel97]), a careful choice of the size for the smoothing function is needed to avoid selection effects in favour of a particular environment. Too large a size could translate into a lack of detections of distant clusters, whose angular sizes are small. A small size could resolve a nearby cluster in many substructures thus leading to the spurious detection of many candidates relative to the actual cluster, or, if the optical surface density excess in each substructure is less than the selected threshold (see below), could lead to a lack of detections. This last case is the most likely for clusters at moderate $z$ having irregular morphologies, like the Abell I types (Abell [@Abell58]), where subclumps in the galaxy distribution are seen. Given the redshift range we expect to cover with our cluster sample, we decided to adopt, for the smoothing of the optical counts, a circular Gaussian function of FWHM$ = 2\arcmin$, which is about half an Abell radius at $z=0.4$.
We then looked for significative excesses in the optical surface galaxy density: for each smoothed matrix, we determined the mode $m_\mathrm{gal}$ and rms $\sigma_\mathrm{gal}$ of the optical surface density.
The threshold we adopted for the detection of density excesses is defined on each smoothed plate in terms of $m_\mathrm{gal}$ and $\sigma_\mathrm{gal}$ as $n_\mathrm{threshold} = m_\mathrm{gal} + 3\sigma_\mathrm{gal}$, that is we consider significative only those peaks where the galaxy surface density exceeds by at least $3\sigma_\mathrm{gal}$ the value of the mode determined over the whole plate. This choice can introduce selection effects in favour of “core-dominated”, regular clusters, and against irregular ones, where galaxies are less concentrated in the cluster core itself.
Finally, from this list of significant peaks we selected only those for which a radiogalaxy belonging to the considered smoothed matrix is found at a maximum distance of $4\arcmin$ from the density peak position. To determine the list of density peaks, no constraint has been set on the number of connected cells above the threshold. Radiogalaxies themselves are not required to belong to pixels whose optical surface density is above $n_\mathrm{threshold}$. Given the definition of the Abell radius, $R_\mathrm{A} = {1.7\arcmin \over z}$, this search distance corresponds to the Abell radius of a cluster at $z \sim 0.45$.
In the case of nearer clusters, this choice will favour the selection of those candidates where the radiogalaxy is located in the central region of the cluster. A larger value of the search distance would however increase the probability of detecting spurious associations between density peaks and radiogalaxies.
In Fig. \[fig1\] we show as an example the smoothed matrix relative to the UKST plate 412: regions with higher surface galaxy density are represented by increasing grey levels; the superimposed contours are given as $2,3,...\times \sigma_\mathrm{gal}$ above $m_\mathrm{gal}$. The $44$ radiogalaxies having $17.5 \le b_\mathrm{J} \le 20.0$ and $d_\mathrm{r-o} \le 7\arcsec$ present in this sky region are shown as well, marked with diamonds, except for the two associated with a known candidate cluster (see Sect. \[sec:clussample\]), that are marked with an asterisk. The $13$ radio–optically selected cluster candidates are marked as small circles around the position of the associated radiogalaxy. For clarity, only the ACO/Abell and EDCC clusters found in this sky region are plotted in Fig. \[fig1\]: they are marked with big circles, whose radius is equal to the cluster Abell radius. The fact that the clusters A2878, 2904 and E536 do not seem very conspicuous in this galaxy density map can be explained in terms of the different optical data set and the different scale used by Abell et al. ([@Abell]) and Lumsden et al. ([@Lumsden]) to look for overdensities in the galaxy distribution.
{width="17cm"}
The candidate cluster sample {#sec:clussample}
============================
By applying the cluster finding method described in the previous Section, we obtained a sample of $171$ candidate clusters associated to NVSS radio sources identified with $d_\mathrm{r-o} \le 7\arcsec$ with EDSGC galaxies of magnitude $17.5 \le b_\mathrm{J} \le 20.0$. Among these candidates, $123$ are associated to NVSS pointlike sources, $23$ to NVSS “close” double sources, and $25$ to NVSS “wide” doubles. The sample covers an area of $\approx 550$ sq. degrees at the South Galactic Pole and the uncertainty on the candidate cluster position is $30\arcsec$.
As a further step in the compilation of the cluster sample we looked for candidates in common with other cluster catalogues, considering a candidate as already “known” if the radiogalaxy and the optical centroid are found inside an Abell radius from the cluster centre.
Out of $171$ candidates, $76$ were found to be associated with a known cluster according to the above definition. In six cases two candidates are associated with the same known cluster. We detected $2$ Abell poor clusters with measured redshift, $40$ out of the $128$ ACO/Abell (Abell et al. [@Abell]) rich clusters present in this sky region ($31$ of them being listed in the EDCC as well), $16$ EDCC (Lumsden et al. [@Lumsden]) clusters, $9$ clusters from the APM catalogue (Dalton et al. [@Dalton94], [@Dalton97]), $2$ groups selected from the ESO Slice Project survey (Ramella et al. [@Ramella]). Finally, in one case the candidate corresponds to a cluster identified with an X-ray source in the Einstein Medium Sensitivity Survey (Stocke et al. [@Stocke]).
To evaluate $R_\mathrm{A}$, we used the measured cluster redshift when available; otherwise, we used the estimated redshift or, in the case of EDCC clusters, values of $R_\mathrm{A}$ estimated by the catalogue authors. In thirty-six cases the association with previously known clusters has been made on the basis of a measured redshift in the literature, while for the other $40$ only an estimated redshift is available.
The detection of these known clusters can be interpreted as a further indication that this radio-optical selection method is powerful in the search of cluster candidates.
Among the $76$ known clusters, $57$ host a pointlike NVSS radio source, while $11$ and $8$ are respectively associated to “close” and “wide” double radio sources.
In Fig. \[fig2\] the sky distribution of the known candidates (empty symbols marked with the cluster name) and of those candidates without a counterpart in the literature (filled symbols) is shown.
The use of this bivariate radio–optical selection method, based on the condition that an optical excess is considered as a reliable candidate cluster only if it is associated with a radiogalaxy, makes it possible to detect cluster candidates whose reliability, in terms of their optical surface density alone, would be in many cases too low to be included in catalogs based on pure optical selection methods.
The total contamination present in the cluster sample, due to the probability of chance coincidence between a radiogalaxy and an excess in the optical surface galaxy density, has been estimated as follows: we repeated the search of cluster candidates coupling the smoothed matrix relative to each UKST plate with the radiogalaxies belonging to another plate. By applying the same cluster finding criteria described above, we found a contamination percentage of $28\%$ in our sample of candidate clusters at intermediate redshifts. As the actual radiogalaxy sample instead of a random-generated one has been used, this is an estimate of the effective contamination, that is also the contamination term due to the presence of spurious radio-optical associations among the radiogalaxies is taken into account.
An assessment of the reliability of the method is provided by the association with known clusters and measurements on new candidates. Therefore, in the next two Sects. we briefly report on those candidates hosting a radiogalaxy whose redshift is known from previous surveys. This search has been made using the NASA Extragalactic Database [^2]. In Sect. \[sec:observations\] and \[sec:results\] the results obtained from a spectroscopic observative run for a first set of new cluster candidates are presented. The whole candidate cluster sample will be presented in a following paper.
Previously known clusters {#sec:knownclus}
-------------------------
The naming convention for cluster candidates in our sample is as follows: first digits identify the number of the UKST plate on which the candidate has been found; letters are used to distinguish among the various radio morphologies (pointlike or double sources) and to identify candidates associated to more than one radiogalaxy. Last digits in the name are the sequential number of the radio source on that UKST plate.
[**295BN07**]{}: this candidate is found to correspond to A2860, whose measured redshifts is $z=0.105800$ (Struble & Rood, [@Struble]). The radiogalaxy lies at $z = 0.10757 \pm 0.00018$ (Vettolani et al. [@Vettolani98]) so that 295BN07 is considered actually coincident with a known cluster.
[**295D24**]{}: the candidate is found inside one Abell radius from the ESP Group 193 (Ramella et al., [@Ramella]). The radiogalaxy has been detected at $z = 0.05547 \pm 0.00007$ in the ESP survey (Vettolani et al. [@Vettolani98]). ESP group 193 lies at redshift $z=0.05444 \pm 0.001$ so that the association is considered real.
[**352N25**]{}: the candidate has been found to be associated with the ACO cluster A2871. In A2871 the presence of $2$ galaxy systems at different redshifts has been detected (Katgert et al., [@Katgert96]) respectively at $<z> = 0.114$ ($14$ galaxies) and $<z> = 0.122$ ($18$ galaxies). The radiogalaxy has measured redshift from the ENACS survey $z = 0.11415 \pm 0.0003$ (Katgert et al. [@Katgert98]) so 352N25 seems to be associated with one of these two substructures.
[**411N35**]{}: the radiogalaxy in this candidate lies inside an Abell radius from the center of the EDCC cluster E482, whose measured redshift is $z = 0.108040$. The radiogalaxy has $z = 0.07673 \pm 0.0003$ (Collins et al. [@Collins95]); 411N35 is thus not actually coincident with E482. However, measured velocities in the E482 field include a set of galaxies with $cz \sim 22700~{\rm km~s}^\mathrm{-1}$, indicating the possible presence of a superimposed structure at about the same redshift as the radio source.
New cluster candidates {#sec:newclus}
----------------------
From the publicly available data of the Las Campanas Redshift Survey (Shectman et al. [@Shectman]) we obtained the redshift of two radiogalaxies associated with the new candidate clusters 297BN04 ($cz = 53460 \pm 58~{\rm km~s^{-1}}$) and 293D22 ($cz = 41695 \pm 95~{\rm km~s^{-1}}$). At $\sim 2\arcmin$ from the radiogalaxy in 297BN04 we found a galaxy with measured velocity $cz = 53319~{\rm km~s^{-1}}$. Finally, a further redshift has been found in the ESP survey for the radiogalaxy associated to the candidate 294N04: $cz = 43097~{\rm km~s^{-1}}$ (Vettolani et al. [@Vettolani98]). No redshift data for other galaxies near these radiogalaxies are available, so these velocity measurements cannot be used to confirm or not the presence of a cluster.
[cc]{}
[cc]{}
Optical observations {#sec:observations}
====================
A first set of $14$ visually good candidate clusters associated with pointlike NVSS radio sources have been observed with the $3.6$m ESO telescope at La Silla, with the EFOSC1 spectrograph in multislit mode. Moreover, photometry of each field in the r-Gunn filter has been acquired during the first night of observation to achieve magnitude $r \simeq 22.5$ – corresponding to roughly $b_\mathrm{J} \sim 24$ – with a photometric accuracy better than $0.2$ mag.
The targets for spectroscopic observations were chosen on the basis of these photometric observations. For $12$ candidate clusters the spectra of the radiogalaxy and about $10 - 14$ companions were acquired. Observations were made with the $B300$ grism, characterized by a wavelength range $3740 - 6950~\mathrm{A}$, central wavelength $\lambda_\mathrm{c} = 5250~\mathrm{A}$ and dispersion $230~\mathrm{A~mm^{-1}}$. The slit width was chosen to be $2.1\arcsec$ while the slit length varied in order to optimize the number of acquired spectra. The resolution on the spectra was about $20~\mathrm{A}$.
Exposures of He-Ar lamps for wavelength calibration have been acquired through the same masks used in the scientific exposures. Spectroscopic dome flats proved not to be useful during the data reduction phase: due to the low quality of the slit profiles achieved with the mask Punching Machine, the flatfielding process did not significantly help in the extraction of spectra. As we did not apply flat field correction, the obtained spectra are not flux calibrated.
Data reduction {#sec:datared}
--------------
Multislit spectroscopic data reduction has been made interactively by means of the IRAF package. Reduction steps involved bias subtraction, spectra extraction, spectra wavelength calibration. We found no need to correct for dark current. This procedure has been applied both to astronomical and to calibration lamps exposures.
We acquired a total number of $129$ galaxy spectra, plus $4$ stars, for $12$ successfully observed candidates, and determined the velocity of galaxies from absorption and, in a few cases, emission lines by means of the RVSAO package. None of the radiogalaxies we observed show emission lines in their spectra. Templates for the cross-correlation consist of $8$ galaxy and $8$ star spectra known from previous observative programs: the $8$ galaxies were observed during the Edinburgh–Milano Cluster Survey (Collins et al. [@Collins95]) with EFOSC1 and with the same spectral resolution as our observations. The $8$ stars come from ESO Slice Project (Vettolani et al. [@Vettolani97]) observations with the fiber spectrograph OPTOPUS.
To allow for cleaning of cosmic rays, multiple exposures were taken for each field, for a total exposure time varying from $40$ to $60$ min. In Fig. \[fig3\] the direct imaging exposures of these $12$ candidates are shown, together with the targets we selected for spectroscopy.
In Table \[tab:allvel\] the r-Gunn magnitudes, the measured velocities and their associated errors are given for each galaxy we observed in the selected candidate clusters. As can be seen from Table \[tab:allvel\], for $22$ out of the $129$ observed galaxies the S/N was not good enough to measure the redshift. The typical $r$ magnitude for galaxies with measured redshift is $18.7$. Stars are marked with an asterisk. The high errors associated to some velocity measurements are mainly due to the low spectral S/N.
[l@l@r@c@c@c]{} NAME & N & $m_\mathrm{r}$ & $v$ & $\sigma_\mathrm{v}$ & Notes\
& & & ${\rm (km~s^{-1})}$ & ${\rm (km~s^{-1})}$ &\
& 1 & 19.69 & 89125 & $\pm$ 86 &\
& 2 & 19.09 & 92061 & $\pm$ 60 &\
& 3 & 17.67 & 19572 & $\pm$ 300 & E\
& 4 & 17.42 & 90363 & $\pm$ 41 & R\
& 5 & 19.78 & 85666 & $\pm$ 122 &\
& 6 & 20.10 & - & - &\
& 7 & 18.55 & 89065 & $\pm$ 50 &\
& 8 & 20.05 & 89540 & $\pm$ 46 &\
& 9 & 15.62 & 21186 & $\pm$ 48 &\
& 10 & 20.53 & 61998 & $\pm$ 100 &\
& 11 & 20.13 & - & - &\
& 12 & 20.33 & 90960 & $\pm$ 259 &\
& 1 & 20.90 & - & - &\
& 2 & 19.50 & 23511 & $\pm$ 130 &\
& 3 & 19.68 & 31000 & $\pm$ 300 &\
& 4 & 19.08 & 79401 & $\pm$ 309 &\
& 5 & 19.53 & 97000 & $\pm$ 300 &\
& 6 & 20.10 & 78568 & $\pm$ 107 &\
& 7 & 17.84 & 79110 & $\pm$ 379 & R\
& 8 & 21.29 & 14800 & - & E\
& 9 & - & - & - &\
& 10 & 20.51 & 98925 & $\pm$ 114 &\
& 11 & 18.38 & 79863 & $\pm$ 198 &\
& 1 & 19.59 & 34237 & $\pm$ 300 &\
& 2 & 18.80 & 29093 & $\pm$ 130 &\
& 3 & 18.81 & 63333 & $\pm$ 85 &\
& 4 & - & - & - &\
& 5 & 18.42 & - & - & $\ast$\
& 6 & 16.03 & 33698 & $\pm$ 49 & R\
& 7 & 18.83 & - & - & $\ast$\
& 8 & 19.69 & 123966 & $\pm$ 300 & E\
& 9 & 20.84 & 146819 & $\pm$ 300 & E\
& 10 & 21.45 & - & - &\
& 1 & 16.97 & 42919 & $\pm$ 74 &\
& 2 & 17.16 & 42586 & $\pm$ 86 &\
& 3 & 20.62 & 80696 & $\pm$ 112 &\
& 4 & 17.73 & 69681 & $\pm$ 169 &\
& 5 & 16.23 & 56290 & $\pm$ 80 &\
& 6 & 18.79 & 56099 & $\pm$ 280 &\
& 7 & 17.80 & 69904 & $\pm$ 109 &\
& 8 & 17.91 & 70484 & $\pm$ 82 & R\
& 9 & 18.44 & 70824 & $\pm$ 119 &\
& 10 & 18.36 & 70060 & $\pm$ 121 &\
& 1 & 19.77 & 51980 & $\pm$ 137 &\
& 2 & 16.68 & 51988 & $\pm$ 114 & R\
& 3 & 19.60 & 81249 & $\pm$ 258 &\
& 4 & 18.06 & 51233 & $\pm$ 150 &\
& 5 & 20.85 & - & - &\
& 6 & 18.70 & 52499 & $\pm$ 150 &\
& 7 & 20.02 & 86464 & $\pm$ 300 &\
& 8 & 19.63 & 63814 & $\pm$ 220 &\
& 9 & 19.90 & 97518 & $\pm$ 172 &\
& 10 & 19.81 & - & - &\
[l@l@r@c@c@c]{} NAME & N & $m_\mathrm{r}$ & $v$ & $\sigma_\mathrm{v}$ & Notes\
& & & ${\rm (km~s^{-1})}$ & ${\rm (km~s^{-1})}$ &\
& 1 & 18.23 & 54587 & $\pm$ 71 &\
& 2 & 19.01 & 55881 & $\pm$ 189 &\
& 3 & 21.28 & - & - &\
& 4 & 20.07 & - & - &\
& 5 & 16.73 & 54698 & $\pm$ 118 &\
& 6 & 18.38 & 54699 & $\pm$ 76 & R\
& 7 & 18.29 & 53486 & $\pm$ 84 &\
& 8 & 18.74 & 55275 & $\pm$ 78 &\
& 9 & 20.60 &- & - &\
& 10 & 20.32 & - & - &\
& 11 & 19.85 & - & - &\
& 1 & 19.57 & - & - &\
& 2 & 18.91 & 40809 & $\pm$ 144 &\
& 3 & 18.41 & 40764 & $\pm$ 62 &\
& 4 & 18.59 & 54726 & $\pm$ 131 &\
& 5 & 19.18 & 55821 & $\pm$ 271 &\
& 6 & 16.15 & 55316 & $\pm$ 123 &\
& 7 & 16.59 & 40492 & $\pm$ 63 & R\
& 8 & 18.63 & 41140 & $\pm$ 68 &\
& 9 & 20.11 & - & - &\
& 10 & 18.04 & - & - &\
& 11 & 18.16 & - & - & $\ast$\
& 12 & 18.43 & 37199 & $\pm$ 151 &\
& 13 & 17.60 & 40377 & $\pm$ 54 &\
& 14 & 20.33 & - & - &\
& 1 & 18.15 & 64532 & $\pm$ 127 &\
& 2 & 17.44 & 47583 & $\pm$ 429 &\
& 3 & 16.38 & 41309 & $\pm$ 50 & R\
& 4 & 20.01 & - & - &\
& 5 & 20.15 & 46345 & $\pm$ 145 &\
& 6 & 17.65 & 46765 & $\pm$ 61 &\
& 7 & 15.89 & 47150 & $\pm$ 55 &\
& 8 & 18.30 & 46879 & $\pm$ 57 &\
& 9 & 20.31 & - & - &\
& 10 & 19.46 & 47460 & $\pm$ 247 &\
& 11 & 19.83 & 46630 & $\pm$ 457 &\
& 12 & 19.62 & - & - &\
& 1 & 20.06 & 97958 & $\pm$ 75 &\
& 2 & 17.44 & 45837 & $\pm$ 39 &\
& 3 & 18.74 & 45687 & $\pm$ 42 &\
& 4 & 18.54 & 45682 & $\pm$ 46 &\
& 5 & 16.62 & 45280 & $\pm$ 31 & R\
& 6 & 20.43 & 85363 & $\pm$ 98 & E\
& 7 & 19.00 & 45343 & $\pm$ 58 &\
& 8 & 20.53 & 38136 & $\pm$ 79 &\
& 9 & 20.67 & 43891 & $\pm$ 100 & E\
& 10 & 19.98 & 78625 & $\pm$ 65 &\
& 11 & 19.59 & 79383 & $\pm$ 165 &\
& 1 & 19.80 & 98238 & $\pm$ 172 &\
& 2 & 20.02 & 69000 & $\pm$ 200 &\
& 3 & 19.91 & 41416 & $\pm$ 158 &\
& 4 & 17.62 & 40147 & $\pm$ 61 & R\
& 5 & 19.69 & 41300 & $\pm$ 300 &\
& 6 & 18.07 & 58828 & $\pm$ 73 &\
& 7 & 17.71 & 40332 & $\pm$ 74 &\
& 8 & 18.18 & 39350 & $\pm$ 42 &\
------ ---- ---------------- --------------------- ------------------------ --------
NAME N $m_\mathrm{r}$ $v$ $\sigma_\mathrm{v}$ Notes
${\rm (km~s^{-1})}$ ${\rm (km~s^{-1})}$
9 18.26 58805 $\pm$ 195
1 20.40 29350 $\pm$ 300
2 19.81 49474 $\pm$ 110
3 17.69 77698 $\pm$ 144 R
4 17.13 48980 $\pm$ 80
5 19.14 32075 $\pm$ 200
6 16.88 30590 $\pm$ 300
7 18.78 76139 $\pm$ 98
8 20.83 - -
9 19.84 47521 $\pm$ 78 E
10 19.07 29934 $\pm$ 57
11 19.52 60417 $\pm$ 300
1 18.99 61683 $\pm$ 84
2 19.86 64065 $\pm$ 300
3 18.24 62720 $\pm$ 300
4 18.59 63050 $\pm$ 300
5 17.98 61318 $\pm$ 94
6 17.34 63758 $\pm$ 130
7 17.98 64280 $\pm$ 173
8 17.49 63160 $\pm$ 108 R
9 20.79 - -
10 19.62 63421 $\pm$ 124
11 19.47 - - $\ast$
12 19.33 64246 $\pm$ 118
------ ---- ---------------- --------------------- ------------------------ --------
: (Continued).
Results {#sec:results}
=======
On the basis of the measured galaxy velocities for each candidate cluster, we verified in which cases the spectroscopic data confirm the presence of a cluster associated to a NVSS radiogalaxy. As can be noticed from Table \[tab:allvel\], for the candidate 349N02 the few available spectroscopic data are not useful for a statistical analysis aimed to assess the presence of a cluster around the radio source.
Among the $11$ fields for which we have sufficient data, in two cases (409N03 and 412N23) the radiogalaxy velocity is significantly different from all the other measured values and we conclude that the radiogalaxy is not associated to a cluster. In both cases the data suggest the presence of a group or cluster, but at a redshift different from that of the radiogalaxy.
For the $9$ remaining candidates, we confirm the presence of a cluster around the radiogalaxy: this corresponds to a positive detection rate of $82\%$. For these $9$ clusters we determined the mean velocity and velocity dispersion by means of the ROSTAT package (Robust Statistics, Beers et al. [@Beers]), which allows robust estimates of central location and scale in data samples affected by the presence of “outliers”. When dealing with small data sets ($n = 5 - 10$) as in our case, the best estimators are the biweight $C_\mathrm{BI}$ (Tukey [@Tukey]) for the central location and the classical standard deviation $S_\mathrm{G}$ for the scale (Beers et al. [@Beers]). The $C_\mathrm{BI}$ estimator is evaluated iteratively, by minimizing a function of the deviations of each observation from the estimate of the central location. It thus requires an additional estimate of this last quantity, which is generally given as the absolute value of the median of the differences between the data and the sample median.
The uncertainties associated to central location and scale have been estimated by the bootstrap method. This technique consists in the generation of a large number of samples, not independent from the original data set, and in the evaluation of the statistical parameters for each of these “bootstrapped” samples.
In Fig. \[fig4\] we show the distributions of measured velocities for the $9$ cluster candidates involved in this statistical analysis: the shadowed regions represent the data sets used as input for the ROSTAT package.
The results of the statistical analysis are shown in Table \[tab:myclus\]: mean cluster velocities vary from $40514~{\rm km~s^{-1}}$ to $90122~{\rm km~s^{-1}}$, corresponding to the redshift range $0.13 \le z \le 0.3$.
Despite the small number of available redshifts for each cluster, which reflects into rather large errors for both the central location and velocity dispersion, an interesting result arises from the velocity dispersions: they range from $210~{\rm km~s^{-1}}$ to $906~{\rm km~s^{-1}}$, that is from values typical of poor clusters or groups of galaxies to those typical of moderately rich clusters.
Following the criteria in Abell ([@Abell58]), we used the EDSGC catalog to get an estimate of the cluster richness for the $9$ confirmed clusters: the background–subtracted galaxy counts in the magnitude range $m_3 \div m_3 +2$ within an Abell radius from the cluster centre range from a minimum of $6$ to a maximum of $23$. These galaxy counts are similar to those found for many of the ACO poor clusters (Abell et al. [@Abell]), and suggest that our radio–optically selected clusters are poorer than Abell richness class $0$. We stress however that these richness estimates must be viewed with caution: first, the values of $m_3 +2$ often fall near or below $b_J=20.5$, where the EDSGC completeness drops significantly, thus seriously biasing the galaxy counts. Second, at our typical $m_3$ the number density of galaxies in the EDSGC is high, about $50$ galaxies per square degree, thus the probability of selecting as the third member of the cluster a galaxy which is actually a background or foreground object seen in projection is not negligible, and this again can alter the richness estimate.
As shown in Fig. \[fig5\], there is no evident correlation between measured velocity dispersion and cluster redshift. The use of radio emission properties of galaxies seems thus a very efficient method to select new candidate clusters samples in a wide range of richness at any redshift.
[cccccrccccccc]{} CLUSTER & & Right Ascension (B1950) & & Declination (B1950) & & n & & & $<v> ~{\rm (km~s^{-1})}$ & & & $\sigma_\mathrm{v} ~{\rm (km~s^{-1})}$\
294N15 & & 00 23 41.0 & & -39 37 15.0 & & 6 & & & 90122 $^{+519}_{-589}$ & & & 906 $^{+227}_{-128}$\
295N35 & & 01 03 11.0 & & -38 47 15.0 & & 4 & & & 79241 $^{+130}_{-270}$ & & & 429 $^{+162}_{-49}$\
350N71 & & 00 35 04.0 & & -34 49 15.0 & & 5 & & & 70180 $^{+293}_{-138}$ & & & 373 $^{~+98}_{~-44}$\
352N47 & & 01 14 13.0 & & -36 44 45.0 & & 4 & & & 51969 $^{+14}_{-309}$ & & & 444 $^{+179}_{-123}$\
352N63 & & 01 19 50.0 & & -33 45 15.0 & & 6 & & & 54844 $^{+497}_{-144}$ & & & 674 $^{+254}_{-127}$\
352N75 & & 01 21 11.0 & & -33 17 15.0 & & 5 & & & 40712 $^{~+66}_{-227}$ & & & 263 $^{~+73}_{~-60}$\
409N15 & & 00 02 36.0 & & -28 20 15.0 & & 5 & & & 45573 $^{+96}_{-254}$ & & & 210 $^{~+41}_{~-16}$\
409N44 & & 23 51 13.0 & & -31 34 15.0 & & 5 & & & 40514 $^{+761}_{-269}$ & & & 757 $^{+184}_{-108}$\
475N50 & & 01 15 13.0 & & -24 09 45.0 & & 10 & & & 63266 $^{+256}_{-414}$ & & & 847 $^{+182}_{-121}$\
If confirmed by future spectroscopic follow-up, this result could be of great interest as our sample would offer the possibility to investigate differences in cluster dynamical properties in a homogeneously selected sample of clusters which spans a wide range in richness, and to improve our knowledge of their number counts, as well as to study the radio emission properties of galaxies residing in different environments.
Conclusions {#sec:concl}
===========
To study the status and the evolution of clusters of galaxies at intermediate redshifts we built a sample of candidate clusters using radiogalaxies in the NRAO VLA Sky Survey as tracers of dense environments.
From the NVSS maps we extracted a catalogue of radio sources over an area of $\approx 550$ square degrees, and made optical identifications with galaxies brighter than $b_\mathrm{J} = 20.0$ in the EDSGC Catalogue, resulting in a sample of $1288$ radiogalaxies (Zanichelli et al. [@Zanichelli], Paper I).
In this paper we have presented the detection technique we applied to select candidate groups and clusters associated to NVSS radio sources. The method is based on the search of excesses in optical surface galaxy density nearby NVSS radiogalaxies. To keep low the probability of spurious radio-optical identifications, as well as to preferentially select clusters at redshifts $z \simgt 0.1$, we restricted the cluster search to the $661$ radiogalaxies having radio-optical distance $\le 7\arcsec$ and magnitude $b_\mathrm{J}\ge 17.5$.
The search of regions having high optical galaxy density has been made using the EDSGC galaxy catalogue, building matrices of galaxy counts down to magnitude $b_\mathrm{J} = 20.5$. This choice allows to find density excesses surrounding the faintest radiogalaxies (identified down to $b_\mathrm{J} = 20$) without introducing significant incompleteness effects in the optical data. Smoothing of galaxy counts has been done using a gaussian filter with FWHM$ = 2\arcmin$. The mode and standard deviation of smoothed galaxy counts have been used to define a detection threshold for the surface density excesses: we selected as cluster candidates those density excesses whose centroid is within $4\arcmin$ from a radiogalaxy. This search radius for candidate clusters associated to NVSS radio sources corresponds to an Abell radius of a cluster at $z \sim 0.45$.
By applying this cluster detection strategy to $661$ radiogalaxies over $\approx 550$ sq. degrees at the South Galactic Pole, we obtained a sample of $171$ cluster candidates. The estimated contamination level is about $28\%$.
Out of these $171$ candidates, $76$ correspond to already known clusters, while $95$ cluster candidates in our list do not have any known counterpart in the literature and have been the subject of subsequent spectroscopic follow-up. The full sample of radio-optically selected cluster candidates will be presented in a following paper. Multi Object Spectroscopy aimed to confirm the detection of clusters has been successfully acquired at the 3.6 m ESO telescope for a subset of $12$ candidates. In $2$ cases the radiogalaxy does not lie at the same redshift as any other observed target, while $9$ candidates have been confirmed as clusters of galaxies in the redshift range $0.13 \le z \le 0.3$, thus confirming that this joint radio-optical cluster selection technique can be used as a powerful tool for the detection of cluster candidates at intermediate redshifts. For one additional candidate, the very low number of measured redshifts does not allow any conclusion on the presence of a cluster surrounding the radiogalaxy, and further observations are needed. Velocity dispersions of the $9$ spectroscopically confirmed clusters vary from values typical of moderately rich clusters to those typical of groups or poor clusters, thus strengthening the assumption that this technique is equally efficient in selecting structures over a wide range of richness at different redshifts. If confirmed by future spectroscopic follow up, this last result could be of great interest as this technique would offer the possibility to study the properties of different environments, such as groups or rich clusters, in a homogeneously selected cluster sample.
The authors acknowledge Marco Mignoli for his valuable help during the observative run.
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|
---
abstract: |
We study a cosmology by considering entropic issues where the emphasis is put on the change of sign of the inhomogeneus term associated to an energy non-conservation equation. We make a bare/effective description of the equation of state of the cosmic fluid through a bare($\omega $)/effective($%
\omega _{eff}$) conservation equation. In the bare case, where we have a non conserved equation of state for the cosmic fluid, we describe the change in the sign of the inhomogeneus term at different times of the cosmic evolution. We show that by a redefinition of the adiabatic $\omega $-parameter we can recover the usual scheme for the cosmic evolution. In the effective case we show also that if the evolution is driven by dust or cosmological constant, the universe evolves on the thermal equilibrium. Additionally, by incorporating a quantum correction only cosmological constant can drive an evolution on thermal equilibrium. The future singularity present in the case when we do not incorporate this correction is avoided if we do it.
author:
- Samuel Lepe
- Francisco Peña
title: '$\Gamma$-sign in entropic cosmology'
---
Introduction
=============
>From the works of E. Verlinde \[1\] and D. A. Easson et al \[2\], entropic gravity is visualized as an alternative to the standard (classical) theory of gravity described by the Einstein theory. In the first formalism the space is generated from the termodynamics on a holographic screen. Here the information is the main ingredient for deriving gravity and the input is the holographic principle; the information is encoded at the boundary. The second formalism is based on the incorporation of surface terms in the gravitacional action but gravity is still a fundamental theory. Both schemes drive to modifications in the usual Friedmann equations of the standard Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmology and so we can have another sand where we can inspect. Whether or not these approaches can give a new scope about gravity is a controversial issue \[3\].
This letter is organized as follows. In Section II we discuss an effective scheme ($\omega _{eff}$) for the equation of state (EoS) of the cosmic fluid through a redefinition of the adiabatic-$\omega $ parameter. In Section III we describe the sign change in the inhomogeneus term (in the bare EoS for the cosmic fluid) associated to the interchange of energy between the bulk and the boundary and we discuss also the absence of future singularities when a quantum correction is incorporated in the field equations. In Section IV we summarize the discussed aspects. We will use the units 8$\pi G=$ $c=1$.
————————————————————————
$\Theta _{eff}$-formulation.
=============================
We discuss the model \[4\] which is described by the following modified Friedmann equations
$$\begin{aligned}
\label{ec1}
3H^{2} &=&\rho +3\alpha H^{2}+3\beta \dot{H},\end{aligned}$$
$$\begin{aligned}
\label{ec2} \frac{\ddot{a}}{a} &=&\dot{H}+H^{2}=-\frac{1}{6}\left(
3\Theta -2\right) \rho +\alpha H^{2}+\beta \dot{H},\end{aligned}$$
where the additional terms, $\alpha H^{2}$ and $\beta \dot{H}$ are originated on surface terms in the gravitational action and $\left\{ \alpha ,\beta \right\} $ are two constant parameters associated to surface curvature where $\alpha =3/2\pi $ and $\beta
=3/4\pi $ and like a generalization this parameters can be bounded by $\alpha <1$ and $0\leq \beta \lesssim 3/4\pi $ in agreement to the observational data \[2\]. The $ \Theta $-quantity is $\Theta
=1+\omega \ $where$\ \omega =p/\rho $ being $p$ the pressure and $\rho $ the energy density.
We will study two approaches in order to manipulate (\[ec1\]-\[ec2\]): the first is based on making a description of the model through an effective EoS ($\Theta _{eff}$). The second approach is based on a non conserved equation for $\rho $. So, we write the scheme (\[ec1\]-\[ec2\]), denoted as effective scheme, in the form
$$\begin{aligned}
\label{ec3}
3H^{2}=\frac{1}{1-\alpha }\left( 1-\frac{3}{2}\beta \Theta \right) \rho ,\end{aligned}$$
$$\begin{aligned}
\label{ec4}
\frac{\ddot{a}}{a}=-qH^{2}=-\left( \frac{3}{2}\Theta _{eff}-1\right) H^{2},\end{aligned}$$
where $\Theta _{eff}$ is defined through
$$\begin{aligned}
\label{ec5}
\Theta _{eff}=\left( 1-\alpha \right) \left( 1-\frac{3}{2}\beta
\Theta \right) ^{-1}\Theta ,\end{aligned}$$
and by using (\[ec4\]) the deceleration parameter $q$ is given by
$$\begin{aligned}
\label{ec6}
1+q=\frac{3}{2}\Theta _{eff},\end{aligned}$$
and the conservation equation becomes
$$\begin{aligned}
\label{ec7}
\dot{\rho}+3\Theta _{eff}H\rho =0.\end{aligned}$$
In the present effective scheme, this equation appears like an usual energy conservation equation and so, the scheme (\[ec3\]-\[ec4\]) can be saw as one standard FLRW scheme in the following sense, from (\[ec3\]-\[ec4\]) we can obtain the solution for the Hubble parameter
$$\begin{aligned}
\label{ec8}
H\left( t\right) =H_{0}\left[ 1+\frac{3}{2}\Theta
_{eff}H_{0}\left( t-t_{0}\right) \right] ^{-1},\end{aligned}$$
and this solution it is a one standard given that $\Theta
_{eff}>0\Longrightarrow $ dark matter-evolution or quintessence-evolution and $\Theta _{eff}<0\Longrightarrow $ phantom-evolution. If we consider early inflation, $\Theta \approx
0$, from (\[ec3\]-\[ec4\]) we obtain $H\approx
const.\rightarrow $ $\rho \approx const.$ and the acceleration is
$$\begin{aligned}
\label{ec9}
\frac{\ddot{a}}{a}=\frac{1}{3}\left( \frac{1}{1-\alpha }\right)
\rho =const.>0\Longrightarrow \alpha <1,\end{aligned}$$
so that if we look (\[ec3\]) we obtain $\beta \Theta <2/3$ and this constraint for $\beta $ will be improved later. Particular cases about bare-effective description $\Theta \left( \omega
\right) -\Theta _{eff}\left( \omega _{eff}\right) $ are:
-cosmological constant
$$\begin{aligned}
\label{ec10}
\Theta =0\left( \omega =-1\right) \Longrightarrow \Theta
_{eff}=0\left( \omega _{eff}=-1\right) ,\end{aligned}$$
and both descriptions, bare and effective, coincide,
-stiff matter
$$\begin{aligned}
\label{ec11}
\Theta =2\left( \omega =1\right) \Longrightarrow \Theta
_{eff}=2\left( 1-\alpha \right) \left( 1-3\beta \right)
^{-1}\left( \omega _{eff}=-1+2\left( 1-\alpha \right) \left(
1-3\beta \right) ^{-1}\right) ,\end{aligned}$$
-dark matter
$$\begin{aligned}
\label{ec12}
\Theta =1\left( \omega =0\right) \Longrightarrow \Theta
_{eff}=\left( 1-\alpha \right) \left( 1-\frac{3}{2}\beta \right)
^{-1}\left( \omega _{eff}=-1+\left( 1-\alpha \right) \left(
1-\frac{3}{2}\beta \right) ^{-1}\right) ,\end{aligned}$$
-phantom dark energy
$$\begin{aligned}
\label{ec13}
\Theta <0\left( \omega <-1\right) \Longrightarrow \Theta
_{eff}<0\left( \omega _{eff}<-1\right) ,\end{aligned}$$
and in this case we are also coincidence in both schemes.
By using (\[ec6\]) we can fit the q-parameter in the following form
$$\begin{aligned}
\label{ec14}
\Theta _{eff}=1\text{ }\left( q=1/2\right) \text{\ \ }and\text{ \
}\Theta =1 \text{ }\Longrightarrow \alpha =\frac{3}{2}\beta ,\end{aligned}$$
so that for dark matter we have $\omega _{eff}=\omega =0$. Thus, we can write
$$\begin{aligned}
\label{ec15}
\Theta _{eff}=\left( 1-\alpha \right) \left( 1-\alpha \Theta
\right) ^{-1}\Theta ,\end{aligned}$$
and we have one parameter ($\alpha $) for fixing. Now, by considering $%
\Theta =2\left( \omega =1\right) $, i.e. stiff matter, we have
$$\begin{aligned}
\label{ec16}
\Theta _{eff}=2\left( 1-\alpha \right) \left( 1-2\alpha \right)
^{-1}\rightarrow \omega _{eff}=-1+\frac{1-\alpha }{1/2-\alpha },\end{aligned}$$
and $1/2<\alpha <1\Longrightarrow \omega _{eff}<-1,$ i.e., something like effective phantom stiff matter! We solve this problem by doing $\alpha <1/2$, and in this case we have a quintessence-evolution driven for stiff matter. If $0<\alpha <<1/2$ we have $\omega _{eff}$ $\approx 1$. We note that $%
\alpha <1/2\Longrightarrow \beta <1/3$ (see (\[ec14\])). So, the present effective scheme result to be on standard... and nothing new under the sun!, that is, we obtain the same as the usual FLRW-scheme by changing $\Theta \rightarrow \Theta _{eff}$.
$\Gamma $-formulation.
======================
We come back to (\[ec7\]) in order to obtain
$$\begin{aligned}
\label{ec17}
\dot{\rho}+3\Theta H\rho =\Gamma ,\end{aligned}$$
where $\Gamma $ is the amount of energy non-conservation which is given by $$\begin{aligned}
\label{ec18}
\Gamma =3\left( \Theta -\Theta _{eff}\right) H\rho =3\alpha \Theta
\left( \frac{1-\Theta }{1-\alpha \Theta }\right) H\rho ,\end{aligned}$$ where we have used (\[ec15\]). This quantity, $\Gamma $, represents the interchange of energy between the bulk (the universe) and the boundary (Hubble horizon). At early times, $\Theta \approx 0\longleftrightarrow H\approx const.$and $\rho
\approx const.$(exponential inflation), so that we have $\Gamma
\approx 0$ and the same occours if we have dark matter $\left(
\Theta =1\right) $. In the case of stiff matter $\left( \Theta
=2\right) $ or radiation $\left( \Theta =4/3\right) $ we obtain $\Gamma <0$ and the same occours at late times if we have the possibility of phantom dark energy $%
\left( \Theta <0\right) $, and we note that $\Gamma >0$ if we are in the quintessence-zone $\left( 0<\Theta <1\right) $. In the case of cosmological constant $\left( \Theta =0\right) $ we have $\Gamma =0$.
By using the eqs. (\[ec15\]), (\[ec8\]) and (\[ec1\]), the expression (\[ec18\]) can be writen in the equivalent forms
$$\begin{aligned}
\label{ec19}
\Gamma \left( \Theta ,\alpha ;t\right) &=&9H_{0}^{3}\alpha \left( 1-\alpha
\right) \frac{\left( 1-\Theta \right) \Theta }{\left( 1-\alpha \Theta
\right) ^{2}}\left[ 1+\frac{3}{2}\frac{\left( 1-\alpha \right) \Theta }{%
\left( 1-\alpha \Theta \right) }H_{0}\left( t-t_{0}\right) \right] ^{-3},\end{aligned}$$
$$\begin{aligned}
\label{ec20}
&=&\frac{8}{3}H_{0}^{3}\frac{\alpha }{\left( 1-\alpha \right) ^{2}}\frac{%
\left( 1-\Theta \right) \left( 1-\alpha \Theta \right) }{\Theta ^{2}}\left[
\frac{2}{3}\frac{\left( 1-\alpha \Theta \right) }{\left( 1-\alpha \right)
\Theta }+H_{0}\left( t-t_{0}\right) \right] ^{-3},\end{aligned}$$
$$\begin{aligned}
\label{ec21}
&=&-\frac{8}{3}\frac{\alpha }{\left( 1-\alpha \right)
^{2}}\frac{\left( 1-\Theta \right) \left( 1-\alpha \Theta \right)
}{\Theta ^{2}}\left( t_{s}-t\right) ^{-3},\end{aligned}$$
where
$$\begin{aligned}
\label{ec22}
t_{s}=t_{0}-\frac{2}{3\Theta _{eff}}H_{0}^{-1}=t_{0}-\frac{2\left(
1-\alpha \Theta \right) }{3\left( 1-\alpha \right) \Theta
}H_{0}^{-1},\end{aligned}$$
and we can see that $\Gamma \left( 1,\alpha ;t\right) =\Gamma
\left( 0,\alpha ;t\right) =\Gamma \left( \alpha ^{-1},\alpha
;t\right) =0$ and we have a future singularity for $\Theta <0$ (this singularity will be removed by the inclusion of quantum corrections on (\[ec1\]- \[ec2\]), see later) and a past singularity if $\Theta >\alpha ^{-1}$. If $0<\alpha
<1/2\Longrightarrow \Theta
>2$, i.e., $\omega >1$ (super stiff matter), but the observational data discards this past singularity. So,$$\begin{aligned}
\label{ec23}
\begin{array}{c}
\Gamma \left( \Theta >1,\alpha ;t\right) <0\Longleftrightarrow \omega >0%
\text{ }\left( beyond\text{ }dust-zone\right) , \\
\Gamma \left( 1,\alpha ;t\right) =0\Longleftrightarrow \omega =0\text{ }%
\left( dust\right) \\
\Gamma \left( 0<\Theta <1,\alpha ;t\right) >0\Longleftrightarrow -1<\omega <0%
\text{ }\left( qu\mathit{in}tessence-zone\right) , \\
\Gamma \left( 0,\alpha ;t\right) =0\Longleftrightarrow \omega =-1\text{ }%
\left( \cos mo\log ical\text{ }cons\tan t\right) \\
\Gamma \left( \Theta <0,\alpha ;t\right) <0\Longleftrightarrow \omega <-1%
\text{ }\left( phantom-zone\right) ,%
\end{array}\end{aligned}$$ and the sign of $\Gamma $ it‘s not always the same during the cosmic evolution. So, the evolution is developed out the thermal equilibrium at exception when we have dust or cosmological constant.
We come back now to (\[ec4\]). By using (\[ec18\]) we write the following expression for the acceleration
$$\begin{aligned}
\label{ec24}
\frac{\ddot{a}}{a}=\frac{3}{2}H^{2}\left( \frac{2}{3}+\frac{\Gamma
}{3H\rho } -\Theta \right) ,\end{aligned}$$
and we note that
$$\begin{aligned}
\label{ec25}
\Theta =\frac{2}{3}\longleftrightarrow \omega
=-\frac{1}{3}\Longrightarrow \frac{\ddot{a}}{a}=\left(
\frac{H}{2\rho }\right) \Gamma \text{ \ \ }and \text{ \ \ }\Gamma
=\left( \frac{\alpha }{3/2-\alpha }\right) H\rho >0.\end{aligned}$$
In standard cosmology, $\Theta =2/3\longleftrightarrow \omega =-1/3$ correspond to a curvature fluid (string gas) and in that case we have $\ddot{%
a}=0$. Nevertheless, in the present case we have $\ddot{a}>0$. What happen today? From WMAP 5-7 \[5\]: $1+\omega \left( 0\right) =\Theta \left( 0\right)
\approx -0.10\pm 0.14$ (for $\omega -$time independent) so that we define $%
\Theta ^{+}\left( 0\right) =0.04$ and $\Theta ^{-}\left( 0\right)
=-0.24$. Thus, in accord to (\[ec18\])
$$\begin{aligned}
\label{ec26}
\Gamma \left( 0\right) >0\longleftrightarrow \Theta \left(
0\right) =\Theta ^{+}\left( 0\right) \text{ \ \ }and\text{ \ \
}\Gamma \left( 0\right) <0\longleftrightarrow \Theta \left(
0\right) =\Theta ^{-}\left( 0\right) ,\end{aligned}$$
and $\Gamma \left( 0\right) =0$ if the evolution is one standard driven by the cosmological constant, and in this case we do not notice today the “presence” of $\Gamma $ (the same happened when $\Theta =1$, evolution driven by dust). So, we can say that $\Gamma \left( 0\right) \approx 0$ and today the cosmic fluid is one conserved (delicate thermal equilibrium between the bulk and the boundary today?). We note that in the quintessence zone $\Theta =1\left( \omega =0\right) $ and $\Theta =0\left( \omega
=-1\right) $ we have energy transference from the boundary to the bulk ($%
\Gamma >0$). If the future is driven for phantom dark energy, then we have $%
\Gamma <0$ plus a singularity and in this case we will have energy going from the bulk to the boundary and the temperature of the boundary ($%
T_{H}=H/2\pi $) will increase and we ask, what will happen when the singularity is nearby?, super hot boundary ($T_{H}\rightarrow
\infty $) and bulk frozen ($T\rightarrow 0$)? We inspect now the phantom zone ($\Theta <0\rightarrow \Gamma <0$). The Hubble temperature is given by $T_{H}\left( t\right) =H\left( t\right)
/2\pi $ and we do the following exercise (in order to have a feeling): by using (\[ec8\]) with $\Theta _{eff}<0$ (phantom zone) we obtain
$$\begin{aligned}
\label{ec27}
H\left( a\right) =H_{0}\left( \frac{a}{a_{0}}\right) ^{3\left\vert
\Theta _{eff}\right\vert /2},\end{aligned}$$
so that the boundary temperature is
$$\begin{aligned}
\label{ec28}
T_{H}\left( a\right) =T_{H}\left( 0\right) \left(
\frac{a}{a_{0}}\right) ^{3\left\vert \Theta _{eff}\right\vert /2}.\end{aligned}$$
For radiation ($\Theta =4/3$ and $\Theta _{eff}=\left( 4/3\right) \left(
1-\alpha \right) /\left( 1-4\alpha /3\right) $) we have
$$\begin{aligned}
\label{ec29}
T_{r}\left( a\right) =T_{r}\left( 0\right) \left(
\frac{a}{a_{0}}\right) ^{-\left( 1-\alpha \right) /\left(
1-4\alpha /3\right) },\end{aligned}$$
and we do now $T_{H}\left( \bar{a}\right) =T_{r}\left( \bar{a}\right) $ (equilibrium) so that
$$\begin{aligned}
\label{ec30}
\bar{a}=a_{0}\left( \frac{T_{r}\left( 0\right) }{T_{H}\left(
0\right) } \right) ^{1/\left( \Delta +3\left\vert \Theta
_{eff}\right\vert /2\right) },\end{aligned}$$
where $\Delta =\left( 1-\alpha \right) /\left( 1-4\alpha /3\right) $. Today $%
T_{r}\left( 0\right) \sim 3K$ and $T_{H}\left( 0\right) \sim 0\left(
10^{-30}\right) K$ and in this case we have thermal equilibrium at $\bar{a}%
\sim 10^{12}a_{0}$! We note that $T_{r}\left( \bar{a}\right) \sim
10^{-12}T_{r}\left( 0\right) $ and $T_{H}\left( \bar{a}\right) \sim
10^{18}T_{H}\left( 0\right) \sim 10^{-12}K$. The time ($t_{eq}$) at which we obtain this thermal equilibrium is given by
$$\begin{aligned}
\label{ec31}
t_{eq}-t_{0}=\left( t_{s}-t_{0}\right) \left( 1-\frac{T_{H}\left( 0\right) }{%
T_{CMB}\left( 0\right) }\right) ^{1/\Omega },\end{aligned}$$
where $\Omega =1+2/3\left\vert \Theta _{eff}\right\vert $. So, we have $%
t_{eq}\sim t_{s}$, and we have thermal equilibrium very near to the singularity.
Now, by adding the quantum corrections $3\gamma H^{4}$ in (\[ec1\]) and $\gamma H^{4}$ in (\[ec2\]) \[4,6\] it is straightforward to obtain
$$\begin{aligned}
\label{ec32}
3H^{2}=\frac{1}{1-\alpha }\left( 1-\frac{3}{2}\beta \Theta \right)
\rho +3 \frac{\gamma }{1-\alpha }H^{4},\end{aligned}$$
and
$$\begin{aligned}
\label{ec33}
\dot{H}+H^{2}=-\left( \frac{3}{2}\tilde{\Theta}-1\right) H^{2}\end{aligned}$$
where
$$\begin{aligned}
\label{ec34}
\tilde{\Theta}=\Theta _{eff}\left( 1-\frac{\gamma }{1-\alpha
}H^{2}\right) ,\end{aligned}$$
and $\Theta _{eff}$ is given by (\[ec5\]). So, the new effective scheme becomes
$$\begin{aligned}
\label{ec35}
\dot{\rho}+3\tilde{\Theta}H\rho =0,\end{aligned}$$
and the new bare scheme is
$$\begin{aligned}
\label{ec36}
\dot{\rho}+3\Theta H\rho =\tilde{\Gamma},\end{aligned}$$
where
$$\begin{aligned}
\label{ec37}
\tilde{\Gamma}=3\Theta \left[ 1-\left( 1-\alpha \right) \left(
1-\frac{3}{2} \beta \Theta \right) ^{-1}\left( 1-\frac{\gamma
}{1-\alpha }H^{2}\right) \right] H\rho .\end{aligned}$$
If we consider $\alpha =\left( 3/2\right) \beta $ we obtain
$$\begin{aligned}
\label{ec38}
\tilde{\Gamma}=3\alpha \Theta \left( \frac{1-\Theta }{1-\alpha
\Theta } \right) \left[ 1+\frac{\gamma }{\alpha
}\frac{H^{2}}{1-\Theta }\right] H\rho\end{aligned}$$
and at difference of $\Gamma \left( \Theta ;t\right) $ (\[ec23\]) we can see that only $\Theta =0\left( \omega
=-1\right) $ implies $\tilde{\Gamma}=0$. So, when the evolution is driven by dust we have not thermal equilibrium if we consider the quantum correction.
Now, from Eqs. (\[ec32\]-\[ec34\]) it is easy to obtain the following implicit solution for the Hubble parameter
$$\begin{aligned}
\label{ec39}
\frac{3}{2}\Theta _{eff}\left( t-t_{0}\right)
=\frac{1}{H}-\frac{1}{H_{0}}+ \frac{1}{2\delta }\ln \left(
\frac{H-\delta }{H_{0}-\delta }\right) \left( \frac{H_{0}+\delta
}{H+\delta }\right) ,\end{aligned}$$
where we are defined
$$\begin{aligned}
\label{ec40}
\delta =\sqrt{\frac{\left( 1-\alpha \right) }{\gamma }}.\end{aligned}$$
If we consider $\Theta _{eff}<0$, Eq. (\[ec40\]) can be written in the form
$$\begin{aligned}
\label{ec41}
\frac{1}{H}-\frac{3}{2}\left\vert \Theta _{eff}\right\vert \left(
t_{s}-t_{0}\right) =\frac{1}{2\delta }\ln \left( \frac{1+\delta
/H}{1-\delta /H}\right) \left( \frac{1-\delta /H_{0}}{1+\delta
/H_{0}}\right) ,\end{aligned}$$
where $t_{s}$ is given in (\[ec22\]) by doing $\Theta
_{eff}=-\left\vert \Theta _{eff}\right\vert $, and it is easy to verify when $t=t_{s}$ there is not future singularity for $H$ (in fact, when $t=t_{s}$ Eq. (\[ec41\]) is satisfied only for $H=H_{s}<H_{0}$). Finally, in accord to Eq. (\[ec32\]) we can visualize the auto-accelerated solution $\rho =0\Longrightarrow
H=\sqrt{\left( 1-\alpha \right) /\gamma }$ and in this case $\tilde{\Gamma}=0$ too and under phantom evolution we have $\tilde{\Gamma}\neq 0$.
Final remarks
=============
We have studied the entropic model given in (\[ec1\]-\[ec2\]) by doing a bare/effective description of the equation of state for the cosmic fluid. The effective description have showed to be an usual, that is, only by redefining the parameter of the equation of state we find a standard FLRW cosmology. In the bare case, we find sign changes in the term which accounts the state of thermal equilibrium, and only when the evolution is driven by dust or cosmological constant we have thermal equilibrium**.** We have showed that during a phantom evolution it is possible to reach the thermal equilibrium between the bulk (radiation) and the boundary (Hubble horizon) in the nearby of the singularity. Finally, by adding a quantum correction to the modified Friedmann ‘equations only cosmological constant can drive the universe on thermal equilibrium and the future singularity, which it is present in absence of the quantum corrections, is avoided. So, under the scope of the entropic cosmology is it possible to have a phantom-free evolution.
acknowledgements
================
This work was supported by FONDECYT Grant No. 1110076 (SL), VRIEA-DI-037.419/2012 (SL), Vice Rectoría de Investigación y Estudios Avanzados, Pontificia Universidad Católica de Valparaíso (SL) and DIUFRO DI12-0006, Dirección de Investigación y Desarrollo, Universidad de La Frontera (FP). One of us (SL) acknowlwdge the hospitality of Departamento de Ciencias Físicas de la Universidad de La Frontera where part of this work was made.
**References**
\[1\] E. P. Verlinde, JHEP **1104** (2011) 029.
\[2\] D. A. Easson, P. H. Frampton and G. F. Smoot, Phys. Lett. **B** **696**: 273 - 277, 2011.
\[3\] A. Kobakhidze, arXiv: 1108.4161 and Phys. Rev. D **83** (2011) 021502,
Shan Gao, Entropy **13** (2011) 936-948,
M. Chaichian, M. Oksanen and A. Tureanu, Phy. Lett. B **702** (2011) 419-421.
\[4\] T. S. Koivisto, D. F. Mota and M. Zumalacarregui, JCAP **1102** (2011) 027.
\[5\] E. Komatsu et al, Astrophys. J. Suppl. **192**:18, 2011.
\[6\] Yi-Fu Cai and E. M. Saridakis, Phys. Lett. B **697** (2011) 280-287.
|
---
abstract: |
In a previous paper \[P.H. Chavanis, Eur. Phys. J. Plus [**130**]{}, 130 (2015)\] we have introduced a new cosmological model that we called the logotropic model. This model involves a fundamental constant $\Lambda$ which is the counterpart of Einstein’s cosmological constant in the $\Lambda$CDM model. The logotropic model is able to account, without free parameter, for the constant surface density of the dark matter halos, for their mass-radius relation, and for the Tully-Fisher relation. In this paper, we explore other consequences of this model. By advocating a form of “strong cosmic coincidence” we predict that the present proportion of dark energy in the Universe is $\Omega_{\rm de,0}=e/(1+e)\simeq 0.731$ which is close to the observed value. We also remark that the surface density of dark matter halos and the surface density of the Universe are of the same order as the surface density of the electron. This makes a curious connection between cosmological and atomic scales. Using these coincidences, we can relate the Hubble constant, the electron mass and the electron charge to the cosmological constant. We also suggest that the famous numbers $137$ (fine-structure constant) and $123$ (logotropic constant) may actually represent the same thing. This could unify microphysics and cosmophysics. We study the thermodynamics of the logotropic model and find a connection to the Bekenstein-Hawking entropy of black holes if we assume that the logotropic fluid is made of particles of mass $m_{\Lambda}\sim
\hbar\sqrt{\Lambda}/c^2=2.08\times 10^{-33}\,
{\rm eV/c^2}$ (cosmons). In that case, the universality of the surface density of the dark matter halos may be related to a form of holographic principle (the fact that their entropy scales like their area). We use similar arguments to explain why the surface density of the electron and the surface density of the Universe are of the same order and justify the empirical Weinberg relation. Finally, we combine the results of our approach with the quantum Jeans instability theory to predict the order of magnitude of the mass of ultralight axions $m\sim 10^{-23}\,
{\rm eV/c^2}$ in the Bose-Einstein condensate dark matter paradigm.
author:
- 'Pierre-Henri Chavanis'
title: New predictions from the logotropic model
---
Introduction
============
The nature of dark matter and dark energy remains one of the greatest mysteries of modern cosmology. Dark matter is responsible for the flat rotation curves of the galaxies and dark energy is responsible for the accelerated expansion of the Universe. It is found that dark energy represents about $70\%$ of the energy content of the present Universe while the proportions of dark matter and baryonic matter are $25\%$ and $5\%$ respectively.
In a previous paper [@epjp] (see also [@lettre; @jcap]) we have introduced a new cosmological model that we called the logotropic model. In this model, there is no dark matter and no dark energy. There is just a single dark fluid. What we call “dark matter” actually corresponds to its rest-mass energy and what we call “dark energy” corresponds to its internal energy.[^1]
Our model does not contain any arbitrary parameter so that it is totally constrained. It involves a fundamental constant $\Lambda$ which is the counterpart of Einstein’s cosmological constant [@einsteincosmo] in the $\Lambda$CDM (cold dark matter) model and which turns out to have the same value. Still the logotropic model is fundamentally different from the $\Lambda$CDM model.
On the large (cosmological) scales, the logotropic model is indistinguishable from the $\Lambda$CDM model up to the present epoch [@epjp; @lettre; @jcap]. The two models will differ in the far future, in about $25\, {\rm Gyrs}$ years, after which the logotropic model will become phantom (the energy density will increase as the Universe expands) and present a Little Rip (the energy density and the scale factor will become infinite in infinite time) contrary to the $\Lambda$CDM model in which the energy density tends towards a constant (de Sitter era).
On the small (galactic) scales, the logotropic model is able to solve some of the problems encountered by the $\Lambda$CDM model [@epjp; @lettre]. In particular, it is able to account, without free parameter, for the constant surface density of the dark matter halos, for their mass-radius relation, and for the Tully-Fisher relation.
In this paper, we explore other consequences of this model. By advocating a form of “strong cosmic coincidence”, stating that the present value of the dark energy density $\rho_{\rm de, 0}$ is equal to the fundamental constant $\rho_\Lambda$ appearing in the logotropic model, we predict that the present proportion of dark energy in the Universe is $\Omega_{\rm
de,0}=e/(1+e)=0.731$ which is close to the observed value $0.691$ [@planck2016]. The consequences of this result, which implies that our epoch is very special in the history of the Universe, are intriguing and related to a form of anthropic cosmological principle [@barrow].
We also remark that the universal surface density of dark matter halos (found from the observations [@donato] and predicted by our model [@epjp; @lettre]) and the surface density of the Universe are of the same order of magnitude as the surface density of the electron. This makes a curious connection between cosmological and atomic scales. Exploiting this coincidence, we can relate the Hubble constant, the electron mass and the electron charge to the cosmological constant $\Lambda$. We also argue that the famous numbers $137$ (fine-structure constant) and $123$ (logotropic constant) may actually represents the same thing. This may be a hint for a theory of unification of microphysics and cosmophysics. Speculations are made in the Appendices to try to relate these interconnections to a form of holographic principle [@bousso] stating that the entropy of the electron, the entropy of dark matter halos, and the entropy of the Universe scales like their area as in the case of the entropy of black holes [@bekenstein; @hawking].
The logotropic model {#sec_lm}
====================
Unification of dark matter and dark energy
------------------------------------------
The Friedmann equations for a flat universe without cosmological constant are [@weinbergbook]: $$\frac{d\epsilon}{dt}+3\frac{\dot a}{a}(\epsilon+P)=0,\quad H^2=\left
(\frac{\dot a}{a}\right )^2=\frac{8\pi
G}{3c^2}\epsilon,
\label{lm1}$$ where $\epsilon(t)$ is the energy density of the Universe, $P(t)$ is the pressure, $a(t)$ is the scale factor, and $H=\dot a/a$ is the Hubble parameter.
For a relativistic fluid experiencing an adiabatic evolution such that $Td(s/\rho)=0$, the first law of thermodynamics reduces to [@weinbergbook]: $$d\epsilon=\frac{P+\epsilon}{\rho}d\rho,
\label{lm2}$$ where $\rho$ is the rest-mass density of the Universe. Combined with the equation of continuity (\[lm1\]), we get $$\frac{d\rho}{dt}+3\frac{\dot a}{a}\rho=0 \Rightarrow \rho=\frac{\rho_0}{a^3},
\label{lm3}$$ where $\rho_0$ is the present value of the rest-mass density (the present value of the scale factor is taken to be $a=1$). This equation, which expresses the conservation of the rest-mass, is valid for an arbitrary equation of state.
For an equation of state specified under the form $P=P(\rho)$, Eq. (\[lm2\]) can be integrated to obtain the relation between the energy density $\epsilon$ and the rest-mass density. We obtain [@epjp]: $$\epsilon=\rho c^2+\rho\int^{\rho}\frac{P(\rho')}{{\rho'}^2}\, d\rho'=\rho
c^2+u(\rho).
\label{lm4}$$ We note that $u(\rho)$ can be interpreted as an internal energy density [@epjp]. Therefore, the energy density $\epsilon$ is the sum of the rest-mass energy $\rho c^2$ and the internal energy $u(\rho)$.
The logotropic dark fluid {#sec_ldf}
-------------------------
We assume that the Universe is filled with a single dark fluid described by the logotropic equation of state [@epjp]: $$P=A\ln\left (\frac{\rho}{\rho_P}\right ),
\label{lm5}$$ where $\rho_P=c^5/\hbar G^2=5.16\times 10^{99}\, {\rm g\, m^{-3}}$ is the Planck density and $A$ is a new fundamental constant of physics, with the dimension of an energy density, which is the counterpart of the cosmological constant $\Lambda$ in the $\Lambda$CDM model (see below). Using Eqs. (\[lm4\]) and (\[lm5\]), the relation between the energy density and the rest-mass density is $$\epsilon=\rho c^2-A\ln \left (\frac{\rho}{\rho_P}\right )-A=\rho c^2+u(\rho).
\label{lm6}$$ The energy density is the sum of two terms: a rest-mass energy term $\rho c^2=\rho_0c^2/ a^{3}$ that mimics the energy density $\epsilon_{\rm m}$ of dark matter and an internal energy term $u(\rho)=-A\ln \left ({\rho}/{\rho_P}\right )-A
=-P(\rho)-A=3A\ln a-A\ln(\rho_0/\rho_P)-A$ that mimics the energy density $\epsilon_{\rm de}$ of dark energy. This decomposition leads to a natural, and physical, unification of dark matter and dark energy and elucidates their mysterious nature.
Since, in our model, the rest-mass energy of the dark fluid mimics dark matter, we identify $\rho_0c^2$ with the present energy density of dark matter. We thus set $\rho_0c^2=\Omega_{\rm m,0}\epsilon_0$, where $\epsilon_0/c^2={3H_0^2}/{8\pi G}$ is the present energy density of the Universe and $\Omega_{\rm m,0}$ is the present fraction of dark matter (we also include baryonic matter). As a result, the present internal energy of the dark fluid, $u_0=\epsilon_0-\rho_0c^2$, is identified with the present dark energy density $\epsilon_{\rm de,0}=\Omega_{\rm de,0}\epsilon_0$ where $\Omega_{\rm de,0}=1-\Omega_{\rm m,0}$ is the present fraction of dark energy. Applying Eq. (\[lm6\]) at the present epoch ($a=1$), we obtain the identity $$A=\frac{\epsilon_{\rm de,0}}{\ln\left(\frac{\rho_Pc^2}{\epsilon_{\rm
de,0}}\right
)+\ln\left (\frac{\Omega_{\rm de,0}}{1-\Omega_{\rm de,0}}\right )-1}.
\label{lm7}$$ At that stage, we can have two points of view. We can consider that this equation determines the constant $A$ as a function of $\epsilon_{0}$ and $\Omega_{\rm de,0}$ that are both obtained from the observations [@planck2016]. This allows us to determine the value of $A$. This is the point of view that we have adopted in our previous papers [@epjp; @lettre] and that we adopt in Sec. \[sec\_B\] below. However, in the following section, we present another point of view leading to an intriguing result.
Strong cosmic coincidence and prediction of $\Omega_{\rm de,0}$
---------------------------------------------------------------
Let us recall that, in our model, $A$ is considered as a fundamental constant whose value is fixed by Nature. As a result, Eq. (\[lm7\]) relates $\Omega_{\rm de,0}$ to $\epsilon_{0}$ for a given value of $A$. A priori, we have two unknowns for just one equation. However, we can obtain the value of $\Omega_{\rm de,0}$ by the following argument.
We can always write the constant $A$ under the form $$A=\frac{\rho_{\Lambda}c^2}{\ln\left(\frac{\rho_P}{\rho_{\Lambda}}\right
)}.
\label{lm8}$$ This is just a change of notation. Eq. (\[lm8\]) defines a new constant, the cosmological density $\rho_{\Lambda}$, in place of $A$. From the cosmological density $\rho_{\Lambda}$, we can define an effective cosmological constant $\Lambda$ by[^2] $$\rho_{\Lambda}=\frac{\Lambda}{8\pi G}.
\label{lm9}$$ Again this is just a change of notation. Therefore, the fundamental constant of our model is either $A$, $\rho_{\Lambda}$ or $\Lambda$ (equivalently). We now advocate a form of “strong cosmic coincidence”. We assume that the present value of the dark energy density is equal to $\rho_{\Lambda}c^2$, i.e., $$\epsilon_{\rm de,0}=\rho_{\Lambda}c^2.
\label{lm10}$$ Since, in the $\Lambda$CDM model, $\epsilon_{\rm de}$ is a constant usually measured at the present epoch our postulate implies that $\rho_{\Lambda}c^2$ coincides with the cosmological density in the $\Lambda$CDM model and that $\Lambda$, as defined by Eq. (\[lm9\]), coincides with the ordinary cosmological constant. This is why we have used the same notations. Now, comparing Eqs. (\[lm7\]), (\[lm8\]) and (\[lm10\]) we obtain $\ln\left
\lbrack \Omega_{\rm de,0}/(1-\Omega_{\rm de,0})\right \rbrack-1=0$ which determines $\Omega_{\rm de,0}$. We find that $$\Omega_{\rm de,0}^{\rm th}=\frac{e}{1+e}\simeq 0.731
\label{lm11}$$ which is close to the observed value $\Omega_{\rm de,0}^{\rm
obs}=0.691$ [@planck2016]. This agreement is puzzling. It relies on the “strong cosmic coincidence” of Eq. (\[lm10\]) implying that our epoch is very special. This is a form of anthropic cosmological principle [@barrow]. This may also correspond to a fixed point of our model. In order to avoid philosophical issues, in the following, we adopt the more conventional point of view discussed at the end of Sec. \[sec\_ldf\].
The logotropic constant $B$ {#sec_B}
---------------------------
We can rewrite Eq. (\[lm8\]) as $$A=B\rho_{\Lambda}c^2\qquad {\rm
with}\qquad B=\frac{1}{\ln\left({\rho_P}/{\rho_{\Lambda}}
\right
)}.
\label{lm12}$$ Again, this is just a change of notation defining the dimensionless number $B$. We shall call it the logotropic constant since it is equal to the inverse of the logarithm of the cosmological density normalized by the Planck density (see Appendix \[sec\_const\]). We note that $A$ can be expressed in terms of $B$ (see below) so that the fundamental constant of our model is either $A$, $\rho_{\Lambda}$, $\Lambda$, or $B$. In the following, we shall express all the results in terms of $B$. For example, the relation (\[lm6\]) between the energy density and the scale factor can be rewritten as $$\frac{\epsilon}{\epsilon_0}=\frac{\Omega_{\rm
m,0}}{a^3}+(1-\Omega_{\rm m,0})(1+3B\ln
a).
\label{lm13}$$ Combined with the Friedmann equation (\[lm1\]) this equation determines the evolution of the scale factor $a(t)$ of the Universe in the logotropic model. This evolution has been studied in detail in [@epjp; @lettre; @jcap].
[*Remark:*]{} Considering Eq. (\[lm13\]), we see that the $\Lambda$CDM model is recovered for $B=0$. According to Eq. (\[lm12\]) this implies that $\rho_P\rightarrow +\infty$, i.e., $\hbar\rightarrow 0$. Therefore, the $\Lambda$CDM model corresponds to the semiclassical limit of the logotropic model. The fact that $B$ is intrinsically nonzero implies that quantum mechanics ($\hbar\neq 0$) plays some role in our model in addition to general relativity. This may suggest a link with a theory of quantum gravity.
The value of $B$ from the observations {#sec_Bobs}
--------------------------------------
The fundamental constant ($A$, $\rho_{\Lambda}$, $\Lambda$, or $B$) appearing in our model can be determined from the observations by using Eq. (\[lm7\]). We take $\Omega_{\rm de,0}=0.6911$ and $H_0=2.195\times 10^{-18}\, {\rm
s^{-1}}$ [@planck2016]. This implies $\epsilon_0/c^2=3H_0^2/8\pi
G=8.62\times 10^{-24}\, {\rm g\, m^{-3}}$ and $\epsilon_{\rm
de,0}/c^2=\Omega_{\rm
de,0}\epsilon_0/c^2=5.96\times 10^{-24}\, {\rm g\, m^{-3}}$. Since $\ln\left
\lbrack \Omega_{\rm de,0}/(1-\Omega_{\rm de,0})\right \rbrack-1=-0.195$ is small as compared to $\ln(\rho_Pc^2/\epsilon_{\rm de,0})=283$, we can write in very good approximation $A$ as in Eq. (\[lm8\]) with $\rho_{\Lambda}\simeq
\epsilon_{\rm de,0}/c^2$ as in Eq. (\[lm10\]). Therefore, $$\rho_{\Lambda}=\frac{3\Omega_{\rm
de,0}H_0^2}{8\pi G}=5.96\times 10^{-24}\, {\rm g\, m^{-3}}
\label{lm14a}$$ and $$\Lambda= 3\Omega_{\rm
de,0}H_0^2=1.00\times 10^{-35}\, {\rm s^{-2}}
\label{lm14b}$$ are approximately equal to the cosmological density and to the cosmological constant in the $\Lambda$CDM model. From Eq. (\[lm12\]) we get $$B=\frac{1}{\ln(\rho_P/\rho_{\Lambda})}\simeq
\frac{1}{123\ln(10)}\simeq 3.53\times 10^{-3}.
\label{lm15}$$ As discussed in our previous papers [@epjp; @lettre; @jcap], $B$ is essentially the inverse of the famous number $123$ (see Appendix \[sec\_const\]). Finally, $$A=B\,\rho_{\Lambda}c^2=1.89\times 10^{-9}
\, {\rm g}\, {\rm m}^{-1}\, {\rm s}^{-2}.
\label{lm16}$$
From now on, we shall view $B$ given by Eq. (\[lm15\]) as the fundamental constant of the theory. Therefore, everything should be expressed in terms of $B$ and the other fundamental constants of physics defining the Planck scales. First, we have $$\frac{\rho_{\Lambda}}{\rho_{P}}=\frac{G\hbar\Lambda}{8\pi
c^5}=e^{-1/B}=1.16\times 10^{-123}.
\label{lm18}$$ Then, $$\frac{A}{\rho_{P}c^2}=Be^{-1/B}=4.08\times 10^{-126}.
\label{lm17}$$ The logotropic equation of state (\[lm5\]) can be written as $P/\rho_Pc^2=Be^{-1/B}\ln(\rho/\rho_P)$. Using Eq. (\[lm10\]) and $\epsilon_{\rm
de,0}=\Omega_{\rm de,0}\epsilon_0$, we get $$\frac{\epsilon_0}{\rho_{P}c^2}=\frac{1}{\Omega_{\rm de,0}}e^{-1/B}=1.67\times
10^{-123}.
\label{lm19}$$ Finally, using Eq. (\[lm1\]), $$t_P H_0=\left (\frac{8\pi}{3\Omega_{\rm de,0}}\right
)^{1/2}e^{-1/2B}=1.18\times 10^{-61},
\label{lm20}$$ where $t_P=(\hbar G/c^5)^{1/2}=5.391\times 10^{-44}\, {\rm s}$ is the Planck time. In the last two expressions, we can either consider that $\Omega_{\rm de,0}$ is “predicted” by Eq. (\[lm11\]) or take its measured value. To the order of accuracy that we consider, this does not change the numerical values.
Previous predictions of the logotropic model
============================================
The interest of the logotropic model becomes apparent when it is applied to dark matter halos [@epjp; @lettre]. We assume that dark matter halos are described by the logotropic equation of state of Eq. (\[lm5\]) with $A=1.89\times 10^{-9} \, {\rm g}\, {\rm m}^{-1}\, {\rm
s}^{-2}$ (or $B=3.53\times 10^{-3}$). At the galactic scale, we can use Newtonian gravity.
Surface density of dark matter halos
------------------------------------
It is an empirical evidence that the surface density of galaxies has the same value $$\Sigma_0^{\rm obs}\equiv \rho_0 r_h\simeq 295\, {\rm g\, m^{-2}}\simeq 141\,
M_{\odot}/{\rm pc^2}
\label{lm21}$$ even if their sizes and masses vary by several orders of magnitude (up to $14$ orders of magnitude in luminosity) [@donato]. Here $\rho_0$ is the central density and $r_h$ is the halo radius at which the density has decreased by a factor of $4$. The logotropic model predicts that the surface density of the dark matter halos is the same for all the halos (because $A$ is a universal constant) and that it is given by [@epjp; @lettre]: $$\Sigma_0^{\rm th}=\left (\frac{A}{4\pi G}\right )^{1/2}\xi_h= \left
(\frac{B}{32}\right
)^{1/2}\frac{\xi_h}{\pi}\frac{c\sqrt{\Lambda}}{G},
\label{lm22}$$ where $\xi_h=5.8458...$ is a pure number arising from the Lane-Emden equation of index $n=-1$ expressing the condition of hydrostatic equilibrium of logotropic spheres.[^3] Numerically, $$\Sigma_0^{\rm th}= 278\, {\rm g\,
m^{-2}}\simeq 133\,
M_{\odot}/{\rm pc^2},
\label{lm23}$$ which is very close to the observational value (\[lm21\]). The fact that the surface density of dark matter halos is determined by the effective cosmological constant $\Lambda$ (usually related to the dark energy) tends to confirm that dark matter and dark energy are just two manifestations of the [*same*]{} dark fluid, as we have assumed in our model.
[*Remark:*]{} The dimensional term $c\sqrt{\Lambda}/G$ in Eq. (\[lm22\]) can be interpreted as representing the surface density of the Universe (see Appendix \[sec\_w\]). We note that this term alone, $c\sqrt{\Lambda}/G=14200\, {\rm g\,
m^{-2}}=6800 M_{\odot}/{\rm pc}^2$, is too large to account precisely for the surface density of dark matter halos so that the prefactor $(B/32)^{1/2}(\xi_h/\pi)=0.01955$ is necessary to reduce this number. It is interesting to remark that the term $c\sqrt{\Lambda}/G$ arises from classical general relativity while the prefactor $\propto B^{1/2}$ has a quantum origin as discussed at the end of Sec. \[sec\_B\]. Actually, we will see that it is related to the fine-structure constant $\alpha$ \[see Eq. (\[lm30\]) below\].
Mass-radius relation
--------------------
There are interesting consequences of the preceding result. For logotropic halos, the mass of the halos calculated at the halo radius $r_h$ is given by [@epjp; @lettre]: $$M_h=1.49\Sigma_0 r_h^2.
\label{lm24a}$$ This determines the mass-radius relation of dark matter-halos. On the other hand, the circular velocity at the halo radius is $v_h^2=GM_h/r_h=1.49\Sigma_0
G r_h$. Since the surface density of the dark matter halos is constant, we obtain $$\frac{M_h}{M_{\odot}}=198 \left (\frac{r_h}{{\rm pc}}\right )^2,\qquad
\left (\frac{v_h}{{\rm
km}\, {\rm s}^{-1}}\right )^2=0.852\, \frac{r_h}{\rm pc}.
\label{lm24}$$ The scalings $M_h\propto r_h^2$ and $v_h^2\propto
r_h$ (and also the prefactors) are consistent with the observations.
The Tully-Fisher relation
-------------------------
Combining the previous equations, the logotropic model leads to the Tully-Fisher [@tf] relation $v_h^4\propto M_h$ or, more precisely, $$\left (\frac{M_b}{v_h^4}\right )^{\rm th}=\frac{f_b}{1.49\Sigma_0^{\rm th}
G^2}=46.4 M_{\odot}{\rm km}^{-4}{\rm s}^4,
\label{lm25}$$ where $f_b=M_b/M_h\sim 0.17$ is the cosmic baryon fraction [@mcgaugh]. The predicted value from Eq. (\[lm25\]) is close to the observed one $\left ({M_b}/{v_h^4}\right )^{\rm obs}=47\pm 6
M_{\odot}{\rm
km}^{-4}{\rm
s}^4$ [@mcgaugh].
[*Remark:*]{} The Tully-Fisher relation is sometimes justified by the MOND (Modification of Newtonian dynamics) theory [@mond] which predicts a relation of the form $v_h^4=Ga_0
M_b$ between the asymptotic circular velocity and the baryon mass, where $a_0$ is a critical acceleration. Our results imply $a_0^{\rm th}=1.62\times
10^{-10}\, {\rm m}\, {\rm s}^{-2}$ which is close to the value $a_0^{\rm obs}=(1.3\pm 0.3)\times
10^{-10}\, {\rm m}\, {\rm s}^{-2}$ obtained from the observations [@mcgaugh]. Combining Eqs. (\[lm24a\]) and (\[lm25\]), we first get $a_0^{\rm
th}=(1.49/f_b)\Sigma_0^{\rm th}G=GM_h/(f_br_h^2)$ which shows that $a_0$ can be interpreted as the surface gravity of the galaxies $G\Sigma_0$ (which corresponds to Newton’s acceleration $GM_h/r_h^2$) or as the surface density of the Universe (see Appendix \[sec\_abh\]). Then, using Eqs. (\[lm14b\]) and (\[lm22\]), we obtain $a_0^{\rm
th}=({1.49}/{f_b})({B}/{32})^{1/2}({\xi_h}/{\pi})c\sqrt{\Lambda}\simeq H_0
c/4$ which explains why $a_0$ is of the order of $H_0 c$. We emphasize, however, that we do not use the MOND theory in our approach and that the logotropic model assumes the existence of a dark fluid.
The mass $M_{300}$
------------------
The logotropic equation of state also explains the observation of Strigari [*et al.*]{} [@strigari] that all the dwarf spheroidals (dSphs) of the Milky Way have the same total dark matter mass $M_{300}$ contained within a radius $r_u=300\, {\rm pc}$, namely $M_{300}^{\rm obs}\simeq 10^7\, M_{\odot}$ The logotropic model predicts the value [@epjp; @lettre]: $$M_{300}^{\rm th}=\frac{4\pi \Sigma_0^{\rm th} r_u^2}{\xi_h\sqrt{2}}=1.82\times
10^{7}\, M_{\odot},
\label{lm26}$$ which is in very good agreement with the observational value.
A curious connection between atomic and cosmological scales {#sec_curious}
===========================================================
The surface density of the electron {#sec_sde}
-----------------------------------
The classical radius of the electron $r_e$ can be obtained qualitatively by writing that the electrostatic energy of the electron, $e^2/r_e$, is equal to its rest-mass energy $m_e c^2$. Recalling the value of the charge of the electron $e=4.80\times 10^{-13}\, {\rm
g^{1/2}\, m^{3/2}\, s^{-1}}$ and its mass $m_e=9.11\times 10^{-28}\, {\rm g}$, we obtain $r_e=e^2/m_ec^2=2.82\times 10^{-15}\, {\rm m}$. As a result, the surface density of the electron is[^4] $$\Sigma_e=\frac{m_e}{r_e^2}=\frac{m_e^3c^4}{e^4}=115\, {\rm g/m^2}= 54.9\,
M_{\odot}/{\rm pc^2},
\label{lm27}$$ which is of the same order of magnitude as the surface density of dark matter halos from Eq. (\[lm21\]). This coincidence is amazing in view of the different scales (atomic versus cosmological) involved. More precisely, we find $\Sigma_e=\sigma\Sigma_0^{\rm th}$ with $\sigma\simeq 0.413$. Of course, the value of $\sigma$ depends on the precise manner used to define the surface density of the electron, or its radius, but the important point is that this number is of order unity.
Relation between $\alpha$ and $B$ {#sec_Ba}
---------------------------------
By matching the two formulae (\[lm22\]) and (\[lm27\]), writing $\Sigma_e=\sigma\Sigma_0^{\rm th}$, we get $$\Lambda=\frac{32\pi^2}{B\xi_h^2\sigma^2}
\frac{m_e^6c^6G^2}{e^8}=\frac{32\pi^2}{B\xi_h^2\sigma^2\alpha^4}
\frac{m_e^6c^2G^2}{\hbar^4},
\label{lm28}$$ where we have introduced the fine-structure constant $\alpha$ in the second equality (see Appendix \[sec\_const\]). This expression provides a curious relation between the cosmological constant, the mass of the electron and its charge. This relation is similar to Weinberg’s empirical relation (see Appendix \[sec\_w\]) which can be written as \[combining Eqs. (\[lm14b\]) and (\[w4\])\] $$\Lambda=192\pi^2\mu^2\Omega_{\rm de,0}\frac{m_e^6c^6G^2}{e^8},
\label{w4b}$$ where $\mu\simeq 3.42$. Note that in our formula (\[lm28\]), $\Lambda$ appears two times: on the left hand side and in $B$ (which depends logarithmically on $\Lambda$). This will have important consequences in the following.
Böhmer and Harko [@bhLambda], by a completely different approach, found a similar relation[^5] $$\Lambda=\nu \frac{\hbar^2 G^2 m_e^6 c^8}{e^{12}}=\frac{\nu}{\alpha^6} \frac{G^2
m_e^6 c^2}{\hbar^{4}},
\label{lm29}$$ where $\nu\simeq 0.816$ is of order unity. Their result can be obtained as follows. They first introduce a minimum mass $m_{\Lambda}\sim\hbar\sqrt{\Lambda}/c^2$ interpreted as being the mass of the elementary particle of dark energy, called the cosmon. Then, they define a radius $R$ by the relation $m_{\Lambda}\sim \rho_{\Lambda} R^3$ where $\rho_{\Lambda}= \Lambda/8\pi G$ is the cosmological density considered as being the lowest density in the Universe. Finally, they remark that $R$ has typically the same value as the classical radius of the electron $r_e=e^2/m_ec^2$. Matching $R$ and $r_e$ leads to the scaling of Eq. (\[lm29\]). We have then added a prefactor $\nu$ and adjusted its value in order to exactly obtain the measured value of the cosmological constant [@planck2016]. Since the approach of Böhmer and Harko [@bhLambda] is essentially qualitative, and depends on the precise manner used to define the radius of the electron, their result can be at best valid up to a constant of order unity.
We would like now to compare the estimates from Eqs. (\[lm28\]) and (\[lm29\]). At that stage, we can have two points of view. If we consider that comparing the prefactors is meaningless because our approach can only provide “rough” orders of magnitude, we conclude that Eqs. (\[lm28\]) and (\[lm29\]) are equivalent, and that they are also equivalent to Weinberg’s empirical relation (\[w4\]). Alternatively, if we take the prefactors seriously into account (in particular the presence of $B$ which depends on $\Lambda$) and match the formulae (\[lm28\]) and (\[lm29\]), we find an interesting relation between the fine-structure constant $\alpha$ and the logotropic constant $B$: $$\alpha=\left (\frac{\nu}{32}\right
)^{1/2}\frac{\xi_h\sigma}{\pi}\sqrt{B}\simeq 0.123 \sqrt{B}.
\label{lm30}$$ Therefore, the fine-structure constant (electron charge normalized by the Planck charge) is determined by the logotropic constant $B$ (cosmological density normalized by the Planck density) by a relation of the form $\alpha\propto B^{1/2}$. This makes a connection between atomic scales and cosmological scales. This also suggests that the famous numbers $137$ and $123$ (see Appendix \[sec\_const\]) are related to each other, or may even represent the same thing. From Eq. (\[lm30\]), we have[^6] $$137\simeq 12.3 \sqrt{123}.
\label{lm31}$$
[*Remark:*]{} the logotropic constant $B$ is related to the effective cosmological constant $\Lambda$ by \[see Eq. (\[lm18\])\] $$B=\frac{1}{\ln\left (\frac{8\pi c^5}{G\hbar\Lambda}\right )}.
\label{lm31b}$$ Using Eqs. (\[lm30\]) and (\[lm31b\]), we can express the fine-structure constant $\alpha$ as a function of the effective cosmological constant $\Lambda$ or, using Eq. (\[lm20\]), as a function of the age of the Universe $t_{\Lambda}=1/H_0$ as $$\alpha=\frac{0.123}{\ln\left (\frac{8\pi c^5}{G\hbar\Lambda}\right
)^{1/2}}=\frac{0.123}{\sqrt{2}\ln\left \lbrack \left (\frac{8\pi}{3\Omega_{\rm
de,0}}\right )^{1/2}\frac{t_{\Lambda}}{t_P}\right\rbrack^{1/2}}.
\label{lm31c}$$ We emphasize the scaling $1/\alpha\propto (\ln t_{\Lambda})^{1/2}$. It is interesting to note that similar relations have been introduced in the past from pure numerology (see [@kragh], P. 428). These relations suggest that the fundamental constants may change with time as argued by Dirac [@dirac1; @dirac2].
The mass and the charge of the electron in terms of $B$
-------------------------------------------------------
Using Eqs. (\[lm9\]), (\[lm18\]), (\[lm28\]) and (\[lm30\]), we find that the mass and the charge of the electron are determined by the logotropic constant $B$ according to $$\begin{aligned}
\frac{m_e}{M_P}=\left (\frac{8\pi}{\nu}\right )^{1/6}\left
(\frac{\nu}{32}\right
)^{1/2}\frac{\xi_h\sigma}{\pi}\sqrt{B}e^{-1/(6B)}\nonumber\\
=0.217\sqrt{B}e^{
-1/(6B) } =4.18\times 10^{-23},
\label{lm32}\end{aligned}$$ $$\frac{e^2}{q_P^2}=\left (\frac{\nu}{32}\right
)^{1/2}\frac{\xi_h\sigma}{\pi}\sqrt{B}=0.123\sqrt{B}=7.29\times 10^{-3},
\label{lm33}$$ where $M_P=(\hbar c/G)^{1/2}=2.18\times 10^{-5}\, {\rm g}$ is the Planck mass and $q_P=(\hbar c)^{1/2}=5.62\times 10^{-12}\, {\rm
g^{1/2}\, m^{3/2}\, s^{-1}}$ is the Planck charge. These relations suggest that the mass and the charge of the electron (atomic scales) are determined by the effective cosmological constant $\Lambda$ or $B$ (cosmological scales). We emphasize the presence of the exponential factor $e^{-1/(6B)}$ in Eq. (\[lm32\]) explaining why the electron mass is much smaller than the Planck mass while the electron charge is comparable to the Planck charge.
A prediction of $B$
-------------------
If we match Eqs. (\[lm22\]) and (\[w3\]), or equivalently Eqs. (\[lm28\]) and (\[w4b\]), we obtain $$B^{\rm app}=\frac{1}{6\lambda^2\xi_h^2\Omega_{\rm
de,0}}.
\label{w5}$$ Taking $\lambda^{\rm app}=1$ (since we cannot predict its value) and $\Omega_{\rm de,0}^{\rm th}=e/(1+e)$ \[see Eq. (\[lm11\])\], we get $B^{\rm
app}=6.67\times 10^{-3}$ instead of $B=3.53\times
10^{-3}$. We recall that the value of $B$ was obtained in Sec. \[sec\_Bobs\] from the observations. On the other hand, Eq. (\[w5\]) gives the correct order of magnitude of $B$ without any reference to observations, up to a dimensionless constant $\lambda\simeq 1.41$ of order unity. Considering that $B$ is predicted by Eq. (\[w5\]) implies that we can predict the values of $\Lambda$, $H_0$, $\alpha$, $m_e$ and $e$ without reference to observations, up to dimensionless constants $\lambda\simeq 1.41$, $\nu\simeq 0.816$ and $\sigma\simeq 0.413$ of order unity. We note, however, that even if these dimensionless constants ($\lambda$, $\nu$, $\sigma$) are of order unity, their precise values are of importance since $B$ usually appears in exponentials like in Eqs. (\[lm18\]), (\[lm20\]) and (\[lm32\]).
Conclusion
==========
In this paper, we have developed the logotropic model introduced in [@epjp; @lettre]. In this model, dark matter corresponds to the rest mass energy of a dark fluid and dark energy corresponds to its internal energy. The $\Lambda$CDM model may be interpreted as the semiclassical limit $\hbar\rightarrow 0$ of the logotropic model. We have first recalled that the logotropic model is able to predict (without free parameter) the universal value of the surface density of dark matter halos $\Sigma_0$, their mass-radius relation $M_h-r_h$, the Tully-Fisher relation $M_b\sim v_h^4$ and the value of the mass $M_{300}$ of dSphs. Then, we have argued that it also predicts the value of the present fraction of dark energy $\Omega_{\rm de,0}$. This arises from a sort of “strong cosmic coincidence” but this could also correspond to a fixed point of the model. Finally, we have observed that the surface density of the dark matter halos $\Sigma_0$ is of the same order as the surface density of the Universe $\Sigma_\Lambda$ and of the same order as the surface density of the electron $\Sigma_e$. This makes an empirical connection between atomic physics and cosmology. From this connection, we have obtained a relation between the fine-structure constant $\alpha\sim
1/137$ and the logotropic constant $B\sim 1/123$. We have also expressed the mass $m_e$ and the charge $-e$ of the electron as a function of $B$ (or as a function of the effective cosmological constant $\Lambda$). Finally, we have obtained a prediction of the order of magnitude of $B$ independent from the observations. In a sense, our approach which expresses the mass and the charge of the electron in terms of the cosmological constant is a continuation of the program initiated by Eddington [@eddington] in his quest for a ‘[*Fundamental Theory*]{}’ of the physical world in which the basic interaction strengths and elementary particle masses would be prediced entirely combinatorically by simple counting processes [@barrow]. In the Appendices, we try to relate these interconnections to a form of holographic principle [@bousso] (of course not known at the time of Eddington) stating that the entropy of the electron, of dark matter halos, and of the Universe scales like their area as in the case of black holes [@bekenstein; @hawking].
This paper has demonstrated that physics is full of “magic” and mysterious relations that are still not fully understood (one of them being the empirical Weinberg relation). Hopefully, a contribution of this paper is to reveal these “mysteries” and propose some tracks so as to induce further research towards their elucidation.
The constants $\alpha$ and $B$ {#sec_const}
==============================
There are two famous numbers in physics, $137$ and $123$, which respectively apply to atomic and cosmological scales.
At the atomic level, the fine-structure constant $\alpha$, also known as Sommerfeld’s constant, is a dimensionless physical constant characterizing the strength of the electromagnetic interaction between elementary charged particles. Its value is $$\alpha=\frac{e^2}{\hbar c}=\frac{e^2}{q_P^2}\simeq \frac{1}{137}\simeq
7.30\times 10^{-3}.
\label{const1}$$ It can be seen as the square of the charge $e=4.80\times 10^{-13}\, {\rm
g^{1/2}\, m^{3/2}\, s^{-1}}$ of the electron normalized by the Planck charge $q_P=(\hbar c)^{1/2}=5.62\times 10^{-12}\, {\rm
g^{1/2}\, m^{3/2}\, s^{-1}}$. The quantum theory does not predict its value. The number $1/\alpha\simeq 137$ intrigued a lot of famous researchers including Eddington, Pauli, Born, Hawking and Feynman among others [@kragh]. Feynman writes [@feynman]: [*It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the “hand of God” wrote that number, and “we don’t know how He pushed his pencil.”* ]{}
At the cosmological level, there is another famous number $$B=\frac{1}{\ln(\rho_P/\rho_{\Lambda})}\simeq
\frac{1}{123\ln(10)}\simeq 3.53\times 10^{-3}.
\label{const2}$$ It can be seen as the logarithm of the cosmological - or dark energy - density $\rho_{\Lambda}=\Lambda/8\pi G=5.96\times 10^{-24}\, {\rm g\, m^{-3}}$ (where $\Lambda=1.00\times 10^{-35}\, {\rm s^{-2}}$ is the cosmological constant), normalized by the Planck density $\rho_P=c^5/\hbar G^2=5.16\times
10^{99}\, {\rm g\, m^{-3}}$. This number appeared in connection to the so-called cosmological constant problem [@weinbergcosmo; @paddycosmo], i.e., the fact that there is a difference of $123$ orders of magnitude between the Planck density and the cosmological density ($\rho_P/\rho_{\Lambda}\sim 10^{123}$) interpreted as the vacuum energy.
We have suggested in this paper that the two dimensionless constants $\alpha$ and $B$, or the two numbers $137$ and $123$, are related to each other \[see Eqs. (\[lm30\]) and (\[lm31\])\] and that, in some sense, they correspond to the same thing. If this idea is correct, it would yield a fascinating connection between atomic and cosmic physics.
Surface density of the Universe, surface density of the electron and Weinberg’s empirical relation {#sec_w}
==================================================================================================
Using qualitative arguments, let us determine the surface density of the Universe. The Hubble time ($\sim$ age of the Universe) is $t_{\Lambda}={1}/{H_0}=14.4$ billion years. The Hubble radius ($\sim$ radius of the visible Universe) is $R_{\Lambda}=ct_{\Lambda}=c/H_0=1.37\times 10^{26}\, {\rm m}$. The present density of the Universe is ${\epsilon_0}/{c^2}={3H_0^2}/{8\pi
G}=8.62\times 10^{-24}\, {\rm g\, m^{-3}}$. The Hubble mass ($\sim$ mass of the Universe) is $M_{\Lambda}=({4}/{3})\pi ({\epsilon_0}/{c^2})
R_{\Lambda}^3={c^3}/{2GH_0}=9.20\times 10^{55}\,
{\rm g}$. Combining these relations, we find that the surface density of the Universe is $$\Sigma_{\Lambda}=\frac{M_{\Lambda}}{4\pi R_{\Lambda}^2}=\frac{cH_0}{8\pi
G}=392\, {\rm g\, m^{-2}}=188\,
M_{\odot}/{\rm pc^2}.
\label{w2}$$ It can be written as $\Sigma_{\Lambda}=cH_0/\kappa c^4$ where $\kappa=8\pi
G/c^4$ is Einstein’s gravitational constant (which includes the $8\pi$ factor). Using Eq. (\[lm14b\]), we obtain $$\Sigma_{\Lambda}=\frac{1}{8\pi\sqrt{3\Omega_{\rm
de,0}}}\frac{c\sqrt{\Lambda}}{G}.
\label{w3}$$ This relation shows that the surface density of the Universe provides the correct scale for the surface density of dark matter halos \[see Eq. (\[lm22\])\]. We have $\Sigma_{\Lambda}=\lambda \Sigma_0^{\rm th}$ with $\lambda\simeq
1.41$.
Therefore, the surface density of the Universe is of the same order as the surface density of the dark matter halos which is also of the same order as the surface density of the electron (as we have previously observed). We have $\Sigma_{\Lambda}=\mu \Sigma_e$ with $\mu=\lambda/\sigma\simeq 3.42$. Matching Eqs. (\[lm27\]) and (\[w2\]), we get $$m_e=\left (\frac{e^4H_0}{8\pi\mu Gc^3}\right
)^{1/3}.
\label{w4}$$ This relation expresses the mass of the electron as a function of its charge and the Hubble constant. This mysterious relation is mentioned in the book of Weinberg [@weinbergbook] where it is obtained from purely dimensional arguments.[^7] He observes that the term in the right hand side of Eq. (\[w4\]) has the dimension of a mass and that this mass, $1.37\times 10^{-27}\, {\rm g}$ (with $\mu^{\rm app}=1$), is of the order of the mass of the electron. The fact that relation (\[w4\]) expresses the commensurability of the surface density of the Universe and the surface density of the electron, as we observe here, may help elucidating its physical meaning (see Appendix \[sec\_post\]).
[*Remark:*]{} If the dark matter halos resulted from the balance between the gravitational attraction and the repulsion due to the dark energy, they would have a typical density $M_h/r_h^3\sim \rho_{\Lambda}$. Actually, such an equilibrium is unstable as is well-known in the case of the Einstein static Universe. Therefore, the radius of dark matter halos must satisfy the constraint $r_h<(M_h/\rho_{\Lambda})^{1/3}$. Now, we have seen that their mass-radius relation scales as $M_h\sim (c\sqrt{\Lambda}/G)r_h^2$. We then find that the constraint $r_h<(M_h/\rho_{\Lambda})^{1/3}$ is satisfied provided that $M_h<c^3/G\sqrt{\Lambda}$. Since the upper bound is of the order of the mass of the Universe, $M_{\Lambda}\sim c^3/G\sqrt{\Lambda}$, we conclude that the size of the dark matter halos is always much smaller than the critical size $(r_h)_{\rm
crit}=(M_h/\rho_{\Lambda})^{1/3}$ as required for stability reasons.
Analogy with black hole thermodynamics {#sec_abh}
======================================
Black hole entropy {#sec_abha}
------------------
The Bekenstein-Hawking [@bekenstein; @hawking] entropy of a Schwarzschild black hole is given by $$S_{\rm BH}=\frac{1}{4}k_B \frac{A}{l_P^2}=\frac{k_B \pi c^3R^2}{G\hbar},
\label{bh1}$$ where $A=4\pi R^2$ is the area of the event horizon of the black hole and $l_P=(G\hbar/c^3)^{1/2}=1.62\times 10^{−35}\, {\rm m}$ is the Planck length. The radius of a Schwarzschild black hole is connected to its mass by $$R=\frac{2GM}{c^2}.
\label{bh2}$$ The Hawking temperature [@hawking] of a Schwarzschild black hole is $$k_B T=\frac{\hbar c^3}{8\pi GM}=\frac{\hbar c}{4\pi R}.
\label{bh3}$$ The black hole entropy (\[bh1\]) can be obtained from the Hawking temperature (\[bh3\]) by using the thermodynamic relation $T^{-1}=dS_{\rm BH}/d(Mc^2)$. If we consider a Planck black hole of radius $l_P$ and mass $M_P$, we find that its temperature is of the order of the Planck temperature $T_P=M_P
c^2/k_B=1.42\times 10^{32}\, {\rm K}$ and its entropy $S_{\rm BH}/k_B\sim 1$.
Analogy between the Universe and a black hole {#sec_abhb}
---------------------------------------------
Using the results of Appendix \[sec\_w\], we note that the radius of the Universe is related to its mass by $$R_{\Lambda}=\frac{2GM_{\Lambda}}{c^2}.
\label{bh4}$$ This expression coincides with the mass-radius relation (\[bh2\]) of a Schwarzschild black hole. This coincidence has sometimes led people to say that the Universe is a black hole, or that we live in a black hole, although this analogy is probably too naive. Nevertheless, at least on a purely dimensional basis, we can use the analogy with black holes to define the entropy and the temperature of the Universe. In this manner, we get a temperature scale (temperature on the horizon) $$k_B T_{\Lambda}=\frac{\hbar c}{4\pi R_{\Lambda}}=\frac{\hbar H_0}{4\pi}\sim
\hbar\sqrt{\Lambda}.
\label{bh5}$$ Its value is $T_{\Lambda}\sim 2.41\times 10^{-29}\, {\rm K}$. The temperature can be written as $$k_B T_{\Lambda}=\frac{2\hbar a_{\Lambda}}{c},
\label{w3b}$$ where $$a_{\Lambda}=G\Sigma_{\Lambda}=\frac{GM_{\Lambda}}{4\pi
R_{\Lambda}^2}=\frac{c^2}{8\pi R_{\Lambda}}=
\frac{cH_0}{8\pi}\sim c\sqrt{\Lambda}
\label{w3bb}$$ is the surface gravity of the Universe (similar relations apply to black holes). We can also write $$k_B T_{\Lambda}=m_{\Lambda}c^2,
\label{bh6}$$ with $$m_{\Lambda}\sim \frac{\hbar\sqrt{\Lambda}}{c^2}=2.08\times 10^{-33}\,
{\rm eV/c^2}.
\label{bh7}$$ This mass scale is often interpreted as the smallest mass of the bosons predicted by string theory [@axiverse] or as the upper bound on the mass of the graviton [@graviton].[^8] It can be contrasted from the mass scale $$M_{\Lambda}\sim \frac{c^3}{G\sqrt{\Lambda}}=7.16\times 10^{88}\,
{\rm eV/c^2},
\label{bh8}$$ which is usually interpreted as the mass of the Universe. Thus $m_{\Lambda}$ and $M_{\Lambda}$ represent fundamental lower and upper mass scales. Their ratio is $$\frac{M_{\Lambda}}{m_{\Lambda}}\sim \frac{c^5}{G\hbar\Lambda}\sim
\frac{\rho_P}{\rho_{\Lambda}}\sim e^{1/B}\sim 10^{123},
\label{bh9}$$ which exhibits the famous number $123$. On the other hand, our analogy between the Universe and a black hole leads to an entropy scale (entropy on the Hubble horizon): $$S_{\Lambda}=\frac{k_B \pi c^3R_{\Lambda}^2}{G\hbar}=\frac{k_B \pi
c^5}{G\hbar H_0^2}\sim \frac{k_B c^5}{G\hbar \Lambda}.
\label{bh10}$$ We note that the entropy of the Universe can be written as $$S_{\Lambda}/k_B\sim \frac{M_{\Lambda}}{m_{\Lambda}}
\sim \frac{\Sigma_{\Lambda}R_{\Lambda}^2}{m_{\Lambda}}
\sim e^{1/B}\sim
10^{123}.
\label{bh11}$$ This entropy may be identified with the total entropy of the logotropic dark fluid (see the Appendix of [@jcap] and Appendix \[sec\_logt\]). It can be compared to the entropy of radiation [@cosmopoly1]: $$\begin{aligned}
S_{\rm rad}/k_B&=&\frac{4}{3}\left (\frac{3\Omega_{\rm rad,0}}{8\pi}\right
)^{3/4}\left (\frac{\pi^2}{15}\right
)^{1/4}\frac{1}{(H_0t_P)^{3/2}}\nonumber\\
&=&5.64\times 10^{87},
\label{bh11b}\end{aligned}$$ obtained by using Eq. (\[t2\]) with $P_{\rm rad}=\epsilon_{\rm rad}/3$, $\epsilon_{\rm rad}=\sigma T^4$ with $\sigma=\pi^2k_B^4/15c^3\hbar^3$ (Stefan-Boltzmann constant), $\epsilon_{\rm rad}=\Omega_{\rm
rad,0}\epsilon_0/a^4$ and $\Omega_{\rm rad,0}=9.24\times 10^{-5}$. They differ by about $36$ orders of magnitude.
[*Remarks:*]{} We note that $T_{\Lambda}S_{\Lambda}=(1/2)M_{\Lambda}c^2$ so the free energy of the Universe is $F_{\Lambda}=M_{\Lambda}c^2-T_{\Lambda}S_{\Lambda}=(1/2)M_{\Lambda}c^2$. On the other hand, using Eqs. (\[lm18\]) and (\[lm32\]), we obtain the relations $$\frac{m_{\Lambda}}{M_P}\sim e^{-1/2B}=3.40\times 10^{-62},
\label{bh12}$$ $$\frac{m_e}{m_{\Lambda}}\sim
\sqrt{B} e^{1/3B}=5.66\times 10^{39}.
\label{bh13}$$ Since $m_{\Lambda}\sim \rho_{\Lambda}r_e^3$ (see Sec. \[sec\_Ba\]) we have $m_e/m_\Lambda\sim \rho_e/\rho_\Lambda$. The gravitational radius of the cosmon is $r_\Lambda=2Gm_{\Lambda}/c^2\sim G\hbar\sqrt{\Lambda}/c^4=2.75\times
10^{-96}\, {\rm m}$.
Entropy of logotropic dark matter halos
---------------------------------------
Let us define the entropy of a logotropic dark matter halo by $$S\sim k_B N\sim k_B
\frac{M}{m_{\Lambda}},
\label{bh14}$$ where $M$ is the halo mass and $m_{\Lambda}$ is the mass of the hypothetical particle composing the logotropic dark fluid. Using the mass-radius relation $M\sim \Sigma_0 R^2$ of a logotropic dark matter halo, where $\Sigma_0\sim c\sqrt{\Lambda}/G$ is the universal surface density given by Eq. (\[lm22\]), we get $$S\sim \frac{k_B c \sqrt{\Lambda}R^2}{Gm_{\Lambda}}.
\label{bh15}$$ Interestingly, the entropy given by Eq. (\[bh15\]) scales like the surface $R^2$ of the object, similarly to the black hole entropy (\[bh1\]).[^9] This may be connected to a form of holographic principle [@bousso]. Matching the formulae (\[bh1\]) and (\[bh15\]), we find that $m_{\Lambda}$ corresponds to the mass given by Eq. (\[bh7\]). Inversely, if we assume from the start that the logotropic dark fluid is composed of particles of that mass (cosmons), we find that the entropy of dark matter halos coincides with the entropy of black holes.[^10] On the other hand, since the surface density of the Universe is of the same order as the surface density of dark matter halos, the previous formulae also apply to the Universe as a whole and return the results of Appendix \[sec\_abhb\]. This may be a form of justification, for reasons of self-consistency, of Eq. (\[bh14\]).
[*Remark:*]{} If we, alternatively, define the entropy of dark matter halos by $S\sim
k_B M/m_e$ where $m_e$ is the electron mass and use $M\sim \Sigma_0 R^2$ with $\Sigma_0\sim
\Sigma_e$ where $\Sigma_e$ is the surface density of the electron given by Eq. (\[lm27\]), we obtain $$S\sim k_B \frac{R^2}{r_e^2},
\label{bh16}$$ which is similar to the black hole entropy formula (\[bh1\]) where the Planck length $l_P$ is replaced by the classical radius of the electron $r_e$. It is not clear, however, if this formula is physically relevant.
Postulates: entropic principles {#sec_post}
-------------------------------
We can find a form of explanation of the different relations found in this paper by making the following two postulates.
[*Postulate 1:*]{} We postulate that the entropy of the electron, the entropy of dark matter halos and the entropy of the Universe (and possibly other objects) is given by $$S\sim \frac{k_B c^3R^2}{G\hbar},
\label{post1}$$ like the Bekenstein-Hawking entropy of black holes, where $R$ is the radius of the corresponding object. This may be connected to a form of holographic principle stating that the entropy is proportional to the area (instead of the volume). Therefore, $$S_e/k_B\sim \frac{c^3r_e^2}{G\hbar} \qquad ({\rm electron})
\label{post2}$$ $$S/k_B \sim \frac{c^3r_h^2}{G\hbar}\qquad ({\rm dark\,\, matter})
\label{post3}$$ $$S_{\Lambda}/k_B \sim \frac{c^3R_{\Lambda}^2}{G\hbar}\qquad ({\rm
Universe})
\label{post4}$$
[*Postulate 2:*]{} We postulate that the entropy of the electron, the entropy of dark matter halos and the entropy of the Universe (and possibly other objects) is also given by[^11] $$S\sim k_B \frac{M}{m_{\Lambda}},
\label{post5}$$ where $M$ is the mass of the corresponding object and $m_{\Lambda}$ is the mass defined by Eq. (\[bh7\]). Therefore, $$S_e/k_B\sim \frac{m_e}{m_{\Lambda}}\sim
\frac{\Sigma_{e}r_{e}^2}{m_{\Lambda}}\sim 10^{39}\qquad ({\rm
electron})
\label{post6}$$ $$S/k_B\sim \frac{M_h}{m_{\Lambda}}\sim
\frac{\Sigma_{0}r_h^2}{m_{\Lambda}}\qquad ({\rm dark \,\, matter})
\label{post7}$$ $$S_{\Lambda}/k_B\sim \frac{M_{\Lambda}}{m_{\Lambda}}\sim
\frac{\Sigma_{\Lambda}R_{\Lambda}^2}{m_{\Lambda}}\sim 10^{123}\qquad ({\rm
Universe})
\label{post8}$$
The comparison of Eqs. (\[post1\]) and (\[post5\]) directly implies that the surface density of the electron, the surface density of [*all*]{} the dark matter halos, and the surface density of the Universe is (approximately) the same and has the typical value $$\Sigma\sim \frac{M}{R^2}\sim \frac{m_{\Lambda}c^3}{G\hbar}\sim
\frac{m_{\Lambda}}{M_P}\Sigma_P\sim \frac{c\sqrt{\Lambda}}{G},
\label{post9}$$ where $\Sigma_P=(c^7/\hbar G^3)^{1/2}=8.33\times 10^{64}\, {\rm g\, m^{-2}}$ is the Planck surface density. Then, comparing this universal value with the surface density of the electron \[see Eq. (\[lm27\])\], we obtain the Weinberg relation $$\Lambda\sim \frac{m_e^6G^2c^6}{e^8}.
\label{post10}$$
[*Remark:*]{} we have introduced the entropy of an electron \[see Eqs. (\[post2\]) and (\[post6\])\] by analogy with the black hole entropy. If these ideas are physically relevant, a notion of thermodynamics for the electron (assuming that it is made of $10^{39}$ subparticles of mass $m_{\Lambda}$) should be developed. Again, the analogy with black holes (although, of course, an electron is not a black hole) might be useful.
Large numbers and coincidences {#sec_ln}
==============================
The ratio between the electric radius of the electron $r_e=e^2/m_ec^2$ and its gravitational radius $r_g=2Gm_e/c^2$ is of the order of $e^2/Gm_e^2=4.17\times 10^{42}$. This dimensionless number was computed by Weyl in 1919 [@barrow]. He was the first to notice the presence of large dimensionless numbers in Nature. This led Eddington [@eddington] and others to try to relate such large numbers to cosmological quantities. In particular, Eddington evaluated the total number of particles in the Universe and found $N\sim 10^{79}$. He then tried to relate the basic interaction strenghts and elementary particle masses to this number. For example, it was observed by different authors that the following quantities are of the same order of magnitude (see Ref. [@barrow], P. 224-231): $$\frac{m_ec^2}{\hbar H_0}\sim
\frac{e^2}{Gm_e^2}\sim \sqrt{N}\sim
\left (\frac{M_P}{m_e}\right )^2\sim 10^{40}.
\label{ln1}$$
These coincidences can be easily understood from our results (\[lm32\]) and (\[lm33\]) which express the mass and the charge of the electron in terms of the cosmological constant. In order to avoid too much digression, we shall replace the Eddington number by[^12] $$N_e=\frac{M_{\Lambda}}{m_e}\sim e^{2/(3B)}\sim 10^{80}.
\label{ln2}$$ On the other hand, combining our results, we find $$\frac{m_e c^2}{\hbar H_0}\sim \frac{m_e}{m_{\Lambda}}\sim
e^{1/(3B)}\sim 10^{40},
\label{ln3}$$ $$\frac{e^2}{Gm_e^2}\sim e^{1/(3B)}\sim 10^{40},
\label{ln4}$$ $$\begin{aligned}
\frac{M_P}{m_e}\sim e^{1/(6B)}\sim 10^{20},
\label{ln5}\end{aligned}$$ leading to the equivalents from Eq. (\[ln1\]). We also note that $$\begin{aligned}
\frac{1}{m_e^4}\left (\frac{\hbar c}{G}\right )^2\sim \left
(\frac{M_P}{m_e}\right )^4\sim 10^{80}\sim N_e,
\label{ln6}\end{aligned}$$ which is one of the “coincidences” pointed out by Chandrasekhar [@chandranature].
In a sense, these results arise from the Weinberg relation (\[w4\]) that has been found by different authors (see footnote 7). Nevertheless we believe that our approach is original and may bring new light on the subject. In particular, we have proposed a form of common explanation of these different “coincidences” in terms of entropic principles (see Appendix \[sec\_post\]).
Thermodynamics of the logotropic dark fluid {#sec_logt}
===========================================
Let us try to relate the results of the previous Appendices to the thermodynamics of the logotropic dark fluid.
We assume that the Universe is filled with a dark fluid at temperature $T$. From the first principle of thermodynamics, one can derive the thermodynamic equation [@weinbergbook]: $$\label{t1}
\frac{dP}{dT}=\frac{1}{T}(\epsilon+P).$$ If the dark fluid is described by a barotropic equation of state of the form $P=P(\epsilon)$, Eq. (\[t1\]) can be integrated to obtain the relation $T=T(\epsilon)$ between the temperature and the energy density. On the other hand, the entropy of the dark fluid in a volume $a^3$ is given by [@weinbergbook]: $$\label{t2}
S=\frac{a^3}{T}(P+\epsilon).$$ From the Friedmann equations, one can show that the entropy of the Universe is conserved: ${dS}/{dt}=0$ [@weinbergbook].
The previous results are general. Let us now apply them to the logotropic dark fluid. According to Eqs. (\[lm5\]) and (\[lm6\]), the equation of state $P=P(\epsilon)$ of the logotropic dark fluid is given by the reciprocal of [@lettre; @jcap]: $$\label{t3}
\epsilon=\rho_P e^{P/A}c^2-P-A.$$ Eq. (\[t1\]) with Eq. (\[t3\]) is easily integrated giving $$\label{t4}
T=\frac{\rho_Pc^2}{K}\left (1-\frac{A}{\rho c^2}\right ),$$ where $K$ is a constant of integration and we have used Eq. (\[lm5\]). Substituting Eqs. (\[t3\]) and (\[t4\]) into Eq. (\[t2\]), and using Eqs. (\[lm3\]) and (\[lm5\]), we find that $$\label{t5}
S=K\frac{\rho_0}{\rho_P}.$$ We explicitly check on this expression that the entropy of the Universe is conserved. Furthermore, since the entropy is positive, we must have $K>0$. Considering Eq. (\[t4\]), we note that the temperature is positive when $\rho>\rho_M=A/c^2$ and negative when $\rho<\rho_M=A/c^2$, that is to say when the Universe becomes phantom [@epjp; @lettre].[^13]
We can determine the constant $K$ by assuming that the entropy of the logotropic dark fluid is given by $$\label{t6}
S\sim k_{B}\frac{M_{\Lambda}}{m_{\Lambda}}\sim 10^{123}\, k_B$$ as in Appendix \[sec\_abh\]. Noting that the “true” entropy is obtained by multiplying Eq. (\[t2\]) by $R_{\Lambda}^3$ (since we have taken $a=1$ at the present time), and comparing Eqs. (\[t5\]) and (\[t6\]), we obtain $$\label{t7}
K\sim k_B \frac{\rho_P}{m_{\Lambda}}.$$ As a result, the temperature of the logotropic dark fluid is given by $$\label{t8}
k_B T\sim m_{\Lambda}c^2\left (1-\frac{B\rho_{\Lambda}}{\rho}\right ),$$ where we have used Eq. (\[lm12\]). In the “early” Universe $\rho\gg\rho_{\Lambda}$ we find that[^14] $$T\simeq
m_{\Lambda}c^2/k_B=2.41\times 10^{-29}\, {\rm K}.$$ In the late Universe $\rho\ll\rho_{\Lambda}$ we find that $$k_B T\sim -m_{\Lambda}c^2B\rho_{\Lambda}/\rho\propto -a^3.$$
[*Remark:*]{} In Ref. [@jcap] we have shown that the logotropic constant $B$ could be interpreted as a dimensionless logotropic temperature $$\label{t9}
B=\frac{k_B T_{\rm L}}{m_{\Lambda}c^2}$$ in a generalized thermodynamical framework [@epjp; @lettre]. This shows that at least two temperatures exist for the logotropic dark fluid, a time-varying temperature $T$ and a constant temperature $T_{\rm L}$. They become equal when $$\label{t10}
\frac{\rho_*}{\rho_{\Lambda}}\sim \frac{B}{1-B}\sim 3.54\times 10^{-3},$$ $$\label{t11}
a_*\sim \left (\frac{\Omega_{\rm m,0}}{\Omega_{\rm de,0}}\frac{1-B}{B}\right
)^{1/3}\sim 5.01.$$
The mass of the bosonic dark matter particle {#sec_mdm}
============================================
It has been suggested that dark matter may be made of bosons (like ultralight axions) in the form of Bose-Einstein condensates (BECs) [^15] We can use the results of the present paper to predict the mass $m$ of the bosonic dark matter particle in terms of the cosmological constant $\Lambda$. We assume that the smallest and most compact dark matter halo that is observed corresponds to the ground state of a self-gravitating BEC (to fix the ideas we assume that this halo is the dSphs Fornax with a mass $M\sim 10^8\, M_{\odot}$ and a radius $R\sim 1\, {\rm kpc}$). For noninteracting bosons, it can be shown by solving the Gross-Pitaevskii-Poisson equations (see, e.g., Sec. III.B.1 of [@becmodel]) that the mass $(M_h)_{\rm min}$, the radius $(r_h)_{\rm min}$ and the central density $(\rho_0)_{\rm max}$ of this ultracompact halo (ground state) are related to each other by the relations $$M_h=1.91\, \rho_0 r_h^3 \qquad {\rm and} \qquad M_h
r_h=1.85\, \frac{\hbar^2}{Gm^2}.
\label{add}$$ As a result, its surface density is given by $$\Sigma_0=0.153\, \frac{G^2m^4M_h^3}{\hbar^4}.
\label{mdm1}$$ On the other hand, the minimum mass of dark matter halos may be obtained from a quantum Jeans instability theory (see, e.g., Ref. [@abriljeans]) giving the result $$M_J=\frac{1}{6}\pi\left
(\frac{\pi^3\hbar^2\rho_{\rm dm,0}^{1/3}}{Gm^2}\right
)^{3/4}.
\label{mdm2}$$ For usually considered values of the boson mass, of the order of $m\sim
10^{-22}\, {\rm eV/c^2}$, the Jeans mass $M_J \sim 10^7\,
M_{\odot}$ is indeed of the order of the minimum mass $(M_h)_{\rm min} \sim
10^8\, M_{\odot}$ of observed dark matter halos. There may be, however, a numerical factor of order $10$ between $M_J$ and $(M_h)_{\rm min}$. For that reason, we introduce a prefactor $\chi$ and write $(M_h)_{\rm min}=\chi
M_J$. Using $\rho_{\rm dm,0}=({\Omega_{\rm dm,0}}/{\Omega_{\rm
de,0}})\rho_{\Lambda}=({\Omega_{\rm dm,0}}/{\Omega_{\rm de,0}})
({\Lambda}/{8 \pi G})$, we get $$(M_h)_{\rm min}=\chi \frac{\pi^3}{6}\left (\frac{\Omega_{\rm
dm,0}}{8\Omega_{\rm
de,0}}\right )^{1/4}\frac{\hbar^{3/2}\Lambda^{1/4}}{Gm^{3/2}}.
\label{mdm3}$$ Then, using Eq. (\[mdm1\]), we obtain $$\Sigma_0= 0.153\,\chi^3 \frac{\pi^9}{216}\left (\frac{\Omega_{\rm
dm,0}}{8\Omega_{\rm de,0}}\right
)^{3/4}\frac{\hbar^{1/2}\Lambda^{3/4}}{Gm^{1/2}}.
\label{mdm4}$$ Comparing this expression with Eq. (\[lm22\]), we predict that the mass of the bosonic particle is given by $$m=\chi^6\frac{0.0234\pi^{20}}{1458B\xi_h^2}\left
(\frac{\Omega_{\rm
dm,0}}{8\Omega_{\rm de,0}}\right
)^{3/2}\frac{\hbar\sqrt{\Lambda}}{c^2}=15397\chi^6\frac{
\hbar\sqrt{\Lambda}}{c^2}.
\label{mdm5}$$ We see that the mass of the bosonic dark matter particle is equal to the mass scale $m_{\Lambda}\sim 10^{-33}\,
{\rm eV/c^2}$ given by Eq. (\[bh7\]) multiplied by a huge numerical factor of order $10^{10}$ (for $\chi\sim 10$). This gives $m\sim
10^{-23}\,
{\rm eV/c^2}$ which is the correct order of magnitude of the mass of ultralight axions usually considered [@marshrevue]. We note that this result has been obtained independently from the observations, except for the value of $\Lambda$ and the other fundamental constants (Planck scales).
[99]{}
[^1]: Many models try to unify dark matter and dark energy. They are called unified dark energy and dark matter (UDE/M) models. However, the interpretation of dark matter and dark energy that we give in Refs. [@epjp; @lettre] is new and original.
[^2]: We stress that our model is different from the $\Lambda$CDM model so that $\Lambda$ is fundamentally different from Einstein’s cosmological constant [@einsteincosmo]. However, it is always possible to introduce from the constant $A$ an effective cosmological density $\rho_{\Lambda}$ and an effective cosmological constant $\Lambda$ by Eqs. (\[lm8\]) and (\[lm9\]).
[^3]: The logotropic spheres [@epjp; @lettre], like the isothermal spheres [@chandra], have an infinite mass. This implies that the logotropic equation of state cannot describe dark matter halos at infinitely large distances. Nevertheless, it may describe the inner region of dark matter halos and this is sufficient to determine their surface density. The stability of bounded logotropic spheres has been studied in [@logo] by analogy with the stability of bounded isothermal spheres and similar results have been obtained. In particular, bounded logotropic spheres are stable provided that the density contrast is not too large.
[^4]: We note that the Thomson cross-section $\sigma=(8\pi/3)(e^2/m_e
c^2)^2$ can be written as $\sigma=(8\pi/3)r_e^2$ giving a physical meaning to the classical electron radius $r_e$. We also note that $r_e$ can be written as $r_e=\alpha\hbar/m_ec$ where $\lambda_C=\hbar/m_e c$ is the Compton wavelength of the electron and $\alpha$ is the fine-structure constant $\alpha$ \[see Eq. (\[const1\])\]. Similarly, $\Sigma_e=(1/\alpha^2)m_e^3c^2/\hbar^2$.
[^5]: A closely related formula, involving the Hubble constant instead of the cosmological constant, was first found by Stewart [@stewart] in 1931 by trial and error.
[^6]: We note that the prefactors in Eqs. (\[lm30\]) and (\[lm31\]) appear to be close to $123/1000$ and $123/10$, where the number $123$ appears again (!). We do not know whether this is fortuitous or if this bears a deeper significance than is apparent at first sight.
[^7]: Weinberg considers this relation as “so far unexplained” and having “a real though mysterious significance”. Similar relations have been obtained in the past by Stewart [@stewart], Eddington [@eddington] and others from purely heuristic arguments or from dimensional analysis [@kragh; @barrow; @calogero]. Their goal was to express the mass of the elementary particles in terms of the fundamental constants of Nature.
[^8]: It is simply obtained by equating the Compton wavelength of the particle $\lambda_c=\hbar/mc$ with the Hubble radius $R_\Lambda=c/H_0$ (the typical size of the visible Universe) giving $m_{\Lambda}=\hbar H_0/c^2$. Using Eq. (\[lm14b\]), we obtain Eq. (\[bh7\]).
[^9]: Inversely, a manner to understand why the surface density of the dark matter halos has a universal value is to argue that their entropy given by Eq. (\[bh14\]) should scale like $R^2$ (see Appendix \[sec\_post\]).
[^10]: Of course, we are not claiming that dark matter halos are black holes since they obviously do not fulfill the Schwarzschild relation (\[bh2\]). However, they may have the same entropy as black holes expressed in terms of $R$ \[see Eq. (\[bh1\])\].
[^11]: Note that this relation is [*not*]{} satisfied by black holes since $M_{\rm
BH}\propto R$ while $S_{\rm BH}\propto R^2$.
[^12]: The Eddington number corresponds typically to the number of protons in the Universe, $N\sim M_{\Lambda}/m_p$, where $m_p$ is the proton mass. This number was introduced before dark matter and dark energy were discovered. If the dark fluid is made of cosmons of mass $m_{\Lambda}$, the number of particles in the Universe is $N_{\Lambda}= M_{\Lambda}/m_{\Lambda}\sim 10^{123}$ giving another interpretation to the famous number $123$. This number should supersede the Eddington number.
[^13]: This is a general result [@cosmopoly3] which can be obtained from Eq. (\[t2\]) using the fact that the entropy is constant and positive. We see that the sign of the temperature coincides with the sign of $P+\epsilon$. As a result, the temperature is positive in a normal Universe ($P>-\epsilon$) and negative in a phantom Universe ($P<-\epsilon$).
[^14]: We recall that the logotropic model, which is a unification of dark matter and dark energy, is not valid in the very early Universe corresponding to the big-bang, the inflation era, and the radiation era. Therefore, the temperature $m_{\Lambda}c^2$ corresponds to the temperature of the dark fluid in the matter era, i.e., when the rest-mass energy of the dark fluid overcomes its internal energy (see Sec. \[sec\_lm\]). We emphasize that the temperature $T$ of the logotropic dark fluid is different from the temperature of radiation and of any other standard temperature. We also note that the corresponding temperature in the $\Lambda$CDM model is not defined since Eq. (\[t1\]) breaks down when $P=0$.
[^15]: See, e.g., the bibliography of Ref. [@abriljeans] for an exhaustive list of references. The possible connections between the BECDM model and the logotropic model will be investigated in a future paper [@forthcoming].
|
---
abstract: 'We present here a completely operatorial approach, using Hilbert-Schmidt operators, to compute spectral distances between time-like separated events , associated with the pure states of the algebra describing the Lorentzian Moyal plane, using the axiomatic framework given by [@Franco1; @Franco2]. The result shows no deformations of non-commutative origin, as in the Euclidean case.'
author:
- |
[**[Anwesha Chakraborty]{}$^{a}$ [^1]**]{}, [**[Biswajit Chakraborty]{}$^{b}$ [^2]**]{}\
$^{a,b}$ [Department of Theoretical Sciences]{}\
[S.N. Bose National Centre for Basic Sciences]{}\
[JD Block, Sector III, Salt Lake, Kolkata 700106, India]{}\
date:
- '24/05/19'
-
title: '[**[Spectral Distance on Lorentzian Moyal Plane]{}**]{}'
---
Introduction
============
In his formulation of Non-commutative geometry (NCG) [@Connes] and its subsequent application to standard model of particle physics [@Suij; @Lizzi], Alain Connes has essentially dealt with spaces with Euclidean signature i.e. Riemannian manifold. This feature of his formulation has remained a sort of a bottle-neck in the further development in its application and eventual reconciliation with the realistic nature of our space-time, which as we all know to be as a manifold with Lorentzian signature. Attempts are being made for quite some time now [@Eli; @Wd; @Kv] and people have devised various ways to circumvent this problem like the so called Wick rotation[@Lizzi] etc. However, the unanimous opinion among the practitioners of NCG and its application to standard model is that one needs to confront this issue of Lorentzian signature head-on. Recent activities in this direction indicates that there is still no consensus in the literature about its axiomatic formulation [@Kopf; @Barrett; @Sitarz; @Stro; @Parf; @Moretti; @Franco1]. Despite the fact, we would like to follow the work of [@Franco1; @Franco2], in our preliminary attempt to compute the spectral distance between a pair of time-like separated events associated with pure states in Lorentzian Moyal plane and show that this axiomatic formulations of [@Franco1; @Franco2], serves our purpose quite adequately. This work can be thought of as a sequel of our earlier work on Euclidean Moyal plane [@Chaoba] and here too we are employing Hilbert-Schmidt operator formulation of NC quantum mechanics [@Scholtz].\
\
The paper is organised as follows: In section 2 after a brief recapitulation of Hilbert-Schmidt approach in quantum mechanics we make use of Fock-Bergman coherent state basis to represent the states of the Hilbert space. The functional differentiation approach is then employed for the extremisation of various functionals arising here naturally in the derivation of spectral distance between a pair of pure states on Euclidean Moyal plane. Section 3 is the brief outline of the construction of Lorentzian spectral triple and emergence of algebraic ball condition as a result of causality following [@Franco1; @Franco2]. The next section consists of the explicit derivation of Dirac operator and Ball condition for two dimensional Lorentzian commutative and non-commutative manifold with (-,+) signature, followed by the calculation of distances in functional derivative approach for the respective manifolds, where we have also discussed the Poincare invariance of distance and non-invariance of vacuumunder Lorentz boost. Finally we conclude in section 5.
Recapitulation of the Computation of Spectral Distance on Moyal Plane with Euclidean Signature
==============================================================================================
Moyal plane is defined through a commutator relation $$[\hat{x}_1,\hat{x}_2]=i\theta;\,\,\,\theta>0\label{algebra}$$ satisfied by the operator valued coordinates $\hat{x}_1$ and $\hat{x}_2$. As it stands, this commutator algebra does not specify the signature of the underlying commutative space (i.e. the one obtained in the commutative limit), in the sense that it can correspond to either Euclidean or Lorentzian signature, as can be seen easily from the fact that this noncommutative algebra (\[algebra\]) is invariant under both SO(2) and SO(1,1) transformation. However, since Connes’ formulation of NCG [@Connes2] requires in the commutative case, the spin manifold to be Euclidean, the initial works on the computation of spectral distances between pure states in the Moyal plane [@Mart] and fuzzy sphere [@Liz] were restricted to the Euclidean signatures only, in the sense that the respective commutative limits yield the corresponding manifold with the associated metrics. In particular , in [@Chaoba] computations were carried out in a complete operatorial level, using Hilbert-Schmidt operator, which requires no use of any star product, thereby bypassing any ambiguities that may result from there [@Prasad]. In the present paper, as mentioned above, we would like to extend our computation of similar distances for the Lorentzian signature as well. For that it will be advantageous to begin by recapitulating briefly the computation of distances with Euclidean signature as in [@Chaoba]. but, in contrast to [@Chaoba], we shall employ the Fock-Bergman coherent states to represent state vectors belonging to the Hilbert space and eventually to extremise the ball functional. It will make the computations much more transparent and can be appreciated by any potential reader. This will help us to set up the basic frame work and introduce the necessary notations, so that the same can be emulated and adapted in the Lorentzian case. In fact as we shall see that there are several points of contacts between these two cases with respective similarities and dissimilarities.\
\
\
The spectral distance on Moyal plane is computed by employing the spectral triple $(\mathcal{A},\mathcal{H},\mathcal{D})$ where $${A}=\mathcal{H}_q, \,\,\, \mathcal{H}=\mathcal{H}_c\otimes \mathbb{C}^2,\,\,\,\mathcal{D}=\sqrt{\frac{2}{\theta}}\begin{pmatrix}
0&\hat{b}^{\dagger}\\\hat{b}&0
\end{pmatrix}\label{o}$$ and $\mathcal{H}_q$ is the space of Hilbert-Schmidt (HS) operators acting on configuration space $\,\mathcal{H}_c $ - which is also a Hilbert space, and defined as, $$\mathcal{H}_c=\textrm{Span}\left\{|n\rangle =\frac{(\hat{b}^{\dagger})^n}{\sqrt{n!}}|0\rangle\right\}; \,\,\,\, \hat{b}=\frac{\hat{x}_1+i\hat{x}_2}{\sqrt{2\theta}}\,\,\,\, \textrm{and}\,\, \hat{b}|0\rangle = 0. \label{k}$$ This furnishes a representation of just the above coordinate algebra (\[algebra\]) and is basically isomorphic to the Hilbert space of 1D harmonic oscillator. $\mathcal{H}_q$ is equipped with the inner product $(\psi,\phi)=tr_{\mathcal{H}_c}(\psi^{\dagger}\phi)$ and as a part of the definition the associated HS norm defined as $\Vert\psi\Vert_{HS}:= \sqrt{tr_{\mathcal{H}_c}(\psi^{\dagger}\psi)} <\infty$ i.e. fulfills finiteness property. In other words, a generic HS operator (denoted sometimes by a round ket $|.)$, in contrast to angular $|.\rangle$ notations used for the elements of $\mathcal{H}_c$ (\[k\])) $\psi :=|\psi)\in\mathcal{H}_q$ can be expanded in the Fock basis (\[k\]) as, $$|\psi) = \sum_{m,n}C_{m,n}|m\rangle\langle n| \label{ju}$$ as $\mathcal{H}_q$ can be identified as $\mathcal{H}_c\otimes\tilde{\mathcal{H}_c}$ with $\tilde{\mathcal{H}_c}$ being the dual Hilbert space. Now making use of the identity $|0\rangle\langle 0| = :e^{-\hat{b}^{\dagger}\hat{b}}:$, where the colons represent normal ordering of the operators [@Itz], one can see that $\psi$ can be recast as a polynomial algebra generated by $\hat{b},\hat{b}^{\dagger}$ as, $$|\psi) =\sum_{m,n} C_{mn}\frac{\hat{b}^{\dagger m}}{\sqrt{m!}}|0\rangle\langle 0| \frac{\hat{b}^n}{\sqrt{n!}} = \sum_{m,n,l}C_{mn}\frac{1}{l!}\frac{(-1)^l}{\sqrt{m!n!}} (\hat{b}^{\dagger} )^{m+l}(\hat{b})^{n+l} \,\,\,\,\in \mathcal{H}_q , \label{poly}$$ which is automatically in the normal-ordered form. Note that $\Vert\psi\Vert_{HS}^2 =\sum_{m,n}|C_{mn}|^2 < \infty $, as required by very definition of HS operator. This $\mathcal{H}_q$, whose generic forms are like in (\[ju\]), has the structure of an algebra. This allows us to identify the algebra $\mathcal{A}$ to $\mathcal{H}_q$ itself: $\mathcal{A}=\mathcal{H}_q$ in (\[o\]). It acts on $\mathcal{H}_c\otimes\mathbb{C}^2$ through the diagonal representation $\pi(a):= \textrm{diag}(a,a)$ from left so that this Hilbert space $\mathcal{H}_c\otimes\mathbb{C}^2$ can be regarded as the left module of the algebra. Also note that the algebra $\mathcal{A}=\mathcal{H}_q$ is a dense subspace of $\mathcal{B}(\mathcal{H}_c)$, where this can be identified with a $C^*$ - algebra with $*$- operation denoting hermitian conjugation which play the role of involution. Finally $\mathcal{D}$ in (\[o\]) is the Dirac operator whose construction has been reviewed in [@Chaoba]. The point that we would like to emphasise is that the structure of the Dirac operator turned out to be such that it can also act on the Hilbert space $\mathcal{H}_c\otimes\mathbb{C}^2$ from left. Consequently, the Hilbert space in the spectral triple is taken to be $\mathcal{H}_c\otimes\mathbb{C}^2$, rather than $\mathcal{H}_q\otimes\mathbb{C}^2$- as would have been expected from the corresponding commutative case where it is taken to be of the form $L^2(\mathbb{R}^2)\otimes\mathbb{C}^2$.\
\
Here one has two kinds of choices of states viz. the normalised coherent state $$|z\rangle =e^{-\bar{z}\hat{b}+z\hat{b}^{\dagger}}|0\rangle \in\mathcal{H}_c ;\,\,\, \langle z|z\rangle =1 \label{coh}$$ and the states $|n\rangle$, as in (\[k\]) . Correspondingly, one can introduce pure states $\rho_z:=|z\rangle\langle z|,\,\,$ and the so called harmonic oscillator states $ \rho_n:=|n\rangle\langle n| \in \mathcal{H}_q$ which are linear functionals of unit norm, acting on the algebra $\mathcal{A}=\mathcal{H}_q$ as $$\rho_z(\hat{a})=\textrm{tr}_{\mathcal{H}_c}(\rho_z\hat{a})=(\rho_z,\hat{a})= \langle z|\hat{a}|z\rangle\,\, ; \rho_n(\hat{a})=tr_{\mathcal{H}_c}(\rho_n \hat{a}) =\langle n|\hat{a}|n\rangle \label{u}$$ These pure states $\rho_z $ and $\rho_n$ can also be regarded as density matrices as viewed from $\mathcal{H}_c$. In the spirit of Gelfand and Naimark, here too one can associate $\rho_z$, in particular, with the point having the complex coordinate $z$ in the Argand diagram with the latter being viewed as smeared Moyal plane. Spectral distance between a generic pair of states $\rho_z$ and $\rho_{w}$ (in this paper we shall be considering only pure states built out of coherent states) *a la* Connes’ is given by $$d(\rho_z,\rho_w)=\sup_{\hat{a}\in \mathcal{B}}\{|\rho_z(\hat{a})-\rho_w(\hat{a})| \}\label{hg}$$ where $\mathcal{B}$ is the Ball defined as $$\mathcal{B}=\{\hat{a}:\Vert[\mathcal{D},\pi(\hat{a})]\Vert_{op}\leq 1 \}\label{ki}$$ Given the structure of $\mathcal{D}$ in (\[o\]) the Ball condition can be written equivalently and more compactly as $$\mathcal{B}=\left\{\hat{a}\,:\, \Vert[\hat{b},\hat{a}]\Vert_{op}=\Vert[\hat{b}^{\dagger},\hat{a}]\Vert_{op}\leq \sqrt{\frac{\theta}{2}}\right\}\label{ballcondition}$$ As has been noted in [@Chaoba] that the optimal algebra element $a_s\in \mathcal{A}$ for which the supremum is attained in (\[hg\]) , yielding the distance i.e. $$d(\rho_z,\rho_w)=|\rho_z(\hat{a}_s)-\rho_w(\hat{a}_s)| \label{hp}$$ must also saturate the Ball condition (\[ballcondition\]) i.e. it should satisfy $$\Vert [\hat{b},\hat{a}_s]\Vert_{op}=\Vert [\hat{b}^{\dagger},\hat{a}_s]\Vert_{op}=\sqrt{\frac{\theta}{2}}\label{A3}$$ Further following [@Mart2], we know that the search of such an optimal algebra element can be restricted to hermitian algebra elements only i.e. we require $\hat{a}_s^{\dagger}= \hat{a}_s$. To that end we take up the simplified Ball condition (\[ballcondition\]) and first try to compute $\Vert[\hat{b},\hat{a}]\Vert_{op}^2$. Making use of the Fock-Bergman basis (\[A67\]) and the associated resolution of identity (\[A68\]) (see Appendix-A), one gets, $$||[\hat{b},\hat{a}]||_{op}^2 = \sup_{||\psi||=1}\langle\psi|[\hat{b},\hat{a}]^{\dagger}[\hat{b},\hat{a}]|\psi\rangle =\sup_{||\psi||=1} \int d\mu({z,\bar{z}})d\mu(w,\bar{w})\psi^*(z)\psi(\bar{w})\langle \bar{z}|[\hat{b},\hat{a}]^{\dagger}[\hat{b},\hat{a}] |w\rangle \label{lom}$$ A simple way to compute the supremum is to employ the method of Lagrange’s undetermined multiplier for extremizing the R.H.S of the above equation subject to the constraint $\langle\psi|\psi\rangle =1$. We therefore extremise the functional $$\textbf{B}[\psi^*(z),\psi(\bar{z});\lambda] := \int d\mu({z,\bar{z}})d\mu(w,\bar{w})\psi^*(z)\psi(\bar{w})\langle \bar{z}|[\hat{b},\hat{a}]^{\dagger}[\hat{b},\hat{a}] |w\rangle-\lambda\int d\mu(\bar{z},z) [\psi^*(z)\psi(\bar{z})-1]$$ with $\lambda$ being a real valued Lagrange’s multiplier enforcing the constraint $\langle\psi|\psi\rangle =1$. By varying $\textbf{B}[\psi^*(z),\psi(\bar{z});\lambda]$ with respect to $\psi(\bar{z}),\psi^*(z)$ we get the following pair of equivalent equations, $$\begin{aligned}
\langle\psi|[\hat{b},\hat{a}]^{\dagger}[\hat{b},\hat{a}]|z\rangle &=\lambda \langle\psi|z\rangle\nonumber\\
\langle\bar{z}|[\hat{b},\hat{a}]^{\dagger}[\hat{b},\hat{a}]|\psi\rangle &=\lambda \langle\bar{z}|\psi\rangle.\label{A9}\end{aligned}$$ In carrying out this functional differentiation we have used the basic relations like (\[A70\]) and (\[A71\]) (see Appendix-A). Now since (\[A9\]) holds for arbitrary $|\psi\rangle$ and $|z\rangle$, we have the matrix elements of the operator $[\hat{b},\hat{a}]^{\dagger}[\hat{b},\hat{a}]$ between any pair of states $|\psi\rangle$ and $|\phi\rangle$ must satisfy $\langle\psi|[\hat{b},\hat{a}]^{\dagger}[\hat{b},\hat{a}]|\phi\rangle =\lambda\langle\psi|\phi\rangle;\,\,\, \forall\,\,\, |\psi\rangle,|\phi\rangle\in\mathcal{H}_c$. Consequently, we must have the following operator identity holding: $$[\hat{b},\hat{a}]^{\dagger}[\hat{b},\hat{a}]=\lambda\label{bp}$$ The structure of this equation apparently indicates that $\hat{a}$ can only be related linearly to $\hat{b}$ and $\hat{b}^{\dagger}$, so that a real number is produced in the R.H.S as can be seen from the general structure of the algebra element $\hat{a}$ (\[poly\]). This, however, is not entirely true! Indeed, as has been shown in [@Chaoba] that there exists finite dimensional matrix solution for $\hat{a}$, so that the positive operator $[\hat{b},\hat{a}]^{\dagger}[\hat{b},\hat{a}]$ is only a diagonal matrix, which is not proportional to the unit matrix, so that one can identify states $|\psi_i\rangle \in \mathcal{H}_c$ which corresponds to local extrema and with the associated eigen value $\lambda_i\geq 0$ as $[\hat{b},\hat{a}]^{\dagger}[\hat{b},\hat{a}]|\psi_i\rangle =\lambda_i|\psi_i\rangle$ . Eventually, one can just read-off the maximum eigenvalue $\lambda_{max}$ and identify $\Vert[\hat{b},\hat{a}]\Vert_{op}=\sqrt{\lambda_{max}}$ (see Appendix B). These finite dimensional matrix solutions of $\hat{a}$ which should result in a distance with $\lambda_{max}=\frac{\theta}{2}$ are employed to compute distances between the harmonic oscillator states $\rho_n$ and $\rho_m$: $d(\rho_n,\rho_m)$, where they serve as an optimal element. Here, on the contrary, we are interested in computing the distance between a pair of coherent states $\rho_z$ and $\rho_w$ (\[hg\]), where we can see from the very definition (\[coh\]) of coherent states itself that we need a non-trivial infinite dimentional solution for ’$\hat{a}$ ’ which should result in a distance, that have a manifest invariance property under ISO(2) [@Chaoba]. Further, in this case, we can also provide an upper bound to the distance (\[hg\]). It will be useful to recapitulate this derivation here very briefly. Here one introduces a one- parameter family of pure states $\rho_{\mu z}:=|\mu z\rangle\langle\mu z |$ with $\mu\in[0,1]$ being a real parameter, interpolating between $\rho_0$ and $\rho_z$ (Note that, we have taken, without loss of generality, $w=0$, by invoking the above mentioned ISO(2) invariance). We then re-write the expression occurring in the RHS of (\[hg\]) as, $$|\rho_z(\hat{a})-\rho_0(\hat{a})|=\left|\int_0^1 d\mu \frac{d\rho_{\mu z}(\hat{a})}{d\mu}\right| \leq \int_0^1 d\mu \left|\frac{d\rho_{\mu z}(\hat{a})}{d\mu}\right| = \int_0^1 d\mu \left|\bar{z}\rho_{\mu z}[\hat{b},\hat{a}]+z\rho_{\mu z}[\hat{b},\hat{a}]^{\dagger}\right|\label{B1}$$ By applying Cauchy-Schwartz inequality, this inequality can further be simplified as , $$|\rho_z(\hat{a})-\rho_0(\hat{a})| \leq \sqrt{2}|z|\int_0^1 d\mu \sqrt{\left|\rho_{\mu z}[\hat{b},\hat{a}]\right|^2+\left|\rho_{\mu z}[\hat{b},\hat{a}]^{\dagger}\right|^2} \leq 2|z|\sqrt{ \Vert [\hat{b},\hat{a}]^{\dagger}[\hat{b},\hat{a}]\Vert_{op}} = 2|z|\,\,\Vert [\hat{b},\hat{a}\Vert_{op}\label{B2}$$ Finally making use of the ball condition (\[A3\]), we get the desired upper bound on the distance as, $$d(\rho_z,\rho_0) \leq \sqrt{2\theta} |z|\label{B3}$$ Equivalently by re-introducing z and $w$ variables again by invoking ISO(2) symmetry we have, $$d(\rho_z,\rho_w) \leq \sqrt{2\theta} |z-w|\label{B4}$$ We can now take the ansatz for $\hat{a}$ as $\hat{a}=\xi\hat{b}+\bar{\xi}\hat{b}^{\dagger}$ with $\xi\in\mathbb{C}$ (as the algebra element should be hermitian). We have from (\[bp\]), $|\xi|^2 =\lambda$. To evaluate the value of $\lambda$ we make use of (\[ballcondition\]), (\[lom\])and (\[bp\]) to have, $$\Vert[\hat{b},\hat{a}]\Vert_{op}= \sup_{\Vert\psi\Vert=1}\sqrt{\langle\psi|[\hat{b},\hat{a}]^{\dagger}[\hat{b},\hat{a}]|\psi\rangle} =\sqrt{\lambda} = \sqrt{\frac{\theta}{2}}$$ So the value of $\lambda$ for which the Ball reaches its supremum value is $\lambda=\frac{\theta}{2}$. So the optimal algebra element $\hat{a}_s$ lies within the one parameter family given by the phase $\alpha$ as, $$\hat{a}_s \in \left\{\sqrt{\frac{\theta}{2}}(\hat{b}e^{-i\alpha}+\hat{b}^{\dagger}e^{i\alpha});\,\, 0\leq\alpha<2\pi\right\}\label{set}$$ where we have taken $\xi =e^{-i\alpha} \sqrt{\frac{\theta}{2}}$. One can further corroborate this by observing that the matrix element of $[\hat{b},\hat{a}_s]$ with $\hat{a}_s$ given by (\[set\]), in the Bergman-Fock basis (\[A67\]) is obtained as, $$_B\langle \bar{z}^{\prime}|[\hat{b},\hat{a}]|z\rangle_B=\sqrt{\frac{\theta}{2}}e^{i\alpha}e^{\bar{z}^{\prime}z}$$ where we recognize the occurrence of Dirac’s delta function in this Bergman-Fock space: $e^{\bar{z}^{\prime}z}=\delta(\bar{z}^{\prime},z) $ (see (\[A71\] in Appendix A) , so that $[\hat{b},\hat{a}_s]$ is proportional to the unit matrix, when written in this continuous Bergman-Fock basis, enabling one to just read-off the operator norm. It should be noted here that, in contrast to the finite dimensional matrix $\hat{a}_s$ (see (\[form3\]) in Appendix-B ) , the $\hat{a}_s$ in (\[set\]) yields $\lambda=\frac{\theta}{2}$ as the unique eigenvalue of $[\hat{b},\hat{a}]^{\dagger}[\hat{b},\hat{a}]$, which is now infinitely degenerate in the eigen-basis provided by whole continuum set of Fock-bergman coherent states $|z\rangle$ (\[A67\]) and therefore is independent of $z$. This plays a vital role in making the distances invariant under translation, in contrast to the distance between the harmonic oscillator states . For example $d(\rho_{n+1},\rho_n)\neq d(\rho_{n+2},\rho_{n+1})$ [@Chaoba; @Mart3]. One can also see that with this identification of $\lambda$, the upper bound on the distance indeed takes the form of (\[B4\]). Now substituting (\[set\]) in (\[hg\]) the distance is essentially found to be given by $$d(\rho_z,\rho_w)=\sup_{\hat{a}\in\mathcal{A}}\,\,\left|\langle z|\hat{a}_s|z\rangle -\langle w|\hat{a}_s|w\rangle\right|= \sqrt{\frac{\theta}{2}}\,\, \sup_{\alpha} \,\,|(z-w)e^{-i\alpha} +(\bar{z}-\bar{w})e^{i\alpha}|$$ Note that as $\rho_z,\,\rho_w$ are normalised pure state density matrices, we have to use the normalised coherent state basis here for distance calculation as defined in (\[coh\]), instead of Fock-Bergman basis[^3]. We can now parametrize the complex number by polar decomposition as $z-w=|z-w|e^{i\beta},\,\,\bar{z}-\bar{w}=|z-w|e^{-i\beta}$. The optimal algebra element for which the supremum value is attained can easily be recognised to be the one for which $\alpha=\beta$ yielding the desired distance between two arbitrary pure states, $$d(\rho_z,\rho_w) = \sqrt{2\theta}\,\,|z-w|\label{mp}$$ which precisely is the upper bound (\[B4\]). This inequality is therefore saturated by $\hat{a}_s$ (\[set\]). Splitting dimensionless complex coordinates $z$ and $w$ into real and imaginary parts involving dimension-ful coordinates, in analogy with the splitting of $\hat{b}$ in (\[k\]) as $$z=\frac{1}{\sqrt{2\theta}}(x_1+ix_2)\,\,\,\textrm{and}\,\,\, w=\frac{1}{\sqrt{2\theta}} (y_1+iy_2)\label{basis}$$ we can rewrite (\[mp\]) as, $$d(\rho_z,\rho_w)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}\label{mmp}$$ reproducing the usual Euclidean distance having the complete ISO(2) symmetry as in[@Mart; @Chaoba], with no NC corrections. Before we conclude this section, we would like to point out that the form of the algebra element, given in (\[set\]), are not the elements of the algebra: $a_s \notin \mathcal{A}$. But these elements can be regarded to belong to an enlarged algebra, called multiplier algebra (For details see [@Mart; @Bimo]). We shall not elaborate on this any further here and just mention that at least in our case the operator product of $\rho_z$ and $\hat{a}_s$ in (\[set\]) is indeed a HS operator, as follows from the fact that $\Vert \rho_z\hat{a}_s\Vert_{HS}^2= tr_{\mathcal{H}_c}((\rho_z\hat{a}_s)^{\dagger}(\rho_z\hat{a}_s))=tr_{\mathcal{H}_c}(\hat{a}_s\rho_z\hat{a}_s) < \infty$. Furthermore, this will be a recurrent feature in the rest of the paper i.e. we will have to consider the multiplier algebra in the Lorentzian case also, which we shall deal with in the sequel.
Lorentzian Spectral Triple
==========================
In this section we would like to use a formulation required in the computation of distances between a pair of time-like separated events in commutative and non-commutative in Lorentzian plane. Although, to the best of our knowledge, a general consensus regarding its recently proposed various axiomatic frameworks is still lacking, we nevertheless try here to follow [@Franco1; @Franco2; @Franco3] and adapt it with our HS - operatorial formulation, recapitulated in the previous section in the Euclidean case and introduce the basic tools like spectral triples for both commutative and non-commutative spaces, apart from the algebraic version of causality required to facilitate a similar computation of distances in commutative and non-commutative (NC) Lorentzian plane to be taken up in the next section.\
\
We begin by providing a brief outline of the very basic ingredients required to construct a Lorentzian spectral triple. This is given by the set of data ($\mathcal{A},\mathcal{H},\mathcal{D},\mathcal{J},\chi$) with
- [A Hilbert space $\mathcal{H}$]{}
- [A non-unital algebra $\mathcal{A}$ with a suitable and faithful representation $\pi$ on $\mathcal{H}$]{}
- [A Dirac operator $\mathcal{D}$ taken to be an unbounded operator, but $[\mathcal{D},\pi(a)]$ is bounded.]{}
- [An operator $\mathcal{J}$ called Fundamental symmetry acting on $\mathcal{H}$ satisfying boundedness condition along with $\mathcal{J}^2=1,\, \mathcal{J}^*=\mathcal{J},\,[\mathcal{J},a]=0,\, \forall a\in \tilde{\mathcal{A}}, \,\, \textrm{where} \,\,\tilde{\mathcal{A}}\,\, \textrm{is a unitized version of} \, \mathcal{A}$ and the Dirac operator fulfills $ \mathcal{D}^*=-\mathcal{J}\mathcal{D}\mathcal{J}$]{}
- [For an even Lorentzian spectral triple we take $\chi$ to be a grading operator fulfilling the following conditions: $$\chi^*=\chi;\, \chi^2=1;\,\,\{\chi,\mathcal{J}\}=0;\,\{\mathcal{D},\chi\}=0$$]{}
The operator $\mathcal{J}$ captures the Lorentzian signature of the space by turning the Hilbert space into a Krein space where the positive definite inner product of the Hilbert space: $\langle\cdot,\cdot\rangle$ turns to an indefinite inner product space :$\,\,(\cdot,\cdot) =\langle\cdot,\mathcal{J}\cdot\rangle$ upon the insertion of the fundamental symmetry operator $\mathcal{J}$, so the Dirac operator in this inner product space becomes a non-hermitian operator and to render it as a skew Krein self-adjoint operator, the condition $\mathcal{D}^*=-\mathcal{J}\mathcal{D}\mathcal{J}$ is imposed. We shall elaborate on this construction in section 4.
### Commutative Lorentzian Spectral Triple and Distance Formula
We now discuss, in this subsection, the example of a commutative Lorentzian space-time taken to be globally hyperbolic $n$ dimensional manifold $\mathcal{M}$ which is described by a commutative spectral triple [@Franco1] constructed with :
- [Hilbert space $\mathcal{H}=L^2(\mathcal{M},S)$ of square integrable spinorial sections over $\mathcal{M}$]{}
- [Dirac operator $\mathcal{D}=-i\gamma^{\mu}\nabla_{\mu}$]{}
- [Algebra $\mathcal{A}=C^{\infty}_0(\mathcal{M})$ of infinitely differentiable smooth real functions (Ideally , one should introduce the unitized version of the algebra- a fact that we are ignoring for the time being. We shall, however, will make some pertinent observations later in the paper. )]{}
- [Fundamental symmetry $\mathcal{J}= i\gamma^0$ where $\gamma^0 $ is the time- component of Dirac’s $\gamma$- matrices $\gamma^{\mu}$ satisfying the fundamental Clifford algebra : $\{\gamma^{\mu},\gamma^{\nu}\}=2\eta^{\mu\nu}\textbf{I}$ and the flat metric $\eta_{\mu\nu}=$diag(-,+,+,+) ]{}
- [Grading operator $\chi=(-i)^{\frac{n}{2}+1}\gamma^0\cdots\gamma^{n-1}$]{}
For an even dimensional Lorentzian manifold i.e. if $n$ is even, then the distance between two points is essentially identified with the distance between the associated pure states given by Dirac’s $\delta$- functional serving as evaluation maps and is given by [@Franco1] $$d(p,q)=\inf_{a\in\mathcal{B}}\{[\delta_q(a)-\delta_p(a)]^+\}=\inf_{a\in \mathcal{B}}\{[a(q)-a(p)]^+ \}\label{f}$$ where $[\alpha]^+=\textrm{max}\{0,\alpha\},$ and $$\mathcal{B}=\{a\in\mathcal{A} :\,\textbf{B}:= \langle\phi,\mathcal{J}([\mathcal{D},a]\pm i\chi)\phi\rangle< 0,\,\,\,\forall\phi\in\mathcal{H}\}\subset \mathcal{A}\label{ballcond}$$ In the next subsection, following [@Franco1; @Eckstein], we shall see how this particular Ball condition essentially captures the conditions of steep functions as a subset of causal functions in an algebraic set-up.
### Non-commutative Lorentzian Spectral Triple and Distance Formula
Now the Spectral triple for Non-commutative Lorentzian space-time, taken to be 2D Moyal plane, can easily be generalised. It is comprised of
- [a Hilbert space $\mathcal{H}=\mathcal{H}_c\otimes\mathbb{C}^2$]{}
- [Non-commutative algebra $\mathcal{A}=\mathcal{H}_q$ with a suitable representation $\pi$, again taken to be diagonal one i.e. $\pi(a):=\textrm{diag}(a,a)$.]{}
- [Dirac operator $\mathcal{D}=-i\gamma^{\mu}\nabla_{\mu}$]{}
The fundamental symmetry operator and grading operator are same as defined in the commutative spectral triple. For non-commutative plane, however, the concept of a point is absent . So the distance will be calculated between two pure states $\rho_1$ and $\rho_2$, just like its Euclidean counter-part in section 2. The Lorentzian distance between two such states $\rho_1$ and $\rho_2$ is defined as $$d(\rho_1,\rho_2)=\inf_{a\in\mathcal{B}}\{[\rho_2(a) - \rho_1(a)]^+\}\label{ld}$$ where $[\alpha]^+ =\textrm{max}\{0,\alpha\}$ and the ball $\mathcal{B}$ is defined as $$\mathcal{B}=\{a\,:\,\textbf{B}=\langle\phi,\mathcal{J}([\mathcal{D},\pi(a)]\pm i\chi)\phi\rangle < 0, \forall \phi\in\mathcal{H} \}\subset\mathcal{A}\label{ball}$$
An Algebraic Construction of Causality
--------------------------------------
In this section we provide a very brief review following [@Franco1; @Franco4; @Eckstein] to show how the effect of causality is captured as an algebraic condition in Lorentzian space.\
**Theorem :** A function $a\in C^1(\mathcal{M},\mathbb{R})$ is causal *iff* $$\forall \phi\in\mathcal{H} ,\,\,\langle\phi,\mathcal{J}[\mathcal{D},a]\phi\rangle < 0 \label{causal}$$ where $\mathcal{J},\mathcal{D},\mathcal{H}$ are defined as above. A sketch of the proof of this algebraic version can be given as following :\
Recall that the causal property of an absolutely continuous real valued $C^1$ function $a\in C^1(\mathcal{M},\mathbb{R})$, can be fully characterised by two conditions i.e. $$\eta(\nabla a,\nabla a)< 0 \Rightarrow a_{,i}^2< a_{,0}^2 \,\,\,\,\,\,\,\textrm{and}\,\,\,\,\,\,\,\, \eta(\nabla a, \nabla T)=-a_{,0}< 0 \label{conditionnn}$$ where $T$ is taken to be a temporal function and is typically taken to be $T=x^0 $ itself so that $\nabla T$ is a time like vector and we have used a $n$-dimensional Lorentzian flat metric $\eta_{\mu\nu}=\textrm{diag}(-,+,+,+)$. If the conditions (\[conditionnn\]) is false at some point of the manifold then from continuity of derivative it will be false in some of its neighbourhood as well. Now using $\mathcal{J}=i\gamma^0$ and $\mathcal{D} = -i\gamma^{\mu}\partial_{\mu}$, as above , we readily obtain, $$\mathcal{J}[\mathcal{D},a]= -a_{,0}+K;\,\,\,\,\,\,\, K:= \gamma^0\gamma^ia_{, i}$$ Now this $K$ can be shown to satisfy $K^2=\eta^{ij} \partial_i a\partial_j a $, so that the spectrum of $K$ is essentially given by Spec$K= \{\pm\Vert\eta^{ij}\partial_i a\partial_j a\Vert^{\frac{1}{2}}\}$. Now since the reduced metric $\eta^{ij}$ is positive definite everywhere (in fact , in our case it is just identity matrix), the matrix $\mathcal{J}[\mathcal{D},a ] $ is point-wise negative and one readily verifies (\[causal\]).\
\
\
An important subspace of the causal function , is given by so called *steep functions* , fulfilling the property, $$\eta(\nabla a,\nabla a) < -1.$$ A corresponding algebraic version of causality of this steep functions can be shown to be given by[^4] $$\forall\phi\in\mathcal{H}, \langle\phi,\mathcal{J}([\mathcal{D},a]\pm i\chi)\phi\rangle < 0\label{ccausal}$$ An elementary proof of this algebraic version has been provided in [@Franco1; @Franco4] , for even dimensional flat Lorentzian manifold. The basic idea behind the proof is to consider, the product manifold $\tilde{\mathcal{M}}=\mathcal{M}\times \mathbb{R}$ with the associated metric $$\tilde{\eta}= \left( \begin{array}{c|c}
\eta & 0 \\
\hline
0&1
\end{array} \right)$$ and introduce the function $\tilde{a}:= a-x^n \in C^1(\mathcal{\tilde{M}},\mathbb{R})$. One can then show that $\tilde{a}$ is causal in $(\mathcal{\tilde{M}},\tilde{\eta})$ implies that $a$ is a steep function in $(\mathcal{M},\mathbb{R})$.\
The stage is therefore ready for its generalisation to NC spaces. All that we need to do there is to introduce an appropriate NC algebra $\mathcal{A}$ and replace the above real function $a$ by suitable hermitian elements $a\in \mathcal{\tilde{A}}$ (where $\tilde{\mathcal{A}}$ is the unitized version of $\mathcal{A}$), fulfilling $$\forall\phi\in\mathcal{H}, \langle\phi,\mathcal{J}([\mathcal{D},\pi (\hat{a})]\pm i\chi)\phi\rangle < 0 \label{equ1}$$ Having in our hand, an algebraic condition (\[equ1\]) to capture the space of steep functions on the manifold we shall use it as *Ball condition* in Lorentzian manifold.\
\
Now we can introduce a convex cone $\mathcal{C}$ to be the set of causal algebra elements satisfying (\[causal\]) fulfilling a further condition , $$\overline{Span_{\mathbb{C}}\mathcal{C}} = \overline{\tilde{\mathcal{A}}}\label{causal2}$$ By this a partial order relation is induced on the space of states $S(\mathcal{A})$, which is called the causal relation [@Wallet], $$\forall \omega_1,\omega_2 \in S(\mathcal{A}), \omega_1 \preceq\omega_2 \,\,\,\textrm{iff}\,\,\, \forall a \in \mathcal{C}, \omega_1(a) \leq \omega_2(a)$$
Construction of Dirac Operator, Krein Space and Ball Condition
==============================================================
In this section we try to mimic our Hilbert-Schmidt formulation as described in section 2 to calculate the distances in Lorentzian commutative and non-commutative spaces. Both of the spaces have the same structure of Dirac operator $\mathcal{D}=\hat{P}_{\mu}\otimes\gamma^{\mu}$, which by default act on the Hilbert spaces $L^2(\mathbb{R}^{1,1})\otimes\mathbb{C}^2$ and $\mathcal{H}_q\otimes\mathbb{C}^2$ respectively for the commutative and non-commutative planes. A generic spinor of this Hilbert space can be written as $\Psi= \begin{pmatrix}
|\psi_1) \\|\psi_2)
\end{pmatrix}$ ,with $|\psi) \in L^2(\mathbb{R}^{1,1})$ and $\mathcal{H}_q $ in the respective cases.\
\
For pseudo-euclidean signature (-,+) of our space-time $\mathbb{R}^{1,1}$, we can take $\gamma^{0}=-i\sigma^1; \gamma^1=\sigma^2$ so that $(\gamma^0)^2=-1$ , $(\gamma^1)^2=1$ and $ \{\gamma^{\mu},\gamma^{\nu}\}=2\eta^{\mu\nu}$. With these the Dirac operator takes the following form:
$$\mathcal{D} =-i\begin{pmatrix}
0& \hat{P}_0+\hat{P}_1\\\hat{ P}_0- \hat{P}_1&0
\end{pmatrix}
\label{d}$$
As it stands, the Dirac operator is not a self adjoint operator. We can however convert the Hilbert space into a Krein space , where the operator will become a skew Krein self-adjoint operator. It will therefore be useful to recall the definition of Krein space first and study a simple example of $\mathbb{C}^2$ , whose Krein counter-part will be constructed and eventually be used to construct the Krein counterpart of the entire Hilbert space $\mathcal{H}=\mathcal{H}_q\otimes \mathbb{C}^2$ occurring in the above spectral triple.\
\
**Def**:\
If an indefinite inner product space $\mathcal{K}$ can be split into two sub-spaces as $\mathcal{K}=\mathcal{K}_+\oplus \mathcal{K}_-$ which are mutually orthogonal, complete (in the norm induced on them) having positive and negative definite inner product respectively and if further the inner product on $\mathcal{K}$ is non degenerate, then $\mathcal{K}$ is called a Krein space [@Franco2; @Franco4].\
\
For every such decomposition we can associate a fundamental symmetry operator $\mathcal{J}$ as $\mathcal{J}=\textrm{id}_+ \oplus (-\textrm{id}_-)$ respecting $\mathcal{J}^2=1$ and whose insertion can turn the subspace with negative inner product $(\cdot,\cdot)$ to the sub-space with a positive definite inner product $\langle\cdot,\cdot\rangle$ and vice-verse with the following operation : $ (\cdot, \mathcal{J}\cdot)=\langle \cdot ,\cdot\rangle$ or equivalently $\langle\cdot,\mathcal{J}\cdot\rangle = (\cdot,\cdot)$.\
\
\
**Converting $\mathbb{C}^2$ and $L^2(\mathbb{R}^{1,1})\otimes\mathbb{C}^2$ and $\mathcal{H}_q\otimes\mathbb{C}^2$ into Krein spaces**\
\
The 2 dimensional complex vector space $\mathbb{C}^2$ can be regarded as direct sum of two sub-spaces $\mathbb{C}^1_{\pm}$ as follows , $$\mathbb{C}^2= \textrm{Span} \left\{\frac{1}{\sqrt{2}}\begin{pmatrix}
1\\1
\end{pmatrix},\frac{1}{\sqrt{2}}\begin{pmatrix}
1\\-1
\end{pmatrix}\right\} =\textrm{Span} \left\{\frac{1}{\sqrt{2}}\begin{pmatrix}
1\\1
\end{pmatrix}\right\} \oplus \textrm{Span} \left\{\frac{1}{\sqrt{2}}\begin{pmatrix}
1\\-1
\end{pmatrix}\right\}:=\mathbb{C}^1_+\oplus \mathbb{C}^1_-$$ with associated projectors $\mathbb{P}^{\pm} = \frac{1}{2}(\textbf{1}\pm \sigma_1)$ projecting a generic two component vector $\begin{pmatrix}
\alpha\\ \beta
\end{pmatrix} \in \mathbb{C}^2 $ to the subspaces $\mathbb{C}^1_+ $and $\mathbb{C}_-^1$ respectively where both the subspaces are naturally endowed with positive definite inner products. Now let us take the fundamental symmetry operator $\mathcal{J}=\sigma_1$ which makes the $\mathbb{C}^1_-$ subspace a negative definite inner product space but retains the positive-definiteness property of $\mathbb{C}^1_+$ space with the following insertion : $$\langle\cdot,\mathcal{J}\cdot\rangle=\langle\cdot,\sigma_1\cdot\rangle =(\cdot,\cdot).\label{inn}$$ So now with the new inner product $(\cdot,\cdot)$ defined in (\[inn\]) we can call our $\mathbb{C}^2$ space as a Krein space.\
\
With this it can now be shown trivially that $\mathcal{H}_q\otimes \mathbb{C}^2$ equipped with the inner product $\langle\cdot,\mathcal{J}\cdot\rangle $, where the used fundamental symmetry operator now is $\mathcal{J}=\textbf{I}\otimes\sigma_1$ ,is an indefinite inner product space or Krein space where $\mathcal{H}_q\otimes\mathbb{C}^2$ splits as $$\mathcal{H}_q\otimes\mathbb{C}^2=( \mathcal{H}_q\otimes\mathbb{C}^1_+)\oplus(\mathcal{H}_q\otimes\mathbb{C}^1_-).$$ With this fundamental symmetry $\mathcal{J}$, it can indeed be checked quite easily that $\mathcal{D}^*=-\mathcal{J}\mathcal{D}\mathcal{J}$ proving $\mathcal{D}$ (\[d\]) to be skew Krein self-adjoint,\
\
The grading operator as defined in section - 3.0.1 comes out to be just $\chi=\textbf{1}\otimes\sigma^3 $. We take a suitable representation of the algebra element $a$ as $\pi(a)=\textrm{diag}(a,a)$. Upto this all the elements of spectral triples are assumed to be quite generic and applicable for both commutative and non-commutative Lorentzian plane. So a general Ball condition can now be derived from (\[ballcond\]) as, $$\textbf{B}=\langle\Psi, \mathcal{J}\{[\mathcal{D},\pi(a)]+i\chi\}\Psi\rangle =\begin{pmatrix}
\langle\psi_1|&\langle\psi_2|\end{pmatrix}
\begin{pmatrix}
-i[\hat{P_0}-\hat{P_1},a]&-i\\ i&-i[\hat{P_0}+\hat{P_1},a]
\end{pmatrix}
\begin{pmatrix}
|\psi_1\rangle\\|\psi_2\rangle
\end{pmatrix}< 0\,\,\, \forall\,\,\, \Psi\in\mathcal{H}\label{pbp}$$ It can be re-cast as, $$\textbf{B}= -i\langle\psi_1|[\hat{P}_0-\hat{P_1},a]|\psi_1\rangle - i\langle\psi_2|[\hat{P}_0+\hat{P}_1,a]|\psi_2\rangle+i[\langle\psi_2|\psi_1\rangle -\langle\psi_1|\psi_2\rangle ] <0 \label{ball2}$$ Above equation can be taken as a master equation or principal ball condition. We now specialise into two separate cases for commutative and non-commutative planes.
Distance in Commutative Lorentzian Plane
----------------------------------------
As a warm up exercise we first undertake the computation of the distance on the (1+1) dimensional flat commutative Lorentzian manifold itself. The spectral triple for commutative Lorentzian manifold $\mathbb{R}^{1,1}$ is defined through the algebra of smooth functions over $\mathbb{R}^{1,1}:\mathcal{A}=C^{\infty}_0(\mathbb{R}^{1,1})$, a Hilbert space [^5] $\mathcal{H}=L^2(\mathbb{R}^{1,1}) \otimes \mathbb{C}^2$, consisting of generic spinor like $
\Psi= \begin{pmatrix}
|\psi_1) \\|\psi_2)
\end{pmatrix}$ where $|\psi_i)$ are abstract kets whose representation in $|t,x\rangle$ basis becomes $L^2(\mathbb{R}^{1,1})$ element. With the Dirac operator already defined in previous section we arrived at a simplified ball condition (\[ball2\]). Inserting completeness relation of $|t,x\rangle \equiv |x\rangle$ basis: $\int dt\,dx\,|t,x\rangle\langle t,x| =\textbf{1}$, (\[ball2\]) can be rewritten as, $$\begin{aligned}
-i\int d^2x\,d^2y\, \psi_1^*(x)\psi_1(y)\langle x|[\hat{P}_0-\hat{P}_1,a]|y\rangle -i\int d^2x\, d^2y\,\psi_2^*(x)\psi_2(y)& \langle x|[\hat{P}_0+\hat{P}_1,a]|y\rangle \nonumber\\
&+i\int d^2x [\psi_2^*(x)\psi_1(x)-\psi_1^*(x)\psi_2(x)]<0\label{ball3}
\end{aligned}$$ Here the action of momentum operator on the $C^{\infty}$ function $a(t,x):=a(x)$ takes place through commutator bracket and it gives, $$\langle x |[\hat{P}_{\mu},a(x)]|y\rangle = -i\partial_{\mu}a(x) \delta^{(2)} (x-y) \label{hj}$$ With this (\[ball3\]) further simplies as $$\int d^2x \,\,[ (\partial_0 a)(-|\psi_1(x)|^2-|\psi_2(x)|^2)+(\partial_1 a)(|\psi_1(x)|^2-|\psi_2(x)|^2)] < i\int d^2x\, [\psi_1^*(x)\psi_2(x)-\psi_2^*(x)\psi_1(x)] \label{hjj}$$ Now this condition is valid for each and every point of the manifold for arbitrary $\psi_1(x)$ and $\psi_2(x)$. So this condition applies for the integrand itself point-wise and enables us to write, $$(\nabla_{\mu}a)V^{\mu} < i(\psi_1^*\psi_2-\psi_2^*\psi_1) \,\,; V_0=-V^0=|\psi_1|^2+|\psi_2|^2, V_1= V^1=|\psi_1|^2-|\psi_2|^2 \label{first}$$
where $V=(V^0,V^1)$ is a time-like two vector and $a\in \mathcal{A}$ belong to the set of steep functions i.e. $\eta(\nabla a,\nabla a) < -1 $ which means that $\nabla a$ too is a time-like vector and $(V^{\mu}\partial_{\mu}a)$ is intrinsically negative[^6], enabling us to recast (\[first\]) as $$\vert V^{\mu}\partial_{\mu}a\vert = -(V^{\mu}\partial_{\mu}a)> i(\psi_1\psi_2^*-\psi_2\psi_1^*)= - 2|\psi_1||\psi_2|\sin(\alpha_1-\alpha_2)\label{eq3}$$ where the phases $\alpha_1$ and $\alpha_2$ are defined as $\psi_1= |\psi_1|e^{i\alpha_1}$ and $\psi_2=|\psi_2|e^{i\alpha_2}$. Now since the computation of Lorentzian distance in (\[f\]) involves the computation of infimum we have to search for the optimal algebra element for which the infimum is attained . For that, it will be advantageous to consider a slight variant of the inequality (\[ball3\],\[eq3\]) as $$\textbf{B}\leq 0\,\, \textrm{i.e.}\,\, \vert V^{\mu}\partial_{\mu}a\vert \geq -2|\psi_1||\psi_2|\sin(\alpha_1-\alpha_2)\label{A1}$$ The inclusion of the equality sign here will not have any effect on the computation of the infimum, as the set determined by (\[ball3\],\[eq3\]) is a dense subset of (\[A1\]). At this stage we we first hold $|\psi_1|$ and $|\psi_2|$ fixed and vary $\alpha_1,\alpha_2$ so that the saturation condition in (\[A1\]) is satisfied. Clearly the maximal value reached by the R.H.S above will correspond to $ 2|\psi_1||\psi_2|$ with the choice $\alpha_2-\alpha_1=\frac{\pi}{2}$ and (\[A1\]) reduces to the following form $$\vert V^{\mu}\partial_{\mu}a\vert\geq 2|\psi_1||\psi_2|\label{A2}$$ Correspondingly the generic form of the spinor $\psi$ which maximises the R.H.S is $\begin{pmatrix}
|\psi_1|\\i|\psi_2|
\end{pmatrix}e^{i\alpha_1}$. On the other hand we can now apply *reverse* Cauchy-Schwarz inequality [@Franco1] to write $$\vert\partial_{\mu}a V^{\mu}\vert \geq \Vert\nabla_{\mu}a\Vert_L \Vert V^{\mu}\Vert_L. \label{eq9}$$ Note that here L in the subscript is a reminder of Lorentzian norm defined as $\Vert v\Vert _L= \sqrt{-\eta(v,v)}$ for a time-like vector $v$. In the next stage , we vary $|\psi_1|$ and $|\psi_2|$, so that $V$ becomes collinear with $\nabla a$ and (\[eq9\]) becomes, $$\vert V^{\mu}\partial_{\mu}a \vert = \Vert\nabla_{\mu}a\Vert_L \Vert V^{\mu}\Vert_L\label{eq2}$$ We now make use of (\[first\]), so that by (\[A1\]) and (\[eq2\]) we finally get $$\Vert \nabla a\Vert_L \geq 1\label{eq1}$$ We therefore look for a solution of $a$ satisfying $$\Vert\nabla a \Vert_L = 1\label{eq12}$$ Equivalently , the search can be restricted to the solution set of the following differential equation satisfied by real functions $a(x,t)$ satisfying $$\left(\frac{\partial a}{\partial t}\right)^2-\left(\frac{\partial a}{\partial x}\right)^2=1\label{eq4}$$ yielding the following one-parameter $(\lambda)$ solution set of $a(t,x)$: $$a(t,x)=t\cosh\lambda +x\sinh\lambda\label{form}$$
By (\[f\]) the distance between two points say $P(t_1,x_1),Q(t_2,x_2)$ in commutative Lorentzian manifold (where $P$ preceeds $Q$ chronologically i.e. P $\prec$ Q) is now given by, $$d(\delta_{P},\delta_{Q})=\inf_{a\in\mathcal{B}}\{[\delta_{Q}(a)-\delta_{P}(a)]^+\} =\inf_{a\in\mathcal{B}}\{[a(t_2,x_2)-a(t_1,x_1)]^+\}\label{m}$$ where $\delta_P,\delta_Q$ are pure states representing the respective events $(t_1,x_1)$ and $(t_2,x_2)$ in the forward light-cone of the commutative plane. Eq (\[m\]) can now be recast as, $$d(\delta_{P},\delta_{Q})= \inf_{\lambda}\{[(t_2-t_1)\cosh\lambda+(x_2-x_1)\sinh\lambda]^+\}\label{dis1}$$ which on minimisation with respect to $\lambda$ (to get the infimum value of the set ) gives , $$d(\delta_{P},\delta_{Q})= \sqrt{(t_2-t_1)^2-(x_2-x_1)^2}\label{dis}$$ Note the resultant distance in (\[dis1\]) automatically contains the causal information. If the events are not causally connected i.e.if $Q$ is not preceeded by $P$ then the quantity in the parenthesis of (\[dis1\]) will definitely be negative [^7] and the distance formula will result in zero value indicating that the distance is not symmetric under the interchange of pair of events.\
### Functional Derivative Approach
In this sub-section we shall try to address the same problem with functional differentiation approach which we will find to be easier to execute and will be easily adaptable to the case of computation of the distance in Lorentzian Moyal plane to be taken up in next sub-section. For this let us consider the left hand side of the ball condition (\[ball2\]) , $\textbf{B}[\psi_1,\psi_1^*,\psi_2,\psi_2^*]$ which can now be regarded as a functional of $\psi_1,\psi_1^*,\psi_2,\psi_2^*$: $$\textbf{B}[\psi_1,\psi_1^*,\psi_2,\psi_2^*]= -\int d^2x \left[ |\psi_1(x)|^2(\partial_0 a-\partial_1 a)+|\psi_2(x)|^2(\partial_0 a+\partial_1 a)+i\{\psi_1^*(x)\psi_2(x)-\psi_2^*(x)\psi_1(x)\}\right] \label{hjp}$$ Clearly the algebra element $a\in\mathcal{A}$ for which the infimum in (\[f\]) is reached giving the distance should be such that $$\sup_{\psi_1,\psi_2,\psi_1^*,\psi_2^*}\,\,\, \textbf{B}[\psi_1,\psi_2,\psi_1^*,\psi_2^*]=0\label{supremum}$$ In other words we want to extract a condition on the algebra element for which the functional $\textbf{B}[\psi_1,\psi_1^*,\psi_2,\psi_2^*]$ reaches its supremum value of zero. All that we need to do first is to maximise **B** when $\psi_1, \psi_2, \psi_1^*, \psi_2^*$ are varied arbitrarily, and show that the maximum of $\textbf{B}[\psi_1,\psi_1^*,\psi_2,\psi_2^*]$ corresponds to the zero value which coincides with the supremum, only with a suitable constraint on the algebra element $a\in\mathcal{A}$, which emerges as an offshoot. When the functional $\textbf{B}[\psi_1,\psi_1^*,\psi_2,\psi_2^*]$ is functionally differentiated with respect to $\psi_1(x),\psi_2(x) $, we get the following matrix equation, $$\begin{pmatrix}
-(\partial_0-\partial_1)a(x)&i\\
-i&-(\partial_0+\partial_1)a(x)
\end{pmatrix}
\begin{pmatrix}
\psi_1^*(x)\\ \psi_2^*(x)
\end{pmatrix}=0\label{matrix}$$ The variation with respect to $\psi_1^*(x)$ and $\psi_2^*(x)$ yields an equivalent matrix equation, which is just the complex conjugate of the above equation. If $\psi_1^*,\psi_2^*$ are zero then the equations are trivially satisfied. For allowing non-vanishing values of $\psi_1^*$ and $\psi_2^*$, albeit a linear and consistent pair of equations relating them, we must have vanishing determinant of the coefficient matrix in (\[matrix\]) yielding a condition on $a(x)$, which is precisely (\[eq4\]) itself, giving (\[form\]) for $a(t,x)$ . Now let us rewrite **B** using (\[form\]) as, $$\textbf{B}=-\int d^2x\,[e^{-\lambda}\psi_1^*(x)\psi_1(x)+e^{\lambda}\psi_2^*(x)\psi_2(x)+i\{\psi_1^*(x)\psi_2(x)-\psi_2^*(x)\psi_1(x)\}]$$ The following scaling transformations, $\psi_1(x)\to \psi_1^{\prime}(x):= e^{-\frac{\lambda}{2}}\psi_1(x);\,\, \psi_2(x)\to \psi_2^{\prime}(x):= e^{\frac{\lambda}{2}}\psi_2(x)$ allows us to write **B** in a simplified form as $$\textbf{B}=-\int d^2x\,[\psi_1^{\prime *}(x)\psi_1^{\prime}(x)+\psi_2^{\prime *}(x)\psi_2^{\prime}(x)+i\{\psi_1^{\prime *}(x)\psi_2^{\prime}(x)-\psi_2^{\prime *}(x)\psi_1^{\prime}(x)\}]$$ We now carry out a linear transformation of the following form which enables us to write **B** in terms of independent functions $\phi_{\pm}(x)$, defined as, $$\begin{pmatrix}
\phi_+(x)\\ \phi_-(x)
\end{pmatrix}=\frac{1}{2}
\begin{pmatrix}
1&i\\-1&i
\end{pmatrix}
\begin{pmatrix}
\psi_1^{\prime }(x)\\ \psi_2^{\prime }(x)
\end{pmatrix}\label{trans5}$$ A simple algebra then shows that **B** is really independent of $\phi_{+}(x)$ and depends on $\phi_-(x)$ only as, $$\textbf{B}=-2\int d^2x |\phi_-(x)|^2 \label{max}$$ Equation (\[max\]) represents an inverted parabola in ($\textbf{B}, |\phi_-|$) plane and clearly implies that , **B** reaches its supremum value zero for $|\phi_-|=0$. So the form of the algebra element (\[form\]) automatically ensures that **B** can not exceed zero whatever be $\phi_+$ and $\phi_-$. We therefore arrive eventually at the same distance function (\[dis\]).
Distance in 2D Lorentzian Moyal Plane
-------------------------------------
We finally take up the computation of the distance in the non-commutative case in this pen-ultimate section. To begin with note that a representation of the non-commutative 2-dimensional Moyal plane algebra : $[\hat{t},\hat{x}]=i\theta$ with Lorentzian signature (-,+) will be furnished again by the same Hilbert space $\mathcal{H}_c$ defined in (\[k\]). Here the creation and annihilation operators along with the *vacuum* $|0\rangle$ are defined analogously as : $$\hat{b}=\frac{\hat{t}+i\hat{x}}{\sqrt{2\theta}},\,\,\hat{b}^{\dagger}=\frac{\hat{t}-i\hat{x}}{\sqrt{2\theta}};\,\,\,\hat{b}|0\rangle =0 \label{b5}$$ However there are certain differences as well. To see this note that although the vacuum transforms under space-time translation as,$$|0\rangle \rightarrow |z\rangle =U(z,\bar{z})|0\rangle;\,\,\,U(z,\bar{z})= e^{-\bar{z}\hat{b}+z\hat{b}^{\dagger}}\label{coh2}$$ along all the raising/lowering operators transforming adjointly : $$\begin{aligned}
&\hat{b}\rightarrow\hat{b}_z=\hat{b}-z=U(z,\bar{z})\hat{b}U^{\dagger}(z,\bar{z});\,\,\,\hat{b}^{\dagger}\rightarrow\hat{b}_z^{\dagger}=\hat{b}^{\dagger}-\bar{z}=U(z,\bar{z})\hat{b}^{\dagger}U^{\dagger}(z,\bar{z});\nonumber\\
&\rho_0\rightarrow\rho_z=|z\rangle \langle z| =U \rho_0 U^{\dagger} ;\,\,\mathcal{D}\rightarrow\mathcal{D}_z= U(z,\bar{z})\mathcal{D}U^{\dagger}(z,\bar{z})\label{trans}
\end{aligned}$$ with $U(z,\bar{z})$ (\[coh2\]) being the same as its Euclidean counter-part, the situation is different when the Lorentz - boost is concerned. In fact the transformation under Lorentz boost, $$\begin{pmatrix}\hat{t}\\\hat{x}\end{pmatrix} \rightarrow \begin{pmatrix} \hat{t}^{\prime}\\\hat{x}^{\prime}\end{pmatrix} = \begin{pmatrix} \cosh\phi&\sinh\phi\\ \sinh\phi&\cosh\phi\end{pmatrix}\begin{pmatrix}\hat{t}\\ \hat{x}\end{pmatrix}\label{E1}$$, is unitarily implemented on Hilbert space $\mathcal{H}_c$ through adjoint action as: $$\begin{aligned}
&\hat{b}\rightarrow\hat{b_{\phi}}=\cosh\phi\,\hat{b}+i\sinh\phi\,\hat{b}^{\dagger}= U(\phi)\hat{b}U^{\dagger}(\phi) ;\,\,\,\,\hat{b}^{\dagger}\rightarrow\hat{b_{\phi}}^{\dagger}=\cosh\phi\,\hat{b}^{\dagger}-i\sinh\phi\,\hat{b}=U(\phi)\hat{b}^{\dagger}U^{\dagger}(\phi)\nonumber\\
& \rho_0\rightarrow\rho_{\phi}=|\phi\rangle\langle\phi |=U(\phi)\rho_0 U^{\dagger}(\phi);\,\,\,\mathcal{D}\rightarrow\mathcal{D}_{\phi}=U(\phi)\mathcal{D}U^{\dagger}(\phi)\label{LT}
\end{aligned}$$ where $U(\phi)= e^{-\frac{i\phi}{2}(\hat{b}^{2}+{\hat{b^{\dagger}}}^2)}$ is now a squeezing operator. Note that the coherent state also transforms by left action as $|z\rangle \rightarrow |z;\phi\rangle = U(\phi)|z\rangle$, like all vectors $|\psi\rangle\in\mathcal{H}_c$ fulfilling $b_{\phi}|z;\phi\rangle =z|z;\phi\rangle $. In particular, the *vacuum* state $|0\rangle\in\mathcal{H}_c$ which also belongs to the family of coherent states with $z=0$ is found to be non-invariant under this boost : $|0;\phi\rangle =U(\phi)|0\rangle \neq |0\rangle$ like its associated pure state $|0;\phi\rangle \langle 0;\phi| \neq |0\rangle\langle 0|$ and this is in contrast to its counter-part in the Euclidean case where the vacuum $|0\rangle$ picks up a simple phase under SO(2) rotation so that the associated pure states $|0\rangle\langle 0|$ remains invariant. And this is despite the fact that the space-time algebra $[\hat{t},\hat{x}]=i\theta$ along with the ball condition is invariant under this Lorentz transformation also. Nevertheless, one can easily prove the following identity: $$\langle z ,\phi|\hat{a}_{w,\phi}|z,\phi\rangle - \langle w,\phi|\hat{a}_{w,\phi}|w,\phi\rangle = \langle z-w|\hat{a}|z-w\rangle -\langle 0|\hat{a}|0\rangle;\,\, \hat{a}_{z,\phi}:=U(\phi)\hat{a}_zU^{\dagger}(\phi)$$ so that the spectral distance is invariant under both translation and Lorentz boost: $$d_{\mathcal{D}}(\rho_0,\rho_{z-w}) = d_{\mathcal{D}_{z,\phi}}(\rho_{w,\phi},\rho_{z,\phi})\label{ISO}$$ proving the invariance of distance under Poincare transformation. In (1+1) dimension we have three generators e.g. two translations and one boost for the transformation which forms a closed ISO(1,1) algebra among themselves.\
\
We finally embark on the explicit computation of the distance. For this we shall essentially follow the same footsteps in HS formalism as shown in section-2 for Euclidean Moyal plane . We construct the spectral triple $(\mathcal{A},\mathcal{H},\mathcal{D})$ here with the algebra $\mathcal{A}=\mathcal{H}_q$, the Hilbert space $\mathcal{H}=\mathcal{H}_q\otimes \mathbb{C}^2$, and the Dirac operator $\mathcal{D}=\hat{P}_{\mu}\otimes \gamma^{\mu}$ (\[d\]) which again by default acts on $\mathcal{H}_q\otimes\mathbb{C}^2$. Here the momentum operators have an adjoint action on an element $|\psi)\in\mathcal{H}_q$ as: $$\hat{P}_i|\psi)=\frac{\epsilon_{ij}}{\theta}|[\hat{x}_j,\psi]); \,\, i,j\in[0,1]\label{adjoint}$$ The commutators occurring in the ball condition (\[ball2\]) now becomes, $$[\hat{P}_0+\epsilon\hat{P}_1,\hat{a}]=\frac{1}{\theta}[\hat{x}-\epsilon\hat{t},\hat{a}];\,\,\,\epsilon=\pm 1,\,\,a\in\mathcal{A}=\mathcal{H}_q \label{adjoint2}$$ It is now evident that instead of $\mathcal{H}_q\otimes\mathbb{C}^2$ as our Hilbert space in the spectral triple , we can again take it to be $\mathcal{H}_c\otimes\mathbb{C}^2$ as $\hat{x},\hat{t}$ has normal left action on $\mathcal{H}_c$, so that the matrix elements corresponding to the first two terms in (\[ball2\]) are now well defined with $|\psi_1\rangle$ and $|\psi_2\rangle\in\mathcal{H}_c$. However in contrast to the commutative case, discussed earlier, we don’t have any concept of a point or rather a well defined event .The best option is to use coherent states (\[coh\]) which enjoys the desirable feature that they are maximally localised space-time event:$\Delta\hat{t}\Delta\hat{x}=\frac{\theta}{2}$. Consequently , it will be advantageous again to take recourse to the Fock-Bergman representation (Appendix A) of coherent state basis. By inserting completeness relation (\[A68\]) in the Ball condition (\[ball2\]) this gives the functional, $$\begin{aligned}
\textbf{B}[\psi_1,\psi_1^*,\psi_2,\psi_2^*]= -i\int d\mu (z,\bar{z})\, d\mu(w,\bar{w})\,[ \psi_1^*(z)\psi_1(\bar{w})\langle \bar{z}|& [\hat{P}_0-\hat{P}_1,a]|w\rangle +\psi_2^*(z)\psi_2(\bar{w}) \langle \bar{z}|[\hat{P}_0+\hat{P}_1,a]|w\rangle]\nonumber\\
&+i\int d\mu (z,\bar{z})\, [\psi_2^*(z)\psi_1(\bar{z}) -\psi_1^*(z)\psi_2(\bar{z})] <0 \label{ball4}\end{aligned}$$ Using (\[adjoint2\]) we have, $$\begin{aligned}
\textbf{B}[\psi_1,\psi_1^*,\psi_2,\psi_2^*]=-\frac{i}{\theta}\int d\mu (z,\bar{z})\, d\mu(w,\bar{w})\,& [\psi_1^*(z)\psi_1(\bar{w})\langle \bar{z}| [\hat{t}+\hat{x},\hat{a}]|w\rangle -\psi_2^*(z)\psi_2(\bar{w}) \langle \bar{z}|[\hat{t}-\hat{x},\hat{a}]|w\rangle] \nonumber\\
&+i\int d\mu (z,\bar{z})\, [\psi_2^*(z)\psi_1(\bar{z})-\psi_1^*(z)\psi_2(\bar{z})] <0 \label{ball9}\end{aligned}$$ We take the left hand side of (\[ball9\]) as a functional $\textbf{B}[\psi_1,\psi_1^*,\psi_2,\psi_2^*]$ of $\psi_1(\bar{z}),\psi_1^*(z),\psi_2(\bar{z}),\psi_2^*(z)$. To evaluate the optimal algebra element which saturates the ball condition we maximise **B** with respect to $\psi_1(\bar{u})$ and $\psi_2(\bar{u})$ using (\[A71\]) and equate them to zero, to arrive respectively at the following pair of equations: $$\begin{aligned}
\frac{1}{\theta}\langle\psi_1|[\hat{t}+\hat{x},\hat{a}]|u\rangle &=\langle\psi_2|u\rangle \nonumber\\
\frac{1}{\theta}\langle\psi_2|[\hat{t}-\hat{x},\hat{a}]|u\rangle &= \langle\psi_1|u\rangle\label{condition}\end{aligned}$$ As can be checked easily that the differentiation w.r.t. $\psi_1^*(u)$ and $\psi_2^*(u)$ just yield an equivalent pair of equations related to (\[condition\]) by just a complex conjugation. Combining this pair of equations by using the completeness relation (\[A68\]), one readily obtains a condition on the algebra element as, $$\frac{1}{\theta^2}[\hat{t}+\hat{x},\hat{a}]\,[\hat{t}-\hat{x},\hat{a}]= 1\label{con3}$$ Like in section-2 here too the general form of the optimal algebra element $\hat{a}_s$ which is an infinite dimensional matrix satisfying (\[con3\]), which can also ensure ISO(1,1) invariance of the resulting distance will clearly depend linearly on $\hat{t}$ and $\hat{x}$ as: $\hat{a}=\alpha\hat{t}+\beta\hat{x}$(with $\alpha,\beta\in\mathbb{R}$ for $\hat{a}$ being hermitian) . Substituting this in (\[con3\]) , we finally arrive at the general form of the algebra element: $$\hat{a}_{\lambda}=\hat{t} \cosh\lambda+\hat{x}\sinh\lambda;\,\,\,\,\lambda\in\mathbb{R}\label{aform2}$$ To see it more concretely, let us we can write (\[con3\]) in form of differential equations.For that we consider the matrix elements of the above operator equation (\[con3\]) in coherent state basis as, $$\int\, d\mu(w,\bar{w}) \langle\bar{z}|[e^{\alpha}\hat{b}+e^{-\alpha}\hat{b}^{\dagger},\hat{a}]|w\rangle\langle\bar{w}|[e^{-\alpha}\hat{b}+e^{\alpha}\hat{b}^{\dagger},\hat{a}]|u\rangle = \theta \int \, d\mu(w,\bar{w}) \langle\bar{z}|w\rangle\langle\bar{w}|u\rangle$$ where $\alpha = -\frac{i\pi}{4}$. Also note that we have inserted the resolution of identity (\[A68\]) here. Now using $$\langle\bar{z}|[\hat{b},\hat{a}]|w\rangle = \partial_{\bar{z}}a(\bar{z},w)-w a(\bar{z},w)$$ $$\langle\bar{z}|[\hat{b}^{\dagger},\hat{a}]|w\rangle = \bar{z}a(\bar{w},z)-\partial_w a(\bar{z},w)$$ and writing $\langle\bar{w}|\hat{a}|z\rangle = a(\bar{w},z) = f(\bar{w},z)e^{\bar{w}z}$ (for normal ordered operator $\hat{a}$, see (\[poly\]) ), we can further simplify the above equation as,
$$\int \, d\mu(\bar{w},w) e^{\bar{z}w+\bar{w}z} \left(e^{\alpha}\partial_{\bar{z}}f(\bar{z},w)-e^{-\alpha} \partial_wf(\bar{z},w)\right)\left(e^{-\alpha}\partial_{\bar{w}}f(\bar{w},u)-e^{\alpha}\partial_uf(\bar{w},u)\right) = \theta \int \, d\mu(w,\bar{w}) e^{\bar{z}w+\bar{w}z}$$
By comparing the coefficients of $e^{\bar{z}w+\bar{w}z}$ in the integrands of either sides of this equation, we can easily see that the following set of differential equations are necessarily satisfied by the function $f(\bar{z},w)$: $$\begin{aligned}
e^{\alpha} \partial_{\bar{z}}f(\bar{z},w)- e^{-\alpha}\partial_w f(\bar{z},w)& = \sqrt{\theta}\,\, pe^{\lambda}\nonumber\\
e^{-\alpha} \partial_{\bar{w}}f(\bar{w},u)-e^{\alpha}\partial_uf(\bar{w},u) &= \sqrt{\theta}\,\, p^*e^{-\lambda}\label{D1}\end{aligned}$$ where $p\in\,\mathbb{C}$ with $|p|=1$ and $\lambda\in\,\mathbb{R}$. This ensures that the product of these two expressions, occurring in the left hand side of this pair of equations is indeed $\theta$. We now replace the variable $\bar{w}$ by $\bar{z}$ and $u$ by $w$ in the second equation of (\[D1\]) and write the equations in matrix form as, $$\begin{pmatrix}
e^{\alpha}&-e^{-\alpha} \\
e^{-\alpha}&-e^{\alpha}
\end{pmatrix} \begin{pmatrix}
\partial_{\bar{z}}f\\ \partial_w f
\end{pmatrix} = \sqrt{\theta} \begin{pmatrix}
pe^{\lambda}\\ p^*e^{-\lambda}
\end{pmatrix}$$ Inverting the above coefficient matrix and solving for $\partial_{\bar{z}}f$ and $\partial_wf$ we get, $$\begin{aligned}
\partial_{\bar{z}}f &= \frac{\sqrt{\theta}}{2} e^{\frac{i\pi}{4}}\left(pe^{\lambda}-ip^* e^{-\lambda}\right) =K_1\nonumber\\
\partial_w f&= \frac{\sqrt{\theta}}{2} e^{\frac{i\pi}{4}}\left(ipe^{\lambda}-p^*e^{-\lambda}\right) =K_2\label{E2}\end{aligned}$$
So from the above equations we can infer that $f(\bar{z},w)=K_1\bar{z}+K_2 w$, which in turn gives the form of $\hat{a}$ as, $\hat{a}= K_2 \hat{b}+K_1\hat{b}^{\dagger}$. Now imposing the hermiticity condition on the algebra element $\hat{a}$, we simply arrive on the conclusion that $K_1=\bar{K}_2$.Equating the real and imaginary part of this equation we find $p=i$. So finally making use of this in (\[E2\]) , we get a one parameter family of $\hat{a}$ dependent on $\lambda\,\in\,\mathbb{R}$ to be given by, $$\hat{a}_{\lambda}=\sqrt{\frac{\theta}{2}} [\hat{b}(\cosh\lambda-i\sinh\lambda)+\hat{b}^{\dagger}(\cosh\lambda+i\sinh\lambda)] = \hat{t}\cosh\lambda+\hat{x}\sinh\lambda$$ which is (\[aform2\]) itself. We now emulate the commutative case and substitute this algebra element (\[aform2\]) in (\[ball9\]) to get, $$\textbf{B}[\psi_1,\psi_1^*,\psi_2,\psi_2^*] = -e^{-\lambda}\int d\mu(z,\bar{z}) \psi_1^*(z)\psi_1(\bar{z})-e^{\lambda}\int d\mu(z,\bar{z})\psi_2^*(z)\psi_2(\bar{z})+i\int d\mu(z,\bar{z})[\psi^*_2(z)\psi_1(\bar{z})-\psi_1^*(z)\psi_2(\bar{z})]\label{kol}$$ Using scaling transformation $\psi_1(\bar{z})\to \psi_1^{\prime}(\bar{z}):= e^{-\frac{\lambda}{2}}\psi_1(\bar{z});\,\, \psi_2(\bar{z})\to \psi_2^{\prime}(\bar{z}):=e^{\frac{\lambda}{2}}\psi_2(\bar{z})$, as before we rewrite (\[kol\]) as, $$\textbf{B}= -\int d\mu(z,\bar{z})[\psi_1^{\prime *}(z)\psi_1^{\prime}(\bar{z})+\psi_2^{\prime *}(z)\psi_2^{\prime}(\bar{z})+i\{\psi_1^{\prime *}(z)\psi_2^{\prime}(z)-\psi_2^{\prime *}(z)\psi_1^{\prime}(z)\}]$$ It seems that the functions on which **B** depends are not all independent. So we again use a transformation similar to (\[trans5\]) as $$\begin{pmatrix}
\psi_1(\bar{z})\\ \psi_2(\bar{z})
\end{pmatrix}\to
\begin{pmatrix}
\phi_+(\bar{z})\\ \phi_-(\bar{z})
\end{pmatrix}
=\frac{1}{2}\begin{pmatrix}
1&i \\ -1&i
\end{pmatrix}
\begin{pmatrix}
\psi_1^{\prime}(\bar{z})\\ \psi_2^{\prime}(\bar{z})
\end{pmatrix}$$ to get **B** as $\textbf{B}[\phi_-,\phi_-^*]$ in the following way, $$\textbf{B}= -2\int d\mu(z,\bar{z}) |\phi_-(z)|^2$$ Above equation clearly shows, **B** indeed is independent of $\phi_+$ and $\phi_+^*$ and **B** reaches its maximum value for $|\phi_-| =0$ which is its supremum value with above choice of $\hat{a}$ (\[aform2\]). The fact that this is indeed the right choice is re-inforced by the fact that analogous to the Euclidean case discussed in section 2, here too we can find a suitable bound to the distance function, except that it has to be lower bound in this case. We shall obtain this bound now in the following sub-section and show that this bound is indeed saturated by the choice of $\hat{a}$ (\[aform2\]).
### Calculation of the lower bound of distance
Like in the Euclidean case, here too we can try to get a bound in the distance between the pure states. Unlike, however, the previous case here we have to get a lower bound as the distance formula includes computation of infimum. Emulating the same procedure as (\[B1\]), we can write, $$d(\rho_0,\rho_z) = \int_0^1 d\mu \frac{d}{d\mu}[tr(\rho_{\mu z}\hat{a})]= \int_0^1 d\mu \, tr\left(\rho_{\mu z}[\bar{z}\hat{b}-z\hat{b}^{\dagger},\hat{a}]\right)=\frac{1}{\sqrt{2\theta}}\int_0^1 d\mu\,\,tr\left(\rho_{\mu z}\{[\hat{t},\hat{a}](\bar{z}-z)+i[\hat{x},\hat{a}](z+\bar{z})\}\right) \label{C1}$$ We can now parametrize $t$ and $x$ as $$t = r \cosh\psi;\,\,\, x = r \sinh\psi,\,\,\,\,\,\,\,\textrm{so that}\,\,\,z = \frac{r(\cosh\psi+i\sinh\psi)}{\sqrt{2\theta}} \label{param}$$ representing the points on a hyperbola $t^2-x^2=r^2$ in the forward light cone and $r$ is a real constant:$0<r<\infty$. Note that with this choice of parametrization we are restricting the points in the causal cone. After some simplification, (\[C1\]) can now be re-written as,$$d(\rho_0,\rho_z) = \frac{ir}{\theta}\int_0^1 d\mu \, tr\left(\rho_{\mu z}[\hat{x}\cosh\psi-\hat{t}\sinh\psi,\hat{a}]\right)\label{C4}$$ But note that, unlike in the Euclidean case (\[B2\]) we *cannot* just write $tr(\rho_{\mu z}[\hat{x},\hat{a}]) \leq \Vert [\hat{x},\hat{a}]\Vert_{op}$, as here the so-called ball condition has a completely different structure. The ball condition (\[ball\]), just implies that the matrix $M:=\mathcal{J}\{[\mathcal{D},\pi(\hat{a})]+i\chi\}$ is a negative operator. Realising its tensor product structure, we can re-write this, by making use of (\[adjoint2\]) as, $$\begin{aligned}
M=\mathcal{J}\{[\mathcal{D},\pi(\hat{a})]+i\chi\} &= -i\begin{pmatrix}
\frac{1}{\theta}[\hat{t}+\hat{x},\hat{a}]& 1\\
-1&\frac{1}{\theta}[\hat{x}-\hat{t},\hat{a}]
\end{pmatrix}\nonumber\\
&=-\frac{i}{\theta}[\hat{x},\hat{a}]\otimes\textbf{1}_2+\textbf{1}\otimes\sigma_2-\frac{i}{\theta}[\hat{t},\hat{a}]\otimes\sigma_3 \label{C2}\end{aligned}$$ where $\textbf{1}_2$ is $2\times 2$ identity matrix. Now carrying out a suitable unitary transformation with $U=U_1\otimes \textbf{1}_2$, we can diagonalise the hermitian operator $-\frac{i}{\theta}[\hat{t},\hat{a}]$, such that, $$U_1^{-1} (-\frac{i}{\theta}[\hat{t},\hat{a}])U_1= \textrm{diag}(\kappa_1,\kappa_2,\cdots ):= \Gamma\,\,;\,\,\,\,\, \kappa_i \in\mathbb{R}\,\,\,\forall \,\,i$$ so that $\Gamma$ is possibly an infinite dimensional diagonal matrix and $$M\to M^{\prime}= U^{-1}MU=U^{-1}_1(-\frac{i}{\theta}[\hat{x},\hat{a}])U_1\otimes \textbf{1}_2+
\textbf{1}\otimes \sigma_2+\Gamma\otimes\sigma_3\label{E5}$$ Now for each entry in the left slot of this total matrix $M^{\prime}$, say the (i,j)-th element, there is a u(2) Lie algebra element and can be written as, $$\left[U^{-1}_1(-\frac{i}{\theta}[\hat{x},\hat{a}])U_1\right]_{ij}\textbf{1}_2+\delta_{ij}\sigma_2+\Gamma_{ij}\sigma_3\in u(2)\label{E4}$$ This can now be subjected to another SU(2) transformation $U=\textbf{1}\otimes U_2$ with $U_2= e^{\frac{i}{2}\alpha_1\sigma_1}$, with a suitable choice of $\alpha_1$, to bring it to the diagonal form $$\left[U^{-1}_1(-\frac{i}{\theta}[\hat{x},\hat{a}])U_1\right]_{ij}\textbf{1}_2+\sqrt{(\delta_{ij})^2+(\Gamma_{ij})^2}\,\,\,\sigma_3\,;\,\,\,\,\,\,\Gamma_{ij}=\kappa_i\delta_{ij}\,\,\,\,(\textrm{no sum})\label{E6}$$ So that after two successive transformation by $(U_1\otimes \textbf{1}_2)$ and ($\textbf{1}\otimes U_2$), the matrix M (\[C2\]), as a whole finally transforms to $$M\to M^{\prime\prime}:= (U_1\otimes U_2)^{-1}M(U_1\otimes U_2) =U_1^{-1} (-\frac{i}{\theta}[\hat{x},\hat{a}])U_1\otimes \textbf{1}_2 +\sqrt{1+\Gamma^{\dagger}\Gamma}\otimes \sigma_3\label{E7}$$ Negativity of this operator implies, $$i\,U_1^{-1} [\hat{x},\hat{a}]U_1\otimes\textbf{1} -\sqrt{\theta^2+U_1^{-1}[\hat{t},\hat{a}]^{\dagger}[\hat{t},\hat{a}]U_1}\,\otimes\,\sigma_3\,\,>0\label{F1}$$ Now since $U_1^{-1}[\hat{t},\hat{a}]^{\dagger}[\hat{t},\hat{a}]U_1$ is a positive operator, we readily obtain, $$i [\hat{x},\hat{a}]\otimes \textbf{1} > \theta \otimes\,\sigma_3$$ which by comparing two sides, further implies that $$i [\hat{x},\hat{a}]\, >\, \theta\label{C3}$$ As an off-shoot, we observe that the saturation condition of the inequality in (\[F1\]), (in case we include the equality sign also in (\[F1\])) will be satisfied by only those algebra elements which commute with $\hat{t}$. This implies, in turn, that $\hat{a}(\hat{t},\hat{x})$ depends on $\hat{t}$ only, so that $[\hat{t},\hat{a}(\hat{t},\hat{x})]=0$. The inequality (\[C3\]) then implies that $\hat{a}(\hat{t},\hat{x})$ should then be linear in $\hat{t}$, so that $i[\hat{x},\hat{a}]$ is just $\theta \textbf{1}$. Thus (\[C3\]) fixes the scale also. One can notice that this feature was absent in the Euclidean case.\
\
At this stage, we observe that this inequality along with the above mentioned observations should be respected in all Lorentz frames, as the original ball condition (\[pbp\]) is a Lorentz invariant quantity. Using the Lorentz boost transformation (\[E1\]), we can therefore write $$i[\hat{x}^{\prime},\hat{a}]= i[\hat{t}\sinh\phi+\hat{x}\cosh\phi,\hat{a}] \,\,>\,\,\theta \,\,\,\,(\forall\,\,\phi)$$ Now by choosing $\phi = -\psi$ and putting the condition in (\[C4\]) we have the following lower bound for the Lorentzian distance: $$d(\rho_0,\rho_z) > r =\sqrt{\theta}\sqrt{z^2+\bar{z}^2}\label{E3}$$
Now for evaluation of distance just like Euclidean Moyal plane, here too the pure states are represented by normalized density matrices . The distance between a pair of pure states (represented in terms of normalised Glauber-Sudarshan coherent state basis (\[coh\]) as) $\rho_0=|0\rangle\langle 0|$ and $\rho_z=|z\rangle\langle z|$ representing two time-like separated events $(0,0), (t_1,x_1)$ in forward light cone with $\rho_0\prec\rho_z$, will be given by, $$\begin{aligned}
d(\rho_0,\rho_z)&=\inf_{\hat{a}\in\mathcal{B}}\{[\rho_z(\hat{a})-\rho_0(\hat{a})]^+\} =\inf_{\lambda} \{[\langle z|\hat{a}_{\lambda}|z\rangle -\langle 0|\hat{a}_{\lambda}|0\rangle ]^+\} \nonumber\\
&=\inf_{\lambda} \sqrt{\frac{\theta}{2}}\{[(z+\bar{z})\cosh\lambda-i(z-\bar{z})\sinh\lambda]^+\}\label{G1}\end{aligned}$$ where again we have defined $z=\frac{t_1+ix_1}{\sqrt{2\theta}}$ in the spirit of (\[basis\]) in section 2 and made use of the form of $\hat{a}_{\lambda}$ given in (\[aform2\]). Now the infimum of the set is attained by minimising the set with respect to $\lambda$ which immediately yields the desired distance between two pure states as, $$d(\rho_z,\rho_w)=\sqrt{\theta}\sqrt{z^2+\bar{z}^2}= \sqrt{t_1^2-x_1^2},\label{dist2}$$ showing that (\[dist2\]) indeed agrees with the lower bound (\[E3\]) .\
\
Finally, note that the minimisation value of $\lambda$, for which the minimum value of the R.H.S of (\[G1\]) is attained here is given by $\lambda=\tilde{\lambda}$, which satisfies $\tanh\tilde{\lambda} = \frac{i(z-\bar{z})}{z+\bar{z}}$. Now putting this value of $\lambda=\tilde{\lambda}$ in the expression of $\hat{a}_s$ (\[aform2\]) we have, $$\hat{a}_s = r\cosh\psi [\hat{b}(\cosh\psi+i\sinh\psi)+\hat{b}^{\dagger}(\cosh\psi-i\sinh\psi)]$$ where we have used the same parametrisation for $z$, as in (\[param\]). Now choosing $\psi=-\phi$ again, we have, using (\[E1\],\[LT\]), $$\hat{a}_s= r\cosh\phi \,\,(\hat{b}_{\phi}+\hat{b}^{\dagger}_{\phi}) = \sqrt{\frac{2}{\theta}}(r\cosh\phi)\, \hat{t}^{\prime}$$ With this form of $\hat{a}_s$ we can explicitly show that in a Lorentz transformed frame (\[E1\]), we have $[\hat{t}^{\prime},\hat{a}_s] = 0$ and $i[\hat{x}^{\prime},\hat{a}_s]=\theta \textbf{1}$, which in turn shows that saturation of the condition (\[F1\],\[C3\]) is truly reached at the same time when infimum of the distance functional is reached in a specific Lorentz frame, with our choice of $\hat{a}_s$ (\[aform2\]). We can therefore really identify the R.H.S of (\[dist2\]) as the true Lorentzian distance between the pure states $\rho_0$ and $\rho_z$ with $\rho_0 \prec \rho_z$.\
\
As the distance is ISO(1,1) invariant (\[ISO\]), we can make a suitable translation to give distance between $\rho_z$ and $\rho_w$ where $\rho_z \prec \rho_w$ as, $$d(\rho_z,\rho_w) = \sqrt{\theta}\sqrt{ (w-z) ^2+(\bar{w}-\bar{z})^2}= \sqrt{(t_2-t_1)^2-(x_2-x_1)^2}\label{E7}$$ where two pure states $\rho_z$ and $\rho_w$ represents the points $(t_1,x_1) ,(t_2,x_2)$ in forward light cone. Like the Euclidean case, (\[E7\]) also does not depend on the non-commutative parameter and it mimics the result of normal geodesic (here straight line) distance which coincides with the flat commutative 1+1 dim space-time.\
\
Before we conclude this pen-ultimate section, we would like to make two pertinent observations. Firstly, like in the Euclidean case, here too we are forced to consider extended multiplier algebra where all the family members of (\[aform2\]) belongs to this . Secondly, note that we have dealt with the non-unital algebra $\mathcal{A}=\mathcal{H}_q$ for both the Euclidean and Lorentzian Moyal plane. However, as mentioned in the beginning of section 3 that the fundamental symmetry $\mathcal{J}$ requires to satisfy $[\mathcal{J},a]=0, \,\forall\,\, a\in\tilde{\mathcal{A}}$, where $\tilde{\mathcal{A}}$ is the unitized version of $\mathcal{A}=\mathcal{H}_q$. This may give rise to the possibility that the bounded-ness of the operator $[\mathcal{D},\pi (a)]$ along with the compact-ness of the operator $\pi (a)(\mathcal{D}-\lambda \textbf{1})^{-1}$ for $a=\textbf{1}$, may get violated without affecting the metric aspect- as happens in the Euclidean case (see [@Mart; @MA] for details). One needs to check this aspect with our skew Krein self-adjoint Dirac operator $\mathcal{D}$ as well. We hope to address this issue in a future publication.
Conclusion and Future Plan
==========================
We have shown in this work that the axiomatic formulation provided in [@Franco1; @Franco4; @Eckstein], serves our purpose adequately and enables us to compute the spectral distance between pair of time-like separated pure states constructed out of Glauber-Sudarshan coherent states on Lorentzian Moyal plane. This computation was made very transparent by making use of un-normalised Fock-Bergman coherent states so that representation of $|.\rangle$ states and its dual $\langle .|$ states are represented by anti-holomorphic and holomorphic functions respectively occurring naturally in our analysis involving Hilbert-Schmidt operators.\
This exercise suggests that we can have our trust in this construction of spectral triple for 2D Lorentzian Moyal plane, as this reproduces the expected results. To proceed further beyond this point we definitely need to replace our left module by a bi-module, so that fluctuations in the Dirac operator can be constructed. This should enable us to introduce prototypes of gauge and Higgs fields. As a first step one may consider doubled Lorentzian Moyal plane for example as a sequel of doubled Euclidean Moyal plane as considered in [@Mart; @KK]. This should pave the way for realistic model building where almost commutative spaces are upgraded to fully non-commutative spaces, so as to provide a glimpse towards Planck scale physics.
Acknowledgement
===============
We would like to thank Prof. Debashish Goswami and Prof. Jyotishman Bhowmick for their useful discussions regarding this paper.
Appendix - A
============
**Fock-Bergman Coherent state basis and their properties:**\
Here we introduce some of the working formulae and notations for the Fock-Bergman coherent state basis. Unlike Glauber-Sudarshan coherent states the Fock-Bergman basis is un-normalised. We define it as, $$|z\rangle_B = e^{z\hat{b}^{\dagger}}|0\rangle,\,\,\,_{B}\langle\bar{z}| =\langle 0| e^{\bar{z}b}, \textrm{giving} \,\,\,_{B}\langle\bar{z}|w\rangle_B =e^{\bar{z}w}\label{A67}$$ and is related to the normalised Glauber-Sudarshan coherent states (\[coh\]) as $|z\rangle = e^{-\frac{|z|^2}{2}}|z\rangle_B$. The advantage of using Fock-Bergman basis is that the representation of vectors belonging to $\mathcal{H}_c$ (\[k\]) (respectively the dual space $\tilde{\mathcal{H}_c}$) are given by anti-holomorphic $\psi(\bar{z}):=_{B}\langle \bar{z}|\psi\rangle$ (resp. holomorphic $\psi^*(z):=\langle\psi|z\rangle_B$) functions. To avoid cluttering of indices we shall drop the $B$ subscript for denoting a Bergman-Fock basis here onwards. The completeness relation is given by $$\int d\mu(z,\bar{z})\,\, |z\rangle \langle \bar{z}| =\textbf{1},\,\,\,\textrm{where}\,\, d\mu(z,\bar{z})= e^{-|z|^2}\frac{Re(z)Im(z)}{\pi}\label{A68}$$ The overlap of this basis with a Fock state in (\[k\]) is given by $$\langle\bar{z}|n\rangle = \frac{\bar{z}^n}{\sqrt{n!}}; \,\,\, \langle n|z\rangle =\frac{z^n}{\sqrt{n!}}$$ Another pair of important identities are, $$\int d\mu(z,\bar{z}) \langle\psi|z\rangle\langle\bar{z}|w\rangle =\langle\psi|w\rangle = \psi^*(w),\,\,\, \int d\mu(z,\bar{z}) \langle\bar{w}|z\rangle\langle\bar{z}|\psi\rangle =\langle\bar{w}|\psi\rangle = \psi (\bar{w})\label{A69}$$ enabling us to identify $\langle \bar{z}|w\rangle = e^{\bar{z}w} = \delta^2(\bar{z},w)$ as a delta function in this space. Indeed, one can introduce the notion of a functional derivative here too. To that end consider the functional $F[\psi(\bar{z})]$, representing a map from the space of anti-holomorphic function $\{\psi(\bar{z})\}$ to complex numbers: $F: \{\psi(\bar{z})\}\,\,\to\,\,\mathbb{C}$\
The functional derivative $\frac{\delta F[\psi(\bar{z})]}{\delta\psi(\bar{w})}$ is then defined by the condition $$\int \frac{\delta F[\psi(\bar{z})]}{\delta\psi(\bar{w})} \phi(\bar{w})d\mu(w) =\left. \frac{dF[\psi(\bar{z})+\lambda\phi(\bar{z})]}{d\lambda}\right\vert_{\lambda =0}\label{A70}$$ where $\phi(\bar{z})$ represents a small perturbaton with a strength $\lambda$. One can then easily evaluate the following functional derivatives: $$\frac{\delta\psi(\bar{z})}{\delta\psi(\bar{w})}=\delta^2(\bar{z},w)\,\,\,\,\textrm{and}\,\,\,\,\frac{\delta\psi^*(z)}{\delta\psi^*(w)}=\delta^2(\bar{w},z)\label{A71}$$ where the second one follows from the first one by taking complex conjugation or alternatively defining analogous functional derivative by introducing appropriate functionals on the space of holomorphic functions.
Appendix - B
============
**On finite dimensional matrix solution of (12) and some related observations**\
In [@Chaoba] it has been shown that there exist a finite dimensional solution of optimal algebra element $\hat{a}_s$ saturating the ball condition $\Vert [\hat{b},\hat{a}]\Vert_{op} \leq \sqrt{\frac{\theta}{2}}$. For computation of distance between harmonic oscillator states $\rho_{n+1}$ and $\rho_n$, the finite dimensional optimal algebra can be taken as $$\hat{a}_s=\frac{\sqrt{\theta}}{2\sqrt{2(n+1)}}\left[ |n+1\rangle\langle n+1|-|n\rangle\langle n|\,\right]\label{form3}$$ If we compute the operator $[\hat{b},\hat{a}]^{\dagger}[\hat{b},\hat{a}] $ with this finite dimensional form of $\hat{a}_s$, we shall get, $$[\hat{b},\hat{a}]^{\dagger}[\hat{b},\hat{a}] = \frac{\theta}{8(n+1)}[4(n+1) |n+1\rangle\langle n+1|+(n+2)|n+2\rangle\langle n+2| +n|n\rangle\langle n|\, ]$$ It is evident from the form of the equation that the right hand side matrix is a diagonal matrix but not proportional to unit matrix. To compute the operator norm we have to take the highest eigenvalue which is $4(n+1)$, giving $\Vert [\hat{b},\hat{a}]\Vert_{op}=\sqrt{\frac{\theta}{2}}$. So we have shown that a finite dimensional form of $\hat{a}_s$ can also saturate the ball condition. But the point to note is that the functional differentiation only provides information about local extrema. It should finally augmented by the task of getting maximal eigenvalue to get the operator norm. In this discrete case operator norm is not translationally invariant as it depends on $n$ and is used to compute the distance between successive harmonic oscillator states $\rho_n$ and $\rho_{n+1}$, which is also, therefore non-invariant under translation and is given by $$d(\rho_n,\rho_{n+1}) = \sqrt{\frac{\theta}{2(n+1)}}$$ See [@Chaoba; @Mart3] for details.
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[^1]: anwesha@bose.res.in
[^2]: biswajit@bose.res.in
[^3]: To avoid cluttering of indices, here after we shall use the same notation $|z\rangle$ to indicate both the normalised coherent state (\[coh\]) and the Fock-Bergman(un-normalised basis) $|z\rangle_B$ in (\[A67\]). It should be clear from the context, which is being used and should not cause any confusion. Particularly, $|z\rangle$ occuring in density matrix $\rho_z=|z\rangle\langle z| $ is clearly the normalised one, otherwise, it is the latter one.
[^4]: Note that the equality signs in the inequalities (\[causal\],\[ccausal\]) have been omitted here.This is in contrast to [@Franco2; @Franco4; @Eckstein]. The reason for this is explained in the sequel (see section 4.1).
[^5]: Here the $L^2(\mathbb{R}^{1,1})$ part of $\mathcal{H}$ is actually analogous to $\mathcal{H}_q$ in our HS formulation.
[^6]: Note that if equality signs were to be included in (\[ccausal\]), $(V^{\mu}\nabla_{\mu}a)$ would not have been intrinsically negative and we would be forced to consider the case $V^{\mu}\nabla_{\mu}a =0$ also. For $V^{\mu}$ time-like, this would have implied that $\nabla_{\mu}a$ space-like - a scenario which we would like to avoid. In any case, the inclusion/exclusion does not impact the computation of supremum/infimum, as one is a dense subspace of the other.
[^7]: For $Q\prec P, \,\, t_1>t_2$ and $ (t_2-t_1)^2>(x_2-x_1)^2$ . Also $ |\cosh\lambda| >|\sinh\lambda|\,\,\, \forall\lambda$. Consequently whole quantity in (\[dis1\]) will always be negative.
|
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abstract: 'Spatially extended localized spins can interact via indirect exchange interaction through Friedel oscillations in the Fermi sea. In arrays of localized spins such interaction can lead to a magnetically ordered phase. Without external magnetic field such a phase is well understood via a “two-impurity” Kondo model. Here we employ non-equilibrium transport spectroscopy to investigate the role of the orbital phase of conduction electrons on the magnetic state of a spin lattice. We show experimentally, that even tiniest perpendicular magnetic field can influence the magnitude of the inter-spin magnetic exchange.'
address:
- 'Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom.'
- 'Department of Physics, Indian Institute of Science, Bangalore 560 012, India.'
author:
- 'C. Siegert'
- 'A. Ghosh'
- 'M. Pepper'
- 'I. Farrer'
- 'D. A. Ritchie'
- 'D. Anderson'
- 'G. A. C. Jones'
title: Sensitivity of the magnetic state of a spin lattice on itinerant electron orbital phase
---
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Intrinsic spin lattice, 2DEG, Spin interaction, Magnetic state, RKKY, Perpendicular magnetic field 72.25.-b ,71.45.Gm ,71.70.Ej
Introduction
============
Ordered arrays of spins, called spin lattices, can form a magnetic state when surrounded by conduction electrons. Such systems are normally characterized by two energy scales. First, the Kondo temperature $T_{\rm K}$, which is the coupling of conduction electrons to localized spins. And second, Ruderman-Kittel-Kasuya-Yosida (RKKY) indirect exchange $J^{\rm RKKY}$, which is the pairwise interaction between localized spins. A mean to investigate the magnetic state of the spin lattice is non-equilibrium transport spectroscopy, where a system with $|J^{\rm RKKY}| = 0$ shows a resonance in differential conductance ${\rm d}I/{\rm d}V$ at zero source-drain bias ($V_{\rm SD}$) [@kouwenhoven2001; @ghosh2005], which we call type-I zero bias anomaly (ZBA). With interacting spins $|J^{\rm RKKY}| > 0$, the resonance is suppressed at $E_{\rm F}$ for $k_{\rm B} T < |J^{\rm RKKY}|$, and $|eV_{\rm SD}| < |J^{\rm RKKY}|$, resulting in a split resonance, which we call type-II ZBA [@heersche2006; @wiel2002; @ghosh2004; @siegert2007]. In two dimensions electron wavevector $k_{\rm F} = \sqrt{2 \pi n_{\rm 2D}}$ and interspin distance $R$ tunes the exchange interaction parameter $J^{\rm RKKY}(k_{\rm F},R) \propto {\rm cos}(2 k_{\rm F} R)/R^2$ for $k_{\rm F}R >> 1$ [@glazman2004; @Beal-Monod1987]. Such a tunig has recently been shown in an $intrinsic$ lattice of spins, that was reported to exist in high mobility modulation doped two dimensional electron gases (2DEG) [@siegert2007]. Theoretically, spin lattices are often analysed in a “two-impurity” Kondo model [@cox1996], where the effect of perpendicular magnetic field, and its effect on the magnetic state via the orbital phase of the electrons that couple the spins, is not included. However, even in two spin systems, $J^{\rm RKKY}$ is strongly influenced by the nature and extent of confinement of the intervening Fermi sea, and the orbital phase of itinerant electrons [@craig2004; @simon2005; @vavilov2005; @utsumi2004]. We investigate such influence by application of a small transverse magnetic field on a quasi-regular two-dimensional array of localized spins, embedded within a sea of conduction electrons. We show that the magnetic state is extremely sensitive to perpendicular field $B_\bot$, and the orbital phase of the itinerant electrons.
Experiment
==========
\[Fig1\]{width="7cm"}
We probe the effect of perpendicular magnetic field on spin ordering in systems with an intrinsic two-dimensional spin lattice by means of non-equilibrium magnetoconductance spectroscopy. We use Si-monolayer-doped Al$_{0.33}$Ga$_{0.67}$As/GaAs heterostructures. Using a 60-80 nm spacer between 2DEG and Si results in as-grown electron density of $n_{\rm 2D} = 1 - 1.5 \times 10^{11}$ cm$^{-2}$ with mobilities $\mu = 1 - 3 \times 10^6$ $\frac{{\rm cm}^2}{\rm Vs}$. We thermally deposit non-magnetic Ti/Au gates on the surface, which allows us to locally tune $n_{\rm 2D}$ of the 2DEG, 300 nm below the surface. Fig. 1(a) shows a typical device, and electrical connections. Voltage $V_{\rm C}$ is used to deplete electrons by application of up to -1 Volt, which defines an 8 $\mu {\rm m}$ wide mesa electrostatically. $V_{\rm C}$ is kept fixed for the duration of the experiment. Alternatively an etched mesa with same width can be used to create lateral confinement. $V_{\rm G}$ is biased such that the devices operate in the range $n_{\rm 2D} = 0.5-2 \times 10^{10}$ cm$^{-2}$ in a $2 \times 8$ $\mu {\rm m}^2$ region. Constant voltage two probe combined AC+DC technique is used with $e V_{\rm AC} \ll k_{\rm B}T$ for all $T$, and source-drain bias $V_{\rm SD}$. Fig. 1(b) shows a typical non-equlibrium spectroscopy image of ${\rm d}I/{\rm d}V (V_{\rm SD},V_{\rm G})$ at $T = 75$ mK and zero external field. Individual ${\rm d}I/{\rm d}V (V_{\rm SD})$ show either a type-I ZBA (as shown in fig. 1(c)), or a type-II ZBA (as shown in fig. 1(d)). For the type-II ZBA, the half width at half depth is defined as $\Delta$ and is proportional to the exchange interaction parameter: $\Delta \propto |J^{\rm RKKY}|$ [@siegert2007; @ghosh2005].
Results and Discussion
======================
\[Fig2\]{width="7cm"}
In figures 2(a), and 2(b) we show the evolution of type-I and type-II ZBA in low parallel magnetic field. Both ZBA’s are not affected by parallel magnetic field in that range, and only show a splitting at higher fields [@siegert2007]. In contrary, perpendicular magnetic field $B_\bot$ forces conduction electrons on paths with cyclotron radii $r_{\rm C} = \frac{\hbar k_{\rm F}}{e B_\bot}$, and electrons aquire additional phase $\theta$, see fig. 2(c). When $R = 2 r_{\rm C}$, electrons from one localized spin are focussed on the next, and vice versa, which can be seen as an enhancement of overall ZBA magnitude, and of fluctuations in $\Delta$ [@usaj2005]. We show such commensurability behaviour in a surface plot of the non-equilibrium conductance over $B_\bot$, up to 105 mT for a typical type-II ZBA, see fig. 2(e). The commensurability condition is indicated by the white line in fig. 2(e) at the field $B_\bot^{\rm C} \approx 65$ mT for interspin distance $R = 630$ nm. The intrinsic interspin distance $R$ is evaluated from Aharonov-Bohm (AB) oscillations in linear ($V_{\rm SD} = 0$) magnetoconductance at low $T$ [@siegert2007]. In fig. 2(d) we show the respective $\Delta (B_\bot)$, and note that above 50 mT we observe strong and irregular fluctuations in $\Delta$, which is substantially different from the low field part.
In fig. 3(a) we show that $\Delta$ is periodically suppressed with a period $\Delta B \approx 9 - 10$ mT at fields $B_\bot \ll B_\bot^{\rm C}$. In the top of the image $\Delta (B_\bot)$ is shown. Since $\Delta \propto J^{\rm RKKY}$, the magnetic state of the spin lattice is directly modulated by perpendicular magnetic field. Theoretically this has been simulated for the two-impurity case, where models link magnetic flux and $J^{\rm RKKY}$ in $B_\bot$, and predict a modulation according to [@utsumi2004; @nitta1999; @schuster1994]: $$\label{deltaphi}
J^{\rm RKKY} (B_\bot \neq 0) \sim (\frac{1}{2} + \frac{1}{2} {\rm cos}(\theta)) \cdot J^{\rm RKKY} (B_\bot = 0),$$ with $\theta(B_\bot)$ being the orbital phase shift in $B_\bot$ from the electron trajectories on their paths from one spin to the next. The periods of AB and $\Delta$ oscillations can be compared. Both, AB oscillations, as well as eq. (\[deltaphi\]) have a $B_\bot$-periodicity of $\Delta \theta = 2 \pi$, and thus the periods are similar. Please note that AB oscillations can not influence $\Delta$, and thus non-equilibrium spectroscopy can be used to distinguish them from $\Delta$ oscillations.
\[Fig3\]{width="7cm"}
An interesting consequence of eq. (\[deltaphi\]) is, that small fields ($B_\bot < B_\bot^{\rm C}$) can only decrease $J^{\rm RKKY}$, but not increase it from the zero field value. A type-I ZBA with $J^{\rm RKKY} = 0$ would then show AB oscillations, but have no $\Delta$ modulation over $B_\bot$. We do indeed observe this, and show in fig. 3(b) a typical surface plot of non-equilibrium low field magnetoconductance of a type-I ZBA, where the linear conductance $G(B_\bot)$ is modulated without a modulation of $\Delta$. Thus, low perpendicular field can not change the magnetic state of a spin lattice with $J^{\rm RKKY} = 0$.
Please note that in our system the background disorder plays a critical role in the arrangement of the localized spins. While the origin of the localized spins is not fully understood yet, periodic tunability of $\Delta$ seems to be only possible in a certain window of disorder. Unintentional disorder can lead to additional scattering of electron trajectories in $B_\bot$, which suppresses clear observation of $B_\bot$-tuning of $\Delta$. Even in the same device, about 25% of electron densities displayed no periodic modulation of $\Delta$ with $B_\bot$, even though the oscillation in $\Delta$ as a function of $k_{\rm F}$ at $B_\bot = 0$ was clearly observable.
Summary
=======
At low temperatures non-equilibrium conductance spectroscopy in low perpendicular magnetic fields is used to analyse the magnetic state of a spin lattice. We show that even tiny fields modulate the indirect exchange interaction $J^{\rm RKKY}$ between localized spins, directly influencing the magnetic state of the spin lattice.
Acknowledgement
===============
This project was supported by EPSRC. CS acknowledges the support of Cambridge European Trust, EPSRC, and Gottlieb Daimler and Karl Benz Foundation.
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---
abstract: 'Although the primary goal of ESA’s [Planck]{} mission is to produce high resolution maps of the temperature and polarization anisotropies of the Cosmic Microwave Background (CMB), its high-sensitivity all-sky surveys of extragalactic sources at 9 frequencies in the range 30–860 GHz will constitute a major aspect of its science products. In particular, [Planck]{} surveys will provide key information on several highly interesting radio source populations, such as Flat Spectrum Radio Quasars (FSRQs), BL Lac objects, and, especially, extreme GHz Peaked Spectrum (GPS) sources, thought to correspond to the very earliest phases of the evolution of radio sources. Above 100 GHz, [Planck]{} will provide the [f]{}[i]{}rst all-sky surveys, that are expected to supply rich samples of highly gravitationally ampli[f]{}[i]{}ed dusty proto-galaxies and large samples of candidate proto-clusters at $z\simeq 2$–3, thus shedding light on the evolution of large scale structure across the cosmic epoch when dark energy should start dominating the cosmic dynamics.'
author:
- 'G. De Zotti'
- 'C. Burigana'
- 'M. Negrello, S. Tinti and R. Ricci'
- 'L. Silva'
- 'J. Gonzalez-Nuevo and L. Toffolatti'
---
Cosmology, galaxy evolution, radio galaxies
Introduction
============
The [Planck]{} satellite will carry our high sensitivity all sky surveys at 9 frequencies in the poorly explored range 30–860 GHz (see the contribution by J. Tauber, this volume). At low frequencies, [Planck]{} will go several times deeper (and will detect about ten times more sources) than WMAP, that has provided the [f]{}[i]{}rst all-sky surveys at frequencies of tens of GHz, comprising about 200 objects (Bennett et al. 2003).
Above 100 GHz, [Planck]{} surveys will be the [f]{}[i]{}rst and will remain the only all sky surveys available for many years to come. They will [f]{}[i]{}ll an order of magnitude gap in our knowledge of the spectrum of bright extragalactic sources and may discover new populations, not represented, or not recognized, in lower or higher frequencies surveys.
Rather than presenting a comprehensive review of the expected scienti[f]{}[i]{}c results from [Planck]{} measurements of extragalactic sources (see, e.g., De Zotti et al. 1999) we will focus on a couple of frequencies, one of the Low Frequency Instrument, namely 30 GHz, and one of the High Frequency Instrument, namely 350 GHz. The relatively shallow but all-sky [Planck]{} surveys will be ideal to study populations which are both very powerful at mm/sub-mm wavelengths, and very rare, such as radio sources with inverted spectra up to $\ge 30\,$GHz \[extreme GHz Peaked Spectrum (GPS) sources or High Frequency Peakers (HFP)\], thought to be the most recently formed and among the most luminous radio sources, and ultra-luminous dusty proto-spheroidal galaxies, undergoing their main and huge episode of star formation at typical redshifts $\ge 2$ (Granato et al. 2001, 2004). And [Planck]{} will observe such sources with an unprecedented frequency coverage.
To estimate the detection limit, and the number of detectable sources, we need to take into account, in addition to the instrument noise, the [fl]{}uctuations due to Galactic emissions, to the Cosmic Microwave Background (CMB), and to extragalactic sources themselves. These [fl]{}uctuations will be brie[f]{}[l]{}y reviewed in Section 2, while in Sections 3 and 4 we will discuss the expected impact of [Planck]{} data on our understanding of HFPs and of ultra-luminous proto-spheroidal galaxies, respectively. Our main conclusions are summarized in Sect. 5.
{width="12cm" height="10cm"}
-5cm
Power spectra of foreground emissions
=====================================
For a very high sensitivity experiment, like [Planck]{}, the main limitation to the capability of mapping the CMB is set by contamination by astrophysical sources (“foregrounds”), while CMB [fl]{}uctuations may be the highest “noise” source for the study of astrophysical emissions at mm wavelengths. The most intense foreground source is our own Galaxy. Because of the different power spectra of the various emission components, the frequency of minimum foreground [fl]{}uctuations depends to some extent on the angular scale (see Fig. \[foreground\], where $\delta T$ are [fl]{}uctuations of the CMB thermodynamic temperature, in $\mu$K, related to the power spectrum $C_\ell$ by $\delta T = [\ell(\ell+1)C_\ell/(2\pi)]^{0.5}(e^x-1)^2/(x^2 e^x)$, with $x=h\nu/kT_{\rm CMB}$). So long as diffuse Galactic emissions dominate the [fl]{}uctuations ($\theta \gsim 30'$; see De Zotti et al. 1999), they have a minimum in the 60–80 GHz range (depending also on Galactic latitude; cf. Bennett et al. 2003).
But the power spectra of diffuse Galactic emissions decline rather steeply with increasing multipole number (or decreasing angular scale). Thus, on small scales, [fl]{}uctuations due to extragalactic sources, whose Poisson contribution has a white-noise power spectrum (on top of which we may have a, sometimes large, clustering contribution) take over, even though their integrated emission is below the Galactic one. At high frequencies, however, Galactic dust may dominate [fl]{}uctuations up to $\ell$ values of several thousands. Unlike the relatively quiescent Milky Way, the relevant classes of extragalactic sources have strong nuclear radio activity or very intense star formation, or both. Thus, although in many cases their SEDs are qualitatively similar to that of the Milky Way, there are important quantitative differences. In particular, dust in active star forming galaxies is signi[fi]{}cantly hotter and the radio to far-IR intensity ratio of the extragalactic background is much higher than that of the Milky Way. Both factors, but primarily the effect of radio sources, cooperate to move the minimum of the SED to 100–150 GHz.
![Predicted 30 GHz differential counts. The left-hand panel shows the counts of all the main populations (see De Zotti et al. 2004 for details). The right-hand panel details the contributions of three sub-classes of canonical radio sources: FSRQs, BL Lac objects, and steep-spectrum sources []{data-label="30GHzcounts"}](dezotti_fig2a.ps "fig:"){width="5.5cm" height="5cm"} ![Predicted 30 GHz differential counts. The left-hand panel shows the counts of all the main populations (see De Zotti et al. 2004 for details). The right-hand panel details the contributions of three sub-classes of canonical radio sources: FSRQs, BL Lac objects, and steep-spectrum sources []{data-label="30GHzcounts"}](dezotti_fig2b.ps "fig:"){width="5.5cm" height="5cm"}
{width="5.5cm" height="5cm"} {width="5.5cm" height="5cm"}
30 GHz counts
=============
Figure \[30GHzcounts\] provides a synoptic view of the contributions of different source classes to the global counts of extragalactic sources. Shallow surveys, such as those by WMAP and [Planck]{}, mostly detect canonical radio sources. As shown by the right-hand panel of Fig. \[30GHzcounts\], detected sources will be mostly [f]{}[l]{}at-spectrum radio quasars (FSRQs), while the second more numerous population are BL Lac objects. [Planck]{} will detect about ten times more sources than WMAP, thus allowing a substantial leap forward in the understanding of evolutionary properties of both populations at high frequencies, only weakly constrained by WMAP data (Fig. \[WMAP\_FSRQ\_BLLacs\]).
[Planck]{} will also provide substantial complete samples of sources not (yet) represented in the WMAP catalog, such as Sunyaev-Zeldovich (1972) signals and extreme GPS sources or HFPs (Dallacasa et al. 2000).
GPS sources are powerful ($\log P_{\rm 1.4\, GHz} \gsim
25\,\hbox{W}\,\hbox{Hz}^{-1}$), compact ($\lsim 1\,$kpc) radio sources with a convex spectrum peaking at GHz frequencies. It is now widely agreed that they correspond to the early stages of the evolution of powerful radio sources, when the radio emitting region grows and expands within the interstellar medium of the host galaxy, before plunging in the intergalactic medium and becoming an extended radio source (Fanti et al. 1995; Readhead et al. 1996; Begelman 1996; Snellen et al. 2000). Conclusive evidence that these sources are young came from measurements of propagation velocities. Velocities of up to $\simeq 0.4c$ were measured, implying dynamical ages $\sim 10^3$ years (Polatidis et al. 1999; Taylor et al. 2000; Tschager et al. 2000). The identi[f]{}[i]{}cation and investigation of these sources is therefore a key element in the study of the early evolution of radio-loud AGNs.
There is a clear anti-correlation between the peak (turnover) frequency and the projected linear size of GPS sources. Although this anti-correlation does not necessarily de[f]{}[i]{}ne the evolutionary track, a decrease of the peak frequency as the emitting blob expands is indicated. Thus high-frequency surveys may be able to detect these sources very close to the moment when they turn on. The self-similar evolution models by Fanti et al. (1995) and Begelman (1996) imply that the radio power drops as the source expands, so that GPS’s evolve into lower luminosity radio sources, while their luminosities are expected to be very high during the earliest evolutionary phases, when they peak at high frequencies. De Zotti et al. (2000) showed that, with a suitable choice of the parameters, this kind of models may account for the observed counts, redshift and peak frequency distributions of the samples then available. The models by De Zotti et al. (2000) imply, for a maximum rest-frame peak frequency $\nu_{p,i}
=200\,$GHz, about 10 GPS quasars with $S_{30{\rm GHz}} > 2\,$Jy peaking at $\geq 30\,$GHz over the 10.4 sr at $|b| >10^\circ$.
Although the number of [*candidate*]{} GPS quasars (based on the spectral shape) in the WMAP survey is consistent with such expectation, when data at additional frequencies (Trushkin 2003) are taken into account the GPS candidates look more blazars caught during a [f]{}[l]{}are optically thick up to high frequencies. Furthermore, Tinti et al. (2004) have shown that most, perhaps two thirds, of the quasars in the sample of HFP candidates selected by Dallacasa et al. (2000) are likely blazars.
Thus, WMAP data are already providing strong constraints on the evolution of HFPs. [Planck]{} will substantially tighten such constraints and may allow us to directly probe the earliest phases (ages $\sim 100\,$yr) of the radio galaxy evolution, hopefully providing hints on the still mysterious mechanisms that trigger the radio activity.
We note, in passing, that contrary to some claims, we do not expect that [Planck]{} can detect the late phases of the AGN evolution, characterized by low accretion/radiative ef[f]{}[i]{}ciency (ADAF/ADIOS sources).
At faint [fl]{}ux densities, other populations come out and are expected to dominate the counts. In addition to SZ effects, we have active star-forming galaxies, seen either through their radio emission, or through their dust emission, if they are at substantial redshift. The latter is the case for the sub-mm sources detected by the SCUBA surveys if they are indeed at high redshifts (see below).
Such sources may be relevant in connection with the interpretation of the excess signal on arc-minute scales detected by CBI (Mason et al. 2003; Readhead et al. 2004) and BIMA (Dawson et al. 2002) experiments at 30 GHz, particularly if, as discussed below, they are highly clustered, so that their contribution to [fl]{}uctuations is strongly super-Poissonian (Toffolatti et al. 2004). In fact, to abate the point source contamination of the measured signals, the CBI and BIMA groups could only resort to existing or new low frequency surveys. But the dust emission is undetectable at low frequencies. Although our reference model (Granato et al. 2004), with its relatively warm dust temperatures yielded by the code GRASIL (Silva et al. 1998), imply dusty galaxy contributions to small scale [fl]{}uctuations well below the reported signals, the (rest-frame) mm emission of such galaxies is essentially unknown and may be higher than predicted, e.g. in the presence of the extended distribution of cold dust advocated by Kaviani et al. (2003) or of a widespread mm excess such as that detected in several Galactic clouds (Dupac et al. 2003) and in NGC1569 (Galliano et al. 2003). This is another instance of the importance of a multifrequency approach, like [Planck]{}’s, capable of keeping under control all the relevant emission components, with their different emission spectra.
![Left-hand panel: contributions of different populations to the 350 GHz counts. Central panel: effect of lensing on counts of proto-spheroidal galaxies. Right-hand panel: estimated counts of “clumps” of proto-spheroids observed with [Planck]{} resolution. []{data-label="lensplusclust"}](dezotti_fig4.ps){width="11.5cm" height="10.5cm"}
-5cm
350 GHz counts
==============
The 350 GHz counts of extragalactic sources have been determined in the range from $\simeq 10\,$mJy to $\simeq 0.25\,$mJy by surveys with the SCUBA camera, covering small areas of the sky (overall, a few tenths of a square degree). These surveys have led to the discovery of very luminous high-$z$ galaxies, with star-formation rates $\sim 10^3\,M_\odot$/yr, a result con[f]{}[i]{}rmed by 1.2mm surveys with MAMBO on the IRAM 30m telescope. These data proved to be extremely challenging for semi-analytic galaxy formation models, and have indeed forced to reconsider the evolution of baryons in dark matter halos.
The bright portion of observed counts appears to be declining steeply with increasing [fl]{}ux density, probably re[fl]{}ecting the exponential decline of the dark-halo mass function at large masses implied by the Press & Schechter formula, so that one would conclude that [Planck]{} cannot do much about these objects, but rather detect brighter sources such as blazars and relatively local star-forming galaxies, or SZ signals. There are, however, two important effects to be taken into account, that may change this conclusion: gravitational lensing and clustering.
We refer here to the model by Granato et al. (2004) according to which SCUBA sources are large spheroidal galaxies in the process of forming most of their stars. Forming spheroidal galaxies, being located at relatively high redshift, have a substantial optical depth for gravitational lensing, and the effect of lensing on their counts is strongly ampli[fi]{}ed by the steepness of the counts. This is illustrated by the left-hand panel of Fig. \[lensplusclust\], based on calculations by Perrotta et al. (2003). Strong lensing is thus expected to bring a signi[fi]{}cant number of high-$z$ forming spheroids above the estimated [Planck]{} $5\sigma$ detection limit.
If indeed SCUBA galaxies are massive spheroidal galaxies at high $z$, they must be highly biased tracers of the matter distribution, and must therefore be highly clustered. There are in fact several, although tentative, observational indications of strong clustering with comoving radius $r_0 \simeq
8\hbox{h}^{-1}\,$Mpc (Smail et al. 2004; Blain et al. 2004; Peacock et al. 2000), consistent with theoretical expectations.
But if massive spheroidal proto-galaxies live in strongly over-dense regions, low resolution experiments like [Planck]{} unavoidably measure not the [fl]{}ux of individual objects but the sum of [fl]{}uxes of all physically related sources in a resolution element.
This is an aspect of the “source confusion" problem, whereby the observed [fl]{}uxes are affected by unresolved sources in each beam. The problem was extensively investigated in the case of a Poisson distribution, particularly by radio astronomers (Scheuer 1957, Murdoch et al. 1973, Condon 1974, Hogg & Turner 1998). The general conclusion is that unbiased [fl]{}ux measurements require a $S/N \ge 5$.
Not much has been done yet on confusion in the presence of clustering (see however Hughes & Gaztanaga 2000). The key difference is that, for a Poisson distribution, a bright source is observed on top of a background of unresolved sources that may be either above or below the all-sky average, while in the case of clustering, sources are preferentially found in over-dense regions.
Clearly, the excess signal (over the [fl]{}ux of the brightest source in the beam) depends on the angular resolution. For a standard $\xi(r) = (r/r_0)^{-1.8}$ the mean clustering contribution is $\propto r_0^{1.8} r_{\rm beam}^{1.2}$. The [Planck]{} beam at this frequency corresponds to a substantial portion of the typical clustering radius at $z\simeq 2$–3, so that [Planck]{} will actually measure a signi[fi]{}cant fraction of the [fl]{}ux of the clump, which may be substantially larger than the [fl]{}ux of any member source. The effect on counts depends on the joint distribution of over-densities and of $M/L$ ratios. The former depends on both the two- and the three-point correlation function, while the latter depends on the luminosity function.
Preliminary estimates of the distribution of excess luminosities due to clustering around bright sources have been obtained by Negrello et al. (2004b). The right-hand panel of Fig. \[lensplusclust\] shows the estimated counts of clumps observed with [Planck]{} resolution for three models for the evolution of the coef[f]{}[i]{}cient $Q$ of the three-point correlation function. Obviously [Planck]{} can provide information only on the brightest clumps, and, except in the extreme case of $Q=1$ at all cosmic times, the clumps will only show up as $< 5\sigma$ [fl]{}uctuations. On the other hand, such [fl]{}uctuations will provide a rich catalogue of candidate proto-clusters at substantial redshifts (typically at $z\simeq
2$–3), very important to investigate the formation of large scale structure and, particularly, to constrain the evolution of the dark energy thought to control the dynamics of the present day universe.
Conclusions
===========
Although extragalactic surveys are not the primary goal of the mission, [Planck]{} will provide unique data for several particularly interesting classes of sources. Examples are the FSRQs, BL Lac objects, but especially extreme GPS sources that may correspond to the earliest phases of the life of radio sources, and proto-spheroidal galaxies. Thus [Planck]{} will investigate not only the origin of the universe but also the origin of radio activity and of galaxies. Sub-mm surveys will provide large samples of candidate proto-clusters, at $z\simeq 2$–3, shedding light on the evolution of the large scale structure (and in particular providing information on the elusive three-point correlation function) and of the dark energy, across the cosmic epoch when it is expected to start dominating the cosmic dynamics.
Work supported in part by MIUR through a PRIN grant and by ASI.
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|
---
abstract: 'Environment-assisted quantum transport (ENAQT) is the possibility of an external environment to enhance transport efficiency of quantum particles. This idea has generated much excitement over recent years, especially due to the experimentally-motivated possibility of ENAQT in photo-synthetic exciton transfer complexes. Many theoretical calculations have shown ENAQT, but the explanations for its origin differ, and a universal explanation has been elusive. Here we demonstrate a universal origin for ENAQT in quantum networks with a dephasing environment, based on a relation between exciton current and occupation within a Markovian open quantum system approach. We show that ENAQT appears due to two competing processes, namely the tendency of dephasing to make the exciton population uniform, and the formation of an exciton density gradient, defined by the source and the sink. Furthermore, we find a geometric condition on the network for the appearance of ENAQT, relevant to natural and artificial systems.'
author:
- 'Elinor Zerah-Harush'
- Yonatan Dubi
title: 'Universal Origin for Environment-Assisted Quantum Transport in Exciton Transfer Networks'
---
Photosynthesis is the process used by plants, algae and some bacteria to convert solar energy into chemical energy. Different species perform photosynthesis with different machinery, but despite the differences, a good model for many photosynthetic complexes is a general three-part structure comprising an antenna, a reaction center, and an exciton transfer complex (ETC), which connects the two. The ETC is a network of chromophores embedded in a protein environment, and its structure and size vary from one organism to another. Its function, however, is similar: transporting the excitation energy from the antenna to the reaction center.[@mohseni2014quantum]
It has been accepted for many years that excitons are transported by an incoherent hopping process, i.e. by classical diffusive dynamics, via the $F\ddot{o}rster$ resonance energy transfer [@govorov2016]. While this seems the case for many molecular aggregates [@levi2015], it is apparently not the whole picture in light-harvesting complexes (LHC). For instance, it cannot explain the high efficiency of those complexes [@fleming2004; @levi2015]. Moreover, it cannot explain results of ultrafast nonlinear spectroscopy experiments [@engel2007; @Calhoun2009; @Collini2010; @Panitchayangkoon2011], showing evidence for long-lived oscillatory signals that were conjectured to be of quantum mechanical origin [@mohseni2014quantum].
In natural transfer complexes, the chromophore network that the exciton is transported through is covered by a large protein structure. Exposed to room temperature, it is far from the “clean” environment in which quantum systems are traditionally described. Consequently, the idea that a biological system, that apparently exploit all of its resources, may be using quantum coherence as a resource to assist energy transport has generated much excitement (see, e.g., reviews in Refs. [@Ishizaki2012; @Collini2013; @Lambert2013; @Pachon2012] and many references therein). The central phenomenon behind this idea is “environment assisted quantum transport” (ENAQT). According to the principle of ENAQT, the environment interrupts the phase-coherent transport of the quantum-mechanical excitations through the transfer complex by dephasing, in a way that [*enhances*]{} the efficiency of the energy transport. While the role of quantum mechanical transport and ENAQT in photosynthetic systems is still under debate[@kassal2013; @Miller2012; @ritschel2011; @duan2017nature], the concept of ENAQT is not limited to photosynthetic complexes, and was expanded to, e.g., electronic and molecular systems [@Semiao2010; @Nalbach2010; @Scholak2011a; @Ajisaka2015; @Lim2014; @leon2015noise], cold atoms [@Scholak2011a] and photonic crystals [@Biggerstaff2016; @viciani2016disorder; @Caruso2016].
ENAQT may arise from different mechanisms, and various suggestions were made in the theoretical literature to explain it. These include dephasing-induced delocalization [@Rebentrost2009; @Manzano2013; @Chin2010; @cao2009optimization], momentum rejuvenation [@Li2015], opening and broadening transport channels [@Plenio2008; @Caruso2009; @wu2010efficient], line-broadening [@Caruso2009], superradiance [@Nesterov2013; @Berman2015], supertransfer and funneling [@Baghbanzadeh2016], trapping-time crossover [@PhysRevLett.110.200402], directed flow [@dubiinterplay2015] and more. Here, we suggest that there is a universal mechanism for the origin of ENAQT in quantum networks which are larger than two chromophores (the realistic situation) and have a broad vibrational spectrum (also a realistic situation), making our theory relevant to many of the models presented in the literature (and, presumably, to natural ETCs). Specifically, we show that exciton transfer enhancement is a result of two competing processes, namely the tendency to uniformly spread the exciton population along the network, and the formation of a uniform population gradient.
Our results and analysis are based on calculation of exciton currents through a quantum network, defined by the general tight-binding Hamiltonian $$\label{eq:Tb_Of_Chain}
H=\sum_{i=1}^{n} \epsilon_i a_i^{\dagger}a_i-\sum_{i,j=1}^{n} t_{i,j} a_i^{\dagger}a_{j}~,$$ where $a^\dagger$ and $a_i$ are creation and annihilation operators of an exciton at chromophore *i*, $\epsilon_i$ are on-site exciton energies, and $t_{i,j}$ are coupling elements between two chromophores. The (Markovian) dynamics are described by the Lindblad equation [@breuer2002theory]: $$\label{eq: 31}
\frac{d\rho_S}{dt}=-i[H_S,\rho_S]+L\rho_s ~~,$$
where $\rho_S$ is the density matrix of the reduced system and $L$ is the Lindbladian, defined as $$L\rho_s=\sum_{k} \Gamma_k\big(V_k\rho_S V_k^\dag-\frac{1}{2}\{V_k^\dag V_k,\rho_S\}\big)~~,$$ where $V_k$ are Lindblad operators describing the action of the environment on the system, and $\Gamma_k$ is the respective rate of the Lindblad operator. The index $k$ represents different environments and/or different processes induced by these environments on the system.
Here, we consider the quantum network to be in contact with a source (the antenna), a sink (reaction center), and a dephasing channel (ETC protein environment), characterized by an exciton injection rate $\Gamma_{inj}$, extraction rate $\Gamma_{ext}$ and dephasing rate $\Gamma_{deph}$, respectively. In the presence of these environments, the Lindblad equation (Eq. \[eq: 31\] ) has the form $$\label{eq:5001}
\frac{d\rho_s}{dt}=-i[H,\rho_s]+L_{inj}\rho_s+L_{ext}\rho_s+L_{dep}\rho_s ~.$$ $L_{inj}$, $L_{ext}$, $L_{dep}$ are the injection, extraction and dephasing elements, respectively, with the corresponding operators $V_{inj}=a^{\dagger}_{inj},~V_{ext}=a_{ext}$ describing creation and anihilation of an exciton in the injection and extraction sites. These operators describe the non-equilibrium condition in which energy is constantly pumped into the system, but not the equilibrium limit [@gelbwaser2017thermodynamic]. Calculations were also performed with a thermodynamically consistent model [@gelbwaser2017thermodynamic; @Dubi2009d], and the results are similar (see SI for details). The dephasing operator is a local measurement, $V_{dep,i}= a^\dagger_i a_i$, and the dephasing part of the Lindbladian is $L_{dep}\rho_s=\sum_i L_{dep,i}\rho_s$, where $L_{dep,i}\rho_s=\Gamma_{deph}\big(V_{dep,i}\rho_S V_{dep,i}^\dag-\frac{1}{2}\{V_{dep,i}^\dag V_{dep,i},\rho_S\}\big)$. This form ensures that the fluctuations (and the ensuing dephasing) are local to each chromophore and not correlated between different chromophores.
We proceed by calculating the exciton transport at the steady-state, which seems to be the relevant state for natural systems[@manzano2012; @dubiinterplay2015]. Bothe energy and exciton currents can be evaluated by noting that the total energy, $\bar{E}=\mathrm{Tr} \left( H\rho_s \right)$ and total exciton number,$\bar{N_e}=\mathrm{Tr} \left( \hat{n} \rho_s \right)$ (where $ \hat{n}=\sum_i a^\dagger_ia_i$ is the total number operator), are time-independent, which allows for a propeper definiton of the energy and exciton currents, $J_q=\mathrm{Tr} \left( H L_{ext} \right),J_p=\mathrm{Tr} \left( \hat{n} L_{ext} \right)$ [@dubiinterplay2015; @manzano2012]. Solving for the steady state, i.e. $\frac{d\rho_s}{dt}=0$, one finds for the heat current $$\label{eq:16}
J_q=\Gamma_{ext}[\epsilon_n\rho_{n,n}+\frac{1}{2}t(\rho_{n,n-1}+\rho_{n-1,n})]~~,$$ and for the exciton current $$\label{eq:3001}
J_p=\Gamma_{ext}\rho_{n,n}~.$$
The relation between the particle current and heat current is evident from the comparison of equations \[eq:16\] and \[eq:3001\]. To plot the currents the steady state solution of Eq. \[eq:5001\] is placed into Eq. \[eq:3001\]. Further details on the calculation are provided in the SI.
We begin our analysis with the simplest symmetric system at hand, namely a uniform chain of *n*-sites and equal on-site-energies, $\epsilon$. Each site interacts with its neighbors via a constant hopping element *t*, as described by the Hamiltonian $$\label{eq:Tb_Of_Chain2}
H=\epsilon\sum_{i=1}^{n} a_i^{\dagger}a_i-t\sum_{i=1}^{n-1} a_i^{\dagger}a_{i+1}+h.c.~.$$ The symmetry of the system is reflected not only in the uniformity of the Hamiltonian, but also by the symmetry of the source and sink. In the case of the linear chain, we require an inversion symmetry between the positions of the source and sink, and place them at the edges of the chain, as depicted in the inset of figure \[fig:SimplestSetup\]a: the excitation takes place at the first site (yellow arrow), travels through the chain, and is extracted from the last site (red arrow).
![The linear symmetric chain: (a) Exciton current as a function of dephasing rate. inset: Schematic description of a symmetric 7-sites-chain with uniform energies and distances. The yellow arrow points to the injection site, and the red arrow marks the extraction site. The dashed line is $\Delta_n$ (see Eq. ). (b) Exciton occupation as a function of site number, different colors mark different dephasing rates (see legend). (c) Exciton occupations as a function of dephasing rate, for 7-sites chain. Different colors mark different sites along the chain (see legend).[]{data-label="fig:SimplestSetup"}](fig1.pdf){width="1\linewidth"}
Figure \[fig:SimplestSetup\]a shows the exciton current for this system, as a function of the dephasing rate $\Gamma_{deph}$. Injection and extraction rates were set to $\Gamma_{inj}=\Gamma_{ext}=5$ ps$^{-1}$ , on-site energies $\epsilon=1.23 \times 10^4$ cm$^{-1}$ and coupling elements $t=60 $ cm$^{-1}$, in line with the realistic parameters estimated for the Fenna-Mathiew-Olson (FMO) ETC [@Cho2005; @Lloyd2011; @dubiinterplay2015]. As could be expected for the linear symmetric chain [@Kassal2012], increasing the rate of dephasing decreases the current. The analytic relation between exciton current and occupation, Eq. \[eq:3001\], is motivation to examine exciton occupations, a quantity rarely addressed in the literature. Figure \[fig:SimplestSetup\]b shows the occupations of the sites (in a linear chain of 7 sites) as a function of site number, for different values of the dephasing rate. In the limit of small dephasing rate (blue spheres), the chain is essentially equally occupied by excitons (with the exception of the extraction site, see SI), reflecting the ballistic nature of the system. With increasing dephasing rate, a density *gradient* gradually forms, with large density at the injection (first) and low density extraction (last) sites. This gradient is most apparent in the fully classical limit of strong dephasing (red triangles). The appearance of a density gradient in the presence of current is a manifestation of Fick’s law which relates current to density gradient [@Meixner1965; @Kubo1966]. In fact, one can [*define*]{} the classical regime as the regime in which the density gradient is fully developed. Note that the occupation of extraction site (site number 7) decreases while the gradient is built, and accordingly, the exciton current decreases. Figure \[fig:SimplestSetup\]c shows the occupations of all sites (each color represent different site number) as a function of dephasing rate; they are uniform in the quantum limit, and spread as the dephasing rate increases.
The formation of a gradient can be understood from looking at the analytic expressions of the occupations of an $L$-site chain with a side-to-side transport, (derivation is detailed in the SI): $$\begin{aligned}
\label{eq:rhoexact}
n_i&=&\frac{m_i}{\sum_i m_i+\frac{\Gamma_{ext}}{\Gamma_{inj}}m_L}~,\nonumber \\
m_i&=&4t^2+(2(L-i)\Gamma_{deph}\Gamma_{ext}+\Gamma_{ext}^2)(1-\delta_{i,L})~~,\end{aligned}$$ where $n_i$ is the occupation of site $i$ (the diagonal element of the density-matrix, $\rho_{ii}$) in a chain of $L$ sites, and $\Gamma_{deph},\Gamma_{ext},\Gamma_{inj}$ are the rate of dephasing, extraction and injection, respectively. The expressions in Eq. \[eq:rhoexact\] reveals the effect of dephasing on the exciton density distribution in the chain; the second expression of $m_i$ shows the formation of the linear slope, with a gradient which is proportional to $\Gamma_{deph}$. Furthermore, it can be seen that the more distant the site is to the extraction point, the more pronounced will the effect of dephasing be on the occupation. It is clear from Eq. \[eq:rhoexact\] that the density at the extraction site is monotonously decreasing, leading to a monotonic decrease in exciton current.
So far as the symmetric linear chain is considered, no ENAQT is observed. However, it appears upon a slight modification of the system [@Kassal2012]. Consider the same uniform linear chain only with a slight change: the extraction site is moved away from the edge of the chain (thus breaking the inversion symmetry), schematically described in the inset of figure \[fig:middlextract\]a. Surprisingly, this seemingly minor difference yields qualitatively different behavior. The exciton current as a function of dephasing (main figure \[fig:middlextract\]a) displays a non-monotonic dependence, with a maximum in the current at a finite $\Gamma_{deph}$[@Kassal2012], signalling the appearances of ENAQT. We stress here that dephasing is a dissipative process, and yet the exction current (and consequently the energy flow) are enhanced in its presence.
![The linear non-symmetric chain: (a)-(c) Same as in Fig. \[fig:SimplestSetup\], for the 7-site linear non-symmetric chain. The non-symmetric chain shows a *qualitative* difference compared to the symmetric chain, manifested through a non-monotonic dependence of the current on dephasing rate, signaling the appearance of ENAQT.[]{data-label="fig:middlextract"}](fig2.pdf){width="1\linewidth"}
Figure \[fig:middlextract\]b shows the exciton occupation along a (non-symmetric) 7-site chain as a function of site number, for different dephasing rates. In contrast with the symmetric setup, here the occupations are not uniform in the quantum limit (blue spheres), reflecting the structure of the wave-functions and their interplay with the source and sink. With the increase in the dephasing rate, a uniform density gradient is formed between the injection and extraction sites. While the gradient is formed, the occupation of the extraction site increases and then decreases, and since the exciton current is proportional to the extraction site occupation, it acts similarly.
This can be seen more clearly in Figure \[fig:middlextract\]c, which shows the exciton occupations as a function of dephasing rate. We observe that the transition from a wide distribution of occupations (at $\Gamma_{deph}=0$) to a linear gradient ($\Gamma_{deph}=100 $ps$^{-1}$) passes through an intermediate stage where occupations along the chain become similar (at $\Gamma_{deph}\approx5 $ps$^{-1}$). This behavior is a result of two competing processes. The first is the direct outcome of dephasing, which can be considered the result of an instantaneous “measurement” of the system by the environment at a random site [@breuer2002theory], implying a full mixing of the system eigen-states. As a result, the real-space occupations tend to average into a narrower distribution [@PhysRevLett.85.812] (see SI for an example). The second process is the formation of the density gradient. While the gradient shape is determined by the positions of the extraction and the injection sites (see figure \[fig:SimplestSetup\]b and \[fig:middlextract\]b), its formation is enabled by the dephasing process. This can be deduced from the dependence of $m_i$ in equation \[eq:rhoexact\] on the dephasing rate. For a small dephasing rate, the position-dependent term (which is responsible for the gradient) is small compared with the first, position-independent part, while for large dephasing rate it is the dominant factor in determining $n_i$. The crossover between these two regimes leads to the non-monotonic shape seen in figure \[fig:middlextract\]c.
Comparing Figure \[fig:middlextract\]c with Figure \[fig:middlextract\]a reveals a correlation between the exciton current and the distribution of the occupations: the maximal current seems to appear at (or close to) the dephasing rate at which the spread of occupations is minimal. To quantify this relation between the distribution of exciton occupations and optimal current, we define the quantity
$$\label{eq:91409}
\Delta_{n}=1-\sqrt{\sum_i \bigg(\langle n_i\rangle - n_{ext}\bigg) ^2}~,$$
where $n_{ext}$ is the occupation of the extraction site, and $\langle n_i \rangle$ is the average occupation of the $i$-th site. $\Delta_{n}$ is a measure of the spread of the occupations, and as such, should exhibit a maximum at the same dephasing rate where the spread is minimal and current is maximal. In fact, this relation can be derived analytically within Lindblad theory under certain limitations (see SI). In Figs. \[fig:SimplestSetup\]a and \[fig:middlextract\]a, $\Delta_n$ is plotted (in arbitrary units) on top of the current (dashed lines). One can clearly see how the behavior of $\Delta_n$ follows that of the exciton current.
To demonstrate the universality of this relation, we examine it in larger and more complex networks. Figure \[fig:Examples\] shows the exciton current $I$ (blue line) and $\Delta_{n}$ (dashed orange line) as a function of dephasing rate, for selected networks of different topologies, dimensions, sizes and symmetries. As seen, the two quantities closely follow each other, and if there is a maximum in the particle current, $\Delta_{n}$ also exhibits a maximum, and at the same dephasing rate.
![Correlation between exciton current and density: Exciton current (Blue lines) and $\Delta_n$ (Orange dashed lines) as a function of dephasing rate for different network geometries and topologies:**(a)** A $5 \times 5$ network of chromophores with inversion symmetry, **b)** A $5 \times 5$ network of chromophores without inversion symmetry, **(c)** a ring of chromophores with uniform energies, **(d)** A ring of chromophores with random energies ($10^2-10^5$ cm$^{-1}$), **(e)** Cube of chromophores in a symmetrical setup, **(f)** Cube of chromophores in a non-symmetric setup, **(g)** full-graph of 16 chromophores with random energies and distances, the energy is extracted from two extraction-sites, **(h)** Biological setup-the FMO exciton network, **(i)** A pyramid-like network of chromophores. Two features are notable: 1) Non-monotonic behavior of the current appears only when there is no inversion symmetry, 2) in all examples, $\Delta_n$ follows the same behavior as the exciton current. []{data-label="fig:Examples"}](fig3.pdf){width="100.00000%"}
The results shown insofar were obtained by evaluating the steady-state solution of the Lindblad equation. We argue that these results do not depend on the calculation method. To show this we have calculated the currents as a function of dephasing rate for the same system (i.e. the symmetric and non-symmetric chains and the examples of Fig. 3) in two additional methods. The first is the full Redfield equation, which takes into account the spectral properties of the environment. The second is the time-dependent Lindblad equation, where a pulse-excitation was considered, and the current as a function of time was evaluated (and integrated to obtain the total current). In both cases we found the same results, namely that non-symmetric networks exhibit ENAQT, and that the behavior of the current correlates with $\Delta_n$, thus supporting our claims (details and results of these calculations are in the SI). We conjecture (and leave the verification to future studies) that these features persist beyond the Markovian limit, as steady-state currents should only be weakly affected by non-Markovianity [@diosi1997; @rebentrost2009non; @Dutta2017].
The question still remains, why does ENAQT only appear in non-symmetric networks, which do not posses an inversion symmetry. In the presence of an inversion symmetry (which includes, as mentioned above, interchanging the source and drain terms), the master equation for inversion points (points connected by inversion, except for the source and drain sites) are exactly the same. It follows that the density matrix itself is symmetric under inversion and every two inversion-related sites will have the same density, thus reducing the density fluctuations. In this case, the dephasing works only to form the density gradient, leading to a monotonic decrease in the sink site density and, respectively, the current. Put simply, ENAQT only occurs if the exciton density is non-uniform in the fully quantum limit, which is never the case in a uniform system with an inversion symmetry.
Comparing between figures \[fig:middlextract\](a) and \[fig:SimplestSetup\](a) one can see that the exciton current is actually higher in the coherent regime for the symmetric system vs the asymmetric system. However, if disorder, asymmetry or a dephasing environment are unavoidably present (which seems to be the case for natural photo-synthetic complexes), the intermediate coherent-dephasing regime delivers better performance. Comparing the transport properties of different geometries can thus be an important tool for understanding transport mechanisms in artificial ETCs [@Eisenberg2014; @eisenberg2017concentration; @banal2017photophysics; @boulais2017programmed], as well as in other systems where dephasing may play an important role, e.g. electronic transport through bio-molecules and molecular junctions [@penazzi2016self; @kocherzhenko2010charge; @nozaki2012disorder; @contreras2014dephasing; @xiang2015intermediate].
We acknowledge support from the Adelis foundation. E.Z-.H. Acknowledges support from the IKI interdisciplinary fellowship.
Detailed methods description. Relation between exciton occupations and current. Derivation of occupations in a linear symmetric chain. Effect of dephasing on occupations in the absence of current. Calculation using the Redfield equation. Detailed examples of exciton occupation as a function of dephasing rate. Calculation using a thermodynamically-consistent Lindblad equation.
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abstract: 'The Transiting Exoplanet Survey Satellite (TESS) has a goal of detecting small planets orbiting stars bright enough for mass determination via ground-based radial velocity observations. Here we present estimates of how many exoplanets the TESS mission will detect, physical properties of the detected planets, and the properties of the stars that those planets orbit. This work uses stars drawn from the TESS Input Catalog Candidate Target List and revises yields from prior studies that were based on Galactic models. We modeled the TESS observing strategy to select approximately 200,000 stars at 2-minute cadence, while the remaining stars are observed at 30-min cadence in full-frame image data. We placed zero or more planets in orbit around each star, with physical properties following measured exoplanet occurrence rates, and used the TESS noise model to predict the derived properties of the detected exoplanets. In the TESS 2-minute cadence mode we estimate that TESS will find $1250\pm70$ exoplanets (90% confidence), including 250 smaller than 2 Earth-radii. Furthermore, we predict an additional 3100 planets will be found in full-frame image data orbiting bright dwarf stars and more than 10,000 around fainter stars. We predict that TESS will find 500 planets orbiting M-dwarfs, but the majority of planets will orbit stars larger than the Sun. Our simulated sample of planets contains hundreds of small planets amenable to radial velocity follow-up, potentially more than tripling the number of planets smaller than 4 Earth-radii with mass measurements. This sample of simulated planets is available for use in planning follow-up observations and analyses.'
author:
- Thomas Barclay
- Joshua Pepper
- 'Elisa V. Quintana'
title: 'A Revised Exoplanet Yield from the Transiting Exoplanet Survey Satellite (TESS)'
---
Introduction
============
While we have known that planets orbit stars other than the Sun since the late 20th Century [@Walker1988; @Latham1989; @Wolszczan1992; @Mayor1995], it is only with the launch of the Kepler spacecraft in 2009 [@Koch2010; @Borucki2010] that we have been able to estimate the occurrence rates of terrestrial worlds. While there is not a firm consensus on the details of how common planets are as a function of size and orbital period [@Howard2010; @Gould2010; @Catanzarite2011; @Youdin2011; @Howard2012; @Traub2012; @Bonfils2013; @Swift2013; @Fressin2013; @Petigura2013a; @Petigura2013b; @Montet2014; @Kane2014; @Foreman2014; @Burke2015; @Clanton2016; @Hsu2018] it is clear that exoplanets overall are fairly commonplace, particularly orbiting the coolest of stars [@Dressing2013; @Dressing2015; @Morton2014; @Mulders2015].
Although we have a fairly large sample of planets with orbital periods of less than a few hundred days, there is still a pressing need to detect planets that are readily characterizable. The primary goal of the Transiting Exoplanet Survey Satellite (TESS), a mission led by the Massachusettes Institute of Technology, is to find small planets that are most amenable for mass measurements through precise radial velocity observations [@Ricker2015; @Ricker2016; @Collins2018]. A secondary, although unofficial, mission goal is to find targets that can be characterized through transmission spectroscopy from the James Webb Space Telescope and other future observatories.
TESS launched on April 18, 2018, and resides in an elliptical 13.7 day high-Earth orbit during a 2-year primary mission. TESS has four cameras, each with a $24^{\circ}\times24^{\circ}$ field of view. The cameras are aligned to provide continuous coverage of $96^{\circ}\times 24^{\circ}$, which is maintained for 27.4 days per pointing (known as a sector). The long axis of the observing region is aligned with a fixed ecliptic longitude, with the boresight of the fourth camera centered on the ecliptic pole, as shown in Figure \[fig:sectors\]. Every two orbits, TESS rotates $\sim$28$^{\circ}$ about the ecliptic pole. In year 1 of the mission, the spacecraft will survey 13 sectors in the southern ecliptic hemisphere, before spending year 2 in the northern ecliptic hemisphere. About 60% of the sky will be covered by a single sector of TESS observations, and a further 15% will be observed over two sectors, located in the overlap areas between two adjacent sectors. Most stars within 12 degrees of the ecliptic poles will be within the TESS continuous viewing zone (CVZ) and observable for more than 300 days (this accounts for approximately 1% of the sky per pole). Over the course of the prime mission, TESS will observe approximately 85% of the sky.
![An illustration showing the first three sectors of the TESS observing plan.[]{data-label="fig:sectors"}](tessectors.pdf){width="45.00000%"}
The TESS mission is focused on detecting small transiting planets that orbit bright stars. Although the dwell time over most of the sky is too short to permit the detection of planets in temperate orbits, that goal can be advanced by discovering planets orbiting cooler stars, especially in the TESS CVZ around the ecliptic poles.
Two observing modes will be initially implemented: the $96^{\circ} \times24^\circ$ full-frame image (FFI) will be recorded every 30-minutes, while approximately 200,000 stars will be preselected to have data recorded at 2-minute cadence. In either case, the system is integrating and reading out every 2 seconds; they differ in the number of coadds.
It is essential that a *reasonable* prediction for the scientific yield of TESS is available because (a) planning follow-up resources requires knowing the properties of the planets we might find [@Louie2018; @Crouzet2017; @Collins2018; @Kempton2018], (b) we can perform trade studies on target prioritization schemes for the 2-minute cadence targets [@Bouma2017; @Stassun2017 Pepper et al. in preparation], and when designing data analysis algorithms [@Kipping2017; @Lund2017; @pyke3; @Oelkers2018], and (c) we can manage the expectations of the scientific community and the public.
A TESS yield simulation created by @Sullivan2015 has been the standard used by both the mission team and the community. Since then, two papers have built on the work of @Sullivan2015 to refine the total mission yield and explore extended mission scenarios [@Bouma2017], and to improve estimates of the planet yield from M-dwarfs [@Ballard2018]. However, @Sullivan2015 simulations were based on a simulated stellar population rather than real stars, and used an earlier hardware configuration that provided for greater storage and downlink limits than the flight hardware being used. Therefore, now is the time to revise the TESS yield estimate using new information. Here we report on a new estimate of the exoplanet yield using the TESS Input Catalog (TIC) Candidate Target List (CTL), the same list that is used by the mission to select stars and perform photometry.
Simulating stars, planets, and detections
=========================================
The process we used to derive a population of planets detectable by TESS uses a Monte Carlo method to (1) simulate the population of stars that TESS will observe, (2) place planets in orbit around these stars, and (3) predict how many of these planets TESS will detect.
Star selection {#sec:starselction}
--------------
The first step was made relatively straightforward by the availability of the CTL - a prioritized list of target stars that the TESS Target Selection Working Group have determined represent the stars most suitable for detection of small planets by TESS. The properties of about 500 million stars were assembled in the TIC [@Stassun2017], and the CTL includes several million of those stars that are most suitable for small transit detection. We used CTL version 6.1[^1], which includes 3.8 million stars with properties such as stellar radii, masses, distances, and apparent brightness in various bandpasses. The CTL stars were then ranked using a simple metric based on stellar brightness and radius, along with the degree of blending and flux contamination (especially important given the large TESS pixels). The CTL does not include all stars. Save for stars on specially curated target lists [e.g. @Muirhead2018], stars with reduced proper motions that indicate they are red giants [@Collier2007], stars with a temperature below 5500 K and a TESS magnitude fainter than 12, or stars with temperature above 5500 K and a TESS magnitude fainter than 13, are excluded from the CTL. Such broad cuts were required in order to assemble a small enough population of stars to practically manage.
We then determined which of these stars are likely to be observed by the mission. We used [*tvguide*]{} [@Mukai2017] on each star to determine whether and for how long it is observable with TESS. We arbitrarily selected a central ecliptic longitude for the first sector of 277$^\circ$ which equates to an antisolar date of June 28 (the precise timing of the first sector is dependent on commissioning duration). Until we have on-orbit measurements of focal plane geometry, [*tvguide*]{} assumes that the cameras are uniform square detectors projected on the sky, placed precisely 24$^\circ$ apart in ecliptic latitude and with identical ecliptic longitude. Gaps between CCDs are assumed to be 0.25$^\circ$. We ended up with a total of 3.18 million individual stars on silicon.
We also needed to simulate which of these stars are likely to be observed at 2-minute cadence and ensure compliance with the TESS mission requirement that states that over the 2-year mission over 200,000 total stars should be targeted, and 10,000 stars should be observed for at least 120 days. It is somewhat less trivial than one would initially assume to simulate this requirement because we could not simply select the top 200,000 stars with the highest priority in the CTL because this would place far too many stars in the CVZ than can actually be observed there at 2-minute cadence. To ensure a realistic distribution of targets, we first divided each ecliptic hemisphere into 15 sections: a polar section with everything within 13 degrees of the pole, representing stars that primarily fall into Camera 4; an ecliptic section including everything within 6 degrees of the ecliptic to represent stars that are not observed in the prime mission; and then the remaining area was divided longitudinally into 13 northern and 13 southern adjacent sections, representing stars observable with Cameras 1–3 in Sectors 1–26. This yielded a total of 28 sections of the sky with observable stars.
A star that fell in a camera overlap region is observed in multiple sectors but only represented one unique target. We found that we could make a reasonable approximation to satisfy the requirements of 200,000 unique targets if in each polar section we selected the 6,000 stars in that region with the highest priority in the CTL, and then for each longitudinal section (representing the footprint of Cameras 1–3 in each sector) we selected the 8,200 highest priority stars in each of the regions. After removing stars that fall into CCD and camera gaps, this yielded 214,000 unique stars. We assumed that any star in an overlap region is observed in every possible sector.
![The number of CTL targets observed for a given number of 27.4-day sectors. FFI targets are shown in blue, and 2-minute cadence targets in red. In total 3.2M CTL targets are observed, of which 214,000 are observed at 2-minute cadence. Roughly three-quarters of targets are only observed for a single sector, with just 2.1% having 12 or 13 sectors of coverage. The 2-minute cadence targets are disproportionately observed for more sectors, with 4.2% of the 2-minute cadence targets receiving 12 or 13 sectors of coverage.[]{data-label="fig:stars-sectors"}](stars-sectors-both.png){width="45.00000%"}
While the CTL includes a great deal of curation, it is not infallible. A particular weakness inherent to stellar catalogs based on photometric colors is in distinguishing between dwarf stars and subgiants [@Huber2014; @Mathur2017]. CTL versions up through 6.2 use parallax information when available to determine stellar radii (and therefore luminosity class), but the vast majority of stars depend on the use of reduced proper motion (RPM) cuts to distinguish dwarfs from giants. While GAIA DR2 will shortly provide reliable parallaxes for most CTL stars [@Huber2017; @Davenport2017; @Stassun2018], the CTL will not be significantly modified until 2019. Furthermore, while the RPM method is highly reliable at distinguishing dwarfs and subgiants as a group from giant stars, it is generally not useful for distinguishing dwarfs from subgiants. Of the CTL stars that are classified as dwarfs based on the RPM cut, about 40% are actually subgiants, although roughly 35% of the CTL stars have parallax measurements confirming their spectral class. To account for this effect, we simulated a misclassified population of subgiants by increasing the stellar radius of 40% of those AFGK stars which had been selected with the RPM cut by a factor 2, with the affected stars drawn at random. That included 25% of all the AFGK stars in the CTL. This approach somewhat overestimates the radii of A-type subgiants but the effect on total planet yield is limited, because A-type stars have large radii, making detecting transiting planets challenging, and thus are already a relatively small fraction of the high-priority CTL stars.
Simulating planets
------------------
To each star in our list we assigned zero or more planets. The number of planets assigned to each star was drawn from a Poisson distribution. The mean (referred to here as $\lambda$) of the Poisson distribution we used differs between AFGK-dwarf stars and M-dwarfs because there is strong evidence that M-dwarfs host more planets on short orbital periods [@Mulders2015; @Burke2015]. For AFGK stars we used the average number of planets per star with orbital periods of up to 85 days of $\lambda=0.689$ [@Fressin2013], while for M-stars $\lambda=2.5$ planets are reported with orbital periods up to 200 days [@Dressing2015].
Each planet was then assigned six physical properties drawn at random: an orbital period ($P$), a radius ($R_p$), an eccentricity, a periastron angle, an inclination to our line of sight ($i$), and a mid-time of first transit. The orbital period and radius were selected using the exoplanet occurrence rate estimate of @Fressin2013 for AFGK stars, and @Dressing2015 for M-stars. Both @Fressin2013 and @Dressing2015 reported occurrence rates in period/radius bins. We drew at random from each of these bins with the probability to draw from a given bin weighted by the occurrence rate in that bin divided by the total occurrence rate of planets. For example, @Dressing2015 reported a 4.3% occurrence rate for planets with radii 1.25–2.0 $R_\oplus$ and orbital period 10–17 days, so in our simulation we drew planets from that bin with a frequency of 4.3 divided by the total occurrence rate in all bins. We normalized by the total occurrence rate of planets since we already took account of systems with zero or multiple planets in the Poisson draw. Once we knew which bin to select a planet from, we drew from a uniform distribution over the bin area to select an orbital period-radius pair, except for the giant planet bin where we draw from a power-law distribution in planet radius with exponent -1.7, which mirrors [@Sullivan2015]. This non-uniform giant planet size distribution reduces the number of nonphysical inflated planets, as discussed by @Mayorga2018. Occurrence rates from both @Fressin2013 and @Dressing2015 are based on Kepler data and are limited in orbital period to 0.5–85 and 0.5–200 days, respectively.
Following @Kipping2014, the orbital eccentricity was selected from a Beta distribution, with parameters $\alpha=1.03$ and $\beta=13.6$, which @Vaneylen2015 found was appropriate for transiting planets. The periastron angle was drawn from a uniform distribution between $-\pi$ and $+\pi$. The cosine of inclination was chosen to be uniform between zero and one. Planets in multiple-planet systems were assumed to be coplanar - i.e. they have the same $\cos{i}$ - which is a reasonable appumption because multiple-exoplanet systems have been found to be highly coplanar [@Xie2016]. Finally, we chose a time of first transit to be uniform between zero and the orbital period – note that this may be greater than the total observation duration, in which case no transit was recorded. We then computed the number of transits observed using the observation duration calculated previously (the number of sectors where a target is observed).
We intentionally kept planets that cross the orbit of other planets in the system because, while they are likely on unphysical orbits, to remove them would change the distribution of the number of planets per star, which is an observed property. We also assumed that none of these planets experience a significant amount of transit timing variations (Hadden et al. in preparation, address transit timing variations and period ratios in detail).
Detection model
---------------
Armed with a sample of planets and host stars, we then determined which planets are detectable. To do this we derived a transit depth modified by several factors: the flux contamination of nearby stars, the number of transits, and the transit duration. It should be noted that flux contamination is significantly more problematic for TESS than with Kepler because TESS has pixels that are 28 times larger than Kepler’s.
The raw transit depth was computed assuming a uniform disk (i.e, transit depth $T_d = (R_p / R_\star)^2$, where $R_\star$ is the stellar radius). That is, we ignored the effects of limb-darkening and grazing transits. We calculated the reduction in transit depth due to dilution from nearby stars using the value of contamination for the CTL as $T_d / (1 + d)$, where $d$ is the dilution, the fraction of light coming from stars that are not the target divided by the total star light. We then multiplied the transit depth by the square root of the transit duration ($T_{\textrm{dur}}$) in hours, with transit duration following @Winn2010 defined as, $$T_{\textrm{dur}} = \frac{P}{\pi} \arcsin\left[{ \frac{R_\star}{a} \frac{\sqrt{(1+R_p/R_\star) - b^2}}{\sqrt{1 - \cos^2{i}}}}\right],$$ where $P$ is the orbital period, $i$ is the orbital inclination relative to our line of sight, $a/R_\star$ is the semimajor axis in units of stellar radius, $b$ is the impact parameter, and $R_p / R_\star$ is the planet to star radius ratio, to derive an effective transit depth. The effective transit depth, $T_d^\prime$, is defined as $$T_d^\prime = (R_p / R_\star)^2 \times \sqrt{T_{\textrm{dur}}} \times \sqrt{N} \times \frac{1}{1 + d},$$ where $N$ is the number of transits observed.
We took the TESS photometric noise level from [@Stassun2017] who used the properties described by @Ricker2016 and tested whether the effective transit depth was greater than the TESS photometric noise at the stellar brightness of the host stars multiplied by 7.3 (i.e. SNR$\ge$7.3). A 7.3-sigma detection is the nominal value used by @Sullivan2015 and is calculated in a similar manner to the detection threshold used by Kepler [@Jenkins2010]. We also required that the impact parameter of the transit is less than 1.0 and that we observed at least 2 transits. Requiring an impact parameter of less than 1 removes a small number of grazing transits but these are difficult to distinguish from eclipsing binaries anyway [@Armstrong2017]. These detection thresholds are relatively aggressive, Section \[sec:conservative\] describes using a more conservative detection thresholds of at least 3 transits and SNR of 10.
Results {#sec:results}
=======
We performed 300 simulations using our nominal planet sample and detection criteria, this enabled us to look at the average and range from our simulations. We predict that TESS will find 4373 planets (median) orbiting stars on the CTL, with the 90% confidence interval ranging from 4260–4510 planets. Henceforth, we designate a simulation that produced the median number of planets as our fiducial simulation and the properties we show come from that simulation. All the stars in the CTL are included in Figure \[fig:skyplot-ffis\] and the detected planets are shown as red dots.
![The spatial distribution of target stars and detected planets from FFI data. The upper panel shows the southern ecliptic hemisphere and the lower panel shows the northern ecliptic hemisphere. Stars observed for 1 sector are shown in blue, two sectors in orange, 3+ sectors in green, and stars in the CVZ are shown in purple. Detected planets are shown as red dots. A total of 4373 planets are shown, of which 54% were only observed for a single sector, and 11% were observed for 12 or 13 sectors. The lower density of stars, offset from the south ecliptic pole, is centered on the south celestial pole, and is due to relatively incomplete proper-motion catalogs in the celestial south.[]{data-label="fig:skyplot-ffis"}](skyplot-ffis-v2.png){width="45.00000%"}
Our fiducial simulation has 1293 planets orbiting 2-minute cadence targets, and the 90% confidence range of planets found in 2-minute data is 1180–1310 planets. The sky distribution is shown in Figure \[fig:skyplot-2min\]. There are clear differences in features between the FFI distributions and the 2-minute cadence distributions. The FFI stars are not evenly distributed, there is a lower density of stars in the southern sky. This is caused by the use of the reduced proper motion cut to identify dwarf stars, since existing proper motion catalogs are less complete below a declination of $-30^\circ$. This low density at southern latitudes is not visible in the 2-minute cadence plots because the high quality AFGK stars chosen for 2-minute cadence observations are bright enough that the proper motion catalogs are essentially complete for them. However, M dwarfs are faint enough that the proper motion catalogs are not complete for even high priority stars below a declination of $-30^\circ$, and they are undersampled among the 2-minute targets in that region.
The Galactic plane is visibly underpopulated in the 2-minute cadence data for two related reasons. Stars near the galactic plane tend to have higher flux contamination, which depressed their calculated priority. Also, photometric catalogs have a great deal of unreliability in the galactic plane in variety of ways, including proper motions, source identification, and the effects of reddening on the stellar temperatures. Therefore the priorities of all CTL stars within 15 degrees of the galactic plane were systematically down-weighted in the CTL, except for a subset of specially identified stars.
For both the 2-minute and the FFI-observed stars, we found planets more frequently closer to the ecliptic poles, where the longer observing baseline makes transit detection easier and where it is possible to find longer-period planets.
![The spatial distribution of target stars and detected planets from 2-minute cadence data. The colors of stars and planets is the same as shown in Figure \[fig:skyplot-ffis\]. The southern hemisphere, and to a lesser extent the northern hemisphere, has a pronounced feature of the Galactic plane running through where priorities are down-weighted because the high stellar density will dilute transit signals making them harder to detect.[]{data-label="fig:skyplot-2min"}](skyplot-2min-v2.png){width="45.00000%"}
As shown in Figure \[fig:planet-radii\], our simulation predicts that TESS will find 41 Earth-sized worlds ($<$1.25 $R_\oplus$), 238 super-Earths (1.25–2.0 $R_\oplus$), 1872 sub-Neptunes (2.0–4.0 $R_\oplus$), and 2222 giant planets ($>$4.0 $R_\oplus$) orbiting stars on the CTL. In total 279 planets smaller than 2.0 $R_\oplus$ were detected in our simulation, 90% of which were orbiting targets observed at 2-minute cadence. The sub-Neptunes were split roughly evenly between those observed at 2-minute cadence and those found only in FFI data, but nearly 90% of giant planets were found in the FFI data.
![Our simulations predict that TESS will detect a total of about 4400 planets orbiting stars on the CTL, of which 1300 will be observed at 2-minute cadence. Roughly 40 Earth-sized planets will be found, almost all of which are on the 2-minute target list. One thousand super-Earths and mini-Neptunes will also be found. Many new giant planets will be discovered, primarily through FFI data. The numbers shown above the FFI bars are total planets, and include the planets found in 2-minute cadence data.[]{data-label="fig:planet-radii"}](planet-radii-both-v2.png){width="45.00000%"}
A summary of the properties of planets detected in FFIs and 2-minute cadence data is given in Table \[tab:summary\]. Full details of every planet detection in our simulation is provided in a machine readable table, with a summary shown in Table \[tab:allplanets\].
-------------------------------- -------- ------------ ------------- -------- ------------ -------------
Property Median 5th pctile 95th pctile Median 5th pctile 95th pctile
Host star radius ($R_\odot$) 1.02 0.23 2.44 1.35 0.32 3.48
Host star mass ($M_\odot$) 0.95 0.20 1.61 1.07 0.32 1.93
Host star temperature (K) 5500 3200 7200 5900 3400 8000
Host star brightness, $Ks$ 9.2 6.7 11.0 10.0 7.4 11.5
Host star brightness, TESS mag 10.4 7.5 13.5 11.0 8.2 13.1
Host star brightness, $V$ 11.3 7.9 16.3 11.7 8.8 15.4
Planet radius ($R_\oplus$) 3.1 1.4 8.9 4.2 1.9 15.1
Planet orbital period (days) 8.2 1.7 34.8 7.0 1.8 29.0
Transit duration (hours) 3.0 1.0 8.7 3.9 1.3 10.4
SNR 13.6 7.7 109 13.3 7.6 93.7
Number of transits 7 2 65 6 2 51
Distance (pc) 140 50 200 260 70 890
-------------------------------- -------- ------------ ------------- -------- ------------ -------------
Num Units Label Explanation
----- --------- --------------- ------------------------------------------------------------------------------------------
1 – TICID TESS Input Catalog ID number of star
2 deg RAdeg Right ascension 2000
3 deg DEdeg Declination 2000
4 deg ELON Ecliptic longitude
5 deg ELAT Ecliptic latitude
6 – Priority CTL v6.1 priority
7 – 2min-target Was this a 2-minute cadence target in our model? 1 = yes, 0 = no
8 – Camera TESS camera number, number between 1–4
9 d Obslen Number of days that target is observed
10 – Num-sectors Number of sectors the target is observed for
11 mag Vmag V-band magnitude
12 mag Kmag Ks-band magnitude
13 mag Jmag J-band magnitude
14 mag Tmag TESS bandpass magnitude
15 solRad Star-radius Stellar radius
16 solMass Star-mass Stellar mass
17 K Star-teff Stellar effecitve temperature
18 pc Distance Distance of the star
19 – Subgiant Was this star randomly selected to be a subgiant? 1 = yes, 0 = no
20 – Detected Was this planet detected? 1 = yes, 0 = no
21 – Detected-cons Was this planet detected using the conservative model? 1 = yes, 0 = no
22 d Planet-period Orbital period of the planet
23 Rgeo Planet-radius Radius of the planet
24 – Ntransits Number of transits the planet has, 0 if planet does not transit
25 – Ars Planet semimajor axis divided by the stellar radius
26 – Ecc Planet orbital eccentricity
27 – Rprs Planet radius divided by the stellar radius
28 – Impact Planet impact parameter
29 h Duration Planet transit duration
30 ppm Depth-obs The observed transit depth, corrected for dilution
31 – Insol Insolation flux the planet receives relative to that received by the Earth from the Sun.
32 ppm Noise-level The one-hour integrated noise level of the star
33 – SNR Combined signal-to-noise ratio of all transits, 0 if planet does not transit
About 75% of stars were observed for a single sector. Unsurprisingly, most planets (2334, 53%) were also only observed for a single sector and three-quarters of planets were observed for one or two sectors. Conversely, while just 2% of CTL stars were observed for 12 or 13 sectors, 11% of all planets detected were found around these stars. The longer observing baseline gave both higher SNR transits, and sensitivity to longer orbital period planets. The number of stars observed at 2-minute cadence for 12 or 13 sectors was fairly heavily constrained in our target selection model, therefore a relatively high fraction (60%) of planets were found in the FFI data for the high latitude fields. Overall 70% of planets were found only in the FFIs, but for stars that were observed between 4–11 sectors, just 40% of planets were found only in the FFI data.
The orbital periods of our planets ranged from 0.5–99 days, which is a somewhat artificial limitation based on the occurrence rates used. The minimum orbital period of the injected transit signals was 0.5 days. While we know of several ultra-short period planets [e.g. @Sanchis2013], they are very rare [@Winn2018] and therefore will not significantly impact the planet yield. On the long period end, we simulated M-dwarf planets with periods up to 200 days, yet no planets with periods longer than 100 days were recovered, so we are confident that few long periods planets were missing here. For hotter stars, we only simulated planets with periods up to 85 days. It is likely we were missing planets orbiting stars with periods longer than 85 days. However, we only found two planets in our M-dwarf sample with periods longer than 85 days, and in the 65–85 day period range for the AFGK sample we had just 17 planets. Since the probability of a planet to transit scales inversely with orbital distance, and the number of stars with a long enough observing baseline to detect at least two transits similarly shrinks, we do not expect more than a handful of additional long period planets. We do caution that our sample should probably not be used to estimate the yield of planets showing a single transit because the 85 day limit becomes more significant. For a study of single transiting planets we point readers to @Villanueva2018.
![The number of sectors that stars with detected planets were observed for, with a sector having an average observing window of 27.4 days. More than half of planets were observed for a single sector, with 10% being observed for 12 or 13 sectors.[]{data-label="fig:planet-sectors"}](planets-sectors-both-v2.png){width="45.00000%"}
In Figure \[fig:hitrate-sectors\] we show the ratio of stars observed to planets detected – which we define as the ‘hit rate’. Overall, the hit rate for 2-minute cadence targets was 0.60%, while for the CTL stars not on the 2-minute cadence list the hit-rate was 0.10%. Hit rate increases with observing duration, from 0.43% for 2-minute cadence targets observed for 1 sector up to 1.8% for 2-minute cadence targets with at least 12 sectors of data.
![The ratio of stars observed to planets detected as a function of the number of sectors a star is observed for. The longer a star was observed, the higher probability a planet would be detected. Targets observed at 2-minute cadence are shown in red, while blue are FFI targets. For 2-minute cadence stars the average hit-rate was 0.60%, while including all stars on the CTL drops this to 0.14%. While observing for a longer baseline increased the number of planets, the increase is not linear. For 2-minute cadence targets, an increase of 12x in observing baseline increased the hit-rate by a factor of just 4.4. There are comparatively few planets in the 12 and 13 sector bins, so we show Poisson uncertainties on these bars demonstrating that there is not a measurable difference between observing for 12 or 13 sectors. Red and blue bars are not stacked, both start at zero.[]{data-label="fig:hitrate-sectors"}](hitrate-sectors-both-v2.png){width="45.00000%"}
We found that the planet host stars range in brightness from $V$-band mag of 4.0–20, with 7 planets predicted to orbit stars brighter than 55 Cnc, currently the brightest transiting planet host [@Winn2011]. As shown in Figure \[fig:tess-mag\], in the TESS bandpass, 90% of planets orbited stars with magnitudes between 8.2–13.1, this compares with Kp=11.9–15.9 for Kepler planet candidates [@Thompson2018]. The simulated planets typically orbited stars 3 magnitudes brighter than Kepler planets. Planets around stars observed at 2-minute cadence were systematically brighter than the planets found orbiting stars observed only in FFI data, with a median TESS magnitude of 10.4 versus 11.0.
With TESS concentrating on finding planets orbiting cool stars, it is unsurprising many planets orbited stars that were bright in the infrared. The median $Ks$-band ($\sim$2.0–2.2$\mu m$) magnitude of planets in 2-minute cadence data was 9.2 and 90% of 2-minute cadence planets were brighter than Ks=10.7. None of the TESS 2-minute planets orbited stars fainter than the median infrared brightness of Kepler planet candidates of Ks=13.0.
![Brightness of the planet host stars in the TESS bandpass magnitude. The median brightness of stars with planets found in 2-minute cadence data was 10.4, with a maximum range of 3.5–15.3. For planets found only in FFI data, the median brightness was 11.3, and a maximum range of 6.1–16.4. []{data-label="fig:tess-mag"}](tess-mag-both-v2.png){width="45.00000%"}
The spectral type distribution of the detected planet host stars is shown in Figure \[fig:spectral-type\]. About a quarter of the planets found in 2-minute cadence data orbited M-dwarfs (371) with the remaining split fairly evenly between K (216), G (351), and F (299) stars. The deficit in planets orbiting K-dwarfs was caused by a deficit in K-dwarfs selected for 2-minute cadence observations. This was a result of the target prioritization strategy employed, and has been noted previously [@Stassun2017]. A few additional planets orbiting cool stars were found in FFI data (only 125 additional M-dwarfs), but 80% of FFI-only planets orbited stars larger than the Sun. In total about 10% of planets in our simulated sample orbited M-dwarfs.
![The spectral type distribution of TESS planet-hosting stars. Our simulations predict that TESS will find 496 planets orbiting M-dwarfs, of which 371 orbit stars observed at 2-minute cadence. About half the simulated planets in 2-minute cadence data orbited stars larger than the Sun, while 80% of planets found only in FFI data orbited stars larger than the Sun.[]{data-label="fig:spectral-type"}](spectral-type-both-v2.png){width="45.00000%"}
Figure \[fig:distance\] shows the distance to the simulated planets[^2]. The closest detected planet in our simulation orbited Lalande 21185, a star 2.5 pc away. We found 46 planets within 50 pc, and 234 within 100 pc, which doubles and quadruples the number of transiting planets known within 50 and 100 pc, respectively [@Akeson2013].
![The distances of planets found in our simulation in parsecs. The upper panel shows both distance and ecliptic latitude of the host stars, and the lower panel is distance plotted against planet radius. Almost all 2-minute cadence planets discovered by TESS will be within 300pc, with 77% within 200pc. FFI planets were found over 1000 pc away but 90% of planets were within 700 pc.[]{data-label="fig:distance"}](distance-both-v2.png){width="45.00000%"}
The circumstellar habitable zone concept has been popular since at least the 1950s [@Strughold1953; @Shapley1953], and refers to the spherical shell around a star where liquid water could be present on a planetary surface. @Kopparapu2013 provided models for an optimistic habitable zone with boundaries of recent Venus and early Mars, which correspond to stellar fluxes of 1.78x and 0.32x the insolation Earth receives from the Sun, respectively. Our simulation contains 69 planets in the optimistic zone, of which 9 are smaller than 2 Earth-radii. All the habitable zone planets orbit M-dwarfs.
Suitable targets for RV follow-up
---------------------------------
For the TESS mission to be successful, it must find planets smaller than 4 Earth-radii with a measurable radial velocity signal. We predict that TESS will find more than 2100 planets smaller than 4 Earth-radii, but many of these will orbit stars whose brightness makes follow-up challenging or impossible with current precision radial velocity facilities. While planets orbiting very faint stars have had their mass determined via radial velocity studies [e.g. @Koppenhoefer2013], it is typically challenging to measure masses of planets around stars fainter than V=12. We predict that TESS will find 1300 planets smaller than four Earth-radii around stars brighter than V=12. Therefore, with more than 1000 potential targets, TESS will have a plethora of targets to choose from when selecting promising RV targets. Even if just 20% are good RV targets, this will more than triple the number of planets smaller than 4 Earth-radii with measured masses.
There are 160 planets in our sample that are smaller than 2 Earth-radii and orbit stars brighter than V=12. We currently have mass and radius constraints on fewer than 60 planets smaller than 2 Earth-radii, so TESS will potentially greatly increase this number, although the precise number will depend on whether individual stars are suitable for precise radial velocity measurements.
Targets for atmospheric characterization
----------------------------------------
A second aim of the TESS mission is to find targets suitable for transmission spectroscopy using the James Webb Space Telescope (JWST). Until on-sky performance is measured, particularly the systematic noise level, there is considerable uncertainty on how JWST will perform [@Batalha2017]. However, we can identify the properties of planets that would make them good JWST targets using a few simple cuts. The host star should be bright in the infrared, and the star should be small. We identified simulated planets whose host stars have Ks$<$10, $T_{\rm eff}<3410 K$ which equates to M3V stars with a radius of approximately 0.37 solar-radii [@Pecaut2013]. In total there were 70 planets fulfilling these criteria. We show in Figure \[fig:planet-radius-insolation-cool-stars\] the simulated small planets we think make interesting candidate JWST targets in terms of insolation fluxes. There are ten planets in the boxed region in Figure \[fig:planet-radius-insolation-cool-stars\] which highlights planets that fell into the optimistic habitable zone [@Kopparapu2013], and had radii between 1.25 and 2.5 Earth-radii, implying a puffed-up atmosphere [@Lopez2014]. These planets, along with those orbiting TRAPPIST-1 [@Gillon2017] and other low mass stars [@Greene2016; @Kreidberg2016; @Morley2017; @Louie2018], will form a reference sample of temperate worlds for observation by JWST.
![Planets make good targets for transmission spectroscopy if they orbit bright, small stars. This plot shows planets that orbit stars with spectral type M3V or later, and that are brighter than Ks=10. The box is an approximate region showing planets that may have somewhat extended atmospheres (i.e. super-Earths) and are in the circumstellar habitable zone. There are 10 planets within this region, making up the prime JWST target sample from TESS.[]{data-label="fig:planet-radius-insolation-cool-stars"}](planet-radius-inxolation-cool-stars-v2.png){width="45.00000%"}
The JWST continuous viewing zone is located within 5$^\circ$ of the ecliptic poles, and is contained within the TESS CVZ, shown in Figure \[fig:sectors\]. However, because of gaps between the TESS CCDs on Camera 4 (each camera is composed of a 2x2 grid of CCDs), the central 2$^\circ$ has limited coverage. In our sample we have 74 planets with ecliptic latitude $\left|b\right|>85^\circ$, of which 29 are 2-minute cadence targets and 11 are smaller than 2 Earth-radii.
Discussion
==========
Alternative selection strategies for the 2-minute cadence targets
-----------------------------------------------------------------
In addition to the nominal 2-minute cadence target selection laid out in Section \[sec:starselction\], we also considered alternative strategies of selecting a higher or lower fraction of targets in the CVZ, which we call scenarios (a) and (b), respectively. There are justifications for both approaches. Placing more of the 2-minute cadence targets in the CVZ increases the overall number of 2-minute targets where TESS is sensitive to long-period planets, and potentially to smaller planets via increased SNR. On the other hand, placing more of the 2-minute targets outside the CVZ should increase the overall number of planets detected, since 13 stars can be observed in regions with single-sector coverage for each target in the CVZ.
To test these scenarios we selected targets in an identical manner to that described in Section \[sec:starselction\] except that in scenario (a) we included 12,000 stars in the CVZ and 2200 stars in the other cameras per sector, while in scenario (b) we select 3000 CVZ targets and 11,200 stars in the remaining cameras.
Under these two different selection strategies, we examined the number of planets found in 2-minute cadence data, compared to our nominal selection strategy. In scenario (a) we found a total of $740\pm50$ planets and in (b) we found $1380\pm60$ planets, which compares with $1250\pm70$ planets in the nominal strategy (where the reported value is the median, and uncertainties are the central 90% of the distribution, calculated by 300 Monte Carlo simulations). These results suggest that the nominal selection strategy was reasonably successful at accomplishing the goal of maximizing the number of planets with 2-minute cadence photometry, which in turn maximizes the number of planets where we can derive precise stellar parameters through asteroseismology [@Campante2016]. Scenario (b) yielded 10% more planets but the results were comparable within uncertainties, and the number of planets with orbital periods beyond 15 days was cut by about 10% in scenario (b). Scenario (a) extended the tail of the orbital period distribution – the 95th percentile shifts from 30 to 42 days – but because of the large decrease in the total number of planets, the absolute number of long period planets was unchanged.
In each scenario the total number of planets detected remained unchanged because almost all planets could be found equally well in 2-minute and FFI data, so the precise stellar selection had limited impact of the primary mission goals.
A more conservative model {#sec:conservative}
-------------------------
Our analysis so far has made two fairly optimistic assumptions, (1) that we can identify a transiting planet by observing just two transits from TESS, and (2) that we can detect all planets with a SNR$\ge$7.3. In actuality, planets with fewer than 3 observed transits are very difficult to uniquely identify using photometric survey data alone [c.f @Thompson2018; @Mullally2018]. Planets have been detected using K2 mission data [@Howell2014] with one [@Vanderburg2015] and two [@Crossfield2015] transits, but these cases occurred in systems where additional space-based follow-up assets were exploited or there were two other planets in the system, so the validity of the planets was less ambiguous [@Lissauer2012]. While with sufficient observing resources characterizing these planets is feasible to identify and confirm, they remain a challenge. Furthermore, analyses of Kepler data have shown that using a detection threshold below 8–10$\sigma$ leads to many spurious detections [@Christiansen2016; @Thompson2018; @Mullally2018]. In K2, a threshold of SNR$>$12 was typically applied [@Crossfield2016] before expending follow-up resources on a candidate planet.
With these limits in mind, we took the fiducial catalog and cut planets that either had fewer than three transits, or had a combined transit SNR$<$10. This resulted in a moderate cut in the total number of planets found to 2609 total planets, of which 820 came from the 2-minute cadence data. This was a 60% overall decrease in the total number of planets detected, but was most significant for small planets. The number of planets with radii below 2 Earth-radii decreased by a factor of two from 279 to 128 planets, with similar fractional losses in the 2–4 Earth-radii bin, but there was only a 25% decrease in detected giant planets.
![The predicted planet radius distribution using our conservative detection model where we required at least 3 transits and a combined SNR of 10. This figure is the counterpart of Figure \[fig:planet-radii\], but using our conservative detection model. The total number of planets shown is 2609, which is roughly 60% lower than our standard detection model. This change is most signification for small planets which saw a factor of two decrease. We have intentionally changed the color scheme from previous figures to differentiate between standard and conservative models.[]{data-label="fig:planet-radii-conservative"}](planet-radii-both-conservative-v2.png){width="45.00000%"}
The decrease in the number of planets amenable to radial velocity follow-up was roughly a factor of two, with planets smaller than 4 Earth-radii orbiting stars with V$<$12 dropping from 1312 to 616, and those smaller than 2 Earth-radii from 151 to 67. The number of habitable zone planets dropped from 69 to 28, and left just four smaller than 2 Earth-radii. The number of premium JWST targets sees a modest decrease. The number of planets orbiting stars cooler than 3410 K, with Ks$<$10 drops from 71 to 58, and the number in the dashed box in Figure \[fig:planet-radius-insolation-cool-stars\] dropped from 10 to 7. While these drops were significant, they are unlikely to seriously impact the primary mission goal, because there were still hundreds of small planets orbiting bright stars in the sample.
Phantom inflated planets
------------------------
This study, and other planet yield simulations [e.g. @Sullivan2015], have not paid particular attention to the physical properties of giant planets, primarily because these are not a focus for the TESS mission team. Nevertheless, we are anticipating groundbreaking scientific advances in our understanding of the atmospheres of giant planets from follow-up observations of planets found by TESS – particularly from Spitzer, HST, and JWST. As pointed out by @Mayorga2018, in the first version of this paper, there were significant numbers of giant planets that were beyond the limit of inflation for their equilibrium temperatures [@Thorngren2018]. The cause of this is that in the occurrence rate estimates of [@Fressin2013] the giant planet bin span 6–22 Earth-radii while temperate planets should rarely be larger than 12 Earth-radii. As a result of this feedback from @Mayorga2018 we changed the selection function in the giant planet bins from a log-normal function to a power law. This reduced the number of phantom planets from 8% of the total population to 1%. We caution giant planets aficionados that there are 45 over-inflated giant planets in our simulation.
The effects of Earth and Moon crossings
---------------------------------------
The nature of the TESS orbit means that a subset of observations will be obscured by the Earth or Moon passing through the field of view. Cameras that receive a significant amount of scattered light from the Earth or Moon will experience larger background flux, and photometry in any camera that receives a large portion of direct light from the Earth or Moon will likely be impossible because of saturation and bleed. However, the Earth and Moon move relatively quickly through the field of view, and Earth or Moon crossings are relatively infrequent [@Ricker2015]. @Bouma2017 estimated that the Earth and Moon will significantly affect photometric performance for 9% of all exposures, although the lost cadences will not be evenly distributed in time or focal plane location. Camera 1, and to a less degree Camera 2, are impacted, but the effect was expected to be limited for Cameras 3 and 4. Estimating how this affects the yield is non-trivial, but we can try by using the @Bouma2017 estimates that 23% of observations in Camera 1, and 12% of observations in Camera 2, will be affected. We can then assume that the SNR of transits will scale with the square root of the number of observations, so Camera 1 targets will have 11% lower SNR, and Camera 2 targets will have 6% lower SNR. This causes a 13% drop in total planets detected in our simulation, and a 9% decrease in the number of planets orbiting 2-minute cadence targets. Early commissioning results have suggested that the effect of the Moon may be more complex than anticipated, and owing to the substantial uncertainty in the impact of Earth and Moon crossings, we have not included Earth and Moon crossings in our yield statistics.
Astrophysical false positives {#sec:falsepositives}
-----------------------------
@Sullivan2015 performed a careful analysis of the sources and rates of false positives expected in the TESS 2-minute cadence data, and we have not reproduced that work here. They estimated that TESS will find over 1000 astrophysical false positives in 2-minute cadence data, but described promising mitigation strategies that utilize follow-up observations and statistical methods to reduce this by a factor of 4 or more.
The ratio of false positives to detected planets will not be uniform over all stars observed by TESS, but will vary as a function of hit-rate. In Section \[sec:results\] we showed that the hit-rate for 2-minute cadence targets is a factor of 5.5 higher than FFI-only stars. Assuming each star has the same chance of yielding a detection of an astrophysical false positive, the fraction of true planets found to false positives will be lower for the FFI-only detections than for 2-minute cadence targets. The reason is that fewer planets are found per stars observed but the same number of false positives are detected. Using the false positive rate from @Sullivan2015 of 1 false positive per 180 stars observed yields one astrophysical false positive per planet detection. However, for the FFI-only targets the ratio of false positives to planets detected increases to more than five per true planet discovered. Furthermore, stars on the CTL that are not included in our 2-minute cadence sample are, on average, 2 magnitudes fainter than the 200,000 stars observed at 2-minute cadence. This means that mitigation strategies that rely on follow-up observations will be significantly more challenging. Given essentially all small planets will be found in the 2-minute cadence data, only the most intrepid of exoplaneteers will want to commit significant resources to discovering and following-up planets in FFI data.
Planets detected around stars not in the CTL {#sec:excluded-stars}
--------------------------------------------
In Section \[sec:starselction\] we simulated planets orbiting stars that are in CTL version 6.1. This totals roughly 3.2 million stars, but includes only those stars that the TESS Target Selection Working Group considered as potential 2-minute cadence targets. The limited number of slots available for 2-minute cadence requires a careful consideration not just of the overall potential for planet detections around a given star, but also comparison of the relative planet detection potential between stars, along with the scientific value of the resulting planets. The CTL was constructed to permit a quantitative relative ranking of the best stars to select for the 2-minute cadence slots, not to identify all stars with detectable planets. While in this work we have adopted the set of several million stars in the CTL as the primary sample to investigate, stars not in the CTL might also yield some planet detections in the FFI data. The reason we adopted this approach is the same reason for the construction of the CTL in the first place – our analysis of planet yield among a population of several million stars is much more tractable than conducting the analysis for all 470 million stars in TIC-6.
Explicitly removed from the CTL are stars with a reduced proper motion that flags them as giants, stars with parallax or other information that flags them as giants or subgiants, dwarf stars that are somewhat hot and relatively faint but not as faint as some dwarf stars that are included, and faint dwarf stars. The magnitude cut used in the CTL is TESS magnitude of 12 for stars hotter than 5500 K, and TESS magnitude 13 for cooler stars, although faint cool dwarfs are explicitly included via a specially curated target list [@Muirhead2018]. The CTL therefore generally excludes hot stars, faint stars, and evolved stars, in favor of bright, cool dwarfs.
Only a handful of transiting planets have been detected around red giants [e.g. @Burrows2000; @Huber2013; @Barclay2015; @Vaneylen2016; @Grunblatt2016; @Grunblatt2017] because finding these planets is extremely challenging. Transit depth scales with the square of the stellar radius, so planets orbiting large stars are hard to find. Therefore, the frequency of planets orbiting giant stars is relatively poorly constrained. However, TESS will observe hundreds of thousands of red giants brighter than 11th magnitude in the TESS bandpass [@Huber2017a] and will certainly detect planets orbiting these stars. However, Kepler observed roughly 16,000 red giants [@Yu2018] and found only a handful of planets. With a factor 20 or so increase in the number of red giants from TESS, we might expect of order 100 new planets. This estimate is comparable to that of @Campante2016, who perform a much more careful analysis and predicted that TESS will find roughly 50 planets orbiting red giants.
The brightness cuts applied to the TIC in creating the CTL have a larger impact on our yield estimates. At 12th magnitude the TESS 1-hour integrated noise level is 600 ppm. This equates to detecting a Neptune-size planet with three transits around a solar radius star, while at 13th mag the noise is 1200 ppm which is equivalent to a 6 Earth-radii planet. So it is certainly the case that many stars not included in the CTL may have planets detectable with TESS. To detect a Jupiter with three transits around a Sun-like star would require a maximum 1-hour integrated noise of approximately 4000 ppm which corresponds to a TESS magnitude of 14.7. The TIC lists 16.0M stars with temperatures above 5500 K, log$g$ above 3.9, and TESS magnitude of 12–14.7, and 4.2M with temperature between 4000–5500 K, $\log{g}$ above 4.2, and brightness between 13–14.7 (where we cut at 4000 K because the cooler stars are included via the cool star curated list). In our fiducial sample, the frequency of detected planets larger than 4 Earth-radii was 0.069%. Assuming an equal detection rate for fainter stars in the 4+ Earth-radii bin as for brighter stars we would expect to find 14,000 additional giant planets. Even under our conservative model, the rate is 0.050%, or 10,000 additional planets.
While these planets will appear in the FFI data, they are not prime targets, hence their exclusion from the CTL, because the planets will be hard to detect and harder to follow up and confirm owing to their faintness and higher crowding. Using the logic described in Section \[sec:falsepositives\], the astrophysical false positive rate in this part of the parameter space is also very high. With a hit-rate around 0.05% and a false positive rate likely to be comparable to that found by @Sullivan2015 of 1 per 180 stars observed, we expect a factor of more than 11-to-1 false positive to true planets detected. Thus we caution that searching for planets in this regime is fraught with challenges.
The omission of these potential host stars from our analysis leads to a large underestimate in the overall planet yield of the mission, although that is almost entirely in the giant planet regime. In Figure \[fig:planet-radii-giant-planets\] we show our final distribution of planet radii and include the sample of giant planets orbiting faint stars, using the conservative yield estimate. This results in a total planet yield of 14,000 transiting planets. However, as discussed, these planets will be resource intensive both to confirm and to meaningfully analyze.
![ The predicted planet radius distribution including large planets orbiting faint stars outside of the CTL. The total number of planets that we predict TESS could find is up to 14,000. This figure is the same as Figure \[fig:planet-radii\] but includes the additional large planets orbiting faint stars. We have intentionally changed the color scheme from previous figures to differentiate from our simulated yield. []{data-label="fig:planet-radii-giant-planets"}](planet-radii-both-giant-planets-v2.png){width="45.00000%"}
One further source of additional planets is from M-dwarfs in the Southern Hemisphere. As mentioned in Section \[sec:starselction\], there is a deficit of cool stars below $-30^\circ$ declination, caused primarily by the lower completion of proper motion catalogs where northern hemisphere telescopes are unable to observe. This manifests in fewer planets detected around cool stars in the south. In the 2-minute cadence data, there are 2.6x as many planets orbiting stars cooler than 3900 K north of declination $30^\circ$ than south of declination $-30^\circ$. Including the FFI planets, this increases to 3x as many northern as southern planets (233 versus 74 planets). With GAIA data release 2 now available, it is probable that new M-dwarfs in the south will be identified. This will help to recover additional planets orbiting cool stars not identified as dwarfs in the CTL. Given that this could potentially yield new candidate planets for JWST there is a pressing need for this work.
Comparisons with earlier estimates
----------------------------------
@Sullivan2015, @Bouma2017, and @Ballard2018 have previously estimated the planet yield from TESS. These previous studies selected stars from a simulated Galactic model rather than real stars, and therefore we expect there are moderate differences between our predicted yields and previous studies. Additionally, we used different selection strategies for both 2-minute cadence targets and for FFI stars. We built a realistic 2-minute cadence star selection model that limits the stars observed at the pole cameras to just 6,000 stars per hemisphere, whereas the previous works assumed TESS can observe many more stars in the CVZ than is possible with the flight hardware configuration used. We also use a different prioritization metric than previous work, which is based on the metric used by the TESS Target Selection Working Group. For the FFI targets we primarily consider those within the CTL, whereas different cuts on brightness are made in earlier works. Therefore, we expect to see significant differences in the planet yield for giant planets.
@Sullivan2015 predicted 1700 planets in 2-minute cadence data, of which 560 are smaller than 2 Earth-radii. @Bouma2017 used the same methodology and software as @Sullivan2015, but fixed a number of software bugs and modified a number of parameters. They also predicted 1700 planets from 2-minute cadence data, of which 430 were smaller than 2 Earth-radii. The total 2-minute cadence planet yield in both these studies was about 30% larger than we have predicted, but the number of planets smaller than 2 Earth-radii in our study is lower by a factor of 1.7 and 2.3 than @Bouma2017 and @Sullivan2015, respectively. However, given the different selection strategies, it may be more reasonable to compare the combined 2-minute cadence and FFI yields. Where @Bouma2017 and @Sullivan2015 differ is in their star selection for FFI targets. @Bouma2017 limit their selection to the top ranked 3.8M stars using a similar priority metric to the one applied in CTL 6.1. This enables easy comparison with our 3.2M star sample. On the other hand, @Sullivan2015 consider all stars brighter than $Ks=15$ totalling 150M stars, which we can compare with our analysis in Section \[sec:excluded-stars\].
Our total simulated yield is remarkably similar to @Bouma2017, with 41 versus 49 Earth-sized planets, 238 versus 390 super-Earths, 1900 versus 2000 mini-Neptunes, and 2200 versus 2500 giant planets, for this work and @Bouma2017 respectively. The only area where we see a significant deviation is for super-Earths, which we attribute to differences between the Galactic model and real stars.
Compared to @Sullivan2015 [@Sullivan2017], we predict lower totals in all bins. However, as mentioned by @Bouma2017, the number of Earths and super-Earths is overestimated by around 30% owing to a bug in their calculation of the dilution from background stars. Taking this into account, our number of Earths matches both @Bouma2017 and @Sullivan2015, while the super-Earths are comparable. Our rate of giant planets predicted in Section \[sec:excluded-stars\] is consistent with @Sullivan2015 @Ballard2018 used the framework and detection rates of @Sullivan2015, but focus entirely on M1–M4 dwarfs, and made significant changes to the occurrence rates to account for covariances between planets in the same systems. In comparison, our analysis of the M-dwarf population is simplistic. @Ballard2018 predicted a 50% increase in the rate of planets orbiting these cool stars compared to the occurrence rates used by @Sullivan2015 (and this work). They predicted $990\pm350$ planets around M1–M4 stars, while we predicted 410 planets orbiting stars with temperatures of 3100–3800 K. If the @Ballard2018 occurrence rate has a similar impact to our yields as it had on @Sullivan2015, and given comparable yields between our studies, we would expect an additional 50% planets in this parameter space, which is 200 more planets orbiting cool stars. Assuming the increase is uniform in planet size, we might expect an increased yield that includes 14 additional Earths, 42 additional super-Earths, and 142 additional mini-Neptunes. The yield could be even higher if we are able to identify additional M-dwarfs in the southern sky, as discussed in Section \[sec:excluded-stars\].
Conclusions
===========
The TESS mission will find a large number of transiting planets. However, up until recently the number and physical properties of the planets that will be discovered has been estimated using simulations performed before the TESS observing strategy, 2-minute target list, and flight hardware had been finalized. Here we simulated TESS detections of transiting planets using the CTL for our star selection. We have estimated that TESS will find more than 14,000 exoplanets, of which $4400\pm110$ orbit stars in the CTL and $1250\pm70$ will be observed at 2-minute cadence. TESS will find over 2100 planets smaller than 4 Earth-radii, of which 280 will be smaller than 2 Earth-radii.
The key design feature that distinguishes TESS from Kepler is that it will observe brighter stars, emphasizing finding planets that can be followed up more readily from the ground. TESS planets range in V-band brightness from 4–20, with 80% of predicted planets orbiting stars brighter than V=13.0. Assuming V=12 as the limit for recovery of a mass via precision radial velocity observations, we predict that TESS will have a sample of 2500 planets for radial velocity observations, of which 1300 will be smaller than 4 Earth-radii, and 150 smaller than 2 Earth-radii. This will provide a plethora of planets to characterize; the TESS follow-up observers should have little problem meeting mission requirements of measuring the masses of 50 planets smaller than 4 Earth-radii. We predict that TESS will find 7 planets orbiting stars brighter than 55 Cnc, the brightest transiting planet host.
There is significant interest in finding habitable zone planets from TESS. We predict around 70 habitable zone planets will be detected and all will orbit M-dwarfs, with 9 habitable zone planets in our simulations with radii smaller than twice that of Earth’s. Our simulations predict that TESS will find 70 planets orbiting bright mid-M-dwarfs (Ks$<$10, M3V or later), 10 of which fall into the optimistic habitable zone, making them prime JWST targets.
We have shown that nearly all planets valuable for contributing to mission goals related to radial velocity and JWST targets will be found in 2-minute cadence data. This is to the great credit of the teams that worked to create the CTL. The availability of 2-minute cadence data will permit more accurate measurements of the radii and orbital configurations of the detected planets. We explored how target selection choices affect the target yield and find that the distribution of targets between the CVZ and shorter observing baseline is well balanced between collecting 2-minute cadence data for the maximum number of planets, and finding long period planets.
There are a large number of stars that are not in the CTL that might host a detectable planet. These stars were intentionally not included in the CTL, and for good reason. They are unlikely to host detectable small planets, and any planets found will be hard to follow up. While there may be as many as 10,000 additional giant planets around the faint stars in the TESS data, we have shown that the astrophysical false positive rate might be as high as 11 false positives per true planet, and there may be as few as one planet detected per 2000 stars searched. While less severe, we anticipate a high astrophysical false positive rate for stars on the CTL but not included in the 2-minute cadence sample because the ratio of detected planets to stars observed is five times lower than for stars observed at 2-minute cadence. The mission’s target of finding planets with SNR$\ge$7.3 and only two transits may be overly aggressive, based upon experience with Kepler and K2 data. We explored an alternative model that applied more conservative detection thresholds of SNR$\ge$10, and requiring three transits. This results in a decrease in the yield estimate of approximately 50% for planets smaller than 4 Earth-radii, and occurs across all parameter spaces considered. However, even if this conservative model is realized, more than enough planets will be found to ensure mission success.
This work builds upon studies by [@Sullivan2015] and [@Bouma2017], and would not be possible without their efforts. We do see a moderate decrease from previous yields estimates, although our numbers are remarkably similar to those @Bouma2017 presented, considering the different stellar selection strategies.
It will not be long before TESS planets are discovered. The real excitement will come from learning about these new worlds using data from ground and space-based facilities. The legacy of TESS will be a catalog of the planets that will be the touchstone planets for years to come. TESS will discover which of our nearest stellar neighbors have transiting planets. The brightest host star in our simulation is 70 Oph A, where we recovered a simulated Earth-sized planet. Were this simulation real, on a clear night from a dark site we could point to this star and tell our friends, “that star there has a planet.”
Planet radius as a function of distance
=======================================
Zach Berta-Thompson created a figure using data from @Sullivan2015 that has been widely shared because it is both informative of TESS’ capabilities and aesthetically pleasing. We have reproduced Berta-Thompson’s plot in Figure \[fig:zachplot\], with our revised TESS yield estimates.
![Orbital distance versus planet radii. This plot updates a widely shared figure created by Z. Berta-Thompson, to now include our new simulation results. Kepler planet candidates from @Thompson2018 are shown in blue, our simulated 2-minute cadence detections in orange, and planets detected using other telescopes in black. The size of the circle is proportional to the transit depth. A subset of nearby planets are marked. Data was extracted from the Exoplanet Archive [@Akeson2013]. Three planets in our simulation orbit stars closer than the nearest known transiting planet system HD 219134. []{data-label="fig:zachplot"}](planet-distance-simulation_V2_lowres.png){width="\textwidth"}
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[^1]: The TIC and CTL are available from the MAST archive at <http://archive.stsci.edu/tess/>.
[^2]: Only about half of the targets in our sample had distances reported in CTL version 6.1, our statistics are based on this sample. Furthermore, a small number of the CTL reported distances were unrealistically large. These issues have been fixed in CTL v6.2.
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abstract: 'We study the transport properties of a quantum dot coupled to a normal and a superconducting lead. The dot is represented by a generalized Anderson model. Correlation effects are taken into account by an appropriate self-energy which interpolates between the limits of weak and strong coupling to the leads. The transport properties of the system are controlled by the interplay between the Kondo effect and Andreev reflection processes. We show that, depending on the parameters range the conductance can either be enhanced or suppressed as compared to the normal case. In particular, by adequately tunning the coupling to the leads one can reach the maximum value $4e^2/h$ for the conductance.'
address: |
$^1$ Departamento de Física Teórica de la Materia Condensada C-V, Universidad Autónoma de Madrid, E-28049 Madrid, Spain\
$^2$ Institut für Theoretische Festkörperphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany
author:
- 'J.C. Cuevas$^{1,2}$, A. Levy Yeyati$^1$, A. Martín-Rodero$^1$'
title: 'Kondo effect in Normal-Superconductor Quantum Dots'
---
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epsf
The Kondo effect is a prototypical correlation effect in Solid State Physics. Although it was first analyzed for the case of magnetic impurities in metals, in the last years there has been a renewed interest in Kondo physics with its observation in a semiconductor quantum dot (QD) [@Goldhaber; @Kowenhoven]. Quantum dots constitute an ideal laboratory for testing the theoretical predictions as they allow to vary the relevant parameters in the problem in a controlled way. This technology also opens some new possibilities like the exploration of the Kondo effect when the dot is connected to superconducting leads. The interesting issue in this case is related to the competition between the strong Coulomb interaction in the quantum dot and the pairing interaction within the leads.
From the theoretical side, the Kondo effect in QDs has been mainly analyzed by means of the single-level Anderson model [@Glazman; @us]. The theory predicts an enhancement of the dot conductance at low temperatures due to the development of the so-called Kondo resonance. The case when one of the leads is superconducting has been recently analyzed by some authors using different theoretical methods [@Fazio; @Kang; @Ambegaokar] assuming a modified Anderson model in which one of the metallic electrodes is substituted by a BCS superconductor. While some authors have predicted an enhancement of the conductance due to Andreev reflection at the superconducting lead [@Kang], others have predicted the opposite effect [@Ambegaokar]. In Refs. [@Fazio; @Ambegaokar] the infinite charging energy limit ($U
\rightarrow \infty$) has been assumed. However, in an actual experiment this assumption may not be completely justified (for instance, in the experiments of Ref. [@Goldhaber] the ratio $U/\Gamma$, $\Gamma$ being the dot tunneling rate, was estimated to be around 6.5). The approach presented in this letter would allow to analyze this problem for a broad range of the different parameters of the model. We will show that, depending on the values of these parameters, one can obtain either an enhancement or a reduction of the conductance with respect to the normal case.
Our approximation scheme is based on the hypothesis that a good approximation to the electron self-energy can be found by interpolating between the limits of weak and strong coupling to the leads. This interpolative method has been applied successfully to analyze different strongly correlated electron systems like the equilibrium [@Alvaro; @Saso], the non-equilibrium [@us] and the multilevel [@us2] Anderson models and the Hubbard model [@Alvaro2; @Kotliar]. In this letter we shall discuss how to extend this method to the superconducting case.
For describing a N-QD-S system we use an Anderson-like Hamiltonian
$$\hat{H} = \hat{H}_N + \hat{H}_S + \sum_{\sigma} \epsilon_0 \hat{n}_{\sigma}
+ U \hat{n}_{\uparrow} \hat{n}_{\downarrow} + \hat{H}_T,$$
where $\hat{n}_{\sigma}=\hat{d}^{\dagger}_{\sigma} \hat{d}_{\sigma}$, $\hat{H}_N$ and $\hat{H}_S$ represent the uncoupled normal and superconducting leads respectively; $\hat{H}_T = \sum_{k \in N,S;\sigma}
t_{0,k} \hat{d}^{\dagger}_{\sigma} \hat{c}_{k,\sigma} + h.c.$ describing the coupling between the dot level and the leads. Within this model the dot is represented by a single spin degenerate level with a repulsive Coulomb interaction described by the U-term in Eq. (1). We shall assume that the superconducting lead is well described by the BCS theory with a superconducting gap $\Delta$ and the normal lead is, as usual, characterized by a flat density of states around the Fermi level, $\rho_F$.
The transport properties of this model can be obtained by means of Green function techniques. In order to analyze the linear regime the main quantity to be determined is the dot retarded Green function, which in a Nambu $2 \times 2$ representation adopts the form
$$\hat{G}^r(\omega) = \left[ \omega \hat{I} - \epsilon_0 \hat{\sigma}_z -
\hat{\Sigma}^r(\omega) - \hat{\Gamma}_N (\omega) - \hat{\Gamma}_S
(\omega) \right]^{-1},$$
where $\hat{\Gamma}_N$ and $\hat{\Gamma}_S$ are the tunneling rates given by $\hat{\Gamma}_N = \Gamma_L \hat{I}$, and the superconducting tunneling rate $\hat{\Gamma}_S$ is given by $\hat{\Gamma}_S = \Gamma_R \hat{g}$, where $\Gamma_{L,R} = \pi t^2_{L,R} \rho_F$, $g_{11} = g_{22} = - \omega/\sqrt{\Delta^2 - \omega^2}$ and $g_{12} = g_{21} = \Delta/\sqrt{\Delta^2 - \omega^2}$ (the chemical potential of the superconducting lead is taken as zero). The self-energy $\hat{\Sigma}^r(\omega)$ takes into account the effect of Coulomb interactions. To the lowest order in $U$ this is given by the Hartree-Fock Bogoliubov approximation: $\hat{\Sigma}^r = U <\hat{n}> \hat{\sigma}_z + \Delta_d
\hat{\sigma}_x$, $\Delta_d$ being the proximity effect induced order parameter in the QD, $\Delta_d = U \langle \hat{d}^{\dagger}_{\uparrow}
\hat{d}^{\dagger}_{\downarrow} \rangle$. The crucial problem is to find a good approximation to include correlation effects beyond this mean field approximation.
Within the spirit of the interpolative method commented above, the self-energy is constructed in such a way as to interpolate between the limits of weak and strong coupling to the leads, for which the exact result are known. Let us first analyze the weak coupling or [*atomic*]{} limit. In this case we have $t_{L,R}/U \rightarrow 0$ and thus $\hat{\Gamma}_N/U$, $\hat{\Gamma}_S/U$ $\rightarrow 0$. In this limit the induced order parameter in the QD vanishes faster than $(t_R/U)^2$ and one can neglect the non-diagonal elements in the self-energy matrix. On the other hand, the diagonal elements can be easily evaluated in this limit using the equation of motion method [@Alvaro] and have the form $$\Sigma^r_{11,22} \rightarrow \pm U <\hat{n}> +
\frac{U^2 <\hat{n}>(1 - <\hat{n}>)}{\omega \mp \epsilon_0
\mp U (1 - <\hat{n}>)}$$
In the opposite limit, $U/t_{L,R} \rightarrow 0$, one can accurately evaluate the self-energy using standard perturbation theory in the Coulomb interaction. The different diagrams contributing to the second order self-energy are depicted in Fig 1. In the superconducting case, there appear additional diagrams to the one in the normal case (diagram a) corresponding to the interaction of an electron with an electron-hole pair in the QD; the remaining diagrams contain at least one anomalous propagator and vanish identically in the normal state. As in the normal case [@us], the non-perturbed one-electron Hamiltonian, over which the diagrammatic series is constructed, is taken as an effective mean field, characterized by an effective dot level $\epsilon_{eff}$, having the same dot charge as the fully interacting problem. As shown in Ref. [@us] this self-consistency condition provides in the normal case a good fulfillment of the Friedel sum rule at zero temperature. The extension of this procedure to the superconducting case requires dressing the propagators in the diagrams of Fig. 1 with the non-diagonal self-energy $\Sigma_{12}$ in order to impose also consistency in the non-diagonal charge $\langle \hat{d}^{\dagger}_{\uparrow} \hat{d}^{\dagger}_{\downarrow}
\rangle$. Notice that although the interaction in the QD is repulsive, there is always some induced paring potential in the dot due to the proximity effect. The inclusion of this effect for finite $U$ is very important for the correct description of the dot electronic properties.
The original interpolative scheme stems from the observation that the second order self-energy ($\Sigma^{(2)}$) has a similar functional form as the atomic self-energy for large frequencies [@Alvaro] thus allowing for a smooth interpolation between the two limits. In the superconducting case, the diagonal elements of the second order self-energy behave as $$\Sigma^{(2)}_{11,22} \sim \frac{U^2 <\hat{n}>(1 - <\hat{n}>)}{\omega \mp
\epsilon_{eff}}$$ for large frequencies, while the non-diagonal elements decay faster than $U^2/\omega$. This behavior permits to define a Nambu $2 \times 2$ interpolative ansatz for the self-energy matrix as: $$\hat{\Sigma}(\omega) = U \langle \hat{n} \rangle \hat{\sigma}_z + \Delta_d
\hat{\sigma}_x + \left[\hat{I} - \alpha \hat{\Sigma}^{(2)}
\hat{\sigma}_z \right]^{-1} \hat{\Sigma}^{(2)}(\omega) ,$$ where $$\alpha = \frac{\epsilon_0 + (1-<\hat{n}>)U -\epsilon_{eff}}{U^2
<\hat{n}>(1-<\hat{n}>)}.$$
and $\hat{\Sigma}^{(2)}$ is the second order self-energy matrix whose elements are given by the diagrams depicted in Fig. 1.
Using this ansatz one recovers the correct behavior of the self-energy both in the weak and strong coupling limits. Moreover, this ansatz satisfies the exact relations between the different matrix elements, i.e $\Sigma_{12}(\omega) = \Sigma_{21}(\omega)$ and $\Sigma_{11}(\omega) =
-\Sigma^{*}_{22}(- \omega)$.
Due to the presence of an additional energy scale fixed by $\Delta$ the number of different physical regimes is larger than in the normal case. We will mainly consider the more interesting physical regime $\Gamma =
\Gamma_L + \Gamma_R \sim \Delta$ [@comment]. In Fig. 2a we show the dot spectral density (LDOS) for a symmetric case ($\epsilon_0 = -U/2$) with $\Gamma_L=\Gamma_R=\Delta$ and increasing values of $U$. As can be observed, when $U \le \Delta$ the LDOS exhibits a double peak around the Fermi energy which is due to the influence of the superconducting electrode by the proximity effect. However, as $U$ increases the double peak is replaced by a single narrow Kondo resonance as in the normal case. The comparison with the normal case reveals that the Kondo resonance gets narrower in the superconducting case and its height increases with $U$ above the normal value. In the limit $U \rightarrow \infty$, this height approaches the value $2/(\pi \Gamma)$, which is twice the value in the normal case at zero temperature, as fixed by the Friedel sum rule. The narrowing of the Kondo resonance gives rise to a lowering of the Kondo temperature with respect to the normal case.
For energies larger than $\Delta$ the differences between the normal and the superconducting LDOS become negligible, with the usual broad resonances at $\epsilon_0$ and $\epsilon_0 + U$ which become more pronounced for increasing $U$.
By varying the dot level position $\epsilon_0$ one can study the transition from the Kondo to the mixed valence regime. The evolution of the dot LDOS is illustrated in Fig. 2b. When approaching the mixed valence regime ($|\epsilon_0| < \Gamma$ or $|\epsilon_0 + U| < \Gamma$) the Kondo resonance is replaced by an asymmetric broad resonance close to the Fermi energy as in the normal case. In the superconducting case, however, the LDOS develops an additional structure associated with the BCS divergencies at the gap edges [@Ambegaokar].
As in any NS contact, transport at low voltages is possible due to Andreev reflection processes. At finite temperature, the linear conductance is given by the expression [@Raimondi]
$$G = \frac{16e^2}{h} \Gamma_L \int^{\infty}_{-\infty} dE \;
\mbox{Im} \left( G^r_{12} G^a_{11} \right) \left( \Gamma_R - \mbox{Re}
\Sigma_{12} \right) \left( - \frac{ \partial f}{\partial E} \right),
\label{conductance}$$
where $f(E)$ is the Fermi function. At zero temperature, $\mbox{Im}\hat{\Sigma}(0)=0$, and Eq. (\[conductance\]) reduces to
$$G = \frac{4e^2}{h} \frac{4 \Gamma_L^2
\tilde{\Gamma}_R^2}{\left[\tilde{\epsilon}^2 + \Gamma_L^2 + \tilde{\Gamma}_R^2
\right]^2},
\label{cond0}$$
where $\tilde{\Gamma}_R = \Gamma_R - \mbox{Re}\Sigma_{12}(0)$ and $\tilde{\epsilon} = \epsilon_0 + \mbox{Re} \Sigma_{11}(0)$. Notice that Eq. (\[cond0\]) coincides at $U=0$ with the well known non-interacting result [@Beenakker].
One would expect that for a dot symmetrically coupled to the leads (i.e. $\Gamma_L = \Gamma_R$) and in the case of electron-hole symmetry ($\epsilon_0 = -U/2$), the conductance should reach its maximum value $4 e^2/h$ [@Kang]. However, the actual situation is more complex due to the reduction of the induced paring amplitude in the dot arising from the repulsive Coulomb interaction. As a consequence the conductance decreases for increasing $U$ even in this case. This decrease is illustrated in Fig. 3a where we plot the conductance as a function of $U$ in the symmetric case for different values of $\Gamma/\Delta$. For large $U/\Gamma$ we find that the conductance decreases roughly as $(\Gamma/U)^4$. This behavior can be understood as follows: in order to have a vanishing pairing amplitude in the $U/\Gamma \rightarrow \infty$ limit, the non-diagonal self-energy $\Sigma_{12}$ should tend to cancel the non-diagonal tunneling rate $(\hat{\Gamma}_S)_{12}$. By analyzing the expression of diagram d in Fig. 1, this requires that $G_{12}$ decays as $(\Gamma/U)^2$ and therefore the conductance given by Eq. (\[conductance\]) in our approximation should decay roughly as $(\Gamma/U)^4$. This decay is probably less pronounced than in the exact solution where one would rather expect an exponential behavior in the Kondo regime.
Although the previous analysis shows that the maximum value for the conductance $4 e^2/h$ can never be reached in the symmetric case for finite $U$, this is not necessarily the case for an asymmetric situation with $\Gamma_L \ne \Gamma_R$. In fact, if the coupling to the electrodes could be tunned in order to reach the condition $\tilde{\Gamma}_R = \Gamma_L$ then, Eq. (\[conductance\]) predicts a maximum in the value of $G$. As shown in Fig. 3b, this condition can be reached by increasing the coupling to the superconducting electrode. The ratio between $\Gamma_R$ and $\Gamma_L$ at the maximum becomes larger for increasing $U$. In a situation with electron hole-symmetry, like the one depicted in Fig. 3b, the conductance at zero temperature reaches its maximum possible value $4e^2/h$.
In normal quantum dots a signature of the Kondo effect is given by an anomalous temperature dependence in the linear conductance [@Goldhaber], which exhibits a continuous transition from a maximum conductance in the Kondo regime to well resolved conductance peaks associated with Coulomb blockade. When one of the electrodes is superconducting there is also a decrease of conductance with temperature in the Kondo regime. However, as depicted in Fig. 4, the conductance already exhibits a double peaked structure at zero temperature when $\Gamma_L = \Gamma_R$. The reduction of conductance with temperature is in this case much faster than in the normal case, as shown in Fig. 4 (inset). This difference is a consequence of the lowering of the Kondo temperature due to the presence of the superconducting electrode.
In conclusion, we have analyzed the electronic transport properties of a quantum dot coupled to a normal and a superconducting lead. For this purpose we have introduced an electron self-energy which interpolates between the limits of weak and strong coupling to the leads, an approach which has been previously used for normal systems [@us; @Alvaro; @Saso; @us2; @Alvaro2; @Kotliar]. This approximation allows to describe a broad range of parameters including the relevant one for an actual experiment. On the other hand, we have shown that for finite charging energy the dot conductance can either be enhanced or suppressed with respect to the normal case. While in a symmetrically coupled dot ($\Gamma_L = \Gamma_R$) an increasing charging energy tends to reduce the conductance, in the asymmetric case it is always possible to reach a maximum in the conductance by fine tunning the coupling to the superconducting electrode. In the case of electron-hole symmetry this maximum reaches the value $4e^2/h$ at zero temperature. The predictions presented in this work could be tested experimentally using similar technologies to those currently used for normal quantum dots [@Goldhaber; @Kowenhoven; @Takayanagi]
We thank Jan von Delft, Hans Kroha, Andrei Zaikin and Gerd Schön for fruitful discussions. This work has been supported by the Spanish CICYT under contract No. PB97-0044 and by the SFB 195 of the German Science Foundation.
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abstract: 'We study if the minimal sneutrino chaotic inflation is consistent with a flavor symmetry of the Froggatt-Nielsen type, to derive testable predictions on the Dirac and Majorana CP violating phases, $\delta$ and $\alpha$. For successful inflation, the two right-handed neutrinos, i.e., the inflaton and stabilizer fields, must be degenerate in mass. First we find that the lepton flavor symmetry structure becomes less manifest in the light neutrino masses in the seesaw mechanism, and this tendency becomes most prominent when right-handed neutrinos are degenerate. Secondly, the Dirac CP phase turns out to be sensitive to whether the shift symmetry breaking depends on the lepton flavor symmetry. When the flavor symmetry is imposed only on the stabilizer Yukawa couplings, distributions of the CP phases are peaked at $\delta \simeq \pm \pi/4, \pm 3\pi/4$ and $\alpha = 0$, while the vanishing and maximal Dirac CP phases are disfavored. On the other hand, when the flavor symmetry is imposed on both the inflaton and stabilizer Yukawa couplings, it is rather difficult to explain the observed neutrino data, and those parameters consistent with the observation prefer the vanishing CP phases $\delta = 0, \pi$ and $\alpha = 0$.'
bibliography:
- 'reference.bib'
---
UT-17-20, TU-1045, IPMU17-0077\
.75in
[ **Neutrino CP phases from Sneutrino Chaotic Inflation** ]{}
.75in
[Kazunori Nakayama$^{a,b}$, Fuminobu Takahashi$^{c,b}$ and Tsutomu T. Yanagida$^{b,d}$]{}
0.25in
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$^a$
[*The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan*]{}
\[.3em\] $^{b}$
[*The University of Tokyo, Kashiwa, Chiba 277-8583, Japan*]{}
\[.3em\] $^{c}$
[*Sendai, Miyagi 980-8578, Japan*]{}
$^{d}$
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.5in
Introduction
============
The chaotic inflation [@Linde:1983gd] is attractive in that it naturally avoids the initial condition problem of the inflation. In addition it generically predicts a large tensor-to-scalar ratio within the reach of the ongoing and future CMB B-mode search experiments. A simple prescription for realizing the chaotic inflation in supergravity was given in Ref. [@Kawasaki:2000yn], where two gauge singlets are required for successful inflation; one is the inflaton and the other is the stabilizer field. See also Refs. [@Kallosh:2010ug; @Nakayama:2013jka; @Nakayama:2013txa; @Nakayama:2014wpa] for variants of the inflation models along the same line.
Recently we proposed a minimal sneutrino chaotic inflation model [@Nakayama:2016gvg], where the two gauge singlets, inflaton and stabilizer fields, are identified with the two right right-handed neutrinos. In particular, the two right-handed neutrinos are almost degenerate in mass, and the mass scale is fixed to be about $10^{13}$[GeV]{} by the CMB normalization [@Ade:2015lrj]. Interestingly, the seesaw mechanism [@Yanagida:1979as; @GellMann:1980vs; @Minkowski:1977sc] as well as leptogenesis [@Fukugita:1986hr] are naturally implemented in the minimal chaotic sneutrino inflation (see Ref. [@Bjorkeroth:2016qsk] for a recent detailed analysis on the leptogenesis in this setup). While the quadratic chaotic inflation is in a strong tension with the observation, there are various extensions with a polynomial potential [@Nakayama:2013jka; @Nakayama:2013txa; @Nakayama:2014wpa; @Saha:2015afa], a periodic potential [@Czerny:2014wza; @Czerny:2014xja; @Czerny:2014qqa; @Nakayama:2016gvg], a modified (running) kinetic term [@Takahashi:2010ky; @Nakayama:2010kt; @Nakayama:2010ga; @Kallosh:2013yoa; @Kallosh:2016sej], which give a better fit to the observed data. See also Refs. [@Murayama:1992ua; @Ellis:2003sq; @Kadota:2005mt; @Antusch:2009ty; @Nakayama:2013nya; @Murayama:2014saa; @Evans:2015mta; @Antusch:2008pn; @Antusch:2010va] for various inflation models with right-handed sneutrinos.
It is known that, in the seesaw mechanism, the number of parameters in high energy is larger than that in low energy. For instance, in the conventional case with three right-handed neutrinos, there are $18$ and $9$ parameters in high and low energy, respectively. In a case of two right-handed neutrinos, they are reduced to $11$ and $7$, respectively. In the minimal sneutrino chaotic inflation, there are only $9$ parameters in high energy, because the right-handed neutrino masses are almost degenerate, and fixed to be about $10^{13}$GeV by inflation. The remaining two degrees of freedom correspond to a complex variable $z$ which parametrizes a complex orthogonal matrix in the Casas-Ibarra parametrization of the neutrino Yukawa matrix [@Casas:2001sr; @Ibarra:2003up]. As we shall see, the imaginary component of $z$ is tightly constrained by the requirement of the stability of the inflationary path. We also note that, in the limit of degenerate right-handed neutrino masses, the real component of $z$ becomes unphysical, further reducing the number of parameters in high energy.
In this Letter we study if the minimal sneutrino chaotic inflation is consistent with a flavor symmetry of the Froggatt-Nielsen (FN) [@Froggatt:1978nt] type, to derive testable predictions on the CP phases. First we note that, in the seesaw mechanism with degenerate right-handed neutrinos, the lepton flavor symmetry structure becomes less manifest in the light neutrino masses. Secondly, we find that the Dirac CP phase is sensitive to whether the shift symmetry breaking depends on the lepton flavor symmetry. The inflaton respects a shift symmetry to ensure the flatness of the inflaton potential beyond the Planck scale, and so, the inflaton Yukawa couplings (as well as the inflaton mass) explicitly break the shift symmetry. Since we do not know the origin of the shift symmetry breaking terms, they may depend on the lepton flavor symmetry. We therefore study two cases in which the flavor structure is imposed on only the stabilzer Yukawa couplings and on both the inflaton and stabilizer Yukawa couplings. In either case, we randomly scan the full theoretically and observationally allowed parameter space of the flavor model, and derive probability distributions of the neutrino Dirac and Majorana CP phases, $\delta$ and $\alpha$.
The rest of the Letter is organized as follows. In Sec. \[sec:2\] we first review the minimal sneutrino chaotic inflation, derive constraint on the complex parameter $z$ from the stability condition of the inflationary path, and explain the flavor model adopted for our analysis. In Sec. \[sec:3\] we show the probability distributions of the CP phases based on the random scan of the allowed parameter space. The last section is devoted for discussion and conclusions.
Models of inflation and flavor symmetry {#sec:2}
=======================================
Minimal sneutrino chaotic inflation
-----------------------------------
The successful chaotic inflation in supergravity requires two gauge singlet superfields, the inflaton and stabilizer fields. Being gauge singlets, they are naturally coupled to leptons and Higgs. Thus, the seesaw mechanism for light neutrino masses is a natural outcome of the chaotic inflation in supergravity [@Nakayama:2016gvg]. Then the inflaton and the stabilizer field are identified with the right-handed sneutrinos.
The relevant Kähler and super-potentials for the sneutrino chaotic inflation are given by $$\begin{aligned}
K &= \frac{1}{2}(N_1+N_1^\dagger)^2 + |N_2|^2 - k_2\frac{|N_2|^4}{M_P^2}, \label{Kapp}\\
W &= M N_1 N_2 + y_{i\alpha} N_i L_\alpha H_u,
\label{Wapp}\end{aligned}$$ where $N_1$ and $N_2$ are the inflaton and stabilizer fields, respectively, $M$ the inflaton mass, $y_{i\alpha}$ the neutrino Yukawa couplings, $L_\alpha$ the lepton doublet, and $H_u$ the up-type Higgs field. Here $i$ runs over $1$ and $2$, and $\alpha = e, \mu, \tau$ is the lepton flavor index. We assume that the $N_1$ respects a shift symmetry along its imaginary component, which is identified with the inflaton, $\varphi \equiv \sqrt{2} {\rm Im} N_1$. The shift symmetry is explicitly broken by both the inflaton mass $M$ and neutrino Yukawa couplings, $y_{1 \alpha}$.
The inflaton mass is fixed to be $M \simeq 2\times 10^{13}$GeV by the CMB normalization, and so, typical values of $y_{i\alpha}$ is of $\mathcal O(0.1)$ for reproducing the observed neutrino masses (see Sec. \[sec:nu\]). Then, if we simply extrapolate the above interactions to $\varphi \sim \mathcal O(10)M_P$, the masses of $L_\alpha H_u$ would exceed the Planck mass during inflation. One can avoid this problem by imposing a discrete shift symmetry on $N_1$ as in Ref. [@Nakayama:2016gvg]. Thanks to the discrete shift symmetry, the actual superpotential is modified near the Planck scale and the masses of $L_\alpha H_u$ become periodic with respect to $\varphi$. Thus their masses remain smaller than the Planck mass, in which case we can safely discuss the inflaton dynamics in the effective field theory. In addition, due to this modification, the prediction of the scalar spectral index and the tensor-to-scalar ratio can give a better fit to the observation.[^1] Alternatively, one may modify the inflaton kinetic term [@Kallosh:2016sej]. In either case, the right-handed neutrino mass scale remains of order $10^{13}$GeV, and the above forms of the Kähler and super-potentials are sufficient for our purpose, since we are interested in the observed neutrino parameters in the present vacuum.
Seesaw mechanism with two right-handed neutrinos {#sec:nu}
------------------------------------------------
Now let us briefly discuss the seesaw mechanism with two right-handed neutrinos [@Frampton:2002qc] based on the superpotential (\[Wapp\]). Here and in what follows we work in the charged lepton mass basis. First we move to the mass eigenbasis in which the right-handed neutrino masses are diagonalized (the parameters in this basis are shown with tildes): $$\begin{aligned}
W = \frac{1}{2}M \widetilde N_i \widetilde N_i + \widetilde y_{i\alpha} \widetilde N_i L_\alpha H_u,\end{aligned}$$ where $$\begin{aligned}
&\widetilde N_1= \frac{1}{\sqrt 2}(N_1+N_2),~~~~~~\widetilde N_2= \frac{i}{\sqrt 2}(N_1-N_2),\\
&\widetilde y_{1\alpha}= \frac{1}{\sqrt 2}(y_{1\alpha}+y_{2\alpha}),~~~~~~\widetilde y_{2\alpha}= \frac{i}{\sqrt 2}(-y_{1\alpha}+y_{2\alpha}).
\label{masseigen}\end{aligned}$$ After integrating out the right-handed neutrinos, we obtain the light neutrino mass matrix as $$\begin{aligned}
m^\nu_{\alpha\beta} = \frac{v_{\rm EW}^2\sin^2\beta}{M} \widetilde y_{i\alpha}\widetilde y_{i\beta},
\label{mnu}\end{aligned}$$ where $v_{\rm EW}=174$GeV and $\tan\beta \equiv \left<H_u^0\right>/\left<H_d^0\right>$. It is diagonalized by the Maki-Nakagawa-Sakata (MNS) matrix as $$\begin{aligned}
m_{\bar \gamma}^\nu \delta_{\bar \gamma \bar \delta} = U_{\bar \gamma\alpha}^{{\rm (MNS)}T} m^\nu_{\alpha\beta} U_{\beta\bar\delta}^{{\rm (MNS)}},\end{aligned}$$ where the subindices with a bar (e.g. $\bar \gamma = 1,2,3$) label the mass eigenstates. Note that the lightest neutrino is massless in the two right-handed neutrino scenario. We take the standard parametrization of the MNS matrix: $$\begin{aligned}
U_{\beta\bar\gamma}^{{\rm (MNS)}} =\begin{pmatrix}
c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta} \\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta} & c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta} & s_{23}c_{13} \\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta} & -c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta} & c_{23}c_{13}
\end{pmatrix}
\times{\rm diag}
\begin{pmatrix}
1 & e^{i\alpha/2} & 1
\end{pmatrix},\end{aligned}$$ where $s_{ij} = \sin \theta_{ij}$, $c_{ij} = \cos \theta_{ij}$, and $\delta$ and $\alpha$ are the Dirac and Majorana phases, respectively. Note that there are 7 observables in low energy: the two mass eigenvalues, three mixing angles and two CP phases. On the other hand, the Yukawa matrix $\widetilde y_{i\alpha}$ contains 9 parameters after rotating away the three phases of $L_\alpha$ (and the right-handed leptons simultaneously to keep the charged lepton masses unchanged). In fact, the Yukawa couplings can be expressed in terms of the 7 observables plus one complex parameter. One can explicitly solve for the Yukawa matrix as [@Casas:2001sr; @Ibarra:2003up] $$\begin{aligned}
\widetilde y_{i\alpha} = \frac{M^{1/2}}{v_{\rm EW}\sin\beta}R_{i \bar\gamma} \sqrt{m_{\bar \gamma}^\nu}\delta_{\bar\gamma\bar\beta} U_{\bar\beta\alpha}^{\rm (MNS)\dagger}.
\label{ytilde}\end{aligned}$$ Here $R_{i \bar\gamma}$ is given by $$\begin{aligned}
R_{i\bar\gamma} = \begin{pmatrix}
0 & \cos z & -\zeta\sin z \\
0 & \sin z & \zeta \cos z
\end{pmatrix}.\end{aligned}$$ for the normal hierarchy and $$\begin{aligned}
R_{i\bar \gamma} = \begin{pmatrix}
-\zeta\sin z & \cos z & 0\\
\zeta \cos z & \sin z & 0
\end{pmatrix}.\end{aligned}$$ for the inverted hierarchy, where $z$ is an arbitrary complex parameter and $\zeta=+1$ or $-1$. In the rest of the paper, we consider only the normal hierarchy because the inverted hierarchy is rarely realized in the flavor model described later.
If there are additional information or constraints on the Yukawa matrix, one may not be allowed to freely choose $z$ and the CP phases. Below we consider two such constraints. One comes from the condition for successful inflation and the other from flavor symmetry. Under these conditions, the parameter ranges of $z$ and CP phases are constrained, which enables us to make testable predictions on these parameters.
Lastly let us comment on the degenerate limit of the right-handed neutrinos. Even though the degeneracy is lifted by a small amount in the actual inflation model and a small breaking of the degeneracy is required for successful leptogenesis, we could study the above mentioned constraints in the limit of degenerate right-handed neutrinos. In this case, one can further rotate $(\tilde{N}_1,\tilde{N}_2)$ by an arbitrary real orthogonal matrix to get rid of the real component of $z$, $z_R\equiv {\rm Re}(z)$. In other words, $z_R$ becomes unphysical in the degenerate limit. Indeed, we have confirmed that our results do not depend on values of $z_R$, and remain valid even if the degeneracy is lifted by a small amount.
Stability condition of the inflationary path
--------------------------------------------
To derive predictions on the CP phases, we need to somehow constrain the neutrino Yukawa couplings in (\[Wapp\]). One important constraint comes from the inflaton dynamics: for large inflaton field values, there could appear a tachyonic direction in the field space of the slepton $L_\alpha$ and Higgs $H_u$ [@Nakayama:2016gvg].
To see this, let us write down the scalar potential of $L_\alpha$ and $H_u$ during inflation when $\varphi = \sqrt{2} {\rm Im} N_1$ takes a large field value, $$\begin{aligned}
V =\left( M\overline{y}_2N_1^* L_2' H_u + {\rm h.c.}\right)
+\overline{y}_1^2 |N_1|^2 \left( \left| L_1' \right|^2 + |H_u|^2\right),\end{aligned}$$ where we have set $N_2=0$, which is ensured by the non-minimal Kähler potential in (\[Kapp\]). We have also defined $$\begin{aligned}
&\overline{y}_1 \equiv \sqrt{ \sum_{\alpha} |y_{1\alpha}|^2 },~~~\overline{y}_2 \equiv \sqrt{ \sum_{\alpha} |y_{2\alpha}|^2 }, \label{ybar}\\
&L_1' \equiv \frac{1}{\overline{y}_1 } \left( \sum_\alpha y_{1\alpha} L_\alpha \right),
~~~L_2' \equiv \frac{1}{\overline{y}_2 } \left( \sum_\alpha y_{2\alpha} L_\alpha \right).\end{aligned}$$ It is clear that $L_1'$ is stabilized at $L_1'=0$ by the huge mass of $\overline{y}_1|N_1|$. On the other hand, the mass matrix of $(H_u, L_2'^*)$ is given by $$\begin{aligned}
m^2_{H L_2'} = \begin{pmatrix}
\overline{y}_1^2 |N_1|^2 & M \overline{y}_2 N_1 \\
M \overline{y}_2 N_1^* & k H^2
\end{pmatrix},\end{aligned}$$ where the Hubble mass correction is included for the mass of $L_2'$ with $\mathcal O(1)$ numerical coefficient $k$. Note that the Hubble mass correction comes from the quartic coupling in the Kähler potential, $\delta K \sim |N_2|^2 |L_2'|^2$. The coefficient of this coupling is considered to be of order unity but remains undetermined in our setup. Since $|N_1|$ takes a large value during inflation and $\overline{y}_1$ is sizable for successful seesaw mechanism, the mass eigenvalues during inflation are well approximated by $$\begin{aligned}
m_{\rm Heavy}^2 \simeq \overline{y}_1^2 |N_1|^2,~~~~~m_{\rm Light}^2\simeq kH^2-\frac{\overline{y}_2^2}{\overline{y}_1^2} M^2.\end{aligned}$$ In order for the inflaton path to be stable, $m_{\rm Light}^2 > 0$ should hold until the end of inflation.[^2] Noting $H \sim M$ at the end of inflation, we have a constraint from the stability of the inflaton path as $$\begin{aligned}
k \gtrsim \frac{\overline{y}_2^2}{\overline{y}_1^2}. \label{k_y1y2}\end{aligned}$$ Since $k$ is considered to be of $\mathcal O(1)$, $\overline{y}_1$ should be comparable to $\overline{y}_2$, which is of ${\cal O}(0.1)$. This provides a non-trivial constraint on the Yukawa couplings in the sneutrino chaotic inflation model. In particular, it implies that the masses of $L_\alpha H_u$ would indeed exceed the Planck mass if one simply extrapolated the inflation model to $\varphi \gtrsim {\cal O}(10)M_P$. As we pointed out in Ref. [@Nakayama:2016gvg], one solution is to assume that the discrete shift symmetry remains unbroken, which can give a better fit to the observation, while avoiding the super-Planckian masses of $L_\alpha H_u$ during inflation.
One can rewrite the stability condition (\[k\_y1y2\]) in terms of the Casas-Ibarra parameter. By an explicit calculation, we find $$\begin{aligned}
\overline{y}_1^2 = \frac{M(m^\nu_2+m^\nu_3)}{2v_{\rm EW}^2 \sin^2\beta}e^{-2z_I},~~~~~~
\overline{y}_2^2 = \frac{M(m^\nu_2+m^\nu_3)}{2v_{\rm EW}^2 \sin^2\beta}e^{2z_I},\end{aligned}$$ where $z_I\equiv{\rm Im}(z)$. Note here that $m_1^\nu = 0$ in our scenario with two right-handed neutrinos. Therefore, $z_I$ can be considered as the order parameter for the shift symmetry breaking in the Yukawa sector, because the larger $z_I$ implies the smaller inflaton Yukawa couplings $|y_{1\alpha}|$, and vice versa. Thus the condition (\[k\_y1y2\]) reads $$\begin{aligned}
k \gtrsim e^{4z_I}. \label{zI}\end{aligned}$$ Specifically, if we impose $k\lesssim 10$, we must have $z_I \lesssim 0.6$, while $z_R$ is not restricted by this condition at all. This is consistent with the fact that $z_R$ becomes unphysical in the degenerate limit.
Flavor symmetry
---------------
Let us now impose a global U(1)$_{\rm FN}$ flavor symmetry of the FN type [@Froggatt:1978nt] on the neutrino Yukawa couplings. We introduce a chiral superfield $\Phi_{\rm FN}$ whose lowest component develops a non-zero vacuum expectation value, $\langle \Phi_{\rm FN} \rangle= v_{\rm FN}$, leading to spontaneous break down of the U(1)$_{\rm FN}$. The observed mass hierarchy is controlled by the FN suppression factor, $\epsilon \equiv v_{\rm FN}/M_P$, where the cut-off scale is set to be the Planck scale.[^3]
The charge assignments are summarized in Table \[FN-charge\], where we assume that both the inflaton and the stabilzer field are singlet under the flavor symmetry, for simplicity. First, we choose the FN charges of lepton doublets as $Q_{\rm FN} (L_1) = q+1$, $Q_{\rm FN} (L_2) = q$ and $Q_{\rm FN} (L_3) = q$ [@Buchmuller:1998zf; @Sato:2000kj] in order to reproduce the observed features of the MNS matrix elements, especially the large $\nu_\mu$-$\nu_\tau$ mixing. Here $q = 0, 1, 2$ is an integer. Next we adopt $\epsilon \simeq 0.2$, for which one can approximately explain the observed charged lepton mass hierarchy by imposing the FN charge on the right-handed lepton superfields as $Q_{\rm FN} ({\overline E}_1) = 4$, $Q_{\rm FN} ({\overline E}_2) = 2$, $Q_{\rm FN} ({\overline E}_3) = 0$. The charge assignment depends on the precise value of $\epsilon$ and $\tan \beta$. In the sneutrino chaotic inflation, the gravitino mass needs to be as heavy as $100$TeV or heavier to avoid the BBN constraint on the gravitino decay [@Murayama:2014saa; @Nakayama:2016gvg], and so, if the sfermion masses are of the same order of the magnitude as the gravitino mass, a relatively small value of $\tan \beta$ is favored by the observed Higgs boson mass. In this case, one may choose $q \geq 1$. One can similarly extend the flavor symmetry to the quark sector in a way consistent with SU(5) GUT, and then, the FN parameter is directly related to the Cabbibo angle. In the following argument, only the FN charges of the lepton doublets and the right-handed neutrinos are relevant.
$L_1$ $L_2$ $L_3$ $N_1$ $N_2$ $\Phi_{\rm FN}$
-------------- ------- ------- ------- ------- ------- -----------------
$Q_{\rm FN}$ $q+1$ $q$ $q$ $0$ $0$ $-1$
: The FN charge assignment of the lepton doublets and right-handed neutrinos[]{data-label="FN-charge"}
With the above flavor symmetry, we expect that there are scaling relations among the Yukawa couplings of the stabilizer/inflaton field. Below we consider two cases separately. The first case (case 1) is that the only Yukawa couplings of the stabilizer field $N_2$ satisfy the following scaling, $$\begin{aligned}
|y_{2 e}|:|y_{2\mu}|:|y_{2 \tau}| \simeq \epsilon : 1 : 1.
\label{FNstructure1}\end{aligned}$$ The second case (case 2) is that both the inflaton ($N_1$) and stabilizer ($N_2$) Yukawa couplings satisfy the scaling relation, $$\begin{aligned}
|y_{i e}|:|y_{i\mu}|:|y_{i \tau}| \simeq \epsilon : 1 : 1~~~~~~{\rm for}~~~~ i=1,2.
\label{FNstructure2}\end{aligned}$$ Note that the inflaton Yukawa couplings break the shift symmetry explicitly and it is not known whether the shift symmetry breaking terms are blind to the lepton flavors or not. To be general, we study both cases in the next section.
Lastly let us comment on the flavor symmetry in the seesaw mechanism with degenerate right-handed neutrino masses. In fact, even if one imposes a certain flavor structure on the neutrino Yukawa couplings, it becomes less manifest in the low-energy neutrino mass matrix, especially when two right-handed neutrino masses are degenerate. This can be seen by rewriting Eq. (\[mnu\]) in terms of the original Yukawa as $$\begin{aligned}
m^\nu_{\alpha\beta} = \frac{v_{\rm EW}^2\sin^2\beta}{M}\left( y_{1\alpha}y_{1\beta} + y_{2\alpha}y_{2\beta}\right).\end{aligned}$$ In the case 2, we impose flavor constraints on the absolute magnitudes of the Yukawa couplings, but their phases are not constrained. Since the two terms in the parenthesis are of the same order in magnitude but with uncorrelated phases, they often add up in a destructive way. Specifically, diagonal elements (i.e. $m^\nu_{\alpha \alpha}$) tend to be less suppressed compared to off-diagonal ones, because the distribution of $|y_{1\alpha}|^2$ is spread more broadly than $|y_{1 \alpha} y_{1 \beta}|$ with $\alpha \ne \beta$ and the cancellation between the two terms occurs less frequently. As a result, the off-diagonal elements tend to be slightly suppressed compared to the diagonal ones, and the distributions of the mixing angles shift to smaller values. Thus the flavor structure of $m^\nu_{\alpha\beta}$ is less manifest with respect to that of the Yukawa sector. In other words, it is difficult for both Yukawa couplings to satisfy the same scaling relation, while keeping the naively expected flavor structure of the light neutrino masses, unless the phases of the Yukawa couplings are aligned. This feature is most prominent in our present model in which the two right-handed neutrinos are degenerate in mass. For a general case where the right-handed neutrinos masses are not degenerate, there is a similar tendency, but the flavor structure of $m^\nu_{\alpha\beta}$ is more retained, since one of the Yukawa couplings tends to give a dominant contribution to each element of the light neutrino mass matrix.
Numerical analysis {#sec:3}
==================
Now let us numerically analyze distributions of the Dirac and Majorana CP phases in the present setup. We adopt the following values of the mixing angles and mass squared differences obtained by a global fit to the experimental data [@Esteban:2016qun]: $$\begin{aligned}
\label{nu-fit}
\sin^2 \theta_{12} &= 0.28-0.33,~~~\sin^2 \theta_{23} = 0.39-0.63,
~~~\sin^2 \theta_{13} = 0.020-0.023,\\
\Delta m^2_{21} &= (7.16-7.88) \times 10^{-5}{\rm\, eV}^2,~~~~\Delta m_{31}^2 = (2.44-2.56)
\times 10^{-3} {\rm\,eV}^2,\end{aligned}$$ where the quoted ranges corresponds to the $2\sigma$ bounds. We generate random numbers of the mixing angles and mass squared differences within this range. The other parameters $z_R$, $z_I$, $\alpha$, and $\delta$ are randomly varied with a flat prior in the following ranges: $$\begin{aligned}
&-\pi < z_R < \pi,~~~0 < z_I < 0.6,~~~-\pi \leq \alpha < \pi,~~~-\pi \leq \delta < \pi.\end{aligned}$$ The range of $z_I$ is restricted because of the inflationary stability condition (\[zI\]). To take account of both positive and negative branches of the Casas-Ibarra parametrization, we also choose the sign of $\zeta (= \pm 1)$ randomly. Then we can calculate Yukawa matrix through Eq. (\[ytilde\]) and compare it with the predictions from the flavor structure of the FN model.
Case 1: flavor structure only on the stabilizer Yukawas
-------------------------------------------------------
First let us study the case 1 in which only the stabilizer Yukawa couplings are required to satisfy the relation (\[FNstructure1\]). To ensure the flavor structure of the Yukawa couplings, we define the normalized Yukawa couplings as $$\begin{aligned}
N \left(\frac{|y_{2e}|}{\epsilon}, ~|y_{2\mu}|, ~|y_{2\tau}| \right),\end{aligned}$$ where $N$ is the normalization factor to set the average of the components equal to unity: $$\begin{aligned}
N \equiv \frac{3}{|y_{2e}|/\epsilon+|y_{2\mu}| +|y_{2\tau}|}.\end{aligned}$$ Then we require that each normalized Yukawa coupling should be within the range of $1\pm \sigma_y$.
In Fig. \[fig:CPphase1\] we show the histograms of the Dirac and Majorana CP phases $\delta$ and $\alpha$, for the FN parameter $\epsilon = 0.2$ with $\sigma_y = 0.2$ (purple) and $\sigma_y=0.1$ (green). One can see that the vanishing Dirac CP violation, $\delta = 0$ and $\pi$, is disfavored, and that the maximal CP violation, $\delta = \pm \pi/2$, is also (mildly) disfavored. The distribution of $\delta$ has peaks located at $\delta \approx \pm \pi/4$ and $\delta \approx \pm 3\pi/4$. We note that those peaks at $\delta < 0$ correspond to the positive branch $\zeta = 1$ (and vice versa). On the other hand, the distribution of the Majorana phase has a broad peak about $\alpha = 0$.
In Fig. \[fig:mee\] we show the histogram of $m_{ee}$ which is relevant for the neutrinoless double beta decay experiments.
![ Histogram of the Dirac and Majorana CP phases $\delta$ and $\alpha$ for case 1, where the flavor structure is imposed only on the stabilizer Yukawa couplings. We have taken the FN parameter $\epsilon = 0.2$ and set $\sigma_y = 0.2$ (purple) and $\sigma_y=0.1$ (green). []{data-label="fig:CPphase1"}](delta_case1_.eps "fig:") ![ Histogram of the Dirac and Majorana CP phases $\delta$ and $\alpha$ for case 1, where the flavor structure is imposed only on the stabilizer Yukawa couplings. We have taken the FN parameter $\epsilon = 0.2$ and set $\sigma_y = 0.2$ (purple) and $\sigma_y=0.1$ (green). []{data-label="fig:CPphase1"}](alpha_case1_.eps "fig:")
![ Histogram of $|m_{ee}^\nu|$ in the unit of eV for the case 1 (left: purple and green) and case 2 (right: light blue). []{data-label="fig:mee"}](m_ee2.eps)
Case 2: flavor structure on both the inflaton and stabilizer Yukawas {#sec:case2}
--------------------------------------------------------------------
Next let us move to the case 2 in which both the inflaton and stabilizer Yukawa couplings are required to satisfy the relation (\[FNstructure2\]). Similarly to the previous case, we define the normalized Yukawa couplings as $$\begin{aligned}
N_i \left(\frac{|y_{ie}|}{\epsilon}, ~|y_{i\mu}|, ~|y_{i\tau}| \right),\end{aligned}$$ where $$\begin{aligned}
N_i \equiv \frac{3}{|y_{ie}|/\epsilon+|y_{i\mu}| +|y_{i\tau}|}\end{aligned}$$ for $i=1,2$. Then we require that each normalized Yukawa coupling should be within the range of $1\pm \sigma_y$. We note however that the number of data points which satisfy the scaling relations are much smaller than the previous case. In fact, with the same FN parameter $\epsilon=0.2$ and the deviation parameter $\sigma_y=0.2$, we could not find any data point satisfying all the constraints in one million generated data points. The main reason for the absence of the allowed parameters is that the observed solar mixing angle, $\theta_{12}$, is somewhat too large with respect to the expected value in the above flavor model. More precisely, the ratio of the observed mass squared differences would prefer a much smaller solar mixing angle in the flavor model. Thus we have relaxed the constraints as $\sigma_y=0.3$ and set $\epsilon=0.25$. In Fig. \[fig:CPphase2\] we show the histograms of the Dirac and Majorana CP phases $\delta$ and $\alpha$ in this case. One can see that, in contrast to the case 1, the vanishing Dirac CP violation, $\delta = 0$ and $\pi$, is favored, and that the maximal CP violation, $\delta = \pm \pi/2$, is strongly disfavored. On the other hand, the distribution of the Majorana phase has a broad peak about $\alpha = 0$.
As we have seen above, it is rather difficult to find data points that satisfy all the flavor conditions, which might imply that it is unlikely that the inflaton Yukawa couplings respect flavor symmetry. In other words, the shift symmetry breaking terms may not be flavor-blind. In fact, the allowed parameter region would increase if the inflaton Yukawa couplings had a milder flavor hierarchy with an effective FN paraemter, $\epsilon_{\rm inf} \sim 0.4 - 0.5$. This result reflects the tendency that the original flavor structures of Yukawa couplings becomes less manifest in the low-energy neutrino mass matrix when two right-handed neutrino masses are degenerate.[^4]
![ Histogram of the Dirac and Majorana CP phases $\delta$ and $\alpha$ for case 2, where the flavor structure is imposed on both the stabilizer and inflaton Yukawa couplings. We have taken the FN parameter $\epsilon = 0.25$ and set $\sigma_y = 0.3$. []{data-label="fig:CPphase2"}](delta_case2_.eps "fig:") ![ Histogram of the Dirac and Majorana CP phases $\delta$ and $\alpha$ for case 2, where the flavor structure is imposed on both the stabilizer and inflaton Yukawa couplings. We have taken the FN parameter $\epsilon = 0.25$ and set $\sigma_y = 0.3$. []{data-label="fig:CPphase2"}](alpha_case2_.eps "fig:")
Comparison with previous works
------------------------------
Before closing, we compare our results with some recent works to clarify the difference. In Ref. [@Rink:2016knw], Rink, Schmitz, and one of the present authors (Yanagida) studied the two right-handed neutrino scenario with a FN symmetry. They introduced the exchange symmetries $\widetilde N_1\leftrightarrow \widetilde N_2$ in the right-handed neutrino mass matrix and $\widetilde N_1\leftrightarrow i\widetilde N_2$ in the neutrino Yukawa couplings to ensure a certain form of the interactions they considered. They also focused on the case of $|z_I| \gtrsim 2$, which corresponds to the so called flavor alignment [@Rink:2016vvl]. We emphasize here that those symmetries are imposed in the mass eigenstate basis of the right-handed neutrinos. Under these conditions, they found that the Dirac phase around $\delta \sim -\pi/2$ is favored. In fact, there are two preferred points around $\delta \sim -\pi/2$, like our result of the case 1 (see Fig. \[fig:CPphase1\]). In our case, the former exchange symmetry is automatically satisfied in the inflation model (\[Wapp\]). The latter symmetry is approximately satisfied in our case 1. The reason is as follows. Although $z_I$ is bounded above by the inflaton stability condition, a mild hierarchy $|y_{2\alpha}| > |y_{1\alpha}|$ is still possible if $k$ takes a value greater than unity. When $|y_{2\alpha}| > |y_{1\alpha}|$, the Yukawa couplings in the mass eigenbasis are dominated by $y_{2\alpha}$, i.e., $\tilde{y}_{1\alpha} \sim y_{2\alpha}/\sqrt{2}$ and $\tilde{y}_{2\alpha} \sim i y_{2\alpha}/\sqrt{2}$ (see Eq. (\[masseigen\])). Such a structure is also obtained by imposing the exchange symmetry, $\widetilde N_1\leftrightarrow i\widetilde N_2$. Therefore, in the limit of flavor alignment, imposing both the flavor structure on $\tilde{y}_{1\alpha}$ and $\tilde{y}_{2\alpha}$ and the exchange symmetry $\widetilde N_1\leftrightarrow i\widetilde N_2$ is essentially equivalent to imposing the flavor structure only on the stabilizer Yukawa coupling, $y_{2\alpha}$. In this sense, the result of Ref. [@Rink:2016knw] is roughly consistent with our case 1, although the imposed symmetries are different.
In Ref. [@Kaneta:2017byo], Kaneta, Tanimoto and one of the present authors (Yanagida) studied the distribution of the CP phases in a model with $\det(m^\nu)=0$ motivated by the two right-handed neutrinos, and found a preference for $\delta = \pm\pi/2$. They randomly scanned parameters in the light neutrino mass matrix, anticipating that the light neutrino mass matrix satisfies the flavor structure after integrating out the heavy right-handed neutrinos. We note however that, as we have explained in Sec. \[sec:case2\], the flavor structure in the light neutrino mass matrix tends to be less manifest compared to that in the neutrino Yukawa couplings, because some cancellation among the elements could happen in the seesaw formula. In particular, this feature becomes prominent if the right-handed neutrinos are degenerate in mass. Therefore, their results can not be directly compared to our case.
Discussion and Conclusions {#discuss}
==========================
In the minimal sneutrino chaotic inflation model, the two gauge singlets required for successful inflation (i.e. the inflaton and stabilizer fields) are identified with the right-handed neutrinos, because they are generically coupled to $L_\alpha H_u$ [@Nakayama:2016gvg]. Interestingly, the seesaw mechanism as well as leptogenesis are natural outcomes of this setup.
In this Letter we have investigated this model further, focusing on the structure of the Yukawa couplings. First, we have derived a constraint on the Yukawa couplings from the stability condition of the inflationary path, which basically states that the inflaton Yukawa cannot be arbitrarily small and they are bounded below. Secondly we have imposed a flavor symmetry of the FN type on only the stabilizer Yukawas (case 1) and on both the inflaton and stabilizer Yukawas (case 2). Under these conditions, we have scanned the parameter space to derive the probability distribution of the CP phases, and found that the Dirac CP phase is sensitive to whether the inflaton Yukawa couplings depend on the lepton flavor symmetry. In the case 1, while the vanishing and maximal Dirac CP phases, $\delta = 0,\pi, \pm\pi/2$, are disfavored, and $\delta \approx \pm \pi/4$ and $\pm 3\pi/4$ are favored. The distribution of the Majorana CP phase has a broad peak at $\alpha = 0$. In the case 2, on the other hand, it is hard to explain the observed data based on the flavor model. This is because of the cancellation in the seesaw formula with degenerate right-handed neutrino masses, which results in smaller mixing angles than naively expected by the flavor symmetry. If we relax the flavor constraint, it is possible to find some parameters consistent with the observation. For those parameters, the vanishing Dirac CP phases $\delta = 0,\pi$ are favored and maximal phase $\delta=\pi/2$ is disfavored. The Majorana CP phase has a broad peak at $\alpha = 0$. We have also clarified differences of our results from those in the literature.
For successful leptogenesis, one needs to lift the degeneracy of the right-handed neutrino masses. In fact, there is a small contribution to the Majorana mass of $N_1$ from the holomorphic terms $N_1^2 + N_1^{\dag 2}$ in the Kähler potential, which can be absorbed into the superpotential by a Kähler transformation [@Bjorkeroth:2016qsk]. The phase of the Majorana mass comes from the constant term in the superpotential, $W \supset m_{3/2} M_P^2$, which is needed to realize the vanishingly small cosmological constant. Note that the lepton asymmetry is independent of the MNS matrix except for the small flavor effects, and in particular, those CP phases relevant for leptogenesis are not observed by low-energy experiments.
Let us here point out somewhat peculiar structure of our sneutrino chaotic inflation. First, the size of the shift symmetry breaking is quite different in the inflaton mass term and the neutrino Yukawa couplings. Namely, while the inflaton mass $M$ is of order $10^{-5}$ in Planck units, the inflaton Yukawa couplings need to be of order ${\cal O}(0.1)$ due to the stability condition (\[k\_y1y2\]). Secondly, we have found that the inflaton Yukawa couplings do not seem to respect the lepton flavor symmetry, and the observed data implies that the hierarchy in the inflaton Yukawa is milder than the flavor model predicts. Such a peculiar pattern of the symmetry breaking is hard to understand from the low energy point of view. However, one may be able to ensure the above features in a set-up with an extra dimension. For instance, let us consider a 5D theory compactified on $S_1/Z_2$. We assume that $N_1$, $L_\alpha$, and $H_u$ reside in the bulk and the other fields (including the FN field $\Phi$ as well as the stabilizer field $N_2$) are in a brane on one of the boundaries where both the flavor and shift symmetries are preserved to a high degree. This ensures the success of the FN model for explaining the charged lepton mass hierarchy as well as the inflaton mass much smaller than the Planck mass. Also the stabilizer Yukawa couplings respect the flavor symmetry. On the other hand, we assume that the inflaton Yukawa couplings are mainly generated on the other brane where both symmetries are largely broken. This explains why the inflaton has large Yukawa couplings which do not faithfully follow the flavor symmetry. We note however that one cannot explain why the Majorana mass term of $N_2$, $W = \frac{1}{2}M_2 N_2^2$, is absent or suppressed in the above set-up. This may be explained by the anthropic reasoning: the inflation would not have occurred unless $M_2$ is smaller than the pseudo Dirac mass $M$. In any case, there may be a deep reason for the suggested structure of the inflaton mass and Yukawa couplings, and it is worth studying the model from this perspective.
Finally we point out that the present sneutrino chaotic inflation model with FN symmetry can solve the cosmological problem of the original model [@Nakayama:2016gvg]. The problem was that the reheating temperature is as high as $T_{\rm R} \sim 10^{14}$GeV because of the sizable Yukawa couplings of the inflaton/stabilizer and it leads to the gravitino overproduction. In the present model, there is a flavon field whose decay can provide additional entropy production to dilute the gravitino abundance. First note that in the exact global U(1)$_{\rm FN}$ limit, there appears a massless Goldstone boson and its scalar partner, sflavon, is also massless up to the soft SUSY breaking [@Ema:2016ops]. However, we can introduce explicit U(1)$_{\rm FN}$ breaking terms to give these light directions sizable masses $m_\Phi$ $(\ll v_{\rm FN})$. Therefore we can consider a situation that the sflavon is initially deviated from the minimum in the low energy by $O(0.1) M_P$ during/after inflation and begins a coherent oscillation when the Hubble parameter $H$ becomes equal to $m_\Phi$. In such a case the (s)flavon dominates the Universe soon after the oscillation and its decay produces a huge amount of entropy. The (s)flavon may dominantly decay into two leptons plus Higgs, and the decay rate is given by $\Gamma_\Phi \sim C\epsilon^\ell m_\Phi^3/M_P^2$ with $C\sim 10^{-4}$ and an integer $\ell$ depending on the detailed FN charge assignments. The decay temperature is thus becomes $T_\Phi \sim 10^8$GeV for $m_\Phi \sim 10^{13}$GeV. (Note that the flavon mass cannot exceed the Hubble parameter during inflation.) The dilution factor of the preexisting gravitino is about $10^{-2}\,T_\Phi/T_{\rm R}$, which is enough to dilute the gravitino to a harmless level. The right amount of baryon asymmetry can be generated by resonant leptogenesis even in the presence of the entropy dilution [@Bjorkeroth:2016qsk].
Acknowledgments {#acknowledgments .unnumbered}
===============
K.N. thanks to So Chigusa for useful discussion. This work is supported by JSPS KAKENHI Grant Numbers JP15H05889 (F.T.), JP15K21733 (F.T.), JP26247042 (F.T), JP17H02875 (F.T.), JP17H02878(F.T. and T.T.Y), JP15H05888 (K.N.), JP26800121 (K.N) and JP26104009 (K.N. and T.T.Y), JP16H02176 (T.T.Y), and by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
[^1]: The quadratic chaotic inflation predicts a too large tensor-to-scalar ratio. To suppress the tensor-to-scalar ratio without changing the prediction of the spectral index significantly, the inflaton potential needs to be flatter and tachyonic (convex up) around $\mathcal O(10)M_P$. This is naturally realized if the potential has a periodicity of $\mathcal O(10)M_P$ [@Nakayama:2016gvg].
[^2]: After inflation ends, $H$ becomes smaller and eventually there appears a tachyonic direction in the zero-temperature potential. However, as shown in Ref. [@Nakayama:2016gvg], the preheating just after inflation is very efficient due to large Yukawa couplings and it leads to effective potential that tends to stabilize the slepton and Higgs fields, although a precise calculation is difficult to perform due to non-perturbative nature of the particle production. We emphasize that, even if a tachyonic direction appears, it is not a serious problem at all: the whole reheating process simply becomes a bit more complicated.
[^3]: We assume that the U(1)$_{\rm FN}$ is already broken during inflation. To this end one may introduce another FN field $\bar \Phi_{\rm FN}$ with $Q_{\rm FN} = 1$ to stabilize the FN fields at non-zero values. Even if the U(1)$_{\rm FN}$ is restored during inflation, our argument is not significantly modified if $q = 0$. If $q \geq 1$, the neutrino Yukawa couplings could vanish during inflation, and in this case, there will be no constraint on $z$ from the stability of the inflationary path.
[^4]: For instance, if one imposes the flavor structure on the light neutrino mass matrix as in Ref. [@Kaneta:2017byo], the observed solar mixing angle is on the upper end of the distribution. In our case, the distribution of the solar mixing angles shifts to smaller values because of the cancellation in the seesaw formula, which results in a strong tension between the flavor model and the observation.
|
---
abstract: 'A microscopic theory of the electromagnetic radiation emitted by a highly excited nucleus is developed on the basis of the Landau theory of a Fermi liquid. Closed formulae are obtained for the mean radiative width and its mean square fluctuation from level to level. The temperatures of many nuclei are found from the observed widths. The relaxation time is estimated from the experimental data on the radiative-width fluctuations. The regions of applicability of the various types of relations between the relaxation time and the lifetime of the compound nucleus, as well as the relevant physical consequences, are discussed.'
author:
- |
V.G. Nosov$^{\dagger}$ and A.M. Kamchatnov$^{\ddagger}$\
$^{\dagger}$[*Russian Research Center Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 Russia*]{}\
$^{\ddagger}$[*Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow Region, 142190 Russia*]{}
title: 'Theory of the radiative width of a highly excited nucleus [^1]'
---
Introduction
============
That there is a gradual unification of the mechanisms underlying the emission of electromagnetic radiation by a not too light nucleus as the excitation energy of the nucleus increases is practically beyond question. Roughly speaking, if an infinite number of the levels of the nucleus as a whole lie below the initial excitation energy, then the system will itself find and prefer the easiest effective way of emitting ${\gamma}$ quanta. In particular, there is below the initial excited level an abundance of levels ${\gamma}$ transitions to which satisfy the most favorable “selection rules", so that from this point of view the process is, in the limit under consideration, practically one with an infinite number of channels. Analysis shows (see \[1\]) that the electric-dipole radiation due to collisions between the (proton and neutron) quasiparticles and the “wall" of the nucleus predominates. The ideas of the Landau theory of the Fermi fluid \[2,3\] allow us to compute in closed form the radiative width $\overline{\Gamma}_\gamma$ and its fluctuations from level to level.
In a sense, spherical nuclei are rather exotic objects: the application to them of the Fermi-fluid concepts requires in each specific case certain precautions. The fact that the spherical configuration is stable is in itself an indication of the essential role played by the “residual interaction" between the quasiparticles, an interaction which blurs the Fermi level: it can be shown that in the scheme without interaction the sphere is absolutely unstable (see \[4\]). Furthermore, analysis of the data on the shell and magic oscillations in the masses of spherical nuclei allows the establishment of the macroscopically ordered structure that this residual interaction possesses in the space of the values of the orbital momentum $l$ of the individual quasiparticles \[5\]. (We shall again touch upon spherical nuclei when we compare below the theoretical results with the experimental data.) Nonspherical nuclei, on the other hand, are easy to investigate, it being apparently necessary to regard their shape as a perfectly natural consequence of the properties of the “normal", disordered nuclear phase, in which the quasiparticles situated near the limit of the Fermi distribution move, in the main, independently of each other.
However, for the theory of radiative widths expounded below the “shape effects", as such, are of no independent importance, and a special allowance for them is not necessary. Indeed, the equilibrium deformation $\alpha$ of a nonspherical nucleus is equal in order of magnitude to $\rho_f^{-1}$ (i.e., $\alpha\sim\rho_f^{-1}$), where $$\label{1}
\rho_f=k_fR\gg 1$$ ($k_f$ is the limiting momentum of the quasiparticle distribution and $R$ is the radius of the nucleus) is an important dimensionless parameter that arises in the most diverse investigations that have as their aim the treatment of the nucleus as a macroscopic body. In view of the scalar nature of the quantity $\Gamma_\gamma$ to be computed, only the squared deformation can enter, so that the relative magnitude $\alpha^2\sim\rho_f^2\ll 1$ of the corresponding corrections is negligibly small and falls outside the limits of accuracy of the theory. In other words, only the possibility of considering the quasiparticles individually is important for what follows, it still being possible in actually occurring deformations to treat the geometry of the motion of each of the quasiparticles as spherically symmetric.
The radiative width of a highly excited nuclei
==============================================
We shall derive the expressions, referred to one quasiparticle, on the basis of the correspondence principle. As applied to radiation processes, this principle asserts total analogy between the formulas of the classical and quantum theories (see, for example \[6\]). The classical intensity $I$ (i.e., the energy emitted per unit time) needs only to be divided by ${\varepsilon}=\hbar{\omega}$ to be converted into the quantity of real interest—the probability of emission of individual ${\gamma}$ quantum. The only remaining difference consists in the following: the spectral component of the multipole moment, which varies according to a classical law, should, generally speaking, be replaced by the corresponding matrix element of its operator. They, however, coincide in the quasiclassical limit (see (1)) \[7\]. Consequently, we can speak of a quasiparticle trajectory: it is in this case a straight line joining two points on the surface of the nucleus.
The basic formula of the classical theory of the electric dipole radiation has the form $$\label{2}
I=\frac{2e_d^2}{3c^3}\,\ddot{\mathbf{r}}^2,$$ where $e_d$ is the charge of the radiating particle (the radiating quasiparticle, in the general case; see below) and $\ddot{\mathbf{r}}$ is its acceleration vector. Before proceeding to the spectral decomposition of $I$, let us note that in the thermal-equilibrium state the quasiparticle motion inside the nucleus does not change its qualitative character in time. Therefore, let us formally carry out the Fourier expansion over an arbitrary, but sufficiently long interval of time $t$ and then take the limit as $t\to\infty$ (see \[8\]): $$\label{3}
\overline{\ddot{\mathbf{r}}^2}=\frac1{\pi t}\int_0^\infty\left|\int_0^t
\mathbf{r}(t)e^{i{\omega}t}dt\right|^2{\omega}^4d{\omega}.$$ In view of the independence of the different chords traced by the quasiparticles in their wall-to-wall motion, we have $$\label{4}
\left|\int_0^t\mathbf{r}e^{i{\omega}t}dt\right|^2=\overline{\left|\int_0^{t_1}
\mathbf{r}e^{i{\omega}t}dt\right|^2}\,n \cong \overline{\left|\int_0^{t_1}
\mathbf{r}dt\right|^2}\,n,$$ where $t_1 = l/v_f$ is the time it takes to travel from one end of a chord to the other, $l = 2\sqrt{R^2 -\rho^2}$ is the length of the chord, and $n$ is the number of chords. Here we have taken into account the fact that in the region of the radiation energy spectrum of interest to us $$\label{5}
{\omega}t_1\ll 1$$ (we shall return to the criterion (5) later). Furthermore, here and below the quasiparticle velocity $v$ is replaced everywhere by its limiting value $v_f$. The point is that because of the Pauli principle only those “elementary emitters" (i.e., quasiparticles) that are situated in the immediate neighborhood of the Fermi level play a role (see below).
The distribution of the chords over the impact parameters $\rho$ is easily found from considerations of isotropy and homogeneity of nuclear matter: $$\label{6}
w(\rho)d\rho=\frac3{R^3}\sqrt{R^2-\rho^2} \rho d\rho;\quad
\int_0^Rw(\rho)d\rho=1.$$ Averaging, in accordance with (4) and (6), the square of the integral over the radius vector, we also express the number $n$ of quasiparticle—nuclear wall collision events in terms of the physical time $t$: $$\label{7}
\overline{\left|\int_0^{t_1}\mathbf{r}dt\right|^2}=\frac4{v_f^2}
\overline{\rho^2(R^2-\rho^2)}=\frac{24}{35}\frac{R^4}{v_f^2},\quad
n=t\frac{v_f}{\bar{l}}=\frac23\frac{v_f}{R}t.$$ Taking into consideration the relations (2)-(7) and the relevant considerations, we obtain $$\label{8}
f({\varepsilon})d{\varepsilon}=\frac{32}{105\pi}\frac{e_d^2}{\hbar^5c^3}\frac{R^3}{v_f}{\varepsilon}^3d{\varepsilon}.$$ This expression gives the probability per unit time of emission by one quasiparticle of a ${\gamma}$ quantum in the interval $d{\varepsilon}$ of its energy values.
According to the theory of the Fermi fluid \[2-3\], the mean occupation numbers of the individual quantum states are given by the standard Fermi distribution $$\label{9}
\bar{n}({\varepsilon}')=\frac1{e^{{\varepsilon}'/T}+1},$$ where ${\varepsilon}'$ is the quasiparticle energy measured relative to the chemical potential and $T$ is the temperature. On the other hand, the number of actual single-quasiparticle states in the volume $V = (4/3)\pi R^3$ is equal to $$\label{10}
d\widetilde{N}=\frac{V}{\pi^2\hbar^3}\frac{p^2}{d{\varepsilon}'/dp}d{\varepsilon}'\cong
\frac{4R^3}{3\pi\hbar^3}\frac{p_f^2}{v_f}d{\varepsilon}'$$ ($p_f =\hbar k_f$ is the limiting momentum in standard units), where allowance has been made for the additional spin doubling. In fact, even in the quasiclassical limiting case, (1) remains an important quantum effect due to the identity, the indistinguishability of identical fermions \[7\]: the above-described classical picture of the process is actually realizable only in the case of radiative transitions that are compatible with the Pauli principle. Therefore, the product of the expressions (8), (9), and (10) should be supplemented by the factor $$1-\bar{n}({\varepsilon}'-{\varepsilon}),$$ which determines the fraction of the transitions admissible by this principle. Then integration over the energies ${\varepsilon}'$ of the radiating quasiparticles reduces to $$\label{11}
\int_{-\infty}^\infty\bar{n}({\varepsilon}')[1-\bar{n}({\varepsilon}'-{\varepsilon})]d{\varepsilon}'=
\frac{\varepsilon}{e^{{\varepsilon}/T}-1}.$$ Integration over the boson energies yields $$\label{12}
\int_0^\infty\frac{{\varepsilon}^4d{\varepsilon}}{e^{{\varepsilon}/T}-1}=24\zeta(5)T^5,\quad
\zeta(z)=\sum_{n=1}^\infty\frac1{n^z},$$ where $\zeta$ is the Riemann zeta function. Finally, the ${\gamma}$-quantum emission probability per unit time, appropriately summed over the entire set of quasiparticles of the same sort will be given by $$\label{13}
W=\frac{1024}{105\pi^2}\zeta(5)\frac{e_d^2m^{*2}}{\hbar^8c^3}R^6T^5,$$ where $m^* = p_f/v_f$ is the effective mass of the quasiparticle.
Above, as the coordinate origin convenient for the calculations, we used the geometrical center of the nucleus. However, the role of the total charge $Ze$ of the whole system in processes induced by the oscillations of the radius vector of the individual nucleons (quasiparticles) is well known. Because of recoil, even the electrically neutral quasiparticles (i.e., the neutron quasiparticles) will appear to emit radiation during their motion relative to the center of the nucleus. The corresponding, well-known, “charge-renormalization" formulae have the form $$\label{14}
e_d^Z=\left(1-\frac{Z}A\right)e,\quad e_d^N=-\frac{Z}A e$$ (the “effective charges," (14), of the two components are correct only for processes induced by the oscillations of the electric dipole moment of the nuclear system (see, for example, \[7\])). Summing, with allowance for (14), the expressions (13) or the proton and neutron components of the nuclear matter, and multiplying them by $\hbar$ in order to convert them into the energy widths of interest, we finally obtain $$\label{15}
\overline{\Gamma}_\gamma=\frac{1024}{105\pi^2}\zeta(5)\frac{e^2m^{*2}}{\hbar^7c^3}
\left[1-2\frac{Z}A\left(1-\frac{Z}A\right)\right]R^6T^5$$ (notice that the numerical factor $(1024/105\pi^2)\zeta(5)\cong 1$ is very close to unity). The law $\overline{\Gamma}_\gamma\propto T^5$ was given in the preceding paper (see \[1\], formula (4)), where it was motivated by semi-phenomenological considerations.
To what extent can the result (15) be identified with the radiative widths of the individual resonance levels of a compound nucleus that is excited, say, in a reaction involving slow-neutron capture? It follows from its derivation that the formula (15) corresponds to a state in which at the temperature $T$ the quasiparticles of the nuclear Fermi liquid are in thermal equilibrium with each other. On the other hand, the width $\Gamma_\gamma$ of a specific level can, reasoning abstractly, be conceived to have been computed from some very complicated, unknown (to us) wave function of the corresponding state of the nucleus as a whole. According to the fundamental principles of statistics, the two approaches lead to results that coincide to within the values of the fluctuations (see, for example, \[3\]).
Let us now rewrite the condition (5) of applicability of the theory in a more concrete form. Owing to the thermal nature of the radiation, the inequality (5) is equivalent to the following inequality: $$\label{16}
T\ll{\varepsilon}_0,$$ where $$\label{17}
{\varepsilon}_0=\hbar v_f/2R\sim 5 \mathrm{MeV}$$ is the characteristic energy corresponding to the reciprocal of the time it takes a quasiparticle to cross the nucleus along a diameter.
It is worth noting that in the opposite limiting case $$\label{18}
T\gg {\varepsilon}_0,$$ because of the oscillations of the exponent $e^{i{\omega}t}$ along the chord traced by the quasiparticle (see (3) and (4)), the energy distribution of the $\gamma$-quantum emission probability acquires the form of the well-known Planck black-body radiation spectrum \[3\]. The radiation width would, accordingly, become proportional to the cube of the temperature in the case of a sufficiently strict fulfilment of the condition (18). However, this “black-body radiation limit" defined by (18) is, in practice, hardly attainable in nuclear physics. At least the temperature of the compound nucleus should not exceed the nucleon binding energy, which is $\sim 8$ MeV—otherwise the neutrons would fly out of the nucleus “instantly," escaping the thermal-equilibrium establishment phase [^2].
The mean-square fluctuation in the radiation width. The role of the relaxation time
===================================================================================
The direct, quantum-mechanical computation of the characteristics of the individual states of the nucleus is inexpedient and practically impossible. Furthermore, as applied to macroscopic bodies (see the criterion (1)), this, as a rule, borders on the theoretical impossibility \[3\]. Therefore, we are obliged here to treat the state of the occupation of the individual quantum states of the quasiparticles of the Fermi liquid as a randomly varying function of the time. We shall calculate the instantaneous “emissive power" $\widetilde{\Gamma}_\gamma$ of the nucleus in a manner completely similar to the computations of the preceding section. The “one-component" variant of the corresponding formula can be represented in the form $$\label{19}
\widetilde{\Gamma}_\gamma=\frac{32}{105\pi}\frac{e_d^2}{\hbar^4c^3}\frac{R^3}{v_f}
\int_0^\infty d{\varepsilon}_1\cdot{\varepsilon}_1^3\sum_{{\varepsilon}'}n({\varepsilon}')[1-n({\varepsilon}'-{\varepsilon}_1)].$$ Here we have, for simplicity and convenience , written the discrete sum $\sum_{{\varepsilon}'}$ over the fermion states. In case of need the transition to integration can easily be accomplished with the aid of (10).
The “instantaneous", physically realizable values $$\label{20}
n_{{\varepsilon}'}=0,\,1$$ of the fermion occupation numbers differ from the mean occupation numbers (9). This circumstance is the obvious cause of fluctuations in Fermi systems. It is convenient to consider them with the aid of the simple relation (see \[3\]) $$\label{21}
\overline{\Delta n'\Delta n''}= \bar{n}'(1-\bar{n}')\delta_{{\varepsilon}',{\varepsilon}''}.$$ Let us find the mean-square fluctuation of the expression (19)—the number of summations and integrations doubles upon squaring. One summation over the quasiparticle states is trivial owing to the presence of the $\delta$ symbol on the right-hand side of (21); the subsequent integration is elementary, although somewhat tedious. Adding, in accordance with (14), the squares of the fluctuations in the proton and neutron components and introducing the dimensionless variables $x_{1,2}={\varepsilon}_{1,2}/T$ in place of the ${\gamma}$-quantum energies, we obtain $$\label{22}
\begin{split}
\overline{(\Delta\widetilde{\Gamma}_{\gamma})^2}=\frac{8192 J}{33075\pi^3}
\frac{e^4m^{*3}}{\hbar^{12}c^6}\left[\frac{(1-Z/A)^4}{\rho_f^Z}+
\frac{(Z/A)^4}{\rho_f^N}\right]R^{10}T^9,\\
J=\iint_0^\infty\left[\frac{x_1e^{x_1}\coth(x_1/2)}{(e^{x_1}-e^{x_2})(e^{x_1}-e^{-x_2})}
+\frac{x_2e^{x_2}\coth(x_2/2)}{(e^{x_2}-e^{x_1})(e^{x_2}-e^{-x_1})}\right]
x_1^3x_2^3dx_1dx_2.
\end{split}$$ The details of the integration over the boson energies are given in the Appendix. The final result has the form $$\label{23}
J=\frac{848}{1575}\pi^8+576\sum_{n=1}^\infty\frac1{n^4}\sum_{k=n+1}^\infty
\frac{k-n}{k^5}.$$ Notice that the term with the double sum is about half percent of the value of the integral, so that in practice we can restrict ourselves to the consideration of only the first term on the right-hand side of (23).
The problems pertaining to the fluctuations are relatively subtle and require a more careful physical treatment. In particular, there is no reason to equate $\overline{(\Delta\widetilde{\Gamma}_{\gamma})^2}$ to the mean square $\overline{(\Delta\Gamma_{\gamma})^2}$ of the actually observable, physical fluctuation in the radiative widths of many close resonance levels. This becomes especially apparent when we consider the most important and interesting case in which thermal equilibrium in the nucleus is established long before the “decay" of the nucleus: $$\label{24}
\Gamma\tau/\hbar\ll 1.$$ Here $\tau$ is the relaxation time (see below) and $\Gamma$ is the total width of the initial state of the nucleus. Taking into account the fact that this quasistationary state decays according to the law $\exp(-\Gamma t/\hbar)$, we express the number $\nu$ of emitted quanta and its fluctuation in terms of the instantaneous emissive power $\widetilde{\Gamma}_{\gamma}$: $$\label{25}
\begin{split}
\nu& =\frac1\hbar\int_0^\infty\widetilde{\Gamma}_{\gamma}(t)e^{-\Gamma t/\hbar}dt,\\
(\Delta\nu)^2& =\frac1{\hbar^2}\iint_0^\infty\Delta\widetilde{\Gamma}_{\gamma}(t)
\Delta\widetilde{\Gamma}_{\gamma}(t')\exp\left[-\Gamma(t+t')/\hbar\right]dtdt',\\
\Delta\widetilde{\Gamma}_{\gamma}(t)&=\widetilde{\Gamma}_{\gamma}(t)-\overline{\Gamma}_{\gamma}.
\end{split}$$ Further, it is convenient to introduce the notation $t' = t + \tau$. According to the thermodynamic theory of non-equilibrium processes (and of the corresponding fluctuations in the thermal-equilibrium state; see, for example, \[3\]), the mean value of the time correlation of the fluctuations is given by the relation $$\label{26}
\overline{\Delta\widetilde{\Gamma}_{\gamma}(t)\Delta\widetilde{\Gamma}_{\gamma}(t+\tau)}=
\overline{(\Delta\widetilde{\Gamma}_{\gamma})^2} \,\exp(-|\tau|/\bar{\tau}),$$ where $\bar{\tau}$ is the relaxation time. With allowance for (24), the substitution of (26) into (25) yields $$\label{27}
\overline{(\Delta\nu)^2}\cong\frac{\overline{(\Delta\widetilde{\Gamma}_{\gamma})^2}}{\hbar^2}
\int_0^\infty dt\exp\left(-\frac{2\Gamma}\hbar t\right)\int_{-\infty}^\infty
\exp\left(-\frac{|\tau|}{\bar{\tau}}\right)d\tau=
\frac{\overline{(\Delta\widetilde{\Gamma}_{\gamma})^2}}{\hbar\Gamma}\bar{\tau}.$$
Let us now consider the ensemble of the large number of close levels of a compound nucleus with radiative width $\Gamma$: owing to the fact that the levels decay according to the single law $\exp(-\Gamma t/\hbar)$, the equilibrium in the ensemble (the equipopulation of the levels) is not destroyed in time. The number of ${\gamma}$ quanta $$\nu=\frac1\hbar\int_0^\infty\Gamma_{\gamma}e^{-\Gamma t/\hbar}dt=\frac{\Gamma_{\gamma}}{\Gamma}$$ has been pre-averaged over a group consisting of many levels with practically the same $\Gamma_{\gamma}$. In the final averaging of the square of the fluctuations $(\Delta\nu)^2$ over the entire ensemble of the groups differing in their radiative widths $\Gamma_{\gamma}$, each group is taken into account with a weight proportional to the number of levels in it: $$\label{28}
\overline{(\Delta\nu)^2}=\overline{(\Delta\Gamma_{\gamma})^2}/\Gamma^2.$$ Equating the right-hand sides of the formulas (27) and (28), we finally obtain $$\label{29}
\overline{(\Delta\Gamma_{\gamma})^2}=(\Gamma\tau/\hbar)\overline{(\Delta\Gamma_{\gamma})^2}.$$ (we shall no longer write the averaging sign over the relaxation time $\tau$). A striking feature of the relation (29) consists in the following: It turns out that the fluctuations in the probability of decay of the compound nucleus via the radiative channel depend on the total decay probability $\Gamma$, including all the generally possible decay channels. The physical meaning of the formula (29) is simple: In the time picture the deviation of $\Delta\overline{\Gamma}_{\gamma}(t)$, the emissive power, from its mean value has time to average out to some extent provided the decaying exponential function varies sufficiently slowly (see the criterion (24)). The small factor $\Gamma\tau/\hbar$ on the right-hand side of (29) is precisely the quantity that determines the fraction of the physical, actually observable effect that remains after such a partial averaging.
Comparison with experiment
==========================
With the aid of the formula (15) we determined the temperatures of compound nuclei from the observed radiative widths of their resonance levels \[9,10\]. The results of such an analysis for two well-known regions of nonspherical nuclei are given in the tables. We assumed in the computations that $$\label{30}
R=1.2\cdot 10^{-13}A^{1/3}\, \mathrm{cm}$$ and $m^* = m_n$, where $m_n$ is the mass of the free nucleon. It is noteworthy that the temperature in the case of the actinide nuclei turns out consistently to be $\sim 100$ keV lower than the characteristic temperature for the lanthanide region. This may be due to both the decrease of the neutron attachment energy toward the end of the Mendeleev periodic table and the difference in the atomic weight $A$. A similar temperature decrease apparently occurs within the nonspherical-lanthanide region.
[|c|c|c|c|]{} Compound nucleus & $E_{max}$, MeV & $\bar{\Gamma}_{\gamma}$, MeV &$T$, MeV\
\
$_{62}$Sm$_{86}^{148}$ & 8.14 & 52 & 0.42\
$_{62}$Sm$_{88}^{150}$ & 7.98 & 64 & 0.44\
$_{62}$Sm$_{91}^{153}$ & 5.89 & 71 & 0.45\
$_{63}$Eu$_{89}^{152}$ & 6.29 & 89 & 0.47\
$_{63}$Eu$_{91}^{154}$ & 6.39 & 102 & 0.48\
$_{64}$Gd$_{92}^{156}$ & 8.53 & 110 & 0.48\
$_{64}$Gd$_{93}^{157}$ & 6.35 & 110 & 0.48\
$_{64}$Gd$_{94}^{158}$ & 7.93 & 89 & 0.46\
$_{64}$Gd$_{95}^{159}$ & 6.03 & 105 & 0.48\
$_{65}$Tb$_{95}^{160}$ & 6.40 & 90 & 0.47\
$_{66}$Dy$_{96}^{162}$ & 8.20 & 122 & 0.49\
$_{66}$Dy$_{97}^{163}$ & 6.25 & 175 & 0.52\
$_{66}$Dy$_{98}^{164}$ & 7.66 & 103 & 0.47\
$_{66}$Dy$_{99}^{162}$ & 5.64 & 166 & 0.51\
$_{66}$Dy$_{96}^{162}$ & 8.20 & 122 & 0.49\
$_{67}$Ho$_{99}^{166}$ & 6.33 & 91 & 0.45\
$_{68}$Er$_{99}^{167}$ & 6.44 & 97 & 0.46\
$_{68}$Er$_{100}^{168}$ & 7.77 & 96 & 0.46\
$_{69}$Tm$_{101}^{170}$ & 6.38 & 86 & 0.44\
$_{70}$Yb$_{102}^{172}$ & 8.14 & 74 & 0.43\
$_{70}$Yb$_{104}^{174}$ & 7.44 & 79 & 0.43\
$_{72}$Hf$_{106}^{178}$ & 7.62 & 64 & 0.41\
$_{73}$Ta$_{109}^{182}$ & 6.06 & 54 & 0.39\
$_{74}$W$_{109}^{183}$ & 6.19 & 58 & 0.40\
$_{74}$W$_{110}^{184}$ & 7.42 & 74 & 0.42\
$_{74}$W$_{111}^{185}$ & 5.75 & 64 & 0.41\
$_{74}$W$_{113}^{187}$ & 5.46 & 62 & 0.40\
$_{75}$Re$_{1131}^{188}$ & 6.24 & 55 & 0.39\
$_{75}$Re$_{113}^{188}$ & 5.73 & 55 & 0.39\
[|c|c|c|c|]{} Compound nucleus & $E_{max}$, MeV & $\bar{\Gamma}_{\gamma}$, MeV &$T$, MeV\
\
$_{90}$Th$_{143}^{233}$ & 4.96 & 21 & 0.30\
$_{91}$Pa$_{141}^{232}$ & 5.52 & 44 & 0.33\
$_{91}$Pa$_{143}^{234}$ & 5.12 & 48 & 0.35\
$_{92}$U$_{142}^{234}$ & 6.78 & 40 & 0.34\
$_{92}$U$_{143}^{235}$ & 5.27 & 25 & 0.31\
$_{92}$U$_{144}^{236}$ & 6.47 & 40 & 0.33\
$_{92}$U$_{145}^{237}$ & 5.30 & 29 & 0.31\
$_{92}$U$_{147}^{239}$ & 4.78 & 23 & 0.30\
$_{93}$Np$_{145}^{238}$ & 5.43 & 34 & 0.32\
$_{94}$Pu$_{146}^{240}$ & 6.46 & 40 & 0.33\
$_{94}$Pu$_{147}^{241}$ & 5.41 & 31 & 0.32\
$_{94}$Pu$_{148}^{242}$ & 6.22 & 37 & 0.33\
$_{95}$Am$_{147}^{242}$ & 5.48 & 42 & 0.33\
$_{95}$Am$_{149}^{244}$ & 5.29 & 50 & 0.35\
$_{96}$Cm$_{148}^{244}$ & 6.72 & 37 & 0.33\
$_{96}$Cm$_{149}^{245}$ & 5.70 & 39 & 0.33\
$_{96}$Cm$_{151}^{247}$ & 5.21 & 35 & 0.32\
Spherical nuclei possess a number of unique features that must be taken into consideration (see the Introduction). However, the question of the applicability to them of the formula (15) is at present difficult to answer categorically. Indeed, nuclei of this sort apparently undergo a phase transition to the “normal", nonspherical state at temperatures $$T\sim \Delta{\varepsilon}',$$ where $\Delta{\varepsilon}'$ is some characteristic width of the diffuse zone of the Fermi distribution, a zone which owes its existence to the residual interaction. Meanwhile, the spectrum of the emitted quanta (it is given by the integrand on the left-hand side of (12)) is such that the energy averaged over it is equal to $$\bar{{\varepsilon}}\cong 5T$$ (see also \[1\], formulae (5) and (6)). Thus, many of the radiative transitions can, roughly speaking, elude that region of the statistical distribution of the quasiparticles where the distribution differs significantly from (9). Therefore, the attempts to apply the formula (15) also to spherical nuclei, though not rigorous, is nevertheless of some interest. It is natural to suppose that spherical nuclei have higher temperatures (and, consequently, relatively low entropies; see also \[1\]). Comparison with the data on the radiative widths apparently corroborates this trend. For example, for the compound nucleus $_{79}$Au$_{119}^{198}$ we obtain $T = 0.45$ MeV, in the case of $_{80}$Hg$_{122}^{202}$ we have $T = 0.56$ MeV, and, finally, $T = 0.62$ MeV for $_{81}$Tl$_{123}^{204}$.
The experimental study of radiative-width fluctuations became possible only recently as a result of an increase in the accuracy of their measurement, and comparison of the theoretical formulas with experiment meets for the present with certain practical difficulties. Let us discuss three specific nuclei, for which a selection of resonance levels with accurately measured radiative widths nevertheless allowed the estimation of the relaxation time $\tau$ from the formula (29) (see also (22) and (23)). Data on two gadolinium isotopes are given in \[11\]; we took into consideration only the levels for which the error in the radiative width is less than $10^{-2}$ eV. In the case of $_{64}$Gd$_{92}^{156}$ (10 levels) $\overline{\Gamma}_{\gamma}= 0.11$ eV, $[\overline{(\Delta\Gamma_{\gamma})^2}]^{1/2} = 0.016$ eV, and $\hbar/\tau = 26$ eV. For $_{64}$Gd$_{94}^{158}$ (11 levels) we have $\overline{\Gamma}_{\gamma}= 0.089$ eV, $[\overline{(\Delta\Gamma_{\gamma})^2}]^{1/2} = 0.0087$ eV, and $\hbar/\tau = 52$ eV. Let us also give the results of a similar analysis of the data on holmium \[12\]: $_{87}$Ho$_{99}^{166}$ (21 levels) $\overline{\Gamma}_{\gamma}= 0.091$ eV, $[\overline{(\Delta\Gamma_{\gamma})^2}]^{1/2} = 0.0099$ eV, and $\hbar/\tau = 39$ eV. Thus, as far as we can judge, $\hbar/\tau \sim 50$ eV and $\tau\sim 10^{-17}$ sec, which is a remarkably long time on the nuclear scale. We must, however, not forget that the longest of the relaxation times $\tau$ of the system enters into the thermodynamic theory (see formula (26)). The “partial equilibrium" at each moment of time was understood to have been established over the significantly shorter relaxation times [^3].
Discussion. Is there enough time for the establishment of thermal equilibrium in a nucleus?
===========================================================================================
The question of relaxation in nuclear matter is of considerable interest. Thus far, as far as we know, it has not been possible to estimate the characteristic time of this process on the basis of any direct analysis of the experimental data. Therefore, the observed fluctuations in the radiative widths can be a valuable source of such information, and even the preliminary, tentative figures ($\tau\sim 10^{-17}$ sec; see the preceding section) need to be discussed. For the above-mentioned particular cases of resonance excitation by neutrons of energy $< 1$ keV, the condition (24) was satisfied with three orders of magnitude to spare—in other words, total thermal equilibrium was attained in the nucleus. The situation can, however, change when we go over to higher kinetic energies of the bombarding particles (see below).
It would, apparently, be somewhat naive to regard the time $t_1 = \hbar/{\varepsilon}_0\sim 10^{-22}$ sec of transit of a quasiparticle through the nucleus as an estimate for the relaxation time. Indeed, for example, the system is not in the least drawn nearer to the state of thermal equilibrium by a coherent, reversible, purely elastic act of collision between a quasiparticle and the surface of the nucleus [^4]. On the other hand, inter-quasiparticle collisions appear to be quite an effective relaxation mechanism; it may well turn out to be the dominant mechanism.
For obvious reasons, the thermal-radiation data used in the present paper actually pertained to an extremely narrow part of the energy spectrum of the system. Let us now qualitatively consider how the real conditions under which the relaxation process proceeds in the nucleus should change upon further increase in the excitation energy of the nucleus. The neutron width $\Gamma_n$ increases first in proportion to the square root of the distance from the neutron-detachment threshold; then there we come into the stage of much more rapid exponential growth. It is well known that owing to this phenomenon the resonance levels merge, forming a continuous spectrum. But then whether the lifetime $\hbar/\Gamma$ of the compound nucleus will be long enough for the establishment of total thermal equilibrium in the nucleus may become doubtful, beginning from [^5] $$\Gamma\cong\Gamma_n\geq 100\,\, \mathrm{eV}.$$
In this connection, it is desirable to try critically reinterpret the method, based on the neutron “evaporation" process, for determining nuclear temperatures. It is difficult for the present to judge how the fact that the state of the neutron-emitting nucleus is not a totally equilibrium state will influence such an analysis. Not much doubt has thus far been expressed about the evaporation temperatures probably because their order of magnitude is quite plausible (and, in so far as we can judge, indeed correct). However, as the experimental investigation of the reactions $(n, n')$ goes on, attention will have to be paid not only to the absolute figures, but also to the behavior of the relevant quantities. Of special interest, in particular, is the case when the temperature of the compound nucleus as a function of its excitation energy is an almost horizontal, non-monotonic in detail, and often simply a decreasing function. Unfortunately, the authors of the corresponding publications give this remarkable circumstance comparatively little consideration (see, for example \[15\]). We could have attempted to interpret the decrease of the temperature with increasing excitation energy as some giant random fluctuation, but it would have been difficult to conceive it as a phenomenon that would occur with any degree of consistency. On the average, however, negative specific heat is impossible for the nucleus. A state with negative specific heat is totally unstable, and cannot be realized in nature \[3\].
We express our thanks to A.I. Baz’, V.K. Voitovetskii, I.I. Gurevich, A.G. Zelenkov, L.P. Kudrin, V.M. Kulakov, A.A. Ogloblin, I.M. Pavlichenkov, N.M. Polievktov-Nikoladze, and K.A. Ter-Martirosyan for a discussion of the results of the paper.
Appendix {#appendix .unnumbered}
========
Let us expound in some detail the integration over the boson energies in the formula (22). The terms of the integral $J$ are identical in form, but individually each of them contains a pole at $x_1 = x_2$. Therefore, it is sufficient to evaluate any of these integrals in the principal value sense: $$\label{a-1}
J=2\int_0^\infty dx_1\cdot x_1^4 e^{x_1}\coth\frac{x_1}2\,\mathrm{P}\!\int_0^\infty
\frac{x_2^3dx_2}{(e^{x_1}-e^{x_2})(e^{x_1}-e^{-x_2})}.$$ Let us transform the inner integral with the aid of the substitution $y = e^{-x_2}$, representing it as a derivative with respect to some parameter: $$\label{a-2}
\begin{split}
\mathrm{P}\!\!\int_0^\infty\frac{x_2^3dx_2}{(e^{x_1}-e^{x_2})(e^{x_1}-e^{-x_2})}&=
e^{-x}\,\mathrm{P}\!\!\int_0^1\frac{(\ln y)^3dy}{(y-e^x)(y-e^{-x})}\\
&=e^{-x}\frac{\partial^3}{\partial\nu^3}\,\mathrm{P}\!\!\int_0^1\frac{y^\nu dy}
{(y-e^x)(y-e^{-x})},\quad \nu\to 0
\end{split}$$ ($x = x_1$). In the decomposition $$\label{a-3}
\mathrm{P}\!\int_0^1\frac{y^\nu dy}
{(y-e^x)(y-e^{-x})}=\frac1{2\sinh x}\left\{J_\nu^+(x)-J_\nu^-(x)\right\},$$ in which the integrand is expressed in partial fractions, the integrals $$\label{a-4}
J_\nu^+(x)=\mathrm{P}\!\!\int_0^1\frac{y^\nu dy}{y-e^x},\quad
J_\nu^-(x)=\mathrm{P}\!\!\int_0^1\frac{y^\nu dy}{y-e^{-x}}$$ can conveniently be expressed as series. For this purpose, let us represent the fractions by the corresponding geometric progressions, and let us also take into consideration the formula $$\sum_{n=-\infty}^\infty\frac1{n+\nu}=\pi\cot\pi\nu.$$ We then have $$\label{a-5}
J_\nu^+(x)=\sum_{n=1}^\infty\frac{e^{-nx}}{n+\nu},\quad
J_\nu^-(x)=-e^{-\nu x}\pi\cot\pi\nu+\frac1\nu-\sum_{n=1}^\infty
\frac{e^{-nx}}{n-\nu}.$$ Taking the limit in accordance with (A.2), we obtain $$\label{a-6}
J=192\int_0^\infty\left\{\frac{\xi^8}3-\frac{\pi^2}3\xi^6-\frac{\pi^4}{90}
\xi^4\right\}\frac{d\xi}{\sinh^2\xi}+6\sum_{n=1}^\infty\frac1{n^4}
\int_0^\infty\frac{\xi^4e^{-n\xi}}{\sinh^2(\xi/2)}d\xi.$$ It is easy to see that the integral standing under the summation sign $$\label{a-7}
\int_0^\infty\frac{\xi^4e^{-n\xi}}{\sinh^2(\xi/2)}d\xi=4\int_0^1
\frac{(\ln y)^4y^n}{(y-1)^2}dy=4\frac{\partial^4}{\partial\nu^4}\,
\mathrm{P}\!\!\int_0^1\frac{y^{n+\nu}dy}{(y-e^x)(y-e^{-x})},\quad \nu\to 0$$ is essentially of the same type as (A.2), except that it is differentiated once more with respect to $\nu$ and that the additional passage to the limit $x\to 0$ will also be necessary. The subsequent simple computations, besides the passages to the limit, also include convenient re-designations of the indices of the double summation. As a result, we obtain the formula (27).
[99]{}
V.G. Nosov and A. M. Kamchatnov, Zh. Eksp. Teor. Fiz. [**65,**]{} 12 (1973) \[Sov. Phys.-JETP [**38,**]{} 6 (1974)\]; arXiv nucl-th/0311045.
L.D. Landau, Zh. Eksp. Teor. Fiz. [**30,**]{} 1058 (1956) \[Sov. Phys.-JETP [**3,**]{} 920 (1957)\].
L.D. Landau and E.M. Lifshitz, Statisticheskaya fizika (Statistical Physics), Fizmatgiz, 1964 (Eng. Transl., Addison-Wesley Publ. Co., Reading, Mass., 1969).
V.G. Nosov, Zh. Eksp. Teor. Fiz. [**57,**]{} 1765 (1969) \[Sov. Phys.-JETP [**30,**]{} 954 (1970)\].
A.M. Kamchatnov and V.G. Nosov, Zh. Eksp. Teor. Fiz. [**63,**]{} 1961 (1972) \[Sov. Phys.-JETP 36, 1036 (1973)\]; arXiv nucl-th/0311011.
V.B. Berestetskii, E.M. Lifshitz, and L.P. Pitaevskii, Relyativistskaya kvantovaya teoriya (Relativistic Quantum Theory), Part. l. Nauka, 1968 (Engl. Transl., Pergamon, New York, 1971).
L.D. Landau and E.M. Lifshitz, Kvantovaya mekhanika (Quantum Mechanics), Fizmatgiz, 1963 (Eng. Transl., Addison-Wesley, Reading, Mass., 1965).
L.D. Landau and E.M. Lifshitz, Teoriya polya (The Classical Theory of Fields), 2nd ed., Goztekhizdat, 1948, (549, pp. 134-135 (Eng. Transl., Addison-Wesley, Reading, Mass., 1962).
M.D. Goldberg, S.F. Mughabghab, S.N. Purohit, B.A. Magurno, and V.M. May, Neutron Cross Sections, Vol. IIC, Brookhaven National Laboratory, 1966.
J.R. Stehn, M.D. Goldberg, R. Wiener-Chasman, S.F. Mughabghab, B.A. Magurno, and V.M. May, Neutron Cross Sections, Vol. Ill, Brookhaven National Laboratory, 1965.
S.J. Friesenhahn, M.P. Fricke, D.G. Costello, W.M. Lopez, and A.D. Carlson, Nucl. Phys. [**A146,**]{} 337 (1970).
M. Asghar, C.M. Chaffey, and M.C. Moxon, Nucl. Phys. [**A108,**]{} 535 (1968).
F. Rahn, H.S. Camarda, G. Hacken, W.W. Havens, Jr., H. I. Liou, J. Rainwater, M. Slagowitz, and S. Wynchank, Phys. Rev. [**C6,**]{} 1854 (1972).
T. Erickson and T. Mayer-Kuckuk, Ann. Rev. Nuc. Sci. Vol. [**16,**]{} 1966 \[Russ. transl. Usp. Fiz. Nauk [**92,**]{} 271 (1967)\].
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[^1]: Zh. Eksp. Teor. Fiz. [**67**]{}, 441–452 (1974) \[Sov. Phys. JETP [**40,**]{} No. 2, 219-224 (1975)\]
[^2]: Notice that the radiation due to collisions between the quasiparticles would then become dominant only at $T\gg\sqrt{\rho_f{\varepsilon}_0}$.
[^3]: From the data of the recent paper \[13\] we find that $\hbar/\tau\sim 20$ eV for the compound nuclei Th$^{233}$ and U$^{239}$.
[^4]: Besides, a characteristic time $\sim t_1$ is quite capable of playing an important role at the earliest stage of the development of the nuclear reaction. Here, however, we are discussing only the late, final phase of the thermal-equilibrium establishment process—see the end of the preceding section. In particular, it is extremely doubtful that there will, at such times, remain reasonable physical criteria for distinguishing the initial bombarding particle (or the corresponding quasiparticle).
[^5]: It is possible that the so-called Erickson fluctuations in the nucleon-nucleus interaction cross sections \[14\] are also due to this circumstance.
|
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abstract: 'A new picture of Quantum Mechanics based on the theory of groupoids is presented. This picture provides the mathematical background for Schwinger’s algebra of selective measurements and helps to understand its scope and eventual applications. In this first paper, the kinematical background is described using elementary notions from category theory, in particular the notion of 2-groupoids as well as their representations. Some basic results are presented, and the relation with the standard Dirac-Schrödinger and Born-Jordan-Heisenberg pictures are succinctly discussed.'
author:
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F. M. Ciaglia$^{1,5}$ [](https://orcid.org/0000-0002-8987-1181), A. Ibort$^{2,3,6}$[](https://orcid.org/0000-0002-0580-5858), G. Marmo$^{4,7}$[](https://orcid.org/0000-0003-2662-2193)\
\
\
\
\
,\
\
title: 'Schwinger’s Picture of Quantum Mechanics I: Groupoids'
---
Introduction
============
Schwinger’s algebraic formulation of Quantum Mechanics
------------------------------------------------------
In an attempt to establish a “*mathematical language that constitutes a symbolic expression of the properties of microscopic measurements*”, J. Schwinger proposed a family of algebraic relations between a set of symbols representing fundamental measurement processes [@Sc70 Chap. 1]. Such symbols can’t consist of the classical representation of physical quantities by numbers, and they should be represented instead in terms of abstract symbols whose composition properties are postulated on the ground of physical experiences. A similar approach was always present in Dirac’s thinking on Quantum Mechanics as it was specifically stated in his lectures at Yeshiva when he dealt with q-numbers [@Di66]. One of the main observations of this paper is that Schwinger’s algebra of symbolic measurements can be thought of as the algebra of a certain groupoid associated with the system.
In Schwinger’s algebraic depiction of the kinematical background of a quantum mechanical system (see [@Sc70 Chap. 1][^1]), the state of a quantum system is established by performing on it a complete selective measurement. A general class of symbols is introduced afterwards, i.e., those denoting compound selective measurements that change a given state in another one. Eventually, a transformation law among generalised selective measurements was postulated in order to introduce a notion of covariance in the theory. A brief account of this perspective was offered in [@Ci18], and we will extend and complete it in the present paper.
In this paper, the algebraic description of Schwinger’s picture of the kinematical background of quantum mechanics systems will be established using abstract categorical notions, leaving the discussion of dynamics to a forthcoming work [@Ib18].
We will depart from an abstract setting inspired by Schwinger’s conceptualisation where the primary notions will be that of ‘outcomes’ of measurements; ‘transitions’, that will refer to physically allowed changes in the outcomes of measurements; and ‘transformations’, that is, relations among transitions as described from different experimental settings or, more generally, transformations of allowed transitions (acquiring a dynamical perspective). These concepts are the abstractions corresponding to Schwinger’s selective measurements, compound measurements and transformation functions, respectively.
We will argue that the mathematical structure behind Schwinger’s algebraic relations encompassed by the notions listed before (outcomes, transitions and transformations) is that of a 2-groupoid, where events are its 0-cells, transitions constitute the 1-cells of the 2-groupoid (and define a groupoid structure themselves), and transformations provide the 2-cells of the structure. It will be shown that such abstract notion can be used to offer an alternative picture for the description of quantum systems that, under some conditions, is equivalent to the standard pictures of Quantum Mechanics.
On the many pictures of Quantum Mechanics
-----------------------------------------
As it is well known, modern Quantum Mechanics was first formulated by Heisenberg as matrix mechanics immediately after Schrödinger formulated his wave mechanics. Both pictures got a better mathematical description by Dirac [@Di81] and Jordan [@Jo34], [@Bo25] with the introduction of the theory of Hilbert spaces and the corresponding theory of transformations, a sound mathematical formulation that was provided by von Neumann [@Ne32].
In all of these pictures, the principle of analogy with classical mechanics, as formulated by Dirac, played a fundamental role. The canonical commutation relations (CCR) were thought to correspond, or to be analogous to, the Poisson Brackets on phase space. Very soon, within the rigorous formulation of von Neumann, domain problems were identified, showing that at least one of position or momentum observables should be represented by an unbounded operator [@Wi47]. Weyl introduced an “exponentiated form” of the commutation relations in terms of unitary operators, i.e., a projective unitary representation of a symplectic Abelian vector group, interpreted also as a phase-space with a Poisson Bracket [@We27]. A $C^*$-algebra, a generalization of the algebraic structure emerging from Heisenberg picture, would be obtained as the group-algebra of the Weyl operators, opening the road to the highly algebraic description of quantum systems provided by $C^*$-algebras.
Thinking of relativistic quantum mechanics, Dirac proposed the introduction of a Lagrangian formulation for quantum dynamics. In his own words: “*...the Lagrangian method can easily be expressed relativistically on the account of the action function being a relativistic invariant; while the Hamiltonian method is essentially non-relativistic in form, since it marks out a particular time variable as the canonical conjugate of the Hamiltonian function.*” This suggestion was taken up by both Feynman and Schwinger, and they developed it in different directions. Feynman’s approach culminated into the path-integral formalism, where the principle of analogy is still present. Schwinger, however, took a different road by introducing the measurement-algebra approach, where the analogy with classical kinematics is much less evident. Indeed, Schwinger’s formulation was written for quantum systems with a finite number of states. In any case, for both approaches, the seed may be found in Dirac’s paper *The Lagrangian in Quantum Mechanics* [@Di33].
While for the various pictures associated with the names of Heisenberg, Dirac, Jordan, Weyl, etc., the intervening algebraic mathematical structures are nowadays clearly identified, it is not the case for the mathematical structure underlying Schwinger’s approach. In this paper, we would like to unveil and identify this structure while postponing a thorough analysis of its implications, most notably for field theories, to forthcoming papers.
Groupoids in Physics {#sec:categories}
--------------------
Groupoids are playing a more relevant role in the description of the structure of physical theories. For instance, the use of groupoids is very convenient to describe systems with internal and external structures (see for instance [@We86], [@La98] and references therein). It should be remarked that a groupoid structure can be identified also in the considerations made by Dirac in the previously quoted paper [@Di33]. Indeed, the composition law of the generating functions representing transformations allows to define a groupoid structure for the latter. Another instance of groupoid is provided by Ritz-Rydberg combination principle of frequencies in spectral lines as observed by Connes [@Co94], where groupoids are connected with the structure of certain measurements, in this case frequencies of the emission spectrum by atoms:
*“The set of frequencies emitted by an atom does not form a group, and it is false that the sum of two frequencies of the spectrum is again one. What experiments dictate is the Ritz–Rydberg combination principle which permits indexing the spectral lines by the set $\Delta $ of all pairs $(i,j)$ of elements of a set $I$ of indices. The frequencies $\nu _{(ij)}$ and $\nu
_{(kl)}$ only combine when $j=k$ to yield $\nu _{(il)}=\nu _{(ij)}+\nu
_{(jl)}$ (...). Due to the Ritz–Rydberg combination principle, one is not dealing with a group of frequencies but rather with a groupoid $\Delta
=\left\{ (i,j);i,j\in I\right\} $ having the composition rule $(i,j)(j,k)=(i,k)$. The convolution algebra still has a meaning when one passes from a group to a groupoid, and the convolution algebra of the groupoid $\Delta $ is none other that the algebra of matrices since the convolution product may be written $(ab)_{(i,k)}=\sum_{n}a_{(i,n)}b_{(n,k)}$ which is identical with the product rule of matrices."*
It is quite convenient to think of groupoids as codifying processes in the sense that the composition law determines the different ways one can use to pass from one base element (the objects of the groupoid) to another one[^2]. Actually, the best way to think about a groupoid is in terms of categories or, put in a different way, a category is a broad generalization of the notion of a group(oid).
The process of abstracting properties of physical systems obtained by their observation, like the properties of measurements on microscopic systems pondered by Schwinger, is extremely useful by itself. However, as it was pointed out by J. Baez, the mathematical language based on set theory is extremely restrictive and limited for many purposes. Physics dealing with processes and relations both at the classical and quantum level, is particularly bad suited to be described by set theory (see for instance [@Ba01]).
On the contrary, category theory is exactly about that, the emphasis is not in the description of the elements of sets, but on the relations between objects, i.e., ‘morphisms’. Therefore, ‘elements’ in set theory correspond to ‘objects’ of a category (and ‘elements’ can be ‘sets’ themselves without incurring in contradictions!) and ‘equations between elements’ correspond to ‘isomorphisms between objects’. ‘Sets’ correspond in this categorification of abstract notions to ‘categories’ and maps between sets to ‘functors’ between categories.
In particular, a representation of a given category is a functor from this category in the category whose objects and morphisms are those mathematical structures we want to use to ‘realise’ our category, e.g., linear spaces and linear maps in the case of linear representations. Finally ‘equations between functions’ will correspond to ‘natural transformations’ between functors.
Considering functors themselves as objects and natural transformations as morphisms led to the notion of higher categories, in particular, the 2-categories and the corresponding notions of 2-groups and 2-groupoids. These abstract notions are gaining more and more interest in the description of physical phenomena (see for instance recent applications to describe topological matter [@Ka17],[@Al17]). Surprisingly enough, we will argue that this highly abstract notions, in particular the notion of 2-groupoid, is just what is needed to provide the formal mathematical background to the kinematics of Schwinger’s algebra of selective measurements.
The structure of this paper is as follows. First, the basic notions of categories and groupoids will be described in a succinct way. Then, it will be sketched how the descripton of physical systems based on groupoids would lead in a natural way to the notion of a 2-groupoid, and eventually, in the finite case, it will be shown how Schwinger’s algebra of selective measurements is an instance of it. In the meantime, a sketch of a theory of representations will be developed and some fair connections with the standard descriptions of Quantum Mechanics will be outlined.
Groupoids and categories
========================
Categories
----------
Using the language of Category Theory a groupoid is a category all of whose morphisms are invertible. Let us recall that a category **C** consists of a family of objects $x,y,\ldots$, denoted collectively as Ob(**C**), and a family of morphisms (or arrows) $\alpha \colon x \to y$, $\beta \colon u \to v$,... denoted collectively as Mor(**C**). Given two objects $x,y$, the family of morphisms from $x$ to $y$ is denoted as Mor($x,y$). The category $\mathbf{C}$ is equipped with a composition law that assigns to any pair of morphisms $\alpha\colon x \to y$ and $\beta\colon y \to z$ a morphism[^3] $\beta \circ \alpha \colon x \to z$. The composition law is associative, that is, $(\alpha \circ \beta) \circ \gamma = \alpha \circ (\beta \circ \gamma)$ whenever $\alpha, \beta$ and $\gamma$ can be composed. Finally it is assumed that there exists a family of morphisms $1_x$ such that $\alpha\circ 1_x = \alpha$ and $1_y \circ \alpha = \alpha$ for any $\alpha \colon x \to y$.
Sometimes it would be convenient to denote the category $\mathbf{C}$ as: Mor(**C**) $\rightrightarrows$ Ob(**C**) where the double arrows denote the assignments to each morphism $\alpha \colon x \to y$ of the ‘source’ object $x$ and the ‘target’ object $y$ respectively. In this sense we will denote $x = s(\alpha)$ and $y = t(\alpha)$ (it will be also denoted sometimes $y = \alpha(x)$). Notice that the morphism $\alpha$ can be composed with the morphism $\beta$ iff $t(\alpha) = s(\beta)$.
A morphism $\alpha\colon x \to y$ is said to be invertible if there exists $\beta\colon y \to x$ such that $\alpha\circ \beta = 1_y$ and $\beta\circ \alpha = 1_x$. Such morphism will be called the inverse of $\alpha$ and will be denoted as $\alpha^{-1}$. An invertible morphism will be called an isomorphism. Notice that for any given category **C** the subcategory of invertible morphisms is a groupoid, called the groupoid of the category.
It will be assumed in what follows that all categories considered are small, that is, their objects and family of objects as well as the family of all morphisms are sets, and morphisms are maps among sets. Then the previous notation for objects and morphisms coincide with the corresponding set-theoretical notions. However it is important to bear in mind that many interesting examples, potentially relevant in the considerations on the foundations of Quantum Mechanics, could involve categories which are not small. Consider for instance the category **Vect** of all vector spaces whose objects are complex linear spaces and morphisms are linear maps between them. Such category is larger than the category **Sets** (whose objects are sets and morphisms are maps among sets) because given a set $S$ we may construct the complex linear space $V(S)$ freely generated by $S$, however the family of all sets is not a set, hence the family of all linear spaces is not a set and the category **Vect** is not small.
Groupoids {#sec:groupoids}
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Even if the notion of groupoid is categorical, along this paper groupoids will be considered to be sets (even finite in Schwinger’s conceptualisation). Thus, in this section abstract groupoids $\mathbf{G}$ will be briefly discussed from a set-theoretical perspective, that is, we will consider groupoids $\mathbf{G}$ whose objects form a set $\Omega$ and whose morphisms are elements $\gamma$ of the set $\mathbf{G}$.
Two maps $s$ (‘source’) and $t$ (‘target’) will be defined from $\mathbf{G}$ onto $\Omega$, such that there is a binary operation $\circ $ (called multiplication) which is defined for pairs $\gamma$ and $\mu$ of elements in $\mathbf{G}$ whenever $t(\gamma) = s(\mu)$ (then $\gamma$ and $\mu$ will be said to be composable) and the resulting element will be denoted $\mu\circ \gamma$ (notice the backwards notation for composition consistent with the convention introduced in the case of categories and the notation used for the composition of maps). We will keep using the diagrammatic notation $\gamma \colon x \to y$ if $s(\gamma) =x$ and $t(\gamma) = y$ as in the abstract categorial setting even if $\gamma$ is not a map between sets.
Moreover for a given groupoid $\mathbf{G}$, the maps $\{ s,t,\circ \} $ must satisfy the following axioms:
1. $s(\mu \circ \gamma) = s(\gamma)$, $t( \mu \circ \gamma) = t(\mu)$ for all composable $\gamma $ and $\mu$.
2. There exists $1_{s(\gamma)}$, and $1_{t(\gamma)}$ elements in $\mathbf{G}$ which are left and right unities for $\gamma$ respectively, i.e., $1_{t(\gamma )}\circ \gamma =\gamma$, $\gamma \circ 1_{s(\gamma)} =\gamma$, for all $\gamma \in \mathbf{G}$.
3. The multiplication $\circ $ is associative: if $(\gamma \circ
\mu )\circ \nu$ is defined, then $\gamma \circ \left(
\mu \circ \nu \right) $ exists and $(\gamma \circ \mu)\circ \nu = \gamma \circ \left( \mu\circ \nu \right) $.
4. Any $\gamma $ has a two-sided inverse $\gamma ^{-1},$ with $\gamma \circ \gamma ^{-1}=1_{t(\gamma )}$ and $\gamma ^{-1}\circ \gamma
=1_{s(\gamma )}$. The map $\mathrm{inv}\colon \gamma \rightarrow \gamma ^{-1}$ is an involution, that is $\left(
\gamma ^{-1}\right) ^{-1}=\gamma $.
There is a natural equivalence relation defined in the space $\Omega$ of objects of a groupoid: $x \sim y$ iff there exists $\gamma \in \mathbf{G}$ such that $\gamma \colon x \to y$. Eleements $y\in \Omega$ equivalent to $x$ form an equivalence class denoted by $\mathcal{O}_x$. Any such equivalence class is called an orbit of $\mathbf{G}$ and $\Omega$ is the union of all these orbits.
The set $G_x = \{ \gamma \in \mathbf{G} \mid s(\gamma )= t(\gamma ) = x \}$ is a group, called the isotropy group of $x \in
\Omega$. Notice that the isotropy groups $G_x$, $G_y$, of objects $x, y$ in the same orbit are isomorphic (even though not canonically).
Two extreme cases regarding the space of objects of a groupoid occur when $\Omega$ is the whole groupoid $\mathbf{G}$, or $\Omega$ consists of just one point. In the first instance $s(\gamma ) = t(\gamma )=\gamma $ for all $\gamma$ and any $\gamma $ can be composed only with itself yielding $\gamma \circ \gamma =\gamma$. The orbits are single elements $\gamma $ and the isotropy group of any $\gamma$ is trivial, containing only $1_\gamma$, the theory is rather dull and becomes just the theory of sets. However, in the latter situation, $\Omega = \left\{ x \right\}$, so that $\mathbf{G}$ is a group, there is only one orbit $
\left\{ x \right\} $, and the isotropy group of $x$ is $\mathbf{G}$.
A groupoid $\mathbf{G}$ is called connected (or transitive) if the map $$\left( s,t \right) \colon \mathbf{G} \rightarrow \Omega \times \Omega \, , \qquad \gamma \mapsto \left(
s\left( \gamma \right) ,t\left( \gamma \right) \right) \, ,$$is onto, which is equivalent to say that $\mathbf{G}$ has only one orbit $\Omega$. Finally a groupoid $\mathbf{G}$ is called principal if $(s,t)$ is one-to-one (we will also say that $\mathbf{G}$ is the groupoid of pairs of the set $\Omega$, see the discussion below, Sect. \[sec:examples\]).
Note that, as a set, any groupoid is the disjoint union of groupoids $\mathbf{G} = \sqcup _{i}\mathbf{G}_{i}$ corresponding to the partition of $\Omega = \sqcup _{i}\mathcal{
O}_{i}$ into orbits $\mathcal{O}_{i}$. Each $\mathbf{G}_{i}$ has only one orbit and elements in $\mathbf{G}_{i}$ cannot be composed with elements in $\mathbf{G}_{k}$ whenever $k\neq i$.
Two simple examples: the groupoid of pairs and the action groupoid {#sec:examples}
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Two simple, but significative, examples of groupoids will be discussed here: the groupoid of pairs of a set and the action groupoid corresponding to the action of a group on a given space. Of course, as it was indicated before, any group $G$ is a groupoid, actually any groupoid with just one object is a group and the isotropy group of such element is the groupoid itself. However other extreme situation happens to be of great importance, specially groupoids such that the isotropy group $G_x$ is trivial for all objects. This correspond to the groupoid of pairs of a set.
### The groupoid of pairs of a set
The groupoid $\boldsymbol{\Gamma} (\Omega)$ of pairs of an arbitrary set $\Omega$ is the groupoid whose objects are the elements $x$ of the set $\Omega$ and whose morphisms are pairs $(x,y)\in \Omega \times \Omega$, that is, a groupoid element $\gamma$ is just a pair $(x,y)$; $y$ will be the source of $\gamma$ and $x$ its target, i.e., any such groupoid element could be writen as $\gamma \colon y \to x$. The composition will follow the standard usage: $(x,y) \circ (y,z) = (x,z)$. The unit morphisms are given by $1_x = (x,x)$ and the inverse of the morphisms $\gamma = (x,y)$ is $\gamma^{-1} = (y,x)$. Notice that the isotropy group of any $x \in \Omega$ is the trivial group $G_x = \{ 1_x\}$. The groupoid of pairs of a finite set $\Omega$ of $n$ elements can be drawn as the complete graph of $n$ vertices (see Fig. \[complete\_graph\]) where links represent both morphisms $(i,j)$ and $(j,i)$.
The groupoid of pairs $\boldsymbol{\Gamma} (\Omega)$ is connected as any element $y$ can be joined to any other $x$ by the morphism $(x,y)$.
### The action groupoid
The action groupoid conveys globally the intuitive notion of a groupoid as acting ‘locally’ on a set. Thus, let $\Phi \colon \Gamma \times \Omega \to \Omega$ denote an action of the group $\Gamma$ on the set $\Omega$, that is, $\Phi(e,x) = x$, $\Phi (g , \Phi(g', x)) = \Phi (gg',x)$ for all $x \in \Omega$ and $g,g' \in \Gamma$. As customary we will denote by $g\cdot x$ (or just $gx$) the action $\Phi (g,x)$ of the element $g$ on $x$. We will denote by $\mathbf{G}(\Gamma,\Omega)$ the groupoid whose morphisms are pairs $(g,x) \in \Gamma \times \Omega$, the source map given by $s(g,x) = x$ and the target map, $t(g,x) = gx$. The composition law is given by $(g',x')\circ (g,x) = (g'g,x)$ provided that $t(g,x) = gx = x' = s(g',x')$. The unit morphisms are given by $1_x = (e,x)$ and the inverse of the morphism $\gamma = (g,x)$ is given by $\gamma^{-1} = (g^{-1}, gx)$.
It is clear that the orbit $\mathcal{O}_x$ of the element $x\in \Omega$ is the set of elements $y$ which are targets of morphisms with source at $x$, that is $y = gx$, $g \in \Gamma$, in other words the standard notion of orbit of an element under the action or the group $\Gamma$.
The groupoid $\mathbf{G}(\Gamma, \Omega)$ is connected iff the action of the group is transitive. The isotropy group of any element $x$ is the set of morphisms $\gamma \in s^{-1}(x) = t^{-1}(x) = \{ g \in G \mid x = gx \}$ which agrees with the standard notion of isotropy group.
It is remarkable that given any subset $S \subset \Omega$, the restriction of the groupoid $\mathbf{G}(\Gamma, \Omega)$ to $S$, that is the set of pairs $(g,x) \in \Gamma \times S$ such that $gx \in S$, is a subgroupoid of $\mathbf{G}(\Gamma, \Omega)$ which, in general, is not the action groupoid of any group acting on $S$.
The groupoid algebra and representations of finite groupoids {#sec:groupoid_algebra}
------------------------------------------------------------
We illustrate the previous ideas by considering finite groupoids and their representations. Thus we will consider a finite groupoid $\mathbf{G}$ of order $N = |\mathbf{G}|$. Thus we can label the morphisms of $\mathbf{G}$ as $\gamma_k$: $$\mathbf{G} =\left\{ \gamma _{k}\mid k = 1\ldots, N\right\} \, .$$ Necessarily the space of objects $\Omega$ will be finite. Denoting by $n$ the number of elements of $\Omega$ we can label them as $\{ x_a \mid a = 1, \ldots, n \}$, $n = |\Omega|$.
The groupoid algebra $\mathbb{C}[\mathbf{G}]$ of the finite groupoid $\mathbf{G}$ is the associative algebra generated by the elements of the groupoid with the corresponding natural composition law, that is, if $\mathbf{a} = \sum_{k} a_k \gamma_k$ and $\mathbf{b} = \sum_{l} b_l \gamma_l$, $a_k,b_l \in \mathbb{C}$, are two finite formal linear combination of elements in $\mathbf{G}$ (hereafter any summation label $j,k$,... ranges from $1$ to $N$), we define its product as: $$\mathbf{a} \cdot \mathbf{b} = \sum_{k,l} a_k\, b_l \, \, \delta(\gamma_k,\gamma_l)\, \, \gamma_k\circ \gamma_l \, ,$$ with the indicator function $ \delta(\gamma_k,\gamma_l)$ defined to be $1$ if $\gamma_k,\gamma_l$ can be composed and zero otherwise. The product defined in this way is clearly associative because the composition law $\circ$ is associative. The groupoid algebra $\mathbb{C}[\mathbf{G}]$ has a unit given by $1 = \sum_a 1_{x_a}$.
Notice that the canonical basis of the groupoid algebra provided by the elements $\gamma_k$ of the groupoid allow to identified the groupoid algebra $\mathbb{C}[\mathbf{G}]$ with the algebra of complex valued functions $\mathcal{F}\left( \mathbf{G}\right) = \{ f \colon \mathbf{G} \to \mathbb{C}\}$ on the groupoid with the convolution product: $$\left( f_{1}\ast f_{2}\right) (\gamma _{i})=\sum_{\gamma
_j \circ \gamma_k = \gamma_i} f_{1}(\gamma _{j})f_{2}(\gamma _{k}) \, ,$$with $f_1, f_2 \colon \mathbf{G} \to \mathbb{C}$ any two such functions. The identification is provided by the correspondence $\mathbb{C}[\mathbf{G}] \to \mathcal{F}(\mathbf{G})$ defined by $\mathbf{a}\mapsto f_\mathbf{a}$, with the function $f_\mathbf{a}$ defined by $f_\mathbf{a} (\gamma_k) = a_k$. The converse map being defined as $f \mapsto \mathbf{a}_f = \sum_k f(k) \gamma_k$. Notice that clearly: $$f_{\mathbf{a}} \ast f_{\mathbf{b}} = f_{\mathbf{a}\cdot \mathbf{b}} \, ,$$ and $$\mathbf{a}_f \cdot \mathbf{a}_g = \mathbf{a}_{f \ast g} \, .$$
The functions $\delta
_{\gamma _{j}},$ defined as$$\delta _{\gamma _{j}}(\gamma _{k})=\left\{
\begin{array}{c}
1\ \mathrm{if}\ \gamma _{j}=\gamma _{k} \\
0\ \ \mathrm{if}\ \gamma _{j}\neq \gamma _{k}\end{array}\right. ,$$determine a basis of the groupoid algebra. Thus for any $ f\in \mathcal{F}\left( \mathbf{G}\right)$, we may write: $$f =\sum\limits_{k}f\left( \gamma _{k}\right) \delta
_{\gamma _{k}} \, .$$Moreover: $$\left( \delta _{\gamma _{j}}\ast f\right) (\gamma _{i})=\sum\limits_{\gamma _{j}\circ \gamma _{k}=\gamma _{i}} f(\gamma _{k}).$$In particular $\delta _{\gamma _j}\ast \delta _{\gamma _h}$ is $1$ on $\gamma _{j}\circ \gamma _{k}$ if $\gamma _{j}$ and $\gamma _{k}$ are composable and $0$ elsewhere; so $$\label{gammajk}
\delta _{\gamma _{j}}\ast \delta _{\gamma _{h}}=\delta _{\gamma _{j}\circ
\gamma _{k}} = \delta(\gamma_j, \gamma_k) \, .$$ The groupoid algebra $\mathbb{C}[\mathbf{G}]$ carries also an involution operator $*$ defined as $\mathbf{a}^* = \sum_k \bar{a}_k \gamma_k^{-1}$ for any $\mathbf{a} = \sum_k a_k \gamma_k$, or, in terms of the isomorphic algebra of functions: $$f^* = \sum_k \overline{f(\gamma_k^{-1})} \delta_{\gamma_k} \, ,$$ for any $f = \sum_k f(\gamma_k) \delta_{\gamma_k}$.
A linear representation of a groupoid $\mathbf{G}$ is a functor $\rho \colon \mathbf{G} \to \mathbf{Vect}$, that is, the functor $\rho$ assigns to any object $x \in \Omega$ a linear space $\rho(x) = V_x$, and to any morphism $\gamma \colon x \to y$, a linear map $\rho(\gamma) \colon V_x \to V_y$ such that $\rho(1_x) = \mathrm{id}_{V_x}$ and $\rho(\gamma \circ \gamma' ) = \rho(\gamma) \rho(\gamma')$ for any $x \in \Omega$ and any composable pair $\gamma, \gamma'$. Thus the notion of linear representation of groupoids extends in a natural way the theory of linear representations of groups.
Notice that given a finite groupoid $\mathbf{G}$ there is a natural identification between linear representations $\rho$ of the groupoid and $\mathbb{C}[\mathbf{G}]$-modules. The correspondence is established as follows. Let $\rho$ be a representation of $\mathbf{G}$, then we define the linear space $V = \bigoplus_a V_{x_a}$ and the map $R \colon \mathbb{C}[\mathbf{G}] \to \mathrm{End}(V)$ as $R(\mathbf{a}) (v) = \sum_k a_k \rho(\gamma_k)(v)$, with $\rho(\gamma_k) (v) = \rho(\gamma_k) (v_a)$ if $\gamma_k \colon x_a \to x_b$ and $v = \oplus_a v_a$. The map $R$ is clearly a homomorphism of algebras, hence we may consider the linear space $V$ as a (left-) $\mathbb{C}[\mathbf{G}]$-module.
The converse of this correspondence is obtained by defining the subspaces $V_a$ of the $\mathbb{C}[\mathbf{G}]$-module $V$ by means of the projectors $P_a = R(1_{x_a})$, that is $V_a = P_a(V)$. Then $\rho(x_a) = V_a$ and $\rho(\gamma_k) (v_a) = R(\gamma_k)(v_a)$ if $\gamma_k \colon x_a \to x_b$.
Any finite groupoid $\mathbf{G}$ possesses two canonical representations: the fundamental and the regular representations. We will describe briefly both of them in what follows.
### The fundamental representation of a finite groupoid {#sec:fundamental}
Given the finite groupoid $\mathbf{G}$ with object space $\Omega$, we define the Hilbert space $\mathcal{H}_\Omega$ as the complex linear space generated by the elements $x \in \Omega$ with inner product: $$\langle \phi, \psi \rangle = \sum_{a = 1}^n \bar{\phi}_a \psi_a \, ,$$ with $\phi = \sum_a \phi_a |x_a\rangle$, $\phi_a \in \mathbb{C}$, and where we have indicated by $| x \rangle$ the vector associated with the element $x \in \Omega$. Notice that with this definition $\langle y, x \rangle = \delta_{xy}$ and the set of vectors $|x_a\rangle$ form an orthonormal basis of $\mathcal{H}_\Omega$. Again, using the previous notation $\mathcal{H}_\Omega = \bigoplus _{a= 1}^n
\mathbb{C} | x_a \rangle$.
The fundamental representation of $\mathbf{G}$ assigns to any object $x\in \Omega$ the linear space $\pi(x) = \mathbb{C}|x \rangle$ and to any groupoid element $\gamma \colon x \to y$, the linear map $\pi(\gamma) \colon \pi(x) \to \pi(y)$, given by $\pi(\gamma) |x\rangle = |y\rangle$.
Because of the one-to-one correspondence between linear representations of groupoids and modules, we may define the fundamental representation by the map $\pi \colon \mathbb{C}[\mathbf{G}] \to \mathrm{End}(\mathcal{H}_\Omega)$, that provides such module structure, given by: $$\pi(\mathbf{a})\phi = \sum_{k,b} a_k\phi_b \, \pi(\gamma_k)|x_b \rangle \, .$$ Introducing the indicator symbol $\delta (\gamma_k,x_b)$ defined as 1 if $s(\gamma_k) = x_b$ and zero otherwise, we can write the previous equation as: $$\pi(\mathbf{a})\phi = \sum_{k,b} a_k\phi_b \, \, \delta (\gamma_k,x_b) \, |t(\gamma_k) \rangle \, .$$ Notice that the fundamental representation is a $*$-representation, that is, $\pi(f^*) = \pi(f)^\dagger$ where $\pi(f)^\dagger$ denotes the adjoint operator with respect to the inner product structure in $\mathcal{H}_\Omega$ (notice that $\langle y , \pi(\gamma) x \rangle = \langle \pi(\gamma^{-1})y , x \rangle$ if $\gamma \colon x \to y$).
The fundamental representation allows us to introduce a natural norm on the groupoid algebra as: $$\label{norm}
|| f || = || \pi(f) || \, , \qquad f \in \mathbb{C}[\mathbf{G}] \, .$$ where the norm in the r.h.s. of Eq. (\[norm\]) is the operator norm. Then it is trivial to check that $|| f^* f || = || f ||^2$, which means that the groupoid algebra $\mathbb{C}[\mathbf{G}]$ is a $C^*$-algebra. The construction of a $C^*$- groupoid algebra can be done in general by selecting a family of left-Haar measures on $\mathbf{G}$ (see for instance [@Renault] for details).
### The regular representation of a groupoid
A representation $R\colon \mathcal{F}(\mathbf{G}) \to \mathrm{End}(\mathcal{F}(\mathbf{G}))$ of the groupoid algebra $\mathbb{C}[\mathbf{G}]$ (identified with $\mathcal{F}(\mathbf{G})$) on its space of functions, is obtained immediately by using the formula: $$R(f) = \sum\limits_{k}f\left( \gamma _{k}\right) D_{\gamma _{k}} \, .
\label{D quant}$$where $$D_\gamma (\cdot) =\delta _{\gamma }\ast \cdot \, , \label{deltagr}$$because from Eq. (\[gammajk\]), we get: $$D_{\gamma _j} D_{\gamma _k} = \left\{
\begin{array}{cl}
D_{\gamma _j\circ \gamma _k} & \mathrm{\,\, if\,}\gamma _{j}\circ
\gamma _{k}\mathrm{\,\, exists,} \\
0 & \mathrm{\,\, if\, }\gamma _{j}\circ \gamma _{k}\mathrm{\,\,
does~not~exist.}\end{array}\right.$$Notice that, consistently, we get: $$\begin{aligned}
R(f_{1}) R(f_{2}) &=&\sum_{j,k}f_{1}(\gamma _{j})f_{2}(\gamma _{k})D_{\gamma
_j \circ \gamma _k} = \sum_{\overset{i,j,k}{{\gamma _{j}\circ \gamma }_{k}{=\gamma }_{i}}}f_{1}(\gamma _{j})f_{2}(\gamma _{k})D_{\gamma _i} \\
&=&\sum_{i}\left( f_{1}\ast f_{2}\right) \left( \gamma _{i}\right) D_{\gamma
_i} =R(f_{1}\ast f_{2}). \notag\end{aligned}$$In other terms, the product of operators corresponds to the convolution product of the associated functions. The representation $R$ will be called the (left) regular representation of the groupoid algebra (there is a similar definition of the groupoid algebra acting as a right-module on the space of functions).
Notice that because the groupoid is finite we may identify the space of functions on it with the space of square integrable functions with respect to the natural inner product defined by the standard basis $\delta_\gamma$. In that case it is easy to check again that the regular representation is a $\ast$-representation.
2-groupoids and quantum systems {#sec:systems}
===============================
The inner groupoid structure: events and transitions {#sec:transitions}
----------------------------------------------------
Given a physical system let us denote by $\mathcal{E}$ an ensemble associated with it, that is a large family of physical systems of the same type and satisfying the same specified conditions [@Es99]. The elements $S$ of the ensenble $\mathcal{E}$, that is, individual systems, are of course noninteracting.
It will be assumed that there is a family of observables $\mathcal{A}$ representing measurable physical quantities, and that the outcomes $a \in \mathbb{R}$ of their individuals $A \in \mathcal{A}$ can be obtained by performing physical measurements on elements $S$ of the ensemble $\mathcal{E}$ (ideally to each $A$ corresponds one particular device by means of which the measurement is made). Any such specific measurement will be denoted as $a = \langle A:S\rangle$ (our notation) and it clearly supposes an idealized simplification of a full fledged statistical interpretation of a quantum mechanical picture.
It will be assumed that the ensemble $\mathcal{E}$ is large enough so that for any observable $A$ and for each possible outcome $a$ of the observable there are elements $S$ of the ensemble such that when $A$ is measured on them the outcome is actually $a$. Under these conditions we will say that the ensemble $\mathcal{E}$ is *sufficient* for the observable $A$[^4].
The measurements we are referring to are assumed to be non-destructive, that is, the act of measuring the observable $A$ on $S$ is separated by the act of registering the outcome. Essentially, if we think to the Stern-Gerlach experiment, we are excluding from the experimental apparatus the screen which the silver atoms hit after experiencing the magnetic field. This instance allows us to define the notion of compatible observables as follows. Two observables $A,B \in \mathcal{A}$ will be said to be compatible if the outcomes of their respective measurements are not affected by the outcomes of the other. Alternatively we may say that their outcomes do not depend on the order in which they are performed. If we denote by $\langle A,B:S\rangle$ the outcome $(a,b)$ obtained of the measurement of the ordered pair of observables $A$ and $B$ on the element $S$ of the ensemble, that is, first $B$ is measured and, after a negligible amount of time, $A$ is measured (notice that the causal relations between the corresponding physical measurement actions depend on the observer performing them), then we will say that the observables $A,B$ are compatible if the outcome of $\langle B,A:S\rangle$ is $(b,a)$. A family of observables $\mathbf{A} = \{ A_1, \ldots, A_N\} \subset \mathcal{A}$ is said to be compatible[^5] if they are compatible among them or, in other words, if the outcomes of their measurements do not depend on the order in which the measurements are performed.
The formulation presented in what follows is inspired by Schwinger’s construction of the “algebra of measurements” [@Sc59], where the starting point of the description of a quantum system is the selection of a family $\mathbf{A}$ of compatible observables, e.g., the $z$-projection of the spin for a two-level system. The outcomes of a measurement of such observables will be called in what follows just ‘outcomes’ or ‘events’[^6], and we will denote them as $a,a',a'',\ldots$, etc.
At this stage, and in what follows, we will not try to make precise the meaning of ‘measurement’ or the nature of the outcomes as we will consider them primary notions determined solely by the experimental setting used to study our system. Neither will we require any particular algebraic structure for the family of observables used in such setting. For instance, the outcomes $a, a', a'',\ldots$, could be just collections of real numbers. We will just be concerned with the structural relations among the various notions that are introduced. A concrete realisation of them will be offered in the next section by adapting Schwinger’s framework to the language developed here.
In the incipit of his first note [@Sc59], Schwinger writes:
*“The classical theory of measurement is implicitly based upon the concept of an interaction between the system of interest and the measuring apparatus that can be made arbitrarily small, or at least precisely compensated, so that one can speak meaningfully of an idealized experiment that disturbs no property of the system. The classical representation of physical quantities by numbers is the identification of all properties with the results of such non-disturbing measurements. It is characteristic of atomic phenomena, however, that the interaction between system and instrument cannot be indefinitely weakened. Nor can the disturbance produced by the interaction be compensated precisely since it is only statistically predictable. Accordingly, a measurement on one property can produce unavoidable changes in the value previously assigned to another property, and it is without meaning to ascribe numerical values to all the attributes of a microscopic system. The mathematical language that is appropriate to the atomic domain is found in the symbolic transcription of the laws of microscopic measurement."*
Inspired by this remark, we postulate that for a given physical system there are transitions among the outcomes of measurements, that is, the outcome $a$ of the measurement of an observable in $\mathbf{A}$ is compatible with other different values of other observables in $\mathbf{A}$ before the act of measurement. Such transitions are determined completely by the intrinsic dynamic of the system and by the interaction of the experimental setting with it.
Consistently with the notation introduced for groupoids, we will denote such transitions using a diagrammatic notation as $\alpha \colon a \to a'$, and we will say that the event $a$ is the source of the transition $\alpha$ and the event $a'$ is its target. We will also say that the transition $\alpha$ transforms the outcome $a$ into the outcome $a'$.
The allowed physical transitions $\alpha$ must satisfy a small number of natural requirements or axioms. The first one is that transitions can be composed, that is, if $\alpha \colon a \to a'$, and $\beta\colon a' \to a''$ denote two allowed transitions, there is a transition $\beta \circ \alpha \colon a \to a''$. Notice that not all transitions can be composed, we may only compose compatible transitions[^7], that is, transitions $\alpha$, $\beta$ such that $\beta$ transforms the target event $a'$ of $\alpha$. This composition law must be associative, that is, if $\alpha$, $\beta$ are transitions as before, and $\gamma \colon a'' \to a'''$, then $$(\gamma \circ \beta) \circ \alpha = \gamma \circ (\beta \circ \alpha ) \, .$$ Moreover we will assume that there are trivial transitions, that is, transitions $1_a \colon a \to a$ such that $1_{a'} \circ \alpha = \alpha$ and $\alpha \circ 1_a = \alpha$ for any transition $\alpha \colon a \to a'$. The physical meaning of transitions $1_a$ is that of manipulations of the system which do not change the outcome of any measurement of $\mathbf{A}$.
Eventually, we will assume a (local) reversibility property of physical systems, that is, transitions are invertible: given any transition $\alpha \colon a \to a'$, there is another one $\alpha' \colon a' \to a$ such that $\alpha \circ \alpha' = 1_{a'}$ and $\alpha' \circ \alpha = 1_a$. We will denote such transition as $\alpha^{-1}$ and it clearly must be unique[^8].
From the previous axioms, it is clear that the collection of transitions $\alpha$ and events $a$ form a groupoid as explained in Sect. \[sec:groupoids\]. Such groupoid will be denoted by $\mathbf{G}_{\mathbf{A}}$, its objects can be understood as the outcomes $a$ provided by measurements of a family of compatible observables $\mathbf{A}$, and its morphisms are the allowed physical transitions among events. We will denote by $\mathbf{G}_\mathbf{A}(a,a')$ the family of transitions $\alpha \colon a \to a'$ between the events $a$ and $a'$.
We will denote by $\Omega$ the collection of all outcomes, and by $s,t$ the natural source and target maps. Notice that so far we are not assuming that the groupoid $\mathbf{G}_{\mathbf{A}}$ has any particular additional property, that is, it could be connected or not, it could possess an Abelian isotropy group or the isotropy group could be trivial (see later on, and Sect. \[sec:examples\] for a few concrete instances).
We will associate with the system a Hilbert space $\mathcal{H}_{A}$. Such Hilbert space $\mathcal{H}_{A}$ is the support of the fundamental representation of the groupoid $\mathbf{G}_\mathbf{A}$. In Sect. \[sec:fundamental\], the case of finite groupoids was discussed. There, the Hilbert space was finite-dimensional and given explicitly as: $$\mathcal{H}_{A} = \bigoplus_{a \in \Omega} \mathbf{C} |a \rangle .$$ In a more general situation, we will assume that the space of events $\Omega$ is a standard Borel space with measure $\mu$. In that case, $\mathcal{H}_{A}$ is the direct integral of the field of Hilbert spaces $\mathbb{C} |a \rangle$: $$\mathcal{H}_{A} = \int_{\Omega}^\oplus \mathbb{C}\, | a \rangle \, d\mu (a) \cong L^2(\Omega, \mu) \, .$$ For instance, in the particular case when $\Omega$ is discrete countable, such measure will be the standard counting measure and $\mathcal{H}_{A}\cong L^2(\mathbb{Z}) = l^2$. Notice that if the space of objects is countable, the unit elements $1_a$ are represented in the fundamental representation as the orthogonal projectors $P_a$ on the subspaces $\mathbb{C} | a \rangle$. Such projectors provide a resolution of the identity for the Hilbert space $\mathcal{H}_{A}$.
The Hilbert space $\mathcal{H}_A$ will allow us to relate the picture provided by the groupoid $\mathbf{G}_\mathbf{A}$ with the standard Dirac-Schrödinger picture and, in the particular instance of a discrete, finite space of events considered by Schwinger, it becomes a finite dimensional space corresponding to a finite-level quantum system.
All the previous arguments can be repeated when considering another system of observables, say $\mathbf{B}$, to describe the transitions of the system. The system $\mathbf{B}$ may be incompatible with $\mathbf{A}$, however, they may have common events, that is events that are outcomes of both $\mathbf{A}$ and $\mathbf{B}$. Thus, in addition to the transitions $\alpha \colon a \to_A a'$ among outcomes of the family $\mathbf{A}$, and $\beta \colon b \to_B b'$ among the outcomes of the family $\mathbf{B}$, new transitions could be added to the previous ones, i.e., those of the form $\gamma = \alpha\circ \beta \colon a \to_A a' = b \to_B b'$ where $a' = b$ is a common outcome for both $\mathbf{A}$ and $\mathbf{B}$.
We may form the groupoid $\mathbf{G}$ consisting of the groupoid generated by the union of all groupoids $\mathbf{G}_{\mathbf{A}}$ over the total space of events $\mathcal{S}$. Each groupoid $\mathbf{G}_\mathbf{A}$ is a subgroupoid of the groupoid $\mathbf{G}$. The construction of $\mathbf{G}$ depends on the possibility of giving a description of a physical systems in terms of two or more maximal families of compatible observables. This instance naturally carries with it the possibility of relating different descriptions, and this, in turns, will lead us to the construction of another layer on our abstract groupoid structure. Specifically, we will build another groupoid $\boldsymbol{\Gamma}$ over $\mathbf{G}$ obtaining what is known as a 2-groupoid. If the groupoid $\mathbf{G}$ describing the system is connected or transitive, we will say that there are no superselection rules. The connected components of the groupoid will determine the different sectors of the theory. In what follows, we will restrict ourselves to consider a connected component of the total groupoid.
Two examples: the qubit and the groupoid of tangles
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### The qubit
Let us start the discussion of some simple examples by considering what is arguably the simplest non-trivial groupoid structure. We call it the extended singleton or groupoid $\mathbf{A}_2$ (see [@Ib19 Chap. 1] for more details), and it is given by the diagram in Fig. \[singleton\] below:
(0,0) circle (0.1); (3.5,0) circle (0.1); (0.3,0.1) arc (107:73:5); (3.2,-0.1) arc (288:253:5); at (-0.1,0) [$+$]{}; at (3.9,0) [$-$]{}; at (2,0.6) [$\alpha$]{}; at (2,-0.6) [$\alpha^{-1}$]{};
This diagram will correspond to a physical system described by a family of observables $\mathbf{A}$ producing just two outputs denoted respectively by $+$ and $-$, and with just one transition $\alpha \colon + \to -$ among them. Notice that the groupoid $\mathbf{G}_{\mathbf{A}}$ associated with this diagram has 4 elements $\{ 1_+, 1_-, \alpha, \alpha^{-1}\}$ and the space of events is just $\Omega_A = \{+,-\}$. The corresponding (non-commutative) groupoid algebra is a complex vector space of dimension 4 generated by $e_1= 1_+$, $e_2= 1_-$, $e_3 = \alpha$ and $e_4= \alpha^{-1}$ with structure constants given by the relations: $$\begin{array}{llll}
e_1^2 = e_1\, , & e_2^2 = e_2 \, , & e_1 e_2 = e_2 e_1 = 0 \, , & e_3 e_4 = e_2\, , \\
e_4 e_3 = e_1\, , & e_3 e_3 = e_4 e_4 = 0\, , & e_1e_3 = 0\, , & e_3e_1 = e_1\, , \\
e_4 e_1 = 0\, , & e_1 e_4 = e_4 \, , & e_3e_2 = 0 \, , & e_2e_3 = e_3 \, .
\end{array}$$ The fundamental representation of the groupoid algebra is supported on the 2-dimensional complex space $\mathcal{H} = \mathbb{C}^2$ with canonical basis $|+\rangle$, $|-\rangle$. The groupoid elements are represented by operators acting on the canonical basis as: $$A_+ |+\rangle = \pi (1_+) |+\rangle = |+\rangle \, , \qquad A_+|-\rangle = \pi(1_+) |- \rangle = 0 \, ,$$ etc., that is, for instance, the operator $A_+$ has associated matrix: $$A_+ = \left[ \begin{array}{cc} 1 & 0 \\ 0 & 0\end{array}\right] \, .$$ Similarly we get: $$A_- = \pi(1_-) = \left[ \begin{array}{cc} 0 & 0 \\ 0 & 1\end{array}\right] \, , \quad A_\alpha = \pi(\alpha) = \left[ \begin{array}{cc} 0 & 0 \\ 1 & 0\end{array}\right] \, , \quad A_{\alpha^{-1}} = \pi(\alpha^{-1}) = \left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}\right] \, ,$$ Therefore, the groupoid algebra can be naturally identified with the algebra of $2\times 2$ complex matrices $M_2(\mathbb{C})$ and the fundamental representation is just provided by the matrix-vector product of matrices and 2-component column vectors of $\mathbb{C}^2$. The dynamical aspects of this system will be described extensively in [@Ib18].
Before discussing the next example it is interesting to observe that if we consider the system without the transition $\alpha$, that is, now the groupoid will consists solely of the elements $\{ 1_+, 1_-\}$, its corresponding groupoid algebra will be the just the 2-dimensional Abelian algebra defined by the relations: $e_1^2 = e_1$, $e_2^2 = e_2$ and $e_1e_2 =e_2 e_1 = 0$, that is, the classical bit.
According to the physical interpretation of transitions given at the beginning of this section, by disregarding the transitions $\alpha$ and $\alpha^{-1}$ we are implicitely assuming that experimental devices do not influence the system, which is an assumption of genuinely classical flavour.
### The homotopy groupoid
An interesting family of groupoids have its origin in topology. Consider a closed (compact without boundary), connected smooth manifold $X$ and the groupoid $\mathbf{G}(X)$ of unparametrised, oriented, piecewise smooth maps $\gamma \colon [0,1] \to X$, that will be called oriented histories on $X$. In other words, a morphism $[\gamma]$ in $\mathbf{G}(X)$ is an equivalence class of piecewise smooth maps up to reparametrizations by positive changes of parameter $dt/ds > 0$. The source and target maps $s,t \colon \mathbf{G}(X) \rightrightarrows X$ are defined as $s([\gamma]) = \gamma(0)$ and $t([\gamma]) = \gamma(1)$. In what follows, we will just denote by $\gamma$ the equivalence class $[\gamma]$. The unit morphisms are defined by the curves $1_x (t) = x$ for all $t\in [0,1]$. The composition law is given by the standard composition of paths, that is $\gamma_1 \circ \gamma_2 (t) = \gamma_1(2t)$, $t\in [0,1/2)$ and $\gamma_1 \circ \gamma_2 (t) = \gamma_2(2t-1)$, $t\in [1/2,1]$.
The groupoid of oriented histories $\mathbf{G}(X)$, used in [@Za83] to provide a universal setting to describe Lagrangian systems with topological obstructions, happens to be too large for the purposes of Topology and is drastically reduced by introducing an equivalence relation on it. A transformation $\varphi \colon \gamma \Rightarrow \gamma'$ between two oriented histories is provided by ambient isotopies, that is, maps $\varphi \colon [0,1]\times M \to X$ such that $\varphi(0,\gamma(t)) = \gamma(t)$, $\varphi(1,\gamma(t)) = \gamma'(t)$, $\varphi(s,t )$ is smooth in the variable $s$ and piecewise smooth on the variable $t$, and $\varphi(s,\cdot)$ is a diffeomorphism for every $s\in [0,1]$.
The notion of the transformation $\varphi$ between groupoid elements will be used in an abstract setting in the following. In the context of the present example, it suffices to use it to introduce an equivalence relation in the groupoid of oriented histories, whose quotient space is a groupoid over $X$, called the homotopy or Poincaré groupoid, whose isotropy group at $x\in X$ are isomorphic to Poincaré’s homotopy groups $\pi_1(X,x)$ (notice that the Poincaré groupoid is connected iff the space $X$ is connected).
Both the groupoid algebra and the fundamental representation of the groupoid of oriented histories $\mathbf{G}(X)$ are hard to describe. The groupoid algebra can be described as the completion with respect to an appropriate norm of the algebra of finite formal linear combinations of oriented histories $\boldsymbol{\gamma} = \sum c_\gamma \gamma$. Assuming that $X$ is orientable choosing an auxiliary volume form on $X$, we may construct a measure on it and define the Hilbert space $\mathcal{H}$ of square integrable functions on $X$. The fundamental representation of this algebra will be suported on the space of distributions on $X$ by means of $\pi(\gamma) (\delta_x) = \delta_y$ if the oriented history $\gamma$ takes $x$ into $y$, where $\delta_x$ is the Dirac’s delta distribution at $x$.
More interesting is the natural generalization of the groupoid $\mathbf{G}(X)$ provided by the groupoid of braids on $X\times [0,1]$, that is, the quotient with respect to the same generalized homotopy equivalence relation of the space of $n$ non-intersecting oriented histories with end points on the boundary $\partial (X\times [0,1]) = X \times \{ 0,1\}$. The corresponding quotient space $\mathbf{B}_n(X)$ with respect to ambient isotopies is again a groupoid over $X^n$. The fundamental representation of this groupoid is supported in the Hilbert space $L^2(X)^{\otimes n}$, and it provides relevant information on the statistics of the system described by it. These and other aspects of the use of groupoids in the description of variational principles in quantum mechanics will be described elsewhere.
The 2-groupoid structure: transformations {#sec:transformations}
-----------------------------------------
The dynamical behaviour of a system is described by a sequence of transitions $w = \alpha_1\alpha_2\cdots \alpha_r$ that will be called histories. In a certain limit[^9], any history would define a one-parameter family $\alpha_t$ of transitions (but we may very well keep working with discrete sequences).
Typically, once the family of observables has been equipped with an algebra structure, these sequences of transitions are generated by a given observable promoted to be the infinitesimal generator of a family of automorphisms. However, we will not enter here in the discussion of this dynamical notions that will be the main subject of [@Ib18]. What we would like to stress here is that the explicit expression of such sequence of transitions depends on the complete measurements chosen to describe the behaviour of the system and it may look very different when observed using two different systems of observables $\mathbf{A}$ and $\mathbf{B}$.
The existence of such alternative descriptions imply the existence of families of ‘transformations’ among transitions that would allow to compare the descriptions of the dynamical behaviour of the system (and its kinematical structure as well) when using different measurements systems. We will also use a diagrammatic notation to denote transformations: $\varphi \colon \alpha \Rightarrow \beta$, or graphicallys as in Fig. \[transformations\] below.
(0.2,0) circle (0.1); (3.5,0) circle (0.1); (0.3,0.2) arc (140:40:2); (0.3,-0.2) arc (40:140:-2); at (1.9,1.3) [$\alpha$]{}; at (1.9,-1.3) [$\beta$]{}; at (1.9,0) [$\bigg\Downarrow$]{}; at (2.3,0) [$\varphi$]{};
The transformations $\varphi$ must satisfy some obvious axioms. First, in order to make sense of the assignment $\alpha \Rightarrow \beta$, the transformation $\varphi$ must be compatible with the source and target maps of the groupoid or, in other words, $\varphi$ must map $\mathbf{G}_\mathbf{A}(a,a')$ to $\mathbf{G}_\mathbf{B}(b,b')$. Moreover, the transformations $\varphi\colon \alpha \Rightarrow \beta$ and $\psi\colon \beta \Rightarrow \gamma$ could be composed providing a new transformation $\varphi\circ_v \psi \colon \alpha \Rightarrow \gamma$, from transition $\alpha$ to $\gamma$. This composition law will be called the ‘vertical’ composition law and denoted accordingly by $\circ_v$ (see Fig. \[vertical\_horizontal\](a) for a diagrammatic representation of the vertical composition of transformations). This vertical composition law of transformations must be associative as there is no a preferred role for the various arrangements of compositions between the various transformations involved. That is, we postulate: $$\varphi \circ_v (\psi \circ_v \zeta) = (\varphi \circ_v \psi) \circ_v \zeta \, ,$$ for any three transformations $\varphi \colon \alpha \Rightarrow \beta$, $\psi \colon \beta \Rightarrow \gamma$, $\zeta \colon \gamma \Rightarrow \delta$.
The transformation $\varphi$ sends the identity transition $1_a$ into the identity transition $1_b$, and there should be transformations $1_\alpha$, $1_\beta$ such that $1_\alpha \circ_v \varphi = \varphi$ and $\varphi \circ_v 1_\beta = \varphi$.
Moreover, it will be assumed that transformations $\varphi \colon \alpha \Rightarrow \beta$ are reversible provided that the physical information determined by using the family of observables $\mathbf{A}$, that is, the groupoid $\mathbf{G}_\mathbf{A}$, is equivalent to that provided by the groupoid $\mathbf{G}_\mathbf{B}$, i.e., by the family of observables $\mathbf{B}$. In other words, the transformations $\varphi \colon \alpha \Rightarrow \beta$ are invertible because there is no natural preference among the corresponding measurements systems[^10]. Thus, under such assumption, for any transformation $\varphi \colon \alpha \Rightarrow \beta$ there exists another one $\varphi^{-1} \colon \beta \Rightarrow \alpha$ such that $\varphi\circ_v \varphi^{-1} = 1_\alpha$ and $\varphi^{-1}\circ_v \varphi = 1_\beta$.
Furthermore, notice that if we have a transformation $\varphi \colon \alpha \Rightarrow \beta$ and another one $\varphi' \colon \alpha' \Rightarrow \beta'$ such that $\alpha$ and $\alpha'$ can be composed, then $\beta$ and $\beta'$ will be composable too because of the consistency condition for transformations, that is, given two transitions $\alpha$, $\alpha'$ that can be composed, if the reference description for them is transformed, then the corresponding description of the transitions $\beta$ and $\beta'$ will be composable too and their composition must be the composition of the transformation of the original transitions (see Fig. \[vertical\_horizontal\](b) for a diagrammatic description). In other words, there will be a natural transformation between the transition $\alpha \circ \alpha'$ to the transition $\beta\circ \beta'$ that will be denoted by $\varphi \circ_h \varphi'$ and called the ‘horizontal’ composition. Figure \[vertical\_horizontal\] provides a diagrammatic representation of both operations $\circ_v$ and $\circ_h$:
(0,0) circle (0.1); (3.3,0) circle (0.1); (0.1,0.2) arc (140:40:2); (0.1,-0.2) arc (40:140:-2); at (1.7,1.3) [$\alpha$]{}; at (1.7,-1.3) [$\gamma$]{}; at (1.7,0) [$\bigg\Downarrow$]{}; at (2.4,0) [$\varphi\circ_v \psi$]{};
at (3.8,0) [$\cong$]{}; at (3.9,-2.7) [(a) Diagrammatic representation of the vertical composition law $\circ_v$.]{};
(4.2,0) circle (0.1); (7.5,0) circle (0.1); (4.3,0.2) arc (160:20:1.7); (4.3,-0.2) arc (20:160:-1.7); (4.4,0) – (7.2,0);
at (5.9,1.6) [$\alpha$]{}; at (5.9,-1.6) [$\gamma$]{}; at (5.9,0.7) [$\Big\Downarrow$]{}; at (6.3,-0.7) [$\beta$]{}; at (5.9,-0.7) [$\Big\Downarrow$]{}; at (6.3,0.7) [$\varphi$]{};
(-1.4,0) circle (0.1); (1.9,0) circle (0.1); (-1.3,0.2) arc (140:40:2); (-1.3,-0.2) arc (40:140:-2); at (0.4,1.3) [$\alpha\circ \alpha'$]{}; at (0.4,-1.3) [$\beta\circ\beta'$]{}; at (0,0) [$\bigg\Downarrow$]{}; at (1,0) [$\varphi\circ_h\varphi'$]{};
at (2.6,0) [$\cong$]{}; at (4.1,-2.7) [(b) Diagrammatic representation of the horizontal composition law $\circ_h$.]{};
(3.2,0) circle (0.1); (6.5,0) circle (0.1); (3.3,0.2) arc (140:40:2); (3.3,-0.2) arc (40:140:-2); at (4.9,1.3) [$\alpha$]{}; at (4.9,-1.3) [$\beta$]{}; at (4.9,0) [$\bigg\Downarrow$]{}; at (5.3,0) [$\varphi$]{};
(6.5,0) circle (0.1); (9.8,0) circle (0.1); (6.6,0.2) arc (140:40:2); (6.6,-0.2) arc (40:140:-2); at (8.2,1.3) [$\alpha'$]{}; at (8.2,-1.3) [$\beta'$]{}; at (8.2,0) [$\bigg\Downarrow$]{}; at (8.6,0) [$\varphi'$]{};
It is clear that the horizontal composition law is associative too, i.e., $$\varphi \circ_h (\varphi' \circ_h \varphi'') = (\varphi \circ_h \varphi') \circ_h \varphi'' \, ,$$ for any three transformations $\varphi \colon \alpha \Rightarrow \beta$, $\varphi' \colon \alpha' \Rightarrow \beta'$, $\varphi'' \colon \alpha'' \Rightarrow \beta''$ such that $\alpha \colon a \to a'$, $\beta \colon b \to b'$, $\alpha' \colon a' \to a''$, $\beta'\colon b' \to b''$ and $\alpha'' \colon a'' \to a'''$, $\beta'' \colon b'' \to b'''$. The horizontal composition rule has natural units, that is, if $\varphi \colon \alpha \Rightarrow \beta$, then the transformation $1_{a'b'} \colon 1_{a'} \Rightarrow 1_{b'}$, if $\alpha \colon a \to a'$ and $\beta \colon b \to b'$, is such that: $\varphi \circ_h 1_{a'b'} = \varphi$ and $1_{ab} \circ_h \varphi = \varphi$.
(0.2,0) circle (0.1); (3.5,0) circle (0.1); (0.3,0.2) arc (160:20:1.7); (0.3,-0.2) arc (20:160:-1.7); (0.4,0) – (3.2,0); at (1.9,1.6) [$\alpha$]{}; at (1.9,-1.6) [$\gamma$]{}; at (1.9,0.7) [$\Big\Downarrow$]{}; at (2.3,-0.7) [$\psi$]{}; at (1.9,-0.7) [$\Big\Downarrow$]{}; at (2.3,0.7) [$\varphi$]{}; at (3,-2) [$\cong$]{};
(3.5,0) circle (0.1); (6.8,0) circle (0.1); (3.6,0.2) arc (160:20:1.7); (3.6,-0.2) arc (20:160:-1.7); (3.7,0) – (6.5,0); at (5.5,1.6) [$\alpha'$]{}; at (5.2,-1.6) [$\gamma'$]{}; at (5.2,0.7) [$\Big\Downarrow$]{}; at (5.6,-0.7) [$\psi'$]{}; at (5.2,-0.7) [$\Big\Downarrow$]{}; at (5.6,0.7) [$\varphi'$]{};
at (7.15,0) [$\cong$]{};
(7.5,0) circle (0.1); (10.8,0) circle (0.1); (7.6,0.2) arc (140:40:2); (7.6,-0.2) arc (40:140:-2); at (9.2,1.3) [$\alpha$]{}; at (9.2,-1.3) [$\gamma$]{}; at (9.2,0) [$\bigg\Downarrow$]{}; at (9.9,0) [$\varphi\circ_v \psi$]{}; at (9.8,-2) [$\cong$]{};
(10.8,0) circle (0.1); (14.1,0) circle (0.1); (10.9,0.2) arc (140:40:2); (10.9,-0.2) arc (40:140:-2); at (12.5,1.3) [$\alpha'$]{}; at (12.5,-1.3) [$\gamma'$]{}; at (12.5,0) [$\bigg\Downarrow$]{}; at (13.3,0) [$\varphi'\circ_v \psi'$]{};
(-0.3,0) circle (0.1); (3,0) circle (0.1); (-0.2,0.2) arc (160:20:1.7); (-0.2,-0.2) arc (20:160:-1.7); (-0.1,0) – (2.7,0); at (1.4,1.6) [$\alpha\circ \alpha'$]{}; at (1.4,-1.6) [$\gamma\circ \gamma'$]{}; at (1.2,0.6) [$\Big\Downarrow$]{}; at (2,-0.5) [$\psi\circ_h \psi'$]{}; at (1.2,-0.6) [$\Big\Downarrow$]{}; at (2.05,0.5) [$\varphi\circ_h \varphi'$]{};
at (3.9,0) [$\cong$]{};
(4.7,0) circle (0.1); (8,0) circle (0.1); (4.8,0.2) arc (140:40:2); (4.8,-0.2) arc (40:140:-2); at (6.4,1.3) [$\alpha\circ \alpha'$]{}; at (6.4,-1.3) [$\beta\circ\beta'$]{}; at (6.4,0) [$\bigg\Downarrow$]{}; at (10.1,-0.5) [$(\varphi\circ_h\varphi') \circ_v (\psi\circ_h\psi')$]{}; at (10.1,0.5) [$(\varphi\circ_v\psi) \circ_h (\varphi'\circ_v\psi')$]{}; at (10.1,0) [$=$]{};
We observe that there must be a natural compatibility condition between the composition rules $\circ_v$ and $\circ_h$. If we have two pairs of vertically composable transformations $\varphi, \psi$ and $\varphi', \psi'$ that can be also pairwise composed horizontally, then the horizontal composition of the previously vertically composed pairs must be the same as the vertical composition of the previously horizontally composed pairs. This consistency condition will be called the exchange identity. Formally, it is written as follows and a diagramatic description is provided in Fig. \[exchange\]: $$\label{exchange_identity}
(\varphi\circ_v \psi) \circ_h (\varphi'\circ_v \psi') = (\varphi \circ_h \varphi') \circ_v (\psi \circ_h \psi') \, .$$
Notice that given a transformation $\varphi \colon \alpha \Rightarrow \beta$, we can define two maps, similar to the source and target defined previously for transitions, that assign $\alpha$ and $\beta$ to $\varphi$, respectively. The family of invertible transformations $\boldsymbol{\Gamma}$ with the vertical composition law and the source and target maps defined in this way form again a groupoid over the space of transitions $\mathbf{G}$. Moreover, the source and target maps are morphisms of groupoids.
This ‘double’ groupoid structure, that is, a groupoid (the family of invertible transformations) whose objects (the family of all transitions) form again a groupoid (whose objects are the events) and such that the source and target maps are groupoid homomorphisms, is called a 2-groupoid.
The morphisms of the first groupoid structure $\boldsymbol{\Gamma}$ (or external groupoid structure) are sometimes called 2-morphisms (or 2-cells). In our case, 2-morphisms correspond to what we have called transformations. The objects $\mathbf{G}$ of the first groupoid structure which are the morphisms of the second groupoid structure (or inner groupoid structure) are called 1-morphisms (or 1-cells). In our setting, they will correspond to what we have called transitions. Eventually, the objects $S$ of the second groupoid structure are called 0-morphisms (or 0-cells) and in the discussion before they correspond to what we have called events.
The set of axioms discussed above can be thus summarised by saying that the notions previously introduced to describe a physical system form a 2-groupoid with 0-, 1- and 2-cells being, respectively, events, transitions and transformations. Therefore, if we denote the total 2-groupoid by $\boldsymbol{\Gamma}$, we will denote its set of 1-morphisms by $\mathbf{G}$, and the outer groupoid structure will be provided by the source and target maps $\boldsymbol{\Gamma} \rightrightarrows \mathbf{G}$. The groupoid composition law in $\boldsymbol{\Gamma}$ will be called the vertical composition law and denoted $\circ_v$. Because $\mathbf{G} \rightrightarrows S$ is itself a groupoid and the source and target maps of $\boldsymbol{\Gamma}$ are themselves groupoid homomorphisms, we can always define a horizontal composition law in a natural way denoted by $\circ_h$ and both composition laws must satisfy the exchange identity above, Eq. (\[exchange\_identity\]).
Consequently, we conclude by postulating that the categorical description of a physical system is provided by a 2-groupoid $\boldsymbol{\Gamma}\rightrightarrows \mathbf{G} \rightrightarrows S$. The 1-cells of the 2-groupoid will be interpreted as the transitions of the system and its 0-cells will be considered as the events or outcomes of measurements performed on the system. The 2-cells will be interpreted as the transformations among transitions of theory providing the basis for its dynamical interpretation.
The identification of these abstracts notions with the corresponding standard physical notions and their relations with other physical notions like quantum states, unitary transformations, etc., will be provided by constructing representations of the given 2-groupoid.
Before elaborating on this, we will briefly discuss a particular instance where all these abstract structures have a specific physical interpretation, albeit not the only possible one, and it corresponds to that proposed by Schwinger in his original presentation of the ‘algebra of measurements’.
Schwinger’s algebra of selective measurements
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Schwinger’s algebra of measurements symbols {#sec:schwinger_algebra}
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As discussed in the introduction in [@Sc59] (see [@Sc70] for a complete presentation) J. Schwinger described the fundamental algebraic relations satisfied by the set of symbols representing fundamental measurement processes of given quantum systems. In this section, we will review such algebraic relations under similar simplifying assumptions used by Schwinger himself from the groupoid perspective discussed in Sect. \[sec:transitions\].
Then following Schwinger and according with the discussion in the beginning of Sect. \[sec:transitions\], we define a selective measurement $M(\mathbf{a})$ associated with the family of compatible observables $\mathbf{A} = \{ A_1, \ldots, A_N\}$, as a process (or device) that rejects all elements $S$ of the ensemble $\mathcal{E}$ whose outcomes $\mathbf{a}' = (a'_1, \ldots, a'_N)$ are different from $\mathbf{a} = (a_1, \ldots, a_N)$ and leaving only the elements $S$ such that the measurement of $\mathbf{A}$ on it gives $\mathbf{a}$. Denoting the outcomes of the measurement of $\mathbf{A}$ on the element $S$ by $\langle \mathbf{A}:S\rangle$, then we may write $M(\mathbf{a})S = S$ if $\mathbf{a} = \langle \mathbf{A}:S\rangle$ and $M(\mathbf{a})S = \emptyset$ if $\mathbf{a} \neq \langle \mathbf{A}:S\rangle$. Hence, using a set-theoretical notation, we may define the subensemble: $$\mathcal{E}_{\mathbf{A}}(\mathbf{a}) = \{ S\in \mathcal{E} \mid \langle \mathbf{A}:S\rangle = \mathbf{a} \} \, ,$$ (in what follows we will just write $\mathcal{E}(\mathbf{a})$ unless risk of confusion). Then, we may rewrite the previous definition as $M(\mathbf{a}) S = S$ if $S \in \mathcal{E}(\mathbf{a})$ and $M(\mathbf{a}) S = \emptyset$ otherwise. The empty set $\emptyset$ will be formally added to the ensemble $\mathcal{E}$ with the physical meaning of the absence of the physical system. Notice, however, that if the ensemble $\mathcal{E}$ is sufficient for $\mathbf{A}$ it cannot happen that $M(\mathbf{a}) S =\emptyset$ for all $S$. Then, with this notation, we have $M(\mathbf{a}) \mathcal{E} = \mathcal{E}(\mathbf{a})$.
In what follows we will assume that the selective measurements $M(\mathbf{a})$ defined by families $\mathbf{A}$ of compatible observables are unique. In other words, suppose that we may prepare a different physical implementation $M'(\mathbf{a})$ of the selective measurement associated with $\mathbf{A}$ that selects the subensemble $\mathcal{E}(\mathbf{a})$ by using different apparatuses, but then $M'(\mathbf{a})$ and $M(\mathbf{a})$ will be identified and denoted collectively as $M(\mathbf{a})$.
Schwinger assumes that the ‘state’ of a quantum system is established by performing on it a maximal selective measurement, and thus the space of states $\mathcal{S}$ of the theory is provided by the space of outcomes of complete selective measurements. According to the general picture introduced in the previous section, we identify Schwinger’s space of states (the space of outcomes of a complete selective measurement) with the space of events[^11]. It is in this sense that Schwinger’s picture produces a concrete realisation of the abstract description of physical systems used in the previous section, Sect. \[sec:systems\].
Again following Schwinger, a more general class of symbols $M(\mathbf{a}',\mathbf{a})$ are introduced. They denote selective measurements that reject all elements of an ensemble whose outcomes are different from $ \mathbf{a}$ and those accepted are changed in such a way that their outcomes are $\mathbf{a}'$. Using the set-theoretical notation introduced above we have: $M(\mathbf{a}',\mathbf{a})S \in \mathcal{E}(\mathbf{a}')$ if $S\in \mathcal{E}(\mathbf{a})$ and $M(\mathbf{a}',\mathbf{a})S = \emptyset$ otherwise. Notice that consistently $M(\mathbf{a},\mathbf{a}) = M(\mathbf{a})$ and $M(\mathbf{a}',\mathbf{a}) \mathcal{E} = \mathcal{E}(\mathbf{a}')$.
It is an immediate consequence of the basic definitions above that if we consider the natural composition law of selective measurements $M(\mathbf{a}'',\mathbf{a}') \circ M(\mathbf{a}',\mathbf{a})$ defined as the selective measurement obtained by performing first the selective measurement $ M(\mathbf{a}',\mathbf{a})$ and immediately afterwards the selective measurement $M(\mathbf{a}'',\mathbf{a}')$, then we get: $$\label{schwinger_groupoid_law}
M(\mathbf{a}'',\mathbf{a}') \circ M(\mathbf{a}',\mathbf{a}) = M(\mathbf{a}'',\mathbf{a}) \, ,$$ and $$\label{schwinger_units}
M(\mathbf{a}')\circ M(\mathbf{a}',\mathbf{a}) = M(\mathbf{a}',\mathbf{a}) \, , \qquad M(\mathbf{a}',\mathbf{a}) \circ M(\mathbf{a})= M(\mathbf{a}',\mathbf{a}) \, ,$$ It is clear that performing two selective measurements $M(\mathbf{a}',\mathbf{a})$, and $M(\mathbf{a}''',\mathbf{a}'')$ one after the other will produce a selective measurement again only if $\mathbf{a}'' = \mathbf{a}'$, otherwise if $\mathbf{a}'' \neq \mathbf{a}'$, then $ M(\mathbf{a}''',\mathbf{a}'') \circ M(\mathbf{a}',\mathbf{a}) S = \emptyset$ for all $S$ which, as indicated before, is not a selective measurement of the form $ M(\mathbf{a}',\mathbf{a})$.
Notice that if we have three selective measurements $M(\mathbf{a},\mathbf{a}')$, $M(\mathbf{a}',\mathbf{a}'')$ and $M(\mathbf{a}'',\mathbf{a}''')$ then, because of the basic definitions, the associativity of the composition law holds: $$\label{schwinger_associative}
M(\mathbf{a},\mathbf{a}') \circ (M(\mathbf{a}',\mathbf{a}'') \circ M(\mathbf{a}'',\mathbf{a}''')) = (M(\mathbf{a},\mathbf{a}') \circ M(\mathbf{a}',\mathbf{a}'')) \circ M(\mathbf{a}'',\mathbf{a}''') \, .$$ Finally, it is worth to observe that given a measurement symbol $M(\mathbf{a}',\mathbf{a})$ the measurement symbol $M(\mathbf{a},\mathbf{a}')$ is such that: $$\label{schwinger_inverse}
M(\mathbf{a}',\mathbf{a}) \circ M(\mathbf{a},\mathbf{a}') = M(\mathbf{a}') \, ,\quad M(\mathbf{a},\mathbf{a}') \circ M(\mathbf{a}',\mathbf{a}) = M(\mathbf{a}) \, .$$
Then, we conclude that the composition law of selective measurements satisfying Eqs. , , , , determines a groupoid law in the collection $\mathbf{G}_\mathbf{A}$ of all measurement symbols $M(\mathbf{a}',\mathbf{a})$ associated with the complete family of observables $\mathbf{A}$, whose objects (the events of the system) are the possible outcomes $\mathbf{a}$ of the observables in $\mathbf{A}$.
In this formulation, the general selective measurements $M(\mathbf{a}', \mathbf{a})$ are the morphisms of the groupoid $\mathbf{G}_\mathbf{A}$ and correspond to the notion of ‘transitions’ introduced in the general setting described in the previous section. Notice that, in this context, ‘transitions’ have not a dynamical meaning but are the consequences of deliberate manipulation of the system by the observers[^12]. Even if at this moment we could change our notation to $\alpha \colon \mathbf{a} \to \mathbf{a}'$ as in the previous section, we will stick with Schwinger’s original notation to facilitate the comparison with the original presentation.
Stern-Gerlach measurements
--------------------------
We have just seen that, in Schwinger’s picture of Quantum Mechanics, with a given physical system we associate a family of groupoids $\mathbf{G}_\mathbf{A}$ for all maximal families of compatible observables $\mathbf{A}$. As it was pointed out before, events are defined as outcomes of maximal selective measurements, thus for each maximal family of compatible observables $\mathbf{A}$ there is a family of events $\mathcal{S}_{\mathbf{A}}$. Given another different maximal family of compatible observables $\mathbf{B}$, it would determine another family of events, denoted now by $\mathbf{b} \in \mathcal{S}_\mathbf{B}$. It could happen that the sub-ensemble determined by the selective measurement $M_\mathbf{A}(\mathbf{a})$ would lie inside that defined by $M_\mathbf{B}(\mathbf{b})$, that is $\mathcal{E}_\mathbf{A}(\mathbf{a}) \subset \mathcal{E}_\mathbf{B}(\mathbf{b})$. We will say that in such case $\mathbf{b}$ is subordinate to $\mathbf{a}$ and we will denote it by $\mathbf{b} \subset \mathbf{a} $. In such case the measurement of $\mathbf{B}$ will not modify the outcomes defined by the sub-ensemble determining the event $\mathbf{a}$.
If it happens that $\mathbf{a}$ is subordinate to $\mathbf{b}$ and viceversa, i.e., $ \mathbf{b} \subset \mathbf{a} $ and $\mathbf{a} \subset \mathbf{b}$, we will consider that both events are the same, $ \mathbf{b} \sim \mathbf{a} $, and we will treat them as the same object. Thus the space of events $\mathcal{S}$ of the system is the collection of equivalence classes of events with respect to equivalence relation “$\sim$” associated with the subordination “$\subset$” relation among them. It is clear now that the relation $\subset$ induces a partial order relation on the set of events $\mathcal{S}$ and can be used to induce a partial order in the space of observables. The implications of such structure will be discussed in [@Ib18].
Notice that the family $\bigcup_{\mathbf{A}}\mathbf{G}_\mathbf{A}$ of all groupoids $\mathbf{G}_{\mathbf{A}}$ associated with all selective measurements is a groupoid over the space of events $\mathcal{S}$. Under such premises two selective measurements $M_\mathbf{A}(\mathbf{a}',\mathbf{a})$ and $M_\mathbf{B}(\mathbf{b}',\mathbf{b})$ can be composed if and only if $\mathbf{a} \sim \mathbf{b'}$ in which case: $M_\mathbf{A}(\mathbf{a}',\mathbf{a}) \circ M_\mathbf{B}(\mathbf{b}',\mathbf{b})$ will correspond to a physical device that will take as inputs elements in the sub-ensemble $\mathcal{E}_\mathbf{B}(\mathbf{b})$ and will return elements in the sub-ensemble $\mathcal{E}_\mathbf{A}(\mathbf{a})$. If we denote it by $M_{\mathbf{AB}}(\mathbf{a}', \mathbf{b})$ (or simply $M(\mathbf{a}', \mathbf{b})$ for short) then, provided that $ \mathbf{a} \sim \mathbf{b'} $ we get again: $$M(\mathbf{a}',\mathbf{a}) \circ M (\mathbf{b}',\mathbf{b}) = M(\mathbf{a}', \mathbf{b}) \, .$$ The operation $M(\mathbf{a}', \mathbf{b}) $ is invertible too. Actually the operation $$M(\mathbf{b}, \mathbf{a}') = M (\mathbf{b},\mathbf{b}') \circ M(\mathbf{a},\mathbf{a}') \, ,$$ is such that $M(\mathbf{b}, \mathbf{a}') \circ M(\mathbf{a}', \mathbf{b}) = M (\mathbf{b},\mathbf{b}') \circ M(\mathbf{a},\mathbf{a}') \circ M(\mathbf{a}',\mathbf{a}) \circ M (\mathbf{b}',\mathbf{b}) = M(\mathbf{b}) = 1_\mathbf{b}$ and $M(\mathbf{a}', \mathbf{b}) \circ M(\mathbf{b}, \mathbf{a}') = 1_{\mathbf{a}'}$.
Notice that this new family of operations does not correspond (in general) to a set of compatible observables $\mathbf{C}$ constructed out of $\mathbf{A}$ and $\mathbf{B}$[^13], however, it is important to introduce them because they correspond to the physical operations of composition of Stern-Gerlach devices (so they could be called *composite Stern-Gerlach measurements* or just $SG$-measurements for short). These extended operations were also introduced by Schwinger as a consistency condition for the relations of the algebra of selective measurements. However, in the reformulation of the basic notions we are presenting here, they will play a more significant role as they would uncover another layer of structure in the algebraic setting for basic measurement operations that, as it was discussed in the previous section, is that of a 2-groupoid.
All together, if we consider the space $\mathbf{G}$ consisting of all complete selective measurements and composite Stern-Gerlarch measurements together with their natural composition law, we see that it has the structure of a groupoid whose space of objects is the space of events $\mathcal{S}$ of the system. Thus we will denote by $\mathbf{G} \rightrightarrows \mathcal{S}$ such groupoid with the source and target maps $s,t$ given by: $$s(M(\mathbf{a}',\mathbf{a}) ) = \mathbf{a} \, , \qquad t(M(\mathbf{a}',\mathbf{a}) ) = \mathbf{a}' \, .$$
The 2-groupoid structure of Schwinger’s algebra of selective measurements
-------------------------------------------------------------------------
It is clear that SG-measurements define transformations among selective measurements. That is, if we consider the transition $M_\mathbf{A}(\mathbf{a}, \mathbf{a}')$ and the SG-measurements $M(\mathbf{a}', \mathbf{b}')$ and $M(\mathbf{b}, \mathbf{a})$ that transform the ensembles $\mathcal{E}_\mathbf{A}(\mathbf{a}')$ in $\mathcal{E}_\mathbf{B}(\mathbf{b}')$, and $\mathcal{E}_\mathbf{B}(\mathbf{b})$ in $\mathcal{E}_\mathbf{A}(\mathbf{a})$ respectively, then the transition $M(\mathbf{b}, \mathbf{a}) \circ M_\mathbf{A}(\mathbf{a}, \mathbf{a}') \circ M(\mathbf{a}', \mathbf{b}')$ must be the transition corresponding to the selective measurement $M_\mathbf{B}(\mathbf{b}, \mathbf{b}')$, that is: $$\label{schwinger_transformation}
M(\mathbf{b}, \mathbf{a}) \circ M_\mathbf{A}(\mathbf{a}, \mathbf{a}') \circ M(\mathbf{a}', \mathbf{b}') = M_\mathbf{B}(\mathbf{b}, \mathbf{b}') \, .$$ Hence, formula (\[schwinger\_transformation\]) defines a transformation $\varphi \colon M_\mathbf{A}(\mathbf{a}, \mathbf{a}') \Rightarrow M_\mathbf{B}(\mathbf{b}, \mathbf{b}')$ in the sense of Sect. \[sec:transformations\]. This transformation could be just denoted as $\varphi(\mathbf{a}, \mathbf{a}'; \mathbf{b}, \mathbf{b}')$ instead of listing the pair of SG-measurements $M(\mathbf{b}, \mathbf{a}) $ and $M(\mathbf{a}', \mathbf{b}')$ involved on its definition in order to avoid a too cumbersome notation.
It is a simple matter to check the axioms introduced in Sect. \[sec:transformations\] for the theory of transformations. The unit transformations are given by the pairs $M(\mathbf{a})$ and $M(\mathbf{a}')$, that is, $1_{(\mathbf{a}, \mathbf{a}')} = \varphi(\mathbf{a}, \mathbf{a}'; \mathbf{a}, \mathbf{a}')$ because $\varphi(\mathbf{a}, \mathbf{a}'; \mathbf{a}, \mathbf{a}')\circ_v \varphi(\mathbf{a}, \mathbf{a}'; \mathbf{b}, \mathbf{b}')$ would transform the transition $M_\mathbf{A}(\mathbf{a}, \mathbf{a}')$ into the transition: $$\begin{aligned}
M_\mathbf{A}(\mathbf{a}, \mathbf{a}') & \overset{\varphi(\mathbf{a}, \mathbf{a}'; \mathbf{a}, \mathbf{a}')}{\Rightarrow} & M(\mathbf{a}) \circ M_\mathbf{A}(\mathbf{a}, \mathbf{a}') \circ M(\mathbf{a}') \\ &\overset{\varphi(\mathbf{a}, \mathbf{a}'; \mathbf{b}, \mathbf{b}')}{\Rightarrow} & M(\mathbf{b}, \mathbf{a}) \circ M_\mathbf{A}(\mathbf{a}, \mathbf{a}') \circ M(\mathbf{a}', \mathbf{b}') = M_\mathbf{B}(\mathbf{b}, \mathbf{b}') \, .\end{aligned}$$
Notice that with this notation the vertical composition law is simply written as: $$\varphi(\mathbf{a}, \mathbf{a}'; \mathbf{b}, \mathbf{b}') \circ_v \varphi(\mathbf{b}, \mathbf{b}'; \mathbf{c}, \mathbf{c}') = \varphi(\mathbf{a}, \mathbf{a}'; \mathbf{c}, \mathbf{c}') \, .$$ as it is shown by the following computation: $$\begin{aligned}
&& M(\mathbf{c}, \mathbf{b}) \circ \left( M(\mathbf{b}, \mathbf{a}) \circ M_\mathbf{A}(\mathbf{a}, \mathbf{a}') \circ M(\mathbf{a}', \mathbf{b}') \right) \circ M(\mathbf{b}', \mathbf{c}') \\ &&=\left (M(\mathbf{c}, \mathbf{b}) \circ M(\mathbf{b}, \mathbf{a}) \right) \circ M_\mathbf{A}(\mathbf{a}, \mathbf{a}') \circ \left( M(\mathbf{a}', \mathbf{b}') \circ M(\mathbf{b}', \mathbf{c}') \right) \\ && = M(\mathbf{c}, \mathbf{a}) \circ M_\mathbf{A}(\mathbf{a}, \mathbf{a}'') \circ M(\mathbf{a}'', \mathbf{c}'') = M_\mathbf{B}(\mathbf{c}, \mathbf{c}'') \, ,\end{aligned}$$
The associativity property of the vertical composition law is easily checked in a similar way.
Regarding the horizontal composition law, we must notice that if $$\varphi(\mathbf{a}, \mathbf{a}'; \mathbf{b}, \mathbf{b}') \colon M_\mathbf{A}(\mathbf{a}, \mathbf{a}') \Rightarrow M_\mathbf{B}(\mathbf{b}, \mathbf{b}') \, ,$$ and $$\varphi(\mathbf{a}', \mathbf{a}''; \mathbf{b}', \mathbf{b}'') \colon M_\mathbf{A}(\mathbf{a}', \mathbf{a}'') \Rightarrow M_\mathbf{B}(\mathbf{b}', \mathbf{b}'') \, ,$$ denote two transformations, then the composition: $$\begin{aligned}
&& \left( M(\mathbf{b}, \mathbf{a}) \circ M_\mathbf{A}(\mathbf{a}, \mathbf{a}') \circ M(\mathbf{a}', \mathbf{b}') \right) \circ \left( M(\mathbf{b}', \mathbf{a}') \circ M_\mathbf{A}(\mathbf{a}', \mathbf{a}'') \circ M(\mathbf{a}'', \mathbf{b}'') \right) \\ &&= M(\mathbf{b}, \mathbf{a}) \circ \left( M_\mathbf{A}(\mathbf{a}, \mathbf{a}') \circ M_\mathbf{A}(\mathbf{a}', \mathbf{a}'')\right) \circ M(\mathbf{a}'', \mathbf{b}'') \\ && = M(\mathbf{b}, \mathbf{a}) \circ M_\mathbf{A}(\mathbf{a}, \mathbf{a}'') \circ M(\mathbf{a}'', \mathbf{b}'') = M_\mathbf{B}(\mathbf{b}, \mathbf{b}'') \, ,\end{aligned}$$ (where we have used that $SG$-measurements are invertible) shows that the pair of $SG$-measurements $M(\mathbf{b}, \mathbf{a})$ and $M(\mathbf{a}'', \mathbf{b}'')$ define a transformation from $M_\mathbf{A}(\mathbf{a}, \mathbf{a}'') $ to $M_\mathbf{B}(\mathbf{b}, \mathbf{b}'') $ or, in other words: $$\varphi(\mathbf{a}, \mathbf{a}'; \mathbf{b}, \mathbf{b}') \circ_h \varphi(\mathbf{a}', \mathbf{a}''; \mathbf{b}', \mathbf{b}'') = \varphi(\mathbf{a}, \mathbf{a}''; \mathbf{b}', \mathbf{b}'') \, .$$
Finally, a simple computation shows that the exchange identity (\[exchange\_identity\]) is satisfied. That is, we compute first: $$\begin{aligned}
&& \left( \varphi(\mathbf{a}, \mathbf{a}'; \mathbf{b}, \mathbf{b}') \circ_h \varphi(\mathbf{a}', \mathbf{a}''; \mathbf{b}', \mathbf{b}'') \right) \circ_v
\left( \varphi(\mathbf{b}, \mathbf{b}'; \mathbf{c}, \mathbf{c}') \circ_h \varphi(\mathbf{b}', \mathbf{b}''; \mathbf{c}', \mathbf{c}'') \right) \\ && =
\varphi(\mathbf{a}, \mathbf{a}''; \mathbf{b}, \mathbf{b}'') \circ_v
\varphi(\mathbf{b}, \mathbf{b}''; \mathbf{c}, \mathbf{c}'') = \varphi(\mathbf{a}, \mathbf{a}''; \mathbf{c}, \mathbf{c}'') \, , \end{aligned}$$ but $$\begin{aligned}
&& \left( \varphi(\mathbf{a}, \mathbf{a}'; \mathbf{b}, \mathbf{b}') \circ_v \varphi(\mathbf{b}, \mathbf{b}'; \mathbf{c}, \mathbf{c}') \right) \circ_h
\left( \varphi(\mathbf{a}', \mathbf{a}''; \mathbf{b}', \mathbf{b}'') \circ_v \varphi(\mathbf{b}', \mathbf{b}''; \mathbf{c}', \mathbf{c}'') \right) \\ && =
\varphi(\mathbf{a}, \mathbf{a}'; \mathbf{c}, \mathbf{c}') \circ_h
\varphi(\mathbf{a}', \mathbf{a}''; \mathbf{c}', \mathbf{c}'') = \varphi(\mathbf{a}, \mathbf{a}''; \mathbf{c}, \mathbf{c}'') \, .\end{aligned}$$ which proves the desired identity.
The previous discussion can be summarised by saying that the family of Schwinger’s maximal selective and Stern-Gerlach measurements have the structure of a 2-groupoid whose 2-cells (corresponding to the notion of transformations discussed in Sect. \[sec:transformations\]) are given by pairs of SG-measurements. The 1-cells, or transitions, are given by maximal measurements, and its 0-cells, or events, are given by the outcomes of maximal selective measurements.
The fundamental representation of Schwinger’s algebra of selective measurements and the standard pictures of QM
---------------------------------------------------------------------------------------------------------------
The fundamental representation of a groupoid was introduced in Sect. \[sec:fundamental\]. We will use this representation to provide an interpretation of Schwinger’s 2-groupoid in terms of the standard pictures of Quantum Mechanics. First we will assume, as Schwinger’s did, that the set of outcomes $\{ \mathbf{a} \}$ of maximal measurements is finite. This restriction can be lifted but we will not worry much about it here. We will select the subgroupoid $\mathbf{G}_{\mathbf{A}}$ corresponding to transitions $M(\mathbf{a},\mathbf{a}')$. The fundamental representation of the groupoid $\mathbf{G}_{\mathbf{A}}$ will take place on the finite dimensional Hilbert space $\mathcal{H}_{A}$ generated by the set of outcomes $\mathbf{a}$. Thinking now of the events $\mathbf{a}$ as labels ordered from 1 to $n$, with $n$ the number of outcomes (and the dimension of $\mathcal{H}_{A}$), the elements in the groupoid algebra $\mathbb{C}[\mathbf{G}_{\mathbf{A}}]$ can be written as: $$A = \sum_{\mathbf{a}, \mathbf{a}' = 1}^n A_{\mathbf{a}, \mathbf{a}'} M(\mathbf{a}, \mathbf{a}') \, ,$$ i.e., they are formal linear combinations of the selective measurements with complex coefficients $A_{\mathbf{a}, \mathbf{a}'}$. Hence, they can be identified with $n\times n$ matrices whose entries are given by $A_{\mathbf{a}, \mathbf{a}'}$. The groupoid algebra composition law is just given by multiplication of matrices, that is: $$\begin{aligned}
A\cdot B &=& \sum_{\mathbf{a}, \mathbf{a}' , \mathbf{a}'', \mathbf{a}''' = 1}^n A_{\mathbf{a}, \mathbf{a}'} B_{\mathbf{a}'', \mathbf{a}'''} \, \, \delta (\mathbf{a}', \mathbf{a}'')\, M(\mathbf{a}, \mathbf{a}') \circ M(\mathbf{a}'', \mathbf{a}''') \\ &=& \sum_{\mathbf{a},\mathbf{a}''' = 1}^n \left( \sum_{\mathbf{a}'' = 1}^n A_{\mathbf{a}, \mathbf{a}''} B_{\mathbf{a}'', \mathbf{a}'''} \right) M(\mathbf{a}, \mathbf{a}''') \\ &=& \sum_{\mathbf{a},\mathbf{a}''' = 1}^n (AB)_{\mathbf{a}, \mathbf{a}'''} M(\mathbf{a}, \mathbf{a}''') \, ,\end{aligned}$$ where in the last row, $AB$ stands for the standard matrix product of the matrices $A$ and $B$.
Continuing with this interpretation, we notice that, using the canonical orthonormal basis provided by the vectors $|\mathbf{a}\rangle$, the vectors $| \psi \rangle$ in the fundamental Hilbert space $\mathcal{H}_{A}$ can be identified with column vectors with components $\psi_{\mathbf{a}}$, that is, $|\psi \rangle = \sum_{\mathbf{a} = 1}^n \psi_{\mathbf{a}} |\mathbf{a}\rangle$. Then, we get: $$\pi (A) |\psi \rangle = \sum_{\mathbf{a}, \mathbf{a}', \mathbf{a}'' = 1}^n A_{\mathbf{a}, \mathbf{a}'} \psi_{\mathbf{a}''}\, \, \delta (\mathbf{a}, \mathbf{a}'')\, \pi (M(\mathbf{a}, \mathbf{a}')) |\mathbf{a}'' \rangle =$$ $$=\sum_{\mathbf{a}, \mathbf{a}' = 1}^n A_{\mathbf{a}, \mathbf{a}'} \psi_{\mathbf{a}}\, |\mathbf{a}' \rangle = \sum_{ \mathbf{a}' = 1}^n (A \psi)_{\mathbf{a}'} \, |\mathbf{a}' \rangle \, ,$$ where $A\psi$ in the last row denotes the standard matrix-vector product. Hence, the fundamental representation becomes just the standar representation of the algebra of $n\times n$ matrices on the corresponding linear space $\mathbb{C}^n$.
Vectors in the fundamental space can be identified with complex linear combinations of events and transitions are represented by rank 1 operators. In other words the transition $M(\mathbf{a}, \mathbf{a}')$ is represented by the operator $|\mathbf{a}' \rangle \langle \mathbf{a} |$ in $\mathcal{H}_{A}$. Elements $A$ in the groupoid algebra are then represented as operators acting on $\mathcal{H}_{A}$. In this sense the units $1_\mathbf{a}$ of the groupoid $\mathbf{G}_{\mathbf{A}}$ are represented by the rank one orthogonal projectors $|\mathbf{a} \rangle \langle \mathbf{a} |$ that provide a resolution of the identity of $\mathcal{H}_{A}$, $ \sum_{\mathbf{a} = 1}^n | \mathbf{a} \rangle \langle \mathbf{a} |= \mathbf{1}$.
We may repeat the same construction starting with another subgroupoid $\mathbf{G}_{B}$ associated with the maximal set of compatible observables $\mathbf{B}$ obtaining a Hilbert space $\mathcal{H}_{B}$ and a representation of $\mathbb{C}[\mathbf{G}_{B}]$ in terms of linear operators on $\mathcal{H}_{B}$. If the space of events $\Omega_{A}$ and $\Omega_{B}$ are assumed to have the same (finite) cardinality (i.e., A and B codify for the same physical information on the system under consideration) we have that $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ are isomorphic as Hilbert spaces. More interestingly, the 2-groupoid structure of Schwinger’s 2-groupoid $\Gamma$ appears represented by the hand of the theory of transformations of Hilbert spaces. Consider a transformation $\varphi (\mathbf{a},\mathbf{a}'; \mathbf{b},\mathbf{b}' )\colon M_\mathbf{A}(\mathbf{a}, \mathbf{a}') \Rightarrow M_\mathbf{B}(\mathbf{b}, \mathbf{b}')$ sending the selective measurement $M_\mathbf{A}(\mathbf{a},\mathbf{a}')$ into the selective measurement $M_\mathbf{B}(\mathbf{b},\mathbf{b}')$. Following the ideas above, the selective measurements $M_\mathbf{A}(\mathbf{a},\mathbf{a}')$ and $M_\mathbf{B}(\mathbf{b},\mathbf{b}')$ will be represented on the corresponding Hilbert spaces $\mathcal{H}_\mathbf{A}$ and $\mathcal{H}_\mathbf{B}$ supporting the fundamental representations of the groupoids $\mathbf{G}_{\mathbf{A}}$ and $\mathbf{G}_{\mathbf{B}}$ respectively. Then, considering the vector $| A \rangle = \sum_{\mathbf{a}, \mathbf{a}' = 1}^n A_{\mathbf{a}, \mathbf{a}'} | M(\mathbf{a}, \mathbf{a}') \rangle$ in the Hilbert space generated by $\mathbf{G}_\mathbf{A}$ and the element $\Phi = \sum T_{\mathbf{a},\mathbf{a}'; \mathbf{b},\mathbf{b}'} \varphi (\mathbf{a},\mathbf{a}'; \mathbf{b},\mathbf{b}' )$, we get (all repeated indexes are summed): $$\begin{aligned}
\pi (\Phi) A & = & \sum T_{\mathbf{a},\mathbf{a}'; \mathbf{b},\mathbf{b}'}
A_{\mathbf{c}, \mathbf{c}'} \, \, \delta(\mathbf{a}, \mathbf{c}) \delta(\mathbf{a}', \mathbf{c}') \, \, \pi (\varphi (\mathbf{a},\mathbf{a}'; \mathbf{b},\mathbf{b}' ))| M(\mathbf{c}, \mathbf{c}') \rangle \\ & = & \sum T_{\mathbf{a},\mathbf{a}'; \mathbf{b},\mathbf{b}'}
A_{\mathbf{a}, \mathbf{a}'} | M(\mathbf{b}, \mathbf{b}') \rangle \, .\end{aligned}$$ But, transformations $\varphi (\mathbf{a},\mathbf{a}'; \mathbf{b},\mathbf{b}' )$ are defined by pairs of SG-measurements, that is the basis for the groupoid algebra $\mathbb{C}[\boldsymbol{\Gamma}]$ consists on pairs of SG-measurements, thus as a linear space it is the tensor product of the linear space generated by SG-measurements. Then, the coefficients $T_{\mathbf{a},\mathbf{a}'; \mathbf{b},\mathbf{b}'}$ can be written as the products $T_{\mathbf{b}, \mathbf{a}} T_{\mathbf{a}', \mathbf{b}'}$, and we conclude: $$\label{matrix_transform}
\pi (\Phi) A = \sum T_{\mathbf{b}, \mathbf{a}} T_{\mathbf{a}', \mathbf{b}'}
A_{\mathbf{a}, \mathbf{a}'} | M(\mathbf{b}, \mathbf{b}') \rangle = \sum (T^\dagger A T')_{\mathbf{b}, \mathbf{b}'} | M(\mathbf{b}, \mathbf{b}') \rangle\, .$$ where $T^\dagger AT'$ stands for the standard matrix multiplication of the matrices defined by $T = \left(\bar{T}_{\mathbf{a}, \mathbf{b}}\right)$, $T' = \left(T_{\mathbf{a}', \mathbf{b}'}\right)$ and $A = \left( A_{\mathbf{a}, \mathbf{a}'} \right)$. The previous formula (\[matrix\_transform\]) shows that the transformation $\varphi (\mathbf{a},\mathbf{a}'; \mathbf{b},\mathbf{b}' )$ is represented in the standard form as an operator between the Hilbert spaces $\mathcal{H}_\mathbf{A}$ and $\mathcal{H}_\mathbf{B}$.
Conclusions and discussion
==========================
A careful analysis of the structure of the algebra of measurements of a quantum system proposed by J. Schwinger reveals that its underlying mathematical structure is that of a 2-groupoid. Using this background, a proposal for the mathematical description of quantum systems based on the primitive notions of *events* or *outcomes*, *transitions* and *transformations* under the mathematical form of a 2-groupoid is stated.
The standard interpretation in terms of vectors and operators in Hilbert spaces is recovered when we consider the fundamental representation of such 2-groupoid. Other representations can be chosen that will reveal different characteristics of the system.
The analysis of the dynamics as well as other aspects of the theory like the reconstruction of the algebra of observables and the states of the quantum system from the 2-groupoid structure, and the statistical interpretation of the theory, have been barely touched and a detailed analysis will be developed in forthcoming works. It suffices to mention here that the algebra of the groupoid, which naturally has the structure of an involution algebra, gives a first insight regarding the connection with the $C^{*}$-algebraic formulation of quantum theories which will be further explored in forthcoming papers. In particular, the projectors $|\mathbf{a} \rangle \langle \mathbf{a} |$ will turn out to be normal pure states of the $C^{*}$-algebra of the system, and the GNS construction applied to any of them will reconstruct the Hilbert space of the fundamental representation.
Composition of systems, symmetries, significant examples and applications begining with the harmonic oscillator and other basic examples of quantum systems, etc., are all aspects that will be discussed in subsequent works.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in RD (SEV-2015/0554). A.I. would like to thank partial support provided by the MINECO research project MTM2017-84098-P and QUITEMAD+, S2013/ICE-2801. G.M. would like to thank partial financial support provided by the Santander/UC3M Excellence Chair Program 2019-2020. G.M. is a member of the Gruppo Nazionale di Fisica Matematica (INDAM),Italy.
[99]{} J. Schwinger. *Quantum Kinematics and Dynamics*. Frontiers in Physics, W.A. Benjamin, Inc., (New York 1970); ibid., *Quantum Kinematics and Dynamics*. Advanced Book Classics, Westview Press (Perseus Books Group 1991); *ibid.,* *Quantum Mechanics Symbolism of Atomic measurements,*, edited by Berthold-Georg Englert. Springer–Verlag, (Berlin 2001). P.A.M. Dirac. *Lectures on Quantum Field Theory*, Belfer Graduate School of Science, Monograph Series, number three (1966).
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[^1]: Throughout this paper we will refer to the original edition of Schwinger’s book whose notation we keep using as it is closer to the spirit of this work.
[^2]: For a friendly introduction to groupoids we refer the reader to [@Ib13], [@Ib19b].
[^3]: This convention mimicks the notation for the composition of functions.
[^4]: Note that we may also proceed by dictating that the observable $A$ is defined by limiting the outcomes to the actual values that can be measured over the elements of the ensemble $\mathcal{E}$, however, the natural notion of sub-ensemble leads immediately to consider maximal ensembles that will be ‘sufficient’ in the previous sense.
[^5]: Notice that such family doesn’t need to be finite even if, for practical purposes, we will be working under this assumption.
[^6]: Not to be confused with space-time events or with Sorkin’s notion of events as subsets of the space of histories of a system [@So16]. Schwinger used the term ‘states’ for such notion, but we rather use a different terminology not to create confusion with the proper notion of states of the system as positive normalized functionals on the algebra of observables of the theory, see [@Ib18].
[^7]: In [@Ib18] it will be discussed the meaning of Schwinger’s compound measurements, that is, the meaning of composing ‘incompatible’ transitions. For the purposes of this paper though, we will restrict ourselves to consider the composition of compatible transitions.
[^8]: Notice that in this picture the reversibility condition could be lifted, in which case we will be dealing with categories, to include open systems.
[^9]: Notice that the limit cannot be obtained by just physically obtaining the outcomes of complete measurements because of Zeno’s effect (see for instance [@Fa00] and references therein).
[^10]: In other words, we are selecting a class of equivalent measurement systems, leaving open the possibility of introducing other classes of measurements subordinate to them. These issues will be discussed in [@Ci19].
[^11]: We prefer the word ‘event’ (or just ‘outcome’) rather than the word ‘state’ in view of the discussion of states as positive normalized functionals on the algebra of observables of the theory, see [@Ib18].
[^12]: Of course, the observers may decide to use the Hamitonian determining the dynamical evolution, and use selective measurements that use the dynamics itself to change the system.
[^13]: Until an algebra structure is introduced in space of observables in which case we can start looking for maximal Abelian subalgebras generated by them [@Ib18].
|
---
abstract: 'Researchers have observed that Visual Question Answering (VQA) models tend to answer questions by learning statistical biases in the data. For example, their answer to the question “What is the color of the grass?" is usually “Green", whereas a question like “What is the title of the book?" cannot be answered by inferring statistical biases. It is of interest to the community to explicitly discover such biases, both for understanding the behavior of such models, and towards debugging them. Our work address this problem. In a database, we store the words of the question, answer and visual words corresponding to regions of interest in attention maps. By running simple rule mining algorithms on this database, we discover human-interpretable rules which give us unique insight into the behavior of such models. Our results also show examples of unusual behaviors learned by models in attempting VQA tasks.'
author:
- |
Varun Manjunatha, Nirat Saini & Larry S. Davis\
Dept. of Computer Science\
University of Maryland, College Park\
`{varunm@cs, nirat@cs, lsd@umiacs}.umd.edu`
bibliography:
- 'egbib.bib'
title: Explicit Bias Discovery in Visual Question Answering Models
---
Introduction {#sec:intro}
============
In recent years, the problem of Visual Question Answering ([[<span style="font-variant:small-caps;">VQA</span> ]{}]{}) - the task of answering a question about an image has become a hotbed of research activity in the computer vision community. While there are several publicly available [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}datasets[@antol2015vqa; @johnson2016clevr; @krishnavisualgenome; @MalinowskiF14], our focus in this paper will be on the dataset provided in [@antol2015vqa] and [@GoyalKSBP17], which is the largest natural image-question-answer dataset and the most widely cited. Even so, the narrowed-down version of the [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}problem on this dataset is not monolithic - ideally, several different skills are required by a model to answer the various questions. In Figure \[fig:types\_of\_vqa\_and\_how\_many\](left) , a question like “What time is it?" requires the acquired skill of being able to read the time on a clock-face, “What is the title of the top book?" requires an OCR-like ability to read sentences, whereas the question “What color is the grass?" can be answered largely using statistical biases in the data itself (because frequently in this dataset, grass is green in color). Many models have attempted to solve the problem of [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}with varying degrees of success, but among them, the vast majority still attempt to solve the [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}task by exploiting biases in the dataset [@KazemiE17; @AndersonTeney; @GVQA; @MCB; @MUTAN etc], while a smaller minority address the individual problem types [@NMN; @InterpretableCounting; @CountingPrith etc].
Keeping the former in mind, in this work, we provide a method to discover and enumerate explicitly, the various biases that are learned by a [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}model. For example, in Figure \[fig:types\_of\_vqa\_and\_how\_many\](right), we provide examples of some rules learned by a strong baseline [@KazemiE17]. The model seems to have learned that if a question contains the words {What, time, day} (Eg : “What time of day is it?") and the accompanying image contains the bright sky ([ ]{}), the model is likely to answer “afternoon". The model answers “night" to the same question accompanied with an image containing a “night-sky" patch ([ ]{}). On the other hand, if it contains a clock face([ ]{}), it tends to answer the question with a time in an “HH:MM" format, while a question like “What time of the year?" paired with leafless trees([ ]{}) prompts “fall" as the answer. The core of our method towards discovering such biases is the classical Apriori algorithm [@AgrawalAPriori] which is used to discover rules in large databases - here the *database* refers to the question-words and model responses on the [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}validation set, which can be mined to produce these rules.
{width="\textwidth"}
. \[fig:types\_of\_vqa\_and\_how\_many\]
Deep learning algorithms reduce training error by learning biases in the data. This is evident from the observation that validation/test samples from the long tail of a data distribution are hard to solve, simply because similar examples do not occur frequently enough in the training set[@WangNIPS; @YangCVPR16 etc]. However, explicitly enumerating these biases in a human-interpretable form is possible only in a handful of problems, such as VQA. [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}is particularly illustrative because the questions and answers are in human language, while the images (and attention maps) can also be interpreted by humans. [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}is also interesting because it is a multi-modal problem - both language and vision are required to solve this problem. The language alone (i.e., an image agnostic model) can generate plausible (but often incorrect) answers to *most* questions (as we show in Section \[sec:language\_only\]), but incorporating the image generates more accurate answers. That the language alone is able to produce plausible answers strongly indicates that [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}models implicitly use simple rules to produce answers - we endeavour in this paper to find an approach that can discover these rules.
Finally, we note that in this work, we do not seek to improve upon the state of the art. We do most of our experiments on the model of [@KazemiE17], which is a strong baseline for this problem. We choose this model because it is simple to train and analyze (Section \[sec:baseline\]). To concretely summarize, our main contribution is to provide a method that can capture macroscopic rules that a [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}model ostensibly utilizes to answer questions. To the best of our knowledge, this is the first detailed work that analyzes the [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}dataset of [@GoyalKSBP17] in this manner.
The rest of this paper is arranged as follows : In Section \[sec:related\], we discuss related work, specifically those which look into identifying pathological biases in several machine learning problems, and “debugging" [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}models. In Section \[sec:method\], we discuss details of our method. In Section \[sec:experiments\], we provide experimental results and list (in a literal sense) some rules we believe the model is employing to answer questions. We discuss limitations of this method in Section \[sec:limitations\] and conclude in Section \[sec:conclusion\].
Background and Related Work {#sec:related}
===========================
The [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}problem is most often solved as a multi-class classification problem. In this formulation, an image(I) usually fed through a CNN, and a question(Q) fed through a language module like an LSTM [@hochreiter1997long] or GRU [@GRU], are jointly mapped to an answer category (“yes", “no", “1", “2", etc). Although the cardinality of the set of all answers given a QI dataset is potentially infinite, researchers have observed that a set of a few thousand (typically 3000 or so) most frequently occurring answers can account for over 90% of all answers in the [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}dataset. Further, the evaluation of [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}in [@antol2015vqa] and [@GoyalKSBP17] is performed such that an answer receives partial credit if at least one human annotator agreed with the answer, even if it might not be the answer provided by the majority of the annotators. This further encourages the use of a classification based [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}system that limits the number of answers to the most frequent ones, rather than an answer generation based [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}system (say, using a decoder LSTM like [@vinyals2015show]).
**On undesirable biases in machine learning models**: Machine learning methods are increasingly being used as tools to calculate credit scores, interest rates, insurance rates, etc, which deeply impact lives of ordinary humans. It is thus vitally important that machine learning models not discriminate on the basis of gender, race, nationality, etc[@equality; @propublica; @DBLP:conf/fat/BuolamwiniG18]. [@CisseECCV2018] focus on revealing racial biases in image-based datasets by using adversarial examples. [@MenShop] explores data as well as models associated with object classification and visual semantic role labeling for identifying gender biases and their amplification. Further, [@Homemaker] shows the presence of gender biases while encoding word embeddings, which is further exacerbated while using those embeddings to make predictions. [@HendrixSnowboard] propose an Equalizer model which ensures equal gender probability when making predictions on image captioning tasks.
**On debugging deep networks**: The seminal work by [@Lipton] suggests that the Machine Learning community does not have a good understanding of what it means to interpret a model. In particular, this work expounds *post-hoc interpretability* - interpretation of a model’s behavior based on some criteria, such as visualizations of gradients [@Gradcam] or attention maps [@Xu2015show], *after* the model has been trained. Locally Interpretable Model Agnostic Explanations (LIME), [@lime:kdd16] explain a classifier’s behavior at a particular point by perturbing the sample and building a linear model using the perturbations and their predictions. A follow up work [@anchors:aaai18] constructs *Anchors*, which are features such that, in an instance where these features hold, a model’s prediction does not change. This work is the most similar prior work to ours, and the authors provide a few results on [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}as well. However, they only assume the existence of a model, and perturb instances of the data, whereas ours assumes the existence of responses to a dataset, but not the model itself. We use standard rule finding algorithms and provide much more detailed results on the [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}problem.
**On debugging [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}**:[@AgrawalBP16] study the behavior of models on the [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}1.0 dataset. Through a series of experiments, they show that [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}models fail on novel instances, tend to answer after only partially reading the question and fail to change their answers across different images. In [@GVQA], recognizing that deep models seem to use a combination of identifying visual concepts and prediction of answers using biases learned from the data, the authors develop a mechanism to disentangle the two. However, they do not explicitly find a way to discover such biases in the first place. In [@GoyalKSBP17], the authors introduce a second, more balanced version of the [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}dataset that mitigates biases (especially language based ones) in the original dataset. The resulting balanced dataset is christened [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}2.0, and is the dataset that our results are reported on. In [@kafle2017analysis], the authors balance yes/no questions (those which indicate the presence or absence of objects), and propose two new evaluation metrics that compensate for forms of dataset bias.
Method {#sec:method}
======
We cast our bias discovery task as an instance of the rule mining problem, which we shall describe below. The connection between discovering biases in [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}and rule mining is as follows : each (Question, Image, Answer) or QI+A triplet can be cast as a transaction in a database, where each word in the question, answer and image patch (or visual word, Section \[sec:codebook\] and \[sec:box\]) is akin to an item. There are now three components to our rule mining operation :
- First, a frequent itemset miner picks out a set of all itemsets which occur at least $s$ times in the dataset where $s$ is the support. Because our dataset has over 200,000 questions (the entire [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}validation set), and the number of items exceeds 40,000 (all question words+all answer words+all visual words), we choose GMiner [@ChonH018] due to its speed and efficient GPU implementation. Examples of such frequent itemsets in the context of [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}include {what, color, red\*}, {what, sport, playing}, where the presence of a \* indicates that the word is an answer-word.
- Next, a rule miner Apriori [@AgrawalAPriori] forms all valid association rules $A \rightarrow C$, such that the rule has a support $>s$ and a confidence $>c$, where the confidence is defined as $\frac{|A \cup C|}{|A|}$. Here, the itemset $A$ is called *antecedent* and the itemset $C$ is called *consequent*. We choose and $c = 0.2$ unless specified otherwise. An example of an association rule is {what, sport, playing, [ ]{}} $\rightarrow$ {tennis\*}, which can be interpreted as “If the question contains the words —what, sport, playing— and the accompanying image contains a tennis player, the answer *could* be tennis".
- Finally, a post-processing step removes obviously spurious rules by considering the causal nature of the [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}problem (i.e., only considering rules that obey : Image/Question $\rightarrow$ Answer). For the purpose of the results in Section \[sec:experiments\], we query these rules with search terms like {What,sport}.
More concretely, let the $i^{th}$ (Image, Question) pair result in the network predicting the answer $a^i$. Let the question itself contain the words $\{w^i_1, w^i_2, ...., w^i_k\}$. Further, while answering the question, let the part of the image that the network shows attention towards correspond to the visual code-word $v^i$ (Section \[sec:codebook\] and \[sec:box\]). Then, this QI+A corresponds to the transaction $\{w^i_1, w^i_2, ...., w^k_k, v^i, a^i\}$. By pre-computing and combining question, answer and visual vocabularies, each item in a transaction can be indexed uniquely. This is shown in Figure \[fig:method\] and explained in greater detail in the following sub-sections.
{width="\textwidth"}
. \[fig:method\]
Baseline Model {#sec:baseline}
--------------
The baseline model we use in this work is from [@KazemiE17], which was briefly a state-of-the-art method, yielding higher performance than other, more complicated models. We choose this model for two reasons : first, its simplicity (in other words, an absence of “bells and whistles") makes it a good test-bed for our method and has been used by other works that explore the behavior of [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}algorithms [@PramodACL; @FengRAWR2018]. The second reason is that the performance of this baseline is within 4% of the state-of-the-art model [@AndersonTeney] without using external data or ensembles. We use the implementation of <https://github.com/Cyanogenoid/pytorch-vqa>. A brief description of this model is as follows : The [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}problem is formulated as a multi-class classification problem (Section \[sec:related\]). The input to the model is an image and a question, while the output is the answer class with the highest confidence (out of 3000 classes). Resnet-152[@Resnet] features are extracted from the image and concatenated with the last hidden state of an LSTM[@hochreiter1997long]. The text and visual features are combined to form attention maps which are fed to the softmax (output) layer through two dense layers. In this work, we focus on the second attention map.
Visual Codebook Generation {#sec:codebook}
--------------------------
We generate the visual codebook using the classical “feature extraction followed by clustering" technique from [@SivicZ03]. First, we use the bounding-box annotations in MSCOCO[@mscoco] and COCO-Stuff[@caesar2018cvpr] to extract 300,000 patches from the MSCOCO training set. After resizing each of the patches to $224 \times 224$ pixels, we extract ResNet-152[@Resnet] features for each of these patches, and cluster them into 1250 clusters using $k$-means clustering[@yinyang]. We note in Figure \[fig:code-words\] that the clusters have both expected and unexpected characteristics beyond “objectness" and “stuffness". Expected clusters include dominant objects in the MSCOCO dataset like zebras, giraffes, elephants, cars, buses, trains, people, etc. However, other clusters have textural content, unusual combinations of objects as well as actions. For example, we notice visual words like “people eating", “cats standing on toilets", “people in front of chain link fences", etc, as shown in Figure \[fig:code-words\]. The presence of these more *eclectic* code-words casts more insight into the model’s learning dynamics - we would prefer frequent itemsets containing the visual code-word corresponding to “people eating" than just “people" for a QA pair of *(what is she doing?, eating)*.
{width="\textwidth"}
From attention map to bounding box {#sec:box}
----------------------------------
In this work, we make an assumption that the network focuses on exactly one part of the image, although our method can be easily extended to multiple parts[@ChenBLL16]. Following the elucidation of our method in Section \[sec:method\] and given an attention map, we would like to compute the nearest visual code-word. Doing so requires making the choice of a bounding box that covers enough of the salient parts of the image, cropping and mapping this patch to the visual vocabulary. While there are trainable (deep network based) methods for cropping attention maps [@WangDeepCropping], we instead follow the simpler formulation suggested by [@ChenBLL16], which states that : within an attention-map $G$, given a percentage ratio $\tau$, find the smallest bounding box $B$ which satisfies : $$\sum_{p\epsilon B}{G(p)} \geq \tau \sum_p {G(p)}, \tau \epsilon [0,1]$$ Since we follow [@KazemiE17] who use a ResNet-152 architecture for visual feature extraction, the attention maps are of size $14 \times 14$. It can be shown easily that given a $m \times n$ grid, the number of unique bounding boxes that can be drawn on this grid, i.e., $num\_bboxes$ = $\frac{m \times n \times (m+1) \times (n+1)}{4}$, and when $m=n=14$, $num\_bboxes$ turns out to be 11,025. Because $m(=n)$ is small and fixed in this case, we pre-compute and enumerate all 11,025 bounding boxes and pick the smallest one which encompasses the desired attention, with $\tau=0.3$. The reason behind a conservatively low choice for $\tau$ is that we do not want to crop large regions of the image, which might contain distractor patches. This part of the pipeline is depicted in Figure \[fig:croppedcode-words\].
{width="\textwidth"}
Pipeline Summarized {#sec:pipeline}
-------------------
Now, the pipeline for the experiments (Figure \[fig:method\]) on the [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}dataset including images is as follows. We provide as input to the network - an image and a question. We observe the second attention map and use the method of Section \[sec:box\] to place a tight-fitting bounding-box around those parts of the image that the model attends to. We then extract features on this bounding-box using a ResNet-152 network and perform a $k$-nearest neighbor search (with $k=1$) to obtain its nearest visual word from the vocabulary. The words in the question, visual code-word and predicted answer for the entire validation set are provided as the database of transactions to the frequent itemset miner [@ChonH018], and rules are then obtained using the Apriori algorithm [@AgrawalAPriori].
Experiments {#sec:experiments}
===========
Language only statistical biases in [[<span style="font-variant:small-caps;">VQA</span> ]{}]{} {#sec:language_only}
----------------------------------------------------------------------------------------------
We show that a large number of statistical biases in VQA are due to language alone. We illustrate this with an obvious example : a language-only model, i.e., one that does not see the image, but still attempts the question, answers about 43% of the questions correctly on VQA 2.0 validation set and 48% of the questions correctly on VQA 1.0 validation set[@GoyalKSBP17]. However, on a random set of 200 questions from VQA 2.0, we observed empirically that the language-only model answers 88.0% of questions with a *plausibly correct* answer even with a harsh metric of what *plausible* means. Some of these responses are fairly sophisticated as can be seen in Table \[table:language\_only\_table\]. We note, for example, that questions containing “kind of bird" are met with a species of bird as response, “What kind of cheese" is answered with a type of cheese, etc. Thus, the model maps out key words or phrases in the question and *ostensibly* tries to map them through a series of rules to answer words. This strongly indicates that these are biases learned from the data, and the ostensible rules can be mined through a rule-mining algorithm.
--------------------------------------------------------- ------------ ------------
Question Predicted G.T Ans.
\[0.5ex\] What kind of bird is perched on this branch ? owl sparrow
What does that girl have on her face ? sunglasses nothing
What kind of cheese is on pizza ? mozzarella mozzarella
What is bench made of ? wood wood
electric LG
--------------------------------------------------------- ------------ ------------
: We run a language-only [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}baseline and note that although only 43% of the questions are answered correctly in VQA 2.0 ([@GoyalKSBP17]), a large number of questions (88%) in our experiments are answered with plausibly correct responses. For example, “Sunglasses" would be a perfectly plausible answer to the question “What does that girl have on her face?" - perhaps even more so than the ground-truth answer (“Nothing"). The shows an implausible answer provided by the model to the question.[]{data-label="table:language_only_table"}
Vision+Language statistical biases in [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}
------------------------------------------------------------------------------------------------
After applying the method of Section \[sec:method\], we will examine some rules that have been learned by our method on some popular question types in [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}. Question types are taken from [@antol2015vqa] and for the purpose of brevity, only a very few instructive rules for each question type are displayed. These question types are : “What is he/she doing?"\[sec:doing\], “Where?" (Figure \[fig:where\]), “How many?" (Section \[sec:many\]), “What brand?" (Figure \[fig:brand\]), and “Why?"(Section \[sec:why\]). The tables we present are to be interpreted thus : A question containing the antecedent words paired with an image containing the antecedent visual words can sometimes (but not always) lead to the consequent answer. Two instances of patches mapping to this visual word (Section \[sec:codebook\]) are provided. The presence of an $*$ after the consequent is to remind the reader that the consequent word came from the set of answers.
{width="\textwidth" height=".27\textheight"}
### What time? {#sec:time}
A selection of rules involving “What time?" questions are provided in Figure \[fig:types\_of\_vqa\_and\_how\_many\](right) which depend on whether the query is for the general time of the day, the current time obtained by reading a clock-face or the time (i.e., season) of the year. The model used in our work, [@KazemiE17], does not have the ability to read the time - it merely guesses a random time in the HH:MM format, as long as this is one of the answer categories. A single antecedent word phrase can be associated with multiple antecedent visual words. Indeed, there are several visual words associated with afternoon and night, but we have provided only two for brevity.
### How many? {#sec:many}
This particular instance of the trained [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}model seems to have learned that giraffes have four legs, stop signs have four letters, kitchen stoves have four burners and zebras and giraffes have several (100) stripes and spots respectively (Figure \[fig:howmany\]). Upon closer examination, we found 33 questions (out of >200k) in the [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}validation set which contain the words {How,many,burners} and the most common answer predicted by our model for these is 4 (which also resembles the ground-truth distribution). However, some of them were along the lines of “How many burners are turned on?", which led to answers different from “4".
### Why? {#sec:why}
Traditionally, “Why?" questions in [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}are considered challenging because they require a reason based answer. We describe some of the rules purportedly learned by our model for answering “Why?" questions, in Figure \[fig:why\]. Some interesting but intuitive beliefs that the model has learned are that movements cause blurry photographs ([why,blurry]{}$\rightarrow$movement), outstretching one’s arms help in balancing ([why,arm]{}$\rightarrow$balance) and that people wear helmets or orange vests for the purpose of safety ([why,helmet/orange]{}$\rightarrow$safety). In many of these cases, no visual element has been picked up by the rule mining algorithm - this strongly indicates that the models are memorizing the answers to the “Why?" questions, and not performing any reasoning. In other words, we could ask the question “Why is the photograph blurry?" to an irrelevant image and obtain “Movement" as the predicted answer.
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### What is he/she doing? {#sec:doing}
More interesting are our results on the “What is he/she doing?" category of questions (Figure \[fig:doing\]). While common activities like “snowboarding" or “surfing" are prevalant among the answers, we noticed a difference in rules learned for male and female pronouns. For the female pronoun (she/woman/girl/lady), we observed only stereotypical outputs like “texting" even for a very low support, as compared to a more diverse set of responses with the male pronoun. This is likely, a reflection on the inherent bias of the MSCOCO dataset which the [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}dataset of [@antol2015vqa; @GoyalKSBP17] is based on. Curiously, another work by [@HendrixSnowboard] had similar observations for image captioning models also based on MSCOCO.
{width="\textwidth" height=".26\textheight"}
{width="\textwidth"}
{width="\textwidth" height=".3\textheight"}
Limitations {#sec:limitations}
===========
While simplicity is the primary advantage of our method, some drawbacks are the following : the exact nature of the rules is limited by the process used to generate the visual vocabulary. In other words, while our method provides a unique insight into the behavior of a [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}model, there surely exist some rules that the models seem to follow which cannot be captured by this method. For example, rules involving colors are difficult to identify because ResNets are trained to be somewhat invariant to colors, so purely color-based visual words are hard to compute. Other examples include inaccurate visual code-words - for example, in rule 4 of Figure \[fig:brand\], the antecedant visual word does show a motorbike, although not a Harley Davidson. Similarly a code-word contains images of scissors and toothbrushes grouped together as part of the (What,brand$\rightarrow$Colgate) associate rule (rule 5 of Figure \[fig:brand\]).
Conclusion {#sec:conclusion}
==========
In this work, we present a simple technique to explicitly discover biases and correlations learned by [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}models. To do so, we store in a database - the words in the question, the response of the model to the question and the portion of the image attended to by the model. Our method then leverages the Apriori algorithm[@AgrawalAPriori] to discover rules from this database. We glean from our experiments that [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}models intuitively seem to correlate *elements* (both textual and visual) in the question and image to answers.
Our work is consistent with prior art in machine learning on fairness and accountability[@HendrixSnowboard], which often shows a skew towards one set of implied factors (like gender), compared to others. It is also possible to use the ideas in this work to demonstrate effectiveness of VQA systems - showing dataset biases presented by a frequent itemset and rule miner is a middle-ground between quantitative and qualitative results. Finally, our method is not limited only to [[<span style="font-variant:small-caps;">VQA</span> ]{}]{}, but any problem with a discrete vocabulary. A possible future extension of this work is to track the development of these rules as a function of training time.
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abstract: 'Inelastic effects arising from electron-phonon coupling in molecular Aharonov-Bohm (AB) interferometers are studied using the nonequilibrium Green’s function method. Results for the magnetoconductance are compared for different values of the electron-phonon coupling strength. At low bias voltages, the coupling to the phonons does not change the lifetime and leads mainly to scattering phase shifts of the conducting electrons. Surprisingly, opposite to the behavior of an electrical gate, the magnetoconductance of the molecular AB interferometer becomes more sensitive to the threading magnetic flux as the electron-phonon coupling is increased. PACS numbers: 73.63.-b, 73.63.Fg, 75.75.+a'
author:
- '$\mbox{Oded Hod}^{1\dagger}$, $\mbox{Roi Baer}^2$, and $\mbox{Eran Rabani}^1$'
title: 'Inelastic effects in Aharonov-Bohm molecular interferometer'
---
Control of conductance in molecular junctions is of key importance in molecular electronics [@Joachim2000; @Nitzan2003_1]. The current in these junctions is often controlled by an electrical gate designed to shift conductance peaks into the low-bias regime. Magnetic fields on the other hand, have been rarely used due to the small magnetic flux captured by molecular conductors (an exception is the Kondo effect in single-molecule transistors [@Park2002; @Liang02]). This is in contrast to a related field, electronic transport through mesoscopic devices, where considerable activity with magnetic fields has led to the discovery of the quantum hall effect [@Klitzing80] and a rich description of transport in such conductors [@Webb1985; @Timp1987; @Yacoby1995; @Oudenaarden1998; @Fuhrer2001]. The scarcity of experimental activity is due to the belief that significant magnetic response is obtained only when the magnetic flux, $\phi$, is on the order of the quantum flux, $\phi_0=h/e$ (where $e$ is the electron charge and $h$ is Planck’s constant). Attaining such a flux for molecular and nanoscale devices requires unrealistically huge magnetic fields [@Hod2004].
Recently, we have described the essential physical requirements necessary for the construction of nanometer scale magnetoresistance devices based on an AB [@Aharonov1959] molecular interferometer [@Hod2004; @Hod2005]. The basic idea was to weakly couple a molecular ring to conducting leads, creating a resonance tunneling junction. The resonant state was tuned by a gate potential to attain maximal conductance in the absence of a magnetic field. The application of a relatively small magnetic field shifts the state out of resonance, and conductance was strongly suppressed within fractions of the quantum flux. The combination of a gate potential and a magnetic field reveals new features and provides additional conductivity control [@Hod2005_1; @Hod2006].
Our previous study has neglected completely inelastic effects arising from electron-phonon interactions [@Ness1999; @Segal2002; @Sutton2002; @Hanggi2003; @Ueba2003; @Galperin2004; @Mitra2004]. Can a relatively small magnetic flux change significantly the conduction in molecular rings when the electron-phonon coupling becomes significant? Or, perhaps inelastic effects will broaden the resonant state and conduction will not be suppressed significantly upon the application of the magnetic field. The decay of the amplitude of the AB oscillations due to electron-phonon coupling has been studied for mesoscopic systems [@Guinea2002]. In this letter we address this problem for molecular rings, where we focus on the low range of the magnetic flux appropriate for molecular rings. Opposite to the effects of an electrical gate, we find that inelastic effect arising from electron-phonon couplings narrows the magnetoconductance peaks.
![A sketch of the AB ring. Each site on the ring can be occupied by a single electron. The ring sites are connect by springs with a frequency $\Omega$. An electron on site $j$ is coupled to the local motion of this site, with a coupling strength $M$.[]{data-label="fig:sketch"}](OdedHodfig1.ps){width="7cm"}
We consider a two terminal junction of an Aharonov-Bohm ring with $N$ sites as sketched in Fig. \[fig:sketch\]. A realization of this model to realistic molecular loops is described elsewhere [@Hod2004; @Hod2005; @Hod2005_1; @Hod2006]. We describe the electronic structure of the ring and the leads using a magnetic extended H[ü]{}ckel model [@Hod2004; @Hodthesis]. The description of the ring also includes local phonons and electron-phonon interactions are approximated to lowest order. The phonon local frequency $\Omega$ and the coupling $M$ of an electron on site $j$ to the local motion of site $j$ are the only two free parameters of the model. The full Hamiltonian in second quantization is given by: $$\begin{split}
&H = \sum_{i,j} t_{i,j}(B) c_{i}^{\dagger} c_{j} + \sum_{m,n \in L,R}
\epsilon_{m,n}(B) d_{m}^{\dagger} d_{n} +\\ &\left(\sum_{m,j}
V_{m,j}(B) d_m^\dagger c_j + H.c.\right) + \sum_{k=0}^{N-1} \hbar
\omega_{k} (b_{k}^{\dagger} b_{k}+1/2) \\ &+ \sum_{j,k=0}^{N-1}
M_{j}^{k} c_{j}^{\dagger} c_{j} (b_{k}^{\dagger} + b_{k}).
\end{split}
\label{eq:hamil}$$ The first two terms on the right hand side (RHS) of Eq. \[eq:hamil\] represent the zero-order electronic Hamiltonian of the ring and leads, respectively. $c_j^\dagger$ ($c_j$) is Fermion creation (annihilation) operators of an electron on site $j$ on the ring, and $d_m^\dagger$ ($d_m$) is Fermion creation (annihilation) operators of an electron on site $m$ on the left (L) or right (R) lead. $t_{i,j}(B)$ and $\epsilon_{i,j}(B)$ are the hopping matrix elements between site $i$ and site $j$ on the ring and lead, respectively. The third term on the RHS of Eq. \[eq:hamil\] corresponds to the coupling between the ring and the leads, where $V_{m,j}(B)$ is the hopping element between site $m$ on the lead and site $j$ on the ring. All hopping elements depend on the magnetic field $B$, which is taken to be uniform in the direction perpendicular to the ring plane. Both linear and quadratic terms in the magnetic field are included in the calculation [@Hod2004]. The last two terms in Eq. \[eq:hamil\] represents the Hamiltonian of the phonons and the electron-phonon interactions. $b_{k}^{\dagger}$ ($b_{k}$) is a boson creation (annihilation) operator of phonon mode $k$ with a corresponding frequency $\omega_{k}$. This set of phonon modes was obtained by a unitary transformation from local to normal coordinates of a one dimensional chain of coupled harmonic oscillators, characterized by a single frequency $\Omega$, as illustrate in Fig. \[fig:sketch\]. These frequencies constitute a band of width proportional to the coupling between the oscillators. The electron-phonon coupling is approximated to lowest order. Each site on the ring is coupled to all phonon modes with a coupling strength $M_{j}^{k}=M
\sqrt{\frac{\Omega}{\omega_{k}}} U_{jk}$, where $U_{jk}$ are the matrix elements of transformation matrix ${\bf U}$ from local to normal modes.
The calculation of the conductance is described within the framework of the nonequilibrium Green’s function (NEGF) method [@Datta_book]. The total current $I=I_{el}+I_{inel}$ is recast as a sum of elastic ($I_{el}$) and inelastic ($I_{inel}$) contributions given by [@Galperin2004; @Mitra2004; @Paulsson2005] $$\begin{split}
I_{el}=\frac{2e}{\hbar}&\int\frac{d\epsilon}{2\pi}
\left[f(\epsilon,\mu_{\mbox \tiny R})-f(\epsilon,\mu_{\mbox \tiny
L})\right]\\& \mbox{Tr}\left[{\bf \Gamma}_{\mbox \tiny
L}(\epsilon){\bf G}^{r}(\epsilon) {\bf \Gamma}_{\mbox \tiny
R}(\epsilon){\bf G}^{a}(\epsilon)\right]
\end{split}
\label{eq:elastic}$$ and $$\begin{split}
I_{inel}=\frac{2e}{\hbar}&\int\frac{d\epsilon}{2\pi}\mbox{Tr}\left[{\bf
\Sigma}_{\mbox \tiny L}^<(\epsilon) {\bf G}^{r}(\epsilon) {\bf
\Sigma}_{ph}^{>}(\epsilon) {\bf G}^{a}(\epsilon)\right.\\&\left.
-{\bf \Sigma}_{\mbox \tiny L}^>(\epsilon) {\bf G}^{r}(\epsilon) {\bf
\Sigma}_{ph}^<(\epsilon) {\bf G}^{a}(\epsilon)\right],
\label{eq:inelastic}
\end{split}$$ respectively. The retarded (advanced) GFs satisfy the Dyson equation $${\bf G}^{r,a}(\epsilon) = \left\{[{\bf g}^{r,a}(\epsilon)]^{-1} - {\bf
\Sigma}_{\mbox \tiny L}^{r,a}(\epsilon) - {\bf \Sigma}_{\mbox \tiny
R}^{r,a}(\epsilon) - {\bf \Sigma}_{ph}^{r,a}(\epsilon)\right\}^{-1},
\label{eq:dyson}$$ where ${\bf g}^{r,a}(\epsilon)$ is the uncoupled retarded (advanced) electronic GF of the ring. The greater (lesser) GFs satisfy the Keldysh equation at steady state (for an initial noninteracting state) $${\bf G}^{\lessgtr}(\epsilon) = {\bf G}^{r}(\epsilon) \left[{\bf
\Sigma}_{\mbox \tiny L}^{\lessgtr}(\epsilon) + {\bf \Sigma}_{\mbox \tiny
R}^{\lessgtr}(\epsilon) + {\bf \Sigma}_{ph}^{\lessgtr}(\epsilon)\right] {\bf
G}^{a}(\epsilon).
\label{eq:keldysh}$$ In the above equations, ${\bf \Sigma}_{\mbox \tiny
L}^{r,a,\lessgtr}(\epsilon)$, ${\bf \Sigma}_{\mbox \tiny
R}^{r,a,\lessgtr}(\epsilon)$, and ${\bf
\Sigma}_{ph}^{r,a,\lessgtr}(\epsilon)$ are the retarded ($r$), advanced ($a$), lesser ($<$) and greater ($>$) self-energies arising from the coupling to the left lead, right lead, and the phonons, respectively, and ${\bf \Gamma}_{\mbox{\tiny
L,R}}(\epsilon)=i\left[{\bf \Sigma}_{\mbox{\tiny
L,R}}^{r}(\epsilon)-{\bf \Sigma}_{\mbox{\tiny
L,R}}^{a}(\epsilon)\right]$, where $${\bf \Sigma}_{\mbox{\tiny L,R}}^{r,a}(\epsilon)= \left(\epsilon {\bf
S}^{*}- {\bf V}^{*}(B)\right) {\bf g}_{\mbox{\tiny
L,R}}^{r,a}(\epsilon) \left(\epsilon {\bf S} - {\bf V}(B)\right).
\label{eq:SigmaLRra}$$ In the above, ${\bf V}(B)$ is the lead-ring hopping matrix with elements $V_{mj}(B)$, ${\bf S}$ is the overlap matrix between the states on the leads and on the ring, and ${\bf g}_{\mbox{\tiny
L,R}}^{r,a}(\epsilon)$ is the retarded (advanced) uncoupled GF of the left or right lead. The corresponding leaser (greater) self-energies are given by $${\bf \Sigma}_{\mbox{\tiny
L,R}}^{\lessgtr}(\epsilon)=\left(\delta_{\lessgtr}-f(\epsilon,\mu_{\mbox{\tiny
\mbox{\tiny L,R}}}) \right) \left[{\bf \Sigma}_{\mbox{\tiny
L,R}}^r(\epsilon)-{\bf \Sigma}_{\mbox{\tiny L,R}}^a(\epsilon)\right],
\label{eq:SigmaLR}$$ where $\delta_{\lessgtr}$ equals $0$ for $<$ and $1$ otherwise, and $f(\epsilon,\mu)=\frac{1}{1+e^{\beta(\epsilon-\mu)}}$. The self-energy arising from the interactions with the phonons is calculated using the first Born approximation (FBA) and is given by [@Galperin2004; @Mitra2004; @Paulsson2005]: $$\begin{split}
{\bf \Sigma}_{ph}^{r}(\epsilon) &= i\sum_{k} \int \frac{d\omega}{2\pi}
{\bf M}^{k} \left\{ D_{k}^{<}(\omega) {\bf g}^{r}(\epsilon-\omega) +
\right.\\ & \left. D_{k}^{r}(\omega){\bf g}^{<}(\epsilon-\omega) +
D_{k}^{r}(\omega) {\bf g}^{r}(\epsilon-\omega) \right\} {\bf M}^{k} ,
\end{split}
\label{eq: SEph}$$ where the Hartree term has been omitted [@Mitra2004]. The lesser and greater self energies arising from the coupling to the phonons are given by: $${\bf \Sigma}_{ph}^\lessgtr (\epsilon) = i\sum_{k} \int
\frac{d\omega}{2\pi} {\bf M}^{k} D_{k}^\lessgtr(\omega) {\bf
g}^\lessgtr(\epsilon-\omega) {\bf M}^{k}.
\label{eq:SEphLG}$$ In the above equations, $D_{k}^{r,a}$ and $D_{k}^{\lessgtr}$ are the uncoupled equilibrium retarded (advanced) and lesser (greater) GFs of phonon mode $k$, respectively, ${\bf g}^{\lessgtr}(\epsilon)$ is the lesser (greater) uncoupled electronic GF of the ring, and ${\bf
M}^{k}$ is the electron-phonon coupling matrix of mode $k$ (diagonal in the $c_{j}$ basis).
![Conduction as a function of the gate voltage at zero magnetic flux with (dashed line) and without (solid line) electron-phonon coupling. Inset: A similar plot for the single resonant level model described in Ref. .[]{data-label="fig:GVgate"}](OdedHodfig2.eps){width="7cm"}
We now turn to discuss the results of a specific realization of the above model. We consider a ring composed of $N=40$ sites. The sites are identical and contribute a single electron which is described by a single Slater $s$-like orbital. The coupling between the ring and the leads is limited to the contact region. For simplicity, the electronic self-energies arising from this coupling are approximated within the wide band limit. Specifically, we neglect the real-part of the electronic self-energy and approximate ${\bf
\Gamma}_{L,R}(\epsilon)$ with matrices that are independent of energy, where the only non-vanishing elements are the diagonal elements ($\Gamma_{L,R}$) corresponding to the two sites coupled to the left or right lead. The local phonon frequency $\Omega=0.0125$eV is characteristic of a low frequency optical phonon in molecular devices. Since our model does not include a secondary phonon bath required to relax the energy from the optical phonons, we include a phonon energy level broadening $\eta = 0.016\Omega$ which is included in the uncoupled GFs of the phonon. The coupling to each of the leads is taken to be $\Gamma_{L}=\Gamma_{R}=4 \Omega$ such that the magnetoconductance switching in the absence of electron-phonon coupling is obtained at $\sim 5$ Tesla.
Before we address the effects of electron-phonon coupling on the magnetoconductance properties of the system described above we will analyze the role of a gate potential on the conductance. In Fig. \[fig:GVgate\] we plot the zero-bias conduction as a function of a gate voltage with ($M/\Gamma=1$, where $\Gamma=\Gamma_L+\Gamma_R$) and without ($M/\Gamma=0$) electron-phonon coupling. The gate voltage was modeled by an additional potential $eV_{g} {\bf S}$, where ${\bf S}$ is the overlap matrix, that was added to the ring hopping matrix element $t_{ij}$. For comparison (inset of Fig. \[fig:GVgate\]) we also include the results of a single resonant level coupled to a single phonon with identical model parameters used by Mitra [*et al*]{}. [@Mitra2004].
The two most significant observations are the expected broadening of the conduction when the electron-phonon coupling is turned on and the value of the zero-bias conduction ($g/g_{0}=1$, where $g_{0}=2e/h$ is the quantum conductance) in the presence of electron-phonon coupling. To better understand these results we rewrite the current for the case that ${\bf \Gamma}_{L}(\epsilon) = {\bf \Gamma}_{R}(\epsilon) \equiv
{\bf \Gamma}(\epsilon)/2$ in the following way [@Jauho1994]: $I=\frac{2e}{h} \int d\epsilon \left(f(\epsilon - \mu_{L}) -
f(\epsilon - \mu_{R})\right) {\cal T}(\epsilon)$ where ${\cal
T}(\epsilon) = \frac{i}{4} \mbox{Tr} {\bf \Gamma}(\epsilon) \left({\bf
G}^{r}(\epsilon) - {\bf G}^{a}(\epsilon) \right)$. Note that ${\cal
T}(\epsilon)$ is the transmission coefficient only when $M=0$. In the wide band limit, for the single resonant level model, ${\cal
T}(\epsilon)$ can be reduced to $\frac{\Gamma}{4} \frac{\Gamma -
2\Sigma_{ph,im}^{r}(\epsilon)} {\left(\epsilon - \epsilon_{0} -
\Sigma_{ph,re}^{r}(\epsilon) \right)^2 + \left(\Gamma/2 -
\Sigma_{ph,im}^{r}(\epsilon) \right)^2}$. As a result of the fact that $\Sigma_{ph,im}^{r}(0)=0$ at zero bias, the only inelastic contribution to the conduction comes from the real part of the phonon self-energy [@Mitra2004]. From this, it follows that even in the presence of electron-phonon coupling, the maximal conduction is $g_{\mbox{\tiny max}}/g_0=1$, as clearly can be seen in Fig. \[fig:GVgate\] for both cases. It also immediately implies that the main contribution to the broadening of the resonant conduction peak comes from the real-part of the phonon self-energy, i.e., from processes that lead to scattering phase shifts, but do not change the lifetime of the state.
So far we have discussed the effect of electron-phonon coupling on the zero-bias conduction as a function of a gate voltage. We now turn to discuss the major result of the present study. In Fig. \[fig:gb\] we plot the magnetoconductance of the AB-ring for several values of the electron-phonon couplings ($M$) and for different temperatures ($T$). We focus on the low value of the magnetic flux $\phi=A B$, where $A$ is the area of the ring and $B$ is taken perpendicular to the ring plane.
![Conductance as a function of magnetic flux for several values of the electron-phonon coupling strength $M$ and for different temperatures.[]{data-label="fig:gb"}](OdedHodfig3.eps){width="9cm"}
The case $M=0$ for different systems was discussed in detail in our previous studies, where the main goal was to establish the conditions required to achieve negative magnetoconductance and magnetic switching at low magnetic fields, despite the relatively large magnetic fields required to complete a full AB period [@Hod2004; @Hod2005; @Hod2006]. The essential procedure described in \[\] was to weakly couple the AB-ring to the conducting leads and at the same time to apply a gate potential to shift the position of the resonance state such that conduction is maximized at $\phi/\phi_0=0$. A manifestation of these ideas is depicted in Fig. \[fig:gb\] for the case that $M=0$ (solid curves), where the conduction is reduced from its maximal value to a small value at a relatively low magnetic flux. As expected, we find that as the temperature is increased the maximal value $g/g_0$ is decreased and the width of the magnetoconductance peaks is increased linearly with $T$ for $M/\Gamma=0$ (with deviations from linearity as $M/\Gamma$ is increased). This increase in the width with temperature is a result of resonant tunneling and the broadening of the Fermi distributions as $T$ is varied.
![Plots of ${\cal T}(\epsilon)$ as a function of energy for $M/\Gamma=0$ (solid curves) and $M/\Gamma=\frac{1}{4}$ (dashed curves) for different values of the magnetic flux. The dotted curve at the lower panel shows $\frac{\partial}{\partial \mu}\Delta
f(\epsilon-\mu)$ at $T/\Omega=0.007$, where $\Delta f(\epsilon-\mu)$ is the difference in the Fermi distribution of the left and right lead.[]{data-label="fig:ene"}](OdedHodfig4.eps){width="7cm"}
Turning to discuss the case of $M \ne 0$, one of the major questions is related to the effects of electron-phonon coupling on the switching capability of small AB-rings. Based on the discussion of the results shown in Fig. \[fig:GVgate\], one might expect that an increase in $M$ will lead to a broadening of the magnetoconductance peaks, thereby increasing the value of the magnetic field required to switch a nanometer AB-ring, and perhaps leads to unphysical values of $B$ required to reduced the conduction significantly. As can be seen in Fig. \[fig:gb\], the numerical solution of the NEGF for $M \ne 0$ leads to a [*reduction*]{} of the width of the magnetoconductance peaks, and the switching of the AB-ring is achieved at lower values of the magnetic flux compared to the case where $M=0$.
This surprising observation can be explained in simple terms. As discussed above, even in the presence of electron-phonon coupling, the maximal conduction at zero bias and zero temperature is $g_{\mbox{\tiny max}}/g_0=1$, as clearly is the case for the results shown in the upper left panel of Fig. \[fig:gb\] for $\phi/\phi_0=0$. For the symmetric ring of $N=4n$ the resonance condition at $\phi/\phi_0=0$ is equivalent to the condition that electrons entering the ring from left [*interfere constructively*]{} when they exit the ring to the right [@Hod2004; @Hod2005]. This picture also holds when $M \ne 0$, and the conduction takes a maximal value at $\phi/\phi_0=0$. The application of a magnetic field leads to destructive interference and increases the back scattering of electrons. This loss of phase is even more pronounced when inelastic effects arising from electron-phonon coupling are included. In the magnetoconductance this is translated to a more rapid loss of conduction as a function of the magnetic field when $M$ is increased.
Mathematically, the rapid decay of the conduction with the magnetic field as the electron-phonon coupling is increased can be explained by analyzing the dependence of ${\cal T}(\epsilon)$. In Fig. \[fig:ene\] we plot ${\cal T}_{el}(\epsilon) =
\mbox{Tr}\left[{\bf \Gamma}_{\mbox \tiny L}(\epsilon){\bf
G}^{r}(\epsilon) {\bf \Gamma}_{\mbox \tiny R}(\epsilon){\bf
G}^{a}(\epsilon)\right]$, which is elastic (and dominant) contribution to ${\cal T}(\epsilon)$, as a function of energy for several values of $\phi/\phi_0$ for $M/\Gamma=0$ or $M/\Gamma=\frac{1}{4}$. In the lower panel we also plot the corresponding Fermi distribution window. At $\phi/\phi_0=0$, ${\cal T}_{el}(\epsilon) \approx 1$ near the Fermi energy ($\epsilon_f$), independent of $M$, and the conduction is $g /
g_{0} \approx 1$. The application of a small magnetic field results in a split of ${\cal T}_{el}(\epsilon)$, where each peak corresponds to a different circular state [@Gefen2002]. The separation between the two peaks in the elastic limit $\Delta = (\epsilon_{2} - \epsilon_{1})
\propto \phi/\phi_0$ is proportional to the magnetic flux, where $\epsilon_{1,2}$ are the corresponding energies of the two circular states. When inelastic terms are included, due to the fact that the imaginary part of ${\bf \Sigma}_{ph}^{r}(\epsilon)$ is negligibly small, the renormalized positions of the two peaks can be approximated by $\epsilon^{*}_{1,2} = \epsilon_{1,2} +
\Sigma_{ph,re}^{r}(\epsilon_{1,2}) = \epsilon_{1,2} \pm
\Sigma_{ph,re}^{r}(\epsilon_{2})$, which implies that the renormalized separation between the two peaks can be approximated by $\Delta^{*} =
(\epsilon^{*}_{2} - \epsilon^{*}_{1}) = \Delta + 2
\Sigma_{ph,re}^{r}(\epsilon_{2})$. Therefore, as $M$ is increased $\Delta^{*}$ is also increased, consistent with the numerical results shown in Fig. \[fig:ene\].
Similarly to the electrical gate, the magnetic field provides means to externally control the conductance of a ring-shaped molecular junction. However, there are striking differences in the properties of these two gauges. This was illustrated previously in a multi-terminal device, where the polarity of the magnetic field, which couples to the electronic angular momentum, played a key role. In the present study we showed that there is also a fundamentally difference with respect to inelastic effects. While the conductance as a function of the gate voltage broadens due to coupling to phonons it actually narrows considerably in response to a magnetic field. This unexpected result was rationalized in terms of a rapid loss of the phase of electrons at the exit channel arising from the coupling to the phonons. Mathematically, this effect was traced to the form of the real part of the phonon self-energy that gives rise to scattering phase shifts, but does not change the lifetime of the resonant level through which conduction takes place.
We thank Abe Nitzan for many fruitful discussions. O.H. would like to thank the generous financial support of the Rothschild and Fulbright foundations. This work was supported by the Israel Science Foundation (grants to E.R. and R.B.). $^{\dagger}$Present address: Department of Chemistry, Rice University, Houston, TX 77251-1892.
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abstract: 'We consider two spin-1/2 particles with isotropic Heisenberg interaction, as the working substance of a quantum heat engine. We observe a frictional effect on the adiabatic branches of the heat cycle, which arises due to an inhomogeneous driving at a finite rate of the external magnetic field. The frictional effect is characterized by entropy production in the system and reduction in the work extracted. Corresponding to a sudden and a very slow driving, we find expressions for the lower and upper bounds of work that can be extracted on the adiabatic branches. These bounds are also confirmed with numerical simulations of the corresponding Liouville-von Neumann equation.'
author:
- 'George Thomas[^1] and Ramandeep S. Johal[^2]'
title: Friction due to inhomogeneous driving of coupled spins in a quantum heat engine
---
Introduction
============
The interplay between thermodynamic properties and quantum behavior has been studied through several models of quantum heat engines [@Scovil; @Alicki1979; @Geva1992; @Kosloff2002; @Kosloff2003; @Scully; @Mahler2004; @Kieu2004; @Kosloff2006; @TZhang; @AJM2008; @Lutz; @Noah2010; @GJ2011; @Abe2011; @Kosloff2012; @Huang2013]. Even though quantum systems show non-classical features like entanglement, coherence and so on, these models are often found to be consistent with thermodynamic interpretations [@Scully; @GJ2011; @Leff]. On the other hand, some models of these thermal machines have been reported to show an unexpected behavior such as extraction of work from a single heat bath [@Scully], exceeding Carnot efficiency [@Lutz] and cooling to absolute zero [@Kolar2012]. In this paper, we focus on the interesting phenomenon of intrinsic friction in quantum engines [@Kosloff2002; @Kosloff2003; @Kosloffarxiv; @Rezek2010]. This effect arises due to non-commutativity of the internal and the external part of the Hamiltonian leading to non-commutativity of the Hamiltonians at different times. Further to reduce friction, an effect called quantum lubrication has also been proposed [@Kosloff2006]. In order to better understand intrinsic friction and its relevance for analysis of dissipation in quantum systems, it seems interesting to look for this effect in other similar models.
In the quantum heat engine that we discuss below, the working medium (system) consists of two spin-half particles with Heisenberg interaction, kept in an external magnetic field. The system is driven by selectively changing the external field in finite time, such that the field on either spin is different (inhomogenous). In this case, we observe the frictional effect. But if the field on both spins remains homogeneous, then friction is absent. As expected, we find that if the driving that creates inhomogeneity of fields on the spins, is performed very slowly, the friction effect is again absent. An analogous system under an inhomogeneous magnetic field plays important role in quantum computing [@sarma2001]. Thermal entanglement of such spin system [@Asoudeh2005] and similar models [@Albayrak2009] have also been studied.
We consider a heat cycle analogous to Otto cycle, consisting of two adiabatic branches and two thermalization branches. On the thermalization branches, the magnetic fields on both spins are kept constant. To simplify our model, we consider the latter branches to take sufficiently long time so that the system attains equilibrium with the bath at the end of the process. Then the system is decoupled from the bath and a thermodynamically adiabatic process is carried out on the system. So the initial state of the system before the adiabatic process is a thermal state, diagonal in the eigenbasis of the Hamiltonian. This will help us to understand the coherence developed in the system during the adiabatic process [@Kosloff2003]. The development of coherence leads to an increase in the entropy of the system which is a signature of the friction observed in our model.
The paper is organised as follows. In section II, we introduce the model of the quantum heat engine. Entropy production in the adiabatic branches of the cycle is discussed in section III. Section IV is devoted for understanding the work extraction in our model. Here we discuss lower and upper bound of the work that can be extracted. Section V is devoted to discussion. We analyse the cycle using numerical simulations by alloting finite time to the adiabatic branches and conclude with a summary and future directions.
Model
=====
We consider two spin-half particles with isotropic exchange interaction, as the working substance for a quantum Otto cycle. In general, the Hamiltonian is written as $H=H_{\rm{int}}+H_{\rm{ext}}$, where $H_{\rm ext}$ is the external Hamiltonian which can be controlled and $H_{\rm{int}}$ is the internal Hamiltonian. In our model, we control in time, the magnetic field applied to particle labeled 2. So we have $$H_{\rm int} = J({\sigma^{(1)}}.{\sigma}^{(2)} + {\sigma}^{(2)}.{\sigma}^{(1)}),$$ $$H_{\rm ext}=B_1 \sigma^{(1)}_z+ B_2(t)\sigma^{(2)}_z,
\label{H}$$ where $\sigma^{(i)}=(\sigma_{x}^{(i)} ,\sigma_{y}^{(i)},\sigma_{z}^{(i)})$ are the Pauli matrices, $J$ is the isotropic exchange constant and $B_1$, $B_2(t)$ are the magnetic fields applied along $z$-axis to the first and the second spin respectively. So the magnetic field applied to the individual spins are not always equal during the adiabatic branch which results in $[H_{\rm ext},H_{\rm int}]\neq0$. This non-commutativity of the external and the internal Hamiltonian when leading to non-commutativity of the Hamiltonian at different times, is the cause of internal friction in our model [@Kosloff2002; @Kosloff2003].
As a special case, we show in Section V that the non-commutative property of external and the internal Hamiltonian by itself is not a sufficent condition for friction.
Now we analyse the system with inhomogeneous magnetic field in more detail. In this case the eigenbasis of the Hamiltonian is $\{|\psi_i\rangle; i=1,..,4\} \equiv$ {$|\psi_1\rangle$, $|00\rangle$, $|\psi_3\rangle$, $|11\rangle$}, where $|\psi_1\rangle$ and $|\psi_3\rangle$ are given by $b|10\rangle-a|01\rangle$ and $a|10\rangle+b|01\rangle$ respectively and {$|00\rangle$, $|10\rangle$, $|01\rangle$, $|11\rangle$} forms the computational basis. Here $a=(y+\sqrt{1+y^2})/N$ and $b=1/N$, where $N=\sqrt{1+(y+\sqrt{1+y^2})^2}$ and $y=(B_1-B_2(t))/4J$. The corresponding eigenvalues are {$-2J-K$, $2J-B_1-B_2(t)$, $-2J+K$, $2J+B_1+B_2(t)$}, where $K=4J(\sqrt{1+y^2})$. The equilibrium density matrix when the system is attached to a bath at temperature $T_e$, is given by $\rho= \exp{(-H/T_e)}/Z$, where $Z={\rm Tr} (\exp{(-H/T_e)})$ is partition function of the system, and we have set Boltzmann’s constant to unity. The eigenvalues of $\rho$, or the occupation probabilities of the energy levels, are given by $$\begin{aligned}
P_1&=&e^{-(-2J-K)/T_e}/Z,\nonumber \\
P_2&=&e^{-(2J-B_1-B_2(t))/T_e}/Z, \nonumber \\
P_3&=&e^{-(-2J+K)/T_e}/Z,\nonumber \\
P_4&=&e^{-(2J+B_1+B_2(t))/T_e}/Z.
\label{probability}\end{aligned}$$ Now we are ready to discuss the quantum heat cycle, which consists of the following four stages:
*Stage 1*: The coupled-spins system is attached to a cold bath with temperature $T_1$. The system attains equilibrium with the bath. The magnetic field applied to the first and second spins are identical ($B_1=B_2$). The density matrix is diagonal in the Hamiltonian’s eigenbasis. Because of the homogeneous magnetic field, the eigenstates $|\psi_1\rangle$ and $|\psi_3\rangle$ are maximally entangled Bell states, with $a=b=1/\sqrt{2}$. The occupation probability $\{p_j\}$ for the state with energy eigenvalue $\{E_j\}$ is calculated from Eq. (\[probability\]) by setting $T_e=T_1$ and $B_2=B_1$. So the mean energy at the end of the first stage is Tr$(\rho H)=\sum_jE_jp_j$.
*Stage 2*: In this stage, the system is isolated from the bath and it can exchange only work with the surroundings. The magnetic field applied to the second spin is changed from $B_2(0)=B_1$ to $ B_2(t)=B_3$ in finite time and the system may undergo a non-adiabatic evolution. By non-adiabatic evolution, we mean that the system may be driven fast enough so that the quantum adiabatic theorem does not hold [@Fock; @Kato]. The density matrix undergoes a unitary evolution. The eigenstates of $H(t)$ are also time dependent. In general, the eigenstates of $\rho(t)$ are not the same as $H(t)$. In the infinitely slow limit ($t\rightarrow \infty$), the adiabatic theorem holds and eigenstates of the density matrix are identical to the eigenvectors of the instantaneous Hamiltonian.
So in case of fast driving, the final state of the system may not be diagonal in the eigenbasis of the final Hamiltonian. When we project the final density matrix onto the eigenbasis of the Hamiltonian, the corresponding occupation probability of the eigenstate of the Hamiltonian with eigenvalue $E_j'$ is given as $p_j'={\rm Tr}\left(|j\rangle\langle j| \rho(t)\right)$, where $|j\rangle$ is the eigenvector of the final Hamiltonian. A pictorial representation is shown in Fig. \[A1\]. At the end of the second stage, the mean energy can be written as Tr$(\rho(t) H(t))= \sum_jE_j'p_j'$. The difference of the initial and the final mean energy during the adiabatic process is equal to the work performed: $W_I=\sum_j E_j p_j-\sum_jE_j' p_j'$.
*Stage 3*: The system under inhomogeneous magnetic field is attached to a hot bath with temperature $T_2$ and it attains equilibrium by absorbing heat from the bath. The occupation probabilities ($q_j$) are calculated from Eq. (\[probability\]) by putting $B_2=B_3$ and $T_e=T_2$. At the end of the third stage, the system is in a thermal state with mean energy $\sum_jE_j'q_j$.
*Stage 4*: The system again undergoes a unitary evolution by a change of the magnetic field of the second spin from $B_3$ to $B_1$, whereby the energy levels change from $E_j'$ back to $E_j$. The occupation probabilities $q_j'$ in the eigenstates of the Hamiltonian are calculated by projecting the density matrix onto the eigenbasis of the Hamiltonian. So the mean energy at the end of the process is $\sum_jE_jq_j'$. The difference in the mean energy due to this process is $W_{II}=\sum_jE_j'q_j-\sum_jE_jq_j'$.
To close the cycle, the system is again brought in contact with cold bath. The system releases on average an amount of heat to cold bath. As we show below, $W_I$ and $W_{II}$ are the work done [*by*]{} and [*on*]{} the system, respectively.
![A pictorial representation of eigenvalues and eigenstates of the Hamiltonian at the end of first adiabatic process (stage 2). The $\{ p_i' \}$ represent the populations in the energy eigenbasis $\{ |\psi_i' \rangle \}$. In the infinite time limit, we get $p_i'=p_i$ and the eigenstates of the density matrix are same as that of the Hamiltonian.[]{data-label="A1"}](fig1.eps)
Dynamics on adiabatic branch and entropy
========================================
Now we analyse the irreversibility associated with the adiabatic branch by quantifying the entropy production. The adiabatic process is represented by a unitary process so that after time $t$, the system-state evolves to $\rho(t)=U(t,0)\rho(0)U^{\dagger}(t,0)$, where $U={\cal T} \exp{(-i\int_0^tH(t')dt')}$. The von Neumann entropy $S_v$ remains constant throughout the process. But energy-entropy $S_e$, defined with the occupational probabilities of the energy levels, changes. $S_e$ in the initial state is given by $ -\sum_ip_i\ln p_i$, where $p_i={\rm Tr}(|\psi_i\rangle \langle \psi_i|\rho(0))$. Since the initial state is a thermal state, we have $S_e=S_v$. But after the finite-time adiabatic step, $S_e$ increases where as $S_v$ remains unchanged. Initially, we have $[H(0),\rho(0)]=0$. Two of the eigenvectors $| 00 \rangle$ and $|11\rangle$ of the Hamiltonian are not functions of the applied magnetic field and hence are independent of time. So if the system is in any of these ${\it two}$ eigenstates, it will remain there during the process. Thus the initial population in theses states remains constant throughout the adiabatic process.
But the eigenvectors $|\psi_1 \rangle$ and $|\psi_3\rangle$ of hamiltonian depend on the magnetic field and hence are time dependent. So if the system is initially in one of these states, then changing the Hamiltonian with a finite rate results in a non-adiabatic evolution. In in other words, the final state of the system is then not an eigenstate of the final Hamiltonian. Let the eigenvectors of the final Hamiltonian be given as $\{ |\psi_1'\rangle$, $|00\rangle$, $|\psi_3'\rangle$, $|11\rangle \}$ and the set of the eigenvectors of the final density matrix is {$|\phi_1'\rangle$, $|00\rangle$, $|\phi_3'\rangle$, $|11\rangle$}. Since $|\phi_1'\rangle$ and $|\phi_3'\rangle$ are orthogonal to each other as well as to $|00\rangle$ and $|11\rangle$, we can express the kets $|\phi_1'\rangle$ and $|\phi_3'\rangle$ as linear combinations of $|\psi_1'\rangle$ and $|\psi_3'\rangle$ as $$\begin{aligned}
|\phi_1'\rangle&=&\cos(\delta/2)|\psi_1'\rangle + \sin(\delta/2)|\psi_3'\rangle, \nonumber \\
|\phi_3'\rangle&=&\sin(\delta/2)|\psi_1'\rangle - \cos(\delta/2)|\psi_3'\rangle,
\label{phipsi}\end{aligned}$$ where $0 \le \delta \le \pi$. Now consider a projection of the system-state on the eigenbasis of Hamiltonian. Two of the populations remain unchanged such that $p_2'=p_2$ and $p_4'=p_4$. The occupation probabilities for the eigenstates $|\phi'_1\rangle$ and $|\phi'_3\rangle$ are $p_1$ and $p_3$ respectively. Now project the density matrix onto the eigenbasis $\{ |\psi_i' \rangle \}$ of the final Hamiltonian. From Eq. (\[phipsi\]), we get the occupation probabilities corresponding to $|\psi'_1\rangle$ and $|\psi'_3\rangle$ as $$\begin{aligned}
p_1'&=&p_1\cos^2(\delta/2)+p_3 \sin^2(\delta/2), \nonumber \\
p_3'&=&p_1\sin^2(\delta/2)+p_3 \cos^2(\delta/2).\end{aligned}$$ Due to $p_1 > p_3$, we can write $$\begin{aligned}
p_1&\geq&p_1'\geq p_3, \nonumber \\
p_1&\geq&p_3'\geq p_3.
\label{p1p3}\end{aligned}$$ As the difference between $p_1'$ and $p_3'$ gets reduced as compared to the one between $p_1$ and $p_3$, and recalling that $p_2'=p_2$ and $p_4'=p_4$, the distribution $\{ p_i' \}$ is more uniform than $\{ p_i \}$, we have $$-\sum_ip_i'\ln p_i'\geq -\sum_i p_i\ln p_i,
\label{entropy}$$ which signifies that the energy-entropy $S_e$ increases in the finite-time adiabatic process. In the infinite time process ($t \rightarrow \infty$), the system undergoes quantum adiabatic evolution and in this limit $S_e$ remains unchanged. The total entropy production versus the total time allocated to adiabatic branch will be discussed in Section V.
Work
====
The work is performed by or on the system only during the adiabatic branches i.e. in stages 2 and 4, when the evolution of the system is governed by Liouville-von Neumann equation (with $\hbar=1$) $$\frac{d\rho(t)}{dt}=-i\left[H(t),\rho(t)\right].
\label{Liouville}$$ The instantaneous mean energy of the system is given by ${\rm Tr}(\rho(t)H(t))$. Differentiating with respect to time we get $${\rm Tr}\left(\frac{d(H(t)\rho(t))}{dt}\right)= {\rm Tr}\left(H(t)
\frac{d\rho(t)}{dt}\right)+{\rm Tr}\left(\frac{dH(t)}{dt}\rho(t)\right),
\label{dmeanE}$$ In general, comparing with the first law of thermodynamics, we identify [@Alicki1979] the first term on the right hand side as the rate of heat flow ($\dot{Q}$) and the second term as the power ($\wp$).
For an adiabatic process, the first term above on the right hand side vanishes due to Eq. (\[Liouville\]).
Upon integrating the power, we get the expression for work as $$\begin{aligned}
W = \int_0 ^{t} \wp dt &=& \int_0^{t}{\rm Tr}\left(\frac{dH(t')}{dt'}\rho(t')\right)dt', \nonumber \\
&=&\int_0^{t}{\rm Tr}\left(\frac{d(H(t')\rho(t'))}{dt'}\right)dt'.
\label{workderivation}\end{aligned}$$ Thus the work performed during the adiabatic process lasting for a time interval $t$, is equal to the change in the mean energy of the system, upto time $t$.
Furthermore, it can be shown that the work done in a infinitely slow process is always higher than the work done in a finite-time process. Thus the lower bound for work extracted is obtained for an extremely fast process ($ t\rightarrow 0$). To evaluate the lower bound, we assume that the density matrix of the system remains unchanged. In case of equilibrium with the cold bath, the initial density matrix is given as $$\rho = p_1\arrowvert\phi_1\rangle\langle\phi_1| + p_2\arrowvert00\rangle\langle00| +
p_3\arrowvert\phi_3\rangle\langle\phi_3| + p_4\arrowvert11\rangle\langle11|,
\label{rho}$$ where $|\phi_{1} \rangle = (\arrowvert10\rangle-\arrowvert01\rangle)/\sqrt{2}$ and $|\phi_{3} \rangle = (\arrowvert10\rangle+\arrowvert01\rangle)/\sqrt{2}$. Since the system is in thermal state, the initial Hamiltonian commutes with the density matrix and both have the same set of eigenvectors. In the sudden limit ($t\rightarrow0$), the density matrix remains the same as the initial, because $U(0,0)=I$. But the Hamiltonian is changed to $$\begin{aligned}
H=&-&(2J+K)\arrowvert\psi'_1\rangle\langle\psi'_1|+(2J-B_1-B_3)\arrowvert00\rangle\langle00| \nonumber \\
&+&(-2J+K)\arrowvert\psi'_3\rangle\langle\psi'_3|+ (2J+B_1+B_3)\arrowvert11\rangle\langle11|, \nonumber \\\end{aligned}$$ where $|\psi'_{1} \rangle =b\arrowvert10\rangle-a\arrowvert01\rangle$ and $|\psi'_{3} \rangle =a\arrowvert10\rangle+b\arrowvert01\rangle$. Now we find the population of the corresponding eigenstates of the Hamiltonian by projecting the density matrix onto the eigenbasis of Hamiltonian as $$\begin{aligned}
p_1'=\langle\psi'_1|\rho|\psi'_1\rangle=\frac{(p_1+p_3)}{2}-ab(p_3-p_1), \\
%
p_3'=\langle\psi'_3|\rho|\psi'_3\rangle=\frac{(p_1+p_3)}{2}+ab(p_3-p_1),
%
\label{p3'p1'}\end{aligned}$$ while $p_2'=p_2$ and $p_4'=p_4$. Similarly for the second adiabatic process where the Hamiltonian is returned to its initial stage with eigenbasis $\{ | \psi_i \rangle \}$, we obtain upon projecting the density matrix $\tilde{\rho}$ for this process, as $$\begin{aligned}
q_1'=\langle\psi_1|\tilde{\rho}|\psi_1\rangle=\frac{(q_1+q_3)}{2}-ab(q_3-q_1), \\
q_3'=\langle\psi_3|\tilde{\rho}|\psi_3\rangle=\frac{(q_1+q_3)}{2}+ab(q_3-q_1),
\label{q3'q1'}\end{aligned}$$ and $q_2'=q_2$ and $q_4'=q_4$.
Now the work extracted in complete cycle ($W=W_I + W_{II}$) with fast adiabatic processes is given by $$W^{\rm fast} =\sum_i p_i E_i - \sum_i p_i' E_i' + \sum_i q_iE_i' - \sum_i q_i'E_i.
\label{Wfast}$$ Using the probabilities calculated above for extremely fast (sudden) processes, we get the lower bound of work $$\begin{aligned}
W_{\rm lb}=& & (B_3-B_1)(q_4-q_2+p_2-p_4) \nonumber \\
&& + (q_3-q_1)(K-8Jab).
\label{wlb} \end{aligned}$$ The upper bound for work is obtained for the slow process ($t\rightarrow \infty$). According to quantum adiabatic theorem the system remains in the instantaneous eigenstate of the Hamiltonian. The work expression is in general written as $$W^{\rm slow} = \sum_i p_i E_i - \sum_i p_i E_i' + \sum_i q_i E_i' - \sum_i q_i E_i,
\label{Wslow}$$ yields the upper bound for the extractable work, $$\begin{aligned}
W_{\rm ub} = && (B_3-B_1)(q_4 - q_2 + p_2 - p_4) \nonumber \\
&& +(q_3-q_1+p_1-p_3)(K-4J).
\label{wub}\end{aligned}$$ These bounds are compared with the finite-time work in Fig. (\[figurework\]).
Discussion
==========
Analytic expressions for work can be derived both in the case of a very slow driving and a sudden one. To estimate the finite-time evolution of the system on the adiabatic branch, we have to integrate the Liouville-von Neumann equation, Eq. (\[Liouville\]). We accomplish this using the fourth-order Runge-Kutta method [@Mahler2006]. In the first adiabatic process, $B(t)$ changes from $B_2(0)$ to $B_3$. This is modeled by applying a pulse $B_2(t)=B_2(0)+ (B_3-B_2(0))\sin{({\pi t}/{\tau})}$ for a time $t=\tau/2$, where $\tau$ is half of the time period. Similarly the second adiabatic process is done by applying a pulse $B_2(t) = B_3 + (B_2(0)-B_3)\sin{({\pi t}/{\tau})}$ for the same time interval. Thus we allot equal time intervals to both the adiabatic branches. The total work performed and the total entropy production due to the finite time process are plotted in Fig. (\[figurework\]). As discussed in the previous section, the work extracted decreases monotonically with a finite rate of driving. This is also reflected in the corresponding increase of entropy production in the finite-time case.
![Work obtained in a cycle versus the total time ($\tau$) allocated for both the adiabatic processes. Here we use $B_1=B_2(0)=3$, $B_3=4$, $J=0.1$, $T_2=2$ and $T_1=1$. The work is bounded above by $W_{\rm ub}$ (Eq. (\[wub\]), dashed line). The thick horizontal line depicts the lower bound $W_{\rm lb}$, Eq. (\[wlb\]), obtained for a sudden adiabatic process ($\tau \to 0$). The inset shows total entropy production on the adiabatic branches versus total time ($\tau$). As $\tau$ is increased, the total entropy production reduces monotonically to zero and the frictional effect vanishes.[]{data-label="figurework"}](w-e.eps){width="7.8cm" height="5cm"}
Let us consider a cycle in which the magnetic fields applied to the first and second spins in [stage 1]{} have different values, $B_1(0)$ and $B_2(0)$ respectively. In this case the internal and external part of the hamiltonian do not commute with each other. Now suppose that during the first adiabatic process, $B_1$ and $B_2$ vary at equal rates so that the difference ($\Delta B = B_1(t) - B_2 (t)$) keeps constant during the process. As we have seen in section II, the parameters $a$ and $b$ appearing in the eigenbasis of the Hamiltonian, are functions of $J$ and $\Delta B$. Since $\Delta B$ remains constant during the adiabatic process, the energy eigenstates of the Hamiltonian become time independent which implies that Hamiltonians at different times commute with each other and so friction is absent in this case. Using the similar argument, no friction is expected on the second adiabatic process, when the magnetic fields are restored to their initial values ($B_1 (0)$ and $B_2(0)$). This serves as an example to appreciate that the non-commutative property of the internal and external Hamiltonian caused by the inhomogeneous magnetic fields may not always lead to non commutativity of Hamiltonian at different times to cause friction. Rather the inhomogeneous driving in which $\Delta B$ changes with time leads to the non commutative property of the Hamiltonian at different times and thereby to frictional effect.
To conclude, we have studied a model of quantum heat engine where the inhomogeneous driving at a finite rate, of the components of the quantum working medium leads to a frictional effect. This effect is characterized by increase in the entropy of the system. As expected of a thermodynamic system, the entropy production leads to decrease in the work obtained from a cycle. The work is plotted versus the time allotted for the adiabatic branches. The amount of work that can be obtained from our model is bounded from both above and below. The upper bound is obtained for a slow process where the frictional effect vanishes and quantum adiabatic theorem holds, while the lower bound is obtained for a sudden process. Some interesting future problems include the study of frictional effect on models with anisotropic interactions and with systems using higher number of spins. The possibility of quantum lubrication [@Kosloff2006] to reduce the intrinsic friction can also be studied.
Acknowledgements {#acknowledgements .unnumbered}
================
GT acknowledges financial support from IISER Mohali. RSJ gratefully acknowledges reearch grant from the Department of Science and Technology, India, under project No. SR/S2/CMP-0047/2010(G) titled, [*Quantum heat engines: work, entropy and information at nanoscale*]{}.
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[^1]: electronic address: george@iisermohali.ac.in
[^2]: electronic address: rsjohal@iisermohali.ac.in
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abstract: 'The survey is devoted to associative $\Z_{\ge0}$-graded algebras presented by $n$ generators and $\frac{n(n-1)}2$ quadratic relations and satisfying the so-called Poincare-Birkhoff-Witt condition (PBW-algebras). We consider examples of such algebras depending on two continuous parameters (namely, on an elliptic curve and a point on this curve) which are flat deformations of the polynomial ring in $n$ variables. Diverse properties of these algebras are described, together with their relations to integrable systems, deformation quantization, moduli spaces and other directions of modern investigations.'
author:
- Alexander Odesskii
title: Elliptic algebras
---
Introduction {#introduction .unnumbered}
============
In the paper \[45\] devoted to study the $XYZ$-model and the representations of the corresponding algebra of monodromy matrices, Sklyanin introduced the family of associative algebras with four generators and six quadratic relations which are nowadays called Sklyanin algebras (see also Appendix D.1). The algebras of this family are naturally indexed by two continuous parameters, namely, by an elliptic curve and a point on this curve, and each of them is a flat deformation of the polynomial ring in four variables in the class of $\Z_{\ge0}$-graded associative algebras. On the other hand, a family of algebras with three generators (and three quadratic relations) with the same properties arose in \[2\], \[34\] (see also \[52\]). In what follows it turned out (see \[10\], \[17\]-\[22\], \[32\]-\[38\]) that such algebras exist for arbitrarily many generators. The algebras in question are associative algebras of the following form. Let $V$ be a linear space of dimension $n$ over the field $\C$. Let $L\subset V\otimes V$ be a subspace of dimension $\frac{n(n-1)}2$. Let us construct an algebra $A$ with the space of generators $V$ and the space of defining relations $L$, that is, $A=T^*V/(L)$, where $T^*V$ is the tensor algebra of the space $V$ and $(L)$ is the two-sided ideal generated by $L$. It is clear that the algebra $A$ is $\Z_{\ge0}$-graded because the ideal $(L)$ is homogeneous. We have $A=\C\oplus A_1\oplus A_2\oplus\dots$, where $A_1=V$, $A_2=V\otimes
V/L$, $A_3=V\otimes V\otimes V/V\otimes L+L\otimes V$, etc.
We say that $A$ is a PBW-algebra (or satisfies the Poincare-Birkhoff-Witt condition) if $\dim A_\alpha=\frac{n(n+1)\dots(n+\alpha-1)}{\alpha!}$.
Thus, a PBW-algebra is an algebra with $n$ generators and $\frac{n(n-1)}2$ quadratic relations for which the dimensions of the graded components are equal to those of the polynomial ring in $n$ variables.
Algebras of this kind arise in diverse areas of mathematics: in the theory of integrable systems \[45\], \[46\], \[28\], \[9\], moduli spaces \[20\], deformation quantization \[12\], \[26\], non-commutative geometry \[2\], \[3\], \[11\], \[27\], \[47\]-\[49\], \[51\], cohomology of algebras \[8\], \[29\], \[41\]-\[44\], \[50\], and quantum groups and $R$-matrices \[45\], \[46\], \[25\], \[16\], \[14\], \[23\], \[31\]. See Appendix D.
Since there are no classification results in the theory of PBW-algebras (for $n>3$), we deal with specific examples only. The known examples can conditionally be divided into two classes, namely, rational and elliptic algebras. Let us present examples of rational algebras.
1\. [*Skew polynomials.*]{} This is the algebra with the generators $\{x_i;i=1,\dots,n\}$ and the relations $x_ix_j=q_{i,j}x_jx_i$, where $i<j$ and $q_{i,j}\ne0$.
One can readily see that the monomials $\{x_1^{\alpha_1}\dots x_n^{\alpha_n};\alpha_1,\dots,\alpha_n\in\Z_{\ge0}\}$ form a basis of the algebra of skew polynomials, which implies the PBW condition. Since $q_{i,j}$ are arbitrary non-zero numbers, we have obtained an $\frac{n(n-1)}2$-parameter family of algebras.
2\. [*Projectivization of Lie algebras.*]{} Let $\g$ be a Lie algebra of dimension $n-1$ with a basis $\{x_1,\dots,x_{n-1}\}$. We construct an algebra with $n$ generators $\{c,x_1,\dots,x_{n-1}\}$ and the relations $cx_i=x_ic$ and $x_ix_j-x_jx_i=c[x_i,x_j]$.
The condition PBW follows from the Poincare-Birkhoff-Witt theorem for the algebra $\g$.
3\. [*Drinfeld algebra.*]{} A new realization of the quantum current algebra $U_q(\wh{\slg}_2)$ was suggested in \[13\] (see also \[25\]). Namely, the generators $x_k^\pm,h_k$ ($k\in\Z$) similar to the ordinary basis of the Lie algebra $\wh{\slg}_2$ were introduced. It is assumed that the elements $x_k^+$ satisfy the quadratic relations $$x_{k+1}^+x_l^+-q^2x_l^+x_{k+1}^+=q^2x_k^+x_{l+1}^+-x_{l+1}^+x_k^+.$$ The elements $x_k^-$ satisfy similar relations. The algebra $\Dr_n(q)\subset U_q(\wh{\slg}_2)$ generated by $x_1^+,\dots,x_n^+$, $n\in\N$, $q\in\C^*$, is a PBW-algebra.
In the elliptic case the algebra depends on two continuous parameters, namely, an elliptic curve $\E$ and a point $\eta\in\E$. Just these algebras are the subject of our survey. Their structure constants are elliptic functions of $\eta$ with modular parameter $\tau$. Our main example is given by the algebras $Q_{n,k}(\E,\eta)$, where $n\ge3$ is the number of generators, $k$ is a positive integer coprime to $n$, and $1\le k<n$. We define the algebra $Q_{n,k}(\E,\eta)$ by the generators $\{x_i;i\in\Z/n\Z\}$ and the relations $$\sum_{r\in\Z/n\Z}\frac{\theta_{j-i+r(k-1)}(0)}
{\theta_{kr}(\eta)\theta_{j-i-r}(-\eta)}x_{j-r}x_{i+r}=0.$$ The structure of these algebras depends on the expansion of the number $n/k$ in the continued fraction, and therefore we first study the simplest case $k=1$ and then pass to the general case. The fact that the algebra $Q_{n,k}(\E,\eta)$ belongs to the class of PBW-algebras is proved only for generic parameters $\E$ and $\eta$ (see §2.6 and §3). However, we conjecture that this holds for any $\E$ and $\eta$. A possible way to prove this conjecture is to produce an analog of the functional realization (see §2.1) for arbitrary $k$ by using the constructions in §5.
As we consider, the algebras $Q_{n,k}(\E,\eta)$ are a typical example of elliptic algebras; however, they are far from exhausting the list of all elliptic algebras. The simplest example of an elliptic algebra that does not belong to this class (and even is not a deformation of the polynomial ring) can be constructed as follows. Let the group $(\Z/2\Z)^2$ with the generators $g_1,g_2$ act by automorphisms on the algebra $Q_4(\E,\eta)$ as follows: $g_1(x_i)=x_{i+2}$, $g_2(x_i)=(-1)^ix_i$. The same group acts on the algebra of ($2\times2$) matrices, $g_1(\gamma)=\begin{pmatrix}-1&0\\0&1\end{pmatrix}\gamma\begin{pmatrix}-1&0\\0&1\end{pmatrix}^{-1}$, $g_2(\gamma)=\begin{pmatrix}0&1\\1&0\end{pmatrix}\gamma\begin{pmatrix}0&1\\1&0\end{pmatrix}^{-1}$. This gives an action on the tensor product of associative algebras $Q_4(\E,\eta)\otimes\Mat_2$. Let $\wt Q_4(\E,\eta)\subset
Q_4(\E,\eta)\otimes\Mat_2$ consist of elements invariant with respect to the group action. One can readily see that the dimension of the graded components of $\wt Q_4(\E,\eta)$ coincide with those of $Q_4(\E,\eta)$, and therefore $\wt Q_4(\E,\eta)$ is a PBW-algebra. For another example of PBW-algebra (with $3$ generators), see the end of §1.
Let us now describe one of the main constructions of PBW-algebras. Let $\lambda(x,y)$ be a meromorphic function of two variables. We construct an associative graded algebra $\F_\lambda$ as follows. Let the underlying linear space of $\F_\lambda$ coincide with $\F_\lambda=\C\oplus F_1\oplus F_2\oplus\dots$, where $F_1=\{f(u)\}$ is the space of meromorphic functions of one variable and $F_\alpha=\{f(u_1,\dots,u_\alpha)\}$ is the space of symmetric meromorphic functions of $\alpha$ variables. The space $F_\alpha$ is a natural extension of the symmetric power $S^\alpha F_1$. The multiplication in the algebra $\F_\lambda$ is defined as follows: for $f\in F_\alpha$, and $g\in F_\beta$ the product $f*g\in
F_{\alpha+\beta}$ is
$f*g(u_1,\dots,u_{\alpha+\beta})=$ $$=\frac1{\alpha!\beta!}\sum_{\sigma\in
S_{\alpha+\beta}}f(u_{\sigma_1},\dots,u_{\sigma_\alpha})
g(u_{\sigma_{\alpha+1}},\dots,u_{\sigma_{\alpha+\beta}})
\prod_{\begin{subarray}{c}1\le i\le\alpha\\\alpha+1\le
j\le\alpha+\beta\end{subarray}}\lambda(u_{\sigma_i},u_{\sigma_j}).$$ In particular, if $f,g\in F_1$, then $$f*g(u_1,u_2)=f(u_1)g(u_2)\lambda(u_1,u_2)+
f(u_2)g(u_1)\lambda(u_2,u_1).$$ One can readily see that the multiplication $*$ is associative for any $\lambda(x,y)$.
We now assume that $\lambda(x,y)=\frac{x-qy}{x-y}$, where $q\in\C^*$. Let $F_1^{(n)}=\{1,u,\dots,u^{n-1}\}\subset F_1$ be the space of polynomials of degree less than $n$. Let $F_\alpha^{(n)}=S^\alpha
F_1^{(n)}\subset F_\alpha$ be the space of symmetric polynomials in $\alpha$ variables of degree less than $n$ with respect to any variable. One can readily see that $F_\alpha^{(n)}*F_\beta^{(n)}\subseteq
F_{\alpha+\beta}^{(n)}$. Therefore, the algebra $\F_\lambda^{(n)}=\oplus_\alpha
F_\alpha^{(n)}$ is a subalgebra of $\F_\lambda$. Moreover, for $q=1$ the algebra $\F_\lambda^{(n)}$ is the polynomial ring $S^*F_1^{(n)}$ because $\lambda(x,y)=1$ in this case. Therefore, the algebra $\F_\lambda^{(n)}$ is a PBW-algebra for generic $q$. This algebra is isomorphic to the Drinfeld algebra $\Dr_n(q)$, and an isomorphism is given by the rule $u^k\mapsto x_{k+1}^+$. The algebra $Q_n(\E,\eta)$ can be obtained in a similar way with the only modification that the polynomials are replaced by theta functions (see §2.1). A similar construction \[38\], \[22\] enables one to construct quantum moduli spaces $\M(\E,B)$ (see Appendix D.3) for any Borel subgroup $B$. The construction of algebras $Q_{n,k}(\E,\eta)$ for $k>1$ (and, more generally, quantum moduli spaces $\M(\E,P)$ for a parabolic subgroup $P$) is more complicated and involves exchange algebras (see §5 and \[21\]) or elliptic $R$-matrices (see §4).
Let us now describe the contents of the survey. In §1 we describe the simplest elliptic PBW-algebras, namely, algebras $Q_3(\E,\eta)$ with three generators. These algebras were studied in many papers, see, for instance, \[2\], \[3\]. The section is of illustrative nature; we intend to explain some methods of studying elliptic algebras by the simplest example. The main attention in the survey is paid to the algebras $Q_n(\E,\eta)$, which are discussed in §2. We give an explicit construction of these algebras, present natural families of their representations (which are studied in \[19\] in more detail), and describe the symplectic leaves of the corresponding Poisson algebra (we recall that $Q_n(\E,0)$ is the polynomial ring in $n$ variables).
The structure of the algebras $Q_{n,k}(\E,\eta)$, $k>1$, is more complicated, and the detailed description of their properties is beyond the framework of the survey (see \[35\], \[20\]). The main properties of these algebras are described in §3. In §4 we explain the relationship between these algebras and Belavin’s elliptic $R$-matrices. In §5 we establish a relation of the algebras $Q_{n,k}(\E,\eta)$ to the so-called exchange algebras (see (24), (25), and also \[36\], \[24\], \[33\]). In Appendices A, B, C we present the notation we need and the properties of theta functions of one and several variables. Appendix D contains a brief survey of relations of elliptic algebras with other areas of mathematics. We tried to make this part independent of the main text.
In conclusion we say a few words concerning the facts that remain outside the survey but are immediately connected with its topic. In \[37\] the algebras $Q_{n,k}(\E,\eta)$ are studied provided that $\eta\in\E$ is a point of finite order. In this case the properties of the algebras $Q_{n,k}(\E,\eta)$ are similar to those of quantum groups if $q$ is a root of unity; in particular, these algebras are finite-dimensional over the centre. In \[32\] we study rational degenerations of the algebras $Q_{n,k}(\E,\eta)$ occurring if the elliptic curve $\E$ degenerates into the union of several copies of $\CP^1$ or into $\CP^1$ with a double point.
The algebras $Q_{n,k}(\E,\eta)$ are obtained when quantizing the components of the moduli spaces $\M(P,\E)$ (see Appendix D.3) that are isomorphic to $\CP^{n-1}$. The quantization of other components leads to elliptic algebras of more general form. These algebras were constructed in \[38\], \[22\] if $P$ is a Borel subgroup of an arbitrary group $G$. The case in which $P\subset GL_m$ is an arbitrary parabolic subgroup of $GL_m$ is studied in \[21\].
The symplectic leaves of a Poisson manifold corresponding to the family of algebras $Q_{n,k}(\E,\eta)$ in a neighbourhood of $\eta=0$ and for a fixed elliptic curve $\E$ were studied in \[20\].
The corresponding Poisson algebras belong to the class of algebras with regular structure of symplectic leaves; these algebras were studied in \[39\].
Algebras with three generators
==============================
In this section we consider the simplest examples of elliptic PBW-algebras, namely, the algebras with three generators. Let us first study the quadratic Poisson structures on $\C^3$. Let $x_0,x_1,x_2$ be the coordinates on $\C^3$ and let there be a Poisson structure that is quadratic in these coordinates. We construct the polynomial $C=x_0\{x_1,x_2\}+x_1\{x_2,x_0\}+x_2\{x_0,x_1\}$. This is a homogeneous polynomial of degree three because the Poisson structure is quadratic. It is clear that the form of this polynomial is preserved under linear changes of coordinates (up to proportionality). Let us restrict ourselves to the non-degenerate case in which the equation $C=0$ defines a non-singular projective manifold. It is clear that this is an elliptic curve. Moreover, by a linear change of variables one can reduce the polynomial $C$ to the form $C=x_0^3+x_1^3+x_2^3+3kx_0x_1x_2$, where $k\in\C$. In this case, as one can readily see by using the definition of $C$ and the Jacoby identity, the Poisson structure must be of the form (up to proportionality): $$\{x_0x_1\}=x_2^2+kx_0x_1,\quad\{x_1x_2\}=x_0^2+kx_1x_2,\quad
\{x_2x_0\}=x_1^2+kx_2x_0.$$ Moreover, $\{x_i,C\}=0$, and every central element is a polynomial in $C$. We recall that each Poisson manifold can be partitioned into the so-called symplectic leaves, which are Poisson submanifolds, and the restrictions of the Poisson structure to these submanifolds are non-degenerate. In our case, the symplectic leaves are as follows:
1\) the origin $x_0=x_1=x_2=0$;
2\) the homogeneous manifold $C=0$ without the origin;
3\) the manifolds $C=\lambda$, where $\lambda\in\C$, $\lambda\ne0$.
It is clear that our Poisson structure admits the automorphisms $x_i\mapsto\epsilon^ix_i$ and $x_i\mapsto x_{i+1}$, where $\epsilon^3=1$, $i\in\Z/3\Z$. It is natural to assume that the quantization of the Poisson structure (see Appendix D.2) is the family of associative algebras with the generators $x_0,x_1,x_2$ and three quadratic relations admitting the same automorphisms. However, each generic three-dimensional space of quadratic relations which is invariant with respect to these automorphisms is of the form $$\begin{aligned}
x_0x_1-qx_1x_0&=px_2^2,\\
x_1x_2-qx_2x_1&=px_0^2,\\
x_2x_0-qx_0x_2&=px_1^2,
\end{aligned}$$ where $p,q\in\C$ are complex numbers. We denote by $A_{p,q}$ the algebra with the generators $x_0,x_1,x_2$ and the defining relations (6). It is clear that the algebra $A_{p,q}$ is $\Z_{\ge0}$-graded, that is, $A_{p,q}=\C\oplus F_1\oplus
F_2\oplus\dots$, where $F_\alpha F_\beta\subseteq F_{\alpha+\beta}$. Here $F_\alpha$ stands for the linear space spanned by the (non-commutative) monomials in $x_0,x_1,x_2$ of degree $\alpha$. It is natural to expect that the dimension of $F_\alpha$ is equal to that of the space of polynomials in three variables of degree $\alpha$, that is, $\dim F_\alpha=\frac{(\alpha+1)(\alpha+2)}2$.
Moreover, the Poisson algebra (5) has a central function $C=x_0^3+x_1^3+x_2^3+3kx_0x_1x_2$, and the centre is generated by the element $C$. Therefore, it is natural to expect that for generic $p$ and $q$ the algebra $A_{p,q}$ has a central element of the form $C_{p,q}=\phi x_0^3+\psi x_1^3+\mu
x_2^3+\lambda x_0x_1x_2$, where $\phi,\psi,\mu$, and $\lambda$ are functions of $p$ and $q$ (one can verify the existence of an element $C_{p,q}$ by the immediate calculation), and the centre is generated by $C_{p,q}$.
The standard technique of proving such assertions (for instance, the Poincare-Birkhoff -Witt theorem for Lie algebras) makes use of the filtration on an algebra and the study of the graded adjoint algebra. In our case the algebra is already graded, and one cannot proceed by the ordinary induction on the terms of lesser filtration; therefore we use another technique. Namely, we shall study a certain class of modules over the algebra $A_{p,q}$ and try to obtain results on the algebra $A_{p,q}$ by using an information on the modules. The following class of modules is useful for our purposes.
A module over a $\Z_{\ge0}$-graded algebra $A$ is said to be linear if it is $\Z_{\ge0}$-graded as an $A$-module, generated by the space of degree $0$, and the dimensions of all components are equal to $1$.
Let us study the linear modules over the algebra $A_{p,q}$. By definition, a linear module $M$ admits a basis $\{v_\alpha,\alpha\ge0\}$ with the following action of the generators: $$x_0v_\alpha=x_\alpha v_{\alpha+1},\quad x_1v_\alpha=y_\alpha
v_{\alpha+1},\quad x_2v_\alpha=z_\alpha v_{\alpha+1},$$ where $x_\alpha,y_\alpha,z_\alpha$ are sequences, and $x_\alpha,y_\alpha,z_\alpha$ do not vanish simultaneously for any $\alpha$ (we want $M$ be generated by $v_0$). A change of the basis of the form $v_\alpha\to\lambda_\alpha v_\alpha$ multiplies the triple $(x_\alpha,y_\alpha,z_\alpha)\in\C^3$ by $\frac{\lambda_{\alpha+1}}{\lambda_\alpha}$, that is, the module $M$ is defined by the sequence of points $(x_\alpha:y_\alpha:z_\alpha)\in\CP^2$ uniquely up to isomorphism of graded modules. It is clear that a sequence of points $(x_\alpha:y_\alpha:z_\alpha)\in\CP^2$ defines a module over the algebra $A_{p,q}$ if and only if the relations (6) hold for the operators on $M$ corresponding to this sequence. This is equivalent to the following relations: $$\begin{aligned}
x_{\alpha+1}y_\alpha-qy_{\alpha+1}x_\alpha&=pz_{\alpha+1}z_\alpha,\\
y_{\alpha+1}z_\alpha-qz_{\alpha+1}y_\alpha&=px_{\alpha+1}x_\alpha,\\
z_{\alpha+1}x_\alpha-qx_{\alpha+1}z_\alpha&=py_{\alpha+1}y_\alpha.
\end{aligned}$$
The relations (7) form a system of linear equations for $x_\alpha,y_\alpha,z_\alpha$ which has a non-zero solution (by the assumption on the module $M$), and therefore the determinant $\begin{vmatrix}-qy_{\alpha+1}&x_{\alpha+1}&-pz_{\alpha+1}\\
-px_{\alpha+1}&-qz_{\alpha+1}&y_{\alpha+1}\\
z_{\alpha+1}&-py_{\alpha+1}&-qx_{\alpha+1}\end{vmatrix}$ must vanish. Similarly, the relations (7) form a system of linear equations on $x_{\alpha+1},y_{\alpha+1},z_{\alpha+1}$ that has a non-zero solution, and therefore $\begin{vmatrix}y_\alpha&-qx_\alpha&-pz_\alpha\\
-px_\alpha&z_\alpha&-qy_\alpha\\
-qz_\alpha&-py_\alpha&x_\alpha
\end{vmatrix}=0$. One can readily see that these determinants give the same cubic polynomial in three variables. We see that for any $\alpha\ge0$ the point with the coordinates $(x_\alpha:y_\alpha:z_\alpha)$ belongs to the cubic $$x_\alpha^3+y_\alpha^3+z_\alpha^3+\frac{p^3+q^3-1}{pq}
x_\alpha y_\alpha z_\alpha=0.$$ Moreover, if a point $(x_\alpha:y_\alpha:z_\alpha)$ belongs to this cubic, then, solving the system of linear equations (7) with respect to $x_{\alpha+1},y_{\alpha+1},z_{\alpha+1}$, we obtain a new point $(x_{\alpha+1}:y_{\alpha+1}:z_{\alpha+1})$ on the same cubic (because the determinant of the system (7) must be equal to $0$). Thus, the system (7) defines an automorphism of the projective manifold (8). Let us choose some $k=\frac{p^3+q^3-1}{pq}$. Then, varying $q$, we obtain a one-parameter family of automorphisms of the projective curve in $\CP^2$ given by the equation $x^3+y^3+z^3+kxyz=0$. As is known, for generic $k$ this equation defines an elliptic curve. Let this curve be $\E=\C/\Gamma$, where $\Gamma$ is an integral lattice generated by $1$ and $\tau$, where $\Im\tau>0$. The parameter $k$ is a function of $\tau$. If $k$ is chosen, then, passing to the limit as $q\to1$, we see that $p\to0$, and the automorphism defined by (7) tends to the identity automorphism. Therefore, our family of automorphisms of the elliptic curve $\E$ given by the equation (8) is a deformation of the identity automorphism. Thus, every automorphism of this family is a translation, of the form $u\to u+\eta$, where $u,\eta\in\E=\C/\Gamma$. Let $u_\alpha\in\E=\C/\Gamma$ be a point with the coordinates $(x_\alpha:y_\alpha:z_\alpha)$. We see that $u_{\alpha+1}=u_\alpha+\eta$, where $\eta$ depends only on the algebra, that is, on $p$ and $q$. Hence, $u_\alpha=u+\alpha\eta$, where $u\in\E$ is the parameter of the module $M$. We have obtained the following result.
The linear modules over the algebra $A_{p,q}$ are indexed by a point of the elliptic curve $\E\subset\CP^2$ given by the equation $x^3+y^3+z^3+k_{p,q}xyz=0$, where $k_{p,q}=\frac{p^3+q^3-1}{pq}$. The module $M_u$ corresponding to a point $u\in\E$ is given by the formulas $$x_0v_\alpha=x_\alpha v_{\alpha+1},\quad x_1v_\alpha=y_\alpha
v_{\alpha+1},\quad x_2v_\alpha=z_\alpha v_{\alpha+1},$$ where $(x_\alpha:y_\alpha:z_\alpha)$ are the coordinates of the point $u+\alpha\eta\in\E$. Here the shift $\eta$ is determined by $p$ and $q$.
We note that, when studying linear modules, for an algebra $A_{p,q}$ we have constructed both an elliptic curve $\E\subset\CP^2$ and a point $\eta\in\E$. In what follows we shall see that, conversely, the algebra $A_{p,q}$ can be reconstructed from $\E$ and $\eta$. Thus, two continuous parameters, $\E$ (that is $\tau$) and $\eta$, give a natural parametrization of the algebras $A_{p,q}$. Therefore, we change the notation and denote the algebra $A_{p,q}$ by $Q_3(\E,\eta)$.
Let us now apply a uniformization of the elliptic curve $\E\subset\CP^2$ given by the equation (8) by theta functions of order three (see Appendix A). A point $u\in\E=\C/\Gamma$ has the coordinates $(\theta_0(u):\theta_1(u):\theta_2(u))\in\CP^2$. In this notation, the module $M_u$ is given by the formulas $$x_0v_\alpha=\theta_0(u+\alpha\eta)v_{\alpha+1},\quad
x_1v_\alpha=\theta_1(u+\alpha\eta)v_{\alpha+1},\quad
x_2v_\alpha=\theta_2(u+\alpha\eta)v_{\alpha+1}.$$ Let $\e$ be the linear operator in the space with basis $\{v_\alpha,\alpha\ge0\}$ given by the formula $\e v_\alpha=v_{\alpha+1}$. Let $u$ be the diagonal operator in the same space such that $\e u=(u-\eta)\e$. We have $uv_\alpha=(u_0+\alpha\eta)v_\alpha$ for some $u_0\in\C$. It is clear that the generators of the algebra $Q_3(\E,\eta)$ in the representation $M_u$ become $$x_0=\theta_0(u)\e,\quad
x_1=\theta_1(u)\e,\quad
x_2=\theta_2(u)\e.$$
This gives the following reformulation of the description of linear modules.
Let us consider the $\Z_{\ge0}$-graded algebra $B(\eta)=\C\oplus
B_1\oplus B_2\oplus\dots$, where $B_\alpha=\{f(u)\e^\alpha\}$, $f$ ranges over all holomorphic functions, and the multiplication is given by the formula *:* $f(u)\e^\alpha\cdot
g(u)\e^\beta=f(u)g(u-\alpha\eta)\e^{\alpha+\beta}$. Then there is an algebra homomorphism $\phi\colon Q_3(\E,\eta)\to B(\eta)$ such that $x_0\to\theta_0(u)\e$, $x_1\to\theta_1(u)\e$, $x_2\to\theta_2(u)\e$.
Proposition 2 provides a lower bound for the dimension $\dim F_\alpha$ of the graded components of the algebra $Q_3(\E,\eta)$. Really, the homomorphism $\phi$ preserves the grading, that is, $\phi(F_\alpha)\subset B_\alpha$. We have $$\phi(x_{i_1}\dots
x_{i_\alpha})=\theta_{i_1}(u)\e\dots\theta_{i_\alpha}(u)\e=
\theta_{i_1}(u)\theta_{i_2}(u-\eta)\dots
\theta_{i_2}(u-(\alpha-1)\eta)\e^\alpha.$$ Thus, $\phi(F_\alpha)$ is the linear space (of holomorphic functions) spanned by the functions $\{\theta_{i_1}(u),\dots,\theta_{i_2}(u-(\alpha-1)\eta)\};
i_1,\dots,i_\alpha=0,1,2$. It is clear that all these functions are theta functions of order $3\alpha$ and belong to the space $\Theta_{3\alpha,\frac{\alpha(\alpha-1)}23\eta}(\Gamma)$. One can readily prove that the image $\phi(F_\alpha)$ coincides with the entire space $\Theta_{3\alpha,\frac{\alpha(\alpha-1)}23\eta}(\Gamma)$, and hence $\dim\phi(F_\alpha)=3\alpha$. We have obtained the bound $\dim F_\alpha\ge3\alpha$. On the other hand, we know that $\dim F_\alpha\le\frac{(\alpha+1)(\alpha+2)}2$ because the relations in $Q_3(\E,\eta)$ are deformations of the relations in the polynomial ring in three variables. We expect that the equality $\dim
F_\alpha=\frac{(\alpha+1)(\alpha+2)}2$ holds for generic $\tau$ and $\eta$. Let us compare these numbers: $$\begin{array}{c|c|c|c|c}
\alpha&1&2&3&4\\
\hline
\dim F_\alpha(\text{conjecture})&3&6&10&15\\
\hline
\dim\phi(F_\alpha)&3&6&9&12
\end{array}$$ We see that the first discrepancy holds for $\alpha=3$; possibly $\phi$ has a one-dimensional kernel on the space $F_3$. It can be shown that, really, there is a cubic element $C\in Q_3(\E,\eta)$ such that $C\ne0$ and $\phi(C)=0$. The element $C$ turns out to be central, that is, $x_\alpha C=Cx_\alpha$ for $\alpha=0,1,2$. Passing to the limit as $\eta\to0$ (for a fixed $\tau$), we see that $C\to x_0^3+x_1^3+x_2^3+kx_0x_1x_2$ because the $\theta_i(u)$s uniformize the elliptic curve, that is, $\theta_0^3+\theta_1^3+\theta_2^3+k\theta_0\theta_1\theta_2=0$. Further, if $C$ is central and is not a zero divisor (the latter obviously holds for generic $\tau$ and $\eta$), then every element $\ker\phi$ must be divisible by $C$ according to the dimensional considerations. Really, the graded linear space $\oplus_{\alpha\ge0}F_\alpha$ turns out to be not smaller than $\left(\oplus_{\alpha\ge0}\Theta_{3\alpha,\frac{\alpha(\alpha-1)}23\eta}\right)\otimes\C[C]$, where $\deg C=3$. One can readily see that the component of degree $\alpha$ of this tensor product of graded linear spaces is of dimension $\frac{(\alpha+1)(\alpha+2)}2$. However, we know that $\dim F_\alpha\le\frac{(\alpha+1)(\alpha+2)}2$, which implies $\dim F_\alpha=\frac{(\alpha+1)(\alpha+2)}2$. We have obtained the following result.
For generic $\tau$ and $\eta$ the algebra $Q_3(\E,\eta)$ has a cubic central element $C$. The quotient algebra $Q_3(\E,\eta)/(C)$ is isomorphic to $\oplus_{\alpha\ge0}\Theta_{3\alpha,\frac{\alpha(\alpha-1)}23\eta}(\Gamma)$, where the product of elements $f\in\Theta_{3\alpha,\frac{\alpha(\alpha-1)}23\eta}(\Gamma)$ and $g\in\Theta_{3\beta,\frac{\beta(\beta-1)}23\eta}(\Gamma)$ is given by the formula $f*g(u)=f(u)g(u-3\alpha\eta)$.
It follows from our description of $Q_3(\E,\eta)/(C)$ that this algebra is centre-free for generic $\eta$. Therefore, the centre of the algebra $Q_3(\E,\eta)$ is generated by the element $C$.
Let us now find the relations in the algebra $Q_3(\E,\eta)$, that is, let us express $p$ and $q$ in term of $\tau$ and $\eta$. We have $x_ix_{i+1}-qx_{i+1}x_i-px_{i+2}^2=0$ (these are the relations in (6)). Applying the homomorphism $\phi$, we obtain $$\theta_i(u)\theta_{i+1}(u-\eta)-q\theta_{i+1}(u)\theta_i(u-\eta)-
p\theta_{i+2}(u)\theta_{i+2}(u-\eta)=0.$$ Hence (see (28) in Appendix A), $q=-\frac{\theta_1(\eta)}{\theta_2(\eta)}$, $p=-\frac{\theta_0(\eta)}{\theta_2(\eta)}$.
The similar investigation of the Sklyanin algebra with four generators (see Appendix D.1) gives the following result.
For a generic Sklyanin algebra $S$ with four generators and the relations (39) one can find an elliptic curve $\E=\C/\Gamma$ defined by two quadrics in $\CP^3$ and a point $\eta\in\E$ such that *:*
*1)* there is a graded algebra homomorphism $\phi\colon
S\to B(\eta)$;
*2)* the image of this homomorphism in $B_\alpha$ is $\Theta_{4\alpha,\frac{\alpha(\alpha-1)}24\eta+\frac\alpha2}(\Gamma)$;
*3)* the kernel of this homomorphism is generated by two quadratic elements $C_1$ and $C_2$.
Thus, $S/(C_1,C_2)=\oplus_{\alpha\ge0}\Theta_{4\alpha,\frac{\alpha(\alpha-1)}24\eta+\frac\alpha2}(\Gamma)$.
The Sklyanin algebra $S$ can be reconstructed from $\E$ and $\eta$. Let us denote this algebra by $Q_4(\E,\eta)$.
The following natural question arises: Does there exist a similar algebra $Q_n(\E,\eta)$ for any $n$?
To answer this question, the information concerning linear modules is insufficient because these modules are too small to reconstruct the algebra $Q_n(\E,\eta)$ for any $n$. Really, the algebra $Q_n(\E,\eta)$ must have the functional dimension $n$, whereas the linear modules are of dimension one. Therefore, these modules can be used only when reconstructing a quotient algebra of $Q_n(\E,\eta)$. To overcome these difficulties, it is natural to study more general modules. Namely, let us study modules over the algebra $Q_3(\E,\eta)$ with a basis $\{v_{i,j};i,j\in\Z_{\ge0}\}$ and such that the generators of the algebra $Q_3(\E,\eta)$ take any element $v_{ij}$ to a linear combination of $v_{i+1,j}$ and $v_{i,j+1}$. Calculations show that every such module is of the form $$x_iv_{\alpha,\beta}=\frac{\theta_i(u_1+(\alpha-2\beta)\eta)}
{\theta(u_1-u_2+3(\alpha-\beta)\eta)}v_{\alpha+1,\beta}+
\frac{\theta_i(u_2+(\beta-2\alpha)\eta)}
{\theta(u_2-u_1+3(\beta-\alpha)\eta)}v_{\alpha,\beta+1},$$ where $i\in\Z/3\Z$, $\alpha,\beta\in\Z_{\ge0}$, and $u_1,u_2\in\C$. Thus, the modules of this kind are indexed by a pair of points $u_1,u_2\in\E$. If we now assume that the algebra $Q_n(\E,\eta)$ has analogous modules (see (15)), then the above information uniquely defines the algebra $Q_n(\E,\eta)$.
1\. One can pose the following more general problem. Let $M\subset\CP^{n-1}$ be a projective manifold and let $T$ be an automorphism of $M$. For a point $u\in M$ we denote by $z_i(u)$ (where $i=0,\dots,n-1$) the homogeneous coordinates of $u$. Does there exist a PBW-algebra with $n$ generators $\{x_i,i=0,\dots,n-1\}$ that has a linear module $L_u$ (for any point $u\in M$) given by the formula $x_iv_\alpha=z_i(T^\alpha
u)v_{\alpha+1}$? Here $T^\alpha u$ stands for $T(T(\dots T(u)\dots)$. The algebras $Q_{n,k}(\E,\eta)$ are a solution of this problem for some $M$ and $T$, namely, if $M=\E^p$ is a power of a curve $\E$ and $T$ a translation (see §5, Proposition 12). Here $p$ stands for the length of the expansion of $n/k=n_1-\frac1{n_2-\ldots-\frac1{n_p}}$ in the continued fraction.
2\. Let [^1] $A_3$ be the algebra with the generators $x,y,z$ and the relations $\epsilon
zx+\epsilon^5y^2+xz=0$, $\epsilon^2z^2+yx+\epsilon^4xy=0$, and $zy+\epsilon^7yz+\epsilon^8x^2=0$, where $\epsilon^9=1$. This PBW-algebra corresponds to the case in which $M\subset\CP^2$ is an elliptic curve given by the equation $x^3+y^3+z^3=0$ and $T$ is an automorphism corresponding to the complex multiplication on $M$. The algebra $A_3$ is not a quantization of any Poisson structure on $\C^3$.
Algebra $Q_n(\E,\eta)$
======================
Construction
------------
For any $n\in\N$, any elliptic curve $\E=\C/\Gamma$, and any point $\eta\in\E$ we construct a graded associative algebra $Q_n(\E,\eta)=\C\oplus F_1\oplus F_2\oplus\dots$, where $F_1=\Theta_{n,c}(\Gamma)$ and $F_\alpha=S^\alpha\Theta_{n,c+(\alpha-1)n}(\Gamma)$. By construction, $\dim F_\alpha=\frac{n(n+1)\dots(n+\alpha-1)}{\alpha!}$. It is clear that the space $F_\alpha$ can be realized as the space of holomorphic symmetric functions of $\alpha$ variables $\{f(z_1,\dots,z_\alpha)\}$ such that $$\begin{aligned}
f(z_1+1,z_2,\dots,z_\alpha)&=f(z_1,\dots,z_\alpha),\\
f(z_1+\tau,z_2,\dots,z_\alpha)&=(-1)^ne^{-2\pi i(nz_1-c-(\alpha-1)n)}f(z_1,\dots,z_\alpha).
\end{aligned}$$
For $f\in F_\alpha$ and $g\in F_\beta$ we define the symmetric function $f*g$ of $\alpha+\beta$ variables by the formula $$\begin{gathered}
f*g(z_1,\dots,z_{\alpha+\beta})=\frac1{\alpha!\beta!}
\sum_{\sigma\in S_{\alpha+\beta}}
f(z_{\sigma_1},\dots,z_{\sigma_\alpha})
g(z_{\sigma_{\alpha+1}}-2\alpha\eta,\dots,z_{\sigma_{\alpha+\beta}}-
2\alpha\eta)\times\\
\times\prod_{\begin{subarray}{c}1\le i\le\alpha\\\alpha+1\le
j\le\alpha+\beta\end{subarray}}
\frac{\theta(z_{\sigma_i}-z_{\sigma_j}-n\eta)}
{\theta(z_{\sigma_i}-z_{\sigma_j})}.\end{gathered}$$ In particular, for $f,g\in F_1$ we have $$f*g(z_1,z_2)=f(z_1)g(z_2-2\eta)\frac{\theta(z_1-z_2-n\eta)}
{\theta(z_1-z_2)}+f(z_2)g(z_1-2\eta)\frac{\theta(z_2-z_1-n\eta)}
{\theta(z_2-z_1)}.$$ Here $\theta(z)$ is a theta function of order one (see Appendix A).
If $f\in F_\alpha$ and $g\in F_\beta$, then $f*g\in F_{\alpha+\beta}$. The operation $*$ defines an associative multiplication on the space $\oplus_{\alpha\ge0}F_\alpha$
Let us show that $f*g\in F_{\alpha+\beta}$. It immediately follows from the assumptions (9) concerning $f$ and $g$ and also from the properties of $\theta(z)$ (see Appendix A) that every summand in the formula for $f*g$ satisfies condition (9) for $F_{\alpha+\beta}$. Hence, $f*g$ is a meromorphic symmetric function satisfying condition (9). This function can have a pole of order not exceeding one on the diagonals $z_i-z_j=0$ and also for $z_i-z_j\in\Gamma$ because $\theta(z)$ has zeros for $z\in\Gamma$. However, the order of a pole of a symmetric function on the diagonal must be even. This implies that the function $f*g$ is holomorphic for $z_i=z_j$, and it follows from (9) that $f*g$ is holomorphic for $z_i-z_j\in\Gamma$ as well.
One can immediately see that the multiplication $*$ is associative.
Main properties of the algebra $Q_n(\E,\eta)$
---------------------------------------------
By construction, the dimensions of the graded components of the algebra $Q_n(\E,\eta)$ coincide with those for the polynomial ring in $n$ variables. For $\eta=0$ the formula for $f*g$ becomes $$f*g(z_1,\dots,z_{\alpha+1})=\frac1{\alpha!\beta!}\sum_{\sigma\in
S_{\alpha+\beta}}f(z_{\sigma_1},\dots,z_{\sigma_\alpha})
g(z_{\sigma_{\alpha+1}},\dots,z_{\sigma_{\alpha+\beta}}).$$ This is the formula for the ordinary product in the algebra $S^*\Theta_{n,c}(\Gamma)$, that is, in the polynomial ring in $n$ variables. Therefore, for a fixed elliptic curve $\E$ (that is, for a fixed modular parameter $\tau$) the family of algebras $Q_n(\E,\eta)$ is a deformation of the polynomial ring. In particular (see Appendix D.2), there is a Poisson algebra, which we denote by $q_n(\E)$. One can readily obtain the formula for the Poisson bracket on the polynomial ring from the formula for $f*g$ by expanding the difference $f*g-g*f$ in the Taylor series with respect to $\eta$. It follows from the semicontinuity arguments that the algebra $Q_n(\E,\eta)$ with generic $\eta$ is determined by $n$ generators and $\frac{n(n-1)}2$ quadratic relations. One can prove (see §2.6) that this is the case if $\eta$ is not a point of finite order on $\E$, that is, $N\eta\not\in\Gamma$ for any $N\in\N$.
The space $\Theta_{n,c}(\Gamma)$ of the generators of the algebra $Q_n(\E,\eta)$ is endowed with an action of a finite group $\wt{\Gamma_n}$ which is a central extension of the group $\Gamma/n\Gamma$ of points of order $n$ on the curve $\E$ (see Appendix A). It immediately follows from the formula for the product $*$ that the corresponding transformations of the space $F_\alpha=S^\alpha\Theta_{n,c}(\Gamma)$ are automorphisms of the algebra $Q_n(\E,\eta)$.
Bosonization of the algebra $Q_n(\E,\eta)$
------------------------------------------
The main approach to obtain representations of the algebra $Q_n(\E,\eta)$ is to construct homomorphisms from this algebra to other algebras with simple structure (close to Weil algebras) which have a natural set of representations. These homomorphisms are referred to as bosonizations, by analogy with the known constructions of quantum field theory.
Let $B_{p,n}(\eta)$ be a $\Z^p$-graded algebra whose space of degree $(\alpha_1,\dots,\alpha_p)$ is of the form $\{f(u_1,\dots,u_p)\e_1^{\alpha_1}\dots\e_p^{\alpha_p}\}$, where $f$ ranges over the meromorphic functions of $p$ variables and $\e_1,\dots,\e_p$ are elements of the algebra $B_{p,n}(\eta)$. Let $B_{p,n}(\eta)$ be generated by the space of meromorphic functions $f(u_1,\dots,u_p)$ and by the elements $\e_1,\dots,\e_p$ with the defining relations $$\begin{gathered}
\e_\alpha
f(u_1,\dots,u_p)=
f(u_1-2\eta,\dots,u_\alpha+(n-2)\eta,\dots,u_p-2\eta)\e_\alpha,\\
\e_\alpha\e_\beta=\e_\beta\e_\alpha,\quad
f(u_1,\dots,u_p)g(u_1,\dots,u_p)=g(u_1,\dots,u_p)f(u_1,\dots,u_p)
\end{gathered}$$
We note that the subalgebra of $B_{p,n}(\eta)$ consisting of the elements of degree $(0,\dots,0)$ is the commutative algebra of all meromorphic functions of $p$ variables with the ordinary multiplication.
Let $\eta\in\E$ be a point of infinite order. For any $p\in\N$ there is a homomorphism $\phi_p\colon Q_n(\E,\eta)\to B_{p,n}(\eta)$ that acts on the generators of the algebra $Q_n(\E,\eta)$ by the formula *:* $$\phi_p(f)=\sum_{1\le\alpha\le
p}\frac{f(u_\alpha)}{\theta(u_\alpha-u_1)\dots\theta(u_\alpha-u_p)}
\e_\alpha.$$ Here $f\in\Theta_{n,c}(\Gamma)$ is a generator of $Q_n(\E,\eta)$ and the product in the denominator is of the form $\prod_{i\ne\alpha}\theta(u_\alpha-u_i)$.
We write $\xi_\alpha=\frac1{\theta(u_\alpha-u_1)\dots\theta(u_\alpha-u_p)}
\e_\alpha$. It is clear that the elements $\xi_1,\dots,\xi_p$ together with the space of meromorphic functions $\{f(u_1,\dots,u_p)\}$ generate the algebra $B_{p,n}(\eta)$. The relations (10) become $$\begin{gathered}
\xi_\alpha
f(u_1,\dots,u_p)=
f(u_1-2\eta,\dots,u_\alpha+(n-2)\eta,\dots,u_p-2\eta)\xi_\alpha\\
\xi_\alpha\xi_\beta=-\frac{e^{2\pi
i(u_\beta-u_\alpha)}\theta(u_\alpha-u_\beta+n\eta)}
{\theta(u_\beta-u_\alpha+n\eta)}\xi_\beta\xi_\alpha\end{gathered}$$ The formula (11) can be represented as $$\phi_p(f)=\sum_{1\le\alpha\le p}f(u_\alpha)\xi_\alpha.$$ Using (12) and the formula for the multiplication in the algebra $Q_n(\E,\eta)$ and assuming that $\phi_p$ is a homomorphism, one can readily evaluate the extension of the map $\phi_p$ to the entire algebra. For instance, in the grading $2$ we have $$\begin{aligned}
\phi_p(f*g)&=\sum_{1\le\alpha\le p}f(u_\alpha)\xi_\alpha\cdot
\sum_{1\le\beta\le p}g(u_\beta)\xi_\beta=
\sum_{1\le\alpha,\beta\le p}f(u_\alpha)\xi_\alpha
f(u_\beta)\xi_\beta=\\
&=\sum_{\begin{subarray}{c}1\le\alpha,\beta\le
p\\\alpha\ne\beta\end{subarray}}
f(u_\alpha)g(u_\beta-2\eta)\xi_\alpha\xi_\beta+
\sum_{1\le\alpha\le p}f(u_\alpha)g(u_\alpha+(n-2)\eta)\xi_\alpha^2.\end{aligned}$$ The first sum is $$\begin{aligned}
\sum_{1\le\alpha<\beta\le
p}&(f(u_\alpha)g(u_\beta-2\eta)\xi_\alpha\xi_\beta+
f(u_\beta)g(u_\alpha-2\eta)\xi_\beta\xi_\alpha)=\end{aligned}$$ $$\begin{aligned}
=\sum_{1\le\alpha<\beta\le
p}\left(f(u_\alpha)g(u_\beta-2\eta)\xi_\alpha\xi_\beta-
f(u_\beta)g(u_\alpha-2\eta)\frac{e^{2\pi
i(u_\alpha-u_\beta)}\theta(u_\beta-u_\alpha-n\eta)}
{\theta(u_\alpha-u_\beta+n\eta)}\xi_\alpha\xi_\beta\right)=\end{aligned}$$ $$\begin{aligned}
=\sum_{1\le\alpha<\beta\le
p}\frac{\theta(u_\alpha-u_\beta)}{\theta(u_\alpha-u_\beta-n\eta)}\times\end{aligned}$$ $$\begin{aligned}
\times\left(f(u_\alpha)g(u_\beta-2\eta)
\frac{\theta(u_\alpha-u_\beta-n\eta)}{\theta(u_\alpha-u_\beta)}+
f(u_\beta)g(u_\alpha-2\eta)\frac{\theta(u_\beta-u_\alpha-n\eta)}
{\theta(u_\beta-u_\alpha)}\right)\xi_\alpha\xi_\beta=\end{aligned}$$ $$\begin{aligned}
=\sum_{1\le\alpha<\beta\le
p}\frac{\theta(u_\alpha-u_\beta)}{\theta(u_\alpha-u_\beta-n\eta)}
f*g(u_\alpha,u_\beta)\xi_\alpha\xi_\beta\quad\text{ }\end{aligned}$$ Moreover, $f(u_\alpha)g(u_\alpha+(n-2)\eta)=
\frac{\theta(-n\eta)}{\theta(-2n\eta)}f*g(u_\alpha,u_\alpha+n\eta)$. We finally obtain $$\phi_p(f*g)=$$ $$=\sum_{1\le\alpha<\beta\le
p}\frac{\theta(u_\alpha-u_\beta)}{\theta(u_\alpha-u_\beta-n\eta)}
f*g(u_\alpha,u_\beta)\xi_\alpha\xi_\beta+\frac{\theta(-n\eta)}
{\theta(-2n\eta)}\sum_{1\le\alpha\le
p}f*g(u_\alpha,u_\alpha+n\eta)\xi_\alpha^2$$ We see that the map $\phi_p$ can be extended to the quadratic part of the algebra $Q_n(\E,\eta)$ because the right-hand side of (13) depends on $f*g$ only but not on $f$ and $g$ separately. Thus implies the assertion for generic $\eta$ because in this case the algebra $Q_n(\E,\eta)$ is defined by quadratic relations. To prove a more exact assertion (for the case in which $\eta$ is a point of infinite order), one must continue the above calculation. We obtain the following formula: if $f\in F_\alpha$, then $$\begin{aligned}
\phi_p(f)&=\sum_{\begin{subarray}{c}i_1,\dots,i_p\ge0,\\ i_1+\ldots+i_p=\alpha\end{subarray}}
B_{i_1,\dots,i_p}f(u_1,u_1+n\eta,\dots,u_1+(i_1-1)n\eta,\\
&\qquad\qquad\qquad\qquad u_2,u_2+n\eta,\dots,u_2+(i_2-1)n\eta,\dots)
\xi_1^{i_1}\dots\xi_p^{i_p},\\
&\text{where}\quad
B_{i_1,\dots,i_p}=\prod_{\begin{subarray}{c}1\le\lambda\le\lambda'\le
p\\0\le\mu<i_\lambda\\0\le\mu'<i_{\lambda'}\\\mu<\mu'\ \text{for}\
\lambda=\lambda'\end{subarray}}
\frac{\theta(u_\lambda+n\mu\eta-u_{\lambda'}-n\mu'\eta)}
{\theta(u_\lambda+n\mu\eta-u_{\lambda'}-n\mu'\eta-n\eta)}.
\end{aligned}$$ This product can be represented as $\prod_{1\le
i<j<p}\frac{\theta(v_i-v_j)}{\theta(v_i-v_j-n\eta)}$, where $(v_1,\dots,v_p)=(u_1,u_1+n\eta,\dots)$ are the arguments of the function $f$ in the formula (14) for $\xi_1^{i_1}\dots\xi_p^{i_p}$.
The formula (14) makes sense if $\eta$ is a point of infinite order, and in this case the direct calculation shows that $\phi_p$ is a homomorphism.
Representations of the algebras $Q_n(\E,\eta)$
----------------------------------------------
The formula (14) shows that the image of the homomorphism $\phi_p$ is contained in the subalgebra $B_{p,n}^{\reg}(\eta)\subset B_{p,n}(\eta)$ consisting of the elements $\sum_{\alpha_1,\dots,\alpha_p}f_{\alpha_1,\dots,\alpha_p}
\e^{\alpha_1}\dots\e_p^{\alpha_p}$, where the functions $f_{\alpha_1,\dots,\alpha_p}$ are holomorphic outside the divisors of the form $u_i-u_j-\lambda n\eta\in\Gamma$, $\lambda\in\Z$.
Let $v_1,\dots,v_p\in\C$ be such that $v_i-v_j-\lambda
n\eta\not\in\Gamma$ for $\lambda\in\Z$. We construct a representation $M_{v_1,\dots,v_p}$ of the algebra $B_{p,n}^{\reg}(\eta)$ as follows. Let the representation $M_{v_1,\dots,v_p}$ have a basis $\{w_{\alpha_1,\dots,\alpha_p};\alpha_1,\dots\alpha_p\in\Z_{\ge0}\}$ in which the elements $\e_1,\dots,\e_p$ act by the rule $\e_iw_{\alpha_1,\dots,\alpha_p}=
w_{\alpha_1,\dots,\alpha_i+1,\dots,\alpha_p}$. Thus, $w_{\alpha_1,\dots,\alpha_p}=\e_1^{\alpha_1}\dots\e_p^{\alpha_p}w$, where $w=w_{0,\dots,0}$. The action of the commutative subalgebra of $B_{p,n}^{\reg}(\eta)$ consisting of the elements of degree $0$ is diagonal in this basis. We set $fw=f(v_1-(n-2)\eta,\dots,v_p-(n-2)\eta)w$, and hence $fw_{\alpha_1,\dots,\alpha_p}=
f(v_1+(2\alpha_1+\ldots+2\alpha_p-n\alpha_1-(n-2))\eta,\dots,
v_p+(2\alpha_1+\ldots+2\alpha_p-n\alpha_p-(n-2))\eta)
w_{\alpha_1,\dots,\alpha_p}$. It is clear that these formulas really define a representation of the algebra $B_{p,n}^{\reg}(\eta)$ in the space $M_{v_1,\dots,v_p}$, and, thanks to the homomorphism $\phi_p$, we have a representation of the algebra $Q_n(\E,\eta)$ as well. One can readily see that the space $M_{v_1,\dots,v_p}$ admits a basis $\{v_{\alpha_1,\dots,\alpha_p};\alpha_1,\dots,\alpha_p\in\Z_{\ge0}\}$, in which the action of the generators of the algebra $Q_n(\E,\eta)$ can be represented in the following form: if $f\in\Theta_n(\Gamma)$, then $$fv_{\alpha_1,\dots,\alpha_p}=\sum_{1\le i\le
p}\frac{f(v_i+(2\alpha_1+\ldots+2\alpha_p-n\alpha_i)\eta)}
{\theta(v_i-v_1-n(\alpha_i-\alpha_1)\eta)\dots
\theta(v_i-v_p-n(\alpha_i-\alpha_p)\eta)}
v_{\alpha_1,\dots,\alpha_i+1,\dots,\alpha_p}.$$ The vectors $v_{\alpha_1,\dots,\alpha_p}$ are proportional to the vectors $w_{\alpha_1,\dots,\alpha_p}$. In particular, for $p=1$ we obtain modules $M_v$ with a basis $\{v_\alpha;\alpha\in\Z_{\ge0}\}$ and the action $fv_\alpha=f(v-(n-2)\alpha\eta)v_{\alpha+1}$. Thus, the algebra $Q_n(\E,\eta)$ has a family of linear modules parametrized by the elliptic curve $\E\subset\CP^{n-1}$, where the embedding is carried out by theta functions of order $n$.
Symplectic leaves
-----------------
We recall that $Q_n(\E,0)$ is the polynomial ring $S^*\Theta_{n,c}(\Gamma)$. For a fixed elliptic curve $\E=\C/\Gamma$ we obtain the family of algebras $Q_n(\E,\eta)$, which is a flat deformation of the polynomial ring. We denote the corresponding Poisson algebra by $q_n(\E)$. We obtain a family of Poisson algebras, depending on $\E$, that is, on the modular parameter $\tau$. Let us study the symplectic leaves of this algebra. To this end, we note that, when passing to the limit as $\eta\to0$, the homomorphism $\phi_p$ of associative algebras gives a homomorphism of Poisson algebras. Namely, let us denote by $b_{p,n}$ the Poisson algebra formed by the elements $\sum_{\alpha_1,\dots,\alpha_p\ge0}
f_{\alpha_1,\dots,\alpha_p}(u_1,\dots,u_p)
\e_1^{\alpha_1}\dots\e_p^{\alpha_p}$, where $f_{\alpha_1,\dots,\alpha_p}$ are meromorphic functions and the Poisson bracket is $$\{u_\alpha,u_\beta\}=\{\e_\alpha,\e_\beta\}=0;\quad
\{\e_\alpha,u_\beta\}=-2\e_\alpha;\quad
\{\e_\alpha,u_\alpha\}=(n-2)\e_\alpha,$$ where $\alpha\ne\beta$.
The following assertion results from Proposition 6 in the limit as $\eta\to0$.
There is a Poisson algebra homomorphism $\psi_p\colon
q_n(\E)\to b_{p,n}$ given by the following formula: if $f\in\Theta_n(\Gamma)$, then $\psi_p(f)=\sum_{1\le\alpha\le
p}\frac{f(u_\alpha)}{\theta(u_\alpha-u_1)\dots\theta(u_\alpha-u_p)}
\e_\alpha$.
Let $\{\theta_i(u);i\in\Z/n\Z\}$ be a basis of the space $\Theta_{n,c}(\Gamma)$ and let $\{x_i;i\in\Z/n\Z\}$ be the corresponding basis in the space of elements of degree one in the algebra $Q_n(\E,\eta)$ (this space is isomorphic to $\Theta_{n,c}(\Gamma)$). For an elliptic curve $\E\subset\CP^{n-1}$ embedded by means of theta functions of order $n$ (this is the set of points with the coordinates $(\theta_0(z):\ldots:\theta_{n-1}(z))$) we denote by $C_p\E$ the variety of $p$-chords, that is, the union of projective spaces of dimension $p-1$ passing through $p$ points of $\E$. Let $K(C_p\E)$ be the corresponding homogeneous manifold in $\C^n$. It is clear that $K(C_p\E)$ consists of the points with the coordinates $x_i=\sum_{1\le\alpha\le
p}\frac{\theta_i(u_\alpha)}{\theta(u_\alpha-u_1)\dots
\theta(u_\alpha-u_p)}\e_\alpha$, where $u_\alpha,\e_\alpha\in\C$.
Let $2p<n$. Then one can show that $\dim K(C_p\E)=2p$ and $K(C_{p-1}\E)$ is the manifold of singularities of $K(C_p\E)$. It follows from Proposition 7 and from the fact that the Poisson bracket is non-degenerate on $b_{p,n}$ for $2p<n$ and $\e_\alpha\ne0$ that the non-singular part of the manifold $K(C_p\E)$ is a $2p$- dimensional symplectic leaf of the Poisson algebra $q_n(\E)$.
Let $n$ be odd. One can show that the equation defining the manifold $K(C_{\frac{n-1}2}\E)$ is of the form $C=0$, where $C$ is a homogeneous polynomial of degree $n$ in the variables $x_i$. This polynomial is a central function of the algebra $q_n(\E)$.
Let $n$ be even. The manifold $K(C_{\frac{n-2}2}\E)$ is defined by equations $C_1=0$ and $C_2=0$, where $\deg C_1=\deg C_2=n/2$. The polynomials $C_1$ and $C_2$ are central in the algebra $q_n(\E)$.
Free modules, generations, and relations
----------------------------------------
Let $\eta$ be a point of infinite order.
Let numbers $v_1,\dots,v_n\in\C$ be in general position. Then the module $M_{v_1,\dots,v_n}$ is generated by $v_{0,\dots,0}$ and is free over $Q_n(\E,\eta)$.
By construction, the dimensions of graded components of $M_{v_1,\dots,v_n}$ coincide with those of the algebra $Q_n(\E,\eta)$. Let us show that the module is generated by the vector $v=v_{0,\dots,0}$. Let
$$f_i=\prod_{\alpha\ne
i}\theta(z-v_\alpha)\cdot(\theta(z+v_1+\ldots+v_n-v_i-c).$$ It is clear that $f_i\in\Theta_{n,c}(\Gamma)$ for $1\le i\le n$. Therefore, the $f_i$s are elements of degree $1$ of the algebra $Q_n(\E,\eta)$. It follows from the formula (15) that $f_iv$ is non-zero and proportional to $v_{0,\dots,1,\dots,0}=\e_iv$. Similarly, one can readily construct elements $f_{i;\alpha_1,\dots,\alpha_n}\in\Theta_{n,c}(\Gamma)$ such that $f_{i;\alpha_1,\dots,\alpha_n}v_{\alpha_1,\dots,\alpha_n}$ is non-zero and proportional to $v_{\alpha_1,\dots,\alpha_i+1,\dots,\alpha_n}$. Namely, $f_{i;\alpha_1,\dots,\alpha_n}=\prod_{\beta\ne
i}\theta(z-v_\beta-(2\alpha_1+\ldots+2\alpha_n-n\alpha_\beta)\eta)
\cdot\theta(z+v_1+\ldots+v_n-v_i+(n-2)(\alpha_1+\ldots+\alpha_n)\eta
-c)$. Thus, all elements $v_{\alpha_1,\dots,\alpha_n}$ are obtained from $v$ by the action of elements of degree one in $Q_n(\E,\eta)$.
The algebra $Q_n(\E,\eta)$ is presented by $n$ generators and $\frac{n(n-1)}2$ quadratic relations.
It follows from the proof of Proposition 8 that the algebra $Q_n(\E,\eta)$ is generated by the elements of degree one. It is clear from the construction of the elements $f_{i;\alpha_1,\dots,\alpha_n}$ that these elements admit quadratic relations of the form $$f_{j;\alpha_1,\dots,\alpha_i+1,\dots,\alpha_n}
f_{i;\alpha_1,\dots,\alpha_n}=
c_{i,j;\alpha_1,\dots,\alpha_n}
f_{i;\alpha_1,\dots,\alpha_j+1,\dots,\alpha_n}
f_{j;\alpha_1,\dots,\alpha_n},$$ where $c_{i,j;\alpha_1,\dots,\alpha_n}\in\C^*$. To prove this relation, one must apply it to the vector $v_{\alpha_1,\dots,\alpha_n}$. Let us show that these quadratic relations imply the other ones. Let a relation be of the form $\sum_\alpha a_t^{(\alpha)}a_{t-1}^{(\alpha)}\dots a_1^{(\alpha)}=0$. We expand the element $a_1^{(\alpha)}$ in the basis $\{f_i\}$. The relation becomes $\sum_{\beta,i}b_t^{(\beta)}\dots b_2^{(\beta)}f_i=0$. Let us now expand $b_2^{(\beta)}$ in the basis $\{f_{i;0,\dots,1,\dots,0}\}$, where $1$ stands at the $i$th place. Continuing this procedure, we eventually represent the relation in the form $\sum
c_{i_1,\dots,i_t}f_{i_1;\alpha_1,\dots,\alpha_n}
f_{i_2;\alpha_1,\dots,\alpha_{i_2}-1,\dots,\alpha_n}\dots f_{i_t}=0$. It is clear that this relation follows from the relations (16).
The relations in the algebra $Q_n(\E,\eta)$ are of the form $$\sum_{r\in\Z/n\Z}\frac1{\theta_{j-i-r}(-\eta)\theta_r(\eta)}x_{j-r}
x_{i+r}=0,\quad\text{сту}\ i\ne j;\ i,j\in\Z/n\Z.$$
Let us apply the formula for the multiplication in the algebra $Q_n(\E,\eta)$ (see §2.1). Since $x_i=\theta_i(z)$, the relations (17) becomes $$\begin{gathered}
\sum_{r\in\Z/n\Z}\frac1{\theta_{j-i-r}(-\eta)\theta_r(\eta)}
\biggl(\theta_{j-r}(z_1)\theta_{i+r}(z_2-2\eta)
\frac{\theta(z_1-z_2-n\eta)}{\theta(z_1-z_2)}+\\
+\theta_{j-r}(z_2)\theta_{i+r}(z_1-2\eta)
\frac{\theta(z_2-z_1-n\eta)}{\theta(z_2-z_1)}\biggr)=0.\end{gathered}$$ This relation immediately follows from the relation (30) (see Appendix A).
Main properties of the algebra $Q_{n,k}(\E,\eta)$
=================================================
We again assume that $\E=\C/\Gamma$ is an elliptic curve and $\eta\in\E$. Let $n$ and $k$ be coprime positive integers such that $1\le k<n$. Let us present the algebra $Q_{n,k}(\E,\eta)$ by the generators $\{x_i;i\in\Z/n\Z\}$ and the relations $$\sum_{r\in\Z/n\Z}\frac{\theta_{j-i+r(k-1)}(0)}
{\theta_{kr}(\eta)\theta_{j-i-r}(-\eta)}x_{j-r}x_{i+r}=0.$$ As is known (see §4), this is a PBW-algebra for generic $\E$ and $\eta$. We conjecture that this holds for any $\E$ and $\eta$. For generic $\E$ and $\eta$ the centre of the algebra $Q_{n,k}(\E,\eta)$ is the polynomial ring in $c=\NOD(n,k+1)$ elements of degree $n/c$ (see \[20\]). Hypothetically, this is the case for any $\E$ and $\eta$, where $\eta$ is a point of infinite order. If $\eta\in\E$ is a point of finite order, then the algebra $Q_{n,k}(\E,\eta)$ is finite-dimensional over its centre (see \[37\]). The following properties can readily be verified:
1\) $Q_{n,k}(\E,0)=\C[x_1,\dots,x_n]$ is commutative;
2\) $Q_{n,n-1}(\E,\eta)=\C[x_1,\dots,x_n]$ is commutative for any $\eta$;
3\) $Q_{n,k}(\E,\eta)\simeq Q_{n,k'}(\E,\eta)$, where $kk'\equiv1\pmod n$;
4\) the maps $x_i\mapsto x_{i+1}$ and $x_i\mapsto\epsilon^ix_i$ (where $\epsilon$ is a primitive root of unity of degree $n$) define automorphisms of the algebra $Q_{n,k}(\E,\eta)$.
It follows from the results of §5 (see Proposition 11) that the space of generators of the algebra $Q_{n,k}(\E,\eta)$ is naturally isomorphic to the space of theta functions $\Theta_{n/k}(\Gamma)$ (see Appendix B). Moreover, this space of generators is dual to the space $\Theta_{n/n-k}(\Gamma)$ (see Proposition 14). For a description of this duality between the spaces of theta functions, see Appendix C.
The algebra $Q_{n,k}(\E,\eta)$ is not a Hopf algebra and admits no comultiplications. However, there are homomorphisms of the algebra $Q_{n,k}(\E,\eta)$ to tensor products of other algebras of this kind (see \[35; §3\]). To describe these homomorphisms, we need the notation of Appendix B. Moreover, we denote by $L_m(\E,\eta)=\C\oplus\Theta_{m,0}(\Gamma)\oplus
\Theta_{2m,m\eta}(\Gamma)\oplus\dots$ the $\Z_{\ge0}$-graded algebra with the multiplication $*$ given by the formula $f*g(z)=f(z+\beta\eta)g(z)$, where $\beta$ is a power of $g$. As we know from §2, $L_n(\E,(n-2)\eta)$ is a quotient algebra of $Q_n(\E,\eta)$.
Let $A$ be an associative algebra and let $G\subset\Aut A$. We denote by $A^G\subset A$ the subalgebra consisting of the elements invariant with respect to $G$.
There are the following algebra homomorphisms:
a\) $Q_{n,k}(\E,\eta)\to
\left(L_{kn}\left(\E,\frac{n-k-1}k\eta\right)\otimes
Q_{k,\l}\left(\E,\frac nk\eta\right)\right)^{\wt{\Gamma_k}}$, where $\l=d(n_3,\dots,n_p)$ and the generators are taken to elements of bidegree $(1,1)$.
b\) $Q_{n,k}(\E,\eta)\to
\left(L_{nk'}\left(\E,\frac{n-k'-1}{k'}\eta\right)\otimes
Q_{k',\l'}\left(\E,\frac
n{k'}\eta\right)\right)^{\wt{\Gamma_{k'}}}$, where $\l'=d(n_1,\dots,n_{p-2})$ and the generators are taken to elements of bidegree $(1,1)$.
c\) $Q_{n,k}(\E,\eta)\to
\left(Q_{a,\alpha}\left(\E,\frac n\alpha\eta\right)\otimes
L_{abn}\left(\E,\frac{n-a-b}{ab}\eta\right)\otimes
Q_{b,\beta}\left(\E,\frac
nb\right)\right)^{\wt{\Gamma_{ab}}}$, where $a=d(n_1,\dots,n_{i-1})$, $b=d(n_{i+1},\dots,n_p)$, $\alpha=d(n_1,\dots,n_{i-2})$, and $\beta=d(n_{i+2},\dots,n_p)$ for some $i$; the generators are taken to elements of multidegree $(1,1,1)$.
Let us describe the map c) geometrically (the description of the maps a) and b) is the same). Let $f(z_1,\dots,z_p)\in\Theta_{n/k}(\Gamma)$. For some $i$ ($1<i<p$) we choose a $z_i$, then $f$ (regarded as a function of $z_1,\dots,z_{i-1}$) belongs to a space isomorphic to $\Theta_{a/\alpha}(\Gamma)$. Similarly, when regarded as a function of $z_{i+1},\dots,z_p$, $f$ belongs to a space isomorphic to $\Theta_{b/\beta}(\Gamma)$. Thus, for a fixed $z_i$ we have $f\in\Theta_{a/\alpha}(\Gamma)\otimes\Theta_{b/\beta}(\Gamma)$. A family of linear maps $\Theta_{n/k}(\Gamma)\to\Theta_{a/\alpha}(\Gamma)\otimes
\Theta_{b/\beta}(\Gamma)$ arises. With regard to the dependence on $z_i$, we obtain a linear map $\Theta_{n/k}(\Gamma)\to\Theta_{a/\alpha}(\Gamma)\otimes
\Theta_{nab,0}(\Gamma)\otimes\Theta_{b/\beta}(\Gamma)$. The homomorphism c) corresponds to this map (the space of generators of the algebra $Q_{n,k}(\E,\eta)$ is $\Theta_{n/k}(\Gamma)$, the space of generators of the algebra $L_{nab}\left(\E,\frac{n-a-b}{ab}\eta\right)$ is $\Theta_{nab,0}(\Gamma)$, etc.).
Belavin elliptic $R$-matrix and the algebra $Q_{n,k}(\E,\eta)$
==============================================================
Let $V$ be a vector space of dimension $n$. For each $u\in\C$ we denote by $V(u)$ a vector space canonically isomorphic to $V$. Let $R$ be a meromorphic function of two variables with values in $\End(V\otimes V)$. It is convenient to regard $R(u,v)$ as a linear map $$R(u,v)\colon V(u)\otimes V(v)\to V(v)\otimes V(u).$$
We recall that by the Yang-Baxter equation one means the condition that the following diagram is commutative:
$$\xymatrix{&V(v)\otimes V(u)\otimes V(w)\ar[r]^{1\otimes R(u,w)}&
V(v)\otimes V(w)\otimes V(u)\ar[dr]^{R(v,w)\otimes1}\\
V(u)\otimes V(v)\otimes V(w)\ar[ur]^{R(u,v)\otimes1}\ar[dr]^{1\otimes
R(v,w)}&&&
V(w)\otimes V(v)\otimes V(u)\\
&V(u)\otimes V(w)\otimes V(v)\ar[r]^{R(u,w)\otimes1}&
V(w)\otimes V(u)\otimes V(v)\ar[ur]_{1\otimes R(u,v)}}$$
A solution of the Yang-Baxter equation is called $R$-matrix.
Let $\{x_i;i=1,\dots,n\}$ be a basis in the space $V$ and let $\{x_i(u)\}$ be the corresponding basis in the space $V(u)$. Let $R_{ij}^{\alpha\beta}(u,v)$ be an entry of an $R$-matrix $R(u,v)$, that is, $R(u,v)\colon x_i(u)\otimes x_j(v)\to
R_{ij}^{\alpha\beta}(u,v)x_\beta(v)\otimes x_\alpha(u)$.
Let an $R$-matrix $R(u,v)$ satisfy the relation $R(u,v)R(v,u)=1$. By the Zamolodchikov algebra $Z_R$ one means the algebra with the generators $\{x_i(u);i=1,\dots,n;u\in\C\}$ and the defining relations $$x_i(u)x_j(v)=R_{ij}^{\alpha\beta}(u,v)x_\beta(v)x_\alpha(u).$$ It is clear that the elements $\{x_{i_1}(u_1)\dots
x_{i_m}(u_m);1\le i_1,\dots,i_m\le n\}$ of the Zamolodchikov algebra are linearly independent for generic $u_1,\dots,u_m$. Thus, Zamolodchikov algebras are infinite-dimensional analogues of PBW-algebras. We recall that by the classical $r$-matrix one means a Poisson structure of the form $\{x_i(u),x_j(v)\}=r_{ij}^{\alpha\beta}(u,v)x_\alpha(u)x_\beta(v)$. It is clear that the Zamolodchikov algebra is a quantization of this Poisson structure if the $R$-matrix depends on an additional parameter $\hbar$ and the relations (19) are of the form $$x_i(u)x_j(v)=x_j(v)x_i(u)+\hbar
r_{ij}^{\alpha\beta}(u,v)x_\beta(v)x_\alpha(u)+o(\hbar).$$
The Yang-Baxter equation has elliptic solutions, which are referred to as Belavin $R$-matrices. Let $n,k\in\N$ be coprime and $1\le k<n$. For any $n$ and $k$ there is a two-parameter family of $R$-matrices $R_{n,k}(\E,\eta)$ depending on an elliptic curve $\E=\C/\Gamma$ and a point $\eta\in\E$. Namely, $$R_{n,k}(\E,\eta)(u-v) (x_i(u)\otimes
x_j(v))=\frac1{p(u-v)}\sum_{r\in\Z/n\Z}
\frac{\theta_{j-i+r(k-1)}(v-u+\eta)}
{\theta_{kr}(\eta)\theta_{j-i-r}(v-u)}x_{j-r}(v)\otimes x_{i+r}(u),$$ where $p(u-v)=
\frac{\theta_1(0)\dots\theta_{n-1}(0)
\theta_0(v-u+\eta)\dots\theta_{n-1}(v-u+\eta)}
{\theta_0(\eta)\dots\theta_{n-1}(\eta)
\theta_0(v-u)\dots\theta_{n-1}(v-u)}$; $i,j\in\Z/n\Z$. One can readily see that $\det
R_{n,k}(\E,\eta)(u-v)=
\left(\frac{\theta_0(v-u-\eta)\dots\theta_{n-1}(v-u-\eta)}
{\theta_0(v-u+\eta)\dots\theta_{n-1}(v-u+\eta)}\right)
^{\frac{n(n-1)}2}$. Thus, the operator $R_{n,k}(\E,\eta)(-\eta)$ has a kernel. Let $L_{n,k}(\E,\eta)\subset
V\otimes V$ and $L_{n,k}(\E,\eta)=\ker R_{n,k}(\E,\eta)(-\eta)$. According to \[10\] we have $\dim(L_{n,k}(\E,\eta))=\frac{n(n-1)}2$, and $L_{n,k}(\E,0)=\Lambda^2V$ for $\eta=0$. Let $Q_{n,k}(\E,\eta)=T^*V/(L_{n,k}(\E,\eta))$ be the algebra with the generators $\{x_i;i\in\Z/n\Z\}$ and the defining relations $L_{n,k}(\E,\eta)$. The dimensions of the graded components of the algebra $Q_{n,k}(\E,\eta)$ coincide with those of the polynomial ring $S^*V$ (see \[10\]). It follows from the formula for $R_{n,k}(\E,\eta)(u-v)$ that the defining relations in the algebra $Q_{n,k}(\E,\eta)$ are $$\sum_{r\in\Z/n\Z}\frac{\theta_{j-i+r(k-1)}(0)}
{\theta_{kr}(\eta)\theta_{j-i-r}(-\eta)}x_{j-r}x_{i+r}=0.$$ In particular, we see that $Q_{n,1}(\E,\eta)=Q_n(\E,\eta)$ if $\eta\in\E$ is a point of infinite order.
It follows from the relations (20) that $Q_{n,k}(\E,\eta)\simeq
Q_{n,k'}(\E,\eta)$, where $kk'\equiv1\pmod n$. Moreover, $Q_{n,n-1}(\E,\eta)$ is commutative for any $\E,\eta$. It is also clear that $Q_{n,k}(\E,0)$ is commutative.
Algebras $Q_{n,k}(\E,\eta)$ and the exchange algebras
=====================================================
Homomorphisms of algebras $Q_{n,k}(\E,\eta)$ into dynamical exchange algebras
-----------------------------------------------------------------------------
The algebras $Q_{n,k}(\E,\eta)$ for arbitrary $k$ have representations similar to the homomorphisms $\phi_p$ related to the case $k=1$ (see §2.3, (11)). However, the structure of the algebra similar to $B_{p,n}(\eta)$ for $k=1$ turns out to be more complicated for $k>1$. Let $n/k=n_1-\frac1{n_2-\ldots-\frac1{n_q}}$ be the expansion of the number $n/k$ in the continued fraction in which $n_\alpha\ge2$ for $1\le\alpha\le q$. It is clear that such an expansion exists and is unique. We recall that $1\le k<n$, where $n$ and $k$ are coprime. Let $d(m_1,\dots,m_t)=\det M$, where $M=(m_{ij})$ be a ($t\times t$) matrix with the entries $m_{ii}=m_i$, $m_{i,i+1}=m_{i+1,i}=-1$, and $m_{ij}=0$ for $|i-j|>1$. For $t=0$ we set $d(\varnothing)=1$. It is clear that $n=d(n_1,\dots,n_q)$ and $k=d(n_2,\dots,n_q)$.
There is an algebra homomorphism of $Q_{n,k}(\E,\eta)$ into the algebra $C_{n,k}(\eta)$ generated by the commutative subalgebra $\{f(y_1,\dots,y_q)\}$ of holomorphic functions of $q$ variables and by an element $\e$ with the defining relations of the form $\e f(y_1,\dots,y_q)=f(y_1+\eta_1,\dots,y_q+\eta_q)\e$, where $\eta_\alpha=(d(n_1,\dots,n_q)-d(n_1,\dots,n_{\alpha-1})-
d(n_{\alpha+1},\dots,n_q))\eta$. Moreover, $x_i\to w_i(y_1,\dots,y_q)\e$, where $w_i\in\Theta_{n/k}(\Gamma)$ *(*see Appendix B).
This is a special case of Proposition 13 below.
The algebra $C_{n,k}(\eta)$ has a family of modules $L_{v_1,\dots,v_q}$ with a basis $\{v_\alpha;\alpha\in\Z_{\ge0}\}$ and with the action given by $\e
v_\alpha=v_{\alpha+1}$ and $f(y_1,\dots,y_q)v_\alpha=
f(v_1-\alpha\eta_1,\dots,v_q-\alpha\eta_q)v_\alpha$. This implies the following assertion.
The algebra $Q_{n,k}(\E,\eta)$ hes a family of modules $L_{u_1,\dots,u_q}$ with a basis $\{v_\alpha;\alpha\in Z_{\ge0}\}$ and with the action *:* $x_iv_\alpha=w_i(u_1-\alpha\eta_1,\dots,u_q-\alpha\eta_q)
v_{\alpha+1}$, where $w_i\in\Theta_{n/k}(\Gamma)$ and $\eta_j=(d(n_1,\dots,n_q)-d(n_1,\dots,n_{j-1})-
d(n_{j+1},\dots,n_q))\eta$.
In particular, we see that the algebra $Q_{n,k}(\E,\eta)$ has a family of linear modules depending on the point of $\E^q$, and the space of generators of the algebra $Q_{n,k}(\E,\eta)$ is isomorphic to $\Theta_{n/k}(\Gamma)$.
Let $C_{m_1,\dots,m_q;n,k}(\E,\eta)$ be the algebra generated by the commutative subalgebra $\{\phi(y_{1,1},\dots,y_{m_1,1};\dots;y_{1,q},\dots,y_{n_q,q})\}$, where $\phi$ are the meromorphic functions with the ordinary multiplication, and by elements $\{\e_{\alpha_1,\dots,\alpha_q};1\le\alpha_t\le m_t\}$. The defining relations for the algebra $C_{m_1,\dots,m_q;n,k}(\E,\eta)$ look as follows: $$\begin{aligned}
\e_{\alpha_1,\dots,\alpha_q}y_{\beta,\nu}&=
(y_{\beta,\nu}-(d(n_1,\dots,n_{\nu-1})+d(n_{\nu+1},\dots,n_q))\eta)
\e_{\alpha_1,\dots,\alpha_q},\quad\ \alpha_\nu\ne\beta,\\
\e_{\alpha_1,\dots,\alpha_q}y_{\alpha_\nu,\nu}&=
(y_{\alpha_\nu,\nu}+(n-d(n_1,\dots,n_{\nu-1})-d(n_{\nu+1},\dots,n_q))
\eta)\e_{\alpha_1,\dots,\alpha_q}.\end{aligned}$$ These relations mean that the $y_{\beta,\nu}$s are dynamical variables. This immediately implies the relations between the elements $\e_{\alpha_1,\dots,\alpha_q}$ and the meromorphic functions of the variables $y_{\beta,\nu}$. The remaining relations are quadratic in $\e_{\alpha_1,\dots,\alpha_q}$ with the coefficients depending on the dynamical variables $y_{\beta,\nu}$. The relations in “general position” are as follows: $$\e_{\alpha_1,\dots,\alpha_q}\e_{\beta_1,\dots,\beta_q}=
\Lambda\e_{\beta_1,\dots,\beta_q}\e_{\alpha_1,\dots,\alpha_q}+
\sum_{1\le t\le q-1}\Lambda_{t,t+1}
\e_{\beta_1,\dots,\beta_t,\alpha_{t+1},\dots,\alpha_q}
\e_{\alpha_1,\dots,\alpha_t,\beta_{t+1},\dots,\beta_q},$$ where $\alpha_1\ne\beta_1$, …, $\alpha_q\ne\beta_q$ and $$\begin{aligned}
\Lambda&=\frac{e^{-2\pi in\eta}\theta(y_{\beta_1,1}-y_{\alpha_1,1})
\theta(y_{\beta_q,q}-y_{\alpha_q,q}+n\eta)}
{\theta(y_{\beta_1,1}-y_{\alpha_1,1}-n\eta)
\theta(y_{\beta_q,q}-y_{\alpha_q,q})},\\
\Lambda_{t,t+1}&=\frac{e^{-2\pi in\eta}\theta(n\eta)
\theta(y_{\beta_1,1}-y_{\alpha_1,1})}
{\theta(y_{\beta_1,1}-y_{\alpha_1,1}-n\eta)}\cdot
\frac{\theta(y_{\beta_t,t}+y_{\beta_{t+1},t+1}-
y_{\alpha_t,t}-y_{\alpha_{t+1},t+1})}
{\theta(y_{\beta_t,t}-y_{\alpha_t,t})
\theta(y_{\beta_{t+1},t+1}-y_{\alpha_{t+1},t+1})}.
\end{aligned}$$ The relations of non-general position occur if some $\alpha_\nu$s are equal to $\beta_\nu$s. These relations exist for any subset of the form $\{\psi+1,\dots,\psi+\phi\}$, where $0\le\psi$, $\psi+\phi\le q$, and $\alpha_\psi=\beta_\psi$ (if $0<\psi$), $\alpha_{\psi+\phi+1}=\beta_{\psi+\phi+1}$ (if $\psi+\phi<q$), and $\alpha_{\psi+1}\ne\beta_{\psi+1}$, …, $\alpha_{\psi+\phi}\ne\beta_{\psi+\phi}$. These relations are of the form $$\begin{gathered}
\e_{\mu_1,\dots,\mu_\psi,\alpha_1,\dots,\alpha_\phi,
\gamma_1,\dots,\gamma_p}
\e_{\mu_1',\dots,\mu_{\psi-1}',\mu_\psi,\beta_1,\dots,\beta_\phi,
\gamma_1,\gamma_2',\dots,\gamma_p'}=\\
=\Lambda\e_{\mu_1,\dots,\mu_\psi,\beta_1,\dots,\beta_\phi,
\gamma_1,\dots,\gamma_p}
\e_{\mu_1',\dots,\mu_\psi,\alpha_1,\dots,\alpha_\phi,
\gamma_1,\dots,\gamma_p'}+\\
+\sum_{1\le t<\phi}\Lambda_{t,t+1}\e_{\mu_1,\dots,\mu_\psi,
\beta_1,\dots,\beta_t,\alpha_{t+1},\dots,\alpha_\phi,
\gamma_1,\dots,\gamma_p}
\e_{\mu_1',\dots,\mu_\psi,\alpha_1,\dots,\alpha_t,
\beta_{t+1},\dots,\beta_\phi,\gamma_1,\dots,\gamma_p'}.\end{gathered}$$ Here $\alpha_1\ne\beta_1$, …, $\alpha_\phi\ne\beta_\phi$, and $\phi+\psi+p=q$. The coefficients $\Lambda,\Lambda_{t,t+1}$ are defined by (21). If $\psi=p=0$, then this relation coincides with (21), that is, becomes a relation in general position.
We note that, if $\eta=0$, then the algebra $C_{m_1,\dots,m_q;n,k}(\E,0)$ is commutative and does not depend on the elliptic curve $\E$. For $q=1$ and $q=2$ the algebra $C_{m_1,\dots,m_q;n,k}(\E,0)$ is the polynomial ring in the variables $\{\e_{\alpha_1,\dots,\alpha_q}\}$ over the field of meromorphic functions of the variables $\{y_{\alpha,\beta}\}$. For $q>2$ the algebra $C_{m_1,\dots,m_q;n,k}(\E,0)$ is no longer a polynomial ring because additional relations occur. Namely, the relations (22) for $\eta=0$ become $$\begin{gathered}
\e_{\mu_1,\dots,\mu_\psi,\alpha_1,\dots,\alpha_\phi,
\gamma_1,\dots,\gamma_p}
\e_{\mu_1',\dots,\mu_{\psi-1}',\mu_\psi,\beta_1,\dots,\beta_\phi,
\gamma_1,\gamma_2',\dots,\gamma_p'}=\\
=\e_{\mu_1,\dots,\mu_\psi,\beta_1,\dots,\beta_\phi,
\gamma_1,\dots,\gamma_p}
\e_{\mu_1',\dots,\mu_{\psi-1}',\mu_\psi,\alpha_1,\dots,\alpha_\phi,
\gamma_1,\gamma_2',\dots,\gamma_p'}.\end{gathered}$$ The algebra $C_{m_1,\dots,m_q;n,k}(\E,\eta)$ is a flat deformation of the function algebra on the manifold defined by the relations (23). Moreover, $y_{\alpha,\beta}$ are dynamical variables. It is clear that the manifold defined by the equations (23) is rational, and the general solution of these equations is $\e_{\alpha_1,\dots,\alpha_q}=\e_{\alpha_1,\alpha_2}^{(1)}
\e_{\alpha_2,\alpha_3}^{(2)}\dots\e_{\alpha_{q-1},\alpha_q}^{(q-1)}$, where $\{\e_{\alpha,\beta}^{(t)}\}$ are independent variables.
There is an algebra homomorphism $\phi\colon Q_{n,k}(\E,\eta)\to
C_{m_1,\dots,m_q;n,k}(\E,\eta)$ that acts on the generators of the algebra $Q_{n,k}(\E,\eta)$ as follows: *:* $$\phi(x_i)=\sum_{\begin{subarray}{c}1\le\alpha_1\le
m_1\\\dots\\1\le\alpha_q\le m_q\end{subarray}}
w_i(y_{\alpha_1,1},\dots,y_{\alpha_q,q})
\e_{\alpha_1,\dots,\alpha_q}.$$ Here $w_i\in\Theta_{n/k}(\Gamma)$.
The algebra $Q_{n,k}(\E,\eta)$ is defined by the relations (18). Let us show that the images $\phi(x_i)$ satisfy the same relations. We have $$\begin{gathered}
\sum_{r\in\Z/n\Z}\frac{\theta_{j-i+r(k-1)}(0)}
{\theta_{kr}(\eta)\theta_{j-i-r}(-\eta)}\phi(x_{j-r})\phi(x_{i+r})=\\
=\sum_{\begin{subarray}{c}r\in\Z/n\Z\\\\1\le\alpha_1,\beta_1\le
m_1\\\dots\\1\le\alpha_q,\beta_q\le m_q\end{subarray}}
\frac{\theta_{j-i+r(k-1)}(0)}{\theta_{kr}(\eta)\theta_{j-i-r}(-\eta)}
w_{j-r}(y_{\alpha_1,1},\dots,y_{\alpha_q,q})
\e_{\alpha_1,\dots,\alpha_q}
w_{i+r}(y_{\beta_1,1},\dots,y_{\beta_q,q})
\e_{\beta_1,\dots,\beta_q}.\end{gathered}$$ Using the relations (21) and (22), we obtain an expression of the form $$\sum_{\begin{subarray}{l}\alpha_1\le\beta_1;\\\alpha_2\le\beta_2\
\text{for}\
\alpha_1=\beta_1;\\
\dots\end{subarray}}
\psi_{\alpha_1,\dots,\alpha_q,\beta_1,\dots,\beta_q}
\e_{\alpha_1,\dots,\alpha_q}\e_{\beta_1,\dots,\beta_q}.$$ We must prove that the coefficients are equal to $0$. Let us restrict ourselves to the case $\alpha_1\ne\beta_1$, …, $\alpha_q\ne\beta_q$. In this case we have by direct calculation $$\begin{gathered}
\psi_{\alpha_1,\dots,\beta_q}=\sum_{r\in\Z/n\Z}
\frac{\theta_{j-i+r(k-1)}(0)}{\theta_{kr}(\eta)\theta_{j-i-r}(-\eta)}
\biggl(w_{j-r}(y_{\alpha_1,1},\dots,y_{\alpha_q,q})
w_{i+r}(y_{\beta_1,1}+t_1,\dots,y_{\beta_q,q}+t_q)+\\
+\frac{e^{-2\pi in\eta}\theta(y_{\alpha_1,1}-y_{\beta_1,1})
\theta(y_{\alpha_q,q}-y_{\beta_q,q}+n\eta)}
{\theta(y_{\alpha_1,1}-y_{\beta_1,1}-n\eta)
\theta(y_{\alpha_q,q}-y_{\beta_q,q})}
w_{j-r}(y_{\beta_1,1},\dots,y_{\beta_q,q})\times\\
\times w_{i+r}(y_{\alpha_1,1}+t_1,\dots,y_{\alpha_q,q}+t_q)+\\
+\sum_{1\le t\le q-1}\frac{e^{-2\pi in\eta}\theta(n\eta)
\theta(y_{\alpha_1,1}-y_{\beta_1,1})}
{\theta_{\alpha_1,1}-y_{\beta_1,1}-n\eta)}\cdot
\frac{\theta(y_{\alpha_t,t}+y_{\beta_{t+1},t+1}-y_{\beta_t,t}-
y_{\alpha_{t+1},t+1})}{\theta(y_{\alpha_t,t}-y_{\beta_t,t})
\theta(y_{\beta_{t+1},t+1}-y_{\alpha_{t+1},t+1})}\times\\
\times w_{j-r}(y_{\beta_1,1},\dots,y_{\beta_t,t},
y_{\alpha_{t+1},t+1},\dots,,y_{\alpha_q,q})\times\\
\times w_{i+r}(y_{\alpha_1,1}+t_1,\dots,y_{\alpha_t}+t_t,
y_{\beta_{t+1}}+t_{t+1},\dots,y_{\beta_q,q}+t_q)\biggr).\end{gathered}$$ Here $t_\alpha=-(d(n_1,\dots,n_{\alpha-1})+d(n_{\alpha+1},\dots,n_q))\eta$.
The equality $\psi_{\alpha_1,\dots,\beta_q}=0$ immediately follows from the identity (35) proved in Appendix B.
Homomorphism of the exchange algebra into the algebra $Q_{n,k}(\E,\eta)$
------------------------------------------------------------------------
Let $q'\in\N$ and $\mu_1,\dots,\mu_{q'},\mu\in\C$. We define an associative algebra $Y_{q'}(\E,\mu;\mu_1,\dots,\mu_{q'})$ as follows. The algebra $Y_{q'}(\E,\mu;\mu_1,\dots,\mu_{q'})$ is presented by the generators $\{\e(u_1,\dots,u_{q'});u_1,\dots,u_{q'}\in\C\}$ and the defining relations $$\begin{gathered}
\frac{\theta(v_1-u_1+\mu)}{\theta(v_1-u_1)} \e(u_1,\dots,u_{q'})
\e(v_1+\mu_1,\dots,v_{q'}+\mu_{q'})=\\
=\sum_{1\le t<q'}\frac{\theta(\mu)\theta(v_t-u_t+u_{t+1}-v_{t+1})}
{\theta(v_t-u_t)\theta(u_{t+1}-v_{t+1})}\times\\
\times\e(v_1,\dots,v_t,u_{t+1},\dots,u_{q'})
\e(u_1+\mu_1,\dots,u_t+\mu_t,v_{t+1}+\mu_{t+1},\dots,v_{q'}+\mu_{q'})
+\\
+\frac{\theta(v_{q'}-u_{q'}+\mu)}{\theta(v_{q'}-u_{q'})}
\e(v_1,\dots,v_{q'})\e(u_1+\mu_1,\dots,u_{q'}+\mu_{q'}).\end{gathered}$$
For $\mu=\mu_1=\ldots=\mu_{q'}=0$ the algebra $Y_{q'}(\E,0;0,\dots,0)$ is the polynomial ring in infinitely many variables $\{\e(u_1,\dots,u_{q'});u_1,\dots,u_{q'}\in\C\}$. One can show that the algebra $Y_{q'}(\E,\mu;\mu_1,\dots,\mu_{q'})$ is a flat deformation of this polynomial ring.
Let $\frac n{n-k}=n_1'-\frac1{n_2'-\ldots-\frac1{n_{q'}'}}$ be an expansion in the continued fraction in which $n_\alpha'\ge2$ for $1\le\alpha\le q'$. It is clear that such an expansion exists and unique. For the relationship between the expansions in continued fractions of the numbers $\frac nk$ and $\frac n{n-k}$, see Appendix C.
There is an algebra homomorphism $$\psi\colon
Y_{q'}(\E,\mu;\mu_1,\dots,\mu_{q'})\to Q_{n,k}(\E,\eta),$$ where $\mu=n\eta$ and $\mu_\alpha=(d(n_1',\dots,n_{\alpha-1}')-d(n_{\alpha+1}',
\dots,n_{q'}')\eta$. This homomorphism is of the form *:* $$\psi\colon\e(u_1,\dots,u_{q'})\to\sum_{\alpha\in\Z/n\Z}
w_\alpha(u_1,\dots,u_{q'})x_{1-\alpha},$$ where $\{x_i;i\in\Z/n\Z\}$ are the generators of the algebra $Q_{n,k}(\E,\eta)$, and $\{w_\alpha;\alpha\in\Z/n\Z\}$ is a basis in the space of theta functions $\Theta_{n/n-k}(\Gamma)$ (see Appendix B).
Let us apply the map $\psi$ to the difference between the left- and right-hand sides of the relations in the algebra $Y_{q'}(\E,\mu;\mu_1,\dots,\mu_{q'})$. We must verify the resulting relation in the algebra $Q_{n,k}(\E,\eta)$. We have $$\begin{gathered}
\frac{\theta(v_1-u_1+\mu)}{\theta(v_1-u_1)}
\psi(\e(u_1,\dots,u_{q'}))\psi(\e(v_1+\mu_1,\dots,v_{q'}+\mu_{q'}))-
\\
-\sum_{1\le t<q'}\frac{\theta(\mu)\theta(v_t-u_t+u_{t+1}-v_{t+1})}
{\theta(v_t-u_t)\theta(u_{t+1}-v_{t+1})}\times\\
\times\psi(\e(v_1,\dots,v_t,u_{t+1},\dots,u_{q'}))
\psi(\e(u_1+\mu_1,\dots,u_t+\mu_t,v_{t+1}+\mu_{t+1},\dots,
v_{q'}+\mu_{q'}))-\\
-\frac{\theta(v_{q'}-u_{q'}+\mu)}{\theta(v_{q'}-u_{q'})}
\psi(\e(v_1,\dots,v_{q'}))\psi(\e(u_1+\mu_1,\dots,u_{q'}+\mu_{q'}))=
\\
=\sum_{\alpha,\beta\in\Z/n\Z}x_{1-\alpha}x_{1-\beta}\times\\
\times\biggl(
\frac{\theta(v_1-u_1+\mu)}{\theta(v_1-u_1)}
w_\alpha(u_1,\dots,u_{q'})
w_\beta(v_1+\mu_1,\dots,
v_{q'}+\mu_{q'})-\\
-\sum_{1\le t<q'}\frac{\theta(\mu)\theta(v_t-u_t+u_{t+1}-v_{t+1})}
{\theta(v_t-u_t)\theta(u_{t+1}-v_{t+1})}\times\\
\times w_\alpha(v_1,\dots,v_t,u_{t+1},\dots,u_{q'})
w_\beta(u_1+\mu_1,\dots,u_t+\mu_t,v_{t+1}+\mu_{t+1},\dots,
v_{q'}+\mu_{q'})-\\
-\frac{\theta(v_{q'}-u_{q'}+\mu)}{\theta(v_{q'}-u_{q'})}
w_\alpha(v_1,\dots,v_{q'})w_\beta(u_1+\mu_1,\dots,u_{q'}+\mu_{q'})
\biggr).\end{gathered}$$ By using the identity (35) in Appendix B together with the relations (18) in the algebra $Q_{n,k}(\E,\eta)$, one can readily see that this expression is equal to $0$.
Let $\Gamma\subset\C$ be an integral lattice generated by $1$ and $\tau\in\C$, where $\Im\tau>0$. Let $n\in\N$ and $c\in\C$. We denote by $\Theta_{n,c}(\Gamma)$ the space of the entire functions of one variable satisfying the following relations: $$f(z+1)=f(z),\quad f(z+\tau)=(-1)^ne^{-2\pi i(nz-c)}f(z).$$ As is known \[30\], $\dim\Theta_{n,c}(\Gamma)=n$, every function $f\in\Theta_{n,c}(\Gamma)$ has exactly $n$ zeros modulo $\Gamma$ (counted according to their multiplicities), and the sum of these zeros modulo $\Gamma$ is equal to $c$. Let $\theta(z)=\sum_{\alpha\in\Z}(-1)^\alpha
e^{2\pi i\left(\alpha
z+\frac{\alpha(\alpha-1)}2\tau\right)}$. It is clear that $\theta(z)\in\Theta_{1,0}(\Gamma)$. It follows from what was said above that $\theta(0)=0$, and this is the only zero modulo $\Gamma$. One can readily see that $\theta(-z)=-e^{-2\pi iz}\theta(z)$. Moreover, as is known, the function $\theta(z)$ can be expanded as the infinite product as follows: $$\theta(z)=\prod_{\alpha\ge1}(1-e^{2\pi
i\alpha\tau})\cdot(1-e^{2\pi
iz})\cdot\prod_{\alpha\ge1}(1-e^{2\pi
i(z+\alpha\tau)})(1-e^{2\pi i(\alpha\tau-z)}).$$
Let us introduce the following linear operators $T_{\frac1n}$ and $T_{\frac1n\tau}$ acting on the space of functions of one variable: $$T_{\frac1n}f(z)=f\left(z+\frac1n\right),\quad
T_{\frac1n\tau}f(z)=e^{2\pi
i\left(z+\frac1{2n}-\frac{n-1}{2n}\tau\right)}
f\left(z+\frac1n\tau\right).$$ One can readily see that the space $\Theta_{n,\frac{n-1}2}(\Gamma)$ is invariant with respect to the operators $T_{\frac1n}$ and $T_{\frac1n\tau}$. Moreover, $T_{\frac1n}T_{\frac1n\tau}=e^{\frac{2\pi i}n}T_{\frac1n\tau}T_{\frac1n}$. The restriction of these operators to the space $\Theta_{n,\frac{n-1}2}(\Gamma)$ satisfy the relations $T_{\frac1n}^n=T_{\frac1n\tau}^n=1$. Let $\wt{\Gamma_n}$ be the group with the generators $a,b,\epsilon$ and the defining relations $ab=\epsilon ba$, $a\epsilon=\epsilon a$, $b\epsilon=\epsilon b$, and $a^n=b^n=\epsilon^n=e$. The group $\wt{\Gamma_n}$ is a central extension of the group $\Gamma_n=\Gamma/n\Gamma\simeq(\Z/n\Z)^2$, namely, the element $\epsilon$ generates a normal subgroup $C_n=\Z/n\Z$, and $\wt{\Gamma_n}/C_n=\Gamma_n$. The formulas $a\mapsto T_{\frac1n}$, $b\mapsto T_{\frac1n\tau}$, and $\epsilon\mapsto{}$(multiplication by $e^{\frac{2\pi i}n}$) define an irreducible representation of the group $\wt{\Gamma_n}$ in the space $\Theta_{n,\frac{n-1}2}(\Gamma)$. Let us choose a basis $\{\theta_\alpha;\alpha\in\Z/n\Z\}$ in the space $\Theta_{n,\frac{n-1}2}(\Gamma)$ in which our operators act as follows: $T_{\frac1n}\theta_\alpha=e^{2\pi i\frac\alpha
n}\theta_\alpha$, and $T_{\frac1n\tau}\theta_\alpha=\theta_{\alpha+1}$. It is clear that this choice can be carried out uniquely up to multiplication by a common constant. The functions $\theta_\alpha(z)$ are of the form $$\theta_\alpha(z)=\theta\left(z+\frac\alpha n\tau\right)
\theta\left(z+\frac1n+\frac\alpha n\tau\right)\dots
\theta\left(z+\frac{n-1}n+\frac\alpha n\tau\right)
e^{2\pi i\left(\alpha
z+\frac{\alpha(\alpha-n)}{2n}\tau+\frac\alpha{2n}\right)}.$$ One can readily see that $\theta_\alpha(z)\in\Theta_{n,\frac{n-1}2}(\Gamma)$, $\theta_{\alpha+n}(z)=\theta_\alpha(z)$, and $$\begin{aligned}
\theta_\alpha\left(z+\frac1n\right)&=e^{2\pi i\frac\alpha
n}\theta_\alpha(z),\\
\theta_\alpha\left(z+\frac1n\tau\right)&=e^{-2\pi
i\left(z+\frac1{2n}-\frac{n-1}{2n}\tau\right)}\theta_{\alpha+1}(z)
\end{aligned}$$ It is clear that the functions $\left\{\theta_\alpha\left(z-\frac1nc-\frac{n-1}{2n}\right);
\alpha\in\Z/n\Z\right\}$ form a basis in the space $\Theta_{n,c}(\Gamma)$.
We need some identities: $$\theta(nz)=\frac{n\theta_0(z)\dots\theta_{n-1}(z)e^{-2\pi
i\frac{n(n-1)}2z}}{\theta_1(0)\dots\theta_{n-1}(0)
\theta\left(\frac1n\right)\dots\theta\left(\frac{n-1}n\right)}.$$
One can readily see by using relations (26) that the functions on both sides of the equation belong to the space $\Theta_{n^2,\frac{n(n-1)}2\tau}(\Gamma)$. Moreover, it is clear that the zeros of both functions coincide, namely, these are $n^2$ points $\left\{\frac\alpha
n+\frac\beta n\tau;\alpha,\beta\in\Z\right\}$ modulo $\Gamma$. Hence, the functions on the left- and right-hand sides of the equation differ by a constant multiple, which can be evaluated by dividing (27) by $\theta(z)$ and passing to the limit as $z\to0$.
Let $\theta_0,\theta_1,\theta_2\in\Theta_{3,0}(\Gamma)$. For $z,\eta\in\C$ and $\alpha\in\Z/3\Z$ we have $$\theta_0(\eta)\theta_\alpha(z+\eta)\theta_\alpha(z)+
\theta_1(\eta)\theta_{\alpha+2}(z+\eta)
\theta_{\alpha+1}(z)+
\theta_2(\eta)\theta_{\alpha+1}(z+\eta)
\theta_{\alpha+2}(z)=0.$$
It is clear that $\theta_\alpha(z+\eta)\theta_\beta(z)\in\Theta_{6,-3\eta}(\Gamma)$ as a function of the variable $z$. There must be three linear relations among these nine functions in a six-dimensional space. With regard to the action of the group $\wt{\Gamma_3}$, we see that the relations must be of the form $a(\eta)\theta_\alpha(z+\eta)\theta_\alpha(z)+
b(\eta)\theta_{\alpha+1}(z+\eta)\theta_{\alpha+2}(z)+
c(\eta)\theta_{\alpha+2}(z+\eta)\theta_{\alpha+1}(z)=0$. Really, every three-dimensional space of relations invariant with respect to the translations $z\to z+\frac13$ and $z\to z+\frac13\tau$ (see (26)) is of this form, where $a,b,c$ do not depend on $\alpha$. By setting $\alpha=1$ and $z=0$, we obtain $\frac{c(\eta)}{a(\eta)}=\frac{\theta_1(\eta)}{\theta_0(\eta)}$. By setting $\alpha=2$ and $z=0$, we obtain $\frac{b(\eta)}{a(\eta)}=\frac{\theta_2(\eta)}{\theta_0(\eta)}$.
Let $\theta_\alpha\in\Theta_{n,c}(\Gamma)$. Then $$\begin{gathered}
\frac{\theta(y-z+nv-nu)}{\theta(y-z)\theta(nv-nu)}
\theta_\alpha(y+u)\theta_\beta(z+v+\eta)+
\frac{\theta(z-y+n\eta)}{\theta(z-y)\theta(n\eta)}
\theta_\alpha(z+u)\theta_\beta(y+v+\eta)=\\
=\frac1n\theta\left(\frac1n\right)\dots\theta\left(\frac{n-1}n\right)
\sum_{r\in\Z/n\Z}\frac{\theta_{\beta-\alpha}(v-u+\eta)}
{\theta_r(\eta)\theta_{\beta-\alpha-r}(v-u)}\theta_{\beta-r}(y+v)
\theta_{\alpha+r}(z+u+\eta).\end{gathered}$$
This is a special case of the relation (31) (for $p=1$) proved in Appendix B.
By setting $u=v+\eta$ in the relation (29) and making the change of variables $y+v\to y$, $z+v\to z$, we obtain $$\begin{gathered}
\frac{\theta(z-y+n\eta)}{\theta(z-y)\theta(n\eta)}
(\theta_\alpha(z+\eta)\theta_\beta(y+\eta)-\theta_\alpha(y+\eta)
\theta_\beta(z+\eta))=\\
=\frac1n\theta\left(\frac1n\right)\dots\theta\left(\frac{n-1}n\right)
\theta_{\beta-\alpha}(0)\sum_{r\in\Z/n\Z}
\frac1{\theta_r(\eta)\theta_{\beta-\alpha-r}(-\eta)}
\theta_{\beta-r}(y)\theta_{\alpha+r}(z+2\eta).\end{gathered}$$
Let $n$ and $k$ be coprime positive integers such that $1\le
k<n$. We expand the ratio $\frac nk$ in a continued fraction of the form: $\frac
nk=n_1-\frac1{n_2-\frac1{n_3-\ldots-\frac1{n_p}}}$, where $n_\alpha\ge2$ for any $\alpha$. It is clear that such an expansion exists and is unique. We denote by $d(m_1, \dots , m_q)$ the determinant of the ($q
\times q$) matrix $(m_{\alpha\beta})$, where $m_{\alpha\alpha}=m_\alpha$, $m_{\alpha,\alpha+1}=m_{\alpha+1,\alpha}=-1$, and $m_{\alpha,\beta}=0$ for $|\alpha-\beta|>1$. For $q = 0$ we set $d(\varnothing)=1$. It follows from the elementary theory of continued fractions that $n = d(n_1, \dots ,
n_p)$ and $k = d(n_2, \dots , n_p)$.
Let $\Gamma\subset\C$ be an integral lattice generated by 1 and $\tau$ again, where $\Im\tau>0$.
We denote by $\Theta_{n/k}(\Gamma)$ the space of entire functions (of $p$ variables) satisfying the following relations: $$\begin{aligned}
f(z_1,\dots,z_\alpha+1,\dots,z_p)&=f(z_1,\dots,z_p),\\
f(z_1,\dots,z_\alpha+\tau,\dots,z_p)&=(-1)^{n_\alpha}e^{-2\pi
i(n_\alpha
z_\alpha-z_{\alpha-1}-z_{\alpha+1}-(\delta_{1,\alpha}-1)\tau)}
f(z_1,\dots,z_p).\end{aligned}$$ Here $1\le\alpha\le p$ and $z_0=z_{p+1}=0$, and $\delta_{1,\alpha}$ stands for the Kronecker delta. Thus, the functions $f\in\Theta_{n/k}(\Gamma)$ are periodic with respect to each of the variables with period 1 and quasiperiodic with period $\tau$. By the periodicity, each function in the space $\Theta_{n/k}(\Gamma)$ can be expanded in a Fourier series of the form $f(z_1,\dots,z_p)=\sum_{\alpha_1,\dots,\alpha_p\in\Z}
a_{\alpha_1\dots\alpha_p}e^{2\pi i(\alpha_1z_1+\ldots+\alpha_pz_p)}$. By the quasiperiodicity, the coefficients satisfy the system of linear equations $$a_{\alpha_1,\dots,\alpha_{\nu-1}-1,\alpha_\nu+n_\nu,
\alpha_{\nu+1}-1,\dots,\alpha_p}=
(-1)^{n_\alpha}e^{2\pi i(\alpha_\nu+\delta_{1,\alpha}-1)\tau}
a_{\alpha_1\dots\alpha_p}.$$
This system clearly has $n = d(n_1, \dots ,
n_p)$ linearly independent solutions each defining (for $\Im\tau>0$; $n_1,\dots,n_p\ge2$) a function in the space $\Theta_{n/k}(\Gamma)$ .
For $k = 1$ we have the space of functions of one variable $\Theta_n(\Gamma)=\Theta_{n,0}(\Gamma)$ (see Appendix A) with a basis $\left\{w_\alpha(z)=\theta_\alpha\left(z+\frac{n-1}2\right),\
\alpha\in\Z/n\Z\right\}$. A similar basis can be constructed in the space $\Theta_{n/k}(\Gamma)$ for an arbitrary $k$. Let us define the operators $T_{\frac1n}$ and $T_{\frac1n\tau}$ in the space of functions of $p$ variables as follows: $$\begin{aligned}
T_{\frac1n}f(z_1,\dots,z_p)&=f(z_1+r_1,\dots,z_p+r_p),\\
T_{\frac1n\tau}f(z_1,\dots,z_p)&=e^{2\pi i(z_1+\phi)}
f(z_1+r_1\tau,\dots,z_p+r_p\tau).\end{aligned}$$ Here $r_\alpha=\frac{d(n_{\alpha+1},\dots,n_p)}{d(n_1,\dots,n_p)}$ and $\phi\in\C$ is a constant.
It is clear that $T_{\frac1n}T_{\frac1n\tau}=e^{2\pi i\frac
kn}T_{\frac1n\tau}T_{\frac1n}$. As in the case of theta functions of one variable, the space $\Theta_{n/k}(\Gamma)$ is invariant with respect to the operators $T_{\frac1n}$ and $T_{\frac1n\tau}$, and the restriction of these operators to $\Theta_{n/k}(\Gamma)$ satisfies the relations $T_{\frac1n}^n=1$ and $T_{\frac1n\tau}^n=\mu$, where $\mu\in\C$. Let us choose a $\phi$ in such a way that $\mu=1$; clearly, this can be done uniquely up to multiplication of $T_{\frac1n\tau}$ by a root of unity of degree $n$.
There is a basis $\bigl\{w_\alpha(z_1,\dots,z_p);\,\alpha\in\Z/n\Z\bigr\}$ in $\Theta_{n/k}(\Gamma)$ such that $$T_{\frac1n}w_\alpha=e^{2\pi i\frac kn\alpha}w_\alpha,\quad
T_{\frac1n\tau}w_\alpha=w_{\alpha+1}.$$ This basis is defined uniquely up to multiplication by a common constant.
Let $f\in\Theta_{n/k}(\Gamma)$ be an eigenvector of the operator $T_{\frac1n}$ with an eigenvalue $\lambda$. Since $T_{\frac1n}^n=1$ on the space $\Theta_{n/k}(\Gamma)$, we have $\lambda^n=1$. Moreover, $T_{\frac1n}T_{\frac1n\tau}f=e^{2\pi
i\frac
kn}T_{\frac1n\tau}T_{\frac1n}f=e^{2\pi i\frac kn}\lambda
T_{\frac1n\tau}f$, and hence $T_{\frac1n\tau}f$ is also an eigenvector with the eigenvalue $e^{2\pi i\frac
kn}\lambda$. Since $n$ and $k$ are coprime, $e^{2\pi i\frac
kn}$ is a primitive root of unity of degree $n$. Thus, the vectors $\bigl\{T_{\frac1n\tau}^\alpha
f;\,\alpha=0,1,\dots,n-1\bigr\}$ are eigenvectors for the operator $T_{\frac1n}$ with different eigenvalues, and every of root of unity of degree $n$ is an eigenvalue for some $T_{\frac1n\tau}^\alpha f$. Let $w_0$ be such that $T_{\frac1n}w_0=w_0$. We set $w_\alpha=T_{\frac1n\tau}^\alpha w_0$. It is clear that $T_{\frac1n}w_\alpha=e^{2\pi i\frac
kn\alpha}w_\alpha$ and $T_{\frac1n\tau}w_\alpha=w_{\alpha+1}$. Moreover, $w_{\alpha+n}=w_\alpha$ because $T_{\frac1n\tau}^n=1$ on the space $\Theta_{n/k}(\Gamma)$.
We note that, as in the case of theta functions of one variable, the group $\wt{\Gamma_n}$ irreducibly acts on the space $\Theta_{n/k}(\Gamma)$ by the rule $a\mapsto T_{\frac1n}$, $b\mapsto
T_{\frac1n\tau}$, and $\epsilon\mapsto\text{(multiplication by $e^{2\pi
i\frac kn}$})$.
Let $L$ be the group of linear automorphisms on the space of functions of $p$ variables of the form $$gf(z_1,\dots,z_p)=e^{2\pi
i(\phi_1z_1+\ldots+\phi_pz_p+\lambda)}f(z_1+\psi_1,\dots,z_p+\psi_p)$$ for $g\in L$. It is clear that $L$ is a $(2p + 1)$-dimensional Lie group. Let $L'\subset L$ be the subgroup of transformations preserving the space $\Theta_{n/k}(\Gamma)$, that is, $L'=\bigl\{g\in
L;\,g(\Theta_{n/k}(\Gamma))=\Theta_{n/k}(\Gamma)\bigr\}$. Let $L''\subset L'$ consist of the elements preserving each point of $\Theta_{n/k}(\Gamma)$, that is, $L''=\bigl\{g\in
L';\,gf=f\text{ for any }f\in\Theta_{n/k}(\Gamma)\bigr\}$. One can see that the quotient group $L'/L''=\wt G_n$ is generated by the elements $T_{\frac1n}$ and $T_{\frac1n\tau}$ and by the multiplications by constants.
We shall use the notation $w_\alpha^{n/k}(z_1,\dots,z_p)$ if it is not clear from the context what are the theta functions in use.
We need the following identity relating theta functions in the spaces $\Theta_{1,0}(\Gamma)$, $\Theta_{n,\frac{n-1}2}(\Gamma)$, and $\Theta_{n/k}(\Gamma)$: $$\begin{aligned}
&\frac{\theta(y_1-z_1+nv-nu)}{\theta(nv-nu)\theta(y_1-z_1)}\nonumber\\
&{}\times w_\alpha(y_1+m_1u,\dots,y_p+m_pu)
w_\beta(z_1+m_1v+l_1,\dots,z_p+m_pv+l_p)\nonumber\\
&+\sum_{1\le t\le p-1}
\frac{\theta(z_t-y_t+y_{t+1}-z_{t+1})}
{\theta(z_t-y_t)\theta(y_{t+1}-z_{t+1})}\nonumber\\
&{}\times w_\alpha(z_1+m_1u,\dots,z_t+m_tu,y_{t+1}+m_{t+1}u,\dots,y_p+m_pu)
\nonumber\\ &{}\times
w_\beta(y_1+m_1v+l_1,\dots,y_t+m_tv+l_t,z_{t+1}+m_{t+1}v+l_{t+1},
\dots,z_p+m_pv+l_p)\nonumber\\
&{}+\frac{\theta(z_p-y_p+n\eta)}{\theta(z_p-y_p)\theta(n\eta)}
w_\alpha(z_1+m_1u_1,\dots,z_p+m_pu)
\nonumber\\ &{}\times w_\beta(y_1+m_1v+l_1,\dots,y_p+m_pv+l_p)\nonumber\\
&{}=\frac1n\theta\left(\frac1n\right)\dots\theta\left(\frac{n-1}n\right)
\nonumber\\ &{}\times
\sum_{r\in\Z/n\Z}\frac{\theta_{\beta-\alpha+ r(k-1)}(v-u+\eta)}
{\theta_{ r k}(\eta)\theta_{\beta-\alpha- r}(v-u)}
w_{\beta- r}(y_1+m_1v,\dots,y_p+m_pv)
\nonumber\\ &{}\times
w_{\alpha+ r}(z_1+m_1u+l_1,\dots,z_p+m_pu+l_p).\end{aligned}$$ Here $m_\alpha=d(n_{\alpha+1},\dots,n_p)$ and $l_\alpha=d(n_1,\dots,n_{\alpha-1})\eta$.
We denote by $\phi_{\alpha,\beta}(\eta,u,v,y_1,\dots,y_p,z_1,\dots,z_p)$ the difference between the right- and left-hand sides of the formula (31). The calculation shows that this function satisfies the following relations: $$\begin{aligned}
\phi_{\alpha,\beta}(\eta,\dots,y_\alpha+1,\dots,z_p)&=
\phi_{\alpha,\beta}(\eta,\dots,z_p),\\
\phi_{\alpha,\beta}(\eta,\dots,y_\alpha+\tau,\dots,z_p)&=
-e^{-2\pi i(n_\alpha y_\alpha-y_{\alpha-1}-y_{\alpha+1}+\delta_{\alpha,1}v)}
\phi_{\alpha,\beta}(\eta,\dots,z_p),\\
\phi_{\alpha,\beta}(\eta,\dots,z_\alpha+1,\dots,z_p)&=
\phi_{\alpha,\beta}(\eta,\dots,z_p),\\
\phi_{\alpha,\beta}(\eta,\dots,z_\alpha+\tau,\dots,z_p)&=
-e^{-2\pi i(n_\alpha z_\alpha-z_{\alpha-1}-z_{\alpha+1}+
\delta_{\alpha,1}u+\delta_{\alpha,p}\eta)}
\phi_{\alpha,\beta}(\eta,\dots,z_p).
\end{aligned}$$ Here $y_0=y_{p+1}=z_0=z_{p+1}=0$, and $\delta_{\alpha,\beta}$ stands for the Kronecker delta. Moreover, an evaluation shows that there are no poles on the divisors $nv-nu\in\Gamma$, $n\eta\in\Gamma$, $y_1-z_1\in\Gamma$, …, $y_p-z_p\in\Gamma$, and hence the function $\phi_{\alpha,\beta}$ is holomorphic everywhere on $\C^{2p+3}$. However, it is clear that the functions $\bigl\{w_\lambda(y_1+m_1v,\dots,y_p+m_pv)
w_\nu(z_1+m_1u+l_1,\dots,z_p+m_pu+l_p);\,
\lambda,\nu\in\Z/n\Z\bigr\}$ form a basis in the space of holomorphic functions (of the variables $y_1,\dots,y_p,z_1,\dots,z_p$) satisfying the conditions (32). Therefore, the function $\phi_{\alpha,\beta}$ is of the form $$\begin{gathered}
\varphi_{\alpha,\beta}(\eta,u,v,y_1,\dots,z_p)=\\
{}=\sum_{\lambda,\nu\in\Z/n\Z}
\psi_{\lambda,\nu}(\eta,u,v)w_\lambda(y_1+m_1v,\dots,y_p+m_pv)
\\ {}\times
w_\nu(z_1+m_1u+l_1,\dots,z_p+m_pu+l_p).\end{gathered}$$
Here the functions $\psi_{\lambda,\nu}(\eta,u,v)$ are holomorphic and satisfy the relations $$\begin{gathered}
\psi_{\lambda,\nu}(\eta+1,u,v)=\psi_{\lambda,\nu}(\eta,u+1,v)=
\psi_{\lambda,\nu}(\eta,u,v+1)=\psi_{\lambda,\nu}(\eta,u,v),\\
\psi_{\lambda,\nu}(\eta+\tau,u,v)=
e^{-2\pi in(v-u)}\psi_{\lambda,\nu}(\eta,u,v),\\
\psi_{\lambda,\nu}(\eta,u+\tau,v)=e^{2\pi in\eta}\psi_{\lambda,\nu}(\eta,u,v),\\
\psi_{\lambda,\nu}(\eta,u,v+\tau)=e^{-2\pi in\eta}\psi_{\lambda,\nu}(\eta,u,v).
\end{gathered}$$
These relations are verified by the immediate calculation, namely, one must compare the multipliers at the translations by 1 and $\tau$ in the formulas (32) and (33).
However, every holomorphic function of the variables $\eta$, $u$ and $v$ that satisfies relations (34) is vanishes. Really, since this function is periodic, it admits the expansion in the Fourier series $$\psi_{\lambda,\nu}(\eta,u,v)=\sum_{\alpha,\beta,\gamma\in\Z}
a_{\lambda,\nu,\alpha,\beta,\gamma}
e^{2\pi i(\alpha\eta+\beta u+\gamma v)}.$$ Further, it follows from the quasiperiodicity that the coefficients $a_{\lambda,\nu,\alpha,\beta,\gamma}$ are equal to 0.
By setting $u=v+\eta$ in the identity (31) and making the change of variables $y_1\to
y_1-m_1v$, $z_1\to z_1-m_1v$, …, $y_p\to y_p-m_pv$, $z_p\to
z_p-m_pv$, we obtain $$\begin{gathered}
\frac{\theta(y_1-z_1-n\eta)}{\theta(-n\eta)\theta(y_1-z_1)}
w_\alpha(y_1+m_1\eta,\dots,y_p+m_p\eta)
\theta_\beta(z_1+\l_1,\dots,z_p+\l_p)+\\
+\sum_{1\le t<p}\frac{\theta(z_t-y_t+y_{t+1}-z_{t+1})}
{\theta(z_t-y_t)\theta(y_{t+1}-z_{t+1})}
w_\alpha(z_1+m_1\eta,\dots,z_t+m_t\eta,
y_{t+1}+m_{t+1}\eta,\dots,y_p+m_p\eta)\times\\
\times
w_\beta(y_1+\l_1,\dots,y_t+\l_t,z_{t+1}+\l_{t+1},\dots,z_p+\l_p)+\\
+\frac{\theta(z_p-y_p+n\eta)}{\theta(z_p-y_p)\theta(n\eta)}
w_\alpha(z_1+m_1\eta,\dots,z_p+m_p\eta)
w_\beta(y_1+\l_1,\dots,y_p+\l_p)=\\
=\frac1n\theta\left(\frac1n\right)\dots\theta\left(\frac{n-1}n\right)
\times\\\times\sum_{r\in\Z/n\Z}\frac{\theta_{\beta-\alpha+r(k-1)}(0)}
{\theta_{rk}(\eta)\theta_{\beta-\alpha-r}(-\eta)}
w_{\beta-r}(y_1,\dots,y_p)
w_{\alpha+r}(z_1+m_1\eta+\l_1,\dots,z_p+m_p\eta+\l_p).\end{gathered}$$
Let us construct a canonical element $\Delta_{n,k}\in\Theta_{n/k}(\Gamma)\otimes\Theta_{n/n-k}(\Gamma)$ carring out the duality between these spaces (see (36)).
Let $$\frac nk=n_1-\frac1{n_2-\ldots-\frac1{n_p}},\quad \frac
n{n-k}=n_1'-\frac1{n_2'-\ldots-\frac1{n_{p'}'}}$$ be the expansions in continued fractions, where $n_\alpha\ge2$ and $n_\beta'\ge2$ for $1\le\alpha\le p$ and $1\le\beta\le p'$, respectively. Here $p$ and $p'$ stand for the lengths of the continued fractions. Then $p'=n_1+\ldots+n_p-2p+1$ and $n_1'+\ldots+n_{p'}'=2(n_1+\ldots+n_p)-3p+1$. Moreover, $n_1'+\ldots+n_\alpha'=2\alpha+\beta$ for $n_1+\ldots+n_\beta-2\beta+1\le\alpha\le
n_1+\ldots+n_{\beta+1}-2\beta-2$. In other words, the Young diagrams for the partitions $(n_1-1,n_1+n_2-3,\dots,n_1+\ldots+n_\alpha-2\alpha+1,\dots)$ and $(n_1'-1,n_1'+n_2'-3,\dots,n_1'+\dots+n_\beta'-2\beta+1,\dots)$ are dual to each other.
For $k = 1$, $p = 1$, and $n_1 = n$ we have $p' = n - 1$ and $n_1'=\ldots=n_{n-1}'=2$. For $p > 1$, if $n_2, \dots , n_{p-1}\ge 3$, then the sequence $(n_1',\dots,n_p')$ becomes $(2^{(n_1-2)},3,2^{(n_2-3)},3,\dots,3,2^{(n_{p-1}-3)},3,2^{(n_p-2)})$. Here $2^{(t)}$, $t\ge0$, stands for a sequence of $t$ twos. This formula remains valid without the assumption that $n_2,\dots,n_p\ge3$ if we agree that the sequence $(m_1,2^{(-1)},m_2)$ is of length $1$ and is equal to $(m_1+m_2-2)$. This rule must be applied in succession to all occurrences $n_\alpha=2$ for $2\le\alpha\le p-1$.
The proof can be carried out by induction on $\min(p,p')$. For $p = 1$, one must prove that $\frac n{n-1}=2-\frac1{2-\ldots-\frac12}$ is of length $n - 1$. Let $p,p'>1$ and let, say, $n_1>2$. We have $\frac
k{d(n_3,\dots,n_p)}=n_2-\frac1{n_3-\ldots-\frac1{n_p}}$. By assumption, $$\frac k{k-d(n_3,\dots,n_p)}=
n_{n_1-1}'-1-\frac1{n_{n_1}'-\frac1{n_{n_1+1}'-\ldots-\frac1{n_{p'}'}}}.$$ and the sequence $(n_1',\dots,n_{n_1-2}')$ is $(2^{(n_1-2)})$. Further, one must show that $n_1'-\frac1{n_2'-\ldots-\frac1{n_{p'}'}}=\frac n{n-k}$. Here it is used that $d(n_1,\dots,n_p)=n$, $d(n_2,\dots,n_p)=k$, $
\frac{d(n_1,\dots,n_p)}{d(n_2,\dots,n_p)}=
n_1-\frac1{n_2-\ldots-\frac1{n_p}},
$ and $d(n_1,\dots,n_p)=n_1d(n_2,\dots,n_p)-d(n_3,\dots,n_p)$.
Let a function $\Delta_{n.k}(z_1,\dots,z_p;z_1',\dots,z_{p'}')$ of $p+p'$ variables $z_1,\dots,z_p,z_1',\dots,z_{p'}'$ be defined by the formula $$\begin{gathered}
\Delta_{n,k}(z_1,\dots,z_p,z_1',\dots,z_{p'}')=\\
{}=e^{2\pi
iz_1'}\theta(z_1-z_1')\theta(z_p+z_{p'}')\cdot\prod_{1\le\alpha\le
p'-1}\theta(z_\alpha'-z_{\alpha+1}'+z_{n_1'+\ldots+n_\alpha'-2\alpha+1})
\\
{}\times\prod_{1\le\beta\le
p-1}\theta(z_\beta-z_{\beta+1}+z_{n_1+\ldots+n_\beta-2\beta+1}').\end{gathered}$$ This function satisfies the following relations: $$\Delta_{n,k}(z_1,\dots,z_\alpha+1,\dots,z_{p'}')=
\Delta_{n,k}(z_1,\dots,z_\beta'+1,\dots,z_{p'}')=
\Delta_{n,k}(z_1,\dots,z_{p'}'),\\$$ $$\begin{gathered}
\Delta_{n,k}(z_1,\dots,z_\alpha+\tau,\dots,z_{p'}')\\
=(-1)^{n_\alpha}e^{-2\pi i
(n_\alpha z_\alpha-z_{\alpha-1}-z_{\alpha+1}-(\delta_{\alpha,1}-1)\tau)}
\Delta_{n,k}(z_1,\dots,z_{p'}'),\end{gathered}$$ $$\begin{gathered}
\Delta_{n,k}(z_1,\dots,z_\beta'+\tau,\dots,z_{p'}')\\
=(-1)^{n_\beta'}e^{-2\pi i
(n_\beta'z_\beta'-z_{\beta-1}'-z_{\beta+1}'-(\delta_{\beta,1}-1)\tau)}
\Delta_{n,k}(z_1,\dots,z_{p'}').\end{gathered}$$ Here $z_0=z_{p+1}=z_0'=z_{p'+1}'=0$ and $\delta_{\alpha,1}$ stands for the Kronecker delta.
The proof immediately follows from our description of the duality between the sequences $(n_1,\dots,n_p)$ and $(n_1',\dots,n_{p'}')$.
$$\Delta_{n,k}(z_1,\dots,z_p;z_1',\dots,z_{p'}')=c_{n,k}
\sum_{\alpha\in\Z/n\Z}w_\alpha^{n/k}(z_1,\dots,z_p)
w_{1-\alpha}^{n/n-k}(z_1',\dots,z_{p'}').$$
Here $c_{n,k}\in\C$ is a constant.
It follows from the previous proposition that the function $\Delta_{n,k}$ belongs to the space $\Theta_{n/k}(\Gamma)$ when regarded as a function of the variables $z_1,\dots,z_p$. Similarly, $\Delta_{n,k}$ belongs to $\Theta_{n/n-k}(\Gamma)$ as a function of $z_1',\dots,z_{p'}'$. Therefore, $$\Delta_{n,k}(z_1,\dots,z_p;z_1',\dots,z_{p'}')=
\sum_{\alpha,\beta\in\Z/n\Z}\lambda_{\alpha,\beta}
w_\alpha^{n/k}(z_1,\dots,z_p)
w_\beta^{n/n-k}(z_1',\dots,z_{p'}').$$ However, one can readily see that $$\Delta_{n,k}(z_1+ r_1,\dots,z_p+ r_p;z_1'+ r_1',\dots,z_{p'}'+ r_{p'}')=
e^{\frac{2\pi i}n}\Delta_{n,k}(z_1,\dots,z_{p'}'),$$ where $ r_\alpha=\frac{d(n_1,\dots,n_{\alpha-1})}n$ and $
r_\beta'=\frac{d(n_1',\dots,n_{\beta-1}')}n$. Hence, $\lambda_{\alpha,\beta}=0$ for $\alpha+\beta\not\equiv1\mmod n$ (because $w_\alpha(z_1+ r_1,\dots,z_p+ r_p)=e^{2\pi i\frac\alpha
n}w_\alpha(z_1,\dots,z_p)$ and $w_\beta(z_1'+ r_1',\dots,z_{p'}'+
r_{p'}')=e^{2\pi i\frac\beta
n}w_\beta(z_1',\dots,z_{p'}')$). Thus, $\lambda_{\alpha,\beta}=\lambda_\alpha\delta_{\alpha+\beta,1}$. Similarly, $$\begin{gathered}
\Delta_{n,k}(z_1+ r_1\tau,\dots,z_p+ r_p\tau;z_1'+ r_1'\tau,
\dots,z_{p'}'+ r_{p'}'\tau)\\
{}=e^{2\pi i\left(\frac1n\tau-z_p-z_{p'}'\right)}
\Delta_{n,k}(z_1,\dots,z_{p'}').\end{gathered}$$ Hence, $\lambda_\alpha=\lambda_{\alpha+1}$, that is, $\lambda_\alpha$ does not depend on $\alpha$.
Integrable system, quantum groups, and $R$-matrices
---------------------------------------------------
One of the main methods in the investigation of exactly solvable models \[6\] in quantum and statistical physics is the inverse problem method (see \[45\]). This method leads to the study of representations of algebras of monodromy matrices, that is, to the study of meromorphic matrix functions $L(u)$ satisfying the relations $$R(u-v)L^1(u)L^2(v)=L^2(v)L^1(u)R(u-v).$$ Here $R(u)$ is a chosen solution of the Yang-Baxter equation in the class of meromorphic matrix-valued functions, $$R^{12}(u-v)R^{13}(u)R^{23}(v)=R^{23}(v)R^{13}(u)R^{12}(u-v).$$ We note that $R(u)$ takes the values in ($n^2\times n^2$) matrices with a fixed decomposition $\Mat_{n^2}=\Mat_n\otimes\Mat_n$. We use the standard notation, namely, $L^1=L\otimes1$, $L^2=1\otimes L$, $R^{12}=R\otimes1$, etc. (see \[45\]).
In \[45\] Sklyanin studies the solutions of the equation (37) for the simplest elliptic solution of the equation (38), that is, for the so-called Baxter $R$-matrix, which is of the form $R(u)=
1+\sum_{\alpha=1}^3W_\alpha(u)\sigma_\alpha\otimes\sigma_\alpha$, where $\sigma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix}$, $\sigma_2=\begin{pmatrix}0&-i\\i&0\end{pmatrix}$, and $\sigma_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ are the Pauli matrices, and the coefficients $W_\alpha(u)$ can be expressed in terms of the Jacobi elliptic functions as follows: $$\begin{aligned}
W_1(u)&=\frac{\sn(i\eta,k)}{\sn(u+i\eta,k)},\quad
W_2(u)=\frac{\dn}{\sn}(u+i\eta,k)\frac{\sn}{\dn}(i\eta,k),\\
W_3(u)&=\frac{\cn}{\sn}(u+i\eta,k)\frac{\sn}{\cn}(i\eta,k).\end{aligned}$$ The functions $W_\alpha(u)$ uniformize the elliptic curve $\frac{W_\alpha^2-W_\beta^2}{W_\gamma^2-1}=J_{\alpha,\beta}$, where the $J_{\alpha,\beta}$s do not depend on $u$ and satisfy the relation $J_{12}+J_{23}+J_{31}+J_{12}J_{23}J_{31}=0$. Here $\alpha,\beta$, and $\gamma$ are pairwise distinct and $J_{\beta,\alpha}=-J_{\alpha,\beta}$.
We note that this elliptic curve is the complete intersection of two quadrics, for instance, $w_1^2-w_2^2=J_{12}(w_3^2-1)$ and $w_2^2-w_3^2=J_{23}(w_1^2-1)$.
Sklyanin discovered that the equation (37) for the Baxter $R$-matrix has a solution of the form $L(u)=S_0+\sum_{\alpha=1}^3W_\alpha(u)S_\alpha$, where $S_0$ and $S_\alpha$ are matrices that do not depend on $u$ and satisfy the following relations: $$\begin{aligned}\relax
[S_\alpha,S_0]_-&=-iJ_{\beta\gamma}[S_\beta,S_\gamma]_+,\\
[S_\alpha,S_\beta]_-&=i[S_0,S_\gamma]_+,\\
\end{aligned}$$ where $[a,b]_\pm=ab\pm ba$.
Sklyanin further studies the algebra with the generators $S_0,S_\alpha$ and the relations (39); he denotes this algebra by $\F_{\eta,k}$. The main assumption concerning this algebra is that it satisfies the PBW condition. Moreover, Sklyanin finds the quadratic central elements of the algebra $\F_{\eta,k}$ and the finite-dimensional representations of the algebra $\F_{\eta,k}$ by difference operators in some function space (see \[46\]).
In our notation, the Sklyanin algebra $\F_{\eta,k}$ is the algebra $Q_4(\E,\eta)$, where $\E$ is an elliptic curve given by the functions $W_\alpha(u)$, that is, a complete intersection of two quadrics in $\C^3$.
The Yang-Baxter equation has other elliptic solutions generalizing the Baxter solution (see \[7\]). The result of \[7\] can be described as follows: for any pair of positive integers $n$ and $k$ such that $1\le k<n$ and $n$ and $k$ are coprime there is a family of solutions $R_{n,k}(\E,\eta)(u)$ of the equation (38). Here $\E$ is an elliptic curve and $\eta\in\E$, as above. The Baxter solution is obtained for $n=2$ and $k=1$.
According to \[10\], the Sklyanin result can be generalized to an arbitrary solution $R_{n,k}(\E,\eta)(u)$. In our notation, the results of \[10\] look as follows: there is a homomorphism of the algebra of monodromy matrices for the $R$-matrix $R_{n,k}(\E,\eta)$ into the algebra $Q_{n^2,nk-1}(\E,\eta)$. Correspondingly, the algebra $Q_{n^2,nk-1}(\E,\eta)$ is a deformation of the projectivization of the Lie algebra $\slg_n$. Moreover, there is a homomorphism of the algebra of monodromy matrices into the algebra $Q_{dn^2,dnk-1}(\E,\eta)$ for any $d\in\N$. It can be conjectured that every finite-dimensional representation of the algebra of monodromy matrices can be obtained from a representation of the algebra $Q_{dn^2,dnk-1}(\E,\eta)$.
Another relationship between the elliptic solutions of the Yang-Baxter equation and the elliptic algebras follows from the results of \[10\]. The multiplication in the algebra $Q_{n,k}(\E,\eta)$ is defined by the so-called Young projections $S^\alpha V\otimes S^\beta V\to S^{\alpha+\beta}V$ corresponding to $R_{n,k}(\E,\eta)(u)$ (see \[10\]). Moreover, $Q_{n,k}(\E,\eta)=\sum_\alpha S^\alpha V$.
We also note that the study of algebras $Q_{n,k}(\E,\eta)$ and their representations led to deeper understanding of the structure of $R$-matrices $R_{n,k}(\E,\eta)(u)$ and of the corresponding algebraic objects (the Zamolodchikov algebra and the algebra of monodromy matrices). For this topic, see \[33\].
Deformation quantization
------------------------
Let $M$ be a manifold ($C^\infty$, analytic, algebraic, etc.) and let $\F(M)$ be a function algebra on $M$. In the “physical” language, $M$ is the state space of the system and $\F(M)$ is the algebra of observables. In \[5\] the following approach to the quantization was suggested: the underlying vector space of the quantum algebra of observables coincide with that of $\F(M)$, but the multiplication is deformed and is no longer commutative (though still associative). Moreover, the multiplication depends on the deformation parameter (Planck constant). For $\hbar=0$ we have the ordinary commutative multiplication. Since the Planck constant is small, we do not notice that the observables in classical mechanics are non-commutative. Expanding the multiplication in the series in powers of $\hbar$ we obtain $f*g=fg+\{f,g\}\hbar+o(\hbar)$. The operation $\{{\cdot},{\cdot}\}\colon\F(M)\otimes\F(M)\to\F(M)$ is bilinear, and, applying the gauge transformations, one can make it anticommutative, $\{f,g\}=-\{g,f\}$. Moreover, since the multiplication $*$ is associative, we see that $\{f,gh\}=\{fg\}h+\{f,h\}g$ (the Leibniz rule) and $\{f,\{g,h\}\}+\{h,\{f,g\}\}+\{g,\{h,f\}\}=0$ (the Jacobi identity). The Leibniz rule means that $\{f,g\}=\langle w,df\wedge dg\rangle$, where $w$ is a bivector field on $M$, and the Jacobi identity means that $[w,w]=0$. Thus, every quantization defines a Poisson structure $w$ (or $\{{\cdot},{\cdot}\}$) on the manifold $M$. The inverse problem arises: construct a quantization $*$ from a manifold $M$ with Poisson structure (a Poisson manifold). This problem was solved in \[26\] at the formal level. Namely, bidifferential operators $B_n$ ($n\ge2$) were constructed on a Poisson manifold $M$ in such a way that the formal series $$f*g=fg+\{f,g\}\hbar+\sum_{n\ge2}B_n(f,g)\hbar^n$$ satisfies the condition $(f*g)*h=f*(g*h)$. Moreover, the operators $B_n$ are constructed from $\{{\cdot},{\cdot}\}$ by an explicit formula. Thus, the problem of formal quantization was solved; however, the problem on the convergence of the series (40) and on its identification remains open. This problem seems to be very complicated. For instance, let $M=\C^n$, $\F(M)=S^*(\C^n)=\C[x_1,\dots,x_n]$, and let the Poisson bracket be quadratic ($\{x_i,x_j\}=\sum_{\alpha,\beta}c_{ij}^{\alpha\beta}x_\alpha
x_\beta$, where $c_{ij}^{\alpha\beta}\in\C$ are symmetric with respect to $\alpha,\beta$ and antisymmetric with respect to $i,j$). In this case, the multiplication $*$ must be homogeneous, that is, $S^\alpha(\C^n)*S^\beta(\C^n)\subset
S^{\alpha+\beta}(\C^n)$. Therefore, according to (40), the structure constants of the multiplication $*$ in the basis $\{x_1^{\alpha_1}\dots
x_n^{\alpha_n};\alpha_1,\dots,\alpha_n\in\Z_{\ge0}\}$ turn out to be formal series in $\hbar$ which baffle the explicit evaluation even in the simplest case $n=2$, $\{x_1,x_2\}=\alpha x_1x_2$. In this case it is natural to assume that the quantum algebra must be defined by the relation $x_1x_2=e^{-\alpha\hbar}x_2x_1$. On the other hand, the algebras $Q_{n,k}(\E,\eta)$ introduced above are examples of the quantization of $M=\C^n$, where $\eta$ plays the role of Planck constant because $Q_{n,k}(\E,0)=\C[x_1,\dots,x_n]$. Moreover, the structure constants of the algebra $Q_{n,k}(\E,\eta)$ turn out to be elliptic functions of $\eta$. There are also rational and trigonometric limits of the algebras $Q_{n,k}(\E,\eta)$ in which the structure constants are rational (trigonometric) functions of $\eta$ (see \[32\]).
Moduli spaces
-------------
Let $G$ be a semisimple Lie group and let $P\subset G$ be a parabolic subgroup. Let $\M(\E,P)$ be the moduli space of the holomorphic $P$-bundles over an elliptic curve $\E$ \[4\]. According to \[20\], every connected component of the space $\M(\E,P)$ admits a natural Poisson structure. The main property of this structure is as follows: the preimages of the natural map $\M(\E,P)\to\M(\E,G)$ corresponding to forgetting the $P$-structure are symplectic leaves of the structure. The quantization problem for the Poisson manifold $\M(\E,P)$ arises. The solution of this problem could establish a relationship between the natural algebro-geometric problem of studying $P$-bundles (and the corresponding $G$-bundles) and the problem to study representations of the quantum function algebra on $\M(\E,P)$ because the representations correspond to symplectic leaves.
In \[38\], \[22\] these quantum algebras were constructed provided that $P=B$ is a Borel subgroup of an arbitrary group $G$. In \[21\] the quantum algebras were constructed in the case of $G=GL_m$ and an arbitrary parabolic subgroup $P$. Here the algebra $Q_n(\E,\eta)$ corresponds to the case $G=GL_2$, and the algebra $Q_{n,k}(\E,\eta)$ to the case $G=GL_{k+1}$, where $P$ consists of upper block triangular matrices of the form $\vcenter{\raisebox{5pt}{\bm{\mathstrut\cr
k&&*&&*\cr\mathstrut\cr1&&0&&*\cr&&k&&1}}
\vspace{20pt}.}$
Non-commutative algebraic geometry
----------------------------------
One of the main ideas of algebraic geometry is to study the geometry of a manifold by using the algebraic properties of a ring of functions on this manifold. The non-commutative algebraic geometry extends these methods and the geometric intuition to an appropriate class of non-commutative rings. In \[51\], \[48\] the non-commutative algebraic geometry is developed in small dimensions. From this point of view, the algebras $Q_{n,k}(\E,\eta)$ give examples of non-commutative vector spaces. Similar examples of non-commutative Grassmannians ans other varieties are also known \[11\], \[19\], \[22\], \[49\].
Cohomology of algebras
----------------------
Cohomology properties of quadratic algebras are studied in \[29\], \[41\]-\[44\], \[50\]. For generic $\eta$, the algebras $Q_{n,k}(\E,\eta)$ are examples of Koszul algebras. One can readily prove this fact for $k=1$ by using the construction of a free module in §2.6. The constructions of dual algebras $Q_{n,k}^!(\E,\eta)$ are given in \[36\].
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Russian Academy of Sciences\
e-mail: odesskii@mccme.ru
[^1]: This example was communicated to the author by Oleg Ogievetsky \[1\], \[15\], \[40\].
|
---
abstract: 'Electric-field-controlled charge transport is a key concept of modern computers, embodied namely in field effect transistors. The metallic gate voltage controls charge population, thus it is possible to define logical elements which are the key to computational processes. Here, we investigate a similar system defined by metallic gates inducing quasi-one-dimensional transport channels on a high-mobility electron system in the presence of a strong perpendicular magnetic field. Firstly, we solve the three-dimensional Poisson equation, self-consistently imposing relevant boundary conditions, and use the output as an initial condition to calculate charge density and potential distribution in the plane of a two-dimensional electron system, in the presence of an external magnetic field. Subsequently, we impose an external current and obtain the spatial distribution of the transport charges, considering various magnetic field and gate voltage strengths at sufficiently low ($<$ 10 Kelvin) temperatures. We show that magnetic field breaks the spatial symmetry of the current distribution, whereas voltage applied to metallic gates determines the scattering processes.'
address:
- 'Vacational School of Health, Yeni Yuzyil University, Istanbul, 34010, Turkey'
- 'Maltepe University, Faculty Engineering and Natural Sciences, Department of Electrics and Electronics, 34857 Istanbul, Turkey'
- 'Ekendiz Tanay Center for Art and Science, Department of Physics, Ula, Mugla, 48650, Turkey'
author:
- Deniz Eksi
- Afif Siddiki
title: 'Investigating the current distribution of parallel-configured quantum point contacts under quantum Hall conditions'
---
and
Quantum Hall Effect, Quantum Point Contact
Introduction
============
Discovery of semiconductor-based electronics stemming from quantum mechanics revolutionized our computational abilities [@Davies]. The basic idea behind this is to confine electrons in the growth direction ($z$) to a plane and control their population by an electric field applied to the metallic gates residing on the surface. These structures are known as the field-effect transistors (FETs). The best known of these semiconductor devices are the metal-oxide-silicon (MOS) heterojunctions, which are the main ingredients of our daily used computers. A similar heterostructure is the GaAs/AlGaAs junction, in which the electron mobility is much higher [@Datta], i.e., scattering due to impurities is reduced. Here, at the initial crystal growth the average electron density $n_{\rm el}$ is fixed by the number of silicon donors $n_0$ which are homogeneously distributed, and electrons are confined to a single quantum well, forming a two-dimensional electron system (2DES). In this paper, we focus on such high-mobility 2DESs, where charge transport is also controlled by surface gates.
The above described 2DESs present peculiar transport properties when they are subject to high and perpendicular magnetic fields $B$, known as quantum Hall effects [@Girvin00:book], the study of which has produced two Nobel prizes. It is observed that the longitudinal resistance vanishes at certain $B$ intervals, whereas the transverse (Hall) resistance assumes quantized values in units of conductance quanta $e^2/h$ [@vK80:494]. Moreover, even in the absence of an external $B$ field, gate-voltage-induced narrow transport channels also present quantized conductance behavior [@Wees88:848]. Such devices are named quantum point contacts (QPCs), which are the main object of our investigation [@Kristensen98:180; @SiddikiMarquardt; @Arslan08:125423]. These devices are claimed to be a key element in developing quantum computers, while coherence is a significant parameter in charge transport, and topologically protected information processing is required [@AdyStern:quantumcomp].
The scope of this paper is to provide a self-consistent calculation scheme which is able to describe electronic transport through QPCs within the local Ohm’s law. In this work, we compute the potential and current distributions of serially connected QPCs, starting from the calculation of bare electrostatic potential assuming a crystal structure which is used experimentally. Next, applying a perpendicular magnetic field to the 2DES, we obtain the spatial distribution of current-carrying channels, depending on field strength. In the final investigation, an external in-plane electric field is taken into account, and the current flow is obtained under certain conditions. The results of this study indicate that minor field variations are robust in determining the current distribution, whereas gate potential $V_{\rm G}$ and temperature $T$ dominate the scattering processes, as expected.
Model
=====
Since the main goal of our study is to obtain the current distribution through the QPCs, one should first obtain the electrostatic potential distribution $V(x,y,z)$ (hence , electron density distribution, $\rho(x,y,z)$) via solving the Poisson equation,
$$\label{key}
\nabla^{2}V(x,y,z)= - 4\pi\rho(x,y,z)$$
in 3D by imposing relevant boundary conditions and using material properties. For this purpose, we utilize a well-developed numerical method called EST3D , which is based on an iterative method to obtain, self-consistently, $V(x,y,z)$ and $\rho(x,y,z)$ [@Arslan08:125423]. An advanced 3D fast-Fourier subroutine is used to calculate the distributions, layer by layer, where all the surfaces (top, side and bottom) are assumed to be under vacuum, silicon-doped (two layers of delta-doping) GaAs/AlGaAs heterostructure is considered (see Fig . \[fig1\]) and metallic gates are defined on the surface, which are kept at $V_{\rm G}$, as in our previous studies [@Salman:13; @Atci:17]. The vacuum, the heterostructure and the metallic gates are defined by their dielectric constants. Initially, the delta-doped silicon layers are charged positively with a fixed number of charges depending on the crystal growth parameters. The metallic gates are taken to be charged positively or negatively. The rest of the heterostructure, i.e., surfaces including vacuum, are neutral in charge. Starting with these boundary conditions, one obtains $V(x,y,z)$ and $\rho(x,y,z)$ depending on the potential on gates, strength of doping and thickness of GaAs and AlGaAs layers.
![\[fig1\] Schematic presentation of the GaAs/AlGaAs heterostructure. The crystal is in vacuum and 2DES is formed at the interface of the junction. The number of donors and the structure geometry are taken from experimental reports [@2013NatSR; @2017NatCo].](fig1.pdf){width="5cm"}
Equipped with the self-consistently calculated potential and charge density distributions for each layer at zero temperature and magnetic field, one can obtain the density and current distributions in the presence of external in-plane electric and off-plane magnetic fields, using the Newton-Raphson iteration [@Eksi:10; @Yildiz:14; @Kilicoglu16:035702]. Our strategy is to use $V(x,y,z)$ as an initial input, obtained in the previous step, and calculate finite temperature and magnetic field reconstructed potential and charge distributions. Considering the experimental values of energies and charge densities, it can be easily seen that our results are viable, such that the typical charge density of the 2DES is similar to $3\times10^{15}$ m$^{-2}$ , corresponding to a Fermi energy ($E_F$) of 13 meV. At $10$ Tesla magnetic energy, $\hbar \omega_{c}(~ \omega_{c}=eB/m^{*}$) is on the order of 17 meV, and thermal energy (T $\leq$ 10 K) is much smaller than the confinement energy (approximately 4 eV) and potential (energy) on metallic gates ($\sim$ -0.2 eV). The details of the calculation procedures and validity of the assumptions are explained in our previous studies [@Arslan08:125423; @Kilicoglu16:035702].
While performing calculations considering an external current, we always stay in the linear-response regime, which essentially imposes that the applied in-plane electric field does not affect the density and potential distributions. This is well justified, as the current amplitudes considered are much smaller than the Fermi energy [@Guven03:115327]. In the following Sections we present our numerical results, first investigating a toy model using cosine-defined QPCs. The rationale is to clarify the effect of scattering processes without including the geometrical dependencies on them and explain the current distribution depending only on $B$ field. Next, we calculate the same quantities for higher gate voltage QPCs at various magnetic fields.
![\[fig2\] (a) The spatial distribution of screened potential at zero temperature and vanishing magnetic field with $V_{g}=-0.1$ V (b) Self-consistent filling factor distribution and (c) The current distribution at $B= 7.5$ T. Gray scale on the left legend denotes the potential strength, whereas right scale shows the filling factor with black indicating $\nu=2$. Small light blue arrows depict local current distribution and large black arrow shows total current direction. In (b) and (c), equilibrium temperature is $7.43$ K. ](fig2.pdf){width="7.5cm"}
![\[fig3\] Current distributions at (a) $8$ T and (b) $8.5$ T, resulting in $T_{\rm E}$ $10.15$ K and $2.0$ K, respectively. At the higher $B$ value, most of the current is owing without scattering; hence, $T_{\rm E}$ is reduced. ](fig3a.pdf "fig:"){width="7cm"} ![\[fig3\] Current distributions at (a) $8$ T and (b) $8.5$ T, resulting in $T_{\rm E}$ $10.15$ K and $2.0$ K, respectively. At the higher $B$ value, most of the current is owing without scattering; hence, $T_{\rm E}$ is reduced. ](fig3b.pdf "fig:"){width="7cm"}
Results and discussion
======================
It is significant to compare our numerical results with already existing ones , to show the consistency between them. The usual approach is to assume that the QPCs generate cosine- or Gaussian-like potentials in the plane of the 2DESs [@Macucci02:39; @Igor07:qpc1; @Igor07:qpc2]. By such modeling, one can obtain reliable results without further computational complications compared to realistically modeled devices. Here, we prefer to use cosine functions because fast-Fourier transformation processes are much faster and more precise compared to other well-defined functions. Also note that we are only interested in the transport properties of the 2DESs; hence, we show our numerical results just for the $z=z_{\rm 2DES}$ layer, i.e., 284 nm below the gate. Therefore, when the electron (number) density $n_{\rm el}(x,y)$ is presented, we take the result of 3D calculation for $\rho(x,y,z_{\rm 2DES})$ . A similar path is taken for the screened potential at zero temperature and vanishing $B$ field, namely $V^{T=0,B=0}_{\rm scr}(x,y,z_{\rm 2DES})$.
As an illuminating example, we define four QPCs on the top surface of our heterostructure, as shown in Fig. \[fig2\]. The corresponding screened potential profile ($V^{T=0,B=0}_{\rm scr})$) is shown in Fig. \[fig2\]a, together with the dimensionless electron density ($\nu(x,y)$, Fig. \[fig2\]b) and current ($j(x,y)$) distribution as a function of position in the plane of the 2DES (i.e. $z=z_{2DES}$), Fig . \[fig2\]c. It is beneficial to parametrize density by normalizing it with the strength of the external magnetic field. The dimensionless electron density is called the filling factor and is given by $\nu(x,y)=2\pi \ell^{2}n_{\rm el}(x,y)$, where $\ell$ is the magnetic length, defined as $\ell^{2}=eB/h$. From Fig. \[fig2\]a, one can see that the potential generated by the surface gates (both QPCs and side gates, dark blue regions in Fig. \[fig2\]b) depletes electrons beneath them ($V_{scr} (x,y)$= -0.1 eV), and the external potential is well screened by the electrons elsewhere ($V_{scr} (x,y)\simeq 0.0$ eV), if a repulsive potential is applied to the gates ($V_{\rm G}=-0.1$ eV). Obviously, the QPCs constrain electron transport together with the side gates which confine them to a quasi-2D channel. The resulting density distribution is shown in Fig. \[fig2\]b, where the color gradient presents the variation, and regions without a gradient (dark blue) indicate the electron depleted zones below the gates.
It is significant to emphasize that integer filling factors play a distinguishing role both in screening and transport properties of the system at hand. Let us consider a situation where the ratio between self-consistent electron density and the magnetic flux density assumes an integer value. In this case, the Fermi energy falls in between the magnetic field quantized density of states (DOS) locally; hence, there are no states available at these regions. This leads to areas of poor screening and constant electron density, called the incompressible strips [@Chklovskii92:4026]. On the other hand, for the very same reason, scattering is suppressed, leading to a highly reduced resistance along the current direction. Essentially, local longitudinal resistance vanishes at the limit of zero temperature . Therefore, we depicted integer filling factors by black color, $\nu=2$, to estimate locations of the incompressible strips.
{width="5cm"} {width="5cm"} {width="5cm"} {width="5cm"} {width="5cm"} {width="5cm"}
Fig. \[fig2\]c shows the distribution of current that is imposed in the positive $y$ direction, with a normalized amplitude of $0.01$. We observe that most of the current exerted is confined to integer-filling-factor regions, namely, $\nu$=2. A closer look at the data indicates that some of the current is backscattered in the proximity of the top- and bottom-most QPCs, indicating that because of finite temperature, a negligible number of transport electrons ($<0.01 \% $) are scattered to compressible (with high DOS, metal-like) regions, where longitudinal resistance is finite. A remarkable feature in the current distribution is the asymmetry between the upper and lower parts of Fig. \[fig2\]c. It is seen that more current flows from the upper half. This is in agreement with the experimental [@afif:njp2] and theoretical [@SiddikiEPL:09] findings reported in the literature, justifying our results. The main reason for such behavior is grounded in the symmetry-breaking external magnetic field, namely, the Lorentz force resulting in induced Hall voltage. Keep in mind that, at relatively low magnetic field , the edge incompressible strips are as narrow as the Fermi wavelength; hence, at this magnetic field the current is mostly driven by the drift velocity. Therefore, one can conclude that scattering is mainly dominated by impurities. At elevated field strengths (Fig. \[fig3\]a and Fig. \[fig3\]b), we observe that current first shifts to the lower part of the sample ($B=8$ T) and then is approximately symmetrically distributed over the sample at $B=8.5$ T. This is mainly due to the enlargement of the incompressible strips while increasing the magnetic field. At $B=8.0$ T, the lower incompressible strip is well developed; i.e., the width of the strips is larger than the Fermi wavelength. Hence, the current is confined mainly to this scattering-free region. Once the magnetic field strength is increased by $0.5$ T, both incompressible strips at the lower and higher parts of the sample become larger than the Fermi wavelength; therefore, current is shared between them in an approximately equal manner. We are able to confirm this behavior just by checking the convergence temperature of the system. It is observed that, for antisymmetric current distributions, the dissipation is higher; hence, equilibrium (convergence) temperature is $7.43$ K and $10.15$ K for $7.5$ T and $8.0$ T, respectively . In accordance with our conclusion, the equilibrium temperature $T_{\rm E}$ is lower once almost all of the current is confined to well-developed incompressible strips, namely, $2.0$ K at $8.5$ T.
Before presenting further results, to summarize the main ow of our understanding: The current is confined to the scattering-free incompressible strips, where dissipation is suppressed, leading to lower equilibrium temperatures. Depending on magnetic field, the existence, location and widths of the strips vary such that at lower fields the upper strip, at intermediate fields the lower strip and at higher fields both strips are well developed.
Next, we present results where the depleting gates are biased with a higher negative voltage of $-0.2$ V, yielding a steeper screened potential profile, which in turn leads to narrower incompressible strips. Fig. \[fig4\]a-f presents our numerical results considering six characteristic $B$ values, in increasing order. At the lowest field value ($6$ T), no incompressible regions are formed; therefore, current is distributed all over the sample with high dissipation yielding a high $T_{\rm E}$ $(= 4.55$ K). Similar to the previous situation, the first incompressible strip is formed at $6.5$ T at the upper edge of the sample, where relatively more current is confined to the strip. This yields less scattering, which ends up with the lowest $T_{\rm E}$. Current is approximately symmetric at $7.0$ T, but interestingly, $T_{\rm E}$ is higher compared to the previous case. One can interpret this behavior as follows: although the current is confined to incompressible strips at both sides of the sample, the remaining current is highly scattered between the QPCs, increasing the dissipation, which then elevates the temperature. Our interpretation is justified when we consider two consecutive field strengths, namely, $7.5$ T and $8.0$ T. In both cases, the current distribution is symmetric; however, at $7.5$ $T_{\rm E}$ the temperature is $5.12$ K, being the highest value of our interval of interest, and then decreases to $3.48$ K at $8.0$ T, where bulk scattering is suppressed, as can be seen clearly from Fig. \[fig4\]e. At the highest $B$ strength, although the bulk scattering is predominantly suppressed, the strong back-scattering at the injection (bottom left of the sample) and the collection (top right) regions dissipation is high , and as a consequence, $T_{\rm E}$ increases to $4.2$ K.
The results shown above give sufficient information about the current distribution in the close vicinity of the QPCs depending on the magnetic field. Moreover, taking into account the scattering processes, one can also comment on the variation of the equilibrium temperature, depending both on $B$ field strength and also on the formation of incompressible strips , strongly bound to the steepness of the potential profile determined by the gate voltage $V_{\rm G}$.
Conclusion
==========
Our investigation focuses on the current distribution of parallel-configured QPCs under QH conditions, utilizing self-consistent numerical calculation schemes. We obtain the potential and charge distribution of a GaAs/AlGaAs heterojunction by solving the Poisson equation in 3D for given boundary conditions. The obtained potential at the layer of 2DES is used to calculate the distribution of scattering-free incompressible strips and hence the current. We found that the location of the transporting channels depends strongly on the applied perpendicular magnetic field strength and switch from asymmetric to symmetric. This behavior is also observed once the gate voltages are varied.
Our main finding is that the variation of the equilibrium temperature is due to (both back- and forward-) scattering affecting the dissipation. In conclusion, low equilibrium temperatures are obtained if the current is mainly confined to the scattering-free incompressible strips.
Our findings are in accordance with previous theoretical and experimental studies. Moreover, we are able to demonstrate that, in a parallel configuration of cosine-defined QPCs, equilibrium temperature is a key indicator to determine coherent transport through such quantum devices. It is admirable to obtain similar results for realistically determined QPCs and to compare them with experimental results. It is also a challenging problem to include (Joule) heating effects, in order to investigate the dissipation processes microscopically.
Acknowledgement {#acknowledgement .unnumbered}
===============
A.S. thanks Mimar Sinan Fine Arts University Physics Department members for fruitful discussions on theoretical issues.
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|
---
abstract: 'In dynamical systems composed of interacting parts, conditional exponents, conditional exponent entropies and cylindrical entropies are shown to be well defined ergodic invariants which characterize the dynamical selforganization and statitical independence of the constituent parts. An example of interacting Bernoulli units is used to illustrate the nature of these invariants.'
author:
- |
R. Vilela Mendes\
Grupo de Física-Matemática\
Complexo Interdisciplinar, Universidade de Lisboa\
Av. Gama Pinto, 2, 1699 Lisboa Codex Portugal\
e-mail: vilela@alf4.cii.fc.ul.pt
title: 'Conditional exponents, entropies and a measure of dynamical self-organization '
---
Conditional exponents
=====================
The notion of conditional Lyapunov exponents (originally called sub-Lyapunov exponents) was introduced by Pecora and Carroll in their study of synchronization of chaotic systems[@Pecora1] [@Pecora2]. It turns out, as I will show below, that, like the full Lyapunov exponent, the conditional exponents are well defined ergodic invariants. Therefore they are reliable quantities to quantify the relation of a global dynamical system to its constituent parts and to characterize dynamical selforganization.
Given a dynamical system defined by a map $f:M\rightarrow M$ , with $%
M\subset R^m$ the [*conditional exponents associated to the splitting* ]{}$%
R^k\times R^{m-k}$ are the eigenvalues of the limit $$\label{1.1}\lim _{n\rightarrow \infty }\left( D_kf^{n*}(x)D_kf^n(x)\right)
^{\frac 1{2n}}$$ where $D_kf^n$ is the $k\times k$ diagonal block of the full Jacobian.
[*Lemma. Existence of the conditional exponents as well defined ergodic invariants is guaranteed under the same conditions that establish the existence of the Lyapunov exponents*]{}
Proof: Let $\mu $ be a probability measure in $M\subset R^m$ and $f$ a measure-preserving $M\rightarrow M$ mapping such that $\mu $ is ergodic. Oseledec’s multiplicative ergodic theorem[@Oseledec], generalized for non-invertible $f$ [@Raghu], states that if the map $T:M\rightarrow M_m$ from $M$ to the space of $m\times m$ matrices is measurable and $$\label{1.2}\int \mu (dx)\log ^{+}\left\| T(x)\right\| <\infty$$ (with $\log ^{+}g=\max \left( 0,\log g\right) $) and if $$\label{1.3}T_x^n=T(f^{n-1}x)\cdots T(fx)T(x)$$ then $$\label{1.4}\lim _{n\rightarrow \infty }\left( T_x^{n*}T_x^n\right) ^{\frac
1{2n}}=\Lambda _x$$ exists $\mu $ almost everywhere.
If $T_x$ is the full Jacobian $Df(x)$ and if $Df(x)$ satisfies the integrability condition (\[1.2\]) then the Lyapunov exponents exist $\mu -$almost everywhere. But if the Jacobian satisfies (\[1.2\]), then the $%
m\times m$ matrix formed by the diagonal $k\times k$ and $m-k\times m-k$ blocks also satisfies the same condition and conditional exponents too are defined a. e.. Furthermore, under the same conditions as for Oseledec’s theorem, the set of regular points is Borel of full measure and $$\label{1.5}\lim _{n\rightarrow \infty }\frac 1n\log \left\|
D_kf^n(x)u\right\| =\xi _i^{(k)}$$ with $0\neq u\in E_x^i/E_x^{i+1}$ , $E_x^i$ being the subspace of $R^k$ spanned by eigenstates corresponding to eigenvalues $\leq \exp (\xi
_i^{(k)}) $.
Conditional entropies and dynamical selforganization
====================================================
For measures $\mu $ that are absolutely continuous with respect to the Lebesgue measure of $M$ or, more generally, for measures that are smooth along unstable directions (SBR measures) Pesin’s[@Pesin] identity holds $$\label{2.1}h(\mu )=\sum_{\lambda _i>0}\lambda _i$$ relating Kolmogorov-Sinai entropy $h(\mu )$ to the sum of the Lyapunov exponents. By analogy we may define the [*conditional exponent entropies*]{} associated to the splitting $R^k\times R^{m-k}$ as the sum of the positive conditional exponents counted with their multiplicity $$\label{2.2}h_k(\mu )=\sum_{\xi _i^{(k)}>0}\xi _i^{(k)}$$ $$\label{2.3}h_{m-k}(\mu )=\sum_{\xi _i^{(m-k)}>0}\xi _i^{(m-k)}$$ The Kolmogorov-Sinai entropy of a dynamical system measures the rate of information production per unit time. That is, it gives the amount of randomness in the system that is not explained by the defining equations (or the minimal model[@Crutchfield]). Hence, the conditional exponent entropies may be interpreted as a measure of the randomness that would be present if the two parts $S^{(k)}$ and $S^{(m-k)}$ were uncoupled. The difference $h_k(\mu )+h_{m-k}(\mu )-h(\mu )$ represents the effect of the coupling.
Given a dynamical system $S$ composed of $N$ parts $\{S_k\}$ with a total of $m$ degrees of freedom and invariant measure $\mu $, one defines a [*measure of dynamical selforganization*]{} $I(S,\Sigma ,\mu )$ as $$\label{2.5}I(S,\Sigma ,\mu )=\sum_{k=1}^N\left\{ h_k(\mu )+h_{m-k}(\mu
)-h(\mu )\right\}$$ Of course, for each system $S$, this quantity will depend on the partition $%
\Sigma $ into $N$ parts that one considers. $h_{m-k}(\mu )$ always denotes the conditional exponent entropy of the complementar of the subsystem $S_k$. Being constructed out of ergodic invariants, $I(S,\Sigma ,\mu )$ is also a well-defined ergodic invariant for the measure $\mu $. $I(S,\Sigma ,\mu )$ is formally similar to a mutual information. However, not being strictly a mutual information, in the information theory sense, $I(S,\Sigma ,\mu )$ may take negative values.
Another ergodic invariant that may be associated to the splitting of a dynamical system into its constituent parts is the notion of [*cylindrical entropies*]{}.
Consider, as before, a $\mu -$preserving and $\mu -$ergodic mapping $%
f:M\rightarrow M$ and a splitting $R^m=R^k\times R^{m-k}$. A measure in $R^m$ induces a measure in $R^k$ by $$\label{2.6}\nu (x)=\int_{R^{m-k}}d\mu (y,x)$$ $x\in R^k$ and $y\in R^{m-k}$.
Given a $\nu -$measurable partition $P(R^k)$ in $R^k$$$\label{2.7}R^k=\cup _iP_i$$ $P_i\in $ $P(R^k)$ , it induces a partition in $R^m$ by the associated cylinder sets $$\label{2.8}R^m=\cup _iP_i^c$$ $P_i^c=P_i\times R^{m-k}\in P^c(R^m)$.
Let $P^c(M)=P^c(R^m)\cap M$ be the corresponding partition of $M$. Denote by $P^c(x)$ the element of $P^c(M)$ that contains $x$. If all powers of $f$ are ergodic, for any nontrivial partition $$\label{2.9}\lim _{n\rightarrow \infty }\mu (P_n^c(x))=0$$ where $$\label{2.10}P_n^c(x)=\cap _{j=0}^nf^{-j}(P^c(f^j(x)))$$ Then, the Shannon-MacMillan-Breiman theorem states that if $$\label{2.11}\sum_i\mu (P_i^c)\log (P_i^c)<\infty$$ the limit $$\label{2.12}h^c(f,P^c,x)=-\lim _{n\rightarrow \infty }\frac 1n\log \mu
(P_n^c(x))$$ exists $\mu $ a. e. and converges in $L^1$. This limit is the entropy at $x$ associated to the cylindrical partition $P^c$. The [*cylindrical entropy relative to the splitting* ]{}$R^k\times R^{m-k}$ may be defined as the integral of the supremum of this limit over all finite cylindrical partitions $$\label{2.13}h^c(f)=-\int_Md\mu (x)\sup _{P^c}\lim _{n\rightarrow \infty
}\frac 1n\log \mu (P_n^c(x))$$ The full Kolmogorov-Sinai entropy is a similar limit where now the supremum would be taken over all finite partitions. Therefore for a smooth measure, if the parts of a composite dynamical system are all uncoupled, the full entropy is simply the sum of the cylindrical entropies. In the uncoupled case each cylindrical entropy is determined by the corresponding conditional exponents. However for coupled mixing systems, the cylindrical partitions may, by themselves, already generate the full entropy of the coupled system. Therefore the relation of the cylindrical entropies to the total entropy is simply a [*measure of the statistical independence*]{} of the constituent parts. The conditional exponent entropies defined in (\[2.2\]-\[2.3\]) seem to be a better quantitative characterization of the dynamical selforganization.
An example
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Consider a fully coupled system defined by $$\label{3.1}x_i(t+1)=(1-c)f(x_i(t))+\sum_{j\neq i}\frac c{N-1}f(x_j(t))$$ with $f(x)=2x$ (mod. $1$).
The Lyapunov exponents are $\lambda _1=\log 2$ and $\lambda _i=\log \left(
2\left( 1-\frac N{N-1}c\right) \right) $ with multiplicity $N-1$.
Therefore, for an absolutely continuous measure $$\label{3.2}
\begin{array}{cclcc}
h(\mu ) & = & \log 2+(N-1)\log \left( 2-\frac{2Nc}{N-1}\right) & \textnormal{for}
& c\leq
\frac{N-1}{2N} \\ & = & \log 2 & \textnormal{for} & c\geq \frac{N-1}{2N}
\end{array}$$ The conditional exponents associated to the splitting $R^1\times R^{N-1}$ are $$\label{3.3}\xi ^{(1)}=\log (2-2c)$$ and $$\label{3.4}
\begin{array}{ccccccccc}
\xi _1^{(N-1)} & = & \log \left( 2-\frac{2c}{N-1}\right) & ; & \xi
_i^{(N-1)} & = & \log \left( 2-\frac{2Nc}{N-1}\right) & \textnormal{with
multiplicity} & N-2
\end{array}$$ Therefore, for a partition $\Sigma $ of the system with $N$ parts one obtains $$\label{3.5}I(S,\Sigma ,\mu )=N\left( \log \left( 1-\frac c{N-1}\right) +\max
\left( \log (2-2c),0\right) -\max \left( \log \left( 2-\frac{2Nc}{N-1}%
\right) ,0\right) \right)$$ which in the limit of large $N$ becomes $$\label{3.6}
\begin{array}{ccccc}
I(S,\Sigma ,\mu ) & = & \frac{c^2}{1-c} & & c\leq
\frac{N-1}{2N} \\ & = & -c & & c\geq \frac 12
\end{array}$$ Fig.1 shows the variation with $c$ of $I(S,\Sigma ,\mu )$ for $N=100$.
At $c=0$, and starting from a random initial condition, the motion of the system is completely disorganized. When $c$ starts to grow the system shows the coexistence of disorganized behavior with patches of synchronized clusters. At the point where $I(S,\Sigma ,\mu )$ is maximum, $c=0.495$, starting from a random initial condition, the system settles rapidly in a state with many different synchronized clusters. Fig.2 shows the first $5000$ time steps. It is indeed at this point that the system shows what intuitively we would call a large organizational structure. Above $c=0.5$, after a short transition period, the system becomes fully synchronized (Fig.3 for $c=0.51$).
Figure captions
===============
Fig.1 - Coupling dependence of the selforganization invariant $I(S,\Sigma
,\mu )$ in the coupled Bernoulli system
Fig.2 - The first $5000$ time steps for $c=0.495$ (maximum $I(S,\Sigma ,\mu )
$). The last column in the right is the color map
Fig.3 - The first $5000$ time steps for $c=0.51$
[9]{} L. M. Pecora and T. L. Carroll; Phys. Rev. Lett. 64 (1990) 821.
L. M. Pecora and T. L. Carroll; Phys. Rev. A44 (1991) 2374.
V. I. Oseledec; Trans. Moscow Math. Soc. 19 (1968) 197.
M. S. Raghunatan; Israel Jour. Math. 32 (1979) 356.
Y. B. Pesin; Russ. Math. Surv. 32 (1977) 55.
J. P. Crutchfield and K. Young; in [*Complexity, Entropy and the Physics of Information*]{}, pag. 223, SFI Studies in the Sciences Complexity, vol. VIII, W. H. Zurek (Ed.), Addison-Wesley 1990.
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abstract: 'In last passage percolation models lying in the KPZ universality class, long maximizing paths have a typical deviation from the linear interpolation of their endpoints governed by the two-thirds power of the interpolating distance. This two-thirds power dictates a choice of scaled coordinates, in which these maximizers, now called polymers, cross unit distances with unit-order fluctuations. In this article, we consider Brownian last passage percolation in these scaled coordinates, and prove that the probability of the presence of $k$ disjoint polymers crossing a unit-order region while beginning and ending within a short distance ${\epsilon}$ of each other is bounded above by ${\epsilon}^{(k^2 - 1)/2 \, + \, o(1)}$. This result, which we conjecture to be sharp, yields understanding of the uniform nature of the coalescence structure of polymers, and plays a foundational role in [@Patch] in proving comparison on unit-order scales to Brownian motion for polymer weight profiles from general initial data. The present paper also contains an on-scale articulation of the two-thirds power law for polymer geometry: polymers fluctuate by ${\epsilon}^{2/3}$ on short scales ${\epsilon}$.'
address: |
A. Hammond\
Department of Mathematics and Statistics\
U.C. Berkeley\
899 Evans Hall\
Berkeley, CA, 94720-3840\
U.S.A.
author:
- Alan Hammond
bibliography:
- 'airy.bib'
title: |
Exponents governing the rarity of disjoint polymers\
in Brownian last passage percolation
---
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\[theorem\][Remark]{}
\[theorem\][Comment]{}
\[theorem\][Definition]{}
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Introduction
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KPZ universality, last passage percolation models, and scaled coordinates
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The $1 + 1$ dimensional Kardar-Parisi-Zhang (KPZ) universality class includes a wide range of random interface models suspended over a one-dimensional domain, in which growth in a direction normal to the surface competes with a smoothening surface tension in the presence of a local randomizing force that roughens the surface. These characteristic features are evinced by many last passage percolation models. Such an LPP model comes equipped with a planar random environment, which is independent in disjoint regions. Directed paths, that are permitted say to move only in a direction in the first quadrant, are then assigned energy via this randomness, by say integrating the environment’s value along the path. For a given pair of planar points, the path attaining the maximum weight over directed paths with such endpoints is called a geodesic.
For LPP models lying in the KPZ class, a geodesic that crosses a large distance $n$ (in say a northeasterly direction) has an energy that grows linearly in $n$ with a standard deviation of order $n^{1/3}$. If the lower geodesic endpoint is held fixed, and the higher one is permitted to vary horizontally, then the geodesic energy as a function of the variable endpoint plays the role of the random interface mentioned at the outset. (With the first endpoint thus fixed, the energy profile goes by the name ‘narrow wedge’.) Non-trivial correlations in the geodesic energy are witnessed when this horizontal variation has order $n^{2/3}$. These assertions have been rigorously demonstrated for only a few LPP models, each of which enjoys an integrable structure: the seminal work of Baik, Deift and Johansson [@BDJ1999] rigorously established the one-third exponent, and the GUE Tracy-Widom distributional limit, for the case of Poissonian last passage percolation, while the two-thirds power law for maximal transversal fluctuation was derived for this model by Johansson [@Johansson2000].
In seeking to understand the canonical structures of KPZ universality, we are led, in view of these facts, to represent the field of geodesics in a scaled system of coordinates, under which a northeasterly displacement of order $n$ becomes a vertical displacement of one unit, and a horizontal displacement of order $n^{2/3}$ becomes a unit horizontal displacement. Moreover, the accompanying system of energies also transfers to scaled coordinates, with the scaled geodesic energy being specified by centring about the mean value and division by the typical scale of $n^{1/3}$.
Probabilistic proof techniques for KPZ, and scaled Brownian LPP
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The theory of KPZ universality has advanced through physical insights, numerical analysis, and several techniques of integrable or algebraic origin. We will not hazard a summary of literature to support this one-sentence history, but refer to the reader to [@IvanSurvey] for a KPZ survey from 2012; in fact, integrable and analytic approaches to KPZ have attracted great interest around and since that time. Now, it is hardly deep or controversial to say that many problems and models in KPZ are intrinsically random. This fact may suggest that it would be valuable to approach the problems of KPZ universality from a predominately probabilistic perspective.
An important illustration is offered by Brownian last passage percolation. This LPP model has very attractive probabilistic features: for example, in the narrow wedge case, the scaled geodesic energy profile may be embedded as the uppermost curve in a Dyson diffusion, namely a system of one-dimensional Brownian motions conditioned on mutual avoidance (with a suitable boundary condition). If we depict Brownian LPP in the scaled coordinates that have been described, may we analyse it with probabilistic tools and thus gain some insight into universal KPZ structures?
The present article forms part of a four-paper answer to this question. The companion papers are [@BrownianReg], [@ModCon] and [@Patch]. Our focus is on scaled coordinates, and we will adopt the terms [*polymer*]{} and [*weight*]{} to refer to scaled geodesics and their scaled energy. The reader may glance ahead to the right sketch in Figure \[f.scaling\] for a depiction of a polymer, bearing in mind that, when the scaling parameter $n$ is high, the microscopic backtracking apparent in the illustration becomes negligible, so that the lifetime $[t_1,t_2]$ of a polymer may be viewed as a vertical interval, with the polymer being a random real-valued function defined on that interval; the range of the polymer thus crosses the planar strip ${\ensuremath{\mathbb{R}}}\times [t_1,t_2]$ from the lower to the upper side.
Principal conclusions and themes in overview
--------------------------------------------
In this article, we will reach three principal conclusions concerning scaled Brownian LPP. As we informally summarise them now, we omit mention of the scaling parameter $n \in {\ensuremath{\mathbb{N}}}$: roughly, our assertions should be understood uniformly in high choices of this parameter.
- Consider the event that there exist $k$ polymers of unit lifetime $[0,1]$, where each begins and ends in a given interval of some given [*small*]{} scaled length ${\epsilon}$. Theorem \[t.disjtpoly.pop\], states an upper bound of ${\epsilon}^{(k^2 - 1)/2 \, + \, o(1)}$ on the probability of this event. The left sketch of the upcoming Figure \[f.triple\] depicts the event with $k=3$.
- Derived as a consequence, Theorem \[t.maxpoly.pop\] asserts that the probability that ${m}$ polymers coexist disjointly in a unit-order region has a superpolynomial decay in ${m}$.
- We have mentioned that the maximal geodesic fluctuation in Poissonian last passage percolation has been shown in [@Johansson2000] to be governed by an exponent of two-thirds. Here is a further expression of this two-thirds power law, showing that this exponent also governs local behaviour of the scaled geodesic: a polymer fluctuates by more than ${\epsilon}^{2/3} r$ on a short duration ${\epsilon}$ with probability at most $\exp \big\{ - O(r^{\alpha}) \big\}$ (for a broad range of values of $r$). In Theorem \[t.polyfluc\], we will prove a Brownian LPP version of this assertion, with $\alpha = 3/4$.
It is the first of these results, namely the exponent bound of the form $(k^2 - 1)/2 \, + \, o(1)$ on the rarity of $k$ disjoint polymers with nearby endpoints, which we regard as the most fundamental. Four reasons, discussed in turn next, will help to explain why.
![ The left and right sketches each depict three aspects of scaled random growth in Brownian LPP, corresponding to two different initial conditions. These initial conditions $f$ are depicted at the bottom. On the left, the narrow wedge case of growth from zero is depicted. The polymer weight profile, top left, is known to be locally Brownian (and globally parabolic). In the middle left, we see the associated system of polymers, forming a tree with root at the origin. On the right, growth is instead initiated from the set of integers. The field of polymers now forms a forest, rather than a tree, and the polymer weight profile, which is depicted in bold, is the maximum of the profiles associated to the roots of the various trees. Indeed, a general polymer weight follows the profile associated to the consecutive trees in the associated polymer forest. This profile may be understood to be locally Brownian as an inference from the Brownian resemblance enjoyed by the ‘narrow-wedge’ profiles associated to the individual trees. To make this idea work, we need to understand that there are not too many trees: their number per unit length must be shown to be tight in the scaling parameter $n$. It is here that Theorem \[t.disjtpoly.pop\], and Theorem \[t.maxpoly.pop\], provide the necessary input.[]{data-label="f.manytrees"}](NonIntPolyManyTrees.pdf){height="12cm"}
[*1. Validating the Brownian regularity of polymer weight profiles begun with general initial data.*]{} Theorem \[t.disjtpoly.pop\] is very useful. In fact, Theorem \[t.maxpoly.pop\] will be derived as a rather direct consequence. More fundamentally, Theorem \[t.disjtpoly.pop\] plays a key role in the principal highway of implications in our four-paper study. In order to indicate why briefly, we mention first a key theme concerning scaled Brownian LPP: that [*narrow wedge polymer weight profiles resemble Brownian motion*]{}. This assertion may be understood in a [*local limit*]{}, when the interval on which comparison is made between the weight profile and Brownian motion shrinks to zero, in which case [@Hagg] and [@CatorPimentel; @Pimentel18] offer rigorous expressions of this statement; they do so respectively by analysing the largest particle in the extended Airy point process and by means of a technique of local comparision to an equilibrium regime. But the assertion may also be understood on a [*unit scale*]{}: indeed, Theorem $2.11$ and Proposition $2.5$ of [@BrownianReg] demonstrate that, after an affine adjustment, narrow wedge profiles withstand a very demanding comparison to Brownian bridge when the profile is restricted to any given compact interval. In the narrow wedge case that these works address, random growth is initiated from a point. Such growth may also begin from much more general initial data. The main conclusion of [@Patch], that article’s Theorem $1.2$, asserts that the corresponding polymer weight profiles enjoy in a uniform sense a strong resemblance to Brownian motion on unit scales. Theorem \[t.disjtpoly.pop\] is an engine that drives the derivation of this conclusion. Figure \[f.manytrees\] gives a very impressionistic account of why the theorem is valuable for this purpose.
[*2. Elucidating fractal geometry in the Airy sheet.*]{} For suitable LPP models, the polymer weight profile in the upper-left sketch of Figure \[f.manytrees\] converges distributionally [@PrahoferSpohn; @Johansson2003] in the high scaling parameter limit $n \to \infty$ to the Airy$_2$ process after the subtraction of a parabola. This process, which has finite-dimensional distributions specified by Fredholm determinants, is a central object in KPZ universality. It has been expected that models in the KPZ universality class share richer universal structure than the Airy$_2$ process, and two significant recent advances have rigorously validated this expectation.
The first of these advances is the recent construction [@MQR17] of the [*KPZ fixed point*]{}. The narrow wedge polymer weight profile may be viewed as a time-one snapshot of a scaled random growth process initiated from the origin at time zero. Growth may be initiated from a much more general initial condition. In [@MQR17], Matetski, Quastel and Remenik have utilized a biorthogonal ensemble representation found by [@Sas05; @BFPS07] associated to the totally asymmetric exclusion process in order to find Fredholm determinant formulas for the multi-point distribution of the height function of this growth process begun from an arbitrary initial condition. Using these formulas to take the KPZ scaling limit, the authors construct a scale invariant Markov process that lies at the heart of the KPZ universality class. The time-one evolution of this Markov process may be applied to very general initial data, and the result is the scaled profile begun from such data, which generalizes the ${\rm Airy}_2$ process seen in the narrow wedge case. Were [@MQR17] to be adapted to Brownian last passage percolation, the present article’s theorems and consequences might become applicable to universal KPZ structure. In particular, this outcome should be rather direct in the case of the consequence [@Patch Theorem $1.2$] concerning the Brownian nature of general initial condition profiles.
The second recent advance [@DOV18] is perhaps of even greater relevance to the results that we present. The [*space-time Airy sheet*]{} is a rich shared scaled structure in the KPZ universality class. It simultaneously encodes the collection of weights of polymers running between any pair of planar endpoint locations. Its existence was mooted in [@CQR2015]. In [@DOV18], the space-time Airy sheet is constructed by use of an extension of the Robinson-Schensted-Knuth correspondence which expresses the construction in terms of a last passage percolation problem whose underlying environment is itself a copy of the distributional limit of the narrow wedge profile in Brownian last passage percolation.
These advances [@MQR17] and [@DOV18] offer new prospects for probabilistic inquiry into KPZ, including applications of the present article. For example, fractal geometry embedded in the Airy sheet has been thus elucidated in [@BGH18]. To explain this application, note that many copies of the Airy$_2$ process are coupled together in the Airy sheet. It is already interesting to consider how two such processes rooted at distinct planar points may be coupled: the random process sending $y \in {\ensuremath{\mathbb{R}}}$ to the [*difference*]{} in weight between polymers beginning at $(-1,0)$ and at $(1,0)$ and ending together at $(y,1)$ is such an example. This [*Airy difference process*]{}, mapping ${\ensuremath{\mathbb{R}}}$ to ${\ensuremath{\mathbb{R}}}$, is a random fractal, being the distribution function of a random Cantor set of Hausdorff dimension one-half. In proving this, [@BGH18] provides the upper bound on Hausdorff dimension as a consequence of Theorem \[t.disjtpoly.pop\] applied in the case of a pair of polymers.
[*3. Polymer surgery as a KPZ tool.*]{} The proof of Theorem \[t.disjtpoly.pop\] has conceptual interest. Two principal probabilistic techniques are at work in the four-paper study, namely [*Brownian Gibbs resamplings*]{} and [*polymer surgery*]{}. The latter is introduced in order to prove Theorem \[t.disjtpoly.pop\]. In order to explain how, it is useful first to discuss the former. This first, resampling, tool is employed to analyse the narrow wedge polymer weight profile by viewing it as the top curve in a Dyson diffusion. Allied with a simple observation from integrable probability, the Karlin-McGregor formula, the technique has been used in [@BrownianReg] to solve a cousin of the problem addressed in Theorem \[t.disjtpoly.pop\]. The solution will later be presented as Theorem \[t.neargeod\]. Crudely, in place of considering, as Theorem \[t.disjtpoly.pop\] does, $k$ disjoint polymers with nearby endpoints, we may instead consider the event that there exists a system of $k$ scaled paths, crossing between two [*given*]{} points at a unit vertical displacement, the paths pairwise disjoint except for these shared endpoints, and with each path coming close to being a polymer, in the sense that each path weight is close to maximal for the given endpoints. Theorem \[t.neargeod\] identifies in a sharp way the decay rate in the probability of this event as a natural parameter that measures this closeness tends to zero: the theorem identifies the exponent $k^2 -1$ as governing this rate.
How then will polymer surgery help to prove Theorem \[t.disjtpoly.pop\]? Faced with $k$ polymers with nearby endpoints, we seek to deform them surgically near their endpoints in order to typically achieve a shared-endpoint outcome whose probability is determined by Theorem \[t.neargeod\]: Figure \[f.triple\] is a one-sketch summary. A probabilistic bridge is being built into an understanding of integrable origin, and the proof technique is thus emblematic of a broader aim of expanding KPZ horizons by probabilistic means.
[*4. The exponent $(k^2 - 1)/2$.*]{} In its key application in [@Patch], Theorem \[t.disjtpoly.pop\] is applied when $k=3$, for a triple of polymers. The form of the exponent is important in this application, dictating that a certain Radon-Nikodym derivative that describes the strength of Brownian comparison lies in $L^{3-}$. Merely knowing that the exponent is at least a positive constant, for example, would not be adequate even for deriving $L^1$-membership.
We believe that the $(k^2 - 1)/2$ exponent in Theorem \[t.disjtpoly.pop\] is sharp and will present a conjecture to this effect in Subsection \[s.maxpoly\]. It is rigorously known that, in Theorem \[t.disjtpoly.pop\]’s proof, nothing is lost in the exponent after the reduction to the shared endpoint Theorem \[t.neargeod\] is made, because the latter result provides a sharp resolution of the cousin problem discussed above in the third point. Moreover, the conjecture of Theorem \[t.disjtpoly.pop\]’s sharpness has recently been validated in [@BGH18 Theorem $2.4$] for the special case $k=2$ as a corollary of the identification of the Hausdorff dimension of the points of increase of the Airy difference process that we have mentioned.
A suggestion for further reading of overview
--------------------------------------------
In seeking to explain the progress made in this article in a few paragraphs, we have also offered a whistle-stop tour of ideas in a broader study of scaled Brownian LPP. Such a brief presentation may well prompt questions, and we refer the reader who wants a more detailed but still informal overview of the main results and ideas in the four-paper study to [@BrownianReg Section $1.2$].
For the reader who wishes to read the broader study, the present article would be read after [@BrownianReg] and [@ModCon] but before [@Patch]. However, the present article has been written so that it may be read on its own. As such, consulting the other articles, including the further reading just suggested, is an optional extra.
Connections: probabilistic tools, geodesics and coalescence
-----------------------------------------------------------
The idea of taking limited integrable inputs, such as control of narrow wedge polymer profiles at given points, and exploiting them via probabilistic means to reach much stronger conclusions is central to this article and the broader four-paper Brownian LPP study. A similar conceptual approach governs many ideas in [@SlowBondSol] and [@SlowBond], where the slow bond conjecture for the totally asymmetric exclusion process, concerning the macroscopic effects of slightly attenuating passage of particles through the origin, is proved, and the associated invariant measures determined. Such a probabilistic technique is also encountered in [@BSS17]. Indeed, in [@BSS17 Theorem $2$], an assertion similar to, and in fact slightly stronger than, Theorem \[t.polyfluc\] has been proved for exponential last passage percolation: in essence, the assertion in the third bullet point in the preceding discussion has been verified for $\alpha = 1$.
Roughly opposite to the phenomenon of geodesic disjointness is the process of geodesic coalescence. An interesting arena of study of this coalescence is the class of first passage percolation models, which are variants of last passage percolation that are not integrable. Although scaling exponents are not rigorously known in this setting, it has been heuristically appreciated for some time that certain circumstances should coincide in this context: the non-existence of bi-infinite geodesics; an exponent of one-third in the standard deviation of geodesic energy; and smoothness of the limiting shape of energy-per-unit-length as a function of angle. A summary of progress until 1995 regarding coalescence of geodesics is offered by Newman in [@Newman95]; this progress included geodesic uniqueness for given direction, subject to a curvature assumption on the limiting shape. The theory of semi-infinite geodesics was developed by [@FerrariPimentel05] and [@Coupier11]. Building on this theory, Pimentel adapted Newman’s technique for uniqueness to the context of exponential LPP, in which the curvature assumption is known to be verified. In this work, it is also proved that, for two geodesics of given direction that begin at points separated by distance $\ell$, it is typical that coalescence of the geodesic occurs by a time of order $\ell^{3/2}$. Moreover, Pimentel conjectured that the probability that coalescence has failed to occur by time $r \ell^{3/2}$ has tail $r^{-2/3}$. In the context of finite geodesics, [@BSS17] has proved an upper bound on this failure probability of the form $r^{-c}$ for some positive $c > 0$, and there seems to be promise in these techniques to obtain a sharper form of the result. Hoffman [@Hoffman08] initiated very fruitful progress on infinite geodesics in first passage percolation by using Busemann functions, with significant geometric information emerging in [@DamronHanson14; @AhlbergHoffman16] and [@DamronHanson17]. Busemann functions have also been studied in non-integrable last passage percolation contexts: see [@GRS15a] and [@GRS15b].
Brownian last passage percolation {#s.brlpp}
---------------------------------
In the two remaining introductory sections, we specify this model precisely, and set up notation for the use of scaled coordinates; and we state our main results.
### The model’s definition.
This model was introduced by [@GlynnWhitt] and further studied in [@O'ConnellYor]; we will call it Brownian LPP. On a probability space carrying a law labelled ${\ensuremath{\mathbb{P}}}$, we let $B:{\ensuremath{\mathbb{Z}}}\times {\ensuremath{\mathbb{R}}}\to {\ensuremath{\mathbb{R}}}$ denote an ensemble of independent two-sided standard Brownian motions $B(k,\cdot):{\ensuremath{\mathbb{R}}}\to {\ensuremath{\mathbb{R}}}$, $k \in {\ensuremath{\mathbb{Z}}}$.
Let $i,j \in {\ensuremath{\mathbb{Z}}}$ with $i \leq j$. We denote the integer interval $\{ i,\cdots , j \}$ by $\llbracket i,j \rrbracket$. Further let $x,y \in {\ensuremath{\mathbb{R}}}$ with $x \leq y$. With these parameters given, we consider the collection of non-decreasing lists $\big\{ z_k: k \in \llbracket i+1,j \rrbracket \big\}$ of values $z_k \in [x,y]$. With the convention that $z_i = x$ and $z_{j+1} = y$, we associate an energy to any such list, namely $\sum_{k=i}^j \big( B ( k,z_{k+1} ) - B( k,z_k ) \big)$. We may then define the maximum energy, $M^1_{(x,i) \to (y,j)}$, to be the supremum of the energies of all such lists.
### A geometric view: staircases. {#s.staircases}
In order to make a study of those lists that attain this maximum energy, we begin by noting that the lists are in bijection with certain subsets of $[x,y] \times [i,j] \subset {\ensuremath{\mathbb{R}}}^2$ that we call [*staircases*]{}. Staircases offer a geometric perspective on Brownian LPP and perhaps help in visualizing the problems in question.
The staircase associated to the non-decreasing list $\big\{ z_k: k \in \llbracket i+1,j \rrbracket \big\}$ is specified as the union of certain horizontal planar line segments, and certain vertical ones. The horizontal segments take the form $[ z_k,z_{k+1} ] \times \{ k \}$ for $k \in \llbracket i , j \rrbracket$. Here, the convention that $z_i = x$ and $z_{j+1} = y$ is again adopted. The right and left endpoints of each consecutive pair of horizontal segments are interpolated by a vertical planar line segment of unit length. It is this collection of vertical line segments that form the vertical segments of the staircase.
The resulting staircase may be depicted as the range of an alternately rightward and upward moving path from starting point $(x,i)$ to ending point $(y,j)$. The set of staircases with these starting and ending points will be denoted by ${SC}_{(x,i) \to (y,j)}$. Such staircases are in bijection with the collection of non-decreasing lists considered above. Thus, any staircase $\phi \in {SC}_{(x,i) \to (y,j)}$ is assigned an energy $E(\phi) = \sum_{k=i}^j \big( B ( k,z_{k+1} ) - B( k,z_k ) \big)$ via the associated $z$-list.
### Energy maximizing staircases are called geodesics.
A staircase $\phi \in {SC}_{(x,i) \to (y,j)}$ whose energy attains the maximum value $M^1_{(x,i) \to (y,j)}$ is called a geodesic from $(x,i)$ to $(y,j)$. It is a simple consequence of the continuity of the constituent Brownian paths $B(k,\cdot)$ that such a geodesic exists for all choices of $(x,y) \in {\ensuremath{\mathbb{R}}}^2$ with $x \leq y$. As we will later explain, in Lemma \[l.severalpolyunique\], the geodesic with given endpoints is almost surely unique.
### The scaling map.
For $n \in {\ensuremath{\mathbb{N}}}$, consider the $n$-indexed [*scaling*]{} map $R_n:{\ensuremath{\mathbb{R}}}^2 \to {\ensuremath{\mathbb{R}}}^2$ given by $$R_n \big(v_1,v_2 \big) = \Big( 2^{-1} n^{-2/3}( v_1 - v_2) \, , \, v_2/n \Big) \, .$$
The scaling map acts on subsets $C$ of ${\ensuremath{\mathbb{R}}}^2$ by setting $R_n(C) = \big\{ R_n(x): x \in C \big\}$.
(Clearly, $n$ must be positive. In fact, ${\ensuremath{\mathbb{N}}}$ will denote $\{1,2,\cdots \}$ throughout.)
### Scaling transforms staircases to zigzags.
The image of any staircase under $R_n$ will be called an $n$-zigzag . The starting and ending points of an $n$-zigzag $Z$ are defined to be the image under $R_n$ of such points for the staircase $S$ for which $Z = R_n(S)$.
Note that the set of horizontal lines is invariant under $R_n$, while vertical lines are mapped to lines of gradient $- 2 n^{-1/3}$. As such, an $n$-zigzag is the range of a piecewise affine path from the starting point to the ending point which alternately moves rightwards along horizontal line segments and northwesterly along sloping line segments, where each sloping line segment has gradient $- 2 n^{-1/3}$; the first and last segment in this journey may be either horizontal or sloping.
### Scaled geodesics are called polymers.
For $n \in {\ensuremath{\mathbb{N}}}$, the image of any geodesic under the scaling map $R_n$ will be called an $n$-polymer, or often simply a polymer . This usage of the term ‘polymer’ for ‘scaled geodesic’ is apt for our study, due to the central role played by these objects. The usage is not, however, standard: the term ‘polymer’ is often used to refer to typical realizations of the path measure in LPP models at positive temperature.
### Some basic notation
For $\ell \geq 1$, we write ${\ensuremath{\mathbb{R}}}^\ell_\leq$ for the subset of ${\ensuremath{\mathbb{R}}}^\ell$ whose elements $(z_1,\cdots,z_\ell)$ are non-decreasing sequences. When the sequences are increasing, we instead write ${\ensuremath{\mathbb{R}}}^\ell_<$. We also use the notation $A^\ell_\leq$ and $A^\ell_<$. Here, $A \subset {\ensuremath{\mathbb{R}}}$ and the sequence elements are supposed to belong to $A$. We will typically use this notation when $\ell=2$.
### Compatible triples.
Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$, which is to say that $n \in {\ensuremath{\mathbb{N}}}$ and $t_1,t_2 \in {\ensuremath{\mathbb{R}}}$ with $t_1 < t_2$. Taking $x,y \in {\ensuremath{\mathbb{R}}}$, does there exist an $n$-zigzag from $(x,t_1)$ to $(y,t_2)$? As far as the data $(n,t_1,t_2)$ is concerned, such an $n$-zigzag may exist only if $$\label{e.ctprop}
\textrm{$t_1$ and $t_2$ are integer multiplies of $n^{-1}$} \, .$$ We say that data $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ is a [*compatible triple*]{} if it verifies the last condition.
An important piece of notation associated to a compatible triple is ${t_{1,2}}$ , which we will use to denote the difference $t_2 - t_1$. The law of the underlying Brownian ensemble $B: {\ensuremath{\mathbb{Z}}}\times {\ensuremath{\mathbb{R}}}\to {\ensuremath{\mathbb{R}}}$ is invariant under integer shifts in the first, curve indexing, coordinate. This translates to a distributional invariance of scaled objects under vertical shifts by multiples of $n^{-1}$, something that makes the parameter ${t_{1,2}}$ of far greater significance than $t_1$ or $t_2$.
![Let $(n,t_1,t_2)$ be a compatible triple and let $x, y \in {\ensuremath{\mathbb{R}}}$. The endpoints of the geodesic in the left sketch have been selected so that, when the scaling map $R_n$ is applied to produce the right sketch, the $n$-polymer $\rho_{n;(x,t_1)}^{(y,t_2)}$ results.[]{data-label="f.scaling"}](NonIntPolyScaling.pdf){height="7cm"}
Supposing now that $(n,t_1,t_2)$ is indeed a compatible triple, the condition $y \geq x - 2^{-1} n^{1/3} {t_{1,2}}$ ensures that the preimage of $(y,t_2)$ under the scaling map $R_n$ lies northeasterly of the preimage of $(x,t_1)$. Thus, an $n$-zigzag from $(x,t_1)$ to $(y,t_2)$ exists in this circumstance. We have mentioned that geodesics exist uniquely for given endpoints almost surely. Taking the image under scaling, this translates to the almost sure existence and uniqueness of the $n$-polymer from $(x,t_1)$ to $(y,t_2)$. This polymer will be denoted $\rho_{n;(x,t_1)}^{(y,t_2)}$: see Figure \[f.scaling\]. This notation has characteristics in common with several later examples, in which objects are described in scaled coordinates. Round bracketed expressions in the subscript or superscript will refer to a space-time pair, with the more advanced time in the superscript. Typically some aspect of the $n$-polymer from $(x,t_1)$ to $(y,t_2)$ is being described when this ${\mathsf{TBA}}_{n;(x,t_1)}^{(y,t_2)}$ notation is used. Appendix \[s.glossary\], which offers a glossary of the principal notation in this article, records several examples in rough accordance with this convention.
Main results, and a conjecture
------------------------------
This article reaches conclusions in two main directions: it establishes upper bounds on the probability of the coexistence of disjoint polymers with nearby endpoints, and it gives expression to the two-thirds power law that dictates polymer geometry, showing that polymers fluctuate by order ${\epsilon}^{2/3}$ on a short scale ${\epsilon}$. In the next two subsections, the principal conclusions in these directions are respectively stated.
### The rarity of many disjoint polymers {#s.maxpoly}
Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ be a compatible triple, and let $I,J \subset {\ensuremath{\mathbb{R}}}$ be intervals. Set ${\mathrm{MaxDisjtPoly}}_{n;(I,t_1)}^{(J,t_2)}$ equal to the maximum cardinality of a [*pairwise disjoint*]{} set of $n$-polymers each of whose starting and ending points have the respective forms $(x,t_1)$ and $(y,t_2)$ where $x$ is some element of $I$ and $y$ is some element of $J$.
(We make two parathetical comments. First, our notation here is an extension of the usage described a moment ago. In this case, the first element in the space-time pairs $(I,t_1)$ and $(J,t_2)$ is an interval, rather than a point. Second, the condition of pairwise disjointness in this definition can be slightly weakened so that our assertions remain valid. We prefer to defer elaborating on this. A remark in Section \[s.closure\] will offer an altered definition of ${\mathrm{MaxDisjtPoly}}_{n;(I,t_1)}^{(J,t_2)}$ whose use results in slightly stronger results.)
Our first main conclusion treats the problem of polymers of shared unit-order lifetimes that cross between intervals that are of a small length ${\epsilon}$, albeit one that is independent of the scaling parameter $n$. Theorem \[t.disjtpoly.pop\] provides an upper bound of ${\epsilon}^{(k^2 - 1)/2 \, + \, o(1)}$ on the probability that $k$ such polymers may coexist disjointly; here, $k \in {\ensuremath{\mathbb{N}}}$ is fixed, and the result is asserted uniformly in high $n$.
\[t.disjtpoly.pop\] There exists a positive constant ${G}$ such that the following holds. Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ be a compatible triple. Let $k \in {\ensuremath{\mathbb{N}}}$, ${\epsilon}> 0$ and $x,y \in {\ensuremath{\mathbb{R}}}$ satisfy the conditions that $k \geq 2$, $${\epsilon}\leq {G}^{-4k^2} \, , \, n {t_{1,2}}\geq {G}^{k^2} \big( 1 + \vert x - y \vert^{36} {t_{1,2}}^{-24} \big) {\epsilon}^{-{G}}$$ and ${t_{1,2}}^{-2/3} \vert x - y \vert \leq {\epsilon}^{-1/2} \big( \log {\epsilon}^{-1} \big)^{-2/3} G^{-k}$.
Setting $I = [x-{t_{1,2}}^{2/3}{\epsilon},x+{t_{1,2}}^{2/3}{\epsilon}]$ and $J = [y-{t_{1,2}}^{2/3}{\epsilon},y+{t_{1,2}}^{2/3}{\epsilon}]$, we have that $${\ensuremath{\mathbb{P}}}\Big( {\mathrm{MaxDisjtPoly}}_{n;(I,t_1)}^{(J,t_2)} \geq k \Big)
\leq
{\epsilon}^{(k^2 - 1)/2} \cdot {R}\, ,$$ where ${R}$ is a positive correction term that is bounded above by ${G}^{k^3} \exp \big\{ {G}^k \big( \log {\epsilon}^{-1} \big)^{5/6} \big\}$.
Theorem \[t.disjtpoly.pop\] is a slightly simplified form of a more detailed result that will be presented later, Theorem \[t.disjtpoly\]. Despite this simpler form, it is worth mentioning that, since a two-thirds power governs the horizontal coordinate under the scaling transformation, there is no generality lost by considering the special case $t_1 = 0$ and $t_2 = 1$, and thus ${t_{1,2}}= 1$, in Theorem \[t.disjtpoly.pop\]. (This point is explained further when the [*scaling principle*]{} is discussed in Section \[s.scalingprinciple\].) The reader is thus encouraged to read this result with $t_1 =0$ and $t_2 = 1$, as well as with $x$ and $y$ both supposed to be at most one say. Indeed, doing so shows that the next result follows immediately.
\[c.forconjecture\] For each $k \in {\ensuremath{\mathbb{N}}}$, $k \geq 2$, we have that $$\liminf_{{\epsilon}\searrow 0} \, \liminf_n \, \frac{\log {\ensuremath{\mathbb{P}}}\Big( {\mathrm{MaxDisjtPoly}}_{n;([- {\epsilon},{\epsilon}],0)}^{([-{\epsilon},{\epsilon}],1)} \geq k \Big)}{\log {\epsilon}} \, \geq \, \frac{k^2 - 1}{2} \, .$$
Note that, despite the $\geq$ symbol, this result is an [*upper bound*]{} on the concerned probability. The formulation permits us to express a conjecture in a symmetric form which indicates a belief that the exponent $(k^2-1)/2$ in Theorem \[t.disjtpoly.pop\] is sharp.
\[c.two\] For each $k \in {\ensuremath{\mathbb{N}}}$, $k \geq 2$, we have that $$\limsup_{{\epsilon}\searrow 0} \, \limsup_n \, \frac{\log {\ensuremath{\mathbb{P}}}\Big( {\mathrm{MaxDisjtPoly}}_{n;([- {\epsilon},{\epsilon}],0)}^{([-{\epsilon},{\epsilon}],1)} \geq k \Big)}{\log {\epsilon}} \, \leq \, \frac{k^2 - 1}{2} \, .$$
If the conjecture is valid, then equality will hold in the corollary and the conjecture. As we have mentioned, Conjecture \[c.two\] has recently been validated in [@BGH18 Theorem $2.4$] for the case that $k=2$.
The second main conclusion is a rather direct consequence of the first. It asserts that, for polymers that cross between unit-length intervals separated by unit-order times, the maximum number of such disjoint polymers has a tail that decays super-polynomially.
\[t.maxpoly.pop\] There exist constants ${G}\geq {g}> 0$ such that the following holds. Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ be a compatible triple. Further let $x,y \in {\ensuremath{\mathbb{R}}}$, $h \in {\ensuremath{\mathbb{N}}}$ and ${m}\in {\ensuremath{\mathbb{N}}}$. Suppose that $${m}\geq ({G}h)^{2{g}^{-1}{G}} \vee \big( \vert x - y \vert {t_{1,2}}^{-2/3} + 2h \big)^3$$ and $n {t_{1,2}}\geq \max \big\{ 1, ( \vert x - y \vert {t_{1,2}}^{-2/3} + 2h )^{36} \big\} {G}{m}^{G}$. Then $${\ensuremath{\mathbb{P}}}\Big( {\mathrm{MaxDisjtPoly}}_{n;(I,t_1)}^{(J,t_2)} \geq {m}\Big) \, \leq \, {m}^{- {g}(\log \log {m})^2} \, ,$$ where $I$ denotes the interval $[x,x+h{t_{1,2}}^{2/3}]$ and $J$ the interval $[y,y+h {t_{1,2}}^{2/3}]$.
Focus on the meaning of this theorem is brought by considering $t_1 =0$, $t_2 = 1$, $h=1$, with $x$ and $y$ chosen to lie in a unit interval about the origin. In this case, the demanded lower bound on ${m}$ merely excludes an initial bounded interval, and the lower bound on $n$ takes the form $n \geq \Theta ({m}^G)$. The latter condition is not demanding given that much of the interest in the use of scaled coordinates lies either in the high $n$ limit or in statements made uniformly for high enough $n$. Theorem \[t.maxpoly.pop\] is such a statement; as seen by these parameter choices, it asserts that there is at most probability ${m}^{-g (\log \log {m})^2}$ of ${m}$ disjoint polymers crossing a bounded region, uniformly in high $n$.
### Polymer fluctuation {#s.polyflucintro}
Our principal conclusion in this regard, Theorem \[t.polyfluc\], will here be stated, after a few paragraphs in which we attend to to setting up and explaining some necessary notation and concepts. The theorem is used to prove Theorem \[t.disjtpoly.pop\] and it is also needed in [@Patch].
Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ be a compatible triple, and let $x,y \in {\ensuremath{\mathbb{R}}}$. The polymer $\rho_{n;(x,t_1)}^{(y,t_2)}$ has been defined to be a subset of ${\ensuremath{\mathbb{R}}}\times [t_1,t_2]$ containing $(x,t_1)$ and $(y,t_2)$, but really as $n$ rises towards infinity, it becomes more natural to seek to view it as a random function that maps its lifetime $[t_1,t_2]$ to the real line. In choosing to adopt this perspective, we will abuse notation: taking $t \in [t_1,t_2]$, we will speak of the value $\rho_{n;(x,t_1)}^{(y,t_2)}(t) \in {\ensuremath{\mathbb{R}}}$, as if the polymer were in fact a function of $[t_1,t_2]$. Some convention must be adopted to resolve certain microscopic ambiguities as we make use of this new notation, however. First, we will refer to $\rho_{n;(x,t_1)}^{(y,t_2)}(t)$ only when $t \in [t_1,t_2]$ satisfies $nt \in {\ensuremath{\mathbb{Z}}}$, a condition that ensures that the intersection of the set $\rho_{n;(x,t_1)}^{(y,t_2)}$ with the line at height $t$ takes place along a horizontal planar interval.
Second, we have to explain which among the points in this interval $\rho_{n;(x,t_1)}^{(y,t_2)} \cap \{ (\cdot,t): \cdot \in {\ensuremath{\mathbb{R}}}\}$ we wish to denote by $\rho_{n;(x,t_1)}^{(y,t_2)}(t)$. To present and explain our convention in this regard, we let $\ell_{(x,t_1)}^{(y,t_2)}$ denote the planar line segment whose endpoints are $(x,t_1)$ and $(y,t_2)$. Adopting the same perspective as for the polymer, we abuse notation to view $\ell_{(x,t_1)}^{(y,t_2)}$ as a function from $[t_1,t_2]$ to ${\ensuremath{\mathbb{R}}}$, so that $\ell_{(x,t_1)}^{(y,t_2)}(t) = {t_{1,2}}^{-1} \big( (t_2 - t) x + (t - t_1)y \big)$.
Our convention will be to set $\rho_{n;(x,t_1)}^{(y,t_2)}(t)$ equal to $z$ where $(z,t)$ is that point in the horizontal segment $\rho_{n;(x,t_1)}^{(y,t_2)} \cap \{ (\cdot,t): \cdot \in {\ensuremath{\mathbb{R}}}\}$ whose distance from $\ell_{(x,t_1)}^{(y,t_2)}(t)$ is maximal. (An arbitrary tie-breaking rule, say $\rho_{n;(x,t_1)}^{(y,t_2)}(t) \geq \ell_{(x,t_1)}^{(y,t_2)}(t)$, resolves the dispute if there are two such points.) The reason for this very particular convention is that our purpose in using it is to explore, in the soon-to-be-stated Theorem \[t.polyfluc\], upper bounds on the probability of large fluctuations between the polymer $\rho_{n;(x,t_1)}^{(y,t_2)}$ and the line segment $\ell_{(x,t_1)}^{(y,t_2)}$ that interpolates the polymer’s endpoints. Our convention ensures that the form of the theorem would remain valid were any other convention instead adopted.
In order to study the intermediate time $(1-a)t_1 + at_2$ (in the role of $t$ in the preceding), we now let $a \in (0,1)$ and impose that $a {t_{1,2}}n \in {\ensuremath{\mathbb{Z}}}$: doing so ensures that, as desired, $t \in n^{-1} {\ensuremath{\mathbb{Z}}}$, where $t = (1-a)t_1 + at_2$.
Consider also $r > 0$. Define the [*polymer deviation regularity*]{} event $$\label{e.pdr}
{\mathsf{PolyDevReg}}_{n;(x,t_1)}^{(y,t_2)}\big(a,r\big) \, = \, \bigg\{ \, \Big\vert \, \rho_{n;(x,t_1)}^{(y,t_2)} \big( (1-a) t_1 + a t_2 \big) - \ell_{(x,t_1)}^{(y,t_2)} \big( (1-a) t_1 + a t_2 \big) \, \Big\vert \, \leq \, r t_{1,2}^{2/3} \big( a \wedge (1-a) \big)^{2/3} \bigg\} \, ,$$ where $\wedge$ denotes minimum. For example, if $a \in (0,1/2)$, the polymer’s deviation from the interpolating line segment, at height $(1-a)t_1 + at_2$ (when the polymer’s journey has run for time $a{t_{1,2}}$), is measured in the natural time-to-the-two-thirds scaled units obtained by division by $(a {t_{1,2}})^{2/3}$, and compared to the given value $r > 0$.
For intervals $I,J \subset {\ensuremath{\mathbb{R}}}$, we extend this definition by setting $${\mathsf{PolyDevReg}}_{n;(I,t_1)}^{(J,t_2)}\big(a,r\big) = \bigcap_{x \in I, y \in J} {\mathsf{PolyDevReg}}_{n;(x,t_1)}^{(y,t_2)}\big(a,r\big) \, .$$ The perceptive reader may notice a problem with the last definition. The polymer $\rho_{n;(x,t_1)}^{(y,t_2)}$ is well defined almost surely for given endpoints, but this property is no longer assured as the parameters vary over $x \in I$ and $y \in J$. In the case of exceptional $(x,y)$ where several $n$-polymers move from $(x,t_1)$ to $(y,t_2)$, we interpret $\rho_{n;(x,t_1)}^{(y,t_2)}$ as the union of all these polymers, for the purpose of defining $\rho_{n;(x,t_1)}^{(y,t_2)}(t)$. This convention permits us to identify worst case behaviour, so that the event $\neg \, {\mathsf{PolyDevReg}}_{n;(I,t_1)}^{(J,t_2)}\big(a,r\big)$ is triggered by a suitably large fluctuation on the part of any concerned polymer. Here, and later, $\neg \, A$ denotes the complement of the event $A$.
We are ready to state our conclusion concerning polymer fluctuation. Bounds in this and many later results have been expressed explicitly up to two positive constants $c$ and $C$. See Section \[s.useful\] for an explanation of how the value of this pair of constants is fixed. We further set $c_1 = 2^{-5/2} c \wedge 1/8$.
\[t.polyfluc\] Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ be a compatible triple, and let $x,y \in {\ensuremath{\mathbb{R}}}$.
1. Let $a \in \big[1 - 10^{-11} c_1^2 , 1 \big)$ satisfy $a {t_{1,2}}\in n^{-1} {\ensuremath{\mathbb{Z}}}$. Suppose that $n \in {\ensuremath{\mathbb{N}}}$ satisfies $$n {t_{1,2}}\geq \max \bigg\{
10^{32} (1-a)^{-25} c^{-18} \, \, , \, \,
10^{24} c^{-18} (1-a)^{-25} \vert x - y \vert^{36} {t_{1,2}}^{-24}
\bigg\} \, .$$ Let $r > 0$ be a parameter that satisfies $$r \geq
\max \bigg\{ 10^9 c_1^{-4/5} \, \, , \, \, 15 C^{1/2} \, \, , \, \, 87(1-a)^{1/3} {t_{1,2}}^{-2/3} \vert x - y \vert \bigg\}$$ and $r \leq 3 (1-a)^{25/9} n^{1/36} {t_{1,2}}^{1/36}$.
Writing $I = \big[ x, x + t_{1,2}^{2/3} (1-a)^{2/3} r \big]$ and $J = \big[ y, y + t_{1,2}^{2/3} (1-a)^{2/3} r \big]$, we have that $${\ensuremath{\mathbb{P}}}\Big( \neg \, {\mathsf{PolyDevReg}}_{n;(I,t_1)}^{(J,t_2)}\big(a,2r\big) \, \Big) \leq
44 C r \exp \big\{ - 10^{-11} c_1 r^{3/4} \big\}
\, .$$
2. The same statement holds verbatim when appearances of $a$ are replaced by $1-a$.
One aspect of our presentation may be apparent from the form of this theorem: we have chosen to be fairly explicit in recording hypothesis bounds in our results. This approach can provide quite lengthy formulas when hypotheses are recorded, and we encourage the reader not to be distracted by this from the essential meaning of results. In the present case, for example, we may set $t_1 = 0$ and $t_1 = 1$, so that ${t_{1,2}}= 1$ results. We may note that $a$ must be a certain small distance from either zero or one, and that the condition $a \in n^{-1}{\ensuremath{\mathbb{Z}}}$ is a negligible constraint, given that we are interested in high choices of $n$. Working in a case where $x$ and $y$ are bounded above by one, we are stipulating that $n \geq n_0(a)$. The parameter $r$, whose role is to gauge scaled fluctuation, is then constrained to lie between a universal constant and a multiple of $n^{1/36}$. We see then that the theorem is asserting that the maximum scaled fluctuation during the first, or last, duration $a$ of time among all polymers beginning and ending in a given unit interval, separated at a unit duration, exceeds $r \in [\Theta(1),\Theta(1)n^{1/36}]$ with probability at most $\exp \{ - O(1) r^{3/4} \}$. Since we are interested in high $n$, and the bound so available when $r = \Theta(1) n^{1/36}$ is rapidly decaying in $n$, the condition imposed on $r$ is quite weak.
If the reader is ever disconcerted by the various conditions imposed in results, it may be helpful to consider the formal case where $n = \infty$, in which upper bounds, such as $r \leq \Theta(1) n^{1/36}$ above, become obsolete.
As a matter of convenience for the upcoming proofs, we also state a version of Theorem \[t.polyfluc\] in which the intervals $I$ and $J$ are singleton sets.
\[p.polyfluc\] Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{R}}}^2_<$ be a compatible triple. Let $x,y \in {\ensuremath{\mathbb{R}}}$ and let $a \in \big[1 - 10^{-11} c_1^2 , 1 \big)$ satisfy $a {t_{1,2}}\in n^{-1} {\ensuremath{\mathbb{Z}}}$. Suppose that $$n {t_{1,2}}\geq \max \bigg\{
10^{32} (1-a)^{-25} c^{-18} \, \, , \, \,
10^{24} c^{-18} (1-a)^{-25} \vert x - y \vert^{36} {t_{1,2}}^{-24}
\bigg\} \, .$$ Let $r > 0$ be a parameter that satisfies $$r \geq
\max \bigg\{ 10^9 c_1^{-4/5} \, \, , \, \, 15 C^{1/2} \, \, , \, \, 87(1-a)^{1/3} {t_{1,2}}^{-2/3} \vert x - y \vert \bigg\}$$ and $r \leq 3 (1-a)^{25/9} n^{1/36} {t_{1,2}}^{1/36}$. Then $${\ensuremath{\mathbb{P}}}\Big( \neg \, {\mathsf{PolyDevReg}}_{n;(x,t_1)}^{(y,t_2)}\big(a,r\big) \, \Big) \leq
22 C r \exp \big\{ - 10^{-11} c_1 r^{3/4} \big\} \, .$$
[*Remark.*]{} The argument leading to the proposition will show that the result equally applies when it is instead supposed that $a \in (0,10^{-11}c_1^2]$, provided that the instances of $1-a$ in the hypothesis conditions are replaced by $a$.
### Acknowledgments.
The author thanks Riddhipratim Basu, Ivan Corwin, Shirshendu Ganguly and Jeremy Quastel for valuable conversations. He thanks a referee for thorough and very useful comments.
A road map for proving the rarity of disjoint polymers with close endpoints
===========================================================================
In this section, we present a very coarse overview of the strategy of the proof of our result, Theorem \[t.disjtpoly.pop\], establishing the improbability of the event that several disjoint polymers begin and end in a common pair of short intervals. (Theorem \[t.polyfluc\]’s proof depends on different ideas and will appear towards the end of the paper.) Before we begin, we first present a key concept: the scaled energy, or [*weight*]{} , of an $n$-zigzag.
Polymer weights
---------------
### Staircase energy scales to zigzag weight.
Let $n \in {\ensuremath{\mathbb{N}}}$ and $(i,j) \in {\ensuremath{\mathbb{N}}}^2_<$. Any $n$-zigzag $Z$ from $(x,i/n)$ to $(y,j/n)$ is ascribed a scaled energy, which we will refer to as its weight, ${\mathsf{Wgt}}(Z) = {\mathsf{Wgt}}_n(Z)$, given by $$\label{e.weightzigzag}
{\mathsf{Wgt}}(Z) = 2^{-1/2} n^{-1/3} \Big( E(S) - 2(j - i) - 2n^{2/3}(y-x) \Big)$$ where $Z$ is the image under $R_n$ of the staircase $S$.
### Maximum weight.
Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ be a compatible triple. Suppose that $x,y \in {\ensuremath{\mathbb{R}}}$ satisfy $y \geq x - 2^{-1} n^{1/3} {t_{1,2}}$. Define $${\mathsf{Wgt}}_{n;(x,t_1)}^{(y,t_2)} \, = \, 2^{-1/2} n^{-1/3} \Big( M^1_{(n t_1 + 2n^{2/3}x,n t_1) \to (n t_2 + 2n^{2/3}y,n t_2)} - 2n {t_{1,2}}- 2n^{2/3}(y-x) \Big) \, .$$ Thus, ${\mathsf{Wgt}}_{n;(x,t_1)}^{(y,t_2)}$ equals the maximum weight of any $n$-zigzag from $(x,t_1)$ to $(y,t_2)$. When the polymer $\rho_{n;(x,t_1)}^{(y,t_2)}$ is well defined, as it is almost surely, we have that ${\mathsf{Wgt}}_{n;(x,t_1)}^{(y,t_2)} = {\mathsf{Wgt}}\big( \rho_{n;(x,t_1)}^{(y,t_2)} \big)$. It is also worth noting, however, that the system of weights ${\mathsf{Wgt}}_{n;(x,t_1)}^{(y,t_2)}$ is well defined almost surely, even as the five concerned parameters vary over admissible choices, despite the possible presence of exceptional parameter choices at which the corresponding polymer is not unique.
The road map {#s.roadmap}
------------
We begin by revisiting a theme from Subsection \[s.polyflucintro\]. When $n$ is large, an $n$-polymer such as $\rho_{n;(x,t_1)}^{(y,t_2)}$ is rather naturally viewed as a random function of its lifetime $[t_1,t_2]$, one that begins at $x$ and ends at $y$. The microscopic ambiguities in the specification of this random function become vanishingly small in the limit of high $n$. Indeed, for the purposes of these paragraphs of overview, we yield to the temptation to set $n=\infty$, and discuss polymers as objects arising after a scaling limit has been taken. In this way, $\rho_{\infty;(x,t_1)}^{(y,t_2)}$ is interpreted as the scaled limit polymer between space-time locations $(x,t_1)$ and $(y,t_2)$. The polymer may be viewed as a subset of ${\ensuremath{\mathbb{R}}}^2$, living inside the strip ${\ensuremath{\mathbb{R}}}\times [t_1,t_2]$, or it may be viewed as a random, real-valued, function of the lifetime $[t_1,t_2]$. Speaking of such objects as $\rho_{\infty;(x,t_1)}^{(y,t_2)}$ raises substantial questions about the uniqueness of a limiting description as $n$ tends to infinity. In this non-rigorous overview, we have no intention of trying to address these questions. We simply use the framework of $n = \infty$ as a convenient device for heuristic discussion, since this framework is unencumbered by microscopic details (such as the structure of zigzags). The term zigzag has become rather inapt in this context; it may be replaced by the term [*path*]{}, a path being any continuous real-valued function defined on a given interval $[t_1,t_2]$. Each path has a weight and a path of maximum weight given its endpoints is a polymer. When these endpoints are $(x,t_1)$ and $(y,t_2)$, this maximum weight is called ${\mathsf{Wgt}}_{\infty;(x,t_1)}^{(y,t_2)}$.
Viewed through the prism of $n = \infty$, the event that Theorem \[t.disjtpoly.pop\] discusses is depicted in the left sketch of Figure \[f.triple\]. Here we take $(n,t_1,t_2) = (\infty,0,1)$. Depicted with ${k}=3$ is the event ${\mathrm{MaxDisjtPoly}}_{\infty;([x-{\epsilon},x+{\epsilon}],0)}^{([y-{\epsilon},y+{\epsilon}],1)} \geq {k}$ that there exist ${k}$ disjoint polymers with lifetime $[0,1]$ each of which begins at distance at most ${\epsilon}$ from $x \in {\ensuremath{\mathbb{R}}}$ and ends at such a distance from $y \in {\ensuremath{\mathbb{R}}}$. It is valuable to bear in mind three general features of polymer geometry and weight that correspond to powers of $2/3$, $1/3$ and $1/2$. The first of these three principles has been articulated rigorously by Theorem \[t.polyfluc\]. For the latter two, we will recall in Section \[s.input\] corresponding results for Brownian LPP from [@ModCon].
[*A power of two-thirds dictates polymer geometry.*]{} A polymer whose lifetime is $[t_1,t_2]$, and thus has duration ${t_{1,2}}$, fluctuates laterally, away from the planar line segment that interpolates its endpoints, by an order of ${t_{1,2}}^{2/3}$.
[*A power of one-third dictates polymer weight.*]{} A polymer whose lifetime is $[t_1,t_2]$ has a weight of order ${t_{1,2}}^{1/3}$. Actually, this is only true if the polymer makes no significant lateral movement. For example, $\rho_{\infty;(x,t_1)}^{(y,t_2)}$ may be expected to have a weight of order ${t_{1,2}}^{1/3}$ provided that the endpoints verify $\vert y - x \vert = O({t_{1,2}}^{2/3})$. The last condition indicates that the endpoint discrepancy is of the order of the polymer fluctuation, so that such a polymer is not deviating more than would be expected by a polymer whose starting and ending points coincide.
[*A power of one-half dictates polymer weight differences.*]{} Consider two polymers of unit duration both of whose endpoints differ by a small quantity ${\epsilon}$. Then the polymers’ weights typically differ by an order of ${\epsilon}^{1/2}$. For example, ${\mathsf{Wgt}}_{\infty;(x + {\epsilon},0)}^{(y + {\epsilon},1)} - {\mathsf{Wgt}}_{\infty;(x ,0)}^{(y ,1)}$ typically has order ${\epsilon}^{1/2}$, at least when $\vert y - x \vert = 0(1)$.
We now introduce an event that has much in common with the event that ${\mathrm{MaxDisjtPoly}}_{\infty;([x-{\epsilon},x+{\epsilon}],0)}^{([y-{\epsilon},y+{\epsilon}],1)}$ is at least ${k}$. Let $\eta > 0$. The [*near polymer*]{} event ${\mathsf{NearPoly}}_{\infty,{k};(x,0)}^{(y,1)}(\eta)$ is said to occur when there exist ${k}$ paths mapping $[0,1]$ to ${\ensuremath{\mathbb{R}}}$, each beginning at $x \in {\ensuremath{\mathbb{R}}}$ and ending at $y \in {\ensuremath{\mathbb{R}}}$, but which are pairwise disjoint except at the endpoint locations, such that the sum of the paths’ weights exceeds ${k}\cdot {\mathsf{Wgt}}_{\infty;(x,0)}^{(y,1)} \, - \, \eta$. To understand the meaning of this definition, note that, since each of these ${k}$ paths has a weight bounded above by that of the polymer with these given endpoints, the sum of the paths’ weights may be at most ${k}\cdot {\mathsf{Wgt}}_{\infty;(x,0)}^{(y,1)}$. Since weights are of unit order, due to a unit duration being considered, we see that, when $\eta > 0$ is fixed to be a small value, the event ${\mathsf{NearPoly}}_{\infty,{k};(x,0)}^{(y,1)}(\eta)$ expresses the presence of a system of ${k}$ near polymers, with shared endpoints but otherwise disjoint, whose sum weight misses the maximum value attainable in principle by a smaller than typical discrepancy of $\eta$.
The backbone of our derivation of Theorem \[t.disjtpoly.pop\] consists of arguing that the occurrence of the event ${\mathrm{MaxDisjtPoly}}_{\infty;([x-{\epsilon},x+{\epsilon}],0)}^{([y-{\epsilon},y+{\epsilon}],1)} \geq {k}$ typically entails that ${\mathsf{NearPoly}}_{\infty,{k};(x,0)}^{(y,1)}(\eta)$ also occurs, where here $\eta$ is set to be of the order of ${\epsilon}^{1/2}$. Before we offer an overview of why this assertion may be expected to be true, we point out that the order of probability of the latter event is understood. Indeed, the use of shared endpoints in the definition of ${\mathsf{NearPoly}}_{\infty,{k};(x,0)}^{(y,1)}(\eta)$ permits the use of integrable techniques arising from the Robinson-Schensted-Knuth correspondence: this event is shown in [@BrownianReg] to have probability of order $\eta^{{k}^2 - 1 + o(1)}$. The result in question will shortly be reviewed as Corollary \[c.neargeod.t\]. In reality, the result actually concerns the events ${\mathsf{NearPoly}}_{n,k;(x,0)}^{(y,1)}(\eta)$ when $n \in {\ensuremath{\mathbb{N}}}$ is finite, and makes an assertion uniformly in high $n$ about them. Anyway, since we are taking $\eta = \Theta({\epsilon}^{1/2})$, we see that the form of Theorem \[t.disjtpoly.pop\] may be expected to follow from such a bound.
Now, how is it that we may argue that the ${k}$ disjoint polymer event ${\mathrm{MaxDisjtPoly}}_{\infty;([x-{\epsilon},x+{\epsilon}],0)}^{([y-{\epsilon},y+{\epsilon}],1)} \geq {k}$ does indeed typically entail ${\mathsf{NearPoly}}_{\infty,{k};(x,0)}^{(y,1)}(\eta)$ with $\eta = \Theta \big( {\epsilon}^{1/2} \big)$? The first event provides much of the structure of the ${k}$ near polymers that would realize the second. But in the second, the near polymers begin and end at precisely the same points, rather than merely nearby one another. What is needed is surgery to alter the ${k}$ polymers that begin and end in $[x-{\epsilon},x + {\epsilon}] \times \{ 0 \}$ and $[y - {\epsilon},y + {\epsilon}] \times \{ 1 \}$ so that their endpoints are actually equal.
Our device enabling this surgery to effect local correction of endpoint locations is called a [*multi-polymer bouquet*]{}. Let $[t_1,t_2]$ be a short real interval, and let $u \in {\ensuremath{\mathbb{R}}}$. Fixing an increasing list of ${k}$ real values, $(v_1,\cdots,v_{k})$, each of unit order, we may consider the problem of constructing a system of ${k}$ paths, each with lifetime $[t_1,t_2]$, with each path beginning at $(u,t_1)$, and with the $i\textsuperscript{th}$ path ending at $\big(u + {t_{1,2}}^{2/3} v_i , t_2 \big)$; crucially, we insist that the paths be disjoint, except for their shared starting location. Such a bouquet of paths is depicted in the middle sketch in Figure \[f.triple\]. These paths are moving on the natural two-thirds power fluctuation scale given their endpoints, so it is a natural belief that they may be selected so that the sum of their weights has order ${t_{1,2}}^{1/3}$, which is the natural polymer weight scale for polymers of duration ${t_{1,2}}$. It is one of the upcoming challenges for the proof of Theorem \[t.disjtpoly.pop\] to prove that the construction of such a path system may be undertaken. We will refer to this challenge as the problem of [*bouquet construction*]{}.
![[*Left:*]{} The three polymers concerned in the formally specified event ${\mathrm{MaxDisjtPoly}}_{\infty;([x-{\epsilon},x+{\epsilon}],0)}^{([y-{\epsilon},y+{\epsilon}],1)} \geq 3$. [*Middle:*]{} A bouquet of three paths of lifetime $[t_1,t_2]$. [*Right:*]{} Two bouquets, facing in opposite directions, are employed to extend the original three polymers into longer paths, each beginning and ending at $(x,-{\epsilon}^{3/2})$ and $(y,1 +{\epsilon}^{3/2})$.[]{data-label="f.triple"}](NonIntPolyTriple.pdf){height="9cm"}
Accepting for now that such a bouquet of ${k}$ paths, tied together at a common base point, may be constructed, it is of course also natural to believe that such a construction may be undertaken where instead it is at the endpoint time that the various paths share a common location. The two constructions may be called a forward and a backward bouquet.
The two bouquets may be used to tie together the nearby endpoints of the ${k}$ polymers that arise in the event ${\mathrm{MaxDisjtPoly}}_{\infty;([x-{\epsilon},x+{\epsilon}],0)}^{([y-{\epsilon},y+{\epsilon}],1)} \geq {k}$. Rather than altering the paths so that they begin and end at $(x,0)$ and $(y,1)$, we will prefer to extend the duration of the paths a little in each direction. Because the spatial discrepancy of endpoint locations is of order ${\epsilon}$, the two-thirds power law for polymer geometry indicates that the natural time scale for our polymer bouquets will be ${\epsilon}^{3/2}$. A forward bouquet of ${k}$ paths will be used at the start and a backward one at the end. The ${k}$ paths in the forward bouquet will each begin at $(x,-{\epsilon}^{3/2})$; their other endpoints, at time zero, will be the ${k}$ starting points of the duration $[0,1]$ polymers. The backward bouquet will take the ${k}$ ending points of the polymers and tie them together at the common location $(y,1+{\epsilon}^{3/2})$.
In this way, a system of ${k}$ paths from $(x,-{\epsilon}^{3/2})$ to $(y,1+{\epsilon}^{3/2})$ will be obtained: see the right sketch in Figure \[f.triple\]. Each is split into three pieces: a short ${\epsilon}^{3/2}$-duration element of the forward bouquet, a long $[0,1]$-time piece given by one of the polymers, and then a further short element of the backward bouquet. This system will be shown to typically realize ${\mathsf{NearPoly}}_{\infty,{k};(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}(\eta)$ with $\eta = \Theta \big( {\epsilon}^{1/2} \big)$. Note that the starting and ending times are $-{\epsilon}^{3/2}$ and $1 + {\epsilon}^{3/2}$, rather than zero and one. We now summarise the challenges involved in proving this. The value $n$ is finite, with statements to be understood uniformly in high choices of this parameter.
1. Polymer weight similarity: if the construction is to result in a collection of ${k}$ paths from $(x,-{\epsilon}^{3/2})$ to $(y,1 + {\epsilon}^{3/2})$ each of whose weights reaches within order ${\epsilon}^{1/2}$ of the maximum achievable value ${\mathsf{Wgt}}_{n;(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}$, then we will want to begin by confirming that the difference in weight between any pair of the ${k}$ polymers moving between $[x - {\epsilon},x+{\epsilon}] \times \{0\}$ and $[y - {\epsilon},y+{\epsilon}] \times \{ 1 \}$ itself has order ${\epsilon}^{1/2}$. This is an instance of the one-half power law principle enunciated above.
2. Bouquet construction: the forward and backward bouquets, defined on the time intervals $[-{\epsilon}^{3/2},0]$ and $[1,1+{\epsilon}^{3/2}]$, must be constructed so that each bouquet has cumulative weight of the desired order ${\epsilon}^{1/2}$.
3. Final polymer comparison: even if our original polymers have similar weights, and they can be altered to run from $(x,-{\epsilon}^{3/2})$ to $(y,1+{\epsilon}^{3/2})$ with weight changes of order ${\epsilon}^{1/2}$, there could still be trouble in principle. We need to know that each of the extended paths has a weight suitably close to ${\mathsf{Wgt}}_{n;(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}$. Thus, we need to verify that the polymer running from $(x,-{\epsilon}^{3/2})$ to $(y,1+{\epsilon}^{3/2})$ has a weight that exceeds the extended paths’ weights by an order of at most ${\epsilon}^{1/2}$.
In a sense, the third of these challenges is simply a restatement of the entire problem. When we discuss it later, however, we will resolve the problem in light of the solutions of the first two difficulties.
This completes our first overview of the road map to the proof of Theorem \[t.disjtpoly.pop\]. We have been left with three challenges to solve. In the next three sections, we will present definitions and tools and cite results that will be used to implement the road map, as well as to prove the polymer fluctuation Theorem \[t.polyfluc\]. After these tools have been described, we will describe at the end of Section \[s.input\] the structure of the rest of the paper. In essence, we will then return to the road map, elaborate further in outline how to resolve its three challenges, and implement it rigorously.
Tools: staircase collections and multi-polymers
===============================================
The definition and study of the event ${\mathsf{NearPoly}}_{n,{k};(x,0)}^{(y,1)}(\eta)$ (for ${k}$ fixed and $n$, although now finite, high in applications), and the problem of bouquet construction, will both involve the use of systems of staircases or zigzags. We begin this discussion of tools by setting up definitions in this regard.
Staircase collections {#s.staircase}
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Two staircases are called horizontally separate if there is no planar horizontal interval of positive length that is a subset of a horizontal interval in both staircases.
We now introduce notation for collections of pairwise horizontally separate staircases. For $k \in {\ensuremath{\mathbb{N}}}$, let $(x_i,s_i)$ and $(y_i,f_i)$, $i \in {\llbracket 1,k \rrbracket}$, be a collection of pairs of elements of ${\ensuremath{\mathbb{R}}}\times {\ensuremath{\mathbb{N}}}$. (The symbols $s$ and $f$ are used in reference to the staircases’ heights at the [*start*]{} and [*finish*]{}.)
Let ${SC}^k_{(\bar{x},\bar{s}) \to (\bar{y},\bar{f})}$ denote the set of $k$-tuples $(\phi_1,\cdots,\phi_k)$, where $\phi_i$ is a staircase from $(x_i,s_i)$ to $(y_i,f_i)$ and each pair $(\phi_i,\phi_j)$, $i \not= j$, is horizontally separate. (This set may be empty; in order that it be non-empty, it is necessary that $y_i \geq x_i$ and $f_i \geq s_i$ for $i \in {\llbracket 1,k \rrbracket}$.) Note also that ${SC}^1_{(x_1,s_1) \to (y_1,f_1)}$ equals ${SC}_{(x_1,s_1) \to (y_1,f_1)}$.
We also associate an energy to each member of ${SC}^k_{(\bar{x},\bar{s}) \to (\bar{y},\bar{f})}$. Each of the $k$ elements of any $k$-tuple in ${SC}^k_{(\bar{x},\bar{s}) \to (\bar{y},\bar{f})}$ has an energy, as we described in Subsection \[s.staircases\]. Define the energy $E\big( \phi \big)$ of any $\phi = \big( \phi_1,\cdots,\phi_k \big) \in {SC}^k_{(\bar{x},\bar{s}) \to (\bar{y},\bar{f})}$ to be $\sum_{j=1}^k E(\phi_j)$.
When ${SC}^k_{(\bar{x},\bar{s}) \to (\bar{y},\bar{f})} \not= \emptyset$, we further define the maximum $k$-tuple energy $$\label{e.mell}
M^k_{(\bar{x},\bar{s}) \to (\bar{y},\bar{f})} = \sup \Big\{ E(\phi): \phi \in {SC}^k_{(\bar{x},\bar{s}) \to (\bar{y},\bar{f})} \Big\} \, .$$ An $k$-tuple of staircases that attains this maximum may be called a multi-geodesic.
Maximizer uniqueness {#s.maxunique}
--------------------
Next is [@Patch Lemma $A.1$]. Since this result is drawn the final article in our four-paper study, it is worth mentioning that the result has a fairly short and self-contained proof in [@Patch Appendix $A$].
\[l.severalpolyunique\] Let $k \in {\ensuremath{\mathbb{N}}}$ and let $(x_i,s_i)$ and $(y_i,f_i)$, $i \in {\llbracket 1,k \rrbracket}$, be a collection of pairs of points in ${\ensuremath{\mathbb{R}}}\times {\ensuremath{\mathbb{N}}}$ such that ${SC}^k_{(\bar{x},\bar{s}) \to (\bar{y},\bar{f})}$ is non-empty. Then, except on a ${\ensuremath{\mathbb{P}}}$-null set, there is a unique element of ${SC}^k_{(\bar{x},\bar{s}) \to (\bar{y},\bar{f})}$ whose energy attains $M^k_{(\bar{x},\bar{s}) \to (\bar{y},\bar{f})}$.
When it exists, we denote the unique maximizer by $$\Big( P^{k}_{(\bar{x},\bar{s}) \to (\bar{y},\bar{f});i} : i \in {\llbracket 1,k \rrbracket} \Big) \in {SC}^k_{(\bar{x},\bar{s}) \to (\bar{y},\bar{f})} \, .$$
Consider the partial order on ${\ensuremath{\mathbb{R}}}^2$ such that $(x_1,x_2)$ is at most $(y_1,y_2)$ if and only if $x_1 \leq x_2$ and $y_1 \leq y_2$.
In the case that $k = 1$, and $(x,s)$ and $(y,f)$ are elements of ${\ensuremath{\mathbb{R}}}\times {\ensuremath{\mathbb{N}}}$ with $(y,f)$ exceeding $(x,s)$ in the partial order, the set ${SC}^1_{(x,s) \to (y,f)}$ is non-empty. The maximizer in Lemma \[l.severalpolyunique\] is in this special case denoted by $P^{1}_{(x,s) \to (y,f)}$.
Maximum weights of zigzag systems {#s.zigzagmax}
---------------------------------
Each $n$-zigzag is the image under the scaling map $R_n$ of a staircase. Two such zigzags will be called horizontally separate if their staircase counterparts have this property. The weight of a collection of zigzags is the sum of the weights of those zigzags.
We wish to record notation for the maximum weight of a pairwise horizontally disjoint collection of $[t_1,t_2]$-lifetime zigzags with prescribed endpoints.
To this end, let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_\leq$ be a compatible triple. Taking $k \in {\ensuremath{\mathbb{N}}}$, a parameter that will denote the number of zigzags, consider $k$ pairs $(x_i,y_i) \in {\ensuremath{\mathbb{R}}}^2$, $i \in {\llbracket 1,k \rrbracket}$, where we suppose that $y_i \geq x_i - 2^{-1} n^{1/3} {t_{1,2}}$ for each $i \in {\llbracket 1,k \rrbracket}$. Further suppose that the vectors $\overline{x} = \big( x_1,\cdots,x_k \big)$ and $\overline{y} = (y_1,\cdots,y_k)$ are non-decreasing. We will write ${\mathsf{Wgt}}_{n,k;(\bar{x},t_1)}^{(\bar{y},t_2)}$ for the maximum weight associated to $k$ horizontally separate zigzags moving consecutively between $(x_i,t_1)$ to $(y_i,t_2)$ for $i \in {\llbracket 1,k \rrbracket}$. Formally, we define $$\label{e.multiweight}
{\mathsf{Wgt}}_{n,k;(\bar{x},t_1)}^{(\bar{y},t_2)} \, = \, 2^{-1/2} n^{-1/3} \Big( M^k_{(n t_1 \bar{\bf 1} + 2n^{2/3}\bar{x},n t_1 \bar{\bf 1}) \to (n t_2 \bar{\bf 1} + 2n^{2/3}\bar{y},n t_2 \bar{\bf 1} )} - 2 k n {t_{1,2}}- 2n^{2/3}\sum_{i=1}^k(y_i-x_i) \Big) \, ,$$ where $\bar{\bf 1}\in {\ensuremath{\mathbb{R}}}^k$ denotes the vector whose components equal $1$.
Multi-polymers and their weights {#s.multipolymer}
--------------------------------
Retaining these parameters, we now may offer a geometric view of this multi-weight. The maximum energy $M^k$ seen on the right-hand side of (\[e.multiweight\]) is attained by a certain multi-geodesic which, after the scaling map is applied, may be viewed as a multi-polymer; the weight specified in (\[e.multiweight\]) is the sum of the weights of the zigzags that constitute this multi-polymer. To set up notation in this regard, note that Lemma \[l.severalpolyunique\] ensures the almost sure existence of the $k$-tuple $$\label{e.ptuple}
\Big( P^{k}_{\big(n t_1 \bar{\bf 1} + 2 n^{2/3} \bar{x}, n t_1 \bar{\bf 1}) \to (n t_2 \bar{\bf 1} + 2n^{2/3} \bar{y} , n t_2 \bar{\bf 1} \big) ;i} : i \in {\llbracket 1,k \rrbracket} \Big) \in {SC}^k_{\big(n t_1 \bar{\bf 1} + 2 n^{2/3} \bar{x}, n t_1 \bar{\bf 1}) \to (n t_2 \bar{\bf 1} + 2n^{2/3} \bar{y} , n t_2 \bar{\bf 1} \big)} \, .$$ Indeed, the right-hand set is clearly non-empty under our hypotheses, permitting the use of the above lemma. We define the multi-polymer $$\Big( \rho_{n,k,i;(\bar{x},t_1)}^{(\bar{y},t_2)} : i \in {\llbracket 1,k \rrbracket} \Big)$$ to be the $k$-tuple of $n$-zigzags whose elements are the respective images under $R_n$ of the elements in the $k$-tuple in (\[e.ptuple\]). The $k$-tuple itself will be denoted by $\rho_{n,k;(\bar{x},t_1)}^{(\bar{y},t_2)}$. In this way, the $i\textsuperscript{th}$ component $\rho_{n,k,i;(\bar{x},t_1)}^{(\bar{y},t_2)}$ is an $n$-zigzag with starting and ending points $(x_i, {t_1})$ and $(y_i, {t_2})$. This zigzag is not necessarily a polymer.
Each of the $k$ zigzag components of $\rho_{n,k;(\bar{x},t_1)}^{(\bar{y},t_2)}$ has a weight via (\[e.weightzigzag\]): the $i\textsuperscript{th}$ weight is ${\mathsf{Wgt}}(Z) = 2^{-1/2} n^{-1/3} \big( E(S) - 2 n {t_{1,2}}- 2n^{2/3}(y_i-x_i) \big)$, where $S$ is the pre-image staircase in question. This $i\textsuperscript{th}$ weight will be denoted by ${\mathsf{Wgt}}_{n,k,i;(\bar{x},t_1)}^{(\bar{y},t_2)}$. The weight of the multi-polymer $\rho_{n,k;(\bar{x},t_1)}^{(\bar{y},t_2)}$ equals the component sum $\sum_{i=1}^k {\mathsf{Wgt}}_{n,k,i;(\bar{x},t_1)}^{(\bar{y},t_2)}$, and is seen to be equal to ${\mathsf{Wgt}}_{n,k;(\bar{x},t_1)}^{(\bar{y},t_2)}$.
An important special case arises when $\bar{x}$ and $\bar{y}$ are both constant vectors, of the form $\overline{x} = x \bar{\bf 1} \in {\ensuremath{\mathbb{R}}}^k$ and $\overline{y} = y \bar{\bf 1} \in {\ensuremath{\mathbb{R}}}^k$. The multi-polymer may be called a [*multi-polymer watermelon*]{} in this case. The cases when merely one of $\bar{x}$ and $\bar{y}$ is a constant vector are also significant: these are the forward and backward bouquets that have been discussed in the road map.
Multi-weight encoding line ensembles: definitions and key properties {#s.multiweight}
====================================================================
As the road map has described, a key input drawing in part on integrable probability, Corollary \[c.neargeod.t\], will assert that ${\mathsf{NearPoly}}_{n,k;(x,0)}^{(y,1)}(\eta)$ has probability $\eta^{k^2 - 1 + o(1)}$ uniformly in high $n$. This assertion about the weight of a multi-polymer watermelon falls within the realm of integrable probability because this weight is the maximum possible attained by a system of $k$ zigzags, with given shared endpoints but otherwise disjoint, and, as such, it may be expressed as the value of the sum of the $k$ uppermost curves in a line ensemble that is naturally associated to Brownian LPP (by means of the RSK correspondence). In this section, we set up the notation for these ensembles of random curves.
Line ensembles that encode polymer weights {#s.encode}
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Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_\leq$ be a compatible triple, and let $x \in {\ensuremath{\mathbb{R}}}$. We define the scaled forward line ensemble $${\mathcal}{L}_{n;(x,t_1)}^{\uparrow;t_2}: {\llbracket 1,n {t_{1,2}}+ 1 \rrbracket} \times \big[ x - 2^{-1} n^{1/3} {t_{1,2}}, \infty\big) \to {\ensuremath{\mathbb{R}}}$$ rooted at $(x,t_1)$ with duration ${t_{1,2}}$ by declaring that, for each $k \in {\llbracket 1,n {t_{1,2}}+ 1 \rrbracket}$ and $y \geq x - 2^{-1} n^{1/3} {t_{1,2}}$, $$\label{e.scaledweight}
\sum_{i=1}^k {\mathcal}{L}_{n;(x,t_1)}^{\uparrow;t_2}(i,y) \, = \,
{\mathsf{Wgt}}_{n,k;(x \bar{\bf 1},t_1)}^{(y \bar{\bf 1},t_2)} \, .$$ The ensemble is called ‘forward’, and the notation is adorned with the symbol $\uparrow$, because it is the spatial location $y$ attached to the more advanced time $t_2$ that is treated as the variable. We stand at $(x,t_1)$ and look forward in time to $t_2$ to witness behaviour as a function of location $y$. When $k=1$, the lowest indexed ensemble curve $y \to {\mathcal}{L}_{n;(x,t_1)}^{\uparrow;t_2}(1,y)$ records the polymer weight profile from $(x,t_1)$ to $(y,t_2)$. The sum of the $k$ lowest indexed curves records the multi-polymer watermelon weight between these locations.
Equally, we may stand at $(y,t_2)$ and look backward in time to a variable location $x \in {\ensuremath{\mathbb{R}}}$ at time $t_1$. Indeed, fixing $y \in {\ensuremath{\mathbb{R}}}$, we define the scaled backward line ensemble $${\mathcal}{L}_{n;t_1}^{\downarrow;(y,t_2)}: {\llbracket 1,n {t_{1,2}}+ 1 \rrbracket} \times \big( - \infty , y + 2^{-1} n^{1/3} {t_{1,2}}\big] \to {\ensuremath{\mathbb{R}}}$$ rooted at $(y,t_2)$ with duration ${t_{1,2}}$ by declaring that, for each $k \in {\llbracket 1,n {t_{1,2}}+ 1 \rrbracket}$ and $x \leq y + 2^{-1} n^{1/3} {t_{1,2}}$, $$\sum_{i=1}^k {\mathcal}{L}_{n;t_1}^{\downarrow;(y,t_2)}(i,x) \, = \,
{\mathsf{Wgt}}_{n,k;(x \bar{\bf 1},t_1)}^{(y \bar{\bf 1},t_2)} \, .$$
It is valuable to hold a vivid picture of the two ensembles. Each is an ordered system of random continuous curves, with the lowest indexed curve uppermost. The ensemble curve of any given index locally resembles Brownian motion but globally follows the contour of the parabola $- 2^{-1/2} (y-x)^2 {t_{1,2}}^{-4/3}$, as a function of $y$ or $x$ in the forward or backward case. Each ensemble adopts values of order ${t_{1,2}}^{1/3}$ when $x$ and $y$ differ by an order of ${t_{1,2}}^{2/3}$. More negative values, dictated by parabolic curvature, are witnessed outside this region. This description holds sway in a region that expands from the origin as the parameter $n$ rises.
Clearly, then, our ensembles have fundamental differences according to the value of ${t_{1,2}}$: sharply peaked ensemble curves when ${t_{1,2}}$ is small, and much flatter curves when ${t_{1,2}}$ is large. A simple further parabolic transformation will serve to put the ensembles on a much more equal footing. Since the ensembles are already scaled objects, we will use the term ‘normalized’ to allude to the newly transformed counterparts. A scaled forward ensemble and its normalized counterpart are depicted in Figure \[f.lineensembles\].
![Let $(n,t_1,t_2)$ be a compatible triple, with ${t_{1,2}}$ some small fixed value, and $n$ large. For $x \in {\ensuremath{\mathbb{R}}}$ given, the two sketches show the highest two curves in the scaled, and normalized, forward line ensembles rooted at $(x,t_1)$ with duration ${t_{1,2}}$.[]{data-label="f.lineensembles"}](NonIntPolyLineEnsembles.pdf){height="10cm"}
That is, we define the [*normalized*]{} forward ensemble $$\label{e.forward}
{\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow;t_2}: {\llbracket 1,n {t_{1,2}}+ 1 \rrbracket} \times \big[- 2^{-1} (n {t_{1,2}})^{1/3} , \infty \big) \to {\ensuremath{\mathbb{R}}}\, ,$$ setting $$\label{e.scaledln}
{\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow;t_2}\big( k, z \big)
= {t_{1,2}}^{-1/3} {\mathcal}{L}_{n;(x,t_1)}^{\uparrow;t_2}\big( k, x + {t_{1,2}}^{2/3} z \big)$$ for each $k \in {\llbracket 1,n {t_{1,2}}+ 1 \rrbracket}$ and $z \geq - 2^{-1} n^{1/3} {t_{1,2}}^{-2/3}$. Similarly, the normalized backward ensemble $$\label{e.backward}
{\mathsf{Nr}{\mathcal}{L}}_{n;t_1}^{\downarrow;(y,t_2)}: {\llbracket 1,n {t_{1,2}}+ 1 \rrbracket} \times \big(-\infty, 2^{-1} (n {t_{1,2}})^{1/3} \big] \to {\ensuremath{\mathbb{R}}}\, ,$$ is specified by setting $${\mathsf{Nr}{\mathcal}{L}}_{n;t_1}^{\downarrow;(y,t_2)}\big( k, z \big)
= {t_{1,2}}^{-1/3} {\mathcal}{L}_{n;t_1}^{\downarrow;(y,t_2)}\big( k, y + {t_{1,2}}^{2/3} z \big)$$ for each $k \in {\llbracket 1,n {t_{1,2}}+ 1 \rrbracket}$ and $z \leq 2^{-1} n^{1/3} {t_{1,2}}^{-2/3}$.
The curves in the new ensembles locally resemble Brownian motion as before, but they have been centred and squeezed so that now the parabola that dictates their overall shape is $- 2^{-1/2} z^2$. This picture is accurate in a region that expands as the parameter $n {t_{1,2}}$ rises.
Brownian Gibbs line ensembles
-----------------------------
The notion of a Brownian Gibbs line ensemble was introduced in [@AiryLE] to capture a system of ordered curves that arise by conditioning Brownian motions or bridges on mutual avoidance. The precise definition is not logically needed in this article, but we offer an informal summary next.
### An overview
Let $n \in {\ensuremath{\mathbb{N}}}$ and let $I$ be a closed interval in the real line. A ${\llbracket 1,n \rrbracket}$-indexed line ensemble defined on $I$ is a random collection of continuous curves ${\mathcal}{L}:{\llbracket 1,n \rrbracket} \times I \to {\ensuremath{\mathbb{R}}}$ specified under a probability measure ${\ensuremath{\mathbb{P}}}$. The $i\textsuperscript{th}$ curve is thus ${\mathcal}{L}(i,\cdot): I \to {\ensuremath{\mathbb{R}}}$. (The adjective ‘line’ has been applied to these systems perhaps because of their origin in such models as Poissonian LPP, where the counterpart object has piecewise constant curves. We will omit it henceforth.) An ensemble is called [*ordered*]{} if ${\mathcal}{L}(i,x) > {\mathcal}{L}(i+1,x)$ whenever $i \in {\llbracket 1,n-1 \rrbracket}$ and $x$ lies in the interior of $I$. The curves may thus assume a common value at any finite endpoint of $I$. We will consider ordered ensembles that satisfy a condition called the Brownian Gibbs property. Colloquially, we may say that an ordered ensemble is called Brownian Gibbs if it arises from a system of Brownian bridges or Brownian motions defined on $I$ by conditioning on the mutual avoidance of the curves at all times in $I$.
### Defining $(c,C)$-regular ensembles
We are interested in ensembles that are not merely Brownian Gibbs but that hew to the shape of a parabola and have one-point distributions for the uppermost curve that enjoy tightness properties. We will employ the next definition, which specifies a $(\bar\phi,{c},{C})$-regular ensemble from [@BrownianReg Definition $2.4$], in the special case where the vector $\bar\phi$ equals $(1/3,1/9,1/3)$.
\[d.regularsequence\] Consider a Brownian Gibbs ensemble of the form $${\mathcal}{L}: {\llbracket 1,{N}\rrbracket} \times \big[ - {z_{\mathcal{L}}}, \infty \big) \to {\ensuremath{\mathbb{R}}}\, ,$$ and which is defined on a probability space under the law ${\ensuremath{\mathbb{P}}}$. The number ${N}= {N}(\mathcal{L})$ of ensemble curves and the absolute value ${z_{\mathcal{L}}}$ of the finite endpoint may take any values in ${\ensuremath{\mathbb{N}}}$ and $[0,\infty)$.
Let ${Q}:{\ensuremath{\mathbb{R}}}\to {\ensuremath{\mathbb{R}}}$ denote the parabola ${Q}(x) = 2^{-1/2} x^2$.
Let ${C}$ and ${c}$ be two positive constants. The ensemble ${\mathcal}{L}$ is said to be $({c},{C})$-regular if the following conditions are satisfied.
1. [**Endpoint escape.**]{} ${z_{\mathcal{L}}}\geq {c}N^{1/3}$.
2. [**One-point lower tail.**]{} If $z \in [ -{z_{\mathcal{L}}}, \infty)$ satisfies $\vert z \vert \leq {c}{N}^{1/9}$, then $${\ensuremath{\mathbb{P}}}\Big( {\mathcal}{L} \big( 1,z\big) + {Q}(z) \leq - s \Big) \leq {C}\exp \big\{ - {c}s^{3/2} \big\}$$ for all $s \in \big[1, {N}^{1/3} \big]$.
3. [**One-point upper tail.**]{} If $z \in [ -{z_{\mathcal{L}}}, \infty)$ satisfies $\vert z \vert \leq {c}{N}^{1/9}$, then $${\ensuremath{\mathbb{P}}}\Big( {\mathcal}{L} \big( 1,z\big) + {Q}(z) \geq s \Big) \leq {C}\exp \big\{ - {c}s^{3/2} \big\}$$ for all $s \in [1, \infty)$.
A Brownian Gibbs ensemble of the form $${\mathcal}{L}: {\llbracket 1,{N}\rrbracket} \times \big( -\infty , {z_{\mathcal{L}}}\big] \to {\ensuremath{\mathbb{R}}}$$ is also said to be $({c},{C})$-regular if the reflected ensemble ${\mathcal}{L}( \cdot, - \cdot)$ is. This is equivalent to the above conditions when instances of $[ - {z_{\mathcal{L}}}, \infty)$ are replaced by $(-\infty, {z_{\mathcal{L}}}]$.
We will refer to these three regular ensemble conditions as ${{\rm Reg}}(1)$, ${{\rm Reg}}(2)$ and ${{\rm Reg}}(3)$.
### The normalized forward and backward ensembles are $(c,C)$-regular
Our reason for invoking the theory of regular Brownian Gibbs ensembles is that the normalized Brownian LPP ensembles verify the definition. This assertion is made by the next result; as we explain shortly, in Section \[s.organization\], its proof has in essence been provided in [@BrownianReg] and will formally be given in Appendix \[s.normal\].
\[p.scaledreg\] Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ be a compatible triple.
1. Let $x \in {\ensuremath{\mathbb{R}}}$. The ensemble ${\mathcal}{L}$ given by $${\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow;t_2}: {\llbracket 1,n {t_{1,2}}+ 1 \rrbracket} \times \big[- 2^{-1} (n {t_{1,2}})^{1/3} , \infty \big) \to {\ensuremath{\mathbb{R}}}\, ,$$ is Brownian Gibbs, where ${N}({\mathcal}{L}) = n {t_{1,2}}+ 1$ and ${z_{\mathcal{L}}}= 2^{-1} (n {t_{1,2}})^{1/3}$.
2. There exist positive constants ${C}$ and ${c}$, which may be chosen independently of all such choices of the parameters $t_1$, $t_2$, $x$ and $n$, such that the ensemble ${\mathcal}{L}$ is $({c},{C})$-regular.
3. If, in place of $x$, we consider $y \in {\ensuremath{\mathbb{R}}}$, and the ensemble ${\mathcal}{L}$ given by $${\mathsf{Nr}{\mathcal}{L}}_{n;t_1}^{\downarrow;(y,t_2)}(1,\cdot) : {\llbracket 1,n {t_{1,2}}+ 1 \rrbracket} \times \big( - \infty , 2^{-1} (n {t_{1,2}})^{1/3} \big] \to {\ensuremath{\mathbb{R}}}\, ,$$ then the two preceding assertions hold, the second now independently of the parameters $t_1$, $t_2$, $y$ and $n$.
Some useful properties of regular ensembles {#s.useful}
-------------------------------------------
We are about to state Proposition \[p.mega\], whose four parts assert the various properties, beyond the regular sequence conditions ${{\rm Reg}}(1)$, ${{\rm Reg}}(2)$ and ${{\rm Reg}}(3)$, of the normalized ensembles that we will be employing.
We fix henceforth the values of the two positive constants ${C}$ and ${c}$, specifying them by Proposition \[p.scaledreg\]. Since bounding the constants would render hypotheses of our results to be explicit, we mention that they are determined in [@BrownianReg Appendix $A.1$] via Ledoux [@Ledoux (5.16)] and Aubrun’s [@Aubrun Proposition $1$] bounds on the lower and upper tail of the maximum eigenvalue of a matrix in the Gaussian unitary ensemble.
We now specify two sequences $\big\{ C_k : k \geq 1 \big\}$ and $\big\{ c_k: k \geq 1 \big\}$, their values expressed in terms of ${C}$ and ${c}$. This usage for $C_k$ and $c_k$ is retained throughout the paper. We set, for each $k \geq 2$, $$\label{e.formere}
{C}_k = \max \Big\{ 10 \cdot 20^{k-1} 5^{k/2} \Big( \tfrac{10}{3 - 2^{3/2}} \Big)^{k(k-1)/2} C \, , \, e^{c/2} \Big\}$$ as well as $C_1 = 140 C$; and $$\label{e.littlec}
c_k = \big( (3 - 2^{3/2})^{3/2} 2^{-1} 5^{-3/2} \big)^{k-1} c_1 \, ,$$ with $c_1 = 2^{-5/2} c \wedge 1/8$. Note that $\limsup {C}_k^{1/k^2} < \infty$ and $\liminf c_k^{1/k} > 0$.
Recall from Definition \[d.regularsequence\] that ${Q}:{\ensuremath{\mathbb{R}}}\to {\ensuremath{\mathbb{R}}}$ denotes the parabola ${Q}(x) = 2^{-1/2} x^2$.
\[p.mega\] Suppose that ${\mathcal}{L} = {\mathcal}{L}_N$, mapping either ${\llbracket 1,N \rrbracket} \times [-{z_{\mathcal{L}}},\infty)$ or ${\llbracket 1,N \rrbracket} \times (-\infty,{z_{\mathcal{L}}}]$, to ${\ensuremath{\mathbb{R}}}$, is a $({c},{C})$-regular ensemble, where $N \in {\ensuremath{\mathbb{N}}}$ and ${z_{\mathcal{L}}}\geq 0$.
1. (Pointwise curve lower bound) Let $k \in {\ensuremath{\mathbb{N}}}$ and $z,s \in {\ensuremath{\mathbb{R}}}$. Suppose that $N \geq k
\vee (c/3)^{-18} \vee 6^{36}$, $\vert z \vert \leq 2^{-1} {c}N^{1/18}$ and $s \in \big[0, 2 N^{1/18} \big]$. Then $${\ensuremath{\mathbb{P}}}\Big( \, {\mathcal}{L}_N\big( k,z\big) + {Q}(z) \leq - s \, \Big) \leq {C}_k \exp \big\{ - c_k s^{3/2} \big\} \, .$$
2. (Uniform curve lower bound) For $k \in {\ensuremath{\mathbb{N}}}$, let ${E}_k = 20^{k-1} 2^{k(k-1)/2} {E}_1$ where ${E}_1 = 10C$. Set $${r_0}= 5 (3 - 2^{3/2})^{-1} \, , \, r_1 = 2^{3/2} \, ,
\, \textrm{ and $r_k = \max \{ 5^3 , {r_0}r_{k-1} \big\}$ for $k \geq 2$} \, .$$ Whenever $k \in {\ensuremath{\mathbb{N}}}$ and $(t,r,y) \in {\ensuremath{\mathbb{R}}}$ satisfy $N \geq k
\vee (c/3)^{-18} \vee 6^{36}$, $t \in \big[ 0 , N^{1/18} \big]$, $r \in \big[ r_k \, , \, 2N^{1/18} \big]$ and $y \in {c}/2 \cdot [- N^{1/18}, N^{1/18}]$, $${\ensuremath{\mathbb{P}}}\Big( \inf_{x \in [y-t,y+t]} \big( {\mathcal}{L}_N(k,x) + {Q}(x) \big) \leq - r \Big) \, \leq \, \Big( t \vee 5 \vee (3 - 2^{3/2})^{-1/2} r_{k-1}^{1/2} \Big)^k \cdot {E}_k \exp \big\{ - c_k r^{3/2} \big\} \, .$$
3. (No Big Max) For $\vert y \vert \leq 2^{-1} c N^{1/9}$, $r \in \big[0,4^{-1} {c}N^{1/9}\big]$, $t \in \big[ 2^{7/2} , 2 N^{1/3} \big]$ and $N \geq c^{-18}$, $${\ensuremath{\mathbb{P}}}\Big( \sup_{x \in [y-r,y+r]} \big( {\mathcal}{L}_N ( 1,x ) + 2^{-1/2}x^2 \big) \geq t \Big) \leq (r + 1) \cdot 6 {C}\exp \big\{ - 2^{-11/2} {c}t^{3/2} \big\} \, .$$
4. (Collapse near infinity) For $\eta \in (0,{c}]$, let $\ell = \ell_\eta:{\ensuremath{\mathbb{R}}}\to {\ensuremath{\mathbb{R}}}$ denote the even function which is affine on $[0,\infty)$ and has gradient $ - 5 \cdot 2^{-3/2} \eta {N}^{1/9}$ on this interval, and which satisfies $\ell(\eta {N}^{1/9}) = \big( - 2^{-1/2} + 2^{-5/2} \big) \eta^2 {N}^{2/9}$. If ${N}\geq 2^{45/4} {c}^{-9}$, then $$\begin{aligned}
& & {\ensuremath{\mathbb{P}}}\Big( {\mathcal}{L}_N \big(1,z\big) > \ell(z) \, \, \textrm{for some} \, \, z \in D \setminus \big[ - \eta {N}^{1/9} , \eta {N}^{1/9} \big] \Big) \\
& \leq &
6C \exp \Big\{ - c \eta^3 2^{-15/4} {N}^{1/3} \Big\} \, .\end{aligned}$$ The set $D$ is the spatial domain of ${\mathcal}{L}$, either $[ - {z_{\mathcal{L}}}, \infty )$ or $(-\infty,{z_{\mathcal{L}}}]$.
These four assertions are proved in [@BrownianReg]. Respectively, they appear as the following results in that article: Proposition $2.7$, Proposition $A.2$, Proposition $2.28$, and Proposition $2.30$. A few words about the meaning of the four parts of this proposition: the first part asserts a lower bound for a curve in a regular ensemble of any given index. This holds by definition when the index is $k=1$ via ${{\rm Reg}}(2)$ but is non-trivial in the other cases. The second part strengthens this conclusion to speak of the minimum value of such a curve on a compact interval. The third similarly strengthens the one-point upper tail ${{\rm Reg}}(3)$. In regard to the fourth, note that ${{\rm Reg}}(2)$ and ${{\rm Reg}}(3)$ do not assert that curves hew to the parabola $-2^{-1/2}z^2$ globally, but only in an expanding region about the origin, of width $2cN^{1/9}$ centred at the origin, where $N$ is the ensemble curve cardinality. Proposition \[p.mega\](4) offers a substitute control on curves far from the origin, showing them to decay at a rapid but nonetheless linear rate in the region beyond scale $N^{1/9}$.
These four assertions from [@BrownianReg] are all consequences of the theory of Brownian Gibbs resampling introduced in [@AiryLE] and developed in [@BrownianReg]. They are all rather simple consequences of this theory, with short proofs in [@BrownianReg]. It is in fact in the upcoming Theorem \[t.neargeod\] that we cite a result from [@BrownianReg] that harnesses the Brownian Gibbs theory from [@BrownianReg] in a substantial way.
Some further key inputs {#s.input}
=======================
The scaling principle {#s.scalingprinciple}
---------------------
Fundamental to the theory of the KPZ fixed point is the triple $(1/3,2/3,1)$ of exponents that reflects the scaling laws for weight, polymer geometry, and polymer lifetime. The triple manifests itself in the context of our use of scaled coordinates. It is a simple consequence of the definition of the scaling transformation $R_n$ and of the energy-weight relationship (\[e.weightzigzag\]) that the following useful fact holds true.
[*The scaling principle.*]{} Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ be a compatible triple. Any statement concerning the system of $n$-zigzags, including weight information, is equivalent to the corresponding statement concerning the system of $n{t_{1,2}}$-zigzags, provided that the following changes are made:
- the index $n$ is replaced by $n{t_{1,2}}$;
- any time is multiplied by ${t_{1,2}}^{-1}$;
- any weight is multiplied by ${t_{1,2}}^{1/3}$;
- and any horizontal distance is multiplied by ${t_{1,2}}^{-2/3}$.
A little more explanation of the scaling principle appears in [@ModCon Section $2.3$].
Several disjoint near polymers with shared endpoints
----------------------------------------------------
Our road map offering a route to Theorem \[t.disjtpoly.pop\] involves specifying and bounding the probability of an event ${\mathsf{NearPoly}}$. This work has been carried out in [@BrownianReg], and we here recall the needed result.
Define $${\mathsf{NearPoly}}_{n,k;(x,t_1)}^{(y,t_2)}(\eta) = \Big\{ \, {\mathsf{Wgt}}_{n,k;(x\bar{\bf 1},t_1)}^{(y\bar{\bf 1},t_2)} \geq k \cdot {\mathsf{Wgt}}_{n;(x,t_1)}^{(y,t_2)} - {t_{1,2}}^{1/3} \eta \, \Big\} \, .$$ The presence of the ${t_{1,2}}^{1/3}$ factor that multiplies $\eta$ is consistent with the scaling principle. It means that $\eta$ has the role of a measure of discrepancy from the maximum weight attainable in principle, independently of the value of ${t_{1,2}}$. This ${\mathsf{NearPoly}}$ may also be expressed using a scaled forward ensemble: namely, $${\mathsf{NearPoly}}_{n,k;(x,t_1)}^{(y,t_2)}(\eta) \, = \, \bigg\{ \,
\sum_{i=1}^k {\mathcal}{L}_{n;(x,t_1)}^{\uparrow;t_2}(i,y) \geq k \cdot {\mathcal}{L}_{n;(x,t_1)}^{\uparrow;t_2}(1,y) \, - \, {t_{1,2}}^{1/3} \eta \, \bigg\} \, .$$
The next result is perhaps the most consequential of this article’s inputs. It is [@BrownianReg Theorem $1.12$] up to a relabelling of one parameter.
\[t.neargeod\] There exist constants $K_0 \geq 1$, $K_1 > 0$, $a_0 \in (0,1)$ and $\eta_0 > 0$, and a positive sequence $\{ \beta_k: k \in {\ensuremath{\mathbb{N}}}\}$ with $\limsup \beta_k^{1/k} < \infty$, such that, for $n,k \in {\ensuremath{\mathbb{N}}}$, $x \in {\ensuremath{\mathbb{R}}}$ and $\eta \in \big(0,(\eta_0)^{k^2}\big)$ satisfying $k \geq 2$, $n \geq k \vee \, (K_0)^{k^2} \big( \log \eta^{-1} \big)^{K_0}$ and $\vert x \vert \leq a_0 n^{1/9}$, $$\eta^{k^2 - 1} \cdot \exp \big\{ - e^{K_1 k} \big\} \, \leq \, {\ensuremath{\mathbb{P}}}\Big(
{\mathsf{NearPoly}}_{n,k;(0,0)}^{(x,1)} \big( \eta \big) \Big) \, \leq \, \eta^{k^2 - 1} \cdot
\exp \Big\{ \beta_k \big( \log \eta^{-1} \big)^{5/6} \Big\} \, .$$
The actual consequence that will be used when we implement the road map is the upper bound in the result now stated.
\[c.neargeod.t\] There exist constants $K_0 \geq 1$, $K_1 > 0$, $a_0 \in (0,1)$ and $\eta_0 > 0$ and a positive sequence $\{ \beta_k: k \in {\ensuremath{\mathbb{N}}}\}$ with $\limsup \beta_k^{1/k} < \infty$ such that, for $n,k \in {\ensuremath{\mathbb{N}}}$, $(t_1,t_2) \in {\ensuremath{\mathbb{R}}}^2_<$, $x,y \in {\ensuremath{\mathbb{R}}}$ and $\eta \in \big(0,(\eta_0)^{k^2}\big)$ satisfying $k \geq 2$, ${t_{1,2}}n \geq k \vee \, (K_0)^{k^2} \big( \log \eta^{-1} \big)^{K_0}$ and ${t_{1,2}}^{-2/3} \vert y - x \vert \leq a_0 n^{1/9}$, $$\eta^{k^2 - 1} \cdot \exp \big\{ - e^{K_1 k} \big\} \, \leq \, {\ensuremath{\mathbb{P}}}\Big(
{\mathsf{NearPoly}}_{n,k;(x,t_1)}^{(y,t_2)} \big( \eta \big) \Big) \, \leq \, \eta^{k^2 - 1} \cdot
\exp \Big\{ \beta_k \big( \log \eta^{-1} \big)^{5/6} \Big\} \, .$$
[**Proof.**]{} This follows from Theorem \[t.neargeod\] by the scaling principle.
A useful tool: tail behaviour for polymer weight suprema and infima {#s.usefultool}
-------------------------------------------------------------------
Short scale rewiring of zigzags is a central part of the plan in the road map. The next definition and result will be used in order to prove that such rewiring occurs at a manageable cost.
For $x,y \in {\ensuremath{\mathbb{R}}}$, $w_1,w_2 \geq 0$ and $r > 0$, let ${{\mathsf{PolyWgtReg}}}_{n;([x,x+w_1],t_1)}^{([y,y+w_2],t_2)}( r )$ denote the [*polymer weight regularity*]{} event that, for all $(u,v) \in [0,w_1] \times [0,w_2]$, $$\Big\vert \, {t_{1,2}}^{-1/3} {\mathsf{Wgt}}_{n;(x+u,t_1)}^{(y+v,t_2)} + 2^{-1/2} {t_{1,2}}^{-4/3} \big(y+v-x - u \big)^2 \, \Big\vert \, \leq \, r \, .$$ When $w_1 = 0$, so that the interval $[x,x+w_1]$ is a singleton, we write $x$ in place of $\{ x \}$ or $[x,x]$ in using this notation.
The next result, [@ModCon Corollary $2.1$], expresses the one-third power law for polymer weight that was discussed in Section \[s.roadmap\].
\[c.maxminweight\] Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ be a compatible triple for which $n{t_{1,2}}\in {\ensuremath{\mathbb{N}}}$ is at least $10^{29} \vee 2(c/3)^{-18}$. Let $x,y \in {\ensuremath{\mathbb{R}}}$, and let $a,b \in {\ensuremath{\mathbb{N}}}$, be such that $\big\vert x - y \big\vert {t_{1,2}}^{-2/3} + \max\{ a,b\} - 1 \leq 6^{-1} {c}(n{t_{1,2}})^{1/18}$. Let $r \in \big[ 34 \, , \, 4 (n {t_{1,2}})^{1/18} \big]$. Then $${\ensuremath{\mathbb{P}}}\Big( \neg \, {{\mathsf{PolyWgtReg}}}_{n;([x,x+a {t_{1,2}}^{2/3}],t_1)}^{([y,y+b{t_{1,2}}^{2/3}],t_2)}(r) \Big) \leq
ab \cdot 400 C \exp \big\{ - c_1 2^{-10} r^{3/2} \big\} \, .$$
Local weight regularity {#s.polyweightreg}
-----------------------
Recall from the road map that a power of one-half dictates the Hölder continuity of polymer weights as the endpoints are varied horizontally. A rigorous version of this assertion has been presented in [@ModCon Theorem $1.1$], and we recall it now, again using the notation $Q:{\ensuremath{\mathbb{R}}}\to {\ensuremath{\mathbb{R}}}$, $Q(u) = 2^{-1/2} u^2$, to denote the parabola that dictates the global shape of the weight profile.
\[t.differenceweight\] Let $n \in {\ensuremath{\mathbb{N}}}$ and $x,y \in {\ensuremath{\mathbb{R}}}$ satisfy $n \geq 10^{32} c^{-18}$ and $\big\vert x - y \big\vert \leq 2^{-2} 3^{-1} {c}n^{1/18}$. Let ${\epsilon}\in (0,2^{-4}]$ and $R \in \big[10^4 \, , \, 10^3 n^{1/18} \big]$. Then $${\ensuremath{\mathbb{P}}}\left( \sup_{\begin{subarray}{c} u_1,u_2 \in [x,x+{\epsilon}] \\
v_1,v_2 \in [y,y+{\epsilon}] \end{subarray}} \Big\vert {\mathsf{Wgt}}_{n;(u_2,0)}^{(v_2,1)} + Q(v_2 - u_2) - {\mathsf{Wgt}}_{n;(u_1,0)}^{(v_1,1)} - Q(v_1 - u_1) \Big\vert \, \geq \, {\epsilon}^{1/2}
R \right)$$ is at most $10032 \, C \exp \big\{ - c_1 2^{-21} R^{3/2} \big\}$.
It is also convenient to record a version of this result in which the parabolic curvature term is absent. Let $I$ and $J$ denote two closed intervals in the real line, each of length ${\epsilon}$. Define the [*local weight regularity*]{} event $${\mathsf{LocWgtReg}}_{n;(I,0)}^{(J,1)}\big( {\epsilon}, r \big) \, = \,
\left\{ \sup_{\begin{subarray}{c} x_1,x_2 \in I \\
y_1, y_2 \in J \end{subarray}} \Big\vert {\mathsf{Wgt}}_{n;(x_2,0)}^{(y_2,1)} - {\mathsf{Wgt}}_{n;(x_1,0)}^{(y_1,1)} \Big\vert \, \leq \, r {\epsilon}^{1/2} \right\} \, .$$
Next is [@ModCon Corollary $6.3$]. The new hypothesis $\vert x - y \vert \leq {\epsilon}^{-1/2}$ controls parabolic curvature and permits comparison of polymer weights without parabolic adjustment.
\[c.ordweight\] Let $n \in {\ensuremath{\mathbb{N}}}$ and $x,y \in {\ensuremath{\mathbb{R}}}$ satisfy $n \geq 10^{32} c^{-18}$ and $\big\vert x - y \big\vert \leq {\epsilon}^{-1/2} \wedge 2^{-2} 3^{-1} {c}n^{1/18}$. Let ${\epsilon}\in (0,2^{-4}]$ and $R \in \big[2 \cdot 10^4 \, , \, 10^3 n^{1/18} \big]$. Then $$\label{e.ordweight}
{\ensuremath{\mathbb{P}}}\, \Big( \, \neg \, {\mathsf{LocWgtReg}}_{n;([x,x+{\epsilon}],0)}^{([y,y+{\epsilon}],1)}\big( {\epsilon}, R \big) \, \Big) \, \leq \,
10032 \, C \exp \big\{ - c_1 2^{-22 - 1/2} R^{3/2} \big\}
\, .$$
Polymer ordering {#s.polyorder}
----------------
An important challenge that lies ahead is bouquet construction, and a key difficulty here will be to ensure the disjointness, except at the shared endpoint, of the concerned zigzags. Some natural monotonicity properties of multi-polymers will be needed in the proof. Lemma \[l.tworelations\] is our result in this regard. It will be proved in Appendix \[s.mgo\].
Let $(n,t_1,t_2)$ be a compatible triple. We introduce two ordering relations, $\prec$ and $\preceq$, on the space of $n$-zigzags with lifetime $[t_1,t_2]$. To define the relations, let $(x_1,x_2),(y_1,y_2) \in {\ensuremath{\mathbb{R}}}^2$ and consider a zigzag $Z_1$ from $(x_1,t_1)$ to $(y_1,t_2)$ and another $Z_2$ from $(x_2,t_1)$ to $(y_2,t_2)$.
[*The $\prec$ relation.*]{} We declare that $Z_1 \prec Z_2$ if $x_1 \leq y_1$, $x_2 \leq y_2$, and the two polymers are horizontally separate.
[*The $\preceq$ relation.*]{} Consider again $Z_1$ and $Z_2$. We declare that $Z_1 \preceq Z_2$ if ‘$Z_2$ lies on or to the right of $Z_1$’: formally, if $Z_2$ is contained in the union of the closed horizontal planar line segments whose left endpoints lie in $Z_1$.
\[l.tworelations\] Let $(n,t_1,t_2)$ be a compatible triple.
1. Let $(x_1,x_2),(y_1,y_2) \in {\ensuremath{\mathbb{R}}}_\leq^2$ and $k \in {\ensuremath{\mathbb{N}}}$. Then the multi-polymer components satisfy $${\ensuremath{\mathbb{P}}}\Big( \, \rho_{n,k,i;(x_1 \bar{\bf 1)},t_1}^{(y_1 \bar{\bf 1},t_2)} \preceq \rho_{n,k,i;(x_2 \bar{\bf 1)},t_1}^{(y_2 \bar{\bf 1},t_2)} \, \, \forall \, \, i \in {\llbracket 1,k \rrbracket} \, \Big) \, = \, 1 \, .$$
2. Let $Z_1$, $Z_2$ and $Z_3$ be $n$-zigzags of lifetime $[t_1,t_2]$ that verify $Z_1 \prec Z_2$ and $Z_2 \preceq Z_3$. Then $Z_1 \prec Z_3$.
A rather simple sandwiching fact about polymers will also be needed.
\[l.sandwich\] Let $(n,t_1,t_2)$ be a compatible triple, and let $(x_1,x_2),(y_1,y_2) \in {\ensuremath{\mathbb{R}}}_\leq^2$. Suppose that there is a unique $n$-polymer from $(x_i,t_1)$ to $(y_i,t_2)$, both when $i=1$ and $i=2$. (This circumstance occurs almost surely, and the resulting polymers have been labelled $\rho_{n;(x_1,t_1)}^{(y_1,t_2)}$ and $\rho_{n;(x_2,t_1)}^{(y_2,t_2)}$.) Now let $\rho$ denote any $n$-polymer that begins in $[x_1,x_2] \times \{ t_1\}$ and ends in $[y_1,y_2] \times \{ t_2 \}$. Then $\rho_{n;(x_1,t_1)}^{(y_1,t_2)} \preceq \rho \preceq \rho_{n;(x_2,t_1)}^{(y_2,t_2)}$.
[**Proof.**]{} When two $n$-zigzags $Z_1$ and $Z_2$ share their starting and ending heights, it is easy enough to define associated minimum and maximum zigzags $Z_1 \wedge Z_2$ and $Z_1 \vee Z_2$. The minimum begins and ends at the leftmost of the starting and ending points of $Z_1$ and $Z_2$; during its lifetime, it follows one or other of the two zigzags, always keeping as far to the left as possible. It is a similar story for the maximum. A formal definition is made for staircases (but this hardly changes for zigzags) in the opening paragraphs of Appendix \[s.mgo\]. The polymer $\rho$ begins and ends its journey to the right of $\rho_{n;(x_1,t_1)}^{(y_1,t_2)}$. If the condition $\rho_{n;(x_1,t_1)}^{(y_1,t_2)} \preceq \rho$ is to be violated, $\rho$ must pass a sojourn in its lifetime to the left of the other polymer. This would make the minimum $\rho \wedge \rho_{n;(x_1,t_1)}^{(y_1,t_2)}$ distinct from $\rho_{n;(x_1,t_1)}^{(y_1,t_2)}$. A moment’s thought shows, however, that this minimum is itself a polymer between $(x_1,t_1)$ and $(y_1,t_2)$. This is a contradiction to polymer uniqueness for these endpoints, which is known as a direct consequence of Lemma \[l.severalpolyunique\] with $\ell = 1$. Thus, $\rho_{n;(x_1,t_1)}^{(y_1,t_2)} \preceq \rho$. The second claimed ordering has a similar derivation.
Operations on polymers: splitting and concatenation {#s.split}
---------------------------------------------------
Two very natural operations are now discussed.
A polymer may be split into two pieces. Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_\leq$ is a compatible triple, and let $(x,y) \in {\ensuremath{\mathbb{R}}}^2$ satisfy $y \geq x - 2^{-1} n^{1/3} {t_{1,2}}$. Let $t \in (t_1,t_2) \cap n^{-1} {\ensuremath{\mathbb{Z}}}$. Suppose that the almost sure event that $\rho_{n;(x,t_1)}^{(y,t_2)}$ is well defined occurs. Select any element $(z,t) \in \rho_{n;(x,t_1)}^{(y,t_2)}$. The removal of $(z,t)$ from $\rho_{n;(x,t_1)}^{(y,t_2)}$ creates two connected components. Taking the closure of each of these amounts to adding the point $(z,t)$ to each of them. The resulting sets are $n$-zigzags from $(x,t_1)$ to $(z,t)$, and from $(z,t)$ to $(y,t_2)$; indeed, it is straightforward to see that these are the unique $n$-polymers given their endpoints. We use a concatenation notation $\circ$ to represent this splitting. In summary, $\rho_{n;(x,t_1)}^{(y,t_2)} = \rho_{n;(x,t_1)}^{(z,t)} \circ \rho_{n;(z,t)}^{(y,t_2)}$. Naturally, we also have ${\mathsf{Wgt}}_{n;(x,t_1)}^{(y,t_2)} = {\mathsf{Wgt}}_{n;(x,t_1)}^{(z,t)} + {\mathsf{Wgt}}_{n;(z,t)}^{(y,t_2)}$. Indeed, the concatenation operation may be applied to any two $n$-zigzags for which the ending point of the first equals the starting point of the second. Since the zigzags are subsets of ${\ensuremath{\mathbb{R}}}^2$, it is nothing other than the operation of union. The weight is additive under the operation.
A closure property for a space of several disjoint polymers {#s.closure}
-----------------------------------------------------------
A rather technical point about stability under closure of systems of disjoint polymers is now addressed.
Let $(n,t_1,t_2)$ be a compatible triple, let $I$ and $J$ be two closed real intervals, and let $k \in {\ensuremath{\mathbb{N}}}$. We now define two random subsets of $I^k \times J^k \subset {\ensuremath{\mathbb{R}}}^{2k}$. Set ${\mathsf{DisjtIndex}}_{n,k;(I,t_1)}^{(J,t_2)}$ to be the collection of vectors $\big( x_1,x_2,\cdots,x_k,y_1,y_2,\cdots,y_k \big) \in I^k \times J^k$ such that there exists a collection of $k$ pairwise disjoint $n$-polymers which consecutively move from $(x_i,t_1)$ to $(y_i,t_2)$. This set is non-empty precisely when ${\mathrm{MaxDisjtPoly}}_{n;(I,t_1)}^{(J,t_2)} \geq k$. We further define ${\mathsf{HorSepIndex}}_{n,k;(I,t_1)}^{(J,t_2)}$, by replacing the condition of pairwise disjoint by that of pairwise horizontally separate in the above.
\[l.disjthorsep\] The closure of ${\mathsf{DisjtIndex}}_{n,k;(I,t_1)}^{(J,t_2)}$, when viewed as a subset of ${\ensuremath{\mathbb{R}}}^{2k}$, is contained in ${\mathsf{HorSepIndex}}_{n,k;(I,t_1)}^{(J,t_2)}$.
[**Proof.**]{} Let $\big( \bar{x}^i, \bar{y}^i \big)$, $i \in {\ensuremath{\mathbb{N}}}$, be elements of ${\mathsf{DisjtIndex}}_{n,k;(I,t_1)}^{(J,t_2)}$ that converge to $\big(\bar{x}, \bar{y} \big) \in {\ensuremath{\mathbb{R}}}^k \times {\ensuremath{\mathbb{R}}}^k$. Associated to index $i$ is a system of disjoint polymers from $\big(\bar{x},t_1\big)$ to $\big(\bar{y},t_2\big)$. It is a simple matter to extract a subsequence of these indices such that all the endpoint locations of the horizontal intervals in each of the $k$ polymers converge pointwise. By considering the $k$ $n$-zigzags with these locations given by the limiting values, we may note that we are dealing with a collection of polymers, due to the continuity of the underlying Brownian motions $B(\ell,\cdot)$, $\ell \in {\ensuremath{\mathbb{Z}}}$, in Brownian LPP. Moreover, these $k$ $n$-zigzags are pairwise horizontally separate: indeed, we are equipping the space of $n$-zigzags of lifetime $[t_1,t_2]$ with the Hausdorff topology, so that it is enough to note that the property of a $k$-tuple of zigzags having a pair that is not horizontally separate is an open condition for the $k$-wise product topology on such tuples of zigzags.
[*Remark.*]{} Theorems \[t.disjtpoly.pop\] and \[t.maxpoly.pop\] make assertions about the random variable ${\mathrm{MaxDisjtPoly}}_{n;(I,t_1)}^{(J,t_2)}$. The value of this random variable cannot decrease if in its definition in Subsection \[s.maxpoly\] we replace the italicized [*pairwise disjoint*]{} by [*pairwise horizontally separate*]{}. If we were to redefine the random variable with this change, then the new versions of the results would imply the old ones. In fact, all proofs in this article are valid for the altered definition, and we take the liberty of adopting it henceforth.
Organization of the remainder of the paper {#s.organization}
------------------------------------------
Our principal results are the disjoint polymer estimates Theorems \[t.disjtpoly.pop\] and \[t.maxpoly.pop\], and the polymer fluctuation bound Theorem \[t.polyfluc\]. A road map has been outlined for the proof of Theorem \[t.disjtpoly.pop\], and, in order to communicate this conceptually central idea with suitable emphasis, the next section, Section \[s.rarity\], is devoted to rigorously formulating it. We mention here that, in fact, this proof will invoke Theorem \[t.polyfluc\]: this aspect of the road map has yet to be explained, but the resolution of the third challenge in the road map will involve polymer fluctuation bounds. Theorem \[t.maxpoly.pop\] is a rather straightforward consequence of Theorem \[t.disjtpoly.pop\], and its proof appears at the end of Section \[s.rarity\].
The proof of Theorem \[t.polyfluc\] has a rather different flavour. We present the proof next, in Section \[s.polyfluc\]. An outline of the proof is offered at the beginning of the section.
The paper ends with four appendices. Appendix \[s.glossary\] recalls in a list some of the article’s principal notation. Appendices \[s.mgo\] and \[s.normal\] present proofs for two remaining tools that we have described.
The multi-polymer ordering Lemma \[l.tworelations\] is proved in Appendix \[s.mgo\]. In our presentation of results, and in the overall perspective of this article, we have been eager to focus attention on scaled coordinates. But of course Lemma \[l.tworelations\] is really simply the scaled counterpart to a result that concerns multi-geodesics. This Lemma \[l.severalpolyorder\] appears late in the paper, in accordance with our wish to encourage the reader to focus on the scaled coordinate interpretation of the subject. It is, however, a basic result that may have an independent interest.
Proposition \[p.scaledreg\], which asserts that our normalized ensembles (\[e.forward\]) and (\[e.backward\]) are regular, has in essence been proved already in [@BrownianReg], on the basis of a fundamental result of O’Connell and Yor describing a counterpart ensemble of curves in unscaled coordinates. Appendix \[s.normal\] explains this connection and provides the proof of this proposition, which involves a mundane argument to reconcile a slight difference in ensemble notation between [@BrownianReg] and the present work.
We close this section by discussing two conventions that will govern the presentation of upcoming proofs.
### Boldface notation for quoted results
During the upcoming proofs, we will naturally be making use of the various tools that we have recalled: the ${{\rm Reg}}$ conditions and the regular ensemble properties Proposition \[p.mega\], and results presented earlier in this section. The statements of such results involve several parameters, in several cases including $(n,t_1,t_2)$, spatial locations $x$ and $y$, and positive real parameters such as $r$. We will employ a device that will permit us to disregard notational conflict between the use of such parameters in the contexts of the ongoing proof in question and the statements of quoted results. When specifying the parameter settings of a particular application, we will allude to the parameters of the result being applied in boldface, and thus permit the reuse of the concerned symbols.
### The role of hypotheses invoked during proofs {#s.rolehyp}
When we quote results in order to apply them, we will take care, in addition to specifying the parameters according to the just described convention, to indicate explicitly what the conditions on these parameters are that will permit the quoted result in question to be applied. Of course, it is necessary that the hypotheses of the result being proved imply all such conditions. The task of verifying that the hypotheses of a given result are adequate for the purpose of obtaining all conditions needed to invoke the various results used during its proof may be called the calculational derivation of the result in question. This derivation is necessary, but also in some cases lengthy and unenlightening: a succession of trivial steps. We have chosen to separate the calculational derivations of most of our results from their proofs. These derivations may be found in Appendix $D$ at the end of the version <http://math.berkeley.edu/~alanmh/papers/NonIntPolymer.pdf> of the paper on the author’s webpage; the latex source code for this version is an ancillary file to the present arXiv submission. Appendix $D$ also contains the proofs of four lemmas that are invoked in the proof of Theorem \[t.polyfluc\]. These proofs are exclusively calculational matters.
Rarity of many disjoint polymers: proofs of Theorems \[t.disjtpoly.pop\] and \[t.maxpoly.pop\] {#s.rarity}
==============================================================================================
Theorems \[t.disjtpoly.pop\] and \[t.maxpoly.pop\] are slightly simplified versions of the next two results, in which hypotheses on parameters are stated in a more explicit form. The results make reference to $\eta_0$, $K_0$ and $a_0$. These are positive constants that are fixed by Theorem \[t.neargeod\]. The sequence of positive constants $\big\{ \beta_i: i \in {\ensuremath{\mathbb{N}}}\big\}$, which verifies $\limsup_{i \in {\ensuremath{\mathbb{N}}}} \beta_i^{1/i} < \infty$, is also supplied by this theorem.
\[t.disjtpoly\] Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ be a compatible triple. Let ${k}\in {\ensuremath{\mathbb{N}}}$, ${\epsilon}> 0$ and $x,y \in {\ensuremath{\mathbb{R}}}$ satisfy the conditions that ${k}\geq 2$, $$\label{e.epsilonbound}
{\epsilon}\leq \min \Big\{ \, (\eta_0)^{4{k}^2} \, , \, 10^{-616} c_{k}^{22} {k}^{-115} \, , \, \exp \big\{ - C^{3/8} \big\} \, \Big\} \, ,$$ $$\label{e.nlowerbound}
n {t_{1,2}}\geq \max \bigg\{ \, 2(K_0)^{{k}^2} \big( \log {\epsilon}^{-1} \big)^{K_0}
\, , \, 10^{606} c_{k}^{-48} {k}^{240} {c}^{-36} {\epsilon}^{-222} \max \big\{ 1 \, , \, \vert x - y \vert^{36} {t_{1,2}}^{-24} \big\} \, , \,
a_0^{-9} \vert y - x \vert^9 {t_{1,2}}^{-6} \, \Bigg\} \, ,$$ as well as $\vert y - x \vert {t_{1,2}}^{-2/3} \leq {\epsilon}^{-1/2} \big( \log {\epsilon}^{-1} \big)^{-2/3} \cdot 10^{-8} c_{k}^{2/3} {k}^{-10/3}$. Then $$\begin{aligned}
& & {\ensuremath{\mathbb{P}}}\bigg( {\mathrm{MaxDisjtPoly}}_{n;\big([x-{t_{1,2}}^{2/3}{\epsilon},x+{t_{1,2}}^{2/3}{\epsilon}],t_1\big)}^{\big([y-{t_{1,2}}^{2/3}{\epsilon},y+{t_{1,2}}^{2/3}{\epsilon}],t_2\big)} \geq {k}\bigg) \\
& \leq &
{\epsilon}^{({k}^2 - 1)/2} \cdot 10^{32{k}^2} {k}^{15{k}^2} c_{k}^{-3{k}^2} C_{k}\big( \log {\epsilon}^{-1} \big)^{4{k}^2} \exp \big\{ \beta_{k}\big( \log {\epsilon}^{-1} \big)^{5/6} \big\} \, .\end{aligned}$$
\[t.maxpoly\] There exists a positive constant ${m}_0$, and a sequence of positive constants $\big\{ {H}_i: i \in {\ensuremath{\mathbb{N}}}\big\}$ for which $\sup_{i \in {\ensuremath{\mathbb{N}}}} {H}_i \exp \big\{ - 2 (\log i)^{11/12} \big\}$ is finite, such that the following holds. Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ be a compatible triple. Let $x,y \in {\ensuremath{\mathbb{R}}}$, $a,b \in {\ensuremath{\mathbb{N}}}$ and ${m}\in {\ensuremath{\mathbb{N}}}$. Write $h = a \vee b$. Suppose that $${m}\geq {m}_0 \vee \big( \vert x - y \vert {t_{1,2}}^{-2/3} + 2h \big)^3$$ and $$\begin{aligned}
n {t_{1,2}}& \geq & \max \bigg\{ \, 2(K_0)^{(12)^{-2} (\log \log {m})^2} \big( \log {m}\big)^{K_0}
\, , \, a_0^{-9} \big( \vert y - x \vert {t_{1,2}}^{-2/3} + 2h \big)^9 \, ,
\label{e.nmaxpoly} \\
& & \qquad \qquad
10^{325} {c}^{-36} {m}^{465} \max \big\{ 1 \, , \, \big(\vert y - x \vert {t_{1,2}}^{-2/3} + 2h \big)^{36} \big\}
\, \bigg\} \, . \nonumber\end{aligned}$$
Then $${\ensuremath{\mathbb{P}}}\bigg( {\mathrm{MaxDisjtPoly}}_{n;\big([x,x+a{t_{1,2}}^{2/3}],t_1\big)}^{\big([y,y+b {t_{1,2}}^{2/3}],t_2\big)} \geq {m}\bigg) \, \leq \, {m}^{- (145)^{-1} ( \log \beta)^{-2} (0 \vee \log \log {m})^2} \cdot h^{(\log \beta)^{-2} (0 \vee \log \log {m})^2/{288} + 3/2} {H}_{m}\, .$$ Here, $\beta$ is specified to be $e \vee \limsup_{i \in {\ensuremath{\mathbb{N}}}} \beta_i^{1/i}$.
[**Proof of Theorems \[t.disjtpoly.pop\] and \[t.maxpoly.pop\].**]{} These results are direct consequences of their just stated counterparts.
In this section, then, our job is to prove Theorems \[t.disjtpoly\] and \[t.maxpoly\]. Our principal task is to prove Theorem \[t.disjtpoly\]. This we do first, by implementing the road map rigorously. Theorem \[t.maxpoly\] then emerges, at the end of Section \[s.rarity\], as a fairly straightforward consequence: after all, many disjoint polymers running between unit intervals entail several disjoint polymers running between short intervals.
Bouquet construction {#s.bouquet}
--------------------
As we noted at the end of Section \[s.roadmap\], there are three main challenges involved in implementing the road map, beyond the ${\mathsf{NearPoly}}$ probability estimate Corollary \[c.neargeod.t\]. The technical input concerning the first of the three challenges, polymer weight similarity, has been cited already, as Corollary \[c.maxminweight\].
We now address the second challenge: the construction of forward and backward bouquets of suitable weight, dictated by the exponent of one-third.
In Section \[s.zigzagmax\], we introduced the maximum weight ${\mathsf{Wgt}}_{n,k;(\bar{x},t_1)}^{(\bar{y},t_2)}$ of a system of $k$ pairwise horizontally separate zigzags that make the journey from $(x_i,t_1)$ to $(y_i,t_2)$ for $i \in {\llbracket 1,k \rrbracket}$. In forward bouquet construction, we specialise to $\bar{x} = x \bar{\bf 1}$ for some $x \in {\ensuremath{\mathbb{R}}}$; in backward bouquet construction, we instead choose $\bar{y} = y \bar{\bf 1}$ for a given $y \in {\ensuremath{\mathbb{R}}}$.
The aim of bouquet construction will be achieved in the guise of Corollary \[c.bouquetreg\], which shows that the cumulative weight of the maximum weight bouquet, when measured in the suitable one-third power units, is tight, uniformly in the scaling parameter $n$. This result is a direct consequence of Proposition \[p.sumweight\], which asserts an upper and a lower bound on the maximum bouquet weight. The first of these bounds is easy, because polymer weights offer a control from above on bouquet weights. The lower bound is more delicate: horizontal separateness of zigzags must be ensured without damaging weight significantly. Our technique will be a diagonal argument, with the ordering Lemma \[l.tworelations\] playing an important role in establishing the necessary form of disjointness.
\[p.sumweight\] Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ be a compatible triple. Let $k \in {\ensuremath{\mathbb{N}}}$, $\bar{u} \in {\ensuremath{\mathbb{R}}}^k_{\leq}$ and $r > 0$.
1. Suppose that $n {t_{1,2}}$ is at least ${t_{1,2}}^{-6} \big\vert u_i - x \big\vert^9 {c}^{-9}$ for $i \in {\llbracket 1,k \rrbracket}$, and that $r \geq k$. Then $${\ensuremath{\mathbb{P}}}\bigg( \, {t_{1,2}}^{-1/3} \cdot {\mathsf{Wgt}}_{n,k;(x \bar{\bf 1},t_1)}^{(\bar{u},t_2)} \geq - \, 2^{-1/2} \sum_{i=1}^k {t_{1,2}}^{-4/3} \big( u_i - x \big)^2 \, \, + \, r \, \bigg) \, \leq \, k \cdot C \exp \Big\{ - c k^{-3/2} r^{3/2} \Big\} \, .$$
2. Suppose that $n {t_{1,2}}$ is at least $\max \big\{ k \, , \, 3^{18} c^{-18} \, , \, 6^{36} \big\}$ and is also bounded below for each $i \in {\llbracket 1,k \rrbracket}$ by $2^{18} {c}^{-18} {t_{1,2}}^{-12} \big\vert u_i - x \big\vert^{18}$. Suppose further that $r \in \big[ 4k^2 , 9 k^2 (n{t_{1,2}})^{1/3} \big]$. Then $${\ensuremath{\mathbb{P}}}\bigg( {t_{1,2}}^{-1/3} \cdot {\mathsf{Wgt}}_{n,k;(x \bar{\bf 1},t_1)}^{(\bar{u},t_2)} \leq - \, 2^{-1/2} \sum_{i=1}^k {t_{1,2}}^{-4/3} \big( u_i - x \big)^2 \, \, - \, r \bigg) \, \leq
\,
3 k^2
C_k \exp \Big\{ - 2^{-3} k^{-3} c_k r^{3/2} \Big\} \, .$$
As we prepare to state our bouquet construction tool, Corollary \[c.bouquetreg\], we mention a further aspect of this construction. It would be natural to suppose, on the basis of the ideas presented in Section \[s.roadmap\], that, on the event ${\mathrm{MaxDisjtPoly}}_{(n;[x-{\epsilon},x+{\epsilon}],0)}^{([y-{\epsilon},y+{\epsilon}],1)} \geq k$, if the $k$ disjoint polymers that move between $[x-{\epsilon},x+{\epsilon}] \times \{ 0 \}$ and $[y-{\epsilon},y+{\epsilon}] \times \{ 1 \}$ have endpoints $(u_i,0)$ and $(v_i,1)$, for $i \in {\llbracket 1,k \rrbracket}$, then the forward and backward bouquets will be chosen to be the multi-polymers whose weights are ${\mathsf{Wgt}}_{n,k;(x \bar{\bf 1},-{\epsilon}^{3/2})}^{(\bar{u},0)}$ and ${\mathsf{Wgt}}_{n,k;(\bar{v},1)}^{(y \bar{\bf 1},1 + {\epsilon}^{3/2})}$. A microscopic detail must be addressed, however. The first of these multi-polymers is measurable with respect to the randomness in the region ${\ensuremath{\mathbb{R}}}\times (-\infty,0]$ and the second to the randomness in ${\ensuremath{\mathbb{R}}}\times [1,\infty)$. Our purpose in the upcoming surgery will be better served were a choice of the bouquet pair to be made for which this assertion is valid with the two [*open*]{} regions ${\ensuremath{\mathbb{R}}}\times (-\infty,0)$ and ${\ensuremath{\mathbb{R}}}\times (1,\infty)$ instead. What is needed is a modified definition in which no use is made of the randomness in the horizontal line with the height of the fixed endpoint: height zero or height one for the forward and backward bouquets.
In the modification, the weight function ${\mathsf{Wgt}}$ will be replaced by a proper weight function ${\mathsf{PropWgt}}$, in a sense we now specify. Recall that if $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ is a compatible triple and $Z$ is an $n$-zigzag with starting point $(x,t_1)$ and ending point $(y,t_2)$, then $Z$ begins with a planar line segment abutting $(x,t_1)$, which is either horizontal or sloping, and likewise ends with such a segment abutting $(y,t_2)$. We will call $Z$ [*backward proper*]{} if the line segment that abuts $(x,t_1)$ is sloping, and [*forward proper*]{} if the line segments that abuts $(y,t_2)$ is sloping.
We now define the [*proper*]{} weight ${\mathsf{PropWgt}}_{n,k;(x \bar{\bf 1},t_1)}^{(\bar{u},t_2)}$ to equal the maximum weight associated to $k$ horizontally separate [*forward proper*]{} zigzags moving consecutively between $(x,t_1)$ to $(u_i,t_2)$ for $i \in {\llbracket 1,k \rrbracket}$. The $k$-tuple maximizer will be denoted by $\rho_{n,k;(x\bar{\bf 1},t_1)}^{{\rm prop};(\bar{u},t_2)}$. (Almost sure uniqueness of the maximizer follows in essence from Lemma \[l.severalpolyunique\] but is not needed for our purpose.) The proper weight definition modifies ${\mathsf{Wgt}}_{n,k;(x \bar{\bf 1},t_1)}^{(\bar{u},t_2)}$ by introducing the insistence that the concerned zigzags be forward proper. Similarly, we define ${\mathsf{PropWgt}}_{n,k;(\bar{v},t_1)}^{(y \bar{\bf 1},t_2)}$ to equal the maximum weight associated to $k$ horizontally separate [*backward proper*]{} zigzags moving consecutively between $(v_i,t_1)$ to $(y,t_2)$ for $i \in {\llbracket 1,k \rrbracket}$. The $k$-tuple maximizer is $\rho_{n,k;(\bar{v},t_1)}^{{\rm prop};(y \bar{\bf 1},t_2)}$.
Define the [*forward bouquet regularity*]{} event ${\mathsf{ForBouqReg}}_{n,k;(x,t_1)}^{(\bar{u},t_2)}( r )$ to equal $$\bigg\{ \, \bigg\vert \, ({t_{1,2}}- n^{-1})^{-1/3} \cdot {\mathsf{PropWgt}}_{n,k;(x \bar{\bf 1},t_1)}^{(\bar{u},t_2)} \, + \, 2^{-1/2} \sum_{i=1}^k ({t_{1,2}}- n^{-1})^{-4/3} \big( u_i + 2^{-1} n^{-2/3} - x \big)^2 \, \bigg\vert \, \leq \, r \bigg\}$$ and the [*backward bouquet regularity*]{} event ${\mathsf{BackBouqReg}}_{n,k;(\bar{v},t_1)}^{(y,t_2)}( r )$ to equal $$\bigg\{ \, \bigg\vert \, ({t_{1,2}}- n^{-1})^{-1/3} \cdot {\mathsf{PropWgt}}_{n,k;(\bar{v},t_1)}^{(y \bar{\bf 1},t_2)} \, + \, 2^{-1/2} \sum_{i=1}^k ({t_{1,2}}- n^{-1})^{-4/3} \big( v_i - 2^{-1} n^{-2/3} - y \big)^2 \, \bigg\vert \, \leq \, r \bigg\} \, .$$
\[c.bouquetreg\] Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ be a compatible triple. Let $k \in {\ensuremath{\mathbb{N}}}$ satisfy $k \leq n$, and let $x,y \in {\ensuremath{\mathbb{R}}}$, $\bar{u},\bar{v} \in {\ensuremath{\mathbb{R}}}^k_{\leq}$ and $r > 0$. Suppose that $$\begin{aligned}
n {t_{1,2}}- 1 & \geq & \max \Big\{ k \, , \, 3^{18} c^{-18} \, , \, 6^{36} \, , \, ({t_{1,2}}- n^{-1})^{-6} \big( \vert u_i + 2^{-1} n^{-2/3} - x \vert \vee \vert v_i - 2^{-1} n^{-2/3} - y \vert \big)^9 {c}^{-9} \, , \\
& & \qquad \qquad \qquad \qquad 2^{18} {c}^{-18} ({t_{1,2}}- n^{-1})^{-12} \big( \vert u_i + 2^{-1} n^{-2/3} - x \vert \vee \vert v_i - 2^{-1} n^{-2/3} - y \vert \big)^{18} \Big\}\end{aligned}$$ where here $i$ varies over ${\llbracket 1,k \rrbracket}$. Suppose also that $r \in \big[ 4k^2 , 9 k^2 ({t_{1,2}}- n^{-1})^{1/3} n^{1/3} \big]$. Then $${\ensuremath{\mathbb{P}}}\Big( \neg \, {\mathsf{ForBouqReg}}_{n,k;(x,t_1)}^{(\bar{u},t_2)} \, \Big) \vee {\ensuremath{\mathbb{P}}}\Big( \, \neg \, {\mathsf{BackBouqReg}}_{n,k;(\bar{v},t_1)}^{(y,t_2)} \Big) \leq
4 k^2
C_k \exp \Big\{ - 2^{-3} k^{-3} c_k r^{3/2} \Big\} \, .$$
The form of the corollary is a little cluttered, not least because the microscopic detail we have introduced forces the bouquets to have lifetime ${t_{1,2}}- n^{-1}$ rather than ${t_{1,2}}$. Setting $n = \infty$ may serve to focus attention on what is essential here, even if it obscures this detail.
[**Proof of Corollary \[c.bouquetreg\].**]{} We begin by claiming that $$\label{e.properweightforward}
{\mathsf{PropWgt}}_{n,k;(x \bar{\bf 1},t_1)}^{(\bar{u},t_2)} = {\mathsf{Wgt}}_{n,k;(x \bar{\bf 1},t_1)}^{(\bar{u} + 2^{-1}n^{-2/3}{\bf 1} , t_2 - n^{-1})} \, .$$ To understand why this is true, we consider the multi-polymer that realizes the left-hand side and apply the inverse scaling map $R_n^{-1}$, so that it is depicted in unscaled coordinates. The result is a $k$-tuple of staircases, each element of which ends in a vertical unit displacement. Omitting these displacements, we obtain a $k$-tuple of staircases which is a multi-geodesic: the sum of the energies of the staircases is maximal given the set of endpoints. Applying the scaling map $R_n$ to reconsider this multi-geodesic in scaled coordinates, we see that its weight is the above right-hand side. Similar considerations yield the formula $$\label{e.properweightbackward}
{\mathsf{PropWgt}}_{n,k;(\bar{v},t_1)}^{(y \bar{\bf 1},t_2)} = {\mathsf{Wgt}}_{n,k;(\bar{v} - 2^{-1}n^{-2/3} \bar{\bf 1} , t_1 + n^{-1})}^{(y \bar{\bf 1},t_2)} \, .$$
The bound on the first probability in the corollary follows from (\[e.properweightforward\]) and an application of the two parts of Proposition \[p.sumweight\] with parameter settings ${\bf t_1} = t_1 + n^{-1}$, ${\bf t_2} = t_2$, ${\bf x} = x$ and ${\bf \bar{u}} = \bar{u} + 2^{-1}n^{-2/3}\bar{\bf 1}$. The bounds $C_k \geq C$ and $c_k \leq c$ are also used. The estimate on the second probability is reduced to this proposition via (\[e.properweightbackward\]) and by noting the half-circle rotational symmetry of Brownian LPP that arises by reindexing its constituent curves $B(k,z)$ in the form $B(-k,-z)$.
[**Proof of Proposition \[p.sumweight\].**]{} The proposition’s first assertion is the simpler of the two to derive. To verify it, note that, since $\rho_{n,k,i ; (x \bar{\bf 1},t_1)}^{(\bar{u},t_2)}$ has starting and ending points $(x,{t_1})$ and $(u_i,{t_2})$, the weight of this $n$-zigzag is at most ${\mathsf{Wgt}}_{n;(x,t_1)}^{(u_i,t_2)} $. Summing over $i \in {\llbracket 1,k \rrbracket}$, we find that $$\label{e.swub}
{\mathsf{Wgt}}_{n,k;(x \bar{\bf 1},t_1)}^{(\bar{u},t_2)}
\leq \sum_{i=1}^k
{\mathsf{Wgt}}_{n;(x,t_1)}^{(u_i,t_2)} \, .$$ Set $w_i = {t_{1,2}}^{-2/3} \big( u_i - x \big)$ and note that $${\mathsf{Wgt}}_{n;(x,t_1)}^{(u_i,t_2)} = {\mathcal}{L}^{\uparrow,t_2}_{n;(x,t_1)}\big( 1 , u_i \big) = {t_{1,2}}^{1/3} \cdot {t_{1,2}}^{-1/3} {\mathcal}{L}^{\uparrow,t_2}_{n;(x,t_1)} \big( 1, x + {t_{1,2}}^{2/3} w_i \big) = {t_{1,2}}^{1/3} \, {\mathsf{Nr}{\mathcal}{L}}^{\uparrow,t_2}_{n;(x,t_1)}(1,w_i) \, .$$ Applying [*one-point upper tail*]{} [[Reg]{}]{}(3) with parameter settings ${\bf z} = w_i$ and ${\bf s} = r k^{-1}$ to the $(n{t_{1,2}}+ 1)$-curve ensemble ${\mathsf{Nr}{\mathcal}{L}}^{\uparrow,t_2}_{n;(x,t_1)}$, we see that the event $$\Big\{ {t_{1,2}}^{-1/3} \cdot {\mathsf{Wgt}}_{n;(x,t_1)}^{(u_i,t_2)} \geq - 2^{-1/2} {t_{1,2}}^{-4/3} \big( u_i - x \big)^2 + r/k \Big\} = \Big\{ {\mathsf{Nr}{\mathcal}{L}}^{\uparrow,t_2}_{n;(x,t_1)}(1,w_i) + 2^{-1/2} w_i^2 \geq r/k \Big\}$$ has ${\ensuremath{\mathbb{P}}}$-probability at most $C \exp \big\{ - c (r/k)^{3/2} \big\}$, provided that the hypotheses $${t_{1,2}}^{-2/3} \big\vert u_i - x \big\vert \leq {c}(n {t_{1,2}})^{1/9} \, \, \textrm{and} \, \, r \geq k$$ are satisfied. Thus, (\[e.swub\]) implies Proposition \[p.sumweight\](1).
To derive the lower-tail bound Proposition \[p.sumweight\](2), we will find $k$ pairwise horizontally separate $n$-zigzags of suitable weight, each of which begins at $(x,{t_1})$ and which end successively at $(u_i,{t_2})$, $i \in {\llbracket 1,k \rrbracket}$.
![The diagonal vector $\big( \rho_{1,1}, \cdots, \rho_{k,k} \big)$ is illustrated in bold for an example with $k=3$. Under the microscope, we see how it is that these three zigzags may remain horizontally separate despite their sharing the birthplace $(x,t_1)$.[]{data-label="f.diag"}](NonIntPolyDiag.pdf){height="9cm"}
Our technique for finding the zigzags is a diagonal argument: see Figure \[f.diag\]. For each $i \in {\llbracket 1,k \rrbracket}$, consider the $k$-tuple multi-polymer watermelon $$\label{e.maximizer}
\bigg( \rho_{n,k,i;(x \bar{\bf{1}} ,t_1)}^{\big(u_i \bar{\bf{1}} , t_2 \big)}: i \in {\llbracket 1,k \rrbracket} \bigg) \, ,
$$ where here recall that $z \bar{\bf{1}} \in {\ensuremath{\mathbb{R}}}^k$ denotes the vector each of whose components equals $z \in {\ensuremath{\mathbb{R}}}$. Our multi-polymer notation used here is specified in Section \[s.multipolymer\]. The object in question is the multi-polymer whose $k$ components are tethered at both endpoints, to $(x,t_1)$ and $(u_i,t_2)$. As we explained when the notation was introduced, it is Lemma \[l.severalpolyunique\] that ensures the almost sure existence and uniqueness of the maximizer (\[e.maximizer\]), which for brevity we will denote by $\big( \rho_{i,1}, \cdots, \rho_{i,k} \big)$.
We now consider the diagonal vector $\big( \rho_{i,i}: i \in {\llbracket 1,k \rrbracket} \big)$. We first verify that this vector’s components are pairwise horizontally separate zigzags that consecutively run from $(x,t_1)$ to $(u_i,t_2)$. The endpoint locations are not in doubt, so it is our task to verify the separateness condition between consecutively indexed zigzags. In the language of Section \[s.polyorder\], we want to check that $\rho_{i,i} \prec \rho_{i+1,i+1}$ for $i \in {\llbracket 1,k-1 \rrbracket}$. We have that $\rho_{i,i} \prec \rho_{i,i+1}$, because these players are components of a given multi-polymer, and we also have $\rho_{i,i+1} \preceq \rho_{i+1,i+1}$ almost surely, by Lemma \[l.tworelations\](1). Thus, Lemma \[l.tworelations\](2) implies the desired fact.
The confirmed property of the diagonal vector implies that $$\label{e.swlb}
{\mathsf{Wgt}}_{n,k;(x \bar{\bf{1}} ,t_1)}^{(\bar{u},t_2)} \geq
\sum_{i=1}^k {\mathsf{Wgt}}\big( \rho_{i,i} \big) \, ,$$ where note that ${\mathsf{Wgt}}\big(\rho_{i,i}\big)$ is equal to ${\mathsf{Wgt}}_{n,k,i;(x \bar{\bf{1}} ,t_1)}^{\big(u_i \bar{\bf{1}} ,t_2\big)}$.
We now present a lemma regarding the values of these weights.
\[l.kweight\] Let $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ be a compatible triple. For $(x,y) \in {\ensuremath{\mathbb{R}}}^2$, $y \geq x - 2^{-1} n^{1/3} {t_{1,2}}$, $k \in {\ensuremath{\mathbb{N}}}$ and $i \in {\llbracket 1,k \rrbracket}$, $$\label{e.kweight}
\mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}(1,y) - \sum_{j=2}^k \Big(\mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}(1,y) - \mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}(j,y) \Big) \leq {\mathsf{Wgt}}_{n,k,i;(x\bar{\bf{1}},t_1)}^{\big(y\bar{\bf{1}},t_2\big)}
\leq \mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}(1,y) \, .$$
[**Proof.**]{} First note that the condition $y \geq x - 2^{-1} n^{1/3} {t_{1,2}}$ is assumed simply in order to assure that the ensemble $\mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}$ is well defined at location $y$. Note that $\mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}(1,y) = {\mathsf{Wgt}}_{n;(x,t_1)}^{(y,t_2)}$. Since $\rho_{n,k,i;(x \bar{\bf{1}} ,t_1)}^{\big(u_i {\bf{1}},t_2 \big)}$ is an $n$-zigzag from $(x,t_1)$ to $(u_i,t_2)$ for $i \in {\llbracket 1,k \rrbracket}$, we have that ${\mathsf{Wgt}}_{n,k,i;(x \bar{\bf{1}} ,t_1)}^{\big(y\bar{\bf{1}},t_2\big)} \leq {\mathsf{Wgt}}_{n;(x,t_1)}^{(y,t_2)}$ for such $i$. Thus, we obtain the second bound in (\[e.kweight\]). As for the first, note that, by (\[e.scaledweight\]), $$\sum_{j=1}^k \mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}(j,y) \, = \,
\sum_{j=1}^k {\mathsf{Wgt}}_{n,k,j;(x \bar{\bf{1}} ,t_1)}^{\big(y\bar{\bf{1}},t_2\big)}
\, ,$$ an equality that we may rewrite $$\label{e.wequal}
k \cdot \mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}(1,y)
\, \, + \, \, \sum_{j=2}^k \Big(\mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}(j,y) - \mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}(1,y) \Big) \, = \, \sum_{j=1}^k {\mathsf{Wgt}}_{n,k,j;(x \bar{\bf{1}} ,t_1)}^{\big(y\bar{\bf{1}},t_2\big)} \ \, .$$ Set $W_j = {\mathsf{Wgt}}_{n,k,j;(x \bar{\bf{1}} ,t_1)}^{\big(y\bar{\bf{1}},t_2\big)}$ for $j \in {\llbracket 1,k \rrbracket}$. Our remaining task is to derive the lower bound on $W_i$ given in the first inequality of (\[e.kweight\]), where $i \in {\llbracket 1,k \rrbracket}$ is given. The right-hand side in the equality (\[e.wequal\]) takes the form $\sum_{j=1}^k W_j$. The summand $W_j$ is at most ${\mathsf{Wgt}}_{n,(x,t_1)}^{(y,t_2)} = \mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}(1,y)$ for any $j \in {\llbracket 1,k \rrbracket}$. Suppose that we push each $W_j$ up to its maximum possible value $\mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}(1,y)$ for every $j \in {\llbracket 1,k \rrbracket}$ with $j \not= i$ – and note that, in so doing, we push the value of $W_i$ down, because the equality (\[e.wequal\]) must be satisfied. That is, the value of the variable $W_i$ in this scenario is determined by the satisfaction of (\[e.wequal\]); and this value offers a lower bound on the actual value of $W_i$. Since the resulting inequality is the first bound in (\[e.kweight\]), the proof of Lemma \[l.kweight\] is completed.
In light of (\[e.swlb\]), Proposition \[p.sumweight\](2) will follow once we show that, under the result’s hypotheses, $$\label{e.sumbound}
{\ensuremath{\mathbb{P}}}\bigg(
{t_{1,2}}^{-1/3} \, \sum_{i=1}^k
{\mathsf{Wgt}}_{n,k,i;(x \bar{\bf{1}} ,t_1)}^{\big(u_i\bar{\bf{1}},t_2\big)} \leq \, - \, 2^{-1/2} \sum_{i=1}^k {t_{1,2}}^{-4/3}(u_i - x)^2 \, \, - \, r \bigg) \leq
3 k^2
C_k \exp \Big\{ - 2^{-3} k^{-3} c_k r^{3/2} \Big\} \, .$$
To begin showing this, note that the occurrence of the left-hand event entails that the bound $${t_{1,2}}^{-1/3} \,
{\mathsf{Wgt}}_{n,k,i;(x \bar{\bf{1}} ,t_1)}^{\big(u_i\bar{\bf{1}},t_2\big)} \leq \, - \, 2^{-1/2} {t_{1,2}}^{-4/3}(u_i - x)^2 \, \, - \, r/k$$ is satisfied for at least one index $i \in {\llbracket 1,k \rrbracket}$. In view of Lemma \[l.kweight\]’s left-hand bound applied with ${\bf y} = u_i$ (for given $i \in {\llbracket 1,k \rrbracket}$), the last inequality implies that either $$\label{e.tsl.one}
{t_{1,2}}^{-1/3} \, \mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}(1,u_i) \leq \, - \, 2^{-1/2} {t_{1,2}}^{-4/3}(u_i - x)^2 \, \, - \, \frac{r}{2k}$$ or at least one among the inequalities $$\label{e.tsl.two}
{t_{1,2}}^{-1/3} \Big( \mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}\big(1,u_i\big) \, - \, \mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}\big(j,u_i\big) \Big) \geq
\frac{r}{2k(k-1)} \, ,$$ indexed by $j \in \llbracket 2,k \rrbracket$, is satisfied.
Condition (\[e.tsl.one\]) asserts that the highest curve of the normalized forward line ensemble, rooted at $(x,t_1)$ and of duration ${t_{1,2}}$, $${\mathsf{Nr}{\mathcal}{L}}^{\uparrow;t_2}_{n;(x,t_1)}\big(1, w_i \big) = {t_{1,2}}^{-1/3} \, \mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}\big(1, x + {t_{1,2}}^{2/3} w_i \big) \, ,$$ evaluated at $w_i = {t_{1,2}}^{-2/3} \big( u_i - x \big)$, is at most $- \, 2^{-1/2} {t_{1,2}}^{-4/3}(u_i - x)^2 \, - \, 2^{-1} r k^{-1}$. Applying one-point lower tail ${{\rm Reg}}(2)$ to the $(n {t_{1,2}}+ 1)$-curve ensemble ${\mathsf{Nr}{\mathcal}{L}}^{\uparrow;t_2}_{n;(x,t_1)}$, with parameter settings ${\bf z} = w_i$ and ${\bf s} = 2^{-1} rk^{-1}$, we find that this eventuality has probability at most $C \exp \big\{ - c \, 2^{-3/2} k^{-3/2} r^{3/2} \big\}$. This application of ${{\rm Reg}}(2)$ may be carried out if the hypotheses $${t_{1,2}}^{-2/3} \vert u_i - x \vert \leq {c}(n {t_{1,2}})^{1/9} \, \, \textrm{and} \, \, r \in \big[2k, 2k (n {t_{1,2}})^{1/3} \big]$$ are satisfied.
When the bound (\[e.tsl.two\]) is rewritten in normalized coordinates, it asserts that $$\begin{aligned}
& & \Big( {\mathsf{Nr}{\mathcal}{L}}^{\uparrow;t_2}_{n;(x,t_1)}\big(1, w_i \big) + 2^{-1/2} w_i^2 \Big) - \Big( {\mathsf{Nr}{\mathcal}{L}}^{\uparrow;t_2}_{n;(x,t_1)}\big(j, w_i \big) + 2^{-1/2} w_i^2 \Big) \\
& = & {t_{1,2}}^{-1/3} \, \mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}\big(1, x + {t_{1,2}}^{2/3} w_i \big) \, - \,
{t_{1,2}}^{-1/3} \, \mathcal{L}^{\uparrow;t_2}_{n;(x,t_1)}\big(j, x + {t_{1,2}}^{2/3} w_i \big) \, ,\end{aligned}$$ when evaluated at $w_i = {t_{1,2}}^{-2/3} \big( u_i - x \big)$, is at least $\frac{r}{2k(k-1)}$. The bound forces at least one of two cases: in the above difference of two terms, either the first term is at least $\frac{r}{4k(k-1)}$ or the second is at most $-\frac{r}{4k(k-1)}$.
The first case is handled by applying one-point upper tail ${{\rm Reg}}(3)$ to ${\mathsf{Nr}{\mathcal}{L}}^{\uparrow;t_2}_{n;(x,t_1)}$, taking ${\bf z} = w_i$ and ${\bf s} = \frac{r}{4k(k-1)}$, with an upper bound of $$C \exp \Big\{ - c \big(\tfrac{r}{4k(k-1)}\big)^{3/2} \Big\}$$ being found on the probability of the event in question. This use of ${{\rm Reg}}(3)$ may be made when $${t_{1,2}}^{-2/3} \big\vert u_i - x \big\vert \leq {c}(n {t_{1,2}})^{1/9} \, \, \textrm{and} \, \, r \geq 4k(k-1) \, .$$ The second case is treated by applying pointwise lower tail Proposition \[p.mega\](1) to the $(n{t_{1,2}}+1)$-curve ensemble ${\mathcal}{L}_n = {\mathsf{Nr}{\mathcal}{L}}^{\uparrow;t_2}_{n;(x,t_1)}$, with parameter settings ${\bf k} = j$, ${\bf z} = w_i$ and ${\bf s} = \frac{r}{4k(k-1)}$. The event in question is thus found to have ${\ensuremath{\mathbb{P}}}$-probability at most $$C_j \exp \Big\{ - c_j \big(\tfrac{r}{4k(k-1)}\big)^{3/2} \Big\} \, .$$ These applications of Proposition \[p.mega\](1) are made for each $j \in \llbracket 2, k \rrbracket$, and the $k-1$ applications may be made provided that $$n {t_{1,2}}\geq k
\vee (c/3)^{-18} \vee 6^{36} \, , \,
{t_{1,2}}^{-2/3} \big\vert u_i - x \big\vert \leq 2^{-1} {c}(n {t_{1,2}})^{1/18} \, \, \textrm{and} \, \, r \leq 8 k (k-1) (n {t_{1,2}})^{1/18} \, .$$ We find then that the probability that (\[e.tsl.two\]) is satisfied for a given pair $(i,j) \in {\llbracket 1,k \rrbracket} \times \llbracket 2, k \rrbracket$ is at most $$C \exp \Big\{ - c \big(\tfrac{r}{4k(k-1)}\big)^{3/2} \Big\} \, + \, C_j \exp \Big\{ - c_j \big(\tfrac{r}{4k(k-1)}\big)^{3/2} \Big\} \, .$$
Gathering together these inferences by means of a union bound over this set of index pairs, we find that the left-hand side of (\[e.sumbound\]) is at most $$C k \exp \big\{ - c 2^{-3/2} k^{-3/2} r^{3/2} \big\} \, + \,
C k^2 \exp \Big\{ - c \big(\tfrac{r}{4k(k-1)}\big)^{3/2} \Big\} \, + \, k \sum_{j=2}^k
C_j \exp \Big\{ - c_j \big(\tfrac{r}{4k(k-1)}\big)^{3/2} \Big\} \, .$$ Noting that $C_k$ is increasing, $c_k$ is decreasing, $C_k \geq C$ and $c_k \leq c$, we verify (\[e.sumbound\]). This completes the proof of Proposition \[p.sumweight\](2).
The road map’s third challenge: the solution in overview {#s.thirdchallenge}
--------------------------------------------------------
We have assembled the elements needed to implement the road map and thus to prove Theorem \[t.disjtpoly\], with the exception of addressing the third challenge, which we labelled [*final polymer comparison*]{} in Section \[s.roadmap\].
Recall that the road map advocates the construction of a system of ${k}$ near polymers, each running from $(x,-{\epsilon}^{3/2})$ to $(y,1+{\epsilon}^{3/2})$, and pairwise disjoint otherwise. These will be zigzags, to be called $\rho_1$, $\rho_2$, $\cdots$, $\rho_{k}$, that during the time interval $[0,1]$ follow the course of the ${k}$ polymers, to be called $\phi_1$, $\phi_2$, $\cdots$, $\phi_{k}$, whose existence is ensured by the occurrence of the event ${\mathrm{MaxDisjtPoly}}_{n;([x-{\epsilon},x+{\epsilon}],0))}^{([y-{\epsilon},y+{\epsilon}],1)} \geq {k}$. They begin by following elements in a forward bouquet of lifetime $[-{\epsilon}^{3/2},0]$ with shared endpoint $(x,-{\epsilon}^{3/2})$ and end by following elements in a backward bouquet of lifetime $[1,1+{\epsilon}^{3/2}]$ with shared endpoint $(y,1+{\epsilon}^{3/2})$.
Now the third challenge involves arguing that each of the $\rho_i$ is indeed a near polymer, with a shortfall in weight from the maximum ${\mathsf{Wgt}}_{n;(x,-{\epsilon}^{3/2})}^{(y,1+{\epsilon}^{3/2})}$ of order ${\epsilon}^{1/2}$. As we pointed out in Section \[s.roadmap\], the third challenge seems merely a restatement of the overall problem. A little more detail is needed to explain even in outline how we propose to solve the third challenge, and we now offer such an outline.
Our task is to establish that, for each $i \in {\llbracket 1,{k}\rrbracket}$, the quantity ${\mathsf{Wgt}}\big( \rho_{n;(x,-{\epsilon}^{3/2})}^{(y,1+{\epsilon}^{3/2})} \big) - {\mathsf{Wgt}}(\rho_i)$, which is necessarily non-negative, is in fact at most $O({\epsilon}^{1/2})$. Now the value ${\mathsf{Wgt}}(\rho_i)$ is naturally written as a sum of three terms: the weight of a forward bouquet element, the weight of the polymer $\phi_i$, which moves from $[x-{\epsilon},x+{\epsilon}] \times \{0\}$ to $[y-{\epsilon},y+{\epsilon}] \times \{ 1 \}$, and the weight of a backward bouquet element. The first and third weights are known to be of order ${\epsilon}^{1/2}$ typically by Corollary \[c.bouquetreg\]. The weight ${\mathsf{Wgt}}\big( \rho_{n;(x,-{\epsilon}^{3/2})}^{(y,1+{\epsilon}^{3/2})} \big)$ may correspondingly be split into three terms, by splitting the polymer in question at the pair of times $(0,1)$. To resolve the third challenge, we want to argue that:
- The first and third pieces of the polymer $\rho_{n;(x,-{\epsilon}^{3/2})}^{(y,1+{\epsilon}^{3/2})}$, with lifetimes $[-{\epsilon}^{3/2},0]$ and $[1,1+{\epsilon}^{3/2}]$, have weight of order ${\epsilon}^{1/2}$.
- The second piece, the grand middle section with lifetime $[0,1]$, has a weight that differs from any of the weights ${\mathsf{Wgt}}(\phi_i)$ for $i \in {\llbracket 1,{k}\rrbracket}$, by an order of ${\epsilon}^{1/2}$.
![Solving the third challenge of the road map. The planar intervals $I \times \{ 0 \}$ and $J \times \{ 1 \}$, with $I = [x-{\epsilon},x+{\epsilon}]$ and $J = [y-{\epsilon},y+{\epsilon}]$, are drawn with thick bold lines. Extended intervals $I^+ \times \{ 0 \}$ and $J^+ \times \{ 1 \}$ are also drawn in solid lines. Here, $I^+ = [x-(r+1){\epsilon},x+(r+1){\epsilon}]$ and $J^+ = [y-(r+1){\epsilon},y+(r+1){\epsilon}]$ with $r > 0$ given. The notation will be used in the upcoming proof of Theorem \[t.disjtpoly\]. The polymer $\rho_{n;(x,-{\epsilon}^{3/2})}^{(y,1+{\epsilon}^{3/2})}$ typically visits these extended intervals at times zero and one, and, in this event, the weight of its $[0,1]$-duration subpath may be closely compared to that of the polymers $\phi_1$ and $\phi_2$. The $\rho_{ij}$ notation will be used in the proof to indicate paths in the forward and backward bouquets.[]{data-label="f.surgery"}](NonIntPolySurgery.pdf){height="9cm"}
Now to reach these conclusions, we will in fact seek control of the geometry of the polymer $\rho_{n;(x,-{\epsilon}^{3/2})}^{(y,1+{\epsilon}^{3/2})}$. This we will do by invoking one of the main results of the present article, Theorem \[t.polyfluc\]. We will learn that, typically, this polymer at time zero is located at distance of order ${\epsilon}$ from $x$, and at time one, at distance of order ${\epsilon}$ from $y$. With this understanding, we see that the polymer’s first and third pieces do not suffer significant lateral shifts, so that Corollary \[c.maxminweight\] will ensure that these pieces have the desired order ${\epsilon}^{1/2}$ weight. The same understanding means that the middle section of $\rho_{n;(x,-{\epsilon}^{3/2})}^{(y,1+{\epsilon}^{3/2})}$ and the middle section $\phi_i$ of any $\rho_i$ have starting and ending points whose locations differ by an order of ${\epsilon}$: close to $x$ or to $y$ respectively. This is crucial information, because it permits us to invoke Theorem \[t.differenceweight\] to conclude that the middle section weights indeed differ by an order of ${\epsilon}^{1/2}$. These ideas are illustrated by Figure \[f.surgery\].
Armed with this elaboration of the third challenge in the road map, we are ready to derive Theorem \[t.disjtpoly\].
Proof of Theorem \[t.disjtpoly\]. {#s.proof}
---------------------------------
By the scaling principle, it suffices to prove the result when $(t_1,t_2) = (0,1)$, and this we now do. Set $I = [x-{\epsilon},x+{\epsilon}]$ and $J = [y - {\epsilon},y+{\epsilon}]$. For a given parameter $r > 0$, we also define the extended intervals $I^+ = [x-(r+1){\epsilon},x+(r+1){\epsilon}]$ and $J^+ = [y - (r+1){\epsilon},y+(r+1){\epsilon}]$ that respectively contain $I$ and $J$.
On the event that ${\mathrm{MaxDisjtPoly}}_{n;(I,0))}^{(J,1)} \geq {k}$, there exist collections of ${k}$ disjoint polymers that make the journey from $I \times \{ 0 \}$ to $J \times \{ 1 \}$. We now seek to choose one of them. Indeed, using the language of Section \[s.closure\], we may choose $\big(\bar{U},\bar{V} \big)$ to be the lexicographically minimal element in the closure of the set ${\mathsf{DisjtIndex}}_{n,{k};(I,0)}^{(J,1)}$. This definition makes sense because we are dealing with a closed set. Invoking Lemma \[l.disjthorsep\], we see that $\big(\bar{U},\bar{V} \big)$ is an element of ${\mathsf{HorSepIndex}}_{n,{k};(I,0)}^{(J,1)}$. That is, we have explicitly selected an ${k}$-tuple of polymers that, while not necessarily pairwise disjoint, is pairwise horizontally separate; this is enough for the upcoming surgery. It is helpful to recall from the road map that it is our aim to show that the occurrence of the event ${\mathrm{MaxDisjtPoly}}_{n;(I,0))}^{(J,1)} \geq {k}$ [*typically*]{} entails ${\mathsf{NearPoly}}_{n,{k};(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}( \eta )$, where the parameter $\eta$ will soon be selected explicitly to have order ${\epsilon}^{1/2}$. What do we mean by typically? Typical behaviour means a collection of provably standard circumstances needed to undertake our rewiring surgery successfully. It has four types:
- ${\mathsf{ForBouqReg}}$/${\mathsf{BackBouqReg}}$: The ${\epsilon}^{1/2}$-order weight of bouquets.
- ${\mathsf{PolyWgtReg}}$: The ${\epsilon}^{1/2}$-order weight of any $[-{\epsilon}^{3/2},0]$- or $[1,1+{\epsilon}^{3/2}]$-lifetime polymer respectively near $x$ and $y$.
- ${\mathsf{PolyDevReg}}$: Control on the geometry of $\rho_{n;(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}$ at times zero and one.
- ${\mathsf{LocWgtReg}}$: The one-half power weight difference for middle section polymers.
We will shortly define a [*favourable surgical conditions*]{} event, which specifies exactly what we mean by ‘typical’, as an intersection of events to which the labels just listed correspond.
Before offering the definition, we introduce a variation of the ${\mathsf{PolyDevReg}}$ notation specified in Subsection \[s.polyflucintro\]. When the event ${\mathsf{PolyDevReg}}_{n;(x,t_1)}^{(y,t_2)}\big(a,r\big)$ is specified in (\[e.pdr\]), the parameter $a$ refers to the proportion of the polymer lifetime $t_{1,2}$ which has elapsed at the intermediate time $(1-a)t_1 + at_2$. We now wish to allude to the intermediate time directly. To do so, we will employ a square bracket notation. Taking $t \in [t_1,t_2] \cap n^{-1}{\ensuremath{\mathbb{Z}}}$, we will write $${\mathsf{PolyDevReg}}_{n;(x,t_1)}^{(y,t_2)}\big[t,r\big] \, = \, \bigg\{ \, \Big\vert \, \rho_{n;(x,t_1)}^{(y,t_2)} ( t ) - \ell_{(x,t_1)}^{(y,t_2)} ( t ) \, \Big\vert \, \leq \, r \big( (t - t_1) \wedge (t_2 - t) \big)^{2/3} \bigg\} \, ,$$ so that ${\mathsf{PolyDevReg}}_{n;(x,t_1)}^{(y,t_2)}\big[t,r\big]$ equals ${\mathsf{PolyDevReg}}_{n;(x,t_1)}^{(y,t_2)}\big( (t-t_1) {t_{1,2}}^{-1} \, , r \big)$. We also extend this notation so that the first square bracket argument $t$ is replaced by a pair of such times. This notation refers to the intersection of the two single-time ${\mathsf{PolyDevReg}}$ events.
We now may define the favourable surgical conditions event ${\mathsf{FavSurCon}}_{n;(I,0)}^{(J,1)}\big({k}; \bar{U},\bar{V} ; {\epsilon},r \big)$. The quantities ${\epsilon}> 0$ and $r > 0$ will retain their roles as parameters in this event for the remainder of the proof of Theorem \[t.disjtpoly\]. The event will only be considered when the event ${\mathrm{MaxDisjtPoly}}_{n;(I,0))}^{(J,1)} \geq {k}$ occurs, so that the random vectors $\bar{U}$ and $\bar{V}$ that are used as parameters in the event’s definition will always make sense. The new event is defined to equal $$\begin{aligned}
& & {\mathsf{LocWgtReg}}_{n;(I^+,0)}^{(J^+,1)}\big(2(r+1){\epsilon},r\big) \cap {\mathsf{PolyDevReg}}_{n;(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})} \big[ \{0,1\} , r \big] \\
& & \quad \cap \, {\mathsf{ForBouqReg}}_{n,{k};(x,-{\epsilon}^{3/2})}^{(\bar{u},0)}(r) \cap {\mathsf{BackBouqReg}}_{n,{k};(\bar{v},1)}^{(y,1+{\epsilon}^{3/2})}(r) \\
& & \quad \cap \, {\mathsf{PolyWgtReg}}_{n;(x,-{\epsilon}^{3/2})}^{(I^+,0)}(r^2) \cap {\mathsf{PolyWgtReg}}_{n;(J^+,1)}^{(y,1+{\epsilon}^{3/2})}(r^2) \, .\end{aligned}$$ The ${\mathsf{LocWgtReg}}$, ${\mathsf{ForBouqReg}}$/${\mathsf{BackBouqReg}}$ and ${\mathsf{PolyWgtReg}}$ events have been defined in Sections \[s.polyweightreg\], \[s.bouquet\] and \[s.usefultool\]. In the use of the new square bracket notation for ${\mathsf{PolyDevReg}}$, the polymer is being controlled at times zero and one. The reason of the choice of parameter $r^2$ in the two ${\mathsf{PolyWgtReg}}$ events will become clearer, and will be discussed, in due course.
Note that the quantity ${\epsilon}^{3/2}$ plays the role of the duration of certain zigzags concerned in this definition, these zigzags having lifetime $[-{\epsilon}^{3/2},0]$ and $[1,1+{\epsilon}^{3/2}]$. In order that these zigzags begin and end at vertical coordinates in the $n^{-1}$-mesh, we will insist throughout that ${\epsilon}> 0$ satisfies ${\epsilon}^{3/2} \in n^{-1} {\ensuremath{\mathbb{Z}}}$. It would seem then that this condition should enter as a hypothesis of Theorem \[t.disjtpoly\], though it does not; we omit it because, in view of the hypotheses (\[e.epsilonbound\]) and (\[e.nlowerbound\]), the condition can be forced by a multiplicative adjustment in ${\epsilon}$ that differs from one by order say $10^{-100}$, and this causes a tiny similar adjustment in the theorem’s conclusion, which the reader may readily confirm is easily absorbed by tightening estimates during the proof. In summary, though ${\epsilon}> 0$ is $n$-dependent, it may in practice be considered to be fixed at a given small positive value, the upper bound specified by several upcoming demands. We may later omit mention of the mesh membership condition ${\epsilon}^{3/2} \in n^{-1} {\ensuremath{\mathbb{Z}}}$.
The next two results assert that favourable surgical conditions are indeed typical, and that, in the presence of these conditions, the occurrence of ${\mathrm{MaxDisjtPoly}}_{n;(I,0))}^{(J,1)} \geq {k}$ indeed forces the occurrence of ${\mathsf{NearPoly}}_{n,{k};(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}( \eta )$ with $\eta = O({\epsilon}^{1/2})$.
\[l.fsc\] Suppose that $n \in {\ensuremath{\mathbb{N}}}$, $k \in {\ensuremath{\mathbb{N}}}$, $x,y \in {\ensuremath{\mathbb{R}}}$, ${\epsilon}> 0$ and $r > 0$ satisfy $$n \geq 10^{40} k c^{-18} {\epsilon}^{-75/2} \max \big\{ 1 \, , \, \vert x - y \vert^{36} \big\} \, \, \, \textrm{and} \, \, \, \vert x - y \vert \leq \big( 2 (r+1) {\epsilon}\big)^{-1/2} \, .$$ Suppose also that $$r \geq \max \Big\{ \, 10^9 c_1^{-4/5} \, , \, 15 C^{1/2} \, , \, 4 k^2 \, , \, 70 {\epsilon}^{1/2} \vert x - y \vert \, \Big\} \, , \, {\epsilon}\leq 2^{-5} (r+1)^{-1} \, \, \textrm{ and } \, \, r \leq 2^{-6} c \, {\epsilon}^{25/6} n^{1/36} \, .$$ Then $$\begin{aligned}
& & {\ensuremath{\mathbb{P}}}\Big( {\mathrm{MaxDisjtPoly}}_{n;(I,0))}^{(J,1)} \geq k \, , \, \neg \, {\mathsf{FavSurCon}}_{n;(I,0)}^{(J,1)}\big(k; \bar{U},\bar{V} ; {\epsilon},r \big) \Big) \\
& \leq & 14062 \, k^2 r C_k \exp \big\{ - 10^{-11} k^{-3} c_k r^{3/4} \big\} \, .\end{aligned}$$
\[p.doubletie\] Let $n,k \in {\ensuremath{\mathbb{N}}}$, $n \geq k \geq 1$. Let ${\epsilon}> 0$ and $x,y \in {\ensuremath{\mathbb{R}}}$ satisfy $n {\epsilon}^{3/2} \geq 10^2$, $\vert x - y \vert \leq {\epsilon}^{-1/2}$, and let $r \geq 1$. Then, whenever $\big\{ {\mathrm{MaxDisjtPoly}}_{n;([x-{\epsilon},x+{\epsilon}],0))}^{([y-{\epsilon},y+{\epsilon}],1)} \geq k \big\} \cap {\mathsf{FavSurCon}}_{n;(I,0)}^{(J,1)}\big(k; \bar{U},\bar{V} ; {\epsilon},r \big)$ occurs, $$\label{e.doubletie}
{\mathsf{Wgt}}_{n,k;(x \bar{\bf 1}, - {\epsilon}^{3/2})}^{(y \bar{\bf 1},1 + {\epsilon}^{3/2})} \geq k \cdot {\mathsf{Wgt}}_{n;(x, - {\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})} \, - \,
15k r^2 {\epsilon}^{1/2}
\, ,$$ which is to say, the event ${\mathsf{NearPoly}}_{n,k;(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}( \eta )$ occurs where $\big(1+2{\epsilon}^{3/2} \big)^{1/3} \eta = 15k r^2 {\epsilon}^{1/2}$.
We now apply these two results to close out the proof of Theorem \[t.disjtpoly\] and then prove them in turn.
Note that, for the value of $\eta$ specified in Proposition \[p.doubletie\], ${\mathsf{NearPoly}}_{n,k;(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}( \eta )$ is a subset of ${\mathsf{NearPoly}}_{n,k;(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}\big( 15k r^2 {\epsilon}^{1/2} \big)$. The proposition thus implies that $$\begin{aligned}
& & {\ensuremath{\mathbb{P}}}\Big( {\mathrm{MaxDisjtPoly}}_{n;(I,0))}^{(J,1)} \geq {k}\, , \, {\mathsf{FavSurCon}}_{n;(I,0)}^{(J,1)}\big( {k}; \bar{U},\bar{V} ; {\epsilon},r \big) \Big) \\
& \leq & {\ensuremath{\mathbb{P}}}\Big( {\mathsf{NearPoly}}_{n,{k};(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}\big( 15 {k}r^2 {\epsilon}^{1/2} \big) \Big) \, ,\end{aligned}$$ provided that $\vert x - y \vert \leq {\epsilon}^{-1/2}$ and $r \geq 1$.
We now apply Corollary \[c.neargeod.t\] to bound above the right-hand probability. Parameter settings are ${\bf t_1} = -{\epsilon}^{3/2}$, ${\bf t_2} = 1 + {\epsilon}^{3/2}$, ${\bf k} = {k}$, ${\bf x} = x$, ${\bf y} = y$ and ${\bm \eta} = 15{k}r^2 {\epsilon}^{1/2}$. Note that, since ${\bm \eta} \geq {\epsilon}$, the corollary’s hypotheses are satisfied when $$15{k}r^2 {\epsilon}^{1/2} < (\eta_0)^{{k}^2} \, , \,
{k}\geq 2 \, , \, (1 + 2{\epsilon}^{3/2}) n/2 \geq {k}\vee \, (K_0)^{{k}^2} \big( \log {\epsilon}^{-1} \big)^{K_0}$$ and $(1 + 2{\epsilon}^{3/2})^{-2/3} \vert y - x \vert \leq a_0 n^{1/9}$. (It is also necessary that $\big(n,1-{\epsilon}^{3/2},1+{\epsilon}^{3/2}\big) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_<$ be a compatible triple, which amounts to the already imposed condition that $n {\epsilon}^{3/2} \in {\ensuremath{\mathbb{N}}}$.) When these conditions are met, the conclusion of the corollary implies that $$\begin{aligned}
& & {\ensuremath{\mathbb{P}}}\Big( {\mathsf{NearPoly}}_{n,{k};(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}\big( 15 {k}r^2 {\epsilon}^{1/2} \big) \Big) \\
& \leq & 15^{{k}^2} {k}^{{k}^2} r^{2{k}^2} {\epsilon}^{({k}^2 - 1)/2} \exp \big\{ \beta_{k}\big( \log {\epsilon}^{-1} \big)^{5/6} \big\}
\, ,\end{aligned}$$ where we again used $\eta \geq {\epsilon}$ to write the final term in the product.
Applying Lemma \[l.fsc\], we find that $$\begin{aligned}
{\ensuremath{\mathbb{P}}}\Big( {\mathrm{MaxDisjtPoly}}_{n;(I,0))}^{(J,1)} \geq {k}\Big)
& \leq & 15^{{k}^2} {k}^{{k}^2} r^{2{k}^2} {\epsilon}^{({k}^2 - 1)/2} \exp \big\{ \beta_{k}\big( \log {\epsilon}^{-1} \big)^{5/6} \big\} \nonumber \\
& & + \, \, 14062 \, {k}^2 r C_{k}\exp \big\{ - 10^{-11} {k}^{-3} c_{k}r^{3/4} \big\} \label{e.explast} \, .\end{aligned}$$ This application requires that $n \geq 10^{40} {k}c^{-18} {\epsilon}^{-75/2} \max \big\{ 1 \, , \, \vert x - y \vert^{36} \big\}$, $\vert x - y \vert \leq \big( 2 (r+1) {\epsilon}\big)^{-1/2}$, $$r \geq \max \Big\{ \, 10^9 c_1^{-4/5} \, , \, 15 C^{1/2} \, , \, 4 {k}^2 \, , \, 70 {\epsilon}^{1/2} \vert x - y \vert \, \Big\} \, , \, {\epsilon}\leq 2^{-5} (r+1)^{-1} \, \, \textrm{ and } \, \, r \leq 2^{-6} c \, {\epsilon}^{25/6} n^{1/36} \, .$$
We now set the parameter $r > 0$ so that $10^{-11} {k}^{-3} c_{k}r^{3/4} = 2^{-1} ({k}^2 - 1) \log {\epsilon}^{-1}$. The choice is made in order that the exponential term in (\[e.explast\]) equal ${\epsilon}^{({k}^2 - 1)/2}$. Thus, $$\label{e.rchoice}
r = 10^{44/3} {k}^4 c_{k}^{-4/3} 2^{-4/3} ({k}^2 - 1)^{4/3} \big( \log {\epsilon}^{-1} \big)^{4/3} \, .$$ Note that $$r \leq 10^{15} c_{k}^{-4/3} {k}^{20/3} \big( \log {\epsilon}^{-1} \big)^{4/3} \, .$$ We then obtain $$\begin{aligned}
& & {\ensuremath{\mathbb{P}}}\Big( {\mathrm{MaxDisjtPoly}}_{n;(I,0))}^{(J,1)} \geq {k}\Big) \\
& \leq & 15^{{k}^2} 10^{30{k}^2} {k}^{43{k}^2/3} c_{k}^{-8{k}^2/3} \big( \log {\epsilon}^{-1} \big)^{4{k}^2} {\epsilon}^{({k}^2 - 1)/2} \exp \big\{ \beta_{k}\big( \log {\epsilon}^{-1} \big)^{5/6} \big\} \\
& & + \, 10^{20} {k}^{26/3} c_{k}^{-4/3} C_{k}\big( \log {\epsilon}^{-1} \big)^2 \cdot {\epsilon}^{({k}^2 - 1)/2} \\
& \leq & 10^{32{k}^2} {k}^{15{k}^2} c_{k}^{-3{k}^2} C_{k}\big( \log {\epsilon}^{-1} \big)^{4{k}^2} {\epsilon}^{({k}^2 - 1)/2} \exp \big\{ \beta_{k}\big( \log {\epsilon}^{-1} \big)^{5/6} \big\} \, ,\end{aligned}$$ where we used $c_{k}\leq c_1$ and ${\epsilon}\leq e^{-1}$ in the first inequality and ${k}\geq 1$, $C_{k}\geq 1$, $c_{k}\leq 1$ and ${\epsilon}\leq e^{-1}$ in the second. This completes the proof of Theorem \[t.disjtpoly\] in the case that $t_1=0$ and $t_1=1$; as we noted at the outset, this special case implies the general one.
[**Proof of Lemma \[l.fsc\].**]{} The ${\mathsf{FavSurCon}}$ event is an intersection of six events. Each of these events is known to be typical, by several results proved or cited in this article. Thus, the present proof need merely gather together the estimates, invoked with suitable parameter settings.
By Corollary \[c.ordweight\] with ${\bm {\epsilon}} = 2(r+1){\epsilon}$, ${\bf R} = r$, ${\bf x} = x - (r+1){\epsilon}$ and ${\bf y} = y - (r+1){\epsilon}$, $${\ensuremath{\mathbb{P}}}\Big( \neg \, {\mathsf{LocWgtReg}}_{n;(I^+,0)}^{(J^+,1)}\big(2(r+1){\epsilon},r\big) \Big) \leq
10032 \, C \exp \big\{ - c_1 2^{-23} r^{3/2} \big\}
$$ when $n \geq 10^{32} c^{-18}$ and $$2(r+1){\epsilon}\in (0,2^{-4}] \, , \, \big\vert x - y \big\vert \leq \big( 2(r+1) {\epsilon}\big)^{-1/2} \wedge 2^{-2} 3^{-1} {c}n^{1/18} \, \, \textrm{and} \, \,
r \in \big[ 2 \cdot 10^4 \, , \, 10^3 n^{1/18} \big] \, .$$
We apply Proposition \[p.polyfluc\], and the remark following this proposition, in order to find that $${\ensuremath{\mathbb{P}}}\Big( \neg \, {\mathsf{PolyDevReg}}_{n;(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})} \big[ \{0,1\} , r \big] \Big) \leq 22 Cr \exp \big\{ - 10^{-11} c_1 r^{3/4} \big\} \, .$$ When the proposition is applied, it is with parameter settings ${\bf x} = x$, ${\bf y} = y$, ${\bf t_1} = -{\epsilon}^{3/2}$, ${\bf t_2} = 1 + {\epsilon}^{3/2}$, ${\bf r} = r$ and with ${\bf a}$ chosen equal to $1 - {\epsilon}^{3/2}(1 + 2{\epsilon}^{3/2})^{-1}$. When the remark is applied, we instead take ${\bf a} = {\epsilon}^{3/2}(1 + 2{\epsilon}^{3/2})^{-1}$. Note that ${\bf a} \wedge (1 - {\bf a} ) \geq {\epsilon}^{3/2}/2$ since ${\epsilon}\leq 2^{-2/3}$. Using ${\bf a}^{-1} \vee (1 - {\bf a} )^{-1} \leq 2 {\epsilon}^{-3/2}$ and ${\bf {t_{1,2}}} \in [1,2]$, we see that the hypotheses of this application are met provided that $$n \geq \max \bigg\{
10^{32} \big({\epsilon}^{3/2}/2 \big)^{-25} c^{-18} \, \, , \, \,
10^{24} c^{-18} \big({\epsilon}^{3/2}/2 \big)^{-25} \vert x - y \vert^{36} (1 + 2{\epsilon}^{3/2})^{-24}
\bigg\} \, ,$$ $$r \geq
\max \bigg\{ 10^9 c_1^{-4/5} \, \, , \, \, 15 C^{1/2} \, \, , \, \, 87 \big({\epsilon}^{3/2}/2 \big)^{1/3} (1 + 2{\epsilon}^{3/2})^{-2/3} \vert x - y \vert \bigg\}$$ and, since ${\epsilon}^{3/2}(1 + 2{\epsilon}^{3/2})^{-1} \geq 2^{-1} {\epsilon}^{3/2}$, $r \leq 3 \big( 2^{-1} {\epsilon}^{3/2} \big)^{25/9} n^{1/36} (1 + 2{\epsilon}^{3/2})^{1/36}$.
Corollary \[c.bouquetreg\] is applied to find that $$\label{e.bouquetapp}
{\ensuremath{\mathbb{P}}}\Big( \neg \, {\mathsf{ForBouqReg}}_{n,k;(x,-{\epsilon}^{3/2})}^{(\bar{U},0)}(r) \Big) \, + \, {\ensuremath{\mathbb{P}}}\Big( \neg \, {\mathsf{BackBouqReg}}_{n,k;(\bar{V},1)}^{(y,1+{\epsilon}^{3/2})}(r) \Big) \leq
8 k^2
C_k \exp \big\{ - 2^{-3} k^{-3} c_k r^{3/2} \big\} \, .$$ Here, parameter settings are ${\bf k} = k$, ${\bf t_1} = - {\epsilon}^{3/2}$, ${\bf t_2} = 0$, ${\bf x} = x$, ${\bf y} = y$, ${\bf \bar{u}} = \bar{U}$, ${\bf \bar{v}} = \bar{V}$ and ${\bf r} = r$. Since $\vert U_i - x \vert \vee \vert V_i - y \vert \leq {\epsilon}$ for each $i \in {\llbracket 1,k \rrbracket}$, this application may be made provided that $$\begin{aligned}
n {\epsilon}^{3/2} - 1 & \geq & \max \Big\{ k \, , \, 3^{18} c^{-18} \, , \, 6^{36} \, , \, {c}^{-9}(1 - n^{-1}{\epsilon}^{-3/2})^{-6}(1 + 2^{-1} n^{-2/3} {\epsilon}^{-1} )^9 \, , \\
& & \qquad \qquad \qquad \qquad \qquad 2^{18} {c}^{-18} (1 - n^{-1}{\epsilon}^{-3/2})^{-12}(1 + 2^{-1} n^{-2/3} {\epsilon}^{-1} )^{18} \Big\} \end{aligned}$$ and $r \in \big[ 4k^2 , 9 k^2 {\epsilon}^{1/2} (1 - n^{-1}{\epsilon}^{3/2})^{1/3} n^{1/3} \big]$.
There is a subtlety in this application of Corollary \[c.bouquetreg\] that deserves mention. In fact, it is in this application that the microscopic detail discussed after Proposition \[p.sumweight\] is implicated. Our parameter choice $\big( \bar{u}, \bar{v} \big) = \big( \bar{U}, \bar{V} \big)$ is a random one. Even if, as is the case, our random variable $\big( \bar{U}, \bar{V} \big)$ almost surely verifies the necessary bounds for this use, it is not formally admissible in this role. However, recall that $\big( \bar{U}, \bar{V} \big)$ has been specified to be measurable with respect to randomness in the region ${\ensuremath{\mathbb{R}}}\times [0,1]$. This random variable is thus independent of the randomness in the open region ${\ensuremath{\mathbb{R}}}\times \big((-\infty,0) \cup (1,\infty) \big)$ that specifies the two events in (\[e.bouquetapp\]) with which the application of the corollary is concerned. It is the use of proper weights in the definition of the events ${\mathsf{ForBouqReg}}$ and ${\mathsf{BackBouqReg}}$ which ensures that these events are measurable with respect to the randomness in ${\ensuremath{\mathbb{R}}}\times \big((-\infty,0) \cup (1,\infty) \big)$. It is this independence which renders the application admissible.
Note that $${\mathsf{PolyWgtReg}}_{n;(x,-{\epsilon}^{3/2})}^{([x - (r+1){\epsilon},x+(r+1){\epsilon}],0)}(r^2) \subseteq {{\mathsf{PolyWgtReg}}}_{n;([x,x+{\epsilon}],-{\epsilon}^{3/2})}^{([y,y+d{\epsilon}],0)}(r^2)$$ where $y = x - (r+1){\epsilon}$ and $d = \lceil 2(r+1) \rceil$, and that $${\mathsf{PolyWgtReg}}_{n;([y - (r+1){\epsilon},y+(r+1){\epsilon}],1)}^{(y,1+{\epsilon}^{3/2})}(r^2) \subseteq {{\mathsf{PolyWgtReg}}}_{n;([z,z+d{\epsilon}],-{\epsilon}^{3/2})}^{([y,y+{\epsilon}],0)}(r^2)$$ where $z = y - (r+1){\epsilon}$. We may apply Corollary \[c.maxminweight\] in order to bound the ${\ensuremath{\mathbb{P}}}$-probability of the two right-hand events. In the first case, we take ${\bf x} = x$, ${\bf y} = x - (r+1){\epsilon}$, ${\bf a} =1$, ${\bf b} = \lceil 2(r+1) \rceil$, ${\bf t_1} = -{\epsilon}^{3/2}$, ${\bf t_2} = 0$ and ${\bf r} = r^2$ to find that $${\ensuremath{\mathbb{P}}}\Big( \neg \,
{{\mathsf{PolyWgtReg}}}_{n;([x,x+{\epsilon}],-{\epsilon}^{3/2})}^{([y,y+d{\epsilon}],0)}(r^2) \Big) \leq
(2r+3) \cdot 400 C \exp \big\{ - c_1 2^{-10} r^3 \big\} \, ,$$ the application being valid provided that $$n {\epsilon}^{3/2} \geq 10^{29} \vee 2(c/3)^{-18} \, , \, 3(r + 1) \leq 6^{-1} {c}n^{1/18} {\epsilon}^{1/12} \, \textrm{ and } \, r \in \big[ 34^{1/2} \, , \, 2 {\epsilon}^{1/24} n^{1/36} \big] \, .$$ In fact, we also need to use that $n {\epsilon}^{3/2}$ is an integer. It was for this reason that we earlier imposed this requirement on ${\epsilon}> 0$.
In the second case, we take ${\bf x} = y - (r+1){\epsilon}$, ${\bf y} = y$, ${\bf a} =\lceil 2(r+1) \rceil$, ${\bf b} = 1$, ${\bf t_1} = 1$, ${\bf t_2} = 1+ {\epsilon}^{3/2}$ and ${\bf r} = r^2$ to find that $${\ensuremath{\mathbb{P}}}\Big( \neg \,
{{\mathsf{PolyWgtReg}}}_{n;([z,z+d{\epsilon}],-{\epsilon}^{3/2})}^{([y,y+{\epsilon}],0)}(r^2) \Big) \leq
(2r+3) \cdot 400 C \exp \big\{ - c_1 2^{-10} r^3 \big\} \, ,$$ the conclusion valid under the same hypotheses as in the first case.
We find then that $${\ensuremath{\mathbb{P}}}\Big( \neg \,
{\mathsf{PolyWgtReg}}_{n;(x,-{\epsilon}^{3/2})}^{([x - (r+1){\epsilon},x+(r+1){\epsilon}],0)}(r^2)
\Big) \, + \, {\ensuremath{\mathbb{P}}}\Big( \neg \, {\mathsf{PolyWgtReg}}_{n;([y - (r+1){\epsilon},y+(r+1){\epsilon}],1)}^{(y,1+{\epsilon}^{3/2})}(r^2) \Big)$$ is at most $2 (2r+3) \cdot 400 C \exp \big\{ - c_1 2^{-10} r^3 \big\}$. Thus, $$\begin{aligned}
& & {\ensuremath{\mathbb{P}}}\Big( {\mathrm{MaxDisjtPoly}}_{n;(I,0))}^{(J,1)} \geq k \, , \, \neg \, {\mathsf{FavSurCon}}_{n;(I,0)}^{(J,1)}\big(m; \bar{U},\bar{V} ; {\epsilon},r \big) \Big) \\
& \leq &
10032 \, C \exp \big\{ - c_1 2^{-23} r^{3/2} \big\}
\, + \, 22 Cr \exp \big\{ - 10^{-11} c_1 r^{3/4} \big\} \\
& & \qquad + \,
8 k^2
C_k \exp \big\{ - 2^{-3} k^{-3} c_k r^{3/2} \big\} \, + \, 2 (2r+3) \cdot 400 C \exp \big\{ - c_1 2^{-10} r^3 \big\}
\\
& \leq & 22
Cr \exp \big\{ - 10^{-11} c_k r^{3/4} \big\} \, + \, (10032 + 8 + 4000 ) k^2 r C_k \exp \big\{ - 2^{-23} k^{-3} c_k r^{3/2} \big\}
\\
& \leq & 14062 \, k^2 r C_k \exp \big\{ - 10^{-11} k^{-3} c_k r^{3/4} \big\} \, .\end{aligned}$$ where the second inequality used that the sequence $c_i$ is decreasing, as well as $C_k \geq C$, $k \geq 1$ and $r \geq 1$. The third used $r \geq 1$, $k \geq 1$, and $C_k \geq C$. This completes the proof of Lemma \[l.fsc\].
[**Proof of Proposition \[p.doubletie\].**]{} In this argument, we rigorously implement the resolution of the third challenge of the road map, expressed in outline in the explanation in Section \[s.thirdchallenge\] that led to the two bullet point comments.
Set $I = [x-{\epsilon},x+{\epsilon}]$ and $J = [y - {\epsilon},y+{\epsilon}]$. When ${\mathrm{MaxDisjtPoly}}_{n;(I,0))}^{(J,1)} \geq k$, recall that $\bar{U} \in I^k_\leq$ and $\bar{V} \in J^k_\leq$ are such that the collection $\big\{ \rho_{n;(U_i,0)}^{(V_i,1)}: i \in \llbracket 1,k \rrbracket \big\}$ of $n$-polymers is pairwise horizontally separate. Set $\phi_i = \rho_{n;(U_i,0)}^{(V_i,1)}$ for $i \in \llbracket 1,k \rrbracket$. It is the task of surgery to tie together the $k$ multi-polymer component starting points $\bar{U} \times \{ 0 \}$ to what we may call the ‘lower knot’, $\big(x, - {\epsilon}^{3/2}\big)$, and the $k$ ending points $\bar{v} \times \{ 1 \}$ to the upper knot $\big(y,1 + {\epsilon}^{3/2}\big)$.
Recall the discussion of bouquets and proper weights that followed Proposition \[p.sumweight\]. In surgery, the maximizer $\rho_{n,k;(x\bar{\bf 1},- {\epsilon}^{3/2})}^{{\rm prop};(\bar{U},0)}$ is selected. (This object is unique, because $\bar{U}$ is measurable with respect to randomness in the region ${\ensuremath{\mathbb{R}}}\times [0,\infty)$, so that Lemma \[l.severalpolyunique\] applies in view of (\[e.properweightforward\]) and (\[e.properweightbackward\]).) The maximizer is the lower bouquet and we denote it by $\big(\rho_{11},\cdots,\rho_{1k}\big)$. Forward bouquet regularity ${\mathsf{ForBouqReg}}$ ensures that the bouquet’s weight ${\mathsf{PropWgt}}_{n,k;(x \bar{\bf 1},- {\epsilon}^{3/2})}^{(\bar{U},0)}$ satisfies $$\bigg\vert \, \big({\epsilon}^{3/2} - n^{-1} \big)^{-1/3} {\mathsf{PropWgt}}_{n,k;(x \bar{\bf 1},- {\epsilon}^{3/2})}^{(\bar{U},0)} \, + \, 2^{-1/2} \big({\epsilon}^{3/2} - n^{-1} \big)^{-4/3} \sum_{i=1}^k \big( U_i - x + 2^{-1} n^{-2/3} \big)^2 \, \bigg\vert \, \leq \, r \, .$$ Since $\vert U_i - x \vert \leq {\epsilon}$ and $n{\epsilon}^{3/2} \geq 10^2$, this leads to the simpler $$\label{e.lowersimple}
{\epsilon}^{-1/2} \, \Big\vert \, {\mathsf{PropWgt}}_{n,k;(x \bar{\bf 1},- {\epsilon}^{3/2})}^{(\bar{U},0)} \, \Big\vert
\leq 2^{1/2} (r + k) \, .$$ The story of the upper bouquet’s construction is no different. This bouquet is the maximizer $\rho_{n,k;(\bar{V},1)}^{{\rm prop};(y \bar{\bf 1} ,1 + {\epsilon}^{3/2})}$, is denoted by $\big(\rho_{21},\cdots,\rho_{2k} \big)$, and has weight that satisfies $$\label{e.uppersimple}
{\epsilon}^{-1/2} \, \Big\vert \, {\mathsf{PropWgt}}_{n,k;(\bar{V},1)}^{(y \bar{\bf 1},1 + {\epsilon}^{3/2})} \, \Big\vert \leq 2^{1/2} (r + k)
\, .$$
Surgery is completed by the construction of a $k$-tuple $\big(\rho_1,\cdots,\rho_k\big)$ of $n$-zigzags each running between $(x,- {\epsilon}^{3/2})$ and $(y,1 + {\epsilon}^{3/2})$, where $$\rho_i = \rho_{1i} \circ \phi_i \circ \rho_{2i} \, \, \textrm{for $i \in \llbracket 1,k \rrbracket$} \, .$$ Given that this $k$-tuple has this set of endpoints, the sum of the weights of its elements offers a lower bound on ${\mathsf{Wgt}}_{n,k;(x \bar{\bf 1}, - {\epsilon}^{3/2})}^{(y \bar{\bf 1},1 + {\epsilon}^{3/2})}$. To show, as Proposition (\[p.doubletie\]) asserts, that the latter weight is at least $k \cdot {\mathsf{Wgt}}_{n;(x, - {\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})} \, - \,
15k r^2 {\epsilon}^{1/2}$, we will now argue that this is true of $\sum_{i=1}^k {\mathsf{Wgt}}(\rho_i)$.
The polymer weight ${\mathsf{Wgt}}_{n;(x,- {\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}$ may be written as a sum of three terms, $\omega_1$, $\omega_2$ and $\omega_3$, these being the weights of the three $n$-polymers formed by intersecting $\rho_{n;(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}$ with the strips ${\ensuremath{\mathbb{R}}}\times [ -{\epsilon}^{3/2} , 0]$, ${\ensuremath{\mathbb{R}}}\times [0,1]$ and ${\ensuremath{\mathbb{R}}}\times [1,1 + {\epsilon}^{3/2}]$.
Note then that $$\begin{aligned}
& & \bigg\vert \, \sum_{i=1}^k {\mathsf{Wgt}}(\rho_i) \, \, - \, k \cdot {\mathsf{Wgt}}_{n;(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})} \, \bigg\vert \label{e.returnbound} \\
& \leq & \Big\vert \, {\mathsf{Wgt}}_{n,k;(x \bar{\bf 1},- {\epsilon}^{3/2})}^{(\bar{u},0)} \, \Big\vert \, + \, \Big\vert \, {\mathsf{Wgt}}_{n,k;(\bar{v},1)}^{(y \bar{\bf 1},1 + {\epsilon}^{3/2})} \, \Big\vert \, + \, \Big\vert \sum_{i=1}^k \big( {\mathsf{Wgt}}(\phi_i) - \omega_2 \big) \Big\vert \, + k \big( \vert \omega_1 \vert + \vert \omega_3 \vert \big) \, , \nonumber\end{aligned}$$ since for example $\sum_{i=1}^k {\mathsf{Wgt}}(\rho_{1i}) = {\mathsf{Wgt}}_{n,k;(x \bar{\bf 1},- {\epsilon}^{3/2})}^{(\bar{u},0)}$ and $\sum_{i=1}^k {\mathsf{Wgt}}(\rho_{2i}) = {\mathsf{Wgt}}_{n,k;(\bar{v},1)}^{(y \bar{\bf 1},1+ {\epsilon}^{3/2})}$. The occurrence of ${\mathsf{PolyDevReg}}_{n;(x,-{\epsilon}^{3/2})}^{y,1+{\epsilon}^{3/2}}\big[ \{ 0,1 \} , r \big]$ implies that $$\bigg\vert \, \rho_{n;(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}(0) - \Big( \big( 1 - \tfrac{{\epsilon}^{3/2}}{1 + 2{\epsilon}^{3/2}} \big) x + \tfrac{{\epsilon}^{3/2}}{1 + 2{\epsilon}^{3/2}} y \Big) \, \bigg\vert \, \leq \, r {\epsilon}(1 + 2{\epsilon}^{3/2})^{-2/3}$$ and $$\bigg\vert \, \rho_{n;(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}(1) - \Big( \tfrac{{\epsilon}^{3/2}}{1 + 2{\epsilon}^{3/2}} x + \big( 1 - \tfrac{{\epsilon}^{3/2}}{1 + 2{\epsilon}^{3/2}} \big) y \Big) \, \bigg\vert \, \leq \, r {\epsilon}(1 + 2{\epsilon}^{3/2})^{-2/3}
\, .$$ Since $\vert y - x \vert \leq {\epsilon}^{-1/2}$, these imply that $$\label{e.pairloc}
\Big\vert \, \rho_{n;(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}(0) - x \, \Big\vert \vee \Big\vert \, \rho_{n;(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})}(1) - y \, \Big\vert \, \leq \, (r+1) {\epsilon}\, .$$
The occurrence of ${\mathsf{PolyWgtReg}}_{n;(x,-{\epsilon}^{3/2})}^{([x - (r+1){\epsilon},x+(r+1){\epsilon}],0)}(r^2) \cap {\mathsf{PolyWgtReg}}_{n;([y - (r+1){\epsilon},y+(r+1){\epsilon}],1)}^{(y,1+{\epsilon}^{3/2})}(r^2)$, which is one of the favourable surgical conditions, then ensures that $$\label{e.omega13}
\max \big\{ \vert \omega_1 \vert , \vert \omega_3 \vert \big\} \leq \big( r^2 + 2^{-1/2}(r+1)^2 \big) {\epsilon}^{1/2} \, .$$ It is by considering this bound that we see the reason that the two concerned ${\mathsf{PolyWgtReg}}$ events have been chosen to have the form ‘$(r^2)$’: parabolic curvature introduces a term $2^{-1/2}(r+1)^2 \big) {\epsilon}^{1/2}$ of the form $\Theta(1)r^2 {\epsilon}^{1/2}$ into the right-hand side of (\[e.omega13\]), and the value $r^2$ has an order which is the highest compatible with this right-hand side maintaining the form $\Theta(1)r^2 {\epsilon}^{1/2}$.
For any given $i \in {\llbracket 1,k \rrbracket}$, the quantity ${\mathsf{Wgt}}(\phi_i) - \omega_2$ is the difference in weight of two $n$-polymers whose lifetime is $[0,1]$. The first polymer begins at a location in $[x-{\epsilon},x+{\epsilon}]$ and ends at one in $[y-{\epsilon},y+{\epsilon}]$ while we see from (\[e.pairloc\]) that the second begins in $[x-(r+1){\epsilon},x+(r+1){\epsilon}]$ and ends in $[y-(r+1){\epsilon},y+(r+1){\epsilon}]$. Thus, ${\mathsf{LocWgtReg}}_{n;(I^+,0)}^{(J^+,1)} \big( 2(r+1){\epsilon},r \big)$ entails that $$\label{e.phiomegadifference}
\vert {\mathsf{Wgt}}(\phi_i) - \omega_2 \vert\, \leq \, r \big(2(r+1) {\epsilon}\big)^{1/2}$$ for each $i \in \llbracket 1,k \rrbracket$.
By revisiting the bound (\[e.returnbound\]) equipped with our knowledge of (\[e.omega13\]) and (\[e.phiomegadifference\]) as well as (\[e.lowersimple\]) and (\[e.uppersimple\]), we come to learn that $$\bigg\vert \, \sum_{i=1}^k {\mathsf{Wgt}}(\rho_i) \, - k \cdot {\mathsf{Wgt}}_{n;(x,-{\epsilon}^{3/2})}^{(y,1 + {\epsilon}^{3/2})} \, \bigg\vert \leq \Big(
2^{3/2} (r + k) + k r 2^{1/2} (r+1)^{1/2} + 2k \big( r^2 + 2^{-1/2}(r+1)^2 \big) \Big) \, {\epsilon}^{1/2} \, .$$ Since $r \geq 1$ and $k \geq 1$, the right-hand side is at most $$\big( 2^{3/2}r + 2^{3/2}k + 2kr^{3/2} + 2 ( 1 + 2^{3/2} ) k r^2 \big) {\epsilon}^{1/2} \leq 15 k r^2 {\epsilon}^{1/2} \, .$$ This completes the proof of Proposition \[p.doubletie\].
Proof of Theorem \[t.maxpoly\].
-------------------------------
By the scaling principle, it suffices to consider the case where $t_1 = 0$ and $t_2 = 1$ (so that ${t_{1,2}}= 1$). On the event that ${\mathrm{MaxDisjtPoly}}_{n;([x,x+a],0)}^{([y,y+b],1)} \geq {m}$, let $\bar{U} \in [x,x+a]^{m}_\leq$ and $\bar{V} \in [y,y+b]^{m}_\leq$ be such that the polymer collection $\big\{ \rho_{n;(U_i,t_1)}^{(V_i,t_2)}: i \in \llbracket 1,{m}\rrbracket \big\}$ is pairwise horizontally separate. At the start of Section \[s.proof\] an explicit means for the selection of a measurable choice of $(\bar{U},\bar{V})$ is offered.
Recall that we denote $h = a \vee b$. Each sequence $\big\{ U_{i+1} - U_i: i \in \llbracket 1, {m}-1 \rrbracket \big\}$ and $\big\{ V_{i+1} - V_i: i \in \llbracket 1, {m}-1 \rrbracket \big\}$ consists of positive terms that sum to at most $h$. Let ${k}\in {\llbracket 1,{m}-1 \rrbracket}$; the value of ${k}$ will be fixed later. Call an index $i \in {\llbracket 1,{m}- 1 \rrbracket}$ [*unsuitable*]{} if $( U_{i+1} - U_i) \vee ( V_{i+1} - V_i) > 2h({k}+1)({m}-1)^{-1}$. Note that the number of unsuitable indices is less than $({m}-1)({k}+1)^{-1}$, and therefore at most $\lfloor ({m}-1)({k}+1)^{-1} \rfloor$. When ${k}\in {\llbracket 1,{m}-1 \rrbracket}$ is chosen so that $5 {k}^2 \leq {m}$, we claim that there exist ${k}$ consecutive indices $i \in {\llbracket 1,{m}-1 \rrbracket}$ such that $( U_{i+1} - U_i) \vee ( V_{i+1} - V_i) \leq 2h({k}+1)({m}-1)^{-1}$. Indeed, were there not such an interval, the number of unsuitable indices would be at least $\lfloor ({m}-1){k}^{-1} \rfloor$, and this is incompatible with our upper bound, since $({m}-1){k}^{-1}$ exceeds $({m}-1)({k}+1)^{-1}$ by at least one when $5 {k}^2 \leq {m}$.
Let $\Theta \in {\llbracket 1,{m}- {k}\rrbracket}$ be chosen so that $\llbracket \Theta, \Theta + {k}- 1 \rrbracket$ is an interval of such not unsuitable indices. The polymers $\rho_{n;(U_i,0)}^{(V_i,1)}$ with index $i \in \llbracket \Theta, \Theta + {k}- 1 \rrbracket$ start and end in the planar intervals $$\big[ U_\Theta, U_\Theta + 2h{k}({k}+1)({m}-1)^{-1} \big] \times \{ 0 \} \, \, \, \,
\textrm{and} \, \, \, \, \big[ V_\Theta, V_\Theta + 2h{k}({k}+1)({m}-1)^{-1} \big] \times \{ 1 \} \, .$$ These planar intervals are contained in $$\big[ U_{\Theta}^-, U_{\Theta}^- + (2h{k}+1)({k}+1)({m}-1)^{-1} \big] \times \{ 0 \} \, \, \, \,
\textrm{and} \, \, \, \, \big[ V^-_{\Theta}, V^-_{\Theta} + (2h{k}+1)({k}+1)({m}-1)^{-1} \big] \times \{ 1 \} \, ,$$ where we set $$U^-_{\Theta} = ({k}+1)({m}-1)^{-1} \big\lfloor ({m}-1)({k}+1)^{-1} U_\Theta
\big\rfloor \, \, \, \textrm{and} \, \, \,
V^-_{\Theta} = ({k}+1)({m}-1)^{-1} \big\lfloor ({m}-1)({k}+1)^{-1} V_\Theta
\big\rfloor$$ to be left-displacements onto a $({k}+1)({m}-1)^{-1}$-mesh. That is, $$\Big\{ {\mathrm{MaxDisjtPoly}}_{n;([x,x+a],0)}^{([y,y+b],1)} \geq {m}\Big\} \, \subseteq \, \bigg\{ {\mathrm{MaxDisjtPoly}}_{n;
\big( \big[ U^-_{\Theta}, U^-_{\Theta} + (2h{k}+1)({k}+1)({m}-1)^{-1} \big] , 0 \big)}^{\big( \big[ V^-_{\Theta}, V^-_{\Theta} + (2h{k}+1)({k}+1)({m}-1)^{-1} \big] , 1 \big)} \geq {k}\bigg\} \, .$$ This is a useful moment to recall that, in a remark in Section \[s.closure\], we permitted that the definition of the ${\mathrm{MaxDisjtPoly}}$ random variable be respecified so that the concerned polymers are merely pairwise horizontally separate, rather than pairwise disjoint. We actually need to use this form of the definition to obtain the preceding inequality. Indeed, our explicit selection of $(\bar{U},\bar{V})$ means that we are forced to deal with polymers that may be merely horizontally separate.
Let $X$ denote the set formed by adding to the mesh points in $(x,x+a]$ the greatest mesh point that is at most $x$, and let $Y$ be the counterpart set where $(y,y+b]$ replaces $(x,x+a]$. The cardinality of $X$ is at most $a({m}-1)({k}+1)^{-1} + 2$ and thus, in view of $a \geq 1$ and ${m}\geq {k}^2$, at most $2a{m}{k}^{-1}$. Similarly, $\vert Y \vert \leq 2b{m}{k}^{-1}$. Since $U^-_\Theta \in U$ and $V^-_\Theta \in V$, we find that $$\begin{aligned}
& & {\ensuremath{\mathbb{P}}}\Big( {\mathrm{MaxDisjtPoly}}_{n;([x,x+a],0)}^{([y,y+b],1)} \geq {m}\Big) \nonumber \\
& \leq & \sum_{u \in X ,v \in Y} {\ensuremath{\mathbb{P}}}\bigg({\mathrm{MaxDisjtPoly}}_{n;
\big( \big[ u , u + (2h{k}+1)({k}+1)({m}-1)^{-1} \big] , 0 \big)}^{\big( \big[ v , v + (2h{k}+1)({k}+1)({m}-1)^{-1} \big] , 1 \big)} \geq {k}\bigg) \nonumber \\
& \leq & 2a{m}{k}^{-1} \cdot 2b{m}{k}^{-1} \cdot
\big( 15h {k}^2/4 \cdot {m}^{-1} \big)^{({k}^2 - 1)/2} \nonumber \\
& & \qquad
\cdot
10^{26{k}^2} {k}^{43{k}^2/3} c_{k}^{-8{k}^2/3} C_{k}\big( \log {m}{k}^{-2} \big)^{4{k}^2} \exp \big\{ \beta_{k}\big( \log {m}{k}^{-2} \big)^{5/6} \big\} \nonumber \\
& \leq & {m}^{5/2 -{k}^2/2} \cdot h^{({k}^2 + 3)/2}
10^{27{k}^2} {k}^{46{k}^2/3} c_{k}^{-8{k}^2/3} C_{k}\big( \log {m}\big)^{4{k}^2} \exp \big\{ \beta_{k}\big( \log {m}\big)^{5/6} \big\} \, . \label{e.expfinalline}\end{aligned}$$
The last inequality is nothing more than a simplifying of terms that uses $h = a \vee b$ and ${k}\geq 1$. In the inequality that precedes it, the summand in the second line is bounded above by using Theorem \[t.disjtpoly\] with parameter settings ${\bf t_1} = 0$, ${\bf t_2} = 1$, ${\bf x} = u + \tfrac{1}{2}(2h{k}+1)({k}+1)({m}-1)^{-1}$, ${\bf y} = v + \tfrac{1}{2}(2h{k}+1)({k}+1)({m}-1)^{-1}$, ${\bf m} = {k}$ and $${\bm {\epsilon}} = \tfrac{1}{2}(2h{k}+1)({k}+1)({m}-1)^{-1} \, .$$
In this second inequality, we also make use of the bounds $$\label{e.eulbounds}
{k}^2 {m}^{-1} \leq {\bm {\epsilon}} \leq \tfrac{15}{4} h {k}^2 {m}^{-1} \, ,$$ where the latter bound is due to ${k}\geq 2$.
Of course, the hypotheses of Theorem \[t.disjtpoly\] must be validated with the above parameter settings for the above bound to hold. In fact, we will now set the value of the parameter ${k}\in {\ensuremath{\mathbb{N}}}$ in terms of ${m}$, and then justify that, for this choice, these hypotheses are indeed validated.
Recall from Theorem \[t.maxpoly\] that the constant $\beta$ is set equal to $e \vee \limsup_{i \in {\ensuremath{\mathbb{N}}}} \beta_i^{1/i}$ and that Corollary \[c.neargeod.t\] implies that $\beta < \infty$. Since $\beta > 1$, we may choose ${k}\in {\ensuremath{\mathbb{N}}}$ to be maximal so that $\beta^{k}\leq \big( \log {m}\big)^{1/12}$. That is, ${k}= \lfloor \tfrac{1}{12 \log \beta} \log \log {m}\rfloor$. Recalling that $\liminf c_i^{1/i}$ is positive, and also that $c_{k}\leq c$, we see that the condition (\[e.epsilonbound\]) on ${\bm {\epsilon}}$ in Theorem \[t.disjtpoly\] is met provided that ${m}\geq {m}_0$, where ${m}_0 \in {\ensuremath{\mathbb{N}}}$ is a certain constant; a suitable choice of ${m}_0$ also ensures that ${k}\geq 2$.
Also note that $\vert {\bf x} - {\bf y} \vert = \vert u - v \vert \leq \vert x - y \vert + 2h$ because $\vert u - v \vert
\leq \vert x - y \vert + h + ({k}+1)({m}-1)^{-1}$ while $h \geq 1 \geq ({k}+1)({m}-1)^{-1}$ due to ${m}\geq {k}+ 2$.
Since ${k}\leq (12)^{-1} \log \log {m}$ (due to $\beta \geq e$), $K_0 \geq 1$, ${\bm {\epsilon}} \geq {m}^{-1}$ and $\liminf c_i^{1/i} > 0$, the bound (\[e.nlowerbound\]) is verified when $n$ is at least $2(K_0)^{(12)^{-2} (\log \log {m})^2} \big( \log {m}\big)^{K_0}$ and $$\max \bigg\{ \,
10^{584} \big( (12)^{-1} \log \log {m}\big)^{240} {c}^{-36} {m}^{225} \max \big\{ 1 \, , \, (\vert y - x \vert + 2h)^{36} \big\} \, , \,
a_0^{-9} ( \vert y - x \vert + 2h ) \, \bigg\} \, ,$$ where we used ${m}\geq {m}_0$ and adjusted the value of ${m}_0$ if need be. Finally, the condition $$\vert {\bf y} - {\bf x} \vert ({\bf {t_{1,2}}})^{-2/3} \leq {\bm {\epsilon}}^{-1/2} \big( \log {\bm {\epsilon}}^{-1} \big)^{-2/3} \cdot 10^{-8} c_{k}^{2/3} {k}^{-10/3}$$ is verified provided that ${m}\geq {m}_0 \vee \big( \vert x - y \vert + 2h \big)^3$ in light of ${\bm {\epsilon}} \leq {m}^{-1 + o(1)}$ (with an increase if necessary in the value of ${m}_0$). This lower bound on $n$ is implied by Theorem \[t.maxpoly\]’s hypothesis (\[e.nmaxpoly\]) due to $\log \log {m}\leq {m}$.
Given the choice of ${k}$, we see that the expression (\[e.expfinalline\]) is at most $${m}^{- (145)^{-1} ( \log \beta)^{-2} (0 \vee \log \log {m})^2} h^{(\log \beta)^{-2} (0 \vee \log \log {m})^2/{288} + 3/2} {H}_{m}\, ,$$ where $\big\{ {H}_i: i \in {\ensuremath{\mathbb{N}}}\big\}$ is a sequence of positive constants such that $\sup_{i \in {\ensuremath{\mathbb{N}}}} {H}_i \exp \big\{ - 2 (\log i)^{11/12} \big\}$ is finite.
This proves Theorem \[t.maxpoly\] when $t_0 = 0$ and $t_1 = 1$; as we noted at the outset, there is no loss of generality in considering this case.
Polymer fluctuation: proving Theorem \[t.polyfluc\]. {#s.polyfluc}
====================================================
In this section, we will derive the polymer fluctuation Theorem \[t.polyfluc\]. The result asserts that any of the polymers that cross between unit-order length intervals separated at unit-order times deviates by a distance $r$ from the line segment that interpolates its endpoints with probability at most $\exp \big\{ - O(1) r^{3/4} \big\}$, uniformly in high choices of the scaling parameter $n$. We may easily reduce to the case where these unit-order intervals are instead singleton sets, however, by a simple application of polymer ordering. In Proposition \[p.polyfluc\], this reduced version of the theorem has been stated. After noting how this proposition implies Theorem \[t.polyfluc\], we will turn to discuss the ideas of the proof of the proposition, and then give the proof itself.
[**Proof of Theorem \[t.polyfluc\].**]{} For $u,v \in {\ensuremath{\mathbb{R}}}$, consider the random variable $$X_{n;(u,t_1)}^{(v,t_2)}(a) = t_{1,2}^{-2/3} \big( a \wedge (1-a) \big)^{-2/3} \Big( \, \rho_{n;(u,t_1)}^{(v,t_2)} \big( (1-a) t_1 + a t_2 \big) - \ell_{(u,t_1)}^{(v,t_2)} \big( (1-a) t_1 + a t_2 \big) \, \Big) \, .$$ Set ${\varphi}= t_{1,2}^{2/3} \big( a \wedge (1-a) \big)^{2/3} r$, and recall from the theorem’s statement that we define $I = [x,x+{\varphi}]$ and $J = [y,y+{\varphi}]$. When the event $\neg \, {\mathsf{PolyDevReg}}_{n;(I,t_1)}^{(J,t_2)}\big(a,2r\big)$ occurs, we may choose $(U,V) \in I \times J$ such that $\big\vert X_{n;(U,t_1)}^{(V,t_2)}(a) \big\vert \geq 2{\varphi}$. It might seem reassuring to know that this selection may be made measurably, and with a modicum of effort we might show this, but in fact this information is not needed.
Recall the polymer sandwich Lemma \[l.sandwich\]. By applying this result with parameter settings ${\bf x_1} = x$, ${\bf x_2} = x + {\varphi}$, ${\bf y_1} = y$, ${\bf y_2} = y + {\varphi}$, we readily find that at least one of $\big\vert X_{n;(x,t_1)}^{(y,t_2)}(a) \big\vert$ and $\big\vert X_{n;(x+{\varphi},t_1)}^{(y+{\varphi},t_2)}(a) \big\vert$ is at least $\big\vert X_{n;(U,t_1)}^{(V,t_2)}(a) \big\vert - {\varphi}$. Thus, $$\begin{aligned}
& & {\ensuremath{\mathbb{P}}}\Big( \neg \, {\mathsf{PolyDevReg}}_{n;(I,t_1)}^{(J,t_2)}\big(a,2r\big) \Big) \\
& \leq &
{\ensuremath{\mathbb{P}}}\Big( \neg \, {\mathsf{PolyDevReg}}_{n;(x,t_1)}^{(y,t_2)}\big(a,r\big) \, \Big) \, + \,
{\ensuremath{\mathbb{P}}}\Big( \neg \, {\mathsf{PolyDevReg}}_{n;(x+{\varphi},t_1)}^{(y+{\varphi},t_2)}\big(a,r\big) \, \Big) \, .\end{aligned}$$ Two applications of Proposition \[p.polyfluc\], with $({\bf x},{\bf y})$ equal to $(x,y)$ and $(x+{\varphi},y+{\varphi})$, complete the proof.
We now overview the ideas of the proof of Proposition \[p.polyfluc\]. Recall from Section \[s.roadmap\] the role of the powers of one-half and one-third. The one-half power law is articulated in Theorem \[t.differenceweight\]: polymer weight has a Hölder exponent of one-half in response to horizontal displacement of endpoints. The one-third power law is articulated by the scaling principle alongside the regular sequence conditions ${{\rm Reg}}(2)$ and ${{\rm Reg}}(3)$: a polymer of lifetime $t$ whose endpoints differ by $r t^{2/3}$ has weight $t^{1/3}\big( U - 2^{-1/2} r^2 \big)$; here, $U$ is a unit-order, random, quantity.
These principles come into conflict unless the two-thirds principle for polymer geometry (which we are trying to prove) also obtains.
To explain this concept in more detail, it is useful to begin by fixing a little notation. We set $Z = \rho_{n;(x,t_1)}^{(y,t_2)} \big( (1-a)t_1 + a t_2 \big)$ and $z = \ell_{(x,t_1)}^{(y,t_2)} \big( (1-a) t_1 + a t_2 \big)$. Recall that $a$ is supposed to be close to one, and that we seek to show that $Z$ is close to $z$. Using notation already seen in the proof of Theorem \[t.polyfluc\], we write ${\varphi}= t_{1,2}^{2/3} (1-a)^{2/3} r$. The factor of $t_{1,2}^{2/3} (1-a)^{2/3}$ is a suitable scale for judging polymer fluctuation between the pairs of times $(1-a)t_1 + a t_2$ and $t_2$, and our task is to show that it is a rare event, with a decaying probability expressed in terms of $r$, that $Z \leq z - {\varphi}$.
![The polymer $\rho = \rho_{n;(x,t_1)}^{(y,t_2)}$ is split at time $(1-a)t_1 + at_2$ and the resulting subpaths are labelled $\rho[1]$ and $\rho[2]$. The polymer $\rho[3]$ has a less hectic journey than does $\rho[2]$ during their shared lifetime. The concatenation $\rho[1] \circ \rho[3]$ will be labelled $\hat\rho$ in the upcoming proof of Proposition \[p.polyfluc\] and shown in cases such as that depicted to have a significantly greater weight than $\rho$.[]{data-label="f.threerho"}](NonIntPolyThreeRho.pdf){height="8cm"}
Abbreviate $\rho = \rho_{n;(x,t_1)}^{(y,t_2)}$ and consult Figure \[f.threerho\]. The polymer $\rho$ may be split into two pieces, $\rho[1]$ and $\rho[2]$, by cutting it at the point $(Z,(1-a)t_1 + a t_2)$. If the big fluctuation event $Z \leq z - {\varphi}$ occurs, then $\rho[2]$ makes a rather sudden deviation, and its weight is dictated by parabolic curvature to be of order $-r^2 (1-a)^{1/3} {t_{1,2}}^{1/3}$. We may consider the polymer $\rho[3]$ whose endpoints are the cut location $(Z,(1-a)t_1 + a t_2)$ and $(z - {\varphi},t_2)$. The latter endpoint being $(z - {\varphi},t_2)$ means that $\rho[3]$ is not making the sudden deviation that $\rho[2]$ does, so that its weight is dictated by local randomness to be a random unit-order multiple of $(1-a)^{1/3} {t_{1,2}}^{1/3}$. Consider now the polymer weight profile ${\mathsf{Wgt}}_{n;(x,t_1)}^{(v,t_2)}$ as a function of $v$. Between the locations $v = y$ and $v = z - {\varphi}$, which is an order of ${\varphi}$ to the left of $y$, the weight profile has risen by an order of least $-r^2 (1-a)^{1/3} {t_{1,2}}^{1/3}$, because the weight profile at $v = z - {\varphi}$ is at least the sum of the weights of $\rho[1]$ and $\rho[3]$, while at $v = z$, it equals the sum of the weights of $\rho[1]$ and $\rho[2]$.
Recalling that ${\varphi}= t_{1,2}^{2/3} (1-a)^{2/3} r$, we see that the weight profile is experiencing a change where a factor $r^{3/2}$ multiplies the square-root $\Theta\big({\varphi}^{1/2}\big)$ of the horizontal displacement in endpoint location. When $r$ is large, this is incompatible with the one-half power law principle recalled above.
This idea is important, but it is not adequate to prove Proposition \[p.polyfluc\]. To understand why the idea is not the end of the story, examine the form of Corollary \[c.ordweight\], which articulates the one-half power law for the polymer weight profile. The quantity ${\epsilon}$, whose role is to measure horizontal displacement, must be at most $2^{-4}$. Indeed, the one-half power law does not govern the weight profile beyond a unit scale. This means that our idea will work only in a case that we may call [*near*]{}: when the quantity $r$ is large, but not so large that the concerned endpoint displacement goes above a unit scale.
When $r$ is large enough that the one-half power law estimate cannot be applied, we will note that the polymer weight profile change is unlikely for other reasons. Indeed, since $r$ is so large, the observed change in the weight profile is very large. In fact, this change is so large that it either forces the profile at value $v = y$ to be abnormally low or the profile at $v = z - {\varphi}$ to be abnormally high. The improbability of these outcomes may then be gauged using the regular ensemble conditions ${{\rm Reg}}(2)$ and ${{\rm Reg}}(3)$.
In fact, even this is not the end of the story. These two regular ensemble conditions capture the parabolic curvature of the weight profile. But this curvature breaks down on an extremely large scale, as witnessed by the upper bounds on $\vert z \vert$ needed in ${{\rm Reg}}(2)$ and ${{\rm Reg}}(3)$. This second argument thus works when $r$ is very large, but not extremely large (where the latter term involves an $n$ dependence). For this reason, this second argument will be said to apply in the [*middle-distant*]{} case.
When $r$ is extremely high, the linear (but not parabolic) collapse near infinity of the polymer weight profile expressed by Proposition \[p.mega\](4) will replace the use of ${{\rm Reg}}(3)$. This third argument applies in what we will call the [*far*]{} case.
This then will be the structure of the argument, with three different estimates treating the near, middle-distant and far cases, in increasing order of the value of $r$.
[**Proof of Proposition \[p.polyfluc\].**]{} Our task is to understand the fluctuation of the polymer $\rho_{n;(x,t_1)}^{(y,t_2)}$ at time $(1-a)t_1 + a t_2$. Note first that, since the endpoints are given, we may harmlessly suppose that this polymer is unique, because, as we discussed in Section \[s.brlpp\], the complementary event has zero probability. Note also that we are working in a case where $a$ is close to one; specifically, $a \geq 1/2$. (The remark following the proposition treats the opposing case.)
For $K$ a real interval, we write ${\mathsf{Fluc}}_{n;(x,t_1)}^{(y,t_2)}\big(a;K \big)$ for the event that $X_{n;(x,t_1)}^{(y,t_2)}(a) \in K$. We omit the parameters that are fixed from this notation, writing for example ${\mathsf{Fluc}}[s,t] = {\mathsf{Fluc}}_{n;(x,t_1)}^{(y,t_2)}\big(a;[s,t]\big)$ for $(s,t) \in {\ensuremath{\mathbb{R}}}^2_<$.
We will argue that, under the proposition’s hypotheses, $$\label{e.fminusinfinityr}
{\ensuremath{\mathbb{P}}}\Big( {\mathsf{Fluc}}(-\infty,-r] \Big) \, \leq \,
22 C r \exp \big\{ - 10^{-11} c_1 r^{3/4} \big\} \, ,$$ and that the same bound holds on ${\ensuremath{\mathbb{P}}}\big( {\mathsf{Fluc}}[r,\infty) \big)$. Indeed, since the event $\neg \, {\mathsf{PolyDevReg}}_{n;(x,t_1)}^{(y,t_2)}\big(a,r\big)$ equals the union of ${\mathsf{Fluc}}(-\infty,-r)$ and ${\mathsf{Fluc}}(r,\infty)$, these bounds suffice to prove Proposition \[p.polyfluc\].
To derive the two bounds, it suffices to prove (\[e.fminusinfinityr\]). The two parts of the next proposition and the following lemma are the three principal tools: they treat the near, middle-distant and far cases. The reader should bear in mind that, in these assertions and later ones, it is supposed, as it was in Proposition \[p.polyfluc\], that $(n,t_1,t_2)$ is a compatible triple, and that $a{t_{1,2}}\in n^{-1} {\ensuremath{\mathbb{Z}}}$.
\[p.feventfirstbound\] Set $\hat{n} = n (1-a) {t_{1,2}}$. Suppose that $$\label{e.firstbound.h1}
\hat{n} \geq 10^{32} c^{-18} \, \, , \, \,
r \in \Big[ 8 \vee 87(1-a)^{1/3} {t_{1,2}}^{-2/3} \vert x - y \vert \, , \, 3 \hat{n}^{1/36} \Big]$$ and $$\label{e.firstbound.h2}
\big\vert x - y \big\vert {t_{1,2}}^{-2/3} \leq 2^{-3} 3^{-1} {c}(n {t_{1,2}})^{1/18} \, .$$
1. Suppose in addition that $$r \in \big[ 2800 \, , \, \tfrac{1}{18} (1-a)^{-2/3} \big]$$ and $a \geq 1 - 10^{-3}$. Then $${\ensuremath{\mathbb{P}}}\Big( {\mathsf{Fluc}}\big[-(r+1),-r\big] \Big) \, \leq \,
(10178 + 3r) \, C \exp \big\{ - c_1 2^{-26} 3^{-3} 5^{3/2} r^{9/4} \big\}
$$
2. Now suppose in addition to (\[e.firstbound.h1\]) and (\[e.firstbound.h2\]) that $r \geq 6(1-a)^{-1/2}$ and $a \geq 1 - 10^{-2}$. Then $${\ensuremath{\mathbb{P}}}\Big( {\mathsf{Fluc}}\big[-(r+1),-r\big] \Big)
\, \leq \,
\big( 148 C + 9r/4 \big) \exp \big\{ - c_1 (1-a)^{1/2} 3^{-3/2} 5^{3/2} 2^{-43/4} r^3 \big\} \, .$$
\[l.feventhigh\] Suppose that $a \in (1/2,1)$, and that $n {t_{1,2}}$ is at least the maximum of $$(1-a)^{-25} {t_{1,2}}^{-24} \vert x - y \vert^{36} \, \, , \, \,
c^{-9} {t_{1,2}}^{-6} \vert x - y \vert^9 \, \, \textrm{ and } \, \, 10^5 (1-a)^{-17} c^{-12}
\, .$$ Then $${\ensuremath{\mathbb{P}}}\Big( {\mathsf{Fluc}}\big(-\infty,-2\hat{n}^{1/36} \big] \Big) \, \leq \, 13C \exp \Big\{ - 2^{-15/4} c (1-a)^{25/12} {t_{1,2}}^{1/12} n^{1/12} \Big\} \, .$$
We now derive (\[e.fminusinfinityr\]) using this proposition and lemma. The interval $(-\infty,-r]$ may be partitioned into a [*far*]{} interval $(-\infty,-2\hat{n}^{1/36}]$ and a collection of disjoint unit intervals of the form $(-t-1,-t]$ where $t - r$ varies over a finite initial sequence of integers; i.e., $t = -r$, then $t=-r-1$, and so on. (In fact, one of the unit intervals may overlap with the far interval.) The unit intervals to the left of $-\tfrac{1}{18} (1-a)^{-2/3}$ will be called [*middle-distant*]{}, and the remaining ones, closer to the origin, will be called [*near*]{}. To each of these intervals $I$ corresponds an event ${\mathsf{Fluc}}\, I$. In the case of the far interval, the event’s probability is bounded above by Lemma \[l.feventhigh\]. Proposition \[p.feventfirstbound\](2) provides the bound for the middle-distant intervals, and Proposition \[p.feventfirstbound\](1) for the near intervals. Note that Proposition \[p.feventfirstbound\](2) is applicable to the middle-distant intervals because the condition $\tfrac{1}{18} (1-a)^{-2/3} \geq 5(1-a)^{-1/2}$, i.e., $a \geq 1 - 90^{-6}$, is implied by Proposition \[p.polyfluc\]’s hypothesis that $a \geq 1 - 10^{-11} c_1^2$ alongside $c_1 \leq 1/8$.
We are summing the right-hand sides of both parts of Proposition \[p.feventfirstbound\]. The next two lemmas provide an upper bound on these sums. The calculational proofs appear in the online Appendix $D$.
\[l.calcone\] Suppose that $r \geq 10^7 c_1^{-4/5}$. We have that $$\sum (10178 + 3r) \, C \exp \big\{ - c_1 2^{-26} 3^{-3} 5^{3/2} s^{9/4} \big\}
\leq 8 C r \exp \big\{ - c_1 10^{-8} r^{9/4} \big\}
\, ,$$ where the sum is taken over values of $s \in {\ensuremath{\mathbb{R}}}$ that satisfy $s \in r + {\ensuremath{\mathbb{N}}}$.
\[l.calctwo\] Suppose now that $r \geq \tfrac{1}{18} (1-a)^{-2/3} \vee 15 C^{1/2}$ as well as $a \geq 1 - 10^{-11} c_1^2$. Setting $\alpha = (5/3)^{3/2} 2^{-43/4}$, we have that $$\begin{aligned}
& & \sum
\big( 148 C + 9s/4 \big) \exp \big\{ - c_1 (1-a)^{3/2} (5/3)^{3/2} \cdot 2^{-43/4} s^3 \big\} \\
& \leq &
\exp \big\{ - 10^{-2} c_1 \alpha r^{9/4} \big\}
\, ,
\end{aligned}$$ where the sum is again over $s \in r + {\ensuremath{\mathbb{N}}}$.
The bound (\[e.fminusinfinityr\]) may now be obtained using Lemma \[l.feventhigh\] alongside the two preceding lemmas. Indeed, we obtain an upper bound on ${\ensuremath{\mathbb{P}}}\big( {\mathsf{Fluc}}(-\infty,-r] \big)$ of the form $$9 C r \exp \big\{ - 10^{-8} c_1 r^{3/4} \big\} \, + \, 13C \exp \Big\{ - 2^{-15/4} c_1 (1-a)^{25/12} {t_{1,2}}^{1/12} n^{1/12} \Big\}$$ because this expression is, in view of $C \geq 1$, $r \geq 1$ and $c_1 \leq c$, an upper bound on the sum of the right-hand sides in Lemmas \[l.feventhigh\], \[l.calcone\] and \[l.calctwo\]. That this expression is under the hypotheses of Proposition \[p.polyfluc\] at most the right-hand side of (\[e.fminusinfinityr\]) is a calculational matter. This assertion is stated as Lemma $D.3$ in the online Appendix $D$ and derived there. This completes the proof of (\[e.fminusinfinityr\]), and Proposition \[p.polyfluc\], subject to confirming Proposition \[p.feventfirstbound\] and Lemma \[l.feventhigh\].
We now prepare to prove this proposition and lemma. Recall throughout that we are supposing that $a \geq 1/2$.
Recalling notation that was used in the overview that preceded the present proof, we write ${\varphi}= t_{1,2}^{2/3} (1-a)^{2/3} r$. We also set $Z = \rho_{n;(x,t_1)}^{(y,t_2)} \big( (1-a)t_1 + a t_2 \big)$ and $z = \ell_{(x,t_1)}^{(y,t_2)} \big( (1-a) t_1 + a t_2 \big)$.
Abbreviate $\rho = \rho_{n;(x,t_1)}^{(y,t_2)}$. We now split $\rho$ at the point $(Z,(1-a)t_1 + a t_2)$ according to polymer splitting as described in Section \[s.split\]. Indeed, we may specify $\rho[1]$ and $\rho[2]$ to be the polymers $\rho_{n;(x,t_1)}^{(Z,(1-a)t_1 + a t_2)}$ and $\rho_{n;(Z, (1-a) t_1 + a t_2)}^{(y,t_2)}$. Thus, $\rho = \rho[1] \circ \rho[2]$. Let $\rho[3]$ denote the polymer $\rho_{n;(Z, (1-a) t_1 + a t_2)}^{(z - {\varphi},t_2)}$. Let $\hat\rho = \rho[1] \circ \rho[3]$. Thus, $\rho$ and $\hat\rho$ are two polymers that begin from $(x,t_1)$ and follow a shared course until time $(1-a) t_1 + a t_2$. They continue on possibly separate trajectories until a shared ending time $t_2$, with $\rho$ then reaching $y$ and $\hat\rho$ reaching $z - {\varphi}= y + (1-a)(x-y) - {r}(1-a)^{2/3} {t_{1,2}}^{2/3}$.
The next lemma is a tool to prove the near and middle-distant Proposition \[p.feventfirstbound\]. It asserts that, when ${\mathsf{Fluc}}\, [-(r+1),-r]$ occurs, the polymer $\rho[3]$ is unlikely to have a significantly negative weight, while $\rho[2]$ is unlikely to have a weight much exceeding a certain constant multiple of $- r^2$; thus, when $r$ is chosen so that either the near and middle-distant case obtains, $\rho[3]$ is typically much heavier than $\rho[2]$ on the event ${\mathsf{Fluc}}\, [-(r+1),-r]$.
\[l.combine\] Suppose that the hypotheses (\[e.firstbound.h1\]) and (\[e.firstbound.h2\]) hold.
1. We have that $${\ensuremath{\mathbb{P}}}\Big( \, {\mathsf{Fluc}}\, [-(r+1),-r] \, , \, t_{1,2}^{-1/3} (1-a)^{-1/3} {\mathsf{Wgt}}\big(\rho[3]\big) \leq - 2^{-1/2} - s \Big) \leq 140 C \, \exp \big\{ - c_1 s^{3/2} \big\} \, .$$
2. We also have $$\begin{aligned}
& & {\ensuremath{\mathbb{P}}}\Big( \, {\mathsf{Fluc}}\, [-(r+1),-r] \, , \, t_{1,2}^{-1/3} (1-a)^{-1/3} {\mathsf{Wgt}}\big(\rho[2]\big) \geq - 2^{-1/2} (7r/8)^2 + s \Big) \\
& \leq & \big(3r/8 + 1 \big) \cdot 6 C \exp \big\{ - 2^{-11/2} c s^{3/2} \big\} \, .
\end{aligned}$$
[**Proof.**]{} Note that $${\mathsf{Wgt}}\big( \rho[3] \big) = {\mathsf{Wgt}}_{n;\big(Z, (1-a) t_1 + a t_2 \big)}^{(z - {\varphi},t_2)} = {\mathcal}{L}_{n;(1-a) t_1 + a t_2}^{\downarrow;(z-{\varphi},t_2)}\big( 1, Z \big) = (1-a)^{1/3} {t_{1,2}}^{1/3} {\mathsf{Nr}{\mathcal}{L}}_{n;(1-a) t_1 + a t_2}^{\downarrow;(z-{\varphi},t_2)}\big( 1, V \big)$$ where $V = (1-a)^{-2/3} {t_{1,2}}^{-2/3} \big( Z- (z-{\varphi}) \big)$. Note that the event ${\mathsf{Fluc}}\, [-(r+1),-r]$ is characterized by $- (r+1) \leq (1-a)^{-2/3} {t_{1,2}}^{-2/3} \big( Z- z) \leq -r$ and thus also by the condition that $V \in [-1,0]$. Thus, $$\begin{aligned}
& & {\ensuremath{\mathbb{P}}}\Big( \, {\mathsf{Fluc}}\, [-(r+1),-r] \, , \, t_{1,2}^{-1/3} (1-a)^{-1/3} {\mathsf{Wgt}}\big(\rho[3]\big) \leq - 2^{-1/2} - s \Big) \nonumber \\
& = &
{\ensuremath{\mathbb{P}}}\Big( \, {\mathsf{Fluc}}\, [-(r+1),-r] \, , \, {\mathsf{Nr}{\mathcal}{L}}_{n;(1-a) t_1 + a t_2}^{\downarrow;(z-{\varphi},t_2)}\big( 1, V \big)
\leq - 2^{-1/2} - s \Big) \nonumber \\
& \leq &
{\ensuremath{\mathbb{P}}}\Big( \, {\mathsf{Fluc}}\, [-(r+1),-r] \, , \, {\mathsf{Nr}{\mathcal}{L}}_{n;(1-a) t_1 + a t_2}^{\downarrow;(z-{\varphi},t_2)}\big( 1, V \big)
+ 2^{-1/2} V^2 \leq - s \Big) \nonumber \\
& \leq &
{\ensuremath{\mathbb{P}}}\bigg( \inf_{-1 \leq v \leq 0} \Big( {\mathsf{Nr}{\mathcal}{L}}_{n;(1-a) t_1 + a t_2}^{\downarrow;(z-{\varphi},t_2)}\big( 1, v \big)
+ 2^{-1/2} v^2 \Big) \leq - s \bigg) \nonumber \\
& \leq & \Big( 1/2 \vee 5 \vee 5^{1/2} (3 - 2^{3/2})^{-1} \Big) \, 10 C \, \exp \big\{ - c_1 s^{3/2} \big\} \leq 140 C \, \exp \big\{ - c_1 s^{3/2} \big\} \, , \nonumber\end{aligned}$$ so that we obtain Lemma \[l.combine\](1). The final inequality was obtained by applying Proposition \[p.mega\](2) with the parameter choices ${\bf k} = 1$, ${\bf y} = -1/2$ , ${\bf t} = 1/2$ and ${\bf r} = s$ to the $(c,C)$-regular ensemble ${\mathsf{Nr}{\mathcal}{L}}_{n;(1-a) t_1 + a t_2}^{\downarrow;(z-r,t_2)}$. The ensemble has $\hat{n} + 1$ curves, where recall that we denote $\hat{n} = n (1-a) {t_{1,2}}$. As such, this application of Proposition \[p.mega\](2) may be made provided that $$1/2 \leq c/2 \cdot \hat{n}^{1/18} \, , \,
1/2 \leq \hat{n}^{1/18} \, , \, s \in \big[ 2^{3/2} \, , \, 2 \hat{n}^{1/18} \big] \, \, \textrm{ and } \, \, \hat{n} \geq 1 \vee (c/3)^{-18} \vee 6^{36} \, .$$
We have the equality $$\label{e.vprime.ensemble}
{\mathsf{Wgt}}\big( \rho[2] \big) = (1-a)^{1/3} {t_{1,2}}^{1/3} {\mathsf{Nr}{\mathcal}{L}}_{n;(1-a) t_1 + a t_2}^{\downarrow;(y,t_2)}\big( 1, V' \big)$$ where we set $V'$ equal $(1-a)^{-2/3} {t_{1,2}}^{-2/3} ( Z- y )$. Indeed, the polymer weight ${\mathsf{Wgt}}\big( \rho[2] \big)$ equals ${\mathsf{Wgt}}_{n;\big(Z, (1-a) t_1 + a t_2 \big)}^{(y,t_2)} = {\mathcal}{L}_{n;(1-a) t_1 + a t_2}^{\downarrow;(y,t_2)}\big( 1, Z \big)$, whence we obtain (\[e.vprime.ensemble\]).
We claim that, when $r \geq 2 \vee 8(1-a)^{1/3} {t_{1,2}}^{-2/3} \vert x - y \vert$, $$\label{e.vprimeclaim}
{\mathsf{Fluc}}\, [-(r+1),-r] \subseteq \Big\{ \,
V' \in r \cdot [-13/8,-7/8] \, \Big\} \, .$$ To see this, note that, when ${\mathsf{Fluc}}[-(r+1),-r]$ occurs, $$Z \in z - (1-a)^{2/3} {t_{1,2}}^{2/3} \cdot [r,r+1]
= y + (1-a)(x-y) - (1-a)^{2/3} {t_{1,2}}^{2/3} \cdot [r,r+1]$$ and, in light of our lower bound on $r$, $$Z \in y - (1-a)^{2/3} {t_{1,2}}^{2/3} \cdot [r -r/8, r + r/8 + 1] \subseteq y \, - \, {\varphi}\cdot [7/8,13/8]$$ where we used $r \geq 2$ in the form $r+1 \leq 3r/2$. Thus, we confirm (\[e.vprimeclaim\]).
We thus obtain Lemma \[l.combine\](2): $$\begin{aligned}
& & {\ensuremath{\mathbb{P}}}\Big( \, {\mathsf{Fluc}}\, [-(r+1),-r] \, , \, t_{1,2}^{-1/3} (1-a)^{-1/3} {\mathsf{Wgt}}\big(\rho[2]\big) \geq - 2^{-1/2} (7r/8)^2 + s \Big) \nonumber \\
& = &
{\ensuremath{\mathbb{P}}}\Big( \, {\mathsf{Fluc}}\, [-(r+1),-r] \, , \, {\mathsf{Nr}{\mathcal}{L}}_{n;(1-a) t_1 + a t_2}^{\downarrow;(y,t_2)}\big( 1, V' \big)
\geq - 2^{-1/2} (7r/8)^2 + s \Big) \nonumber \\
& \leq &
{\ensuremath{\mathbb{P}}}\Big( \, {\mathsf{Fluc}}\, [-(r+1),-r] \, , \, {\mathsf{Nr}{\mathcal}{L}}_{n;(1-a) t_1 + a t_2}^{\downarrow;(y,t_2)}\big( 1, V' \big)
+ 2^{-1/2} (V')^2 \geq s \Big) \nonumber \\
& \leq &
{\ensuremath{\mathbb{P}}}\bigg( \sup_{-13r/8 \leq v \leq -7r/8} \Big( {\mathsf{Nr}{\mathcal}{L}}_{n;(1-a) t_1 + a t_2}^{\downarrow;(y,t_2)}\big( 1, v \big)
+ 2^{-1/2} v^2 \Big) \geq s \bigg) \nonumber \\
& \leq & \big(3r/8 + 1 \big) \cdot 6 C \exp \big\{ - 2^{-11/2} c s^{3/2} \big\}
\, . \nonumber\end{aligned}$$ Here, the final inequality is due to an application of Proposition \[p.mega\](3) to the $(c,C)$-regular ensemble ${\mathsf{Nr}{\mathcal}{L}}_{n;(1-a) t_1 + a t_2}^{\downarrow;(y,t_2)}$ which shares a curve cardinality of $\hat{n} + 1$ with the ensemble that we considered a few moments ago. Our parameter choice for the application is ${\bf y} = - 5r/4$, ${\bf r} = 3r/8$ and ${\bf t} = s$. This application may be made provided that $$5r/4 \leq c/2 \cdot \hat{n}^{1/9} \, , \, 3r/8 \leq {c}/4 \cdot \hat{n}^{1/9} \, , \, s \in \big[ 2^{7/2} , 2 \hat{n}^{1/3} \big] \, \, \textrm{ and } \, \, \hat{n} \geq c^{-18} \, .$$ This completes the proof of Lemma \[l.combine\].
[**Proof of Proposition \[p.feventfirstbound\](1).**]{} Here we implement the [*near*]{} case argument outlined before the proof of Proposition \[p.polyfluc\]. Consider the event $$\begin{aligned}
& & {\mathsf{Fluc}}\, [-(r+1),-r] \, \cap \, \Big\{ t_{1,2}^{-1/3} (1-a)^{-1/3} {\mathsf{Wgt}}\big(\rho[3]\big) > - 2^{-1/2} - s \Big\} \label{e.ftriple} \\
& & \qquad \cap \, \, \Big\{ t_{1,2}^{-1/3} (1-a)^{-1/3} {\mathsf{Wgt}}\big(\rho[2]\big) < - 2^{-1/2} (7r/8)^2 + s \Big\} \, . \nonumber\end{aligned}$$ Note that $ {\mathsf{Wgt}}\big(\rho[3]\big) - {\mathsf{Wgt}}\big(\rho[2]\big) = {\mathsf{Wgt}}\big(\hat\rho\big) - {\mathsf{Wgt}}\big(\rho \big)$.
Note that $- 2^{-1/2} - s = \big( - 2^{-1/2} (7r/8)^2 + s \big) + s$ is solved by $s = 3^{-1} \cdot 2^{-1/2} \big( (7r/8)^2 - 1 \big)$. Setting the value of $s$ in this way, note that $s \geq 5/3 \cdot 2^{-7/2} r^2$ since $r \geq 4$. We find then that the ${\ensuremath{\mathbb{P}}}$-probability of the above event is bounded above by $$\label{e.rhohatrho}
{\ensuremath{\mathbb{P}}}\bigg( t_{1,2}^{-1/3} (1-a)^{-1/3} \Big( {\mathsf{Wgt}}\big(\hat\rho\big) - {\mathsf{Wgt}}\big(\rho \big) \Big) \geq d r^2 \bigg) \, ,$$ where here we write $d = 5/3 \cdot 2^{-7/2}$.
As a temporary notation, we denote the [*maximum weight difference*]{} ${\mathsf{Max}\Delta\mathsf{Wgt}}_{n;(I,t_1)}^{(J,t_2)}$ to be the supremum over $x_1,x_2 \in I$ and $y_1,y_2 \in J$ of $\big\vert {\mathsf{Wgt}}_{n;(x_1,t_1)}^{(y_1,t_2)} - {\mathsf{Wgt}}_{n;(x_2,t_1)}^{(y_2,t_2)} \big\vert$.
Set $I = \{ x \}$ and $J = [y-9{\varphi}/8,y]$. We claim that, when $r \geq 8(1-a)^{1/3} {t_{1,2}}^{-2/3} \vert x - y \vert$, $$\label{e.incverify}
\Big\{ t_{1,2}^{-1/3} (1-a)^{-1/3} \big( {\mathsf{Wgt}}(\hat\rho) - {\mathsf{Wgt}}(\rho) \big) \geq d r^2 \Big\} \, \subseteq \, \Big\{ {\mathsf{Max}\Delta\mathsf{Wgt}}_{n;(I,t_1)}^{(J,t_2)} \geq d t_{1,2}^{1/3} (1-a)^{1/3} r^2 \Big\} \, .$$ Indeed, note that ${\mathsf{Wgt}}_{n;(x,t_1)}^{(z - {\varphi},t_2)} \geq {\mathsf{Wgt}}(\hat{\rho})$ while ${\mathsf{Wgt}}_{n;(x,t_1)}^{(y,t_2)} = {\mathsf{Wgt}}(\rho)$. Moreover, the lower bound on $r$ ensures that $z - {\varphi}\geq y - 9{\varphi}/8$, so that ${\mathsf{Max}\Delta\mathsf{Wgt}}_{n;(I,t_1)}^{(J,t_2)} \geq {\mathsf{Wgt}}(\hat\rho) - {\mathsf{Wgt}}(\rho)$. This implies (\[e.incverify\]).
Since ${\varphi}= t_{1,2}^{2/3} (1-a)^{2/3} r$, the right-hand event in (\[e.incverify\]) equals $$\label{e.mdw.bound}
t_{1,2}^{-1/3} {\mathsf{Max}\Delta\mathsf{Wgt}}_{n;(I,t_1)}^{(J,t_2)} \geq (8/9)^{1/2} d r^{3/2} \cdot \big(9
t_{1,2}^{-2/3}{\varphi}/8 \big)^{1/2} \, .$$ We seek to bound above the probability of this event by using Theorem \[t.differenceweight\]. Two obstacles are that this result is stated for the special case where the start time is zero and the end time is one, and the presence of parabolic terms $Q$ in its statement. Regarding the first difficulty, we use the scaling principle from Section \[s.scalingprinciple\] to obtain a general form for Theorem \[t.differenceweight\] which will be applicable to the time pair $(t_1,t_2)$ with which we work. The conclusion of this general form of the result is that $${\ensuremath{\mathbb{P}}}\left( \sup_{\begin{subarray}{c} u_1,u_2 \in [x,x+{\epsilon}{t_{1,2}}^{2/3}] \\
v_1,v_2 \in [y,y+{\epsilon}{t_{1,2}}^{2/3}] \end{subarray}} \left\vert \, {t_{1,2}}^{-1/3} {\mathsf{Wgt}}_{n;(u_2,t_1)}^{(v_2,t_2)} + Q\left(\frac{v_2 - u_2}{{t_{1,2}}^{2/3}}\right) - {t_{1,2}}^{-1/3} {\mathsf{Wgt}}_{n;(u_1,t_1)}^{(v_1,t_2)} - Q\left(\frac{v_1 - u_1}{{t_{1,2}}^{2/3}}\right) \, \right\vert \, \geq \, {\epsilon}^{1/2}
R \right)$$ is at most $10032 \, C \exp \big\{ - c_1 2^{-21} R^{3/2} \big\}$; while the new result’s hypotheses are those of Theorem \[t.differenceweight\] after making the replacements $n \to n{t_{1,2}}$ and $(x,y) \to {t_{1,2}}^{-2/3} \cdot (x,y)$.
We will find an upper bound on the event (\[e.mdw.bound\]) by applying this general version of Theorem \[t.differenceweight\], with parameter choices with ${\bm {\epsilon}} = 9
t_{1,2}^{-2/3} {\varphi}/8$, which also equals $9(1-a)^{2/3}r/8$; as well as ${\bf x} = x$, ${\bf y} = y - 9{\varphi}/8$, and ${\bf R} =
2^{-1}(8/9)^{1/2} d r^{3/2}$. In order that this indeed provide such an upper bound, we must first overcome the second obstacle just mentioned, confirming that parabolic curvature is suitably controlled. The relevant bound is $$\sup_{\begin{subarray}{c} u_1,u_2 \in [{\bf x},{\bf x}+ {\bm {\epsilon}} {t_{1,2}}^{2/3}] \\
v_1,v_2 \in [{\bf y}, {\bf y}+ {\bm {\epsilon}}{t_{1,2}}^{2/3}] \end{subarray}} \big\vert Q \big( {t_{1,2}}^{-2/3} (v_2 - u_2) \big) - Q \big( {t_{1,2}}^{-2/3} (v_1 - u_1) \big) \big\vert \leq 2^{-1} (8/9)^{1/2} d r^{3/2} \cdot \big(9
t_{1,2}^{-2/3}{\varphi}/8 \big)^{1/2} \, .$$ This bound holds because this supremum is at most $$2{\bm {\epsilon}}{t_{1,2}}^{2/3} \cdot 2^{1/2} \big( \vert {\bf y} - {\bf x} \vert + 2{\bm {\epsilon}} {t_{1,2}}^{2/3} \big) {t_{1,2}}^{-4/3} = 2^{-3/2} \cdot 9 (1-a)^{2/3} r \Big( \vert x - y \vert {t_{1,2}}^{-2/3} + 9 (1 - a)^{2/3} r/4 \Big) \, .$$ Expanding the right-hand bracket, the resulting two terms are both at most $4^{-1} (8/9)^{1/2} d r^{3/2} \cdot \big(9
t_{1,2}^{-2/3}{\varphi}/8 \big)^{1/2}$ (which suffices for our purpose) respectively provided that $$r \geq 3^3 2^4 5^{-1} (1-a)^{1/3} \vert x - y \vert {t_{1,2}}^{-2/3} \, \, \, \textrm{and} \, \, \, 1 - a \leq 5 \cdot 3^{-5} 2^{-2}$$ since ${t_{1,2}}^{-2/3} {\varphi}= (1-a)^{2/3}r$ and $d = 5/3 \cdot 2^{-7/2}$.
We further mention that, for the application in question to be made, it suffices that $$\begin{aligned}
& & 9(1-a)^{2/3}r/8 \leq 2^{-4} \, , \,
n{t_{1,2}}\geq 10^{32} c^{-18} \, , \, \\
& & \big\vert x - y \big\vert {t_{1,2}}^{-2/3} + 9(1-a)^{2/3}r/8 \leq 2^{-2} 3^{-1} {c}(n{t_{1,2}})^{1/18} \, \, \textrm{ and } \, \, 2^{1/2}/3 \cdot d r^{3/2} \in \big[ 10^4 \, , \, 10^3 (n {t_{1,2}})^{1/18} \big] \, ,\end{aligned}$$ as well as $n {t_{1,2}}\in {\ensuremath{\mathbb{N}}}$.
With these pieces in place, we apply the general version of Theorem \[t.differenceweight\], learning that the event in (\[e.mdw.bound\]) has probability at most $$10032 \, C \exp \big\{ - c_1 2^{-26} 3^{-3} 5^{3/2} r^{9/4} \big\} \, .$$ Recall that this quantity is thus an upper bound on the probability of the event (\[e.ftriple\]). Combining this information with the two parts of Lemma \[l.combine\], we find that $$\begin{aligned}
{\ensuremath{\mathbb{P}}}\Big( \, {\mathsf{Fluc}}\, [-(r+1),-r] \, \Big) & \leq &
140 C \, \exp \big\{ - c_1 s^{3/2} \big\} \, + \,
\big(3r/8 + 1 \big) \cdot 6 C \exp \big\{ - 2^{-11/2} c s^{3/2} \big\} \\
& & \qquad \qquad \qquad \qquad
+ \,
10032 \, C \exp \big\{ - c_1 2^{-26} 3^{-3} 5^{3/2} r^{9/4} \big\} \, .\end{aligned}$$
Recalling that $s \geq 5/3 \cdot 2^{-7/2} r^2$, we obtain $$\begin{aligned}
{\ensuremath{\mathbb{P}}}\Big( \, {\mathsf{Fluc}}\, [-(r+1),-r] \, \Big) & \leq &
\big( 146 + 9r/4 \big) C \exp \big\{ - c_1 2^{-43/4} 3^{-3/2} 5^{3/2} r^3 \big\} \\
& & \qquad + \,
10032 \, C \exp \big\{ - c_1 2^{-26} 3^{-3} 5^{3/2} r^{9/4} \big\} \\
& \leq &
(10178 + 3r) \, C \exp \big\{ - c_1 2^{-26} 3^{-3} 5^{3/2} r^{9/4} \big\} \, ,\end{aligned}$$ where we used $c_1 \leq c$ in the first inequality and $r \geq 1$ in the second. This completes the proof of Proposition \[p.feventfirstbound\](1).
[**Proof of Proposition \[p.feventfirstbound\](2).**]{} We now provide an alternative upper bound on the probability of the event in the display beginning at (\[e.ftriple\]), one that treats the middle-distant case. Recalling that this probability is at most (\[e.rhohatrho\]), we will find a new upper bound on the probability (\[e.rhohatrho\]). Recalling that we set $d = 5/3 \cdot 2^{-7/2}$, the occurrence of the event whose probability is in question entails that $$\begin{aligned}
\textrm{either} & & t_{1,2}^{-1/3} {\mathsf{Wgt}}\big(\hat\rho\big) \geq 2^{-1} d (1-a)^{1/3} r^2 \, - \, 2^{-1/2}(x-y)^2 {t_{1,2}}^{-4/3} \\
\textrm{or} & &
t_{1,2}^{-1/3} {\mathsf{Wgt}}(\rho) \leq - 2^{-1} d (1-a)^{1/3} r^2 \, - \, 2^{-1/2}(x-y)^2 {t_{1,2}}^{-4/3} \, .\end{aligned}$$
We now state and prove a claim regarding the probabilities of these two outcomes.
[**Claim: (1).**]{} The former outcome has ${\ensuremath{\mathbb{P}}}$-probability at most ${C}\exp \big\{ - {c}(1-a)^{1/2} 5^{3/2} 3^{-3/2} 2^{-33/4} r^3 \big\}$. [**(2).**]{} The latter outcome has ${\ensuremath{\mathbb{P}}}$-probability at most ${C}\exp \big\{ - {c}(1-a)^{1/2} 5^{3/2} 3^{-3/2} 2^{-27/4} r^3 \big\}$.
First we prove Claim (2). Since ${\mathsf{Wgt}}_{n;(x,t_1)}^{(y,t_2)} = {\mathsf{Wgt}}(\rho)$, the latter outcome entails $${\mathsf{Nr}{\mathcal}{L}}_{n;t_1}^{\downarrow;(y,t_2)}\big( 1, \omega' \big) +2^{-1/2} \omega'^2 \leq - 2^{-1} d (1 - a)^{1/3} r^2$$ where we set $\omega' = {t_{1,2}}^{-2/3} ( x - y )$. The probability of the outcome is thus bounded above by $$\begin{aligned}
& & {\ensuremath{\mathbb{P}}}\Big( {\mathsf{Nr}{\mathcal}{L}}_{n;t_1}^{\downarrow;(y,t_2)}\big( 1, \omega' \big) + 2^{-1/2} \omega'^2 \leq - 5/3 \cdot 2^{-9/2} (1 - a)^{1/3} r^2 \Big) \\
& \leq & {C}\exp \big\{ - {c}(1-a)^{1/2} (5/3)^{3/2} \cdot 2^{-27/4} r^3 \big\} \, ,\end{aligned}$$ where the displayed inequality is a consequence of an application of one-point lower tail ${{\rm Reg}}(2)$ with the ensemble equalling the $(n{t_{1,2}}+ 1)$-curve ${\mathsf{Nr}{\mathcal}{L}}_{n;t_1}^{\downarrow;(y,t_2)}$, and where the parameters are chosen to be ${\bf z} = \omega'$ and ${\bf s} = 5/3 \cdot 2^{-9/2} (1-a)^{1/3} r^2$. This application may be made when $\vert \omega' \vert \leq c (n {t_{1,2}})^{1/9}$ and $5/3 \cdot 2^{-9/2} (1-a)^{1/3} r^2 \in \big[ 1,(n {t_{1,2}})^{1/3} \big]$.
We now derive Claim $(1)$. This is an assertion about $\hat{\rho}$, an $n$-zigzag with starting and ending points $(x,t_1)$ and $(z - {\varphi},t_2)$. Note that the quadratic correction term in the assertion is $2^{-1/2}(y - x)^2 {t_{1,2}}^{-4/3}$, whereas we would like to instead work with $2^{-1/2}(z-{\varphi}- x)^2 {t_{1,2}}^{-4/3}$, this being the natural quadratic correction associated to the starting and ending points of $\hat\rho$. We begin then by noting that the difference is suitably small: $$\label{e.diffsuit}
\Big\vert \, 2^{-1/2}(y-x)^2 {t_{1,2}}^{-4/3} \, - \, 2^{-1/2}(z-{\varphi}- x)^2 {t_{1,2}}^{-4/3} \, \Big\vert \, \leq \, 4^{-1} d (1-a)^{1/3} r^2 \, .$$ An explicit condition that suffices to ensure this bound is that $r \geq 11 (1-a)^{1/3} \vert y -x \vert {t_{1,2}}^{-2/3}$ and $a \geq 1 - 10^{-2}$. See Lemma $D.4$ in the online Appendix $D$. We see from (\[e.diffsuit\]) and ${\mathsf{Wgt}}_{n;(x,t_1)}^{(z - {\varphi},t_2)} \geq {\mathsf{Wgt}}(\hat{\rho})$ that the outcome that Claim $(1)$ concerns entails that $${\mathsf{Nr}{\mathcal}{L}}_{n;t_1}^{\downarrow;(z-{\varphi},t_2)} \big( 1, \omega \big) + 2^{-1/2} \omega^2 \geq 4^{-1} d (1-a)^{1/3} r^2 \, \,$$ where now we set $\omega = {t_{1,2}}^{-2/3} ( x - z + {\varphi})$. Next we apply one-point upper tail ${{\rm Reg}}(3)$ to the $(n{t_{1,2}}+ 1)$-curve ensemble ${\mathsf{Nr}{\mathcal}{L}}_{n;t_1}^{\downarrow;(z-{\varphi},t_2)}$ with parameter settings ${\bf z} = \omega$ and ${\bf s} = 4^{-1} d (1-a)^{1/3} r^2$. We learn that $${\ensuremath{\mathbb{P}}}\Big( {\mathsf{Nr}{\mathcal}{L}}_{n;t_1}^{\downarrow;(z-{\varphi},t_2)} \big( 1, \omega \big) + 2^{-1/2} \omega^2 \geq 4^{-1} d (1-a)^{1/3} r^2 \Big) \, \leq \, {C}\exp \Big\{ - {c}(1-a)^{1/2} (5/3)^{3/2} \cdot 2^{-33/4} r^3 \Big\} \, .$$ This application of ${{\rm Reg}}(3)$ may be made when $\vert \omega \vert \leq c (n {t_{1,2}})^{1/9}$ and $(1-a)^{1/3} \cdot 5/3 \cdot 2^{-11/2} r^2 \geq 1$. We have proved the Claim.
The sum of the two expressions in Claim $(1)$ and $(2)$ the sought alternative upper bound on the probability of the event in the display beginning at (\[e.ftriple\]). Combining again with the two parts of Lemma \[l.combine\], we find that $$\begin{aligned}
{\ensuremath{\mathbb{P}}}\Big( {\mathsf{Fluc}}\, [-(r+1),-r] \Big) & \leq &
140 C \, \exp \big\{ - c_1 s^{3/2} \big\} \, + \,
\big(3r/8 + 1 \big) \cdot 6 C \exp \big\{ - 2^{-11/2} c s^{3/2} \big\} \\
& & \qquad \qquad \qquad \qquad \, + \,
2{C}\exp \big\{ - {c}(1-a)^{1/2} 5^{3/2} 3^{-3/2} 2^{-33/4} r^3 \big\}
\, . \end{aligned}$$
Recalling that $s \geq 5/3 \cdot 2^{-7/2} r^2$, we obtain $$\begin{aligned}
{\ensuremath{\mathbb{P}}}\Big( {\mathsf{Fluc}}\, [-(r+1),-r] \Big) & \leq &
\big( 146 + 9r/4 \big) C \exp \big\{ - c_1 2^{-43/4} 3^{-3/2} 5^{3/2} r^3 \big\} \\
& & \qquad \, + \, 2 {C}\exp \big\{ - {c}(1-a)^{1/2} 3^{-3/2} 5^{3/2} 2^{-33/4} r^3 \big\} \\
& \leq &
\big( 148 C + 9r/4 \big) \exp \big\{ - c_1 (1-a)^{1/2} 3^{-3/2} 5^{3/2} 2^{-43/4} r^3 \big\} \, ,\end{aligned}$$ where we used $c_1 \leq {c}$ in both inequalities. This completes the proof of Proposition \[p.feventfirstbound\](2).
[**Proof of Lemma \[l.feventhigh\].**]{} Now we treat the far case. In outline, the one-point lower tail ${{\rm Reg}}(2)$ shows that neither the weight of $\rho[1]$ nor of $\rho[2]$ can be extremely low, except with a tiny probability. In the opposing case, the polymer weight ${\mathsf{Wgt}}(\rho) = {\mathsf{Wgt}}\big( \rho[1] \big) + {\mathsf{Wgt}}\big( \rho[2] \big)$ is not highly negative. However, this eventuality is unlikely in the far case, where $Z$ is very much less than $z$, due to collapse near infinity Proposition \[p.mega\](4).
Naturally, here we are continuing to denote $Z = \rho\big( (1-a) t_1 + a t_2 \big)$ and $z = (1-a)x + ay$. To begin the formal argument, note that the far case event ${\mathsf{Fluc}}\big(-\infty,-2\hat{n}^{1/36} \big]$ occurs if and only if $Z - z \leq - {t_{1,2}}^{2/3} (1 -a)^{2/3} \cdot 2 \hat{n}^{1/36}$. Note that $\hat{n}^{1/36} \geq (1 -a)^{1/3} {t_{1,2}}^{-2/3} \vert x - y \vert$ implies that $\vert z - y \vert \leq \hat{n}^{1/36} \cdot {t_{1,2}}^{2/3} (1 -a)^{2/3}$, so that the occurrence of ${\mathsf{Fluc}}\big(-\infty,-2\hat{n}^{1/36} \big]$ entails that $Z - y \leq - {t_{1,2}}^{2/3} (1 -a)^{2/3} \hat{n}^{1/36}$. Using again the notation $V' = (1-a)^{-2/3} {t_{1,2}}^{-2/3} (Z-y)$, the latter condition is given by $V' \leq - \hat{n}^{1/36}$.
Recall that in (\[e.vprime.ensemble\]), the polymer weight ${\mathsf{Wgt}}\big( \rho[2] \big)$ is expressed in terms of ${\mathsf{Nr}{\mathcal}{L}}_{n;(1-a) t_1 + a t_2}^{\downarrow;(y,t_2)} \big( 1,V' \big)$. We thus find that $$\begin{aligned}
& & {\ensuremath{\mathbb{P}}}\Big( {\mathsf{Fluc}}\big(-\infty,-2\hat{n}^{1/36} \big] \, , \, {t_{1,2}}^{-1/3} (1-a)^{-1/3} {\mathsf{Wgt}}\big( \rho[2] \big) \geq \big( - 2^{-1/2} + 2^{-5/2} \big) (1-a)^{1/18} {t_{1,2}}^{1/18} n^{1/18}
\Big) \nonumber \\
& \leq & {\ensuremath{\mathbb{P}}}\Big( \sup_{x \leq - \hat{n}^{1/36}} {\mathsf{Nr}{\mathcal}{L}}_{n;(1-a) t_1 + a t_2}^{\downarrow;(y,t_2)} \big( 1,x \big) \geq - \big( 2^{-1/2} - 2^{-5/2} \big) (1-a)^{1/18} {t_{1,2}}^{1/18} n^{1/18}
\Big) \, . \label{e.ensemblecollapse}\end{aligned}$$
Recall that $\hat{n} = n (1-a) {t_{1,2}}$. To find an upper bound on the probability (\[e.ensemblecollapse\]), we now apply collapse near infinity Proposition \[p.mega\](4) to the $(c,C)$-regular $(\hat{n} + 1)$-curve ensemble ${\mathsf{Nr}{\mathcal}{L}}_{n;(1-a) t_1 + a t_2}^{\downarrow;(y,t_2)}$. We set the parameter ${\bm \eta}$ so that ${\bm \eta}( \hat{n} + 1) ^{1/9} = \hat{n}^{1/36}$. For the application to be made, it is enough that $$\big(n (1-a) {t_{1,2}}\big)^{-1/12} \leq {c}\, \, \textrm{ and } \, \, n (1-a) {t_{1,2}}\geq \big( 2^{5/4} {c}^{-1} \big)^{9} \, .$$ Note then that $$\ell\big(-\hat{n}^{-1/12} (\hat{n} + 1)^{1/9} \big) = \big( - 2^{-1/2} + 2^{-5/2} \big) \big( \hat{n}^{-1/12} \big)^2 (\hat{n} + 1)^{2/9} = \big( - 2^{-1/2} + 2^{-5/2} \big) ( \hat{n}^{-1/12})^2 (\hat{n} + 1)^{2/9} \, .$$ Since $\ell(x)$ is increasing for $x \leq 0$, we find that $\ell\big(-\hat{n}^{-1/36} \big) \leq - \big( 2^{-1/2} - 2^{-5/2} \big) \hat{n}^{1/18}$. Thus, the expression (\[e.ensemblecollapse\]) is found to be at most $$\label{e.firstub}
6C \exp \Big\{ - c \eta^3 2^{-15/4} \big( \hat{n} + 1 \big)^{1/3} \Big\} =
6C \exp \Big\{ - 2^{-15/4} c \hat{n}^{1/12} \Big\} \, .$$
Set $\tilde{V} = a^{-2/3} {t_{1,2}}^{-2/3} ( Z - x)$. Note that $$\label{e.weightrhoone}
{\mathsf{Wgt}}\big( \rho[1] \big) = a^{1/3} {t_{1,2}}^{1/3} {\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow; (1-a)t_1 + at_2} \big( 1, \tilde{V} \big) \, ,$$ since both of these quantities equal ${\mathsf{Wgt}}_{n;(x,t_1)}^{\big(Z, (1-a)t_1 + at_2 \big)}$.
We claim that $$\label{e.vtilde.claim}
{\mathsf{Fluc}}\big(-\infty,-2\hat{n}^{1/36} \big] \,\subseteq \, \Big\{ \tilde{V} \leq - (1 -a)^{2/3} \hat{n}^{1/36} \Big\} \, .$$ Indeed, note that the left-hand event entails that $$Z - x \leq a \vert x -y \vert - {t_{1,2}}^{2/3} (1 -a)^{2/3} \cdot 2 \hat{n}^{1/36} \leq - {t_{1,2}}^{2/3} (1 -a)^{2/3} \hat{n}^{1/36}$$ since $\vert x - y \vert \leq {t_{1,2}}^{2/3} (1 -a)^{2/3} \hat{n}^{1/36}$ and $a \leq 1$. From another use of $a \leq 1$, we obtain (\[e.vtilde.claim\]).
From (\[e.weightrhoone\]) and (\[e.vtilde.claim\]), we find that $$\begin{aligned}
& & {\ensuremath{\mathbb{P}}}\Big(
{\mathsf{Fluc}}\big(-\infty,-2\hat{n}^{1/36} \big] \, , \, {t_{1,2}}^{-1/3} a^{-1/3} {\mathsf{Wgt}}\big( \rho[1] \big) \geq - \big( 2^{-1/2} - 2^{-5/2} \big) (1-a)^{25/18} {t_{1,2}}^{1/18} n^{1/18}
\Big) \nonumber \\
& \leq & {\ensuremath{\mathbb{P}}}\Big( \sup_{x \leq - (1 -a)^{2/3} \hat{n}^{1/36}} {\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow; (1-a)t_1 + at_2} \big( 1,x \big) \geq - \big( 2^{-1/2} - 2^{-5/2} \big)
(1-a)^{25/18} {t_{1,2}}^{1/18} n^{1/18} \Big) \, . \label{e.hatprob}\end{aligned}$$
To find an upper bound on the latter probability, we apply Proposition \[p.mega\](4) to the $(c,C)$-regular ensemble ${\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow; (1-a)t_1 + at_2}$. This ensemble has $\tilde{n} + 1$ curves, where here we introduce the shorthand $\tilde{n} = na{t_{1,2}}$. We set the Proposition \[p.mega\](4) parameter ${\bm \eta}$ so that ${\bm \eta} ( \tilde{n} + 1 )^{1/9} = (1 -a)^{2/3} \hat{n}^{1/36}$. Recalling that $\hat{n} = n (1-a) {t_{1,2}}$, we see that $\eta \leq n^{-1/12} a^{-1/9} (1-a)^{25/36} {t_{1,2}}^{-1/12}$. For Proposition \[p.mega\](4) to be applied in this way, it suffices then that $$n^{-1/12} a^{-1/9} (1-a)^{25/36} {t_{1,2}}^{-1/12} \leq c \, \, \textrm{ as well as } \, \, n a {t_{1,2}}\geq \big( 2^{5/4} {c}^{-1} \big)^{9} \, .$$ Note then that $$\begin{aligned}
\ell\big(- (1 -a)^{2/3} \hat{n}^{1/36} \big) & = & - \big( 2^{-1/2} - 2^{-5/2} \big) \big( (1 -a)^{2/3} \hat{n}^{1/36} ( \tilde{n} + 1 )^{-1/9} \big)^2 (\tilde{n} + 1)^{2/9} \\
& = & - \big( 2^{-1/2} - 2^{-5/2} \big) (1-a)^{25/18} {t_{1,2}}^{1/18} n^{1/18} \, .\end{aligned}$$ Also using that $\ell(x)$ is increasing for $x \leq - (1 -a)^{2/3} \hat{n}^{1/36}$, we see that our application of Proposition \[p.mega\](4) implies that the probability (\[e.hatprob\]) is at most $$\label{e.secondub}
6C \exp \Big\{ - c \eta^3 2^{-15/4} (\tilde{n} + 1)^{1/3} \Big\} =
6C \exp \Big\{ - 2^{-15/4} c (1-a)^{25/12} {t_{1,2}}^{1/12} n^{1/12} \Big\} \, .$$ Note that ${t_{1,2}}^{1/3} {\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow;t_2}\big(1, {t_{1,2}}^{-2/3}(y-x)\big) = {\mathcal}{L}_{n;(x,t_1)}^{\uparrow;t_2}(1,y) = {\mathsf{Wgt}}(\rho) = {\mathsf{Wgt}}\big(\rho[1] \big) + {\mathsf{Wgt}}\big(\rho[2] \big)$. Thus, $$\begin{aligned}
& & {\ensuremath{\mathbb{P}}}\Big( {t_{1,2}}^{-1/3} a^{-1/3} {\mathsf{Wgt}}\big( \rho[1] \big) \leq - \big( 2^{-1/2} - 2^{-5/2} \big) (1-a)^{25/18} {t_{1,2}}^{1/18} n^{1/18} \, , \, \nonumber \\
& & \qquad \qquad {t_{1,2}}^{-1/3} (1-a)^{-1/3} {\mathsf{Wgt}}\big( \rho[2] \big) \leq - \big( 2^{-1/2} - 2^{-5/2} \big) (1-a)^{1/18} {t_{1,2}}^{1/18} n^{1/18}
\Big) \nonumber \\
& \leq & {\ensuremath{\mathbb{P}}}\bigg( {\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow}\big(1, {t_{1,2}}^{-2/3}(y-x)\big) \leq - \big( 2^{-1/2} - 2^{-5/2} \big) \Big( (1-a)^{25/18} a^{1/3} + (1-a)^{1/18 + 1/3} \Big) {t_{1,2}}^{1/18} n^{1/18} \bigg) \nonumber \\
& \leq & C \exp \bigg\{ - c \big( 2^{-1/2} - 2^{-5/2} \big)^{3/2} \Big( (1-a)^{25/18} a^{1/3} + (1-a)^{17/18} \Big)^{3/2} {t_{1,2}}^{1/12} n^{1/12} \bigg\} \, , \label{e.thirdub}\end{aligned}$$ where in the latter inequality, we applied one-point lower tail ${{\rm Reg}}(2)$ to the $(n{t_{1,2}}+ 1)$-curve ensemble ${\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow;t_2}$. In this application, we take $${\bf z} = {t_{1,2}}^{-2/3}(y-x) \, \, \textrm{ and } \, \, {\bf s} = \big( 2^{-1/2} - 2^{-5/2} \big) \big( (1-a)^{25/18} a^{1/3} + (1-a)^{17/18} \big) {t_{1,2}}^{1/18} n^{1/18} \, .$$ The application may be made provided that $$\vert y - x \vert {t_{1,2}}^{-2/3} \leq {c}(n {t_{1,2}})^{1/9} \, \,
\textrm{ and } \, \,
\big( 2^{-1/2} - 2^{-5/2} \big) \big( (1-a)^{25/18} a^{1/3} + (1-a)^{17/18} \big) {t_{1,2}}^{1/18} n^{1/18} \in \big[ 1, (n{t_{1,2}})^{1/3} \big] \, .$$
Recalling the upper bounds on the probabilities in the first lines of the displays (\[e.ensemblecollapse\]) and (\[e.hatprob\]) offered by (\[e.firstub\]) and (\[e.secondub\]), and combining these with the bound (\[e.thirdub\]), we find that ${\ensuremath{\mathbb{P}}}\big( {\mathsf{Fluc}}(-\infty,-2\hat{n}^{1/36} ] \big)$ is at most $$\begin{aligned}
& &
6C \exp \Big\{ - 2^{-15/4} c (1-a)^{1/12} {t_{1,2}}^{1/12} n^{1/12} \Big\}
\, + \,
6C \exp \Big\{ - 2^{-15/4} c (1-a)^{25/12} {t_{1,2}}^{1/12} n^{1/12} \Big\} \\
& & + \, C \exp \bigg\{ - c \big( 2^{-1/2} - 2^{-5/2} \big)^{3/2} \Big( (1-a)^{25/18} a^{1/3} + (1-a)^{17/18} \Big)^{3/2} {t_{1,2}}^{1/12} n^{1/12} \bigg\} \\
& \leq & 13C \exp \Big\{ - 2^{-15/4} c (1-a)^{25/12} {t_{1,2}}^{1/12} n^{1/12} \Big\} \, .\end{aligned}$$ This completes the proof of Lemma \[l.feventhigh\].
Glossary of notation {#s.glossary}
====================
This article uses quite a lot of notation. Each line of the list that we now present recalls one of the principal pieces of notation; provides a short summary of its meaning; and gives the number of the page at which the notation is either introduced or formally defined. The summaries are, of course, imprecise: phrases in quotation marks, such as ‘disjointly travel’, are merely verbal approximations of a precise meaning. When a quantity is said to be ‘$\leq r$’, it is in fact the absolute value of the roughly recalled quantity which must be at most $r$.
Multi-geodesic ordering {#s.mgo}
=======================
The multi-geodesic with given endpoints was defined in Section \[s.staircase\] and further discussed in Section \[s.maxunique\], where its almost surely uniqueness was stated in Lemma \[l.severalpolyunique\]. The principal aim of this appendix is to establish in Lemma \[l.severalpolyorder\] that this multi-geodesic satisfies a natural monotonicity property, moving to the right in response to rightward displacement of endpoints. The multi-polymer counterpart, Lemma \[l.tworelations\](1), will follow immediately.
We introduce two ordering relations on sets of staircases that share their starting and ending heights.
Let $(i,j) \in {\ensuremath{\mathbb{Z}}}^2_<$ and $(x_1,y_1),(x_2,y_2) \in {\ensuremath{\mathbb{R}}}^2_<$ satisfy $x_1 \leq x_2$ and $y_1 \leq y_2$. Consider two staircases. $\phi_1 \in {SC}_{i,j}(x_1,y_1)$ and $\phi_1 \in {SC}_{i,j}(x_2,y_2)$. We define two relations, $\prec$ and $\preceq$, that $\phi_1$ and $\phi_2$ may satisfy.
- We say that $\phi_1 \prec \phi_2$ if, whenever $(z,t) \in {\ensuremath{\mathbb{R}}}\times [i,j]$ is an element of $\phi_2$, the open planar line segment that emanates rightwards from $(z,t)$, namely $(z,\infty) \times \{ t \}$, is disjoint from $\phi_1$.
- We say that $\phi_1 \preceq \phi_2$ if, whenever $(z,t) \in {\ensuremath{\mathbb{R}}}\times [i,j]$ is an element of $\phi_1$, the closed planar line segment running rightwards from $(z,t)$, $[z,\infty) \times \{ t \}$, intersects $\phi_2$.
In fact, each of these two conditions is equivalent to the conditions seen in Section \[s.polyorder\] when the statements there are made for zigzags, rather than staircases.
Two basic properties are readily verified:
- $\phi_1 \prec \phi_2$ implies $\phi_1 \preceq \phi_2$;
- and $\phi_1 \preceq \phi_2$ and $\phi_2 \prec \phi_3$, or for that matter $\phi_1 \prec \phi_2$ and $\phi_2 \preceq \phi_3$, imply that $\phi_1 \prec \phi_3$.
When two staircases $\phi_1$ and $\phi_2$ share their pair $(i,j) \in {\ensuremath{\mathbb{Z}}}^2_<$ of starting and ending heights, we may define their maximum $\phi_1 \vee \phi_2$. Indeed, suppose that $\phi_1 \in {SC}_{i,j}(x_1,y_1)$ and $\phi_2 \in {SC}_{i,j}(x_2,y_2)$ where $(x_1,y_1),(x_2,y_2) \in {\ensuremath{\mathbb{R}}}^2_\leq$; contrary to a moment ago, we permit the cases that $x_2 < x_1$ or $y_2 < y_1$. The maximum $\phi_1 \vee \phi_2$ is a staircase from $\big( x_1 \vee x_2 , i \big)$ to $\big( y_1 \vee y_2 , j \big)$. It is equal to the unique staircase with these starting and ending points which is contained in the union of the horizontal and vertical segments of $\phi_1$ and $\phi_2$ and which dominates these two staircases in the $\preceq$ ordering. To these two staircases is also associated a minimum $\phi_1 \wedge \phi_2$, a staircase that makes its way from $\big( x_1 \wedge x_2 , i \big)$ to $\big( y_1 \wedge y_2 , j \big)$. It is analogously defined and is instead dominated by $\phi_1$ and $\phi_2$ according to $\preceq$.
\[l.severalpolyorder\] For $k \in {\ensuremath{\mathbb{N}}}$, let $\bar{u},\bar{v},\bar{x},\bar{y} \in {\ensuremath{\mathbb{R}}}_\leq^k$ be four non-decreasing lists such that $\bar{v} - \bar{u} \in [0,\infty)^k$ and $\bar{y} - \bar{x} \in [0,\infty)^k$. Let $(j_1,j_2) \in {\ensuremath{\mathbb{Z}}}^2_<$. Suppose that these parameters are such that the two maximizers $$\Big( P^k_{( \bar{u} , j_1 {\bf 1}) \to (\bar{x},j_2 {\bf 1}) ; i} : i \in {\llbracket 1,k \rrbracket} \Big)
\, \, \, \textrm{ and } \, \, \, \Big( P^k_{( \bar{v} , j_1 {\bf 1}) \to (\bar{y},j_2 {\bf 1}); i} : i \in {\llbracket 1,k \rrbracket} \Big)$$ exist uniquely. Then $$P^k_{( \bar{u} , j_1 {\bf 1}) \to (\bar{x},j_2 {\bf 1}) ; i} \preceq P^k_{( \bar{v} , j_1 {\bf 1}) \to (\bar{y},j_2 {\bf 1}); i } \, \, \textrm{for $i \in {\llbracket 1,k \rrbracket}$} \, .$$
(In this appendix, we write ${\bf 1}$, rather than $\bar{\bf 1}$, for a $k$-tuple whose components equal one.)
This result permits us to derive Lemma \[l.tworelations\].
[**Proof of Lemma \[l.tworelations\]: (1).**]{} This assertion is the multi-polymer counterpart to Lemma \[l.severalpolyorder\] and follows immediately from this lemma.
[**(2).**]{} This result is a direct consequence of the definitions of the relations $\prec$ and $\preceq$.
[**Proof of Lemma \[l.severalpolyorder\].**]{} For convenience, denote the two maximizers by $\big( {P}_{1,i}: i \in {\llbracket 1,k \rrbracket} \big)$ and $\big( {P}_{2,i}: i \in {\llbracket 1,k \rrbracket} \big)$ in such a way that we seek to show that ${P}_{1,i} \preceq {P}_{2,i}$ for $i \in {\llbracket 1,k \rrbracket}$.
Suppose that this ordering property fails. Our plan is to argue that this assumption leads to a violation of the almost sure uniqueness of multi-geodesics with given endpoints asserted by Lemma \[l.severalpolyunique\].
Let $i \in {\llbracket 1,k \rrbracket}$ be maximal such that ${P}_{1,i} \npreceq {P}_{2,i}$. Let $j \in {\llbracket 1,i \rrbracket}$ be minimal such that ${P}_{1,j} \npreceq {P}_{2,i}$. (The stroke through the symbol, $\preceq$, and later $\prec$, is used to indicate that the relation does not hold.)
We now set the value of a parameter $L$ in the interval $\llbracket 0, j-1 \rrbracket$. If ${P}_{1,j-1} \prec {P}_{2,i}$, then $L$ is set equal to zero. In the other case, we set $L \in {\llbracket 1,j-1 \rrbracket}$ so that $${P}_{1,j-m-1} \nprec {P}_{2,i-m} \, \, \, \textrm{for each $m \in \llbracket 0, L - 1 \rrbracket$,}$$ and, if $L \not= j-1$, then ${P}_{1,j-L-1} \prec {P}_{2,i-L}$.
Whatever the value of $L$, we now specify two $k$-vectors of staircases, $\big( {P}'_{1,\ell} : \ell \in {\llbracket 1,k \rrbracket} \big)$ and $\big( {P}'_{2,\ell} : \ell \in {\llbracket 1,k \rrbracket} \big)$, according to the formulas $${P}'_{1,\ell} =
\begin{cases}
\, {P}_{1,\ell} & \ell \in {\llbracket 1,k \rrbracket} \setminus \llbracket j - L, j \rrbracket \, , \\
\, {P}_{1,j - m} \wedge {P}_{2,i-m} & \ell = j - m \, \, \textrm{with $m \in \llbracket 0,L \rrbracket$} \, ,
\end{cases}$$ and $${P}'_{2,\ell} =
\begin{cases}
\, {P}_{2,\ell} & \ell \in {\llbracket 1,k \rrbracket} \setminus \llbracket i - L, i \rrbracket \, , \\
\, {P}_{1,j - m} \vee {P}_{2,i-m} & \ell = i - m \, \, \textrm{with $m \in \llbracket 0,L \rrbracket$} \, .
\end{cases}$$ Using such notation as $P'_{1,\cdot} = \big( P'_{1,i}: i \in {\llbracket 1,k \rrbracket} \big)$, we now make a claim with four parts.
[*Claim.*]{}
1. ${P}'_{1,\cdot}$ does not equal ${P}_{1,\cdot}$, and nor does ${P}'_{2,\cdot}$ equal ${P}_{2,\cdot}$.
2. The vector ${P}'_{1,\cdot}$ is an element of $D^k_{(x {\bf 1},j_1 {\bf 1}) \to (\bar{u},j_2{\bf 1})}$.
3. The vector ${P}'_{2,\cdot}$ belongs to $D^k_{(x {\bf 1},j_1 {\bf 1}) \to (\bar{v},j_2{\bf 1})}$.
4. These two vectors are maximizers of their respective sets.
We verify the four assertions in turn.
![Illustrating the proof of Lemma \[l.severalpolyorder\]. In this example, $\ell = 4$. On the left, the consecutive staircases $\big\{ {P}_{1,i}: i \in {\llbracket 1,4 \rrbracket} \big\}$ are indicated in unbroken black. The four components of ${P}_{2,\cdot}$ are shown in dashed red. On the right, the ${P}'$ counterparts are similarly indicated. We have $i=3$, $j=2$ and $L =1$. Substaircases that change hands are highlighted. There are two such rearrangements. One leads to $P'_{1,1} = P_{1,1} \wedge P_{2,2}$ and $P'_{2,2} = P_{1,1} \vee P_{2,2}$, and the other to $P'_{1,2} = P_{1,2} \wedge P_{2,3}$ and $P'_{2,3} = P_{1,2} \vee P_{2,3}$. Otherwise, $P'$ components coincide with their $P$ counterparts. []{data-label="f.switch"}](switch.pdf){width="90.00000%"}
[*Verifying Claim (1):*]{} Since ${P}'_{1,j} = {P}_{1,j} \wedge {P}_{2,i}$ and ${P}_{1,j} \not\preceq {P}_{2,i}$, we see that, although ${P}'_{1,j} \preceq {P}_{1,j}$, we also have ${P}'_{1,j} \not= {P}_{1,j}$. Similarly, since ${P}'_{2,i} = {P}_{1,j} \vee {P}_{2,i}$ while ${P}_{1,j} \not\preceq {P}_{2,i}$, we see that, although ${P}'_{2,i} \preceq P_{2,i}$, we also have ${P}'_{2,i} \not= {P}_{2,i}$.
[*Verifying Claim (2):*]{} For each $i \in {\llbracket 1,k \rrbracket}$, the staircase ${P}_{2,i}$ staircase begins and ends at or to the right of ${P}_{1,i}$. For this reason, each staircase ${P}'_{1,i}$ shares its starting and ending points with ${P}_{1,i}$. In light of this, it is enough in order to show that ${P}'_{1,\cdot}\in
D^k_{(x {\bf 1},j_1 {\bf 1}) \to (\bar{u},j_2{\bf 1})}$ that it be demonstrated that ${P}'_{1,\ell} \prec {P}'_{1,\ell+1}$ for each $\ell \in \llbracket 0,k-1 \rrbracket$. This statement follows from the ordering under $\prec$ of the components of ${P}_{1,\cdot}$ in the case that $\ell \not\in \llbracket j-L-1,j \rrbracket$.
We have that ${P}'_{1,j} \prec {P}'_{1,j+1}$ since ${P}_{1,j} \prec {P}_{1,j+1}$; and, for $m \in {\llbracket 1,L \rrbracket}$, that ${P}'_{1,j - m} \prec {P}'_{1,j-m+1}$ since ${P}_{1,j-m} \prec {P}_{1,j-m+1}$ and ${P}_{2,i-m} \prec {P}_{2,i-m+1}$. Finally, when $L < j-1$, we have that ${P}'_{1,j - L - 1} \prec {P}'_{1,j-L}$ since ${P}_{1,j-L-1} \prec {P}_{1,j-L}$ and ${P}_{1,i-L-1} \prec {P}_{2,i-L}$ (an inequality which is indeed valid when $L \not= j-1$). Note that when $L = j-1$, $j - L - 1$ equals zero, so that the verification in the preceding sentence is not made in this case.
Thus, we verify that ${P}'_{1,\cdot}\in
D^k_{(x {\bf 1},j_1 {\bf 1}) \to (\bar{u},j_2{\bf 1})}$.
[*Verifying Claim (3):*]{} For each $i \in {\llbracket 1,k \rrbracket}$, ${P}'_{2,i}$ shares its starting and ending points with ${P}_{2,i}$, for the same reason that ${P}'_{1,i}$ and ${P}_{1,i}$ do.
It is thus enough to verify that ${P}'_{2,\cdot}\in D^k_{(x {\bf 1},j_1 {\bf 1}) \to (\bar{v},j_2{\bf 1})}$. This is confirmed in view of:
- ${P}'_{2,\ell} \prec {P}'_{2,\ell+1}$ for $\ell \in {\llbracket 1,k \rrbracket} \setminus \llbracket i-L , i \rrbracket$ since ${P}_{2,\ell} \prec {P}_{2,\ell+1}$;
- ${P}'_{2,i} \prec {P}'_{2,i+1}$ since ${P}_{2,i} \prec {P}_{2,i+1}$ and, by $j < i+1$ and then $i$’s maximality, ${P}_{1,j} \prec {P}_{1,i+1} \preceq {P}_{2,i+1}$;
- ${P}'_{2,i - m - 1} \prec {P}'_{2,i - m}$ for $m \in {\llbracket 1,L-1 \rrbracket}$ since ${P}_{1,j - m -1} \prec {P}_{1,j -m}$ and ${P}_{2,i - m -1} \prec {P}_{2,i -m}$;
- and ${P}'_{2,i - L - 1} \prec {P}'_{2,i - L}$ since ${P}_{2,i - L - 1} \prec {P}_{2,i - L}$.
[*Verifying Claim (4):*]{} To argue that the two $k$-vectors ${P}'_{1,\cdot}$ and ${P}'_{2,\cdot}$ are maximizers of their respective sets, we first note that $$W \big( {P}'_1 \big) +
W \big( {P}'_2 \big) =
W \big( {P}_1 \big) +
W \big( {P}_2 \big) \, .$$ Since ${P}'_1 \in D^k_{(x {\bf 1},j_1 {\bf 1}) \to (\bar{u},j_2{\bf 1})}$ and ${P}_1$ is a maximizer of this set, $W\big( {P}'_1 \big) \leq W\big( {P}_1 \big)$. Likewise, $W \big( {P}'_2 \big) \leq W \big( {P}_2 \big)$. This circumstance forces $W \big( {P}'_1 \big)$ to equal $W \big( {P}_1 \big)$ and $W \big( {P}'_2 \big)$ to equal $W \big( {P}_2 \big)$. Thus, ${P}'_1$ and ${P}'_2$ is each a maximizer of the set to which it belongs.
The proof of the claim complete, we need now merely invoke the maximizer uniqueness Lemma \[l.severalpolyunique\] to learn that ${P}'_1 = {P}_1$, and also that ${P}'_2 = {P}_2$. However, we have seen that neither of these equalities holds. From this contradiction, we learn that ${P}_{1,i} \preceq {P}_{2,i}$ for $i \in {\llbracket 1,k \rrbracket}$, as we sought to do.
Normalized ensembles are regular: deriving Proposition \[p.scaledreg\] {#s.normal}
======================================================================
In order to prove Proposition \[p.scaledreg\], it is convenient to expand a little on the review in Section \[s.encode\] of the definition of the scaled forward ensemble ${\mathcal}{L}_{n;(x,t_1)}^{\uparrow;t_2}$ and its normalized counterpart. It is natural to view the first object as the scaled counterpart to an unscaled forward ensemble which we now specify.
To do so, let $(m_1,m_2) \in {\ensuremath{\mathbb{N}}}^2_\leq$ and $u \in {\ensuremath{\mathbb{R}}}$. Define the unscaled forward ensemble $$L_{(u,m_1)}^{\uparrow;m_2}: {\llbracket 1,m_2 - m_1 + 1 \rrbracket} \times [u,\infty) \to {\ensuremath{\mathbb{R}}}$$ with base-point $(u,m_1)$ and end height $m_2$ by insisting that, for each $k \in {\llbracket 1,m_2 - m_1 + 1 \rrbracket}$ and $v \geq u$, $$\sum_{i=1}^k L_{(u,m_1)}^{\uparrow;m_2}(i,v) = M^k_{(u,m_1) \to (v,m_2)} \, .$$
The relation of this object to its scaled counterpart may be described by letting $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_\leq$ be a compatible triple, and $x \in {\ensuremath{\mathbb{R}}}$. The scaled forward ensemble $${\mathcal}{L}_{n;(x,t_1)}^{\uparrow;t_2}: {\llbracket 1,n {t_{1,2}}+ 1 \rrbracket} \times \big[ x - 2^{-1} n^{1/3} {t_{1,2}}, \infty\big) \to {\ensuremath{\mathbb{R}}}$$ is seen to satisfy, for each $k \in {\llbracket 1,n {t_{1,2}}+ 1 \rrbracket}$ and $y \geq x - 2^{-1} n^{1/3} {t_{1,2}}$, the identity $$\label{e.ln}
{\mathcal}{L}_{n;(x,t_1)}^{\uparrow;t_2}(k,y) = 2^{-1/2} n^{-1/3} \Big( L_{(n t_1 + 2n^{2/3}x,n t_1)}^{\uparrow;n t_2}\big( k, n t_2 + 2n^{2/3} y \big) - 2 n t_{1,2} - 2n^{2/3}(y-x) \Big) \, .$$ A further, parabolic, change of coordinates (\[e.scaledln\]) then specifies the normalized forward ensemble ${\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow;t_2}$ in terms of the scaled one ${\mathcal}{L}_{n;(x,t_1)}^{\uparrow;t_2}$.
A similar story holds for the backward ensemble. Indeed, considering now $(m_1,m_2) \in {\ensuremath{\mathbb{N}}}^2_\leq$ and $v \in {\ensuremath{\mathbb{R}}}$, we may define the unscaled backward ensemble $$L_{m_1}^{\downarrow;(v,m_2)}: {\llbracket 1,m_2 - m_1 + 1 \rrbracket} \times (-\infty, v ] \to {\ensuremath{\mathbb{R}}}$$ with base-point $(v,m_2)$ and end height $m_1$ by insisting that, for each $k \in {\llbracket 1,m_2 - m_1 + 1 \rrbracket}$ and $u \leq v$, $$\sum_{i=1}^k L_{m_1}^{\downarrow;(v,m_2)}(i,u) = M^k_{(u,m_1) \to (v,m_2)} \, .$$
Again fixing a compatible triple $(n,t_1,t_2) \in {\ensuremath{\mathbb{N}}}\times {\ensuremath{\mathbb{R}}}^2_\leq$, and now letting $y \in {\ensuremath{\mathbb{R}}}$, we note that the scaled backward ensemble $${\mathcal}{L}_{n;t_1}^{\downarrow;(y,t_2)}: {\llbracket 1,n {t_{1,2}}+ 1 \rrbracket} \times \big( - \infty , y + 2^{-1} n^{1/3} {t_{1,2}}\big] \to {\ensuremath{\mathbb{R}}}$$ satisfies, for each $k \in {\llbracket 1,n {t_{1,2}}+ 1 \rrbracket}$ and $x \leq y + 2^{-1} n^{1/3} {t_{1,2}}$, $${\mathcal}{L}_{n;t_1}^{\downarrow;(y,t_2)}(k,x) = 2^{-1/2} n^{-1/3} \Big( L_{n t_1}^{\downarrow;( n t_2 + 2n^{2/3}y , n t_2)}\big( k, n t_1 + 2n^{2/3} x \big) - 2 n {t_{1,2}}- 2n^{2/3}(y-x) \Big) \, .$$
Now consider again a compatible triple $(n,t_1,t_2)$ as well as $x \in {\ensuremath{\mathbb{R}}}$, and write $N = n {t_{1,2}}+ 1 \in {\ensuremath{\mathbb{N}}}$. Consider the ensemble $L_N: {\llbracket 1,N \rrbracket} \times [ 0 , \infty ) \to {\ensuremath{\mathbb{R}}}$, given by $$L_N(i,u) = L_{(n t_1 + 2n^{2/3}x, n t_1)}^{\uparrow; n t_2}\big( i ,
n t_1 + 2n^{2/3}x + u \big) \qquad \textrm{for $(i,u) \in {\llbracket 1,N \rrbracket} \times [ 0 , \infty )$} \, .$$ By [@O'ConnellYor Theorem $7$], this ensemble has the law of an $N$-curve Dyson’s Brownian motion each of whose curves begins at the origin at time zero. The ensemble has a scaled counterpart $\mathcal{L}_N^{{\rm sc}}: {\llbracket 1,N \rrbracket} \times [-2^{-1} N^{1/3},\infty ) \to {\ensuremath{\mathbb{R}}}$, given by $$\mathcal{L}_N^{{\rm sc}}(i,u) = 2^{-1/2} N^{-1/3} \Big( L_N\big( i,N + 2N^{2/3} u \big) - 2N - 2 N^{2/3} u \Big) \, .$$ By [@BrownianReg Proposition $2.5$], we have the
[*Key fact:*]{} there exist constants ${C},{c}\in (0,\infty)$ such that, for all choices of the parameters $(n,t_1,t_2,x)$, the ensemble $\mathcal{L}_N^{{\rm sc}}$ is $(c,C)$-regular.
There is only one, very mundane, difference between the ensemble $\mathcal{L}_N^{{\rm sc}}$ and the normalized forward ensemble ${\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow;t_2}$ that is the subject of Proposition \[p.scaledreg\]. With our choice of $N = n{t_{1,2}}+ 1$, the latter ensemble is simply equal to $\mathcal{L}_{N-1}^{{\rm sc}}$. The key fact thus bears the essence of the proof of Proposition \[p.scaledreg\]. The formal proof, which will now be given, is merely a matter of verifying that the adjustment of the parameter value $N \to N -1$ is inconsequential for the purpose of checking the conditions enjoyed by a $(c,C)$-regular ensemble.
[**Proof of Proposition \[p.scaledreg\].**]{} Note from (\[e.ln\]) and (\[e.scaledln\]) that ${\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow;t_2}(k,z)$ equals $$2^{-1/2} (n {t_{1,2}})^{-1/3} \Big( L_{(n t_1 + 2n^{2/3}x, n t_1 )}^{\uparrow; n t_2}\big( k, n t_2 + 2n^{2/3} x + 2(n {t_{1,2}})^{2/3} z \big) - 2 n t_{1,2} - 2 (n {t_{1,2}})^{2/3}z \Big) \, .$$ Note that $$\begin{aligned}
& & L_{(n t_1 + 2n^{2/3}x, n t_1)}^{\uparrow; n t_2}\big( k, n t_2 + 2n^{2/3} x + 2(n {t_{1,2}})^{2/3} z \big) - 2 n t_{1,2} - 2 (n {t_{1,2}})^{2/3}z \\
& = & L_N \big( k , n {t_{1,2}}+ 2(n {t_{1,2}})^{2/3} z \big) - 2 n t_{1,2} - 2 (n {t_{1,2}})^{2/3}z \\
& = & L_N \big( k , N + 2 N^{2/3} z (n {t_{1,2}}N^{-1})^{2/3} + n {t_{1,2}}- N \big) \\
& & \qquad - 2N - 2 N^{2/3}z - 2 (n t_{1,2} - N) - 2 \big( (n {t_{1,2}})^{2/3} - N^{2/3}\big) z \\
& = & L_N \big( k , N + 2 N^{2/3} \tilde{z} \big) - 2N - 2 N^{2/3} \tilde{z} + \phi_1 \, ,
\end{aligned}$$ where $\tilde{z} = z (n {t_{1,2}}N^{-1})^{2/3} - 2^{-1} N^{-2/3}$ satisfies $\tilde{z} = z \big( 1 + O(N^{-1}) \big) + O(N^{-2/3})$. Recalling that $N = n {t_{1,2}}+ 1$, the error $\phi_1$ is seen to equal $2 - 2 \big( (N - 1)^{2/3} - N^{2/3}\big) z - 2 N^{2/3}(z - \tilde{z})$ and thus to satisfy $$\phi_1 = O(1) + \vert z \vert O(N^{-1/3}) + \vert z \vert O(N^{-1/3}) + O(1) = \vert z \vert O(N^{-1/3}) + O(1) \, .$$
We see then that ${\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow;t_2}(k,z)$ equals $$2^{-1/2} (n {t_{1,2}})^{-1/3} \Big( L_N \big( k , N + 2 N^{2/3} \tilde{z} \big) - 2N - 2 N^{2/3} \tilde{z} + \phi_1 \Big) \, = \, \mathcal{L}_N^{{\rm sc}}\big( k, \tilde{z} \big) \phi_2 \, - \, 2^{-1/2} (n {t_{1,2}})^{-1/3} \phi_1 \, ,$$ where $\phi_2 = (n {t_{1,2}})^{-1/3} N^{1/3} = 1 + O(N^{-1})$. We find then that $${\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow;t_2}(k,z) = \mathcal{L}_N^{{\rm sc}}\big( k, \tilde{z} \big) \big( 1 + O(N^{-1}) \big) + \vert z \vert O(N^{-2/3}) + O(N^{-1/3}) \, ,$$ where $\tilde{z}$ differs from $z$ by $\vert z \vert O(N^{-1}) + O(N^{-2/3})$. We may thus note that $$\begin{aligned}
& & {\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow;t_2}(k,z) + 2^{-1/2} z^2 \\
& = & \mathcal{L}_N^{{\rm sc}}\big( k, \tilde{z} \big) \big( 1 + O(N^{-1}) \big) + 2^{-1/2} \tilde{z}^2 + \vert z \vert^2 O(N^{-1}) + \vert z \vert O(N^{-2/3}) + \vert z \vert O(N^{-2/3}) + O(N^{-1/3}) \\
& = & \mathcal{L}_N^{{\rm sc}}\big( k, \tilde{z} \big) \big( 1 + O(N^{-1}) \big) + 2^{-1/2} \tilde{z}^2 + \vert z \vert^2 O(N^{-1}) + \vert z \vert O(N^{-2/3}) + O(N^{-1/3}) \, . \end{aligned}$$ When $z = O(N^{1/2})$, the triple sum of error terms in the last expression is $O(1)$. The three ${{\rm Reg}}$ properties hold for $\mathcal{L}_N^{{\rm sc}}$ by the key fact stated before the proof. We want to learn that these hold also for ${\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow;t_2}$. The bound on the triple sum error permits the passage of ${{\rm Reg}}(2)$ and ${{\rm Reg}}(3)$ from $\mathcal{L}_N^{{\rm sc}}$ to ${\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow;t_2}$, provided that ${C}< \infty$ is increased, and ${c}> 0$ decreased, in order that the case of small $N$ is included. The property ${{\rm Reg}}(1)$ also makes the passage, because the left endpoint modulus ${z_{\mathcal{L}}}$ differs by a factor of $\big( n {t_{1,2}}N^{-1} \big)^{1/3} = 1 + O(N^{-1})$ between the two ensembles; again, ${c}> 0$ is being decreased in order to capture the case of small $N$. Thus, the ensemble ${\mathsf{Nr}{\mathcal}{L}}_{n;(x,t_1)}^{\uparrow;t_2}$ is $({c},{C})$-regular, as we sought to show.
[^1]: The author is supported by NSF grant DMS-$1512908$.
|
---
abstract: 'We study the renormalized energy-momentum tensor (EMT) of the inflaton fluctuations in rigid space-times during the slow-rollover regime for chaotic inflation with a mass term. We use dimensional regularization with adiabatic subtraction and introduce a novel analytic approximation for the inflaton fluctuations which is valid during the slow-rollover regime. Using this approximation we find a scale invariant spectrum for the inflaton fluctuations in a rigid space-time, and we confirm this result by numerical methods. The resulting renormalized EMT is covariantly conserved and agrees with the Allen-Folacci result in the de Sitter limit, when the expansion is exactly linearly exponential in time. We analytically show that the EMT tensor of the inflaton fluctuations grows initially in time, but saturates to the value $H^2 H^2_0$, where $H$ is the Hubble parameter and $H_0$ is its value when inflation has started. This result also implies that the quantum production of light scalar fields (with mass smaller or equal to the inflaton mass) in this model of chaotic inflation depends on the duration of inflation and is larger than the usual result extrapolated from the de Sitter result.'
address: |
$^1$ Department of Physics, Purdue University, West Lafayette, IN 47907, USA\
$^2$ Dipartimento di Fisica, Università degli Studi di Bologna and I.N.F.N.,\
via Irnerio, 46 – 40126 Bologna – Italy\
$^3$ H. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149,\
D-33761 Hamburg, Germany
author:
- 'F. Finelli [^1] $^{\,1,2}$, G. Marozzi$^\dag \ ^{\,2}$, G. P. Vacca$^\ddag \ ^{\,3,2}$ and G. Venturi$^\S \ ^{\,2}$'
title: 'Energy-Momentum Tensor of Field Fluctuations in Massive Chaotic Inflation'
---
epsf
Introduction
============
Particle production in expanding universe, pionereed by L. Parker [@parker1], is an essential ingredient of inflationary cosmology [@BOOKS]. The nearly scale invariant spectrum of density perturbations predicted by inflationary models [@all] is at present inextricably related to the concept of amplification of vacuum fluctuations by the geometry. The scale invariant spectrum for massless minimally coupled fields during a de Sitter era was indeed computed before inflation was suggested.
The calculation of the energy carried by these amplified fluctuations is then [*the*]{} natural question. To answer this question a renormalization scheme is necessary, as in ordinary Minkowski space-time. Ultraviolet divergences due to fluctuations on arbitrary short scales are common in field theory. In Minkowski space-time, infinities in a free theory are removed by the subtraction of the vacuum expectation value of the energy, also called normal ordering, the physical justification being that these vacuum contributions are unobservable.
A similar prescription is used in order to regularize infinities in cosmological space-times. However, one additional problem is the absence of an unambiguous choice of vacuum, because of the absence of a class of privileged observers, which are the inertial observers in the Minkowski space-times. The idea is then to subtract the energy associated with a vacuum for which the effects of particle production by the time-dependence of the metric are minimized. This vacuum is determined by the assumption of an adiabatic expansion of the metric. This procedure is therefore called [*adiabatic subtraction*]{} [@adiabatic; @birrell]. In this way the infinities of field theory are removed.
Even with a minimal prescription such as the above, there are several surprising effects accompanying the renormalized energy-momentum tensor (henceforth EMT) of a test field in cosmological space-times. To name a few, there could be the avoidance of singularities due to quantum effects [@parkerfulling], conformal anomalies which break classical conformal symmetries at the quantum level [@duff], violation of the various energy conditions [@visser]. One of the first models of inflation proposed by Starobinsky [@rsquare] was indeed based on the role of the conformal anomaly, which both avoids the singularity and produces an inflationary phase.
While the adiabatic vacuum for a test field - and its associated EMT - can be computed for generic cosmological space-times, the unrenormalized EMT can be calculated analytically only if exact analytic solutions for the field Fourier modes are available. Because of this, space-times such as de Sitter have been throughly investigated [@ren_desitter; @birrell], since analytic solutions for a scalar field with generic mass and coupling are available. In the absence of analytic solutions, numerical schemes are implemented [@numerical]. In a de Sitter space-time, the back-reaction of a test field seems important only if $m^2+\xi R = 0$, with $m$ and $\xi$ separately different from zero, or $m^2 + \xi R < 0$ ($m$ is the mass of the test field and $\xi$ its coupling to the curvature $R$) [@HMPM; @attractor]. In the former case, the EMT of the test field grows linearly in time, while in the latter case it grows exponentially. The EMT of a massless minimally coupled field is constant in de Sitter space-time [@AF].
Inflationary models based on the use of scalar fields, have an accelerated stage, usually called [*slow rollover*]{}, in which the Hubble parameter is almost frozen. During this stage it is rare to have exact solutions for the fluctuations. However, these inflationary models are more attractive than the de Sitter space-time one in furnishing phenomenological models whose predictions can be tested against observations. Chaotic models [@chaoticlinde] are the simplest among these. In this paper we study [*analytically*]{} the problem of back-reaction of inflaton fluctuations during the regime of slow-rollover for the case of a massive inflaton. We consider inflaton fluctuations in rigid space-times, i.e. we neglect metric perturbations coupled to them, as pioneeringly investigated by Abramo, Brandenberger and Mukhanov [@ABM]. We plan to come back to this issue in a future work [@new].
The plan of the paper is as follows: in section II we describe the background classical dynamics for a massive inflaton and the novel analytical approximation for its fluctuations during the slow-rollover regime. In section III we discuss the normalization of quantum fluctuations and in section IV we compare the numerical evaluation of the spectrum with the analytic approximation. We discuss the EMT of inflaton fluctuations, the adiabatic subtraction and the renormalization in sections V-VII, respectively. In section VIII the problem of the back-reaction of the EMT of inflaton fluctuations is addressed and in section IX our novel analytic approximation is compared with the slow-rollover technique [@SL]. In section X we analyze the production of a secondary massive field $\chi$ lighter than the inflaton and we show that its production depends on the duration of inflation. In section XI we conclude and in the two appendices we relegate useful formulae for the adiabatic expansion and the dimensional regularization with cut-off.
Analytic approximation
======================
We consider inflation driven by a classical minimally coupled massive scalar field. The action is: S d\^4x [L]{} = d\^4x \[action\] where ${\cal L}$ is the lagrangian density and $m$ is the mass of the field $\phi$. Further we consider the Robertson-Walker line element with flat spatial section:
ds\^2 = g\_ dx\^ dx\^ = - dt\^2 + a\^2(t) d\^2 \[metric\]
The scalar field is separated in its homogeneous component and the fluctuations around it, $\phi(t, {\bf x}) = \phi(t) +
\varphi (t, {\bf x})$. During slow rollover the potential energy dominates and therefore H\^2 = \^2 \[hubble\] where $\kappa^2 = 8 \pi G = 8 \pi/M_{\rm pl}^2$. The Hubble parameter evolves as: H = - \^2 \[hubbleder\] On using the equation of motion for the scalar field: + 3 H + m\^2 = 0 \[scalar\_hom\] and neglecting the second derivative with respect to time we have $3 H
\dot \phi \simeq - m^2 \phi$, and from Eq. (\[hubbleder\]) one obtains: H - H\_0 \[hdot\] which leads to a linearly decreasing Hubble parameter and correspondingly to an evolution for the scale factor which is not exponentially linear in time, i.e.: H(t) H\_0 + H\_0 t a(t) \[evolution\] In Fig. \[fig:hubble\] the comparison of the analytic approximation with the numerical evolution for the Hubble parameter $H$ is displayed.
=2.5in
We now consider the equations of motion for the inflaton fluctuations $\varphi$ in a rigid space-time (i.e. without metric perturbations):
\_[**k**]{} + 3 H \_[**k**]{} + \_k\^2 \_[**k**]{} = 0 , \[scalar1\] where $\omega_k^2 = k^2/a^2 + m^2$ and the $\varphi_{\bf k}$ are the Fourier modes of the inflaton fluctuations, (t, [**x**]{}) = \_[**k**]{} \[scalarFourier\]
As already emphasized in the introduction, exact solutions for scalar fields in an expanding universe are rare, and indeed we do not have exact solutions for Eq. (\[scalar1\]) with the time evolution given by Eq. (\[evolution\]). We therefore introduce an approximation scheme based on an analogy with de Sitter space-time, where exact solution for scalar fields with arbitrary mass and coupling to the curvature do exist. We introduce $\psi_{\bf k} = a^{3/2} \varphi_{\bf k}$ and we split the time dependence in $\psi_{\bf k}$ as follows: \_[**k**]{} = \_[**k**]{} (, H) = . \[zdef\] The equation for $\psi_{\bf k}$ is \_[**k**]{} + \_[**k**]{} = 0 . \[psi\] We now make the ansatz $\psi_{\bf k} = \zeta^{\mu} Z_\nu (\lambda \zeta)$ with $\mu$, $\nu$ and $\lambda$ functions of $H$. On expressing the first and second time derivatives as derivatives with respect to $(\zeta,H)$ and using $\ddot H \simeq 0$, as follows from Eq. (\[hdot\]), we obtain from Eq. (\[psi\]) after a little algebra: $$\begin{aligned}
\zeta^\mu && \left[ \zeta^2 \frac{\partial^2}{\partial \zeta^2} +
\zeta
\frac{\partial}{\partial \zeta} + (\lambda^2 \zeta^2 - \nu^2 ) \right]
Z_\nu + \nonumber \\
&&
{\rm Res}_1 \, \zeta^{\mu +1} \, \frac{\partial Z_\nu}{\partial \zeta} +
{\rm Res}_2 \, \zeta^{\mu+2} \, Z_\nu + {\rm Res}_3 \,
\zeta^{\mu} \, Z_\nu
+ {\cal O} (\frac{\dot H^2}{H^4}) = 0 \,
\label{maineq} \end{aligned}$$ where we have neglected quadratic and higher order terms in $\dot{H}/H^2$. Indeed, in order to have a value of density perturbations compatible with observations, $m$ is constrained to be ${\cal O} (10^{-5}-10^{-6} M_{\rm pl})$: from Eqs. (\[hubble\],\[hdot\]) one can see that working to first order in $\dot{H}/H^2$ during slow-rollover ($\phi \sim {\rm few}
M_{\rm pl}$) is a good approximation. On considering $H$ and $\zeta$ as independent variables, the first term vanishes if $Z_\nu$ is a Bessel function of argument $\lambda \zeta$ and index $\nu$. On requiring that the residual functions ${\rm Res}_i , \, i=1-3$ vanish individually, the parameters $\lambda, \mu$ and $\nu$ are determined to be: = 1 - , = \[parameters1\] \^2 = - - 3 . \[parameters2\] Hence the general solution to Eq. (\[scalar1\]) is: \_[**k**]{} = \^ \[solution\] where $H_\nu^{(1,2)}$ are the Hankel functions of first and second kind respectively, and $A, B$ are time-independent coefficients to the order of our approximation.
We note that in the de Sitter limit ($\dot H/H^2 \rightarrow 0$) the solution in Eq. (\[solution\]) tends to the de Sitter solution [@ren_desitter; @sol_desitter; @BD], since $\lambda = 1 \,, \mu=0$.
On using Eq. (\[hdot\]) the value for the index $\nu$ in Eq. (\[parameters2\]) corresponds to an exact scale invariant spectrum for the inflaton fluctuations $\varphi$, i.e. $\nu = 3/2$. We shall show numerically in section IV that this analytic approximation is very good for the relevant spectrum range. This numerical analysis agrees with a previous numerical estimate of the same spectral index [@kuztka]. This scale invariant spectral index could seem a little surprising, since in de Sitter space-time, a mass term would lead to a spectrum, which is slightly blue shifted with respect to scale invariance ($\nu < 3/2$). To see this, it is sufficient to put $\dot H = 0$ in Eqs. (\[psi\],\[parameters2\]). On considering $\dot H
\ne 0$ (and negative), it appears to give a positive contribution to the mass term in Eq. (\[psi\]), instead it compensates the mass term in Eq. (\[parameters2\]). The interpretation is the following: on considering a de Sitter stage in which $H$ slowly decreases, a fluctuation freezes when it crosses the Hubble radius, with an amplitude determined by the value of the Hubble radius at the horizon crossing. However $H$ decreases, therefore if $k_1 > k_2$, this effect implies that the amplitude for the mode $k_1$ is smaller than the one for the mode $k_2$, since the latter crosses the Hubble radius first. This effect is a red tilt of the de Sitter scale invariant spectrum. For the case of slow-rollover in a chaotic inflationary model with a massive inflaton, these red and blue shifts exactly compensate, leading to a scale-invariant spectrum for the inflaton fluctuations $\varphi$ in rigid space-time.
Quantized fluctuations
======================
We now consider quantized fluctuations of the inflaton. This means that Eq. (\[scalarFourier\]) is promoted to an operator form: (t, [**x**]{}) = \_[**k**]{} \[quantumFourier\] where the $\hat{b}_k$ are time-independent Heisenberg operators (also called time independent invariants in [@desitter; @nonzero]). In order to have the usual commutation relations among the $\hat{b}_k$: \[\_[**k**]{}, \_[[**k**]{}’]{}\] = \[\^\_[**k**]{}, \^\_[[**k**]{}’]{}\] = 0 \[\_[**k**]{}, \^\_[[**k**]{}’]{}\] = \^[(3)]{} ([**k**]{} - [**k**]{}’) one must normalize the solution to the equations of motion through the Wronskian condition: \_[**k**]{} \^\*\_[**k**]{} - \_[**k**]{} \^\*\_[**k**]{} = . \[wronskian\] This normalization condition yields to the following relation among the coefficients $A, B$ of Eq. (\[solution\]): |A|\^2 - |B|\^2 = \^[-2]{} . The fact that $A, B$ depend on time should not surprise. In time-dependent perturbation theory, these coefficients, which would be time independent for exact solutions, acquire a time dependence [@sakurai], just as the Wronskian of the solutions. In our case, this time dependence is consistent with the approximation, i.e. self-consistent to (including) order $\dot H/H^2$.
The solution corresponding to the Bunch-Davies vacuum [@BD] in de Sitter space-time, that is the adiabatic vacuum for $k \rightarrow \infty$ during the slow-rollover phase, corresponds to choosing $A = (\pi \lambda/4H)^{1/2} \zeta^{-\mu}
\,, B=0$. With this choice, for $\lambda \zeta \rightarrow \infty$, the solution (\[solution\]) becomes: \_[**k**]{} - e\^[+i ]{} \[adiabatic\_limit\] Let us now discuss the behaviour for $\lambda \zeta \ll 1$. On using Eqs. (\[zdef\],\[parameters1\]) one sees that $\lambda \zeta \ll 1$ implies $k \ll a H$, i.e. wavelengths which are much larger than the Hubble radius. In this limit, the solution is [@AS]: \_[**k**]{} - i ( )\^[1/2]{} ( )\^[-]{}. \[longw\] In order to compute expectation values of operators with respect to states in the time-independent invariant $b$ [@desitter; @nonzero] basis it is useful to introduce the modulus of mode functions $x_k =
|\varphi_{\bf k}/\sqrt{2}|$. The variable $x_k$ satisfies the following Pinney equation: x\_k + 3 H x\_k + \_k\^2 x\_k = . \[pinneyx\] We now rescale $x_k = \rho_k/a^{3/2}$ to eliminate the damping term and obtain: \_k + \_k = . \[pinneyrho\] The general solution to Eq. (\[pinneyrho\]) is given as a nonlinear combination of two independent solutions $y_1, y_2$ to the linear part of Eq. (\[pinneyrho\]). From Eq. (\[psi\]) we can use the following solutions: $$\begin{aligned}
y_1 &=& \zeta^\mu J_\nu(\lambda \zeta) \nonumber \\
y_2 &=& \zeta^\mu N_\nu(\lambda \zeta) \label{newsolution2B}
\end{aligned}$$ Therefore the solution to Eq. (\[pinneyrho\]) is \_k = ( L y\_1\^2 + M y\_2\^2 +2 N y\_1 y\_2 )\^ where the coefficients satisfy $LM - N^2 = 1/\bar{W}^2$, with $\bar{W}$ the (time-dependent) Wronskian of $y_1, y_2$. The choice of initial conditions for $\rho_k$ which corresponds to the adiabatic vacuum for $k \rightarrow \infty$ is $N=0$ and $L=M=\lambda \pi/(2 H
\zeta^{2\mu})$. The solution for $x_k$ is: x\_k = ( )\^[1/2]{} \^[1/2]{} \^[1/2]{} \[xsolution\] which coincides with the Bunch-Davies choice in the de Sitter limit [@desitter].
Numerical Analysis
==================
In this section we present the numerical analysis of the time evolution of the $\varphi$ modes. Besides checking the validity of the analytical approximation introduced in section II, this analysis is useful in order to understand how a natural infrared cut-off emerges in the problem, on assuming that inflation is not eternal in the past, but starts at some finite time. This infrared cut-off becomes relevant when the $\varphi$ fluctuations are generated in an infrared state [@fordparker], as occurs for $\nu \ge 3/2$ (see Eq. (\[parameters2\])).
First we want to analyze the properties of the spectrum of the inflaton fluctuations. We numerically evolve Eqs. (\[hubble\],\[scalar\_hom\]) and Eq. (\[pinneyx\]). We present numerical data for an interval of comoving wavenumbers for which $1 \le k/m \le 10^5$ at the initial time $t_0$ ($a(t_0) = 1$). The initial conditions are $\phi(t_0)= 4.5 M_{\rm pl}$, $\dot \phi (t_0) = 0$ for the inflaton. If we consider the vacuum state for each mode of the field fluctuations to be the initial condition, one has
$$\begin{aligned}
x_k(t_0) &=& \frac{1}{a^{3/2}(t_0) \omega_k^{1/2}(t_0)} \nonumber \\
\dot{x}_k(t_0) &=& 0
\label{initpinney}
\end{aligned}$$
This fact can be easily seen in terms of the invariant operators introduced to quantize time dependent harmonic oscillators [@desitter; @nonzero; @chaotic].
As a second initial condition, we consider the limit for large $k$ of the conditions in (\[initpinney\]), which correspond to setting the mass equal to zero. A third set of initial conditions is related to the adiabatic expansion in conformal time, (see later Eq. (\[conformal\_eq\])): $$\begin{aligned}
x_k(t_0) &=& \frac{1}{a(t_0) \Omega_k^{1/2}(t_0)} \nonumber \\
\dot x_k(t_0) &=& - H(t_0) x_k(t_0)
\end{aligned}$$ where $\Omega_k$ will be defined in Eq. (\[conformal\_freq\]). Let us note that for the last case the frequency becomes imaginary below a certain threshold and so we shall consider the region above it. In Fig. \[fig:comparison\] we exhibit the three cases. The first two initial conditions lead to spectra practically equal for $k$ of order $m$ and above, the third set instead has a spectrum which joins the others at values of $k$ of the order of $H_0$.
=2.5in
The spectrum of the fluctuations, related to the initial conditions in (\[initpinney\]), are displayed, over a broader range, in Fig. \[fig:spettro\].
=2.5in
Figs. \[fig:comparison\] and \[fig:spettro\] display the spectrum at $t=10/m$ (for this case inflation lasts a period of time $\sim 27/m$). We note that this scale invariant spectrum extends only up to a certain scale, $\ell$, which is of the same order as the initial Hubble radius $H(t_0)$ (we have checked this by changing the initial conditions for the homogeneous mode of the inflaton). For comoving modes smaller than $\ell$ the spectrum oscillates and bends towards the initial conformal adiabatic vacuum state, as shown in Fig. (\[fig:spettro\]). For all practical purposes we can therefore safely consider a scale invariant spectrum for $k > l$, where $\ell = C H (t_0)$, with $C$ a numerical coefficient ${\cal O} (1)$. This numerical evidence favours the picture in which the amplification of the modes occurs mainly at the crossing of Hubble radius. Since $m << H$ during inflation, this means that all the modes for which $m \le k \lesssim H(t_0)$, are not stretched by the geometry.
Secondly, we wish to show how accurate the approximation (\[xsolution\]) is mode by mode. In order to do this we numerically solve Eq. (\[pinneyrho\]) and compare it to the approximation employed. The agreement is very good up to times very close to the end of inflation. In Fig. \[fig:error\] we show for the mode with $k=10 \, m$ the time evolution of the relative error $(\rho_k^{(num)}-\rho_k^{(approx)})/\rho_k^{(num)}$. The larger $k$ is the better the agreement. We are therefore allowed to use the approximation for the inflaton dynamics almost til the end of inflation.
=2.5in
Let us now consider the correlator $\langle \varphi^2 \rangle$ (similar considerations are valid for other quantities bilinear in the field). We note from the spectral analysis that the integral over the modes can be split into two parts \^2 = d\^3 [**k**]{} |\_[**k**]{}|\^2 = d\^3 [**k**]{} x\_k\^2 = \[fisquare\]
Below the scale $\ell$ both the interval with the oscillations and the tilt around the scale invariant spectrum shown in Fig. 1 are included. The analytic treatment of the far infrared modes which contribute to the first integral would need an analytic approximation for the modes which at the beginning of inflation are outside the Hubble radius. This would amount to knowing the phase and the initial quantum states which precede the inflationary phase. Therefore we shall proceed by considering only the second integral and we shall neglect the first one in the far infrared. Even on neglecting the first integral which would require extra assumptions, the correct leading behaviour of the renormalized quantities is obtained [@explanation].
We conclude this section by noting that other methods to deal with infrared states are present in the literature. Infrared states were studied by Ford and Parker [@fordparker] for massless fields in Robertson-Walker space-times with a power-law expansion of a particular kind. By matching an earlier static space-time with a space-time with a scale factor which expands in time with a power law, they noted that an infrared finite state cannot evolve to an infrared divergent state [@fordparker]. The same scheme was also used by Vilenkin and Ford [@VF] for the problem of massless minimally coupled scalar fields in de Sitter space-time. An earlier radiation dominated phase was matched to the de Sitter metric. Through this matching the infrared tail becomes suppressed leading to an infrared finite state. As we shall see both the calculations performed by eliminating the infrared tail (i.e. working with the cut-off) and suppressing the infrared tail (through the Bogoliubov coefficients obtained by the matching prescription) lead to the same result to leading order. Indeed, their physical motivation is the same: inflation is not eternal, but starts at a finite time. However, the two methods treat the infrared tail in a different way: the agreement to leading order implies that the relevant contribution to the correlator $\langle \varphi^2 \rangle$ - and to the EMT - comes from intermediate modes, and not from the furthest infrared modes.
The energy-momentum tensor {#four}
==========================
The classical energy-momentum tensor (henceforth EMT) of inflaton fluctuations is: T\_ = \_\_ + g\_ \[emt\] and its operator form is simply obtained by promoting $\varphi$ to an operator as in Eq. (\[quantumFourier\]).
When averaged over the vacuum state annihilated by $b$: \_[**k**]{} | 0 = 0 the energy-momentum tensor assumes a perfect fluid form because of the symmetries of the RW background [@guven]: T\_ = [diag]{} (, a\^2 p \_[ij]{}) , where $\epsilon$ and $p$ are the energy density and the pressure density respectively.
In the following we consider, according to the previous sections, $\nu=3/2$, employing the dimensional regularization [@birrell] to treat the UV behaviour. Therefore the integrands will be in $3$ dimensions and the integration measure analitically continued in $d$ dimension.
The energy density is $$\begin{aligned}
\epsilon = \langle T_{00} \rangle &=&
\frac{\hbar}{2 (2 \pi)^d} \int_{|k|>\ell} d^d {\bf k} \left[
|\dot\varphi_{\bf k}|^2
+ \frac{k^2}{a^2} |\varphi_{\bf k}|^2 + m^2 |\varphi_{\bf k}|^2
\right] \nonumber \\
&=& \frac{\hbar}{4 (2 \pi)^d}
\int_{|k|>\ell}
d^d {\bf
k} \left[ {\dot x}_k^2 + \frac{1}{a^6 x_k^2} + \left( \frac{k^2}{a^2} +
m^2 \right) x_k^2 \right] \,,
\label{endensity}
\end{aligned}$$ and the pressure density $p$, related to the space-space component of the EMT, is: p = = \_[|k|>]{} d\^d [**k**]{} . \[pressdensity\]
On using Eqs. (\[xsolution\]) and (\[prud1\]) in Appendix B with $\alpha = d-1$, we may now compute the second part of the integral (\[fisquare\]): $$\begin{aligned}
\langle \varphi^2 \rangle &=&
\frac{\hbar}{2 (2 \pi)^d} \int_{|k|>\ell} d^d {\bf k} \, x_k^2
\label{newphiB} \\
&=&\frac{\hbar}{16\pi^2}
H^2 \left( 1-\frac{2}{3}\frac{m^2}{H^2}\right) \Bigl\{ 2-4 \ln 2-
2 \left(\frac{\ell}{2 \pi^{1/2}} \right)^{d-3}
\Gamma \left(\frac{1}{2}-\frac{d}{2}\right) + \nonumber \\
& &
+{\cal O} \left (
\frac{1}{a^2} \right )\Bigr\} +{\cal O} (d-3)
\label{fisquare_paper}
\end{aligned}$$
Analogously, the energy and pressure densities in Eqs. (\[endensity\],\[pressdensity\]), with the help of the formulae in Appendix B, are: $$\begin{aligned}
\epsilon=\langle T_{0 0} \rangle = \frac{\hbar}{16\pi^2} H^4 &\Bigl\{&
4 \frac{m^2}{H^2}
\left( \frac{1}{2 \pi^{1/2}} \right)^{d-3}
\ell^{d-3} \Gamma(1-d) \nonumber \\
& &+\frac{m^2}{H^2} \Bigl[ -1
+\gamma -2 \ln 2 \Bigr] +{\cal O}
\left( \frac{1}{a^2} \right) \Bigr\} +{\cal O} (d-3)
\label{bare_energy}
\end{aligned}$$ $$\begin{aligned}
p = \frac{\langle T_{i i} \rangle}{a^2} =
\frac{\hbar}{16\pi^2} H^4 &\Bigl\{&
- 4 \frac{m^2}{H^2} \left (\frac{1}{2 \pi^{1/2}} \right
)^{d-3}\ell^{d-3} \Gamma(1-d) \nonumber \\
& & +\frac{m^2}{H^2} \Bigl[ +1 -\gamma +2 \ln 2 \Bigr] +{\cal O} \left (
\frac{1}{a^2} \right )\Bigr\} +{\cal O} (d-3)
\label{bare_pressure}
\end{aligned}$$
The poles given by the negative values of the argument of the $\Gamma$ function in Eqs. (\[fisquare\_paper\],\[bare\_energy\],\[bare\_pressure\]) represent part of the ultraviolet infinities of field theory.
The adiabatic subtraction {#five}
=========================
In order to remove the divergent quantities which appear in the integrated quantities as poles in the $\Gamma$ functions, we shall employ the method of [*adiabatic subtraction*]{} [@adiabatic]. Such a method consists in replacing $x_k$ with an expansion in powers of derivatives of the logarithm of the scale factor in Eqs. (\[endensity\]-\[newphiB\]). This expansion coincides with the adiabatic expansion introduced by Lewis in [@lewis] for a time dependent oscillator.
Usually it is more convenient to formulate the adiabatic expansion by using the conformal time $\eta$ [@adiabatic] ($d \eta = d t/a $). We follow this procedure and write an expansion in derivatives with respect to the conformal time (denoted by $'$) for $x_k$. Then go back to the cosmic time and we insert the expansion in the expectation values we wish to compute. Adiabatic expansion in cosmic time and conformal time lead to equivalent results, because of the explicit covariance under time reparametrization [@HMPM].
We rewrite Eq. (\[pinneyx\]) in conformal time in the following way: (a x\_k)” + \_k\^2 (a x\_k) = \[conformal\_eq\] where \_k\^2 = k\^2 + m\^2 a\^2 - a\^2 R \[conformal\_freq\] and $R$ is the Ricci curvature: R = 6 . \[ricci\_d\]
The fourth order expansion for $\langle \varphi^2\rangle$, the energy and pressure densities are therefore (as before $\nu=3/2$, integrands in $3$ space dimensions and the measure is analytically continued in $d$ dimensions), using the expression in (\[fourth\_d\]) and the results of appendix A and B,
$$\begin{aligned}
\langle \varphi^2\rangle_{(4)} &=&\frac{\hbar}{16\pi^2} H^2
\Bigl\{ \left[ -2+\frac{4}{3}\frac{m^2}{H^2}\right]
\left(\frac{a m}{2 \pi^{1/2}} \right)^{d-3} \Gamma
\left(\frac{1}{2}-\frac{d}{2}\right) +\frac{2}{9}
\frac{m^2}{H^2}-\frac{4}{3} + \nonumber \\
& &
+\frac{1}{m^2}\left[\frac{7}{45}m^2+\frac{29}{15}H^2\right]+{\cal O} \left
( \frac{1}{a^3} \right )\Bigr\} +{\cal O} (d-3)
\label{fisquare_appendix2}
\end{aligned}$$
$$\begin{aligned}
\epsilon_{(4)}=\langle T_{0 0} \rangle_{(4)} = \frac{\hbar}{16\pi^2} H^4 &\Bigl\{&
4 \frac{m^2}{H^2} \left (\frac{1}{2 \pi^{1/2}}
\right )^{d-3}a^{d-3}m^{d-3} \Gamma(1-d) \nonumber \\
& & + \frac{119}{60}
+\frac{m^2}{H^2} \left [ -\frac{33}{10}
+\gamma \right ] +{\cal O} \left
( \frac{1}{a^3} \right )\Bigr\} +{\cal O} (d-3)
\label{en_fourth}
\end{aligned}$$
$$\begin{aligned}
p_{(4)}=\frac{\langle T_{i i}\rangle_{(4)}}{a^2} = \frac{\hbar}{16\pi^2} H^4 &\Bigl\{&
-4 \frac{m^2}{H^2} \left (\frac{1}{2 \pi^{1/2}}
\right )^{d-3}a^{d-3}m^{d-3} \Gamma(1-d) \nonumber \\
& & - \frac{119}{60} +
\frac{m^2}{H^2} \left [ +\frac{1309}{270}
-\gamma \right ] +{\cal O} \left
( \frac{1}{a^3} \right )\Bigr\} +{\cal O} (d-3)
\label{press_fourth}
\end{aligned}$$
The conserved renormalized EMT
==============================
On subtracting the adiabatic part given in Section \[five\] from the bare integrated contribution given in Section \[four\] and taking the limit $d \rightarrow 3$ one obtains the finite renormalized expectation value for the correlator and for the energy-momentum tensor.
The renormalized expectation value of $\langle \varphi^2 \rangle$ is therefore, neglecting terms of order $1/a^3$ and for $a> H/m$ [@explanation], $$\begin{aligned}
\langle \varphi^2 \rangle_{REN} &=& \langle \varphi^2\rangle-
\langle \varphi^2 \rangle_{(4)} \nonumber \\
&=& \frac{\hbar}{16\pi^2} H^2
\Bigl\{ \left( 4-\frac{8}{3}\frac{m^2}{H^2}\right) \left( \ln a-
\ln \frac{C H(t_0)}{m} \right) -\left( 1-\frac{2}{3}\frac{m^2}{H^2}\right)
4 \ln 2- \nonumber \\
& & -\frac{14}{9}\frac{m^2}{H^2}+\frac{10}{3}
- \frac{1}{m^2}\left[\frac{7}{45}m^2+\frac{29}{15}H^2\right]
+ {\cal O} \left (\frac{1}{a^3} \right )\Bigr\}\; .
\label{fisquare_ren}
\end{aligned}$$
Considering (\[fisquare\_ren\]), we note that for late times it resembles more a massless, than a massive, field in de Sitter space-time. This is a consequence of the scale invariant spectrum (\[parameters2\]) of inflaton fluctations (the same spectrum occurs for massless minimally coupled fields in de Sitter space-time). The leading behaviour for $\langle \varphi^2 \rangle_{REN}$ agrees for late times with the de Sitter result [@AF; @VF; @cutoff] when $\dot H=0$ and $a(t) = a_0 e^{H_{\rm DS} t}$: \^2 \_[REN]{}\^[DS]{} \~ H\_[DS]{}\^3 t . \[afleading\] At earlier times, $\langle \varphi^2 \rangle_{REN}$ is dominated by a contribution ${\cal O} (H^4/m^2)$, but then the time dependent piece ${\cal O}(\ln a)$ takes over. We warn the reader about the massless limit taken at face value of Eq. (\[fisquare\_ren\]). In the massless limit, as discussed in the appendix B, one has to use a different analytic continuation for the adiabatic part, related to the expression in (\[intadiamassless\]) and the result for $\langle \varphi^2
\rangle_{REN}$ will be finite, different from Eq. (\[fisquare\_ren\]), but with the same leading contribution (\[afleading\]). However, this massless limit is just of academic interest, since for $m=0$ inflation would not happen.
Even if both inflaton fluctuations for a massive inflaton and a massless minimally coupled scalar fields in de Sitter share the same scale invariant spectrum, the energy and pressure carried by fluctuations for these two cases are very different. For the latter case, a linear growth in time is present only for the correlator; the EMT does not contain the correlator, but only bilinear quantities less infrared than $\varphi^2$ (for a nice explanation of this difference see [@HMPM]). For the case of inflaton fluctuations, the correlator appears directly in the EMT because of the nonvanishing mass. The kinetic and gradient terms should be smaller than the potential term, as for the massless minimally coupled case. Therefore for the case of inflation driven by a massive inflaton, the EMT of inflaton fluctuations should grow in time. This is what we shall show in the following.
On subtracting Eq. (\[en\_fourth\]) from Eq. (\[bare\_energy\]), the renormalized energy density $\epsilon_{\rm REN}$ is: $$\begin{aligned}
\epsilon_{\rm REN} &=& \langle T_{00} \rangle_{\rm REN} =
\langle T_{00} \rangle - \langle T_{00} \rangle_{(4)} = \nonumber \\
&=&\frac{\hbar}{16\pi^2} H^4 \Bigl\{
-2 \frac{m^2}{H^2} \left [ \ln \frac{C H(t_0)}{m}-\ln a(t)\right ]-
\frac{119}{60}
+\frac{m^2}{H^2} \Bigl[\frac{23}{10}
- 2 \ln 2 \Bigr] +{\cal O} \left
( \frac{1}{a^2} \right )\Bigr\}\,.
\label{energy_ren}
\end{aligned}$$ Similarly, by subtracting Eq. (\[press\_fourth\]) from Eq. (\[bare\_pressure\]), the renormalized pressure density $p_{\rm REN}$ is: $$\begin{aligned}
p_{\rm REN} &=& \frac{\langle T_{i i} \rangle_{\rm REN}}{a^2} =
\frac{ \langle T_{ii} \rangle - \langle T_{ii} \rangle_{(4)} }{a^2}
\nonumber \\
&=&\frac{\hbar}{16\pi^2} H^4 \Bigl\{
2 \frac{m^2}{H^2} \left [ \ln \frac{C H(t_0)}{m}-\ln a(t)\right ]+
\frac{119}{60}
+\frac{m^2}{H^2} \Bigl[ -\frac{1039}{270}
+ 2 \ln 2 \Bigr] + {\cal O} \left
( \frac{1}{a^2} \right )\Bigr\}
\label{pressure_ren}
\end{aligned}$$
We note that this result does not agree for $\dot H=0$ (and therefore for $m=0$ because of Eq. (\[hdot\])) and $\xi=0$ with the de Sitter result [@birrell; @guven] obtained with the Bunch-Davies vacuum: $$\begin{aligned}
T_{\mu \nu \, {\rm REN}}^{{\rm DS}\,{\rm BD}} &=& - \frac{g_{\mu
\nu}}{64
\pi^2} \left[ m^2
\left[ m^2 + (\xi - \frac{1}{6}) R \right]
\left[ \psi(\frac{3}{2} + \nu) + \psi(\frac{3}{2} - \nu) -
\ln \frac{12 m^2}{R} \right] - \right. \nonumber \\
&& \left. - m^2 (\xi - \frac{1}{6}) R
- \frac{1}{18} m^2 R - \frac{1}{2} (\xi - \frac{1}{6})^2 R^2
+ \frac{1}{2160} R^2 \right]
\label{desitter}
\end{aligned}$$ where $R$ is the curvature in de Sitter ($R = 12 H^2_{\rm DS}$ with $H_{\rm DS}$ as the Hubble parameter in de Sitter space-time) [@largem; @senzacutoff]. Indeed, the limit of Eq. (\[desitter\]) for vanishing mass and coupling is finite and is [@AF]: T\_[ [REN]{}]{}\^[[DS]{} [BD]{}]{} = - g\_ \[desitter\_BD\] Instead, the result (\[energy\_ren\],\[pressure\_ren\]) agrees with the Allen-Folacci [@AF] result for $m=0$: T\_[ [REN]{}]{}\^[[DS]{} [AF]{}]{} = g\_ \[desitter\_AF\]
However, the renormalized EMT of inflaton fluctuations in Eqs. (\[energy\_ren\],\[pressure\_ren\]) grows in time, as the logarithm of the scale factor. This feature is due both to the fact that the inflaton is massive and to the infrared state in which its fluctuations are generated. For a test field in de Sitter space-time a linear growth in time of the EMT is possible, only for $m^2 + \xi R = 0$, with $m$ and $\xi$ both different from zero.
The renormalized EMT in de Sitter space-time in the Bunch-Davies vacuum (\[desitter\]) and in the Allen-Folacci vacuum corresponds to a perfect fluid with an equation of state $w = p/\epsilon
=-1$, which is identical to the background driven by a cosmological constant. The conservation of the renormalized EMT (\[desitter\],\[desitter\_AF\]) is direct consequence of its symmetries: T\_[ [REN]{}]{}\^[DS]{} g\_ \^T\_[ [REN]{}]{}\^[DS]{} = 0 since $\nabla^\mu g_{\mu \nu} = 0$.
The renormalized EMT in an inflationary stage driven by a massive inflaton corresponds to a perfect fluid, [*but*]{} with an equation of state which differs from $-1$ by terms ${\cal O} (m^2/H^2)$, as one can see from Eqs. (\[energy\_ren\],\[pressure\_ren\]).
The derivation of the conservation of the renormalized EMT in chaotic inflation is also straightforward. The renormalized EMT is conserved consistently with the approximation used, i.e. to the order ${\cal O}
(m^2/H^2)$. The conservation can be easily checked mode by mode, i.e. by considering the covariant derivative inside the integrals in $\bf k$ in the difference between the bare value and the fourth order adiabatic value and using the equations of motion for the field modes (\[scalar1\]). The conservation of the final renormalized value EMT can also be checked by inserting the expressions given by Eqs. (\[energy\_ren\]) and (\[pressure\_ren\]) in + 3 H (\_[REN]{} + p\_[REN]{} ) = 0 . \[cons\_ren\] and retaining only the terms up to and including ${\cal O} (m^2/H^2)$.
Back-reaction on the geometry
=============================
We now discuss the back-reaction of the amplified $\varphi$ fluctuations on the geometry.
We consider the back-reaction equations perturbatively: we evaluate the higher order geometrical terms [@birrell] generated by renormalization as their background value and we do not use them to generate higher order differential equations. We note that, in accord with the approximations used, many higher order derivative terms are implicitly absent in Eqs. (\[energy\_ren\],\[pressure\_ren\]), since they would be of higher order in powers of ${\dot H}/H^2$ and because $\ddot H \simeq 0$. We estimate the back-reaction effects without changing the structure of the left hand side of Einstein equations. Hence the back-reaction equations we consider are: $$\begin{aligned}
H^2 &=& \frac{8 \pi G}{3} \left[ \frac{\dot \phi^2}{2} + \frac{m^2}{2}
\phi^2
+ \epsilon_{\rm REN} \right] \label{hubble_back} \\
\dot H &=& - 4 \pi G ( \dot \phi^2 + \epsilon_{\rm REN} + p_{\rm REN})
\label{hdot_back} \,.
\end{aligned}$$
The main point is that the energy and pressure of inflaton fluctuations grows in time as the logarithm of the scale factor, while the Hubble parameter driven by the background energy density decreases linearly in time. However, the approximation we use, i.e. Eqs. (\[hdot\],\[evolution\]), is valid for a certain time interval, $\Delta t$ given by:
t \~ \[interval\]
Therefore the term which grows in the renormalized EMT will saturate at the value: \_[REN]{} (t) \~- p\_[REN]{} (t) \~ H\^2 H\_0\^2 . \[maximum\]
The term (\[maximum\]) is larger than the Allen-Folacci value $\sim H^4$ in Eq. (\[desitter\_AF\]) and has the opposite sign. Therefore, the energy density of fluctuations starts with a negative value and changes sign to a positive value when the logarithm takes over. This behaviour is shown in Figs. \[fig:eren\] and \[fig:ratio\_br\].
Such a value leads to a contribution to the Einstein equations of the order $H^2 H^2_0/M_{\rm pl}^2$. The importance of back-reaction is therefore related to the ratio $H^2_0/M_{\rm pl}^2$. If inflation starts at a Planckian energy density then back-reaction during inflation cannot be neglected.
A sligthly different conclusion can be reached on considering the variation in time of the Hubble parameter, i.e. Eq. (\[hdot\_back\]). The important point to note is that the leading contribution in $\epsilon_{\rm REN}, p_{\rm
REN}$, i.e. the terms $\sim H^4$ and $\sim m^2 H^2 \log a$, do not contribute to $\dot H$ since these terms have an equation of state $p_{\rm REN}/\epsilon_{\rm REN}=-1$. The contribution of the inflaton fluctuations to $\dot H$ is therefore of order $m^2 H^2/M_{\rm pl}^2$, which is suppressed with respect to the classical value by the factor $H^2/M_{\rm pl}^2$.
=2.5in
=2.5in
Comparison with the slow-rollover calculation
=============================================
We now compare the approximation which lead us to Eq. (\[solution\]) with the slow-rollover technique introduced by Stewart and Lyth [@SL]. The latter technique was developed directly for scalar and tensor metric perturbations [@SL], and not for field perturbations in rigid space-times, as treated here. However, the equation for gravitational waves differ from Eq. (\[scalar1\]) only by the presence of the mass term $m$. Therefore, as a first check we note from Eq. (\[parameters2\]) that \^2\_[GW]{} = - 3 \_[GW]{} - \[gw\] where the second relation holds when ${\dot H}/H^2$ is small. The value $\nu_{\rm GW}$ in Eq. (\[gw\]) coincides with the index $\mu$ of Eq. (41) of [@SL]. On the other hand, if one applies the Stewart-Lyth procedure to Eq. (\[scalar1\]) one would obtain an index for Hankel functions which is precisely $\nu$ in Eq. (\[parameters2\]).
A natural question is to ask whether inflaton fluctuations are generated in infrared states also for other models of chaotic inflation. Analytic approximations, such as the one presented in Section 2, are very difficult to obtain. However, since our calculation agrees with the slow-rollover result [@SL] for inflaton fluctuations in rigid space-times and gravitational waves in the case of a massive inflaton, one can use the latter technique to estimate the spectral index of fluctuations in a generic chaotic model with potential $V(\phi) = \lambda \phi^n/n$ (here $\lambda$ has the dimensions of a mass elevated to the power $4 - n$). We follow Stewart and Lyth and we study the equation: (a \_[**k**]{})” + ( k\^2 + m\^2\_[eff]{} a\^2 - ) (a \_[**k**]{}) = 0 , where $m^2_{\rm eff} = V,_{\phi \phi} = \lambda (n-1) \phi^{n-2}$. On using $a(\eta) = - 1/[ H \eta (1 - \epsilon)]$, after some algebra one obtains the following spectral index for the constant mode: = + \[general\_nu\] where the slow-rollover parameter $\epsilon$ is taken as constant and is defined as: = - = The slow-rollover approximation is better for large values of the inflaton. The result (\[general\_nu\]) agrees for $n=2$ with the quadratic case. However, for $n > 2$ the spectral index $\nu$ is smaller than the critical value $3/2$, leading to inflaton fluctuations which are infrared finite.
Although the approximation presented here gives the same result as the slow-rollover technique to the lowest order [@SL] for gravitational waves and inflaton fluctuations in rigid space-times for a massive chaotic model, it will not be so when one includes metric perturbations [@new].
Moduli Production
=================
Let us now discuss the quantum production of a light scalar field $\chi$ in this model of massive chaotic inflation. By the term light, we mean a scalar field with a mass $M$, which is smaller than (or equal to) the inflaton one, $m$. We assume a vanishing homogeneous component for $\chi$. Therefore, for $\chi$ the approximation of a rigid space-time is correct.
With $M < m$, we see from Eqs. (\[hdot\],\[parameters2\]) that the index $\nu_\chi$ for the Hankel functions, which are involved in the solutions for the $\chi$ modes, is larger than $3/2$. The leading contribution for the renormalized value of $\langle \chi^2 \rangle$ is: \^2 \_[REN]{} \~ H\^2 \[chisquare\_ren\] where $\beta$ is a numerical coefficient. This formula generalizes Eq. (\[fisquare\_ren\]) to the case of a mass $M < m$. The case $M=m$ represents a limiting case, for which a logarithm appears. Analogously, in de Sitter space-time, a limiting case is also present for $\nu = 3/2$ [@HMPM], and a logarithm of the scale factor appears instead of a power.
For a massive $\chi$ the following relation holds: \_[GW]{} > \_> = where $\nu_{\rm GW}$ is defined in Eq. (\[gw\]) and, owing to the smallness of the ratio $m^2/H^2$, $\nu_\chi$ is very close to $3/2$. However, the growth in time of $\langle \chi^2
\rangle_{\rm REN}$ is more rapid than the growth of $\langle \varphi^2 \rangle_{\rm REN}$.
This fact can be checked numerically: for example, on taking the initial condition for inflation already used in the previous sections for the other numerical checks, one finds for $M=0$, $\nu_\chi \simeq 1.5043$ and for $M=0.5m$, $\nu_\chi \simeq 1.5032$. Such a value is time independent after a very short transient phase needed for the modes to freeze and corresponds to the one given in (\[gw\]) for $H$ computed for a time very close to the beginning of the inflation. We can therefore substitute $H(t \simeq 0) \simeq H_0$ in place of $H(t)$ in $\nu$. We have checked this relation for different initial $H_0$. This fact is in agreement with the fact that the approximate solution, which we employed for the inflaton field, can be used for other fields with different masses, but only at the beginning of the inflation period. Thus we can control the spectral behavoiur of the moduli field, but not its normalization.
To study the behaviour in (\[chisquare\_ren\]) it is convenient to rewrite the evolution of the scale factor $a$ and its exponent $(2\nu_\chi-3)$ as a(t)= , 2\_-3 .
Therefore the main result (let us only write the dominant contribution) is that for $M < m$ and at the end of inflation \^2 \_[REN]{} \~H\^2 H\_0\^2 { } > \^2 \_[ REN]{} \~H\^2 \[result1\] and for the renormalized EMT associated with the $\chi$ field \_[REN]{}\^[()]{} \~ H\^2 H\_0\^2 { } . \[result2\] The results in Eqs. (\[maximum\],\[result1\], \[result2\]) are very interesting. They show that the production of a scalar field $\chi$ with mass $M$ smaller or equal to the inflaton mass $m$ depends on the duration of inflation and is larger than the usual extrapolation of the de Sitter result ($\langle \chi^2 \rangle \sim H^4/m^2$ and $\epsilon_\chi
\sim (M^2/m^2) H^4\;$). This result implies that the quantum production of light scalar fields in chaotic inflation with a mass term is even greater than expected on extrapolating the de Sitter result, and depends on the duration of inflation, as is also stated in [@kuztka; @moduli]. Again, if inflation starts at a Planckian energy density, the back-reaction of light scalar fields cannot be neglected during inflation.
We note that the factor in curly brackets in (\[result1\], \[result2\]) is larger than the logarithmic term one has for the $\nu=3/2$ case, by a factor of $2$.
We see that on computing the exact numerical solution of the Pinney equation for the modes of the inflaton with mass $m$ and for a moduli field with $M=0.5m$, one obtains for the same momenta amplitudes more than one order of magnitude larger for the latter at the end of inflation, again in agreement with [@kuztka]. We therefore expect an enhancement of 2-3 orders of magnitude for its backreaction with respect to the inflaton case.
Discussion and Conclusions
==========================
We have computed the renormalized conserved EMT of the inflaton fluctuations $\varphi (t, {\bf x})$ in rigid space-times during the inflationary stage driven by a mass term. The method of dimensional regularization has been applied by using an analytic approximation valid during the slow-rollover regime. All the results agree with the Allen-Folacci results for $T_{\mu\nu}$ of a test field in de Sitter space-time [@AF], in the limit for which the Hubble parameter is constant (which is also the massless limit because of Eq. (\[hdot\])).
We find that the EMT of inflaton fluctuations grows in time. The reason for this behaviour is that chaotic inflation driven by a massive scalar field produces a scale invariant spectrum of fluctuations even if the field is massive. This effect is due to the decrease of the Hubble parameter during the slow-rollover regime.
In de Sitter space-time, the renormalized EMT of a quantum field grows linearly in time [*only*]{} if $m^2 + \xi R_{\rm DS} = 0$ with $m$ and $\xi$ different from zero [@HMPM]. A massless minimally coupled scalar field in de Sitter space-time, characterized by a scale invariant spectrum of fluctuations, leads to a correlator which grows in time. However, only bilinear quantities less infrared than the correlator appear in the EMT, and therefore the expectation value of the EMT of a massless minimally coupled scalar field is constant in time [@HMPM]. In massive chaotic inflation, inflaton fluctuations are generated with a scale invariant spectrum. Since the correlator appears directly in the EMT because of the nonvanishing mass, then the renormalized EMT grows in time just as the correlator does.
We find that the growth of the EMT of inflaton fluctuations during slow-rollover leads to a positive energy density which reaches a maximum value ${\cal O} (H^2 H^2_0)$, where $H_0$ is the Hubble radius at the beginning of inflation. This value exceeds the usual value ${\cal O} (H^4)$, which is of the same order of magnitude as the conformal anomaly. These values also show that back-reaction effects cannot be neglected if inflation starts at Planckian energies, i.e. at $H_0 \sim
M_{\rm pl}$. If inflation started at Planckian energies, although the contribution of the terms ${\cal O} (H^2 H^2_0)$ and of the conformal anomaly would be of the same order of magnitude, we think that the two contributions could be different because of the different signs and of the different behaviours in time.
In this model of chaotic inflation, we have also analyzed the geometric production of an additional field $\chi$ with mass $M$ smaller than the inflaton mass $m$. Of course, on considering the normalization, we have found that $\epsilon_\chi >
\epsilon_\varphi \sim H^2 H_0^2$. This result implies that the quantum production of light fields depends on the duration of inflation and it is greater than expected on extrapolating the de Sitter result (in de Sitter $\epsilon_\chi \sim H^4$). As in the case of inflaton fluctuations, the energy density of light scalar fields could be comparable to the background one at the end of inflation, if inflation started at Planckian energy densities. Also in this case, the back-reaction of $\chi$ fluctuations does not appear to be negligible during inflation.
One may then ask whether this behaviour of the back-reaction due to the fluctuations in rigid space-time is common to other inflationary chaotic models. Analytic approximations, such as the one presented in Section 2, are very difficult to obtain. However, since our calculation agrees with the slow-rollover result [@SL] for the massive case, we have used the latter technique to estimate the spectrum of inflaton fluctuations in rigid space-time for a generic inflaton potential $V(\phi) = \lambda \phi^n/n$. We have found that for $n > 2$ the inflaton fluctuations are generated in an infrared finite state, leading to a back-reation which does not increase in time. However, we think that we must address the problem while including metric perturbations, in order to fully understand this issue. It is known that chaotic inflationary models predict a spectrum of curvature perturbations which is red tilted [@SL] - i.e. with a spectrum more infrared than the scale invariant one -, a result which does not hold for field perturbations in rigid space-time, as we have shown. Since infrared states could lead to a correlator which grows in time, the possibility exists that a back-reaction growing in time is common to all the chaotic inflationary models once metric perturbations are included.
Other important issues are whether an eventual self-consistent scheme to include the back-reaction would prevent the development of infrared states. The effect of the self-consistent inclusion of back-reaction effects on the spectrum of fluctuations during inflation is, to our knowledge, an issue still to be fully explored. It would be interesting also to investigate the effect of the inclusion of the higher order terms [@simon] in the back-reaction equations (\[hubble\_back\],\[hdot\_back\]). Obviously, the Starobinsky model [@rsquare] is a surprising example of the importance of higher order terms.
**Acknowledgments**
We would like to thank Raul Abramo, Robert Brandenberger, Sergei Khlebnikov and Igor Tkachev for discussions and comments on the manuscript. One of us (F. F.) would like to thank Salman Habib and Katrin Heitmann for many important discussions on renormalization in curved space-times and for warm hospitality at Los Alamos Laboratories, where part of this work was written.
Appendix A: The adiabatic fourth order expansion
================================================
From Eqs. (\[conformal\_eq\],\[conformal\_freq\],\[ricci\_d\]) one obtains the expansion for $x_k$ up to the fourth adiabatic order: x\_[k]{}\^[(4)]{}= ( 1- \_2+\_2\^2- \_4 ) \[fourth\_1\] where $\Omega_k$ is defined in Eq. (\[conformal\_freq\]) and $\epsilon_2 \,, \epsilon_4$ are given by: $$\begin{aligned}
\epsilon_2&=&-\frac{1}{2}\frac{\Omega_k^{''}}{\Omega_k^3}+\frac{3}{4}
\frac{\Omega_k^{'2}}{\Omega_k^4} \nonumber \\
\epsilon_4 &=& \frac{1}{4}\frac{\Omega_k^{'}}{\Omega_k^3}\epsilon_2^{'}-
\frac{1}{4}\frac{1}{\Omega_k^2}\epsilon_2^{''}
\end{aligned}$$
The solution in Eq. (\[fourth\_1\]) must be expanded again since the Ricci curvature is of adiabatic order 2. Therefore $x_k^{(4)}$ is: $$\begin{aligned}
x_{k (4)} &=& \frac{1}{c^{1/2}}\frac{1}{\Sigma^{1/2}} \Bigl\{
1+\frac{1}{4}c\frac{R}{6}\frac{1}{\Sigma_k^2}+\frac{5}{32}c^2\frac{R^2}{36}
\frac{1}{\Sigma_k^4}+ \nonumber \\
& & +\frac{1}{16}\frac{1}{\Sigma_k^4}\left[c^{''} \left(m^2-\frac{R}{6}
\right)-2 c^{'} \frac{R^{'}}{6}-c\frac{R^{''}}{6}\right]- \nonumber \\
& & -\frac{5}{64}\frac{1}{\Sigma_k^6}\left[ c^{'2}m^4-2c^{'2}m^2
\frac{R}{6}-2 c^{'}m^2c\frac{R^{'}}{6}\right]+ \nonumber \\
& & +\frac{9}{64}\frac{1}{\Sigma_k^6} c \frac{R}{6} c^{''}m^2-\frac{65}{256} \frac{1}{\Sigma_k^8}
c \frac{R}{6} c^{'2}m^4+\frac{5}{32}\epsilon_{2*}^2-
\frac{1}{4}\epsilon_{4*} \Bigr\}\end{aligned}$$
where $c=a^2$ and $$\begin{aligned}
\Sigma_k&=&(k^2 + a^2 m^2)^{1/2} \nonumber \\
\epsilon_{2*}&=&-\frac{1}{2}\frac{\Sigma_k^{''}}{\Sigma_k^3}+\frac{3}{4}
\frac{\Sigma_k^{'2}}{\Sigma_k^4} \nonumber \\
\epsilon_{4*} &=&
\frac{1}{4}\frac{\Sigma_k^{'}}{\Sigma_k^3}\epsilon_2^{'}-
\frac{1}{4}\frac{1}{\Sigma_k^2}\epsilon_2^{''}
\end{aligned}$$
Appendix B: Dimensional Regularization with cut-off
===================================================
We start with $\varphi^2$ as an example. According to the discussion following Eq. (\[fisquare\]), we neglect the infrared piece of the integral since it gives a small finite part. Therefore the relevant integral $$\begin{aligned}
\langle \varphi^2\rangle &=& \frac{1}{(2 \pi)^3}\frac{\hbar}{2} \frac{2
\pi^{3/2}}{\Gamma
(3/2)} \int_\ell^{+\infty} dk \, k^{2}
x_k^2 \nonumber \\
&=&
\frac{\hbar}{(2 \pi)^3}\frac{\lambda}{a^3}
\frac{\pi}{4 H} \frac{2 \pi^{3/2}}{\Gamma
(3/2)} \int_\ell^{+\infty} dk \, k^{2} \left[J_\nu^2
\left(\frac{\lambda k}{a H}\right)+
N_\nu^2\left(\frac{\lambda k}{a H} \right) \right]
\end{aligned}$$ on extending it to $d$-dimensions (integrands in $3$ space dimensions and analytic continuation of the measure to $d$ dimensions) :
$$\begin{aligned}
\langle \varphi^2\rangle &=&
\frac{\hbar}{(2 \pi)^d}\frac{\lambda}{a^3} \frac{\pi}{4 H}
\frac{2 \pi^{d/2}}{\Gamma
(d/2)} \int_\ell^{+\infty} dk \, k^{d-1}
\left[J_\nu^2\left(\frac{\lambda k}{a H}\right)+
N_\nu^2\left(\frac{\lambda k}{a H} \right) \right] \,.
\end{aligned}$$
The two-point function can be computed by using the following integral [@prudnikov]: $$\begin{aligned}
& & \int_\frac{\lambda \ell}{a H}^{+\infty}dx
x^{\alpha}[J_\nu^2(x)+N_\nu^2(x)]
= \nonumber \\
& & \frac{1}{\pi^2}\Bigl\{\pi^{1/2}\left[
\cos \left(\frac{\pi}{2}(\alpha+1-2\nu)\right)
\frac{\Gamma ((\alpha+1)/2) \Gamma ((\alpha+1)/2-\nu)}{\Gamma
(1+\alpha/2)}
+\pi \frac{\Gamma (-\alpha/2)}{\Gamma
((1-\alpha)/2) \Gamma ((1-\alpha)/2+\nu)} \right] \nonumber \\
& & \Gamma (\frac{\alpha+1}{2}+\nu)+\frac{1}{(\alpha+1)\nu}\Bigl[ 2 \pi
\left( \frac{\lambda \ell}{a H}\right)^d \cot (\pi \nu)
_2F_3 \Bigl(\frac{1}{2},\frac{\alpha+1}{2};\frac{\alpha+3}{2},1-\nu,
\nonumber \\
& & 1+\nu;-\left(
\frac{\lambda \ell}{a H}\right)^2\Bigr)\Bigr]+\frac{1}{2 \nu- \alpha -1}
\Bigr[ 4^{\nu}\left(
\frac{\lambda \ell}{a H}\right)^{\alpha+1-2 \nu} \Gamma (\nu)^2 \nonumber \\
& & _2F_3 \left( \frac{1}{2}-\nu,\frac{\alpha+1}{2}-\nu;1-2\nu,1-\nu,
\frac{\alpha+3}{2}-\nu;
-\left(\frac{\lambda \ell}{a H}\right)^2\right)\Bigr]- \nonumber \\
& & -\frac{1}{(\alpha+1+2\nu)\Gamma (1+\nu)^2}\Bigl[ 4^{-\nu} \left(
\frac{\lambda \ell}{a H}\right)^{\alpha+1+2 \nu}
(2 \pi^2+\cos(2 \nu \pi) \Gamma (-\nu)^2 \Gamma (1+\nu)^2 )
\nonumber \\
& & _2F_3 \left(\frac{1}{2}+\nu,\frac{\alpha+1}{2}+\nu;1+\nu,
\frac{\alpha+3}{2}+\nu,1+2 \nu;
-\left(\frac{\lambda \ell}{a H}\right)^2\right) \Bigr] \Bigr\}
\label{prud1}
\end{aligned}$$
With $\nu=3/2$ and $\alpha = d -1$, and on using: $$\begin{aligned}
_2F_3 \left( b,c;d,e,f;-\left(\frac{\lambda \ell}{a H}\right)^2 \right)
&=& 1+ {\cal O} \left( \frac{1}{a(t)^2}
\right)
\label{fact}
\end{aligned}$$ the expectation value for $\langle \varphi^2 \rangle$ is: $$\begin{aligned}
\langle \varphi^2\rangle &=&\frac{\hbar}{16\pi^2}
H^2 \left( 1-\frac{2}{3}\frac{m^2}{H^2}\right) \Bigl\{ 2-4 \ln 2-
2 \left(\frac{\ell}{2 \pi^{1/2}} \right)^{d-3}
\Gamma \left(\frac{1}{2}-\frac{d}{2}\right) + \nonumber \\
& &
+{\cal O} \left (
\frac{1}{a^2} \right )\Bigr\} +{\cal O} (d-3)
\end{aligned}$$
Similarly, the integral used for the fourth order adiabatic quantities in $d$-dimensions is (integrands in $3$ space dimensions and analytic continuation of the measure to $d$ dimensions): $$\begin{aligned}
& & \frac{2 \pi^{\alpha/2+1/2}}{\Gamma ((\alpha+1)/2)}
\int_\ell^{+\infty}
dk k^{\alpha}\frac{1}{(k^2+a^2m^2)^{n/2}} =
\nonumber \\ & & =\frac{2 \pi^{(\alpha+1)/2}}{\Gamma ((\alpha+1)/2)}
\int_0^{+\infty} dk k^{\alpha}
\frac{1}{(k^2+a^2m^2)^{n/2}}-
\frac{2 \pi^{\alpha/2+1/2}}{\Gamma ((\alpha+1)/2)}
\int_0^{\ell} dk k^{\alpha}\frac{1}{(k^2+a^2m^2)^{n/2}} \nonumber \\
& & =\pi^{\alpha/2+1/2} (a^2 m^2)^{\alpha/2+1/2-n/2}\frac{\Gamma
(n/2-\alpha/2-1/2)}{\Gamma (n/2)}-
\frac{2 \pi^{(\alpha+1)/2}}{\Gamma
((\alpha+1)/2)} \frac{1}{1+\alpha} (a^2 m^2)^{-n/2} \ell^{1+\alpha} \nonumber \\
& & _2F_1\left( \frac{n}{2},\frac{1+\alpha}{2};\frac{3+\alpha}{2};-\left(
\frac{\ell^2}{a^2 m^2}\right) \right)
\label{fourth_d}
\end{aligned}$$ Let us note that on taking the massless limit one can analytically continue the hypergeometric function and after straightforward calculations one gets - \^[1+-n]{}, \[intadiamassless\] which is of course the result one would obtain setting $m=0$ from the beginning. Therefore all the massless singularities in (\[fourth\_d\]) correctly cancel. We note that in the massless limit the analytic continuation misses the UV divergencies stronger than the logarithmic ones.
On again considering the case $m \ne 0$ such that $a(t)>H/m$ and using the result (\[fact\]), which also holds also for $_2F_1$, one obtains, to the fourth adiabatic order, for $\langle \varphi^2 \rangle$: $$\begin{aligned}
\langle \varphi^2\rangle_{(4)} &=&\frac{\hbar}{16\pi^2} H^2
\Bigl\{ \left[ -2+\frac{4}{3}\frac{m^2}{H^2}\right]
\left(\frac{a m}{2 \pi^{1/2}} \right)^{d-3} \Gamma
\left(\frac{1}{2}-\frac{d}{2}\right) +\frac{2}{9}
\frac{m^2}{H^2}-\frac{4}{3} + \nonumber \\
& &
+\frac{1}{m^2}\left[\frac{7}{45}m^2+\frac{29}{15}H^2\right]+{\cal O} \left
( \frac{1}{a^3} \right )\Bigr\} +{\cal O} (d-3)
\label{fisquare_appendix}
\end{aligned}$$
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[^1]: e-mail: fabio@physics.purdue.edu\
$ \hspace*{0.2cm}^\dag \!\!$ e-mail: marozzi@bo.infn.it\
$\hspace*{0.2cm}^\ddag \!\!$ e-mail: vacca@bo.infn.it\
$\hspace*{0.2cm}^\S \!\!$ e-mail: armitage@bo.infn.it
|
---
abstract: 'Previous research shows that eye-tracking data contains information about the lexical and syntactic properties of text, which can be used to improve natural language processing models. In this work, we leverage eye movement features from three corpora with recorded gaze information to augment a state-of-the-art neural model for named entity recognition (NER) with gaze embeddings. These corpora were manually annotated with named entity labels. Moreover, we show how gaze features, generalized on word type level, eliminate the need for recorded eye-tracking data at test time. The gaze-augmented models for NER using token-level and type-level features outperform the baselines. We present the benefits of eye-tracking features by evaluating the NER models on both individual datasets as well as in cross-domain settings.'
author:
- |
Nora Hollenstein\
ETH Zurich\
[noraho@ethz.ch]{}\
Ce Zhang\
ETH Zurich\
[ce.zhang@inf.ethz.ch]{}\
bibliography:
- 'naaclhlt2019.bib'
title: |
Entity Recognition at First Sight:\
Improving NER with Eye Movement Information
---
Introduction
============
The field of natural language processing includes studies of tasks of different granularity and depths of semantics: from lower level tasks such as tokenization and part-of-speech tagging up to higher level tasks of information extraction such as named entity recognition, relation extraction, and semantic role labeling [@collobert2011natural]. As NLP systems become increasingly prevalent in society, how to take advantage of information passively collected from human readers, e.g. eye movement signals, is becoming more interesting to researchers. Previous research in this area has shown promising results: Eye-tracking data has been used to improve tasks such as part-of-speech tagging [@barrett2016weakly], sentiment analysis [@mishra2017leveraging], prediction of multiword expressions [@rohanian2017using], and word embedding evaluation [@sogaard2016evaluating].
However, most of these studies focus on either relatively lower-level tasks (e.g. part-of-speech tagging and multiword expressions) or relatively global properties in the text (e.g. sentiment analysis). In this paper, we test a hypothesis on a different level: [*Can eye movement signals also help improve higher-level semantic tasks such as extracting information from text?*]{}
The answer to this question is not obvious. On one hand, the quality improvement attributed to eye movement signals on lower-level tasks implies that such signals do contain linguistic information. On the other hand, it is not clear whether these signals can also provide significant improvement for tasks dealing with higher-level semantics. Moreover, even if eye movement patterns contain signals related to higher-level tasks, as implied by a recent psycholinguistic study [@tokunaga2017eye], noisy as these signals are, it is not straightforward whether they would help, if not hurt, the quality of the models.
In this paper, we provide the first study of the impact of gaze features to automatic named entity recognition from text. We test the hypothesis that eye-tracking data is beneficial for entity recognition in a state-of-the-art neural named entity tagger augmented with embedding layers of gaze features. Our contributions in the current work can be summarized as follows:
1. First, we manually annotate three eye-tracking corpora with named entity labels to train a neural NER system with gaze features. This collection of corpora facilitates future research in related topics. The annotations are publicly available.
2. Beyond that, we present a neural architecture for NER, which in addition to textual information, incorporates embedding layers to encode eye movement information.
3. Finally, we show how gaze features generalized to word types eliminate the need for recorded eye-tracking data at test time. This makes the use of eye-tracking data in NLP applications more feasible since recorded eye-tracking data for each token in context is not required anymore at prediction time. Moreover, type-aggregated features appear to be particularly useful for cross-domain systems.
Our hypotheses are evaluated not only on the available eye-tracking corpora, but also on an external benchmark dataset, for which gaze information does not exist.
Related Work
============
[|l|c|c|c||c|]{} **** & **Dundee** & **GECO** & **ZuCo** & **Total**\
domain(s) & news articles & literature &
--------------------
movie reviews,
Wikipedia articles
--------------------
& -\
number of sentences & 2367 & 5424 & 700 & 8491\
mean sentence length & 24.75 & 12.65 & 22.12 & 19.84\
number of words & 58598 & 68606 & 15237 & 142441\
unique word types & 9131 & 5283 & 4408 & 13937\
mean word length & 4.29 & 3.76 & 4.44 & 4.16\
fixation duration (ms) & 202 & 214 & 226 & 214\
gaze duration (ms) & 237 & 232 & 265 & 244.7\
-------------- ---------- --------- ---------- --------- ---------- --------- ---------- ----------
****
all unique all unique all unique all unique
PERSON 732 415 1870 108 657 446 3259 955
ORGANIZATION 475 261 26 12 156 95 657 364
LOCATION 431 177 101 23 366 155 898 1646
**total** **1638** **853** **1997** **143** **1179** **696** **4814** **1646**
52% 7% 59% 34%
-------------- ---------- --------- ---------- --------- ---------- --------- ---------- ----------
The benefits of eye movement data for machine learning have been assessed in various domains, including NLP and computer vision. Eye-trackers provide millisecond-accurate records on where humans look when they are reading, and they are becoming cheaper and more easily available by the day [@san2009low; @sewell2010real]. Although eye-tracking data is still being recorded in controlled experiment environments, this will likely change in the near future. Recent approaches have shown substantial improvements in recording gaze data while reading by using cameras of mobile devices [@gomez2016evaluation; @papoutsaki2016webgazer]. Hence, eye-tracking data will probably be more accessible and available in much larger volumes in due time, which will facilitate the creation of sizable datasets enormously.
@tokunaga2017eye recently analyzed eye-tracking signals during the annotation of named entities to find effective features for NER. Their work proves that humans take into account a broad context to identify named entities, including predicate-argument structure. This further strengthens our intuition to use eye movement information to improve existing NER systems. And going even a step further, it opens the possibility for real-time entity annotation based on the reader’s eye movements.
The benefit of eye movement data is backed up by extensive psycholinguistic studies. For example, when humans read a text they do not focus on every single word. The number of fixations and the fixation duration on a word depends on a number of linguistic factors [@clifton2007eye; @demberg2008data]. First, readers are more likely to fixate on open-class words that are not predictable from context [@rayner1998eye]. Reading patterns are a reliable indicator of syntactical categories [@barrett2015reading]. Second, word frequency and word familiarity influence how long readers look at a word. The frequency effect was first noted by @rayner1977visual and has been reported in various studies since, e.g. @just1980theory and @cop2017presenting. Moreover, although two words may have the same frequency value, they may differ in familiarity (especially for infrequent words). Effects of word familiarity on fixation time have also been demonstrated in a number of recent studies [@juhasz2003investigating; @williams2004eye]. Additionally, the positive effect of fixation information in various NLP tasks has recently been shown by @barrett2018sequence, where an attention mechanism is trained on fixation duration.
#### State-of-the-art NER
Non-linear neural networks with distributed word representations as input have become increasingly successful for any sequence labeling task in NLP [@huang2015bidirectional; @chiu2016named; @ma2016end]. The same applies to named entity recognition: State-of-the-art systems are combinations of neural networks such as LSTMs or CNNs and conditional random fields (CRFs) [@strauss2016results]. @lample2016neural developed such a neural architecture for NER, which we employ in this work and enhance with eye movement features. Their model successfully combines word-level and character-level embeddings, which we augment with embedding layers for eye-tracking features.
Eye-tracking corpora
====================
For our experiments, we resort to three eye-tracking data resources: the *Dundee corpus* [@kennedy2003dundee], the *GECO corpus* [@cop2017presenting] and the *ZuCo corpus* [@hollenstein2018zuco]. For the purpose of information extraction, it is important that the readers process longer fragments of text, i.e. complete sentences instead of single words, which is the case in all three datasets.
Table \[corpora\] shows an overview of the domain and size of these datasets. In total, they comprise 142,441 tokens with gaze information. Table \[corpora\] also shows the differences in mean fixation times between the datasets (i.e. fixation duration (the average duration of a single fixation on a word in milliseconds) and gaze duration (the average duration of all fixations on a word)).
#### Dundee Corpus
The gaze data of the Dundee corpus [@kennedy2003dundee] was recorded with a *Dr. Bouis Oculometer Eyetracker*. The English section of this corpus comprises 58,598 tokens in 2,367 sentences. It contains eye movement information of ten native English speakers as they read the same 20 newspaper articles from *The Independent*. The text was presented to the readers on a screen five lines at a time. This data has been widely used in psycholinguistic research to analyze the reading behavior of subjects while reading sentences in context under relatively naturalistic conditions.
#### GECO Corpus
The Ghent Eye-Tracking Corpus [@cop2017presenting] is a more recent dataset, which was created for the analysis of eye movements of monolingual and bilingual subjects during reading. The data was recorded with an *EyeLink 1000* system. The text was presented one paragraph at a time. The subjects read the entire novel *The Mysterious Affair at Styles* by Agatha @christie containing 68,606 tokens in 5,424 sentences. We use only the monolingual data recorded from the 14 native English speakers for this work to maintain consistency across corpora.
**Basic**
-------------------------------- ------------------------------------------------------------------
*n* fixations total number of fixations on a word *w*
fixation probability the probability that a word *w* will be fixated
mean fixation duration mean of all fixation durations for a word *w*
**Early**
first fixation duration duration of the first fixation on a word *w*
first pass duration sum of all fixation durations during the first pass
**Late**
total fixation duration sum of all fixation durations for a word *w*
*n* re-fixations number of times a word *w* is fixated (after the first fixation)
re-read probability the probability that a word *w* will be read more than once
**Context**
total regression-from duration combined duration of the regressions that began at word *w*
*w-2* fixation probability fixation probability of the word before the previous word
*w-1* fixation probability fixation probability of the previous word
*w+1* fixation probability fixation probability of the next word
*w+2* fixation probability fixation probability of the word after the next word
*w-2* fixation duration fixation duration of the word before the previous word
*w-1* fixation duration fixation duration of the previous word
*w+1* fixation duration fixation duration of the next word
*w+2* fixation duration fixation duration of the word after the next word
#### ZuCo Corpus
The Zurich Cognitive Language Processing Corpus [@hollenstein2018zuco] is a combined eye-tracking and EEG dataset. The gaze data was also recorded with an *EyeLink 1000* system. The full corpus contains 1,100 English sentences read by 12 adult native speakers. The sentences were presented at the same position on the screen one at a time. For the present work, we only use the eye movement data of the first two reading tasks of this corpus (700 sentences, 15,237 tokens), since these tasks encouraged natural reading. The reading material included sentences from movie reviews from the Stanford Sentiment Treebank [@socher2013recursive] and the Wikipedia dataset by @culotta2006integrating.\
For the purposes of this work, all datasets were manually annotated with named entity labels for three categories: PERSON, ORGANIZATION and LOCATION. The annotations are available at <https://github.com/DS3Lab/ner-at-first-sight>.
The datasets were annotated by two NLP experts. The IOB tagging scheme was used for the labeling. We followed the ACE Annotation Guidelines [@linguistic2005ace]. All conflicts in labelling were resolved by adjudication between both annotators. An inter-annotator reliability analysis on 10,000 tokens (511 sentences) sampled from all three datasets yielded an agreement of 83.5% on the entity labels ($\kappa$ = 0.68).
Table \[annot-entities\] shows the number of annotated entities in each dataset. The distribution of entities between the corpora is highly unbalanced: Dundee and ZuCo, the datasets containing more heterogeneous texts and thus, have a higher ratio of unique entity occurrences, versus GECO, a homogeneous corpus consisting of a single novel, where the named entities are very repetitive.
Eye-tracking features {#features}
=====================
The gaze data of all three corpora was recorded for multiple readers by conducting experiments in a controlled environment using specialized equipment. It is important to consider that, while we extract the same features for all corpora, there are certainly practical aspects that differ across the datasets. The following factors are expected to influence reading: experiment procedures; text presentation; recording hardware, software and quality; sampling rates; initial calibration and filtering, as well as *human factors* such as head movements and lack of attention. Therefore, separate normalization for each dataset should better preserve the signal within each corpus and for the same reason the type-aggregation was computed on the normalized feature values. This is especially relevant for the type-aggregated features and the cross-corpus experiments described below.
In order to add gaze information to the neural network, we have selected as many features as available from those present in all three corpora. Previous research shows benefits in combining multiple eye-tracking features of different stages of the human reading process [@barrett2016weakly; @tokunaga2017eye].
The features extracted follow closely on @barrett2016weakly. As described above, psycho-linguistic research has shown how fixation duration and probability differ between word classes and syntactic comprehension processes. Thus, the features focus on representing these nuances as broadly as possible, covering the complete reading time of a word at different stages. Table \[feat\_table\] shows the eye movement features incorporated into the experiments. We split the 17 features into 4 distinct groups (analogous to @barrett2016weakly), which define the different stages of the reading process:
1. *BASIC* eye-tracking features capture characteristics on word-level, e.g. the number of all fixations on a word or the probability that a word will be fixated (namely, the number of subjects who fixated the word divided by the total number of subjects).
2. *EARLY* gaze measures capture lexical access and early syntactic processing and are based on the first time a word is fixated.
3. *LATE* measures reflect the late syntactic processing and general disambiguation. These features are significant for words which were fixated more than once.
4. *CONTEXT* features capture the gaze measures of the surrounding tokens. These features consider the fixation probability and duration up to two tokens to the left and right of the current token. Additionally, regressions starting at the current word are also considered to be meaningful for the syntactic processing of full sentences.\
The eye movement measurements were averaged over all native-speaking readers of each dataset to obtain more robust estimates. The small size of eye-tracking datasets often limits the potential for training data-intensive algorithms and causes overfitting in benchmark evaluation [@xu2015turkergaze]. It also leads to sparse samples of gaze measurements. Hence, given the limited number of observations available, we normalize the data by splitting the feature values into quantiles to avoid sparsity issues. The best results were achieved with 24 bins. This normalization is conducted separately for each corpus.
Moreover, special care had to be taken regarding tokenization, since the recorded eye-tracking data considers only whitespace separation. For example, the string *John’s* would constitute a single token for eye-tracking feature extraction, but would be split into *John* and *’s* for NER, with the former token holding the label PERSON and the latter no label at all. Our strategy to address this issue was to assign the same values of the gaze features of the originating token to split tokens.
Type aggregation
----------------
@barrett2015using showed that type-level aggregation of gaze features results in larger improvements for part-of-speech tagging. Following their line of work, we also conducted experiments with type aggregation for NER. This implies that the eye-tracking feature values were averaged for each word type over all occurrences in the training data. For instance, the sum of the features of all *n* occurrences of the token “island” are averaged over the number of occurrences *n*. As a result, for each corpus as well as for the aggregated corpora, a lexicon of lower-cased word types with their averaged eye-tracking feature values was compiled. Thus, as input for the network, either the type-level aggregates for each individual corpus can be used or the values from the combined lexicon, which increases the number of word types with known gaze feature values.
The goal of type aggregation is twofold. First, it eliminates the requirement of eye-tracking features when applying the models at test time, since the larger the lexicon, the more tokens in the unseen data receive type-aggregated eye-tracking feature values. For those tokens not in the lexicon, we assign a placeholder for unknown feature values. Second, type-aggregated features can be used on any dataset and show that improvements can be achieved with aggregated gaze data without requiring large quantities of recorded data.
Model
=====
The experiments in this work were executed using an enhanced version of the system presented by @lample2016neural. This hybrid approach is based on bidirectional LSTMs and conditional random fields and relies mainly on two sources of information: character-level and word-level representations.
{width="73.00000%"}
For the experiments, the originally proposed values for all parameters were maintained. Specifically, the bidirectional LSTMs for character-based embeddings are trained on the corpus at hand with dimensions set to 25. The lookup table tor the word embeddings was initialized with the pre-trained GloVe vectors of 100 dimensions [@pennington2014glove]. The model uses a single layer for the forward and backward LSTMs. All models were trained with a dropout rate at 0.5. Moreover, all digits were replaced with zeros.
The original model[^1] was modified to include the gaze features as additional embedding layers to the network. The character-level representation, i.e. the output of a bidirectional LSTM, is concatenated with the word-level representation from a word lookup table. In the augmented model with eye-tracking information, the embedding for each discrete gaze feature is also concatenated to the input. The dimension of the gaze feature embeddings is equal to the number of quantiles. This architecture is shown in Figure \[architecture\]. Word length and word frequency are known to correlate and interact with gaze features [@tomanek2010cognitive], which is why we selected a base model that allows us to combine the eye-tracking features with word- and character-level information.
Results
=======
Our main finding is that our models enhanced with gaze features consistently outperform the baseline. As our baseline, we trained and evaluated the original models with the neural architecture and parameters proposed by @lample2016neural on the GECO, Dundee, and ZuCo corpora and compared it to the models that were enriched with eye-tracking measures. The best improvements on F$_1$-score over the baseline models are significant under one-sided t-tests (p<0.05).
All models were trained with 10-fold cross validation (80% training set, 10% development set, 10% test set) and early stopping was performed after 20 epochs of no improvement on the development set to reduce training time.
First, the performance on the individual datasets is tested, together with the performance of one combined dataset consisting of all three corpora (consisting of 142,441 tokens). In addition, we evaluate the effects of the type-aggregated features using individual type lexicons for each datasets, and combining the three type lexicons of each corpus. Finally, we experiment with cross-corpus scenarios to evaluate the potential of eye-tracking features in NER for domain adaptation. Both settings were also tested on an external corpus without eye-tracking features, namely the CoNLL-2003 dataset [@tjong2003introduction].
Individual dataset evaluation
-----------------------------
First, we analyzed how augmenting the named entity recognition system with eye-tracking features affects the results on the individual datasets. Table \[results\] shows the improvements achieved by adding all 17 gaze features to the neural architecture, and training models on all three corpora, and on the combined dataset containing *all* sentences from the Dundee, GECO and ZuCo corpora. Noticeably, adding token-level gaze features improves the results on all datasets individually *and* combined, even on the GECO corpus, which yields a high baseline due to the homogeneity of the contained named entities (see Table \[annot-entities\]).
**P** **R** **F**
----------------- ----------- ----------- -------------
**Dundee** **** **** ****
baseline 79.29 78.56 78.86
with gaze 79.55 79.27 79.35
type individual **81.05** **79.37** **80.17**\*
type combined 80.27 79.26 79.67
**Geco** **** **** ****
baseline 96.68 97.24 96.95
with gaze **98.08** **97.94** **98.01**\*
type individual 97.72 97.42 97.57\*
type combined 97.76 97.16 97.46\*
**ZuCo** **** **** ****
baseline 84.52 81.66 82.92
with gaze **86.19** **84.28** **85.12**\*
type individual 84.21 82.61 83.30
type combined 83.26 83.37 83.31
**All** **** **** ****
baseline 86.92 86.58 86.72
with gaze 88.72 89.39 89.03\*
type combined **89.04** **89.52** **89.26**\*
: Precision (P), recall (R) and F$_1$-score (F) for all models trained on individual datasets (best results in bold; \* indicates statistically significant improvements on F$_1$-score). *With gaze* are models trained on the original eye-tracking features on token-level, *type individual* are the models trained on type-aggregated gaze features of this corpus only, while *type combined* are the models trained with type-aggregated features computed on all datasets.[]{data-label="results"}
Furthermore, Table \[results\] also presents the results of the NER models making use of the type-aggregated features instead of token-level gaze features. There are two different experiments for these type-level features: Using the features of the word types occurring in the corpus only, or using the aggregated features of all word types in the three corpora (as describe above). As can be seen, the performance of the different gaze feature levels varies between datasets, but both the original token-level features as well as the individual and combined type-level features achieve improvements over the baselines of all datasets.
To sum up, the largest improvement with eye-tracking features is achieved when combining all corpora into one larger dataset, where an additional 4% is gained in F$_1$-score by using type-aggregated features. Evidently, a larger mixed-domain dataset benefits from the type aggregation, while the original token-level gaze features achieve the best results on the individual datasets. Moreover, the additional gain when training on all datasets is due to the higher signal-to-noise ratio of type-aggregated features from multiple datasets.
**CoNLL-2003** **P** **R** **F**
---------------- ----------- ----------- -------------
baseline 93.89 94.16 94.03
type combined **94.38** **94.32** **94.35**\*
: Precision (P), recall (R) and F$_1$-score (F) for using type-aggregated gaze features on the CoNLL-2003 dataset (\* marks statistically significant improvement).[]{data-label="conll-ind"}
------------ ---------- ----------- ------------ ----------- ----------- ----------- ------------- ----------- ----------- -----------
**Dundee** **GECO** **ZuCo**
**P** **R** **F** **P** **R** **F** **P** **R** **F**
baseline 74.20 70.71 72.40 75.36 75.62 75.44
**Dundee** token 75.68 71.54 73.55\* **78.85** 74.51 77.02
type **76.44** **77.09** **76.75**\* 78.33 **76.49** **77.35**
baseline 58.91 34.91 43.80 68.88 42.49 52.38
**GECO** token **59.61** 35.62 **44.53** **69.18** **44.22** **53.81**
type 58.39 **35.99** 44.44 67.69 42.36 52.01
baseline 65.85 **54.01** 59.34 83.00 **78.11** **80.48**
**ZuCo** token **72.62** 50.76 59.70 82.92 75.35 78.91
type 69.21 53.05 **59.95** **83.68** 74.57 78.85
------------ ---------- ----------- ------------ ----------- ----------- ----------- ------------- ----------- ----------- -----------
#### Evaluation on CoNLL-2003
Going on step further, we evaluate the type-aggregated gaze features on an external corpus with no eye movement information available. The CoNLL-2003 corpus [@tjong2003introduction] has been widely used as a benchmark dataset for NER in different shared tasks. The English part of this corpus consists of Reuters news stories and contains 302,811 tokens in 22,137 sentences. We use this dataset as an additional corpus without gaze information. Only the type-aggregated features (based on the combined eye-tracking corpora) are added to each word. Merely 76% of the tokens in the CoNLL-2003 corpus also appear in the eye-tracking corpora described above and thus receive type-aggregated feature values. The rest of the tokens without aggregated gaze information available receive a placeholder for the unknown feature values.
Note that to avoid overfitting we do not train on the official train/test split of the CoNLL-2003 dataset, but perform 10-fold cross validation. Applying the same experiment setting, we train the augmented NER model with gaze features on the CoNLL-2003 data and compare it to a baseline model without any eye-tracking features. We achieve a minor, but nonetheless significant improvement (shown in Table \[conll-ind\]), which strongly supports the generalizability effect of the type-aggregated features on unseen data.
Cross-dataset evaluation
------------------------
In a second evaluation scenario, we test the potential of eye-tracking features for NER across corpora. The goal is to leverage eye-tracking features for domain adaptation. To show the robustness of our approach across domains, we train the models with token-level and type-level features on 100% of corpus A and a development set of 20% of corpus B and test on the remaining 80% of the corpus B, alternating only the development and the test set for each fold.
Table \[cross\] shows the results of this cross-corpus evaluation. The impact of the eye-tracking features varies between the different combinations of datasets. However, the inclusion of eye-tracking features improves the results for all combinations, except for the models trained on the ZuCo corpus and tested on the GECO corpus. Presumably, this is due to the combination of the small training data size of the ZuCo corpus and the homogeneity of the named entities in the GECO corpus.
**CoNLL-2003** **P** **R** **F**
---------------- ----------- ----------- -------------
baseline 72.80 56.97 63.92
type combined **74.56** **60.20** **66.61**\*
: Precision (P), recall (R) and F$_1$-score (F) for using type-aggregated gaze features trained on all three eye-tracking datasets and tested on the CoNLL-2003 dataset (\* marks statistically significant improvement).[]{data-label="all-conll"}
#### Evaluation on CoNLL-2003
Analogous to the individual dataset evaluation, we also test the potential of eye-tracking features in a cross-dataset scenario on an external benchmark dataset. Again, we use the CoNLL-2003 corpus for this purpose. We train a model on the Dundee, GECO *and* ZuCo corpora using type-aggregated eye-tracking features and test this model on the ConLL-2003 data. Table \[all-conll\] shows that compared to a baseline without gaze features, the results improve by 3% F$_1$-score. These results underpin our hypothesis of the possibility of generalizing eye-tracking features on word type level, such that no recorded gaze data is required at test time.
![Results per class for the models trained on all gaze datasets combined.[]{data-label="class"}](class2.png)
Discussion
==========
The models evaluated in the previous section show that eye-tracking data contain valuable semantic information that can be leveraged effectively by NER systems. While the individual datasets are still limited in size, the largest improvement is observed in the models making use of *all* the available data.
At a closer look, the model leveraging gaze data yield a considerably higher increase in recall when comparing to the baselines. In addition, a class-wise analysis shows that the entity type benefiting the most from the gaze features over all models is ORGANIZATION, which is the most difficult class to predict. Figure \[class\] illustrates this with the results per class of the models trained on all three gaze corpora jointly.
In the individual dataset evaluation setting, the combined type-level feature aggregation from all datasets does not yield the best results, since each sentence in these corpora already has accurate eye-tracking features on toke-level. Thus, it is understandable that in this scenario the original gaze features and the gaze features aggregated only on the individual datasets result in better models. However, when evaluating the NER models in a cross-corpus scenario, the type-aggregated features lead to significant improvements.
Type aggregation evidently reduces the fine-grained nuances contained in eye-tracking information and eliminates the possibility of disambiguation between homographic tokens. Nevertheless, this type of disambiguation is not crucial for named entities, which mainly consist of proper nouns and the same entities tend to appear in the same context. Especially noteworthy is the gain in the models tested on the CoNLL-2003 benchmark corpus, which shows that aggregated eye-tracking features from other datasets can be applied to any unseen sentence and show improvements, even though more than 20% of the tokens have unknown gaze feature values. While the high number of unknown values is certainly a limitation of our approach, it shows at once the possibility of not requiring original gaze features at prediction time. Thus, the trained NER models can be applied robustly on unseen data.
Conclusion
==========
We presented the first study of augmenting a NER system with eye-tracking information. Our results highlight the benefits of leveraging cognitive cues such as eye movements to improve entity recognition models. The manually annotated named entity labels for the three eye-tracking corpora are freely available. We augmented a neural NER architecture with gaze features. Experiments were performed using a wide range of features relevant to the human reading process and the results show significant improvements over the baseline for all corpora individually.
In addition, the type-aggregated gaze features are effective in cross-domain settings, even on an external benchmark corpus. The results of these type-aggregated features are a step towards leveraging eye-tracking data for information extraction at training time, without requiring real-time recorded eye-tracking data at prediction time.
[^1]: https://github.com/glample/tagger
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abstract: 'We use a statistical mechanical model to study nonthermal denaturation of DNA in the presence of protein-mediated loops. We find that looping proteins which randomly link DNA bases located at a distance along the chain could cause a first-order phase transition. We estimate the denaturation transition time near the phase transition, which can be compared with experimental data. The model describes the formation of multiple loops via dynamical (fluctuational) linking between looping proteins, that is essential in many cellular biological processes.'
author:
- 'K.G. Petrosyan'
- 'Chin-Kun Hu'
title: 'Protein-mediated Loops and Phase Transition in Nonthermal Denaturation of DNA'
---
Denaturation of DNA is a fundamental biological process before the transcription stage [@watson]. Thermal denaturation of DNA [@wartell] has been modelled in many ways, including the ladder [@lavis], Poland-Scheraga [@scheraga] and Peyrard-Bishop [@peyrard] models. The process still attracts attention of theoreticians in an attempt to describe it most efficiently [@weber]. Besides the melting, DNA also denatures under the influence of other factors such as pH value, salt concentration, other chemical factors, and mechanical forces. One example of the latter is the DNA denaturation induced by an externally applied torque. The experiments with single DNA molecules under torsional stress were reported in [@strick; @bryant] that shed more light on the mechanical properties of DNA molecules in connection with their functioning in living cells. [*In vivo*]{} the torque is exerted by the RNA polymerase that causes transcription-generated torsional stress [@kouzine] (see also [@harada] where a direct observation of DNA rotation during transcription by [*Escherichia coli*]{} RNA polymerase was reported). A theoretical study of torque-induced DNA denaturation was presented by Cocco and Monasson [@cocco] and a thorough investigation of the effect of mechanical forces and torques on DNA and its denaturation was done by Marko [@marko].
Here we are interested in [*nonthermal*]{} denaturation of DNA that precedes the transcription process. Transcription regulation typically involves the binding of proteins over long distances on multiple DNA sites which are then brought close to each other to form DNA loops [@saiz]. The DNA loops can be formed by protein complexes, e.g., by the regulators of bacterial operons, such as ara, gal, and lac, and human proteins involved in cancer, such as retinoic X receptor. The presence of protein-mediated loops is also important for many other cellular processes, including DNA replication, recombination, and nucleosome positioning as was extensively discussed in [@sv].
Recently Vilar and Saiz [@vilar] studied multiprotein DNA looping. They developed a model of formation of a single loop via connection of an arbitrary large number of proteins. Their model describes a switchlike transition between looped and unlooped phases, and has been extended to account for multiple loops [@sv]. Dynamic protein-mediated loops within the framework of molecular systems biology were considered in [@saiz] for the cases of the lac operon and phage $\lambda$ induction switches. Here we consider a different model to describe the denaturation of DNA, which has loops formed by proteins that link bases randomly located along the molecular chain. Thus our model accounts for formation of multiple loops that is essential in cellular biological processes like pre-mRNA splicing [@watson]. Yet another important feature of our model is that it presents a dynamic rather than static picture of formation of loops as the protein-mediated links between the base pairs fluctuate, [*i.e.*]{} the proteins couple and decouple in the course of time. This demonstrates a connection between formation of the structure of protein-mediated loops for the particular DNA-protein node-link interaction network and co-evolutionary complex networks [@statnets; @dorogov]. We show below that looping proteins can make the nonthermal denaturation process to be a first-order phase transition. It is due to the effective long-range interactions by the mediating proteins. We are primarily interested in the phase transition, in the metastability phenomenon that we have found and in the kinetics of the denaturation. We then calculate the transition time from the double-helix state to the coil state, which can be compared with experimental data.
2 mm
[**The Model.**]{} Lattice models proved to be useful in studies of the phenomenology of DNA denaturation [@palmeri]. Here we consider a simple statistical mechanical model defined on an one-dimensional lattice with each site corresponding to a rung of the ladder [@lavis]. A spin variable $\sigma_i$ is associated with each site $i$ where $\sigma_i=-1$ when the corresponding $H$-bond is intact and $\sigma_i = +1$ when it is broken. We assume an arbitrary folding of the DNA molecule so that any two base pairs may get connected via the looping proteins. The proposed model has the following Hamiltonian $$\begin{aligned}
H= &-& g\sum_{i=1}^{N-1}\sigma_i\sigma_{i+1} -\gamma\sum_{i<j}
J_{ij} p_i p_j \sigma_i \sigma_j \nonumber \\
&-& h\sum_{i=1}^N\sigma_i - \epsilon\sum_{i=1}^N p_i +
\alpha\sum_{i<j} J_{ij} \label{dna}\end{aligned}$$ where $g>0$ is the coupling parameter of nearest-neighbor interactions; $J_{ij}$ are the link variables, taking values $0$ and $1$ when the $i$ and $j$ nodes are uncoupled or coupled by the proteins, correspondingly; the absence or presence of proteins at site $i$ is defined by the variable $p_i$ that takes values 0 or 1, respectively; $\gamma>0$ is the energy of interaction between the base pair sites coupled via an appeared link caused by the on-site proteins; $h$ is the binding energy between base pairs that includes the energy of the hydrogen bonds; $\epsilon$ is the energy of binding of a protein at the site $i$; $\alpha$ is the energy of formation of a link connecting $i$ and $j$ sites; $N$ is the number of base pairs.
The first term assures that broken pairs tend to break pairs next to them and in the same way it makes to pair up bases next to paired ones. The second term describes creation of links between proteins bound to bases at random sites of the molecule. These links form protein-mediated loops. The links actually fluctuate as the proteins at different sites may couple and decouple in the course of time. In general, $\gamma$ may depend on the length of the loop. However, such a dependence is a higher order effect and we do not consider that. In the third term the energy $h$ depends on the external parameters that are determined by environmental conditions such as temperature $T$, pH value, salt concentration, and other chemical as well as mechanical factors. Change in $h$ may cause openings and closings of base pairs. As an example we will consider its dependence on an externally applied torque. The energy $h$ is a sum of contributions from the base pairing energy $h_0<0$ and from the torsional energy $h_{\tau}$, associated with a change in the local twist, that is $h=h_0+h_{\tau}$, where $h_{\tau}=(1/2)C(\Delta\omega)^2$, with $C$ being the twisting elastic constant (torsional stiffness) and $\Delta\omega=\omega-\omega_0$ being the deviation of the spatial angular frequency $\omega$ (change of the rung angle around the axis per unit length along the chain) from its unstressed value $\omega_0$ [@marko; @siggia]. The torsional energy can also be represented via the torque $\tau$ as $h_{\tau}=\tau\varphi_0$, where $\varphi_0=2\pi/10.5=0.6$ radians per base pair (double helix contains about 10.5 base pairs per helical turn). The fourth term is the energy of binding of proteins. The last (fifth) term is the energy of formation of a link between base pairs mediated by the looping proteins. We will use another parameter $c$ defined via ${c}/{(N-c)} = e^{-\alpha\beta}$, where $\beta$ is the inverse temperature. The ratio can be roughly treated as the probability of a link formation (see [@statnets] for more rigorous formulations and details of a related model that describes a network of fluctuating links). We will assume sparse connectivity ${c}/{N} \ll 1$ with the number of looping proteins much less than the number of bases. The Hamiltonian may also include long-range direct H-bond interactions between open base pairs via a term proportional to $\sum_{ij} A_{ij} (1+\sigma_i)(1+\sigma_j)$ with an interaction matrix $A_{ij}$. However we neglect these interactions assuming that their contribution is smaller compared to the interactions mediated by proteins [@energy; @values].
2 mm
[**Effective Hamiltonian and Free Energy.**]{} The small number of proteins compared to the number of base pairs allows us to reduce (\[dna\]) to an effective mean-field type Hamiltonian. For that purpose we eliminate consequently $J_{ij}$ and $p_i$ variables while calculating the partition function $Z=Tr_{J, p,
\sigma}e^{-\beta H}$, where the trace means summing up over the corresponding variables. Taking the trace over $J_{ij}$’s [@statnets] we arrive at the partition function $Z \propto Tr_{p,
\sigma}e^{-\beta H'}$ with the following effective Hamiltonian $$\begin{aligned}
H' = - g\sum_{i=1}^{N-1}\sigma_i\sigma_{i+1} - \gamma' \sum_{i<j}
p_i p_j \sigma_i \sigma_j \nonumber \\
- \lambda \sum_{i<j}p_i p_j - h\sum_{i=1}^N\sigma_i -
\epsilon\sum_{i=1}^N p_i . \label{effective}\end{aligned}$$ Here $\gamma' = (c/N)\sinh\beta\gamma$ and $\lambda =
(c/N)(\cosh\beta\gamma - 1)$. The Hamiltonian (\[effective\]) describes a system consisting of two interacting subsystems, DNA and proteins. Different time scales and different temperatures for two subsystems may lead to novel phenomena [@anomalous]. However, here we assume that DNA and proteins are in contact with the same heat bath at temperature $T$.
![Double helix fraction $\theta = (1-\mu)/{2}$ vs the base pair binding energy $h$ for the parameters $g = 8.5$ kcal/mol, $\gamma = 0.02$ kcal/mol, $c = 10$, $\epsilon = 7.2$ kcal/mol and $k_B T = 0.6$ kcal/mol. The first-order denaturation phase transition occurs at the critical value $h_c=0$. The critical torque is $\tau_c=1.6 k_B T$ for AT-rich and $\tau_c=7
k_B T$ for GC-rich chains. The double helix (coil) becomes metastable for $h>0$ ($h<0$) as indicated by dashes.[]{data-label="fig1"}](fig1.eps){width="0.95\linewidth"}
For the case of strong binding energies $\epsilon\gg\lambda$,$\gamma'$, we can make a mean-field approximation and replace $p_i$’s by their mean values $\langle p
\rangle = {e^{\beta\epsilon}}/{(1+e^{\beta\epsilon})}$, the proposed model is then reduced to the following effective Hamiltonian $$\begin{aligned}
H_{eff} = - g\sum_{i=1}^{N-1}\sigma_i\sigma_{i+1} -
\gamma''\sum_{i<j} \sigma_i \sigma_j - h\sum_{i=1}^N\sigma_i
\label{Heff}\end{aligned}$$ where $\gamma'' = \frac{c}{N} \sinh\beta\gamma \cdot
\left(\frac{e^{\beta\epsilon}}{1+e^{\beta\epsilon}}\right)^2$ represents the effective coupling between base pairs mediated by proteins. Notice that we have neglected the effect caused by the presence of the persistence length $l_0$ that would require to take into account only the terms for which one has $|i-j|>l_0$ as the correction would be of order $O(\frac{l_0}{N})$ and would go to zero in the thermodynamic limit. The coupling in (\[Heff\]) is similar to that of a synchronization model with small world coupling [@gade].
To calculate the partition function $Z \propto Tr_{\sigma}
e^{-\beta H_{eff}}$ for the Hamiltonian (\[Heff\]), we use the relationship $\sum_{i<j} \sigma_i \sigma_j = \frac{1}{2}(\sum
\sigma_i)^2 - \frac{1}{2}N$, the Hubbard-Stratonovich transformation $e^{\frac{1}{2}a(\sum_{i=1}^N
\sigma_i)^2}=\int^{+\infty} _{-\infty} \frac{d \mu}{\sqrt{2\pi/a}}
e^{-\frac{1}{2}a\mu^2 + a\mu\sum_{i=1}^N \sigma_i}$ and the expression for the partition function of the one-dimensional (1D) Ising model [@lavis]. Then the partition function takes the form $Z \propto \int^{+\infty} _{-\infty} d\mu e^{- \beta N
f(\mu)}$ with the effective free energy $f(\mu)$ given by $$\begin{aligned}
f(\mu) &=& \frac{1}{2}b \mu^2 - \beta^{-1}\ln [\cosh \beta (h + b \mu)\nonumber \\
&+& \sqrt{\sinh^2 \beta (h + b \mu) + e^{-4\beta g}}]. \label{free
energy}\end{aligned}$$ Here $b = c \cdot \sinh\beta\gamma \cdot
\left(\frac{e^{\beta\epsilon}}{1+e^{\beta\epsilon}}\right)^2$ and $\mu$ is the order parameter for the denaturation process. For the double helix state with all base pairs bound, $\mu=-1$; for the completely denaturated state, $\mu=1$. The values of $\mu$, which determine the state of the molecule, are obtained via $f'(\mu)=0$ that leads to the equation $$\begin{aligned}
\mu = \frac{\sinh\beta (h + b \mu)}{\sqrt{\sinh^{2}\beta (h + b
\mu) + e^{-4\beta g}}}. \label{mean}\end{aligned}$$
The model is an effective Ising model with 1D nearest-neighbor and global (all-to-all) interactions. It can be shown that the model goes through a phase transition provided $\beta b e^{2\beta g}
\geq 1$. That gives the necessary condition for the model parameters, e.g., the temperature. The sufficient condition for the phase transition would be the sign change of $h$. Thus $h_c=0$ or $\tau_c=h_0/\varphi_0$ defines the critical point for the first-order phase transition if necessary condition $\beta b
e^{2\beta g} \geq 1$ is satisfied for the given parameters. The critical torque $\tau_c$ ranges from $1.6 k_B T$ for weakly bound (AT-rich) sequences to $7 k_B T$ for the most strongly bound (GC-rich) sequences [@torque; @values].
![Free energy vs the order parameter $\mu$ at the base pair binding energy value $h = -0.05$kcal/mol for the parameters $g = 8.5$ kcal/mol, $\gamma = 0.02$ kcal/mol, $c = 10$, $\epsilon = 7.2$ kcal/mol and $k_BT = 0.6$ kcal/mol. Double helix is stable and coil is metastable for $h<0$.[]{data-label="fig2"}](fig2.eps){width="0.95\linewidth"}
In order to quantify the degree of denaturation we introduce the parameter $\theta = (1-\mu)/2$ that is the fraction of bound base pairs. The parameter takes the value $\theta=1$ for the double helix state and the value $\theta=0$ for the denaturated coiled state. The dependence of the double helix fraction $\theta$ on $h$ is presented in Fig.\[fig1\]. There is a metastability in a range of the controlling external parameter $h$. This effect is illustrated in Fig.\[fig2\] where the free energy with two minima is presented. Notice that there is no phase transition if the proteins do not interact ($\gamma=0$) and thus the protein-mediated loops are absent. These are the looping proteins which provide with the long-range interactions that make it possible to obtain a phase transition for the effectively 1D lattice model.
2 mm
[**Transition time.**]{} The kinetics of the denaturation transition can be treated via the Langevin equation $\dot \mu = -\Gamma\frac{\partial f(\mu)}{\partial \mu} + \xi (t)$, where $\Gamma$ defines the inverse relaxation time, $\xi (t)$ is the while noise satisfying the relation $\langle\xi(t)\xi(t')\rangle = D\delta(t-t')$ with the diffusion coefficient $D$ determined by the fluctuation-dissipation relation $D=2\Gamma k_B T$. The corresponding Fokker-Planck equation (FPE) for the probability distribution function $P(\mu)$ of the order parameter $\mu$ is $\dot P = \frac{\partial}{\partial \mu}A(\mu)P
+ \frac{1}{2}D\frac{\partial^2}{\partial \mu^2}P$, where $A(\mu) =
-\Gamma\frac{\partial f(\mu)}{\partial \mu}$. Making the transformations $P \rightarrow Pe^{\frac{f(\mu)}{D}}$, $D
\rightarrow D\Gamma$ and $t \rightarrow t/\Gamma$, we can rewrite the FPE as $-\dot P = HP$ with the Hamiltonian $H =
-\frac{1}{2}D\frac{\partial^2}{\partial \mu^2} +\frac{1}{2D}\Phi^2
+ \frac{1}{2}\frac{\partial \Phi}{\partial \mu}$ where $\Phi(\mu)=-f'(\mu)$. One can exactly solve the FPE to obtain $P_t(\mu, \mu_0) = |\psi_0(\mu)|^2 +
\frac{\psi_0(\mu)}{\psi_0(\mu_0)} \sum_{n=1}^{\infty}
e^{-\frac{\lambda_n t}{D}}\psi_n(\mu)\psi_n(\mu_0)$, where $\psi_0(\mu)\propto e^{-f(\mu)/2k_B T}$ and $\mu_0$ is the initial value. The decay rates $\lambda_n$ and the eigenfunctions $\psi_n(\mu)$ can be, in principle, derived exactly [@junker]. However, we are not considering here the dynamics of the probability distribution function. Our goal is to analyze the dependence of the transition rate on the model parameters, such as temperature $T$. Therefore we are only interested in the first eigenvalue given by $\lambda_1 \simeq \frac{D}{\pi}
\sqrt{f''(\mu_{min})|f''(\mu_{max})|} \cdot
e^{-\frac{2}{D}[f(\mu_{max}) - f(\mu_{min})]}$ which governs the dynamics for long times.
Let us consider the transition from the left minimum of the free energy in Fig. 2, corresponding to the native double helix state of DNA, to the right minimum representing denaturated state at the critical value $h_c$. At this value, that corresponds to the first-order phase transition point, the free energy is a symmetric curve with two equal minima and the maximum located at $\mu=0$. The transition time $\Omega^{-1}$ (the inverse transition frequency) is twice the time needed to achieve the top of barrier $\mu_{max}$ from the minimum $\mu_{min}$ which is obtained from $\lambda_1$ $$\begin{aligned}
\Omega^{-1} \simeq \frac{2\pi}{\Gamma} \frac{e^{\beta[f(\mu_{max})
- f(\mu_{min})]}}{\sqrt{f''(\mu_{min})|f''(\mu_{max})|}}.
\label{time}\end{aligned}$$ This is a standard expression for the Kramers problem [@gardiner]. However there is a qualitative difference since the potential $f(\mu)$ itself depends on temperature. The behavior of the denaturation transition time versus temperature drastically differs from the conventional Arrhenius case. Although the transition time first decreases at very low temperatures (frozen DNA) it begins to increase at high enough (physiological) temperatures. The reason is that the second derivative present in the denominator of Eq.(\[time\]) at the point $\mu_{max}=0$ diverges since $f''(0)=b(1-\beta be^{2\beta g})$ and $\beta
be^{2\beta g} \rightarrow 1$ ($f''(0) \rightarrow 0$) at the critical temperature defined by the above mentioned necessary condition of denaturation. For the set of parameters given in Figs. 1 and 2, the transition time is $\Omega^{-1}=2.35 \cdot
10^{-5} \Gamma^{-1}$. The kinetics of pH-driven denaturation of DNA was studied experimentally in [@ageno], where the transition time for single molecule denaturation was estimated to be of order of $1 \div 10$ seconds. Taken these values we come up with the inverse relaxation time $\Gamma$ to be of order of $10^{-6}$ Hz. However we believe that modern measurements in experiments with single molecules are needed to find precise values of the quantities.
2 mm
[**Discussion.**]{} In summary, we have introduced and studied a model of nonthermal denaturation of DNA that can be induced by chemical factors, such as pH value or salt concentration, or by externally applied mechanical forces and torques (as an example we considered the case of torque-induced denaturation) in the presence of protein-mediated loops. The model accounts for proteins that bind to the DNA molecule. The bound proteins are then allowed to interact in a random way with each other thus creating the loops. We have found a first-order denaturation phase transition that is caused by the looping proteins, the proteins that connect base pairs that are at a distance along the chain. The model possesses a metastability region provided that the necessary and sufficient conditions are satisfied. The kinetics of the denaturation phase transition was described by a stochastic dynamics for the order parameter that is, in principle, exactly solvable. However we have been mainly interested in obtaining the transition rate in the vicinity of the first-order phase transition. It has the standard form by Kramers with the associated potential being temperature-dependent. This leads to deviation from the Arrhenius law at physiological temperatures. In particular, the transition time becomes extremely large when the temperature approaches its critical value that is defined by the necessary condition for the denaturation phase transition. The DNA denaturation kinetics considered here can be extended spatially to describe a front propagation process in the presence of protein-mediated loops and the noise that corresponds to the [*in vivo*]{} case.
Finally, we have revealed a new purpose of the protein-mediated looping that is to facilitate [*in vivo*]{} denaturation of DNA needed to take it to the next transcription stage. The model also describes the formation of multiple loops via dynamical (fluctuational) linking between looping proteins, that is essential in cellular biological processes such as the pre-mRNA splicing [@wang; @cooper] and the phenomenon of genomic plasticity [@alberts]. It can mimick, e.g., the coevolutionary networks of splicing cis-regulatory elements [@xiao] having the loops to splice out introns thus defining the exons within the DNA molecule. The presented theory can be applied in studies of the above enumerated [*in vivo*]{} processes as well as for description of [*in vitro*]{} experiments with single DNA molecules. Yet another application of this or a generalized statistical mechanics model would be an investigation of dynamic genome architecture in eukaryotic cells [@nicodemi].
We thank A.E. Allahverdyan, D. Mukamel, E.I. Shakhnovich, and M.C. Williams for comments and discussions. This work was supported by Grants NSC 96-2911-M 001-003-MY3, NSC 96-2811-M 001-018, NSC 97-2811-M-001-055 & AS-95-TP-A07, and by the National Center for Theoretical Sciences in Taiwan.
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abstract: |
We investigate the uniform approximation provided by least squares polynomials on the unit Euclidean sphere $\SS^q$ in $\RR^{q+1}$, with $q\ge 2$. Like any other polynomial projection, the study concerns the growth, as the degree $n$ tends to infinity, of the associated Lebesgue constant, i.e., of the uniform norm of the least squares operator. If the least squares polynomial of degree $n$ is based on a set of points, which are nodes of a positive weighted quadrature rule of degree of exactness $2n$, then we state two different sufficient conditions for having an optimal Lebesgue constant that increases with $n$ at the minimal projections order. Hence, under our assumptions least squares and hyperinterpolation polynomials provide a comparable approximation with respect to the uniform norm.\
[**keywords:**]{} polynomial approximation on the (hyper)sphere, least squares polynomial, hyperinterpolation, uniform approximation, Lebesgue constant, tensor product quadrature rules.\
[**MSC2010:**]{} 41-A10, 65-D99, 33-C45.
author:
- 'Woula Themistoclakis[^1]'
- 'Marc Van Barel[^2]'
bibliography:
- 'longstrings.bib'
- 'TOTAL.bib'
title: 'Optimal Lebesgue constants for least squares polynomial approximation on the (hyper)sphere'
---
Introduction
============
In recent decades, the polynomial approximation on the sphere $$\SS^q:=\left\{\x=(x_0,\ldots,x_{q})\in\RR^{q+1}\ : \sum_{i=0}^q x_i^2=1\right\},\qquad q\ge 2,$$ by using function values at a discrete point set $X_N:=\{\xi_1,\ldots,\xi_N\}\subset \SS^q$, has received more and more interest by many authors motivated by the wide field of applications in geophysics, biology and engineering (see, e.g., [@b509; @W-geoBook; @W-GiaMha; @W-ReBook; @W-GiaSlo] and the references therein).
Limiting our concern to polynomial projections, besides the classical Lagrange interpolation [@W-Xu], we recall the hyperinterpolation polynomials firstly introduced by Sloan in [@W-SloFirst] supposing that the point set $X_N$ consists of nodes of a positive weighted quadrature rule of suitable degree of precision. By means of this quadrature rule, hyperinterpolants approximate Fourier orthogonal projections w.r.t. the scalar product $$\label{prod}
<f,g> :=\int_{\SS^2}f(\x)g(\x)d\sigma(\x),$$ where $d\sigma$ denotes the usual surface measure on $\SS^q$.
It is known [@W-Re; @W4] that hyperinterpolation polynomials provide an optimal approximation w.r.t. the uniform norm, but for their construction we need to explicitly know the quadrature weights. This is not necessary if we consider the least squares polynomials, defined as the orthogonal projections w.r.t. the discrete scalar product $$\label{prod-dis}
<f,g>_N :=\sum_{i=1}^Nf(\xi_i)g(\xi_i).$$ In the case $q=2$, it has been proved by the authors [@W-Marc1 Th. 2.3] that similarly to hyperinterpolation, also least squares projections have optimal Lebesgue constants w.r.t. the uniform norm, provided that the nodes $\{\xi_j\}$ support a quadrature rule with positive weights (required for hyperinterpolation too) and they are well separated on the sphere.
In this paper we are going to extend this result to the hypersphere case $q\ge 2$ (cf. Theorems \[th-LS\] and \[th-LSequi\]).
Moreover, when $q=2$ we focus on the special case of tensor product Gauss–Legendre quadrature rules nodes [@W4 Example 6.1]. These nodes do not satisfy the assumption to be well separated on the sphere. Nevertheless, from our numerical experiments an optimal behavior of the associated Lebesgue constants comes out.
This is justified by a second theorem (cf. Theorem \[th-tensor\]) that we state in $\SS^q$ with $q\ge 2$, where the assumption on the well separated nodes is replaced by an hypothesis on the quadrature weights, which is certainly satisfied by the tensor product Gaussian quadrature rules.
In the next section we briefly recall some basic properties of Fourier and hyperinterpolation projections. The main theorems are given in Section 3, where some numerical experiments are also given. The proofs are left to Section 4 and Section 5 summarizes the obtained results.
Basic properties of Fourier and hyperinterpolation projections
==============================================================
Let $\PP_n$ be the space of all spherical polynomials (i.e., polynomials of $q+1$ variables restricted to the sphere $\SS^q$) of degree at most $n$. It is well–known (see, e.g., [@b509]) that $$\dim \PP_n= \frac{(2n+q)\Gamma(n+q)}{\Gamma(q+1)\Gamma(n+1)}=:d_n$$ and spherical harmonics (i.e., harmonic homogeneous polynomials restricted to $\SS^q$) of degree at most $n$ provide a basis of $\PP_n$, which is orthonormal w.r.t. the scalar product (\[prod\]).
Moreover, spherical harmonics are related to ultraspherical polynomials of index $q/2-1$ by an addition formula (cf. [@b509 (1.6.7)]), which allows us to write the associated Fourier orthogonal projection $\S_n:L^2(\SS^q)\rightarrow \PP_n$ as follows $$\label{Fourier}
\S_nf(\x)= \frac 1{|\SS^{q-1}|}\int_{\SS^q}K_n(\x\cdot\y)f(\y)d\sigma(\y),\qquad \x\in\SS^q,$$ where $|\SS^{q-1}|$ is the surface area of $\SS^{q-1}$, $\x\cdot\y$ denotes the Euclidean scalar product in $\RR^{q+1}$, and $$\label{Darboux}
K_n(t):=K_n(t,1),\qquad t\in [-1,1],$$ is the $n$–th Darboux kernel related to the weight function $w(x)=(1-x^2)^{\frac q2-1}$, as defined in [@b210].
Fourier projection, as any other projection onto $\PP_n$, satisfies for all functions $f$ s.t. $\|f\|_\infty:=\sup_{\x\in\SS^q}|f(\x)|<\infty$, the following error estimate $$\label{err}
E_n(f)\le\|f-\S_n f\|_\infty\le \left(1+\|\S_n\|_\infty\right) E_n(f),$$ where $E_n(f)$ is the error of best polynomial approximation w.r.t. the uniform norm, i.e., $$E_n(f):=\inf_{P\in\PP_n}\|f-P\|_\infty,$$ and $\|\S_n\|_\infty$ denotes the so–called Lebesgue constant of $\S_n$, given by $$\label{Leb-Fou}
\|\S_n\|_\infty=\frac 1{|\SS^{q-1}|}\sup_{\x\in\SS^q}\int_{\SS^q}|K_n(\x\cdot\y)|d\sigma(\y).$$ More generally, we recall that the Lebesgue constant of any projection $T_n$ is defined as the following operator norm $$\|T_n\|_\infty:=\sup_{\|f\|_\infty\le 1}\|T_nf\|_\infty,$$ and its behaviour as $n\rightarrow + \infty$ strongly influences the quality of the approximation.
It is known (see, e.g., [@b509; @W-Da; @W-ReBook]) that the previous Fourier projection $\S_n$ is the projection onto $\PP_n$ having minimal Lebesgue constant. More precisely, if we denote by ${\cal T}_n$ the class of all the polynomial projections onto $\PP_n$, then for sufficiently large $n$, we have $$\label{min-norm}
\|T_n\|_\infty\ge \|\S_n\|_\infty\sim n^\frac{q-1}2,\qquad \forall T_n\in{\cal T}_n,$$ where throughout the paper by $a_n\sim b_n$ we mean that $c_1 a_n\le b_n\le c_2 a_n$ being $c_1,c_2>0$ independent of $n$.
However, the approximation $\S_nf$ requires the computation of the Fourier coefficients that are integrals of the unknown function $f$. If we suppose to know $f$ only at a discrete point set $X_N:=\{\xi_1,\ldots,\xi_N\}$ such that the quadrature rule $$\label{quad-2n}
\int_{\SS^q} f(\x)d\sigma(\x)=\sum_{i=1}^N\lambda_i f(\xi_i),\qquad \lambda_i>0,\qquad\quad\forall f\in\PP_{2n},$$ holds, then we can discretize $\S_nf$ by applying (\[quad-2n\]) to (\[Fourier\]). In this way, we get the following polynomial of degree at most $n$ [@W-SloFirst; @W4] $$\label{hyper}
L_{n}f(\x)=\frac 1{|\SS^{q-1}|}\sum_{i=1}^N\lambda_{i}f(\xi_i)K_{n}(\xi_i\cdot \x),\qquad \x\in\SS^q,$$ which is usually called [*hyperinterpolation polynomial*]{} because it is based on the function values at a number of nodes $N$ that is greater than $d_n$, the dimension of $\PP_n$ [@W-Ba; @W-Re].
The double degree of exactness in (\[quad-2n\]) assures that $L_{n}$ is a discrete polynomial projection onto $\PP_{n}$, namely $$\label{inva-hyper}
L_nP=P,\qquad \forall P\in\PP_n.$$ Moreover, it is known that the Lebesgue constants $\|L_{n}\|_\infty$ increase with $n$ at the order of the minimal projections, i.e., for all sufficiently large $n\in\NN$, we have [@W-Re; @W4] $$\label{hyper-norm}
\|L_{n}\|_\infty\sim\|\S_n\|_\infty\sim n^\frac{q-1}2.$$
On least squares polynomial approximation
=========================================
A different kind of discrete polynomial projection is given by the least squares approximations $\tilde\S_nf\in \PP_n$, defined by $$\label{LS-min}
\sum_{i=1}^N[f(\xi_i)-\tilde\S_nf(\xi_i)]^2=\min_{P\in\PP_n}
\sum_{i=1}^N[f(\xi_i)-P(\xi_i)]^2.$$ In explicit form, for all $\x\in\SS^q$, the least squares polynomial $\tilde\S_nf(\x)$ related to the point set $X_N=\{\xi_1,\ldots,\xi_N\}\subset\SS^q$ is given by $$\label{LS-sum}
\tilde\S_{n}f(\x)=\sum_{i=1}^Nf(\xi_i)H_{n}(\x, \xi_i),\qquad H_n(\x,\y):=\sum_{r=1}^{d_n}I_r(\x)I_r(\y),$$ where $\{I_r : r=1,\ldots,d_n\}$ is a basis of $\PP_n$ orthonormal w.r.t. the discrete scalar product defined in (\[prod-dis\]). Moreover, we observe that $$\label{inva}
P(\x)=\sum_{i=1}^NP(\xi_i)H_{n}(\x, \xi_i),\qquad \forall P\in\PP_n, \qquad \forall\x\in\SS^q .$$ With respect to the hyperinterpolation $L_nf$, the least squares polynomial $\tilde\S_nf$ does not require to know any quadrature weight, neither any quadrature rule is indeed necessary for its definition.
Concerning the Lebesgue constant $\|\tilde\S_n\|_\infty$, for the $2$–sphere case (i.e., $q=2$) in [@W-Marc1] it has been proved that $\|\tilde\S_n\|_\infty\sim \|L_n\|_\infty$ holds if the point set $X_N=\{\xi_1,\xi_2,\ldots,\xi_N\}\subset \SS^2$ is such to support a positive weighted quadrature rule of degree of exactness $2n$ and if the following Marcinkievicz type inequality holds $$\frac 1{n^2}\sum_{i=1}^N |Q(\xi_i)|\le\C \|Q\|_{L^1(\SS^2)},\qquad \forall Q\in\PP_{n},\qquad \C\ne\C(n,N,Q),$$ where throughout the paper we denote by $\C$ a positive constant, which can take different values at the different occurrences, and we write $\C\ne\C(n,N,Q,..)$ to mean that $\C$ is independent of $n,N,Q,...$
The next theorem generalizes [@W-Marc1 Th. 2.3] to any dimension $q\ge 2$.
\[th-LS\] Let the point set $X_N=\{\xi_1,\xi_2,\ldots,\xi_N\}\subset \SS^q$ and $n\in\NN$ be such that (\[quad-2n\]) holds. Moreover, suppose that $$\label{Marci-q}
\frac 1{n^q}\sum_{i=1}^N |Q(\xi_i)|\le\C \|Q\|_{L^1(\SS^q)},\qquad \forall Q\in\PP_{n},\qquad \C\ne\C(n,N,Q).$$ Then for all sufficiently large $n\in\NN$, the Lebesgue constant of the least squares polynomial of degree $n$ associated to $X_N$, satisfies $$\label{Leb-LS}
\|\tilde\S_n\|_\infty\sim n^{\frac{q-1}2}.$$
Let $card(A)$ denote the cardinality of the set $A$ and let $d(\x,\y):=\arccos(\x\cdot\y)$ be the geodesic distance of $\x,\y\in\SS^q$. In [@W-Dai Th. 2.1] it has been proved that $$\label{hp-sep}
\sup_{\x\in\SS^q} card\left(\left\{\xi_i\in X_N : d(\xi_i, \x)\le \frac 1n\right\}\right)\le \C, \qquad \C\ne\C(n,N),$$ is a necessary and sufficient condition for (\[Marci-q\]), so that the previous theorem is equivalent to the following
\[th-LSequi\] Let the point set $X_N=\{\xi_1,\xi_2,\ldots,\xi_N\}\subset \SS^q$ be such that (\[hp-sep\]) and (\[quad-2n\]) holds. Then for sufficiently large $n\in\NN$, we have $$\|\tilde\S_n\|_\infty\sim n^{\frac{q-1}2}.$$
The assumption (\[hp-sep\]) is also required to state the existence of positive weighted quadrature rules (see, e.g., [@W-Dai; @r938; @W-Po]). Nevertheless there exist several positive quadrature rules not satisfying (\[hp-sep\]). This is the case of tensor product Gauss–Legendre quadrature rules deduced in [@W4 Example 6.1] for $q=2$, by combining the trigonometric rectangular rule (exact for trigonometric polynomials of degree $\le 2n+1$) $$\int_0^{2\pi}g(\Phi)d\Phi=\frac\pi{n+1}\sum_{k=0}^{2n+1}g\left(\Phi_k\right),\qquad \Phi_k:=\frac{k\pi}{n+1},$$ and the $(n+1)$–point Gauss–Legendre quadrature rule $$\label{GL-quadrule}
\int_{-1}^1G(z)dz=\sum_{j=1}^{n+1}\nu_jG(z_j). $$ The resulting tensor product rule has degree of precision $2n+1$ and it is based on $N=2(n+1)^2$ points. It looks like $$\label{tensor}
\int_{\SS^2}f(\x)d\sigma(\x)=\sum_{k=0}^{2n+1}\sum_{j=1}^{n+1} \frac {\pi \nu_j}{n+1} f(\xi_{j,k}),
\qquad \forall f\in\PP_{2n+1},$$ where each node $\xi_{j,k}$ has azimuthal angle $\Phi_k$ and polar angle $\theta_j=\arccos z_j$.
The main advantage of tensor product rules is the explicit knowledge of the quadrature weights and nodes, but the latter have the disadvantage of not being well–separated on the sphere.
This can be seen by Figure \[fig001\], which shows how the nodes $$\label{tensor-pt}
\tilde X_N:=\{\xi_{j,k} : k=0,...,2n+1, \ j=1,...,n+1\}$$ are distributed on the sphere $\SS^2$ for degree of precision $31$, i.e., $n=15$, $N=512$, and degree of precision $51$, i.e., $n=25$, $N = 1352$.
![Examples of the tensor product Gauss–Legendre quadrature nodes related to degrees of precision $31$ and $51$, i.e., having $N=512$ (left) and $N=1352$ (right) points.\[fig001\]](Figures/fig003.pdf "fig:") ![Examples of the tensor product Gauss–Legendre quadrature nodes related to degrees of precision $31$ and $51$, i.e., having $N=512$ (left) and $N=1352$ (right) points.\[fig001\]](Figures/fig004.pdf "fig:")
We recall (see, e.g., [@HardMichSaff2016]) that a measure of the uniformity of a sampling set $X_N=\{\xi_1,\ldots,\xi_N\}$ is given by the mesh norm $\delta_{X_N}$ and separation distance $\gamma_{X_N}$ defined by $$\begin{aligned}
\label{delta}
\delta_{X_N}&:=&\max_{\x\in\SS^2}\min_{1\le i\le N} d(\x,\xi_i),\\
\label{gamma}
\gamma_{X_N}&:=&\min_{ i\ne j}d(\xi_i,\xi_j),\end{aligned}$$ and a sequence of point configurations $\{X_N\}_N$ is said to be quasi–uniform if the mesh ratio $\delta_{X_N}/\gamma_{X_N}$ is bounded as $N\rightarrow +\infty$.
Figure \[fig011\] displays some values of the mesh norm, separation distance and mesh ratio for the point set $\tilde X_N$ in (\[tensor-pt\]). To estimate the mesh norm $\delta_{X_N}$, instead of taking the maximum over the set of all points of the sphere, the maximum is computed over a point set with a number of points considerably larger than the number of points for which we want to approximate the mesh norm. To this end we consider the “spiral points” as defined in [@RakhSaffZhou1995] (see also the overview paper [@HardMichSaff2016]). In the sequel we refer to this point set as the point set of second type. These can be computed very efficiently and seem to be uniformly distributed over the unit sphere where each of the points seems to be well separated from the others. To estimate $\delta_{\tilde X_N}$ we considered a point set of the second type having $16$ times more points.
![The values of the separation distance $\gamma_{\tilde X_N}$ and the mesh-norm $\delta_{\tilde X_N}$ (left) and the mesh ratio $\delta_{\tilde X_N} / \gamma_{\tilde X_N}$ (right) for point sets $\tilde X_N$ with $N = 2(n+1)^2$ for $n = 5, 10, 15,\ldots,50$. \[fig011\]](Figures/fig011.pdf "fig:") ![The values of the separation distance $\gamma_{\tilde X_N}$ and the mesh-norm $\delta_{\tilde X_N}$ (left) and the mesh ratio $\delta_{\tilde X_N} / \gamma_{\tilde X_N}$ (right) for point sets $\tilde X_N$ with $N = 2(n+1)^2$ for $n = 5, 10, 15,\ldots,50$. \[fig011\]](Figures/fig015.pdf "fig:")
It is evident that the distribution of the nodes in $\tilde X_N$ is not uniform and indeed it turns out that $$\delta_{\tilde X_N}\le\frac\C n, \qquad \mbox{but}\qquad \gamma_{\tilde X_N}\ge\frac \C{n^2},\qquad\qquad \C\ne\C(n,N).$$ Hence, we can say that the previous tensor product nodes provide an optimal hyperinterpolation polynomial, but from a theoretical point of view, up to now nothing can be said regarding the least squares polynomial, since the assumption (\[hp-sep\]) of the previous theorem is not satisfied.
Now we investigate numerically the behaviour of the Lebesgue constants of both least squares and hyperinterpolation polynomials of degree $n$ related to the previous point set $\tilde X_N$. To this end, we’ll estimate the uniform norm of the corresponding operators by taking a larger point set of the second type containing $4N$ points. Figure \[fig021\] shows the results. The circles and squares indicate the Lebesgue constant for the least squares operator and the hyperinterpolation operator, respectively, when we take for degree $n$ on the horizontal axis the corresponding point set $\tilde X_N$ related to the degree of exactness $2n+1$, i.e., having $N = 2(n+1)^2$ points.
![The values of the Lebesgue constants of the least squares operator $\tilde\S_n$ and the hyperinterpolation operator $L_n$ for the degrees $n=10,20,\ldots,80$ with corresponding point set $\tilde X_N$ having $N = 2(n+1)^2$ points.\[fig021\]](Figures/fig028b.pdf)
The figure shows for $\|\tilde\S_n\|_\infty$ the same optimal behavior as for $\|L_n\|_\infty$, i.e., the $\sqrt{n}$ behaviour as indicated by the dotted line.
In order to explain such a numerical output in the case of the Gauss–Legendre tensor product rule (\[tensor\]), we recall that the Legendre zeros $$z_0:=-1<z_1<\ldots <z_{n+1}<1=:z_{n+2}$$ are arcsin distributed on $[-1,1]$ and that for the weights $\nu_i$ of the Gauss-Legendre quadrature rule (\[GL-quadrule\]), $\nu_i\sim (z_{i+1}-z_i)$ holds uniformly w.r.t. $i$ and $n$ [@W-Ne]. Consequently $$\nu_i\sim \nu_{i+ 1},\qquad i=1,\ldots, n,
$$ holds uniformly w.r.t. $i$ and $n$. This is indeed the property replacing (\[Marci-q\]) or (\[hp-sep\]), in order to get the same result as in the previous theorems.
\[th-tensor\] Let $X_N=\{\xi_1,\ldots,\xi_N\}\subset \SS^q$ and $\lambda_1\ge\lambda_2\ge\ldots\ge\lambda_N>0=:\lambda_{N+1}$ be such that $$\int_{\SS^q} f(\x)d\sigma(\x)=\sum_{i=1}^N\lambda_i f(\xi_i),\qquad\forall f\in\PP_{2n},$$ holds for $n\in\NN$. Moreover, suppose that the quadrature weights are such that $$\label{hp-tensor}
\lambda_i\le \C \lambda_{i+ 1},\qquad i=1,\ldots, N-1, \qquad \C\ne \C(n,N,i).$$ Then for all sufficiently large $n$, we have $\displaystyle
\|\tilde\S_n\|_\infty\sim n^\frac{q-1}2.
$
In contrast to the previous two theorems, Theorem \[th-tensor\] results to be applicable to all tensor product rules that usually have the nodes very close to each other near the poles, so (\[hp-sep\]) as well as (\[Marci-q\]) do not generally hold, but (\[hp-tensor\]) holds.
Proofs
======
Proof of Theorem \[th-LS\].
---------------------------
From (\[LS-sum\]) we deduce that $$\|\tilde\S_n\|_\infty=\sup_{\x\in\SS^q} \left[\sum_{k=1}^N|H_{n}(\x, \xi_k)|\right].$$ Hence, due to (\[min-norm\]), it is sufficient to prove that $$\label{eq-th}
\sum_{k=1}^N|H_n(\x,\xi_k)|\le\C n^{\frac{q-1}2}, \qquad \forall\x\in\SS^q, \qquad \C\ne\C(n,N,\x).$$ Let us first prove (\[eq-th\]) when $\x\in X_N$.
To this aim we observe that $P(\xi_k):=H_n(\x,\xi_k)$ is a spherical polynomial of degree $n$ w.r.t. the variable $\xi_k$. Consequently, recalling that $L_nP=P$, we get $$\label{trans1}
H_n(\x,\xi_k)=\frac 1{|\SS^{q-1}|}\sum_{i=1}^N\lambda_i H_n(\x,\xi_i)
K_n(\xi_k\cdot\xi_i),\qquad k=1,\ldots,N,$$ where without losing the generality, we assume that $\lambda_i$ are labeled in non increasing order, namely $$\lambda_1\ge\lambda_2\ge\ldots \ge\lambda_N>\lambda_{N+1}:=0.$$ Then, by applying the following summation by parts formula $$\label{sum-part}
\sum_{i=1}^Na_ib_i=a_N\sum_{i=1}^N b_i+\sum_{i=1}^{N-1}(a_i-a_{i+1})\sum_{j=1}^ib_j,$$ we get $$\label{eq-start1}
H_n(\x,\xi_k)= \frac 1{|\SS^{q-1}|}\sum_{i=1}^{N}(\lambda_i-\lambda_{i+1})\sum_{j=1}^iH_n(\x,\xi_j)
K_n(\xi_k\cdot\xi_j).$$ Consequently, by taking into account that $\lambda_i-\lambda_{i+1}\ge 0$, we have $$\begin{aligned}
\sum_{k=1}^N|H_n(\x,\xi_k)|&\le&\frac 1{|\SS^{q-1}|} \sum_{k=1}^N\sum_{i=1}^{N}(\lambda_i-\lambda_{i+1})
\left|\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|\\
&=&\frac 1{|\SS^{q-1}|}
\sum_{i=1}^{N}(\lambda_i-\lambda_{i+1})\sum_{k=1}^N\left|\sum_{j=1}^iH_n(\x,\xi_j)
K_n(\xi_k\cdot\xi_j)\right|\\
&\le&\frac 1{|\SS^{q-1}|}
\sup_{1\le i\le N}\left(\sum_{k=1}^N\left|\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|\right)
\sum_{i=1}^{N}(\lambda_i-\lambda_{i+1})
\\
&=& \frac{\lambda_1}{|\SS^{q-1}|}\sup_{1\le i\le N}\sum_{k=1}^N
\left|\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|,\end{aligned}$$ and recalling that [@b509 Lemma 5.4.3] $$\label{lambda}
\lambda_i\le \frac\C{n^q},\qquad i=1,\ldots, N,\qquad \C\ne \C(n,N,i),$$ we obtain $$\sum_{k=1}^N|H_n(\x,\xi_k)|\le \frac \C{n^q}\sup_{1\le i\le N} \sum_{k=1}^N
\left|\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|,\qquad \C\ne\C(n,N,\x).$$ Hence, set for any $n\in\NN$ and $\x\in X_N$ $$A_i:=\frac 1{n^q} \sum_{k=1}^N
\left|\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|, \qquad i=1,\ldots,N,$$ to get the statement when $\x\in X_N$, we are going to prove that $$\label{tesi}
A_i\le\C n^{\frac{q-1}2}, \qquad i=1,\ldots,N,\qquad \C\ne\C(n,N,\x),$$ holds for all sufficiently large $n\in\NN$ and any $\x\in X_N$.
For the case $i=N$, note that by (\[inva\]) we get $$A_N:=\frac 1{n^q} \sum_{k=1}^N
\left|\sum_{j=1}^NH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|= \frac 1{n^q} \sum_{k=1}^N
\left|K_n(\xi_k\cdot\x)\right|.$$ Moreover, (\[Marci-q\]) and (\[Leb-Fou\]) imply $$\frac 1{n^q} \sum_{k=1}^N
\left|K_n(\xi_k\cdot\xi)\right|\le \C\int_{\SS^q}\left|K_n(\y\cdot\xi)\right|d\sigma(\y)\le\C \|\S_n\|_\infty,\qquad \forall\xi\in\SS^q,$$ and hence by (\[min-norm\]) we have $$\label{eq-darboux}
\frac 1{n^q} \sum_{k=1}^N
\left|K_n(\xi_k\cdot\xi)\right|\le \C \|\S_n\|_\infty\le\C n^{\frac{q-1}2},\qquad \forall\xi\in\SS^q,
\qquad \C\ne\C(n,N,\xi).$$ So, we conclude that $$\label{AN}
A_N\le\C n^{\frac{q-1}2}, \qquad \forall \x\in X_N,\qquad \C\ne\C(n,N,\x) .$$ As regards the case $1\le i<N$, we observe that for any pair of nodes $\xi_l,\xi_j\in X_N$, we have $$\label{CSW}
|H_n(\xi_l,\xi_j)|\le\sum_{r=1}^{d_n}|I_r(\xi_l)I_r(\xi_j)|\le \left(\sum_{r=1}^{d_n}|I_r(\xi_l)|^2\right)^\frac 12
\left(\sum_{r=1}^{d_n}|I_r(\xi_j)|^2\right)^\frac 12.$$ On the other hand, we point out that the existence of (\[quad-2n\]) implies that $d_n< N$ for sufficiently large $n$ (see, e.g., [@W-Re p. 274]). Consequently, the matrix consisting of the orthonormal columns $[I_k(\xi_1),\ldots I_k(\xi_N)]^T$, $k=1,\ldots, d_n$, namely the matrix $$I := [I_k(\xi_h)]_{h=1,...,N}^{k=1,\ldots,d_n}$$ is rectangular, but it can be extended by additional columns to form a square orthogonal matrix $$Q=[Q_{h,k}]_{h=1,...,N}^{k=1,...,N}, \qquad \mbox{such that}\qquad Q_{h,k}=I_k(\xi_h),\quad \forall k\le d_n.$$ Thus we have $$\label{Q}
\sum_{k=1}^{d_n}|I_k(\xi_j)|^2\le \sum_{k=1}^{N}|Q_{j,k}|^2 =1,\qquad j=1,\ldots,N,
$$ and assembling (\[CSW\]) and (\[Q\]), we conclude that $$\label{ls-ker1}
|H_n(\xi_l,\xi_j)|\le 1,\qquad \forall \xi_l,\xi_j\in X_N.$$ By means of (\[ls-ker1\]), we deduce $$A_1:=\frac 1{n^q} \sum_{k=1}^N
\left|H_n(\x,\xi_1)K_n(\xi_k\cdot\xi_1)\right|\le \frac 1{n^q} \sum_{k=1}^N
\left|K_n(\xi_k\cdot\xi_1)\right|,\qquad \forall \x\in X_N,$$ and using (\[eq-darboux\]), we get $$\label{A1}
A_1\le\C n^{\frac{q-1}2}, \qquad \forall \x\in X_N,\qquad \C\ne \C(n,N,\x) .$$ Similarly, for any $1\le i<N$ and $\x\in X_N$, by (\[ls-ker1\]) and (\[eq-darboux\]), we have $$\begin{aligned}
\nonumber
|A_{i+1}- A_i| &\le&\frac 1{n^q} \sum_{k=1}^N
\left|H_n(\x,\xi_{i+1})K_n(\xi_k\cdot\xi_{i+1})\right|\\
\nonumber
&\le& \frac 1{n^q} \sum_{k=1}^N\left|K_n(\xi_k\cdot\xi_{i+1})\right|\\
\label{Ai}
&\le&\C n^{\frac{q-1}2},\qquad \C\ne\C(n,N,\x, i).\end{aligned}$$ In conclusion, let us show that (\[AN\]), (\[A1\]) and (\[Ai\]) imply that as $n\rightarrow + \infty$ (\[tesi\]) holds for all $x\in X_N$.
Indeed, if ad absurdum there exists $\x\in X_N$ s.t. for some index $l$ we have that $$\limsup_{n\rightarrow +\infty}\ n^\frac{1-q}2 A_l=+\infty,$$ then (\[AN\]) and (\[A1\]) imply $1<l<N$, and from (\[Ai\]) we deduce that we also have $$\limsup_{n\rightarrow +\infty}\ n^\frac{1-q}2 A_{l\pm 1} =+\infty.$$ Thus, by iterating the reasoning, we arrive to contradict (\[AN\]) or (\[A1\]).
Hence, we conclude that (\[eq-th\]) holds for all $\x\in X_N$.
For arbitrary $\x\in\SS^q$, we reason analogously, but we start applying the invariance property $L_nP=P$ to the polynomials $P(\x)=H_n(\x,\xi_k)$, with $k=1,\ldots,N$. Hence, instead of (\[trans1\]) we get $$\label{trans2}
H_n(\x,\xi_k)=\frac 1{|\SS^{q-1}|}\sum_{i=1}^N\lambda_i H_n(\xi_i,\xi_k)
K_n(\x\cdot\xi_i),\qquad k=1,\ldots,N,$$ which differs from (\[trans1\]) by the exchanged position of the variables $\x$ and $\xi_k$ at the right–hand sides.
Consequently, by using (\[sum-part\]) and (\[lambda\]) as before, we deduce $$\begin{aligned}
\sum_{k=1}^N|H_n(\x,\xi_k)|&=&\frac 1{|\SS^{q-1}|} \sum_{k=1}^N\left|\sum_{i=1}^N\lambda_i H_n(\xi_i,\xi_k)
K_n(\x\cdot\xi_i)\right|\\
&\le& \frac\C{n^q}\max_{1\le i\le N}\left(\sum_{k=1}^N\left|\sum_{j=1}^i H_n(\xi_j,\xi_k)
K_n(\x\cdot\xi_j)\right|\right), \qquad \C\ne\C(n,N,\x).\end{aligned}$$ Then, for arbitrarily fixed $n\in\NN$ and $\x\in\SS^q$, we set $$B_i:=\frac 1{n^q}\sum_{k=1}^N\left|\sum_{j=1}^i H_n(\xi_j,\xi_k)K_n(\x\cdot\xi_j)\right|,\qquad i=1,\ldots,N.$$ When $i=N$, by virtue of (\[inva\]) and (\[eq-darboux\]), we have $$\label{BN}
B_N=\frac 1{n^q}\sum_{k=1}^N\left|K_n(\x\cdot\xi_k)\right|\le\C n^{\frac{q-1}2},\qquad \forall \x\in\SS^q,\qquad \C\ne\C(n,N,\x).$$ Moreover, recalling that (see, e.g., [@r938; @b210]) $$\label{sup-Darboux}
|K_n(\x\cdot\y)|\le \sup_{|t|\le 1}|K_n(t)|= K_n(1)\sim n^q,\qquad \forall\x,\y\in\SS^q,$$ and taking into account that we have already proved (\[eq-th\]) in $X_N$, for all $\x\in\SS^q$ we get $$\begin{aligned}
\nonumber
B_1&:=&\frac 1{n^q}\sum_{k=1}^N\left| H_n(\xi_1,\xi_k)K_n(\x\cdot\xi_1)\right|\le
\C\sum_{k=1}^N\left|H_n(\xi_1\cdot\xi_k)\right|\\
\label{B1}
&\le&\C n^{\frac{q-1}2},\qquad\qquad\C\ne\C(n,N,\x,\xi_1),\end{aligned}$$ as well as, for all $i=1,\ldots,N-1$, and any $\x\in\SS^q$, we deduce $$\begin{aligned}
\nonumber
|B_{i+1}- B_i|&\le&\frac 1{n^q}\sum_{k=1}^N\left|H_n(\xi_{i+1},\xi_k)K_n(\x\cdot\xi_{i+1})\right|\\
\nonumber
&\le& \C\sum_{k=1}^N\left|H_n(\xi_{i+1},\xi_k)\right|\\
\label{Bi}
&\le& \C n^{\frac{q-1}2},\qquad\qquad\C\ne\C(n,N,\x,i).\end{aligned}$$ In conclusion, similarly to the case $\x\in X_N$, from (\[BN\]), (\[B1\]) and (\[Bi\]) we deduce that for all sufficiently large $n\in\NN$, and any $\x\in \SS^q$, we have $$B_i\le \C n^{\frac{q-1}2},\qquad i=1,\ldots,N,\quad\qquad\C\ne\C(n,N,\x),$$ and the statement follows in the case $\x\in\SS^q$ too.
Proof of Theorem \[th-tensor\].
-------------------------------
Following the same reasoning of the previous proof, we arrive to say that it is sufficient to state that for all sufficiently large $n\in\NN$, we have $$\sum_{k=1}^N|H_n(\x,\xi_k)|\le\C n^{\frac{q-1}2}, \qquad \forall\x\in X_N ,\qquad \C\ne\C(n,N,\x).$$ Note that, by using (\[eq-start1\]) and $\lambda_i-\lambda_{i+1}\ge 0$, we get $$\begin{aligned}
|\SS^{q-1}|\sum_{k=1}^N|H_n(\x,\xi_k)|&=&\sum_{k=1}^N\left|\sum_{i=1}^{N}(\lambda_i-\lambda_{i+1})
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|
\\
&\le&\sup_{1\le r\le N}\left(\sum_{k=1}^N\left|\sum_{i=r}^{N}(\lambda_i-\lambda_{i+1})
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|\right)
\\
&\le&\sup_{1\le r\le N}\left(\sum_{k=1}^N\sum_{i=r}^{N}(\lambda_i-\lambda_{i+1})
\left|\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|\right)
\\
&=&\sup_{1\le r\le N}\left(\sum_{i=r}^{N}(\lambda_i-\lambda_{i+1})\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|\right)
\\
&\le&\sup_{1\le r\le N}\left(\sum_{i=r}^{N}(\lambda_i-\lambda_{i+1})\sup_{r\le i\le N}\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|\right)
\\
&=&\sup_{1\le r\le N}\left(\lambda_r\sup_{r\le i\le N}\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|\right).\end{aligned}$$ Hence, set $$A_r(\x):=\lambda_r\sup_{r\le i\le N}\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|, \qquad r=1,\ldots,N,$$ we are going to prove that as $n\rightarrow +\infty$, we have $$\label{eq-th-tens}
\sup_{x\in X_N}A_r(\x)=O( n^{\frac{q-1}2}),\qquad r=1,\ldots,N .$$
First of all, we prove (\[eq-th-tens\]) for $r=N$. Indeed, from (\[inva\]) and $\lambda_N=\min_{1\le k\le N} \lambda_k$, we deduce that for all $\x\in X_N$ $$A_N(\x):=\sum_{k=1}^N\lambda_{N}\left|\sum_{j=1}^NH_n(\x,\xi_j)
K_n(\xi_k\cdot\xi_j)\right|=\sum_{k=1}^N\lambda_{N}|K_n(\x\cdot\xi_k)|\le \sum_{k=1}^N\lambda_{k}|K_n(\x\cdot\xi_k)|.$$ On the other hand, it is known [@W-Dai; @W-Re] that the following Marcinkiewicz inequality follows from the existence of the quadrature rule (\[quad-2n\]) $$\label{Marci}
\sum_{i=1}^N \lambda_i |Q(\xi_i)|\le \C \|Q\|_{L^1(\SS^q)}, \qquad \forall Q\in\PP_{n}, \qquad \C\ne\C(n,N,Q).$$ Hence, by using (\[Marci\]), (\[Leb-Fou\]) and (\[min-norm\]), the previous estimate continues as follows $$A_N(\x)\le \sum_{k=1}^N\lambda_{k}|K_n(\x\cdot\xi_k)|
\le\C\int_{\SS^q}|K_n(\x\cdot\y)|d\sigma(\y)\le\C\|\S_n\|_\infty \le \C n^{\frac{q-1}2},$$ i.e., we get $$\label{AN-1}
\sup_{\x\in X_N} A_N(\x)\le \C n^{\frac{q-1}2}, \qquad \C\ne\C(n,N).$$ Now, for any $1\le r< N$ let us prove that the assumption $$\label{hp}
\lambda_{r+1}\le \lambda_r\le\C\lambda_{r+1}, \qquad \C\ne\C(n,N,r),$$ implies $$\label{th-ind}
\sup_{\x\in X_N} A_{r+1}(\x)\le \sup_{\x\in X_N} A_r(\x)\le 2 \C \sup_{\x\in X_N} A_{r+1}(\x),$$ where the constant $\C$ in (\[th-ind\]) is the same of that in (\[hp\]).
Indeed for any $\x\in X_N$, by the first inequality in (\[hp\]), we get $$\begin{aligned}
A_{r+1}(\x)&:=& \lambda_{r+1}\sup_{r+1\le i\le N}\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|\\
&\le& \lambda_{r}\sup_{r+1\le i\le N}\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|\\
&\le& \lambda_{r}\sup_{r\le i\le N}\sum_{k=1}^N\left|\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|= A_{r}(\x),\end{aligned}$$ which yields the first inequality in (\[th-ind\]).
In order to state the second inequality in (\[th-ind\]), we distinguish two cases.
[*Case 1:*]{} $\displaystyle \sup_{r+1\le i\le N}\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|= \sup_{r\le i\le N}\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|$.
In this case, by the second inequality in (\[hp\]), we get $$\begin{aligned}
A_{r}(\x)&:=& \lambda_{r} \sup_{r\le i\le N}\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|\\
&=& \lambda_{r} \sup_{r+1\le i\le N}\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|\\
&\le& \C \lambda_{r+1} \sup_{r+1\le i\le N}\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|=\C A_{r+1}(\x).\end{aligned}$$ [*Case 2:*]{} $\displaystyle \sup_{r+1\le i\le N}\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|< \sup_{r\le i\le N}\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|$.
In this case, by taking into account that $$\begin{aligned}
&&\sup_{r\le i\le N}\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|\\
&=& \max\left\{\sup_{r+1\le i\le N}\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|, \ \sum_{k=1}^N\left|
\sum_{j=1}^{r}H_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|\right\},\end{aligned}$$ we can say that $$\sup_{r\le i\le N}\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|= \sum_{k=1}^N\left|
\sum_{j=1}^{r}H_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|.$$ Consequently, by the second inequality in (\[hp\]), we get $$\begin{aligned}
A_{r}(\x)&=&\lambda_{r}\sum_{k=1}^N\left|
\sum_{j=1}^rH_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|\\
&\le & \C \lambda_{r+1}\sum_{k=1}^N\left|
\sum_{j=1}^{r+1}H_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)- H_n(\x,\xi_{r+1})K_n(\xi_k\cdot\xi_{r+1})\right|\\
&\le& \C \lambda_{r+1}\sum_{k=1}^N\left|
\sum_{j=1}^{r+1}H_n(\x,\xi_j)K_n(\xi_k\cdot\xi_j)\right|+
\C\lambda_{r+1}\sum_{k=1}^N\left|
H_n(\x,\xi_{r+1})K_n(\xi_k\cdot\xi_{r+1})\right|\\
&=& \C A_{r+1}(\x) + \C \lambda_{r+1} \sum_{k=1}^N\left|
H_n(\x,\xi_{r+1})K_n(\xi_k\cdot\xi_{r+1})\right|.\end{aligned}$$ Moreover, by means of (\[ls-ker1\]) and (\[inva\]), we observe that $$\begin{aligned}
\lambda_{r+1} \sum_{k=1}^N\left|H_n(\x,\xi_{r+1})K_n(\xi_k\cdot\xi_{r+1})\right|&\le&
\lambda_{r+1} \sum_{k=1}^N\left|K_n(\xi_k\cdot\xi_{r+1})\right|\\
&=& \lambda_{r+1} \sum_{k=1}^N\left|\sum_{j=1}^N H_n(\xi_{r+1}, \xi_j)K_n(\xi_k\cdot\xi_j)\right|\\
&\le& \lambda_{r+1}\sup_{r+1\le i\le N}\sum_{k=1}^N\left|
\sum_{j=1}^iH_n(\xi_{r+1}, \xi_j)K_n(\xi_k\cdot\xi_j)\right|\\
&=& A_{r+1}(\xi_{r+1}).\end{aligned}$$ Hence, in the second case we conclude that $$A_r(\x)\le \C A_{r+1}(\x)+\C A_{r+1}(\xi_{r+1}).$$ Summing up, in both the previous cases, for all $\x\in X_N$, we can say that $$A_{r}(\x)\le \C A_{r+1}(\x)+ \C A_{r+1}(\xi_{r+1})\le 2\C\sup_{\xi\in X_N} A_{r+1}(\xi),$$ which yields the second inequality in (\[th-ind\]).
In conclusion, let us prove that (\[th-ind\]) and (\[AN-1\]) imply (\[eq-th-tens\]).
Indeed, set for brevity $$A_r:=\sup_{\x\in X_N} A_r(\x),\qquad r=1,\ldots,N,$$ we have to prove that $$\limsup_{n\rightarrow +\infty}\ n^{\frac{1-q}2}A_r < +\infty,\qquad r=1,\ldots,N.$$ But if ad absurdum, for some index $l$ we have $$\label{hp-abs}
\limsup_{n\rightarrow +\infty}\ n^{\frac{1-q}2}A_l= +\infty,$$ then by virtue of (\[AN-1\]) it will be $1\le l <N$. Consequently, since (\[th-ind\]) yields $$\frac{1}{2\C}\ n^{\frac{1-q}2} A_l\le n^{\frac{1-q}2} A_{l+1}\le n^{\frac{1-q}2} A_l,\qquad \C\ne \C(n,N,l),$$ from (\[hp-abs\]) we deduce that $$\limsup_{n\rightarrow +\infty}\ n^{\frac{1-q}2}A_{l+1}= +\infty$$ holds too. Then by iterating the reasoning, we arrive to say that $$\limsup_{n\rightarrow +\infty}\ n^{\frac{1-q}2} A_N = +\infty,$$ which contradicts (\[AN-1\]).
Hence, due to (\[th-ind\]) and (\[AN-1\]), we conclude that (\[eq-th-tens\]) necessarily holds.
Conclusions
===========
On the unit sphere $\SS^q\subset\RR^{q+1}$, with $q\ge 2$ we studied the approximation provided by least squares polynomials, $\tilde\S_nf$ defined in (\[LS-min\]), w.r.t. the uniform norm. To this aim, we estimated the behaviour of the associated Lebesgue constants as the polynomial degree $n$ tends to infinity.
Similarly to the hyperinterpolation approximation, for all the polynomial degrees $n$, we supposed that the underlying point set $X_N=\{\xi_1,\ldots,\xi_N\}$ consists of nodes of a positive weighted quadrature rule of degree of precision $2n$. Then, for least squares polynomial approximation, we stated an optimal behaviour of Lebesgue constants by proving that they grow at the minimal projection order (namely as $n^{\frac{q-1}2}$) under two different additional hypotheses:
- In a first case (cf. Theorems \[th-LS\] and \[th-LSequi\]) we supposed that the Marcinkiewicz type inequality (\[Marci-q\]) holds. This is equivalent to requiring that the nodes $\xi_i$ in $X_N$ are well–separated, namely (\[hp-sep\]) holds.
- In the second case (cf. Theorem \[th-tensor\]), the nodes can be also not well–separated, but we require that the quadrature weights $\lambda_i$, labeled in non increasing order, satisfy (\[hp-tensor\]).
We remark that in the literature one can find a variety of quadrature nodes fitting into the first or the second case (see, e.g., [@W4; @r939; @W-Po; @W-SloBook; @W-Xu1] ). In particular, a point set satisfying Theorem \[th-LSequi\] can be selected from any sufficiently dense set of points on the sphere [@W2; @W-GiaMha; @W-NPW]. Moreover, Theorem \[th-tensor\] can be applied to the tensor product Gauss–Legendre nodes in [@W4].
In conclusion, under our assumptions, we can say that the approximation provided by least squares and hyperinterpolation polynomials are comparable w.r.t. the uniform norm, having in both cases optimal Lebesgue constants.
From a computational point of view, least squares polynomials depend only on the function values at the nodes, while hyperinterpolation polynomials also require a preliminary knowledge of the quadrature weights. Hence, the choice of hyperinterpolation or least squares polynomial approximation depends on the specific problem at hand.
[^1]: C.N.R. National Research Council of Italy, Istituto per le Applicazioni del Calcolo “Mauro Picone”, via P. Castellino, 111, 80131 Napoli, Italy. woula.themistoclakis@cnr.it. Partially supported by GNCS-INDAM.
[^2]: KU Leuven, Department of Computer Science, KU Leuven, Celestijnenlaan 200A, B-3001 Leuven (Heverlee), Belgium. marc.vanbarel@cs.kuleuven.be. Supported by the Research Council KU Leuven, C1-project (Numerical Linear Algebra and Polynomial Computations), and by the Fund for Scientific Research–Flanders (Belgium), EOS Project no 30468160.
|
---
abstract: 'We consider the homotopy types of $PD_4$-complexes $X$ with fundamental group $\pi$ such that $c.d.\pi=2$ and $\pi$ has one end. Let $\beta=\beta_2(\pi;\mathbb{F}_2)$ and $w=w_1(X)$. Our main result is that (modulo two technical conditions on $(\pi,w)$) there are at most $2^\beta$ orbits of $k$-invariants determining “strongly minimal" complexes (i.e., those with homotopy intersection pairing $\lambda_X$ trivial). The homotopy type of a $PD_4$-complex $X$ with $\pi$ a $PD_2$-group is determined by $\pi$, $w$, $\lambda_X$ and the $v_2$-type of $X$. Our result also implies that Fox’s 2-knot with metabelian group is determined up to homeomorphism by its group.'
address: |
School of Mathematics and Statistics\
University of Sydney, NSW 2006\
Australia
author:
- 'Jonathan A. Hillman'
title: 'Strongly minimal $PD_4$-complexes'
---
It remains an open problem to give a homotopy classification of closed 4-manifolds, or more generally $PD_4$-complexes, in terms of standard invariants such as the fundamental group, characteristic classes and homotopy intersection pairings. The class of groups of cohomological dimension at most 2 seems to be both tractable and of direct interest to geometric topology, as it includes all surface groups, knot groups and the groups of many other bounded 3-manifolds. In our earlier papers we have shown that this case can largely be reduced to the study of “strongly minimal" $PD_4$-complexes $Z$ with trivial intersection pairing on $\pi_2(Z)$. If $X$ is a $PD_4$-complex with fundamental group $\pi$, $k_1(X)=0$ and there is a 2-connected degree-1 map $p:X\to{Z}$, where $Z$ is strongly minimal then the homotopy type of $X$ is determined by $Z$ and the intersection pairing $\lambda_X$ on the “surgery kernel" $K_2(p)=\mathrm{Ker}(\pi_2(p))$, which is a finitely generated projective left $\mathbb{Z}[\pi]$-module [@Hi06]. Here we shall attempt to determine the homotopy types of such strongly minimal $PD_4$-complexes, under further hypotheses on $\pi$ and the orientation character.
The first two sections review material about generalized Eilenberg-Mac Lane spaces and cohomology with twisted coefficients, the Whitehead quadratic functor and $PD_4$-complexes. We assume thereafter that $X$ is a $PD_4$-complex, $\pi=\pi_1(X)$ and $c.d.\pi=2$. Such complexes have strongly minimal models $p:X\to{Z}$. In §3 we show that the homotopy type of $X$ is determined by its first three homotopy groups and the second $k$-invariant $k_2(X)\in H^4(L_\pi(\pi_2(X),2);\pi_3(X))$.
The key special cases in which the possible strongly minimal models are when:
1. $\pi\cong {F(r)}$ is a finitely generated free group;
2. $\pi=F(r)\rtimes{Z}$; or
3. $\pi$ is a $PD_2$-group.
We review the first two cases briefly in §4, and in §5 we outline an argument for the case of $PD_2$-groups, which involves cup product in integral cohomology. (This is a model for our later work in Theorem 13.) In Theorem 8 we show that the homotopy type of a $PD_4$-complex $X$ with $\pi$ a $PD_2$-group is determined by $\pi$, $w=w_1(X)$, $\lambda_X$ and the $v_2$-type of $X$. (The corresponding result was already known for $\pi$ free and in the Spin case when $\pi$ is a $PD_2$-group.) In §6 we assume further that $\pi$ has one end, and give a partial realization theorem for $k$-invariants (Theorem 9); we do not know whether the 4-complexes we construct all satisfy Poincaré duality. In §7 and §8 we extend the cup product argument sketched in §5 to a situation involving local coefficient systems, to establish our main result (Theorem 13). Here we show that the number of homotopy types of minimal $PD_4$-complexes for $(\pi,w)$ is bounded by the order of $H^2(\pi;\mathbb{F}_2)$, provided that $(\pi,w)$ satisfies two technical conditions. (However we do not have an explicit invariant.) One of these conditions fails for $\pi$ a $PD_2$-group and $w_1(\pi)$ or $w$ nontrivial, and thus our result is far from ideal. Nevertheless it holds in other interesting cases, notably when $\pi=Z*_m$ (with $m$ even) and $w=1$. (See §9.) In the final section we show that if $\pi$ is the group of a fibred ribbon 2-knot $K$ the knot manifold $M(K)$ is determined up to TOP $s$-cobordism by $\pi$, while Example 10 of Fox’s “Quick Trip Through Knot Theory" [@Fo62] is determined up to TOP isotopy and reflection by its group.
generalities
============
Let $X$ be a topological space with fundamental group $\pi$ and universal covering space $\widetilde{X}$, and let $f_{X,k}:X\to{P_k(X)}$ be the $k^{th}$ stage of the Postnikov tower for $X$. We may construct $P_k(X)$ by adjoining cells of dimension at least $k+2$ to kill the higher homotopy groups of $X$. The map $f_{X,k}$ is then given by the inclusion of $X$ into $P_k(X)$, and is a $(k+1)$-connected map. In particular, $P_1(X)\simeq{K=K(\pi,1)}$ and $c_X=f_{X,1}$ is the classifying map for the fundamental group $\pi=\pi_1(X)$.
Let $[X;Y]_K$ be the set of homotopy classes over $K$ of maps $f:X\to{Y}$ such that $c_X=c_Yf$. If $M$ is a left $\mathbb{Z}[\pi]$-module let $L_\pi(M,n)$ be the generalized Eilenberg-Mac Lane space over $K$ realizing the given action of $\pi$ on $M$. Thus the classifying map for $L=L_\pi(M,n)$ is a principal $K(M,n)$-fibration with a section $\sigma:K\to{L}$. We may view $L$ as the $ex$-$K$ loop space $\overline\Omega{{L_\pi}(M,n+1)}$, with section $\sigma$ and projection $c_L$. Let $\mu:L\times_KL\to {L}$ be the (fibrewise) loop multiplication. Then $\mu(id_L,\sigma{c_L})=\mu(\sigma{c_L},id_L)=id_L$ in $[L;L]_K$. Let $\iota_{M,n}\in{H^n}(L;M)$ be the characteristic element. The function $\theta:[X,L]_K\to{H^n}(X;M)$ given by $\theta(f)=f^*\iota_{M,n}$ is a isomorphism with respect to the addition on $[X,L]_K$ determined by $\mu$. Thus $\theta(id_L)=\iota_{M,n}$, $\theta(\sigma{c_X})=0$ and $\theta(\mu(f,f'))=\theta(f)+\theta(f')$. (See Definition III.6.5 of [@Ba].)
Let $\Gamma_W$ be the quadratic functor of J.H.C.Whitehead and let $\gamma_A:A\to\Gamma_W(A)$ be the universal quadratic function, for $A$ an abelian group. The natural epimorphism from $A$ onto $A/2A=\mathbb{F}_2\otimes{A}$ is quadratic, and so induces a canonical epimorphism from $\Gamma_W(A)$ to $A/2A$. The kernel of this epimorphism is the image of the symmetric square $A\odot{A}$. If $A$ is a $\mathbb{Z}$-torsion-free left $\mathbb{Z}[\pi]$-module the sequence $$0\to{A}\odot{A}\to\Gamma_W({A})\to{A}/2{A}\to0$$ is an exact sequence of left $\mathbb{Z}[\pi]$-modules, when ${A}\odot{A}$ and $\Gamma_W({A})$ have the diagonal left $\pi$-action. Let $A\odot_\pi{A}=\mathbb{Z}\otimes_\pi(A\odot{A})$.
The natural map from $\Pi\odot\Pi$ to $\Gamma_W(\Pi)$ is given by the Whitehead product $[-,-]$, and there is a natural exact sequence of $\mathbb{Z}[\pi]$-modules $$\begin{CD}
\pi_4(X)@> hwz_4>>H_4(\widetilde{X};\mathbb{Z})\to\Gamma_W(\Pi)\to
\pi_3(X)@> hwz_3>>H_3(\widetilde{X};\mathbb{Z})\to0,
\end{CD}$$ where $hwz_q$ is the Hurewicz homomorphism in dimension $q$. (See Chapter 1 of [@Ba'].)
Let $w:\pi\to\{\pm1\}$ be a homomorphism, and let $\varepsilon_w:\mathbb{Z}[\pi]\to\mathbb{Z}^w$ be the $w$-twisted augmentation, given by $w$ on elements of $\pi$. Let $I_w=\mathrm{Ker}(\varepsilon_w)$. If $N$ is a right $\mathbb{Z}[\pi]$-module let $\overline{N}$ denote the conjugate left module determined by $g.n=w(g)n.g^{-1}$ for all $g\in\pi$ and $n\in{N}$. If $M$ is a left $\mathbb{Z}[\pi]$-module let $M^\dagger=\overline{Hom_\pi(M,\mathbb{Z}[\pi])}$. The higher extension modules are naturally right modules, and we set $E^iM=\overline{Ext^i_{\mathbb{Z}[\pi]}(M,\mathbb{Z}[\pi])}$. In particular, $E^0M=M^\dagger$ and $E^i\mathbb{Z}=\overline{H^i(\pi;\mathbb{Z}[\pi])}$.
Let $M$ be a $\mathbb{Z}[\pi]$-module with a finite resolution of length $n$ and such that $E^iM=0$ for $i<n$. Then $Aut_\pi(M)\cong{Aut_\pi(E^nM)}$.
Since $c.d.\pi\leq2$ and $E^iM=0$ for $i<n$ the dual of a finite resolution for $M$ is a finite resolution for $E^nM$. Taking duals again recovers the original resolution, and so $E^nE^nM\cong{M}$. If $f\in{Aut(M)}$ it extends to an endomorphism of the resolution inducing an automorphism $E^nf$ of $E^nM$. Taking duals again gives $E^nE^nf=f$. Thus $f\mapsto{E^nf}$ determines an isomorphism $Aut_\pi(M)\cong{Aut_\pi(E^nM)}$.
In particular, if $\pi$ is a duality group of dimension $n$ over $\mathbb{Z}$ and $\mathcal{D}=H^n(G;\mathbb{Z}[G])$ is the dualizing module then $\overline{\mathcal{D}}=E^n\mathbb{Z}$ and $Aut_\pi(\overline{\mathcal{D}})=\{\pm1\}$. Free groups are duality groups of dimension 1, while if $c.d.\pi=2$ then $\pi$ is a duality group of dimension 2 if and only if it has one end ($E^1\mathbb{Z}=0$) and $E^2\mathbb{Z}$ is $\mathbb{Z}$-torsion-free. (See Chapter III of [@Bi].)
$PD_4$-complexes
================
We assume henceforth that $X$ is a $PD_4$-complex, with orientation character $w=w_1(X)$. Then $\pi$ is finitely presentable and $X$ is homotopy equivalent to $X_o\cup_\phi{e^4}$, where $X_o$ is a complex of dimension at most 3 and $\phi\in\pi_3(X_o)$ [@Wa]. In [@Hi04a] and [@Hi04b] we used such cellular decompositions to study the homotopy types of $PD_4$-complexes. Here we shall follow [@Hi06] instead and rely more consistently on the dual Postnikov approach.
If $\pi$ is infinite the homotopy type of $X$ is determined by $P_3(X)$.
If $X$ and $Y$ are two such $PD_4$-complexes and $h:P_3(X)\to{P_3(Y)}$ is a homotopy equivalence then $hf_{X,3}$ is homotopic to a map $g:X\to{Y}$. Since $\pi$ is infinite $H_4(\widetilde{X};\mathbb{Z})=H_4(\widetilde{Y};\mathbb{Z})=0$. Since $g$ is 4-connected any lift to a map $\tilde{g}:\widetilde{X}\to\widetilde{Y}$ is a homotopy equivalence, by Whitehead’s Theorem, and so $g$ is a homotopy equivalence.
Let $\Pi=\pi_2(X)$, with the natural left $\mathbb{Z}[\pi]$-module structure. In Theorem 11 of [@Hi06] we showed that two $PD_4$-complexes $X$ and $Y$ with the same strongly minimal model and with trivial first $k$-invariants ($k_1(X)=k_1(Y)=0$ in $H^3(\pi;\Pi)$) are homotopy equivalent if and only if $\lambda_X\cong\lambda_Y$. The appeal to [@Ru92] in the second paragraph of the proof is inadequate. Instead we may use the following lemma. (In its application we need only $P_2(X)\simeq{P_2(Y)}$, rather than $k_1(X)=k_1(Y)=0$).
Let $P=P_2(X)$ and $Q=P_2(Z)$, and let $f,g:P\to{Q}$ be $2$-connected maps such that $\pi_i(f)=\pi_i(g)$ for $i=1,2$. Then there is a homotopy equivalence $h:P\to{P}$ such that $gh\sim{f}$.
This follows from Corollaries 2.6 and 2.7 of Chapter VIII of [@Ba].
Let $Z$ be a $PD_4$-complex with a finite covering space $Z_\rho$. Then $Z$ is strongly minimal if and only if $Z_\rho$ is strongly minimal.
Let $\pi=\pi_1(Z)$, $\rho=\pi_1(Z_\rho)$ and $\Pi=\pi_2(Z)$. Then $\pi_2(Z_\rho)\cong\Pi|_\rho$, and so the lemma follows from the observations that since $[\pi:\rho]$ is finite $H^2(\pi;\mathbb{Z}[\pi])|_\rho\cong{H^2}(\rho;\mathbb{Z}[\rho])$ and $Hom_{\mathbb{Z}[\pi]}(\Pi,\mathbb{Z}[\pi])|_\rho\cong
{Hom}_{\mathbb{Z}[\rho]}(\Pi|_\rho,\mathbb{Z}[\rho])$, as right $\mathbb{Z}[\rho]$-modules.
In particular, if $v.c.d.\pi\leq2$ and $\rho$ is a torsion-free subgroup of finite index then $c.d.\rho\leq2$, and so $\chi(Z_\rho)=2\chi(\rho)$, by Theorem 13 of [@Hi06]. Hence $[\pi:\rho]$ divides $2\chi(\rho)$, thus bounding the order of torsion subgroups of $\pi$ if $\chi_{virt}(\pi)=\chi(\rho)/[\pi:\rho]\not=0$.
The next theorem gives a much stronger restriction, under further hypotheses.
Let $Z$ be a strongly minimal $PD_4$-complex and $\pi=\pi_1(Z)$. Suppose that $\pi$ has one end, $v.c.d.\pi=2$ and $E^2\mathbb{Z}$ is free abelian. If $\pi$ has nontrivial torsion then it is a semidirect product $\kappa\rtimes(Z/2Z)$, where $\kappa$ is a $PD_2$-group.
Let $G$ be a torsion-free subgroup of finite index in $\pi$. Then $H^2(\pi;\mathbb{Z}[\pi])|_G=H^2(G;\mathbb{Z}[G])$, by Shapiro’s Lemma, and so $Aut_\pi(E^2\mathbb{Z}))\leq{Aut_G}(E^2\mathbb{Z}))=\{\pm1\}$, by Lemma 1. Therefore the kernel $\kappa$ of the natural action of $\pi$ on $\Pi=\pi_2(Z)\cong{E}^2\mathbb{Z}$ has index $[\pi:\kappa]\leq2$. Suppose that $g\in\pi$ has prime order $p>1$. Then $H_{s+3}(Z/pZ;\mathbb{Z})\cong{H_s}(Z/pZ;\Pi)$ for $s\geq4$, by Lemma 2.10 of [@Hi]. In particular, $Z/pZ\cong{H_4(Z/pZ;\Pi)}$. If $g$ acts trivially on $\Pi$ then $H_4(Z/pZ;\Pi)=0$. Thus we may assume that $\kappa$ is torsion-free, $p=2$, $g$ acts via multiplication by $-1$ and $\pi\cong\kappa\rtimes(Z/2Z)$. Moreover $H_4(Z/pZ;\Pi)=\Pi/2\Pi\cong{Z/2Z}$, and so the free abelian group $E^2\mathbb{Z}\cong\Pi$ must in fact be infinite cyclic. Hence $\kappa$ is a $PD_2$-group [@Bo].
This result settles the question on page 67 of [@Hi].
If $X$ is a $PD_4$-complex with $\pi_1(X)\cong{Z*_m\rtimes{Z/2Z}}$ and $m>1$ then ${\chi(X)>0}$.
Let $\rho=Z*_m$. Then $\chi(X)=\frac12\chi(X_\rho)$. Hence $\chi(X)\geq0$, with equality if and only if $X_\rho$ is strongly minimal, by Theorem 13 of [@Hi06]. In that case $X$ would be strongly minimal, by Lemma 4. Since $\pi$ is solvable $E^2\mathbb{Z}$ is free abelian [@Mih]. Therefore $X$ is not strongly minimal and so $\chi(X)>0$.
$c.d.\pi\leq2$
==============
We now assume that $c.d.\pi\leq2$. In this case we may drop the qualification “strongly", for the following three notions of minimality are equivalent, by Theorem 13 of [@Hi06]:
1. $X$ is strongly minimal;
2. $X$ is minimal with respect to the partial order determined by 2-connected degree-1 maps;
3. $\chi(X)=2\chi(\pi)\leq\chi(Y)$ for $Y$ any $PD_4$-complex with $(\pi_1(Y),w_1(Y))\cong(\pi,w)$.
We have $\Pi\cong{E^2\mathbb{Z}}\oplus{P}$, where $P$ is a finitely generated projective left $\mathbb{Z}[\pi]$-module, and $X$ is minimal if and only if $P=0$. The first $k$-invariant is trivial, since $H^3(\pi;\Pi)=0$, and so $P_2(X)\simeq{L}=L_\pi(\Pi,2)$. Let $\sigma$ be a section for $c_L$. The group $E_\pi(L)$ of based homotopy classes of based self-homotopy equivalences of $L$ which induce the identity on $\pi$ is the group of units of $[L,L]_K$ with respect to composition, and is isomorphic to a semidirect product $H^2(\pi;\Pi)\rtimes{Aut}_\pi(\Pi)$. (See Corollary 8.2.7 of [@Ba].)
The homotopy type of $X$ is determined by $\pi$, $\Pi$, $\pi_3(X)$ and the orbit of $k_2(X)\in{H^4(L;\pi_3(X))}$ under the actions of $E_\pi(L)$ and $Aut_\pi(\pi_3(X))$.
Since these invariants determine $P_3(X)$ this follows from Lemma 2.
It follows from the Whitehead sequence (1) that $H_3(\widetilde{L};\mathbb{Z})=0$ and $H_4(\widetilde{L};\mathbb{Z})\cong\Gamma_W(\Pi),$ since $\widetilde{L}\simeq{K(\Pi,2)}$. Hence the spectral sequence for the universal covering $p_L:\widetilde{L}\to{L}$ gives exact sequences $$0\to{Ext^2_{\mathbb{Z}[\pi]}(\mathbb{Z},\Pi)}={H^2(\pi;\Pi)}\to
{H^2(L;\Pi)}\to
{Hom_{\mathbb{Z}[\pi]}(\Pi,\Pi)}=End_\pi(\Pi)\to0,$$ which is split by $H^2(\sigma;\Pi)$, and $$\begin{CD}
0\to{Ext^2_{\mathbb{Z}[\pi]}(\Pi,\pi_3(X))}\to{H^4(L;\pi_3(X))}@>p_L^*>>
Hom_{\mathbb{Z}[\pi]}(\Gamma_W(\Pi),\pi_3(X))\to0,
\end{CD}$$ since $c.d.\pi\leq2$. The right hand homomorphisms are the homomorphisms induced by $p_L$, in each case. (There are similar exact sequences with coefficients any left $\mathbb{Z}[\pi]$-module $\mathcal{A}$.) The image of $k_2(X)$ in ${Hom}(\Gamma_W(\Pi),\pi_3(X))$ is a representative for $k_2(\widetilde{X})$, and determines the middle homomorphism in the Whitehead sequence (1). If $p_L^*k_2(X)$ is an isomorphism its orbit under the action of $Aut_\pi(\pi_3(X))$ is unique. If $\pi$ has one end the spectral sequence for $p_X:\widetilde{X}\to{X}$ gives isomorphisms $Ext^2_{\mathbb{Z}[\pi]}(\Pi,\mathcal{A}))\cong{H^4(X;\mathcal{A})}$ for any left $\mathbb{Z}[\pi]$-module $\mathcal{A}$, and so $f_{X,2}$ induces splittings $H^4(L;\mathcal{A})\cong{H^4(X;\mathcal{A})}
\oplus{H^4(\Pi,2;\mathcal{A})}^\pi$.
We wish to classify the orbits of $k$-invariants for minimal $PD_4$-complexes. We shall first review the known cases, when $\pi$ is a free group or a $PD_2$-group.
the known cases: free groups and semidirect products
====================================================
The cases with fundamental group a free group are well-understood. A minimal $PD_4$-complex $X$ with $\pi\cong {F(r)}$ free of rank $r$ is either $\#^r(S^1\times{S^3})$, if $w=0$, or $\#^r(S^1\tilde\times{S^3})$, if $w\not=0$. In [@Hi04a] this is established by consideration of the chain complex $C_*(\widetilde{X})$, using the good homological properties of $\mathbb{Z}[F(r)]$. From the present point of view, if $X$ is strongly minimal $\Pi=0$, so $L=K(\pi,1)$, $H^4(L;\pi_3(X))=0$ and $k_2(X)$ is trivial.
If $X$ is not assumed to be minimal $\Pi$ is a free $\mathbb{Z}[\pi]$-module of rank $\chi(X)+2r-2$ and the homotopy type of $X$ is determined by the triple $(\pi,w,\lambda_X)$ [@Hi04a].
The second class of groups for which the minimal models are known are the extensions of $Z$ by finitely generated free groups. If $\pi=F(s)\rtimes_\alpha{Z}$ the minimal models are mapping tori of based self-homeomorphisms of closed 3-manifolds $N=\#^s(S^1\times{S^2})$ (if $w|_\nu=0$) or $\#^s(S^1\tilde\times{S^2})$ (if $w|_\nu\not=0$). (See Chapter 4 of [@Hi].) Two such mapping tori are orientation-preserving homeomorphic if the homotopy classes of the defining self-homeomorphisms are conjugate in the group of based self homotopy equivalences $E_0(N)$. There is a natural representation of $Aut(F(s))$ by isotopy classes of based homeomorphisms of $N$, and $E_0(N)$ is a semidirect product $D\rtimes{Aut(F(s))}$, where $D$ is generated by Dehn twists about nonseparating 2-spheres [@He77]. We may identify $D$ with $(Z/2Z)^s=H^1(F(s);\mathbb{F}_2)$, and then $E_0(N)=(Z/2Z)^s\rtimes{Aut(F(s))}$, with the natural action of $Aut(F(s))$.
Let $f$ be a based self-homeomorphism of $N$, and let $M(f)$ be the mapping torus of $f$. If $f$ has image $(d,\alpha)$ in $E_0(N)$ then $\pi=\pi_1(M(f))\cong{F(s)}\rtimes_\alpha{Z}$. Let $\delta(f)$ be the image of $d$ in $H^2(\pi;\mathbb{F}_2)=
H^1(F(s);\mathbb{F}_2)/(\alpha-1)H^1(F(s);\mathbb{F}_2)$. If $g$ is another based self-homeomorphism of $N$ with image $(d',\alpha)$ and $\delta(g)=\delta(f)$ then $d-d'=(\alpha-1)(e)$ for some $e\in{D}$ and so $(d,\alpha)$ and $(d',\alpha)$ are conjugate. In fact this cohomology group parametrizes such homotopy types; see Theorem 13 for a more general result (subject to some algebraic hypotheses). However in this case we do not yet have explicit invariants enabling us to decide which are the possible minimal models for a given $PD_4$-complex. (It is a remarkable fact that if $\pi=F(s)\rtimes_\alpha{Z}$ and $\beta_1(\pi)\geq2$ then $\pi$ is such a semidirect product for infinitely many distinct values of $s$ [@Bu]. However this does not affect our present considerations.)
It can be shown that if $N$ is a $PD_3$-complex with fundamental group $\nu$ and $\pi=\nu\rtimes_\alpha{Z}$ for some automorphism $\alpha$ the strongly minimal $PD_4$-complexes with fundamental group $\pi$ are the mapping tori of based self homotopy equivalences $h$ of $N$ which induce $\alpha$. However if $\nu$ is not free $\alpha$ may not be nonrealizable, and there may be $PD_4$-complexes with group $\pi$ having no strongly minimal model. (See Theorem 6 of [@Hi06] and the subsequent construction, for the aspherical case.)
the known cases: $PD_2$-groups
==============================
The cases with fundamental group a $PD_2$-group are also well understood, from a different point of view. A minimal $PD_4$-complex $X$ with $\pi$ a $PD_2$-group is homotopy equivalent to the total space of a $S^2$-bundle over a closed aspherical surface. Thus there are two minimal models for each pair $(\pi,w)$, distinguished by their second Wu classes. This follows easily from the fact that the inclusion of $O(3)$ into the monoid of self-homotopy equivalences $E(S^2)$ induces a bijection on components and an isomorphism on fundamental groups. (See Lemma 5.9 of [@Hi].) However it is instructive to consider this case from the present point of view, in terms of $k$-invariants, as we shall extend the following argument to other groups in our main result.
When $\pi$ is a $PD_2$-group and $X$ is minimal $\Pi$ and $\Gamma_W(\Pi)$ are infinite cyclic. The action $u:\pi\to{Aut}(\Pi)$ is given by $u(g)=w_1(\pi)(g)w(g)$ for all $g\in\pi$, by Lemma 10.3 of [@Hi], while the induced action on $\Gamma_W(\Pi)$ is trivial.
Suppose first that $\pi$ acts trivially on $\Pi$. Then $L\simeq{K\times{CP^\infty}}$. Fix generators $t$, $x$, $\eta$ and $z$ for $H^2(\pi;\mathbb{Z})$, $\Pi$, $\Gamma_W(\Pi)$ and $H^2(CP^\infty;\mathbb{Z})=Hom(\Pi,\mathbb{Z})$, respectively, such that $z(x)=1$ and $2\eta=[x,x]$. (These groups are all infinite cyclic, but we should be careful to distinguish the generators, as the Whitehead product pairing of $\Pi$ with itself into $\Gamma_W(\Pi)$ is not the pairing given by multiplication.) Let $t,z$ denote also the generators of ${H^2}(L;\mathbb{Z})$ induced by the projections to $K$ and $CP^\infty$, respectively. Then $H^2(\pi;\Pi)$ is generated by $t\otimes{x}$, while $H^4(L;\Gamma_W(\Pi))$ is generated by $tz\otimes\eta$ and $z^2\otimes\eta$. (Note that $t$ has order 2 if $w_1(\pi)\not=0$.)
The action of $[K,L]_K=[K,CP^\infty]\cong{H^2(\pi;\mathbb{Z})}$ on ${H^2}(L;\mathbb{Z})$ is generated by $t\mapsto{t}$ and $z\mapsto{z+t}$. The action on $H^4(L;\Gamma_W(\Pi))$ is then given by $tz\otimes\eta\mapsto{tz}\otimes\eta$ and $z^2\otimes\eta\mapsto{z^2\otimes\eta+2tz\otimes\eta}$. There are thus two possible $E_\pi(L)$-orbits of $k$-invariants, and each is in fact realized by the total space of an $S^2$-bundle over the surface $K$.
If the action $u$ is nontrivial these calculations go through essentially unchanged with coefficients $\mathbb{F}_2$ instead of $\mathbb{Z}$. There are again two possible $E_\pi(L)$-orbits of $k$-invariants, and each is realized by an $S^2$-bundle space. (See §4 of [@Hi04b] for another account.)
In all cases the orbits of $k$-invariants correspond to the elements of $H^2(\pi;\mathbb{F}_2)=Z/2Z$. In fact the $k$-invariant may be detected by the Wu class. Let $[c]_2$ denote the image of a cohomology class under reduction [*mod*]{} (2). Since $k_2(X)=\pm(z^2\otimes\eta+mtz\otimes\eta)$ has image 0 in $H^4(X;\Pi)$ it follows that $[z]_2^2\equiv{m[tz]_2}$ in $H^4(X;\mathbb{F}_2)$. This holds also if $\pi$ is nonorientable or the action $u$ is nontrivial, and so $v_2(X)=m[z]_2$ and the orbit of $k_2(X)$ determine each other.
If $X$ is not assumed to be minimal its minimal models may be determined from Theorem 7 of [@Hi04b]. The enunciation of this theorem in [@Hi04b] is not correct; an (implicit) quantifier over certain elements of $H^2(X;\mathbb{Z}^u)$ is misplaced and should be “there is" rather than “for all". More precisely, where it has
“[*and let $x\in H^2(X;\mathbb{Z}^u)$ be such that $(x\cup c_X^*\omega_F)[X]=1$. Then there is a $2$-connected degree-$1$ map $h:X\to E$ such that $c_E=c_Xh$ if and only if $(c_X^*)^{-1}w_1(X)$ $=(c_E^*)^{-1}w_1(E)$, $[x]_2^2=0$ if $v_2(E)=0$ and $[x]_2^2=[x]_2\cup c_X^*[\omega_F]_2$ otherwise*]{}"
[it should read]{}
“[*Then there is a $2$-connected degree-$1$ map $h:X\to E$ such that $c_E=c_Xh$ if and only if $(c_X^*)^{-1}w_1(X)=(c_E^*)^{-1}w_1(E)$ and there is an $x\in H^2(X;\mathbb{Z}^u)$ such that $(x\cup c_X^*\omega_F)[X]=1$, with $[x]_2^2=0$ if $v_2(E)=0$ and $[x]_2^2=[x]_2\cup c_X^*[\omega_F]_2$ otherwise*]{}".
[The]{} argument is otherwise correct. Thus if $v_2(\widetilde{X})=0$ the minimal model $Z$ is uniquely determined by $X$; otherwise this is not so. Nevertheless we have the following result. It shall be useful to distinguish three “$v_2$-types" of $PD_4$-complexes:
1. $v_2(\widetilde{X})\not=0$ (i.e., $v_2(X)$ is not in the image of $H^2(\pi;\mathbb{F}_2)$ under $c_X^*$);
2. $v_2(X)=0$;
3. $v_2(X)\not=0$ but $v_2(\widetilde{X})=0$ (i.e., $v_2(X)$ is in $c_X^*(H^2(\pi;\mathbb{F}_2))-\{0\}$);
(This trichotomy is due to Kreck, who formulated it in terms of Stiefel-Whitney classes of the stable normal bundle of a closed 4-manifold.)
If $\pi$ is a $PD_2$-group the homotopy type of $X$ is determined by the triple $(\pi,w,\lambda_X)$ together with its $v_2$-type.
Let $t_2$ generate $H^2(\pi;\mathbb{F}_2)$. Then $\tau=c_X^*t_2\not=0$. If $v_2(X)=m\tau$ and ${p:X\to{Z}}$ is a $2$-connected degree-$1$ map then $v_2(Z)=mc_Z^*t_2$, and so there is an unique minimal model for $X$. Otherwise $v_2(X)\not=\tau$, and so there are elements $y,z\in{H^2}(X;\mathbb{F}_2)$ such that $y\cup\tau\not=y^2$ and $z\cup\tau\not=0$. If $y\cup\tau=0$ and $z^2\not=0$ then $(y+z)\cup\tau\not=0$ and $(y+z)^2=0$. Taking $x=y,z$ or $y+z$ appropriately, we have $x\cup\tau\not=0$ and $x^2=0$, so there is a minimal model $Z$ with $v_2(Z)=0$. In all cases the theorem now follows from the main result of [@Hi06].
In particular, if $C$ is a smooth projective complex curve of genus $\geq1$ and $X=(C\times{S^2})\#\overline{CP^2}$ is a blowup of the ruled surface $C\times{CP^1}=C\times{S^2}$ each of the two orientable $S^2$-bundles over $C$ is a minimal model for $X$. In this case they are also minimal models in the sense of complex surface theory. (See Chapter VI.§6 of [@BPV].) Many of the other minimal complex surfaces in the Enriques-Kodaira classification are aspherical, and hence strongly minimal in our sense. However 1-connected complex surfaces are never minimal in our sense, since $S^4$ is the unique minimal 1-connected $PD_4$-complex and $S^4$ has no complex structure, by a classical result of Wu. (See Proposition IV.7.3 of [@BPV].)
realizing $k$-invariants
========================
We assume now that $\pi$ has one end. Then $c.d.\pi=2$. If $X$ is a $PD_4$-complex with $\pi_1(X)=\pi$ then $H_3(\widetilde{X};\mathbb{Z})=H_4(\widetilde{X};\mathbb{Z})=0$. Hence $k_2(\widetilde{X}):\Gamma_W(\Pi)\to\pi_3(X)$ is an isomorphism, by the Whitehead sequence (1), while $E_\pi(L)\cong{H^2}(\pi;\Pi)\rtimes\{\pm1\}$, by Corollary 8.2.7 of [@Ba] and Lemma 3. Thus if $X$ is minimal its homotopy type is determined by $\pi$, $w$ and the orbit of $k_2(X)$. We would like to find more explicit and accessible invariants that characterize such orbits. We would also like to know which $k$-invariants give rise to $PD_4$-complexes.
Let $P(k)$ denote the Postnikov 3-stage determined by $k\in{H^4}(L;\mathcal{A})$.
Let $\pi$ be a finitely presentable group with $c.d.\pi=2$ and one end, and let $w:\pi\to\{\pm1\}$ be a homomorphism. Let $\Pi=E^2\mathbb{Z}$ and let $k\in{H^4}(L;\Gamma_W(\Pi))$. Then
1. There is a finitely dominated $4$-complex $Y$ with $H_3(\widetilde{Y};\mathbb{Z})=H_4(\widetilde{Y};\mathbb{Z})=0$ and Postnikov $3$-stage $P(k)$ if and only if $p_L^*k$ is an isomorphism and $P(k)$ has finite $3$-skeleton. These conditions determine the homotopy type of $Y$.
2. If $\pi$ is of type $FF$ we may assume that $Y$ is a finite complex.
3. $H_4(Y;\mathbb{Z}^w)\cong\mathbb{Z}$ and there are isomorphisms $\overline{H^p(Y;\mathbb{Z}[\pi])}\cong{H_{4-p}}(Y;\mathbb{Z}[\pi])$ induced by cap product with a generator $[Y]$, for $p\not=2$.
Let $Y$ be a finitely dominated 4-complex with $H_3(\widetilde{Y};\mathbb{Z})=H_4(\widetilde{Y};\mathbb{Z})=0$ and Postnikov $3$-stage $P(k)$. Since $Y$ is finitely dominated it is homotopy equivalent to a 4-complex with finite 3-skeleton, and since $P(k)\simeq{Y}\cup{e^{q\geq5}}$ may be constructed by adjoining cells of dimension at least 5 we may assume that $P(k)$ has finite 3-skeleton. The homomorphism $p_L^*k$ is an isomorphism, by the exactness of the Whitehead sequence (1).
Suppose now that $p_L^*k$ is an isomorphism and $P(k)$ has finite $3$-skeleton. Let $P=P(k)^{[4]}$ and let $C_*=C_*(\widetilde{P})$ be the equivariant cellular chain complex for $\widetilde{P}$. Then $C_q$ is finitely generated for $q\leq3$. Let $B_q\leq{Z_q}\leq{C_q}$ be the submodules of $q$-boundaries and $q$-cycles, respectively. Clearly $H_1(C_*)=0$ and $H_2(C_*)\cong\Pi$, while $H_3(C_*)=0$, since $p_L^*k$ is an isomorphism. Hence there are exact sequences $$0\to{B_1}\to{C_1}\to{C_0}\to\mathbb{Z}\to0$$ and $$\quad 0\to{B_3}\to{C_3}\to{Z_2}\to\Pi\to0.$$ Schanuel’s Lemma implies that $B_1$ is projective, since $c.d.\pi=2$. Hence $C_2\cong{B_1}\oplus{Z_2}$ and so $Z_2$ is finitely generated and projective. It then follows that $B_3$ is also finitely generated and projective, and so $C_4\cong{B_3}\oplus{Z_4}$. Thus $H_4(C_*)=Z_4$ is a projective direct summand of $C_4$.
After replacing $P$ by $P\vee{W}$, where $W$ is a wedge of copies of $S^3$, if necessary, we may assume that $Z_4=H_4(P;\mathbb{Z}[\pi])$ is free. Since $\Gamma_W(\Pi)\cong\pi_3(P)$ the Hurewicz homomorphism from $\pi_4(P)$ to $H_4(P;\mathbb{Z}[\pi])$ is onto. (See Chapter I§3 of [@Ba'].) We may then attach 5-cells along maps representing a basis to obtain a countable 5-complex $Q$ with 3-skeleton $Q^{[3]}=P(k)^{[3]}$ and with $H_q(\widetilde{Q};\mathbb{Z})=0$ for $q\geq3$. The inclusion of $P$ into $P(k)$ extends to a 4-connected map from $Q$ to $P(k)$. Now $C_*(\widetilde{Q})$ is chain homotopy equivalent to the complex obtained from $C_*$ by replacing $C_4$ by $B_3$, which is a finite projective chain complex. It follows from the finiteness conditions of Wall that $Q$ is homotopy equivalent to a finitely dominated complex $Y$ of dimension $\leq4$ [@Wa66]. The homotopy type of $Y$ is uniquely determined by the data, as in Lemma 1.
If $\pi$ is of type $FF$ then $B_1$ is stably free, by Schanuel’s Lemma. Hence $Z_2$ is also stably free. Since dualizing a finite free resolution of $\mathbb{Z}$ gives a finite free resolution of $\Pi=E^2\mathbb{Z}$ we see in turn that $B_3$ must be stably free, and so $C_*(\widetilde{Y})$ is chain homotopy equivalent to a finite free complex. Hence $Y$ is homotopy equivalent to a finite 4-complex [@Wa66].
Let $D_*$ and $E_*$ be the subcomplexes of $C_*$ corresponding to the above projective resolutions of $\mathbb{Z}$ and $\Pi$. (Thus $D_0=C_0$, $D_1=C_1$, $D_2=B_1$ and $D_q=0$ for $q\not=0,1,2$, while $E_2=Z_2$, $E_3=C_3$, $E_4=B_3$ and $E_r=0$ for $r\not=2,3,4$.) Then $C_*(\widetilde{Y})\simeq{D_*}\oplus{E}_*$. (The splitting reflects the fact that $c_Y$ is a retraction, since $k_1(Y)=0$.) Clearly $H^p(Y;\mathbb{Z}[\pi])=H_{4-p}(Y;\mathbb{Z}[\pi])=0$ if $p\not=2$ or 4, while $H^4(Y;\mathbb{Z}[\pi])=E^2\Pi\cong\mathbb{Z}$ and $H_4(Y;\mathbb{Z}^w)=Tor_2(\mathbb{Z}^w;\Pi)\cong
\mathbb{Z}^w\otimes_\pi\mathbb{Z}[\pi]\cong\mathbb{Z}.$ The homomorphism $\varepsilon_{w\#}:H^4(Y;\mathbb{Z}[\pi])\to{H^4}(Y;\mathbb{Z}^w)$ induced by $\varepsilon_w$ is surjective, since $Y$ is 4-dimensional, and therefore is an isomorphism. Hence $-\cap[Y]$ induces isomorphisms in degrees other than 2.
Since $\overline{H^2(Y;\mathbb{Z}[\pi])}\cong{E^2\mathbb{Z}}$, $H_2(Y;\mathbb{Z}[\pi])=\Pi$ and $Hom_\pi(E^2\mathbb{Z},\Pi)\cong{End_\pi}(E^2\mathbb{Z})$ $=\mathbb{Z}$, cap product with $[Y]$ in degree 2 is determined by an integer, and $Y$ is a $PD_4$-complex if and only if this integer is $\pm1$. The obvious question is: what is this integer? Is it always $\pm1$? The complex $C_*$ is clearly chain homotopy equivalent to its dual, but is the chain homotopy equivalence given by slant product with $[Y]$?
There remains also the question of characterizing the $k$-invariants corresponding to Postnikov 3-stages with finite 3-skeleton.
If $\pi$ is either a semidirect product $F(s)\rtimes{Z}$ or the fundamental group of a Haken $3$-manifold $M$ then $\widetilde{K}_0(\mathbb{Z}[\pi])=0$, i.e., projective $\mathbb{Z}[\pi]$-modules are stably free [@Wd78]. (This is not yet known for all torsion-free one relator groups.) In such cases finitely dominated complexes are homotopy finite.
a lemma on cup products
=======================
In our main result (Theorem 13) we shall use a “cup-product" argument to relate cohomology in degrees 2 and 4. Let $G$ be a group and let $\Gamma=\mathbb{Z}[G]$. Let $C_*$ and $D_*$ be chain complexes of left $\Gamma$-modules and $\mathcal{A}$ and $\mathcal{B}$ left $\Gamma$-modules. Using the diagonal homomorphism from $G$ to $G\times{G}$ we may define [*internal products*]{} $$H^p(Hom_\Gamma(C_*,\mathcal{A}))\otimes
{H^q}(Hom_\Gamma(D_*,\mathcal{B}))\to
{H^{p+q}}(Hom_\Gamma(C_*\otimes{D_*},
\mathcal{A}\otimes\mathcal{B}))$$ where the tensor products of $\Gamma$-modules are taken over $\mathbb{Z}$ and have the diagonal $G$-action. (See Chapter XI.§4 of [@CE].) If $C_*$ and $D_*$ are resolutions of $\mathcal{C}$ and $\mathcal{D}$, respectively, we get pairings $$Ext^p_\Gamma(\mathcal{C},\mathcal{A})\otimes
{Ext^q_\Gamma}(\mathcal{D},\mathcal{B})\to
{Ext^{p+q}_\Gamma}
(\mathcal{C}\otimes\mathcal{D},\mathcal{A}\otimes\mathcal{B}).$$ When $\mathcal{A}=\mathcal{B}=\mathcal{D}=\Pi$, $\mathcal{C}=\mathbb{Z}$ and $q=0$ we get pairings $$H^p(\pi;\Pi)\otimes{End}_\pi(\Pi)\to
{Ext}^p_{\mathbb{Z}[\pi]}(\Pi,\Pi\otimes\Pi).$$ If instead $C_*=D_*=C_*(\widetilde{S})$ for some space $S$ with $\pi_1(S)\cong{G}$ composing with an equivariant diagonal approximation gives pairings $$H^p(S;\mathcal{A})\otimes{H^q}(S;\mathcal{B})\to
{H^{p+q}}(S;\mathcal{A}\otimes\mathcal{B}).$$ These pairings are compatible with the universal coefficient spectral sequences $Ext^q_\Gamma(H_p(C_*),\mathcal{A})\Rightarrow
{H^{p+q}}(C^*;\mathcal{A})=H^{p+q}(Hom_\Gamma(C_*,\mathcal{A}))$, etc. We shall call these pairings “cup products", and use the symbol $\cup$ to express their values.
We wish to show that if $c.d.\pi=2$ and $\pi$ has one end the homomorphism $c^2_{\pi,w}:H^2(\pi;\Pi)\to{Ext^2_{\mathbb{Z}[\pi]}(\Pi,\Pi\otimes\Pi)}$ given by cup product with $id_{\Pi}$ is an isomorphism. The next lemma shows that these groups are isomorphic; we state it in greater generality than we need, in order to clarify the hypotheses on the group.
Let $G$ be a group for which the augmentation (left) module $\mathbb{Z}$ has a finite projective resolution $P_*$ of length $n$, and such that $H^j(G;\Gamma)=0$ for $j<n$. Let $\mathcal{D}=H^n(G;\Gamma)$, $w:G\to\{\pm1\}$ be a homomorphism and $\mathcal{B}$ be a left $\Gamma$-module. Then there are natural isomorphisms
1. $h_{\mathcal{B}}:H^n(G;\mathcal{B})\to\mathcal{D}\otimes_G\mathcal{B}$; and
2. $e_{\mathcal{B}}:Ext^n_\Gamma(\overline{\mathcal{D}},\mathcal{B})\to
\mathbb{Z}^w\otimes_G\mathcal{B}=\mathcal{B}/I_w\mathcal{B}$.
Hence $\theta_{\mathcal{B}}=
e_{\overline{\mathcal{D}}\otimes\mathcal{B}}^{-1}h_{\mathcal{B}}:
H^n(G;\mathcal{B})\cong
Ext^n_\Gamma(\overline{\mathcal{D}},
\overline{\mathcal{D}}\otimes\mathcal{B})$ is an isomorphism;
We may assume that $P_0=\Gamma$. Let $Q_j=Hom_\Gamma(P_{n-j},\Gamma)$ and $\partial_i^Q=Hom_\Gamma(\partial^P_{n-j},\Gamma)$. This gives a resolution $Q_*$ for $\mathcal{D}$ by finitely generated projective right modules, with $Q_n=\Gamma$. Let $\eta:{Q}_0\to{\mathcal{D}}$ be the canonical epimorphism. Tensoring $Q_*$ with $\mathcal{B}$ gives (1). Conjugating and applying $Hom_\Gamma(-,\mathcal{B})$ gives (2). Since we may identify ${\mathcal{D}\otimes_G\mathcal{B}}$ with $\mathbb{Z}^w\otimes_G(\overline{\mathcal{D}}\otimes\mathcal{B})$, composition gives an isomorphism $\theta_{\mathcal{B}}=
e_{\overline{\mathcal{D}}\otimes\mathcal{B}}^{-1}h_{\mathcal{B}}:
H^n(G;\mathcal{B})\cong
Ext^n_\Gamma(\overline{\mathcal{D}},
\overline{\mathcal{D}}\otimes\mathcal{B})$.
If $\mathcal{D}$ is $\mathbb{Z}$-torsion free then $G$ is a duality group of dimension $n$, with dualizing module $\mathcal{D}$. (See [@Bi].) It is not known whether all the groups considered in the lemma are duality groups, even when $n=2$.
Let $A:Q_0\otimes_G\overline{\mathcal{D}}\to
{Hom}_\Gamma(P_n,\overline{\mathcal{D}})$ be the homomorphism given by ${A(q\otimes_G\delta)(p)}\!=q(p)\delta$ for all $p\in{P_n}$, $q\in{Q_0}$ and $\delta\in\overline{\mathcal{D}}$, and let $[\xi]\in H^n(G;\overline{\mathcal{D}})$ be the image of $\xi\in{Hom}_\Gamma(P_n,\overline{\mathcal{D}})$. If $\xi=A(q\otimes_G\delta)$ then $h_{\overline{\mathcal{D}}}([\xi])=\eta(q)\otimes\delta$ and $\xi\otimes\eta:
P_n\otimes\overline{Q}_0\to\overline{\mathcal{D}}\otimes\overline{\mathcal{D}}$ represents $[\xi]\cup{id_{\overline{\mathcal{D}}}}$ in $Ext^n_\Gamma(\overline{\mathcal{D}},
\overline{\mathcal{D}}\otimes\overline{\mathcal{D}})$. There is a chain homotopy equivalence $j_*:\overline{Q}_*\to{P_*\otimes\overline{Q}_*}$, since $P_*$ is a resolution of $\mathbb{Z}$. Given such a chain homotopy equivalence, $e_{\overline{\mathcal{D}}\otimes\overline{\mathcal{D}}}
([\xi]\cup{id_{\overline{\mathcal{D}}}})$ is the image of $(\xi\otimes\eta)(j_n(1^*))$, where $1^*$ is the canonical generator of $\overline{Q}_n$, defined by $1^*(1)=1$.
Let $\tau$ be the ($\mathbb{Z}$-linear) involution of $H^n(G;\overline{\mathcal{D}})$ given by $\tau(h_{\overline{\mathcal{D}}}^{-1}(\rho\otimes_G\alpha))=
h_{\overline{\mathcal{D}}}^{-1}(\alpha\otimes_G\rho))$. If $G$ is a $PD_n$-group then $H^n(G;\overline{\mathcal{D}})\cong {Z}$ (if $w=w_1(\pi)$) or $Z/2Z$ (otherwise), and so $\tau$ is the identity.
Suppose now that $c.d.G=2$ and $G$ has one end (i.e., $n=2$). In order to make explicit calculations we shall assume there is a finite 2-dimensional $K(G,1)$-complex with corresponding presentation $\langle{X}\mid{R}\rangle$. Then the free differential calculus gives a free resolution $$0\to{P_2}=\Gamma\langle{p_r^2;r\in{R}}\rangle\to
{P_1}=\Gamma\langle{p_x^1;x\in{X}}\rangle\to{P_0}=\Gamma\to\mathbb{Z}\to0$$ in which $\partial{p_r^2}=\Sigma_{x\in{X}}r_xp_x^1$, where $r_{x} =\frac{\partial{r}}{\partial{x}}$ and $\partial{p_x^1}=x-1$, for $r\in{R}$ and $x\in{X}$. Let $\{q_x^1\}$ and $\{q_r^0\}$ be the dual bases for $\overline{Q}_1$ and $\overline{Q}_0$, respectively. (Thus $q_x^1(p_y^1)=1$ if $x=y$ and 0 otherwise, and $q_r^0(p_s^2)=1$ if $r=s$ and 0 otherwise.) For simplicity of notation we shall write $\bar{g}=w(g)g^{-1}$ for $g\in{G}$. Then $\partial 1^*=\Sigma_{x\in{X}}(\overline{x}-1)q_x^1$ and $\partial{q_x^1}=\Sigma_{r\in{R}}\overline{r_x}q_r^0$. We may write $\overline{r_x}=\Sigma_k{e_{rxk}}r_{xk}$, where $e_{rxk}=\pm1$ and $r_{xk}\in{G}$. Then $r_{xk}-1=\partial(\Sigma_{y\in{X}}\frac{\partial{r_{xk}}}{\partial{y}}p_y^1)$. Define $j_*$ in degrees 0 and 1 by setting $$j_0(q_r^0)=1\otimes{q_r^0}\quad\mathrm{ for }\quad{r\in{R}}\quad\mathrm
{and}$$ $$j_1(q_x^1)=1\otimes{q_x^1}+
\Sigma_{r,k,y}e_{rxk}(\frac{\partial{r_{xk}}}{\partial{y}}p_y^1\otimes
{r_{xk}q_r^0)}
\quad\mathrm{ for }\quad{x\in{X}}.$$ At this point we must specialize further. We shall give several simple examples, where we have managed to determine $j_2(1^*)$. (We do not need formulae for the higher degree terms.) The evidence suggests that if $w$ is trivial we should expect $$j_2(1^*)=1\otimes1^*-\Sigma_{x\in{X}}x^{-1}(p_x^1\otimes{q_x^1})-\Psi$$ where $\Psi=\Sigma_{r\in{R}}u_r(p_r^2\otimes{q_r^0})$ with $u_r$ the inverse of a segment of $r$ and such that $$\partial\Psi=1\otimes\partial1^*-
\Sigma_{x\in{X}}{x}^{-1}((x-1)\otimes{q_x^1})
+\Sigma_{x\in{X}}{x}^{-1}(p_x^1\otimes\Sigma_{r\in{R}}\overline{r_x}q_r^0)
-j_1(\partial1^*)$$ $$=\Sigma_{x,r,k}e_{rxk}[x^{-1}
((p_x^1-\Sigma_y\frac{\partial{r_{xk}}}{\partial{y}}p_y^1)\otimes{r_{xk}}q_r^0)
+(\Sigma_y\frac{\partial{r_{xk}}}{\partial{y}}p_y^1)\otimes{r_{xk}}q_r^0)].$$
[**Examples.**]{}
1. Let $G=F(X)\times{Z}$, with presentation $\langle{t,X}\mid{txt^{-1}x^{-1}~\forall{x}\in{X}}\rangle$. Then we may take $$j_2(1^*)=1\otimes1^*-t^{-1}(p_t^1\otimes{q_t^1})-\Sigma_{x\in{X}}x^{-1}(p_x^1\otimes{q_x^1})
-\Sigma_{x\in{X}}x^{-1}t^{-1}(p_x^2\otimes{q_x^0}).$$
2. Let $G$ be the group with presentation $\langle{a,b}\mid{a^mb^{-n}}\rangle$. Then we may take $$j_2(1^*)=1\otimes1^*-a^{-1}(p_a^1\otimes{q_a^1})-b^{-1}(p_b^1\otimes{q_b^1})
-a^{-m}(p^2\otimes{q^0}).$$
3. Let $G$ be the orientable $PD_2$-group of genus 2, with presentation
$\langle{a,b,c,d}\mid{aba^{-1}b^{-1}cdc^{-1}d^{-1}}\rangle$. Then we may take $$j_2(1^*)=1\otimes1^*-\Sigma_{x\in{X}}
\overline{x}(p_x^1\otimes{q_x^1})-bab^{-1}a^{-1}(p^2\otimes{q^0}).$$
4. Let $G=Z*_m$ be the group with presentation $\langle{a,t}\mid{tat^{-1}a^{-m}}\rangle$, for $m\not=0$. Then we may take $$j_2(1^*)=1\otimes1^*-
a^{-1}(p_a^1\otimes{q_a^1})-t^{-1}(p_t^1\otimes{q_t^1})
-a^{-1}t^{-1}(p^2\otimes{q^0}).$$
In each of these cases we find that $[\xi]\cup{id_{\overline{\mathcal{D}}}}=
-\theta_{\overline{\mathcal{D}}}(\tau([\xi]))$ for $\xi\in{H^2}(\pi;\overline{\mathcal{D}})$. Similar formulae apply for $n\leq1$, i.e., for free groups of finite rank $r\geq0$.
If $H$ is a subgroup of finite index in $G$ and $\mathcal{A}$ is a left $\mathbb{Z}[G]$-module then Shapiro’s Lemma gives isomorphisms $H^n(G;\mathcal{A})\cong{H^n}(H;\mathcal{A}|_H)$. Thus if $G$ satisfies the hypotheses of Lemma 10 $\overline{\mathcal{D}}|_H$ is the corresponding module for $H$. Further applications of Shapiro’s Lemma imply that cup product with $id_{\overline{\mathcal{D}}}$ is an isomorphism for $(G,w)$ if and only if it is so for $(H,w|_H)$. In particular, Examples (1)–(3) imply that $c^2_{\pi,w}$ is an isomorphism for all torus knot groups and $PD_2$-groups, and all orientation characters $w$.
the action of $E_\pi(L)$
========================
In this section we shall attempt to study the action of $E_\pi(L)$ on the set of possible $k$-invariants for a minimal $PD_4$-complex by extending the argument sketched above for the case of $PD_2$-groups. We believe that the restrictions we impose here on the pair $(\pi,w)$ shall ultimately be seen to be unnecessary.
Our argument shall involve relating the algebraic and homotopical (obstruction-theoretic) interpretations of cohomology classes. We shall use the following special case of a result of Tsukiyama [@Tsu]; we give only the part that we need below.
There is an exact sequence $0\to{H^2}(\pi;\Pi)\to{E_\pi(L)}\to{Aut_\pi(\Pi)}\to0.$
Let $\theta:[K,L]_K\to{H^2}(\pi;\Pi)$ be the isomorphism given by $\theta(s)=s^*\iota_{\Pi,2}$, and let $\theta^{-1}(\phi)=s_\phi$ for $\phi\in{H^2}(\pi;\Pi)$. Then $s_\phi$ is a homotopy class of sections of $c_L$, $s_0=\sigma$ and $s_{\phi+\psi}=\mu(s_\phi,s_\psi)$, while $\phi=s_\phi^*\iota_{\Pi,2}$. (Recall that $\mu:L\times_KL\to {L}$ is the fibrewise loop multiplication.)
Let $h_\phi=\mu(s_\phi{c_L},id_L)$. Then $c_Lh_\phi=c_L$ and so $h_\phi\in[L;L]_K$. Clearly $h_0=\mu(\sigma{c_L},id_L)=id_L$ and $h_\phi^*\iota_{\Pi,2}=\iota_{\Pi,2}+c_L^*\phi\in{H^2}(L;\Pi)$. We also see that $$h_{\phi+\psi}=\mu(\mu(s_\phi,s_\psi){c_L},id_L)=
\mu(\mu(s_\phi{c_L},s_\psi{c_L}),id_L)=
\mu(s_\phi{c_L},\mu(s_\psi{c_L},id_L))$$ (by homotopy associativity of $\mu$) and so $$h_{\phi+\psi}=\mu(s_\phi{c_L},h_\psi)=
\mu(s_\phi{c_L}h_\psi,h_\psi)=h_\phi{h_\psi}.$$ Therefore $h_\phi$ is a homotopy equivalence for all $\phi\in{H^2}(\pi;\Pi)$, and $\phi\mapsto{h_\phi}$ defines a homomorphism from ${H^2}(\pi;\Pi)$ to ${E_\pi(L)}$.
The lift of $h_\phi$ to the universal cover $\widetilde{L}$ is (non-equivariantly) homotopic to the identity, since the lift of $c_L$ is (non-equivariantly) homotopic to a constant map. Therefore $h_\phi$ acts as the identity on $\Pi$. The homomorphism $h:\phi\mapsto{h_\phi}$ is in fact an isomorphism onto the kernel of the action of ${E_\pi(L)}$ on $\Pi=\pi_2(L)$ [@Tsu].
Note also that we may view elements of $[K,L]_K$ (etc.) as $\pi$-equivariant homotopy classes of $\pi$-equivariant maps from $\widetilde{K}$ to $\widetilde{L}$.
There is an exact sequence $\Pi\odot_\pi\Pi\to
\mathbb{Z}\otimes_\pi\Gamma_W(\Pi)\to{H^2}(\pi;\mathbb{F}_2)\to0.$ If $\Pi\odot_\pi\Pi$ is $2$-torsion-free this sequence is short exact.
Since $\pi$ is finitely presentable $\Pi$ is $\mathbb{Z}$-torsion-free [@GM], and so the natural map from $\Pi\odot\Pi$ to $\Gamma_W(\Pi)$ is injective. Applying $\mathbb{Z}\otimes_\pi-$ to the exact sequence $$0\to\Pi\odot\Pi\to\Gamma_W(\Pi)\to\Pi/2\Pi\to0$$ gives the above sequence, since $\mathbb{Z}\otimes_\pi\Pi/2\Pi\cong\Pi/(2,I_w)\Pi\cong
{H^2}(\pi;\mathbb{F}_2)$. The kernel on the left in this sequence is the image of the 2-torsion group $Tor_1^{\mathbb{Z}[\pi]}(\mathbb{Z},\Pi/2\Pi)$.
Let $\pi$ be a finitely presentable group such that $c.d.\pi=2$ and $\pi$ has one end. Let $\Pi=E^2\mathbb{Z}$ and $\beta=\beta_2(\pi;\mathbb{F}_2)$. Assume that $c^2_{\pi,w}$ is surjective and $\mathbb{Z}^w\otimes_\pi\Gamma_W(\Pi)$ is $2$-torsion-free. Then there are at most $2^\beta$ orbits of $k$-invariants of minimal $PD_4$-complexes with Postnikov $2$-stage $L$ under the actions of $E_\pi(L)$ and $Aut_\pi(\Gamma_W(\Pi))$.
Let $\phi\in{H^2}(\pi;\Pi)$ and let $s_\phi\in[K,L]_K$ and $h_\phi\in[L,L]_K$ be as defined in Lemma 11. Let $M={L_\pi}(\Pi,3)$ and let $\overline\Omega:{[M,M]_K}\to[L,L]_K$ be the loop map. Since $c.d.\pi=2$ we have $[M,M]_K\cong{H^3}(M;\Pi)=End_\pi(\Pi)$. Let $g\in[M,M]_K$ have image $[g]=\pi_3(g)\in{End_\pi(\Pi)}$ and let $f=\overline\Omega{g}$. Then $\omega([g])=f^*\iota_{\Pi,2}$ defines a homomorphism $\omega:End_\pi(\Pi)\to{H^2}(L;\Pi)$ such that $p_L^*\omega([g])=[g]$ for all $[g]\in{End_\pi(\Pi)}$. Moreover $f\mu=\mu(f,f)$, since $f=\overline\Omega{g}$, and so $fh_\phi=\mu(fs_\phi{c_L},f)$. Hence $h_\phi^*\xi=\xi+c_L^*s_\phi^*\xi$ for $\xi=\omega([g])=f^*\iota_{\Pi,2}$.
Naturality of the isomorphisms $H^2(X;\mathcal{A})\cong[X,L_\pi(\mathcal{A},2)]_K$ for $X$ a space over $K$ and $\mathcal{A}$ a left $\mathbb{Z}[\pi]$-module implies that $$s_\phi^*\omega([g])=[g]_\#s_\phi^*\iota_{\Pi,2}=[g]_\#\phi$$ for all $\phi\in{H^2}(\pi;\Pi)$ and $g\in[M,M]_K$. (See Chapter 5.§4 of [@Ba0].) If $u\in{H^2}(\pi;\mathcal{A})$ then $h_\phi^*c_L^*(u)=c_L^*(u)$, since $c_Lh_\phi=c_L$. The homomorphism induced on the quotient $H^2(L;\mathcal{A})/c_L^*H^2(\pi;\mathcal{A})\cong
{Hom_{\mathbb{Z}[\pi]}(\Pi,\mathcal{A})}$ by $h_\phi$ is also the identity, since the lifts of $h_\phi$ are (non-equivariantly) homotopic to the identity in $\widetilde{L}$.
Taking $\mathcal{A}=\Pi$ we obtain a homomorphism $\delta_\phi:End_\pi(\Pi)\to{H^2}(\pi;\Pi)$ such that $h_\phi^*(\xi)=\xi+c_L^*\delta_\phi(p_L^*\xi)$ for all $\xi\in{H^2}(L;\Pi)$. Since $p_L^*\delta_\phi=0$ and $h_{\phi+\psi}=h_\phi{h_\psi}$ it follows that $\delta_\phi$ is additive as a function of $\phi$. If $g\in[M,M]_K$ and $\phi=\rho\otimes_\pi\alpha\in{H^2}(\pi;\mathbb{Z}[\pi])\otimes_\pi\Pi$ then $$\delta_\phi([g])=\delta_\phi(p_L^*\omega([g]))=
s_\phi^*\omega[g]=\rho\otimes_\pi[g](\alpha).$$ The automorphism of $H^4(L;\mathcal{A})$ induced by $h_\phi$ preserves the subgroup $Ext^2_{\mathbb{Z}[\pi]}(\Pi,\mathcal{A})$ and induces the identity on the quotient $Hom_\pi(\Gamma_W(\Pi),\mathcal{A})$. Taking $\mathcal{A}=\Gamma_W(\Pi)$ we obtain a homomorphism $f_\phi=h_\phi^*-id$ from $H^4(L;\Gamma_W(\Pi))$ to $Ext^2_{\mathbb{Z}[\pi]}(\Pi,\Gamma_W(\Pi))$.
When $S=L$, $\mathcal{A}=\mathcal{B}=\Pi$, and $p=q=2$ the construction of §7 gives a cup product pairing of $H^2(L;\Pi)$ with itself with values in ${H^4(L;\Pi\otimes\Pi)}$. Since $c.d.\pi=2$ this pairing is trivial on the image of $H^2(\pi;\Pi)\otimes{H^2(\pi;\Pi)}$. The maps $c_L$ and $\sigma$ induce a splitting $H^2(L;\Pi)\cong{H^2(\pi;\Pi)}\oplus{End_\pi(\Pi)}$, and this pairing restricts to the cup product pairing of $H^2(\pi;\Pi)$ with $End_\pi(\Pi)$ with values in ${Ext^2_{\mathbb{Z}[\pi]}}(\Pi,\Pi\otimes\Pi)$. We may also compose with the natural homomorphisms from $\Pi\otimes\Pi$ to $\Pi\odot\Pi$ and $\Gamma_W(\Pi)$ to get pairings with values in $H^4(L;\Pi\odot\Pi)$ and $H^4(L;\Gamma_W(\Pi))$.
Since $h_\phi^*(\xi\cup\xi')=h_\phi^*\xi\cup{h_\phi^*}\xi'$ we have also $$f_\phi(\xi\cup\xi')=
(c_L^*\delta_\phi(p_L^*\xi'))\cup\xi
+(c_L^*\delta_\phi(p_L^*\xi))\cup\xi'$$ for all $\xi,\xi'\in{H^2}(L;\Pi)$. In particular, if $\xi\in{H^2(\pi;\Pi)}$ then $f_\phi(\xi\cup\xi')=0$, and so $f_\phi(c^2_{\pi,w}(\xi))=0$. Since $c^2_{\pi,w}$ is surjective and the quotient of $Ext^2_{\mathbb{Z}[\pi]}(\Pi,\Gamma_W(\Pi))$ by the image of $Ext^2_{\mathbb{Z}[\pi]}(\Pi,\Pi\otimes\Pi)$ has exponent 2, by Lemma 12, it follows that $2f_\phi=0$ on $Ext^2_{\mathbb{Z}[\pi]}(\Pi,\Gamma_W(\Pi))$.
On passing to $\widetilde{L}\simeq{K(\Pi,2)}$ we find that $$p_L^*(\xi\cup\xi')(\gamma_\Pi(x))=p_L^*\xi(x)\odot{p_L^*}\xi'(x)$$ for all $\xi,\xi'\in{H^2(L;\Pi)}$ and $x\in\Pi$. (To see this, note that the inclusion of $x$ determines a map from $CP^\infty$ to $K(\Pi,2)$, since $[CP^\infty,K(\Pi,2)]=Hom(\mathbb{Z},\Pi)$. Hence we may use naturality of cup products to reduce to the case when $K(\Pi,2)=CP^\infty$ and $x$ is a generator of $\Pi=\mathbb{Z}$.) In particular, if $\Xi=\sigma^*id_\Pi\cup\sigma^*id_\Pi$ then $$p_L^*(\Xi)=2id_{\Gamma_W(\Pi)}\quad\mathrm{ and}\quad
f_\phi(\Xi)=2(c_L^*\phi)\cup{id_\Pi}=2c^2_{\pi,w}(\phi).$$
If $k=k_2(X)$ for some minimal $PD_4$-complex $X$ with $\pi_1(X)\cong\pi$ then $p_L^*k$ is an isomorphism. After composition with an automorphism of $\Gamma_W(\Pi)$ we may assume that $p_L^*k=id_{\Gamma_W(\Pi)}$, and so $p_L^*(2k-\Xi)=0$. Therefore $4(f_\phi(k)-c^2_{\pi,w}(\phi))=2f_\phi(2k-\Xi)=0$. Since $\mathbb{Z}^w\otimes_\pi\Gamma_W(\Pi)$ is 2-torsion-free $f_\phi(k)=c^2_{\pi,w}(\phi)$. Since $c^2_{\pi,w}$ is surjective the orbit of $k$ under the action of $E_\pi(L)$ corresponds to an element of $Ext^2_{\mathbb{Z}[\pi]}(\Pi,\Pi/2\Pi)\cong{H^2}(\pi;\mathbb{F}_2)$, and so there are at most $2^\beta$ possibilities.
If $\pi$ is a $PD_2$-group and $w=w_1(\pi)$ then $L=K\times{CP^\infty}$, $p_L^*k=z^2\otimes\eta$ and $f_{t\otimes{x}}(k)=2tz\otimes\eta
=c^2_{\pi,w}(t\otimes{x})$. (However $\mathbb{Z}^w\otimes_\pi\Gamma_W(\Pi)=Z/2Z$ if $w\not=1$.) The hypotheses also hold if $\pi\cong {Z*_m}$ for $m$ even and $w=1$. (See §7 and §9.)
If we could show that $h_\phi$ is the identity on the image of $Ext^2_{\mathbb{Z}[\pi]}(\Pi,\Gamma_W(\Pi))$ it would follow that $f_\phi$ is additive as a function of $\phi$. We could then relax the hypothesis on 2-torsion to require only that the image of $\Pi\odot_\pi\Pi$ in $\mathbb{Z}^w\otimes_\pi\Gamma_W(\Pi)$ be $2$-torsion-free. (The latter condition holds for all $PD_2$-groups and orientation characters $w$, and is easier to check; see Lemma 16 below for the case $\pi=Z*_m$.)
If $H^2(\pi;\mathbb{F}_2)=0$, $c^2_{\pi,w}$ is surjective and $\Pi\odot_\pi\Pi$ is $2$-torsion-free there is an unique minimal $PD_4$-complex realizing $(\pi,w)$. Hence two $PD_4$-complexes $X$ and $Y$ with $\pi_1(X)\cong\pi_1(Y)\cong\pi$ are homotopy equivalent if and only if there is an isomorphism $\theta:\pi_1(X)\to\pi_1(Y)$ such that $w_1(X)=w_1(Y)\circ\theta$ and an isometry of homotopy intersection pairings $\lambda_X\cong\theta^*\lambda_Y$.
We note that we do not yet have explicit invariants that might distinguish two such minimal $PD_4$-complexes when $\beta>0$. Does $v_2(X)$ suffice when $\beta=1$?
verifying the torsion condition for $Z*_{m}$
============================================
If $\pi$ is finitely presentable and $c.d.\pi=2$ but $\pi$ is not a $PD_2$-group then $H^2(\pi;\mathbb{Z}[\pi])$ is not finitely generated [@Fa74]. Whether it must be free abelian remains an open question. We shall verify this for the groups of most interest to us here.
Let $\pi$ have one end, and be either a semidirect product ${F(s)\rtimes{Z}}$, a torsion-free one-relator group or the fundamental group of a $3$-manifold $M$ with nonempty aspherical boundary. Then there is a finite $2$-dimensional $K(\pi,1)\!$-complex and $\Pi=E^2_\pi\mathbb{Z}$ is free abelian. In particular, $\pi$ is a $2$-dimensional duality group.
If $\pi=\nu\rtimes{Z}$, where $\nu\cong{F(s)}$ is a nontrivial finitely generated free group, then $s\geq1$, since $\pi$ has one end. We may realize $K(\pi,1)$ as a mapping torus of a self-map of $\vee^sS^1$. This is clearly a finite aspherical 2-complex. An LHS spectral sequence argument shows that $\Pi|_\nu=E^2_\pi\mathbb{Z}|_\nu\cong{E^1_\nu\mathbb{Z}}$, which is free abelian.
If $\pi$ has a one-relator presentation and is torsion-free the 2-complex associated to the presentation is aspherical (and clearly finite). It is shown in [@MT] that one-relator groups are semistable at infinity and hence that $\Pi$ is free abelian.
Let $M$ be a $3$-manifold. If $\pi=\pi_1(M)$ has one end then $H_2(\widetilde{M},\partial\widetilde{M};\mathbb{Z})=
H^1(M;\mathbb{Z}[\pi])=0$, by Poincaré duality. Hence $H_1(\partial\widetilde{M};\mathbb{Z})=0$. If $\partial{M}$ is a union of aspherical surfaces it follows that $H_2(\partial\widetilde{M};\mathbb{Z})=0$. Hence $H_*(\widetilde{M};\mathbb{Z})=0$ for $*>0$ and so $M$ is aspherical. If moreover $\partial{M}$ is nonempty $M$ retracts onto a finite 2-complex. The group $\Pi=\overline{H^2(M;\mathbb{Z}[\pi])}$ is free abelian since ${H^2}(M;\mathbb{Z}[\pi])\cong
{H_1}(\widetilde{M},\partial\widetilde{M};\mathbb{Z})$ is the kernel of the augmentation $H_0(\partial\widetilde{M};\mathbb{Z})\to{H_0}(\widetilde{M};\mathbb{Z})$.
Since $H^s(\pi;\mathbb{Z}[\pi])=0$ for $s\not=2$ and $H^2(\pi;\mathbb{Z}[\pi])$ is torsion-free $\pi$ is a 2-dimensional duality group [@Bi].
The class of groups covered by this lemma includes all $PD_2$-groups, classical knot groups and solvable $HNN$ extensions $Z*_m$ other than $Z$. Whether every finitely presentable group $\pi$ of cohomological dimension 2 has a finite 2-dimensional $K(\pi,1)$-complex and is semistable at infinity remain open questions.
Let $\pi=Z*_m$, $w=1$ and $\Pi=E^2\mathbb{Z}$. Then $\Pi\odot_\pi\Pi$ is torsion-free.
The group $\pi=Z*_m$ has a one-relator presentation $\langle{a,t}\mid{ta=a^mt}\rangle$ and is also a semidirect product $Z[\frac1m]\rtimes{Z}$. Let $R=\mathbb{Z}[\pi]$ and $D=\mathbb{Z}[a_n]/(a_{n+1}-a_n^m)$, where $a_n=t^nat^{-n}$ for $n\in\mathbb{Z}$. Then $R=\oplus_{n\in\mathbb{Z}}{t^n}D$ is a twisted Laurent extension of the commutative domain $D$.
On dualizing the Fox-Lyndon resolution of the augmentation module we see that ${H^2}(\pi;\mathbb{Z}[\pi])\cong{R}/(a^m-1, t-\mu_m)R$ and so $\Pi\cong{R}/R(a^m-1,t\mu_m-1)$, where $\mu_m=\Sigma_{i=0}^{i=m-1}a^i$. Let $E=D/(a^m-1)$ and let $a_{k/m^n}$ be the image of $a_{-n}^k$ in $E$. Then $E$ is freely generated as an abelian group by $\{a_x\mid{x}\in{J}\}$, where $J=\{\frac{k}{m^n}\mid0<n,~0\leq{k}<m^{n+1}\}$. Since $ta_{1-n}^k=a_{-n}^kt$ we have $\Pi\cong\oplus_{n\in\mathbb{Z}}t^nE/\sim$, where $t^ma_{x}\sim{t^m}a_xt\mu_m=t^{m+1}\mu_ma_{x/m}$.
Therefore $\Pi\odot\Pi\cong\oplus_{m\in\mathbb{Z}}(t^mE\odot{t^mE})/\sim$, where $$t^ma_x\odot{t^ma_y}\sim{t^{m+1}}\mu_ma_{x/m}\odot{t^{m+1}}\mu_ma_{y/m}.$$ Setting $z=y-x$ this gives $$t^ma_x(1\odot{a_z})\sim{t^{m+1}}a_{x/m}(\mu_m\odot\mu_m{a_{z/m}})
={t^{m+1}}a_{x/m}
(\Sigma_{i,j=0}^{i,j=m-1}a^i(1\odot{a^{j-i}a_{z/m}})).$$ Define a function $f:E\to\Pi\odot\Pi$ by $f(e)=1\odot{e}=e\odot1$ for $e\in{E}$. Then $f(a_x)=a_xf(a_{m-x})$ for all $x$, since $a_x\odot1=a_x(1\odot{a_{m-x}})$. On factoring out the action of $\pi$ we see that $$\Pi\odot_\pi\Pi\cong{E}/
(a_z-a_{m-z},a_z-m(\Sigma_{k=0}^{k=m-1}a^ka_{z/m})~\forall{z\in{J}}).$$ (In simplifying the double sum we may set $k=j-i$ for $j\geq{i}$ and $k=j+m-i$ otherwise, since $a^ma_{z/m}=a_{z/m}$ for all $z$.) Thus $\Pi\odot_\pi\Pi$ is a direct limit of free abelian groups and so is torsion-free.
If moreover $\mathbb{Z}\otimes_\pi\Pi/2\Pi=H^2(\pi;\mathbb{F}_2)=0$ then $\Pi\odot_\pi\Pi\cong\mathbb{Z}\otimes_\pi\Gamma_W(\Pi)$. Thus if $\pi=Z*_m$ with $m$ even $\mathbb{Z}\otimes_\pi\Gamma_W(\Pi)$ is torsion-free. This group is also torsion-free for $Z*_1=Z^2$; does this hold for all $m$?
applications to 2-knots
=======================
Suppose that $\pi$ is either the fundamental group of a finite graph of groups, with all vertex groups $Z$, or is square root closed accessible, or is a classical knot group. (This includes all $PD_2$-groups, semidirect products $F(s)\rtimes{Z}$ and the solvable groups $Z*_m$.) Then $L_5(\pi,w)$ acts trivially on the $s$-cobordism structure set $S^s_{TOP}(M)$ and the surgery obstruction map $\sigma_4(M):[M,G/TOP]\to{L_4(\pi,w)}$ is onto, for any closed 4-manifold $M$ realizing $(\pi,w)$. (See Lemma 6.9 and Theorem 17.8 of [@Hi].) Thus there are finitely many $s$-cobordism classes within each homotopy type of such manifolds.
In particular, $Z*_m$ has such a graph-of-groups structure and is solvable, so the 5-dimensional TOP $s$-cobordism theorem holds. Thus if $m$ is even the closed orientable 4-manifold $M$ with $\pi_1(M)\cong{Z*_m}$ and $\chi(M)=0$ is unique up to homeomorphism. If $m=1$ there are two such homeomorphism types, distinguished by the second Wu class $v_2(M)$.
Let $\pi$ be a finitely presentable group with $c.d.\pi=2$. If $H_1(\pi;\mathbb{Z})=\pi/\pi'\cong{Z}$ and $H_2(\pi;\mathbb{Z})=0$ then $\mathrm{def}(\pi)=1$, by Theorem 2.8 of [@Hi] If moreover $\pi$ is the normal closure of a single element then $\pi$ is the group of a 2-knot $K:S^2\to{S^4}$. (If the Whitehead Conjecture is true every knot group of deficiency 1 has cohomological dimension at most 2.) Since $\pi$ is torsion-free it is indecomposable, by a theorem of Klyachko. Hence $\pi$ has one end.
Let $M=M(K)$ be the closed 4-manifold obtained by surgery on the 2-knot $K$. Then $\pi_1(M)\cong\pi=\pi{K}$ and $\chi(M(K))=\chi(\pi)=0$, and so $M$ is a minimal model for $\pi$. If $\pi=F(s)\rtimes{Z}$ the homotopy type of $M$ is determined by $\pi$, as explained in §4 above. If $K$ is a ribbon 2-knot it is -amphicheiral and is determined (up to reflection) by its exterior. It follows that a fibred ribbon 2-knot is determined up to $s$-concordance and reflection by its fundamental group together with the conjugacy class of a meridian. (This class of 2-knots includes all Artin spins of fibred 1-knots.)
A stronger result holds for the group $\pi=Z*_2$. This is the group of Fox’s Example 10, which is a ribbon 2-knot [@Fo62]. In this case $\pi$ determines the homotopy type of $M(K)$, by Theorem 13. Since metabelian knot groups have an unique conjugacy class of normal generators (up to inversion) Fox’s Example 10 is the unique 2-knot (up to TOP isotopy and reflection) with this group. This completes the determination of the 2-knots with torsion-free elementary amenable knot groups. (The others are the unknot and the Cappell-Shaneson knots. See Chapter 17.§6 of [@Hi] for more on 2-knots with $c.d.\pi=2$.)
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---
abstract: 'A complete set of QCD sum rules for the magnetic moments of decuplet baryons are derived using the external field method. They are analyzed thoroughly using a Monte-Carlo based procedure. Valid sum rules are identified under the criteria of OPE convergence and ground state dominance and their predictions are obtained. The performances of these sum rules are further compared and a favorable sum rule is designated for each member. Correlations between the input and the output parameters are examined and large sensitivities to the quark condensate magnetic susceptibility $\chi$ are found. Using realistic estimates of the QCD input parameters, the uncertainties on the magnetic moments are found relatively large and they can be attributed mostly to the poorly-known $\chi$. It is shown that the accuracy can be improved to the 30% level, provided the uncertainties in the QCD input parameters can be determined to the 10% level. The computed magnetic moments are consistent with existing data. Comparisons with other calculations are made.'
address: |
Nuclear Physics Laboratory, Department of Physics, University of Colorado,\
Boulder, CO 80309-0446
author:
- 'Frank X. Lee'
title: |
Determination of Decuplet Baryon Magnetic Moments\
from QCD Sum Rules
---
Introduction {#intro}
============
The QCD sum rule method [@SVZ79] has proven a powerful tool in revealing the deep connection between hadron phenomenology and QCD vacuum structure via a few condensate parameters. The method has been successfully applied to a variety of problems to gain a field-theoretical understanding into the structure of hadrons. Calculations of the nucleon magnetic moments in the approach were first carried out in Refs. [@Ioffe84] and [@Balitsky83]. They were later refined and extended to the entire baryon octet in Refs. [@Chiu86; @Pasupathy86; @Wilson87; @Chiu87]. On the other hand, the magnetic moments of decuplet baryons were less well studied within the same approach. There were previous, unpublished reports in Ref. [@Bely84] on $\Delta^{++}$ and $\Omega^-$ magnetic moments. The magnetic form factor of $\Delta^{++}$ in the low $Q^2$ region was calculated based on a rather different technique [@Bely93]. In recent years, the magnetic moment of $\Omega^-$ has been measured with remarkable accuracy [@Wallace95]: $\mu_{\scriptscriptstyle \Omega^-}=-2.02\pm0.05\;\mu_{\scriptscriptstyle N}$. The magnetic moment of $\Delta^{++}$ has also been extracted from pion bremsstrahlung [@Bosshard91]: $\mu_{\scriptscriptstyle \Delta^{++}}=4.5\pm 1.0\;\mu_{\scriptscriptstyle N}$. In an earlier work [@Heller87], the magnetic moment of $\Delta^{0}$ extracted from $\pi^-p$ bremsstrahlung data was found to be consistent with $\mu_{\scriptscriptstyle \Delta^{0}}=0$. The experimental information provides new incentives for theoretical scrutiny of these observables.
In this work, we present a systematic, independent calculation of the magnetic moments for the entire decuplet family in the QCD sum rule approach. The goal is two-fold. First, we want to find out if the approach can be successfully applied to these observables by carrying out an explicit calculation. Second, we want to achieve some realistic understanding of the uncertainties involved in such a determination by employing a Monte-Carlo based analysis procedure. This would help us assess the limitations and find ways for improvements.
We will show that both goals are achieved in this work. The entire calculation is more challenging than the octet case due to the more complex spin structure of spin-3/2 particles. One has to overcome enormous amount of algebra to arrive at the final results. But conceptually it presents no apparent difficulties. Particular attention is paid to the complete treatment of the phenomenological representation, which leads to the isolation of the tensor structures from which the QCD sum rules for the magnetic moments can be constructed. Flavor symmetry breakings in the strange quark are treated consistently across the decuplet family. The success also hinges upon a new analysis of the two-point functions [@Lee97a], which provides more accurately determined current couplings for normalization. Part of the results on $\Delta^{++}$ and $\Omega^-$ have been communicated in a letter [@Lee97b].
Magnetic moments of decuplet baryons have also been studied in various other methods, including lattice QCD [@Derek92], chiral perturbation theory [@Butler94], Bethe-Salpeter formalism [@Mitra84], non-relativistic quark model [@PDG92], relativistic quark models [@Schlumpf93; @Capstick96; @Linde95; @Chao90; @Georgi83], chiral quark-soliton model [@Kim97], chiral bag model [@Hong94], cloudy bag model [@Kriv87], Skyrme model [@Kim89]. A comparison will be made with some of the calculations and with existing data.
Sec. \[meth\] deals with the derivation of the QCD sum rules. Sec. \[ana\] discusses the Monte-Carlo analysis procedure. Sec. \[res\] gives the results and discussions. Sec. \[sumcon\] contains the conclusions. The Appendix collects the QCD sum rules derived.
Method {#meth}
======
Consider the time-ordered two-point correlation function in the QCD vacuum in the presence of a [*constant*]{} background electromagnetic field $F_{\mu\nu}$: $$\Pi_{\alpha\beta}(p)=i\int d^4x\; e^{ip\cdot x}
\langle 0\,|\, T\{\;\eta_{\alpha}(x)\,
\bar{\eta}_{\beta}(0)\;\}\,|\,0\rangle_F,
\label{cf2pt}$$ where $\eta_{\alpha}$ is the interpolating field for the propagating baryon. The subscript $F$ means that the correlation function is to be evaluated with an electromagnetic interaction term added to the QCD Lagrangian: $${\cal L}_I = - A_\mu J^\mu,$$ where $A_\mu$ is the external electromagnetic potential and $J^\mu=e_q \bar{q} \gamma^\mu q$ the quark electromagnetic current.
Since the external field can be made arbitrarily small, one can expand the correlation function $$\Pi_{\alpha\beta}(p)=\Pi^{(0)}_{\alpha\beta}(p)
+\Pi^{(1)}_{\alpha\beta}(p)+\cdots.$$ Here $\Pi^{(0)}_{\alpha\beta}(p)$ is the correlation function in the absence of the field, and gives rise to the mass sum rules of the baryons. The magnetic moments will be extracted from the QCD sum rules obtained from the linear response function $\Pi^{(1)}_{\alpha\beta}(p)$.
The action of the external electromagnetic field is two-fold: it couples directly to the quarks in the baryon interpolating fields, and it also polarizes the QCD vacuum. The latter can be described by introducing new parameters called vacuum susceptibilities.
The interpolating field is constructed from quark fields, and has the quantum numbers of the baryon under consideration. We use the following interpolating fields for the baryon decuplet family: $$\begin{array}{l}
\eta_{\alpha}^{\Delta^{++}}=
\epsilon^{abc}\left(u^{aT}C\gamma_\alpha u^b\right) u^c,
\\
\eta_{\alpha}^{\Delta^+}=
\sqrt{1/3}\;\epsilon^{abc}
\left[2\left(u^{aT}C\gamma_\alpha d^b\right) u^c
+\left(u^{aT}C\gamma_\alpha u^b\right) d^c\right],
\\
\eta_{\alpha}^{\Delta^0}=
\sqrt{1/3}\;\epsilon^{abc}
\left[2\left(d^{aT}C\gamma_\alpha u^b\right) d^c
+\left(d^{aT}C\gamma_\alpha d^b\right) u^c\right],
\\
\eta_{\alpha}^{\Delta^-}=
\epsilon^{abc}\left(d^{aT}C\gamma_\alpha d^b\right) d^c,
\\
\eta_{\alpha}^{{\Sigma^*}^+}=
\sqrt{1/3}\;\epsilon^{abc}
\left[2\left(u^{aT}C\gamma_\alpha s^b\right) u^c
+\left(u^{aT}C\gamma_\alpha u^b\right) s^c\right],
\\
\eta_{\alpha}^{{\Sigma^*}^0}=
\sqrt{2/3}\;\epsilon^{abc}
\left[2\left(u^{aT}C\gamma_\alpha d^b\right) s^c
+\left(d^{aT}C\gamma_\alpha s^b\right) u^c
+\left(s^{aT}C\gamma_\alpha u^b\right) d^c\right],
\\
\eta_{\alpha}^{{\Sigma^*}^-}=
\sqrt{1/3}\;\epsilon^{abc}
\left[2\left(d^{aT}C\gamma_\alpha s^b\right) d^c
+\left(d^{aT}C\gamma_\alpha d^b\right) s^c\right],
\\
\eta_{\alpha}^{{\Xi^*}^0}=
\sqrt{1/3}\;\epsilon^{abc}
\left[2\left(s^{aT}C\gamma_\alpha u^b\right) s^c
+\left(s^{aT}C\gamma_\alpha s^b\right) u^c\right],
\\
\eta_{\alpha}^{{\Xi^*}^-}=
\sqrt{1/3}\;\epsilon^{abc}
\left[2\left(s^{aT}C\gamma_\alpha d^b\right) s^c
+\left(s^{aT}C\gamma_\alpha s^b\right) d^c\right],
\\
\eta_{\alpha}^{\Omega^-}=
\epsilon^{abc}\left(s^{aT}C\gamma_\alpha s^b\right) s^c.
\end{array}$$ Here implicit function forms $\eta(x)$ and $q(x)$ (q=u,d,s) are assumed. $C$ is the charge conjugation operator. The superscript $T$ means transpose. The indices a, b and c are color indices running from one to three. The antisymmetric tensor $\epsilon^{abc}$ ensures the three quarks form a color singlet state. The normalization factors are chosen so that correlation functions of these interpolating fields coincide with each other under SU(3)-flavor symmetry (see Eqs. (\[maso\]) to (\[mass0\])).
The interpolating field excites (or annihilates) the ground state as well as the excited states of the baryon from the QCD vacuum. The ability of a interpolating field to annihilate the [*ground state*]{} baryon into the QCD vacuum is described by a phenomenological parameter $\lambda_B$ (called current coupling or pole residue), defined by the overlap $$\langle 0\,|\,\eta_{\alpha}\,|\,Bps\rangle
=\lambda_B\,u_{\alpha}(p,s),$$ where $u_{\alpha}$ is the Rarita-Schwinger spin-vector [@Rarita41].
Phenomenological Representation {#rhs}
-------------------------------
On the hadronic level, let us consider the linear response defined by $$\Pi^{(1)}_{\alpha\beta}(p) =
i\int d^4x\; e^{ip\cdot x}\langle 0\,|\,
\eta_{\alpha}(x)\,
\left[ -i \int d^4y\;A_\mu(y) J^\mu(y)\right]
\bar{\eta}_{\beta}(0)\,|\,0\rangle.
\label{phen1}$$ After inserting two complete sets of physical intermediate states, it becomes $$\begin{aligned}
\Pi^{(1)}_{\alpha\beta}(p) & = &
\int d^4x \int d^4y {d^4 k \over (2\pi)^4} {d^4 k^\prime \over (2\pi)^4}
\sum_{BB^\prime} \sum_{ss^\prime}
{-i \over k^2-M^2_B -i\epsilon}\;
{-i \over {k^\prime}^2-M^2_{B^\prime} -i\epsilon}
\nonumber \\ & &
e^{ip\cdot x} A_\mu(y)
\langle 0 | \eta_{\alpha}(x) | ks \rangle
\langle ks | J^\mu(y) | k^\prime s^\prime \rangle
\langle k^\prime s^\prime | \bar{\eta}_{\beta}(0) | 0 \rangle.
\label{phen2}\end{aligned}$$ QCD sum rule calculations are most conveniently done in the fixed-point gauge. For electromagnetic field, it is defined by $x_\mu A^\mu(x)=0$. In this gauge, the electromagnetic potential is given by $$A_\mu(y)=-{1\over 2} F_{\mu\nu} y^\nu.$$ The electromagnetic vertex of spin-3/2 baryons is defined by the current matrix element [@Derek92] $$\langle ks | J^\mu(0) | k^\prime s^\prime \rangle =
\bar{u}_\alpha(k,s)\;{\cal O}^{\alpha \mu \beta}(P,q)\;
u_\beta(k^\prime,s^\prime).$$ The Lorentz covariant tensor $${\cal O}^{\alpha \mu \beta}(P,q) \equiv
-g^{\alpha\beta} \left( a_1 \gamma^\mu + {a_2 \over 2M_B} P^\mu \right)
- {q^\alpha q^\beta \over (2M_B)^2}
\left( c_1 \gamma^\mu + {c_2 \over 2M_B} P^\mu \right),$$ where $P=k+k^\prime$ and $q=k-k^\prime$, satisfies the standard requirements of invariance under time reversal, parity, $G$ parity, and gauge transformations. The parameters $a_1$, $a_2$, $c_1$ and $c_2$ are independent covariant vertex functions. They are related to the multipole form factors by$$\begin{array}{l}
G_{E0}(q^2) = (1+{2\over 3} \tau) \left[ a_1 + (1+\tau) a_2 \right]
-{1\over 3} \tau (1+\tau) \left[ c_1 + (1+\tau) c_2 \right]
\\
G_{E2}(q^2) = \left[ a_1 + (1+\tau) a_2 \right]
-{1\over 2} (1+\tau) \left[ c_1 + (1+\tau) c_2 \right]
\\
G_{M1}(q^2) = (1+{4\over 5}\tau) a_1 - {2\over 5}\tau (1+\tau) c_1
\\
G_{M3}(q^2) = a_1 - {1\over 2} (1+\tau) c_1.
\end{array}
\label{multipole}$$ where $\tau=-q^2/(2M_B)^2 (\ge 0)$. They are referred to as charge (E0), electric quadrupole (E2), magnetic dipole (M1), and magnetic octupole (M3) form factors. The magnetic moment is related to the magnetic dipole form factor $G_{M1}(q^2)$ at zero momentum transfer. From Eq. (\[multipole\]), it is clear that $$G_{M1}(0)=a_1 \equiv \mu_{\scriptscriptstyle B},$$ where the magnetic moment $\mu_{\scriptscriptstyle B}$ is in units of particle’s natural magneton: $e\hbar/(2c M_B)$. So the goal is to isolate terms in Eq. (\[phen2\]) that involve only $a_1$.
The ground state contribution to Eq. (\[phen2\]) can be written as $$\begin{aligned}
\Pi^{(1)}_{\alpha\beta}(p) & = & {i\over 2} \lambda^2_B F_{\mu\nu}
\int d^4x {d^4 k \over (2\pi)^4}
e^{i(p-k)\cdot x} {1 \over k^2-M^2_B -i\epsilon}\;
\sum_{s} u_\alpha(k,s) \bar{u}_\rho(k,s)
\nonumber \\ & &
{\partial \over \partial q^\nu} \left[
{1 \over (k-q)^2-M^2_B -i\epsilon}
O^{\rho\mu\lambda}(2k-q,q)
\sum_{s^\prime} u_\lambda(k-q,s^\prime) \bar{u}_\beta(k-q,s^\prime)
\right] \bigg|_{q=0}.
\label{phen3}\end{aligned}$$ In arriving at Eq. (\[phen3\]), we have used a number of steps: the translation invariance on $\eta_\alpha(x)$ and $J^\mu(y)$, a change of variable from $k^\prime$ to $q$, the relation $$\int d^4y e^{iq\cdot y} y^\nu = -i (2\pi)^4
{\partial \over \partial q^\nu} \delta^4(q),$$ integration by parts, and the Rarita-Swinger spin sum [@Rarita41] $$\sum_{s} u_\alpha (p,s) \bar{u}_\beta (p,s) =
-(\hat{p} + M_B) \left(
g_{\alpha\beta} - {1\over3} \gamma_\alpha \gamma_\beta
- {2 p_\alpha p_\beta \over 3 M^2_B}
+ {p_\alpha \gamma_\beta - p_\beta \gamma_\alpha \over 3 M_B} \right),$$ with normalization $\bar{u}_\alpha u_\alpha = 2M_B$. The hat notation denotes $\hat{p}=p^\alpha\,\gamma_\alpha$.
Direct evaluation of Eq. (\[phen3\]) leads to numerous tensor structures, not all of them are independent of each other. The dependencies can be removed by ordering the gamma matrices in a specific order. Here we choose to order in $ \hat{p} \gamma_\alpha \gamma_\mu \gamma_\nu \gamma_\beta $. After a lengthy calculation, 18 tensor structures which involve only $a_1$ are isolated. They can be organized as $$\begin{array}{ll}
\Pi^{(1)}_{\alpha\beta}(p) =
\\
\;\;\; \mbox{WE}_1(p^2)\;\hat{p} F^{\mu\nu} \sigma_{\mu\nu} g_{\alpha\beta}
&
+ \mbox{WO}_1(p^2)\;F^{\mu\nu} \sigma_{\mu\nu} g_{\alpha\beta}
\\
+ \mbox{WE}_2(p^2)\;\hat{p} p_{\alpha} F^{\mu\nu} \sigma_{\mu\nu} p_{\beta}
&
+ \mbox{WO}_2(p^2)\;p_{\alpha} F^{\mu\nu} \sigma_{\mu\nu} p_{\beta}
\\
+ \mbox{WE}_3(p^2)\;\hat{p} \gamma_{\alpha} F^{\mu\nu} \sigma_{\mu\nu} \gamma_{\beta}
&
+ \mbox{WO}_3(p^2)\;\gamma_{\alpha} F^{\mu\nu} \sigma_{\mu\nu} \gamma_{\beta}
\\
+ \mbox{WE}_4(p^2)\; p_{\alpha} F^{\mu\nu} \sigma_{\mu\nu} \gamma_{\beta}
&
+ \mbox{WO}_4(p^2)\;\hat{p} p_{\alpha} F^{\mu\nu} \sigma_{\mu\nu} \gamma_{\beta}
\\
+ \mbox{WE}_5(p^2)\; \gamma_{\alpha} F^{\mu\nu} \sigma_{\mu\nu} p_{\beta}
&
+ \mbox{WO}_5(p^2)\;\hat{p} \gamma_{\alpha} F^{\mu\nu} \sigma_{\mu\nu} p_{\beta}
\\
+ \mbox{WE}_6(p^2)\;\hat{p} \gamma_\alpha F^{\mu\nu}
(\gamma_\mu g_{\beta\nu} -\gamma_\nu g_{\beta\mu})
&
+ \mbox{WO}_6(p^2)\;\gamma_{\alpha}F^{\mu\nu}
(\gamma_\mu g_{\beta\nu} -\gamma_\nu g_{\beta\mu})
\\
+ \mbox{WE}_7(p^2)\;\hat{p} F^{\mu\nu}
(\gamma_\mu g_{\alpha\nu} - \gamma_\nu g_{\alpha\mu}) \gamma_\beta
&
+ \mbox{WO}_7(p^2)\;F^{\mu\nu}
(\gamma_\mu g_{\alpha\nu} - \gamma_\nu g_{\alpha\mu}) \gamma_\beta
\\
+ \mbox{WE}_8(p^2)\; p_\alpha F^{\mu\nu}
(\gamma_\mu g_{\beta\nu} - \gamma_\nu g_{\beta\mu})
&
+ \mbox{WO}_8(p^2)\;\hat{p} p_\alpha F^{\mu\nu}
(\gamma_\mu g_{\beta\nu} - \gamma_\nu g_{\beta\mu})
\\
+ \mbox{WE}_9(p^2)\; F^{\mu\nu}
(\gamma_\mu g_{\alpha\nu} - \gamma_\nu g_{\alpha\mu}) p_\beta
&
+ \mbox{WO}_9(p^2)\;\hat{p} F^{\mu\nu}
(\gamma_\mu g_{\alpha\nu} - \gamma_\nu g_{\alpha\mu}) p_\beta
+ \cdots.
\end{array}
\label{phen4}$$ The tensor structures associated with WE$_i$ have odd number of gamma matrices, while those associated with WO$_i$ have even number of gamma matrices. Apart from a common factor $i{\lambda}^2_B\;\mu_{\scriptscriptstyle B}/(p^2-M^2_B)^2$, the invariant functions are given by $$\begin{array}{ll}
\mbox{WE}_1={1\over 2}, &
\mbox{WO}_1={1\over 2}M_B,\\
\mbox{WE}_2={-1\over 9M^2_B}, &
\mbox{WO}_2={-1\over 9M_B},\\
\mbox{WE}_3={-7\over 18},&
\mbox{WO}_3={-7\over 18} M_B,\\
\mbox{WE}_4={ 7\over 18},&
\mbox{WO}_4={ 7\over 18 M_B},\\
\mbox{WE}_5={-7\over 18}, &
\mbox{WO}_5={-7\over 18 M_B},\\
\mbox{WE}_6={ 2\over 3},&
\mbox{WO}_6={ 2\over 3} M_B,\\
\mbox{WE}_7={-2\over 3}, &
\mbox{WO}_7={-2\over 3} M_B,\\
\mbox{WE}_8={-2\over 3}, &
\mbox{WO}_8={-2\over 3 M_B},\\
\mbox{WE}_9={-2\over 3},&
\mbox{WO}_9={-2\over 3 M_B}.
\end{array}$$
In addition to the ground state contribution, there exist also excited state contributions. For a generic invariant function, the pole structure has the form $${\lambda^2_B\;\mu_{\scriptscriptstyle B} \over (p^2-M_B^2)^2}
+ \sum_{B^*}{C_{B\leftrightarrow B^*} \over (p^2-M_B^2)(p^2-M_{B^*}^2)}
+ \cdots.
\label{pole}$$ where $C_{B\leftrightarrow B^*}$ are constants. The first term is the ground state double pole which contains the desired magnetic moment of the baryon, the second term represents the non-diagonal transitions between the ground state and the excited states caused by the external field, and the ellipses represent pure excited state contributions. Upon Borel transform, one has $${\lambda^2_B\;\mu_{\scriptscriptstyle B} \over M^2}\;e^{-M_B^2/M^2}
+ e^{-M_B^2/M^2} \left[\sum_{B^*}
{C_{B\rightarrow B^*} \over M_{B^*}^2 - M_B^2}
\left(1-e^{-(M_{B^*}^2-M_B^2)/M^2}\right)\right]
+ \cdots.
\label{pole-borel}$$ We see that the transitions give rise to a contribution that is not exponentially suppressed relative to the ground state. This is a general feature of the external-field technique. The strength of such transitions at each structure is [*a priori*]{} unknown and is an additional source of contamination in the determination of $\mu_B$ not found in mass sum rules. The usual treatment of the transitions is to approximate the quantity in the square brackets by a constant, which is to be extracted from the sum rule along with the ground state property of interest. Inclusion of such contributions is necessary for the correct extraction of the magnetic moments. The pure excited state contributions are exponentially suppressed relative to the ground state and can be modeled in the usual way by introducing a continuum model and threshold parameter.
Calculation of the QCD Side {#lhs}
---------------------------
On the quark level, one evaluates the correlation function in Eq. (\[cf2pt\]) using Operator Product Expansion (OPE). The calculation is most readily done in coordinate space. To arrive at the final sum rules, one needs a subsequent Fourier transform, followed by a Borel transform.
We decide to carry out four separate calculations for $\Omega^-$(sss), ${\Sigma^*}^+$(uus), ${\Xi^*}^0$(uss), and ${\Sigma^*}^0$(uds). They have distinct strange quark content, which requires special treatment. The QCD sum rules for other members can be obtained by appropriate substitutions in those for these four members.
The master formula, which is obtained from contracting out the quark pairs in the correlation function, is given by, for $\Omega^-$: $$\begin{array}{rll}
\langle 0\,|\, T\{\;\eta^{\Omega^-}_{\alpha}(x)\,
\bar{\eta}^{\Omega^-}_{\beta}(0)\;\}\,|\,0\rangle_F =
\\
2 \epsilon^{abc}\epsilon^{a^\prime b^\prime c^\prime}
\{ &
S^{aa^\prime}_s \mbox{Tr} \left[ \gamma_\beta C {S^{bb^\prime}_s}^T
C \gamma_\alpha S^{cc^\prime}_s \right]
&
+ 2 S^{aa^\prime}_s \gamma_\beta C {S^{bb^\prime}_s}^T
C \gamma_\alpha S^{cc^\prime}_s
\},
\end{array}
\label{maso}$$ for ${\Sigma^*}^+$: $$\begin{array}{lll}
\langle 0\,|\, T\{\;\eta^{{\Sigma^*}^+}_{\alpha}(x)\,
\bar{\eta}^{{\Sigma^*}^+}_{\beta}(0)\;\}\,|\,0\rangle_F =
&
{2 \over 3} \epsilon^{abc}\epsilon^{a^\prime b^\prime c^\prime}
\{ & \\
\;\;\; S^{aa^\prime}_u \mbox{Tr} \left[ \gamma_\beta C {S^{bb^\prime}_u}^T
C \gamma_\alpha S^{cc^\prime}_s \right]
&
+ S^{aa^\prime}_u \mbox{Tr} \left[ \gamma_\beta C {S^{bb^\prime}_s}^T
C \gamma_\alpha S^{cc^\prime}_u \right]
&
+ S^{aa^\prime}_s \mbox{Tr} \left[ \gamma_\beta C {S^{bb^\prime}_u}^T
C \gamma_\alpha S^{cc^\prime}_u \right]
\\
+ 2 S^{aa^\prime}_u \gamma_\beta C {S^{bb^\prime}_u}^T
C \gamma_\alpha S^{cc^\prime}_s
&
+ 2 S^{aa^\prime}_u \gamma_\beta C {S^{bb^\prime}_s}^T
C \gamma_\alpha S^{cc^\prime}_u
&
+ 2 S^{aa^\prime}_s \gamma_\beta C {S^{bb^\prime}_u}^T
C \gamma_\alpha S^{cc^\prime}_u
\},
\end{array}
\label{massp}$$ for ${\Xi^*}^0$: $$\begin{array}{lll}
\langle 0\,|\, T\{\;\eta^{{\Xi^*}^0}_{\alpha}(x)\,
\bar{\eta}^{{\Xi^*}^0}_{\beta}(0)\;\}\,|\,0\rangle_F =
&
{2 \over 3} \epsilon^{abc}\epsilon^{a^\prime b^\prime c^\prime}
\{ & \\
\;\;\; S^{aa^\prime}_s \mbox{Tr} \left[ \gamma_\beta C {S^{bb^\prime}_s}^T
C \gamma_\alpha S^{cc^\prime}_u \right]
&
+ S^{aa^\prime}_s \mbox{Tr} \left[ \gamma_\beta C {S^{bb^\prime}_u}^T
C \gamma_\alpha S^{cc^\prime}_s \right]
&
+ S^{aa^\prime}_u \mbox{Tr} \left[ \gamma_\beta C {S^{bb^\prime}_s}^T
C \gamma_\alpha S^{cc^\prime}_s \right]
\\
+ 2 S^{aa^\prime}_s \gamma_\beta C {S^{bb^\prime}_s}^T
C \gamma_\alpha S^{cc^\prime}_u
&
+ 2 S^{aa^\prime}_s \gamma_\beta C {S^{bb^\prime}_u}^T
C \gamma_\alpha S^{cc^\prime}_s
&
+ 2 S^{aa^\prime}_u \gamma_\beta C {S^{bb^\prime}_s}^T
C \gamma_\alpha S^{cc^\prime}_s
\},
\end{array}
\label{masx0}$$ and for ${\Sigma^*}^0$: $$\begin{array}{lll}
\langle 0\,|\, T\{\;\eta^{{\Sigma^*}^0}_{\alpha}(x)\,
\bar{\eta}^{{\Sigma^*}^0}_{\beta}(0)\;\}\,|\,0\rangle_F =
&
{2 \over 3} \epsilon^{abc}\epsilon^{a^\prime b^\prime c^\prime}
\{ & \\
\;\;\; S^{aa^\prime}_u \mbox{Tr} \left[ \gamma_\beta C {S^{bb^\prime}_d}^T
C \gamma_\alpha S^{cc^\prime}_s \right]
&
+ S^{aa^\prime}_d \mbox{Tr} \left[ \gamma_\beta C {S^{bb^\prime}_s}^T
C \gamma_\alpha S^{cc^\prime}_u \right]
&
+ S^{aa^\prime}_s \mbox{Tr} \left[ \gamma_\beta C {S^{bb^\prime}_u}^T
C \gamma_\alpha S^{cc^\prime}_d \right]
\\
+ S^{aa^\prime}_u \gamma_\beta C {S^{bb^\prime}_d}^T
C \gamma_\alpha S^{cc^\prime}_s
&
+ S^{aa^\prime}_d \gamma_\beta C {S^{bb^\prime}_s}^T
C \gamma_\alpha S^{cc^\prime}_u
&
+ S^{aa^\prime}_s \gamma_\beta C {S^{bb^\prime}_u}^T
C \gamma_\alpha S^{cc^\prime}_d
\\
+ S^{aa^\prime}_u \gamma_\beta C {S^{bb^\prime}_s}^T
C \gamma_\alpha S^{cc^\prime}_d
&
+ S^{aa^\prime}_d \gamma_\beta C {S^{bb^\prime}_u}^T
C \gamma_\alpha S^{cc^\prime}_s
&
+ S^{aa^\prime}_s \gamma_\beta C {S^{bb^\prime}_d}^T
C \gamma_\alpha S^{cc^\prime}_u
\}.
\end{array}
\label{mass0}$$ In the above equations, $$S^{ab}_q (x,0;F) \equiv
\langle 0\,|\, T\{\;q^a(x)\,
\bar{q}^b(0)\;\}\,|\,0\rangle_F,
\hspace{3mm} q=u, d, s,$$ is the fully interacting quark propagator in the presence of the electromagnetic field. To first order in $F_{\mu\nu}$ and $m_q$ (assume $m_u=m_d=0, m_s\neq 0$), and order $x^4$, it is given by [@Ioffe84; @Pasupathy86; @Wilson87]: $$\begin{aligned}
S^{ab}_q(x,0;Z) &\equiv&
{i \over 2\pi^2} {\hat{x}\over x^4} \delta^{ab}
- {m_q \over 4\pi^2 x^2} \delta^{ab}
- {1\over 12}\langle\bar{q}q\rangle \delta^{ab}
+ {im_q \over 48} \langle\bar{q}q\rangle \hat{x} \delta^{ab}
\nonumber \\ & &
+ {1\over 192} \langle\bar{q}g_c\sigma\cdot Gq\rangle x^2 \delta^{ab}
- {im_q\over 1152} \langle\bar{q}g_c\sigma\cdot Gq\rangle
\hat{x} x^2 \delta^{ab}
- {1\over 3^3 2^{10}} \langle\bar{q}q\rangle \langle g^2_c G^2\rangle
x^4 \delta^{ab}
\nonumber \\ & &
+ {i\over 32\pi^2} (g_cG^n_{\alpha\beta})
{ \hat{x} \sigma^{\alpha\beta} +\sigma^{\alpha\beta} \hat{x}\over x^2 }
\left({\lambda^n\over 2}\right)^{ab}
+ {1\over 48} {i\over 32\pi^2} \langle g^2_c G^2\rangle
{\hat{x} \sigma^{\alpha\beta} +\sigma^{\alpha\beta} \hat{x}\over x^2}
\left({\lambda^n\over 2}\right)^{ab}
\nonumber \\ & &
+ {1\over 3^2 2^{10}} \langle\bar{q}q\rangle \langle g^2_c G^2\rangle
x^2 \sigma^{\alpha\beta} \left({\lambda^n\over 2}\right)^{ab}
- {1\over 192}\langle\bar{q}g_c\sigma\cdot Gq\rangle
\sigma^{\alpha\beta} \left({\lambda^n\over 2}\right)^{ab}
\nonumber \\ & &
+ {im_q\over 768}\langle\bar{q}g_c\sigma\cdot Gq\rangle
\left( \hat{x} \sigma^{\alpha\beta} +\sigma^{\alpha\beta} \hat{x} \right)
\left({\lambda^n\over 2}\right)^{ab}
+ {i e_q\over 32\pi^2} F_{\alpha\beta}
{ \hat{x} \sigma^{\alpha\beta} +\sigma^{\alpha\beta} \hat{x} \over x^2 }
\delta^{ab}
\nonumber \\ & &
- {e_q\over 24} \chi \langle\bar{q}q\rangle
F_{\alpha\beta} \sigma^{\alpha\beta} \delta^{ab}
+ {ie_q m_q\over 96} \chi \langle\bar{q}q\rangle F_{\alpha\beta}
\left( \hat{x} \sigma^{\alpha\beta} +\sigma^{\alpha\beta} \hat{x} \right)
\delta^{ab}
\nonumber \\ & &
+ {e_q \over 288} \langle\bar{q}q\rangle F_{\alpha\beta}
\left( x^2 \sigma^{\alpha\beta} - 2 x_\rho x^\beta \sigma^{\beta\alpha}
\right) \delta^{ab}
\nonumber \\ & &
+ {e_q \over 576} \langle\bar{q}q\rangle F_{\alpha\beta}
\left[ x^2 (\kappa+\xi) \sigma^{\alpha\beta}
- x_\rho x^\beta (2\kappa-\xi) \sigma^{\beta\alpha} \right] \delta^{ab}
\nonumber \\ & &
- {e_q \over 16} \langle\bar{q}q\rangle
\left( \kappa F_{\alpha\beta}
- {i\over 4} \xi \epsilon_{\alpha\beta\mu\nu} F^{\mu\nu} \right)
\left({\lambda^n\over 2}\right)^{ab}
+ \mbox{higher order terms}.
\label{prop}\end{aligned}$$ We use the convention $\epsilon^{0123}=+1$ in this work. The vacuum susceptibilities are defined by $$\begin{array}{r}
\langle\bar{q} \sigma_{\mu\nu} q\rangle_F \equiv
e_q \chi \langle\bar{q}q\rangle F_{\mu\nu}, \\
\langle\bar{q} g_c G_{\mu\nu} q\rangle_F \equiv
e_q \kappa \langle\bar{q}q\rangle F_{\mu\nu}, \\
\langle\bar{q} g_c \epsilon_{\mu\nu\rho\lambda} G^{\rho\lambda} \gamma_5
q\rangle_F \equiv
i e_q \xi \langle\bar{q}q\rangle F_{\mu\nu}.
\end{array}$$ Note that $\chi$ has the dimension of GeV$^{-2}$, while $\kappa$ and $\xi$ are dimensionless.
The calculation proceeds by substituting the quark propagator into the master formulae, keeping terms to first order in the external field and in the strange quark mass. Terms up to dimension 8 are considered. The various combinations can be represented by diagrams. Fig. \[xmag\] shows the basic diagrams considered for the decuplet baryon magnetic moments. Fig. \[xmagm\] shows the diagrams considered for the strange quark mass corrections. Note that each diagram is only generic. All possible color permutations are understood. Numerous tensor structures emerge from the calculations. Upon ordering the gamma matrices in the same order as in the phenomenological side, 18 invariant functions are obtained at the corresponding tensor structures. By equating them with those in Eq. (\[phen4\]), QCD sum rules are constructed. These invariant functions can be classified by the chirality of the vacuum condensates they contain. Eight of them , denoted by $\mbox{WE}_i$, involve only dimension-even condensates, thus we call the corresponding sum rules chiral-even. The other eight, denoted by $\mbox{WO}_i$, involve only dimension-odd condensates, and we call the corresponding sum rules chiral-odd. Note that previous works such as Refs. [@Ioffe84; @Chiu86] use the chirality of the tensor structures to refer to the sum rules. The two are opposite.
To keep the presentation smooth, the complete set of sum rules (a total of 160 for the decuplet family) obtained in this work are given in the Appendix in a highly condensed form. As it turns out, the validity of a particular sum rule depends on the input parameter set. Sum rules that are valid for one set may become invalid for another, and [*vice versa*]{}. For this reason, it is useful to present all of the sum rules. Another benefit is that it provides a basis for other authors to check the calculation. Sufficient detail is given in this work for that purpose.
The various symbols in the sum rules are explained in the following. The condensate parameters are denoted by $$a=-(2\pi)^2\,\langle\bar{u}u\rangle,
\hspace{2mm}
b=\langle g^2_c\, G^2\rangle,
\hspace{2mm}
\langle\bar{u}g_c\sigma\cdot G u\rangle=-m_0^2\,\langle\bar{u}u\rangle.$$ The re-scaled current coupling $$\tilde{\lambda}_B=(2\pi)^2\lambda_B.$$ The quark charge factors $e_q$ are given in units of electric charge $$e_u=2/3,
\hspace{4mm}
e_d=-1/3,
\hspace{4mm}
e_s=-1/3.$$ Note that we choose to keep the quark charge factors explicit in the sum rules. The advantage is that it can facilitate the study of quark effective magnetic moments. The parameters $f$ and $\phi$ account for the flavor symmetry breaking of the strange quark in the condensates and susceptibilities: $$f={ \langle\bar{s}s\rangle \over \langle\bar{u}u\rangle }
={ \langle\bar{s}g_c\sigma\cdot G s\rangle \over
\langle\bar{u}g_c\sigma\cdot G u\rangle },
\hspace{4mm}
\phi={ \chi_s \over \chi }={ \kappa_s \over \kappa }={ \xi_s \over \xi }.$$ The four-quark condensate is parameterized by the factorization approximation $$\langle\bar{u}u\bar{u}u\rangle=\kappa_v\langle\bar{u}u\rangle^2,$$ and we will investigate its possible violation via the parameter $\kappa_v$. The anomalous dimension corrections of the currents and various operators are taken into account in the leading logarithmic approximation via the factor $$L^\gamma=\left[{\alpha_s(\mu^2) \over \alpha_s(M^2)}\right]^\gamma
=\left[{\ln(M^2/\Lambda_{QCD}^2) \over \ln(\mu^2/\Lambda_{QCD}^2)}
\right]^\gamma,$$ where $\mu=500$ MeV is the renormalization scale and $\Lambda_{QCD}$ is the QCD scale parameter. As usual, the excited state contributions are modeled using terms on the OPE side surviving $M^2\rightarrow \infty$ under the assumption of duality, and are represented by the factors $$E_n(x)=1-e^{-x}\sum_n{x^n \over n!}, \hspace{3mm} x=w_B^2/M_B^2,$$ where $w_B$ is an effective continuum threshold. Note that $w_B$ is in principle different for different sum rules and we will treat it as a free parameter in the the analysis.
The coefficients for other members of the decuplet family can be obtained by appropriate replacements of quark contents. They are:
1. for $\Delta^{++}$, replace s quark by u quark in $\Omega^-$,
2. $\Delta^+$: replace s quark by d quark in ${\Sigma^*}^+$,
3. for $\Delta^0$: replace s quark by d quark in ${\Xi^*}^0$,
4. for $\Delta^-$, replace s quark by d quark in $\Omega^-$,
5. for ${\Sigma^*}^-$, replace u quark by d quark in ${\Sigma^*}^+$,
6. for ${\Xi^*}^-$, replace u quark by d quark in ${\Xi^*}^0$.
Here the conversions between u and d quarks are achieved by simply switching their charge factors $e_u$ and $e_d$. The conversions from s quark to u or d quarks involve setting $m_s=0$, $f=\phi=1$, in addition to the switching of charge factors.
Furthermore, in the course of collecting the coefficients for the four selected members ${\Sigma^*}^+$, ${\Sigma^*}^0$, ${\Xi^*}^0$, $\Omega^-$, we discovered some relations among them that allow one to write down one set of $c_i$ staring from another. The relations are given as follows.
1. From ${\Sigma^*}^+$ to ${\Sigma^*}^0$: simply replace every occurrence of $e_u$ by $(e_u + e_d)/2$.
2. From ${\Xi^*}^0$ to $\Omega^-$ involves converting the u quark to s quark. This is achieved by collapsing each coefficient into a single term that has the maximum number of $e_s$, $f$, $\phi$ in that coefficient. The numerical factor of it is the sum of the numerical factors in front each of the terms in the coefficient. For example, $(2e_s+e_u)$ goes to $3e_s$, $(2 e_s f \phi + e_u)$ goes to $3e_s f \phi$, $(2e_s f - 3e_s - 3e_u f + e_u)$ goes to $-3e_s f$, etc..
These relations were also used as consistency checks of the calculation.
From the above discussions, we see that it is possible to write down the coefficients for all other members of the decuplet family starting just from those for ${\Sigma^*}^+$ and ${\Xi^*}^0$. In the sum rule from WE$_1$ in Eq. (\[we1\]), the complete sets of $c_i$ are given for the four selected members ${\Sigma^*}^+$, ${\Sigma^*}^0$, ${\Xi^*}^0$ and $\Omega^-$. They are intended as examples for the reader to get familiar with the relations. The rest of the sum rules are presented with $c_i$ only given for ${\Sigma^*}^+$ and ${\Xi^*}^0$.
Finally, let us point out some exact relations among the OPE sides of the sum rules: $$\mbox{OPE}_{\scriptscriptstyle \Delta^+}=
{1\over 2} \, \mbox{OPE}_{\scriptscriptstyle \Delta^{++}},
\label{exact1}$$ $$\mbox{OPE}_{\scriptscriptstyle \Delta^0}=0,
\label{exact2}$$ $$\mbox{OPE}_{\scriptscriptstyle \Delta^-}=
-\, \mbox{OPE}_{\scriptscriptstyle \Delta^+}.
\label{exact3}$$ $$\mbox{OPE}_{\scriptscriptstyle {\Sigma^*}^0}={1\over 2} \,
( \mbox{OPE}_{\scriptscriptstyle {\Sigma^*}^+}
+ \mbox{OPE}_{\scriptscriptstyle {\Sigma^*}^-} ).
\label{exact4}$$ These results are consequences of symmetries in the correlation functions. As an example, let us examine Eq. (\[exact1\]). For a given diagram, the master formula for $\Delta^{++}$ can be written as $2(e_uC_1 + e_uC_2)={4\over 3}(C_1+C_2)$ where $C_1$ has the trace dependence, while $C_2$ not. On the other hand, the master formula for $\Delta^{+}$ can be written as ${2\over 3}[(2e_u+e_d)C_1 + (2e_u+e_d)C_2]={2\over 3}(C_1+C_2)$, hence the factor of 2. The key here is: a) each term is proportional to a quark charge factor; b) SU(2) flavor symmetry in u and d quarks; c) it is the same $C_1$ and $C_2$ that appear in both cases. The argument can be generalized to any diagrams, only with $C_1$ and $C_2$ different from diagram to diagram. Thus the factor of 2 will survive, regardless of the number of diagrams considered. One can argue for the rest of the relations by the same token. The above results have been explicitly verified using the calculated coefficients in the sum rules. They also provided a set of highly non-trivial checks of the calculation. A number of hard-to-detect errors have been eliminated this way.
Now let us consider the phenomenological side of Eq. (\[exact1\]). Since the continuum is modeled using terms on the OPE side, the continuum contributions also differ by a factor of 2. Assuming the transitions, which are modeled by a constant, also differ by a factor of 2, then Eq. (\[exact1\]) can be extended to the magnetic moments. This assumption was confirmed by numerical analysis. The same is true for Eq. (\[exact2\]) and Eq. (\[exact3\]). The situation for Eq. (\[exact4\]) is a little different. The convergence properties may change when two OPE series are added up. Numerical analysis confirmed that fewer sum rules are valid for ${\Sigma^*}^0$ than for ${\Sigma^*}^+$ and ${\Sigma^*}^-$.
Monte-Carlo Analysis {#ana}
====================
To analyze the sum rules, we use a Monte-Carlo based procedure recently developed in Ref. [@Derek96]. The basic steps are as follows. First, the uncertainties in the QCD input parameters are assigned. Then, randomly-selected, Gaussianly-distributed sets for these uncertainties are generated, from which an uncertainty distribution in the OPE, $\sigma^2_{\scriptscriptstyle OPE}(M_j)$ where $M_j$ are evenly distributed points in the desired Borel window, can be constructed. Next, a $\chi^2$ minimization is applied to the sum rule by adjusting the phenomenological fit parameters. Note that the uncertainties in the OPE are not uniform throughout the Borel window. They are larger at the lower end where uncertainties in the higher-dimensional condensates dominate. Thus, it is crucial that the appropriate weight is used in the calculation of $\chi^2$. For the OPE obtained from the k’th set of QCD parameters, the $\chi^2$ per degree of freedom is $${\chi^2_k\over N_{DF}}={1\over n_B-n_p}
\sum^{n_{\scriptscriptstyle B}}_{j=1}
{ [\Pi^{\scriptscriptstyle OPE}_k(M_j)
-\Pi^{\scriptscriptstyle Phen}_k(M_j;\lambda_k,m_k,w_k)]^2
\over \sigma^2_{\scriptscriptstyle OPE}(M_j) },$$ where $n_p$ is the number of phenomenological search parameters, and $\Pi^{\scriptscriptstyle Phen}$ denotes the phenomenological representation. In practice, $n_B$=51 points were used along the Borel axis. The procedure is repeated for many QCD parameter sets, resulting in distributions for phenomenological fit parameters, from which errors are derived. Usually, 200 such configurations are sufficient for getting stable results. We generally select 1000 sets which help resolve more subtle correlations among the QCD parameters and the phenomenological fit parameters.
The Borel window over which the two sides of a sum rule are matched is determined by the following two criteria. First, [*OPE convergence*]{}: the highest-dimension-operators contribute no more than 10% to the QCD side. Second, [*ground-state dominance*]{}: excited state contributions should not exceed more than 50% of the phenomenological side. The first criterion effectively establishes a lower limit, the second an upper limit. Those sum rules which do not have a Borel window under these criteria are considered invalid.
QCD Input Parameters {#input}
--------------------
The QCD input parameters and their uncertainty assignments are given as follows. The condensates are taken as $$a=0.52\pm0.05 \;GeV^3, \;\;
b=1.2\pm0.6 \;GeV^4, \;\;
m^2_0=0.72\pm0.08 \;GeV^2.$$ For the factorization violation parameter, we use $$\kappa_v=2\pm 1 \;\;\mbox{and}\;\; 1\leq \kappa_v \leq 4.$$ The QCD scale parameter is restricted to $\Lambda_{QCD}$=0.15$\pm$0.04 GeV. The vacuum susceptibilities have been estimated in studies of nucleon magnetic moments [@Ioffe84; @Balitsky83; @Chiu86], but the values vary in a wide range depending on the method used. Here we take some median values with 50% uncertainties: $$\chi=-6.0\pm 3.0\; GeV^{-2}
\;\; \mbox{and}\;\; 0 \; GeV^{-2}\leq \chi \leq -10 \;GeV^{-2},$$ and $$\kappa=0.75\pm 0.38,\;\; \xi=-1.5\pm 0.75.$$ Note that $\chi$ is almost an order of magnitude larger than $\kappa$ and $\xi$, and is the most important of the three. The strange quark parameters are placed at [@Pasupathy86; @Lee97a] $$m_s=0.15\pm 0.02 \;GeV, \;\;
f=0.83\pm0.05, \;\;
\phi=0.60\pm0.05.$$ These uncertainties are assigned conservatively and in accord with the state-of-the-art in the literature. While some may argue that some values are better known, others may find that the errors are underestimated. In any event, one will learn how the uncertainties in the QCD parameters are mapped into uncertainties in the phenomenological fit parameters. In the numerical analysis below, we will also examine how the spectral parameters depend on different uncertainty assignments in these input parameters.
Search Procedure {#search}
----------------
To extract the magnetic moments, a two-stage fit was performed. First, the corresponding chiral-odd mass sum rule, as obtained previously in Ref. [@Lee97a], was fitted to get the mass $M_B$, the coupling $\tilde{\lambda}_B^2$ and the continuum threshold $w_1$. Then, $M_B$ and $\tilde{\lambda}_B^2$ were used in the magnetic moment sum rule for a three-parameter fit: the transition strength $A$, the continuum threshold $w_2$, and the magnetic moment $\mu_{\scriptscriptstyle B}$. Note that $w_1$ and $w_2$ are not necessarily the same. We impose a physical constraint on both $w_1$ and $w_2$ requiring that they are larger than the mass, and discard QCD parameter sets that do not satisfy this condition. In the actual analysis of the sum rules, however, we found that a full search was not always successful. In such cases, the search algorithm consistently returned $w_2$ either zero or smaller than $M_B$. This signals insufficient information in the OPE to completely resolve the spectral parameters. To proceed, we fixed $w_2$ at $w_1$, which is a commonly-adopted choice in the literature, and searched for $A$ and $\mu_{\scriptscriptstyle B}$. The two-stage fit incorporates the uncertainties from the two-point functions in a correlated fashion into the three-point functions, and represents a more realistic scenario.
To illustrate how well a sum rule works, we first cast it into the subtracted form, $$\Pi_S=\tilde{\lambda}^2_B\, \mu_{\scriptscriptstyle B}\, e^{-M^2_B/M^2},$$ then plot the logarithm of the absolute value of the two sides against the inverse of $M^2$. In this way, the right-hand side will appear as a straight line whose slope is $-M_B^2$ and whose intercept with the y-axis gives some measure of the coupling strength and the magnetic moment. The linearity (or deviation from it) of the left-hand side gives an indication of OPE convergence, and information on the continuum model and the transitions. The two sides are expected to match for a good sum rule. This way of matching the sum rules is similar to looking for a ‘plateau’ as a function of Borel mass in the conventional analysis, but has the advantage of not restricting the analysis regime in Borel space to the valid regimes common to [*both*]{} two-point and three-point correlation functions.
Results and Discussions {#res}
=======================
We have analyzed all of the sum rules for the entire decuplet family. We confirmed the three relations among magnetic moments as extended from Eqs. (\[exact1\]) to (\[exact3\]). So we will only present results for seven members. Valid sum rules were identified using the criteria discussed earlier. The results are given in three tables: Tables \[tabdo\] to \[tabxi\]. The corresponding overlap plots are given in seven figures: Figs. \[rhslhs-delpp\] to \[rhslhs-xim\]. These plots show how well a sum rule performs in the entire Borel region. Such information is absent in the tables. From the results, the following observations are in order.
In general, more chiral-even sum rules are valid than chiral-odd ones. This is consistent with previous findings for the octet baryon magnetic moments. It was argued in Ref. [@Ioffe84] that the interval of dimensions (not counting the dimension of $F_{\mu\nu}$) in the chiral-even sum rules (0 to 8) is larger than that in the chiral-odd sum rules (1 to 7). Indeed, more chiral-even sum rules (WE$_2$, WE$_4$, WE$_5$, WE$_6$, WE$_8$, WE$_9$) have power corrections up to $1/M^4$, than chiral-odd ones (WO$_2$, WO$_4$, WO$_8$, WO$_9$). Because of the additional terms in the OPE series, these sum rules are expected to be more reliable than the other sum rules. The situation here is almost opposite to that for the two-point functions [@Lee97a]. It was pointed out in Ref. [@Jin97] that chiral-odd sum rules are more reliable than chiral-even sum rules for baryon two-point functions. The reason could be traced to the fact that even and odd parity excited states contribute with different signs. In the three-point functions, however, the statement is no longer valid due to the appearance of transitions and vacuum susceptibilities. Therefore, caution should be used when applying the chirality argument to determine the reliability of a sum rule in three-point functions. In addition, numerical analysis showed that the sum rules from WE$_1$, WE$_3$, WO$_1$ are valid for the standard input parameter set, despite the absence of $1/M^4$ terms. We have varied the central values of the input parameters and discovered that sum rules that were valid for one set of input parameters became invalid for another, and [*vice versa*]{}. Thus the situation with three-point functions is more complicated. Our experience is that each sum rule should be examined individually in order to find out its reliability.
It turns out that for most of the valid sum rules, a full search was unsuccessful, except for three sum rules: WE$_5$, WE$_6$ and WE$_8$ for ${\Sigma^*}^{+}$. Of the three, only WE$_6$ returned a continuum threshold with reasonable error. The other two returned it with large errors. The large errors indicate that the sum rules are not very stable: they contain barely enough information to completely resolve the spectral parameters.. The important point is that the results with the continuum threshold searched or not are almost the same. This suggests that fixing it to that of the corresponding two-point function seems a good approximation.
It is gratifying to observe that the valid sum rules for most members give consistent predictions for the magnetic moments in terms of the sign, except for ${\Sigma^*}^{0}$ and ${\Xi^*}^0$ whose magnitudes are small. The magnitudes for the magnetic moments are consistent within errors for the most part, with only a few exceptions. The performances of the sum rules are quite different within each member. This is best displayed in the overlap plots. In some sum rules, the overlap is poor, as evidenced by the deviation from linearity (dotted lines). It signals poor OPE convergence in these sum rules. As expected, the deviation is more severe in the lower end of the Borel region where nonperturbative physics dominates. These sum rules will more likely suffer from uncertainties associated with the selection of the Borel window. As a result, the spectral parameters extracted from them are less reliable. One way to alleviate the problem is to increase the lower end of the Borel window to values where the overlap is good, even to extend the upper end to ensure the existence of a window. This was not attempted in this work because we feel the results obtained this way are somewhat misleading. The reason is that the sum rules in these windows will be dominated mostly by perturbative physics. It is common knowledge that if one goes deep enough into the Borel space, one can always find a match in a QCD sum rule. But such practice is against the philosophy of the QCD sum rule approach, which relies upon the power corrections to resolve the spectral properties. Therefore, some standard is necessary to emphasize such physics, and we feel the 10%-50% criteria adopted here are a reasonable choice.
Based on the quality of the overlap, the broadness of the Borel window and its reach into the lower end, the size of the continuum contribution, and the standard QCD input parameter set, we designate one sum rule for each member as the most favorable. They are WE$_5$ for $\Delta^{++}$, WE$_5$ for ${\Sigma^*}^{+}$, WO$_8$ for ${\Sigma^*}^{0}$, WE$_5$ for ${\Sigma^*}^{-}$, WO$_8$ for ${\Xi^*}^0$, WE$_5$ for ${\Xi^*}^-$, and WE$_5$ for $\Omega^{-}$. The selection is undoubtedly subjective. The reader may find a different set that have equal or comparable performance. We want to stress that such a selection depends on the QCD input parameters. It is possible that the ones selected here become invalid for a different set of input parameters, in which case a new set should be selected.
Relatively large errors were found in the valid sum rules using the standard QCD input parameter set: from 50% to 100% in the magnetic moments. But in most cases, the sign and order of magnitudes are unambiguously predicted when compared to the measured values. The situation is similar to a previous finding on $g_A$ [@Lee97]. To gain some idea on how the uncertainties depend on the input, we also analyzed the sum rules by adjusting the error estimates individually. We found large sensitivities to the quark condensate magnetic susceptibility $\chi$. In fact, most of the errors came from the uncertainties in $\chi$. We also tried with reduced error estimates on all the QCD input parameters: 10% relative errors uniformly. It leads to about 30% accuracy on the magnetic moments in the favorable sum rules. Further improvement of the accuracy by reducing the errors in the input is beyond the capability of these sum rules as the $\chi^2/N_{DF}$ becomes unacceptably large, signaling internal inconsistency of the sum rules. For that purpose, one would have to resort to finding sum rules that have better convergence properties and depend less critically on the poorly-known $\chi$.
To get a different perspective on how the spectral parameters depend on the input parameters, we study correlations among the parameters by way of scatter plots. In the Monte-Carlo analysis, all the parameters are correlated. Therefore, one can study the correlations between any two parameters by looking at their scatter plots. Such plots are useful in revealing how a particular sum rule resolves the spectral properties. We have examined numerous such plots. Here we focus on the favorable sum rules as selected earlier. To conserve space, we only give two examples. Fig. \[corr-omeg-we5\] shows the scatter plot for correlations between $\Omega^-$ magnetic moment and the QCD input parameters for the sum rule from WE$_5$. Fig. \[corr-sig0-wo8\] shows a similar plot for ${\Sigma^*}^0$ and the sum rule from WO$_8$. Perhaps the most interesting feature is the strong correlations with $\chi$ in both sum rules. This is the reason for the large sensitivities to this parameter as alluded to earlier. Precise determination of $\chi$ is crucial for keeping the uncertainties in the spectral parameters under control. Other charged members (all use sum rules from WE$_5$) display qualitatively the same patterns for parameters other than $\chi$ and the factorization violation parameter $\kappa_v$. For $\chi$, positively-charged members ($\Delta^{++}$ and ${\Sigma^*}^+$) show negative correlations. The opposite is true for negatively-charged members ($\Omega^-$, ${\Sigma^*}^-$ and ${\Xi^*}^-$): they show positive correlations with $\chi$. The patterns for $\kappa_v$ essentially follow those for $\chi$, although the correlations are weaker. The correlation patterns for ${\Xi^*}^0$ are qualitatively the same as those for ${\Sigma^*}^0$.
Table \[comp\] shows a comparison of the magnetic moments from various calculations and existing experimental data. The results with 10% errors from the QCD sum rule method are used in the comparison. Note that the central values are slightly different from those in Tables \[tabdo\] to \[tabxi\] where conservative uncertainties were used. The reason is that the resultant distributions vary with input errors and are not Gaussian in this case. In such event the median and the average of the asymmetric errors are quoted. The QCDSR results are consistent with data, although the central value for $\Omega^-$ is slightly underestimated. We would like to point out that it is possible to reproduce the central value for $\Omega^-$ (using it as input) by fine-tuning of the susceptibility $\chi$ alone, given the sensitivity to this parameter and the large freedom at the present time on its value. However, we feel that such an attempt is not very meaningful given the accuracy of the method. A more meaningful practice would be to re-analyze the octet baryon magnetic moments by the same method as employed here, and obtain a best fit on the the susceptibilities using their accurately measured values, then use them to predict the decuplet magnetic moments. It would yield valuable information on these important quantities and on the consistency of the approach. From the table, it is fair to say that the QCDSR approach is at least competitive with other calculations. The results came about from a rather different perspective: the nonperturbative structure of the QCD vacuum. The results from various calculations roughly agree, except for the charge-neutral resonances $\Delta^{0}$, ${\Sigma^*}^{0}$, and ${\Xi^*}^0$ for which both the sign and the magnitude vary. It would be helpful to have experimental information on the other members of the decuplet, although such measurements appear difficult.
Conclusion {#sumcon}
==========
It has been demonstrated in this work that the magnetic moments of decuplet baryons can be successfully computed in the QCD sum rule approach. A complete set of QCD sum rules are derived using the external field technique. They are analyzed extensively with a comprehensive Monte-Carlo based procedure which , in our opinion, provides the most realistic estimates of the uncertainties present in the approach.
Valid sum rules are identified using criteria established by OPE convergence and ground-state dominance. For each member, usually several sum rules are valid, but not all of them perform equally well. This was best displayed by the overlap plots. Some have large deviations in the lower end of the Borel window, signaling insufficient convergence in the OPE. These sum rules are less reliable. Based on overall performance, a favorable sum rule was selected for each member. They are WE$_5$ for charged members, WO$_8$ for charge-neutral members. We also found the following relations between the magnetic moments: $\mu_{\scriptscriptstyle \Delta^+}=
{1\over 2} \, \mu_{\scriptscriptstyle \Delta^{++}}$, $\mu_{\scriptscriptstyle \Delta^0}=0$, and $\mu_{\scriptscriptstyle \Delta^-}=
-\, \mu_{\scriptscriptstyle \Delta^+}$, and approximately $\mu_{\scriptscriptstyle {\Sigma^*}^0}={1\over 2}
(\mu_{\scriptscriptstyle {\Sigma^*}^r+}
+ \mu_{\scriptscriptstyle {\Sigma^*}^-} )$. .
Using conservative estimates of the QCD input parameters, the uncertainties in the extracted magnetic moments are found relatively large as compared to the two-point functions. We found that the results are sensitive to the quark condensate magnetic susceptibility $\chi$. In fact, most of the uncertainties could be attributed to $\chi$. Better estimate of this parameter is clearly needed. By varying the uncertainty estimates in the input parameters, we found that a 30% accuracy can be achieved with the designated sum rules if the QCD input parameters could be determined to the 10% accuracy level.
It is a pleasure to thank D.B. Leinweber for providing an original version of his Monte-Carlo analysis program and for helpful discussions. This work was supported in part by U.S. DOE under Grant DE-FG03-93DR-40774.
QCD Sum Rules for Magnetic Moments of Decuplet Baryons {#qcdsr}
======================================================
Here we give the complete set of QCD sum rules derived in this work. Foe each member, there are 18 sum rules, 9 chiral-even, 9 chiral-odd. It turns out that the sum rules from WE$_6$ and WE$_7$ are degenerate, so are those from WO$_4$ and WO$_5$. So the number of independent sum rules is 16 for each member. The total number for the entire decuplet family is 160. They are given in the following in a highly compact form. The explanation on how to obtain a sum rule for a particular member is discussed in the main text.
The sum rule from WE$_1$: $$\begin{aligned}
& &
c_1 L^{4/27} E_1 M^4
+ c_2 m_s \chi a L^{-12/27} E_0 M^2
+ c_3 b L^{4/27}
+ c_4 \chi a^2 L^{12/27}
+ (c_5+c_6) m_s a L^{4/27}
\nonumber \\ & &
+ (c_7+c_8) a^2 L^{28/27} {1\over M^2}
+ c_9 \chi m^2_0 a^2 L^{-2/27} {1\over M^2}
+ c_{10} m_s m^2_0 a L^{-10/27} {1\over M^2}
\nonumber \\ & &
={1\over 2} \tilde{\lambda}^2_B
\left( {\mu_{\scriptscriptstyle B}\over M^2} + A \right) e^{-M^2_B/M^2},
\label{we1}\end{aligned}$$ where the coefficients for ${\Sigma^*}^+$ are: $$\begin{array}{ll}
c_1={1\over 8}(e_s + 2e_u), &
c_2={-7\over 18}(e_s f \phi + 2e_u), \\
c_3={-1\over 72}(e_s + 2e_u), &
c_4={-1\over 9}(e_s f \phi + e_u f + e_u),\\
c_5={1\over 18}(-2e_s f + 9e_s + 9e_u f + 5e_u), &
c_6={-1\over 18}(e_s f \phi + 2e_u)(7\kappa+\xi), \\
c_7={1\over 27}(-e_s f + 3e_s + 5e_u f - e_u)\kappa_v,&
c_8={-1\over 54}(e_s f \phi + e_u f + e_u)(7\kappa+\xi),\\
c_9={7\over 216}(e_s f \phi + e_u f + e_u),&
c_{10}={-5\over 72}(e_s + e_u f + e_u),
\end{array}$$ for ${\Sigma^*}^0$: $$\begin{array}{ll}
c_1={1\over 8}(e_s + e_u + e_d), &
c_2={-7\over 18}(e_s f \phi + e_u + e_d), \\
c_3={-1\over 72}(e_s + e_u + e_d), &
c_4={-1\over 9}(e_s f \phi + (e_u+e_d)(f+1)/2),\\
c_5={1\over 18}(-2e_s f + 9e_s + (e_u+e_d)(9f+5)/2), &
c_6={-1\over 18}(e_s f \phi + e_u + e_d)(7\kappa+\xi), \\
c_7={1\over 27}(-e_s f + 3e_s + (e_u+e_d)(5f-1)/2)\kappa_v, &
c_8={-1\over 54}(e_s f \phi + (e_u+e_d)(f+1)/2)(7\kappa+\xi),\\
c_9={7\over 216}(e_s f \phi+ (e_u+e_d)(f+1)/2),&
c_{10}={-5\over 72}(e_s + (e_u+e_d)(f+1)/2),
\end{array}$$ for ${\Xi^*}^0$: $$\begin{array}{ll}
c_1={1\over 8}(2e_s + e_u), &
c_2={-7\over 18}(2e_s f \phi + e_u), \\
c_3={-1\over 72}(2e_s + e_u), &
c_4={-1\over 9} f (e_s f \phi + e_s \phi + e_u),\\
c_5={1\over 18}(5e_s f + 9e_s + 9e_u f - 2e_u), &
c_6={-1\over 18}(2e_s f \phi + e_u)(7\kappa+\xi), \\
c_7={1\over 27} f (-e_s f + 5e_s + 3e_u f - e_u)\kappa_v, &
c_8={-1\over 54} f (e_s f \phi + e_s \phi + e_u)(7\kappa+\xi),\\
c_9={7\over 216} f (e_s f \phi + e_s \phi + e_u),&
c_{10}={-5\over 72}(e_s f + e_u f + e_s),
\end{array}$$ for $\Omega^-$: $$\begin{array}{ll}
c_1={3\over 8}e_s, &
c_2={-7\over 6}e_s f \phi, \\
c_3={-1\over 24}e_s, &
c_4={-1\over 3}e_s f^2 \phi,\\
c_5={7\over 6}e_s f, &
c_6={-1\over 6}e_s f \phi(7\kappa+\xi), \\
c_7={2\over 9}e_s f^2\kappa_v, &
c_8={-1\over 17}e_s f^2 \phi(7\kappa+\xi),\\
c_9={7\over 72}e_s f^2 \phi,&
c_{10}={-5\over 24}e_s f.
\end{array}$$ The sum rule from WE$_2$: $$\begin{aligned}
& &
c_1 L^{4/27} E_0 M^2
+ c_2 m_s \chi a L^{-12/27}
+ c_3 b L^{4/27} {1\over M^2}
+ c_4 m_s a L^{4/27} {1\over M^2}
+ c_5 m_s m^2_0 a L^{-10/27} {1\over M^4}
\nonumber \\ & &
={-1\over 9} \tilde{\lambda}^2_B
\left( {\mu_{\scriptscriptstyle B}\over M^2_B\,M^2} + A \right) e^{-M^2_B/M^2},
\label{we2}\end{aligned}$$ where the coefficients for ${\Sigma^*}^+$ are: $$\begin{array}{l}
c_1={-1\over 12}(e_s + 2e_u),
c_2={-1\over 9}(e_s f \phi + 2e_u),
c_3={-1\over 48}(e_s + 2e_u),
c_4={ 1\over 3}(e_s + e_u f + e_u), \\
c_5={1\over 18}(e_s + e_u f + e_u),
\end{array}$$ for ${\Xi^*}^0$: $$\begin{array}{l}
c_1={-1\over 12}(2e_s + e_u),
c_2={-1\over 9}(2e_s f \phi + e_u),
c_3={-1\over 48}(2e_s + e_u),
c_4={ 1\over 3}(e_s f + e_u f + e_s), \\
c_5={1\over 18}(e_s f + e_u f + e_s).
\end{array}$$
The sum rule from WE$_3$: $$\begin{aligned}
& &
c_1 L^{4/27} E_1 M^4
+ c_2 m_s \chi a L^{-12/27} E_0 M^2
+ c_3 b L^{4/27}
+ c_4 \chi a^2 L^{12/27}
+ (c_5+c_6) m_s a L^{4/27}
\nonumber \\ & &
+ (c_7+c_8) a^2 L^{28/27} {1\over M^2}
+ c_9 \chi m^2_0 a^2 L^{-2/27} {1\over M^2}
+ c_{10} m_s m^2_0 a L^{-10/27} {1\over M^2}
\nonumber \\ & &
={-7\over 18} \tilde{\lambda}^2_B
\left( {\mu_{\scriptscriptstyle B}\over M^2} + A \right) e^{-M^2_B/M^2},
\label{we3}\end{aligned}$$ where the coefficients for ${\Sigma^*}^+$ are: $$\begin{array}{ll}
c_1={-1\over 24}(e_s + 2e_u), &
c_2={ 5\over 24}(e_s f \phi + 2e_u), \\
c_3={ 1\over 576}(e_s + 2e_u), &
c_4={ 1\over 18}(e_s f \phi + e_u f_s + e_u),\\
c_5={1\over 36}(2e_s f - 3e_s - 3e_u f + e_u), &
c_6={ 1\over 36}(e_s f \phi + 2e_u)(4\kappa+\xi), \\
c_7={1\over 54}( e_s f - 3e_s/2 - 2e_u f + e_u)\kappa_v &
c_8={ 1\over 108}(e_s f \phi + e_u f + e_u)(4\kappa+\xi),\\
c_9={-7\over 432}(e_s f \phi + e_u f + e_u),&
c_{10}={ 1\over 48}(e_s + e_u f + e_u),
\end{array}$$ for ${\Xi^*}^0$: $$\begin{array}{ll}
c_1={-1\over 24}(2e_s + e_u), &
c_2={ 5\over 24}(2e_s f \phi + e_u), \\
c_3={ 1\over 576}(2e_s + e_u), &
c_4={ 1\over 18} f (e_s f \phi + e_s \phi + e_u),\\
c_5={1\over 36}(e_s f - 3e_s - 3e_u f + 2e_u), &
c_6={ 1\over 36}(2e_s f \phi + e_u)(4\kappa+\xi), \\
c_7={1\over 54} f ( e_s f - 2e_s - 3e_u f/2 + e_u)\kappa_v &
c_8={ 1\over 108} f (e_s f \phi + e_s \phi + e_u)(4\kappa+\xi),\\
c_9={-7\over 432} f (e_s f \phi + e_s \phi + e_u),&
c_{10}={ 1\over 48}(e_s f + e_u f + e_s).
\end{array}$$
The sum rule from WE$_4$: $$\begin{aligned}
& &
c_1 L^{4/27} E_1 M^4
+ c_2 m_s \chi a L^{-12/27} E_0 M^2
+ c_3 b L^{4/27}
+ (c_4+c_5) m_s a L^{4/27}
\nonumber \\ & &
+ (c_6+c_7) a^2 L^{28/27} {1\over M^2}
+ c_8 m^2_0 a^2 L^{14/27} {1\over M^4}
\nonumber \\ & &
={7\over 18} \tilde{\lambda}^2_B
\left( {\mu_{\scriptscriptstyle B}\over M^2} + A \right) e^{-M^2_B/M^2},
\label{we4}\end{aligned}$$ where the coefficients for ${\Sigma^*}^+$ are: $$\begin{array}{ll}
c_1={1\over 24}(e_s + 2e_u), &
c_2={-5\over 36}(e_s f \phi + 2e_u), \\
c_3={-1\over 96}(e_s + 2e_u), &
c_4={1\over 18}(e_s f + 6e_s + 6e_u f + 8e_u), \\
c_5={-1\over 72}(e_s f \phi + 2e_u)(10\kappa+\xi),&
c_6={1\over 27}( e_s f + 3e_s/2 + 4e_u f + e_u)\kappa_v, \\
c_7={-1\over 108}(e_s f \phi + e_u f + e_u)(4\kappa+\xi), &
c_8={-7\over 648}( e_s f + e_u f + e_u),
\end{array}$$ for ${\Xi^*}^0$: $$\begin{array}{ll}
c_1={1\over 24}(2e_s + e_u), &
c_2={-5\over 36}(2e_s f \phi + e_u), \\
c_3={-1\over 96}(2e_s + e_u), &
c_4={1\over 18}(8e_s f + 6e_s + 6e_u f + e_u), \\
c_5={-1\over 72}(2e_s f \phi + e_u)(10\kappa+\xi),&
c_6={1\over 27} f ( e_s f + 4e_s + 3e_u f/2 + e_u)\kappa_v, \\
c_7={-1\over 108} f (e_s f \phi + e_s \phi + e_u)(4\kappa+\xi), &
c_8={-7\over 648} f ( e_s f + e_s + e_u).
\end{array}$$
The sum rule from WE$_5$: $$\begin{aligned}
& &
c_1 L^{4/27} E_1 M^4
+ c_2 m_s \chi a L^{-12/27} E_0 M^2
+ c_3 b L^{4/27}
+ (c_4+c_5) m_s a L^{4/27}
+ c_6 \chi a^2 L^{12/27}
\nonumber \\ & &
+ (c_7+c_8) a^2 L^{28/27} {1\over M^2}
+ c_9 \chi m^2_0 a^2 L^{-2/27} {1\over M^2}
+ c_{10} m_s m^2_0 a L^{-10/27} {1\over M^2}
+ c_{11} m^2_0 a^2 L^{14/27} {1\over M^4}
\nonumber \\ & &
={-7\over 18} \tilde{\lambda}^2_B
\left( {\mu_{\scriptscriptstyle B}\over M^2} + A \right) e^{-M^2_B/M^2},
\label{we5}\end{aligned}$$ where the coefficients for ${\Sigma^*}^+$ are: $$\begin{array}{ll}
c_1={-1\over 24}(e_s + 2e_u), &
c_2={ 5\over 18}(e_s f \phi + 2e_u), \\
c_3={-1\over 144}(e_s + 2e_u), &
c_4={ 1\over 6}(e_s f + e_s + e_u f + 3e_u), \\
c_5={ 1\over 24}(e_s f \phi + 2e_u)(2\kappa+\xi), &
c_6={ 1\over 9}(e_s f \phi + e_u f + e_u), \\
c_7={2\over 27}( e_s f + e_u f + e_u)\kappa_v, &
c_8={ 1\over 108}(e_s f \phi + e_u f + e_u)(4\kappa+\xi), \\
c_9={-7\over 216}( e_s f \phi + e_u f + e_u), &
c_{10}={ 1\over 24}(e_s + e_u f + e_u), \\
c_{11}={-7\over 648}(e_s f + e_u f + e_u),
\end{array}$$ for ${\Xi^*}^0$: $$\begin{array}{ll}
c_1={-1\over 24}(2e_s + e_u), &
c_2={ 5\over 18}(2 e_s f \phi + e_u), \\
c_3={-1\over 144}(2e_s + e_u), &
c_4={ 1\over 6}(3e_s f + e_s + e_u f + e_u), \\
c_5={ 1\over 24}(2e_s f \phi + e_u)(2\kappa+\xi), &
c_6={ 1\over 9} f (e_s f \phi + e_s \phi + e_u), \\
c_7={2\over 27} f ( e_s f + e_s + e_u)\kappa_v, &
c_8={ 1\over 108} f (e_s f \phi + e_s\phi + e_u)(4\kappa+\xi), \\
c_9={-7\over 216} f ( e_s f \phi + e_s \phi + e_u), &
c_{10}={ 1\over 24}(e_s f + e_u f + e_s), \\
c_{11}={-7\over 648} f (e_s f + e_s + e_u).
\end{array}$$
The sum rule from WE$_6$: $$\begin{aligned}
& &
c_1 m_s \chi a L^{-12/27} E_0 M^2
+ (c_2+c_3) m_s a L^{4/27}
+ (c_4+c_5) a^2 L^{28/27} {1\over M^2}
\nonumber \\ & &
+ c_6 m_s m^2_0 a L^{-10/27} {1\over M^2}
+ c_7 m^2_0 a^2 L^{14/27} {1\over M^4}
\nonumber \\ & &
={2\over 3} \tilde{\lambda}^2_B
\left( {\mu_{\scriptscriptstyle B}\over M^2} + A \right) e^{-M^2_B/M^2},
\label{we6}\end{aligned}$$ where the coefficients for ${\Sigma^*}^+$ are: $$\begin{array}{ll}
c_1={-2\over 3}(e_s f \phi + 2e_u),&
c_2={1\over 18}(e_s f + 2e_u), \\
c_3={ 1\over 72}(e_s f \phi + 2e_u)(2\kappa-\xi), &
c_4={1\over 27} (e_s f + e_u f + e_u)\kappa_v,\\
c_5={ 1\over 108} (e_s f \phi + e_u f + e_u)(2\kappa-\xi),&
c_6={1\over 36}(e_s + e_u f + e_u),\\
c_7={-7\over 648}(e_s f + e_u f + e_u),
\end{array}$$ for ${\Xi^*}^0$: $$\begin{array}{ll}
c_1={-2\over 3}(2 e_s f \phi + e_u),&
c_2={1\over 18}(2e_s f + e_u), \\
c_3={ 1\over 72}(2e_s f \phi + e_u)(2\kappa-\xi), &
c_4={1\over 27} f (e_s f + e_s + e_u)\kappa_v,\\
c_5={ 1\over 108} f (e_s f \phi + e_s \phi + e_u)(2\kappa-\xi),&
c_6={1\over 36}(e_s f + e_u f + e_s),\\
c_7={-7\over 648} f (e_s f + e_s + e_u).
\end{array}$$
The sum rule from WE$_7$ is identical to that from WE$_6$ after multiplying an overall sign on both sides.
The sum rule from WE$_8$: $$\begin{aligned}
& &
c_1 m_s \chi a L^{-12/27} E_0 M^2
+ c_2 b L^{4/27}
+ c_3 \chi a^2 L^{12/27}
+ (c_4+c_5) m_s a L^{4/27}
\nonumber \\ & &
+ (c_6+c_7) a^2 L^{28/27} {1\over M^2}
+ c_8 \chi m^2_0 a^2 L^{-2/27} {1\over M^2}
+ c_9 m_s m^2_0 a L^{-10/27} {1\over M^2}
+ c_{10} m^2_0 a^2 L^{14/27} {1\over M^4}
\nonumber \\ & &
={-2\over 3} \tilde{\lambda}^2_B
\left( {\mu_{\scriptscriptstyle B}\over M^2} + A \right) e^{-M^2_B/M^2},
\label{we8}\end{aligned}$$ where the coefficients for ${\Sigma^*}^+$ are: $$\begin{array}{ll}
c_1={1\over 3}(e_s f \phi + 2e_u), &
c_2={ 5\over 144}(e_s + 2e_u), \\
c_3={-2\over 9}(e_s f \phi + e_u f + e_u), &
c_4={-1\over 9}(5e_s f + 12e_s + 12e_u f + 22e_u),\\
c_5={1\over 36}(e_s f \phi + 2e_u)(14\kappa-\xi), &
c_6={-1\over 27}(8e_s f + 6e_s + 20e_u f + 8e_u)\kappa_v,\\
c_7={ 1\over 54}(e_s f \phi + e_u f + e_u)(4\kappa+\xi),&
c_8={ 7\over 108}(e_s f \phi + e_u f + e_u), \\
c_9={1\over 36}( e_s + e_u f + e_u),&
c_{10}={ 7\over 108}(e_s f + e_u f + e_u),
\end{array}$$ for ${\Xi^*}^0$: $$\begin{array}{ll}
c_1={1\over 3}(2e_s f \phi + e_u), &
c_2={ 5\over 144}(2e_s + e_u), \\
c_3={-2\over 9} f (e_s f \phi + e_s \phi + e_u), &
c_4={-1\over 9}(22e_s f + 12e_s + 12e_u f + 5e_u),\\
c_5={1\over 36}(2e_s f \phi + e_u)(14\kappa-\xi), &
c_6={-1\over 27} f (8e_s f + 20e_s + 6e_u f + 8e_u)\kappa_v,\\
c_7={ 1\over 54} f (e_s f \phi + e_s \phi + e_u)(4\kappa+\xi),&
c_8={ 7\over 108} f (e_s f \phi + e_s\phi + e_u), \\
c_9={1\over 36}(e_s f + e_u f + e_s),&
c_{10}={ 7\over 108} f (e_s f + e_s + e_u).
\end{array}$$
The sum rule from WE$_9$ has the same form as that from WE$_8$, only with different $c_i$: $$\begin{array}{ll}
c_1=(e_s f \phi + 2e_u), &
c_2={-5\over 144}(e_s + 2e_u), \\
c_3={ 2\over 9}(e_s f \phi + e_u f + e_u), &
c_4={ 1\over 9}(4e_s f + 12e_s + 12e_u f + 20e_u),\\
c_5={1\over 18}(e_s f \phi + 2e_u)(-8\kappa+\xi), &
c_6={1\over 9}(2e_s f + 2e_s + 6e_u f + 2e_u)\kappa_v,\\
c_7={-1\over 9}(e_s f \phi + e_u f + e_u)\kappa,&
c_8={-7\over 108}(e_s f \phi + e_u f + e_u), \\
c_9={-1\over 12}(e_s + e_u f + e_u),&
c_{10}={-7\over 162}(e_s f + e_u f + e_u),
\end{array}$$ for ${\Xi^*}^0$: $$\begin{array}{ll}
c_1=(2e_s f \phi + e_u), &
c_2={-5\over 144}(2e_s + e_u), \\
c_3={ 2\over 9} f (e_s f \phi + e_s \phi + e_u), &
c_4={ 1\over 9}(20e_s f + 12e_s + 12e_u f + 4e_u),\\
c_5={1\over 18}(2 e_s f \phi + e_u)(-8\kappa+\xi), &
c_6={1\over 9} f (2e_s f + 6e_s + 2e_u f + 2e_u)\kappa_v,\\
c_7={-1\over 9} f (e_s f \phi + e_s \phi + e_u)\kappa,&
c_8={-7\over 108} f (e_s f \phi + e_s \phi + e_u), \\
c_9={-1\over 12}(e_s f + e_u f + e_s),&
c_{10}={-7\over 162} f (e_s f + e_s + e_u).
\end{array}$$
The sum rule from WO$_1$: $$\begin{aligned}
& &
c_1 \chi a E_1 M^4
+ c_2 m_s L^{16/27} E_1 M^4
+ (c_3+c_4) a L^{16/27} E_0 M^2
+ c_5 m^2_0 a L^{2/27}
\nonumber \\ & &
+ c_6 \chi a b
+ c_7 m_s \chi a^2
+ c_8 a b L^{16/27} {1\over M^2}
+ (c_9+c_{10}) m_s a^2 L^{16/27} {1\over M^2}
\nonumber \\ & &
={1\over 2} \tilde{\lambda}^2_B
\left( {\mu_{\scriptscriptstyle B}\,M_B\over M^2} + A \right) e^{-M^2_B/M^2},
\label{wo1}\end{aligned}$$ where the coefficients for ${\Sigma^*}^+$ are: $$\begin{array}{ll}
c_1={-1\over 6}(e_s f \phi + 2e_u), &
c_2={ 1\over 6}(e_s + 2e_u), \\
c_3={-5\over 27}(e_s f + 2e_u), &
c_4={ 5\over 432}(e_s f \phi + 2e_u)(-8\kappa+7\xi),\\
c_5={1\over 12}(e_s + e_u f + e_u), &
c_6={ 1\over 96}(e_s f \phi + 2e_u), \\
c_7={-2\over 9}(e_s f \phi + e_u f + e_u), &
c_8={-1\over 216}(e_s + e_u f + e_u),\\
c_9={1\over 27}(-2e_s f + 3e_s + 4e_u f - 2e_u)\kappa_v,&
c_{10}={-5\over 432}(e_s f \phi + e_u f + e_u)(8\kappa+11\xi),
\end{array}$$ for ${\Xi^*}^0$: $$\begin{array}{ll}
c_1={-1\over 6}(2e_s f \phi + e_u), &
c_2={ 1\over 6}(2 e_s + e_u), \\
c_3={-5\over 27}(2e_s f + e_u), &
c_4={ 5\over 432}(2e_s f \phi + e_u)(-8\kappa+7\xi),\\
c_5={1\over 12}(e_s f + e_u f + e_s), &
c_6={ 1\over 96}(2e_s f \phi + e_u), \\
c_7={-2\over 9} f (e_s f \phi + e_s \phi + e_u), &
c_8={-1\over 216}(e_s f + e_u f + e_s),\\
c_9={1\over 27} f (-2e_s f + 4e_s + 3e_u f - 2e_u)\kappa_v,&
c_{10}={-5\over 432} f (e_s f \phi + e_s \phi + e_u)(8\kappa+11\xi).
\end{array}$$ The sum rule from WO$_2$: $$\begin{aligned}
& &
c_1 \chi a E_0 M^2
+ (c_2+c_3) a L^{16/27}
+ c_4 m_s \chi a^2
+ c_5 \chi a b {1\over M^2}
+ (c_6+c_7) m_s a^2 L^{16/27} {1\over M^4}
\nonumber \\ & &
={-1\over 9} \tilde{\lambda}^2_B
\left( {\mu_{\scriptscriptstyle B}\over M_B\,M^2} + A \right) e^{-M^2_B/M^2},
\label{wo2}\end{aligned}$$ where the coefficients for ${\Sigma^*}^+$ are: $$\begin{array}{ll}
c_1={ 1\over 9}(e_s f \phi + 2e_u), &
c_2={ 5\over 27}(e_s f + 2e_u), \\
c_3={ 1\over 54}(e_s f \phi + 2e_u)(5\kappa+2\xi), &
c_4={ -2\over 9}(e_s f \phi + e_u f + e_u),\\
c_5={1\over 216}(e_s f \phi + 2e_u), &
c_6={-2\over 9}(e_s f + e_u f + e_u)\kappa_v, \\
c_7={-1\over 9}(e_s f \phi + e_u f + e_u)\kappa,
\end{array}$$ for ${\Xi^*}^0$: $$\begin{array}{ll}
c_1={ 1\over 9}(2e_s f \phi + e_u), &
c_2={ 5\over 27}(2e_s f + e_u), \\
c_3={ 1\over 54}(2e_s f \phi + e_u)(5\kappa+2\xi), &
c_4={ -2\over 9} f (e_s f \phi + e_s \phi + e_u),\\
c_5={1\over 216}(2e_s f \phi + e_u), &
c_6={-2\over 9} f (e_s f + e_s + e_u)\kappa_v, \\
c_7={-1\over 9} f (e_s f \phi + e_s\phi + e_u)\kappa.
\end{array}$$
The sum rule from WO$_3$: $$\begin{aligned}
& &
c_1 \chi a E_1 M^4
+ (c_2+c_3) a L^{16/27} E_0 M^2
+ c_4 m^2_0 a L^{2/27}
+ c_5 \chi a b
+ c_6 m_s\chi a^2
\nonumber \\ & &
+ c_7 a b L^{16/27} {1\over M^2}
+ (c_8+c_9) m_s a^2 L^{16/27} {1\over M^2}
\nonumber \\ & &
={-7\over 18} \tilde{\lambda}^2_B
\left( {\mu_{\scriptscriptstyle B}\,M_B\over M^2} + A \right) e^{-M^2_B/M^2},
\label{wo3}\end{aligned}$$ where the coefficients for ${\Sigma^*}^+$ are: $$\begin{array}{ll}
c_1={ 5\over 72}(e_s f \phi + 2e_u), &
c_2={1\over 72}(7e_s f + 6 e_s + 6 e_u f + 20e_u), \\
c_3={ 1\over 288}(e_s f \phi + 2e_u)(2\kappa-11\xi), &
c_4={-1\over 12}(e_s + e_u f + e_u),\\
c_5={-11\over 1728}(e_s f \phi + 2e_u), &
c_6={ 1\over 12}(e_s f \phi + e_u f + e_u), \\
c_7={5\over 1728}(e_s + e_u f + e_u),&
c_8={-1\over 36}(e_s f + 3e_s + 7e_u f + e_u)\kappa_v,\\
c_9={ 1\over 432}(e_s f \phi + e_u f + e_u)(12\kappa+7\xi),
\end{array}$$ for ${\Xi^*}^0$: $$\begin{array}{ll}
c_1={ 5\over 72}(2e_s f \phi + e_u), &
c_2={1\over 72}(20e_s f + 6 e_s + 6 e_u f + 7e_u), \\
c_3={ 1\over 288}(2e_s f \phi + e_u)(2\kappa-11\xi), &
c_4={-1\over 12}(e_s f + e_u f + e_s),\\
c_5={-11\over 1728}(2e_s f \phi + e_u), &
c_6={ 1\over 12} f (e_s f \phi + e_s\phi + e_u), \\
c_7={5\over 1728}(e_s f + e_u f + e_s),&
c_8={-1\over 36} f (e_s f + 7e_s + 3e_u f + e_u)\kappa_v,\\
c_9={ 1\over 432} f (e_s f \phi + e_s \phi + e_u)(12\kappa+7\xi).
\end{array}$$
The sum rule from WO$_4$: $$\begin{aligned}
& &
c_1 m_s L^{-8/27} E_0 M^2
+ (c_2+c_3) a L^{16/27}
+ c_4 m_s\chi a^2
+ c_5 m^2_0 a L^{2/27} {1\over M^2}
+ c_6 \chi a b {1\over M^2}
\nonumber \\ & &
+ c_7 a b L^{16/27} {1\over M^4}
+ (c_8+c_9) m_s a^2 L^{16/27} {1\over M^4}
\nonumber \\ & &
={7\over 18} \tilde{\lambda}^2_B
\left( {\mu_{\scriptscriptstyle B}\over M_B\,M^2} + A \right) e^{-M^2_B/M^2},
\label{wo4}\end{aligned}$$ where the coefficients for ${\Sigma^*}^+$ are: $$\begin{array}{ll}
c_1={1\over 6}(e_s + 2e_u), &
c_2={ 1\over 6}(e_s + e_u f + e_u), \\
c_3={-1\over 144}(e_s f \phi + 2e_u)(12\kappa+\xi), &
c_4={ 1\over 9}(e_s f \phi + e_u f + e_u),\\
c_5={-1\over 12}(e_s + e_u f + e_u), &
c_6={-1\over 288}(e_s f \phi + 2e_u), \\
c_7={1\over 432}(e_s + e_u f + e_u),&
c_8={-1\over 18}(e_s + 2e_u f)\kappa_v,\\
c_9={1\over 432}(e_s f \phi + e_u f + e_u)(12\kappa+\xi),
\end{array}$$ for ${\Xi^*}^0$: $$\begin{array}{ll}
c_1={1\over 6}(2e_s + e_u), &
c_2={ 1\over 6}(e_s f + e_u f + e_s), \\
c_3={-1\over 144}(2e_s f \phi + e_u)(12\kappa+\xi), &
c_4={ 1\over 9} f (e_s f \phi + e_s \phi + e_u),\\
c_5={-1\over 12}(e_s f + e_u f + e_s), &
c_6={-1\over 288}(2e_s f \phi + e_u), \\
c_7={1\over 432}(e_s f + e_u f + e_s),&
c_8={-1\over 18} f (2e_s + e_u f)\kappa_v,\\
c_9={1\over 432} f (e_s f \phi + e_s \phi + e_u)(12\kappa+\xi).
\end{array}$$
The sum rule from WO$_5$ is identical to that from WO$_4$ after multiplying an overall sign on both sides.
The sum rule from WO$_6$: $$\begin{aligned}
& &
c_1 \chi a E_1 M^4
+ c_2 m_s L^{-8/27} E_1 M^4
+ (c_3+c_4) a L^{16/27} E_0 M^2
\nonumber \\ & &
+ c_5 m^2_0 a L^{2/27}
+ c_6 \chi a b
+ c_7 a b L^{16/27} {1\over M^2}
+ (c_8+c_9) m_s a^2 L^{16/27} {1\over M^2}
\nonumber \\ & &
={2\over 3} \tilde{\lambda}^2_B
\left( {\mu_{\scriptscriptstyle B}\,M_B\over M^2} + A \right) e^{-M^2_B/M^2},
\label{wo6}\end{aligned}$$ where the coefficients for ${\Sigma^*}^+$ are: $$\begin{array}{ll}
c_1={-1\over 9}(e_s f \phi + 2e_u), &
c_2={-1\over 3}(e_s + 2e_u), \\
c_3={-1\over 54}(11e_s f + 18e_s + 18e_u f + 40e_u), &
c_4={ 1\over 216}(e_s f \phi + 2e_u)(14\kappa+23\xi), \\
c_5={ 1\over 6}(e_s + e_u f + e_u),&
c_6={5\over 432}(e_s f \phi + 2e_u), \\
c_7={-1\over 432}(e_s + e_u f + e_u), &
c_8={1\over 27}(5e_s f + 3e_s + 11e_u f + 5e_u)\kappa_v,\\
c_9={ 1\over 54}(e_s f \phi + e_u f + e_u)(2\kappa-3\xi),
\end{array}$$ for ${\Xi^*}^0$: $$\begin{array}{ll}
c_1={-1\over 9}(2e_s f \phi + e_u), &
c_2={-1\over 3}(2e_s + e_u), \\
c_3={-1\over 54}(40e_s f + 18e_s + 18e_u f + 11e_u), &
c_4={ 1\over 216}(2e_s f \phi + e_u)(14\kappa+23\xi), \\
c_5={ 1\over 6}(e_s f + e_u f + e_s),&
c_6={5\over 432}(2 e_s f \phi + e_u), \\
c_7={-1\over 432}(e_s f + e_u f + e_s), &
c_8={1\over 27} f (5e_s f + 11e_s + 3e_u f + 5e_u)\kappa_v,\\
c_9={ 1\over 54} f (e_s f \phi + e_s\phi + e_u)(2\kappa-3\xi).
\end{array}$$
The sum rule from WO$_7$ has the same form as that from WO$_6$ after multiplying an overall sign on both sides. They only differ in $c_4$ and $c_9$ for ${\Sigma^*}^+$: $$c_4={ 1\over 216}(e_s f \phi + 2e_u)(14\kappa+29\xi),
\hspace{2mm}
c_9={ 1\over 108}(e_s f \phi + e_u f + e_u)(-4\kappa+5\xi),$$ for ${\Xi^*}^0$: $$c_4={ 1\over 216}(2e_s f \phi + e_u)(14\kappa+29\xi),
\hspace{2mm}
c_9={ 1\over 108} f (e_s f \phi + e_s\phi + e_u)(-4\kappa+5\xi).$$
The sum rule from WO$_8$: $$\begin{aligned}
& &
c_1 \chi a E_0 M^2
+ c_2 m_s L^{-8/27} E_0 M^2
+ (c_3+c_4) a L^{16/27}
+ c_5 m^2_0 a L^{2/27} {1\over M^2}
\nonumber \\ & &
+ c_6 \chi a b {1\over M^2}
+ c_7 a b L^{16/27} {1\over M^4}
+ (c_8+c_9) m_s a^2 L^{16/27} {1\over M^4}
\nonumber \\ & &
={-2\over 3} \tilde{\lambda}^2_B
\left( {\mu_{\scriptscriptstyle B}\over M_B\,M^2} + A \right) e^{-M^2_B/M^2},
\label{wo8}\end{aligned}$$ where the coefficients for ${\Sigma^*}^+$ are: $$\begin{array}{ll}
c_1={-1\over 9}(e_s f \phi + 2e_u), &
c_2={-1\over 3}(e_s + 2e_u), \\
c_3={-1\over 27}(5e_s f + 9e_s + 9e_u f + 19e_u), &
c_4={ 1\over 108}(e_s f \phi + 2e_u)(8\kappa-\xi), \\
c_5={ 1\over 6}(e_s + e_u f + e_u),&
c_6={1\over 432}(e_s f \phi + 2e_u), \\
c_7={-1\over 216}(e_s + e_u f + e_u), &
c_8={1\over 9}(2e_s f + e_s + 4e_u f + 2e_u)\kappa_v, \\
c_9={ 1\over 108}(e_s f \phi + e_u f + e_u)(6\kappa-\xi),
\end{array}$$ for ${\Xi^*}^0$: $$\begin{array}{ll}
c_1={-1\over 9}(2e_s f \phi + e_u), &
c_2={-1\over 3}(2e_s + e_u), \\
c_3={-1\over 27}(19e_s f + 9e_s + 9e_u f + 5e_u), &
c_4={ 1\over 108}(2e_s f \phi + e_u)(8\kappa-\xi), \\
c_5={ 1\over 6}(e_s f + e_u f + e_s),&
c_6={1\over 432}(2e_s f \phi + e_u), \\
c_7={-1\over 216}(e_s f + e_u f + e_s), &
c_8={1\over 9} f (2e_s f + 4e_s + e_u f + 2e_u)\kappa_v, \\
c_9={ 1\over 108} f (e_s f \phi + e_s \phi + e_u)(6\kappa-\xi).
\end{array}$$
The sum rule from WO$_9$ has the same form as that from WO$_8$. They only differ in $c_4$ and $c_9$ for ${\Sigma^*}^+$: $$c_4={ 1\over 27}(e_s f \phi + 2e_u)(2\kappa-\xi),
\hspace{2mm}
c_9={ 1\over 18}(e_s f \phi + e_u f + e_u)\kappa,$$ for ${\Xi^*}^0$: $$c_4={ 1\over 27}(2e_s f \phi + e_u)(2\kappa-\xi),
\hspace{2mm}
c_9={ 1\over 18} f (e_s f \phi + e_s\phi + e_u)\kappa.$$
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----------------------- --------------- ------ ------- ------------------ --------------------------------
Sum Rule Region Cont $w$ A $\mu_{\scriptscriptstyle B}$
(GeV) (%) (GeV) (GeV$^{-2}$) $(\mu_{\scriptscriptstyle N})$
$\Delta^{++}$: WE$_1$ 0.70 to 1.53 1.7 1.65 -0.28 $\pm$ 0.52 7.76 $\pm$ 2.67
WE$_3$ 1.04 to 1.42 19 1.65 0.20 $\pm$ 0.20 3.06 $\pm$ 1.14
WE$_4$ 0.675 to 1.56 5 1.65 -0.35$\pm$ 0.37 3.34 $\pm$ 1.44
WE$_5$ 0.765 to 1.47 8.5 1.65 0.53$\pm$ 0.81 3.56$\pm$ 3.49
$\Omega^{-}$: WE$_1$ 0.592 to 1.70 2.3 2.30 -0.12$\pm$ 0.11 -2.66 $\pm$ 0.88
WE$_2$ 0.872 to 1.53 20 2.30 -0.26$\pm$ 0.20 -5.31 $\pm$ 3.66
WE$_3$ 0.885 to 1.68 8.5 2.30 -0.09$\pm$ 0.04 -1.24 $\pm$ 0.51
WE$_4$ 0.60 to 1.72 2 2.30 -0.03$\pm$ 0.05 -1.24 $\pm$ 0.24
WE$_5$ 0.747 to 1.66 7.4 2.30 -0.14$\pm$ 0.14 -1.32 $\pm$ 1.08
WE$_6$ 0.59 to 2.32 0.86 2.30 -0.01 $\pm$ 0.02 -1.14 $\pm$ 0.40
WE$_8$ 0.69 to 2.60 3.3 2.30 0.03 $\pm$ 0.03 -0.65 $\pm$ 1.22
WO$_1$ 0.663 to 1.26 12 2.30 -0.32$\pm$ 0.42 -0.65 $\pm$ 1.22
WO$_2$ 1.06 to 1.43 31 2.30 -0.62$\pm$ 0.18 -4.94 $\pm$ 5.58
WO$_4$ 0.836 to 2.22 7.4 2.30 -0.03$\pm$ 0.01 -0.70 $\pm$ 0.24
----------------------- --------------- ------ ------- ------------------ --------------------------------
: Monte-Carlo analysis of the QCD sum rules for the magnetic moment of $\Delta^{++}$ and $\Omega^{-}$. The six columns correspond to, from left to right: the sum rule that has a valid Borel region, the Borel region determined by the 10%-50% criteria, the percentage contribution of the excited states and transitions to the phenomenological side at the lower end of the Borel region (it increases to 50% at the upper end), the continuum threshold, the transition strength, the magnetic moment in nuclear magnetons. The uncertainties in each sum rule were obtained from consideration of 1000 QCD parameter sets.[]{data-label="tabdo"}
-------------------------- ---------------- ------ ----------------- ------------------ --------------------------------
Sum Rule Region Cont $w$ A $\mu_{\scriptscriptstyle B}$
(GeV) (%) (GeV) (GeV$^{-2}$) $(\mu_{\scriptscriptstyle N})$
${\Sigma^*}^{+}$: WE$_1$ 0.853 to 1.445 11 1.80 0.28$\pm$ 0.18 2.96 $\pm$ 1.41
WE$_3$ 0.996 to 1.39 23 1.80 0.23$\pm$ 0.08 1.49 $\pm$ 0.79
WE$_4$ 0.622 to 1.61 1 1.80 -0.05$\pm$ 0.10 1.74 $\pm$ 0.42
WE$_5$ 0.715 to 1.45 10 1.80 0.34$\pm$ 0.35 1.82 $\pm$ 1.94
0.715 to 1.45 6 2.65 $\pm$ 5.96 0.26$\pm$ 0.49 1.71 $\pm$ 1.96
WE$_6$ 0.575 to 1.96 0.2 1.80 -0.01 $\pm$ 0.08 2.10 $\pm$ 0.79
0.575 to 1.96 0.9 1.56$\pm$ 0.11 -0.06 $\pm$ 0.04 2.00 $\pm$ 0.68
WE$_8$ 0.79 to 2.36 13 1.80 -0.17 $\pm$ 0.08 1.09 $\pm$ 0.71
0.79 to 2.36 15 1.52$\pm$ 5.39 -0.21$\pm$ 0.08 1.08 $\pm$ 0.67
WO$_4$ 0.89 to 1.46 23 1.80 0.07 $\pm$ 0.06 0.39 $\pm$ 0.48
${\Sigma^*}^{0}$: WE$_5$ 0.577 to 1.95 2.8 1.80 0.01 $\pm$ 0.01 0.19 $\pm$ 0.13
WE$_8$ 0.639 to 1.70 9.4 1.80 0.03 $\pm$ 0.01 -0.18 $\pm$ 0.06
WO$_2$ 0.846 to 1.38 18 1.80 0.11 $\pm$ 0.09 1.00 $\pm$ 0.96
WO$_8$ 0.662 to 1.66 5 1.80 -0.01$\pm$ 0.01 -0.30 $\pm$ 0.18
WO$_9$ 0.627 to 1.73 3.4 1.80 -0.01 $\pm$ 0.01 -0.33 $\pm$ 0.19
${\Sigma^*}^{-}$: WE$_1$ 0.662 to 1.54 1 1.80 -0.05$\pm$ 0.19 -3.34 $\pm$ 1.33
WE$_3$ 0.926 to 1.42 16 1.80 -0.17$\pm$ 0.08 -1.42 $\pm$ 0.71
WE$_4$ 0.602 to 1.61 1.3 1.80 0.06 $\pm$ 0.10 -1.70 $\pm$ 0.38
WE$_5$ 0.735 to 1.37 13 1.80 -0.33$\pm$ 0.36 -1.40 $\pm$ 1.74
WE$_6$ 0.588 to 1.97 0.2 1.80 0.01 $\pm$ 0.07 -1.72 $\pm$ 0.63
WE$_8$ 0.71 to 2.51 9.4 1.80 0.15 $\pm$ 0.07 -1.22 $\pm$ 0.65
WO$_1$ 0.618 to 1.05 10 1.80 -0.34 $\pm$ 0.77 -0.66 $\pm$ 1.45
WO$_4$ 0.89 to 1.57 19 1.80 -0.08 $\pm$ 0.05 -0.54 $\pm$ 0.39
-------------------------- ---------------- ------ ----------------- ------------------ --------------------------------
: Same as Table \[tabdo\], but for ${\Sigma^*}^{+}$, ${\Sigma^*}^{0}$ and ${\Sigma^*}^{-}$. The presence of a second row in a specific sum rule indicates that the continuum threshold was successfully searched.[]{data-label="tabsig"}
--------------------- --------------- ------ ------- ------------------- --------------------------------
Sum Rule Region Cont $w$ A $\mu_{\scriptscriptstyle B}$
(GeV) (%) (GeV) (GeV$^{-x}$) $(\mu_{\scriptscriptstyle N})$
${\Xi^*}^0$: WE$_8$ 0.636 to 1.55 13 2.00 0.08 $\pm$ 0.03 -0.35 $\pm$ 0.12
WO$_2$ 0.977 to 1.25 13 2.00 0.11$\pm$ 0.13 2.25 $\pm$ 1.92
WO$_8$ 0.654 to 1.85 3.7 2.00 -0.02$\pm$ 0.02 -0.62 $\pm$ 0.34
WO$_9$ 0.621 to 1.91 2.8 2.00 -0.02$\pm$ 0.02 -0.69 $\pm$ 0.35
${\Xi^*}^-$: WE$_1$ 0.628 to 1.61 1.3 2.00 -0.07$\pm$ 0.14 -2.88 $\pm$ 1.02
WE$_2$ 0.898 to 1.12 37 2.00 -0.30$\pm$ 0.83 -3.65 $\pm$ 5.68
WE$_3$ 0.906 to 1.53 12 2.00 -0.12$\pm$ 0.05 -1.25 $\pm$ 0.55
WE$_4$ 0.6 to 1.66 0.3 2.00 0.0004$\pm$ 0.07 -1.38 $\pm$ 0.27
WE$_5$ 0.74 to 1.50 10 2.00 -0.22$\pm$ 0.21 -1.27 $\pm$ 1.30
WE$_6$ 0.59 to 2.11 0.5 2.00 -0.006 $\pm$ 0.04 -1.38 $\pm$ 0.48
WE$_8$ 0.70 to 2.54 6.3 2.00 0.07 $\pm$ 0.05 -0.88 $\pm$ 0.47
WO$_1$ 0.641 to 1.12 13 2.00 -0.35$\pm$ 0.57 -0.58 $\pm$ 1.26
WO$_4$ 0.863 to 1.86 12 2.00 -0.05$\pm$ 0.02 -0.60 $\pm$ 0.29
--------------------- --------------- ------ ------- ------------------- --------------------------------
: Same as Table \[tabdo\], but for ${\Xi^*}^0$ and ${\Xi^*}^-$.[]{data-label="tabxi"}
Baryon Exp. QCDSR Latt $\chi$PT RQM NQM $\chi$QSM
------------------ -------------------- ------------------ ------------------ ------------------ ------- ------- -----------
$\Delta^{++}$ 4.5 $\pm$ 1.0 4.13 $\pm$ 1.30 4.91 $\pm$ 0.61 4.0 $\pm$ 0.4 4.76 5.56 4.73
$\Delta^{+}$ 2.07 $\pm$ 0.65 2.46 $\pm$ 0.31 2.1 $\pm$ 0.2 2.38 2.73 2.19
$\Delta^{0}$ $\approx$ 0 0.00 0.00 -0.17 $\pm$ 0.04 0.00 -0.09 -0.35
$\Delta^{-}$ -2.07 $\pm$ 0.65 -2.46 $\pm$ 0.31 -2.25 $\pm$ 0.25 -2.38 -2.92 -2.90
${\Sigma^*}^{+}$ 2.13 $\pm$ 0.82 2.55 $\pm$ 0.26 2.0 $\pm$ 0.2 1.82 3.09 2.52
${\Sigma^*}^{0}$ -0.32 $\pm$ 0.15 0.27 $\pm$ 0.05 -0.07 $\pm$ 0.02 -0.27 0.27 -0.08
${\Sigma^*}^{-}$ -1.66 $\pm$ 0.73 -2.02 $\pm$ 0.18 -2.2 $\pm$ 0.2 -2.36 -2.56 -2.69
${\Xi^*}^0$ -0.69 $\pm$ 0.29 0.46 $\pm$ 0.07 0.1 $\pm$ 0.04 -0.60 0.63 0.19
${\Xi^*}^-$ -1.51$\pm$ 0.52 -1.68 $\pm$ 0.12 -2.0 $\pm$ 0.2 -2.41 -2.2 -2.48
$\Omega^{-}$ -2.024 $\pm$ 0.056 -1.49 $\pm$ 0.45 -1.40 $\pm$ 0.10 -2.48 -1.84 -2.27
: Comparisons of decuplet baryon magnetic moments from various calculations: this work (QCDSR), lattice QCD (Latt) [@Derek92], chiral perturbation theory ($\chi$PT) [@Butler94], light-cone relativistic quark model (RQM) [@Schlumpf93], non-relativistic quark model (NQM) [@PDG92], chiral quark-soliton model ($\chi$QSM) [@Kim97]. All results are in units of nuclear magnetons.[]{data-label="comp"}
|
---
abstract: 'In the framework of minimal flavor violation (MFV), we discuss the decay properties of a supersymmetric scalar top (stop) in the presence of a light gravitino. Given a small mass difference between the lighter stop and lightest neutralino and an otherwise sufficiently decoupled spectrum, the stop may be long–lived and thus can provide support to MFV at hadron colliders. For a bino–like lightest neutralino, we apply bounds from searches in the $\gamma\gamma{\ensuremath{\slashed{E}_T}}$ channel (ATLAS with [$\unit[1]{fb^{-1}}$]{} and [D$\slashed{\text{0}}$]{} with [$\unit[6.3]{fb^{-1}}$]{}) and give a [$\unit[5]{fb^{-1}}$]{} projection for the ATLAS search.'
author:
- 'J. S. Kim'
- 'H. Sedello'
title: 'Probing Minimal Flavor Violation with Long–Lived Stops and Light Gravitinos at Hadron Colliders'
---
=1
Introduction {#sec:intro}
============
Supersymmetry (SUSY) [@Ferrara:1974ac] is an attractive extension of the standard model (SM). However the simplest version of a supersymmetric SM, the minimal supersymmetric standard model (MSSM) [@Haber:1997if], does not predict a specific flavor structure; all superrenormalizable soft supersymmetry breaking terms allowed by gauge and Lorentz symmetry as well as $R$ parity are present in its Lagrange density [@Haber:1997if]. However, it is clear that a supersymmetric extension of the standard model must have a non–generic flavor structure to be compatible with experimental results [@Gabbiani:1996hi; @Amsler:2008zzb].
The way the standard model flavor structure is extended to the MSSM is not unique, yet a widely discussed flavor scheme is minimal flavor violation [@D'Ambrosio:2002ex] (MFV). In MFV the standard model Yukawa couplings are promoted to spurion fields transforming under the SM flavor group to restore the SM’s flavor symmetry. If all additional flavor structure of a new physics model can be understood as higher dimensional flavor invariant operators including these spurions and the model’s fields, the model is called MFV.
As the LHC is running and eventually will find supersymmetry, it will be challenging to investigate the flavor structure at a hadron collider due to the detectors’ limited flavor identification abilities and the complexity of the recorded events. In [@Hiller:2008wp] it was pointed out that the third generation’s squarks decouple from the first two generations in MFV. As a result, a light stop can be long–lived decaying through the flavor changing neutral current channel [@Hikasa:1987db; @Muhlleitner:2011ww] $${\ensuremath{\tilde{t}_1}}\rightarrow c{\ensuremath{\tilde{\chi}_1^0}},
\label{eq:mfv-decay}$$ if all flavor diagonal channels are kinematically closed. (${\ensuremath{\tilde{t}_1}}$ denotes the light stop, ${\ensuremath{\tilde{\chi}_1^0}}$ the lightest neutralino and $c$ a charm quark.) An observation of long–living light stops thus would hint in the direction of MFV.
In MFV, the coupling $Y$ between ${\ensuremath{\tilde{t}_1}}$, $c$ and ${\ensuremath{\tilde{\chi}_1^0}}$ is $$Y\propto\lambda_b^2 V_{cb}V_{tb}^*,
\label{eq:defY}$$ where $\lambda_b$ and $V_{ij}$ denotes the bottom Yukawa coupling and elements of the Cabibbo Kobayashi Maskawa (CKM) matrix respectively. The precise value of $Y$ depends on the stop left–right composition, the neutralino decomposition, and on a numeric factor stemming from the MFV expansion; see Ref. [@Hiller:2008wp] for details.
In [@Hiller:2009ii] it is shown that the average transverse impact parameters for the stop decay products can be expected to be ${\mathcal{O}(\unit[1800]{\mu m})}$ for stop lifetimes of the order of ten ps in the production channel $pp\rightarrow\bar t\bar t
{\ensuremath{\tilde{t}_1}}{\ensuremath{\tilde{t}_1}}$ [@Kraml:2005kb] . When both top quarks in this channel decay leptonically, the pair of same signed leptons in the final state allows to separate the signal process from its SM background; however, the small leptonic branching ratio of top quarks suppresses this process so that, after applying all kinematic cuts, only few events are left in this channel. Ref. [@Carena:2008mj] proposes an alternative collider signature assuming stop pair production in association with one hard jet. Demanding a minimum transverse momentum of 1 TeV for the additional jet, the whole parameter region consistent with electroweak baryogenesis can be probed. Ref. [@Bornhauser:2010mw] considers an analogous process, stop pair production in association with two b–jets. However, these two studies do not consider stops in the MFV framework.
If we consider local SUSY instead of a global implementation of SUSY, we can have distinct collider signal signatures with little SM background: In local SUSY, a massive gravitino emerges in the supersymmetric mass spectrum [@Nilles:1983ge]. Its interactions with other particles are severely suppressed by the reduced Planck mass $${\ensuremath{m_{\rm{Pl}}}}=(8\pi G_N)^{-\frac{1}{2}}=2.4\times{\unit[10^{18}]{GeV}},$$ where $G_N$ is Newton’s constant. Depending on the exact breaking mechanism in the hidden sector, the gravitino can be very light. A light gravitino interacts through its goldstino components with couplings proportional to $$({\ensuremath{m_{3/2}}}{\ensuremath{m_{\rm{Pl}}}})^{-1},$$ where ${\ensuremath{m_{3/2}}}$ is the gravitino mass.
In models of gauge mediation [@Dine:1981za; @Dine:1981gu; @Dine:1993yw; @Meade:2008wd; @Buican:2008ws], ${\ensuremath{m_{3/2}}}$ is generally much smaller than the sparticle mass scale; thus, the gravitino is the lightest supersymmetric particle (LSP). Its goldstino interactions are enhanced and can be of the electroweak order.
Consequently, the lightest neutralino decays via $${\ensuremath{\tilde{\chi}_1^0}}\rightarrow X{\ensuremath{\tilde{G}}},
\label{eq:neutr-decay}$$ where $X$ denotes a photon, $Z$, or a Higgs boson; ${\ensuremath{\tilde{G}}}$ denotes a gravitino [@Ambrosanio:1996jn]. If $X$ is a photon ($\gamma$), this decay leads to very clear collider signatures with high $p_T$ isolated photons plus missing transverse energy (${\ensuremath{\slashed{E}_T}}$) stemming from gravitinos leaving the detector unseen.
Several studies with light gravitinos at hadron colliders were performed in the past. Ref. [@Shirai:2009kn; @Baer:1998ve; @Ambrosanio:1996jn] consider the diphoton plus ${\ensuremath{\slashed{E}_T}}$ channel at hadron colliders. In Ref. [@Hamaguchi:2006vu; @Ellis:2006vu] the authors examine a stau NLSP and a gravitino LSP. A sneutrino NLSP and a gravitino LSP scenario is investigated in Ref. [@Katz:2009qx; @Santoso:2009qa]. Ref. [@Meade:2009qv] considers the discovery potential of a neutralino NLSP and a gravitino LSP at the Tevatron, where they consider a general decomposition of the lightest neutralino. In Ref. [@Ambrosanio:1996jn] the authors consider a light stop NNLSP and a light neutralino NLSP and a gravitino LSP. Ref. [@Kats:2011it] investigate a stop NLSP and a gravitino LSP scenario for the Tevatron as well as the LHC. A chargino NLSP and a gravitino LSP is considered in Ref. [@Kribs:2008hq]. Depending on the size of the gravitino mass, non–pointing photons can be measured. The discovery potential of sparticle decays with a finite decay length are investigated in [@Hamaguchi:2007ji; @Feng:2010ij; @Meade:2010ji]. A recent experimental search for sparticles with finite decay lengths is published in [@Aad:2011zb].
In this paper we investigate the parameter region where the decay in Eq. is dominant in the context of a light gravitino, and how the stop masses are constrained in this framework by recent collider searches, assuming that ${\ensuremath{\tilde{t}_1}}$, ${\ensuremath{\tilde{\chi}_1^0}}$, and ${\ensuremath{\tilde{G}}}$ are the only light supersymmetric particles. We discuss the stop and ${\ensuremath{\tilde{\chi}_1^0}}$ decay patterns in section \[sec:patterns\]. In section \[sec:collider\] we apply collider bounds in the $\gamma\gamma{\ensuremath{\slashed{E}_T}}$ channel [@Abazov:2010us; @arXiv:1111.4116]. The cuts adapted from the experimental studies are discussed in the two appendices.
Decay patterns {#sec:patterns}
==============
Light stop decays {#sec:stop}
-----------------
Here we discuss possible decay patterns of light stops and their implications on the parameter space. We start with decays of the light stop via Yukawa and gauge couplings and then discuss direct stop decays into a gravitino.
The lighter stop mass eigenstate ${\ensuremath{\tilde{t}_1}}$ is the lightest squark state in many supersymmetry breaking scenarios. On one hand, the large top Yukawa coupling can induce a sizable left-right mixing in the stop sector leading to light ${\ensuremath{\tilde{t}_1}}$ masses. On the other hand, if the soft squark mass terms are unified at a high scale, the stop mass terms are prominently reduced by the top Yukawa coupling in the running of the renormalization group equations to the electroweak scale [@Ibanez:1984vq]. In this case, ${\ensuremath{\tilde{t}_1}}$ is mostly a $SU(2)$ singlet state since its mass term does not receive any contributions from $SU(2)$ gaugino loops. In addition, right–handed stop loop contributions to the rho parameter are sufficiently suppressed [@Drees:1990dx].
Surprisingly light masses of ${\mathcal{O}({\unit[100]{GeV}})}$ are still consistent with experimental searches for stops decaying to $c{\ensuremath{\tilde{\chi}_1^0}}$ at the Tevatron if the mass splitting $$\Delta m = m_{{\ensuremath{\tilde{t}_1}}}-m_{{\ensuremath{\tilde{\chi}_1^0}}}
\label{eq:massSplitting}$$ is smaller than $\approx{\unit[30]{GeV}}$ [@CDFexotic].
Since we want the flavor changing decay in Eq. to be the dominant decay and the stop to be long–lived, we must ensure that the potentially dominant decays ${\ensuremath{\tilde{t}_1}}\rightarrow t {\ensuremath{\tilde{\chi}_1^0}}$, ${\ensuremath{\tilde{t}_1}}\rightarrow b{\ensuremath{\tilde{\chi}_1}}^\pm$, and ${\ensuremath{\tilde{t}_1}}\rightarrow b{\ensuremath{\tilde{\chi}_1^0}}W$ are kinematically closed or sufficiently suppressed, $$m_{{\ensuremath{\tilde{t}_1}}}<m_b+m_{{\ensuremath{\tilde{\chi}_1^0}}}+ m_{W},\quad m_{{\ensuremath{\tilde{t}_1}}}<m_b+m_{{\ensuremath{\tilde{\chi}_1}}^\pm}.$$
Depending on the chargino and slepton masses , the four–body decay ${\ensuremath{\tilde{t}_1}}\rightarrow b \ell \nu{\ensuremath{\tilde{\chi}_1^0}}$ may be dominant if $\Delta m$ exceeds a few . For small stop neutralino mass splittings, the tree–level four–body decay is strongly phase space suppressed and Eq. remains the dominant decay mode [@Hiller:2008wp].
The MFV decay width can be written as $$\label{eq:sigdecay}
\Gamma({\ensuremath{\tilde{t}_1}}\rightarrow c{\ensuremath{\tilde{\chi}_1^0}}) = \frac{m_{{\ensuremath{\tilde{t}_1}}}
Y^2}{4\pi}\left(\frac{\Delta
m}{m_{{\ensuremath{\tilde{t}_1}}}}\right)^2$$ in the limit of $\Delta m \ll m_{{\ensuremath{\tilde{t}_1}}}$ [@Hiller:2008wp]. Thus, if this decay is dominant, the stop lifetime is governed by $\Delta m
Y$. The kinetic energy of the hadronic stop decay remnants depends on the size of the mass splitting $\Delta m$. In a study with same-sign leptons [@Hiller:2009ii], a major reduction of event numbers due to a minimal $p_T$ cut on these low energy decay products has been found.
In the presence of a light gravitino, restrictions on $\Delta m$ and $Y$ are not sufficient to guarantee the dominance of the FCNC (flavor changing neutral current) decay mode Eq. . If $\Delta m Y$ is too small, the two–body decays of the stop into the gravitino, $${\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{G}}}c,\quad{\ensuremath{\tilde{t}_1}}\rightarrow {\ensuremath{\tilde{G}}}t,
\label{eq:2bGrav}$$ and—if the resonant decay to tops is kinematically closed—the three–body decay mode of the stop, $${\ensuremath{\tilde{t}_1}}\rightarrow {\ensuremath{\tilde{G}}}b W,
\label{eq:3bGrav}$$ can have a significant contribution to the full stop decay width. These decays can invalidate our assumption, that the stop dominantly decays via a FCNC process with a finite impact parameter.
For the flavor diagonal two and three–body decay channels, the decay rates are [@Ambrosanio:1996jn; @Sarid:1999zx]
\[eq:gravdecay\] $$\begin{aligned}
\label{eq:grav2bdecay}
\Gamma({\ensuremath{\tilde{t}_1}}\rightarrow t{\ensuremath{\tilde{G}}}) & =
\frac{1}{48\pi}\frac{m_{{\ensuremath{\tilde{t}_1}}}^5}{{\ensuremath{m_{\rm{Pl}}}}^2{\ensuremath{m_{3/2}}}^2}
\left(1-x_t^2\right)^4\\
\label{eq:grav3bdecay}
\Gamma({\ensuremath{\tilde{t}_1}}\rightarrow W^+b{\ensuremath{\tilde{G}}}) & =
\frac{V_{tb}^2\alpha_{em}}{384\pi^2\sin^2{\theta_W}}
\frac{m_{{\ensuremath{\tilde{t}_1}}}^5}{{\ensuremath{m_{\rm{Pl}}}}^2{\ensuremath{m_{3/2}}}^2}\nonumber\\
\cdot\left[|c_L|^2\right.I\left(\right.&\!\!\!\left.\left.x_W^2,x_t^2\right)
+|c_R|^2J\left(x_W^2,x_t^2\right)\right],
\end{aligned}$$
where the gravitino mass is neglected in the phase space integrals. $\alpha_{em}$, and $\theta_W$ denote the fine-structure constant and the Weinberg angle, respectively. Further $x_W=m_W/m_{{\ensuremath{\tilde{t}_1}}}$ and $x_t=m_t/m_{{\ensuremath{\tilde{t}_1}}}$ with the top mass $m_t$. $c_L$ and $c_R$ parametrize the $\tilde{t}_L$ and $\tilde{t}_R$ contribution to ${\ensuremath{\tilde{t}_1}}$; due to the 3rd generation’s decoupling in MFV the other squarks’ admixture is small, *i.e.* $|c_L|^2+|c_R|^2\approx 1$. The functions $I(x_W^2,x_t^2)$ and $J(x_W^2,x_t^2)$ are phase space integrals and can be found in [@Sarid:1999zx]. Note that Eq. does not comprise the finite top width and diverges at the top mass threshold. We use this formula for $m_{{\ensuremath{\tilde{t}_1}}}<m_t$ only. As the three–body decay proceeds trough a virtual top quark, its rate is largest for a right–handed stop, because the chirality flipping top mass dominates the propagator.
For a bino–like ${\ensuremath{\tilde{\chi}_1^0}}$, the flavor structure of the ${\ensuremath{\tilde{t}_1}}-{\ensuremath{\tilde{G}}}-c$ coupling stemming from the MFV expansion is the same as in the ${\ensuremath{\tilde{t}_1}}-{\ensuremath{\tilde{\chi}_1^0}}-c$ coupling, thus the decay rate for ${\ensuremath{\tilde{t}_1}}\rightarrow {\ensuremath{\tilde{G}}}c$ can be written as [@Hiller:2009ii] $$\label{eq:cgravdecay}
{\ensuremath{\Gamma({\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{G}}}c)}}=
\frac{Y^{\prime^2}}{48\pi}\frac{m_{{\ensuremath{\tilde{t}_1}}}^5}{m_{\rm{Pl}}^2{\ensuremath{m_{3/2}}}^2},$$ where $Y'$ and $Y$ are related by a factor dependent on the stop composition as $Y$ comprises the hypercharges of the left- and right–handed stop fields. The factor is $$\left|\frac{Y'}{Y}\right|=\frac{1}{\sqrt{2}g'Y_Q}\approx
\begin{cases}
3&\text{(right--handed ${\ensuremath{\tilde{t}_1}}$)}\\
12&\text{(left--handed ${\ensuremath{\tilde{t}_1}}$)}
\end{cases}$$ with the SM ${U}(1)$ coupling $g'$ and the left–handed (right–handed) stop hypercharge $Y_Q=\frac{1}{6}\left(\frac{2}{3}\right)$.
We show the branching ratio ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{t}_1}}\rightarrowc{\ensuremath{\tilde{\chi}_1^0}})}}}$ and the stop lifetime in the $m_{{\ensuremath{\tilde{G}}}}$–$Y$ plane for three different masses of a right–handed stop in Fig. \[fig:brs\] using the decay rates , , and . To generate the plots, we keep $\Delta m$ fixed at ${\unit[10]{GeV}}$ and use $\sin^2\theta_W=0.23$, $\alpha_{em}=1/128$, and $m_t={\unit[173]{GeV}}$. The plot for left–handed stops does not differ significantly from the one shown.
Due to the $m_{{\ensuremath{\tilde{t}_1}}}^5$ dependence of the decay widths in Eqs , and the weaker $m_{{\ensuremath{\tilde{t}_1}}}^{-1}$ dependence of $\Gamma({\ensuremath{\tilde{t}_1}}\rightarrow c{\ensuremath{\tilde{\chi}_1^0}})$, the minimal gravitino mass necessary to account for a sizable increases with larger stop masses. The $m_{{\ensuremath{\tilde{t}_1}}}^{-1}$ dependence of $\Gamma({\ensuremath{\tilde{t}_1}}\rightarrow c{\ensuremath{\tilde{\chi}_1^0}})$ also causes the smallness of the shifts of the lifetime regions in Fig. \[fig:brs\] to larger $Y$ values when the stop mass is increased.
As it is clearly visible from Fig. \[fig:brs\], very small values of $Y\lesssim{\mathcal{O}(10^{-5})}$ and at least gravitino masses of ${\mathcal{O}({\unit[0.1-1]{keV}})}$ are required in addition to the mass hierarchy $${\ensuremath{m_{3/2}}}\ll m_{{\ensuremath{\tilde{\chi}_1^0}}}\leq m_{{\ensuremath{\tilde{t}_1}}}\leq m_{{\ensuremath{\tilde{\chi}_1}}^\pm}
\label{eq:massOrder2}$$ for the stop to be long–lived and to decay dominantly to $c{\ensuremath{\tilde{\chi}_1^0}}$.
As the charmed hadron produced in the decay has a macroscopic lifetime of ${\mathcal{O}({\unit[1]{ps}})}$ itself, however, a macroscopic stop lifetime might turn out to be accessible experimentally only if it exceeds this timescale.
NLSP neutralino composition and decays {#sec:neutralino}
--------------------------------------
Neutralinos are mass eigenstates of the $U(1)$ gauge fermion (bino), the neutral $SU(2)$ gauge fermion (wino), and the neutral up– and down–type Higgs fermion (higgsino). The fields’ individual contributions to ${\ensuremath{\tilde{\chi}_1^0}}$ as well as the neutralino mass spectrum depends on the bino mass $M_1$, the wino mass $M_2$, the Higgs mixing parameter $\mu$ and the ratio between the up–type and down–type Higgs vacuum expectation value (VEV) $\tan\beta$. If ${\ensuremath{\tilde{\chi}_1^0}}$ is the NLSP and ${\ensuremath{\tilde{G}}}$ the LSP, ${\ensuremath{\tilde{\chi}_1^0}}$ decays via ${\ensuremath{\tilde{\chi}_1^0}}\rightarrow X
{\ensuremath{\tilde{G}}}$, where $X$ is either the photon, the Z boson, or a neutral Higgs boson. Branching ratios into the various decay channels are fixed by phase space suppression factors and the decomposition of the lightest neutralino. General formulae for the decay widths are given in Refs [@Ambrosanio:1996jn; @Meade:2009qv].
In the previous subsection, we argued that the mass of the light chargino ${\ensuremath{\tilde{\chi}_1}}^\pm$, a mass eigenstate of charged winos and higgsinos, has to be larger than the light stop mass in order to suppress the flavor diagonal stop decay to ${\ensuremath{\tilde{\chi}_1}}^\pm b$. This requirement cannot be satisfied if the ${\ensuremath{\tilde{\chi}_1^0}}$ is wino–like, *i.e.* if $M_2\ll M_1,|\mu|$, as in this case both the ${\ensuremath{\tilde{\chi}_1^0}}$ and the ${\ensuremath{\tilde{\chi}_1}}^\pm$ mass are $\approx M_2$. The mass splitting between wino–like ${\ensuremath{\tilde{\chi}_1}}^\pm$ and ${\ensuremath{\tilde{\chi}_1^0}}$ is of the order of $\frac{m_Z^5}{\mu^4}$ [@Martin:1993ft], given $M_1 \ll |\mu|$, and thus is extremely suppressed for $|\mu|\gtrsim\text{few }{\unit[100]{GeV}}$.
Similarly, if ${\ensuremath{\tilde{\chi}_1^0}}$ is higgsino–like, ${\ensuremath{\tilde{\chi}_1^0}}$ and ${\ensuremath{\tilde{\chi}_1}}^\pm$ have masses of the same order of magnitude given by $\mu$. The mass splitting $\Delta m_{{\ensuremath{\tilde{\chi}_1}}^\pm{\ensuremath{\tilde{\chi}_1^0}}}$ is of the order of $\frac{m_Z^2}{M_2}$ [@Martin:1993ft] for $|\mu|\ll M_1, M_2$.
If ${\ensuremath{\tilde{\chi}_1^0}}$ is bino–like, its mass is $M_1$ approximately, while the ${\ensuremath{\tilde{\chi}_1}}^\pm$ mass is given by $|\mu|$ or $M_2$. As the mass gap depends on two different supersymmetric mass parameters, it can be sizable depending on the details of high scale physics.
While in a mass region close to the $Z$ mass, also a higgsino–like ${\ensuremath{\tilde{\chi}_1^0}}$ may respect the anticipated mass hierarchy in Eq. , we focus on a bino–like ${\ensuremath{\tilde{\chi}_1^0}}$ in discussing experimental bounds as 1) in the bino case the mass hierarchy can exist over a large stop mass scale and 2) binos have a large branching fraction to photons. For $m_{{\ensuremath{\tilde{\chi}_1^0}}}>m_{Z}$ and negligible phase space suppression, the branching ratio is ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{\chi}_1^0}}\rightarrow\gamma{\ensuremath{\tilde{G}}})}}}\approx\cos^2\theta_W$. This value is obvious as ${\ensuremath{\tilde{G}}}$ is a gauge singlet and $\gamma$ a mixed state of the hypercharge gauge boson and the neutral $SU(2)$ gauge boson where the mixture is parametrized by the Weinberg angle. Including the phase space suppression from $m_{Z}$ and assuming that the higgsino sector is decoupled, the bino decay rates are [@Ambrosanio:1996jn; @hep-ph/9609434]
\[eq:neutrdecay\] $$\begin{aligned}
\Gamma({\ensuremath{\tilde{\chi}_1^0}}\rightarrow \gamma{\ensuremath{\tilde{G}}}) &=
\cos^2{\theta_W}\frac{m_{{\ensuremath{\tilde{\chi}_1^0}}}^5}{48\pi{\ensuremath{m_{3/2}}}^2{\ensuremath{m_{\rm{Pl}}}}^2}\\ \Gamma({\ensuremath{\tilde{\chi}_1^0}}\rightarrow
Z{\ensuremath{\tilde{G}}}) &=\nonumber\\
\sin^2&\theta_W\frac{m_{{\ensuremath{\tilde{\chi}_1^0}}}^5}{48\pi{\ensuremath{m_{3/2}}}^2{\ensuremath{m_{\rm{Pl}}}}^2}\left(1-\frac{m_{Z}^2}{m_{{\ensuremath{\tilde{\chi}_1^0}}}^2}\right)^4.
\end{aligned}$$
For reference we show the bino–neutralino lifetime as a function of the lightest neutralino mass for gravitino masses 1, 10, 100, 1000 eV in Fig. \[fig:neutr-ltime\].
Collider bounds {#sec:collider}
===============
When ${\ensuremath{\tilde{t}_1}}$, ${\ensuremath{\tilde{\chi}_1^0}}$, and ${\ensuremath{\tilde{G}}}$ are the only light supersymmetric particles, stops are dominantly produced in pairs, both at $\bar pp$ and $pp$ colliders,
$$\bar pp\rightarrow{\ensuremath{\tilde{t}_1}}{\ensuremath{\tilde{t}_1}}^*,\qquad pp\rightarrow{\ensuremath{\tilde{t}_1}}{\ensuremath{\tilde{t}_1}}^*.$$
The production cross sections are given in Fig. \[fig:xsec\] as a function of the stop mass for the LHC at as well as for Tevatron and are calculated with [`Prospino`]{} [@Beenakker:1997ut] using the built-in CTEQ6.6M [@Nadolsky:2008zw] parton distribution functions (PDFs). We also show the next–to–leading order uncertainty by varying the factorization scale ($\mu_F$) and the renormalization scale ($\mu_R$) between $\frac{1}{2}m_{{\ensuremath{\tilde{t}_1}}}$ and $2m_{{\ensuremath{\tilde{t}_1}}}$ while keeping $\mu_R$ equal to $\mu_F$.
Given a bino–like ${\ensuremath{\tilde{\chi}_1^0}}$, the final state signatures of a decay chain via Eq. and Eq. are $$\gamma\gamma c\bar c{\ensuremath{\tilde{G}}}{\ensuremath{\tilde{G}}},\quad
\gamma Z c\bar c{\ensuremath{\tilde{G}}}{\ensuremath{\tilde{G}}},\quad
Z Z c\bar c{\ensuremath{\tilde{G}}}{\ensuremath{\tilde{G}}}.$$ In general, with a small mass gap $\Delta m$, the charm jets are too soft to be useful for event selection. Thus constrains on our parameter space can stem from searches for an excess in the $\gamma\gamma{\ensuremath{\slashed{E}_T}}$, $\gamma Z{\ensuremath{\slashed{E}_T}}$ and $ZZ{\ensuremath{\slashed{E}_T}}$ channels. As binos dominantly decay to $\gamma{\ensuremath{\tilde{G}}}$, the SM background is negligible for energetic photons, and large ${\ensuremath{\slashed{E}_T}}$ and high $p_T$ photons are efficiently identified in multipurpose detectors, we concentrate on the $\gamma\gamma{\ensuremath{\slashed{E}_T}}$ channel in this work. Several experimental searches for the diphoton and ${\ensuremath{\slashed{E}_T}}$ channel have been published [@Aad:2010qr; @Aaltonen:2009tp; @Chatrchyan:2011wc; @Abazov:2010us; @arXiv:1111.4116]. So far, no excess above the SM expectation has been found.
In the following, we present exclusion limits in the stop–gravitino mass plane derived from the latest search in the $\gamma\gamma{\ensuremath{\slashed{E}_T}}$ channel for a luminosity ($\mathcal L$) of ${\ensuremath{\unit[1.07]{fb^{-1}}}}$ [@arXiv:1111.4116]. We derive also bounds from the [D$\slashed{\text{0}}$]{} search with $\mathcal L={\ensuremath{\unit[6.3]{fb^{-1}}}}$ [@Abazov:2010us], and give a $\mathcal L={\ensuremath{\unit[5]{fb^{-1}}}}$ projection for the bound.
The dominant SM background with ${\ensuremath{\slashed{E}_T}}$ originating from the hard process stems from $W+\gamma$, $W+{\rm jets}$, and $W/Z\gamma(e)
\gamma(e)$ production. Here, electrons/jets are misidentified as photons. SM background with ${\ensuremath{\slashed{E}_T}}$ from mismeasurements emerges from multijet production and direct photon production.
A supersymmetric background can only arise from ${\ensuremath{\tilde{\chi}_1^0}}{\ensuremath{\tilde{\chi}_1^0}}$ pair production as all other colored sparticles, sleptons, heavier neutralinos and charginos are assumed to be heavy and thus will have a negligible contribution. However, also the ${\ensuremath{\tilde{\chi}_1^0}}{\ensuremath{\tilde{\chi}_1^0}}$ production cross section is severely suppressed. In the limiting case of vanishing higgsino admixture to ${\ensuremath{\tilde{\chi}_1^0}}$, the cross section vanishes even at ${\mathcal{O}(\alpha_{\text{EW}}^2\alpha_s)}$. Consequently we do not take ${\ensuremath{\tilde{\chi}_1^0}}{\ensuremath{\tilde{\chi}_1^0}}$ pair production into account in the following.
Calculation of exclusion limits
-------------------------------
To constrain the stop mass, the gravitino mass, and the MFV coupling $Y$, we calculate $\sigma_{{\ensuremath{\tilde{t}_1}}{\ensuremath{\tilde{t}_1}}^*}$, the total cross section for ${\ensuremath{\tilde{t}_1}}{\ensuremath{\tilde{t}_1}}^*$ production, in a grid of the light stop mass $m_{{\ensuremath{\tilde{t}_1}}}$ using [`Prospino`]{}. Note here that the light stop mass is the dominant SUSY parameter in the cross section [@Beenakker:1997ut], both for $p\bar p$ and $pp$ initial states.
For each stop mass in the grid, we generate 100$\,$000 ${\ensuremath{\tilde{t}_1}}{\ensuremath{\tilde{t}_1}}^*$ pair events with [`pythia 6.4.25`]{} [@Sjostrand:2006za] using the CTEQ6.6M [@Nadolsky:2008zw] parton distribution functions. With the hadron level events we simulate the efficiency/acceptance for the $\gamma\gamma c\bar c{\ensuremath{\tilde{G}}}{\ensuremath{\tilde{G}}}$ final state in the and [D$\slashed{\text{0}}$]{} analyses employing a slightly modified version of [`Delphes 1.9`]{} [@Ovyn:2009tx][^1]. The photon energy, and therefore our signal’s detection efficiency, depends on the mass splitting $\Delta m$. As in the previous sections, we fix $\Delta m ={\unit[10]{GeV}}$. In appendices \[sec:dzero\] and \[sec:atlas\], we give details on the cuts adopted from the experimental studies and on further simulation parameters. As a result of this simulation step, we obtain efficiencies $\epsilon_n$ in bins of ${\ensuremath{\slashed{E}_T}}$ and can calculate a signal cross section in bin $n$ from $$\sigma^{sig}_{n}=\epsilon_n{\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{\chi}_1^0}}c)}}}^2{\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{\chi}_1^0}}\rightarrow{\ensuremath{\tilde{G}}}\gamma)}}}^2\sigma_{{\ensuremath{\tilde{t}_1}}{\ensuremath{\tilde{t}_1}}^*}.$$
Using Eqs to calculate ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{\chi}_1^0}}\rightarrow{\ensuremath{\tilde{G}}}\gamma)}}}$, we finally employ the $\mathrm {CL}_s$ method [@Junk:1999kv; @599622][^2] to calculate the 95% exclusion limits for ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{\chi}_1^0}}c)}}}$. Those are depicted in Fig. \[fig:br-excl\], where we use the measurements plus background predictions of the experimental studies as enlisted in Tab. \[tab:data\]. When calculating the exclusion limit, we treat the errors on the luminosity and the background as Gaussian nuisance parameters, see Tab. \[tab:data\], but do not take into account theory uncertainties stemming from scale variations and the choice of PDF sets.
The projection for the ATLAS study with a luminosity of ${\ensuremath{\unit[5]{fb^{-1}}}}$ is calculated using the prescription of [@Conway:2000ju].
${\ensuremath{\slashed{E}_T}}$ Bin \[\]
------------------------------------------------ ----------------------------------------- ------ --------------
[D$\slashed{\text{0}}$]{} [@Abazov:2010us] $35-50$ $18$ $11.9\pm2.0$
${\ensuremath{\unit[(6.3\pm0.4)]{fb^{-1}}}}$ $50-75$ $3$ $5.0\pm0.9$
$>75$ $1$ $1.9\pm0.4$
ATLAS [@arXiv:1111.4116] $>125$ $5$ $4.1\pm0.6$
${\ensuremath{\unit[(1.07\pm0.04)]{fb^{-1}}}}$
: ATLAS and [D$\slashed{\text{0}}$]{}measurements and background (bgd) predictions.[]{data-label="tab:data"}
Numerical analysis and discussion
---------------------------------
As can be seen in Fig. \[fig:br-excl\], the ATLAS search (solid pink curve) gives a bound on ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{\chi}_1^0}}c)}}}$ for stop masses up to ${\unit[560]{GeV}}$. For a luminosity of ${\ensuremath{\unit[5]{fb^{-1}}}}$, this mass is projected to ${\unit[660]{GeV}}$. For larger masses, a dominant tree level FCNC decay ${\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{\chi}_1^0}}c$ is in agreement with the measurement. As can be seen from Fig. \[fig:br-excl\], the [$\unit[1.07]{fb^{-1}}$]{} ATLAS data already excludes a larger parameter region than the [$\unit[6.3]{fb^{-1}}$]{} [D$\slashed{\text{0}}$]{} data [^3].
For each stop mass, the bound in Fig. \[fig:br-excl\] can be mapped to a bound on the maximal gravitino mass for given values of $Y$ using Eqs , , and . We plot these bounds for $Y=10^{-7}$, $10^{-6}$, $10^{-5}$ in Fig. \[fig:excl\]. As Eqs do not provide the correct stop width for masses in the threshold region close to the top mass, we exclude this region from the mapping and interpolate our result in the region $m_t\pm{\unit[30]{GeV}}$ ($m_t={\unit[173]{GeV}}$).
The mass difference $\Delta m$ enters the bounds in Fig. \[fig:excl\] through Eq. and through the hardness of the photon $p_T$ spectrum. For small changes in $\Delta
m$, the latter dependency can be neglected, and the bounds depend on the product $(Y\Delta m)$ only. Therefore the bounds for other viable values of $\Delta m$ can be derived from those shown for $\Delta m
={\unit[10]{GeV}}$ by rescaling $Y$.
Obviously, the smaller we choose $Y$ the larger ${\ensuremath{m_{3/2}}}$ can be to generate a branching ratio below a certain value. The bound on ${\ensuremath{m_{3/2}}}$ resulting from the mapping in Fig. \[fig:excl\] varies over a wide mass range and thus potentially implies different regimes of ${\ensuremath{\tilde{t}_1}}$ and ${\ensuremath{\tilde{\chi}_1^0}}$ lifetimes; therefore, we show slopes of fixed values: For the ${\ensuremath{\tilde{\chi}_1^0}}$ lifetime, we show slopes for , PS. [1]{}, and (black dotted) following approximately ${\ensuremath{m_{3/2}}}\propto
m_{{\ensuremath{\tilde{t}_1}}}^{5/2}$. For ${\ensuremath{\tilde{t}_1}}$, we show slopes for ${\unit[1/5]{ps}}$, ${\unit[1]{ps}}$, and ${\unit[5]{ps}}$ (black solid). At $Y=10^{-5}$, the stop lifetime is below PS. [1]{} already induced by ${\ensuremath{\tilde{t}_1}}\rightarrow {\ensuremath{\tilde{\chi}_1^0}}c$ irrespective of the gravitational decay channels; therefore, only the ${\unit[1/5]{ps}}$ slope can be drawn here. Similarly, the stop lifetime drops below ${\unit[1/5]{ps}}$ for stop masses smaller than $\approx{\unit[230]{GeV}}$ for $Y=10^{-5}$, as $\Gamma({\ensuremath{\tilde{t}_1}}\rightarrow c{\ensuremath{\tilde{\chi}_1^0}})\propto m_{{\ensuremath{\tilde{t}_1}}}^{-1}$ for fixed $\Delta
m$. For both smaller values of $Y$, the stop lifetime slopes shown only depend weakly on $Y$ because the corresponding total stop widths are dominated by the gravitational decay ${\ensuremath{\tilde{t}_1}}\rightarrow t{\ensuremath{\tilde{G}}}$ resp. ${\ensuremath{\tilde{t}_1}}\rightarrow W^+b{\ensuremath{\tilde{G}}}$.
Fig. \[fig:excl\] shows that the gravitino mass region promoted in [@Hiller:2009ii] as the region where the decay ${\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{\chi}_1^0}}c$ dominates for stop masses between $100$ and ${\unit[150]{GeV}}$ is now disfavored. More generally, Fig. \[fig:excl\] allows to discuss two regimes of different orders of ${\ensuremath{\tilde{t}_1}}$ and ${\ensuremath{\tilde{\chi}_1^0}}$ lifetimes:
- In the stop mass region where ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{\chi}_1^0}}c)}}}$ is bounded, *i.e.* for $m_{{\ensuremath{\tilde{t}_1}}}\lesssim{\unit[500]{GeV}}$, the gravitino channels have a significant contribution to the stop decay. If this contribution is dominant, both—the ${\ensuremath{\tilde{\chi}_1^0}}$ and the ${\ensuremath{\tilde{t}_1}}$ decay—are governed by the same coupling $\sim1/{{\ensuremath{m_{3/2}}}^2}$. As the stop decay width is suppressed by phase space ($t{\ensuremath{\tilde{G}}}$ channel) or top propagator (${\ensuremath{\tilde{G}}}W^+b$ channel), the stop lifetime is expected to be larger than, or at the same order of magnitude as, the neutralino lifetime in this region.
- In the mass region where ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{\chi}_1^0}}c)}}}$ is allowed to dominate, *i.e.* for $m_{{\ensuremath{\tilde{t}_1}}}\gtrsim{\unit[500]{GeV}}$, the phase space suppression of the stop decay width Eq. is less pronounced; thus, the stop and neutralino gravitational partial decay widths are of the same order of magnitude. Consequently, if ${\ensuremath{\tilde{t}_1}}\rightarrow c{\ensuremath{\tilde{\chi}_1^0}}$ is the dominant stop decay in this mass region, the stop lifetime is significantly smaller than the neutralino lifetime.
The relation between the stop and neutralino lifetimes ($\tau_{{\ensuremath{\tilde{t}_1}}}$ and $\tau_{{\ensuremath{\tilde{\chi}_1^0}}}$) described above is summarized in Fig. \[fig:lt-ratios\] where we plot the allowed ratio of both lifetimes within the bounds. Along the solid black line bounding the grey area, the contribution of Eq. to the stop decay width vanishes; thus, this line represents the smallest value the ratio can have. The plot is generated for $Y=10^{-6}$; however, only the leftmost parts of the exclusion areas depend weakly on the specific value of $Y$. For smaller masses, the bounds are driven by the phase space dependence of Eqs and .
Note that for large ${\ensuremath{\tilde{t}_1}}$ or ${\ensuremath{\tilde{\chi}_1^0}}$ lifetimes, care must be taken in the interpretation of the bounds in Figs \[fig:excl\] and \[fig:lt-ratios\]: The selection criteria for photon candidates in the ATLAS publication are chosen for prompt photons [@arXiv:1012.4389; @atlas-conf-2010-005]. Also, more explicitly, the [D$\slashed{\text{0}}$]{} study requires that photon candidates point to a reconstructed primary vertex. We assumed in our calculation that all signal photons fulfill these criteria as if they were prompt. In a more realistic simulation, for increasing neutralino lifetimes, the signal photons’ selection efficiency should decrease. For longitudinal neutralino decay lengths of ${\mathcal{O}(\unit[1000]{mm})}$ ATLAS simulations show that for several photon selection criteria [^4] used in [@arXiv:1111.4116] the efficiency drops from ${\mathcal{O}({)}85\%}$ to ${\mathcal{O}({)}55\%}$ [@Aad:2009wy]. Consequently, for lifetimes $\gtrsim{\mathcal{O}({\unit[]{ns}})}$, we overestimate the number of photons accepted, and the bounds on ${\ensuremath{m_{3/2}}}$ presented should be regarded with care in this lifetime region.
While for a bino–like neutralino, the $\gamma\gamma{\ensuremath{\slashed{E}_T}}$ channel offers the highest sensitivity for setting mass limits, it may be difficult to measure the neutralino lifetime in this channel due to lack of photon tracks. To construct the photons’ impact parameters, CMS focuses on converted photons in a [$\unit[2.1]{fb^{-1}}$]{} search in the $\gamma\gamma{\ensuremath{\slashed{E}_T}}+\text{jets}$ channel in [@CMS-PAS-EXO-11-067]. As pointed out in [@Meade:2010ji], the ${\ensuremath{\tilde{\chi}_1^0}}\rightarrow Z{\ensuremath{\tilde{G}}}$ channel may be used to investigate the neutralinos’ lifetimes, as the $Z$’s decay products allow to reconstruct the ${\ensuremath{\tilde{\chi}_1^0}}$’s trajectory.
Summary {#sec:summary}
=======
In SUSY models with MFV, the third generation of squarks decouples from the other two generations. This decoupling opens the opportunity to support the MFV hypotheses with the measurement of a macroscopic stop decay length if the stop decay can only proceed through a generation–changing channel due to kinematic constraints [@Hiller:2008wp]. In this work, we investigate the decay of a stop into a charm and a bino–like lightest neutralino with a subsequent bino decay to a photon and a light gravitino, $${\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{\chi}_1^0}}(\rightarrow\gamma{\ensuremath{\tilde{G}}}) c,
\label{eq:complete_decay}$$ given a sufficiently small mass difference between ${\ensuremath{\tilde{t}_1}}$ and ${\ensuremath{\tilde{\chi}_1^0}}$.
While a macroscopic stop decay length serves as a hint for MFV, the neutralino’s decay to a photon leaves a distinct signature in LHC and Tevatron detectors offering a good signal isolation.
Assuming that the remainder of the SUSY spectrum is decoupled, we find that the ATLAS search in the $\gamma\gamma{\ensuremath{\slashed{E}_T}}$ channel based on a luminosity of ${\ensuremath{\unit[1]{fb^{-1}}}}$ [@arXiv:1111.4116] implies a bound on ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{t}_1}}\rightarrowc{\ensuremath{\tilde{\chi}_1^0}})}}}$ for stop masses up to ${\unit[560]{GeV}}$, see Fig. \[fig:br-excl\]. In a ${\ensuremath{\unit[5]{fb^{-1}}}}$ projection, the bound is raised to ${\unit[660]{GeV}}$.
We find that stops lighter than $\sim{\unit[400]{GeV}}$ are still compatible with the $\gamma\gamma{\ensuremath{\slashed{E}_T}}$ searches; however, in this region, a significant fraction of the stops decays into gravitinos and quarks of the third generation. Here the stops are expected to have larger lifetimes than the lightest neutralinos; though, a macroscopic stop decay length is governed by the gravitino mass scale ${\ensuremath{m_{3/2}}}$ and is no hint for a decoupled stop flavor mixing structure.
Stops heavier than $\sim{\unit[500]{GeV}}$ can dominantly decay through the decay chain in Eq. and eventually support MFV if they are a long–lived. As the lifetime depends on both the stops’s flavor mixing and the gravitino mass, it can serve as a hint to MFV only if the contribution of the gravitino modes to the total decay width is negligible. This is assured if the neutralino lifetime is much larger than the stop lifetime.
We have discussed a split spectrum where the only light particles are a light gravitino, a light bino, and a light right–handed stop. It may be possible, however, to separate supersymmetric background processes in a scenario with additional light sparticles by vetoing on additional jets and/or isolated leptons.
We thank Manuel Drees, Sebastian Grab, and Gudrun Hiller for discussions in the initial phase of this project and for the comments on the manuscript. JSK also thanks the Bethe Center of Theoretical Physics and the Physikalisches Institut at the University of Bonn for their hospitality. This work has been supported in part by the Initiative and Networking Fund of the Helmholtz Association, Contract No. HA-101 (“Physics at the Terascale”) and by the ARC Centre of Excellence for Particle Physics at the Terascale.
D0 cuts {#sec:dzero}
=======
Irrespective of the very clear signal event structure, we employ the fast detector simulation [`Delphes`]{} as a framework to calculate the signal efficiency. For [D$\slashed{\text{0}}$]{}we use a simplified calorimeter layout composed from cells of dimension $0.1\times\frac{2\pi}{64}$ in $\eta\times\phi$ space covering $|\eta|\le 4.2$ and $\phi\in[0,2\pi[$ where $\eta$ denotes pseudorapidity and $\phi$ the azimuthal angle.
Jets are constructed employing the iterative midpoint algorithm with $R=0.5$.
We adopt the cuts of [@Abazov:2010us] by requiring
- at least two isolated photons with $p_T>{\unit[25]{GeV}}$ and $|\eta| <
1.1$,
- the azimuthal angle between $\vec{\ensuremath{\slashed{E}_T}}$ and the hardest jet, if existent, is $< 2.5$,
- the azimuthal angles between $\vec{\ensuremath{\slashed{E}_T}}$ and both photons are $> 0.2$,
- ${\ensuremath{\slashed{E}_T}}>{\unit[35]{GeV}}$.
Photons must have 95% of their energy deposited in the electromagnetic calorimeter. For a photon to be isolated, the calorimetric isolation variable $I$ defined in [@Abazov:2010us] must fulfill $I<0.1$ and the scalar sum of transverse momenta of the tracks in a distance of $0.05<R<0.4$ from the photon must be smaller than . Here is $R=\sqrt{\Delta \phi^2+\Delta\eta^2}$, where $\Delta \phi$ is a track’s azimuthal distance from the photon and $\Delta \eta$ is the corresponding distance in pseudorapidity.
Note that nearly all signal events fulfill the isolation criteria hinting that the hadronic stop decay products are well separated from the photon stemming from the subsequent neutralino decay. This is expected as the neutralino decay products $\gamma$ and ${\ensuremath{\tilde{G}}}$ are massless and thus the photon can have a large $p_T$ relative to the stop flight direction.
Also note that we assume that a primary vertex can be reconstructed, and that the photon trajectories point to this vertex. The latter assumption should be regarded with care when the neutralino lifetime is large.
ATLAS cuts {#sec:atlas}
==========
We use the default detector layout implemented in [`Delphes`]{} and apply the following simplified cuts:
- At least two isolated photons exist with $p_T>{\unit[25]{GeV}}$ and $|\eta| < 1.81$, but outside the transition region $1.37 < |\eta|
< 1.52$,
- ${\ensuremath{\slashed{E}_T}}>{\unit[125]{GeV}}$
The study employs a tight photon selection criterion on photon candidates, where the efficiency to identify a true (prompt) photon is approximately 85% in the kinematic region considered [@arXiv:1012.4389; @atlas-conf-2010-005]. We mimic this selection criterion by removing photons from our Monte Carlo sample with a probability of 15%.
We consider a photon to be isolated if in a cone of width $R=0.2$, the scalar $E_T$ sum is less than . Here we sum the $E_T$ of the [`Delphes`]{} calorimeter cells and exclude the cell the photon is mapped to.
As for the [D$\slashed{\text{0}}$]{}case, nearly all signal photons fulfill the isolation requirement. For larger ${\ensuremath{\tilde{\chi}_1^0}}$ masses, the main reduction of signal event numbers stems from the 85% photon selection efficiency.
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[^1]: We modify [`Delphes`]{} slightly to simulate a [D$\slashed{\text{0}}$]{}-like calorimeter with $>40$ segments in $\eta$ direction and flag gravitinos as undetectable particles.
[^2]: We use the implementation in the [`TLimit`]{} class of [`ROOT 5.28.00b`]{}[@Brun:1997pa].
[^3]: We also performed the calculation for the [$\unit[36]{pb^{-1}}$]{} CMS $\gamma\gamma{\ensuremath{\slashed{E}_T}}$ results [@Chatrchyan:2011wc]. These impose a weaker bound on ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{t}_1}}\rightarrowc{\ensuremath{\tilde{\chi}_1^0}})}}}$ than the [$\unit[6.3]{fb^{-1}}$]{} [D$\slashed{\text{0}}$]{} data [@Abazov:2010us].
[^4]: The ratio of the energy deposits in $3\times7$ and $7\times7$ cells ($\eta\times\phi)$ in the electromagnetic calorimeter contributing to the photon cluster, and the shower’s lateral width.
|
---
author:
- 'Changhong Li ,'
- 'Yeuk-Kwan E. Cheung'
bibliography:
- 'dual-reference.bib'
title: 'The Scale-invariant Power Spectrum of Primordial Curvature Perturbation in CSTB Cosmos'
---
Introduction {#sec:intro}
============
In accordance with observations of Cosmic Microwave Background (CMB) anisotropies [@Komatsu:2010fb; @Planck:2013kta] the scale-invariance of power spectrum of the primordial curvature perturbations serves as a crucial criterion for testing the validity of early-universe models. A well-known example is the exponential inflation driven by a slowly-rolling scalar field, the spectrum of curvature perturbations of which could be nearly scale-invariant by fine-tuning the flatness of the inflaton potential. However–besides this fine-tuning problem of the flatness of potential[^1]–slow-roll inflation suffers another severe problem: the singularity of its initial conditions, [*i.e.*]{} the Big Bang Singularity [@Borde:1993xh].
Although proven challenging many bounce/cyclic universe models[^2] have been proposed in an effort to address the problems of the Big Bang Singularity [@Novello:2008ra; @Brandenberger:2012zb; @Bassett:2005xm; @Lehners:2011kr; @Khoury:2001wf; @Khoury:2001bz; @Cai:2007qw; @Cai:2009zp; @Gasperini:2002bn; @Creminelli:2006xe; @Creminelli:2007aq; @Abramo:2007mp]. Moreover, to get a scale-invariant curvature spectrum in bounce universe scenario, Wands introduced a remarkable mechanism in [@Wands:1998yp] (see also [@Finelli:2001sr; @Durrer:1995mz]): the scale invariant spectrum of a single scalar field perturbation can–seemingly–be generated during a matter-dominated contraction phase. The idea of Wands has since been warmly embraced in many variations on the theme of the matter-bounce universe models [@Brandenberger:2009yt; @Peter:2008qz; @Lidsey:1999mc; @Cai:2009in; @Allen:2004vz; @Wang:2009rw; @Cai:2008qw; @Cai:2011zx; @Lin:2010pf; @Piao:2009ku; @Cai:2012va; @WilsonEwing:2012pu]. However, as it was first pointed out by [@Gratton:2003pe], the perturbation modes grew out of the horizon indicating an unstable cosmological background.
In order to achieve a physically scale-invariant perturbation spectrum which does not renders its own cosmological background unstable, in either inflationary or cyclic/bounce cosmos, we extended–and analyzed in detail–the parameter space governing the equations of motion of the cosmological perturbations [@Li:2012vi]. With the standard assumption that Equation of State(EoS) of the cosmological background being constant in the period of perturbation generation[^3], the scale factor of the cosmological background is a power law in conformal time, $a\propto \eta^\nu$. On other hand, relaxing the conventional assumption that the universe must be driven by one single canonical scalar field[^4], the Hubble friction term (or the red/blue-shift term as it is sometimes called) in the equation of the perturbation mode becomes $mH\dot{\chi}_k$ and $m$ becomes a free parameter[^5]. In this framework different values of $\nu$ and $m$ indicate, respectively, different cosmological background and different Hubble friction terms in the equations of perturbations. Therefore, for one given model, it can be characterized by a point, $(\nu, m)$ , in the $\nu$–$m$ parameter space. In particular, $(\nu, m)=(-1,3)$ for a slow-roll model, $(\nu, m)=(2,3)$ for Wands’s matter-dominated contraction [@Wands:1998yp; @Finelli:2001sr; @Durrer:1995mz] , and $(\nu, m)=(2,0)$ for the Coupled-Scalar-Tachyon Bounce model [@Li:2011nj]. In general there are two groups of scale-invariant solutions generated in an expanding or a contracting background, as tabulated in Table \[table1\]. Outside of the horizon, solutions belong to Group I and Group II are stable while those in Group III and Group IV grow and render the background unstable [@Li:2012vi]. It is easy to check that the slow-roll inflation, $(\nu, m)=(-1,3)$, belongs to Group I–scale-invariant and stable solutions in an expanding background. And Wands’s model of matter dominated contraction, $(\nu, m)=(2,3)$, belongs to Group III which is scale-invariant but not stable in a contracting background. It is worth emphasize that a physically acceptable bounce universe model with its spectrum of density perturbation being generated in the pre-bounce contraction should be scale-invariant and stable and [*i.e.*]{} satisfying the conditions $m = -\frac{2}{\nu}+1$, $\nu>0$ of Group II.
Stable Unstable
------------------- -------------------------------------- -------------------------------------
Expanding Phase I: $m = -\frac{2}{\nu}+1$, $\nu<0$ IV: $m = \frac{4}{\nu}+1$, $\nu<0$
Example: Example:
Slow-roll inflation $(\nu,m)=(-1,3)$ Unknown yet
Contracting Phase II: $m = -\frac{2}{\nu}+1$, $\nu>0$ III: $m = \frac{4}{\nu}+1$, $\nu>0$
Example: CSTB $(\nu,m)=(2,0)$ Example: Wands $(\nu,m)=(2,3)$
: Four groups of scale invariant solutions in the $(\nu, m)$ parameter space.[]{data-label="table1"}
\[default\]
In this paper we undertake a thorough study of the cosmological perturbations generated in the recently proposed Coupled-Scalar-Tachyon Bounce (CSTB) model [@Li:2011nj]. As a bounce universe model the spectrum of density perturbations in CSTB cosmos is generated during its contracting phase before the bounce point; the spectrum of density perturbations is assumed to be unperturbed throughout the non-singular bounce as well as its re-entry in recent expansion phase.
In the pre-bounce contraction, CSTB cosmos enjoys the following two properties
- $\nu=2$: The pre-bounce contraction is dominated by the tachyon matter which behaves like cold dust. Thus we have $\omega=0$ and $\nu=2$ in this phase of tachyon matter dominated contraction;
- $m= 0$: The Hubble friction term in the equation of tachyon perturbations is that $mH=3\sqrt{1-\dot{T}^2}H$. During the tachyon matter dominated contraction the tachyon field has already condensed and $\dot{T}^2\rightarrow 1$, therefore, we have $m=3\sqrt{1-\dot{T}^2}\rightarrow 0$ .
In other words, the tachyon field perturbations in the CSTB cosmos correspond to a point, $(\nu,m)=(2, 0)$, in the $(\nu, m)$ parameter space. This solution clearly belongs to Group II. It indicates that the power spectrum of density perturbations due to the tachyon field is scale-invariant and stable. Furthermore during the pre-bounce contracting phase–in which the perturbations are outside of the horizon–the spectrum of curvature perturbation, in the long wavelength limit, is related to the spectrum of tachyon perturbation by a factor $(H/\dot{T}_c)^2$, with $T_c$ being the vacuum expectation value of the tachyon field in this phase. Because of the following characteristic
- $\dot{T}_c\propto H$: During the contracting phase with tachyon matter domination the vacuum expectation value of tachyon field is proportional to the number of e-foldings of the background, $T_c\propto N\equiv \int H dt$. It follows that the factor relating the spectrum of curvature perturbations and that of the tachyon perturbations is a constant, $(H/\dot{T}_c)^2 = \kappa^2$,
we conclude that the curvature spectrum of CSTB cosmos is also scale-invariant and stable.
This paper is organized as follows: in Section 2 we review the cosmology of CSTB model; in Section 3 we calculate the primordial spectral index of curvature perturbation of CSTB cosmos, and show that its power spectrum of curvature perturbation is nearly scale-invariant and stable. We discuss the “dualities”[^6] between slow-roll inflation model, Wands’s matter-dominated contraction and our CSTB cosmos. We compare their stability properties in Section 4, and close with a conclusion and prospects in Section 5.
Cosmological Background in CSTB Cosmos
======================================
In this section, we review the cosmology of a string-inspired bounce universe model driven by a canonical scalar field coupled with a tachyon field, for short the CSTB cosmos [@Li:2011nj]. The Lagrangian density is comprised of three parts, $$\label{eq-totl}
\mathcal{L}(\phi, T)=\mathcal{L}(T)+\mathcal{L}(\phi) -\, \lambda\phi^{2} T^{2}~,$$ where $\mathcal{L}(\phi)$ is the Lagrangian for a massive (no further assumption on the value of the mass is made) canonical scalar field, $$\label{eq-sl}
\mathcal{L}(\phi)=-\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}m_\phi^2\phi^2~.$$ The dynamics of the tachyon field is governed by $$\mathcal{L}(T)=-V(T)\sqrt{1+\partial_\mu T\partial^\mu T}~,\quad V(T)=V_0\left[\cosh\left(\frac{T}{\sqrt{2}}\right)\right]^{-1}~, \label{eq-tl}$$ describing the annihilation process of a pair of D3–anti-D3 branes [@Sen:2002in; @Sen:2002nu; @Gibbons:2002md]. In effective string theory, $\phi$ can simply be viewed as the distance between the two stacks of D-branes and anti-D-branes. The scalar field sector describing an attraction between the pair of D3-brane and anti-D3-brane at long distance [@HenryTye:2006uv; @Dvali:1998pa; @Dvali:2001fw; @Burgess:2001fx] has an effective coupling with the tachyon, $$\mathcal{L}_{int}\, =\, -\lambda\, \phi^2\, T^2,$$ where $\lambda$ is the coupling constant.
#### The single tachyon cosmological model
The tachyon cosmology model was first proposed by [@Sen:2002in; @Sen:2002nu] and, independently, by Gibbons [@Gibbons:2002md]. It depicts the picture that a pair of static D3-anti-D3 branes lay over each other [^7] and annihilate into closed string vacuum [@Sen:2004nf]. In an effective field theory language, the potential of the tachyon field has a maximum at $T=\dot{T}=0$. During the annihilation process of the brane–anti-brane pair the tachyon field rolls down the potential hill and condenses. Right after the tachyon condensation starts $\dot{T}\rightarrow1$ and $T\rightarrow\infty$ and the tachyon field behaves like cold matter $$\rho_{T}=\frac{V(T)}{\sqrt{1-\dot{T}^2}}\propto a^{-3}~,\quad w_T=-\left(1-\dot{T}^2\right)=0~.$$ This single tachyon field cosmological model has various applications, for example see [@Gibbons:2002md; @Fairbairn:2002yp; @Feinstein:2002aj; @Mazumdar:2001mm; @Bagla:2002yn; @Shiu:2002qe]. However, for the purpose of constructing bounce universe, such a lone tachyon does not suffice, since the tachyon’s vacuum expectation value increases monotonically after condensation. In particular the universe driven by a single tachyon will contract to a cosmic singularity in a closed FLRW background [@Sen:2003mv].
#### The coupled scalar and tachyon bounce (CSTB) model
In the presence of a canonical scalar and its coupling with the tachyon we take $\phi=\phi_0$ and $\dot{\phi}=T=\dot{T}=0$ as the initial conditions for the system.
The picture of the CSTB model is that, at the beginning, a stack of D3-branes and another stack of anti-D3-branes are separated by a long distance, $\phi=\phi_0$. The coupling term, $~\lambda T^2\phi_0^2$, stabilizes the system at $T=\dot{T}=0$. Due to a weak attractive force between D-brane and anti-D-brane [@HenryTye:2006uv; @Dvali:1998pa; @Dvali:2001fw; @Burgess:2001fx], modeled by the term, $-m^2_\phi\phi$, in the Lagrangian, the two stacks will eventually encounter each other, $\phi\rightarrow 0$, and (some of the D-anti-D-brane pairs) annihilate into the closed string vacuum at the end of the tachyon condensation, $T\rightarrow \infty$.
Furthermore, the CSTB model suggests a novel property: the vacuum expectation value acquired by the tachyon is finite but it never reaches infinite in our construct (as shown in [Fig. \[fig-vacuum.pdf\]]{}) [@Li:2011nj]. The tachyon always gets pulled back and up the potential hill–due to the coupling with the scalar–before its condensation is completed. This property is key to the existence of the contraction–bounce–expansion cycles in the CSTB cosmos.
Dynamically, the tachyon field and scalar field oscillate swiftly around $(T_c, 0)$ along the $T$-direction and the $\phi$-direction in $(T,\phi)$ field space, respectively, with the commencement of the tachyon condensation. During these oscillations, the average EoS of tachyon is $$\label{eq-awt}
\left\langle w_T \right\rangle=-\left(1-\langle\dot{T}^2\rangle\right)\simeq 0~,$$ [*[i.e.]{}*]{} the tachyon field acts like a form of cold matter once condensed, where $\langle * \rangle$ denotes the averaged value of the field over a few oscillations.
![A sketch of the effective potential of the scalar and tachyon fields, $V(T, \phi)$, in the $(T,\phi)$ field space. The effective vacuum of CSTB cosmos is located at $(T,\phi)=(T_c,0)$. During the tachyon matter dominated phases of the CSTB cosmos, the tachyon and the scalar swiftly oscillate around $(T,\phi)=(T_c,0)$ along the $T$-direction and the $\phi$-direction, respectively.[]{data-label="fig-vacuum.pdf"}](vacuum.pdf){width="80.00000%"}
With [(Eq.\[eq-totl\])]{}, one can study the cosmology of a universe governed by the coupled scalar-tachyon fields in the closed FLRW background. A cosmological solution with bounce/cyclic behaviour was obtained in [@Li:2011nj]. One typical cycle of the cosmological evolution comprises the following three phases,
1. [**Tachyon-matter-dominated contraction phase**]{}[^8][**:**]{} After the D3–anti-D3-brane pair annihilate, the tachyon field condenses to “tachyon matter,” the Equation of State(EoS) of which is equal to zero, $\langle w\rangle_T=0$. In a closed FLRW background the universe undergoes a matter-dominated contraction [@Sen:2003mv; @Li:2011nj].
2. [**The bounce phase:**]{} A pair of D3–anti-D3-branes can be pair-created, again, from the open string vacuum by vacuum fluctuations. The tension of these two branes behaves like a cosmological constant, $w_{branes}=-1$, the universe bounces smoothly from the pre-bounce contraction to a post-bounce expansion in the closed FLRW background[^9].
3. [**Post-bounce Expansion Phases:**]{} After the bounce the universe experiences an expansion driven by the tension of the branes preceding another expansion phase driven by the tachyon matter. One of these cycles can possibly evolute into today’s universe.
In the bounce/cyclic universe scenario the primordial perturbations are generated and their subsequent exit of the effective horizon all during the pre-bounce contraction. To study the power spectrum of the primordial perturbations in CSTB cosmos we, therefore, focus on the physics of tachyon-matter-dominated contraction.
During the tachyon matter domination, according to the analytical results and numerical simulations presented in [@Li:2011nj], the vacuum expectation value of tachyon field, $T_c$ , is proportional to the number of e-foldings of the cosmological background during contraction, $N_p$, $$T_c\equiv\langle T \rangle\propto N_p~,\quad N_p\equiv\int H dt~.$$ This is shown in [Fig. \[fig-Tcn.pdf\]]{} below.
![A schematic plot of the evolution of the tachyon field against the number of e-foldings, $N_p$, in the contraction phase ([**Right $\rightarrow$ Left**]{}) in which the expectation value of tachyon field, $T_c$, evolves toward $0$. Notable is the linear dependence of $ \langle T_c \rangle$ on $N_p$.[]{data-label="fig-Tcn.pdf"}](Tcn.pdf){width="70.00000%"}
Therefore $\dot{T}_c$ is linear in the Hubble parameter $H$, $$\dot{T}_c=\kappa H~, \label{eq-tch}$$ where we have decomposed $T_c$ as $T_c= \kappa N+\theta$, and both of $\kappa$ and $\theta$ are nearly constant. This property is, in turn, crucial to the successful generation of scale invariant power spectrum of curvature perturbations in CSTB cosmos, to which we will now turn our attention.
Scale invariant Power Spectrum in CSTB Cosmos
=============================================
To pave the road for the study of power spectrum of the primordial curvature perturbations generated by the tachyon field perturbations during the tachyon-matter-dominated contraction we derive the equations of motion for $\delta\phi$ and $\delta T$. In Newtonian gauge [@Mukhanov:1990me], with $g_{\mu\nu}=diag \{-1-2\psi, a^2(1+2\psi))\delta_{ij}\}$ [^10], we obtain $$\label{eq-dpe}
-\ddot{\delta \phi_k}+2\psi\ddot{\phi}-k^{2}a^{-2}\delta \phi_k+\left(-3H\dot{\delta \phi_k}-4\dot{\psi}\dot{\phi}+6H\psi\dot{\phi}\right)-\left(m^2_\phi+2\lambda T^2\right)\delta \phi_k=0~;$$ $$\begin{aligned}
\label{eq-dte}
&\ddot{\delta T_k}& -\, 2\, \psi\, \ddot{T}+k^2\, \delta T_k\, a^{-2}\, +\, (2\, \psi\, \dot{T}^2-2\dot{\delta T_k}\, \dot{T})\, \left(-\frac{1}{\sqrt{2}}+3H\dot{T}+2\lambda\phi^2T \frac{\sqrt{1-\dot{T}^2}}{V(T)}\right)\\ \nonumber
&+& (1-\dot{T}^2) \left[4\, \dot\psi\,\dot{T}\, +3H\, \delta\dot{T}\, -6\, H\, \psi\,\dot{T}\right]\\ \nonumber
&+& (1-\dot{T}^2)
\left[2\lambda\, \sqrt{1-\dot{T}^2}\, e^{T/\sqrt{2}}\, \phi\, \left(2\, T\, \delta \phi_k\, +\, \frac{\sqrt{2}+1}{\sqrt{2}}\phi\, \delta T_k + \psi\, \dot{T}^2 \, \phi -\dot{T}\, \phi\, \dot{\delta T_k}\right)\right]=0~,\end{aligned}$$ where $\delta \phi_k$ and $\delta T_k$ are the Fourier modes of scalar and tachyon perturbations respectively.
According to [(Eq.\[eq-dpe\])]{} the effective mass of $\delta\phi_k$, $M_{eff}=(m_\phi^2+2\lambda T^2)^{\frac{1}{2}}$, is very large during contraction and thus $\delta \phi$ is highly suppressed and can be safely neglected. Let us now turn our attention to the perturbations of the tachyon field. In general the background fields and its derivatives, which appear in [(Eq.\[eq-dte\])]{}, can be viewed as the external currents for the equation of motion for tachyon’s perturbations. Taking the time-average of these fast-varying external currents [(Eq.\[eq-dte\])]{} can be simplified. During the contraction phase the background fields $\phi$ and $T$ oscillate swiftly around the effective vacuum of the system [^11], $(T,\phi)=(T_c,0)$ (as shown in [Fig. \[fig-vacuum.pdf\]]{}. See [@Li:2011nj] for a detailed analysis.). Over a few complete oscillations simplification is achieved because $$\label{eq-ap}
\langle \phi\rangle=\langle \dot{\phi}\rangle=\langle \ddot{\phi}\rangle=0$$ and $$\label{eq-at}
T_c\equiv\langle T\rangle~,\quad r_1\equiv \langle1-\dot{T}^2\rangle\simeq 0,\quad \langle \ddot{T}\rangle=0~.$$ Furthermore the “effective driving force” for the perturbations $$r_2\equiv\left\langle-\frac{1}{\sqrt{2}}+3H\dot{T}+2\lambda\phi^2T \frac{\sqrt{1-\dot{T}^2}}{V(T)}\right\rangle\simeq0~, \label{eq-atf}$$ in $T$-direction also vanishes [@Li:2011nj]. Substituting [(Eq.\[eq-ap\])]{}, [(Eq.\[eq-at\])]{} and[(Eq.\[eq-atf\])]{} into [(Eq.\[eq-dte\])]{} we obtain a simplified equation of motion for the tachyon perturbations, $$\label{eq-pts}
\ddot{\delta T_k}+\frac{k^2}{a^2}\delta T_k=0~.$$ We have taken $r_1 \sim r_2 \sim 0$ to the lowest order approximation; but we will put them back when we calculate the primordial spectral index.
Well before effective horizon exit at $|aH|\sim k$, each perturbation mode, $\delta T_k$, with wavenumber $k/a$ is evolving independently, and it is negligible at the classical level as the “vacuum fluctuations”. However, after the effective horizon exit, $k\eta\rightarrow 0$, it grows to be a classical perturbation, which, in turn, determines the curvature perturbations evaluated on spatially flat slices. In the tachyon-matter-dominated contraction phase of CSTB cosmos, the cosmological background is evolving by a power-law, $$a\propto \eta^2~,\quad \eta\rightarrow 0\label{eq-aeta}$$ with $\eta$ being the conformal time, $d\eta=a^{-1}dt$. Solving [(Eq.\[eq-pts\])]{} with [(Eq.\[eq-aeta\])]{} in the limit $k\eta\rightarrow 0$, we obtain the solution of each tachyon perturbation mode after horizon exit $$\delta T_k\propto k^{-\frac{3}{2}} \eta^0$$ at leading order. And the power spectrum of tachyon perturbations becomes $$\label{eq-spt}
\mathcal{P}_{\delta T}\equiv\frac{k^3|\delta T_k|^2}{2\pi}\propto k^0\eta^0$$ which is time-independent as well as scale-invariant.
Turning to Wands’s model of matter dominated contraction, the power spectrum of primordial perturbation can be re-casted into a simple form [@Wands:1998yp; @Li:2012vi] , $$\mathcal{P}_{\delta \Phi}\propto k^0\eta^{-6}.$$ During a perfectly matter-dominated contraction as proposed by Wands, $a\propto \eta^2$ and $\eta\rightarrow 0$, the total energy density of the cosmological background evolves as $\rho_b\propto \eta^{-6}$. The energy density of the perturbations and that of the background field grow with the same rate, $\eta^{-6}$. However, in a realistic case that the cosmological background has a small departure from the perfectly matter-dominated contraction, $a\propto\eta^{2+\delta\nu}$ and $\delta \nu>0$, the energy density of field perturbations in a model like Wands’s, $$\rho_{\delta \Phi}\propto \mathcal{P}_{\delta \Phi}\propto\eta^{-6-4\delta \nu},$$ grows faster than the energy density of the background field, $\rho_{\Phi}\propto \eta^{-6-2\delta \nu}$. It implies that the energy density of perturbations would become dominated during a contraction $(\eta\rightarrow 0)$ and henceforth rendering the cosmological evolution unstable [^12]. Moreover the time-dependence in the power spectrum of density perturbations derived from Wands’s model also implies an implicit $k$-dependence when the perturbation modes exit the horizon at different moments, so that the power spectrum is not truly scale-invariant.
To the contrary, the CSTB cosmos does not suffer these two problems. The time independence of power spectrum of CSTB cosmos guarantee that the perturbations are always sub-dominated during a contraction phase and would not destabilize the background. Furthermore, perturbing the matter-dominated background, $a\propto\eta^{2+\delta\nu}$, the power spectrum of the tachyon field is still scale invariant and time independent, $\mathcal{P}_{\delta T}\propto k^0\eta^0$ . Once again the time independence ensures the power spectrum of tachyon field perturbations in the CSTB cosmos is explicitly scale invariant even through each perturbation mode exits the horizon at a different moment. Therefore we conclude that the power spectrum of tachyon field perturbation in CSTB cosmos is truly scale-invariant and stable under time evolution.
#### Curvature perturbation of CSTB cosmos:
To make contact with observations we compute the power spectrum of curvature perturbations evaluated on spatially flat slices in the long wavelength limit [@Senatore:2012tasi; @Maldacena:2002vr], $$\zeta=\frac{\delta a}{a}=H\delta t=\frac{H}{\dot{T}_c}\, \delta T~,$$ where we have used the relation, $\delta T=\frac{d\langle T\rangle}{dt} \delta t=\dot{T}_c\, \delta t$. The power spectrum of curvature perturbations generated during tachyon-matter-dominated phase in CSTB cosmos then follows $$\label{eq-spct}
\mathcal{P}_\zeta=\left(\frac{H}{\dot{T_c}}\right)^2\mathcal{P}_{\delta T}~.$$ With [(Eq.\[eq-spt\])]{} at hand we need to compute the $k$-dependence and time-dependence of the factor, $(H/\dot{T_c})^2$, in [(Eq.\[eq-spct\])]{} to determine whether or not the power spectrum of curvature perturbations is stable and truly scale invariant in the cosmological sense. Using [(Eq.\[eq-tch\])]{} we obtain the power spectrum of curvature perturbations $$\mathcal{P}_\zeta=\kappa^{-2}\mathcal{P}_{\delta T}\propto k^0\eta^0.$$ All in all we conclude that the power spectrum of curvature perturbations is stable and is cosmologically scale-invariant.
#### Primordial spectral index:
We will now present the computation of the primordial spectral index of the curvature perturbations in the CSTB cosmos. In the last section we take $r_1\,=\,r_2\,=\,0$ in the discussion of scale-invariance of power spectrum at the lowest order. We shall hereby take their small values into account. By [(Eq.\[eq-dte\])]{} we obtain the equation of motion for the perturbations of the tachyon, $$\label{eq-dtns}
\ddot{\delta T_k}+\delta mH\dot{\delta T_k}+\frac{k^2_{e}}{a^2}\delta T_k=0~,$$ where $\delta m=-r_12\dot{T}+r_2^2\left[2\lambda\, \rho_T \, \phi\, \left( 3H-\dot{T}\, \phi \right)\right] $, and $k_e^2\equiv k^2+a^2 m_e^2$ with $m_e^2$ being the effective mass of tachyon perturbations given by $m_e^2=r_2^2 (2+\sqrt{2})\lambda \rho_T\, \phi^2$. Solving [(Eq.\[eq-dtns\])]{} in the cosmological background, $$a\propto \eta^{2+\delta \nu}~,$$ with $\delta \nu$ denoting the small deviation of CSTB cosmos background from a perfectly matter-dominated background, we obtain the power spectrum of tachyon field perturbation as $$\mathcal{P}_{\delta T_k}=k^3k_e^{-3+2\delta m-\delta \nu}\eta^0~.$$ Therefore the spectral index of curvature perturbation is $$\begin{aligned}
\nonumber n_s-1\equiv \frac{d \ln \mathcal{P}_\zeta}{d \ln k}=-2\frac{d\ln \kappa}{d\ln k}+\frac{d \ln \mathcal{P}_{\delta T_k}}{d \ln k}\qquad\qquad\\
=-2\frac{d\ln \kappa}{d\ln k}+2\delta m-\delta\nu+3\left(\frac{a^2m_e^2}{k^2}-\frac{d(a^2m_e^2)}{2k d k}\right)~. \label{eq-sindex}\end{aligned}$$ The first term in the last line is from the factor relating curvature perturbations to field perturbations. The value of the quantity, $\kappa$, is determined by the dynamics of background fields and principally independent of $k$. The second term relates the time-averaged quantities which nearly canceled during each oscillation cycle. The third term is derived from the small deviation of CSTB cosmos background from a perfectly matter-dominated background. And the last term indicates the influence of the effective mass on the tachyon field perturbations, which is negligible for the range of $k$ that we are interested in. All of these terms are not sensitive to the choices of initial conditions and the small changes in the shape of potential in CSTB cosmos. In other words, in contrast to the slow-roll inflation model’s need of fine-tuning the initial conditions and extreme flatness of potential to arrive at a small spectral index, the CSTB cosmos is much more stable as well as natural in having the value of $n_s-1$ around a few percents.
CSTB cosmos [*versus*]{} Slow-roll Inflation
============================================
In this section we show the underlying “dualities” relating the slow-roll inflation [@Liddle:2000cg; @Dodelson:2003ft], Wands’s model [@Wands:1998yp; @Finelli:2001sr; @Durrer:1995mz] and CSTB cosmos [@Li:2011nj] in the $(\nu, m)$ parameter space. And we analyze the stabilities of the power spectra for these three models.
The equation of field perturbations of slow-roll inflation, Wands’s model and CSTB cosmos–generally speaking–can be written as $$\ddot{\chi}_k+mH\dot{\chi}_k+\frac{k^2}{a^2}\chi_k=0~,\quad a\propto \eta^\nu~, \label{eq-gdet}$$ where $\chi_k$ denotes the Fourier mode of each field perturbation. For each model $(\nu, m)$ takes constant value in the parameter space: $(\nu, m)=(-1,3)$ for the slow-roll inflation, $(\nu, m)=(2,3)$ for Wands’s model, and $(\nu, m)=(2,0)$ for the CSTB cosmos.
#### Parameter space of different cosmoses:
In a model independent approach, solving [(Eq.\[eq-gdet\])]{}, one can obtain the power spectrum of $\chi_k$ , $$\label{eq-gsp}
\mathcal{P}_\chi\equiv\frac{k^3|\chi_k|^2}{2\pi}\sim k^{2L(\nu,m)+3}\eta^{2W(\nu,m)}~,$$ out of the effective horizon, $k\eta\rightarrow 0$ . The indices, $L$ and $W$, are functions of $\nu$ and $m$: $$L(\nu,m)=-\frac{1}{2}|(m-1)\nu-1|,~~~
W(\nu,m)=-\frac{1}{2}\{ [(m-1)\nu-1]+|(m-1)\nu-1| \}~.$$ With [(Eq.\[eq-gsp\])]{} at hand we can obtain all scale-invariant solutions and time-independent solutions in the $(\nu, m)$ parameter space by solving the $k$-independence condition, $2L(\nu, m)+3=0$, and the time-independence condition, $W(\nu, m)=0$ . We plot the solutions in the $(\nu, m)$ parameter space in [Fig. \[fig-htvt.pdf\]]{}.
#### Time-independent solutions:
To avoid the severe problem that increasingly growing perturbation modes may destabilize the cosmological background, each stable power spectrum of density perturbations should be time-independent, $W(\nu, m)=0$. In the $(\nu, m)$ parameter space, we find that all solutions satisfying $$(m-1)\,\nu-1 < 0$$ are time-independent, [*i.e.*]{} stable. In [Fig. \[fig-htvt.pdf\]]{}, the shaded region includes all time-independent solutions satisfying $(m-1)\,\nu-1 < 0$ whose boundaries are defined by $(m-1)\,\nu-1= 0$ and are drawn with thin dash lines.
![A parameter space $(\nu,m)$ to classify the power spectra of density perturbations. $\nu$ is the power law index (horizontal axis) and $m$ is the red/blue-shift index (vertical axis). The shaded region includes all time-independent solutions satisfying $(m-1)\,\nu-1 < 0$ whose boundaries are defined by $(m-1)\,\nu-1= 0$ and are drawn with thin dash lines. The purple dot-dash lines obeying $(m-1)\,\nu =-2$ represent scale-invariant as well as time-independent solutions. Another set of scale invariant solutions given by $(m-1)\,\nu =4 $ (violet solid lines) have Fourier modes varying with time and therefore are not truly scale-invariant in a physical sense.[]{data-label="fig-htvt.pdf"}](htvt.pdf){width="80.00000%"}
#### Scale-invariant solutions:
The scale-invariance condition, $2L(\nu, m)+3=0$, yields four groups of scale-invariant solutions. Two of them, I and II, generated in expansion and contraction respectively, are stable (time-independent), $$\label{eq-ssis}
\left\{
\begin{array} {l}
{\displaystyle I: m = -\frac{2}{\nu}+1~, \quad \nu<0} \\
\\
{\displaystyle II: m = -\frac{2}{\nu}+1~, \quad \nu>0} \\
\end{array}
\right. ~.$$ In [Fig. \[fig-htvt.pdf\]]{} they are drawn with dot-dash lines. The other two groups, III and IV, also generated in expansion and contraction phase respectively, are unstable (time-dependent), $$\label{eq-isis}
\left\{
\begin{array} {l}
{\displaystyle III: m = \frac{4}{\nu}+1~, \quad \nu>0} \\
\\
{\displaystyle IV: m = \frac{4}{\nu}+1~, \quad \nu<0} \\
\end{array}
\right. ~.$$ In [Fig. \[fig-htvt.pdf\]]{} they are drawn with solid lines.
In particular the slow-roll inflation, $(\nu, m)=(-1, 3)$ , and CSTB cosmos, $(\nu, m)=(2, 0)$ , $$\mathcal{P}_{slow-roll}\propto k^0\eta^0,~~~\mathcal{P}_{CSTB}\propto k^0\eta^0~. \label{eq-slcs}$$ belong to the class of stable scale-invariant solutions, I and II, respectively. And the Wands’s model, $(m,\nu)=(3, 2)$ , $$\mathcal{P}_{Wands}\propto k^0\eta^{-6}~.$$ belongs in the group of the unstable scale-invariant solutions, III. Now we turn our attention to the “duality” transformations which would connect these three models.
#### Duality transformations:
In the $(\nu, m)$ parameter space shown in [Fig. \[fig-htvt.pdf\]]{} there are two kinds of transformations connecting the stable scale-invariant solutions and unstable scale-invariant solutions, Horizontal Transformation (HT, relating cosmos of which perturbations with the same Hubble friction term but different background time evolution, [*i.e.*]{} “iso-damping transformation" ), $$(m,\nu)\rightarrow (m, -\nu+\frac{2}{m-1} )~,$$ and Vertical Transformation (VT, relating cosmos of which perturbations with different Hubble terms but same background time evolution, [*i.e.*]{} “iso-temporal transformation"), $$(m,\nu)\rightarrow (2-m+\frac{2}{\nu}, \nu )~.$$ Under a HT, the two group of stable scale-invariant solutions, I and II , are respectively mapped to the two group of unstable scale-invariant solutions, III and IV , in the horizontal direction. Under a VT, I and II are mapped to IV and III respectively in vertical direction.
The solution of the slow-roll inflation, $(\nu, m)=(-1, 3)$ , is connected horizontally to Wands’s Case, $(\nu, m)=(2,3)$ . Clearly this duality, which connects Wands’s matter-dominated contraction and slow-roll inflation shown in [@Wands:1998yp] [^13], is a special case of all possible HT’s with $m=3$. On the other hand, under a VT, the Wands’s Case, $(\nu, m)=(2,3)$ , is dual to CSBT cosmos, $(\nu, m)=(2,0)$ .
However, neither a HT nor a VT is a complete operation since each of them only maps a stable scale-invariant solution to an unstable scale-invariant solution and [*vice verse*]{}. We are looking for a complete duality transformation connecting a stable scale-invariant solution in an expansion phase to another stable and scale invariant solution in a contraction phase. The simplest way is to perform HT and VT consecutively, $$(m,\nu)\xrightarrow{HT} (m, -\nu+\frac{2}{m-1} )\xrightarrow{VT} (2-m+\frac{2(m-1)}{-(m-1)\nu+2}, -\nu+\frac{2}{m-1} )~,$$ as shown in [Fig. \[fig-htvt.pdf\]]{}. We call this transformation a [*[complete]{}*]{} duality transformation (HTVT in [Fig. \[fig-htvt.pdf\]]{}). It can connect all stable scale-invariant solutions in an expansion lying on Line I to Line II of all possible stable and scale invariant solutions in a contraction. Interestingly the slow-roll inflation, $(\nu, m)=(-1,3)$ , is related to the CSTB cosmos, $(\nu, m)=(2, 0)$, in this way–both possess stable and scale invariant spectra with the former generated in an exponential expansion while the latter in a tachyon-matter-dominated contraction.
#### Stability Analysis:
Noting that both slow-roll inflation and CSTB cosmos can produce power spectra of density perturbations satisfying current cosmological constraints. And they are “dual” to each other in the $(\nu, m)$ parameter space. However hidden in their curvature perturbation spectra there is a significant difference in fine-tuning. The curvature perturbations of slow-roll inflation model are $$\mathcal{P}_{s-\zeta}=\left(\frac{H}{\dot{\Phi}}\right)^2\mathcal{P}_{\delta \Phi}~,
\label{eq-rcfs}$$ and those of the CSTB cosmos are $$\mathcal{P}_{c-\zeta}=\left(\frac{H}{\dot{T}_c}\right)^2\mathcal{P}_{\delta T}~,
\label{eq-rcft}$$ where $\mathcal{P}_{s-\zeta}$ and $\mathcal{P}_{\delta \Phi}$ being the spectra of the curvature perturbations and field perturbations for slow-roll inflation while those for CSTB cosmos being $\mathcal{P}_{c-\zeta}$ and $\mathcal{P}_{\delta T}$. $\Phi$ and $T$ are the scalar field and the tachyon field driving slow-roll inflation model and CSTB cosmos respectively.
Given that $\mathcal{P}_{\delta \Phi}$ and $\mathcal{P}_{\delta T}$ are scale-invariant and stable[^14], to ensure the spectra of their curvature perturbations to be also stable and scale-invariant, both $\left(H/\dot{\Phi}\right)^2$ and $\left(H/\dot{T}_c\right)^2$ are required to be nearly constant. In slow-roll inflation models, to make $\left(H/\dot{\Phi}\right)^2$ nearly constant one needs to fine-tune the extreme flatness of the scalar potential. However in the case of CSTB cosmos, $\dot{T}_c\propto H$, is a dynamical attractor solution of the background field [@Li:2011nj]. According to [(Eq.\[eq-tch\])]{} $\left(H/\dot{T}_c\right)^2\sim \kappa^{-2}$ is automatically–and always will be–nearly constant. Therefore we can conclude that the scale-invariance of curvature perturbation spectrum in the CSTB cosmos is more stable and free of fine tuning problem in contrast to that in the slow-roll inflation model.
Summary and Prospects
=====================
In this paper we present a string-inspired bounce universe model–utilizing the coupling of a canonical scalar with the ubiquitous string tachyon field to realize the bounce (CSTB cosmos). We obtain a spectrum of curvature perturbations that is a stable and nearly scale-invariant. The big bang singularity problem is resolved in CSTB cosmos universe by the 5-D completion of the D3-anti-D3-brane annihilation and creation processes [@Li:2011nj], which is no longer singular event when viewed from one dimension higher.
The pre-bounce contraction phase of CSTB cosmos–during which the cosmological perturbations are generated–is dominated by the condensing tachyon field, [*i.e.*]{} tachyon matter. Because of the Hubble friction term of tachyon field perturbation vanishes after tachyon condensation, $mH=3H\sqrt{\langle1-\dot{T}^2\rangle}\rightarrow 0$ , a time-independent and scale-invariant power spectrum of tachyon field is generated in this contraction phase.
Furthermore, the power spectrum of curvature perturbation is related to that of tachyon field by a factor, $\left(H/\dot{T}_c\right)^2$, in long wavelength limit, $k\eta\rightarrow 0$ . According to the background evolution of CSTB cosmos [@Li:2011nj] , this factor is a constant in both time and scale–independent of $k$ and $\eta$. Therefore, the power spectrum of curvature perturbations is also stable and is scale-invariant in the cosmological sense.
We present a detailed study of the spectral index of primordial curvature perturbations. We find that each term of this spectral index is insensitive to choices of initial conditions and/or the slight changes of cosmological background in CSTB cosmos. It indicates that CSTB cosmos is stable as well as natural in having the value of $n_s-1$ around a few percents consistent with observations. This may serve as an explicit model for constructing a bounce universe in which the scale invariance of the power spectrum is generated during the pre-bounce contraction; and it is then preserved by the smooth bounce process in the long wavelength limit and subsequently becomes the density perturbations of the obervable universe.
To gain a deeper understanding of how stable scale-invariance emerges in the contraction phase of CSTB cosmos, we studied all scale-invariant and/or time-independent solutions in the unified parameter space of inflationary-bouncing cosmologies, [*i.e.*]{} the $(\nu, m)$ parameter space in [Fig. \[fig-htvt.pdf\]]{}. We find a [*complete*]{} duality transformation–iso-background and followed by an iso-temporal transformation–connecting all stable scale-invariant solutions in an expansion to all stable scale-invariant solutions in a contracting phase. Interestingly the CSTB cosmos, $(\nu, m)=(2, 0)$, is related to the slow-roll inflation, $(\nu, m)=(-1,3)$ , in this way–both possess stable and scale invariant spectra with the former generated in a tachyon matter dominated contraction while the latter in a well-known exponential expansion.
Summing up our discussion, we note several issues for further studies. In this paper our study focuses on the generation and the evolution of perturbations during the pre-bounce contraction phase in CSTB cosmos. Though it can be expected that the long wavelength modes of perturbations at the classical level would not be perturbed significantly through the smooth bounce[^15], , a full investigation, of how these perturbations going through the smooth bounce and re-entering horizon in post-bounce expansion in CSTB cosmos, is a worthwhile exercise.
On other hand, the perturbation spectra of slow-roll inflation and CSTB cosmos are (almost) identical, at the leading order, in [(Eq.\[eq-slcs\])]{}. To distinguish CSTB cosmos from the famous slow-roll inflation, the bispectrum of CSTB cosmos should be computed to extract a specific prediction of the shape of bispectrum in the CSTB cosmos. To conclude, we remark that the unified parameter we introduced in this paper is not only useful for proving the stability and scale invariance of the slow roll and CSTB models, it also enlighten the search for such spectra from other early universe models.
Acknowledgments
===============
Useful discussions with Robert Brandenberger, Yifu Cai, Jin U Kang, Konstantin Savvidy, Henry Tye and Lingfei Wang are gratefully acknowledged.
This research project has been supported in parts by the Jiangsu Ministry of Science and Technology under contract BK20131264 and by the Swedish Research Links programme of the Swedish Research Council (Vetenskapsradets generella villkor) under contract 348-2008-6049.
We also acknowledge 985 Grants from the Ministry of Education, and the Priority Academic Program Development for Jiangsu Higher Education Institutions (PAPD).
[^1]: With a slight change of the potential of the scalar field, the background is no longer an exponential expansion. And the spectrum of curvature perturbation becomes time-dependent, which in turn renders it scale-variant implicitly [@Liddle:2000cg; @Dodelson:2003ft].
[^2]: The idea of a collapsing phase preceding a phase of expansion could be traced back to three giants Tolman, Einstein and Lemaitre who, independently, proposed the idea in the early 1930s, for example see [@Tolman:1931fi].
[^3]: Relaxing the constant EoS assumption, however, the scale-invariant power spectrum can also be generated in a slowly contracting Ekpyrotic background [@Khoury:2009my; @Khoury:2010gw; @Joyce:2011ta]. See [@Linde:2009mc] for critiques of this category of models.
[^4]: There are many extensions for non-canonical and/or multi-fields cosmological models, for example, see [@Garriga:1999vw; @Senatore:2010wk; @Langlois:2008qf; @Shiu:2011qw].
[^5]: For the single canonical scalar field models we always have $m=3$, where $3$ comes from the spatial dimensions of our presently observable universe. However, in the non-canonical single/multi-field cases, $mH=3f(\phi_i,\dot{\phi}_i)H$ generically with $f$ being a function determined by the underlying models. For instance, in the tachyon field model, one gets $mH=3\sqrt{1-\dot{T}^2}H$. And in the tachyon matter condensation phase, $\dot{T}\rightarrow 1$, [*i.e.*]{} $f(T,\dot{T})=\sqrt{1-\dot{T}^2}\rightarrow 0$, so that we have $m$ approach $0$ rather than $3$ in this case. Without loss of generality we can for the time being take $m$ to be constant for any given model.
[^6]: Following the same abuse of language in the bounce literatures by which it merely indicates the possible existence of a scale invariant spectrum obtained from models other than slow roll inflation.
[^7]: In the single tachyon field case, the dynamics of tachyon only describes the annihilation process of D-anti-D-brane pair, but does not include the issue that how these branes move to collide. To see how these branes move to collide with a weak attractive force between them, we will turn our attention to a coupled scalar-tachyon field model soon.
[^8]: In the CSTB cosmos, as an auxiliary field, $\phi$ sector is always sub-dominated, and during this contraction, the averaged EoS of $\phi$ is also equal to zero, $w_\phi=0$. For simplicity, we call this phase as “tachyon matter dominated contraction phase”.
[^9]: One, if preferred, can picture a stack of D-branes and anti-D-branes, some of them undergo pair annihilation while some remain intact in each collision.
[^10]: During the tachyon-matter-dominated contraction in CSTB cosmos the curvature term of FLRW metric, $K a^{-2}, K=1$, is well sub-dominated. For simplicity we use the flat FLRW metric and ignore tensor modes in the perturbation study.
[^11]: An analytic study of the dynamics of two coupled scalar fields can be found in [@Wang:2011ed].
[^12]: A similar analysis and conclusion have been made in [@Gratton:2003pe].
[^13]: This duality is also discussed in various gauge choice with taking account in subdominated modes of perturbations [@Boyle:2004gv; @Piao:2004uq]
[^14]: They only include the small derivations in their cosmological backgrounds from a perfectly exponential expansion or a perfectly matter-dominated contraction respectively, which is not related to the fine-tuning problem. Therefore, we take them as perfect scale-invariant here for simplicity.
[^15]: Recently many great progresses have been made on studying the evolution of perturbations going through a bounce, for example, see [@Deruelle:1995kd; @Finelli:2001sr; @Allen:2004vz; @Creminelli:2007aq; @Lin:2010pf; @Xue:2011nw].
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-1.0cm 14.5 cm 22.5 cm 0.7cm \#1\#2\#3[Nucl. Phys. B [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[Phys. Lett. B [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[Phys. Lett. [**\#1B**]{} (19\#2) \#3]{} \#1\#2\#3[Phys. Rev. D [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[Phys. Rev. Lett. [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[Phys. Rep. [**\#1**]{} C (19\#2) \#3]{} \#1\#2\#3[Ann. Rev. Astron. Astrophys. [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[Ann. Rev. Nucl. Part. Sci. [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[Mod. Phys. Lett. A [**\#1**]{} (19\#2) \#3]{}
hep-ph/9507402\
HD-THEP-95-28\
IOA-226-95\
[**E. G. Floratos $^{1,*}$, G. K. Leontaris $^{2,**}$**]{} and [**S. Lola$^{3}$**]{}
----------------------------------------------------------------------
$^{1}$ [Institute of Nuclear Physics, NSRC Demokritos,]{}
[Athens, Greece]{}
$^{2}$[Centre de Physique Théorique, Ecole Polytechnique]{},
[F-91128 Palaiseau, France]{}
$^{3}$[Institut für Theoretische Physik, Univerisität Heidelberg,]{}
[Philosophenweg 16, 69120 Heidelberg, Germany ]{}
----------------------------------------------------------------------
[**ABSTRACT**]{}
We discuss aspects of the low energy phenomenology of the MSSM, in the large $\tan {\beta} $ regime. We explore the regions of the parameter space where the $h_t$ and $h_b$ Yukawa couplings exhibit a fixed point structure, using previous analytic solutions for these couplings. Expressions for the parameters $A_{t}$ and $A_{b}$ and the renormalised soft mass terms are also derived, making it possible to estimate analytically the sparticle loop – corrections to the bottom mass, which are important in this limit.
July 1995
------------------------------------------------------------------------
. The masses in the tables are given in GeV.
As we increase the supersymmetry breaking scale, $tan\beta$ slightly increases, in order to get the same low energy parameters. (At the same time the unification scale drops slightly, while the inverse gauge coupling at the unification scale increases, by a small amount).
[|c|c|c|c|c|c|c|]{} $tan\beta$ & $m_0=m_{1/2}$ & $\mu$ & $A_{t}$ & $I_1 (10^{-6})$ & $I_2(10^{-6})$ & $\delta m_b$\
58.1 & 100 & 124 & -147 & 12.8 & 18.1 & 0.40\
59.3 & 150 & 192 & -208 & 6.2 & 8.5 & 0.41\
60.0 & 200 & 253 & -264 & 3.8 & 5.1 & 0.42\
60.3 & 250 & 310 & -316 & 2.6 & 3.4 & 0.43\
60.6 & 300 & 365 & -364 & 1.9 & 2.5 & 0.43\
[[**Table 2 :**]{} [Bottom mass corrections for $h_G\sim 2.0$.]{}]{}
\[table:2\]
[|c|c|c|c|c|c|c|]{} $tan\beta$ & $m_0=m_{1/2}$ & $\mu$ & $A_{t}$ & $I_1 (10^{-6})$ & $I_2(10^{-6})$ & $\delta m_b$\
55.5 & 100 & 119 & -153 & 12.1 & 18.0 & 0.34\
56.6 & 150 & 184 & -216 & 5.9 & 8.4 & 0.35\
57.1 & 200 & 243 & -274 & 3.5 & 5.0 & 0.36\
57.4 & 250 & 297 & -328 & 2.4 & 3.4 & 0.37\
57.5 & 300 & 348 & -377 & 1.8 & 2.5 & 0.37\
[[**Table 3 :** ]{} [Bottom mass corrections for $h_G\sim 1.0$.]{}]{}
\[table:3\]
Conclusions
===========
In this letter we have used simplified analytic solutions for the $h_t , h_b$ Yukawa couplings in order to study the MSSM in the large $tan\beta$ regime. We have explored the regions of the parameter space which lead to a fixed point structure and derived the evolution of the Yukawas towards these fixed points. Using this information, one may identify the regions for the initial values of the Yukawa couplings which lead to the strongest attraction towards these infrared fixed points. Under these considerations, top-bottom Yukawa coupling equality, and values of the couplings close to the non-perturbative regime seem to be favoured. Finally, we obtained corrections on the renormalised soft mass terms due to the evolution of the trilinear parameters $A_{t}$ and $A_{b}$. Using these results, we estimated analytically the sparticle loop – corrections to the bottom mass, which are important in the large tan$\beta$ scenario. In agreement with previous calculations we find that the maximal corrections arise at the fixed point.
[*G.K.L. would like to thank the group of Centre de Physique Theorique de l’ Ecole Polytechnique for kind hospitality.*]{}
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author:
- |
[Maria Apostolaki]{}\
ETH Z[ü]{}rich
- |
[Ankit Singla ]{}\
ETH Z[ü]{}rich
- |
[Laurent Vanbever]{}\
ETH Z[ü]{}rich
bibliography:
- 'main.bib'
title: 'Performance-Driven Internet Path Selection'
---
|
---
abstract: 'The advent of quantum computing has challenged classical conceptions of which problems are efficiently solvable in our physical world. This motivates the general study of how physical principles bound computational power. In this paper we show that some of the essential machinery of quantum computation – namely reversible controlled transformations and the phase kick-back mechanism – exist in any operational-defined theory with a consistent notion of information. These results provide the tools for an exploration of the physics underpinning the structure of computational algorithms. We use these results to investigate the relationship between interference behaviour and computational power, demonstrating that non-trivial interference behaviour is a general resource for post-classical computation. In proving the above, we connect post-quantum interference – the higher-order interference of Sorkin – to the existence of post-quantum particle types, potentially providing a novel experimental test for higher-order interference. Finally, we conjecture that theories with post-quantum interference can solve problems intractable even on a quantum computer.'
author:
- 'Ciar[á]{}n M. Lee'
- 'John H. Selby'
title: 'Generalised phase kick-back: the structure of computational algorithms from physical principles'
---
Introduction
============
One of the major conceptual breakthroughs in physics over the past thirty years was the realisation that quantum theory offers dramatic advantages [@Nielsen] for various information-processing tasks – computation in particular [@shor; @arkhipov; @Nielsen]. This raises the general question of how physical principles bound computational power. Moreover, what broad relationships exist between such principles and computation? A major roadblock to such an investigation is that quantum computation is phrased in the language of Hilbert spaces, which lacks direct physical or operational significance.
In contrast, the framework of operationally-defined theories [@Pavia1; @Pavia2; @Hardy-2011; @Barrett-2007; @LB-2014] provides a clear-cut operational language in which to investigate this problem. Theories within this framework can differ [@Barrett-2007] from classical and quantum theories. Whilst many of them may not correspond to descriptions of our physical world, they make good operational sense and allow one to assess how computational power depends on the physical principles underlying them in a systematic manner.
Previous investigations into computation within this framework have taken a high-level approach using the language of complexity classes to derive general bounds on the power of computation [@LB-2014; @Proofs; @landscape]. However, much of quantum computing is concerned not so much with this high-level view, but instead with the construction of concrete algorithms to solve specific problems. A deeper understanding of the general structure of computational algorithms in this framework has so far remained illusive. Here we take this low-level algorithmic view and ask which physical principles are required to allow for some of the common machinery of quantum computation in this context.
In this paper we show that three physical principles, *causality* (which roughly states that information propagates from present to future), *purification* (roughly, that information is fundamentally conserved) and *strong symmetry* (all information carriers of the same size are equivalent) – which are necessary for a well defined notion of information – are sufficient for the existence of reversible controlled transformations (Thm. (\[Reversible-Control\]), Sec. (\[Control2\])) and a generalised *phase kick-back mechanism* (Thm. (\[ALL\]), Sec. (\[Control2\])). In the quantum case, the phase kick-back mechanism [@kickback] plays a vital role in almost all algorithms – notably the Deutsch-Jozsa algorithm, Grover’s search algorithm and Simon’s algorithm – whilst reversible controlled transformations are central components of most well-studied universal gate sets and fundamental for the definition of computational oracles.
One might ask how the computational power of theories with these crucial algorithmic components depends on their underlying physical properties. One such property – currently under both theoretical [@Higher-order-reconstruction; @Niestegge-2012; @Henson-2015; @ududec2011three] and experimental [@sinha2008testing; @park2012three] investigation – is the existence of *higher-order interference*.
Sorkin [@sorkin1994quantum; @sorkin1995quantum] has introduced a hierarchy of mathematically conceivable *higher-order* interference behaviours and shown that quantum theory is limited to having only second-order interference. Informally, this means that the interference pattern created in a three – or more – slit experiment can be written in terms of the two and one slit interference patterns obtained by blocking some of the slits; no genuinely new features result from considering three slits instead of two. This is in contrast to the existence of second-order interference where the two slit interference cannot be reproduced from that of single slits. Informally, theories are said to have higher-order interference if irreducible interference patterns can be created in multi-slit experiments.
Second-order interference between quantum computational paths appears to be a resource for non-classical computation [@Interference-speed-up; @Nielsen]. It therefore seems prudent to investigate how different interference behaviour is related to computation in general. In quantum theory there is an intimate connection between phase transformations – such as those used in the kick-back mechanism – and interference. Motivated by this, in Sec. (\[int\]), we introduce a framework that relates higher-order interference to *phase transformations* in operationally-defined theories.
We show that the generalised phase kick-back mechanism allows one to access any ‘higher-order phase’ in a controlled manner. Using this, in Sec. (\[oracle\]), we show that the existence of non-trivial interference behaviour allows for the solution of problems intractable on a classical computer. We also conjecture that these higher-order phase kick-backs allow for the solution of computational problems intractable even on a *quantum* computer. Additionally, in Sec. (\[Exchange\]), we show that higher-order phases lead to new particle types that exhibit both qualitatively and quantitatively different behaviour to fermions, bosons and anyons. Thus potentially providing a new experimental test of higher-order interference.
The framework
=============
Operational physical theories
-----------------------------
We work in the circuit framework for operationally-defined theories developed in [@Hardy-2011; @Pavia1; @Pavia2]. An operational theory specifies a set of physical processes that can be connected together to form experiments and assigns probabilities to different experimental outcomes. A process has input ports, output ports, and a classical pointer. When a process is used in an experiment, the pointer comes to rest in one of a number of positions, indicating an outcome has occurred. Intuitively, one can think of *physical systems* as passing between the ports of these processes. These systems come in different types, denoted $A,B..$. In an experiment these processes can be composed both sequentially and in parallel, and when composed sequentially, types must match.
In this framework, closed circuits define probabilities. Processes that yield the same probabilities in all closed circuits are identified. The set of equivalence classes of processes with no input ports are called *states*, no output ports *effects* and both input and output ports *transformations*. The set of all states of system $A$ is denoted $\Omega_A$, the set of all effects on $B$ is denoted $\mathcal{E}_B$ and the set of *reversible* transformations between systems $A$ and $B$ is denoted $\mathcal{R}^A_B$ [^1]. Note that $\mathcal{R}^A_B$ has a group structure. A state is *pure* if it does not arise as a *coarse-graining* of other states [^2]; a pure state is one for which we have maximal information. A state is *mixed* if it is not pure. We assume for this paper that the composite of two pure states is itself pure [^3]. Similarly, one says a transformation is pure if it does not arise as a coarse-graining of other transformations. It can be shown that reversible transformations preserve pure states.
The ‘Dirac-like’ notation $_A|s)$ is used to represent a state of system $A$, and $(e_{r}|_B$ to represent an effect on $B$. Here $r$ is the position of the classical pointer, which can be thought of as the outcome of the measurement defined by $\{(e_r|\}_r$. States, effects and transformations can be represented diagrammatically: $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=cpoint] (0) at (-0.5, -0) {$s$};
\node [style={small box}] (1) at (1.5, -0) {$T$};
\node [style=cocpoint] (2) at (3.5, -0) {$e_r$};
\node [style=none] (3) at (4.5, -0) {$=$};
\node [style=none] (4) at (6.5, -0) {$(e_r|_BT_A|s)$};
\node [style=none] (5) at (0.5, 0.5) {$A$};
\node [style=none] (6) at (2.5, 0.5) {$B$};
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\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1);
\draw (1) to (2);
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\end{tikzpicture}$$ This diagrammatic approach was inspired by the categorical formalism of quantum mechanics [@cqm1; @cqm2].
A theory is said to be *causal* if there exists a unique deterministic effect ${(\begin{tikzpicture}
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\node [style=detEff] (0) at (0, -0) {};
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\end{tikzpicture}|}$ for every system, such that $\sum_r (e_r|={(\begin{tikzpicture}
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\node [style=detEff] (0) at (0, -0) {};
\node [style=none] (1) at (0.15, -0) {};
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\end{tikzpicture}|}$ for all measurements, $\{(e_r|\}_r$.
Mathematically, causality is equivalent to the statement: “Probabilities of present experiments are independent of future measurement choices”. In causal theories, all states are *normalised* [@Pavia1]. That is, ${(\begin{tikzpicture}
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\node [style=detEff] (0) at (0, -0) {};
\node [style=none] (1) at (0.15, -0) {};
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\end{tikzpicture}|}s)=1$ for all $|s)$. The deterministic effect allows one to define a notion of *marginalisation* for multi-partite states.
\[Pure\] Given a state $_A|s)$ there exists a system $B$ and a pure state $_{AB}|\psi)$ on $AB$ such that $_A|s)$ is the marginalisation of $_{AB}|\psi)$: $$\begin{tikzpicture}
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\end{tikzpicture}$$ Moreover, the purification $_{AB}|\psi)$ is unique up to reversible transformations on the purifying system, $B$ [^4].
While the purification principle appears to only concern states, it can be leveraged to prove somewhat analogous results about transformations [@Pavia1 Thm. 15]: let $T,T'$ be reversible transformations. If, $$\begin{tikzpicture}
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\node [style=cpoint] (26) at (6, -0.25) {s'};
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\draw (7) to (6.center);
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\draw (9.center) to (8.center);
\draw (8.center) to (2.center);
\draw (2.center) to (3.center);
\draw (15.center) to (17.center);
\draw (26) to (20.center);
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\end{tikzpicture} ,$$ then there exists a reversible transformation $G$ such that $$\begin{tikzpicture}
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\node [style=cpoint] (0) at (6, 1) {$\sigma$};
\node [style=none] (1) at (8, 1.5) {};
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\node [style=none] (28) at (12, -0) {$\forall |\sigma)$};
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\end{tikzpicture} .\label{Dilation}$$ Eq. (\[Dilation\]), above, depicts the equivalence of purifications of a transformation up to a local reversible transformation, as opposed to the purification of states mentioned in Def. (\[Pure\]). These can in fact been shown to be equivalent [@Pavia1; @Pavia2], and so we will use the term *purification* when referring to either notion.
Pure states $\{|s_i)\}_{i=1}^n$ are *perfectly distinguishable* if there exists a measurement, corresponding to effects $\{(e_j|\}_{j=1}^n$, such that $(e_j|s_i)=\delta_{ij}$ for all $i,j$. Note that an $n$-tuple of pure and perfectly distinguishable states can reliably encode an $n$-level classical system.
A theory satisfies *strong symmetry* if for any two $n$-tuples of pure and perfectly distinguishable states $\{|\rho_i)\},\{|\sigma_i)\},$ there exists a reversible transformation $T$ such that $T|\rho_i)=|\sigma_i)$ for $i=1,\dots,n$.
Informally, the purification principle says that information is fundamentally conserved, strong symmetry states that all information carriers of the same size are equivalent and causality implies that information propagates from present to future. Note that standard quantum theory, real vector space quantum theory and the classical theory of pure states satisfy all of the above principles. These principles will be shown to be a primer for interesting and consistent computation. In Sec. \[oracle\], we shall investigate the change in computational power as one varies the interference behaviour in theories satisfying causality, purification and strong symmetry.
Higher-order interference via phase transformations
---------------------------------------------------
### A quantum example {#Quantum-Example}
Perhaps the cleanest example of interference in quantum theory is exhibited by the Mach-Zehnder interferometer, illustrated below: $$\begin{tikzpicture}
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\node [style=sbox, scale=1] (11) at (3.5, -3.75) {\small $P_{\Delta\phi}$};
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\end{tikzpicture}$$ There are three parts to this:
1. Prepare a state as a superposition of paths: $$|s)={|+\rangle}{\langle +|}:=\rho_+$$
2. Apply a ‘phase transformation’: $$P_{\Delta\phi}|s)=R_z^{\Delta\phi}\rho_+R_z^{\Delta\phi\dagger},$$ with $R_z^{\Delta\phi}$ a rotation by $\Delta\phi$ about the $z$ axis of the Bloch ball.
3. Measure in a superposition of paths: $$\begin{aligned} (e|P_{\Delta\phi}|s)&=\mathrm{Tr}\left(\rho_+R_z^{\Delta\phi}\rho_+R_z^{\Delta\phi\dagger}\right)=\cos^2\left(\frac{\Delta\phi}{2}\right). \end{aligned}$$
The observed interference pattern is therefore a map from the group of ‘phase transformations’, parametrised by $\Delta\phi$, to the unit interval (i.e. probabilities), $$\label{Quantum-Pattern}
P_{\Delta\phi}\mapsto \cos^2\left(\frac{\Delta\phi}{2}\right).$$
The existence of interference in quantum theory is encapsulated in the statement: “the interference pattern observed for a particular superposition measurement cannot be reproduced by the statistics generated by ‘which path’ measurements”. In the above example this translates to: $$\label{Quantum-Interference}
\cos^2\left(\frac{\Delta\phi}{2}\right)\neq \sum_{i=0}^1 q_i \mathrm{Tr}\left({|i\rangle}{\langle i|}R_z^{\Delta\phi}\rho_+R_z^{\Delta\phi\dagger}\right),$$ where $q_i$ is an arbitrary constant. Eq. (\[Quantum-Interference\]) is to be interpreted as an inequality of the functions defined on the right and left hand side. That is, these functions do not coincide on all phase transformations. This follows from the fact that: $$\label{Phase}
R_z^{\Delta\phi\dagger}{|i\rangle}{\langle i|}R_z^{\Delta\phi}={|i\rangle}{\langle i|}, \quad \forall i\in\{0,1\}.$$ That is, the left hand side of Eq. (\[Quantum-Interference\]) depends on $\Delta\phi$ whilst the right hand side does not.
### Operational theories \[int\]
The quantum example from Sec. (\[Quantum-Example\]) illustrates the key components necessary to discuss interference:
- a notion of ‘path’,
- a notion of ‘superposition of paths’,
- transformations that leave the statistics of ‘which path’ measurements invariant, i.e. ‘phase transformations’,
- a notion of ‘interference pattern’, i.e. a way of associating phase transformations with probabilities.
These points will now be discussed in the context of arbitrary operationally-defined theories. We then use this framework to link higher-order interference and phase transformations. Our approach is similar in spirit to that of Garner et al. [@Garner], with the caveat that they have not considered higher-order interference.
#### —(i) Paths: {#i-paths .unnumbered}
A path is defined by a state and effect pair, where we view the state as ‘preparing a state which belongs to the path’ and the effect as ‘measuring whether the state belongs to the path’ and so we demand the probability of the state-effect pair to be one.
[Paths, $p$:]{} $$p:=(|s),(e|) \text{ s.t. } (e|s)=1.$$
In our quantum example, the paths were $p_0=\left({|0\rangle}{\langle 0|}, {|0\rangle}{\langle 0|}\right)$ and $p_1=\left({|1\rangle}{\langle 1|}, {|1\rangle}{\langle 1|}\right)$.
Paths are disjoint if the state defining one path has zero probability of belonging to the other, and vice versa.
[Disjoint paths, $p_1\perp p_2$:]{} $$p_1\perp p_2 \iff (e_i|s_j)=\delta_{ij}.$$
An $n$-path experiment is defined by $n$ mutually disjoint paths such that the set consisting of the effects from each path forms a measurement.
[$n$-path experiment, $\mathds{P}$:]{} $$\mathds{P}:= \{p_i\} \text{ s.t. } p_i\perp p_j \ \forall i\neq j, \ \mathrm{and} \ \sum_i (e_i|={(\begin{tikzpicture}
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In the quantum case, an $n$-path experiment would correspond to a multi-arm interferometer.
#### —(ii) Superposition of paths: {#ii-superposition-of-paths .unnumbered}
A superposition of paths will be defined relative to some $n$-path experiment $\mathds{P}$ via the notion of *support*. We say that a state (or effect) has support on a path if the effect (or state) associated to that path gives a non-zero probability.
[Support of a state or effect, $Supp[|s)]$ or $Supp[(e|]:$]{} $$\begin{aligned}
Supp[|s)]:=\{p_i\in \mathds{P} \ | \ (e_i|s)\neq 0\},\\
Supp[(e|]:=\{p_i\in \mathds{P} \ | \ (e|s_i)\neq 0\}.
\end{aligned}$$
If the support of a state consists of more than one path this does not guarantee that it is a superposition of paths – it could equally well be a classical mixture of paths. A superposition state must therefore lie outside the convex hull of the states which have support only on a single path. In our quantum example, the state ${|+\rangle}{\langle +|}$ – introduced in point (1) of Sec. (\[Quantum-Example\]) – was a superposition of paths.
We can define set of states (or effects) with support on some subset of paths $I\subseteq \mathds{P}$ as: $$\begin{aligned}
\Omega_I&:=\{|s)\in \Omega \ | \ Supp[|s)]=I\}, \\
\mathcal{E}_I&:=\{(e|\in \mathcal{E} \ | \ Supp[(e|]=I\}.
\end{aligned}$$
#### —(iii) Phase transformations: {#iii-phase-transformations .unnumbered}
A phase transformation – relative to some $\mathds{P}$ – is any transformation that leaves the statistics of ‘which path’ measurements invariant.
[Phase group, $\mathcal{P}$:]{} $$\mathcal{P}:=\{T\in \mathcal{R} \ | \ (e_i|T=(e_i|, \ \forall i\in \mathds{P}\}$$
In the quantum example, the phase transformation was the rotation $R_z^{\Delta\phi}$ introduced in point (2) of Sec. (\[Quantum-Example\]).
#### —(iv) Interference patterns: {#iv-interference-patterns .unnumbered}
We now generalise the quantum interference pattern of Eq. (\[Quantum-Pattern\]) to arbitrary operational theories.
[Interference pattern, $\mathcal{C}_{s,e}:$]{} $$\mathcal{C}_{s,e}: \ \mathcal{P}\to[0,1] \ :: \ T\mapsto (e|T|s)$$
Given this definition, Eq. (\[Quantum-Interference\]) translates into the existence of $(e|\in\mathcal{E}_{\{0,1\}}$ – that is, an effect with support on path $0$ and path $1$ – and $|s)\in\Omega_{\{0,1\}}$ such that $$\label{General-Interference}
{C}_{s,e}\neq \sum_{i=0}^1 \mathcal{C}_{s,e_i}$$ for all possible choices of $(e_i|\in\mathcal{E}_{\{i\}}$ including subnormalised effects, this is the analogue of the $q_i$’s in Eq. (\[Quantum-Interference\]). In other words, there is some choice of superposition state and effect such that their interference pattern cannot be reproduced by the statistics generated by effects with support on a single path.
Other approaches to defining higher-order interference in operational theories (for example [@Higher-order-reconstruction]) have additional structure such that one can define a set of ‘filters’, $\{F_I\}$, for the theory. These are transformations that represent the action of leaving open some subset of paths $I$ whilst blocking the others. In this case one can define $(e_I|=(e|F_{I}$ giving a specific set of effects. However, arbitrary theories do not have sufficient structure to define filters and so one must consider all possible choices $(e_I|$ with the correct support. Otherwise [@Thesis; @LS-2015] one can – even in quantum and classical theory – choose a specific set of $(e_I|$ to give the artificial appearance of higher-order interference.
It follows that the existence of a non-trivial phase group implies the existence of interference in a general theory. Indeed, the left hand side of Eq. (\[General-Interference\]) depends on the phase group element, whilst the right hand side does not – the analogue of Eq. (\[Phase\]) from the quantum example. We now use our framework to discuss *higher-order* interference.
### Higher-order interference and phase
Adapting Sorkin’s original definition of higher-order interference [@sorkin1994quantum] to our framework results in: the existence of $n$th-order interference in an $n$-path experiment corresponds to the existence of an effect $|e)$ and a state $(s|$ such that $$\label{Higher-order}
\mathcal{C}_{s,e}\neq \sum_{I\subset\mathds{P}}(-1)^{n-|I|+1}\mathcal{C}_{s,e_I},$$ for all $|e_I)\in\mathcal{E}_I.$ As in Eq. (\[Quantum-Interference\]), Eq. (\[Higher-order\]) is to be interpreted as an inequality of the functions defined on the right and left. See Appendix (\[Higher-order-App\]) for an in-depth discussion of Eq. (\[Higher-order\]).
Motivated by Eq. (\[Higher-order\]), we wish to determine if particular phase transformations give rise to higher-order interference. The defining feature of phase transformations is that they leave the statistics of effects with support on single paths – that is, effects in $\bigcup_i\mathcal{E}_{\{i\}}$ – invariant. The natural generalisation of this is to consider transformations that not only leave the statistics of effects on single paths invariant, but also superposition effects. This motivates the following definition.
A transformation $T$ is *$n$-undetectable* if: $(e|T=(e|, \ \forall (e|\in\bigcup_{I:|I|\leq n}\mathcal{E}_I$.
Together with its natural converse.
A transformation $T$ is *$m$-detectable* if there exists $(e|\in\bigcup_{I:|I|\leq m}\mathcal{E}_I$, such that $(e|T\neq(e|$.
We can now link higher-order interference to certain types of phase transformations, which we call *higher-order phases*.
\[Higher-Phase\] A transformation $T$ that is $n$ detectable and $n\mathrm{-}1$ undetectable implies the existence of $n$th-order interference.
Choose $|s)$ and $(e|$ such that $T$ is detected. It is then clear that the left hand side of Eq. (\[Higher-order\]) is dependent on $T$, whilst – due to undetectability – the right hand side is not. They are thus distinct functions.
In our quantum example, the phase transformation was $2$-detectable, but $1$-undetectable.
Controlled transformations and a generalised phase kick-back {#Control2}
============================================================
Given a set of pure and perfectly distinguishable states $\{|i)\}$ and a set of transformations $\{T_i\}$, we define a controlled transformation $C\{T_i\}$ as: $$\label{Control}
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\end{tikzpicture}$$ The top system and lower systems are referred to as the *control* and *target* respectively.
Note that classical controlled transformations – where the control is measured and conditioned on the outcome a transformation is applied to the target – exist in any causal theory [@Pavia1] with sufficient distinguishable states. However, such transformations are in general not reversible and do not offer an advantage over classical computation [@Rev]. Moreover, the existence of reversible controlled transformations appears to be a rare property of operational theories [@Rev]. The following states that in theories satisfying our assumptions, there exist reversible controlled transformations. The proof is in contained in Appendix (\[CT\]).
\[Reversible-Control\] In any theory satisfying i) causality, ii) purification, iii) strong symmetry, there exists a *reversible* controlled transformation for all sets of reversible transformations $\{T_i\}$.
Moreover, the following theorem states that any controlled transformation in such theories ‘preserves superpositions’. Where ‘superposition’ is meant in the sense of Sec. (\[int\]) part (ii) and ‘preserves superposition’ means that the probability of detecting the system in each path of the superposition is preserved by the transformation. See Appendix (\[SuperpositionProof\]) for the proof.
Superpositions are preserved on the control input: $$\label{SupPres}
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\end{tikzpicture}$$ where $\{(i|\}$ is the measurement that perfectly distinguishes the control states $\{|i)\}$ [^5].
Every controlled transformation in quantum theory has a *phase kick-back* mechanism [@Nielsen]. Such mechanisms form a vital component of most quantum algorithms. We now show the existence of a *generalised* phase kick-back mechanism in any theory satisfying our assumptions.
\[Generalised-Kick-Back\] Given an $|s)$ such that $T_i|s)=|s), \ \forall{i}$, there exists a reversible transformation $Q_s$ such that $$\label{KB}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
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\node [style=none] (8) at (0.75, 1.5) {};
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\end{tikzpicture}$$ Moreover, $Q_s$ is phase transformation: $$\begin{tikzpicture}
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\node [style={small box}] (0) at (1, -0) {$Q_s$};
\node [style=none] (1) at (5.5, -0) {};
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\node [style=none] (3) at (8, -0) {$\forall i$};
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\begin{pgfonlayer}{edgelayer}
\draw (5) to (0);
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\end{tikzpicture}$$
$$\begin{tikzpicture}
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\node [style=none] (0) at (9.25, -0.75) {};
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\node [style=cpoint] (8) at (7, -0.25) {$s$};
\node [style=none] (9) at (1.5, 1) {$C$};
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\node [style=cpoint] (20) at (0, -0.25) {$s$};
\node [style=none] (21) at (0.75, -0.75) {};
\node [style=none] (22) at (11.5, 0.5) {$=$};
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\node [style=none] (27) at (14.75, -0) {};
\node [style=none] (28) at (5.75, 0.5) {$\sum_i$};
\node [style=none] (29) at (8.5, -0.25) {$\{T_i\}$};
\node [style=none] (30) at (8.5, 1) {$C$};
\node [style=none] (31) at (1.5, -0.25) {$\{T_i\}$};
\node [style=none] (32) at (2.25, 1) {};
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\end{tikzpicture}$$ The first equality follows from causality and the second from Eq. (\[SupPres\]) and the definition of $|s)$. Eq. (\[Dilation\]) then implies the existence of a reversible $Q_s$ such that: $$\begin{tikzpicture}
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\end{tikzpicture}$$ Note that $Q_s$ depends on both the controlled transformation and the joint eigenstate $|s)$. Note that: $$\begin{tikzpicture}
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\end{tikzpicture}$$ Causality – via state normalisation – then gives: $$\begin{tikzpicture}
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In quantum theory, it is possible to achieve any phase transformation via a kick-back mechanism. However, Thm. (\[Generalised-Kick-Back\]) only implies the existence of at least one phase that can be ‘kicked-back’. We now show that all phases arise via the generalised mechanism. Consider the set of pure and perfectly distinguishable states $\{|s_i)\}$ and let $\{T_i\}$ be elements of their phase group, i.e. $T_i|s_j)=|s_j),$ $\forall i,j$. Construct the controlled transformation $C\{T_i\}$. The designation of control and target for $C\{T_i\}$ is symmetric: $$\begin{tikzpicture}
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\node [style=cpoint] (30) at (11.25, -0.25) {$s_i$};
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\end{tikzpicture}$$ Thus any $C\{T_i\}$ is control-target symmetric if the $T_i$ are elements of a phase group. The transformations on the target are given by the kicked-back phases, $\{Q_i\}$. Given an arbitrary $W_i$, construct the transformation $\{W_i\}C$ and note that via control-target symmetry it is equivalent to $C\{G_i\}$, for some $\{G_i\}$. The controlled transformation $C\{G_i\}$ thus gives rise to the kicked-back phase $W_i$ and we have:
\[ALL\] Every phase transformation can arise via a generalised phase kick-back mechanism
Particle exchange experiments {#Exchange}
-----------------------------
Dahlsten et al. [@Oscar] have shown that there is a close connection between particle exchange statistics and the phase group in operational theories. We use the framework and results presented in this paper to expand upon and formalise these connections. Motivated by the quantum case, place a pair of indistinguishable particles in superposition by inputting them to an interferometer, as shown in the following diagram. $$\begin{tikzpicture}
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\end{tikzpicture}$$ On the upper path the two particles are swapped using some operation ‘$S$’, whilst on the lower path they are left invariant – that is, the identity operation $\mathds{1}$ is applied. The entire physical set-up is described by a bipartite state, one partition of which corresponds to the state of the particles, $|s)\in\Omega_{P'cles}$, and the other to the ‘which path’ information embodied in the interferometer, $|s')\in\Omega_{Path}$. The entire scenario thus takes place in the state space $\Omega_{Path}\otimes \Omega_{P'cle}$. In the quantum case, the phase transformation generated by this procedure corresponds to the type of indistinguishable particle employed in the experiment.
The whole experiment can be described via a controlled transformation, with path information as the control and particle state as the target. Via Thm. (\[Reversible-Control\]), such an experiment exists in theories satisfying our assumptions. Applying operation $S$ to the particle state corresponds to swapping a pair of indistinguishable particles and so must leave the statistics of any measurement invariant. Therefore $S|s)=|s)$, where $|s)$ is the initial particle state. $$\begin{tikzpicture}
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\end{tikzpicture}$$ Thm. (\[Generalised-Kick-Back\]) tells us that the above diagram corresponds to a kicked-back phase on the control system, as in Eq. (\[KB\]). Thus, to every particle type, there exists a corresponding phase transformation, which was the connection discussed in [@Oscar]. But, in an arbitrary theory, the converse is not necessarily true. The quantum phase group is $U(1)$ and, fixing its representation to be $\{e^{i\theta}\}$, bosons kick-back the transformation corresponding to $\theta=0$, fermions $\theta=\pi$ and anyons any arbitrary $\theta$. Thus, to every particle type in quantum theory there is an associated phase, and vice versa.
To generalise this to theories satisfying our three assumptions, we must connect the operational description of these theories to the more physical notion of particles. Towards this end, we make the following two assumptions:
1. Every operational state $|s)$ corresponds to the state of some collection of indistinguishable particles,
2. Every transformation that leaves the operational state $|s)$ invariant corresponds to a (possibly trivial) permutation of the collection of indistinguishable particles.
Given the above, Thm. (\[ALL\]) tells us that to every phase transformation there exists a corresponding particle type. Therefore, to each higher-order phase – described in Thm. (\[Higher-Phase\]) – there is associated a particle type that should be observable through a generalisation of the above experiment.
Consider $|s)$, which corresponds to the state of some collection of indistinguishable particles, and a permutation operation $\pi$ which leaves $|s)$ invariant. Note that, for a given permutation, there may be multiple topologically distinct ways of performing it, particularly in two dimensions or topologically non-trivial spaces. Now consider the $n$-path experiment, illustrated below, where on each path some distinct permutation operation $\pi_i,\ i=1,\dots,n,$ takes place. $$\begin{tikzpicture}
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\end{tikzpicture}$$ This can be described by a controlled transformation $C\{\pi_i\}$. Given the above two assumptions, any $n$th-order phase – if they exist in the operational theory – can be kicked-back by such an experiment.
Recall that $n$th-order phases are $n$-detectable, but $n\mathrm{-}1$-undetectable. That is, the action of such phases cannot be detected by any effect with support on less than $n$ paths. Which is in stark contrast to quantum, or $2$nd-order, phases, which can always be detected by an effect with support on two paths. Thus, permutations of particles, whose type all correspond to an $n$th-order phase, can only be detected by recombining *all* paths in an $n$-path experiment. In some sense then, $n$th-order phases encode holistic information about all paths in an $n$-path experiment.
Computational oracles {#oracle}
---------------------
Oracles play a vital role in quantum computing, forming the basis of most known speed-ups over classical computation [@Nielsen]. Despite their importance, defining a general notion of oracle – that reduces to the standard notion in the quantum case – in operationally-defined theories has proven difficult [@LB-2014]. A particular example of a quantum oracle is the following controlled unitary: $$\label{Quantum-Oracle}
U_f= {|0\rangle}{\langle 0|}\otimes Z^{f(0)} + {|1\rangle}{\langle 1|}\otimes Z^{f(1)},$$ with $Z$ a Pauli matrix, $f:\{0,1\}\rightarrow\{0,1\}$ a function encoding some decision problem and $Z^{0}:=\mathbb{I}$. The quantum phase kick-back for $U_f$ amounts to $$\label{Quantum-kick-back}
U_f=\mathbb{I}\otimes{|0\rangle}{\langle 0|}+Z^{f(0)\oplus f(1)}\otimes{|1\rangle}{\langle 1|}.$$ One can see that inputting ${|+\rangle}{|1\rangle}$ and measuring the first qubit in the $\{{|+\rangle},{|-\rangle}\}$ basis reveals the value of $f(0)\oplus f(1)$ in a single query of the oracle – a feat impossible on a classical computer [@Nielsen].
The results of Thm. (\[Reversible-Control\]) provide a way to define computational oracles in any theory satisfying our three assumptions. An oracle in such theories corresponds to a reversible controlled transformation [^6] where the set of transformations $\{T_{i,f(i)}\}$ being controlled depend on a function $f:\{i\}\rightarrow\{0,1\}$ encoding a decision problem of interest. As the transformations $T_{i,f(i)}$ depend on the value of $f(i)$, so does the controlled transformation and the kicked-back phase. That is, in theories with a non-trivial phase group, the phase kick-back of an oracle encodes information about the value $f(i)$ for all $i$. In such theories, there is thus a non-zero probability of extracting such global information. Non-trivial interference behaviour can thus be seen as a general resource for non-classical computation.
In the quantum case, there is a limit to how much global information one can obtain in a single oracle query. In the situation where $f:\{0,\dots, n\mathrm{-}1\}\rightarrow\{0,1\}$ a quantum oracle can only extract the value of $f(i)\oplus f(j)$, for some $i,j$, in a single query without error [@Nielsen]. Can theories with higher-order interference reliably extract more global information about $f$ – without error – in a single query? The results of Sec. (\[Exchange\]) appear to suggest that $n$th-order phases encode information about all paths in an $n$-path experiment, as opposed to $2$nd-order, or quantum, phases which only encode information about at most two paths. Based on this fact – that higher-order phases encode more holistic information that quantum ones – and the result of Thm. (\[ALL\]), we conjecture that theories with higher-order interference can solve problems intractable on a quantum computer. To prove such a conjecture however, a concrete example of such a theory is needed. While there are partial examples in the literature [@DensityCube; @Thesis], none of these are complete [@LS-2015]. Answering such a conjecture in the affirmative is thus not yet possible, although some evidence was provided in [@LS-2015].
Conclusion
==========
The key result of this paper was to provide a set of physical principles that are sufficient for the existence of reversible controlled transformations. Such transformations are central to our understanding of quantum computing, information processing and thermodynamics. Moreover, these were shown to guarantee the existence of a generalised phase kick-back mechanism, which, in the quantum case, forms a fundamental component of almost all algorithms. These physical principles are defining characteristics of information: independence of encoding medium; propagation from present to future; and conservation at a fundamental level. It would therefore be surprising if these principles were not necessary primers for information processing. These results provide the tools for an exploration of the structure of computational algorithms – and how they connect to physical principles – in operational theories.
We developed a framework that connects higher-order interference and phase transformations, generalising the intimate connection between phase and interference witnessed in quantum theory. These ‘higher-order’ phases are accessible via our generalised kick-back mechanism. Given two assumptions which connect the operational theory to a physical description of particles, these higher-order phases were shown to give rise to exotic particle types. Additionally, using the controlled transformations to define an oracle model of computation, we conjectured that these higher-order phases may allow for the solution of problems intractable even on a quantum computer. Computational problems that may be susceptible to efficient solution by generalised phase kick-back include the $n$-collision problem, and the non-abelian hidden subgroup problem. Discovering that higher-order interference leads to ‘unreasonable’ computational power may provide a reason ‘why’ quantum theory is limited in its interference behaviour – in the same way that implausible communication complexity is thought to limit quantum non-locality [@PR-van-Dam]. In Sec. (\[Exchange\]) it was shown that to observe the exotic particle types corresponding to higher-order phases, there must be distinct ways to swap particles. As we live in a topologically trivial three dimensional space, there is only one topologically distinct way to swap point particles. This can either be seen as evidence of *why* quantum theory is limited to only second-order interference, or evidence that such particle types must have non-trivial structure, similar to toroidal anyons [@torons] – which are constructed from a solenoid ring with an attached charge – or closed strings [@Barton].
Finally, reference [@work] has shown that thermodynamic work can be extracted from quantum coherences – $2$nd-order phases in our language. This raises this the question of whether one can extract work more efficiently using higher-order phases? If such efficiencies are in contention with thermodynamic principles this could provide a reason ‘why’ quantum theory has limited interference. Initial investigations into formulating a consistent thermodynamics in operational theories have been reported in [@Thermo; @Thermo1; @Thermo2]. The framework and results presented here may therefore have implications for thermodynamics, information processing and how each arises in a unified manner from physical principles.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank Oscar Dahlsten, Terry Rudolph, Jon Barrett and Matty Hoban for useful discussions. Matty Hoban is also thanked for proof reading a draft of the current paper. The authors also thank Carlo Maria Scandolo for a careful reading of the appendices in a previous draft of the current paper. This work was supported by EPSRC through the Controlled Quantum Dynamics Centre for Doctoral Training and the Oxford Department of Computer Science. CML also acknowledges funding from University College, Oxford.
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General results following from causality purification and strong symmetry
=========================================================================
Uniqueness of distinguishing measurement {#Sym}
----------------------------------------
Strong symmetry (together with the no restriction hypothesis, which says that all mathematically well-defined measurements are physical) implies that, given any set of pure and perfectly distinguishable states $\{|i)\}$, there exists a unique measurement $\{(j|\}$ such that, $$(i|j)=\delta_{ij}.$$ See [@MU; @Higher-order-reconstruction] for details. Moreover if there is a set $\{(e_j|\}$ such that $(e_j|i)=\alpha_j \delta_{ij}$ then, $$(e_j|=\alpha_j(j|.$$
Existence of a maximally mixed state
------------------------------------
Purification implies that there is a unique *completely mixed* state ${|\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=maxMix] (0) at (0, -0) {};
\node [style=none] (1) at (0.1, -0) {};
\end{pgfonlayer}
\end{tikzpicture})}$ defined by, $$T{|\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=maxMix] (0) at (0, -0) {};
\node [style=none] (1) at (0.1, -0) {};
\end{pgfonlayer}
\end{tikzpicture})}={|\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=maxMix] (0) at (0, -0) {};
\node [style=none] (1) at (0.1, -0) {};
\end{pgfonlayer}
\end{tikzpicture})},\quad \forall T\in\mathcal{R}$$ Any state is a ‘refinement’ of this state. See [@Pavia1; @Pavia2] for details.
Purification of the maximally mixed state is dynamically faithful
-----------------------------------------------------------------
Purification implies that there exists a state $|\psi)$ that purifies the completely mixed state: $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (1, -0.25) {};
\node [style=none] (1) at (1, 1.25) {};
\node [style=none] (2) at (1, 1.75) {};
\node [style=none] (3) at (1, -0.75) {};
\node [style=trace] (4) at (2.5, -0.25) {};
\node [style=none] (5) at (2.5, 1.25) {};
\node [style=none] (6) at (1, -0.75) {};
\node [style=none] (7) at (0.5, 0.5) {$\psi$};
\node [style=none] (8) at (3.75, 0.75) {$=$};
\node [style=traceState] (9) at (5, 0.75) {};
\node [style=none] (10) at (6.5, 0.75) {};
\node [style=none] (11) at (7.25, -0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=90, looseness=1.25] (2.center) to (3.center);
\draw (2.center) to (3.center);
\draw (1.center) to (5.center);
\draw (0.center) to (4);
\draw (9) to (10.center);
\end{pgfonlayer}
\end{tikzpicture}$$ This is unique up to reversible transformation. We denote a particular choice of this purification as, $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (4.75, -0.25) {};
\node [style=none] (1) at (4.75, 1.25) {};
\node [style=none] (2) at (4.75, 1.75) {};
\node [style=none] (3) at (4.75, -0.75) {};
\node [style=none] (4) at (6.25, -0.25) {};
\node [style=none] (5) at (6.25, 1.25) {};
\node [style=none] (6) at (4.75, -0.75) {};
\node [style=none] (7) at (4.25, 0.5) {$\psi$};
\node [style=none] (8) at (2.5, 0.5) {$:=$};
\node [style=none] (9) at (0.75, 1.25) {};
\node [style=none] (10) at (1.25, -0.25) {};
\node [style=none] (11) at (0.75, -0.25) {};
\node [style=none] (12) at (1.25, 1.25) {};
\node [style=none] (13) at (7, -0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=90, looseness=1.25] (2.center) to (3.center);
\draw (2.center) to (3.center);
\draw (1.center) to (5.center);
\draw (0.center) to (4.center);
\draw (9.center) to (12.center);
\draw (11.center) to (10.center);
\draw [bend right=90, looseness=1.75] (9.center) to (11.center);
\end{pgfonlayer}
\end{tikzpicture}$$ Purifications of the completely mixed state are called *dynamically faithful* states [@Pavia1; @Pavia2] and satisfy the following important condition [@Pavia1; @Pavia2]: $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (5.5, 7.25) {$=$};
\node [style=none] (1) at (0.7499999, 7) {};
\node [style=none] (2) at (1.25, 5.25) {};
\node [style=none] (3) at (0.7499997, 5.25) {};
\node [style=none] (4) at (1.25, 7) {};
\node [style=none] (5) at (1.25, 9) {};
\node [style=none] (6) at (1.25, 6) {};
\node [style=none] (7) at (2.75, 6) {};
\node [style=none] (8) at (2.75, 9) {};
\node [style=none] (9) at (2, 7.5) {$T$};
\node [style=none] (10) at (1.25, 8.5) {};
\node [style=none] (11) at (1.25, 7.5) {};
\node [style=none] (12) at (0, 8.5) {};
\node [style=none] (13) at (3.75, 7) {};
\node [style=none] (14) at (3.75, 8.5) {};
\node [style=none] (15) at (2.75, 8.5) {};
\node [style=none] (16) at (2.75, 7) {};
\node [style=none] (17) at (3.75, 5.25) {};
\node [style=none] (18) at (8.25, 6) {};
\node [style=none] (19) at (8.25, 5.25) {};
\node [style=none] (20) at (9.75, 7) {};
\node [style=none] (21) at (8.25, 8.5) {};
\node [style=none] (22) at (8.25, 9) {};
\node [style=none] (23) at (9.75, 8.5) {};
\node [style=none] (24) at (10.75, 5.25) {};
\node [style=none] (25) at (9.75, 6) {};
\node [style=none] (26) at (9.000001, 7.5) {$T'$};
\node [style=none] (27) at (6.999999, 8.5) {};
\node [style=none] (28) at (7.75, 7) {};
\node [style=none] (29) at (8.25, 7.5) {};
\node [style=none] (30) at (10.75, 8.5) {};
\node [style=none] (31) at (9.75, 9) {};
\node [style=none] (32) at (8.25, 7) {};
\node [style=none] (33) at (7.75, 5.25) {};
\node [style=none] (34) at (10.75, 7) {};
\node [style=none] (35) at (2, 3.5) {$\implies$};
\node [style=none] (36) at (3.75, -0.7499997) {};
\node [style=cpoint] (37) at (2.749999, -0) {$\sigma$};
\node [style=none] (38) at (13.25, 1.75) {};
\node [style=none] (39) at (11.5, 0.7499999) {$T'$};
\node [style=none] (40) at (12.25, 0.25) {};
\node [style=none] (41) at (5.25, 1.75) {};
\node [style=none] (42) at (5.25, 2.25) {};
\node [style=none] (43) at (12.25, -0.7499999) {};
\node [style=none] (44) at (3.75, 1.75) {};
\node [style=none] (45) at (3.75, -0) {};
\node [style=none] (46) at (12.25, 2.25) {};
\node [style=none] (47) at (9.500001, 1.75) {};
\node [style=none] (48) at (3.75, 0.7499997) {};
\node [style=none] (49) at (10.75, 1.75) {};
\node [style=none] (50) at (5.25, 0.25) {};
\node [style=none] (51) at (2.5, 1.75) {};
\node [style=none] (52) at (3.75, 2.25) {};
\node [style=none] (53) at (8, 0.5000001) {$=$};
\node [style=none] (54) at (13.25, 0.2499996) {};
\node [style=none] (55) at (10.75, -0) {};
\node [style=cpoint] (56) at (9.750001, -0) {$\sigma$};
\node [style=none] (57) at (6.25, 1.75) {};
\node [style=none] (58) at (4.5, 0.7499999) {$T$};
\node [style=none] (59) at (12.25, 1.75) {};
\node [style=none] (60) at (10.75, -0.7499997) {};
\node [style=none] (61) at (6.249999, 0.2499996) {};
\node [style=none] (62) at (5.25, -0.7499999) {};
\node [style=none] (63) at (10.75, 0.7499997) {};
\node [style=none] (64) at (10.75, 2.25) {};
\node [style=none] (65) at (15, -0) {$\forall \sigma$};
\node [style=none] (66) at (2.749999, -0) {};
\node [style=none] (67) at (15.75, -0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (4.center);
\draw (3.center) to (2.center);
\draw [bend right=90, looseness=1.75] (1.center) to (3.center);
\draw (5.center) to (6.center);
\draw (6.center) to (7.center);
\draw (7.center) to (8.center);
\draw (8.center) to (5.center);
\draw (12.center) to (10.center);
\draw (15.center) to (14.center);
\draw (16.center) to (13.center);
\draw (2.center) to (17.center);
\draw (28.center) to (32.center);
\draw (33.center) to (19.center);
\draw [bend right=90, looseness=1.75] (28.center) to (33.center);
\draw (22.center) to (18.center);
\draw (18.center) to (25.center);
\draw (25.center) to (31.center);
\draw (31.center) to (22.center);
\draw (27.center) to (21.center);
\draw (23.center) to (30.center);
\draw (20.center) to (34.center);
\draw (19.center) to (24.center);
\draw (37) to (45.center);
\draw (52.center) to (36.center);
\draw (36.center) to (62.center);
\draw (62.center) to (42.center);
\draw (42.center) to (52.center);
\draw (51.center) to (44.center);
\draw (41.center) to (57.center);
\draw (50.center) to (61.center);
\draw (64.center) to (60.center);
\draw (60.center) to (43.center);
\draw (43.center) to (46.center);
\draw (46.center) to (64.center);
\draw (47.center) to (49.center);
\draw (56) to (55.center);
\draw (59.center) to (38.center);
\draw (40.center) to (54.center);
\end{pgfonlayer}
\end{tikzpicture}$$
Existence of controlled transformations\[CT\]
=============================================
Recalling that in this work we assume that the composite of pure states is pure, we can define two sets of pure and perfectly distinguishable states: $$\mathcal{B}_1 :=
\left\{\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (1, -0) {};
\node [style=none] (1) at (2, -0) {};
\node [style=none] (2) at (2, -1) {};
\node [style=none] (3) at (2, 1) {};
\node [style=none] (4) at (2, -1) {};
\node [style=cpoint] (5) at (0.4999999, 0.9999999) {$i$};
\node [style=none] (6) at (1, -1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=90, looseness=2.25] (0.center) to (6.center);
\draw (5) to (3.center);
\draw (6.center) to (2.center);
\draw (0.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}\right\} ,\quad \text{and}\quad\mathcal{B}_2 := \left\{\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (1, -0) {};
\node [style={small box}] (1) at (1.75, -0) {$T_i$};
\node [style=none] (2) at (2.75, -0) {};
\node [style=none] (3) at (2.75, -0.9999999) {};
\node [style=none] (4) at (2.75, 0.9999999) {};
\node [style=none] (5) at (2.75, -0.9999999) {};
\node [style=cpoint] (6) at (0.4999999, 0.9999999) {$i$};
\node [style=none] (7) at (1, -1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=90, looseness=2.25] (0.center) to (7.center);
\draw (6) to (4.center);
\draw (7.center) to (3.center);
\draw (0.center) to (1);
\draw (1) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}\right\} .$$
Strong symmetry implies that there exists a reversible transformation between these two sets, $T:\mathcal{B}_1\to \mathcal{B}_2$. $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (7.75, 0.7499997) {};
\node [style=none] (1) at (1.75, 0.7499997) {$T$};
\node [style=none] (2) at (2.25, -0.7499997) {};
\node [style=none] (3) at (1.25, 1.75) {};
\node [style=none] (4) at (5.25, 0.7499997) {$=$};
\node [style=none] (5) at (1.25, 0.7499997) {};
\node [style=none] (6) at (3.000001, -0.2499996) {};
\node [style=none] (7) at (1.25, 2.25) {};
\node [style=none] (8) at (1.25, -0.2499996) {};
\node [style=none] (9) at (0.7499997, -0.2499996) {};
\node [style=none] (10) at (3.000001, 0.7499997) {};
\node [style=none] (11) at (3.000001, -0.2499996) {};
\node [style=none] (12) at (2.25, -0.2499996) {};
\node [style={small box}] (13) at (8.499999, 0.7499997) {$T_i$};
\node [style=none] (14) at (9.499999, 0.7499997) {};
\node [style=none] (15) at (9.499999, -0.2499996) {};
\node [style=none] (16) at (9.499999, 1.75) {};
\node [style=cpoint] (17) at (0.2499996, 1.75) {$i$};
\node [style=none] (18) at (0.7499997, 0.7499997) {};
\node [style=none] (19) at (9.499999, -0.2499996) {};
\node [style=none] (20) at (3.000001, 1.75) {};
\node [style=none] (21) at (2.25, 2.25) {};
\node [style=cpoint] (22) at (7.25, 1.75) {$i$};
\node [style=none] (23) at (2.25, 1.75) {};
\node [style=none] (24) at (2.25, 0.7499997) {};
\node [style=none] (25) at (7.75, -0.2499996) {};
\node [style=none] (26) at (1.25, -0.7499997) {};
\node [style=none] (27) at (10, -0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (7.center) to (21.center);
\draw (21.center) to (2.center);
\draw (2.center) to (26.center);
\draw [bend right=90, looseness=2.25] (18.center) to (9.center);
\draw (7.center) to (26.center);
\draw (17) to (3.center);
\draw (18.center) to (5.center);
\draw (9.center) to (8.center);
\draw (12.center) to (11.center);
\draw (24.center) to (10.center);
\draw (23.center) to (20.center);
\draw [bend right=90, looseness=2.25] (0.center) to (25.center);
\draw (22) to (16.center);
\draw (25.center) to (15.center);
\draw (0.center) to (13);
\draw (13) to (14.center);
\end{pgfonlayer}
\end{tikzpicture}$$ This result, together with the existence of dynamically faithful states, will be used to show the existence of a reversible controlled transformation $C\{T_i\}$ for an arbitrary set of reversible transformations $\{T_i\}$.
‘Superposition preservation’\[SP\] $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (7.75, 0.7499997) {};
\node [style=none] (1) at (1.75, 0.7499997) {$T$};
\node [style=none] (2) at (2.25, -0.7499997) {};
\node [style=none] (3) at (1.25, 1.75) {};
\node [style=none] (4) at (5.25, 0.7499997) {$=$};
\node [style=none] (5) at (1.25, 0.7499997) {};
\node [style=none] (6) at (3.000001, -0.2499996) {};
\node [style=none] (7) at (1.25, 2.25) {};
\node [style=none] (8) at (1.25, -0.2499996) {};
\node [style=none] (9) at (0.7499997, -0.2499996) {};
\node [style=none] (10) at (3.000001, 0.7499997) {};
\node [style=none] (11) at (3.000001, -0.2499996) {};
\node [style=none] (12) at (2.25, -0.2499996) {};
\node [style={small box}] (13) at (8.499999, 0.7499997) {$T_i$};
\node [style=none] (14) at (9.499999, 0.7499997) {};
\node [style=none] (15) at (9.499999, -0.2499996) {};
\node [style=cocpoint] (16) at (9.499999, 1.75) {$i$};
\node [style=none] (17) at (0.2499996, 1.75) {};
\node [style=none] (18) at (0.7499997, 0.7499997) {};
\node [style=none] (19) at (9.499999, -0.2499996) {};
\node [style=cocpoint] (20) at (3.000001, 1.75) {$i$};
\node [style=none] (21) at (2.25, 2.25) {};
\node [style=none] (22) at (7.25, 1.75) {};
\node [style=none] (23) at (2.25, 1.75) {};
\node [style=none] (24) at (2.25, 0.7499997) {};
\node [style=none] (25) at (7.75, -0.2499996) {};
\node [style=none] (26) at (1.25, -0.7499997) {};
\node [style=none] (27) at (10, -0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (7.center) to (21.center);
\draw (21.center) to (2.center);
\draw (2.center) to (26.center);
\draw [bend right=90, looseness=2.25] (18.center) to (9.center);
\draw (7.center) to (26.center);
\draw (17.center) to (3.center);
\draw (18.center) to (5.center);
\draw (9.center) to (8.center);
\draw (12.center) to (11.center);
\draw (24.center) to (10.center);
\draw (23.center) to (20);
\draw [bend right=90, looseness=2.25] (0.center) to (25.center);
\draw (22.center) to (16);
\draw (25.center) to (15.center);
\draw (0.center) to (13);
\draw (13) to (14.center);
\end{pgfonlayer}
\end{tikzpicture}$$
Firstly we prove a weaker condition which is superposition preservation for pure local effects, $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=cpoint] (0) at (0, 10.25) {$i$};
\node [style=none] (1) at (2.250001, 10.25) {};
\node [style=none] (2) at (2.250001, 10.75) {};
\node [style=none] (3) at (3.75, 10.75) {};
\node [style=none] (4) at (2.250001, 8.5) {};
\node [style=none] (5) at (2.250001, 6.5) {};
\node [style=none] (6) at (3.75, 6.5) {};
\node [style=none] (7) at (3.75, 8.5) {};
\node [style=none] (8) at (3.75, 10.25) {};
\node [style=cocpoint] (9) at (4.75, 10.25) {$j$};
\node [style=trace] (10) at (4.75, 8.5) {};
\node [style=none] (11) at (3, 8.5) {$T$};
\node [style=none] (12) at (0.7499999, 11.25) {};
\node [style=none] (13) at (0.7499999, 6) {};
\node [style=none] (14) at (5.499999, 6) {};
\node [style=none] (15) at (5.499999, 11.25) {};
\node [style=none] (16) at (6.499999, 9.25) {$=$};
\node [style=none] (17) at (12.25, 9) {$\delta_{ij}$};
\node [style={small box}] (18) at (9.25, 9.75) {$T_j$};
\node [style=trace] (19) at (10.5, 9.75) {};
\node [style=none] (20) at (4.5, 5) {Strong symmetry $\implies$};
\node [style=trace] (21) at (4.75, 7) {};
\node [style=none] (22) at (8.500001, 8.25) {};
\node [style=none] (23) at (8.500001, 9.75) {};
\node [style=trace] (24) at (10.5, 8.25) {};
\node [style=none] (25) at (13.25, 1.5) {};
\node [style=trace] (26) at (14.5, -0) {};
\node [style=none] (27) at (13.25, -0) {};
\node [style=none] (28) at (10.5, 1.75) {$=$};
\node [style=trace] (29) at (14.5, 1.5) {};
\node [style=cocpoint] (30) at (14.5, 3) {$j$};
\node [style=none] (31) at (12, 3) {};
\node [style=none] (32) at (2.250001, 7) {};
\node [style=none] (33) at (3.75, 7) {};
\node [style=none] (34) at (4.25, 4) {};
\node [style=none] (35) at (7.25, -0.7499999) {};
\node [style=cocpoint] (36) at (8.25, 3) {$j$};
\node [style=none] (37) at (5.75, 3.5) {};
\node [style=none] (38) at (7.25, 3) {};
\node [style=trace] (39) at (8.25, -0.25) {};
\node [style=none] (40) at (9.000001, 4) {};
\node [style=none] (41) at (5.75, -0.25) {};
\node [style=none] (42) at (7.25, -0.25) {};
\node [style=none] (43) at (5.75, -0.7499999) {};
\node [style=none] (44) at (5.75, 1.25) {};
\node [style=none] (45) at (7.25, 1.25) {};
\node [style=trace] (46) at (8.25, 1.25) {};
\node [style=none] (47) at (4.25, -1.25) {};
\node [style=none] (48) at (5.75, 3) {};
\node [style=none] (49) at (7.25, 3.5) {};
\node [style=none] (50) at (9.000001, -1.25) {};
\node [style=none] (51) at (6.499999, 1.25) {$T$};
\node [style=none] (52) at (3.5, 3) {};
\node [style=none] (53) at (16.5, -0) {};
\node [style=none] (54) at (14, 9) {$=$};
\node [style=none] (55) at (15.75, 9) {$\delta_{ij}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (5.center);
\draw (5.center) to (6.center);
\draw (6.center) to (3.center);
\draw (3.center) to (2.center);
\draw (0) to (1.center);
\draw (7.center) to (10);
\draw (8.center) to (9);
\draw [style={thick gray dashed edge}] (12.center) to (15.center);
\draw [style={thick gray dashed edge}] (15.center) to (14.center);
\draw [style={thick gray dashed edge}] (14.center) to (13.center);
\draw [style={thick gray dashed edge}] (13.center) to (12.center);
\draw (18) to (19);
\draw [bend right=90, looseness=1.50] (23.center) to (22.center);
\draw (23.center) to (18);
\draw (22.center) to (24);
\draw [bend right=90, looseness=1.50] (25.center) to (27.center);
\draw (27.center) to (26);
\draw (31.center) to (30);
\draw (33.center) to (21);
\draw [bend left=90, looseness=2.00] (32.center) to (4.center);
\draw (37.center) to (43.center);
\draw (43.center) to (35.center);
\draw (35.center) to (49.center);
\draw (49.center) to (37.center);
\draw (45.center) to (46);
\draw (38.center) to (36);
\draw [style={thick gray dashed edge}] (34.center) to (40.center);
\draw [style={thick gray dashed edge}] (40.center) to (50.center);
\draw [style={thick gray dashed edge}] (50.center) to (47.center);
\draw [style={thick gray dashed edge}] (47.center) to (34.center);
\draw (42.center) to (39);
\draw [bend left=90, looseness=2.00] (41.center) to (44.center);
\draw (52.center) to (48.center);
\draw (25.center) to (29);
\end{pgfonlayer}
\end{tikzpicture}$$ the implication follows from the uniqueness of the maximally distinguishing measurement up to normalisation. Then purification implies, $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (6.75, 2.5) {};
\node [style=none] (1) at (7.25, 0.25) {};
\node [style=none] (2) at (5.5, 2) {$=$};
\node [style=cocpoint] (3) at (8.75, 2.5) {$i$};
\node [style=none] (4) at (7.25, 1.25) {};
\node [style=none] (5) at (7.25, 1.25) {};
\node [style=none] (6) at (7.25, 1.75) {};
\node [style=none] (7) at (8.25, 1.75) {};
\node [style=none] (8) at (8.25, -0.25) {};
\node [style=none] (9) at (7.25, -0.25) {};
\node [style=none] (10) at (7.25, 0.25) {};
\node [style=none] (11) at (7.25, 1.25) {};
\node [style=none] (12) at (8.25, 1.25) {};
\node [style=none] (13) at (8.25, 0.25) {};
\node [style=none] (14) at (9.000001, 1.25) {};
\node [style=none] (15) at (9.000001, 0.25) {};
\node [style=none] (16) at (7.75, 0.7499999) {$T'_i$};
\node [style=none] (17) at (0, 2.25) {};
\node [style=none] (18) at (2.5, -0.7499999) {};
\node [style=none] (19) at (2.5, 2.25) {};
\node [style=none] (20) at (0.9999999, 2.75) {};
\node [style=none] (21) at (0.9999999, 0.9999999) {};
\node [style=none] (22) at (0.9999999, -0.25) {};
\node [style=none] (23) at (2.5, 0.9999999) {};
\node [style=none] (24) at (2.5, -0.25) {};
\node [style=none] (25) at (0.9999999, 2.25) {};
\node [style=none] (26) at (2.5, 2.75) {};
\node [style=none] (27) at (0.9999999, -0.7499999) {};
\node [style=cocpoint] (28) at (3.5, 2.25) {$i$};
\node [style=none] (29) at (2.5, 2.25) {};
\node [style=none] (30) at (3.5, 0.9999999) {};
\node [style=none] (31) at (3.5, -0.25) {};
\node [style=none] (32) at (1.75, 0.9999999) {$T$};
\node [style=none] (33) at (9.75, -0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=90, looseness=2.00] (4.center) to (1.center);
\draw (0.center) to (3);
\draw (4.center) to (5.center);
\draw (6.center) to (9.center);
\draw (9.center) to (8.center);
\draw (8.center) to (7.center);
\draw (7.center) to (6.center);
\draw (12.center) to (14.center);
\draw (13.center) to (15.center);
\draw (20.center) to (27.center);
\draw (27.center) to (18.center);
\draw (18.center) to (26.center);
\draw (26.center) to (20.center);
\draw (17.center) to (25.center);
\draw [bend right=90, looseness=2.00] (21.center) to (22.center);
\draw (29.center) to (28);
\draw (23.center) to (30.center);
\draw (24.center) to (31.center);
\end{pgfonlayer}
\end{tikzpicture}$$ Now consider, $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.9999999, 7) {};
\node [style=none] (1) at (0.9999999, 8.25) {};
\node [style=none] (2) at (2.5, 8.25) {};
\node [style=none] (3) at (3.5, 8.25) {};
\node [style=none] (4) at (2.5, 6.5) {};
\node [style=none] (5) at (0.9999999, 6.5) {};
\node [style=none] (6) at (0.9999999, 10) {};
\node [style=cocpoint] (7) at (3.5, 9.5) {$i$};
\node [style=none] (8) at (0.9999999, 9.5) {};
\node [style=cpoint] (9) at (0, 9.5) {$i$};
\node [style=none] (10) at (2.5, 9.5) {};
\node [style=none] (11) at (2.5, 10) {};
\node [style=none] (12) at (5, 8.5) {$=$};
\node [style=none] (13) at (4.75, 5.5) {${\rotatebox{-45}{$\,=$}}$};
\node [style=none] (14) at (7.75, 9.5) {};
\node [style=none] (15) at (7.25, 7) {};
\node [style=cpoint] (16) at (6.75, 9.5) {$i$};
\node [style=none] (17) at (9.000001, 8) {};
\node [style=cocpoint] (18) at (8.75, 9.5) {$i$};
\node [style=none] (19) at (7.25, 8) {};
\node [style=none] (20) at (8.25, 8) {};
\node [style=none] (21) at (7.25, 8.5) {};
\node [style=none] (22) at (8.25, 8.5) {};
\node [style=none] (23) at (8.25, 6.5) {};
\node [style=none] (24) at (7.25, 6.5) {};
\node [style=none] (25) at (7.75, 7.5) {$T'_i$};
\node [style=none] (26) at (9.000001, 7) {};
\node [style=none] (27) at (8.25, 7) {};
\node [style=cocpoint] (28) at (8.000001, 4.75) {$i$};
\node [style=none] (29) at (7.25, 2.75) {};
\node [style=none] (30) at (8.000001, 2.75) {};
\node [style=none] (31) at (6.25, 2.75) {};
\node [style=none] (32) at (6.999999, 4.75) {};
\node [style=cpoint] (33) at (5.999999, 4.75) {$i$};
\node [style={small box}] (34) at (6.999999, 3.75) {$T_i$};
\node [style=none] (35) at (6.25, 3.75) {};
\node [style=none] (36) at (8.000001, 3.75) {};
\node [style=none] (37) at (0.9999999, 1.25) {$\implies$};
\node [style=none] (38) at (3.5, 0.9999999) {};
\node [style=none] (39) at (5, 0.9999999) {};
\node [style=none] (40) at (5, -0) {};
\node [style={small box}] (41) at (4, 0.9999999) {$T_i$};
\node [style=none] (42) at (3.5, -0) {};
\node [style=none] (43) at (6.499999, 0.4999999) {$=$};
\node [style=none] (44) at (8.500001, 0.9999999) {};
\node [style=none] (45) at (9.500001, 1.5) {};
\node [style=none] (46) at (10.25, -0) {};
\node [style=none] (47) at (9.500001, -0) {};
\node [style=none] (48) at (9.500001, -0.4999999) {};
\node [style=none] (49) at (8.500001, -0.4999999) {};
\node [style=none] (50) at (8.500001, 1.5) {};
\node [style=none] (51) at (9.500001, 0.9999999) {};
\node [style=none] (52) at (9.000001, 0.4999999) {$T'_i$};
\node [style=none] (53) at (10.25, 0.9999999) {};
\node [style=none] (54) at (8.500001, -0) {};
\node [style=none] (55) at (8.25, -0) {};
\node [style=none] (56) at (8.25, 0.9999999) {};
\node [style=none] (57) at (2.5, 7) {};
\node [style=none] (58) at (3.5, 7) {};
\node [style=none] (59) at (1.75, 8.25) {$T$};
\node [style=none] (60) at (3.75, 0.9999999) {};
\node [style=none] (61) at (3.5, -0) {};
\node [style=none] (62) at (3.5, 0.9999999) {};
\node [style=none] (63) at (8.25, -0) {};
\node [style=none] (64) at (8.25, 0.9999999) {};
\node [style=none] (65) at (11, -0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (6.center) to (5.center);
\draw (5.center) to (4.center);
\draw (4.center) to (11.center);
\draw (11.center) to (6.center);
\draw (10.center) to (7);
\draw (9) to (8.center);
\draw [bend right=90, looseness=2.00] (1.center) to (0.center);
\draw (2.center) to (3.center);
\draw (14.center) to (18);
\draw [bend right=90, looseness=2.00] (19.center) to (15.center);
\draw (20.center) to (17.center);
\draw (21.center) to (24.center);
\draw (24.center) to (23.center);
\draw (23.center) to (22.center);
\draw (22.center) to (21.center);
\draw (16) to (14.center);
\draw (27.center) to (26.center);
\draw (32.center) to (28);
\draw [bend right=90, looseness=2.00] (35.center) to (31.center);
\draw (34) to (36.center);
\draw (33) to (32.center);
\draw (29.center) to (30.center);
\draw (31.center) to (29.center);
\draw (35.center) to (34);
\draw (41) to (39.center);
\draw (42.center) to (40.center);
\draw (38.center) to (41);
\draw (51.center) to (53.center);
\draw (50.center) to (49.center);
\draw (49.center) to (48.center);
\draw (48.center) to (45.center);
\draw (45.center) to (50.center);
\draw (47.center) to (46.center);
\draw (56.center) to (44.center);
\draw (55.center) to (54.center);
\draw (57.center) to (58.center);
\draw [bend right=90, looseness=2.00] (62.center) to (61.center);
\draw [bend right=90, looseness=2.00] (64.center) to (63.center);
\end{pgfonlayer}
\end{tikzpicture}$$ where the above follows the fact that $(i|i)=1$. This, in conjunction with the previous results, gives: $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (6.75, 2.25) {};
\node [style=none] (1) at (7.25, -0) {};
\node [style=none] (2) at (5.499999, 1.75) {$=$};
\node [style=cocpoint] (3) at (8.75, 2.25) {$i$};
\node [style=none] (4) at (7.25, 0.9999999) {};
\node [style=none] (5) at (7.25, 0.9999999) {};
\node [style=none] (6) at (7.25, 1.5) {};
\node [style=none] (7) at (8.25, 1.5) {};
\node [style=none] (8) at (8.25, -0.4999999) {};
\node [style=none] (9) at (7.25, -0.4999999) {};
\node [style=none] (10) at (7.25, -0) {};
\node [style=none] (11) at (7.25, 0.9999999) {};
\node [style=none] (12) at (8.25, 0.9999999) {};
\node [style=none] (13) at (8.25, -0) {};
\node [style=none] (14) at (9.000001, 0.9999999) {};
\node [style=none] (15) at (9.000001, -0) {};
\node [style=none] (16) at (7.75, 0.4999999) {$T'_i$};
\node [style=none] (17) at (0, 2) {};
\node [style=none] (18) at (2.5, -0.9999999) {};
\node [style=none] (19) at (2.5, 2) {};
\node [style=none] (20) at (0.9999999, 2.5) {};
\node [style=none] (21) at (0.9999999, 0.7499999) {};
\node [style=none] (22) at (0.9999999, -0.4999999) {};
\node [style=none] (23) at (2.5, 0.7499999) {};
\node [style=none] (24) at (2.5, -0.4999999) {};
\node [style=none] (25) at (0.9999999, 2) {};
\node [style=none] (26) at (2.5, 2.5) {};
\node [style=none] (27) at (0.9999999, -0.9999999) {};
\node [style=cocpoint] (28) at (3.5, 2) {$i$};
\node [style=none] (29) at (2.5, 2) {};
\node [style=none] (30) at (3.5, 0.7499999) {};
\node [style=none] (31) at (3.5, -0.4999999) {};
\node [style=none] (32) at (1.75, 0.7499999) {$T$};
\node [style=none] (33) at (12.25, -0) {};
\node [style=none] (34) at (14, 0.9999999) {};
\node [style=none] (35) at (12.25, -0) {};
\node [style=none] (36) at (14, -0) {};
\node [style=cocpoint] (37) at (13.75, 2.25) {$i$};
\node [style=none] (38) at (12.25, 0.9999999) {};
\node [style={small box}] (39) at (13, 0.9999999) {$T_i$};
\node [style=none] (40) at (11.75, 2.25) {};
\node [style=none] (41) at (12.25, 0.9999999) {};
\node [style=none] (42) at (13.25, -0) {};
\node [style=none] (43) at (12.25, 0.9999999) {};
\node [style=none] (44) at (10.5, 1.75) {$=$};
\node [style=none] (45) at (14.75, -0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=90, looseness=2.00] (4.center) to (1.center);
\draw (0.center) to (3);
\draw (4.center) to (5.center);
\draw (6.center) to (9.center);
\draw (9.center) to (8.center);
\draw (8.center) to (7.center);
\draw (7.center) to (6.center);
\draw (12.center) to (14.center);
\draw (13.center) to (15.center);
\draw (20.center) to (27.center);
\draw (27.center) to (18.center);
\draw (18.center) to (26.center);
\draw (26.center) to (20.center);
\draw (17.center) to (25.center);
\draw [bend right=90, looseness=2.00] (21.center) to (22.center);
\draw (29.center) to (28);
\draw (23.center) to (30.center);
\draw (24.center) to (31.center);
\draw [bend right=90, looseness=2.00] (38.center) to (33.center);
\draw (40.center) to (37);
\draw (38.center) to (43.center);
\draw (42.center) to (36.center);
\draw (38.center) to (39);
\draw (39) to (34.center);
\draw (33.center) to (42.center);
\end{pgfonlayer}
\end{tikzpicture}$$
$\exists\ T'$ such that, $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (6.75, 0.7499999) {};
\node [style=none] (1) at (1.5, 0.7499999) {$T$};
\node [style=none] (2) at (2, -0.7499999) {};
\node [style=none] (3) at (0.9999999, 1.75) {};
\node [style=none] (4) at (4.5, 0.7499999) {$=$};
\node [style=none] (5) at (0.9999999, 0.7499999) {};
\node [style=none] (6) at (2.750001, -0.25) {};
\node [style=none] (7) at (0.9999999, 2.25) {};
\node [style=none] (8) at (0.9999999, -0.25) {};
\node [style=none] (9) at (0.7499999, -0.25) {};
\node [style=none] (10) at (2.750001, 0.7499999) {};
\node [style=none] (11) at (2.750001, -0.25) {};
\node [style=none] (12) at (2, -0.25) {};
\node [style=none] (13) at (8.000001, 0.7499999) {};
\node [style=none] (14) at (8.75, -0.25) {};
\node [style=none] (15) at (0.7499999, 0.7499999) {};
\node [style=none] (16) at (8.25, -0.25) {};
\node [style=none] (17) at (2, 2.25) {};
\node [style=none] (18) at (2, 1.75) {};
\node [style=none] (19) at (2, 0.7499999) {};
\node [style=none] (20) at (6.75, -0.25) {};
\node [style=none] (21) at (0.9999999, -0.7499999) {};
\node [style=cpoint] (22) at (0, 1.75) {$\sigma$};
\node [style=none] (23) at (2.750001, 1.75) {};
\node [style=none] (24) at (6.999999, 0.25) {};
\node [style=none] (25) at (8.000001, 0.25) {};
\node [style=cpoint] (26) at (5.999999, 1.75) {$\sigma$};
\node [style=none] (27) at (8.000001, 1.75) {};
\node [style=none] (28) at (6.999999, 2.25) {};
\node [style=none] (29) at (8.000001, 2.25) {};
\node [style=none] (30) at (8.75, 1.75) {};
\node [style=none] (31) at (8.75, 0.7499999) {};
\node [style=none] (32) at (6.999999, 0.7499999) {};
\node [style=none] (33) at (6.999999, 1.75) {};
\node [style=none] (34) at (7.499999, 1.25) {$T'$};
\node [style=none] (35) at (10.5, -0) {$\forall |\sigma)$};
\node [style=none] (36) at (11.25, -0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (7.center) to (17.center);
\draw (17.center) to (2.center);
\draw (2.center) to (21.center);
\draw [bend right=90, looseness=2.25] (15.center) to (9.center);
\draw (7.center) to (21.center);
\draw (15.center) to (5.center);
\draw (9.center) to (8.center);
\draw (12.center) to (11.center);
\draw (19.center) to (10.center);
\draw [bend right=90, looseness=2.25] (0.center) to (20.center);
\draw (20.center) to (14.center);
\draw (28.center) to (24.center);
\draw (24.center) to (25.center);
\draw (25.center) to (29.center);
\draw (29.center) to (28.center);
\draw (26) to (33.center);
\draw (0.center) to (32.center);
\draw (13.center) to (31.center);
\draw (27.center) to (30.center);
\draw (22) to (3.center);
\draw (18.center) to (23.center);
\end{pgfonlayer}
\end{tikzpicture}$$
$$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (10.75, 1.25) {};
\node [style=none] (1) at (1.5, 11.25) {$T$};
\node [style=none] (2) at (2, 9.75) {};
\node [style=none] (3) at (1, 12.25) {};
\node [style=none] (4) at (4.5, 11) {$=$};
\node [style=none] (5) at (1, 11.25) {};
\node [style=trace] (6) at (2.75, 12.25) {};
\node [style=none] (7) at (1, 12.75) {};
\node [style=none] (8) at (1, 10.25) {};
\node [style=none] (9) at (0.5000001, 10.25) {};
\node [style=trace] (10) at (2.75, 11.25) {};
\node [style=none] (11) at (2, 12.25) {};
\node [style=none] (12) at (12.25, 1.25) {};
\node [style=none] (13) at (13, 0.2500001) {};
\node [style=none] (14) at (0.5000001, 11.25) {};
\node [style=none] (15) at (12.5, 0.2500001) {};
\node [style=none] (16) at (2, 12.75) {};
\node [style=none] (17) at (2, 10.25) {};
\node [style=none] (18) at (2, 11.25) {};
\node [style=none] (19) at (10.75, 0.2500001) {};
\node [style=none] (20) at (1, 9.75) {};
\node [style=none] (21) at (0.5000001, 12.25) {};
\node [style=none] (22) at (2.75, 10.25) {};
\node [style=none] (23) at (11.25, 0.7500002) {};
\node [style=none] (24) at (12.25, 0.7500002) {};
\node [style=none] (25) at (10.75, 2.25) {};
\node [style=none] (26) at (12.25, 2.25) {};
\node [style=none] (27) at (11.25, 2.75) {};
\node [style=none] (28) at (12.25, 2.75) {};
\node [style=none] (29) at (13, 2.25) {};
\node [style=none] (30) at (13, 1.25) {};
\node [style=none] (31) at (11.25, 1.25) {};
\node [style=none] (32) at (11.25, 2.25) {};
\node [style=none] (33) at (11.75, 1.75) {$T'$};
\node [style=cocpoint] (34) at (15.5, 12.75) {$i$};
\node [style=none] (35) at (15.75, 10.5) {};
\node [style=none] (36) at (16, 9.5) {};
\node [style=none] (37) at (16.25, 10.5) {};
\node [style=trace] (38) at (16, 11.5) {};
\node [style=none] (39) at (14, 11.5) {};
\node [style=none] (40) at (14, 10.5) {};
\node [style={small box}] (41) at (14.75, 11.5) {$T_i$};
\node [style=none] (42) at (0.5000001, 9.25) {};
\node [style=none] (43) at (0.5000001, 12.25) {};
\node [style=none] (44) at (2.75, 9.25) {};
\node [style=none] (45) at (14, 9.5) {};
\node [style=none] (46) at (14, 12.75) {};
\node [style=none] (47) at (8.5, 11.25) {$T$};
\node [style=none] (48) at (8.999999, 13) {};
\node [style=none] (49) at (7.999999, 13) {};
\node [style=trace] (50) at (9.75, 11.25) {};
\node [style=none] (51) at (8.999999, 12.5) {};
\node [style=none] (52) at (9.75, 10.25) {};
\node [style=none] (53) at (8.999999, 11.25) {};
\node [style=cocpoint] (54) at (9.75, 12.5) {$i$};
\node [style=none] (55) at (9.75, 9.25) {};
\node [style=none] (56) at (8.999999, 10.25) {};
\node [style=none] (57) at (7.999999, 12.5) {};
\node [style=none] (58) at (7.5, 11.25) {};
\node [style=none] (59) at (7.5, 12.5) {};
\node [style=none] (60) at (7.999999, 11.25) {};
\node [style=none] (61) at (7.5, 10.25) {};
\node [style=none] (62) at (8.999999, 9.75) {};
\node [style=none] (63) at (7.999999, 10.25) {};
\node [style=none] (64) at (7.999999, 9.75) {};
\node [style=none] (65) at (7.5, 12.5) {};
\node [style=none] (66) at (7.5, 9.25) {};
\node [style=none] (67) at (5.499999, 11) {$\sum_i$};
\node [style=none] (68) at (12, 11) {$\sum_i$};
\node [style=none] (69) at (11, 11) {$=$};
\node [style=none] (70) at (13, 6.75) {};
\node [style=none] (71) at (13, 5.75) {};
\node [style=trace] (72) at (14, 6.75) {};
\node [style=none] (73) at (14, 5.75) {};
\node [style=none] (74) at (11, 6.75) {$=$};
\node [style=none] (75) at (3, 3.75) {Purification $\implies$};
\node [style=none] (76) at (5.75, 2.25) {};
\node [style=none] (77) at (5.75, 1.25) {};
\node [style=none] (78) at (4.75, 2.75) {};
\node [style=none] (79) at (6.5, 2.25) {};
\node [style=none] (80) at (5.75, 0.2500001) {};
\node [style=none] (81) at (4.75, 1.25) {};
\node [style=none] (82) at (4.75, -0.2499996) {};
\node [style=none] (83) at (4.25, 1.25) {};
\node [style=none] (84) at (5.75, -0.2499996) {};
\node [style=none] (85) at (4.75, 0.2500001) {};
\node [style=none] (86) at (4.25, 0.2500001) {};
\node [style=none] (87) at (6.5, -0.7499997) {};
\node [style=none] (88) at (6.5, 0.2500001) {};
\node [style=none] (89) at (5.25, 1.25) {$T$};
\node [style=none] (90) at (4.25, 2.25) {};
\node [style=none] (91) at (4.25, 2.25) {};
\node [style=none] (92) at (4.25, -0.7499997) {};
\node [style=none] (93) at (6.5, 1.25) {};
\node [style=none] (94) at (4.75, 2.25) {};
\node [style=none] (95) at (5.75, 2.75) {};
\node [style=none] (96) at (8.000001, 1.25) {$=$};
\node [style=none] (97) at (10.75, 2.25) {};
\node [style=none] (98) at (10.75, -0.7499997) {};
\node [style=none] (99) at (10.75, 2.25) {};
\node [style=none] (100) at (13, -0.7499997) {};
\node [style=none] (101) at (13.75, 0.5000001) {};
\node [style=none] (102) at (13, 7.75) {};
\node [style=none] (103) at (14, 4.75) {};
\node [style=none] (104) at (13, 4.75) {};
\node [style=trace] (105) at (14, 7.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (7.center) to (16.center);
\draw (16.center) to (2.center);
\draw (2.center) to (20.center);
\draw [bend right=90, looseness=2.25] (14.center) to (9.center);
\draw (7.center) to (20.center);
\draw (14.center) to (5.center);
\draw (9.center) to (8.center);
\draw (18.center) to (10);
\draw [bend right=90, looseness=2.25] (0.center) to (19.center);
\draw (19.center) to (13.center);
\draw (27.center) to (23.center);
\draw (23.center) to (24.center);
\draw (24.center) to (28.center);
\draw (28.center) to (27.center);
\draw (25.center) to (32.center);
\draw (0.center) to (31.center);
\draw (12.center) to (30.center);
\draw (26.center) to (29.center);
\draw (21.center) to (3.center);
\draw (17.center) to (22.center);
\draw (11.center) to (6);
\draw [bend right=90, looseness=2.25] (39.center) to (40.center);
\draw (40.center) to (37.center);
\draw (39.center) to (41);
\draw (41) to (38);
\draw [bend left=90, looseness=1.75] (42.center) to (43.center);
\draw (42.center) to (44.center);
\draw [bend left=90, looseness=1.50] (45.center) to (46.center);
\draw (46.center) to (34);
\draw (45.center) to (36.center);
\draw (49.center) to (48.center);
\draw (48.center) to (62.center);
\draw (62.center) to (64.center);
\draw [bend right=90, looseness=1.75] (58.center) to (61.center);
\draw (49.center) to (64.center);
\draw (58.center) to (60.center);
\draw (61.center) to (63.center);
\draw (53.center) to (50);
\draw (59.center) to (57.center);
\draw (56.center) to (52.center);
\draw (51.center) to (54);
\draw [bend left=90, looseness=1.25] (66.center) to (65.center);
\draw (66.center) to (55.center);
\draw [bend right=90, looseness=2.25] (70.center) to (71.center);
\draw (70.center) to (72);
\draw (71.center) to (73.center);
\draw (78.center) to (95.center);
\draw (95.center) to (84.center);
\draw (84.center) to (82.center);
\draw [bend right=90, looseness=2.25] (83.center) to (86.center);
\draw (78.center) to (82.center);
\draw (83.center) to (81.center);
\draw (86.center) to (85.center);
\draw (77.center) to (93.center);
\draw (91.center) to (94.center);
\draw (80.center) to (88.center);
\draw (76.center) to (79.center);
\draw [bend left=90, looseness=1.75] (92.center) to (90.center);
\draw (92.center) to (87.center);
\draw [bend left=90, looseness=1.75] (98.center) to (99.center);
\draw (98.center) to (100.center);
\draw [bend right=90, looseness=1.50] (102.center) to (104.center);
\draw (102.center) to (105);
\draw (104.center) to (103.center);
\end{pgfonlayer}
\end{tikzpicture}$$ Dynamic faithfulness then gives the result.
$T'$ is a controlled transformation, $T'=C\{T_i\}$.\[Existance\]
$$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.5000004, 5.75) {};
\node [style=none] (1) at (7.249999, 5.75) {$T$};
\node [style=none] (2) at (7.749999, 4.25) {};
\node [style=none] (3) at (6.750001, 6.75) {};
\node [style=none] (4) at (6.750001, 5.75) {};
\node [style=none] (5) at (6.750001, 7.25) {};
\node [style=none] (6) at (6.750001, 4.75) {};
\node [style=none] (7) at (6.25, 4.75) {};
\node [style=none] (8) at (7.749999, 6.75) {};
\node [style=none] (9) at (2, 5.75) {};
\node [style=none] (10) at (2.75, 4.75) {};
\node [style=none] (11) at (6.25, 5.75) {};
\node [style=none] (12) at (2.25, 4.75) {};
\node [style=none] (13) at (7.749999, 7.25) {};
\node [style=none] (14) at (7.749999, 4.75) {};
\node [style=none] (15) at (7.749999, 5.75) {};
\node [style=none] (16) at (0.5000004, 4.75) {};
\node [style=none] (17) at (6.750001, 4.25) {};
\node [style=cpoint] (18) at (5.75, 6.75) {$i$};
\node [style=none] (19) at (8.500001, 4.75) {};
\node [style=none] (20) at (0.9999995, 5.25) {};
\node [style=none] (21) at (2, 5.25) {};
\node [style=cpoint] (22) at (0, 6.75) {$i$};
\node [style=none] (23) at (2, 6.75) {};
\node [style=none] (24) at (0.9999995, 7.25) {};
\node [style=none] (25) at (2, 7.25) {};
\node [style=none] (26) at (2.75, 6.75) {};
\node [style=none] (27) at (2.75, 5.75) {};
\node [style=none] (28) at (0.9999995, 5.75) {};
\node [style=none] (29) at (0.9999995, 6.75) {};
\node [style=none] (30) at (1.5, 6.25) {$T'$};
\node [style=none] (31) at (4.249999, 6) {$=$};
\node [style=none] (32) at (8.500001, 6.75) {};
\node [style=none] (33) at (8.500001, 5.75) {};
\node [style=none] (34) at (10.25, 6) {$=$};
\node [style=none] (35) at (14.5, 4.75) {};
\node [style=cpoint] (36) at (11.75, 6.75) {$i$};
\node [style=none] (37) at (14.5, 6.75) {};
\node [style=none] (38) at (12.25, 4.75) {};
\node [style=none] (39) at (14.5, 5.75) {};
\node [style=none] (40) at (12.25, 5.75) {};
\node [style={small box}] (41) at (13.25, 5.75) {$T_i$};
\node [style=none] (42) at (5.000001, 2.75) {Dynamically faithful state $\implies$};
\node [style=none] (43) at (2.25, 1.25) {};
\node [style=none] (44) at (1.25, -0.7500005) {};
\node [style=none] (45) at (1.25, 1.25) {};
\node [style=none] (46) at (1.25, -0.2500002) {};
\node [style=none] (47) at (2.25, 1.75) {};
\node [style=none] (48) at (1.25, 1.75) {};
\node [style=none] (49) at (1.75, 0.5000004) {$T'$};
\node [style=none] (50) at (2.25, -0.7500005) {};
\node [style=none] (51) at (3, -0.2500002) {};
\node [style=none] (52) at (2.25, -0.2500002) {};
\node [style=none] (53) at (3, 1.25) {};
\node [style=cpoint] (54) at (0.2500002, -0.2500002) {$\sigma$};
\node [style=cpoint] (55) at (0.2500002, 1.25) {$i$};
\node [style=none] (56) at (8.75, -0.2500002) {};
\node [style={small box}] (57) at (7.5, -0.2500002) {$T_i$};
\node [style=none] (58) at (8.75, 1.25) {};
\node [style=cpoint] (59) at (5.999999, -0.2500002) {$\sigma$};
\node [style=cpoint] (60) at (5.999999, 1.25) {$i$};
\node [style=none] (61) at (4.5, 0.5000004) {$=$};
\node [style=none] (62) at (10.75, -0) {$\forall |\sigma)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (5.center) to (13.center);
\draw (13.center) to (2.center);
\draw (2.center) to (17.center);
\draw [bend right=90, looseness=2.25] (11.center) to (7.center);
\draw (5.center) to (17.center);
\draw (11.center) to (4.center);
\draw (7.center) to (6.center);
\draw [bend right=90, looseness=2.25] (0.center) to (16.center);
\draw (16.center) to (10.center);
\draw (24.center) to (20.center);
\draw (20.center) to (21.center);
\draw (21.center) to (25.center);
\draw (25.center) to (24.center);
\draw (22) to (29.center);
\draw (0.center) to (28.center);
\draw (9.center) to (27.center);
\draw (23.center) to (26.center);
\draw (18) to (3.center);
\draw (14.center) to (19.center);
\draw (8.center) to (32.center);
\draw (15.center) to (33.center);
\draw [bend right=90, looseness=2.25] (40.center) to (38.center);
\draw (36) to (37.center);
\draw (38.center) to (35.center);
\draw (40.center) to (41);
\draw (41) to (39.center);
\draw (48.center) to (44.center);
\draw (44.center) to (50.center);
\draw (50.center) to (47.center);
\draw (47.center) to (48.center);
\draw (55) to (45.center);
\draw (54) to (46.center);
\draw (52.center) to (51.center);
\draw (43.center) to (53.center);
\draw (60) to (58.center);
\draw (59) to (57);
\draw (57) to (56.center);
\end{pgfonlayer}
\end{tikzpicture}$$ which is the defining characteristic of $C\{T_i\}$.
Superposition preservation\[SuperpositionProof\]
================================================
Lem. \[SP\] already gives some notion of superposition preservation, we can use our other results above to extend this. $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.7500001, 0.5000003) {};
\node [style=none] (1) at (7.999999, 0.5000003) {$T$};
\node [style=none] (2) at (8.5, -1.000001) {};
\node [style=none] (3) at (7.5, 1.5) {};
\node [style=none] (4) at (7.5, 0.5000003) {};
\node [style=none] (5) at (7.5, 2) {};
\node [style=none] (6) at (7.5, -0.5000003) {};
\node [style=none] (7) at (7, -0.5000003) {};
\node [style=none] (8) at (8.5, 1.5) {};
\node [style=none] (9) at (2.5, 0.5000003) {};
\node [style=none] (10) at (3, -0.5000003) {};
\node [style=none] (11) at (7, 0.5000003) {};
\node [style=none] (12) at (2.5, -0.5000003) {};
\node [style=none] (13) at (8.5, 2) {};
\node [style=none] (14) at (8.5, -0.5000003) {};
\node [style=none] (15) at (8.5, 0.5000003) {};
\node [style=none] (16) at (0.7500001, -0.5000003) {};
\node [style=none] (17) at (7.5, -1.000001) {};
\node [style=none] (18) at (9.25, -0.5000003) {};
\node [style=none] (19) at (1.000001, -0) {};
\node [style=none] (20) at (2.5, -0) {};
\node [style=none] (21) at (2.5, 1.5) {};
\node [style=none] (22) at (1.000001, 2) {};
\node [style=none] (23) at (2.5, 2) {};
\node [style=none] (24) at (3, 0.5000003) {};
\node [style=none] (25) at (1.000001, 0.5000003) {};
\node [style=none] (26) at (1.000001, 1.5) {};
\node [style=none] (27) at (1.749999, 1.5) {$C$};
\node [style=none] (28) at (4.75, 0.7500001) {$=$};
\node [style=none] (29) at (9.25, 0.5000003) {};
\node [style=none] (30) at (11.25, 0.7500001) {$=$};
\node [style=none] (31) at (15, -0.5000003) {};
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\end{tikzpicture}$$ The first equality uses Thm. \[Existance\] and the second Lem. \[SP\]. Using the result of dynamically faithful states then implies, $$\begin{tikzpicture}
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This actually only proves the existence of a controlled transformation that preserves superpositions, it is simple to show that it must be true for all controlled transformations using an argument analogous to Lem. \[SP\].
Definition of higher-order interference {#Higher-order-App}
=======================================
The original definition of higher-order interference was in the framework of quantum measure theory [@sorkin1994quantum; @sorkin1995quantum]. The definition revolves around the concepts of ‘histories’ and the ‘quantum measure’. Histories correspond to paths through space-time, a set of histories $A$ is any collection of these paths. We will be concerned with sets of histories with some initial condition $s$ and some final condition $e$ along with an intermediate condition $i$ that ‘the history passes through slit $i$’. We label these sets of histories $A^{se}_i$. In an interference experiment it is common to have some way to create a path difference between the different slits, either by introducing some ‘phase shifter’ or by moving the final detection point, we label this data $t$.
The quantum measure $\mu$ associates some probability to each set of histories, which should be thought of as the probability that a particle ‘has a history from that set’. In general $\mu$ will depend on the experimental control $t$.
The existence of higher-order interference in this framework is as follows:
$n$-th order interference $\iff$ $\exists s,e$ s.t.$$\mu\left[\bigcup_{i\in\mathds{P}}A^{se}_i\right](t) \neq \sum_{I\subset \mathds{P}}(-1)^{n-|I|+1}\mu\left[\bigcup_{i\in I}A^{se}_{i}\right](t)$$
Given this definition we can provide a translation to the operational definition that we are using,
QMT GPT
--------------------------- ------------------------------------ -------------------------------
Initial condition $s$ $|s)$
Final condition $e$ $(e|$
Experimental control $t$ $T$
Probability of path $i$ $\mu[A^{se}_i](t)$ $\mathcal{C}_{se_{\{i\}}}(T)$
Probability of subset $I$ $\mu[\bigcup_{i\in I}A^{se}_i](t)$ $\mathcal{C}_{se_{I}}(T)$
Note that some ambiguity is introduced in switching to the operational framework, which $e_I\in\mathcal{E}_I$ should be picked? Other approaches to defining higher-order interference in operational theories [@Higher-order-reconstruction] have required sufficient structure to define a set of ‘filters’, $\{F_I\}$, for the theory. These are transformations that represent the action of leaving open some subset of slits $I$ whilst closing the others, in which case $(e_I|=(e|F_{I}$. However, arbitrary theories do not have sufficient structure to define filters and so one must consider all possible choices $(e_I|$ with the correct support. This leads to the following definition of $n$th-order interference,
$n$-th order interference $\iff$ $\exists s,e$ s.t.$$\mathcal{C}_{s,e}(T)\neq \sum_{I\subset\mathds{P}}(-1)^{n-|I|+1}\mathcal{C}_{s,e_I}(T),$$ $\forall (e_I|\in \mathcal{E}_I$.
The introduction of the ‘$\forall$’ statement compared to the original definition is due to the ambiguity in choosing which effect corresponds to blocking some subset of paths. In the main text the explicit dependence on $T$ in the above equation will be suppressed, as $\mathcal{C}_{se}$ has already been defined as a function from the phase group to probabilities.
For example, the existence of second-order interference implies that there exists $|s)$ and $(e|$ such that $$\mathcal{C}_{s,e}\neq \mathcal{C}_{s,e_{\{0\}}}+\mathcal{C}_{s,e_{\{1\}}},$$ $\forall\ |e_{\{i\}})\in\mathcal{E}_i$. Whilst the existence of third-order interference corresponds to the existence of, $|s)$ and $(e|$ such that $$\mathcal{C}_{s,e}\neq \mathcal{C}_{s,e_{\{0,1\}}}+\mathcal{C}_{s,e_{\{1,2\}}}+\mathcal{C}_{s,e_{\{2,0\}}}-\mathcal{C}_{s,e_{\{0\}}}-\mathcal{C}_{s,e_{\{1\}}}-\mathcal{C}_{s,e_{\{2\}}},$$ $\forall\ |e_{I})\in\mathcal{E}_I$.
We consider the above for the case of quantum theory to provide some intuition for the definitions. Firstly, we show the existence of second-order interference. Define our paths by $p_i:=({|i\rangle}{\langle i|},{|i\rangle}{\langle i|})$, then choose $|s)={|+\rangle}{\langle +|}=(e|$. Then $(e_{\{i\}}|\in \{r_i {|i\rangle}{\langle i|}\}$ where $r_i$ is an arbitrary positive real number. The phase group is given by $\mathcal{P}:=\{e^{i\theta_0}{|0\rangle}{\langle 0|}+e^{i\theta_1}{|1\rangle}{\langle 1|}\}$. It is then simple to show that, $$\mathcal{C}_{s,e}(T) = \cos^2\left(\frac{\theta_0-\theta_1}{2}\right),$$ whilst, $$\mathcal{C}_{s,e_{\{0\}}}(T)+\mathcal{C}_{s,e_{\{1\}}}(T)= \frac{r_0+r_1}{\sqrt{2}}.$$ It is then simple to see that, as functions of $\theta_i$, $$\cos^2\left(\frac{\theta_0-\theta_1}{2}\right)\neq \frac{r_0+r_1}{\sqrt{2}},$$ for any choice of $r_i$, i.e. $(e_{\{i\}}|$. Therefore – by our definition – quantum theory has second-order interference as we would expect.
Next we consider our definition of third-order interference for quantum theory. We consider a specific choice of $|s)$ and $(e|$, and note that this can be simply – but tediously – generalised to all choices. Consider $|s)=\frac{1}{3}({|0\rangle}+{|1\rangle}+{|2\rangle})({\langle 0|}+{\langle 1|}+{\langle 2|})=(e|$, and the phase group, $\mathcal{P}=\{e^{i\theta_0}{|0\rangle}{\langle 0|}+e^{i\theta_1}{|1\rangle}{\langle 1|}+e^{i\theta_2}{|2\rangle}{\langle 2|}\}$. Then let $(e_{\{i,j\}}|=\frac{1}{3}({|i\rangle}+{|j\rangle})({\langle i|}+{\langle j|})$ and $(e_{\{i\}}|=\frac{1}{3}{|i\rangle}{\langle i|}$. Note that these are sub-normalised effects. It is then simple to check our definition for this particular choice of $|s)$ and $(e|$. The interference patterns can be written as,
1. $$\begin{aligned}
\mathcal{C}_{s,e}(T) &= \frac{1}{9}|e^{i\theta_0}+e^{i\theta_1}+e^{i\theta_2}|^2 \\ &=\frac{1}{9}\left(3+\sum_{i>j}e^{i(\theta_i-\theta_j)}+e^{i(\theta_j-\theta_i)}\right), \end{aligned}$$
2. $$\begin{aligned} \mathcal{C}_{s,e_{\{i,j\}}}(T) &= \frac{1}{9}|e^{i\theta_i}+e^{i\theta_j}|^2 \\ &=\frac{1}{9}\left(2+e^{i(\theta_i-\theta_j)}+e^{i(\theta_j-\theta_i)}\right),\end{aligned}$$
3. $$\begin{aligned} \mathcal{C}_{s,e_{\{i\}}}(T) = \frac{1}{9}|e^{i\theta_i}|^2=\frac{1}{9} \end{aligned}$$
and so, $$\mathcal{C}_{s,e}(T)=\sum_{i>j}\mathcal{C}_{s,e_{\{i,j\}}}(T)-\sum_i \mathcal{C}_{s,e_{\{i\}}}(T).$$ This proves that the particular choice of state $|s)$ and effect $(e|$ do not give higher-order interference for quantum theory. This can, however, be readily generalised to hold for any choice, and so demonstrates that quantum theory does not exhibit higher-order interference.
[^1]: The set of states, effects and transformations each give rise to a vector space and transformations and effects act linearly on the vector space of states. We assume in this work that all vector spaces are finite dimensional.
[^2]: The process $\{\mathcal{U}_j\}_{j\in{Y}}$, where $j$ index the positions of the classical pointer, is a coarse-graining of the process $\{\mathcal{E}_i\}_{i\in{X}}$ if there is a disjoint partition $\{X_j\}_{j\in{Y}}$ of $X$ such that $\mathcal{U}_j=\sum_{i\in{X_j}}\mathcal{E}_i$.
[^3]: Note that this is not the case for every generalised probabilistic theories. For example, the theories based on Euclidean-Jordan algebras presented in [@barnum2015some] do not satisfy this requirement.
[^4]: That is if two states $|\psi)_{AB}$ and $|\psi')_{AB}$ purify $|s)_A$, then there exists a reversible transformation $T_B$ on system $B$ such that $|\psi)_{AB}=(\mathbb{I}\otimes{T_B})|\psi)_{AB}$.
[^5]: In theories satisfying strong symmetry the measurement $\{(i|\}$ is unique up to normalisation, see Appendix (\[Sym\]).
[^6]: There could be many distinct transformations that have the same behaviour on a set of control states. As long as one fixes which transformation corresponds to the oracle, this is not a problem.
|
---
abstract: 'The entropic force has been recently argued to be responsible for dissociation of heavy quarkonia. In this paper, we analyze $R^2$ corrections and $R^4$ corrections to the entropic force, respectively. It is shown that for $R^2$ corrections, increasing $\lambda_{GB}$ (Gauss-Bonnet factor) leads to increasing the entropic force. While for $R^4$ corrections, increasing $\lambda$ (’t Hooft coupling) leads to decreasing the entropic force. Also, we discuss how the entropic force changes with the shear viscosity to entropy density ratio, $\eta/s$, at strong coupling.'
author:
- 'Zi-qiang Zhang'
- 'Zhong-jie Luo'
- 'De-fu Hou'
title: Higher derivative corrections to the entropic force from holography
---
Introduction
============
The experimental programs at LHC and RHIC have produced a new state of matter so-called “strongly coupled quark-gluon plasma (sQGP)” [@JA; @KA; @EV]. One of the main experimental signatures for sQGP formation is dissociation of quarkonia [@KM]. It was suggested earlier that the color screening is the main mechanism responsible for this suppression [@TMA]. Subsequently, some authors argued that the imaginary part of the heavy quark potential may be a more important reason than screening [@ML; @AB; @NB]. Recently, it was proposed by D. E. Kharzeev [@DEK] that the entropic force would be responsible for melting the quarkonia as well.
The entropic force is related to the increase of the entropy with the separation between the constituents of the bound state. It is an emergent force and does not describe other fundamental interactions. Based on the second law of thermodynamics, it stems from multiple interactions that drive the system toward the state with a larger entropy. The entropic force was developed in [@KHM] to explain the elasticity of polymer strands in rubber. Subsequently, Verlinde argued [@EPV] that it would be responsible for gravity, but this interesting idea may be controversial (see for [@DC]) and will not be discussed here. Recently, it was argued [@DEK] that the entropic force can drive the dissociation process if one considers the process of deconfinement as an entropic self-destruction. This argument is based upon the Lattice results that show that there is a peak in the heavy quark entropy around the crossover region of the sQGP [@DKA1; @DKA2; @PPE]. However, it should be noticed that the entropic force cannot be taken as a fundamental property of the system, but it allows us to understand the behavior of complicated microscopic systems not amenable to microscopic treatment. In this work, we will restrict ourselves to its application in dissociation of quarkonia in the sQGP.
AdS/CFT [@Maldacena:1997re; @Gubser:1998bc; @MadalcenaReview], the duality between a string theory in AdS space and a conformal field theory in the physical space-time, has yielded many important insights for studying different aspects of the sQGP. In this approach, K. Hashimoto et al have carried out the entropic force associated with a heavy quark pair for $\mathcal N=4$ SYM theory in their seminal work [@KHA]. There, it is found that the peak of the entropy near the transition point is related to the nature of deconfinement and the growth of the entropy with the distance can yield the entropic force. Soon after [@KHA], investigations of the entropic force with respect to a moving quarkonium appeared in [@KBF]. It is shown that the velocity has the effect of increasing the entropic force thus enhancing the quarkonia dissociation. Recently, we have studied the effect of chemical potential on the entropic force and observed that the chemical potential increases the entropic force implying that the quarkonia dissociation is enhanced at finite density [@ZQ].
In general, string theory contains higher derivatives corrections due to the presence of stringy effects. Although very little is known about the forms of higher derivative corrections in string theory, given the vastness of the string landscape one may expect that generic corrections do occur [@MRD]. As a concrete example, type IIB string theory on $AdS_5\times S^5$ is dual to $\mathcal N=4$ SYM theory. Using the relation $\sqrt{\lambda}=\frac{L^2}{\alpha^\prime}$ ($L$ is the radius of $AdS_5$ and $\alpha^\prime$ the reciprocal of the string tension), the $\mathcal{O}(\alpha^\prime)$ expansion in type IIB string theory becomes the $\frac{1}{\sqrt{\lambda}}$ expansion in SYM theory. The leading order corrections in $1/\lambda$ ($R^4$ corrections) come from stringy corrections to the type IIB tree level effective action of the form $\alpha^{\prime 3}R^4$. It was argued [@ABR1; @PB] that $R^4$ corrections to $\eta/s$ are positive, consistent with the viscosity bound [@ABR2; @RCM]. On the other hand, curvature squared interactions (corresponding to $R^2$ corrections) can be induced in the gravity sector in $AdS_5$ by including the world-volume action of D7 branes [@ABR3; @OA; @OA1]. It was shown [@MB; @MB1; @YK] that in the five dimensional gravity theories with $R^2$ corrections $\eta/s$ can be lower than $1/(4\pi)$. Also, there are other observables or quantities that have been studied in theories with higher derivative corrections, see e.g. [@JN; @KB1; @JN1; @ZQ2].
In this paper, we study $R^2$ corrections and $R^4$ corrections to the entropic force. More specially, we would like to see how these corrections affect the enropic force as well as the quarkonia dissociation. On the other hand, $\eta/s$ is different than $1/(4\pi)$ in the theories with higher derivative corrections, so the connection between $\eta/s$ and the entropic force in these theories may be an interesting fact that comes for free in holography. These are the main motivations of the present work.
The organization of the paper is as follows. In section 2, we analyze $R^2$ corrections to the entropic force and explore how these corrections affect the quarkonia dissociation. Also, we discuss how the entropic force changes with $\eta/s$ in this case. In section 3, we investigate $R^4$ corrections to the entropic force as well. Finally, we provide a concluding discussion in section 4.
$R^2$ corrections
=================
In string theory, the $R^2$ interactions is argued to arise from the world-volume action of D7 branes [@ABR3; @OA; @OA1]. Restricting to the gravity sector in $AdS_5$, the effective gravity action to leading order can be written as [@MB; @MB1] $$I=\frac{1}{16\pi G_5}\int
d^5x\sqrt{-g}[R+\frac{12}{L^2}+L^2(c_1R^2+c_2R_{\mu\nu}R^{\mu\nu}+c_3R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma})],\label{action}$$ where $G_5$ is the 5 dimensional Newton constant, $R_{\mu\nu\rho\sigma}$ is the Riemann tensor, $R$ is the Ricci scalar, $R_{\mu\nu}$ is the Ricci tensor, $L$ is the radius of $AdS_5$ at leading order in $c_i$ with $\lim_{\lambda\rightarrow\infty}c_i=0$. Other terms with additional derivatives or factors of $R$ are suppressed by higher powers of $\frac{\alpha\prime}{L^2}$. However, at this order only $c_3$ is unambiguous while $c_1$ and $c_2$ can be arbitrarily altered by a field redefinition [@MB; @MB1; @YK]. To avoid this issue, one applies the Gauss-Bonnet (GB) gravity, a special case of the action (\[action\]), in which $c_i$ are fixed in terms of a single parameter $\lambda_{GB}$. The GB gravity gives the following action [@BZ1] $$I=\frac{1}{16\pi G_5}\int
d^5x\sqrt{-g}[R+\frac{12}{L^2}+\frac{\lambda_{GB}}{2}L^2(R^2-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma})],\label{action1}$$ where $\lambda_{GB}$ is constrained in $$-\frac{7}{36}<\lambda_{GB}\leq\frac{9}{100},$$ where the lower bound originates from requiring the boundary energy density to be positive-definite [@DM] and the upper bound comes from avoiding causality violation in the boundary [@MB1].
The black brane solution of GB gravity can be written as [@RG] $$ds^2=-a^2\frac{r^2}{L^2}f(r)dt^2+{\frac{r^2}{L^2}}d\vec{x}^2+\frac{L^2}{r^2}\frac{dr^2}{f(r)}
\label{metric},$$ with $$f(r)=\frac{1}{2\lambda_{GB}}\big{[}1-\sqrt{1-4\lambda_{GB}(1-\frac{r_h^4}{r^4})}\big{]}\label{f1},$$ and $$a^2=\frac{1}{2}(1+\sqrt{1-4\lambda_{GB}}),$$ where $\vec{x}=x_1,x_2,x_3$ represent the boundary coordinates and $r$ denotes the coordinate of the 5th dimension. The boundary is located at $r=\infty$. The horizon is located at $r=r_h$. Moreover, the temperature is $$T=\frac{ar_h}{\pi L^2}. \label{T}$$
It was argued [@MB; @MB1; @YK] that $$\frac{\eta}{s}=\frac{1}{4\pi}(1-4\lambda_{GB}),\label{eta}$$ one can see that $\eta/s\geq\frac{1}{4\pi}$ can be violated for $\lambda_{GB}>0$. And, by increasing $\lambda_{GB}$, $\eta/s$ decreases.
We now follow the calculations of [@KHA] to analyze the entropic force for the background metric (\[metric\]). The entropic force is defined as [@DEK] $$\mathcal{F}=T\frac{\partial S}{\partial x},\label{f}$$ where $T$ is the temperature of the plasma, $S$ represents the entropy, $x$ denotes the inter-quark distance.
The Nambu-Goto action is $$S_{NG}=-\frac{1}{2\pi\alpha^\prime}\int d\tau d\sigma\mathcal
L=-\frac{1}{2\pi\alpha^\prime}\int d\tau
d\sigma\sqrt{-detg_{\alpha\beta}}, \label{S}$$ with $$g_{\alpha\beta}=g_{\mu\nu}\frac{\partial
X^\mu}{\partial\sigma^\alpha} \frac{\partial
X^\nu}{\partial\sigma^\beta},$$ where $g_{\alpha\beta}$ is the induced metric and parameterized by $(\tau,\sigma)$ on the string world-sheet. $g_{\mu\nu}$ is the metric, $X^\mu$ is the target space coordinate.
For our purpose, we take the static gauge $$t=\tau, \qquad x_1=\sigma,$$ and assume that $r$ depends only on $\sigma$, $$r=r(\sigma).$$
Under this assumption, the lagrangian density is found to be $$\mathcal L=a\sqrt{\frac{f(r)r^4}{L^4}+\dot{r}^2},\label{L}$$ with $\dot{r}=\frac{\partial r}{\partial\sigma}$.
Note that $\mathcal L$ does not depend on $\sigma$ explicitly, so the corresponding Hamiltonian is a constant, $$\mathcal L-\frac{\partial\mathcal
L}{\partial\dot{r}}\dot{r}=constant.$$
Imposing the boundary condition at $\sigma=0$, $$\dot{r}=0,\qquad r=r_c \qquad (r_h<r_c)\label{con},$$ where $r=r_c$ is the deepest point of the U-shaped string.
One finds $$\frac{f(r)r^4}{\sqrt{f(r)r^4+L^4\dot{r}^2}}=\sqrt{f(r_c)r_c^4},$$ with $$f(r_c)=\frac{1}{2\lambda_{GB}}\big{[}1-\sqrt{1-4\lambda_{GB}(1-\frac{r_h^4}{r_c^4})}\big{]},$$ results in $$\dot{r}=\frac{dr}{d\sigma}=\sqrt{\frac{r^4f(r)[r^4f(r)-r_c^4f(r_c)]}{L^4r_c^4f(r_c)}}\label{dotr}.$$
By integrating (\[dotr\]) the inter-quark distance of $Q\bar{Q}$ is obtained $$x=2\int_{r_c}^{\infty}dr\sqrt{\frac{L^4r_c^4f(r_c)}{r^4f(r)[r^4f(r)-r_c^4f(r_c)]}}\label{x}.$$
To study the effect of $R^2$ corrections on the inter-distance, we plot $xT$ versus $\varepsilon$ with $\varepsilon\equiv r_h/r_c$ for different $\lambda_{GB}$ in the left panel of Fig.1. In the plots from top to bottom $\lambda_{GB}=-0.1,0.01,0.06$ respectively. We can see that for each plot there exists a maximum value of $xT$, and that $xT$ is an increasing function of $\varepsilon$ for $xT<xT_{max}$ but a decreasing one for $xT>xT_{max}$. In fact, for the later case, one needs to consider some new configurations [@DB] which are not solutions of the Nambu-Goto action. Here we are interested mostly in the region of $xT<xT_{max}$. For convenience, we write $c\equiv xT_{max}$. From the left panel of Fig.1, one also finds that increasing $\lambda_{GB}$ leads to decreasing $c$. Therefore, one concludes that with increasing $\lambda_{GB}$ the inter-distance decreases, similarly to what occurred in [@JN1].
{width="8cm"} {width="8cm"}
The next step is to calculate the entropy $S$, given by $$S=-\frac{\partial F}{\partial T},\label{s}$$ where $F$ is the free energy of $Q\bar{Q}$. This quantity has been studied from the AdS/CFT correspondence, see e.g. [@JMM; @ABR; @SJR]. There are two cases for the free energy.
1\. If $x>\frac{c}{T}$, the fundamental string breaks in two disconnected strings implying the quarks are completely screened. In this case, the free energy is $$F^{(1)}=\frac{a}{\pi\alpha^\prime}\int_{r_h}^{\infty}dr.$$
In terms of (\[s\]), one gets $$S^{(1)}=a\sqrt{\lambda}\theta(L-\frac{c}{T})\label{S2}.$$
2\. If $x<\frac{c}{T}$, the fundamental string is connected. In this case, the free energy is actually the total energy of the quark pair which can be derived from the on-shell action of the fundamental string in the dual geometry. Plugging (\[dotr\]) into (\[S\]), one finds $$F^{(2)}=\frac{1}{\pi\alpha^\prime}\int_{r_c}^{\infty} dr
\sqrt{\frac{a^2r^4f(r)}{r^4f(r)-r_c^4f(r_c)}}.$$
As $r_h$ is related to $T$, one can rewrite (\[s\]) as $$S=-\frac{\partial F}{\partial T}=-\frac{\partial F}{\partial
r_h}\frac{\partial r_h}{\partial T}=-\frac{\pi
L^2}{a}\frac{\partial F}{\partial r_h}.\label{s1}$$
By virtue of (\[s1\]), one gets $$\frac{S^{(2)}}{\sqrt{\lambda}}=-\frac{1}{2a}\int_{r_c}^{\infty}
dr\frac{[a^\prime(r)b(r)+a(r)b^\prime(r)][a(r)-a(r_c)]-a(r)b(r)[a^\prime(r)-a^\prime(r_c)]}{\sqrt{a(r)b(r)[a(r)-a(r_c)]^3}}\label{S21},$$ with $$a(r)=\frac{a^2f(r)r^4}{L^4}, \qquad
a(r_c)=\frac{a^2f(r_c)r_c^4}{L^4},\qquad b(r)=a^2,$$ where we have used the relation $\alpha^\prime=\frac{L^2}{\sqrt{\lambda}}$. Also, the derivatives in the above equation are with respect to $r_h$.
To proceed further we have to resort to numerical methods. In the right panel of Fig 1, we plot $S^{(2)}/\sqrt{\lambda}$ versus $xT$ for different $\lambda_{GB}$. In the plots from right to left $\lambda_{GB}=-0.1,0.01,0.06$, respectively. From the figures, one can see that increasing $\lambda_{GB}$ leads to larger entropy at small distances. On the other hand, from (\[f\]) one finds that the entropic force is related to the growth of the entropy with the distance. As a result, increasing $\lambda_{GB}$ leads to increasing the entropic force. Since the entropic force is responsible for dissociating the quarkonia, one concludes that increasing $\lambda_{GB}$ the quarkonia dissociation is enhanced. Interestingly, it was argued [@JN1] that increasing $\lambda_{GB}$ leads to increasing the imaginary potential thus making the quakonia melt easier, consistent with the findings here.
Also, it follows from (\[eta\]) that increasing $\lambda_{GB}$ leads to decreasing $\eta/s$. While increasing $\lambda_{GB}$ leads to enhancing the quarkonia dissociation. Thus, one concludes that in the case of $R^2$ corrections the quarkonia dissociation is enhanced as $\eta/s$ decreases.
$R^4$ corrections
=================
In this section we study $R^4$ corrections to the entropic force. These corrections are related to $\alpha^\prime$ corrections on the string theory side [@JP] and correspond to leading order correction in $1/\lambda$ on the gauge theory side. The $\alpha^\prime$-corrected metric is given by [@GB1] $$ds^2=G_{tt}dt^2+G_{xx}d\vec{x}^2+G_{rr}dr^2 \label{metric2},$$ with $$G_{tt}=-r^2(1-w^{-4})T(w),\qquad G_{xx}=r^2X(w),\qquad
G_{rr}=r^{-2}(1-w^{-4})^{-1}U(w),$$ where $$\begin{aligned}
T(w)&=&1-k(75w^{-4}+\frac{1225}{16}w^{-8}-\frac{695}{16}w^{-12})+...,\nonumber\\
X(w)&=&1-\frac{25k}{16}w^{-8}(1+w^{-4})+...,\nonumber\\
U(w)&=&1+k(75w^{-4}+\frac{1175}{16}w^{-8}-\frac{4585}{16}w^{-12})+...,\end{aligned}$$ with $w=\frac{r}{r_h}$.
The parameter $k$ is related to $\lambda$ by $$k=\frac{\zeta(3)}{8}\lambda^{-3/2} \sim 0.15\lambda^{-3/2}.$$
The horizon is $r=r_h$ and the temperature is $$T=\frac{r_h}{\pi L^2(1-k)}.$$
In addition, it was argued [@ABR1; @PB] that $$\frac{\eta}{s}=\frac{1}{4\pi}(1+\frac{135}{8}\zeta(3)\lambda^{-3/2}),\label{eta1}$$ one can see that $\eta/s\geq1/4\pi$ remains valid in theories with $R^4$ corrections. Also, decreasing $\lambda$ leads to increasing $\eta/s$.
The next analysis is almost parallel to the previous section, so we present the final results here. One finds $$x=2\int_{r_c}^{\infty}dr\sqrt{\frac{a(r_c)b(r)}{a^2(r)-a(r)a(r_c)}}\label{x1},$$ with $$a(r)=r^4(1-w^{-4})T(w)X(w), \qquad
a(r_c)=r_c^4(1-w_1^{-4})T(w_1)X(w_1),\qquad b(r)=T(w)U(w),$$ where $w_1=\frac{r_c}{r_h}$, $T(w_1)=T(w)|_{w=w_1}$, $X(w_1)=X(w)|_{w=w_1}$.
Likewise, to analyze $R^4$ corrections to the inter-distance, we plot $xT$ versus $\varepsilon$ with different $\lambda$ in the left panel of Fig.2. Note that the behavior of $R^4$ corrections is not the same as the $R^2$ corrections. Here one can see that the value of $xT_{max}$ increases as $\lambda$ increases, in agreement with the findings of [@KB1].
On the other hand, the free energy $F^{(2)}$ is found to be $$F^{(2)}=\frac{1}{\pi\alpha^\prime}\int_{r_c}^{\infty} dr
\sqrt{\frac{a(r)b(r)}{a(r)-a(r_c)}}.$$ After some manipulations, one finds $$\frac{S^{(2)}}{\sqrt{\lambda}}=\frac{k-1}{2}\int_{r_c}^{\infty}
dr\frac{[a^\prime(r)b(r)+a(r)b^\prime(r)][a(r)-a(r_c)]-a(r)b(r)[a^\prime(r)-a^\prime(r_c)]}{\sqrt{a(r)b(r)[a(r)-a(r_c)]^3}}\label{S22},$$ with $$\begin{aligned}
a^\prime(r)&=&r^4(T^\prime(w)X(w)+T(w)X^\prime(w)+4w^{-5}w^\prime T(w)X(w)-w^{-4}T^\prime(w)X(w)-w^{-4}T(w)X^\prime(w)),\nonumber\\
a^\prime(r_c)&=&r_c^4(T^\prime(w_1)X(w_1)+T(w_1)X^\prime(w_1)+4w_1^{-5}w_1^\prime T(w_1)X(w_1)-w_1^{-4}T^\prime(w_1)X(w_1)-w_1^{-4}T(w_1)X^\prime(w_1)),\nonumber\\
b^\prime(r)&=&T^\prime(w)U(w)+T(w)U^\prime(w),\end{aligned}$$ and $$\begin{aligned}
T^\prime(w)&=&300kw^{-5}w^\prime+\frac{1225}{2}kw^{-9}w^\prime-\frac{2085}{4}kw^{-13}w^\prime,\nonumber\\
T^\prime(w_1)&=&300kw_1^{-5}w_1^\prime+\frac{1225}{2}kw_1^{-9}w_1^\prime-\frac{2085}{4}kw_1^{-13}w_1^\prime,\nonumber\\
X^\prime(w)&=&\frac{25}{2}w^{-9}w^\prime+\frac{75}{4}w^{-13}w^\prime,\nonumber\\
X^\prime(w_1)&=&\frac{25}{2}w_1^{-9}w_1^\prime+\frac{75}{4}w_1^{-13}w_1^\prime,\nonumber\\
U^\prime(w)&=&-300kw^{-5}w^\prime-\frac{1175}{2}kw^{-9}w^\prime+\frac{13755}{4}kw^{-13}w^\prime,\end{aligned}$$ where the derivatives are with respect to $r_h$.
Note that (\[S22\]) is complicated and one needs to resort to numerical methods. In the right panel of Fig.2, we plot $S^{(2)}/\sqrt{\lambda}$ versus $xT$ for different $\lambda$. From the figures, one can see that increasing $\lambda$ leads to smaller entropy at small distances, which means the entropic force decreases as $\lambda$ increases. In other words, decreasing $\lambda$ enhances quarkonia dissociation. Interestingly, it was argued that [@KB1] decreasing $\lambda$ leads to larger imaginary potential or smaller dissociation length, consistent with the findings here. Moreover, it follows from (\[eta1\]) that decreasing $\lambda$ leads to increasing $\eta/s$. Thus, one concludes that in the case of $R^4$ corrections the quarkonia dissociation is enhanced as $\eta/s$ increases.
However, it should be emphasized that without computing at least the next-to-leading order corrections in the ’t Hooft coupling one can not assure that it is physically meaningful to go all the way down from infinity coupling to $\lambda \sim 5.5$ by just considering $R^4$ corrections.
{width="8cm"} {width="8cm"}
conclusion
==========
The entropic force may represent a mechanism for melting the heavy quarkonia. In this paper, we studied the effects of higher derivative corrections to the entropic force and discussed how the entropic force changes with $\eta/s$ at strong coupling. It is shown that for $R^2$ corrections, increasing $\lambda_{GB}$ leads to increasing the entropic force thus enhancing the quarkonia dissociation. While for $R^4$ corrections, increasing $\lambda$ leads to decreasing the entropic force thus suppressing the quarkonia dissociation. It is found that for $R^2$ corrections the entropic force is enhanced as $\eta/s$ decreases, while for $R^4$ corrections the entropic force is enhanced as $\eta/s$ increases. Namely, $R^2$ corrections affect the entropic force in the opposite way of $R^4$ corrections. This is conceivable, because $R^2$ corrections are of different nature than $R^4$ corrections (for the origin of the two corrections, see [@ABR2; @ABR3]). In fact, a similar problem has been explained in the study of $\eta/s$ [@ABR3]. Therein, it was argued that in certain regimes of the parameter space, i.e., $\lambda=6\pi, N_c=3$, it is not unreasonable to include both $R^2$ corrections and $R^4$ corrections as making independent and comparable contributions to the CFT properties.
Certainly, one may doubt why the entropic force is the correct (and useful) approach to understand dissociation of quarkonia. Although we cannot provide a clear interpretation at present, we believe that the entropic self destruction is an intriguing idea and worth studying. Actually, in some sense one can check the effectiveness of this idea by comparing the same effect on the entropic force and with that on the imaginary potential. To our knowledge, the velocity effect [@KBF; @JN2; @MAL], the chemical potential effect [@ZQ], $R^2$ corrections [@JN1] and $R^4$ corrections [@KB1] on the two quantities give consistent results regarding the quarkonia dissociation. These agreements support that if the imaginary potential is right the entropic force may be also effective.
Finally, it should be noticed that the background considered here is a purely gravitational background with no matter fields in the bulk and no dynamical breaking of the conformal symmetry. It is of great interest to pursue in the investigations performed on top of phenomenologically realistic gauge/gravity backgrounds. We hope to report our progress in this regard in the future.
Acknowledgments
===============
This work is partly supported by the Ministry of Science and Technology of China (MSTC) under the ¡°973¡± Project No. 2015CB856904(4). Z-q Zhang is supported by the NSFC under Grant No. 11705166. D-f. Hou is supported by the NSFC under Grants Nos. 11735007, 11521064.
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|
---
abstract:
- 'The angular dependence of the nonlinear transverse magnetic moment of untwinned high-quality single crystals of YBa$_{2}$Cu$_{3}$O$_{6.95}$ has been studied at a temperature of 2.5K using a low frequency AC technique. The absence of any signature at angular period [*2*]{}${\em \pi }$[*/4* ]{}is analyzed in light of the numerical predictions of such a signal for a pure [*d*]{}$_{{\em x}^{2}\text{-}{\em y}^{2}}$ order parameter with line nodes. Implications of this null result for the existence of a non-zero gap at all angles on the Fermi surface are discussed.'
-
address:
- |
School of Physics and Astronomy, University of Minnesota, Minneapolis, MN\
55455, USA
- 'Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439.'
author:
- 'Anand Bhattacharya, Igor Zutic, O.T.Valls and A.M.Goldman'
- Ulrich Welp and Boyd Veal
date:
- 'Sept. 25, 1998'
- The Date
title:
- 'Angular Dependence of the Nonlinear Transverse Magnetic Moment of YBa$_{2}$Cu$_{3}$O$_{6.95}$ in the Meissner State.'
-
---
Measurements of the low energy quasiparticle excitation spectrum of high temperature superconductors (HTS) are crucial in elucidating the nature of the pairing state and gap function in these materials. For a [*d*]{}$%
_{x^{2}-y^{2}}$ order parameter, the gap function varies as ${\it \Delta }$($%
{\it \varphi }$) =[* * ]{}${\it \Delta }_{{\it 0}}$[*cos(2*]{}${\it \varphi }$[*)* ]{}, riding on a cylindrical Fermi surface in momentum space, with zeros or line nodes at ${\it \varphi }$[* = (*]{}${\it \pi }$[*/4 + n*]{}${\it \pi }$[*/2)*]{}. Here,[* *]{}${\it \Delta }_{{\it 0}}$ is about 25meV [@photo] for YBa$_{2}$Cu$_{3}$O$_{6.95}$ (YBCO), and ${\it \varphi }$ is the angle measured from the crystallographic [*a*]{} or [*b*]{} directions. The density of states for quasiparticle excitations near these nodes on the Fermi surface increases linearly with energy. In a two fluid model, this results in the superconducting condensate density [*n*]{}$_{s}$[*(T)*]{} decreasing linearly with increasing temperature from its zero temperature value [*n*]{}$_{s}$[*(0)*]{} [@annett]. In conventional [*s-wave* ]{}superconductors, there is a non-zero gap everywhere on the Fermi surface for quasiparticle excitations, and the depletion of the condensate is exponentially small at the lowest temperatures
The temperature dependence of the in-plane London penetration depth [@hardy93][* *]{}${\it \lambda }$[*(T)*]{} in these materials has been widely cited as evidence in support of the presence of line nodes in the gap function, though at the lowest temperatures there is some deviation from the expected linear behavior, even in the best samples [@hardy98]. Measurements of the magnetic field dependence of the low temperature specific heat have also been interpreted as evidence for the presence of nodes, but the analysis is subject to the limitations of a many-parameter fit[@spheat1]. Data from later experiments, taken over a wider range of temperature and field do not yield the same conclusions upon analysis [@spheat2]. Recent studies on the scaling of specific heat data has also been cited as evidence in support of nodes [@scaling], but the scaling works poorly, and only over a limited temperature range. Amongst experiments that provide angular information, angle resolved photemission experiments (ARPES) can be interpreted as showing a minimum in the energy gap in the (110) direction in YBCO and BSCCO [@photo; @shen], but have a resolution of only a few meV (1meV = 11.62K) and thus cannot resolve excitations or an underlying [*s-wave*]{} gap at the lowest energies. Inelastic neutron scattering shows evidence in support of minima [@neutrons1], but there too, the resolution is 1meV at best. Thus, one can safely say that the overall picture is quite ambiguous regarding the possibility of an underlying gap below 1meV.
In this Letter, we describe an experimental technique for distinguishing nodes from deep minima (quasinodes) in the order parameter, that also provides information about the angular position of these nodes or quasinodes, by probing the existence of low lying excitations in response to an applied magnetic field in the Meissner regime. Here, the kinetic energy of the superflow of the screening currents provides the energy for quasiparticle excitations. Our null result for this probe rules out the existence of nodes, and allowing for quasinodes, sets a lower bound on the size of the underlying gap.
For a type II superconductor in a magnetic field in the Meissner regime, screening supercurrents flow in a volume near the surface given approximately by the penetration depth ${\em \lambda }$. For the condensate participating in these currents, the quasiparticle excitation spectrum is modified by a semiclassical ‘Doppler shift’ to [*E(k) =* ]{}$\sqrt{(\Delta
_{k}^{2}+\epsilon _{k}^{2})}$[* +* ]{}$\widehat{{\it v}}_{s}\cdot \widehat{%
{\it v}}_{F}$ [@bardeen], where[** **]{}$\widehat{{\it v}}_{s}$ is the superfluid flow field and $\widehat{{\it v}}_{F}$ is the Fermi velocity. For a gap function with nodes, this leads to quasiparticle excitations even at zero temperature [@ys]. Due to the linear density of states near a line node, the depletion of the condensate due to quasiparticles is proportional to ($\widehat{{\it v}}_{s}\cdot \widehat{{\it v}}_{F}$)$^{2}$. These quasiparticles create a ‘backflow’ which is then responsible for a nonlinear contribution to the magnetization that goes as [*H*]{}$^{2}{\it \lambda }%
^{2} $, where $\widehat{{\it H}}$ is the applied magnetic field, and [*v*]{}$%
_{s}$[* *]{}$\sim $ [*H*]{}${\it \lambda }$. In the case of a [*d*]{}$%
_{x^{2}-y^{2}}$ order parameter, the gap function has a four-fold angular symmetry, and this gives rise to an intrinsic nonlinear [*transverse*]{} magnetization, superimposed on the nominal diamagnetic response[@zv1]. Since this effect is felt only by a fraction of the condensate within a thickness of ${\it \lambda }$ of the surface, the nonlinear magnetic moment is proportional to the surface area of the sample, and not the full sample volume. Also, since ${\it \lambda }$ ${\it \xi }$, where ${\it \xi }$ is the in-plane coherence length, this is a bulk effect.
We are interested in the nature of the gap function for supercurrents flowing within the Cu-O planes of the superconductor, which consists of layers of these planes. The YBCO crystals are flat with the [*c*]{}-axis oriented perpendicular to the crystal plane which is the [*a-b*]{} plane. The magnetic field is applied parallel to the crystal plane and the transverse magnetic moment is measured in the crystal plane perpendicular to the applied field. As the crystal is rotated in the applied magnetic field about the [*c*]{}-axis, the screening currents flow in different directions relative to the in-plane gap function, which is pinned to the crystal lattice. This leads to an angular modulation of the nonlinear transverse magnetic moment that provides [*angle-resolved*]{} information about the low lying excitations in YBCO.
Numerical calculations of the nonlinear transverse magnetic moment[* *]{}([*m*]{}$_{T}$) of a sample with a finite disk shaped geometry have been carried out for pure [*d-wave*]{} and mixed order parameters. For pure [*d-wave*]{}, the calculation predicts the amplitude of the expected four-fold modulation of [*m*]{}$_{T}$ with angular period ${\em \pi /2}$ [@zv1], a consequence of the symmetry of the gap and angular position of the nodes. The presence of a small [*s-wave*]{} component as in [*d+s*]{} changes the angular position of the line nodes, and introduces a component in the angular modulation of [*m*]{}$_{T}$ with period ${\em \pi }$ [@zv2]. However, the angular period ${\em \pi /2}$ component is not adversely affected by small additions of [*s-wave*]{}. Anisotropy of the Fermi surface and anisotropy in ${\em \lambda }_{a}$ and ${\em \lambda }_{b}$ have also been considered, and have very similar consequences. For experimentally determined values of such anisotropy[@abaniso], there is a small additional angular period ${\em \pi }$ component, but the period ${\em \pi /2%
}$ component is essentially unchanged.[* *]{}On the other hand, for a[*d-wave*]{} like order parameter, with varying levels of a non-zero quasinode, as in [*d*]{}$_{x^{2}-y^{2}}$+ i[*s*]{} and [*d*]{}$_{x^{2}-y^{2}}$ + i[*d*]{}$_{xy}$ symmetries, the amplitude of the period ${\em \pi /2}$ component is suppressed (see Fig.2 inset). In addition, to prevent the nonlinear Meissner effect (NLME) from being thermally washed out, the temperature has to be such that $\frac{T}{{\it \Delta }_{0}}$ $\frac{H}{H_{o}}$, where [*H*]{}$_{o}$= $\frac{\phi _{o}}{\pi ^{2}\lambda
\xi }$, and ${\it \phi }_{o}$ is the flux quantum. For ${\it \lambda }_{ab}$ = 1400Å and ${\it \xi }_{ab}$ = 20 Å, which are typical values, [*H*]{}$_{o}\approx $ 8000 Oe. For measurements in a field of amplitude 300 Oe, this yields $\frac{H}{H_{o}}$ = 0.0375, and requires that T $\lesssim $ 10K.
To measure [*m*]{}$_{T}$, an AC magnetic field is applied in the [*a-b*]{} plane of a mm size single crystal of YBCO. The field is modulated at 12Hz, and the transverse magnetic moment at 36Hz is detected by the transverse coil of a Quantum Design SQUID susceptometer. The signal is expected at the third harmonic because [*m*]{}$_{T}$ $\sim $ [*sgn(H)H*]{}$^{2}$. The sample is maintained at 2.5K with continuous cooling throughout the measurements. The superconducting magnet is operated with its persistent-current switch open, and the AC magnetic field is generated by driving a very low distortion (-110dB in higher harmonics) and low noise AC current through the magnet coil, with a maximum amplitude of 300 Oe. The sample is placed in the center of the magnet coil and aligned optimally with the transverse flux detection coils. The analog output from the transverse SQUID is fed directly to a phase sensitive detection system. The large 12Hz background from the fringing field of the magnet and geometric demagnetization fields from the sample are rejected by a high-pass filter section. The signal at 36Hz is then detected with a phase sensitive detector locked to [*3f*]{} of the current generator. The sample is rotated [*in situ*]{} between measurements in steps of 6$^{o}$ using a custom built sample holder that allows for [*in situ* ]{}detection of the angular position of the sample at low temperatures. This system has been described in detail elsewhere [@rsi]. It allows for precision in angular positioning of about 0.1$^{o}$ between steps and accuracy of better than 1$^{o}$ in an entire rotation. The sample holder and associated angular detection systems are designed to provide minimal magnetic background.
The detection setup has been calibrated and tested by ‘simulating’ a real magnetic moment in an environment identical to that of the actual experiment. This is done by running an AC current through a copper coil mounted on the sample stage at a frequency [*3f*]{}, referred to the frequency [*f* ]{}=12 Hz of the oscillating magnetic field, which has an amplitude of 300 Oe. The AC moment of 1.5 x 10$^{-8}$ emu rms amplitude at [*3f*]{} is detected in the presence of the large background signal at frequency [*f*]{} due to the fringing field of the magnet. This calibration was done at 2.5K, with just the bare sample holder. The measured amplitude is shown in Fig.1 as a function of the angular orientation of the coil. The angular Fourier transform of this data (inset) shows the amplitude at angular period [*2*]{}${\it \pi }$. The ‘noise floor’ in the Fourier transform of the calibration signal at higher angular frequencies corresponds to an amplitude less than 5 x 10$^{-10}$ emu.
The angular dependence of the NLME has been measured in a number of untwinned single crystals of YBCO, including a very high quality crystal (UWHC), grown in a YSZ crucible by the Argonne group. This crystal, which is 99.95% pure, has been cut into a disk shape with diameter 1.5mm, and is 67$%
\mu m$[* *]{}thick. The high quality of the crystal is borne out by a rocking curve with full width at half maximum of 0.08$^{0}$ for the 006 peak of a high resolution x-ray scan, and there were no twins observable in repeated area scans over different parts of the crystal. The superconducting transition of this crystal has an onset of 93K and a width of less than 2.5K as measured magnetically with a 10 Oe field applied in the [*a-b*]{} plane. The field of first flux entry was measured to be about 300 Oe in the [*a-b*]{} plane at 2.5K.
Measurements of [*m*]{}$_{T}$ are shown in Fig.2 as a function of the angular orientation of the sample with respect to the applied magnetic field, with the[* ‘a’*]{} direction being initially oriented along the field. The sine Fourier transform of the data is shown in the inset. For a pure [*d-wave*]{} order parameter, we are interested in the angular period [*2*]{}${\it \pi }$[*/4*]{} component of [*m*]{}$_{T},$ the predicted value for which is 1.7 x 10$^{-9}$emu according to the calculations in Ref.[@zv1]. This component is clearly in the noise of the data, below 5 x 10$%
^{-10}$ emu. This ’noise’ is in part due to residual trapped flux, and also due to the noise floor of the measurement apparatus, as has been verified in repeated measurements. This imposes an [*upper bound*]{} on the size of the nonlinear Meissner effect, which if there, is less than 30% of the predicted value. Identical measurements have also been carried out on a very high quality rectangular crystal (UBCyca) grown by the UBC group in a BaSZ crucible[@hardy98]. It has the same surface area as our disk shaped crystal (UWHC) and the results were essentially the same.
This null result has to be examined in the light of measurements of the penetration depth. Scrutiny of data on the temperature dependence of the penetration depth ${\it \lambda }$[*(T)*]{} indicates that there is always some curvature away from the pure [*d-wave*]{} result at the lowest temperatures. This is found in all published data on the cleanest bulk single crystals [@hardy93; @hardy98], and has been attributed to the presence of unitary scattering centers, that do not affect T$_{c}\cite
{zn,unitary}$. However, even with the concentration of unitary scatterers needed to produce the required curvature, the nonlinear transverse magnetic moment effect may only be reduced to about 90% of the full value [@ys], and our data indicates a much stronger suppression. This same data for ${\it %
\lambda }$[*(T)* ]{}for UBCyca [@chris] can be fit at the lowest temperatures by a model which contains quasinodes, i.e., where the $\frac{%
\Delta _{\min }}{\Delta _{o}}$ is 2.5%, ${\it \Delta }_{\min }$ being the residual gap in the nodal direction. Such a fit is significantly better at the lowest temperatures than the fit for a pure [*d-wave*]{} order parameter. The suppression of the nonlinear transverse magnetic moment in all our measurements is consistent with $\frac{\Delta _{\min }}{\Delta _{o}}$ between 2 - 3%. In the event that the ${\it \lambda }$[*(T)* ]{}and photoemision results can be explained by something other than a minimum in the gap, our experiment is also consistent with a[* d+s*]{} order parameter where $\Delta _{s}$ $\Delta _{d}.$
Recently, other experiments that have attempted [@giannetta; @hardy] to see the NLME in measurements of the field dependence of ${\it \lambda }$ at low temperatures, but have not obtained conclusive results. According to theory[@ys], for a pure [*d-wave*]{} order parameter, ${\em \lambda }$ should vary linearly with [*H*]{} at the lowest temperatures where $\frac{T}{%
{\it \Delta }_{0}}$ $\frac{H}{H_{o}}.$ These most recent measurements were done at a very high level of sensitivity, about 100 times better than an earlier effort [@maeda]. In one of these experiments done at UIUC [@giannetta], a significant field dependence in ${\em \lambda }$ was observed, but this was attributed to trapped flux as the field dependence was closer to $\sqrt{H}$and the temperature dependence was not as predicted by theory. The experimental effort at UBC[@hardy] observed a field dependence in ${\em %
\lambda },$ and also measured the effect as a function of the in-plane angular orientation of the applied field. However, they report that the field dependence measured at $\pm $ 45$^{o}$ to the crystal axes were not identical as would have been expected from theory, and the temperature and field dependence of the effect could not be understood within the current picture for the NLME [@ys].
Earlier attempts at measuring the angular dependence of the nonlinear transverse magnetic moment using a DC technique [@buan] were less sensitive by more than two orders of magnitude than the present work due to the very large linear background in a DC measurement, and the analysis overestimated the size of the effect significantly. In our experiments, trapped flux may produce a signal, but the angular dependence of this effect has the largest amplitude at angular period [*2*]{}${\it \pi }$, and is easily distinguished from the period [*2*]{}${\it \pi }$[*/4* ]{} modulation for which we are searching. The angular modulation arising from the geometric demagnetization factor is linear in the field below the field of first flux entry, and is minimized by looking at only the nonlinear component which is extracted directly in our work. Thus, artifacts due to trapped flux and geometry are both minimized by our technique.
There have been some calculations of the effects of nonlocal electrodynamics [@leggett; @hirschfeld]. These are motivated by the idea that in a BCS like picture, when the gap ${\it \Delta }_{k}$ goes to zero, then ${\it \xi }
$ = $\hbar $[*v*]{}$_{F}$/${\it \pi \Delta }_{k}$ diverges, and one might no longer be in the local limit where ${\it \xi }$ [* *]{}${\it \lambda }$ . However, it turns out that these considerations are relevant only when $\widehat{{\it H}}$[* * ]{}$\widehat{{\it c}}$. In our experiment, $\widehat{{\it H}}$ is applied in the [*a-b*]{} plane, and the volume within a depth ${\it \lambda }$ of the [*a-b*]{} plane crystal face that is responsible for the nonlinear Meissner effect is not affected by nonlocal effects. Even if we consider a ‘weakly 3D’ case [@hirschfeld], the effect is at a field scale of about 20 Oe, far too small to suppress the NLME, whose characteristic field in our measurements is 300 Oe.
In summary, in view of the results of this paper, and in light of the evidence for a predominantly [*d-wave*]{} order parameter [@squids] from other experiments, we suggest that the order parameter in YBCO may still be [*d-wave*]{} like, but have quasinodes instead of line nodes, with $\frac{%
\Delta _{\min }}{\Delta _{o}}$ between 2 - 3%. Measurements of the penetration depth to lower temperatures or photoemisssion experiments at higher resolution may confirm this.
We’d like to acknowledge R.Giannetta, J.Buan, B.P.Stojkovic, R.Klemm, and D.E.Grupp for helpful discussions. We are also extremely grateful to C.Bidinosti, D.Bonn, R.Liang, W.Hardy of the UBC group for letting us confirm our results with one of their very high quality crystals of YBCO (UBCyca), and also for freely sharing their data on $\lambda (T)$[* *]{}and [* *]{}$\lambda (H)$ for this and other crystals with us before publication. The early stages of this work were supported in part by the AFOSR under grant F49620-96-0043. One of us (A.B.) would like to acknowledge Foster Wheeler and Graduate Dissertation Fellowships from the Graduate School of the University of Minnesota.
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|
---
abstract: 'We introduce the notion of *virtual Bergman kernel* and study some of its applications.'
address: 'Nevski prospekt 113/4-53, 191024 St Petersburg, Russian Federation'
author:
- Guy Roos
date: 'August 25, 2004'
title: Weighted Bergman kernels and virtual Bergman kernels
---
[^1]
[ subjclass[32A36, 32M15]{} ifundefined[subjclassname@2000]{}[ ]{}[ xpxpsubjclassname@2000]{}]{}
Introduction {#introduction .unnumbered}
============
Let $\Omega\subset\mathbb{C}^{n}$ be a domain and $p:\Omega\rightarrow
]0,+\infty\lbrack$ a weight function on $\Omega$. Consider the inflated domains$$\begin{aligned}
\widehat{\Omega}_{1} & =\left\{ \left( z,\zeta\right) \in\Omega
\times\mathbb{C}\mid\left\vert \zeta\right\vert ^{2}<p(z)\right\} ,\\
\widehat{\Omega}_{m} & =\left\{ \left( z,Z\right) \in\Omega
\times\mathbb{C}^{m}\mid\left\Vert Z\right\Vert ^{2}<p(z)\right\} ,\end{aligned}$$ where $\left\Vert ~\right\Vert $ is the standard Hermitian norm on $\mathbb{C}^{m}$.
In our joint work [@YinRoos2003] with Yin Weiping, we computed explicitly the Bergman kernel of some egg domains; among them $\widehat{\Omega}_{1}$, when $\Omega$ is a bounded symmetric domain and $p$ a real power of the generic norm of $\Omega$. We then obtained the Bergman kernel of the corresponding $\widehat{\Omega}_{m}$ by using the *inflation principle* of [@BoasFuStraube1999], which allows to deduce (for any weight function $p$) the Bergman kernel of $\widehat{\Omega}_{m}$ from the Bergman kernel of $\widehat{\Omega}_{1}$. The inflation principle says that if the Bergman kernel of $\widehat
{\Omega}_{1}$ is [$$\widehat{\mathcal{K}}_{1}(z,\zeta)=\mathcal{L}_{1}\left( z,\left\vert
\zeta\right\vert ^{2}\right) ,$$ ]{}then the Bergman kernel of $\widehat{\Omega}_{m}$ is $$\widehat{\mathcal{K}}_{m}(z,Z)=\frac{1}{m!}\left. \frac{\partial^{m-1}}{\partial r^{m-1}}\mathcal{L}_{1}(z,r)\right\vert _{r=\left\Vert Z\right\Vert
^{2}}. \label{VB0}$$
It appears that the two previous steps can be unified in the following way. There exists a function $\mathcal{L}_{0}(z,r)$, defined in a neighborhood of $\Omega\times\{0\}$ in $\Omega\times\lbrack0,+\infty\lbrack$, such that for all $m\geq1$, the Bergman kernel of $\widehat{\Omega}_{m}$ is $$\widehat{\mathcal{K}}_{m}(z,Z)=\frac{1}{m!}\left. \frac{\partial^{m}}{\partial r^{m}}\mathcal{L}_{0}(z,r)\right\vert _{r=\left\Vert Z\right\Vert
^{2}}.$$ We call $\mathcal{L}_{0}(z,r)$ the *virtual Bergman kernel* of $\left(
\Omega,p\right) $. Its existence is closely related to the Forelli-Rudin construction ([@ForelliRudin1974], [@Ligocka1989], [@Englis2000]).
In this talk, we investigate the properties of this virtual Bergman kernel. We then show how it can be explicitly computed on bounded symmetric domains, for a special but natural choice of the weight function $p$: $p$ is taken to be a real power of the generic norm of the bounded symmetric domain. The explicit computation of the virtual Bergman kernel is then related to properties of the Hua integral.
Virtual Bergman kernels
=======================
Notations
---------
Let $V\cong\mathbb{C}^{n}$ be a Hermitian vector space, with Hermitian norm $\left\Vert ~\right\Vert _{V}$ and volume form $\omega_{V}(z)=\left(
\frac{\operatorname*{i}}{2\pi}\partial\overline{\partial}\left\Vert
z\right\Vert ^{2}\right) ^{n}$. Let $\Omega$ be a domain in $V$ and $p:\Omega\rightarrow]0,+\infty\lbrack$ a continuous function on $\Omega$. The space of holomorphic functions on $\Omega$ is denoted by $\operatorname{Hol}(\Omega)$. We denote by $H(\Omega)$ the Bergman space[$$H(\Omega)=H\left( \Omega,\omega_{V}\right) =\left\{ f\in\operatorname{Hol}(\Omega)\mid\left\Vert f\right\Vert _{\Omega}^{2}=\int_{\Omega}\left\vert
f(z)\right\vert ^{2}\omega_{V}(z)<\infty\right\}$$ and by ]{}$H(\Omega,p)$ the weighted Bergman space[$$H(\Omega,p)=H\left( \Omega,p\omega_{V}\right) =\left\{ f\in
\operatorname{Hol}(\Omega)\mid\left\Vert f\right\Vert _{\Omega,p}^{2}=\int_{\Omega}\left\vert f(z)\right\vert ^{2}p(z)\omega_{V}(z)<\infty\right\}
.$$ The Hilbert products on these spaces are denoted respectively ]{}$\left(
~\mid~\right) _{\Omega}$ and $\left( ~\mid~\right) _{\Omega,p}$. The Bergman kernel of $\Omega$ (reproducing kernel of $H(\Omega)$) is denoted by $K_{\Omega}(z,t)$; it is fully determined by $\mathcal{K}_{\Omega}$:$$\mathcal{K}_{\Omega}(z)=K_{\Omega}(z,z)\qquad\left( z\in\Omega\right) ,$$ which we call also Bergman kernel of $\Omega$. In the same way, the weighted Bergman kernel of $\left( \Omega,p\right) $ (reproducing kernel of $H(\Omega,p)$) is denoted by $K_{\Omega,p}(z,t)$ and is determined by $\mathcal{K}_{\Omega,p}(z)=K_{\Omega,p}(z,z)$.
Let $$\widehat{\Omega}_{1}=\left\{ \left( z,\zeta\right) \in\Omega\times
\mathbb{C}\mid\left| \zeta\right| ^{2}<p(z)\right\} .$$ We endow $\widehat{\Omega}_{1}$ with the volume form$$\omega_{V}(z)\wedge\omega_{1}(\zeta),$$ where $\omega_{1}(\zeta)=\frac{\operatorname*{i}}{2\pi}\partial\overline
{\partial}\left| z\right| ^{2}$. The Bergman space $H\left( \widehat
{\Omega}_{1}\right) $ will be defined with respect to this volume form.
Consider a holomorphic function $f\in\operatorname{Hol}\left( \widehat
{\Omega}_{1}\right) $; such a function may be written $$f(z,\zeta)=\sum_{k=0}^{\infty}f_{k}(z)\zeta^{k},$$ with $f_{k}\in\operatorname{Hol}\Omega$. We compute $\left\| f\right\|
_{\widehat{\Omega}_{1}}^{2}$:$$\begin{aligned}
\left\| f\right\| _{\widehat{\Omega}_{1}}^{2} & =\int_{\widehat{\Omega
}_{1}}\left| f(z,\zeta)\right| ^{2}\omega_{V}(z)\wedge\omega_{1}(\zeta)\\
& =\int_{\Omega}\omega_{V}(z)\left( \int_{\left| \zeta\right| ^{2}<p(z)}\left| f(z,\zeta)\right| ^{2}\omega_{1}(\zeta)\right) \\
& =\int_{\Omega}\omega_{V}(z)\left( \sum_{k=0}^{\infty}\left|
f_{k}(z)\right| ^{2}\frac{p^{k+1}(z)}{k+1}\right) ,\end{aligned}$$ which gives$$\left\| f\right\| _{\widehat{\Omega}_{1}}^{2}=\sum_{k=0}^{\infty}\frac
{1}{k+1}\left\| f_{k}\right\| _{\Omega,p^{k+1}}^{2}. \label{VB1}$$
Let $\widehat{\mathcal{K}}_{1}$ be the Bergman kernel of $\widehat{\Omega}_{1}$ and $\mathcal{K}_{\Omega,p^{k}}$ the weighted Bergman kernel of $\Omega$ for the weight function $p^{k}$. Then $$\widehat{\mathcal{K}}_{1}(z,\zeta)=\sum_{k=0}^{\infty}\left( k+1\right)
\mathcal{K}_{\Omega,p^{k+1}}(z)\left\vert \zeta\right\vert ^{2k}. \label{VB2}$$
For each $k\in\mathbb{N}$, let $\left( \varphi_{jk}\right) _{j\in
J_{k}}$ be a Hilbert basis (complete orthonormal system) of $H\left(
\Omega,p^{k}\right) $. Then it follows from (\[VB1\]) that $$\left( \left( k+1\right) ^{1/2}\varphi_{j,k+1}(z)\zeta^{k}\right)
_{k\in\mathbb{N},\ j\in J_{k+1}}$$ is a Hilbert basis of $H\left( \widehat{\Omega}_{1}\right) $. From the classical properties of Bergman kernels, we get $$\begin{aligned}
\mathcal{K}_{\Omega,p^{k}}(z) & =\sum_{j\in J_{k}}\left\vert \varphi
_{jk}(z)\right\vert ^{2},\\
\widehat{\mathcal{K}}_{1}(z,\zeta) & =\sum_{k\in\mathbb{N},\,j\in J_{k}}\left( k+1\right) \left\vert \varphi_{j,k+1}(z)\right\vert ^{2}\left\vert
\zeta\right\vert ^{2k}\\
& =\sum_{k=0}^{\infty}\left( k+1\right) \mathcal{K}_{\Omega,p^{k+1}}(z)\left\vert \zeta\right\vert ^{2k}.\end{aligned}$$
This leads to the following definition.
Let $\Omega$ be a domain in $V$ and $p:\Omega\rightarrow]0,+\infty\lbrack$ a continuous function on $\Omega$. Denote by $K_{\Omega,p^{k}}(z,w)$ ($\mathcal{K}_{\Omega,p^{k}}(z)$) the weighted Bergman kernel of $\Omega$ w.r. to $p^{k}$. The *virtual Bergman kernel* of $\left( \Omega,p\right) $ is defined by $$L_{\Omega,p}\left( z,w;r\right) =L_{0}\left( z,w;r\right) =\sum
_{k=0}^{\infty}K_{\Omega,p^{k}}(z,w)r^{k}. \label{VB3'}$$ The function $\mathcal{L}_{0}(z,r)=L_{0}(z,z;r)$, i.e.$$\mathcal{L}_{\Omega,p}\left( z,r\right) =\mathcal{L}_{0}\left( z,r\right)
=\sum_{k=0}^{\infty}\mathcal{K}_{\Omega,p^{k}}(z)r^{k} \label{VB3}$$ will also be called virtual Bergman kernel of $\left( \Omega,p\right) $.
With these definitions, the relation (\[VB2\]) may be rewritten and the reproducing kernel of $\widehat{\Omega}_{1}$ is given by $$\begin{aligned}
\widehat{K}_{1}\left( \left( z,\zeta\right) ,\left( w,\eta\right)
\right) & =L_{1}\left( z,w;\zeta\overline{\eta}\right) ,\label{VB4'}\\
\widehat{\mathcal{K}}_{1}(z,\zeta) & =\mathcal{L}_{1}\left( z,\left\vert
\zeta\right\vert ^{2}\right) , \label{VB4}$$ with$$\begin{aligned}
L_{1}(z,w;r) & =\frac{\partial}{\partial r}L_{0}\left( z,w;r\right) ,\\
\mathcal{L}_{1}(z,r) & =\frac{\partial}{\partial r}\mathcal{L}_{0}\left(
z,r\right) .\end{aligned}$$
From the virtual Bergman kernel of $\left( \Omega,p\right) $, it is easy to recover the weighted Bergman kernels of $\left( \Omega,p^{k}\right) $ ($k\in\mathbb{N}$):$$\mathcal{K}_{\Omega,p^{k}}(z)=\frac{1}{k!}\left. \frac{\partial^{k}}{\partial
r^{k}}\mathcal{L}_{\Omega,p}\left( z,r\right) \right\vert _{r=0}.
\label{VB13}$$
Let us recall some facts about harmonic analysis in the Hermitian unit ball $B_{m}$. Let $H\left( B_{m}\right) =H\left( B_{m},\omega_{m}\right) $ be the Bergman space of $B_{m}$; let $K_{B_{m}}(Z,T)$, $\mathcal{K}_{B_{m}}(Z)=K_{B_{m}}(Z,Z)$ be the Bergman kernel. It is well known (using for instance the automorphisms of $B_{m}$) that $$\mathcal{K}_{B_{m}}(Z)=\frac{1}{\left( 1-\left\| Z\right\| ^{2}\right)
^{m+1}}. \label{VB6}$$ This may also be written$$\mathcal{K}_{B_{m}}(Z)=\frac{1}{m!}\left. \frac{\partial^{m}}{\partial r^{m}}\left( \frac{1}{1-r}\right) \right| _{r=\left\| Z\right\| ^{2}}.
\label{VB8}$$
Let $f\in H\left( B_{m}\right) $; then $f$ may be written$$f(Z)=\sum_{k=0}^{\infty}f_{k}(Z), \label{VB11}$$ where the $f_{k}$ are $k$-homogeneous polynomials, which can be obtained through$$f_{k}(Z)=\int_{0}^{1}f\left( \operatorname*{e}\nolimits^{2\pi
\operatorname*{i}\theta}Z\right) \operatorname*{e}\nolimits^{-2\pi
\operatorname*{i}k\theta}\operatorname{d}\theta.$$ The expansion (\[VB11\]) converges uniformly on each compact of $B_{m}$.
For $k\neq\ell$, $f_{k}$ and $f_{\ell}$ are orthogonal in $H\left(
B_{m}\right) $. This implies that $H\left( B_{m}\right) $ is the Hilbert direct sum$$H\left( B_{m}\right) =\widehat{\bigoplus}_{k\geq0}H_{k}\left( B_{m}\right)
,$$ where $H_{k}\left( B_{m}\right) $ is the space of $k$-homogeneous polynomials, endowed with the scalar product of $H\left( B_{m}\right) $. For each $k\in\mathbb{N}$, let $\left( \phi_{k,j}\right) _{j\in J(k)}$ be an orthonormal basis of $H_{k}\left( B_{m}\right) $; then$$\left( \phi_{k,j}\right) _{k\in\mathbb{N},j\in J(k)}$$ is a Hilbert basis of $H\left( B_{m}\right) $ and $$\mathcal{K}_{B_{m}}(Z)=\sum_{k\in\mathbb{N},j\in J(k)}\left| \phi
_{k,j}(Z)\right| ^{2}=\sum_{k\geq0}\mathcal{K}_{B_{m},k}(Z), \label{VB9}$$ where$$\mathcal{K}_{B_{m},k}(Z)=\sum_{j\in J(k)}\left| \phi_{k,j}(Z)\right| ^{2}$$ is the reproducing kernel of $H_{k}\left( B_{m}\right) $. The expansions (\[VB9\]) also converge uniformly on compact subsets of $B_{m}$. Clearly, $\mathcal{K}_{B_{m},k}$ is a real polynomial, homogeneous of bidegree $\left(
k,k\right) $. Comparing with the expansion of (\[VB8\])$$\mathcal{K}_{B_{m}}(Z)=\sum_{k=0}^{\infty}\binom{k+m}{m}\left\| Z\right\|
^{2k},$$ we conclude that $$\mathcal{K}_{B_{m},k}(Z)=\binom{k+m}{m}\left\| Z\right\| ^{2k}. \label{VB7}$$
More generally, consider the Hermitian ball $B_{m}(\rho)$ of radius $\rho$. Its Bergman kernel w.r. to *the same* $\omega_{m}$ is $$\mathcal{K}_{B_{m}(\rho)}(Z)=\frac{1}{\rho^{2m}}\mathcal{K}_{B_{m}}\left(
\frac{Z}{\rho}\right) ;$$ the component of bidegree $\left( k,k\right) $ is $$\mathcal{K}_{B_{m}(\rho),k}(Z)=\frac{1}{\rho^{2m+2k}}\binom{k+m}{m}\left\|
Z\right\| ^{2k}. \label{VB7'}$$ If $f\in H\left( B_{m}(\rho)\right) $, its component of degree $k$ is then given by $$f_{k}(Z)=\int_{B_{m}(\rho)}\frac{1}{\rho^{2m+2k}}\binom{k+m}{m}\left\langle
Z,W\right\rangle ^{k}f(W)\omega_{V}(W). \label{VB10}$$
Now we show that the virtual Bergman kernel of $\left( \Omega,p\right) $ allows us to compute the Bergman kernel of any inflated domain $$\widehat{\Omega}_{m}=\left\{ \left( z,Z\right) \in\Omega\times
\mathbb{C}^{m}\mid\left\| Z\right\| ^{2}<p(z)\right\} .$$ Here $\widehat{\Omega}_{m}$ is endowed with the volume form $$\omega_{V}(z)\wedge\omega_{m}(Z),$$ where $\omega_{m}(Z)=\left( \frac{\operatorname*{i}}{2\pi}\partial
\overline{\partial}\left\| Z\right\| ^{2}\right) ^{m}$.
\[TH1\]The Bergman kernel $\widehat{K}_{m}$ ($\widehat{\mathcal{K}}_{m}$) of $\widehat{\Omega}_{m}$ is $$\begin{aligned}
\widehat{K}_{m}\left( (z,Z),(w,W)\right) & =L_{m}\left( z,w;\left\langle
Z,W\right\rangle \right) ,\label{VB5'}\\
\widehat{\mathcal{K}}_{m}(z,Z) & =\mathcal{L}_{m}\left( z,\left\Vert
Z\right\Vert ^{2}\right) ,\quad\label{VB5}$$ where $$\begin{aligned}
L_{m}(z,w;r) & =\frac{1}{m!}\frac{\partial^{m}}{\partial r^{m}}L_{0}(z,w;r),\label{VB12'}\\
\mathcal{L}_{m}(z,r) & =\frac{1}{m!}\frac{\partial^{m}}{\partial r^{m}}\mathcal{L}_{0}(z,r). \label{VB12}$$
Note that the relation between $\widehat{\mathcal{K}}_{1}$ and $\widehat
{\mathcal{K}}_{m}$, deduced from (\[VB5\]) and (\[VB12\]), is nothing else that the inflation principle (\[VB0\]).
We have $$L_{m}\left( z,w;r\right) =\sum_{k=0}^{\infty}\binom{k+m}{m}K_{\Omega
,p^{k+m}}(z,w)r^{k}.$$ So we want to prove that the Bergman kernel of $\widehat{\Omega}_{m}$ is $$\widehat{K}_{m}\left( (z,Z),(w,W)\right) =\sum_{k=0}^{\infty}\binom{k+m}{m}K_{\Omega,p^{k+m}}(z,w)\left\langle Z,W\right\rangle ^{k}.$$
Let $H_{k}\left( \widehat{\Omega}_{m}\right) $ be the subspace of functions $f(z,Z)$ in $H\left( \widehat{\Omega}_{m}\right) $, which are $k$-homogeneous polynomial w.r. to the variable $Z$. For $k\neq\ell$, $f\in
H_{k}\left( \widehat{\Omega}_{m}\right) $, $g\in H_{\ell}\left(
\widehat{\Omega}_{m}\right) $, we have $$\begin{aligned}
\left( f\mid g\right) _{\widehat{\Omega}_{m}} & =\int_{\widehat{\Omega
}_{m}}f(w,W)\overline{g(w,W)}\omega_{V}(w)\wedge\omega_{m}(W)\\
& =\int_{w\in\Omega}\left( f(w,\ )\mid g(w,\ )\right) _{B_{m}\left(
p(w)^{1/2}\right) }\omega_{V}(w)=0.\end{aligned}$$ This implies that $H\left( \widehat{\Omega}_{m}\right) $ is the Hilbert direct sum $$H\left( \widehat{\Omega}_{m}\right) =\widehat{\bigoplus_{k\geq0}}H_{k}\left( \widehat{\Omega}_{m}\right) . \label{VB16}$$
Fix $k\in\mathbb{N}$. Let $f\in H_{k}\left( \widehat{\Omega}_{m}\right) $. For almost all $w\in\Omega$, the function $W\mapsto f(w,W)$ belongs to $H\left( B_{m}\left( p(w)^{1/2}\right) \right) $; by (\[VB10\]),$$p(w)^{m+k}f(w,Z)=\int_{\left\Vert W\right\Vert ^{2}<p(w)}\binom{k+m}{m}\left\langle Z,W\right\rangle ^{k}f(w,W)\omega_{m}(W).$$ By the reproducing property of $K_{\Omega,p^{k+m}}$, we have$$f\left( z,Z\right) =\int_{w\in\Omega}K_{\Omega,p^{k+m}}(z,w)p(w)^{m+k}f(w,Z)\omega_{V}(w).$$ These relations imply$$f(z,Z)=\int_{\widehat{\Omega}_{m}}K_{\Omega,p^{k+m}}(z,w)\binom{k+m}{m}\left\langle Z,W\right\rangle ^{k}f(w,W)\omega_{V}(w)\wedge\omega_{m}(W),$$ which means that $$K_{\Omega,p^{k+m}}(z,w)\binom{k+m}{m}\left\langle Z,W\right\rangle ^{k}$$ is the reproducing kernel of $H_{k}\left( \widehat{\Omega}_{m}\right) $.
From (\[VB16\]), we deduce that $$\sum_{k=0}^{\infty}K_{\Omega,p^{k+m}}(z,w)\binom{k+m}{m}\left\langle
Z,W\right\rangle ^{k}$$ is the reproducing kernel of $H\left( \widehat{\Omega}_{m}\right) $.
Virtual Bergman kernels for bounded symmetric domains
=====================================================
In this section, we compute the virtual Bergman kernel $\mathcal{L}_{\Omega
,p}\left( z,r\right) =\mathcal{L}_{0}\left( z,r\right) $ when $\Omega$ is an irreducible bounded circled symmetric domain and $p$ is a power of the generic norm of $\Omega$.
Let $V$ be a complex finite-dimensional vector space and $\Omega\subset V$ an irreducible bounded circled symmetric domain. Then $V$ is endowed with a canonical structure of *positive Hermitian Jordan triple*. The numerical invariants of $V$ (or of $\Omega$) are the *rank* $r$ and the *multiplicities* $a$ and $b$ ($b=0$ iff the domain is of tube type); the *genus* is defined by $$g=2+a(r-1)+b.$$ The *generic minimal polynomial* $m\left( T,x,y\right) $ and the *generic norm* $N(x,y) $ of $\Omega$ are written as $$\begin{aligned}
m\left( T,x,y\right) & =T^{r}-T^{r-1}m_{1}(x,y)+\cdots+(-1)^{r}m_{r}(x,y),\\
N\left( x,y\right) & =m\left(
1,x,y\right)=1-m_{1}(x,y)+\cdots+(-1)^{r}m_{r}(x,y),\end{aligned}$$ where $m_{1},\ldots,m_{r}$ are polynomials on $V\times\overline{V}$, homogeneous of bidegrees $\left( 1,1\right) ,\ldots,$ $\left( r,r\right) $ respectively. In particular, $m_{1}$ is an Hermitian inner product on $V$; we endow $V$ with the Kähler form $$\alpha(z)=\frac{\operatorname{i}}{2\pi}\partial\overline{\partial}m_{1}(z,z)$$ and with the volume form $$\omega=\alpha^{n},$$ where $n$ is the complex dimension of $V$.
(See [@YinRoos2003] for a review of the above properties).
The *Bergman kernel* of $\Omega$ is then $$\mathcal{K}(z)=\frac{1}{\operatorname{vol}\Omega}\frac{1}{N\left( z,z\right)
^{g}},$$ with $\operatorname{vol}\Omega=\int_{\Omega}\omega$.
More generally, consider the *weighted Bergman space* of $\Omega$ with respect to a power of the generic norm:$$H^{(\mu)}(\Omega)=\left\{ f\in\operatorname{Hol}\Omega\mid\int_{\Omega
}\left\vert f(z)\right\vert ^{2}N(z,z)^{\mu}\omega(z)<\infty\right\} .$$ For $\mu>-1$, the space $H^{(\mu)}(\Omega)$ is non-zero and is a Hilbert space of holomorphic functions. Its reproducing kernel is $$\mathcal{K}^{(\mu)}(z)=\frac{1}{\int_{\Omega}N(z,z)^{\mu}\omega(z)}N\left(
z,z\right) ^{-g-\mu}. \label{WeightBK}$$
The denominator $\int_{\Omega}N(z,z)^{\mu}\omega(z)$ of the previous formula is called the *Hua integral*. It has been computed for the four series of classical domains (with different normalizations of the volume element) by Hua L.K. [@Hua1963].
\[TH2\][@YinRoos2003]Let $\Omega$ be an irreducible bounded circled symmetric domain. The value of the Hua integral is given by$$\int_{\Omega}N(z,z)^{s}\omega(z)=\frac{\chi(0)}{\chi(s)}\int_{\Omega}\omega,
\label{HuaInt}$$ where $\chi$ is the polynomial of degree $n=\dim_{\mathbb{C}}\Omega$, related to the numerical invariants of $\Omega$ by $$\chi(s)=\prod\limits_{j=1}^{r}\left( s+1+(j-1)\frac{a}{2}\right) _{1+b+(r-j)a}. \label{HuaPol}$$
Here $\left( s\right) _{k}$ denotes the *raising factorial*$$(s)_{k}=s(s+1)\cdots(s+k-1)=\frac{\Gamma(s+k)}{\Gamma(s)}.$$
The proof uses the polar decomposition in positive Hermitian Jordan triples (which generalizes the polar decomposition of matrices) and the following generalization, due to Selberg [@Selberg1944], of the Beta integral:$$\begin{aligned}
& \int_{0}^{1}\cdots\int_{0}^{1}{\displaystyle\prod\limits_{j=1}^{n}}
t_{j}^{x-1}\left( 1-t_{j}\right) ^{y-1}{\displaystyle\prod\limits_{1\leq j<k\leq n}} \left\vert
t_{j}-t_{k}\right\vert ^{2z}\operatorname{d}t_{1}\cdots
\operatorname{d}t_{n}\nonumber\\
& ={\displaystyle\prod\limits_{j=1}^{n}}
\frac{\Gamma(x+(j-1)z)\Gamma(y+(j-1)z)\Gamma(jz+1)}{\Gamma(x+y+(n+j-2)z)\Gamma
(z+1)},\end{aligned}$$ for $\operatorname{Re}x>0$, $\operatorname{Re}y>0$, $\operatorname{Re}z>\min\left( \frac{1}{n},\frac{\operatorname{Re}x}{n-1},\frac
{\operatorname{Re}y}{n-1}\right) $.
The Bergman kernel of $H^{(\mu)}(\Omega)$ is $$\mathcal{K}^{(\mu)}(z)=\frac{\chi(\mu)}{\chi(0)}N\left( z,z\right) ^{-\mu
}\mathcal{K}(z), \label{WeightBK2}$$ where $\mathcal{K}=\mathcal{K}^{(0)}$ is the Bergman kernel of $\Omega$ .
\[TH3\]Let $\Omega$ be an irreducible bounded circled symmetric domain. The virtual Bergman kernel of $\left( \Omega,N(z,z)^{\mu}\right) $ is $$\mathcal{L}^{(\mu)}\left( z,r\right) =\mathcal{K}(z)F_{\chi,\mu}\left(
\frac{r}{N(z,z)^{\mu}}\right) , \label{VBKern}$$ where $\mathcal{K}$ is the Bergman kernel of $\Omega$, $\chi$ the polynomial defined by (\[HuaPol\]) and $F_{\chi,\mu}$ is the rational function $$F_{\chi,\mu}(t)=\frac{1}{\chi(0)}{\displaystyle\sum\limits_{k=0}^{\infty}}
\chi(\mu k)t^{k}. \label{VBRat}$$
The proof of (\[VBKern\]) is straightforward, using (\[VB3\]) and (\[WeightBK2\]):$$\begin{aligned}
\mathcal{L}^{(\mu )}\left( z,r\right) & =\sum_{k=0}^{\infty}\mathcal{K}^{(k\mu)}(z)r^{k}\\
& =\sum_{k=0}^{\infty}\frac{\chi(k\mu)}{\chi(0)}N\left( z,z\right) ^{-k\mu
}\mathcal{K}(z)r^{k}\\
& =\frac{\mathcal{K}(z)}{\chi(0)}\sum_{k=0}^{\infty}\chi(k\mu)\left(
\frac{r}{N(z,z)^{\mu}}\right) ^{k}.\end{aligned}$$
If the polynomial $k\mapsto\chi(k\mu)$ is decomposed as $$\frac{\chi(k\mu)}{\chi(0)}=\sum_{j=0}^{n}c_{\mu,j}\frac{\left( k+1\right)
_{j}}{j!}, \label{VBComb}$$ the function defined by (\[VBRat\]) is $$F_{\chi,\mu}(t)=\sum_{j=0}^{n}c_{\mu,j}\left( \frac{1}{1-t}\right) ^{j}.$$
Tables for bounded symmetric domains
====================================
Hereunder we give the results for each type of irreducible bounded symmetric domains, which allow the reader to apply Theorems \[TH1\] and \[TH3\] to special cases.
Classification of irreducible circled bounded symmetric domains
---------------------------------------------------------------
Here is the complete list of irreducible circled bounded symmetric domains, up to linear isomorphisms. There is some overlapping between the four infinite families, due to a finite number of isomorphisms in low dimemsions.
### *Type I*$_{m,n}$ ** $\left( 1\leq m\leq
n\right) $ {#type-i_mn-left-1leq-mleqnright .unnumbered}
$V=\mathcal{M}_{m,n}(\mathbb{C})$ (space of $m\times n$ matrices with complex entries). $$\Omega=\left\{ x\in V\mid I_{m}-x^{t}\overline{x}\gg0\right\} \mathit{.}$$
### *Type II*$_{n}$ ** $\left( n\geq2\right) $ {#type-ii_n-left-ngeq2right .unnumbered}
$V=\mathcal{A}_{n}(\mathbb{C})$ (space of $n\times n$ alternating matrices). $$\Omega=\left\{ x\in V\mid I_{n}+x\overline{x}\gg0\right\} .$$
### *Type III*$_{n}$ ** $\left( n\geq1\right) $ {#type-iii_n-left-ngeq1right .unnumbered}
$V=\mathcal{S}_{n}(\mathbb{C})$ (space of $n\times n$ symmetric matrices). $$\Omega=\left\{ x\in V\mid I_{n}-x\overline{x}\gg0\right\} .$$
### *Type IV*$_{n}$ ** $\left( n\neq2\right) $ {#type-iv_n-left-nneq2right .unnumbered}
$V=\mathbb{C}^{n}$, $q(x)=\sum x_{i}^{2},$ $q(x,y)=2\sum x_{i}y_{i}$. The domain $\Omega$ is defined by $$1-q(x,\overline{x})+\left\vert q(x)\right\vert ^{2}>0,\quad2-q(x,\overline
{x})>0.$$
### *Type V* {#type-v .unnumbered}
$V=\mathcal{M}_{2,1}(\mathbb{O}_{\mathbb{C}})\simeq\mathbb{C}^{16}$, exceptional type.
### *Type VI* {#type-vi .unnumbered}
$V=\mathcal{H}_{3}(\mathbb{O}_{\mathbb{C}})\simeq\mathbb{C}^{27}$, exceptional type.
Numerical invariants
--------------------
### *Type I*$_{m,n}$ ** $\left( 1\leq m\leq
n\right) $ {#type-i_mn-left-1leq-mleqnright-1 .unnumbered}
$$r=m,\quad a=2,\quad b=n-m,\quad g=m+n.$$
### *Type II*$_{2p}$ $(p\geq1)$ {#type-ii_2p-pgeq1 .unnumbered}
$$r=\frac{n}{2}=p,\quad a=4,\quad b=0,\quad g=2\left( n-1\right) .$$
### *Type II*$_{2p+1}$ $(p\geq1)$ {#type-ii_2p1-pgeq1 .unnumbered}
$$r=\left[ \frac{n}{2}\right] =p,\quad a=4,\quad b=2,\quad g=2(n-1).$$
### *Type III*$_{n}$ ** $\left( n\geq1\right) $ {#type-iii_n-left-ngeq1right-1 .unnumbered}
$$r=n,\quad a=1,\quad b=0,\quad g=n+1.$$
### *Type IV*$_{n}$ ** $\left( n\neq2\right) $ {#type-iv_n-left-nneq2right-1 .unnumbered}
$$r=2,\quad a=n-2,\quad b=0,\quad g=n.$$
### *Type V* {#type-v-1 .unnumbered}
$$r=2,\quad a=6,\quad b=4,\quad g=12.$$
### *Type VI* {#type-vi-1 .unnumbered}
$$r=3,\quad a=8,\quad b=0,\quad g=18.$$
Generic norm
------------
### *Type I*$_{m,n}$ ** $\left( 1\leq m\leq
n\right) $ {#type-i_mn-left-1leq-mleqnright-2 .unnumbered}
$V=\mathcal{M}_{m,n}(\mathbb{C})$ (space of $m\times n$ matrices with complex entries). $$N(x,y)=\operatorname*{Det}(I_{m}-x^{t}\overline{y}).$$
### *Type II*$_{n}$ ** $\left( n\geq2\right) $ {#type-ii_n-left-ngeq2right-1 .unnumbered}
$V=\mathcal{A}_{2p}(\mathbb{C})$ (space of $n\times n$ alternating matrices). $$N(x,y)^{2}=\operatorname*{Det}(I_{n}+x\overline{y}).$$
### *Type III*$_{n}$ ** $\left( n\geq1\right) $ {#type-iii_n-left-ngeq1right-2 .unnumbered}
$V=\mathcal{S}_{n}(\mathbb{C})$ (space of $n\times n$ symmetric matrices). $$N(x,y)=\operatorname*{Det}(I_{n}-x\overline{y}).$$
### *Type IV*$_{n}$ ** $\left( n\neq2\right) $ {#type-iv_n-left-nneq2right-2 .unnumbered}
$V=\mathbb{C}^{n}$. $$N(x,y)=1-q(x,\overline{y})+q(x)q(\overline{y}).$$
### *Type V* {#type-v-2 .unnumbered}
$V=\mathcal{M}_{2,1}(\mathbb{O}_{\mathbb{C}})$.$$N(x,y)=1-(x|y)+(x^{\sharp}|y^{\sharp}).$$
### *Type VI* {#type-vi-2 .unnumbered}
$V=\mathcal{H}_{3}(\mathbb{O}_{\mathbb{C}})$. $$N(x,y)=1-(x|y)+(x^{\sharp}|y^{\sharp})-\det x\det\overline{y}.$$
The polynomial $\chi$ for the Hua integral\[HuaPolList\]
--------------------------------------------------------
Recall that for $s>-1$, $$\chi(s)\int_{\Omega}N(x,x)^{s}\omega=\chi(0)\int_{\Omega}\omega.$$
### *Type I*$_{m,n}$ {#type-i_mn .unnumbered}
$${\chi(s)=\prod_{j=1}^{m}(s+j)_{n}.}$$
### *Type II*$_{2p}$ {#type-ii_2p .unnumbered}
$${\chi(s)=\prod_{j=1}^{p}(s+2j-1)_{2p-1}.}$$
### *Type II*$_{2p+1}$ {#type-ii_2p1 .unnumbered}
$${\chi(s)=\prod_{j=1}^{p}(s+2j-1)_{2p+1}.}$$
### *Type III*$_{n}$ {#type-iii_n .unnumbered}
$${\chi(s)=\prod_{j=1}^{n}\left( s+\frac{j+1}{2}\right) _{1+n-j}.}$$
### *Type IV*$_{n}$ {#type-iv_n .unnumbered}
$${\chi(s)=\left( s+1\right) _{n-1}\left( s+\frac{n}{2}\right) .}$$
### *Type V* {#type-v-3 .unnumbered}
$${\chi(s)=(s+1)_{8}(s+4)_{8}.}$$
### *Type VI* {#type-vi-3 .unnumbered}
$${\chi(s)=(s+1)_{9}(s+5)_{9}(s+9)_{9}.}$$
Open problems
=============
Understand the rational function$$F_{\chi,\mu}(t)=\frac{1}{\chi(0)}{\displaystyle\sum\limits_{k=0}^{\infty}}
\chi(\mu k)t^{k}=\sum_{j=0}^{n}c_{\mu,j}\left( \frac{1}{1-t}\right) ^{j},$$ when $\chi$ is a polynomial from the list of subsection \[HuaPolList\]. Recall that the coefficients $c_{\mu,j}$ are given by $$\frac{\chi(k\mu)}{\chi(0)}=\sum_{j=0}^{n}c_{\mu,j}\frac{\left( k+1\right)
_{j}}{j!}.$$
The virtual Bergman kernel $$\mathcal{L}_{\Omega,p}\left( z,r\right) =\sum_{k=0}^{\infty}\mathcal{K}_{\Omega,p^{k}}(z)r^{k}$$ is suitable for the computation of the Bergman kernel of inflated domains by Hermitian balls$$\widehat{\Omega}_{m}=\left\{ \left( z,Z\right) \in\Omega\times
\mathbb{C}^{m}\mid\left\Vert Z\right\Vert ^{2}<p(z)\right\} ,$$ which may also be written$$\widehat{\Omega}_{m}=\left\{ \left( z,Z\right) \in\Omega\times
\mathbb{C}^{m}\mid Z\in\left( p(z)\right) ^{1/2}B_{m}\right\} .$$
If $F$ is any circled domain in $\mathbb{C}^{m}$, what can be said about the Bergman kernel of $$\widehat{\Omega}_{F}=\left\{ \left( z,Z\right) \in\Omega\times
\mathbb{C}^{m}\mid Z\in\left( p(z)\right) ^{1/2}F\right\} ,$$ for example when $\Omega$ is a bounded symmetric domain, $p=N_{\Omega
}(z,z)^{\mu}$ and $F$ another bounded symmetric domain?
For some families $\left\{ F\right\} $ other than the family $\left\{
B_{m}\right\} $ of Hermitian balls, is it possible to define an analogous of the virtual Bergman kernel $\mathcal{L}_{\Omega,p}$ and obtain an analogous of Theorem \[TH1\]?
It would also be interesting to replace the weighted Bergman space $H(\Omega,p)$ by more general Hilbert spaces of holomorphic functions; for example, when $\Omega$ is a bounded symmetric domain, the spaces with reproducing kernel $N(z,z)^{\mu}$, where $\mu$ is in the Berezin-Wallach set of $\Omega$ (see [@FarautKoranyi1990]).
[99]{} Yin Weiping, Lu Keping, Roos Guy, New classes of domains with explicit Bergman kernel, *Science in China* Ser. A Mathematics, **47**(3) (2004), 352–371.
Boas H., Fu Siqi, Straube E., The Bergman kernel function: explicit formulas and zeroes, *Proc. Amer. Math. Soc.*, **127**(3) (1999), 805–811.
Forelli F., Rudin W., Projections on spaces of holomorphic functions in balls, *Indiana Univ. Math. J.*, **24**(1974), 593–602.
Ligocka E., On the Forelli-Rudin construction and weighted Bergman projections, *Studia Math.*, **94**(1989), 257-272.
Engliš M., A Forelli-Rudin construction and asymptotics of weighted Bergman kernels, *J. Funct. Analysis*, **177**(2000), 257–281.
Hua L.K., *Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains*, Amer. Math. Soc., Providence, RI, 1963.
Selberg A., Bemerkninger om et multiplet integral, *Norske Mat. Tidsskr.,* **26**(1944), 71–78.
Faraut J., Korányi A., Function spaces and reproducing kernels on bounded symmetric domains. *J. Funct. Anal.* **88**(1990), 64–89.
Faraut J., Korányi A., *Analysis on Symmetric Cones*, Clarendon Press, Oxford, 1994.
Helgason S., *Differential Geometry, Lie Groups, and Symmetric Spaces*, Academic Press, New York, 1978.
Korányi A., Function spaces on bounded symmetric domains, pp. 183–282, in *J.Faraut, S.Kaneyuki, A.Korányi, Q.-k.Lu, G.Roos, Analysis and Geometry on Complex Homogeneous Domains,* Progress in Mathematics, Birkhäuser, Boston, 1999.
Loos, Ottmar, *Jordan Pairs,* Lecture Notes in Mathematics, **460**, Springer-Verlag, Berlin-Heidelberg-New York, 1975.
Loos, Ottmar,* Bounded symmetric domains and Jordan pairs*, Math. Lectures, Univ. of California, Irvine, 1977.
Roos, Guy, Algèbres de composition, Systèmes triples de Jordan exceptionnels, pp. 1–84, in *G.Roos, J.P.Vigué, Systèmes triples de Jordan et domaines symétriques*, Travaux en cours, **43**, Hermann, Paris, 1992.
Roos, Guy, Jordan triple systems, pp. 425–534, in *J.Faraut, S.Kaneyuki, A.Korányi, Q.-k.Lu, G.Roos, Analysis and Geometry on Complex Homogeneous Domains,* Progress in Mathematics, Birkhäuser, Boston, 1999.
Satake I., *Algebraic Structures of Symmetric Domains*, Iwanami Shoten, Tokyo, and Princeton Univ. Press, Princeton, NJ, 1980.
[^1]: One hour lecture for graduate students, SCV2004, Beijing
|
---
abstract: 'We study the growth of a directed transportation network, such as a food web, in which links carry resources. We propose a growth process in which new nodes (or species) preferentially attach to existing nodes with high indegree (in food-web language, number of prey) and low outdegree (or number of predators). This scheme, which we call [ *inverse preferential attachment*]{}, is intended to maximize the amount of resources available to each new node. We show that the outdegree (predator) distribution decays at least exponentially fast for large outdegree and is continuously tunable between an exponential distribution and a delta function. The indegree (prey) distribution is poissonian in the large-network limit.'
author:
- 'Volkan Sevim$^{1}$'
- 'Per Arne Rikvold$^{1,2}$'
title: Network Growth with Preferential Attachment for High Indegree and Low Outdegree
---
Introduction
============
Directed networks that transport a resource, such as energy, from one or several sources to a large number of consumers are important in many areas of science [@Banavar:1999]. Among such networks, food webs provide an example of great interest, both from a purely scientific point of view, and because of their importance for nature-conservation efforts [@Rooney2006]. Although knowledge is rapidly accumulating about the [*structure*]{} of food webs [@Rooney2006; @CommunityFoodWebs; @Camacho:2002B; @Camacho:2002A; @Stouffer:2005; @Camacho:2006], and static models have been developed to describe some of their statistical properties [@CommunityFoodWebs; @Camacho:2002B; @Camacho:2002A; @Martinez:2000], we are as yet only beginning to develop an understanding of the processes by which such networks are formed and evolve under the influence of speciation, invasion, and extinction of interacting species [@DROS01B; @DROS04; @Quince2005; @Rossberg:2006; @Rikvold:2007; @Rikvold:2007b].
An important aspect of the network structure of food webs is that they have degree distributions that generally decay quite fast with increasing degree – in most cases at least exponentially [@Camacho:2002B; @Camacho:2002A; @Martinez:2000; @Rikvold:2007]. This is in sharp contrast to the class of networks known as scale-free, which have power-law degree distributions [@Barabasi:1999; @Barabasi:2002]. While there has been a veritable explosion of research on scale-free networks, there has been no similar surge of interest in networks with rapidly convergent degree distributions. Most food webs, and some (but not all [@Guimera:2004]) transportation networks, such as the North American power grid [@Albert:2004], and the European railway network [@Kurant:2006] belong to this class. Much work remains to be done before a comprehensive understanding of the mechanisms by which such networks evolve is reached.
As a step toward the development of such an understanding, we here propose a network growth scheme that produces a poissonian indegree distribution (in food-web language: prey distribution) and an outdegree (predator) distribution that is continuously tunable between an exponential distribution and a delta function. We note that these degree distributions do not agree with current food-web theory. In particular, the indegree distribution produced by the niche model [@Martinez:2000] has an exponential tail [@Camacho:2002B]. It has been claimed that models with an exponentially decaying probability of preying on a given fraction of species with lower or equal niche values are capable of producing food webs that are structurally in agreement with empirical data [@Stouffer:2005]. However, it is not clear why this condition is necessary or why schemes that invoke no physical mechanisms (as in the niche model) are able to produce such webs. Therefore, other plausible schemes should also be explored.
Our model employs a scheme in which new nodes (species) attach to existing nodes with a preference for nodes $i$ with high indegree $k'_i$ and low outdegree $k_i$. In food-web terms, this corresponds to a prospective predator choosing prey that have a large number of resources (represented by the large indegree), while the competition from previously established predators should be as small as possible (low outdegree). Among the influences on the network growth process mentioned above (speciation, invasion, and extinction), we have thus chosen to focus on invasion and/or speciation. By ignoring extinction, we essentially model the early phase of steady network growth. The proposed growth process corresponds to a probability of attachment, $$\Pi(k'_{i},k_{i})\propto(k'_{i}/k_{i})^{\gamma}
\label{eq:one}$$ with $\gamma \ge 0$. This attachment scheme is the direct opposite of the “rich get richer" scheme of preferential attachment with an attachment probability proportional to the total degree $l_i = k'_i+k_i$, which is known to produce scale-free networks and has been studied in a myriad of variations over the last decade [@Barabasi:1999; @Barabasi:2002]. To emphasize this difference, we shall call the scheme proposed here [*inverse preferential attachment*]{}. \[We note that nonlinear forms of the “rich get richer" scheme with a probability of attachment proportional to $l_{i}^{\alpha}$ with $\alpha>0$ have also been studied. However, except for $\alpha=1$, no $\alpha>0$ leads to a power-law degree distribution [@Krapivsky:2000].\]
In a previous paper [@Sevim:2006], we studied a simplified version of the scheme proposed here, in which a new node makes a constant number of incoming links (${k'_i}=\mathrm{const}.)$, and $\gamma=1$. In this simplified version, the probability of attachment, $\Pi(k_{i})\propto1/k_{i}$, depends only on the outdegree (the number of predators). We calculated the outdegree distribution for this simplified model both analytically and by Monte Carlo simulations. It is given by the self-consistent equation $$n_{k}^{*}=(k+1)({m/z_m^*})^{k}\frac{\Gamma\big(1+{m/z_m^*}\big)}
{\Gamma\big(k+2+{m/z_m^*}\big)} \;,
\label{eq:nk}$$ where $$z_{m}^{*}=\sum_{j=0}^{\infty}{\frac{n_{j}^{*}}{j+1}}
\;,
\label{eq:zstar}$$ and $\Gamma(x)$ represents the Gamma function.
Model and results
=================
In the present paper, we investigate the general form of the attachment probability presented in Eq. (\[eq:one\]). With this form we relax both of the restrictions of the simplified model: we do not fix the indegree (number of prey) for the new nodes, so that each can make a different number of links, and we also vary the exponent $\gamma$. This makes full analytical treatment much harder, and the results for the outdegree distribution presented here are therefore only numerical. The indegree distribution, however, is found analytically to be a poissonian in the large-network limit.
The generalized attachment process proceeds as follows. We start the growth process with $N_{0}$ nodes and assign each initial node an indegree $m\le N_{0}$. These nodes act as source nodes as they are not connected to any other node at the beginning. Actually, the attributes of the initial nodes have no significance for the statistics because the final size of the network, $N_{\mathrm{max}}+N_{0}$, is much larger than $N_{0}$. In each time step, we add a new isolated node. Then, we give the new node $m$ chances to establish a link to an existing node with probability $$\Pi({k'_i},{k_i})=\frac{1}{Z}\left(\frac{{k'_i}}{{k_i}+1}\right)^{\gamma}
\;,
\label{eq:attachprob}$$ where $$Z=\sum_{i=1}^{N}\left({\frac{{k'_i}}{{k_i}+1}}\right)^{\gamma}
\label{eq:Z}$$ with $\gamma \ge 0$. Here, $N$ denotes the number of existing nodes at that time step. We use ${k_i}+1$ in the denominator to prevent a divergence for ${k_i}=0$. Multiple links between two nodes are not allowed. The direction of a link is from the old node to the new one.
We implement the growth process in Monte Carlo simulations as follows. We seed the system with $N_{0}$ source nodes, each with indegree $m$, and introduce a new node in each Monte Carlo step. To create links between the new node and the existing ones, we pick existing nodes, $i,$ one by one and calculate the probabilities of attachment, $\Pi({k'_i},{k_i})$. Then, we generate a random number, $r,$ and attach the new node to node $i$ if $r<\Pi$. We repeat this procedure until all existing nodes in the network are tested, i.e., till the sweep is completed. Since $\sum_{i}\Pi(k'_{i},k_{i})=1$, a new node makes on average one connection per sweep. We sweep the whole network $m$ times, so that $\langle{k'_i}\rangle=\langle{k_i}\rangle=m$. The new node is kept in the system, even if it does not acquire any links. However, a node with $k'=0$ stays isolated throughout the growth since the probability of attachment to it is zero. We stop the growth when the network size $N$ reaches $N_{\mathrm{max}}+N_{0}$ nodes with $N_{\mathrm{max}}=10^{5}$. We average over fifteen independent runs for each value of $m$ and $\gamma$.
We first tested the case of $\gamma=1$ to compare the outdegree distribution of the full $k'/k$ model to the outdegree distribution of our simplified $1/k$ model, Eqs. (\[eq:nk\]-\[eq:zstar\]), which, for large $k$, decays like $k\mu^{k}/\Gamma(k)$, where $\mu$ is a constant. As seen in Fig. \[fig:gamma1-out\], the outdegree distribution for the $k'/k$ model also decays faster than exponentially for large $k$. However, the dependence on the variable indegree leads to a broadening of the outdegree distribution: decreased probabilities for $k\approx m$, and compensating increased probabilities for $k\gg m$ and $k\ll m$. In the limit of large $m$, the central part of the outdegree distribution of the $k'/k$ model approaches that of the $1/k$ model.
![\[fig:gamma1-out\]Outdegree distributions for $\gamma=1$ and $m=1,$ 3, and 5 with $N_{0}=10$ shown on linear (a) and log-linear (b) scales. The simulations were stopped when the network size reached $N_{0}+10^{5}$ nodes. Each curve (with symbols) represents an average over fifteen runs. The curves without symbols are the theoretical outdegree distributions for our simplified model with $\Pi(k)\propto1/k$. Both the $k'/k$ and $1/k$ models yield the same distribution for $m\gg1$. As $k$ is a discrete variable, lines connecting the symbols are merely guides to the eye.](x-varindegree--out-linear "fig:")\
![\[fig:gamma1-out\]Outdegree distributions for $\gamma=1$ and $m=1,$ 3, and 5 with $N_{0}=10$ shown on linear (a) and log-linear (b) scales. The simulations were stopped when the network size reached $N_{0}+10^{5}$ nodes. Each curve (with symbols) represents an average over fifteen runs. The curves without symbols are the theoretical outdegree distributions for our simplified model with $\Pi(k)\propto1/k$. Both the $k'/k$ and $1/k$ models yield the same distribution for $m\gg1$. As $k$ is a discrete variable, lines connecting the symbols are merely guides to the eye.](x-varindegree--out "fig:")
![\[fig:vargamma-out\]Outdegree distributions for $\gamma=0.5,$ 1, 2, and 3, with network size $N_{0}+10^{5}$ nodes, and $m=5.$ Each curve is averaged over fifteen runs. Inset: The same distributions shown on a log-linear scale. The lines connecting the symbols are guides to the eye.](varm-vargamma-out)
The outdegree distribution of the general model also varies with $\gamma.$ Higher values of $\gamma$ sharpen the peak of the distribution around the mean outdegree, $m$, as it increases the tendency of the new nodes to prefer existing nodes with a higher value of $k'/k$ (Fig. \[fig:vargamma-out\]). In the limit $\gamma\rightarrow\infty$ one should obtain a delta function at $k=m$. Similarly, lower values of $\gamma$ relax the constraint and flatten the outdegree distribution. The limiting case, $\gamma=0$, corresponds to growth without preferential attachment, which yields an exponential outdegree distribution of mean $m$ [@Barabasi:2002; @Sevim:2006]. The lines connecting the symbols are guides to the eye.
![\[fig:gamma1-in\]Indegree distributions for $\gamma=1$ and $m=1,$ 3, and 5 with $N_{0}=10$. The simulations were stopped when the network size reached $N_{0}+10^{5}$ nodes. Each curve represents an average over fifteen runs. The error bars are smaller than the symbol sizes. Inset: The same distributions shown on a log-linear scale. The symbols $\ast,\times,$ and + show the Poisson distribution, Eq. , for $m=1,$ 3, and 5, respectively. The lines connecting the symbols are guides to the eye. See text for details. ](x-varindegree--in-w-binom)
In contrast to the outdegree distribution, the indegree distribution of the generalized model in the $N\gg m,k'$ limit can be described analytically and is extremely well approximated by a Poisson distribution with mean $m$, independent of $\gamma$ (Fig. \[fig:gamma1-in\]). This can be shown as follows. Each new node makes one link per sweep on average. When $N\gg m$, each existing node has a probability equal to $1/N$ of acquiring a new link per sweep on average, independent of the history of the network. Therefore, the probability of acquiring $k'$ links (after $m$ sweeps) for the node added at time $t$, when the total number of nodes is $N,$ is a binomial, $$P_{k'}(N)={N \choose k'}p^{k'}(1-p)^{N-k'}
\label{eq:binom}$$ with $p=m/N$. Thus, the final indegree distribution is an average (ignoring the $N_{0}$ initial nodes), $${n_{k'}^*}=\frac{1}{N_{\mathrm{max}}}
\sum_{N=N_{0}}^{N_{0}+N_{\mathrm{max}}}P_{k'}(N)\; .
\label{eq:nktimeaverage}$$ In general, this cannot be calculated exactly. However, for $N\gg m,k'$, $P_{k'}(N)$ can be approximated by a poissonian of mean $m$ [@RohatgiProb], $$P_{k'}=\frac{m^{k'}\exp(-m)}{k'!} \;,
\label{eq:poisson}$$ which is independent of $N$. The convergence with $N$ to this result is fast, so that for $N_{\mathrm{max}}\gg N_{0},m,$ the sum in Eq. is dominated by the $N$-independent terms. As a result, $${n_{k'}^*}\approx\frac{m^{k'}\exp(-m)}{k'!}$$ is an excellent approximation, as shown in Fig. \[fig:gamma1-in\]. Computer simulations confirm the $\gamma$-independent form of the indegree distribution, as shown in Fig. \[fig:vargamma-in\].
Discussion
==========
The mechanism that we propose and study in this paper, growth by preference for high indegree (number of prey) and low outdegree (number of predators), is intended to simulate the early stages of the development of food webs and other transportation networks. To distinguish it from the more commonly studied “rich get richer" schemes that produce scale-free networks, we call it [*inverse preferential attachment*]{}.
![\[fig:vargamma-in\]Indegree distributions for $\gamma=0.5,$ 1, 2, and 3 with $m=5,$ and network size $N_{0}+10^{5}$ nodes. Each curve is averaged over fifteen runs. The distributions are identical and they all practically overlap with the Poisson distribution, Eq. , with $m=5.$ Inset: The same distributions shown on a log-linear scale. The lines connecting the symbols are guides to the eye.](varm-vargamma-in-w-binom)
The outdegree distribution obtained using the generalized form of the probability of attachment, $\Pi(k'_{i},k_{i})\propto(k'_{i}/k_{i})^{\gamma},$ (with $\gamma=1)$ is broader than the one obtained from the simplified form, $\Pi(k_{i})\propto1/k_{i}$. However, both decay faster than exponentially for large $k_{i}$. The shape of the outdegree distribution is continuously tunable by $\gamma$, from a exponential distribution for $\gamma=0$ to a delta function for $\gamma \rightarrow \infty$. The indegree distribution does [*not*]{} depend on $\gamma$, but only on $m$, the mean number of links per node. In the limit $N\gg k,m$, the indegree distribution is a poissonian.
Some features of the networks produced by this model, like the outdegree distribution, which decays faster than exponentially, resemble those of some empirical and model food webs [@Camacho:2002B; @Camacho:2002A; @Martinez:2000; @Rossberg:2006; @Rikvold:2007]. In this, they differ sharply from the scale-free networks generated by the conventional “rich get richer" preferential-attachment schemes [@Barabasi:1999; @Barabasi:2002]. However, differences from real food webs remain, such as the correlation between the in- and outdegrees of a node. Our model produces webs with a positive in-outdegree correlation, whereas most empirical and model webs have a negative correlation [@Stouffer:2005; @Rikvold:2007]. This may be due to the unrestrained growth of our networks, which would require an extinction process to achieve a steady state [@Rossberg:2006]. Also, this growth scheme is not designed to produce loops. We intend to include such features in future versions of the model, thus enabling modeling of mature, steady-state networks.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was supported by U.S. National Science Foundation Grant Nos. DMR-0240078 and DMR-0444051, and by Florida State University through the Department of Physics, the School of Computational Science, the Center for Materials Research and Technology, and the National High Magnetic Field Laboratory.
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|
---
abstract: |
Machine Learning has become very famous in today’s world which assist to identifying the patterns from the raw data. Technological advancement has led to substantial improvement in Machine Learning which, thus helping to improve prediction. Current Machine Learning models are based on Classical Theory, which can be replaced by Quantum Theory to improve the effectiveness of the model.
In the previous work, we developed binary classifier inspired by Quantum Detection Theory. In this extended abstract, our main goal is to develop multi class classifier. We generally use the terminology multinomial classification or multi-class classification when we have a classification problem for classifying observations or instances into one of three or more classes.
bibliography:
- 'refs.bib'
title: 'Multi-class Classification Model Inspired by Quantum Detection Theory'
---
Introduction
============
Quantum Mechanics has already shown its effectiveness in many fields so there is a good possibility that it will prove to be useful in as well. Quantum theory can open a new way towards quantum inspired which might outperform traditional machine learning if used properly. The theory of has been implemented in several domain of IR recently by [@li2018quantum; @zhang2018quantum; @zhang2016quantum; @zhang2018end; @wang2016exploration; @zhang2018quantum1; @li2018quantum; @li2015modeling] . Quantum Probability theory is the quantum generalization of classical probability theory, which was developed by [@vonNeumann55]. Classical probability theory provides that a system can have either state 0 or 1 and quantum probability comes into existence to go beyond classical theory and describe states in between 0 and 1 with classical states.
The effectiveness of the state-of-the-art classification algorithms relies on logical theory of sets, theory of probability and the algebra of vector spaces. For example, the most straightforward technique is called Naive Bayes, which considers objects (e.g. documents) as elements of sets and applies basic probability measures to these sets for selecting classes. Another effective classification technique called Support Vector Machines considers objects as points of a multi-dimensional space and aims to select subspaces as classes. However, an effective combination of techniques stemming from different theories is still missing, although it has been investigated in IR since the book on the Geometry of IR by [@vanRijsbergen79a].
Despite its effectiveness in some domains, classification effectiveness is still unsatisfactory in a number of application domains due to a variety of reasons, such as the number of categories and the nature of data. The number of categories may be so large that a classification technique that is effective for a few categories may be ineffective when thousands of categories are required; moreover, the nature of data may be so complex that the techniques that are effective for simple objects may prove to be ineffective for complex objects. A sensible approach to addressing the problems caused by unconventional categorical systems or complex data is to adapt well-known and effective techniques to these new contexts. Another approach, which is indeed the focus of this paper, is to radically change paradigm and to investigate whether a new theoretical framework may be beneficial and be a new research direction.
Quantum Theory may provide a theoretical model for classification. To our knowledge no work has been done on quantum based classification so this paper is a first step to enter into quantum inspired machine learning and prove the effectiveness of quantum theory in classification.
Proposed Methodology and Discussion
===================================
In the previous work, we developed a binary classifier inspired by Quantum Detection Theory by [@di2018binary]. The main task was to identify whether a document belongs to a given topic or not. We used Reuters21578[^1] in order to check the performance of our model. Our proposed binary classifier model inspired from quantum detection theory performed very well in terms of recall and F-measures for most of the topics. An experiment was done on the small dataset so work is still in progress. Our algorithm works as follows: it starts by computing the density operators $\rho_1$ and $\rho_0$ from positive and negative samples, respectively. In order to achieve this, for a particular feature, we first compute the number of documents with non-zero values in the feature. In this way, one vector is generated from each class, thus obtaining two vectors which are respectively denoted as $\vert v_1 \rangle$ and $\vert v_0
\rangle$. These vectors can be regarded as a representation of the feature statistics among a class. We normalize the vectors and compute the outer spaces in order to obtain the density operators $\rho_1$ and $\rho_0$:
$$\label{eq:rhos}
\rho_1 = \frac{\vert v_1 \rangle \langle v_1 \vert}{tr(\vert v_1 \rangle \langle v_1 \vert)} \qquad
\rho_0 = \frac{\vert v_0 \rangle \langle v_0 \vert}{tr(\vert v_0 \rangle \langle v_0 \vert)}$$
We computed the projection operator $P$ according to the eigen decomposition described in [@Melucci15b], that is, $$\label{eq:decomposition}
\rho_1 - \lambda \rho_0 = \eta\, P + \beta\,P^\perp \qquad \eta > 0
\qquad \beta < 0 \qquad P\,P^\perp = 0$$ where $\xi$ is the prior probability of the negative class and $
\lambda = {\xi}\,/\,(1-\xi) $. Moreover, $\eta$ is the positive eigenvalue corresponding to $P$ which represents the subspaces of the vectors representing the documents to be accepted in the target class.
In this extended abstract, our main goal is to develop multi class classifier. We generally use the terminology “multinomial classification or multi-class classification” when we have a classification problem for classifying observations or instances into one of three or more classes.
The main theory behind quantum inspired multi-class classification is as follows: The choice among the $N$ hypotheses, which the $k^{th}$ asserts “The system has the density operator $\rho_k$,” in which $k =1, 2, 3, ....., N$ can be based on the result of the measurement of $N$ commuting operators $P_1, P_2, ...., P_N$, making a resolution of identity operator $1$:
$$P_1+P_2+ ....+P_N = 1$$
Our problem is getting the set of projectors so that the choice among the $N$ hypothesis can be made with the minimum average cost. It will assist in the event of Quantum Detection Theory, in constructing and estimating the best receiver for the communications system. In this, messages are coded into an alphabets of 3 or more symbols, and a distinct signal is transmitted for each.
Assume $\xi_k$ is the prior probability of the hypothesis $H_k$, and $K_{ij}$ is the cost for choosing $H_i$ when $H_j$ is correct. So the average cost per decision can be described as, $$\bar{K} = \sum^N _{i=1} \sum^N _{j=1} \xi_j K_{ij} \mbox Tr(\rho_j P_i) ,$$ which has to be minimized by the set of given commuting projection operators $P_k$. In particular, $K_{ii}=0$, $K_{ij}=1$, $i\neq j$, $\bar{k}$ can be approximate to the average probability of error.
In each hypotheses, the state of the system is in pure state $\rho_k = \vert \psi_k \rangle \langle \psi_k \vert$, so the projection operator will have such form, $P_j =\vert \eta_j \rangle \langle \eta_j \vert$. Here $\vert \eta_j \rangle$ is the linear combination of the the given state $\vert \psi_k \rangle$. The main problem is to find the set of projectors minimizing the average cost when more than two categories or hypotheses are available; the solution can be a generalization of the solution of the problem of finding the set of projectors in the event of two categories.
Conclusion and Future Works
===========================
Although research work is still in progress we are testing a multi class classifier based on Quantum Detection Theory and we expect that it is possible to develop such a model. In order to learn about and classification tasks in more detail, these works may be beneficial and a basis for developing this model. [@helstrom1971quantum; @yuen1975optimum; @helstrom1969quantum; @helstrom1972vii; @eldar2001quantum; @helstrom1974noncommuting; @helstrom1968detection; @holevo1998capacity; @vilnrotter2001quantum; @di2016evaluation; @melucci2015introduction; @melucci2012contextual; @melucci2011quantum; @melucci2011advanced; @melucci2018efficient; @melucci2017algorithm; @nanni2016combination]
Acknowledgement {#acknowledgement .unnumbered}
===============
This work is supported by the Quantum Access and Retrieval Theory (QUARTZ) project, which has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 721321.
[^1]: <http://www.daviddlewis.com/resources/testcollections/reuters21578/>
|
---
abstract: 'Thermal Radiative Transfer (TRT) is the dominant energy transfer mechanism in high-energy density physics with applications in inertial confinement fusion and astrophysics. The stiff interactions between the material and radiation fields make TRT problems challenging to model. In this study, we propose a multi-dimensional extension of the deterministic particle (DP) method. The DP method combines aspects from both particle and deterministic methods. If the emission source is known [*a priori*]{}, and no physical scattering is present, the intensity of a particle can be integrated analytically. This introduces no statistical noise compared to Monte-Carlo methods, while maintaining the flexibility of particle methods. The method is closely related to the popular method of long characteristics. The combination of the DP-method with a discretely-consistent, nonlinear, gray low-order system enables an efficient solution algorithm for multi-frequency TRT problems. We demonstrate with numerical examples that the use of a linear-source approximation based on spatial moments improves the behavior of our method in the thick diffusion limit significantly.'
address:
- |
Fluid Dynamics and Solid Mechanics (T-3)\
Los Alamos National Laboratory\
Los Alamos, NM, 87545
- |
Applied Mathematics and Plasma Physics (T-5)\
Los Alamos National Laboratory\
Los Alamos, NM, 87545
author:
- Hans Hammer
- HyeongKae Park
- Luis Chacón
bibliography:
- 'literature.bib'
title: 'A multi-dimensional, moment-accelerated deterministic particle method for time-dependent, multi-frequency thermal radiative transfer problems'
---
Thermal Radiative Transfer ,HOLO algorithm ,Deterministic particle method
Introduction {#sec:introduction}
============
Many applications in astrophysics and plasma physics require the simulation of high-energy density phenomena. In these regimes, thermal X-ray radiation is the dominant mechanism of energy transfer. Thermal radiative transfer (TRT) is described by a system of stiff, nonlinear equations, and therefore difficult to model.
Two distinct methods are typically used, namely, the deterministic transport method and stochastic particle methods. Both methods have their advantages and disadvantages. A popular particle method is Implicit Monte Carlo (IMC)[@fleck_implicit_1971], which is based on a linearization of the reemission physics. The particle approach gives IMC flexibility for complicated geometries. The most common deterministic methods are the discrete ordinance or [$S_N$]{}method and the method of characteristics (MOC). Both are frequently used for neutron transport calculations in reactor physics. The advantage of [$S_N$]{}and MOC over IMC is that they do not show statistical noise, and they are able to obtain the asymptotic diffusion limit [@larsen_asymptotic_1987; @larsen_asymptotic_1989; @larsen_asymptotic_1983] in a relatively straightforward manner [@adams_discontinuous_2001; @adams_characteristic_1998].
A newly proposed deterministic particle (DP) method [@park_multigroup_2019] combines these two approaches. If the emission source is known [*a priori*]{}, the intensity of a particle can be integrated analytically. In contrast to Monte-Carlo (MC) methods, our method does not feature any randomness (with the possible exception of particle initialization), and hence does not show stochastic noise, while maintaining the flexibility of particle methods. This method is similar to MOC in space and time proposed by @pandya_method_2009. However, our method is particle based, and therefore does not require fixed tracks.
When this DP solver is cast in a high-order, low-order (HOLO) algorithmic framework [@chacon_multiscale_2017], a discretely-consistent, gray, nonlinear low-order (LO) system provides a well-informed emission source, acts as an interface with other physics in a multi-physics setting, and provides algorithmic acceleration [@chacon_multiscale_2017]. Because of the presence of the consistent, nonlinear LO system, the emission source is known and the contribution from the absorption-emission physics can be analytically integrated along a particle’s trajectory. This implicit treatment of the absorption-emission physics removes the explicit time step constraint, allowing us to choose the time step size based on accuracy, not stability.
The DP-HOLO method has been recently demonstrated in one dimension [@park_multigroup_2019]. In this work we present the extension of the DP-HOLO method to multiple dimensions using a ray-tracing approach and a linear-source reconstruction scheme that preserves the diffusion limit. The flat-source approximation requires many cells in optically thick materials to model the changes in the temperature and the emission source sufficiently well. Introducing a linear reconstruction reduces the number of cells necessary to obtain a good representation of the source. We use the method proposed by @ferrer_linear_2016 [@ferrer_linear_2018] for MOC to obtain the linear representation of tallies. @adams_characteristic_1998 showed that linear source representation on a triangular mesh, and bi-linear on orthogonal, rectangular meshes are required to obtain the thick diffusion limit. For general polygons, piecewise linear basis functions are required as shown by @pandya_long_2011, which again reduce to linear functions on triangles. Therefore, a linear-source reconstruction is also necessary to obtain the asymptotic diffusion limit [@larsen_asymptotic_1987; @larsen_asymptotic_1989] with our method.
The reminder of the paper is structured as follows. We first introduce our method in \[sec:method\], with the LO solver first followed by the HO solver. We then present numerical results to show the capabilities of this method in \[sec:results\]. We chose the Tophat [@gentile_implicit_2001] problem to demonstrate the effects of the linear source approximation and a planar Hohlraum problem [@mcclarren_robust_2010; @brunner_forms_2002] to demonstrate the effects of different quadratures and random particle initialization. We compare our results to Capsaicin (which employs an $S_N$ implementation) [@thompson_capsaicin:_2006]. Finally, we show results for runtime and convergence studies of the algorithm before we conclude.
Method {#sec:method}
======
Thermal radiative transfer without physical scattering can be described by the following system of equations $$\begin{aligned}
\frac{1}{c} {{\frac{\partial I}{\partial t}}} + {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}{\mathbf{\cdot}}{{\bm{\nabla}}}I + {\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}I &= {\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}B \\
{\ensuremath{\rho}\xspace}{\ensuremath{c_{v}}\xspace}{{\frac{\partial T}{\partial t}}} &= \int_{0}^{\infty}{\int_{4\pi}}\left({\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}I - {\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}B\right) {{\,d\Omega}}{\,d{\ensuremath{\nu}\xspace}}
\end{aligned}$$ where $I\left({\ensuremath{{\bm{x}}}\xspace}, {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}, {\ensuremath{\nu}\xspace}, t\right)$ is the specific radiation intensity at position [${\bm{x}}$]{}traveling in along direction [$\hat{{\bm{\Omega}}}$]{}with the speed of light $c$ and frequency [$\nu$]{}at time $t$. Here, is the frequency integrated (gray) radiation energy density, ${\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}\left({\ensuremath{{\bm{x}}}\xspace}, {\ensuremath{\nu}\xspace}, T\right)$ is the opacity at temperature $T$, [$\rho$]{}is the host material’s density, and [$c_{v}$]{}is its specific heat capacity. The emission spectrum is defined by the Planck function $$B\left({\ensuremath{\nu}\xspace}, T\right) \equiv \frac{2{\ensuremath{h}\xspace}{\ensuremath{\nu}\xspace}^3}{c^2}\frac{1}{{\mathrm{e}^{{{\ensuremath{h}\xspace}{\ensuremath{\nu}\xspace}}/{{\ensuremath{k_\mathrm{B}}\xspace}T}}} - 1},$$ where [$h$]{}denotes the Planck constant and [$k_\mathrm{B}$]{}the Boltzmann constant.
Solving the TRT system is difficult due to the high dimensionality of the phase-space and the stiff, nonlinear coupling between the radiation field and the material temperature described by the absorption and emission physics.
The DP algorithm uses a high-order (HO), multi-frequency transport solve combined with a gray low-order solver (LO). For simplicity, we will describe the basic algorithm for the gray case and will extend it to the multi-frequency case afterwards. The gray TRT equation is $$\label{eq:method:transport}
\frac{1}{c} {{\frac{\partial I}{\partial t}}} + {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}{\mathbf{\cdot}}{{\bm{\nabla}}}I + {\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}I = Q^{\mathrm{\,{ LO}}}\left({\ensuremath{{\bm{x}}}\xspace}, t\right).$$ The DP method requires an [*a priori*]{}known emission source $$\label{eq:method:emission_source}
Q^{\mathrm{\,{ LO}}}\left({\ensuremath{{\bm{x}}}\xspace},t\right) = \frac{{\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}acT^4\left({\ensuremath{{\bm{x}}}\xspace}, t\right)}{4\pi}$$ for the integration along the particle trajectory, where $a$ is the radiation constant. Here, the LO superscript indicates that the source is evaluated from low-order quantities, obtained from a recently developed iterative, moment-based HOLO algorithm [@park_consistent_2012; @park_efficient_2013; @chacon_multiscale_2017]. The LO system is defined by taking the first two angular moments of \[eq:method:transport\] together with the material temperature equation,
\[eq:method:lo\_system\_consistent\] $$\begin{aligned}
\label{eq:method:lo_system_consistent:balance}
{{\frac{\partial E^{\mathrm{\,{ LO}}}}{\partial t}}} + {{{\bm{\nabla}}}{\mathbf{\cdot}}}{\bm{F}}^{\mathrm{\,{ LO}}}+ {\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}c E^{\mathrm{\,{ LO}}}&= {\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}ac T^{4} + {\ensuremath{\mathcal{S}^{\mathrm{\,{ HO}}}}\xspace}\\
\label{eq:method:lo_system_consistent:flux}
\frac{1}{c} {{\frac{\partial {\bm{F}}^{\mathrm{\,{ LO}}}}{\partial t}}} + \frac{c}{3} {{\bm{\nabla}}}E^{\mathrm{\,{ LO}}}+ {\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}{\bm{F}}^{\mathrm{\,{ LO}}}&= {\ensuremath{{\bm{\gamma}}^{\mathrm{\,{ HO}}}}\xspace}cE^{\mathrm{\,{ LO}}}\\
\label{eq:method:lo_system_consistent:temperature}
{\ensuremath{\rho}\xspace}{\ensuremath{c_{v}}\xspace}{{\frac{\partial T}{\partial t}}} + {\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}ac T^4 &= {\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}cE^{\mathrm{\,{ LO}}},
\end{aligned}$$
where the HO superscript indicates that the quantity is evaluated using HO quantities, $$\label{eq:method:radiation_energy}
E\left({\ensuremath{{\bm{x}}}\xspace}, t\right) \equiv \frac{1}{c} \int_{0}^{\infty} {\int_{4\pi}}I {{\,d\Omega}}{\,d{\ensuremath{\nu}\xspace}}$$ is the frequency integrated (gray) radiation energy density, and $$\label{eq:method:flux}
{\bm{F}}\left({\ensuremath{{\bm{x}}}\xspace}, t\right) \equiv \int_{0}^{\infty} {\int_{4\pi}}{\ensuremath{\hat{{\bm{\Omega}}}}\xspace}I {{\,d\Omega}}{\,d{\ensuremath{\nu}\xspace}}$$ is the gray radiative flux. We used the standard ${\ensuremath{P_1}\xspace}$ closure in \[eq:method:lo\_system\_consistent:flux\] instead of the more consistent Eddington tensor closure [@goldin_quasi-diffusion_1964]. This introduces inconsistencies between the HO and LO descriptions, in addition to inconsistencies in the discretization. Adding the consistency term ${\ensuremath{{\bm{\gamma}}^{\mathrm{\,{ HO}}}}\xspace}$ to \[eq:method:lo\_system\_consistent:flux\] will correct for transport effects and the mismatch in truncation errors. While \[eq:method:lo\_system\_consistent:balance\] is exact in the continuum, we advance that we will need to correct for mismatches between HO and LO temporal discretizations, and add the residual source term $${\ensuremath{\mathcal{S}^{\mathrm{\,{ HO}}}}\xspace}= {{\frac{\partial E^{\mathrm{\,{ HO}}}}{\partial t}}} \Big|_\mathrm{LO} - {{\frac{\partial E^{\mathrm{\,{ HO}}}}{\partial t}}} \Big|_\mathrm{HO}.$$ Here, the subscript LO indicates that we apply the LO discrete temporal derivative scheme, and similarly with the HO subscript. Both the LO and the HO methods conserve energy.
Both [${\bm{\gamma}}^{\mathrm{\,{ HO}}}$]{}and [$\mathcal{S}^{\mathrm{\,{ HO}}}$]{}are evaluated from the HO solution, and their specific form depends on the discretization used. Details will be discussed in the next section.
Low-order solver {#sec:method:lo}
----------------
The integral balance equation for \[eq:method:lo\_system\_consistent:balance\] obtained by integrating in time and cell volume is $$\label{eq:method:lo:balance}
\frac{{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ LO}}}_{i,n+{\frac{1}{2}}} - {\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ LO}}}_{i,n-{\frac{1}{2}}}}{\Delta t_n} + \sum_{j \in i} {\ensuremath{\hat{{\bm{n}}}}\xspace}_{ij} {\mathbf{\cdot}}{\ensuremath{\hat{{\bm{n}}}}\xspace}_{j}\frac{{\ensuremath{\overline{F}}\xspace}^{\mathrm{\,{ LO}}}_{j,n} A_j}{V_i} + {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} c {\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ LO}}}_{i,n} - {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} ac{\ensuremath{\overline{T_{i,n}^4}}\xspace} = 0$$ where $\Delta t_n = t_{n+{\frac{1}{2}}} - t_{n-{\frac{1}{2}}}$ is the time step size for step $n$, $V_i$ the cell volume, $A_j$ the surface area, ${\ensuremath{\hat{{\bm{n}}}}\xspace}_j$ is the global unit normal of surface $j$ and ${\ensuremath{\hat{{\bm{n}}}}\xspace}_{ij}$ is the unit normal of surface $j$ pointing outwards from cell $i$. The sign of the product ${\ensuremath{\hat{{\bm{n}}}}\xspace}_j {\mathbf{\cdot}}{\ensuremath{\hat{{\bm{n}}}}\xspace}_{ij}$ indicates if the flux is outgoing or incoming. The bar notation denotes a spatial average over a cell or surface. Note that we use half indices for end-of-time-step variables and integer indices for time averaged quantities. Therefore, we find
\[eq:method:lo:ho\_quantities\] $$\begin{aligned}
\label{eq:method:lo:ho_quantities:E_end}
{\ensuremath{\overline{E}}\xspace}_{i,n+{\frac{1}{2}}} &\equiv \frac{1}{cV_i} \int_{V_{i}} {\int_{4\pi}}I({\ensuremath{{\bm{x}}}\xspace}, {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}, t_{n+{\frac{1}{2}}}) {{\,d\Omega}}{\,dV} \\
\label{eq:method:lo:ho_quantities:E_avg}
{\ensuremath{\overline{E}}\xspace}_{i,n} &\equiv \frac{1}{c\Delta t_{n}V_i} \int_{t_{n-{\frac{1}{2}}}}^{t_{n+{\frac{1}{2}}}} \int_{V_{i}} {\int_{4\pi}}I\left({\ensuremath{{\bm{x}}}\xspace}, {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}, t\right) {{\,d\Omega}}{\,dV} {{\,dt}}\\
\label{eq:method:lo:ho_quantities:F}
{\ensuremath{\overline{F}}\xspace}_{j,n} &\equiv \frac{1}{\Delta t_{n}A_j} \int_{t_{n-{\frac{1}{2}}}}^{t_{n+{\frac{1}{2}}}} \int_{A_j} {\int_{4\pi}}{\ensuremath{\hat{{\bm{n}}}}\xspace}_j {\mathbf{\cdot}}{\ensuremath{\hat{{\bm{\Omega}}}}\xspace}I\left({\ensuremath{{\bm{x}}}\xspace}, {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}, t\right) {{\,d\Omega}}{\,dA} {{\,dt}}\\
{\ensuremath{\overline{T^4}}\xspace}_{i,n} &\equiv \frac{1}{\Delta t_{n}V_i} \int_{t_{n-{\frac{1}{2}}}}^{t_{n+{\frac{1}{2}}}} \int_{V_{i}} T^4\left({\ensuremath{{\bm{x}}}\xspace}, t\right) {\,dV} {{\,dt}}\end{aligned}$$
The time-discrete LO equation cannot update simultaneously quantities defined at $n$ and $n+{\frac{1}{2}}$, and therefore we replace end-of-time-step quantities with time step average quantities by adding a residual source term: $$\label{eq:method:lo:balance_discrete}
\frac{{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ LO}}}_{i,n} - {\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ LO}}}_{i,n-1}}{\Delta t_n} + \sum_{j \in i} {\ensuremath{\hat{{\bm{n}}}}\xspace}_{ij} {\mathbf{\cdot}}{\ensuremath{\hat{{\bm{n}}}}\xspace}_{j} \frac{{\ensuremath{\overline{F}}\xspace}^{\mathrm{\,{ LO}}}_{j,n} A_j}{V_i} + {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} c {\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ LO}}}_{i,n} - {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} ac{\ensuremath{\overline{T^4_{i,n}}}\xspace} = {\ensuremath{\mathcal{S}^{\mathrm{\,{ HO}}}}\xspace}_{i,n},$$ where the residual source is defined as: $$\label{eq:method:lo:residual_source}
{\ensuremath{\mathcal{S}^{\mathrm{\,{ HO}}}}\xspace}_{i,n} \equiv \frac{{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{i,n} - {\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{i,n-1}}{\Delta t_n} - \frac{{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{i,n+{\frac{1}{2}}} - {\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{i,n-{\frac{1}{2}}}}{\Delta t_n}.$$ The discrete equation for the flux ${\ensuremath{\overline{F}}\xspace}_{j,n}$ across surface $j$ using the method proposed by @park_multigroup_2019 can be written as $$\frac{1}{c} \frac{{\ensuremath{\overline{F}}\xspace}^{\mathrm{\,{ LO}}}_{j,n} - {\ensuremath{\overline{F}}\xspace}^{\mathrm{\,{ LO}}}_{j,n-1}}{\Delta t_{n}} + \frac{c}{3}\frac{{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ LO}}}_{j+{\frac{1}{2}},n} - {\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ LO}}}_{j-{\frac{1}{2}},n}}{\Delta x_{j}} + {\ensuremath{\sigma_{{\ifthenelse{\isempty{j,n}}{}{_{j,n}}}}}\xspace} {\ensuremath{\overline{F}}\xspace}^{\mathrm{\,{ LO}}}_{j,n} = \gamma^{+{\mathrm{\,{ HO}}}}_{j,n} c{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ LO}}}_{j+{\frac{1}{2}},n} - \gamma^{-{\mathrm{\,{ HO}}}}_{j,n} c{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ LO}}}_{j-{\frac{1}{2}},n}.$$ where the indices $j\pm{\frac{1}{2}}$ denote the cells adjacent to surface $j$, $\Delta x_j$ is the characteristic length between these cells, and [$\sigma_{{\ifthenelse{\isempty{j,n}}{}{_{j,n}}}}$]{} is a weighted opacity at the surface. The consistency terms are given by [@park_consistent_2012]
\[eq:method:lo:consistency\] $$\begin{aligned}
\label{eq:method:lo:consistency:+}
\gamma^{+{\mathrm{\,{ HO}}}}_{j,n} &= \frac{1}{c {\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{j+{\frac{1}{2}},n}}
\left(\frac{1}{c} \frac{f^{+{\mathrm{\,{ HO}}}}_{j,n} - f^{+{\mathrm{\,{ HO}}}}_{j,n-1}}{\Delta t_n} + \frac{c}{6}\frac{{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{j+{\frac{1}{2}},n} - {\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{j-{\frac{1}{2}},n}}{\Delta x_j} + {\ensuremath{\sigma_{{\ifthenelse{\isempty{j,n}}{}{_{j,n}}}}}\xspace} f^{+{\mathrm{\,{ HO}}}}_{j,n}\right) \\
\label{eq:method:lo:consistency:-}
\gamma^{-{\mathrm{\,{ HO}}}}_{j,n} &= \frac{1}{ c {\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{j-{\frac{1}{2}},n}}
\left(\frac{1}{c} \frac{f^{-{\mathrm{\,{ HO}}}}_{j,n} - f^{-{\mathrm{\,{ HO}}}}_{j,n-1}}{\Delta t_n} - \frac{c}{6}\frac{{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{j+{\frac{1}{2}},n} - {\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{j-{\frac{1}{2}},n}}{\Delta x_j} + {\ensuremath{\sigma_{{\ifthenelse{\isempty{j,n}}{}{_{j,n}}}}}\xspace} f^{-{\mathrm{\,{ HO}}}}_{j,n} \right)
\end{aligned}$$
where the partial fluxes are defined from the HO solution with respect to the global surface normal ${\ensuremath{\hat{{\bm{n}}}}\xspace}_j$ as
\[eq:method:lo:partial\_flux\] $$\begin{aligned}
\label{eq:method:lo:partial_flux:+}
f^+_{j,n} &\equiv \frac{1}{\Delta t_n} \int_{t_{n-{\frac{1}{2}}}}^{t_{n+{\frac{1}{2}}}} \int_{A_j} \int_{{\ensuremath{\hat{{\bm{\Omega}}}}\xspace}{\mathbf{\cdot}}{\ensuremath{\hat{{\bm{n}}}}\xspace}_j > 0} {\ensuremath{\hat{{\bm{n}}}}\xspace}_j{\mathbf{\cdot}}{\ensuremath{\hat{{\bm{\Omega}}}}\xspace}I\left({\ensuremath{{\bm{x}}}\xspace}, {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}, t\right) {{\,d\Omega}}{\,dA} {{\,dt}}\\
\label{eq:method:lo:partial_flux:-}
f^-_{j,n} &\equiv - \frac{1}{\Delta t_n} \int_{t_{n-{\frac{1}{2}}}}^{t_{n+{\frac{1}{2}}}} \int_{A_j} \int_{{\ensuremath{\hat{{\bm{\Omega}}}}\xspace}{\mathbf{\cdot}}{\ensuremath{\hat{{\bm{n}}}}\xspace}_j < 0} {\ensuremath{\hat{{\bm{n}}}}\xspace}_j{\mathbf{\cdot}}{\ensuremath{\hat{{\bm{\Omega}}}}\xspace}I\left({\ensuremath{{\bm{x}}}\xspace}, {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}, t\right) {{\,d\Omega}}{\,dA} {{\,dt}}.
\end{aligned}$$
Finally, the flux across the boundary surface $j$ is $${\ensuremath{\overline{F}}\xspace}_{j,n} = f^+_{j,n} - f^-_{j,n}
= \left(1 - \alpha_{j}\right) f^+_{j,n} - f'^-_{j,n}$$ where we assume the normal points outwards, and $\alpha_{j} \in \left[0,1\right]$ is the reflection or albedo factor. It allows the user to define boundaries as vacuum, partially or fully reflecting. The incoming boundary flux without reflection is $$f'^-_{j,n} = f^-_{j,n} - \alpha_{j} f^+_{j,n} = \frac{acT_{\mathrm{BC},j}^4}{4}.$$ Defining the boundary factors
$$\begin{aligned}
\kappa^{+{\mathrm{\,{ HO}}}}_{j,n} &\equiv \frac{f^{+{\mathrm{\,{ HO}}}}_{j,n}}{c{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{j-{\frac{1}{2}},n}} \\
\kappa^{-{\mathrm{\,{ HO}}}}_{j,n} &\equiv
\begin{cases}
0 & T_{\mathrm{BC},j} = 0 \\
\frac{f^{-{\mathrm{\,{ HO}}}}_{j,n} - \alpha_{j}f^{+{\mathrm{\,{ HO}}}}_{j,n}}{acT_{\mathrm{BC},j}^4} & T_{\mathrm{BC},j} > 0
\end{cases}
\end{aligned}$$
gives the LO boundary condition $${\ensuremath{\overline{F}}\xspace}^{\mathrm{\,{ LO}}}_{j,n} = \left(1 - \alpha_{j}\right) \kappa^{+{\mathrm{\,{ HO}}}}_{j,n} c{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ LO}}}_{i-{\frac{1}{2}},n} -\kappa^{-{\mathrm{\,{ HO}}}}_{j,n} acT_{\mathrm{BC}, j}^4.$$ In these equations, [$\mathcal{S}^{\mathrm{\,{ HO}}}$]{}, $\gamma^{\pm{\mathrm{\,{ HO}}}}_{j,n}$ and $\kappa^{\pm{\mathrm{\,{ HO}}}}_{j,n}$ are evaluated from the HO system using \[eq:method:lo:ho\_quantities,eq:method:lo:partial\_flux\]. This gives discrete consistency between the LO and HO system. The LO-system is solved using a Newton-Krylov method with non-linear elimination [@park_consistent_2012]. The details are given in \[sec:appendix:lo\_solution\].
Finally, the temperature equation, \[eq:method:lo\_system\_consistent:temperature\], is discretized as $${\ensuremath{\rho}\xspace}{\ensuremath{c_{v}}\xspace}\frac{{\ensuremath{\overline{T}}\xspace}^{\mathrm{\,{ LO}}}_{i,n} - {\ensuremath{\overline{T}}\xspace}^{\mathrm{\,{ LO}}}_{i,n-1}}{\Delta t_n} + {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} ac {\ensuremath{\overline{T^4_{i,n}}}\xspace} - {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} c {\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ LO}}}_{i,n} = 0.$$
High-order solver {#sec:method:ho}
-----------------
The high-order solver is a particle-based ray-tracing algorithm. In particle-based methods (e.g. Monte Carlo) the angular intensity $I$ is represented as a collection of $P$ particles with their specific intensity $I^{\mathrm{\,{ HO}}}_p$ as $$\label{eq:method:ho:particle_field}
I^{\mathrm{\,{ HO}}}\left({\ensuremath{{\bm{x}}}\xspace}, {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}, t\right) = \sum_{p = 1}^{P} {\ensuremath{\mathrm{w}{\ifthenelse{\isempty{p}}{}{_{p}}}}\xspace} I^{\mathrm{\,{ HO}}}_p\left(t\right) \delta\left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\bm{x}}}\xspace}_p\left(t\right)\right)\delta\left({\lVert{\ensuremath{\hat{{\bm{\Omega}}}}\xspace}- {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_{p}\left(t\right)\rVert_{{}}}\right)$$ where ${\ensuremath{{\bm{x}}}\xspace}_p\left(t\right)$, ${\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_{p}\left(t\right)$ are the spatial position and direction of particle $p$ at time $t$. The particle phase-space volume [$\mathrm{w}{\ifthenelse{\isempty{p}}{}{_{p}}}$]{} is the analog to the track width of MOC methods, an integral weight factor. Details of its calculation are given later. The evolution equation of the particle intensity can by found by multiplying \[eq:method:transport\] by $\delta\left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\bm{x}}}\xspace}_p\left(t\right)\right)\delta\left({\lVert{\ensuremath{\hat{{\bm{\Omega}}}}\xspace}- {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_{p}\left(t\right)\rVert_{{}}}\right)$ and integrating over the phase-space to obtain $$\label{eq:method:ho:evolution}
\frac{1}{c}\frac{{\,dI^{\mathrm{\,{ HO}}}_p}}{{{\,dt}}} + {\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}I^{\mathrm{\,{ HO}}}_p = Q^{\mathrm{\,{ LO}}}\left(x_p, t\right)$$ where $Q^{\mathrm{\,{ HO}}}\left(x, t\right)$ is the [*a priori*]{}known emission source \[eq:method:emission\_source\]. The formal solution for $I^{\mathrm{\,{ HO}}}_p$ along the characteristic for particle $p$ in cell $i$ at time step $n$ is $$\label{eq:method:ho:characteristic}
I^{\mathrm{\,{ HO}}}_{p,i,n}\left(t\right) = I^{\mathrm{\,{ HO}}}_{p,i,n}\left(t_0\right) {\mathrm{e}^{-\int_{t_0}^{t} {\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}c{{\,dt}}'}}
+ \int_{t_0}^{t} {\mathrm{e}^{-\int_{t'}^{t}{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} c{{\,dt}}''}} Q^{\mathrm{\,{ LO}}}_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}_{p}\left(t'\right), t'\right) c{{\,dt}}'.$$ This equation, and its first spatial moment, can be integrated analytically when one considers a linear emission source (as will be the case here), and that particle trajectories are straight. We consider the particle initialization and trajectory computation next.
### Particle initialization {#sec:appendix:particle_initalization}
The particles are initialized on a per-cell basis. In each cell $i$, a number of points are selected as starting points for the particles. The points are found by increasingly refining the cell up to a specified level. Each triangle or rectangle cell is divided into four subcells, until the level of requested refinement is reached. After the cell is refined, the particles are initialized at the center point of each subcell $\zeta$ with the volume $V_\zeta$. The particles at each point are launched in the direction of a given angular quadrature with corresponding weights ${{\left\{{\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_m, \omega_m \right\}}}_{m = 1}^{M}$. This quadrature can be deterministic or random. The particle phase-space factor is $${\ensuremath{\mathrm{w}{\ifthenelse{\isempty{p}}{}{_{p}}}}\xspace} \equiv V_\zeta \omega_m$$ and must satisfy $$\sum_{p \in i} {\ensuremath{\mathrm{w}{\ifthenelse{\isempty{p}}{}{_{p}}}}\xspace} = 4\pi V_i.$$
In this work, we assume initial conditions that are isotropic in angle and Planckian in frequency. Therefore the initial radiation energy density can be described with $${\ensuremath{\overline{E}}\xspace}_{i,{\frac{1}{2}}} = aT_{i,{\frac{1}{2}}}^4$$ where $T_{i,{\frac{1}{2}}}$ is the initial temperature at $t = 0$. Using \[eq:method:ho:particle\_field\] and the definition of the radiation energy density, \[eq:method:radiation\_energy\], we find $$I^{\mathrm{\,{ HO}}}_{p,i,{\frac{1}{2}}} = \frac{acT_{i,{\frac{1}{2}}}^4}{4\pi}.$$
### Particle trajectory
Since no accelerating forces affect the particles, their trajectory can be simply expressed as a straight ray $$\label{eq:method:ho:ray}
{\ensuremath{{\bm{x}}}\xspace}_{p}\left(t\right) = {\ensuremath{{\bm{x}}}\xspace}_{0,p} + ct{\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_{p}$$ from ${\ensuremath{{\bm{x}}}\xspace}_{0,p} = {\ensuremath{{\bm{x}}}\xspace}_{p}\left(t_0\right)$ in direction ${\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_{p}$. In curvilinear geometry, local orthogonal coordinates can be used [@park_multigroup_2018-1]. The distance a particle travels within a timestep is $s_p = c \Delta t$. The particle movement is subdivided by intersections with cell surfaces. Each subdivision produces a straight track, contained within one cell with constant material properties. After each track, either the cell changes, the particle is reflected at a boundary or the end of the time step is reached. The calculation of intersections is a common problem in computational geometry or graphics applications and many efficient algorithms for all types of surfaces can be found in literature [@glassner_introduction_1989]. We limit our mesh to cells that are strictly convex with planar surfaces. In this case, the intersection between a ray originating from within the cell and the surface of the cell is the shortest positive distance to all of the surfaces. This approach allows us to avoid costly vertex comparisons to determine which surface the particle goes through.
An infinite, planar surface in three dimensions is fully described by the implicit definition $$\label{eq:method:ho:surface}
{\ensuremath{\hat{{\bm{n}}}}\xspace}\cdot {\ensuremath{{\bm{x}}}\xspace}- b = 0$$ where $ {\ensuremath{\hat{{\bm{n}}}}\xspace}\in \mathbb{R}^3$ is the normal of the surface and $b \in \mathbb{R}$ is the offset. With \[eq:method:ho:ray\] the distance the particle has to travel to cross surface $j$ from its origin ${\ensuremath{{\bm{x}}}\xspace}_{0,p,i,n}$ in cell $i$ at time step $n$ can be found as $$\label{eq:method:ho:intersection}
s_{j,p,i,n} = \frac{b_{j} - {\ensuremath{\hat{{\bm{n}}}}\xspace}_{j} \cdot {\ensuremath{{\bm{x}}}\xspace}_{0,p,i,n}}{{\ensuremath{\hat{{\bm{n}}}}\xspace}_{j} \cdot {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_p}.$$ The surfaces are extended to infinity beyond the limits of the cell. Therefore, there is an intersection with the ray, if $${\ensuremath{\hat{{\bm{n}}}}\xspace}_j \cdot {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_p \ne 0,$$ otherwise the surface and the ray are parallel. The relevant intersection of the ray is then the intersection with the smallest positive distance $$s_{p,i,n} = \min\limits_{j} s_{j,p,i,n} \qquad \text{for}~ s_{j,p,i,n} > 0.$$ Special care is necessary if the particle hits a corner as detailed in \[sec:appendix:particle\_corner\]. The time it takes the particle to reach the surface is $$\Delta t_{s,i} = \frac{s_{p,i,n}}{c}$$ and it must be smaller than the remaining time $\Delta t_p$ in the time step. Otherwise the particle cannot reach the surface within the time step. In this case, the particle is simply moved to its end of time step position $${\ensuremath{{\bm{x}}}\xspace}_{p} = {\ensuremath{{\bm{x}}}\xspace}_{0,p} + c\Delta t_p{\ensuremath{\hat{{\bm{\Omega}}}}\xspace}.$$ The remaining time of the particle is updated by $$\Delta t'_p = \Delta t_p - \Delta t_{s,i}.$$
If a particle crosses a surface that is part of the boundary, it is always reflected back into the domain. Therefore, the number of particles remain constant throughout the calculation. The new direction is found by the reflection law $$\label{eq:method:ho:reflection}
{\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_{p}' = {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_{p} - 2\left({\ensuremath{\hat{{\bm{n}}}}\xspace}_{j} {\mathbf{\cdot}}{\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_{p}\right) \cdot {\ensuremath{\hat{{\bm{n}}}}\xspace}_{j}$$ and the intensity is $${I'}^{\mathrm{\,{ HO}}}_{p,j,n} = \alpha_{j} I^{\mathrm{\,{ HO}}}_{p,j,n} + \frac{ac T_{\mathrm{BC},j}^4}{4\pi}$$ where $\alpha_{j}$ is the reflection factor. This provides both vacuum ($\alpha = 0$) and reflective ($\alpha = 1$) conditions, and in between. For vacuum boundaries with no influx, the intensity is set to ${I'}^{{\mathrm{\,{ HO}}}}_{p,j,n} = 0$, but it evolves according to \[eq:method:ho:characteristic\].
### Linear source approximation and tallying
To solve the characteristic equation, \[eq:method:ho:characteristic\], effectively, we must be able to evaluate the source term $Q^{LO}\left({\ensuremath{{\bm{x}}}\xspace}_{p}\left(t\right), t\right)$ given LO quantities. A key consideration for the source evaluation is the need to capture the asymptotic diffusion limit (a critical numerical property [@larsen_asymptotic_1987; @larsen_asymptotic_1989; @larsen_asymptotic_1983]), for which a linear (or higher order) representation of the relevant quantities $E$ and $T$ is needed [@ferrer_linear_2012; @ferrer_linear_2016; @ferrer_linear_2018; @wollaeger_implicit_2016]. Here, we consider a linear source reconstruction. However, linear descriptions are not without issues. They increase the computational cost per cell and, depending on the slope, may violate positivity (see \[sec:appendix:negative\_temperatures\] for our treatment to enforce positivity of the source). This may occur for cells with low temperatures and steep gradients, e.g., at boundary layers and at thermal fronts. Finally, not all linear source reconstructions capture correctly the asymptotic diffusion limit. In what follows, following @ferrer_linear_2016 [@ferrer_linear_2018], we first outline a general linear-reconstruction procedure for an arbitrary function that will yield a method able to capture the asymptotic diffusion limit. Later, we use this reconstruction for the emission source in the HO solver, and derive the corresponding moment tallies needed.
#### Linear reconstruction procedure
Let $\phi_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right)$ be a quantity evaluated by scoring of particles in cell $i$ and timestep $n$, which can be an arbitrary function in ${\ensuremath{{\bm{x}}}\xspace}\in V_i$ . We seek its linear representation $\psi_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right)$ within the same cell $i$. For simplicity, let $\phi_{i,n}$ be angle-independent, to focus on the spatial aspect. Its zeroth spatial moment is $$\begin{aligned}
\label{eq:method:ho:linear:phi_zeroth}
{\ensuremath{\overline{\phi}}\xspace}_{i,n} &= \frac{1}{V_{i}}\int_{V_{i}} \phi_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right) {\,dV} \notag \\
&= \frac{1}{4\pi V_{i}} \sum_{p = 1}^{P_{i}} {\ensuremath{\mathrm{w}{\ifthenelse{\isempty{p}}{}{_{p}}}}\xspace} \int_{0}^{s_{p,i,n}} \phi_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\left(s'\right)\right) {\,ds'},
\end{aligned}$$ and its first spatial moment is $$\begin{aligned}
\label{eq:method:ho:linear:phi_first}
{\ensuremath{{\bm{\widetilde{\phi}}}}\xspace}_{i,n} &= \frac{1}{V_{i}}\int_{V_{i}} {\ensuremath{{\bm{x}}}\xspace}\phi_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right) {\,dV} \notag \\
&= \frac{1}{4\pi V_{i} } \sum_{p = 1}^{P_{i}} {\ensuremath{\mathrm{w}{\ifthenelse{\isempty{p}}{}{_{p}}}}\xspace} \int_{0}^{s_{p,i,n}} {\ensuremath{{\bm{x}}}\xspace}\left(s'\right) \phi_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\left(s'\right)\right) {\,ds'},
\end{aligned}$$ where $s_{p,i,n}$ is the track of particle $p$ in cell $i$ at time step $n$, [$\mathrm{w}{\ifthenelse{\isempty{p}}{}{_{p}}}$]{} denotes the particle’s phase-space volume, and $P_i$ is the number of particles in cell $i$.
We consider the following ansatz for the linear representation of $\phi_{i,n}$ in cell ${i}$ $$\label{eq:method:ho:linear:model}
\psi_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right) = {\ensuremath{\overline{\psi}}\xspace}_{i,n} + {\ensuremath{{\bm{\widetilde{\psi}}}}\xspace}_{i,n} \cdot \left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right)$$ where ${\ensuremath{\overline{\psi}}\xspace}_{i,n}$ is the constant part and the vector ${\ensuremath{{\bm{\widetilde{\psi}}}}\xspace}_{i,n}$ is its gradient. We will adopt this notation also for other linear quantities throughout this paper. In \[eq:method:ho:linear:model\], $${\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}= \frac{1}{V_{i}} \int_{V_{i}} {\ensuremath{{\bm{x}}}\xspace}{\,dV}.$$ is the center of mass of the cell $i$. The following property follows: $$\label{eq:method:ho:linear:psi_average}
\frac{1}{V_{i}} \int_{V_{i}} \psi_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right) {\,dV} = {\ensuremath{\overline{\psi}}\xspace}_{i,n}$$ Also, by definition: $$\frac{1}{V_{i}} \int_{V_i}\left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right) {\,dV} = 0.$$ Since the spatial moments of the linear representation must equal the scored spatial moments, the zeroth moment must satisfy: $$\frac{1}{V_{i}} \int_{V_{i}} \psi_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right) {\,dV} = {\ensuremath{\overline{\phi}}\xspace}_{i,n}$$ and hence with \[eq:method:ho:linear:psi\_average\] $${\ensuremath{\overline{\psi}}\xspace}_{i,n} = {\ensuremath{\overline{\phi}}\xspace}_{i,n}.$$ To approximate the gradient, we compute the first spatial moment as: $$\frac{1}{V_{i}} \int_{V_{i}} \left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right) \psi_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right) {\,dV} = \frac{1}{V_{i}} \int_{V_{i}} \left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right) \phi_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right) {\,dV},$$ which with \[eq:method:ho:linear:phi\_zeroth,eq:method:ho:linear:phi\_first,eq:method:ho:linear:model\] gives the equation system $${\ensuremath{{\bm{\widetilde{\psi}}}}\xspace}_{i,n} \frac{1}{V_{i}}\int_{V_{i}} \left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right) \otimes \left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right) {\,dV}
= {\ensuremath{{\bm{\widetilde{\phi}}}}\xspace}_{i,n} - {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}{\ensuremath{\overline{\phi}}\xspace}_{i,n}$$ where $\otimes$ denotes the tensor product. Written algebraically, the solution is $$\label{eq:method:ho:linear:matrix_eq}
{\ensuremath{{\bm{\widetilde{\psi}}}}\xspace}_{i,n} = {\ensuremath{\mathbf{M}}}_{i}^{-1}\left({\ensuremath{{\bm{\widetilde{\phi}}}}\xspace}_{i,n} - {\ensuremath{\overline{\phi}}\xspace}_{i,n} {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right)$$ where the matrix $${\ensuremath{\mathbf{M}}}_{i} = \frac{1}{V_{i}}\int_{V_{i}} \left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right) \otimes \left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right) {\,dV}$$ only contains geometric information [@steger_calculation_1996], and can be precomputed for each cell.
#### Linear reconstruction of the emission source
Using the linear approximation, \[eq:method:ho:linear:model\], the source in cell $i$ for time step $n$ has the form $$Q^{\mathrm{\,{ LO}}}_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right) = {\ensuremath{\overline{Q}}\xspace}^{\mathrm{\,{ LO}}}_{i,n} + {\ensuremath{{\bm{\widetilde{Q}}}}\xspace}^{\mathrm{\,{ LO}}}_{i,n} \,{\mathbf{\cdot}}\left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right).$$ where ${\ensuremath{\overline{Q}}\xspace}^{\mathrm{\,{ LO}}}_{i,n} $ is the average and ${\ensuremath{{\bm{\widetilde{Q}}}}\xspace}^{\mathrm{\,{ LO}}}_{i,n}$ the source gradient. We begin by introducing the auxiliary variable $$\label{eq:method:temperature:theta}
\Theta_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right) = T_{i,n}^4\left({\ensuremath{{\bm{x}}}\xspace}\right)$$ so that the source term can be written as $$\begin{aligned}
Q^{\mathrm{\,{ LO}}}_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right) &= {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}ac\Theta_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right) \notag \\
&= {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}ac\left({\ensuremath{\overline{\Theta}}\xspace}_{i,n} + {\ensuremath{{\bm{\widetilde{\Theta}}}}\xspace}_{i,n}\cdot\left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right)\right).
\end{aligned}$$ The temperature is linearized by expanding $\Theta_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right)$ using a Taylor series $$\begin{aligned}
\label{eq:method:temperature:temp_linear}
T_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right) &= \sqrt[4]{{\ensuremath{\overline{\Theta}}\xspace}_{i,n}} + \frac{1}{4} {\ensuremath{\overline{\Theta}}\xspace}_{i,n}^{\,-\frac{3}{4}} {\ensuremath{{\bm{\widetilde{\Theta}}}}\xspace} {\mathbf{\cdot}}\left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right) \notag \\
&= {\ensuremath{\overline{T}}\xspace}_{i,n} + {\ensuremath{{\bm{\widetilde{T}}}}\xspace}_{i,n} {\mathbf{\cdot}}\left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right).
\end{aligned}$$ The cell-average temperature, ${\ensuremath{\overline{T}}\xspace}_{i,n}$, is updated according to the evolution equation obtained by integrating \[eq:method:lo\_system\_consistent:temperature\] over time step $n$ and cell $i$ \[and using \[eq:method:ho:linear:psi\_average\]\]: $$\label{eq:method:temperature:average}
\frac{{\ensuremath{\rho}\xspace}{\ensuremath{c_{v}}\xspace}}{\Delta t_n} \left({\ensuremath{\overline{T}}\xspace}_{i,n} - {\ensuremath{\overline{T}}\xspace}_{i,n-1}\right) + {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} ac{\ensuremath{\overline{\Theta}}\xspace}_{i,n} - {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} c{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ LO}}}_{i,n} = 0$$ For the temperature gradient, $ {\ensuremath{{\bm{\widetilde{T}}}}\xspace}_{i,n}$, we use the first spatial moment of \[eq:method:lo\_system\_consistent:temperature\], $$\frac{1}{V_{i}\Delta t_n} \int_{\Delta t_{n}} \int_{V_i}\left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right) \left[{\ensuremath{\rho}\xspace}{\ensuremath{c_{v}}\xspace}{{\frac{\partial T_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right)}{\partial t}}} + {\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}ac\Theta_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right) - {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} cE^{\mathrm{\,{ LO}}}_{i,n}\left({\ensuremath{{\bm{x}}}\xspace}\right)\right] {\,dV} {{\,dt}}= 0$$ to obtain $$\label{eq:method:linear:temperature_first_moment}
\frac{1}{V_{i}} \int_{V_i}\left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right) \otimes \left({\ensuremath{{\bm{x}}}\xspace}- {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right) {\,dV}
\cdot \left[\frac{{\ensuremath{\rho}\xspace}{\ensuremath{c_{v}}\xspace}}{\Delta t_n} \left({\ensuremath{{\bm{\widetilde{T}}}}\xspace}_{i,n} - {\ensuremath{{\bm{\widetilde{T}}}}\xspace}_{i,n-1}\right) + {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} ac{\ensuremath{{\bm{\widetilde{\Theta}}}}\xspace}_{i,n} - {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} c{\ensuremath{{\bm{\widetilde{ E}}}}\xspace}^{\mathrm{\,{ LO}}}_{i,n}\right]
= 0.$$ Using \[eq:method:temperature:temp\_linear\], we can solve this equation for the gradient of $\Theta_{i,n}$ as a function of the gradient of $E^{LO}_{i,n}$: $${\ensuremath{{\bm{\widetilde{\Theta}}}}\xspace}_{i,n} = \frac{\frac{{\ensuremath{\rho}\xspace}{\ensuremath{c_{v}}\xspace}}{4\Delta t_{n}}{\ensuremath{\overline{\Theta}}\xspace}_{i,n-1}^{\,-\frac{3}{4}} {\ensuremath{{\bm{\widetilde{\Theta}}}}\xspace}_{i,n-1} + {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}c{\ensuremath{{\bm{\widetilde{E}}}}\xspace}^{\mathrm{\,{ LO}}}_{i,n}} {\frac{{\ensuremath{\rho}\xspace}{\ensuremath{c_{v}}\xspace}}{4\Delta t_{n}}{\ensuremath{\overline{\Theta}}\xspace}_{i,n}^{\,-\frac{3}{4}} + {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}ac}.$$ For simplicity, the LO gradient of $E_{i,n}$ is found in this study by scaling its HO gradient as: $${\ensuremath{{\bm{\widetilde{E}}}}\xspace}^{\mathrm{\,{ LO}}}_{i,n} = {\ensuremath{{\bm{\widetilde{E}}}}\xspace}^{\mathrm{\,{ HO}}}_{i,n} \frac{{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ LO}}}_{i,n}}{{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{i,n}}$$ A better choice would be to discretize the LO system with Discontinuous Galerkin (DG), but we leave this for future work. Thus, all that remains is to tally the HO average and gradient components of $E_{i,n}$. We explain next how this is done.
#### Tallying of HO moments
Combining the linearized emission source, \[eq:method:temperature:temp\_linear\], with the particle equation of motion, \[eq:method:ho:ray\], yields the emission source for a specific particle $p$: $$Q^{\mathrm{\,{ LO}}}_{p,i,n}\left(s\right) = {\ensuremath{\overline{Q}}\xspace}^{\mathrm{\,{ LO}}}_{i,n} + {\ensuremath{{\bm{\widetilde{Q}}}}\xspace}^{\mathrm{\,{ LO}}}_{i,n} {\mathbf{\cdot}}\left({\ensuremath{{\bm{x}}}\xspace}_{0,p,i,n} - {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}+ s{\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_{p}\right) \notag \\
= {\ensuremath{\overline{q}}\xspace}^{\mathrm{\,{ LO}}}_{p,i,n} + s~ {\ensuremath{\widetilde{q}}\xspace}^{\mathrm{\,{ LO}}}_{p,i,n},$$ which is a linear function of the orbit distance, $s$. Within a cell, the material properties and opacities are assumed to be constant and given by $${\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} = {\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}\left({\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}, {\ensuremath{\overline{T}}\xspace}_{i,n-1} \right).$$ We can now analytically solve the integral in \[eq:method:ho:characteristic\] and obtain the intensity function $$\label{eq:method:ho:intensity}
I^{\mathrm{\,{ HO}}}_{p,i,n}\left(s\right) = I^{\mathrm{\,{ HO}}}_{p,i,n}\left(0\right) {\mathrm{e}^{-{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} s}}
+ \left({\ensuremath{\overline{q}}\xspace}^{\mathrm{\,{ LO}}}_{p,i,n} - \frac{{\ensuremath{\widetilde{q}}\xspace}^{\mathrm{\,{ LO}}}_{p,i,n}}{{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}}\right)\frac{G\left({\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} s\right)}{{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}} +s \frac{{\ensuremath{\widetilde{q}}\xspace}^{\mathrm{\,{ LO}}}_{p,i,n}}{{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}},$$ with $G\left(\tau\right) = \left(1 - {\mathrm{e}^{-\tau}}\right)$. Note that, with the known emission source, the particle intensity asymptotes to the equilibrium solution (instead of zero) in optically thick regimes, which results in much improved behavior compared to MC.
With the intensity analytically known, we can analytically tally the contribution of particle $p$ in cell $i$ to the average and gradient of the radiation energy density. The average radiation energy density per particle is found as: $$\begin{aligned}
\label{eq:method:ho:delta_E_avg}
{\ensuremath{\overline{\delta E}}\xspace}^{\mathrm{\,{ HO}}}_{p,i,n}
&= \int_{0}^{s_{p,i,n}} I^{\mathrm{\,{ HO}}}_{p,i,n}\left(s'\right) {\,ds'} \notag \\
&= I^{\mathrm{\,{ HO}}}_{p,i,n}\left(0\right)\frac{G\left({\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} s_{p,i,n}\right) }{{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}} + \frac{1}{{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}}\left({\ensuremath{\overline{q}}\xspace}^{\mathrm{\,{ LO}}}_{p,i,n} - \frac{{\ensuremath{\widetilde{q}}\xspace}^{\mathrm{\,{ LO}}}_{p,i,n}}{{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}}\right) \notag \\
&\qquad\cdot \left(s_{p,i,n} - \frac{G\left({\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} s_{p,i,n}\right)}{{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}}\right)
+ \frac{s_{p,i,n}^2}{2{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}} {\ensuremath{\widetilde{q}}\xspace}^{\mathrm{\,{ LO}}}_{p,i,n}.
\end{aligned}$$ The average radiation energy density in cell $i$ is found as the sum of all particle contributions $$\label{eq:method:ho:E_avg}
{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{i,n} = \frac{1}{V_{i} c^2\Delta t_n}\sum_{p = 1}^{P_{i}} {\ensuremath{\mathrm{w}{\ifthenelse{\isempty{p}}{}{_{p}}}}\xspace} {\ensuremath{\overline{\delta E}}\xspace}^{\mathrm{\,{ HO}}}_{p,i,n}.$$
Per \[eq:method:ho:linear:phi\_first\], the gradient of the radiation energy density is calculated from the first spatial moment of the intensity function, with a single particle contribution given by: $$\begin{aligned}
\label{eq:method:ho:delta_E_linear}
{ {\ensuremath{{\bm{\widetilde{\delta E}}}}\xspace}}^{\mathrm{\,{ HO}}}_{p,i,n} &= \int_{0}^{s_{p,i,n}} {\ensuremath{{\bm{x}}}\xspace}\left(s'\right) I^{\mathrm{\,{ HO}}}_{p,i,n}\left(s'\right) {\,ds'}
\notag \\
&= {\ensuremath{{\bm{x}}}\xspace}_{0,p,i,n} \int_{0}^{s_{p,i,n}} I^{\mathrm{\,{ HO}}}_{p,i,n}\left(s'\right) {\,ds'}
+ {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_{p} \int_{0}^{s_{p,i,n}} s' I^{\mathrm{\,{ HO}}}_{p,i,n}\left(s'\right) {\,ds'},
\end{aligned}$$ where we used \[eq:method:ho:ray\] for the particle orbit. The first integral in \[eq:method:ho:delta\_E\_linear\] is $ {\ensuremath{\overline{\delta E}}\xspace}_{p,i,n}$, \[eq:method:ho:delta\_E\_avg\], and the second integral gives $$\begin{gathered}
\int_{0}^{s_{p,i,n}} s' I^{\mathrm{\,{ HO}}}_{p,i,n}\left(s'\right) {\,ds'}
= \left(\frac{G\left({\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} s_{p,i,n}\right)}{{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}^2} - \frac{s_{p,i,n}}{{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}}{\mathrm{e}^{-{\ensuremath{\sigma_{{\ifthenelse{\isempty{_{i,n}}}{}{_{_{i,n}}}}}}\xspace} s_{p,i,n}}} \right) I^{\mathrm{\,{ HO}}}_{p,i,n}\left(0\right)
+ \frac{s_{p,i,n}^3}{3{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}} {\ensuremath{\widetilde{q}}\xspace}^{\mathrm{\,{ LO}}}_{p,i,n}
\\
+ \left({\ensuremath{\overline{q}}\xspace}^{\mathrm{\,{ LO}}}_{p,i,n} - \frac{{\ensuremath{\widetilde{q}}\xspace}^{\mathrm{\,{ LO}}}_{p,i,n}}{{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}}\right)\left(\frac{s_{p,i,n}^2}{2{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}} + \frac{s_{p,i,n}}{{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}^2}{\mathrm{e}^{-{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} s_{p,i,n}}} - \frac{G\left({\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} s_{p,i,n}\right)}{{\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}^3}\right).
\end{gathered}$$ Projecting the gradient according to \[eq:method:ho:linear:matrix\_eq\], the linear reconstruction of the gradient of the radiation energy density reads: $$\label{eq:method:ho:E_linear}
{\ensuremath{{\bm{\widetilde{E}}}}\xspace}^{\mathrm{\,{ HO}}}_{i,n} = {\ensuremath{\mathbf{M}}}^{-1} \left[\frac{1}{V_{i} c^2\Delta t_{n}}\sum_{p=1}^{P_{i}} {\ensuremath{\mathrm{w}{\ifthenelse{\isempty{p}}{}{_{p}}}}\xspace}{{\ensuremath{{\bm{\widetilde{\delta E}}}}\xspace}}^{\mathrm{\,{ HO}}}_{p,i,n} - {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{i,n}\right].$$
It is useful to point out that, in voids (${\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} = 0$) \[eq:method:ho:intensity,eq:method:ho:delta\_E\_avg,eq:method:ho:delta\_E\_linear\] simplify to
$$\begin{aligned}
I^{\mathrm{\,{ HO}}}_{p,i,n}\left(s\right) &= I^{\mathrm{\,{ HO}}}_{p,i,n} \left(0\right), \\
{\ensuremath{\overline{\delta E}}\xspace}^{\mathrm{\,{ HO}}}_{p,i,n} &= s I^{\mathrm{\,{ HO}}}_{p,i,n} \left(0\right), \\
{\ensuremath{{\bm{\widetilde{\delta E}}}}\xspace}^{\mathrm{\,{ HO}}}_{p,i,n} &= {\frac{s^2}{2}} I^{\mathrm{\,{ HO}}}_{p,i,n} \left(0\right) {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_{p},
\end{aligned}$$
while \[eq:method:ho:E\_avg,eq:method:ho:E\_linear\] remain the same.
Other required tallies include the end-of-time-step radiation energy density, which is the census of particles within a cell: $${\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{i,n+{\frac{1}{2}}} = \frac{1}{cV_{i}} \sum_{p=1}^{P_{i}} {\ensuremath{\mathrm{w}{\ifthenelse{\isempty{p}}{}{_{p}}}}\xspace} I^{\mathrm{\,{ HO}}}_{p,i,n+{\frac{1}{2}}},$$ and the partial fluxes across surface $j$, which are the sum of intensities of all particles crossing it:
$$\begin{aligned}
f^{+{\mathrm{\,{ HO}}}}_{j,n} &= \frac{1}{A_j c\Delta t_n} \sum_{p=1}^{P_{j}} {\ensuremath{\mathrm{w}{\ifthenelse{\isempty{p}}{}{_{p}}}}\xspace} I^{\mathrm{\,{ HO}}}_{p,j,n} \qquad \text{for} \quad {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_{p} {\mathbf{\cdot}}{\ensuremath{\hat{{\bm{n}}}}\xspace}_{j} > 0,
\\
f^{-{\mathrm{\,{ HO}}}}_{j,n} &= \frac{1}{A_j c\Delta t_n} \sum_{p=1}^{P_{j}} {\ensuremath{\mathrm{w}{\ifthenelse{\isempty{p}}{}{_{p}}}}\xspace} I^{\mathrm{\,{ HO}}}_{p,j,n} \qquad \text{for} \quad {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_{p} {\mathbf{\cdot}}{\ensuremath{\hat{{\bm{n}}}}\xspace}_{j} < 0.
\end{aligned}$$
Multi-frequency extension {#sec:method:mf}
-------------------------
For frequency-dependent problems, we employ the standard multi-frequency discretization. The specific angular intensity $I_g$ for group $g$ is defined as $$I^{\mathrm{\,{ HO}}}_g\left({\ensuremath{{\bm{x}}}\xspace}, {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}, t\right) = \int_{{\ensuremath{\nu}\xspace}_{g - {\frac{1}{2}}}}^{{\ensuremath{\nu}\xspace}_{g+{\frac{1}{2}}}} I\left({\ensuremath{{\bm{x}}}\xspace}, {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}, {\ensuremath{\nu}\xspace}, t\right) {\,d{\ensuremath{\nu}\xspace}}.$$ This leads to the multi-group TRT equation $$\frac{1}{c} {{\frac{\partial I^{\mathrm{\,{ HO}}}_g}{\partial t}}} + {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}{\mathbf{\cdot}}{{\bm{\nabla}}}I^{\mathrm{\,{ HO}}}_g + {\ensuremath{\sigma_{{\ifthenelse{\isempty{g}}{}{_{g}}}}}\xspace} I^{\mathrm{\,{ HO}}}_g = \frac{{\ensuremath{\sigma_{{\ifthenelse{\isempty{g}}{}{_{g}}}}}\xspace} b_g ac T^4}{4\pi},$$ where $$b_g\left(T\right) = \frac{\int_{\nu_{g - {\frac{1}{2}}}}^{\nu_{g+{\frac{1}{2}}}} B\left(\nu, T\right) {\,d\nu}}{\int_{0}^{\infty} B\left(\nu, T\right) {\,d\nu}}
= \frac{\int_{\nu_{g-{\frac{1}{2}}}}^{\nu_{g+{\frac{1}{2}}}} B\left(\nu, T\right) {\,d\nu}}{\frac{1}{4\pi}acT^4}$$ is the Planck Spectrum factor. The characteristic solution along a particle trajectory, \[eq:method:ho:characteristic\], can be written as $$\label{eq:method:mf:characteristic}
I^{\mathrm{\,{ HO}}}_{g,p}\left(t\right) = I^{\mathrm{\,{ HO}}}_{g,p}\left(t_0\right) {\mathrm{e}^{-\int_{t_0}^{t} {\ensuremath{\sigma_{{\ifthenelse{\isempty{g}}{}{_{g}}}}}\xspace} c{{\,dt}}'}}
+ \int_{t_0}^{t} {\mathrm{e}^{-\int_{t'}^{t}{\ensuremath{\sigma_{{\ifthenelse{\isempty{g}}{}{_{g}}}}}\xspace} c{{\,dt}}''}} Q^{\mathrm{\,{ LO}}}_g\left({\ensuremath{{\bm{x}}}\xspace}_{p}\left(t'\right), t'\right) c{{\,dt}}'$$ with $$Q^{\mathrm{\,{ LO}}}_g\left({\ensuremath{{\bm{x}}}\xspace}, t'\right) = \frac{b_g a c T^4\left({\ensuremath{{\bm{x}}}\xspace}, t\right)}{4\pi} .$$ Using the same procedure as in \[sec:method:ho\], the group-wise quantities $I^{\mathrm{\,{ HO}}}_{p,i,n}\left(s\right)$, ${\ensuremath{\overline{\delta E}}\xspace}^{\mathrm{\,{ HO}}}_{g,p,i,n} $, ${{\ensuremath{{\bm{\widetilde{\delta E}}}}\xspace}}^{\mathrm{\,{ HO}}}_{g,p,i,n}$ are calculated the same as the gray counterparts in \[eq:method:ho:intensity,eq:method:ho:delta\_E\_avg,eq:method:ho:delta\_E\_linear\], respectively, using $I^{\mathrm{\,{ HO}}}_{g,p,i,n}$, ${\ensuremath{\overline{q}}\xspace}^{\mathrm{\,{ LO}}}_{g,p,i,n}$, ${\ensuremath{\widetilde{q}}\xspace}^{\mathrm{\,{ LO}}}_{g,p,i,n}$ and [$\sigma_{{\ifthenelse{\isempty{g,i,n}}{}{_{g,i,n}}}}$]{}. The tallied quantities then become $$\begin{aligned}
{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{i,n} &= \frac{1}{V_{i} c^2\Delta t_n}\sum_{p = 1}^{P_{i}} {\ensuremath{\mathrm{w}{\ifthenelse{\isempty{p}}{}{_{p}}}}\xspace} \sum_{g=1}^{G} {\ensuremath{\overline{\delta E}}\xspace}^{\mathrm{\,{ HO}}}_{g,p,i,n} \\
{\ensuremath{{\bm{\widetilde{E}}}}\xspace}^{\mathrm{\,{ HO}}}_{i,n} &= {\ensuremath{\mathbf{M}}}^{-1} \left[\frac{1}{V_{i} c^2\Delta t_{n}}\sum_{p=1}^{P_{i}} {\ensuremath{\mathrm{w}{\ifthenelse{\isempty{p}}{}{_{p}}}}\xspace} \sum_{g=1}^{G} {{\ensuremath{{\bm{\widetilde{\delta E}}}}\xspace}}^{\mathrm{\,{ HO}}}_{g,p,i,n} - {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{i,n}\right] \\
{\ensuremath{\overline{E}}\xspace}^{\mathrm{\,{ HO}}}_{i,n+{\frac{1}{2}}} &= \frac{1}{cV_{i}} \sum_{p=1}^{P_{i}} {\ensuremath{\mathrm{w}{\ifthenelse{\isempty{p}}{}{_{p}}}}\xspace} \sum_{g=1}^{G} I^{\mathrm{\,{ HO}}}_{g,p,i,n+{\frac{1}{2}}} \\
f^{\pm{\mathrm{\,{ HO}}}}_{j,n} &= \frac{1}{A_j c\Delta t_n} \sum_{p=1}^{P_{j}} {\ensuremath{\mathrm{w}{\ifthenelse{\isempty{p}}{}{_{p}}}}\xspace} \sum_{g=1}^{G} I^{\mathrm{\,{ HO}}}_{g,p,j,n}
\qquad \text{for} \quad {\ensuremath{\hat{{\bm{\Omega}}}}\xspace}_{p} {\mathbf{\cdot}}{\ensuremath{\hat{{\bm{n}}}}\xspace}_{j} \gtrless 0
\end{aligned}$$ with the only difference being the summation over the groups.
The LO-system remains gray in the multi-frequency case. However, the opacities must be changed to weighted opacities [@yee_stable_2017]. The LO-system becomes
$$\begin{aligned}
{{\frac{\partial E^{\mathrm{\,{ LO}}}}{\partial t}}} + {{{\bm{\nabla}}}{\mathbf{\cdot}}}{\bm{F}}^{\mathrm{\,{ LO}}}+ {\ensuremath{\sigma_{\mathrm{E}{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}^{\mathrm{\,{ HO}}}c E^{\mathrm{\,{ LO}}}- {\ensuremath{\sigma_{\mathrm{P}{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}ac T^{4} &= {\ensuremath{\mathcal{S}^{\mathrm{\,{ HO}}}}\xspace}, \\
\frac{1}{c} {{\frac{\partial {\bm{F}}^{\mathrm{\,{ LO}}}}{\partial t}}} + \frac{c}{3}{{\bm{\nabla}}}E^{\mathrm{\,{ LO}}}+ {\ensuremath{\sigma_{\mathrm{R}{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}{\bm{F}}^{\mathrm{\,{ LO}}}&= {\bm{\gamma}}^{\mathrm{\,{ HO}}}cE^{\mathrm{\,{ LO}}}, \\
{\ensuremath{\rho}\xspace}{\ensuremath{c_{v}}\xspace}{{\frac{\partial T}{\partial t}}} + {\ensuremath{\sigma_{\mathrm{P}{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}ac T^4 - {\ensuremath{\sigma_{\mathrm{E}{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}^{\mathrm{\,{ HO}}}cE^{\mathrm{\,{ LO}}}&= 0,
\end{aligned}$$
where
$${\ensuremath{\sigma_{\mathrm{E}{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}^{\mathrm{\,{ HO}}}\equiv \frac{\int_{0}^{\infty} {\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}\int_{4\pi} I^{\mathrm{\,{ HO}}}{{\,d\Omega}}{\,d{\ensuremath{\nu}\xspace}}}{\int_{0}^{\infty} \int_{4\pi} I^{\mathrm{\,{ HO}}}{{\,d\Omega}}{\,d{\ensuremath{\nu}\xspace}}}
= \frac{\sum_{g=1}^{G} {\ensuremath{\sigma_{{\ifthenelse{\isempty{g}}{}{_{g}}}}}\xspace}\int_{4\pi} I^{\mathrm{\,{ HO}}}_g{{\,d\Omega}}}{cE^{\mathrm{\,{ HO}}}}$$
is the radiation weighted opacity evaluated from the HO system, $${\ensuremath{\sigma_{\mathrm{P}{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}\equiv \frac{\int_{0}^{\infty} {\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}B\left({\ensuremath{\nu}\xspace}, T\right) {\,d{\ensuremath{\nu}\xspace}}}{\int_{0}^{\infty} B\left({\ensuremath{\nu}\xspace}, T\right) {\,d{\ensuremath{\nu}\xspace}}}
= \sum_{g=1}^{G} {\ensuremath{\sigma_{{\ifthenelse{\isempty{g}}{}{_{g}}}}}\xspace} b_g$$
is the Planck weighted opacity, and $${\ensuremath{\sigma_{\mathrm{R}{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}= \frac{\int_{0}^{\infty} {\frac{\partial B}{\partial T}}\big\vert_{T} {\,d{\ensuremath{\nu}\xspace}}}
{\int_{0}^{\infty} \frac{1}{{\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}\left({\ensuremath{\nu}\xspace}, T\right)}{\frac{\partial B}{\partial T}}\big\vert_{T} {\,d{\ensuremath{\nu}\xspace}}}
= \frac{\sum_{g=1}^{G} {\frac{\partial B_g}{\partial T}}\big\vert_{T}}{\sum_{g=1}^{G} \frac{1}{{\ensuremath{\sigma_{{\ifthenelse{\isempty{g}}{}{_{g}}}}}\xspace}}{\frac{\partial B_g}{\partial T}}\big\vert_{T}}$$ is the Rosseland weighted opacity. It is important to note that we use the most up-to-date quantities $E^{\mathrm{\,{ HO}}}$, $I^{\mathrm{\,{ HO}}}_g$ and $b_g$ to evaluate the weighted opacities, regardless the temporal centering of the multi-frequency opacities [$\sigma_{{\ifthenelse{\isempty{g}}{}{_{g}}}}$]{}. The solution strategy for the LO system remains the same as described for the gray case, \[sec:method:lo\].
We emphasize here that there is a significant advantage of the DP method compared to IMC or [$S_N$]{}for multi-frequency problems in that the particle trajectory, \[eq:method:ho:ray\], is independent of frequency and thus all frequency information is carried by each particle. Thus, we can provide the same phase-space resolution with the same number of particles regardless of the number of frequency groups. Although each particle carries more information, and therefore the memory requirements will increase, we can use a single ray-tracing step per particle, thus reducing the computational effort per group.
Numerical Results {#sec:results}
=================
In the following section, we present multi-dimensional numerical results for the Tophat and Hohlraum problems. One-dimensional demonstrations for the DP method can be found in earlier publications [@hammer_multi-dimensional_2018; @park_multigroup_2019].
Tophat problem
--------------
The Tophat, or crooked pipe, problem is a two-dimensional problem in which a region of dense, opaque material (${\ensuremath{\rho}\xspace}= \SI{10}{g\per\centi\metre\cubed}$, ${\ensuremath{c_{v}}\xspace}= \SI{1e12}{erg\per\gram\per\electronvolt}$, ${\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}= \SI{2000}{\per\centi\metre}$) is embedded into a channel of thin material (${\ensuremath{\rho}\xspace}= \SI{0.1}{g\per\centi\metre\cubed}$, ${\ensuremath{c_{v}}\xspace}= \SI{1e12}{erg\per\gram\per\electronvolt}$, ${\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}= \SI{0.2}{\per\cm}$). The channel itself is surrounded by the opaque material. While the original definition was in cylindrical coordinates [@gentile_implicit_2001], we have adapted the problem to Cartesian coordinates. The problem is by with a reflective boundary condition at $x = 0$, and vacuum on all other sides. shows the geometry of the problem with the corresponding measurements. The grayed regions contain the opaque material, while the white region is optically thin. The mesh used for the calculation is rectangular with $\Delta x = \Delta y = \SI{0.05}{\cm}$, which gives by cells. In each cell, particles were initialized at the four centers of the corner subcells, using a [$S_N$]{}standard Gauss-Chebychev product quadrature with 8 polar and 24 azimuthal angles (see \[sec:appendix:particle\_initalization\] for details). This results in a total of particles.
At the beginning, the problem is in thermal equilibrium at the initial temperature $T_0 = \SI{50}{\electronvolt}$ and a temperature source $T_\mathrm{inc} = \SI{500}{\electronvolt}$ is applied to the bottom of the thin channel at $y=\SI{0}{\centi\metre}$, $x < \SI{0.5}{\centi\metre}$. The problem was run with an initial time step of $\Delta t_0 = \SI{1e-12}{\second}$, which increased by a factor of 1.1 each step up to a maximum of $\Delta t_\mathrm{max} = \SI{5e-11}{\second}$, for a total time of $t_\mathrm{end} = \SI{1e-6}{\second}$. We used a tolerance of $\tau_\mathrm{HOLO} = \SI{1e-4}{}$ for the HOLO solver with the convergence criteria $$\frac{{\lVert{\bm{E}}^{\mathrm{\,{ HO}}}_{n} - {\bm{E}}^{\mathrm{\,{ LO}}}_{n}\rVert_{\infty}}}{{\bm{E}}^{\mathrm{\,{ LO}}}_{n}} < \tau_\mathrm{HOLO},$$ $\tau_\mathrm{P1} = \SI{1e-8}{}$ for the P1 solver with the error calculated as the relative infinity norm of the nonlinear Newton-update, and $\tau_T = \SI{1e-12}{}$ for the temperature solver with the relative Newton-update as error.
With time, the radiation travels along the thin channel. The problem cannot be solved accurately with diffusion alone, since diffusion cannot model the flow of radiation around the corners. On the other hand, it is necessary for the algorithm to respect the asymptotic diffusion limit, or the radiation will diffuse too fast into the thick material.
We use six points to track the temperature evolving over time, five in the thin material ($X_1 - X_5$), plus one in the thick material ($X_6$). The points and their corresponding coordinates are shown in \[fig:results:tophat:design\].
![Material temperature for the Tophat problem at using a square mesh and flat- and linear-source approximations[]{data-label="fig:results:tophat:square_temp"}](graphics/tophat_flat_square_final_T_mat.png "fig:"){width="\textwidth"} \[fig:results:tophat:square\_temp:flat\]
![Material temperature for the Tophat problem at using a square mesh and flat- and linear-source approximations[]{data-label="fig:results:tophat:square_temp"}](graphics/tophat_linear_square_final_T_mat.png "fig:"){width="\textwidth"} \[fig:results:tophat:square\_temp:linear\]
The material temperature at the final time is shown in \[fig:results:tophat:square\_temp\]. These calculations use an orthogonal or square mesh. \[fig:results:tophat:square\_temp\] clearly shows the effect of the linear source approximation. Without a linear source, the radiation diffuses too fast into the thick material (\[fig:results:tophat:square\_temp:flat\]). Using the linear-source representation, the temperature only heats the first line of cells in the thick material except at the corners of the channel, giving a highly improved solution. The excessive numerical diffusion at the corners is caused by the lack of the bi-linear term in the source shape, which is necessary to recover the asymptotic diffusion limit on rectangular meshes [@adams_characteristic_1998]. The error shows preferentially at the corners of the channel, because the linear representation cannot describe the temperature profile sufficiently there.
![Material temperature for the Tophat problem at using a triangular sub-mesh and flat- and linear-source approximations[]{data-label="fig:results:tophat:tri_temp"}](graphics/tophat_flat_final_T_mat.png "fig:"){width="\textwidth"} \[fig:results:tophat:tri\_temp:flat\]
![Material temperature for the Tophat problem at using a triangular sub-mesh and flat- and linear-source approximations[]{data-label="fig:results:tophat:tri_temp"}](graphics/tophat_linear_final_T_mat.png "fig:"){width="\textwidth"} \[fig:results:tophat:tri\_temp:linear\]
The linear-source approximation is sufficient for the asymptotic diffusion limit on triangular meshes. To show this, we used a triangular mesh where four cells of the square mesh were combined and then divided into triangles using the center of these four cells as shown in \[fig:results:tophat:mesh\]. This approach maintained the number of cells, particles and the area per cell, without introducing any preferred directionality. The results for the flat source do not show any significant change compared to the square mesh (\[fig:results:tophat:tri\_temp:flat\]), however the linear case does not exhibit the excessive numerical diffusion at the corner of the channel (\[fig:results:tophat:tri\_temp:linear\]). This demonstrates that the corner problem is caused by a lack of preservation of the asymptotic limit on the quadrilateral elements.
\[fig:results:tophat:time\_material:x1\]
\[fig:results:tophat:time\_material:x2\]
\[fig:results:tophat:time\_material:x3\]
\[fig:results:tophat:time\_material:x4\]
\[fig:results:tophat:time\_material:x5\]
\[fig:results:tophat:time\_material:x6\]
The time-dependent material temperature for the tracking points is shown in \[fig:results:tophat:time\_material\]. The further down the channel the tracking point is, the larger is the difference between the flat- and linear-source results (), with the flat-source cases showing a much slower increase of temperature. The linear-source cases show also differences between the triangular and the square mesh in regard to how fast the heating occurs. While the square mesh shows faster heating, the final temperature is approximately the same as for the triangular mesh once an equilibrium is reached.
The results for $X_6$ (\[fig:results:tophat:time\_material:x6\]) show the material temperature away from the channel at a corner. It confirms the previous finding with regard to the diffusion limit. The cases using a flat-source approximation show a strong increase of the material temperature. The linear-source case on the square mesh also shows an increase in the temperature for later times, but it is significantly smaller than for the flat-source cases. The linear-source case on the triangular mesh, preserving the asymptotic diffusion limit, shows no increase at all for point $X_6$.
We have compared our results to Capsaicin [@thompson_capsaicin:_2006]. Capsaicin shows faster transients than our code with the linear source, especially for points $X_4$ and $X_5$. But both codes show the same asymptotic solution for later times. The differences in the transients arise from the mesh, which is not sufficiently refined to resolve the boundary layer at the interface between the channel and the thick material.
Hohlraum problem
----------------
The second problem we present in this paper is the heating of a cavity from a radiation source. Similar problems have been studied in literature [@mcclarren_robust_2010; @brunner_forms_2002], but with different geometry and materials. The layout is shown in \[fig:results:hohlraum:design\]. The problem is by , with a square mesh of $39\times84$ cells. The walls of the cavity have a frequency-dependent opacity $$\label{eq:results:hohlraum:opacity}
{\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}\left(\nu, T\right) = {\ensuremath{\rho}\xspace}\alpha\frac{1 - {\mathrm{e}^{-\frac{{\ensuremath{h}\xspace}\nu}{{\ensuremath{k_\mathrm{B}}\xspace}T}}}}{\left({\ensuremath{h}\xspace}\nu\right)^3},$$ with the opacity factor $\alpha = \SI{1e12}{ \electronvolt\cubed\centi\meter\squared\per\gram }$, density ${\ensuremath{\rho}\xspace}= \SI{1.0}{\gram\per\centi\metre\cubed}$ and the heat capacity ${\ensuremath{c_{v}}\xspace}= \SI{3e12}{erg\per\gram\per\electronvolt }$, while the cavity is filled with a material that is almost a vacuum (${\ensuremath{\rho}\xspace}= \SI{1e-3}{\gram\per\centi\metre\cubed}$, ${\ensuremath{c_{v}}\xspace}= \SI{1e12}{ erg\per\gram\per\electronvolt}$) and highly transparent (${\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}= \SI{1e-8}{\per\cm}$). The frequency was discretized into 100 uniform, logarithmic groups between ${\ensuremath{\nu}\xspace}_\mathrm{min} = \SI{1e-3}{\electronvolt}$ and ${\ensuremath{\nu}\xspace}_\mathrm{max} = \SI{1e6}{\electronvolt}$. The left side of the problem has a reflective boundary condition, all other sides are vacuum conditions. At the beginning, the problem is in thermal equilibrium at $T_0 = \SI{1.0}{\electronvolt}$, with the temperature at $y=\SI{0}{\centi\metre}$ set to $T_\mathrm{inc} = \SI{300}{\electronvolt}$. The problem was run with an initial time step of $\Delta t_0 = \SI{1e-12}{\second}$, increased by a factor of 1.1 each step up to a maximum of $\Delta t_\mathrm{max} = \SI{5e-12}{\second}$, for a total time of $t_\mathrm{end} = \SI{1e-8}{\second}$. We used a tolerance of $\tau_\mathrm{HOLO} = \SI{1e-4}{}$ for the HOLO solver, $\tau_\mathrm{P1} = \SI{1e-8}{}$ for the P1 solver and $\tau_T = \SI{1e-12}{}$ for the temperature solver.
![Radiation temperature for the Hohlraum problem at using the different quadrature types.[]{data-label="fig:results:hohlraum:temp_rad"}](graphics/hohlraum_gauss_final_T_rad.png "fig:"){width="\textwidth"} \[fig:results:hohlraum:temp\_rad:gauss\]
![Radiation temperature for the Hohlraum problem at using the different quadrature types.[]{data-label="fig:results:hohlraum:temp_rad"}](graphics/hohlraum_aligned_final_T_rad.png "fig:"){width="\textwidth"} \[fig:results:hohlraum:temp\_rad:aligned\]
![Radiation temperature for the Hohlraum problem at using the different quadrature types.[]{data-label="fig:results:hohlraum:temp_rad"}](graphics/hohlraum_random_final_T_rad.png "fig:"){width="\textwidth"} \[fig:results:hohlraum:temp\_rad:random\]
![Radiation temperature for the Hohlraum problem at using the different quadrature types.[]{data-label="fig:results:hohlraum:temp_rad"}](graphics/capsaicin_T_rad.png "fig:"){width="\textwidth"} \[fig:results:hohlraum:temp\_rad:capsaicin\]
![Material temperature for the Hohlraum problem at using the different quadrature types.[]{data-label="fig:results:hohlraum:temp_mat"}](graphics/hohlraum_gauss_final_T_mat.png "fig:"){width="\textwidth"} \[fig:results:hohlraum:temp\_mat:gauss\]
![Material temperature for the Hohlraum problem at using the different quadrature types.[]{data-label="fig:results:hohlraum:temp_mat"}](graphics/hohlraum_aligned_final_T_mat.png "fig:"){width="\textwidth"} \[fig:results:hohlraum:temp\_mat:aligned\]
![Material temperature for the Hohlraum problem at using the different quadrature types.[]{data-label="fig:results:hohlraum:temp_mat"}](graphics/hohlraum_random_final_T_mat.png "fig:"){width="\textwidth"} \[fig:results:hohlraum:temp\_mat:ranomd\]
![Material temperature for the Hohlraum problem at using the different quadrature types.[]{data-label="fig:results:hohlraum:temp_mat"}](graphics/capsaicin_T_mat.png "fig:"){width="\textwidth"} \[fig:results:hohlraum:temp\_mat:capsaicin\]
The radiation temperature at $t = \SI{1e-8}{\second}$ is shown in \[fig:results:hohlraum:temp\_rad:gauss\] using a standard [$S_N$]{}Gauss-Chebychev product quadrature with 8 polar and 24 azimuthal angles. This quadrature results a total number of particles used in the calculation, where we initialize 4 particles per cell. Note that the number of particles remains constant once initialized, as no particles are killed or created during the calculation. The plot clearly shows ray effects, with heating preferentially occurring along directions included in the quadrature set, while the temperature between these directions stays unphysically cold. They can be seen especially well in the upper region, where the wall sees only a highly localized deposition of energy (as shown by the material temperature in \[fig:results:hohlraum:temp\_mat:gauss\]), whereas it should be a much wider deposition on the part that is not shadowed. The lower wall shows a smooth material temperature profile, while the lower side of the center block already shows indications of ray effects with localized temperature extrema.
It is customary not to include the axis of the coordinate system in [$S_N$]{}quadrature sets. However, having a quadrature set that is axis-aligned improves the results at the top wall as shown in \[fig:results:hohlraum:temp\_rad:aligned\]. The axis aligned quadrature has 9 polar angles and 20 azimuthal angles, which results in a total of particles. We chose these settings to be closer to the total number of angles of the standard Gauss-Chebychev quadrature, and to avoid angles, which cause many particles to hit the corners of the square-mesh cells. While our implementation is capable of handling particles hitting cell corners, it fails for large numbers of particles because the corner case is highly ill-conditioned due to very short particle tracks. Even though the axis-aligned quadrature improves the results for the top wall, we still see strong ray effects.
To further ameliorate this problem, we switched to a random quadrature with a total number of 96 angles (which is the same as for the Gauss-Chebychev quadrature, since we are two-dimensional). These random angles are different for each particle starting point (4 per cell). After the particles are initialized, they maintain their direction, except when they are reflected at a boundary by the reflection law \[eq:method:ho:reflection\]. This approach results in a lot more angles covered by the particles, but introduces random noise. The radiation temperature does not show ray-effects, and a smooth radiation field develops in the optically thin material, as can be seen in \[fig:results:hohlraum:temp\_rad:random\]. However, we see strong differences in the thick material between neighboring cells caused by noise. The material temperature in \[fig:results:hohlraum:temp\_mat:ranomd\] shows this well for the lower wall. Note that the HOLO solver did not converge for the random quadrature case due to abrupt changes in particle surface fluxes when particles cross cells, stalling after about two iterations at a residual magnitude of approximately . Therefore we limited the number of HOLO iterations to 5 per time step for the random cases. A test with an increased number of particles per cell showed a significantly improved HOLO convergence. Future work will explore higher-order particle interpolation to ameliorate this problem.
The results obtained with Capsaicin [@thompson_capsaicin:_2006] are shown in \[fig:results:hohlraum:temp\_rad:capsaicin,fig:results:hohlraum:temp\_mat:capsaicin\]. We see good agreement for the radiation temperature in the optically thin material between Capsaicin and the standard Gauss-Chebychev quadrature, \[fig:results:hohlraum:temp\_rad:gauss\]. The ray effects are more smeared out in Capsaicin but clearly visible. The material temperature was higher and more evenly distributed in the upper part of the bottom wall. We believe these differences are caused by Capsaicin’s linearization of the $T^4$ nonlinearity in the emission source.
shows the radiation wave front at the time $t = \SI{4e-11}{\second}$ along the dashed line in \[fig:results:hohlraum:design\]. The exact location should be $c t = \SI{1.1991}{\centi\metre}$. The results show that our code is within reasonable range of this analytical value. Deviations from this value come from the different angles contained in the quadrature used, i.e., the Gauss quadrature has no direction going perpendicular to the wave front, which results in a slower propagation. The steps in the temperature profile for both the Gauss quadrature and the axis-aligned quadrature are related to the discrete propagation angles considered. The Capsaicin wave front has propagated much further compared to both our results and the analytical value, a consequence of the backward Euler time discretization introducing excessive numerical diffusion.
Runtime and Convergence
-----------------------
To control the runtime necessary for the convergence studies, we limited our study to the one-dimensional Marshak-wave problem. The implementation is the same as in two dimensions.
The Marshak-wave problem propagates a radiation wave through a material with a temperature dependent opacity $${\ensuremath{\sigma_{{\ifthenelse{\isempty{}}{}{_{}}}}}\xspace}\left(T\right) = \frac{{\ensuremath{\rho}\xspace}\alpha}{T^3}$$ with the opacity factor $\alpha = \SI{1e6}{\electronvolt\cubed\centi\meter\squared\per\gram}$. The problem is long divided into $n$ mesh cells, with vacuum boundaries on both sides. In the beginning, the problem is in thermal equilibrium at $T_0 = \SI{0.025}{\electronvolt}$, and on the left side a temperature of $T_\mathrm{inc} = \SI{150}{\electronvolt}$ is applied. The problem was run with an initial time step of $\Delta t_0 = \SI{1e-12}{\second}$, which increased by a factor of 1.1 each step up to a prescribed maximum $\Delta t_\mathrm{max}$, for a total time of $t_\mathrm{end} = \SI{5e-8}{\second}$.
The total runtime is a function of the maximum time step size $\Delta t_\mathrm{max}$, as shown in \[fig:results:convergence:dt\_dx\] for different mesh sizes. There are three major effects influencing the runtime. The first factor is the number of time steps required to reach the final time, which decreases inversely with increasing $\Delta t_\mathrm{max}$. The second is the number of cells a particle crosses on average during one time step, which increases proportionally with the time step size. The last is the number of HOLO iterations necessary to converge the residual. This number increases for large time step sizes, while it remains almost constant for small ones. The combination of these leads to an optimal time step size for which the total run time is minimal. shows that this optimum can be found for different mesh sizes for an almost constant ratio of $c\Delta t_\mathrm{max} / \Delta x$, corresponding to a maximum number of cells crossings between and . The actual number of cells crossings is a function of the particle’s direction.
While the method is stable for large time steps [@park_multigroup_2019], there are upper and lower limits to consider. If the time step is too small, no particle crosses the cell surfaces. This results in zero fluxes for the LO solver, and a decoupling of the cells. This can also occur in a later time step due to alignment of the particles resulting in no surface crossings. The upper time step limit is imposed by the dynamical time scale. If the time step size is too large, the wave front stalls as shown in \[fig:results:convergence:dt\_dx\_result\]. For $\Delta t_\mathrm{max} \le \SI{2e-11}{\second}$, or $c\Delta t_\mathrm{max} / \Delta x \lesssim 480 $, the wave front reaches the correct final position, but for larger time steps the final position of the wave front lags behind. We observed the same value of the time step threshold for different meshes.
The DP method is very well suited for multi-frequency calculations as mentioned before. All frequency information is carried by each particle, and the group iteration is the innermost loop during the tracking step. Therefore, the runtime is proportional to the number of frequency groups with a factor significantly less than unity. shows the runtime of the Marshak wave problem as a function of group number for several mesh sizes using $\Delta t_\mathrm{max} = \SI{1e-12}{\second}$. The slope is dependent on the ratio of work done within the inner loop to the work outside of it. A finer mesh results in more cell crossings, increasing the ratio and hence increasing the slope. However, the slope will always remain less than unity.
We present next results on the convergence properties of the algorithm. To reduce the runtime for the convergence studies, we limited the simulation time to $t_\mathrm{end} = \SI{2e-9}{\second}$, and the problem size to . To avoid problems with the upper time step-size limit, we used a time step of $\Delta t =\SI{1e-12}{\second}$. For the spatial convergence, we use a mesh with $n = 3200$ cells as reference to calculate the error. The results shown in \[fig:results:convergence:spatial\] indicate an approximately first-order convergence rate with spatial refinement. The convergence in angle is second order, as can be seen in \[fig:results:convergence:polar\], where the reference solution used $M_\mathrm{P} = 64$ polar angles. shows the convergence with the number of particle starting points per cell (each point uses all directions of the angular quadrature). The error with respect to a reference solution using 32 starting points is much lower than for the other cases, \[fig:results:convergence:spatial,fig:results:convergence:polar\], and converges with first order. However, there seem to be some fluctuations.
Conclusion
==========
We have extended the moment accelerated, multi-frequency deterministic particle method proposed by @park_multigroup_2019 to two dimensional thermal radiative transfer problems using a fast ray-tracing algorithm. A linear reconstruction of the emission source, obtained from a discretely consistent, moment-based low-order solver, allows the analytical integration of the characteristic equation along a particle track, resulting in an improved solution in optically thick materials compared to other particle methods. In contrast to Monte-Carlo (MC) methods, our method does not feature randomness (with the possible exception of particle initialization and random quadrature to ameliorate ray effects), therefore the solution does not contain stochastic noise. We showed how we can obtain a linear reconstruction of the energy deposition, material temperature and emission source using spatial moments. With this, we were able to demonstrate that our HO system features the asymptotic diffusion limit, at least on triangular meshes.
We further showed that, using the flexibility of a particle method, we can reduce ray-effects by using random quadrature sets, but it introduces noise in the solution.
Future work will include the implementation of a bi-linear source reconstruction for rectangular meshes, the development of a DG discretization scheme for the LO system, which self-consistently solves for the LO slope, and the extension to cylindrical geometries.
Acknowledgment
==============
This work was supported by the US Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001). Research presented in this article was supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory under project number 20160448ER.
References
==========
Appendix
========
Low-Order solution strategy {#sec:appendix:lo_solution}
---------------------------
The LO system is solved using a Newton-Krylov method with nonlinear elimination [@park_consistent_2012]. Given an iterate for the radiation energy density ${\ensuremath{\overline{E}}\xspace}_{i,n,\ell}$ at LO iteration $\ell$, the radiative flux can be calculated from $${\ensuremath{\overline{F}}\xspace}_{j,n,\ell+1} = \frac{1}{\frac{1}{c\Delta t_n} + {\ensuremath{\sigma_{{\ifthenelse{\isempty{j,n}}{}{_{j,n}}}}}\xspace}} \left(
\left(\gamma^{+}_{j,n} + \frac{1}{3 \Delta x_{j}} \right) c{\ensuremath{\overline{E}}\xspace}_{j-{\frac{1}{2}},n,\ell}
- \left(\gamma^{-}_{j,n} + \frac{1}{3\Delta x_{j}}\right) c{\ensuremath{\overline{E}}\xspace}_{j+{\frac{1}{2}},n,\ell}
+ \frac{{\ensuremath{\overline{F}}\xspace}_{j,n-1}}{c\Delta t_n} \right)$$ and the material temperature $${\ensuremath{\rho}\xspace}{\ensuremath{c_{v}}\xspace}\frac{T_{i,n,\ell+1} - T_{i,n-1}}{\Delta t_n} + {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} ac T_{i,n,\ell+1}^4 - {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} c{\ensuremath{\overline{E}}\xspace}_{i,n,\ell} = 0.$$ using a cell-wise Newton solve. With the new iterates, we can update the radiation energy density with a Newton-Krylov step. The residual vector ${\bm{R}}_{n, \ell+1}$
$$R_{i,n,\ell+1} = \frac{{\ensuremath{\overline{E}}\xspace}_{i,n,\ell} - {\ensuremath{\overline{E}}\xspace}_{i,n-1}}{\Delta t_n} + \sum_{j \in i}{\ensuremath{\hat{{\bm{n}}}}\xspace}_{ij} {\mathbf{\cdot}}{\ensuremath{\hat{{\bm{n}}}}\xspace}_{j}\frac{{\ensuremath{\overline{F}}\xspace}_{j,n,\ell+1} A_j}{V_{i}} + {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} c {\ensuremath{\overline{E}}\xspace}_{i,n,\ell} - {\ensuremath{\sigma_{{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} ac T_{i,n,\ell+1}^{4} - \mathcal{R}_{i,n}$$
is used to solve for the Newton update $$\label{eq:mathod:lo:newton_update}
\delta_{n, \ell+1} = -\tilde{{{\ensuremath{\mathbf{\mathbb{J}}}}}}_{n, \ell+1}^{-1} {\bm{R}}_{n,\ell+1}$$ and to update the solution vector for the radiation energy density $${\bm{E}}_{n,\ell+1} = {\bm{E}}_{n,\ell} + \delta_{n,\ell+1}.$$
The Jacobian in \[eq:mathod:lo:newton\_update\] is given by $$\tilde{{{\ensuremath{\mathbf{\mathbb{J}}}}}} = {{\ensuremath{\mathbf{\mathbb{J}}}}}_{EE} - {{\ensuremath{\mathbf{\mathbb{J}}}}}_{ET} {{\ensuremath{\mathbf{\mathbb{J}}}}}_{TT}^{-1} {{\ensuremath{\mathbf{\mathbb{J}}}}}_{TE} - {{\ensuremath{\mathbf{\mathbb{J}}}}}_{EF} {{\ensuremath{\mathbf{\mathbb{J}}}}}_{FF}^{-1} {{\ensuremath{\mathbf{\mathbb{J}}}}}_{FE}$$ which is found via Gauss Block elimination of the LO Jacobian $${{\ensuremath{\mathbf{\mathbb{J}}}}}= \left[
\begin{matrix}
{{\ensuremath{\mathbf{\mathbb{J}}}}}_{EE} & {{\ensuremath{\mathbf{\mathbb{J}}}}}_{EF} & {{\ensuremath{\mathbf{\mathbb{J}}}}}_{ET} \\
{{\ensuremath{\mathbf{\mathbb{J}}}}}_{FE} & {{\ensuremath{\mathbf{\mathbb{J}}}}}_{FF} & 0 \\
{{\ensuremath{\mathbf{\mathbb{J}}}}}_{TE} & 0 & {{\ensuremath{\mathbf{\mathbb{J}}}}}_{TT}
\end{matrix}
\right].$$ The submatrices can be split into two groups, diagonal matrices
$$\begin{aligned}
{{\ensuremath{\mathbf{\mathbb{J}}}}}_{EE,ii} &= \left(\frac{1}{\Delta t_n} + {\ensuremath{\sigma_{\mathrm{E}{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} c \right) \\
{{\ensuremath{\mathbf{\mathbb{J}}}}}_{ET,ii} &= -4{\ensuremath{\sigma_{\mathrm{P}{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} ac T_{i,n,\ell+1}^{3} \\
{{\ensuremath{\mathbf{\mathbb{J}}}}}_{TT,ii} &= \left(\frac{{\ensuremath{\rho}\xspace}_{i} c_{v,i}}{\Delta t_n} + 4{\ensuremath{\sigma_{\mathrm{P}{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace}acT_{i,n,\ell+1}^{3} \right) \\
{{\ensuremath{\mathbf{\mathbb{J}}}}}_{TE,ii} &= -{\ensuremath{\sigma_{\mathrm{E}{\ifthenelse{\isempty{i,n}}{}{_{i,n}}}}}\xspace} c \\
{{\ensuremath{\mathbf{\mathbb{J}}}}}_{FF,ii} &= \frac{1}{c\Delta t_{n}} + {\ensuremath{\sigma_{\mathrm{R}{\ifthenelse{\isempty{j,n}}{}{_{j,n}}}}}\xspace}
\end{aligned}$$
and matrices with off-diagonal parts $$\begin{aligned}
{{\ensuremath{\mathbf{\mathbb{J}}}}}_{EF,ij} &= {\ensuremath{\hat{{\bm{n}}}}\xspace}_{ij} \cdot {\ensuremath{\hat{{\bm{n}}}}\xspace}_{j} \frac{ A_{j}}{V_{i}} \\
{{\ensuremath{\mathbf{\mathbb{J}}}}}_{FE,ji} &= c{\ensuremath{\hat{{\bm{n}}}}\xspace}_{j} \cdot \left(\frac{{\ensuremath{\hat{{\bm{n}}}}\xspace}_{ij}}{3\Delta x_{j}} + \gamma_{ij,n} \right).
\end{aligned}$$ The boundary Jacobian is (also with off-diagonal parts) $${{\ensuremath{\mathbf{\mathbb{J}}}}}_{FE,ji}^\mathrm{bc} = -\left(1-\alpha_{j}\right) \kappa^\mathrm{out}_{j,t}.$$
Particle corner case {#sec:appendix:particle_corner}
--------------------
Special caution is necessary if a particle leaves a cell at a corner. This is indicated by two surfaces with the same distance. Even though this case seems unlikely, it has been observed during calculations. If a particle crosses a cell corner, the next cell it enters is ambiguous. If not handled correctly, the particle can enter a cell without actually being within the cell boundaries. Our mitigation strategy is to move the intersection into one of the connected surfaces, resolving all ambiguity. Due to floating-point round-off issues, this is triggered if the particle crosses a surface in a small circle around a corner, with a fraction of the distance from the surface to the cell center as radius. Our solution addresses round-off issues and is valid for all mesh sizes.
Negative temperatures {#sec:appendix:negative_temperatures}
---------------------
A linear source representation can lead to negative values. To prevent this we check all corners of a cell, and adjust, if necessary the slope. For all vertices $v$ of cell $i$, perform the test $${\ensuremath{\overline{\Theta}}\xspace}_{i,n} + {\ensuremath{{\bm{\widetilde{\Theta}}}}\xspace}_{i,n} \cdot \left({\ensuremath{{\bm{x}}}\xspace}_{v} - {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right) < \Theta_{\min}$$ where $\Theta_{\min}$ is the lowest allowed temperature. If the inequality is true, we will adjust the slope such that the value at the vertex is $\Theta_{\min}$. For this, we use the equation $${\ensuremath{\overline{\Theta}}\xspace}_{i,n} + \beta_v {\ensuremath{{\bm{\widetilde{\Theta}}}}\xspace}_{i,n} \cdot \left({\ensuremath{{\bm{x}}}\xspace}_{v} - {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right) = \Theta_{\min}$$ and solve for $$\beta_v = \frac{ \Theta_{\min} - {\ensuremath{\overline{\Theta}}\xspace}_{i,n}}{{\ensuremath{{\bm{\widetilde{\Theta}}}}\xspace}_{i,n} {\mathbf{\cdot}}\left({\ensuremath{{\bm{x}}}\xspace}_{v} - {\ensuremath{{\ensuremath{{\bm{x}}}\xspace}_{\mathrm{C},i}}\xspace}\right)}.$$ The slope is corrected as $${\ensuremath{{\bm{\widetilde{\Theta}}}}\xspace}'_{i,n} = \beta_v{\ensuremath{{\bm{\widetilde{\Theta}}}}\xspace}_{i,n}$$ This will reduce the slope so that all corners with the negative values are reset to the minimum temperature or above, while preserving the cell average.
|
---
abstract: 'Almost all novel observable phenomena in quantum optics are related to the quantum coherence. The coherence here is determined by the relative phase inside a state. Unfortunately, so far all the relevant experimental results in quantum optics are insensitive to the phase information of the coherent state. Lack of phase information may cause serious consequences in many problems in quantum optics. For example, an ensemble of two mode squeezed states is a classical ensemble if the phase in each state is totally random; but it is a non-classical ensemble if the phase in each state is fixed. As a timly application, verification of this type of phase information in an ensemble of two mode squeezed states from the conventional laser is crucial to the validity of the continuous variavable quantum teleportation(CVQT) experiment. Here we give a simple scheme to distinguish two different ensemble of states: the Rudolph-Sanders ensemble, by which each squeezed states emitted has a uniform distribution from $0-2\pi$ on the phase value; and the van Enk-Fuchs ensemble, which emmits identical states with a fixed(but unknown) phase for every state. We believe our proposal can help to give a clear picture on whether the existing two mode squeezed states so far are indeed non-classical states which can be used as the entanglement resource.'
author:
- |
Wang Xiangbin[^1], Matsumoto Keiji[^2] and Tomita Akihisa[^3]\
Imai Quantum Computation and Information project, ERATO, Japan Sci. and Tech. Corp.\
Daini Hongo White Bldg. 201, 5-28-3, Hongo, Bunkyo, Tokyo 113-0033, Japan
title: On the coherence verification of the continuous variable state in Fock space
---
\[theorem\][Corollary]{}
Quantum coherence of different states plays a fundamentally important role in the whole subject of quantum optics. The squeezed states have been widely used to demonstrate various novel non-classical properties in the past. In particular, a two mode squeezed state $$\begin{aligned}
|r,\phi>=e^{re^{i\phi}(a_1a_2-a_1^\dagger a_2^\dagger)}|00>=\sum_{k=0}^\infty \lambda^ke^{ik\phi}|00\rangle,\label{pure}\end{aligned}$$ where $a_i^{\dagger}$ and $a_i$ are bosonic creation and annihilation operators respectively in the two mode Fock space, $|00>$ is the vaccum state, is hopefully to be used as the entangled resource to carry out various novel tasks in the creteria of quantum information[@chuang] such as the quantum teleportation[@ben1; @pan; @frusawa]. Recently, this state has been used as the enatnglement resource to teleport a quantum state between two spatially separated parties[@frusawa]. However, as it is pointed out by Rudolph and Sanders[@barry] that the the phase information $\phi$ here has never been tested, therefore states as defined by the above equation from certain source could have different phase $\phi$ for each different wavepackets. The lack of the phase information causes the loss of coherence of the state. Although lots of experiments have been done sucessfully to reconstruct the continuous variable states in the past, none of them is related to the phase information. That is to say, a pure state is not the only possible state that is compatible with the observed results. As it is noted[@barry] that, a conventional meassurement on optical fields, such as homodyne detection using lasers, involving mixing of defferent incoherent fields and subsquent detection by energy absorption in photodetectors: all such meassurements are completely insensitive to any optical coherence. Actually, the state could be in arbitrary classical probabilistic distribution via the different phases. If we use the random uniform distribution, the observed state is actually a mixed diagonal state in Fock space[@barry; @molmer], which is given by $$\begin{aligned}
\frac{1}{2\pi}\int^{2\pi}_0|r,\phi><r,\phi|d\phi
=\sum^\infty_0 \lambda^{2n} |nn><nn|.\label{mix}\end{aligned}$$ Definitely, there is neither quantum coherence nor quantum entanglement in the above state. The state defined by eq(\[mix\]) is a totally classical state. If the phase $\phi$ of each states from certain source are indeed uniformly distributed from zero to $2\pi$, the so called squeezed states will have little novel properties in the practical use because in such a case all quantum coherence has been lost and all the observable phenomena are just the same as that given by classical optics. For example, we can never really take the advantage of its squeezing property for certain quadrature variable( although a certain quadrature operator ie indeed squeezed on a single wavepacket).
Due to the lack of the phase information, the observable quantum coherence property have never been verified on an ensemble of squeezed states. To answer the question whether the conventional source can produce the nontrivil quantum coherent state we must have a way to detect the phase information $\phi$.
A timly application for this type detection is on the validity of the CVQT experiment[@frusawa]. Recently, the phase information of the two mode squeezed states has drawn much attention of the physicists due to the issue of quantum teleportation of the continuous variable state in Fock space[@frusawa]. Since the tomography result is independent of the phase information $\phi$, it is also possible that the the state used as the entanglement source in the quantum teleportation experiment is a mixture of the states with different $\phi$, as defined by eq.(\[mix\]). If this is the case, then no quantum state can be teleported by such a separable state. Definitely, an ensemble of $N$ identical copies of pure states as defined by $|r,\phi><r,\phi|^{\otimes N}$ with the unknown $\phi$(but all states in the ensemble have the same $\phi$) is totally different from the ensemble as defined by eq.(\[mix\]). Unfortunetely, so far there has been no way to distinguish these two totally different cases.
Due to the unclearity of the phase $\phi$, there is a very hot discussions on the validity of the entanglement resource used in the CVQT experiment[@barry; @fuchs; @fuchs1; @barry1]. The discussions can be sumarrized as the following:
1\. Rudolph and Sanders: The phase $\phi$ in each of the squeezed states from a conventional laser source are uniformly distributed from zero to $2\pi$. The correct form of the quantum state for the ensemble is given by eq(\[mix\])
2\. van Enk and Fuchs: The traditional formalism is insufficient to describe the meachanism two mode squeezed states produced by the conventional laser source. Using quantum de Finetti theorem one can see that the two mode squeezed states produced by a conventional laser is essentially an ensemble of many coppies of identical two mode squeezed states with a fixd $\phi$, though the value of $\phi$ is unknown.
We may see that the validity of the CVQT experiment is now reduced to which of the above statements correctly describe the property of the source which produses the two mode squeezed states in the CVQT experiment. That is to say, to know the validity of CVQT experiment, we have to distinguish the Rudolph-Sanders source and the van Enk-Fuchs source. In this letter we give a scheme to detect the phase information $\phi$ in the two mode squeezed state. That is to say, the meassurement result by our scheme is $sensitive$ to the quantum coherence. Using our scheme, the ensemble of states defined by eq(\[pure\]) or eq.(\[mix\]) can be easily distinguished. Obviously, the scheme has broad potential applications in the whole subject of quantum optics. For example, given many copies of pure squeezed states, we can verify that they are indeed pure states with the same phase $\phi$. For another example, given two Fuch sources, each source emmits many identical two mode squeezed states, the phase of the state from each source $\phi_1$ and $\phi_2$ respectively, they are fixed and unknown, our scheme can be used detect the value $\phi_1-\phi_2$. As an immediate application, the scheme can be used to judge whether the states used as the entangled resource in a recent quantum teleportation[@frusawa] is the pure state as defined in eq.(\[pure\]) or a mixed state defined by eq.(\[mix\]). Consequently, this detection could give a a clear judgement on whether the experiment done in ref.[@frusawa] is essentially a quantum teleportation or a classical simulation of quantum teleportation.
Now we show how to distinguish a Roudolph-Sanders source and a van Enk-Fuchs source. Lets first see what happens to the van Enk-Fuchs source, i.e. a photon source emmits the identical pure squeezed states, all of them have the same but unknown phase $\phi$ . As it is noted in ref[@bow] that a two mode squeezed state can be produced with the specifically chosen polarization for each modes as the following $$\begin{aligned}
|\psi_1(\phi)>=\sum_{l=0}^{\infty} \lambda^l e^{il\phi}|l>_{ah}|l>_{bv},\end{aligned}$$ where the subscripts $a$, $b$ are for the mode $a$ and $b$ respectively, $h$ and $v$ represents the horizontal and vertical polarizations respectively. Now we consider another wave packet from the same source, of which the mode $a$ and $b$ is exchanged. The quantum state for this wavepacket is $|\psi_2(\phi)>=\sum_{m=0}^{\infty}\lambda^m e^{im\phi}|m>_{av}|m>_{bh}$. The total state is then given by $$\begin{aligned}
|\Psi>=|\psi_1>|\psi_2>=\sum_{l=0}^{\infty}\sum_{m=0}^{\infty}\lambda^{l+m}e^{i(l+m)\phi}
|l>_{ah}|m>_{av}|l>_{bv}|m>_{bh}.\end{aligned}$$ After taking a meassurement on the photon number of each mode, i.e., on the quantity $n=l+m$, the state $|\Psi>$ is collapsed to a specific entangled state. As it has been demonstrated in ref[@bow], this type of meassurement cab be carried out by either a quantum non-demolition meassurement[@duan] or by a more feasible way, the destructive photon counting with post selection. Suppose the meassurement result is $n$ for each mode, then the state after the meassurement is $\sum_{m=0}^{n}e^{i[(n-m)\phi+m\phi]}|n-m>_{ah}|m>_{av}|n-m>_{bv}|m>_{bh}$. The phase information is clearly included in the state after the meassurement. For simplicity, we consider only the cases of $n=1$ only. In the real experiment we can set an appropriate value of $\lambda$ so that we have a significant probability to get the result of $n=1$. With such a setting, given many coppies of state $|\Psi>$, we can always have a significant number of states with $n=l+m=1$ after the meassurement. In the case of $n=1$, the state is $$\begin{aligned}
|\Psi_1>=e^{2i\phi}(|1>_{ah}|0>_{av}|1>_{bv}|0>_{bh})+|0>_{ah}|1>_{av}|0>_{bv}|1>_{bh}.\end{aligned}$$ There is only one photon in each mode for the satte defined above. Obviously, the state defined above can be rewritten in the following way $$\begin{aligned}
|\Psi_1>=e^{2i\phi}(|H>_a|V>_b+|V>_a|H>_b).\end{aligned}$$ Where the states $|H>$, $|V>$ are for a horizontal or a vertical polarized photon states respectively. For this state, meassurement results for the polarization of the two modes are always $different$. Clearly, if the states given from the source are indeed pure states, i.e., they are many coppies of $|\Psi_1>$, then we can rotate the polarizers by the same angle to both modes, and in principle we can find certain angle($\pi/4$) by which the meassurement result of the polarization in the two modes are always $same$ ! On the other hand, if the given states from the source are mixed states as defined in eq(\[mix\]), i.e., phase $\phi$ in each wave packet can be different, we will have a clearly different observation results. In such a case, by the same operation, for all cases we get the meassurement of $l+m=1$, the state is $\frac{1}{ 2}(|HV><HV|+|VH><VH|)$. The correlation between the two modes will linearly decreased with $\cos^2\beta$ when the polarizers are rotated, here $\beta$ is the angle that is rotated. Consequently, a van Enk-Fuchs source and a Rudolph-Sanders source can be distinguished in the following way:
Initially, in both cases they are totally negatively correlated, i.e., whenever a photon is detected in mode $a$, there most be no photon detected in mode $b$, and vice versa. We rotate the polarizers by $\pi/4$, the van Enk-Fuchs source(see eq(\[pure\])) will give a totally positive correlation for the meassurement result, while the Rudolph-Sanders source(see eq(\[mix\])) will give no correlation for the meassurement. Also, in rotating the polarizers, if the correlation does not decrease linearly with $\cos^2\beta$, the source must be not a Rudolph-Sanders one.
Obviously, the above scheme can be also used to detect the phase difference for two van Enk-Fuchs sources. Now $|\psi_1>$ and $|\psi_2>$ are collected from two different sources whose unknown fixed phase are denoted by $\phi_1$ and $\phi_2$ respectively. After the meassurement on the photon number basis of each mode is done, the state up to a global phase factor is $$\begin{aligned}
|\Psi_1>=|H>|V>+e^{i(\phi_2-\phi_1)}|V>|H>.\end{aligned}$$ We rotate the polarizers continuously and test the correlations at each stage. There must be certain angle $\beta_0$ at which we can observe that the meassurement results of the two modes are totally positively correlated. This angle $\beta_0$ determines the phase difference $\phi_2-\phi_1$.
Note that in a real experiment, we actually do not need to take a meassurement on the basis of of the photon numbers of each mode, i.e. $\sum_{n=0}^\infty|n><n|$, where $n=l+m$. The meassure on such a basis could be difficult by our current technology. The only subtle task for us is to collect the wave packet $|\psi_1>$ and $|\psi_2>$ so that they are indistiuguishable. Once we can make $|\psi_1>$ $|\psi_2>$ indistinguishable we can simply carry out the detection scheme by the usual photon counting and obtain the correct result to a good approximation. Suppose we use port A and port B to detect the photons from mode A and B respectively. A polarizer is placed before each port. The polarizers are placed horizontally in the begining. We choose a very small $r$ so that $r^2<<1$ for the photon source. In the photon counting, we only record the data of those events where at least one port detects one photon $and$ no port detects more than one photon. For simplicity, we name the events satisfying this condition as “good events”. We will analyse the correlation between the two modes only using the data of the good events. In the whole process, there is only a very small probability that a good event is caused by a state with $l+m\not=1$, i.e., the wave packet has been collapsed to a state of which the photon number of each mode is not 1. It’s easy to see that this type of event can only happen with a small ptobability. First, $l+m=0$ is impossible, because whenever we observed a good event, we have observed one photon at least in one port, so the photon number of that mode must not be 0, consequently $l+m\not=0$. Second, $l+m=2$ or a larger number is possible but the chance is negligible. We have already set $r$ to be very small. Since $r^2<<1$, the probability that the state $|\Psi>$ collapsed to a state with $l+m=1$ is much larger than that of a state with $l+m>1$. For example, taking $r=0.01$, in average, we can have one good event from 10000 events, and the probability that this good event is caused by a stete with $l+m=1$ is several thousands times larger than that by a state with $l+m\not=1$. Actually, to a good approximation, we even do not have to require the photon detector to distinguish 1 photon case from many photon case in the detection if $r$ is small enough. Because once the detector detects photons, the probability of 1 photon is much higher than other cases. For clarity, we insist on using the term “1 photon” in the rest parts of the letter. However, whenever the term “one photon” is used, it can be approximately understood as “any number of photons”.
Initially, all good evnts must be the events on which 1 photon detected in certain mode and 0 photon is detected in another mode. If the source is the van Enk-Fuchs source, when the polarizers are rotated by $\pi/4$, a photon in each mode must be detected to all good events. If the source is the Rudolph-Sanders source, when the polarizers are rotated by $\pi/4$, to all good events, half of them are $(1_A,0_B)$ or $(0_A,1_B)$ and half of them are $(1_A,1_B)$, where the 0 or 1 represents the number of photons detected, subscripts $A,B$ represent the the port $A,B$(or the mode $a,b$) respectively.
The phase difference can also be detected for two Enk-Fuchs sources. In this case, we have to rotate the polarizers continuously. The phase difference $\phi_2-\phi_1$ is determined by the angle $\beta_0$ we observed 1 photon in each mode to all good events. Thus we see, to a good approximation, our scheme can carried out by the normal photon counting technique.
[**Acknowledgement:**]{} We thank Prof Imai for support.
[99]{}
Nielsen M A and Chuang I L, Quantum computation and information,Cambridge university press, 2000. C.H. Bennette et al, Teleporting an unknown quantum state via dual classical and EPR channels. Phys. Rev. Lett., 70:1895-1999(1993). Bowmeester D et al, Experimental quantum teleportation, Nature, 390, 6660(1997). A. Furusawa et al, Unconditional quantum teleportation. Science 282, 706(1998). T. Rudolph and Barry C. Sanders., Requirement of optical coherence for continuous-variable quantum teleportation, Phys. Rev. Lett., 87, 077903(2001). K. Molmer, Phys. Rev. A 55, 3195(1997). S.J. van Enk and C.A. Fuchs, Quantum state of an ideal propagating laser field, Phys. Rev. Lett., 88, 027902(2002). S.J. van Enk and C.A. Fuchs, Quantum state of an ideal propagating laser field, arXiv: quant-ph/0111157. Rudolph T and Barry C.S., Comment on “Quantum state of an ideal propagating laser field”, arXiv:quant-ph/0112020. Gabirel A Durkin, C. Simon and D. Bouwmeester, Multi-photon Entanglement Concentration and Quantum Cryptography. arXiv: quant-ph/0109132 Duan L M et al, Entanglement purification of Gaussian continuous variable quantum states, Phys. Rev. Lett., 84, 4002(2000).
[^1]: email: wang@qci.jst.go.jp
[^2]: email: keiji@qci.jst.go.jp
[^3]: email: a-tomita@az.jp.nec.com
|
---
author:
- 'C. Bonatto , E. Bica'
- 'D. B. Pavani'
date: 'Received –; accepted –'
title: 'NGC2180: a disrupting open cluster'
---
Introduction {#intro}
============
Open clusters are formed along the gas and dust-rich Galactic plane and contain from tens to a few thousands of stars distributed in an approximately spherical structure of up to a few parsecs in radius. This loose condition makes them potentially short-lived stellar systems, since disruptions may occur by the cumulative effect of passages near interstellar clouds and/or by shocks with the Galactic disk. Cumulative orbital perturbations may lead to more internal orbits, enhancing such effects Bergond et al. ([@Bergond2001]). Consequently, most of the open clusters in the Galaxy evaporate completely in less than 1Gyr. Indeed, the open cluster catalogue of Lyngå ([@Lyngaa1987]) indicated about 70 objects older than 1Gyr ($\approx6\%$ of the total number).
The dynamical evolution of an open cluster depends both on internal and external factors. Internal factors are: [*(i)*]{} after successive 2-body encounters with more massive stars, less-massive stars may acquire velocities larger than the cluster’s escape velocity, and [*(ii)*]{} the normal stellar evolution via mass-loss. The external factors are: [*(i)*]{} large-scale encounters with giant molecular clouds (Wielen [@Wielen1991]), and [*(ii)*]{} tidal stripping by the Galactic gravitational field. A typical open cluster at the solar radius will cross the Galactic plane 10–20 times before being disrupted and leaving an open cluster remnant (de la Fuente Marcos [@delaF1998]). Bergond et al. ([@Bergond2001]) estimate the destruction time-scale for open clusters in the solar neighbourhood at about 600Myr. Consequently, it is expected that only those open clusters which are born with the largest masses or those located at large Galactic radii will survive longer than a few Gyr (Friel [@Friel1995]).
The numerical simulations of de la Fuente Marcos ([@delaF1996]) have shown that the final cluster remnant composition depends on the initial mass function, fraction of primordial binaries and galactocentric distance. The resulting cluster remnants are rich in binaries and do not appear to contain collapsed objects. Remnants of poorly populated clusters are expected to contain early-type stars, while those of more massive clusters contain late-type stars (de la Fuente Marcos [@delaF1996]), owing to different evolutionary time-scales. In the central region of the more evolved clusters, mass segregation should deplete the low main-sequence (MS) stars thus creating a core rich in compact and giant stars (Takahashi & Portegies Zwart [@TakaP2000]).
Mass segregation in a star cluster scales with the relaxation time, defined as $t_{relax}=\frac{N}{8\ln N}t_{cross}$, where $t_{cross}=R/v$ is the crossing time (Binney & Tremaine [@BinTre1987]). For a typical cluster radius of $R\sim5$pc and velocity dispersion $v\sim1$kms$^{-1}$, $t_{relax}\sim13$Myr for a cluster with $N=10^2$ stars, and $t_{relax}\sim90$Myr for $N=10^3$ stars.
Recently, several studies called attention to the possibility of detecting open cluster remnants in the Galaxy, e.g. Bica et al. ([@Bica2001]), Carraro ([@Carraro2002]), Pavani et al. ([@Pavani2002], [@Pavani2003]). A fundamental question to dynamical evolution studies is whether any open cluster can be observed right at the disrupting phase, when the remaining low-mass stars in the cluster’s halo get dispersed into the background and the corresponding mass function becomes eroded.
Depletion of low-MS stars in the central parts of a cluster is a sign of advanced dynamical evolution. This has been detected e.g. in NGC3680 (Anthony-Twarog et al. [@Twa1991]) and M67 (Bonatto & Bica [@BB2003]), in which the presence of a corona rich in low-mass stars has been confirmed with 2MASS photometry.
The Two Micron All Sky Survey (hereafter 2MASS, Skrutskie et al. [@2mass1997]) has proven to be a powerful tool in the analyses of the structure and stellar content of open clusters (e.g. Bonatto & Bica [@BB2003], Bica et al. [@BBD2003]). Indeed, the uniform and essentially complete sky coverage provided by 2MASS allows one to properly take into account background regions with suitable star count statistics, which is fundamental in order to correctly identify and characterize the stellar content of clusters, since their ages and distances can be determined by fitting isochrones to their colour-magnitude diagrams (CMDs), with a precision depending on the depth of the photometry and field contamination.
In the present study we address the actual dynamical state of NGC3680 analysing a large spatial area in the direction of the cluster, which 2MASS can provide. In particular, we search for the presence of a low-mass star-rich corona. In addition, we discuss NGC2180, an overlooked open cluster. This cluster appears to be in a more advanced dynamical evolutionary stage than NGC3680, and thus might be a missing link between evolved open clusters with a corona and final remnants.
In Section \[targets\] we provide available information on NGC2180 and the intermediate-age open cluster NGC3680. In Sect. \[2massPh\] we obtain the 2MASS photometry and introduce the $\jj\times\jh$ CMDs. In Sect. \[StructAnal\] we discuss the radial density distribution of stars and derive structural parameters for the clusters. In Sect. \[Fund\_par\] we fit isochrones to the near-infrared CMDs and derive cluster parameters. In Sect. \[LumFunc\] we derive the luminosity and mass functions (hereafter LF and MF) and estimate the stellar masses of each cluster. In Sect. \[Comp\] we compare NGC2180 and NGC3680 with well-known dynamically evolved open clusters and open cluster remnants. Concluding remarks are given in Sect. \[Conclu\].
The target clusters {#targets}
===================
NGC2180: a W. Herschell’s overlooked open cluster {#N2180}
-------------------------------------------------
NGC2180 has been first observed and described as a cluster by W. Herschell (Dreyer [@Dreyer1888]). In modern catalogues of open clusters, it is not included (Alter et al. [@Alter1970]; Lyngå [@Lyngaa1987]). In Dias et al. ([@Dias2002]) and in a revision of the NGC catalogue (Sulentic & Tifft [@Sulentic1973]), NGC2180 is indicated as non-existing. Houston ([@Houston1976]) included it in a list of possible clusters. The only information currently available in the WEBDA open cluster database (Mermilliod [@Merm1996] — [*http://obswww.unige.ch/webda*]{} is the object’s designation.
The original coordinates of NGC2180 precessed to J2000 are $\alpha=06^h\,09^m\,36^s$ and $\delta=+04^\circ\,43\arcmin$. However, in what follows we revised the central coordinates to $\alpha=06^h\,09^m\,48^s$and $\delta=+04^\circ\,48\arcmin\,26\arcsec$ based on bright star membership (Sect. \[2massPh\]). The latter coordinates convert to $\ell=203.85^\circ$and $b=-7.01^\circ$. A Digitized Sky Survey (XDSS) R image of NGC2180, centered on the revised coordinates is given in Fig. \[fig1\].
Visually, the cluster is sparse and poorly populated, but a few bright stars are present in the area.
The dynamically evolved open cluster NGC3680 {#N3680}
--------------------------------------------
The intermediate-age open cluster NGC3680 has already been extensively studied in the past. Its slightly supersolar metallicity has been determined by Eggen ([@Eggen1969]) and through the photoelectric [*uvby*]{}-H$_\beta$ photometry of Nissen ([@Nissen1988]). CCD [*uvby*]{} photometry of the central parts of NGC3680 has been published by Anthony-Twarog et al. ([@Twa1989]), and CCD and photographic BV photometry by Anthony-Twarog et al. ([@Twa1991]).
Previous age estimates for NGC3680 varied from $\age\sim1.0$Gyr (Mazzei & Pigatto [@MP1988]) to $\age\sim4.5$Gyr (Anthony-Twarog et al. [@Twa1991]). The Strömgren photometry of Bruntt et al. ([@Bruntt1999]) resulted in an age of $\age=1.45\pm0.15$Gyr, comparable to the value $\age=1.6\pm0.5$Gyr derived by Kozhurina-Platais et al. ([@Kozhu1997]).
With respect to the dynamical state, Nordström et al. ([@Nordstrom1996]), based on [*by*]{} CCD photometry of 310 stars as well as radial velocity and proper motion data, concluded that NGC3680 should be the last remnant of a cluster in an advanced state of dissolution, almost lost in the foreground field of similar stars. They estimate the present stellar mass as $\mobs\sim100\,\ms$, and an initial total mass of $\sim1200\,\ms$.
The central coordinates of this intermediate-age open cluster, estimated from the XDSS image, are $\alpha=11^h\,25^m\,38^s$ and $\delta=-43^\circ\,14\arcmin\,30\arcsec$. These values agree with the WEBDA coordinates. The new coordinates convert to $\ell=286.76^\circ$ and $b=16.92^\circ$. A Digitized Sky Survey (XDSS) R image of NGC3680 is given in Fig. \[fig2\], in which a concentration of bright stars is present in the central $8\arcmin\times8\arcmin$.
The 2MASS photometry {#2massPh}
====================
We investigate the nature and structure of both clusters using J and H photometry obtained from the 2MASS All Sky data release, which is available at [*http://www.ipac.caltech.edu/2mass/releases/allsky/*]{}. 2MASS photometric errors typically attain 0.10mag at $\jj\approx16.2$ and $\hh\approx15.0$, see e.g. Soares & Bica ([@SB2002]). Star extractions have been performed using the VizieR tool at [*http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=2MASS*]{}. For each cluster we made circular extractions centered on the coordinates given in Sect. \[targets\]. In order to maximize the statistical significance and representativity of background star counts, we decided to use an external ring (same area as the cluster) as offset field. This offset field will be used to represent the stellar background contribution to the cluster. We used an extraction radius of 40 for NGC2180 and 30 for NGC3680.
In Fig. \[fig3\] we show the $\jj\times\jh$ CMDs for each cluster (left panels) along with the corresponding (same area) offset fields (right panels). In order to maximize the cluster/background contrast, the CMDs have been built with stars extracted within 10 and 15, respectively for NGC2180 and NGC3680. According to the analysis in Sect. \[StructAnal\], these dimensions are smaller than the limiting radius of each cluster.
NGC2180 can be recognized as a cluster by the presence of the MS and a group of bright giants. These features are not present in its comparison field (top-right panel). Since the 6 giant stars are located within the central 10, we recalculated the central coordinates of NGC2180 as the average values for these stars (Sect. \[targets\]).
The MS in the CMD of NGC3680 is well-defined, particularly near the turnoff, and the presence of giants is also clear.
In all CMDs shown in Fig. \[fig3\], the contribution of disk stars is seen as nearly vertical sequences at $\jh\approx0.6$ and $\jh\approx0.35$. These contributions, as well as that of faint and spurious detections, has to be properly taken into account in order to isolate the cluster members. Indeed, it is interesting to note that the number of sources in the CMD in the direction of NGC2180 is 1410, while in the CMD of its comparison field, this number is 1399. For NGC3680, the above numbers are 1887 and 1890, respectively for cluster and comparison field.
Cluster structure {#StructAnal}
=================
The overall cluster structure is analysed by means of the star density radial distribution, defined as the number of stars per area in the direction of a cluster, which is shown in Fig. \[fig4\] for NGC2180 (top panel) and NGC3680 (bottom panel).
Before counting stars, we applied a cutoff ($\jj<15.0$) to both clusters and corresponding offset fields to avoid oversampling, i.e. to avoid spatial variations in the number of faint stars which are numerous, affected by large errors, and may include spurious detections. Colour filters in the CMDs have also been applied to both clusters and corresponding offset fields, in order to account for the contamination of the Galaxy. This procedure has been applied in the analysis of the open cluster M67 (Bonatto & Bica [@BB2003]). As a result of the filtering process, the number of stars in the direction of NGC2180 turns out to be 253, compared to 238 in its offset field. For NGC3680, these numbers are 755 and 668, respectively for cluster and comparison field. The radial distributions have been determined by counting stars inside concentric annuli with a step of 2.0 in radius for NGC2180 and 2.5 for NGC3680, and dividing the number of stars by the respective annulus area. The background contribution level, shown in Fig. \[fig4\] as shaded rectangles, corresponds to the average number of stars present in the external annuli.
The radial density profile of NGC2180 (top panel) is not smooth, presenting bumps and dips with respect to the background level. The relatively small number of stars in this cluster is reflected by the large $1-\sigma$ Poissonian error bars. Even so, NGC2180 still presents a central concentration of stars for $R<5\arcmin$ as well as an excess in the corona at $14\arcmin\le R\le18\arcmin$. Considering the profile fluctuations with respect to the background level, most of the cluster’s stars can be considered to be contained within a radius of $\approx10\arcmin$. However, a corona (Sect. \[Fund\_par\]) is detected, and we adopt as limiting radius $\rlim\approx18\arcmin$. The limiting radius corresponds to the radius at which the cluster’s profile merges into the background level (Fig. \[fig4\]). On the other hand, the radial density profile of NGC3680 (bottom panel) is smooth and presents a well-defined central concentration of stars, as well as an excess in the corona at $17\arcmin\le R\le22\arcmin$. Its limiting radius lies at $\rlim\approx22\arcmin$.
Although the spatial shape of the clusters may not be perfectly spherical, King law ([@King1966]) is still useful to derive first order structural parameters. A cluster core radius can be calculated by fitting a King surface density profile $\sigma(R)=
\frac{\sigma_0}{1+\left(R/R_{core}\right)^2}$ to the background-subtracted radial distribution of stars. The fits have been performed using a non-linear least-squares fit routine which uses the error bars as weights. The resulting fits are shown in Fig. \[fig4\], as dashed lines. The radial density profile of NGC3680 (bottom panel) follows well King law, with a resulting core radius $\rc=2.3\pm0.4\arcmin$. The large $1-\sigma$ Poissonian errors and non-uniform density profile of NGC2180 produce a significant uncertainty in the resulting core radius, $\rc=2.6\pm1.0\arcmin$. Using the cluster distances derived in Sect. \[Fund\_par\] below, the linear core radii turn out to be $\rc=0.7\pm0.3$pc and $\rc=0.7\pm0.1$pc, respectively for NGC2180 and NGC3680. Finally, the angular diameters ($2\times\rlim$) of $36\pm2\arcmin$ and $44\pm2\arcmin$, convert to linear limiting diameters of $9.5\pm1.2$pc and $12.8\pm1.3$pc, respectively for NGC2180 and NGC3680.
Fundamental parameters {#Fund_par}
======================
To maximize cluster membership probability, the analyses in the following two sections will be restricted to stars extracted within (Sect. \[StructAnal\]). Cluster parameters will be derived using solar metallicity Padova isochrones from Girardi et al. ([@Girardi2002]) computed with the 2MASS J, H and K$_S$ filters (available at [*http://pleiadi.pd.astro.it*]{}). The solar metallicity isochrones have been selected to be consistent with the results of Eggen ([@Eggen1969]) and Nissen ([@Nissen1988]), at least for NGC3680. The 2MASS transmission filters produced isochrones very similar to the Johnson ones, with differences of at most 0.01 in (Bonatto et al. [@BBG2004]). For reddening and absorption transformations we use R$_V$ = 3.2, and the relations A$_J = 0.276\times$A$_V$ and $\ejh=0.33\times\ebv$, according to Dutra et al. ([@DSB2002]) and references therein.
We show in Fig. \[fig5\] the isochrone fits to the $\mj\times\jh$ CMD of NGC2180 (left panel) and NGC3680 (right panel). values are obtained after applying the distance modulus derived below for each cluster.
The upper MS of NGC2180 is relatively depleted of stars, with a single star near the turnoff at $\mj\approx0.0$. Thus, the solar-metallicity fit which best matches the MS features has been obtained with the 710Myr isochrone. This solution is shown as a solid line in Fig. \[fig5\]. The isochrone fit and related uncertainties result in a distance modulus $\mM=10.10\pm0.20$, $\ebv=0.0$ and a distance to the Sun $\ds=1.05\pm0.08$kpc. However, the above isochrone solution fails to reproduce the 6 giants at $\mj\approx-2$ and $\jh\approx0.5$, which might be accounted for by differences in metallicity. Accordingly, we present an alternative fit with the subsolar metallicity ($\zz=-0.38$), 710Myr isochrone, which results in a distance modulus $\mM=9.40\pm0.20$, $\ebv=0.18$ and $\ds=0.76\pm0.06$kpc. According to this solution (dashed line), the 6 bright stars would be, in fact, the giant clump of NGC2180. The giant clump in NGC2180 is similar to that of the Hyades cluster (e.g. Perryman et al. [@Perryman1998]). Both clusters have similar ages. Since the CMD features below the turnoff are equally well reproduced by both isochrones, the intrinsic metallicity of NGC2180 is probably in the range $-0.4\leq\zz\leq0.0$. In the same way, we adopt as distance to the Sun the average of the values derived for both isochrones, i.e. $\ds=0.91\pm0.15$kpc. With the above distance to the Sun, the galactocentric distance of NGC2180 becomes $8.8\pm0.1$kpc. In the subsequent calculations we will adopt the solar metallicity solution as reference, particularly to derive the LFs and MFs.
The presence of a well-defined turnoff and giant stars in the CMD of NGC3680 constrain the isochrone fits to the $\age=1.6$Gyr solution. For this cluster we derive $\mM=10.00\pm0.20$, $\ebv=0.00$ and $\ds=1.00\pm0.09$kpc. The galactocentric distance of NGC3680 is $7.8\pm0.1$kpc. The present age estimate, using 2MASS photometry, is in close agreement with those derived by Kozhurina-Platais et al. ([@Kozhu1997]) and Bruntt et al. ([@Bruntt1999]).
Luminosity and mass functions {#LumFunc}
=============================
In this section we analyze the observed star counts as a function of magnitude (LF) and mass (MF) as well as their spatial dependence.
The accurate determination of a cluster’s LF (or MF) suffers from some problems, in particular [*(i)*]{} the contamination of cluster members by field stars, [*(ii)*]{} the observed incompleteness at low-luminosity (or low-mass) stars, and [*(iii)*]{} the mass segregation, which may affect even poorly populated, relatively young clusters (Scalo [@Scalo1998]). The 2MASS uniform sky coverage allows one to overcome, at least in part, points [*(i)*]{} — since suitable offset fields can be selected around the cluster and [*(iii)*]{} — the entire cluster area can be included in the analyses. Thus, advanced stages of mass segregation would affect more significantly the analysis of very old, dynamically evolved clusters (e.g. M67, Bonatto & Bica [@BB2003]).
In Fig. \[fig6\] we show the LFs ($\phi(\mj)$) in the J filter (shaded area) for the two clusters, built as the difference of the number of stars in a given magnitude bin between object (continuous line) and average offset field (dotted line). The LFs are given in terms of the absolute magnitude , obtained after applying the distance modulus derived in Sect. \[Fund\_par\] for each cluster, and the bin in magnitude used is $\Delta\mj=0.50$mag. We remind that the LFs in Fig. \[fig6\] are built after applying magnitude ($\jj<15.0$) and colour cutoffs to the objects and their offset fields (Sect. \[Fund\_par\]). The LFs for different spatial regions of NGC2180 are in the left panels, and those of NGC3680 are in the right panels.
To search for spatial variations in the stellar content, we built LFs for different regions in and around the clusters, according to the structures present in the radial density profiles (Fig. \[fig4\]). Thus, the first LF encompasses the central region of each cluster (panels (d)), $0.0\arcmin\le R\le 5.0\arcmin$ and $0.0\arcmin\le
R\le 7.5\arcmin$, respectively for NGC2180 and NGC3680; the second one corresponds to the outskirts (panels (c)), $5.0\arcmin\le R\le 10.0\arcmin$ and $7.5\arcmin\le R\le
15.0\arcmin$, and the third LF to the corona (panels (b)). In panels (a) we show the overall LFs ($0.0\arcmin\le R\le\rlim$). The background $\phi(\mj)$ has been scaled to match the projected area of each region. Representative MS spectral types (adapted from Binney & Merrifield [@Binney1998]) are shown in the top panels. The turnoff is indicated in all panels. Giants are contained within 10 in NGC2180 and 15 in NGC3680.
The severe depletion of MS stars in NGC2180 can be clearly seen in the internal background-subtracted LFs (Fig. \[fig6\], left panels (c) and (d)) which, in fact, are similar to each other. The above spatial properties of the LFs probably reflect mass segregation followed by advanced and significant Galactic tidal stripping effects (severe depletion of stars in the corona). Indeed, the overall cluster and offset field LFs (left panel (a)) are very similar, indicating that the original stellar content of NGC2180 is already nearly dispersed into the background population. In order to reach such an advanced dynamical state in a time span of $\sim700$Myr it is reasonable to assume that the initial cluster was not massive.
On the other hand, the central LF of NGC3680 (Fig. \[fig6\], right panel (d)) is nearly flat from the turnoff (spectral type $\sim$A0) to the turnover ($\sim$G1), confirming the central depletion of lower-MS stars found by Anthony-Twarog et al. ([@Twa1997]). This effect may be accounted for by mass segregation alone. Indeed, the $7.5\arcmin\le R\le15\arcmin$ and corona regions (right panels (c) and (b)) are still well populated by low-mass ($\sim$G0) stars, suggesting that the Galactic tidal stripping has not yet been effective in severely depleting this cluster. In this sense, NGC3680 is very similar to the $\age\approx3.2$Gyr, mass-segregated open cluster M67, as spatially analysed in Bonatto & Bica ([@BB2003]).
The overall LFs in Fig. \[fig6\] (top panels), restricted to the region between the turnoff and turnover, have been converted to MFs according to $\phi(m)=
\phi(\mj)\left|\frac{dm}{d\mj}\right|^{-1}$. We used the stellar mass-luminosity relations taken from the Padova isochrones (Sect. \[Fund\_par\]), 710Myr for NGC2180 and 1.6Gyr for NGC3680. Since we are interested in obtaining only an estimate of the mass in NGC2180, we used the solar metallicity isochrone to derive the mass–luminosity relation. The resulting MFs, including $1-\sigma$ error bars, are shown in Fig. \[fig7\].
Despite the non-uniform overall MF of NGC3680, we applied a fit of $\phi(m)\propto m^{-(1+\chi)}$ to the turnoff ($\approx1.8\,\ms$, $\approx$A0) — turnover ($\approx1.0\,\ms$, $\approx$G1) region, which resulted in a slope $\chi=2.06\pm1.08$, a steeper MS than a Salpeter’s $\chi=1.35$. Numerical integration of the MF in the above mass interval, taking into account the uncertainties, resulted in a MS mass of $\mMs=111\pm24\,\ms$. We estimate the mass stored in the giants () by counting the number of stars brighter than the turnoff present in the overall LFs, and multiplying this number by the mass at the turnoff, resulting in $\mg\approx19\,\ms$ for NGC3680. Thus, the present observed stellar mass in NGC3680 is $\mobs=130\pm24\,\ms$.
Assuming that the very low-mass stars have not yet been stripped away from the cluster, the total mass locked up in stars can be estimated by directly extrapolating the present MF fit down to the theoretical stellar low-mass end $m_{low}=0.08\,\ms$. The resulting total stellar mass in NGC3680 is $\sim(2.4\pm1.2)\times10^3\,\ms$ which, within errors, is similar to the value found by Nordström et al. ([@Nordstrom1996]). This agreement is interesting, considering the different methods of taking into account the stellar background. On the other hand, Kroupa, Tout & Gilmore ([@KTG1991]) and Kroupa ([@Kroupa2001]) presented evidence that the MFs of most globular and open clusters flatten below $\sim0.5\,\ms$. As a consequence, the total stellar mass in NGC3680 would be less than the value derived above, since most of the stars in a cluster are expected to be found in the low-mass range. Accordingly, we derive a more conservative total mass value for NGC3680 assuming the universal IMF of Kroupa ([@Kroupa2001]), in which $\chi=0.3\pm0.5$ for $0.08\,\ms - 0.50\,\ms$, and $\chi=1.3\pm0.3$ for $0.50\,\ms - 0.80\,\ms$. For $0.80\,\ms -
1.80\,\ms$ we adopt the value derived in this paper, $\chi=2.06\pm1.08$. As expected, the resulting total stellar mass decreases to $546\pm206\,\ms$.
Similar to NGC3680, we derive a giant mass of $\mg\approx16\,\ms$ for NGC2180. However, the considerable variations in the MF do not allow a statistically significant fit. Consequently, the present observed stellar mass in NGC2180 has been estimated by numerically integrating its MF (Fig. \[fig7\], top panel) and adding to this value. The resulting observed stellar mass is $\mobs\sim47\pm7\,\ms$. Considering a severe low-mass star depletion (Bergond et al. [@Bergond2001], de la Fuente Marcos [@delaF1996]) for NGC2180, it would certainly be less massive than NGC3680, which is older and at a somewhat smaller Galactocentric distance (Sect. \[Comp\]).
Taking into account the mass estimates, the derived ages of both clusters are more than an order of magnitude larger than the corresponding $t_{relax}$, which is consistent with the significant presence of mass-segregation effects.
Comparison with other dynamical states {#Comp}
======================================
At this point, it may be useful to see where NGC2180 and NGC3680 fit in the context of [*(i)*]{} well-known dynamically evolved open clusters, and [*(ii)*]{} poorly populated remnants of open clusters. In particular, we will pay attention to parameters intimately associated to dynamical evolution, i.e., age, core and limiting radii and stellar mass. The remnants are NGC1252 (Pavani et al. [@Pavani2002]), Ruprecht3 (Pavani et al. [@Pavani2003]), NGC7036 and NGC7772 (Carraro [@Carraro2002]). The old open clusters are M67 (Bonatto & Bica [@BB2003]) and NGC188 (Bonatto, Bica & Santos [@BBJF2004]), both more massive than NGC2180 and NGC3680. All these objects have been analysed by means of 2MASS photometry following the same methods as in the case of NGC2180 and NGC3680, thus ensuring homogeneity in terms of analysis and derived parameters. Relevant parameters for the objects are given in Table \[tab1\]. The results are shown in Fig. \[fig8\]. Part of the parameters for NGC1252, Ruprecht3, NGC7036 and NGC7772 have been derived using 2MASS data, following a similar analysis as that used for NGC2180 and NGC3680.
As expected, the structural parameters of the intrinsic open clusters present a strong dependence with time. This is particularly true for the limiting radius (panel (c)), which increases linearly with time, probably as a consequence of the Galactic tidal pull. The observed mass, on the other hand, initially increases linearly with time and then reaches a saturation threshold (panel (a)). This threshold is defined by M67 and NGC188, both dynamically advanced open clusters, more massive than NGC2180 and NGC3680, with strong effects of mass segregation. The relations of the core radius to the limiting radius (panel (d)), and of to (panel (b)), present a similar pattern, linear increase first followed by a saturation. This behaviour may be accounted for by a combination of the Galactic tidal pull (increasing with time) and mass segregation (initial core formation and growth, followed by core stabilization (de la Fuente Marcos [@delaF1996])). In the far future, these massive open clusters will leave behind only a core, with most of the low-mass stars dispersed into the background (de la Fuente Marcos [@delaF1996]).
\[tab1\]
----------- ------------ ------------ ---------- --------------- --------------- --------------- -------------- -- --
Cluster $\ell$ $b$ d$_{GC}$ Age
($^\circ$) ($^\circ$) (kpc) (Gyr) (pc) (pc) ()
NGC2180 203.85 -7.01 9.0$^a$ $0.7\pm0.1^a$ $0.7\pm0.3^a$ $4.7\pm0.6^a$ $47\pm7^a$
NGC3680 286.76 16.92 8.8$^a$ $1.6\pm0.1^a$ $0.7\pm0.1^a$ $6.4\pm0.6^a$ $130\pm24^a$
M67 215.68 31.93 8.6$^b$ $3.2\pm0.1^b$ $1.2\pm0.1^b$ $8.7\pm0.8^b$ $344\pm40^b$
NGC188 122.85 22.39 8.9$^c$ $7.1\pm0.1^c$ $1.3\pm0.1^c$ $9.7\pm0.4^c$ $380\pm12^c$
NGC1252 274.08 -50.83 7.8$^d$ $3.0\pm1.0^d$ — $0.7\pm0.1^a$ $10\pm5^a$
NGC7772 102.73 -44.27 8.4$^d$ $1.5\pm0.5^e$ — $0.9\pm0.1^g$ $10\pm5^a$
NGC7036 64.54 -21.44 7.6$^d$ $3.5\pm0.5^e$ — $1.0\pm0.1^g$ $10\pm5^a$
Ruprecht3 238.77 -14.81 8.4$^d$ $1.5\pm0.5^f$ — $0.2\pm0.0^a$ $10\pm5^a$
----------- ------------ ------------ ---------- --------------- --------------- --------------- -------------- -- --
: Open cluster and remnant parameters.
[Table Notes.]{}
Column (4): Galactocentric distance; Data sources are: (a) - this paper, using 2MASS; (b) - Bonatto & Bica ([@BB2003]); (c) - Bonatto, Bica & Santos ([@BBJF2004]) ; (d) - Pavani et al. ([@Pavani2002]); (e) - Carraro ([@Carraro2002]); (f) - Pavani et al. ([@Pavani2003]); (g) - this paper, using Carraro ([@Carraro2002]).
With respect to the structural parameters, the open clusters seem to follow a well-defined sequence (panels (a) – (d)). The remnants, however, follow this sequence only for $\rlim\times\mobs$ (panel (b)), since most of their stellar content may have already been dispersed into the background.
NGC2180 is consistently closer than NGC3680 to the loci occupied by the remnants in panels (a), (b) and (c). Despite being younger than NGC3680, the less-massive nature of NGC2180 probably accelerated its evolutionary time-scale, setting it in a more dynamically advanced state than NGC3680.
Interestingly, the remnants dealt with in this paper have Galactocentric distances smaller that those of the evolved open clusters (Column 4 of Table \[tab1\]). Again, this suggests the effects of the Galactic tidal pull (external evolution). However, to draw objective conclusions with respect to open cluster evolution it is necessary to take separately into account selection and intrinsic evolution effects. To further explore these issues we intend to analyze a larger sample of clusters which span a variety of photometric, structural and dynamical properties, using the same methods employed in the present paper.
Concluding remarks {#Conclu}
==================
As a consequence of the internal dynamical evolution and the relentless Galactic tidal pull, most open clusters are expected to evaporate completely in less than 1Gyr, leaving behind a remnant with characteristics which depend on the cluster’s initial conditions. Only the more massive open clusters may survive to old ages. In this context, the observation of an actually dissolving open cluster becomes very interesting to check existing theories on dynamical evolution of stellar systems, N-body codes in particular, as well as to test stellar evolution theories.
In the present work we analyse the physical structure, stellar content and dynamical state of the overlooked open cluster NGC2180. We also examine in detail the intermediate-age open cluster NGC3680, formerly considered to be in the last stages of its dynamical evolution. The present analyses make use mostly of J and H 2MASS All Sky data release photometry.
NGC2180 presents a non-uniform radial distribution of stars (Sect. \[StructAnal\]), with significant $1-\sigma$ Poissonian error bars due to the small number of member stars. Its radial density profile has a central concentration of stars, as well as a corona (Fig. \[fig4\]). From a King law fit we estimate $\rc=0.7\pm0.3$pc and a linear limiting diameter of $9.5\pm1.2$pc. Its $\mj\times\jh$ CMD (Fig. \[fig5\]) is depleted of stars near the turnoff, and can be fitted with 710Myr isochrones of solar and sub-solar metallicity. The $\zz=0.0$ solution results in $\mM=10.10\pm0.20$, $\ebv=0.0$ and a distance to the Sun $\ds=1.05\pm0.08$kpc, while the $\zz=-0.4$ solution gives $\mM=9.40\pm0.20$, $\ebv=0.18$ and $\ds=0.76\pm0.06$kpc. Thus, we adopt as distance to the Sun $\ds=0.91\pm0.15$kpc, which puts NGC2180 at a galactocentric distance of $8.8\pm0.1$kpc. Mass segregation and advanced Galactic tidal stripping on NGC2180 are reflected on the spatial properties of its LFs (Fig. \[fig6\]). Low-mass stars are severely depleted from the MS in each region, from the center to the cluster’s limiting radius. In addition, the MS of the corona, although depleted as well, is slightly more populated of low-mass stars than the MS of more internal regions. The observed stellar mass in NGC2180 is $\sim47\pm7\,\ms$.
NGC3680 presents a uniform radial distribution of stars, with a well-defined core and a corona, with $\rc=0.7\pm0.1$pc and a linear limiting diameter of $12.8\pm1.3$pc (Sect. \[StructAnal\]). Its $\mj\times\jh$ CMD (Fig. \[fig5\]) presents a nearly complete MS, including the turnoff and giants. We derive an age of $\age\approx1.6$Gyr, $\mM=10.00\pm0.20$, $\ebv=0.00$ and $\ds=1.00\pm0.09$kpc, in reasonable agreement with previous works. The LF of the central region of NGC3680 (Fig. \[fig6\]) is depleted of low-mass stars which, contrary to what is observed in NGC2180, are still present in the external region and corona. Thus, Galactic tidal stripping has not yet been effective in severely depleting NGC3680 of stars. For NGC3680, a MF fit $\phi(m)\propto m^{-(1+\chi)}$ resulted in a slope $\chi=2.06\pm1.08$, and in an observed stellar mass (MS and giants) of $\approx130\pm24\,\ms$. Extrapolating the MF fit down to the theoretical low-mass end $m_{low}=0.08\,\ms$, the total stellar mass in NGC3680 turns out to be $\sim(2.4\pm1.2)\times10^3\,\ms$, which agrees with previous estimates, within uncertainties. Assuming a more representative IMF (Kroupa [@Kroupa2001]), which flattens for masses below $\sim0.5\,\ms$, the total stellar mass in NGC3680 turns out to be $546\pm206\,\ms$.
Finally, comparing NGC2180 and NGC3680 with clusters in other dynamical states, [as well as open cluster remnants]{}, we found that the less-massive nature of NGC2180 put it closer to cluster remnants than NGC3680.
The above arguments lead us to conclude that NGC2180 is in a more advanced dynamical state than NGC3680, on its way to become a fossil cluster. Thus, NGC2180 may be the missing link between evolved open clusters and remnants.
We thank the referee, Dr. R. de Grijs, for interesting remarks. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. We employed catalogues from CDS/Simbad (Strasbourg) and Digitized Sky Survey images from the Space Telescope Science Institute (U.S. Government grant NAG W-2166) obtained using the extraction tool from CADC (Canada). We also made use of the WEBDA open cluster database. We acknowledge support from the Brazilian Institution CNPq.
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|
---
abstract: 'Models of the decoupling of baryons and photons during the recombination epoch predict the existence of a large-scale velocity offset between baryons and dark matter at later times, the so-called streaming velocity. In this paper, we use high resolution numerical simulations to investigate the impact of this streaming velocity on the spin and shape distributions of high-redshift minihalos, the formation sites of the earliest generation of stars. We find that the presence of a streaming velocity has a negligble effect on the spin and shape of the dark matter component of the minihalos. However, it strongly affects the behaviour of the gas component. The most probable spin parameter increase from $\sim$0.03 in the absence of streaming to $\sim$0.15 for a run with a streaming velocity of three times $\sigma_{\rm rms}$. The gas within the minihalos becomes increasingly less spherical and more oblate as the streaming velocity increases, with dense clumps being found at larger distances from the halo centre. The impact of the streaming velocity is also mass-dependent: less massive objects are influenced more strongly, on account of their shallower potential wells. The number of halos in which gas cooling and runaway gravitational collapse occurs decreases substantially as the streaming velocity increases. However, the spin and shape distributions of gas that does manage to cool and collapse are insensitive to the value of the streaming velocity and uncorrelated with the same properties measured on the scale of the halo as a whole.'
author:
- |
Maik Druschke$^{1}$[^1], Anna T. P. Schauer$^{1,2}$[^2], Simon C. O. Glover$^{1}$, Ralf S. Klessen$^{1}$\
$^{1}$ Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany\
$^{2}$ Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA
bibliography:
- 'sample.bib'
title: 'Shape and Spin of Minihaloes. II: The Effect of Streaming Velocities'
---
\[firstpage\]
early universe – dark ages, reionisation, first stars – stars: Population III.
Introduction
============
The currently most favoured cosmological model for describing the evolution of our Universe since the Big Bang is the so-called $\Lambda$CDM (Lambda Cold Dark Matter) model. This is the simplest model that is consistent with existing cosmological measurements and is thus considered the standard model of Big Bang cosmology. In this model, most of the matter is dark and interacts with baryonic matter solely through gravity. $\Lambda$CDM is a hierarchical model in which structure forms first on the smallest scales, with larger bound structures forming later via mergers and accretion. The first stars to form in the Universe – the so-called Population III (Pop. III) – therefore form in small-scale, low-mass bound structures known as dark matter minihaloes. Understanding the properties of the dark matter and gas making up these minihaloes is hence of great importance for our understanding of the onset of Pop. III star formation.
In an earlier paper (@Druschke; hereafter, Paper I), we used a high resolution cosmological simulation to investigate the distribution of shapes and spins of a large sample of dark matter minihaloes. In the subset of minihaloes in which the gas cools and undergoes runaway gravitational collapse (i.e. the minihaloes in which Pop. III star formation occurs), we also investigated whether the spin of the dense, cooling gas was correlated with the spin of the halo on large scales, and showed that there was no significant correlation.
However, the simulation analyzed in Paper I started from initial conditions in which there was no relative velocity or “streaming velocity” between the dark matter and the gas. In reality we do not expect this to be the case. Before recombination, baryons and photons were tightly coupled by Compton scattering and behaved as if they were a single fluid. On the other hand, the dark matter was not directly coupled to this baryon-photon fluid and interacted with it only via gravity. Consequently, even though fluctuations in the density of the baryon-photon fluid and the dark matter are correlated, one nevertheless expects some motion of the fluid relative to the dark matter. In an influential paper, @Tseliakhovich pointed out that an imprint of this motion survives in the baryons even after recombination, in the form of a large-scale streaming motion of the baryons with respect to the dark matter. The resulting relative velocity (distributed like a multivariate Gaussian, with a standard deviation $\sigma \approx 30$kms$^{-1}$ at recombination) is highly supersonic shortly after recombination, but decays with the expansion of the Universe. It therefore plays little role in the formation of galaxies at low redshifts. However, several authors have shown that at high redshift, streaming velocities have a significant effect on the creation and formation of dark matter minihaloes and so-called atomic cooling haloes.[^3] Amongst other effects, a non-zero streaming velocity suppresses the formation of minihaloes [@Tseliakhovich; @naoz12] and increases the minimum halo mass required for efficient gas cooling and consequent Pop. III star formation [@Dalal; @greif11; @stacy11a; @schauer18] This leads to a delayed onset of Population III star formation [@greif11; @maio11].
In this paper, we therefore extend the analysis from Paper I to the case where the initial streaming velocity of the baryons relative to the dark matter is non-zero. We analyze several high-resolution simulations carried out with different streaming velocities and explore whether the value of the streaming velocity affects the spin or the shape distributions of the gas or the dark matter in minihaloes. The paper is structured as follows. In Section \[Simulation\], we give a short overview of the set of simulations from [@schauer18] that we use for our analysis, and in Section \[AnalysisMethods\] we present our analysis methods. In Section \[Plots\], we look in detail at the changes that occur in a representative halo as we increase the streaming velocity, while in Section \[Results\] we present the results of our analysis of the full set of haloes. Finally we summarize our findings and results in Section \[Conclusion\].
Simulation {#Simulation}
==========
The cosmological simulations that we use for our analysis were previously described in @anna17b and @schauer18, and full details can be found in those papers. We therefore give here only a brief overview of the most important properties of the simulations.
The simulations were carried out using the moving mesh code [arepo]{} [@arepo] and include both gas and dark matter. The chemical and thermal evolution of the gas were treated using a primordial chemistry network and cooling function based on the one presented in @cgkb11, but updated as described in @schauer18. Here, we choose to work with the four simulations from @schauer18 that have a box size of 1cMpc/$h$, where the ‘c’ denotes comoving units and $h$ is the value of the Hubble parameter in units of $100 \, {\rm km \, s^{-1} \, Mpc^{-1}}$. These simulations have a particle mass of $\sim 100$[$\rm{M_{\odot}}~$]{}for dark matter and a target mass of $\sim 20$[$\rm{M_{\odot}}~$]{}for the gas cells[^4]. After creating the initial conditions with MUSIC [@hahn11] at $z=200$ with Planck parameters [@Planck15], the dark matter and gas are followed to redshift $z=14$. In Paper I, we found that the minihalo spin and shape distributions evolve only weakly with redshift, and so in this paper we focus on the properties of the haloes at the final output time.
In order to mimic different regions of the Universe with different streaming velocities, a constant offset velocity term was added to the initial conditions. This offset velocity was arbitrarily chosen to point in the positive $x$ direction, but our results are independent of this choice, owing to the large-scale isotropy of the Universe. For the four different boxes, the amplitude of the streaming velocity was set to 0, 1, 2 and 3 times $\sigma_\mathrm{rms}$, corresponding to 0, 6, 12 and 18kms$^{-1}{}$ at $z=200$. The no streaming run was previously analyzed in Paper I but is included here for the purposes of comparison.
Analysis {#AnalysisMethods}
========
This paper focuses primarily on the spin and the shape of minihaloes at different streaming velocities. Before we discuss our results, we introduce some important physical quantities. A more detailed description can be found in Paper I.
We start with the angular momentum $\vec{J}(R){}$, which is defined for each minihalo by summing up the values of every dark matter particle or gas cell within a distance $R$ from the most bound cell, which we take to define the centre of the halo: $$\begin{split}
\vec{J}(R) = \sum\limits_{r_{i}<R} m_{i} \vec{r}_{i} \times \vec{v}_{i}.
\end{split}$$ Here, $m_i$ is the mass of the particle or gas cell, $\vec{r}_i$ the distance from the particle or gas cell to the center of the halo and $\vec{v}_i$ the velocity relative to the center of the halo. In the special case where $R = R_{\rm vir}$ (the virial radius), this equation yields the angular momentum of the halo as a whole.
Additionally, we compute the inertia tensor in order to calculate the side lengths of a halo [@Springel2004]: $$I_{jk} = \sum_{i=1}^N m_{i}(|\vec{r}_{i}|^{2}\delta_{jk}-r_{i,j}r_{i,k}) .$$ Here, $\delta_{jk} $ represents the Kronecker delta, $m_{i}{}$ is the mass of the $i$-th gas cell or dark matter particle, $\vec{r}_{i}$ is its distance from the halo centre, and $r_{i,j}$ and $r_{i,k}$ and the $j$-th and $k$-th components of $\vec{r}_{i}$. Using the eigenvalues $I_{1}$, $I_{2}$, $I_{3}$ of this tensor and the halo mass, the side lengths $a \geq b \geq c{}$ of an ellipsoid can be calculated (see Paper I), which can then be used to compute the sphericity and triaxiality of the halo, as outlined in Section \[sec:shape\] below.
Spin
----
We are interested in how fast a halo is rotating, independent of its mass. We therefore choose to work with the spin parameter $\lambda^{\prime}$, and follow the definition of [@Bullock]: $$\label{LambdaGasDM}
\lambda^{\prime}_{i}(R)= \dfrac{\vert J \vert_{i}(R)}{\sqrt{2} R\, M_{i}(R)V_{\mathrm{circ}}(R)}.$$ As before, $J_{i}$ describes the angular momentum and $M_{i}$ the mass. This time, the index $i{}$ stands for the components (gas, dark matter, or the total matter content of the halo) that we are interested in. Furthermore, $R$ is the radius up to which the particles or gas cells are considered and $V_{\mathrm{circ}}$ is the circular velocity which is given by $V_{\mathrm{circ}}(R)^{2} =R^{-1} G M(R) $.
The spin parameter can take values between $\lambda^{\prime} = 0$ and $\lambda^{\prime} = 1$. A spin parameter of 0 corresponds to a non-rotating halo, while a spin parameter of 1 corresponds to Keplerian rotation of all particles and gas cells.
Statistically, the spin of a set of haloes can be approximated by a log-normal distribution [@Warren; @Mo]: $$P(\lambda^{\prime}) = \dfrac{1}{\lambda^{\prime} \sqrt{2\pi}\sigma_{0}} \mathrm{exp\left(-\dfrac{ln^{2}\left(\frac{\lambda^{\prime}}{\lambda_{0}}\right)}{2\sigma_{0}^{2}}\right)} .$$\[Lambda\] The most probable value of this well-known log-normal distribution is referred as the peak value in the following. In regions of the Universe without streaming velocity, previous studies (@Sasaki, Paper I) find peak values of $\lambda^{\prime} \simeq 0.03$. The distribution of the spin parameter of all haloes with a mass of at least $\mathrm{M_{min}} = 10^{5}\ \mathrm{M_{\odot}}$ (corresponding to roughly 1000 gas and dark matter particles) at $z=14$ can be seen in Figure 1 (left panel).
![Distribution of the halo properties for all haloes with $M \geq M_{\rm min}$ in a simulation without streaming velocity at redshift $z=14$. Left panel: spin parameter. The red curve is a log-normal fit to the data, with parameters as given in the text. Middle panel: triaxiality. The red curve is a beta-distribution fit to the data, with parameters as given in the text. Right panel: sphericity. Again, the red curve is a beta-distribution fit to the data, with parameters given in the text.[]{data-label="fig:LambdaTriaxSphaereVsPDF"}](figures/LambdaTriaxSphaereVsPDF.pdf){width="0.99\columnwidth"}
Shape {#sec:shape}
-----
In order to quantify the shape of the haloes, we follow the definition of [@Springel2004]. The triaxiality and sphericity can be determined using the side lengths $a \geq b \geq c{}$ of the ellipsoid that we determined above using the inertia tensor.
### Triaxiality {#Triaxiality}
The triaxiality can be calculated as follows [@Franx]: $$T = \dfrac{a^{2}-b^{2}}{a^{2}-c^{2}} . \label{triax-eq}$$ This definition classifies haloes from oblate (low triaxiality, $\mathrm{T}=0$) to prolate (high triaxiality, $\mathrm{T}=1$). In the middle panel of Figure \[fig:LambdaTriaxSphaereVsPDF\], we show the triaxiality distribution of all haloes with masses $M > M_{\rm min}$ at redshift $z=14$ in the simulation without streaming velocities. It can be described by a beta distribution displayed in red (see also Paper I): $$P(T)=\dfrac{\Gamma(a+b)T^{a-1}(1-T)^{b-1}}{\Gamma(a)\Gamma(b)}$$ Since the gamma function $\Gamma$ serves only for normalization, the shape of the distribution is described by the two fit variables $a = 3.355$ and $b = 1.892$. The peak of the best fitting distribution is at $\mathrm{T_{peak}=0.700}$. Other studies [@Jang; @Sasaki] have already shown that most haloes in our mass range are prolate, while haloes are more oblate at higher masses [see e.g. @Warren; @Allgood].
### Sphericity
The sphericity is defined as the ratio of the smallest to the largest side length of the halo ellipsoid: $$S = \dfrac{c}{a} . \label{sphericity-eq}$$ A sphericity of $S=1$ describes a perfectly spherical halo, while a lower value represents a stronger deviation from a perfect sphere. Analogously to the triaxiality, the sphericity distribution of all haloes is well described by a beta-distribution, which is shown in the right panel of Figure \[fig:LambdaTriaxSphaereVsPDF\] for the simulation without streaming velocity at redshift $z=14$. Here, the shape parameters are $a=6.582$ and $b=5.750$ and the most probable value of $S$ is $S=0.685$.
Impact of streaming on individual haloes {#Plots}
========================================
{width="1.99\columnwidth"}
Before examining the effect that a non-zero streaming velocity has on the full distribution of haloes, it is informative to study its effects in detail on a representative halo. In Figure \[fig:SubPlot\_Nr13\_Final\_0\_25\_1\_5\_hot\], we show slices of density in the $x$-$y$ and $y$-$z$ planes through the centre of a halo taken from the simulation with 2$\sigma$ streaming. This halo has a mass $M = 9.57\times 10^{6}\, \mathrm{M}_\odot$ and a spin parameter $\lambda^{\prime} = 0.0483$. For a better overview, the halo is shown at two different scales. In the upper panels, the slice has a side length of 0.5 ckpc/h, showing the inner core and thus the cold dense gas associated with the halo. In the lower panels, the side length is 3.0 ckpc/h, which allows a more general overview of the position and shape of the entire halo. The origin of the coordinate system is taken to be at the center of the halo (defined as the most bound particle), and the coordinate system is aligned so that the halo angular momentum vector points in the position $z$ direction. The colour-coding indicates the number density of the gas, ranging from $n_\mathrm{min}=0.001$cm$^{-3}{}$ (dark red) to $n_\mathrm{max}=250$cm$^{-3}{}$ (bright yellow). In addition, we have added contours for density thresholds of $n_\mathrm{thres}=1$cm$^{-3}{}$ (black), $10$cm$^{-3}{}$ (blue) and $100$cm$^{-3}{}$ (red). The white arrows indicate the direction of the gas velocity and their length indicates the magnitude of the velocity with respect to the centre of the halo. For comparison, the black arrow on a white background in the top of the image is normalized to a velocity of 10 km/s. Furthermore, the black circle on top right in the Figure indicates the radius of the gravitational softening length.
In comparison to Paper I, where similar plots are shown for a pair of haloes taken from a run with no streaming, we see that here, the center of rotation can deviate from the halo center. For example, the halo shown in Figure \[fig:SubPlot\_Nr13\_Final\_0\_25\_1\_5\_hot\] has a rotational centre at $x \approx y \approx$0.5ckpc/$h$, which can be seen in the lower left panel.
{width="1.99\columnwidth"}
In Figure \[fig:PlotAlleSigmaNummer13\], we show the same minihalo in all four simulations with different streaming velocities. Since we use the same initial conditions for all four simulations (other than the gas velocity offset), we can identify the same dark matter halo in all four simulations and compare its properties.
We see immediately that the higher the streaming velocity, the lower the maximum density in the minihalo. The core of the halo in the 0$\sigma$ run contains a lot of cold dense gas. Near the center, several clumps with number densities exceeding $50$cm$^{-3}{}$ have formed and the innermost clump contains gas at a density of more than $100$cm$^{-3}{}$. Even in the 1$\sigma$ streaming run, there is still a lot of dense gas visible. With 2$\sigma$ streaming, however, the halo contains no gas denser than $50$cm$^{-3}{}$, while at 3$\sigma$ the densest gas is $\sim 1 \, {\rm cm}^{-3}$, only a factor of a few larger than the mean halo density.
As we will see later, the behaviour of this particular halo is quite typical. In the no streaming run, there are 206 haloes at $z=14$ that contain cold gas that is denser than $100 \, {\rm cm^{-3}}$, but in the 2$\sigma$ streaming run this number has dropped to 15 haloes, while in the 3$\sigma$ case we find only a single halo with cold gas above this density.
Figure \[fig:PlotAlleSigmaNummer13\] also illustrates another important phenomenon related to strong streaming, namely that the peak gas overdensity can become strongly offset from the halo center. There is a hint of this effect in the run with 2$\sigma$ streaming, but in this case the offset between the density peak and the halo center (which is always located at the origin) is small. In the case of 3$\sigma$ streaming, however, the offset between the highest gas densities and the halo center is clearly apparent. In extreme cases, this offset can result in the formation of baryon-dominated clouds outside of the virial radius of the closest dark matter halo [@nn14; @pnmv16; @Hirano2018b].
Statistical analysis of the full minihalo sample {#Results}
================================================
As a next step, we study the distribution of spins and shapes in a statistical manner. For this part of our analysis, we selected all haloes with a mass of at least $M_{\rm min} = 10^{5} \ \mathrm{M_{\odot}}{}$, as less massive haloes are not well resolved in our cosmological simulations (@schauer18; Paper I). This mass limit leads to the selection of 7982 haloes for no streaming, 6282 haloes for 1$\sigma$ streaming, 4798 haloes for 2$\sigma$ streaming and 4263 haloes for 3$\sigma$ streaming. The different number of haloes results from the delaying effect that streaming motions have on the formation and growth of minihaloes [see e.g. @Tseliakhovich; @greif11; @mkc11; @stacy11a].
Spin and shape distribution {#Spin and shape properties}
---------------------------
![The distributions of the spin parameters of all haloes for gas (upper panel), dark matter (middle panel) and total matter (lower panel). The blue solid ($0\sigma$), red dashed ($1\sigma$), black dash and dotted ($2\sigma$) and purple dotted ($3\sigma$) lines indicate the respective streaming velocities.[]{data-label="fig:LambdaGASdmTOT0123Subplot"}](figures/LambdaGASdmTOT0123Subplot.pdf){width="0.99\columnwidth"}
In Figure \[fig:LambdaGASdmTOT0123Subplot\], we show the spin parameter distribution for gas (top panel), dark matter (middle panel) and the total halo (bottom panel) for all four streaming velocity simulations.[^5] As we can see, the distribution for the dark matter does not change significantly as we vary the streaming velocity. The same is true for the spin parameter distribution for the halo as a whole, since this is dominated by the dark matter contribution. The peak value varies between $\lambda^{\prime}_\mathrm{dm} = 0.030$ and $\lambda^{\prime}_\mathrm{dm} = 0.033$ for the dark matter and between $\lambda^{\prime}_\mathrm{tot} = 0.029$ and $\lambda^{\prime}_\mathrm{tot} = 0.033$ for the total mass.
For gas on the other hand, the distribution is strongly affected by the size of the streaming velocity. Increasing the streaming velocity flattens the spin parameter distribution and shifts its peak toward much higher values of $\lambda^{\prime}$: it changes from $\lambda^{\prime}_\mathrm{gas} = 0.026$ with no streaming to $\lambda^{\prime}_\mathrm{gas} = 0.138$ for 3$\sigma$ streaming. These results are broadly consistent with the recent study of [@Chiou], who examine the minihalo spin parameter distribution in simulations with no stream and 2$\sigma$ streaming. They also find a significant shift in the peak of the distribution with increasing streaming velocity, with $\lambda^{\prime}_\mathrm{gas}$ increasing from 0.04 in the case without streaming at $z = 10$ to 0.12 in the case of 2$\sigma$ streaming.
![Same as in Figure \[fig:LambdaGASdmTOT0123Subplot\], but for the triaxiality of all haloes.[]{data-label="fig:TriaxGASdmTOT0123Subplot"}](figures/TriaxGASdmTOT0123Subplot.pdf){width="0.99\columnwidth"}
We have also examined how the shape of the minihaloes changes due to the streaming velocity. For this purpose, we show beta distribution fits to the triaxiality and sphericity distributions in Figures \[fig:TriaxGASdmTOT0123Subplot\] and \[fig:SpaereGASdmTOT0123Subplot\], respectively. Again, we see that there are major changes in the distribution of gas, while the distribution of dark matter and total matter remains virtually unchanged. As the streaming velocity increases, both the sphericity and the triaxiality of the gas distribution decrease (i.e. it becomes less spherical and more oblate). The peak values of $\lambda^{\prime}$, $S$ and $T$ at $z=14$ for each component in each simulation are listed in Table \[Table:ValuesTabelle\].
-- ----- -------- ------- -------
gas 0.0256 0.768 0.744
dm 0.0300 0.688 0.668
tot 0.0294 0.700 0.685
gas 0.0672 0.634 0.683
dm 0.0305 0.679 0.688
tot 0.0301 0.676 0.700
gas 0.1193 0.678 0.637
dm 0.0326 0.684 0.691
tot 0.0323 0.683 0.696
gas 0.1381 0.661 0.642
dm 0.0331 0.679 0.687
tot 0.0330 0.677 0.691
-- ----- -------- ------- -------
: List of all calculated peak values at $z=14$ for the spin ($\lambda^{\prime}$), triaxiality (T) and sphericity (S) distributions, for our four simulations and for all components: gas, dark matter (dm) and total matter (tot).[]{data-label="Table:ValuesTabelle"}
![Same as Figure \[fig:LambdaGASdmTOT0123Subplot\], but for the sphericity of all haloes.[]{data-label="fig:SpaereGASdmTOT0123Subplot"}](figures/SpaereGASdmTOT0123Subplot.pdf){width="0.99\columnwidth"}
### Mass dependence {#Mass dependence}
In order to understand why the spin and shape parameters for the gas change as the streaming velocity increases, we have explored how they vary as a function of the halo mass. In Figure \[fig:Mass\_vs\_Lambda\_0Sig\_013\], we show 2D histograms of the spin parameter as a function of the halo mass $M$ for the whole halo (left-hand panels) and for the gas component (right-hand panels) for each of the simulations. In these histograms, the spin is plotted against the logarithmically scaled mass using a $40 \times 40$ pixel grid. The colour of the pixels indicates the number of haloes contained in each. We also show the most probable spin parameter (i.e. the peak in the distribution) for each halo mass (solid line). The shaded region around this line is an estimate of how accurately we are able to determine the peak in the distribution, computed using the bootstrap method.
![Two dimensional histograms for the total spin parameter (left) and spin parameter of the gas component (right), as a function of the total mass of the minihalo. The solid lines show the peak value of the spin parameter distribution in that halo mass bin, and the shaded regions indicate the error in the determination of this peak value, estimated using the bootstrap method.[]{data-label="fig:Mass_vs_Lambda_0Sig_013"}](figures/logMass_vs_Lambda_0Sig_013log.pdf){width="0.99\columnwidth"}
We can clearly see that in the case of no streaming velocity, there is no correlation between the spin parameter and the halo mass, neither for the gas nor for the total halo. Instead, the peak value of the spin parameter distribution remains approximately constant with halo mass. Although there are haloes with relatively large values for the spin parameter ($\lambda^{\prime} > 0.15$), these are all found in the lowest mass bins with $M \sim 10^{5}\ \mathrm{M_{\odot}}$. Additionally, these haloes represent the high $\lambda^{\prime}$ tail of the distribution and may be missing in the higher mass bins simply because there are far fewer minihaloes in total present in those bins. We note that this is not a new result: other studies of the minihalo spin parameter distribution in the absence of streaming have also found it to be independent of halo mass [@Hirano; @Sasaki].
On the other hand, when the streaming velocity is non-zero, we find a clear difference in behaviour. While for total matter the peak value of the spin parameter remains independent of halo mass for all streaming velocities, this is no longer true for the gas component. Instead, we see an increase in the peak value with decreasing halo mass. This can be seen more clearly in Figure \[fig:Mass\_vs\_Lambda\_AlleInEinemPlotMedian\_Balken\], where we plot only the peak value of the spin parameter distribution for each mass bin (the red and blue lines from Figure \[fig:Mass\_vs\_Lambda\_0Sig\_013\]), plus the same quantity for the dark matter (black lines).
In the 1$\sigma$ streaming run, we see that $\lambda^{\prime}_{\rm gas} \simeq \lambda^{\prime}_{\rm tot}$ in the highest mass bin, but that it systematically increases above this value for decreasing halo mass, so that in the lowest mass bin it is almost three times as large. In the runs with even stronger streaming, we not only see a similar dependence on halo mass but also a clear offset between $\lambda^{\prime}_{\rm gas}$ and $\lambda^{\prime}_{\rm tot}$ even in the highest mass bin.
![The peak value of the spin parameter distribution, plotted as a function of halo mass, for the gas (red), dark matter (black) and total matter (blue). The four panels show results for our four different simulations, as indicated in the top left corner of each panel.[]{data-label="fig:Mass_vs_Lambda_AlleInEinemPlotMedian_Balken"}](figures/Mass_vs_Lambda_AlleInEinemPlotLOG_balken.pdf){width="0.99\columnwidth"}
We have also investigated the mass dependence of the shape parameters (triaxiality $T{}$ and sphericity $S{}$), as shown in Figures \[fig:Mass\_vs\_Triax\_AlleInEinemPlotMedian\_Balken\] and \[fig:Mass\_vs\_Sphaere\_AlleInEinemPlotMedian\_Balken\]. In this case, the beta distribution is not always a good description of the distribution of $T$ and $S$ in each bin (especially in the low mass bins) and so the representative value we plot for each bin is the median value.
We see from the Figures that the triaxiality is largely independent of the halo mass, regardless of the streaming velocity, although there is a hint that in the low streaming runs, higher mass haloes are slightly more prolate than lower mass haloes.
On the other hand, the sphericity does show a more pronounced mass dependence, with lower mass haloes being more spherical than higher mass haloes. When the streaming velocity is low, there is little difference between the sphericity of the gas distribution and that of the total mass, but when the streaming velocity is high there is a clear offset between the two, with the gas having a systematically less spherical distribution than the dark matter or the total mass.
![Same as Figure \[fig:Mass\_vs\_Lambda\_AlleInEinemPlotMedian\_Balken\], but for the triaxiality of all haloes.[]{data-label="fig:Mass_vs_Triax_AlleInEinemPlotMedian_Balken"}](figures/Mass_vs_Triax_AlleInEinemPlotMedian_Balken.pdf){width="0.99\columnwidth"}
![Same as Figure \[fig:Mass\_vs\_Lambda\_AlleInEinemPlotMedian\_Balken\], but for the sphericity of all haloes.[]{data-label="fig:Mass_vs_Sphaere_AlleInEinemPlotMedian_Balken"}](figures/Mass_vs_Sphaere_AlleInEinemPlotMedian_Balken.pdf){width="0.99\columnwidth"}
### Radial dependence
To further explore the impact of the streaming velocity on the spin parameter of the gas, we have also examined how this varies as a function of radius in the different simulations.
![Peak value of the spin parameter distribution as a function of radius (for shells with a width of $d=0.1 R_{\rm vir}$). We use the same color scheme as in Figure \[fig:Mass\_vs\_Lambda\_AlleInEinemPlotMedian\_Balken\], i.e. gas (red), dark matter (black) and total matter (blue).[]{data-label="fig:LambdaSlicesAllSigma"}](figures/LambdaSlicesAllSigma.pdf){width="0.99\columnwidth"}
To do this, we split up each halo into ten separate shells of thickness 0.1 $R_\mathrm{Vir}$ and calculated the spin and shape parameters for each shell individually.[^6]
Comparison of the spins of shells with the same $R/R_{\rm Vir}$ in different haloes shows that the distribution can once again be represented as a log-normal. In Figure \[fig:LambdaSlicesAllSigma\] we show the peak value of this log-normal distribution as a function of $R/R_{\rm Vir}$ for our four different runs. The uncertainty in this peak value was calculated using the bootstrap method and is indicated by the shaded region. As before, the calculation was carried out with the various components and plotted in the usual colors gas (red), dark matter (black) and total matter (blue).
We can immediately see from Figure \[fig:LambdaSlicesAllSigma\] that the spin parameter of the gas component changes significantly with the streaming velocity, while the distributions for dark matter and for the entire halo only change slightly. The distribution of $\lambda^{\prime}$ with radius for the dark matter remains fairly flat in all four simulations, with a slight rise to $\lambda^{\prime}_\mathrm{dm} \sim 0.1$ in the outer third of the halo, and the distribution of $\lambda^{\prime}$ for the total mass behaves similarly. For the gas, we also find a flat distribution in the no streaming run. However, in the runs with streaming, we find qualitatively different behaviour. In this case, $\lambda^{\prime}$ is higher close to the centre of the halo than at the virial radius, and hence largely decreases with $R / R_{\rm Vir}$. However, the largest value of $\lambda^{\prime}$ is not found at the halo centre, but instead is offset to $R/R_{\rm Vir} \sim 0.2$–0.3. It is also clear that the degree to which $\lambda^{\prime}$ drops between this peak and the virial radius also depends on the streaming velocity: with a low streaming velocity, $\lambda^{\prime}$ decreases by almost a factor of two between $R = 0.3 R_{\rm Vir}$ and $R = R_{\rm Vir}$, while in the 3$\sigma$ streaming run, the drop is much smaller.
![Same as Figure \[fig:LambdaSlicesAllSigma\], but for the median value of the specific angular momentum of all haloes.[]{data-label="fig:JSlicesAllSigmaMedianBootstrap"}](figures/JSlicesAllSigmaMedianBootstrap.pdf){width="0.99\columnwidth"}
To further explore this behaviour, we have also calculated the specific angular momentum $J_\mathrm{spec} = J / M{}$ as a function of radius for the different streaming velocities and for gas (red), dark matter (black) and total matter (blue). As the peak value, we choose the median of the specific angular momentum in each shell. The results are shown in Figure \[fig:JSlicesAllSigmaMedianBootstrap\]. In every case, the specific angular momentum increases with increasing radius. As before, there is little change of the dark matter component and hence the combined halo when including streaming velocities. The gas component shows a higher specific angular momentum for the entire radius range for higher streaming velocities, consistent with Figure \[fig:LambdaSlicesAllSigma\]. This is also in agreement with Figure \[fig:PlotAlleSigmaNummer13\], where we see clumps of high density gas offset from the gravitational centre of the minihalo [compare also @Chiou; @Chiou19]. For minihaloes in the 1$\sigma{}$ streaming velocity simulation, the increase is stronger at smaller radii than at larger radii, which we interpret as a smaller offset of the denser gas from the centre than in the higher streaming velocity simulations. Further investigations have shown that in the run without streaming, any offset between the highest density gas and the halo centre is small ($\ll 0.1 R_{\rm vir}$), but that it increases significantly as we increase the streaming velocity. We therefore conclude that the higher spin parameter of the gas found in the higher streaming velocity simulations is not because the gas forms a larger or more rapidly rotating disk, but is instead due to the presence of dense clumps of gas at large distances from the halo centre with significant tangential velocities.
![Distribution of the angles between the gas and total halo components of the specific angular momentum (red) and between the specific gas angular momentum and the vector of the streaming velocity flow direction (blue).[]{data-label="fig:AnglecorrelationAlleInEinem2"}](figures/AnglecorrelationAlleInEinem2.pdf){width="0.99\columnwidth"}
Angle Correlations {#Angle Correlations}
------------------
From the previous section, it is clear that minihaloes formed in runs with higher streaming velocities tend to have gas components with higher spin parameters. We can gain some insight into why this is so if we examine how the angular momentum of the gas is oriented with respect to the angular momentum of the dark matter. In Figure \[fig:AnglecorrelationAlleInEinem2\], we show the distribution of the angle $\alpha$ between the angular momentum vector for the total halo mass distribution and that for the gas.
In the run without streaming, we see that in most haloes the direction of the two vectors is highly correlated, with $\alpha$ peaking close to zero. The median angle is $\sim 23$ degrees, in good agreement with the value of around 30 degrees found in previous work [@Bosch; @Liao]. However, as the streaming velocity increases, this correlation disappears: the distribution of $\alpha$ flattens and becomes consistent with a purely isotropic distribution [@Chiou]. We can see why this happens if we examine the alignment between the direction of the gas angular momentum vector and the direction of the streaming velocity in these runs (blue curves in Figure \[fig:AnglecorrelationAlleInEinem2\]). This peaks around an angle of 90 degrees, with this peak becoming increasingly pronounced as the streaming velocity increases.
If the motion of the gas were purely due to the streaming, we would expect to recover a perfect 90 degree alignment between the streaming direction and the angular momentum (since $\vec{J} = \vec{r} \times \vec{v}$). In reality, the gas also has motions due to its infall into the dark matter potential well and due to the tidal torque acting on it from the surrounding distribution of matter. However, the fact that we nevertheless recover a clear peak in the alignment at 90 degrees in the runs with streaming demonstrates that in the majority of haloes it is the streaming that dominates the large-scale motion of the gas within the virial radius, particularly in the runs with high sigma streaming. Moreover, since the angular momentum of the dark matter is uncorrelated with the streaming, its alignment is random with respect to the streaming direction and hence also with respect to the gas in the high sigma streaming runs. On the other hand, in the run with no streaming, tidal torques dominate and so we recover a good correlation between the directions of the gas and dark matter angular momentum vectors since the same torques act on both components.
Dense gas {#cold}
---------
{width="1.99\columnwidth"}
It is also interesting to examine the impact of streaming on the spin of the cold dense gas found at the centre of the subset of minihaloes that are capable of forming stars. In Paper I, we showed that in the absence of streaming, there is no correlation between the spin of this gas and that of the halo as a whole, implying that the latter cannot be used to predict the former, contrary to previous conjectures in the literature [e.g. @Souza]. Does this result still hold in runs that include the effects of streaming?
To investigate this, we have examined the behaviour of gas above two different density thresholds, $n_{\rm thres} = 1 \, {\rm cm^{-3}}$ and $n_{\rm thres} = 100 \, {\rm cm^{-3}}$. In each halo, we first calculate the distance from the centre of the halo to the farthest cell with a density above $n_{\rm thres}$. We then calculate the spin, sphericity and triaxiality of the gas contained within a sphere with a radius equal to this distance. In order to allow us to make a meaningful statement about the spin and the shape of the haloes, we require a minimum number of 25 cold dense cells per halo. Haloes that do not satisfy this requirement are not included in the analysis. Since streaming hampers gas cooling, particularly in low mass haloes [see e.g. @schauer18], the number of haloes considered here decreases for high streaming velocities. The number of haloes included in the analysis for each combination of simulation and $n_{\rm thres}$ is summarized in Table \[Table:Anzahl\]. Note that in the 3$\sigma$ streaming simulation, there are no haloes with 25 gas cells denser than $n_{\rm thres} = 100 \: {\rm cm^{-3}}$, preventing us from analyzing the properties of the dense gas in this case.
In Figure \[fig:LaTrSpCgasAllpart\], the histograms (blue) and their distributions (red) are shown for both the spin and the shape of the dense gas in our four simulations. The different columns show $\lambda^{\prime}$, $T$ and $S$, respectively, for simulations with no streaming (first two rows), 1$\sigma$ streaming (third and fourth rows), 2$\sigma$ streaming (fifth and sixth rows) and 3$\sigma$ streaming (last row). The value of $n_{\rm thres}$ considered in each case is indicated in the plot. In the simulations with higher streaming velocities, the spin parameter, triaxiality and sphericity histograms are no longer well fit by log-normal or beta distributions, respectively, particularly when $n_{\rm thres} = 100 \, {\rm cm^{-3}}$. This is likely a consequence of the small number of haloes we are dealing with in these cases. As a result, we cannot easily calculate meaningful peak values for all of the histograms. Nevertheless, some basic trends are clear.
$n_\mathrm{thres}\ [\mathrm{cm}^{-3}]$ 0$ \sigma$ 1$ \sigma$ 2$ \sigma$ 3$ \sigma$
---------------------------------------- ------------ ------------ ------------ ------------
1 2604 831 199 64
100 206 87 15 -
: Number of haloes in each simulation with at least 25 gas cells with densities greater than the specified threshold density.[]{data-label="Table:Anzahl"}
Figure \[fig:LaTrSpCgasAllpart\] demonstrates that in gas denser than $n_{\rm thres} = 1 \: {\rm cm^{-3}}$ (roughly a factor of ten higher than the virial density at this redshift), we recover a very similar result to the one we found in Section \[Spin and shape properties\] for the total gas content, namely that as the streaming velocity increases, so does the spin parameter, while the sphericity decreases. However, if we turn our attention to gas denser than $n_{\rm thres} = 100 \: {\rm cm^{-3}}$, we see that in this case, neither the spin parameter nor the shape of gas distribution show any clear dependence on the streaming velocity.
![Total spin parameter ($\lambda^{\prime}_{\rm tot}$) and spin parameter of gas denser than $n_{\rm thres} = 100 \: {\rm cm^{-3}}$ ($\lambda^{\prime}_{100}$), shown for each halo with at least 25 gas cells denser than $n_{\rm thres}$ and for the four different runs. The points are color-coded by the total mas of the halo. In the run with $3 \sigma$ streaming, there are no minihaloes with 25 cells above the density threshold.[]{data-label="fig:CrossLambda"}](figures/CrossLambda.pdf){width="0.99\columnwidth"}
We have also examined whether there is any correlation between the spin parameter of the dense gas and that for the halo as a whole (Figure \[fig:CrossLambda\]). In Paper I, we showed that in the absence of streaming, these two quantities are not correlated. Figure \[fig:CrossLambda\] demonstrates that this important result continues to hold in runs that include streaming.
Both of these results suggest that the spin and shape of the dense, gravitationally-collapsing gas are determined primarily by the details of the collapse itself, and preserve little or no memory of the state of the gas on large scales.
Conclusion {#Conclusion}
==========
In this paper, we have investigated the spin and shape distributions of a large sample of minihaloes formed in simulations with streaming velocities ranging from zero to $3\sigma$. We examine the state of the simulations at a redshift of $z = 14$ and only consider haloes with a minimum mass of at least $\mathrm{M_{min}} = 10^{5}\ \mathrm{M_{\odot}}$. This results in 7982 haloes for the simulation with no streaming, 6282 haloes for the case of 1$ \sigma$ streaming, 4798 haloes for 2$ \sigma$ streaming and 4263 haloes for 3$ \sigma$ streaming. As well as measuring the spin and shape distributions for the full sample of minihaloes, we have also explored how these properties vary as a function of halo mass. In a subset of the full minihalo sample, gas cools and undergoes runaway gravitational collapse. In these haloes, we have quantified the spin and shape distributions of the dense gas and have examined whether the spin and shape of the dense gas are correlated with the same properties measured on the scale of the halo as a whole. Below, we summarize our main results.
- Streaming velocities only affect the spin and shape distributions of the gas component in the minihaloes. Their effect on the dark matter component is negligibly small.
- As the streaming velocity increases, the spin parameter of the gas component increases. The gas component of the halo is less spherical and less prolate for a non-zero streaming velocity than for the case of no streaming velocity [compare @Druschke].
- The spin parameter of minihaloes in a region of the Universe with no streaming velocity is independent of mass. However, the minihalo shape has a slight dependence on mass: more massive minihaloes tend to be slightly more prolate and less spherical.
- In regions with streaming, the spin parameter of the gas in the minihaloes becomes mass dependent. Low-mass minihaloes develop higher spin parameters than higher-mass minihaloes. The shape parameters, on the other hand, become completely independent of mass. The effect on dark matter is once again negligible.
- The center of rotation of the halo and the position of the most bound particle can deviate significantly from each other under the influence of streaming velocities. This can leads to an increase of the spin parameter outside the centre of the minihalo. Figure \[fig:PlotAlleSigmaNummer13\] shows an example of a rotational offset that is very strong for a streaming velocity of 3$ \sigma$. More quantitatively, Figure \[fig:JSlicesAllSigmaMedianBootstrap\] shows that the specific angular momentum of the gas component increases with increasing streaming velocities at all radii. However, for a streaming velocity of 1$\sigma$, the change is more significant at the centre than for higher streaming velocity simulations, showing that the gas clumps causing the large spin move to larger radii.
- The streaming velocity also affects the orientation the angular momentum of the gas component. As the streaming velocity increases, it becomes aligned increasingly strongly in a direction perpendicular to the streaming motion. Since the dark matter haloes themselves have angular momenta that are aligned randomly with respect to the streaming motion, the result is that the alignment between the gas and dark matter angular momentum in any given minihalo becomes increasingly random.
- The spin and shape distributions of dense, gravitationally collapsing gas within the minihaloes are uncorrelated with the values on the scale of the virial radius and unaffected by the strength of the streaming velocity.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Volker Bromm, Mattis Magg and Naoki Yoshida for fruitful discussions. The authors acknowledge support from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007 - 2013) via the ERC Advanced Grant “STARLIGHT: Formation of the First Stars" (project number 339177). SCOG and RSK also appreciate support from the Deutsche Forschungsgemeinschaft via SFB 881, “The Milky Way System” (sub-projects B1, B2 and B8) and SPP 1573 , “Physics of the Interstellar Medium” (grant number GL 668/2-1). The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre. Support for this work was provided by NASA through the NASA Hubble Fellowship grant HST-HF2-51418.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. The authors acknowledge support by the state of Baden-Württemberg through bwHPC and the German Research Foundation (DFG) through grant INST 35/1134-1 FUGG. The authors would like to thank the Cox Fund for providing a travel grant which lead to the completion of this manuscript.
\[lastpage\]
[^1]: E-mail: vu412@ix.urz.uni-heidelberg.de
[^2]: Hubble Fellow
[^3]: Dark matter haloes with virial temperatures $T \sim 10^{4} \: {\rm K}$ or above which are cooled by Lyman-$\alpha$ emission from atomic hydrogen.
[^4]: [arepo]{} refines or de-refines gas cells as required to ensure that their masses stay within a factor of two of the specified target mass; see @arepo for more details.
[^5]: To improve the visibility, we only show the fit of the log-normal distribution (compare Figure \[fig:LambdaTriaxSphaereVsPDF\]), and not the entire histogram.
[^6]: We chose this particular shell width to ensure that each shell would contain enough dark matter particles and gas cells to enable an accurate calculation of the spin and shape parameters.
|
---
abstract: 'Superconducting nanowires used in single-photon detectors have been realized on amorphous or poly-crystalline films. Here, we report the use of single-crystalline NbN thin films for superconducting nanowire single-photon detectors (SNSPDs). Grown by molecular beam epitaxy (MBE) at high temperature on nearly lattice-matched AlN-on-sapphire substrates, the NbN films exhibit high degree of uniformity and homogeneity. Even with relatively thick films, the fabricated nanowire detectors show saturated internal efficiency at near-IR wavelengths, demonstrating the potential of MBE-grown NbN for realizing large arrays of on-chip SNSPDs and their integration with AlN-based $\chi^{(2)}$ quantum photonic circuits.'
author:
- Risheng Cheng
- John Wright
- 'Huili G. Xing'
- Debdeep Jena
- 'Hong X. Tang'
bibliography:
- 'My\_reference.bib'
title: 'Epitaxial niobium nitride superconducting nanowire single-photon detectors'
---
[^1]
[^2]
Superconducting nanowire single-photon detectors (SNSPDs)[@goltsman_2001_first_SNSPD; @Hadfield_2009_SPD_review] have become an indispensable resource for a range of quantum and classical applications due to their high detection efficiency over a broad spectrum[@nist_2013_93p_efficiency; @nist_2019_mid-ir_detector_spectroscopy; @berggren_2012_mid-ir_detector; @nist_2019_98p_snspd], ultra-fast speed[@pernice_2018_2DPC; @simit_2019_16_pixel_detector], exceptional timing performance[@Delft_2018_10ps_jitter_detector; @jpl_2020_3ps_jitter; @delft_2020_platform_snspd], and ultra-low dark count noise[@schuck_2013_mHz_dark_count; @NTT_2015_ultimate_darkcounts]. Two categories of superconducting materials have so far been used for the fabrication of high-efficiency SNSPDs – poly-crystalline nitride, and amorphous alloy superconductors. SNSPDs patterned with thin-film amorphous superconducting materials, such as WSi[@nist_2013_93p_efficiency; @nist_2011_first_wsi_detector] and MoSi[@switzerland_2018_mosi_detector; @goltsman_2014_mosi_detector; @hadfield_2016_MoSi_waveguide_detector], have exhibited excellent homogeneity over a large device area[@nist_2020_wsi_microwire; @Charaev2020; @nist_2019_kilopixel_snspd] due to the absence of grain boundaries. However, they require relatively lower operation temperature and have lower maximum counting rates, resulting from longer hot spot relaxation time in comparison with the SNSPDs made from nitride superconductors, such as NbN[@simit_2017_92p_nbn_detector; @pernice_2015_waveguide_snspd; @pernice_2012_waveguide_SNSPD; @cheng_2019_snspd_ald] and NbTiN[@delft_2017_92p_nbn_detector; @cheng_2017_multiple_SNAP; @cheng_2016_self_aligned_detector; @nict_2017_nbtin_snap; @Machhadani2019]. On the other hand, Nb(Ti)N-based detectors have shown relatively superior timing performance, demonstrating <3ps jitter measured with a short straight nanowire[@jpl_2020_3ps_jitter] and <8ps with a large-area meandered nanowires[@EsmaeilZadeh2020]. Despite these advantages, the homogeneity of Nb(Ti)N-SNSPDs are ultimately limited by the poly-crystalline nature of the sputtered Nb(Ti)N films, which could lead to non-uniform distribution of critical currents in a large array of single-photon detectors required for future integrated quantum photonic circuits.
![ (a) *In-situ* RHEED pattern measured after the film growth demonstrating the epitaxial nature of the NbN film. The streakiness of the pattern evidences that the surface is effectively 2D. (b) AFM surface height map of the NbN thin film exhibiting $R\textsubscript{rms}$=0.29nm. (c) RHEED intensity monitored throughout the NbN thin film growth. The exhibited oscillations of the specular spot brightness indicates the 2D layer-by-layer growth mode of the NbN. (d) Cross-sectional sketch of the thin film layer structure. (e) Measured sheet resistance of the NbN thin film versus temperature with the inset showing the $T_\mathrm{c}$ of 12.1K. []{data-label="fig:material"}](growth_figure_final.pdf){width="1\linewidth"}
{width="1\linewidth"}
In this Letter, we demonstrate SNSPDs made from single-crystal NbN thin films grown by molecular beam epitaxy (MBE)[@Yan2018] on nearly lattice-matched AlN-on-sapphire substrates. This substrate platform is attractive for the integration of SNSPDs with several other elements of nitride-based photonic integrated circuits [@pernice_2018_waveguide_snspd_review; @cheng_2019_broadband_spectrometer; @italy_2019_amplitude_multiplex; @berggren_2015_on_chip_detector; @cheng_2020_perfect_absorber]. The epitaxial NbN films exhibit a high degree of thickness uniformity and structural perfection owing to the 2D layer-by-layer growth unique to MBE technique. The fabricated device consisting of 20nm-wide and 7nm-thick nanowire shows saturated internal efficiency at the wavelength of 780nm and 1050nm, while further reduction in achievable thin film thickness holds promise for saturating the efficiency at longer wavelength with more relaxed wire width. We expect MBE-NbN on AlN-on-sapphire substrate shown here could provide a scalable material platform for realizing large array of on-chip SNSPDs and integration with nitride-based photonic circuits.
As illustrated in \[fig:material\], epitaxial NbN films are grown by radio-frequency plasma-assisted MBE on a commercial 2inch-diameter c-plane sapphire substrate with a 3$\mu$m-thick AlN film grown by hydride vapor phase epitaxy (HVPE). A 240nm-thick AlN film of Al-polar orientation is grown by MBE, followed by the growth of NbN as shown in \[fig:material\](d). During the growth of the films, the reactive nitrogen is generated using a radio-frequency plasma source fed by ultrahigh-purity N$_2$ gas, which is further purified by an in-line purifier. Aluminum (99.9999$\%$ purity) is supplied using a Knudsen effusion cell. The Nb flux is generated using an *in situ* electron-beam evaporator source with 3N5-pure (excluding tantalum, Ta) Nb pellets in a tungsten hearth liner. The NbN films are grown at the temperature of 1100C measured by a thermo-couple behind the substrate, at a growth rate of approximately 1.0nm/min.
The MBE film growth is monitored *in situ* using a reflection high-energy electron diffraction (RHEED) system operated at 15kV voltage and 1.5A current. (a) shows sharp and streaky patterns formed by electron diffraction from the smooth surface of the NbN film, indicating the epitaxial nature of the single-crystal NbN film. As shown in \[fig:material\](c), the *in situ* observation of oscillations of the RHEED intensity versus the growth time confirms that the NbN grows in a 2D layer-by-layer growth mode on the AlN surface. The thickness of the NbN film is 7.0nm, measured by X-ray reflectivity (XRR) with a Rigaku SmartLab diffractometer using CuK$\alpha$1 radiation. (b) shows the morphology of the NbN film surface characterized employing tapping mode atomic force microscopy (AFM); the root-mean-square roughness ($R\textsubscript{rms}$) of the film surface is less than 0.3nm within a scan size of 1$\mu$m$\times$1$\mu$m. In addition, the crystal orientation of the NbN is determined using RHEED and X-ray diffraction (XRD), which indicates the cubic NbN grows with the $\{$1 1 1$\}$ crystal axis aligned to the c-axis of AlN.
\(e) shows the temperature dependence of the sheet resistance of the MBE-NbN thin film with the inset showing a zoom-in view of the superconducting transition region. The transition temperature of the film is measured to be $T_\mathrm{c}=$12.1K, defined as the temperature where the resistance of the film drops to 50% of the normal-state resistance measured at 20K. This value is among one of the highest results reported so far for NbN thin films of a few nanometer thickness. The high $T_\mathrm{c}$ value is also consistent with the significantly low resistivity of the film, which is calculated to be only $\sim$100$\,\mu\Omega\cdot\mathrm{cm}$ obtained by multiplying the thickness with the room-temperature sheet resistance.
We fabricate SNSPD devices by patterning the 7nm-thick MBE-NbN film. The nanowires are defined by the exposure of negative-tone 6% hydrogen silsesquioxane (HSQ) resist using $100\,\mathrm{kV}$ electron-beam lithography (Raith EBPG5200) and the subsequent development in 25% tetramethylammonium hydroxide (TMAH) for 2minutes at room temperature. The HSQ resist is spun at the speed of 4000rpm, resulting in an approximate thickness of 90nm. In a second electron-beam lithography step, contact electrodes are defined using double-layer polymethyl methacrylate (PMMA) positive-tone resist. After the development in the mixture of methyl isobutyl ketone (MIBK) and isopropyl alcohol (IPA), we liftoff electron-beam evaporated 10$\,$nm-thick Cr adhesion layer and 100$\,$nm-thick Au in acetone overnight to form the contact pads. The HSQ nanowire pattern is then transferred to the NbN layer in a timed reactive-ion etching (RIE) step employing CF~4~ chemistry and 50W RF power. The HSQ resist is left on top of the NbN nanowires after fabrication, serving as a barrier to oxidation.
For initial tests, we fabricate short-nanowire detectors with widths ranging from 20nm to 100nm for comparison of the internal efficiencies. As shown in \[fig:device\](a) and (b), the active detection parts of the devices are made of 20$\mu$m-long straight nanowires which are suitable for future waveguide integration. All the nanowires are serially connected to long 1$\mu$m-wide meandered wires to prevent the detector latching at high bias currents. The sheet resistance of the devices are measured to be around 180$\ohm$/sq, which slightly increase compared to the value measured on the bare film prior to fabrication.
In order to characterize the optical response of the fabricated detectors, the detector chip containing multiple devices is mounted on a 3-axis stack of Attocube stages inside a closed-cycle refrigerator and cooled down to 1.7K base temperature. Continuous wave (CW) laser light with varied wavelength is attenuated to the single-photon level and sent to the detector chip via a standard telecommunication fiber (SMF-28) installed in the refrigerator. The detectors are flood-illuminated by fixing the fiber tip far away from the surface of the detector chip. We control the Attocube stages to move the detector chip at low temperature and make the electrical contact between the RF probes and the gold pads of the detectors. The RF probes are connected to a semi-rigid coaxial cable installed in the refrigerator, while the room-temperature end of the cable is attached to a bias-tee (Mini-Circuits ZFBT-6GW+) to separate the DC bias current and RF output pulses for the detectors. The bias current is supplied by a programmable sourcemeter (Keithley 2401) in conjunction with a low-pass filter (1kHz cut-off frequency). The output pulses of the detectors are amplified by a low-noise RF amplifier (RF bay LNA-650) and sent to a 4GHz oscilloscope for the pulse observation or a pulse counter (PicoQuant PicoHarp 300) for the photon counting measurement. (c) shows a single-shot trace measurement of the output voltage pulse from the 20nm-wide detector. The decay time constant extracted from the exponential fitting (red dashed line) is 5.4ns, which translates into 24pH/sq sheet kinetic inductance of the NbN film, assuming 50$\ohm$ input impedance of the readout amplifier.
![ Normalized photon counting rates (PCR) versus the relative bias current ($I_\mathrm{bias}/I_\mathrm{SW}$) measured with the 20nm-wide and 30nm-wide nanowire detectors for varying wavelength of photons. $I_\mathrm{SW}$ of the nanowires are measured to be 25.5$\mu$A and 38.8$\mu$A, respectively. []{data-label="fig:efficiency"}](efficiency.pdf){width="1\linewidth"}
demonstrates the normalized photon counting rates (PCR) as a function of the relative bias current to the switching current ($I_\mathrm{bias}/I_\mathrm{SW}$) for 20nm-wide and 30nm-wide nanowire detectors. $I_\mathrm{SW}$ of the devices are measured to be 25.5$\mu$A and 38.8$\mu$A, respectively. As expected, detectors made of narrower nanowires with reduced $I_\mathrm{SW}$ show better saturated internal efficiencies at shorter wavelength. For 20nm-wide nanowire detector, we observe a broad saturation plateau at 780nm wavelength, while the efficiency is only nearly saturated at 1050nm wavelength. The minor fluctuation in the curve corresponding to 780nm wavelength is due to the polarization instability of the laser, since the photon absorption of the nanowire is significantly dependent on the polarization status of the incident photons. Neither the 20nm-wide nor the 30nm-wide nanowires show saturation behavior at 1550nm wavelength. We attribute the inefficiency of the detectors to the significantly low electrical resistivity of the MBE-NbN material. In comparison with the SNSPDs made from sputtered[@simit_2017_92p_nbn_detector] or atomic-layer-deposited (ALD)[@cheng_2019_snspd_ald; @sayem_2019_ln_snspd] NbN film of comparable thickness, the nanowires shown in this work demonstrate approximately 3 times reduced sheet resistance as well as sheet kinetic inductance, and in the meantime, 3-5 times improved critical current density. All of these results are in agreement with the 2-3 times lower resistivity of MBE-NbN compared to sputtered or ALD-NbN. Thus, we expect that by further reducing the MBE NbN film thickness down to 2-3nm, saturated efficiency can be obtained in longer wavelengths with relaxed nanowire widths up to 100nm. The growth of high crystalline quality NbN films of 3nm thick or less are achievable by MBE, although a method to protect such thin films from oxidation upon exposure to the ambient is necessary and under investigation. Future work will explore the suitability of such ultra-thin films for SNSPDs.
In summary, we have demonstrated the first SNSPDs patterned from MBE-grown single-crystal NbN thin films on AlN. The 20nm-wide SNSPDs show saturated internal efficiency at the wavelength of 780 nm and 1050 nm. It is worth mentioning that the AlN-on-sapphire substrate, which the epitaxial growth of NbN relies on, is particularly attractive due to its potential of the on-chip integration of SNSPDs with versatile AlN nanophotonic circuits. The excellent optical functionalities of AlN, such as strong $\chi^{(2)}$/$\chi^{(3)}$ nonlinearity[@guo_2017_photon_pair] and large electro-optic effect[@fan_2018_eo_converter; @xiong_2012_aln_review], renders NbN on AlN-on-sapphire a very attractive material platform for realizing fully integrated quantum photonic circuits with the generation, routing, active manipulation and the final detection of single photons on a single chip.
Acknowledgments {#acknowledgments .unnumbered}
===============
This project is funded by Office of Naval Research grants (N00014-20-1-2126 and N00014-20-1-2176) monitored by Dr. Paul Maki. D.J. acknowledges funding support from NSF RAISE TAQs Award No. 1839196 monitored by Dr. D. Dagenais. H.X.T. acknowledges funding support from DARPA DETECT program through an ARO grant (No: W911NF-16-2-0151), NSF EFRI grant (EFMA-1640959) and the Packard Foundation. The authors would like to thank Sean Rinehart, Kelly Woods, Dr. Yong Sun, and Dr. Michael Rooks at Yale University for their assistance provided in the device fabrication, and Dr. Scott Katzer and Dr. David Meyer at the Naval Research Laboratory for useful discussions. The epitaxial growth was performed at Cornell University, and material characterization used resources made available by the NSF CCMR MRSEC Award No. 1719875. The fabrication of the devices was done at the Yale School of Engineering & Applied Science (SEAS) Cleanroom and the Yale Institute for Nanoscience and Quantum Engineering (YINQE).
[^1]: These authors contributed equally to this work.
[^2]: These authors contributed equally to this work.
|
---
abstract: 'We prove the generalized induction equation and the generalized local induction equation (GLIE), which replaces the commonly used local induction approximation (LIA) to simulate the dynamics of vortex lines and thus superfluid turbulence. We show that the LIA is, without in fact any approximation at all, a general feature of the velocity field induced by any length of a curved vortex filament. Specifically, the LIA states that the velocity field induced by a curved vortex filament is asymmetric in the binormal direction. Up to a potential term, the induced incompressible field is given by the Biot-Savart integral, where we recall that there is a direct analogy between hydrodynamics and magnetostatics. Series approximations to the Biot-Savart integrand indicate a logarithmic divergence of the local field in the binormal direction. While this is qualitatively correct, LIA lacks metrics quantifying its small parameters. Regardless, LIA is used in vortex filament methods simulating the self-induced motion of quantized vortices. With numerics in mind, we represent the binormal field in terms of incomplete elliptic integrals, which is valid for $\mathbb{R}^{3}$. From this and known expansions we derive the GLIE, asymptotic for local field points. Like the LIA, generalized induction shows a persistent binormal deviation in the local-field but unlike the LIA, the GLIE provides bounds on the truncated remainder. As an application, we adapt formulae from vortex filament methods to the GLIE for future use in these methods. Other examples we consider include vortex rings, relevant for both superfluid $^4$He and Bose-Einstein condensates.'
author:
- 'Scott A. Strong and Lincoln D. Carr'
bibliography:
- 'GLIE\_arXiv.bib'
title: 'Generalized Local Induction Equation, Elliptic Asymptotics, and Simulating Superfluid Turbulence'
---
Introduction
============
The term superfluid denotes a phase of matter whose dynamical flows can be described, at finite non-zero temperature, by a two-component macroscopic field with well-defined properties. [@PhysRev.60.356; @1959flme.book.....L; @2009JPCM...21p4220G] One component is a purely classical field, while what remains is called the superfluid component. The superfluid is ideal in the sense that it is inviscid and has infinite heat capacity provided by its lack of classical entropy. Rotation enters the superfluid component in quantized vortex filaments[@Feynman195517; @springerlink:10.1007/BF02780991] that, for example in $^{4}$He, transmit thermal information acoustically and are detected by this second-sound. [@2005qvhi.book.....D] Superfluid dynamics can be generated by the introduction of a small heat flux. [@2001LNP...571....3B] Conservation of mass requires that the classical movement away from the heat flux be offset by a counterflow of the superfluid component. If this heat flux is not small, then the quantized vortices tangle, indicating the onset of superfluid turbulence.[@1957RSPSA.240..114V; @1957RSPSA.240..128V; @1957RSPSA.242..493V; @1958RSPSA.243..400V; @1977PhRvL..38..551S; @Schwarz1988] For large heat fluxes, the superfluid transitions into a purely classical phase. In classical turbulence, vorticity can concentrate into complicated geometries. For this reason, large-scale simulation of classical vortex dominated flows is computationally costly.
Vortex line structures are most appropriate to superfluid models of $^4$He where the quantized filaments have radii of a few angstroms.[@2005qvhi.book.....D; @2001LNP...571...97S] These quantized vortices provide a coherent structure for aggressive analytical and numerical study unavailable to classical fluids. Vortex line structures can also be used to model atomic Bose-Einstein condensates, where there has recently been a revival of interest in superfluid turbulence due in part to a series of remarkable experiments in the Bagnato group.[@Henn2009] Our study begins with simplifications to the Biot-Savart representation of the field induced by a vortex line $$\begin{aligned}
\label{BS}
\textbf{v}(\textbf{x})=
\frac{\Gamma}{4\pi}\int_{D\subset \mathbb{R}^{3}}
\frac{(\textbf{x}-\bm{\omega})\times d\bm{\omega}}
{|\textbf{x}-\bm{\omega}|^{3}} =
\frac{\Gamma}{4\pi}\int_{D\subset \mathbb{R}}
\frac{(\textbf{x}-\bm{\xi})\times d\bm{\xi}}
{|\textbf{x}-\bm{\xi}|^{3}}.
\end{aligned}$$ The associated reduction of dimension aids analytic calculation and reduces numerical cost. When such filaments are considered initial-data to the Navier-Stokes problem, then global well-posedness results. [@Cottet1988234; @Newton2001] The use of these data to approximate self-induced vortex motion is the backbone of the *vortex filament method*.[@1980JCoPh..37..289L; @1985AnRFM..17..523L] The question of numerical convergence and accuracy of such vortex filament techniques has been addressed affirmatively in the literature. [@1982MaCom..39....1B] Vortex filament methods reduce the cost of large-scale simulations by restricting analysis to the local field. Due to the complexity of classical vortical flows, interest in these methods waned during the 1980s.[@1980JCoPh..37..289L; @1985AnRFM..17..523L; @chorin:1; @Couet1981305; @Ashurst1988] However, filament methods are highly appropriate for the constrained vortex structure associated with a superfluid. Although (\[BS\]) provides a straightforward starting point for numerical computations, vortex filament methods avoid numerical integration altogether by replacing (\[BS\]) with a local induction approximation.
The local induction approximation (LIA) is a result from classical fluid dynamics, which states that a space-curve vortex defect of an incompressible fluid field with nontrivial curvature generates a binormal asymmetry in the local velocity field. That is, the field local to a length of curved vortex filament induces a flow, which generates filament dynamics. This result, known by Tullio Levi-Civita and his student Luigi Sante Da Rios in the early 1900’s, [@1991Natur.352..561R] was rediscovered by various post World-War II groups. [@1965JFM....22..471B; @1965PhFl....8..553A; @2000ifd..book.....B; @1978SJAM...35..148C] Together, Ricca [@1996FlDyR..18..245R] and Hama [@1988FlDyR...3..149H] provide an excellent chronology of LIA, a topic now common in vortex dynamics texts.[@1996QJRMS.122.1015C; @2001vif..book.....M; @Newton2001] Exploration of LIA occurs in various settings including differential geometry,[@DaRios1906; @DaRios1909; @DaRios1910; @DaRios1911; @DaRios1916a; @DaRios1916b; @DaRios1916c; @DaRios1930; @DaRios1931a; @DaRios1931b; @DaRios1931c; @DaRios1933a; @DaRios1933b] differential equations[@1965JFM....22..471B] and limits of matched asymptotic expansions of vortex tubes. [@1965PhFl....8..553A; @1978SJAM...35..148C; @1991JFM...222..369F]
Our derivation avoids the complications of matched asymptotic expansions by treating vorticity concentrated to an arc. Under this geometry Eq. (\[BS\]) can be reduced to a canonical elliptic representation without the use of power-series approximation to the Biot-Savart integrand.[@2000ifd..book.....B; @1986LPB.....4..316R] While Taylor approximation quickly reveals binormal flow as a dominant feature of the induced field, it lacks error bounds and is often restricted to a two-dimensional subspace of $\mathbb{R}^{3}$. This paper resolves both issues by recasting Eq. (\[BS\]), for a vortex-arc, into an elliptic form valid for all field points. Using this form, one can then use a known asymptotic formula to represent the local field. We offer that this should be adopted, instead of LIA, for use in vortex filament methods and Schwarz’s description of the Magnus force.[@2001LNP...571...97S; @Schwarz1982] Specifically, we will prove the following results:
\[thrm:1\] **Generalized Induction Equation**\
Let $\bm{\omega} = \nabla \times \textbf{v}$ be localized to an arbitrary arc with parameterization $\bm{\xi} =
(R\sin(\theta),R-R\cos(\theta),0)$, where $R\in\mathbb{R}^{+}$ and $\theta \in D_{L}= (-L,L]$ for some $L\in(-\pi,\pi]$. Then there exists bounded functions $\textbf{V}_{1}$, $\alpha_{1}$, $\alpha_{2}$, $L_{\pm}$ and $k$, of $\varepsilon=|\textbf{x}|/R= \kappa |\textbf{x}|$, such that the induced velocity field is given by $$\begin{aligned}
\label{V}
\textbf{v}(\textbf{x}) = \textbf{V}_{1}(\varepsilon)
\left(\alpha_{1}(\varepsilon)
\left[F(L_{+},k)-F(L_{-},k)\right] + \alpha_{2}(\varepsilon)
\left[\frac{dF(L_{+},k)}{d\varepsilon} -
\frac{dF(L_{-},k)}{d\varepsilon}\right]\right)
\end{aligned}$$ where $F$ is an incomplete elliptic integral of the first kind. Moreover, there exist constants $\beta_{1}, \beta_{2},
\beta_{3}, \beta_{4}$ such that $\textbf{V}_{1}$ can be written as $$\begin{aligned}
\label{V1}
\textbf{V}_{1}(\varepsilon) =
\varepsilon\beta_{1} \hat{\textbf{t}} -
\varepsilon \beta_{2}\hat{\textbf{n}} +
\left(\varepsilon\beta_{2} +
\varepsilon \beta_{3} +
\beta_{4}
\right) \hat{\textbf{b}}
\end{aligned}$$ where $\hat{\textbf{t}},\hat{\textbf{n}},\hat{\textbf{b}},$ are the tangent, normal and binormal vectors of the local coordinate system.
\[thrm:2\] **Generalized Local Induction Equation**\
Under the same hypotheses of theorem \[thrm:1\] and for $\varepsilon
\ll 0$ the induced velocity field is dominated by the binormal flow, $$\begin{aligned}
\label{Ve}
\textbf{v}_{\varepsilon}(\textbf{x})=
4 \kappa x_{2} \left(\alpha_{1}(\varepsilon)
\left[F(L_{+},k)-F(L_{-},k)\right] + \alpha_{2}(\varepsilon)
\left[\frac{dF(L_{+},k)}{d\varepsilon} -
\frac{dF(L_{-},k)}{d\varepsilon}\right]\right)\hat{\textbf{b}}.
\end{aligned}$$ where $x_{2}$ is a dimensionless angular component of the spherical decomposition of $\textbf{x}$ and $\kappa$ is the curvature of the vortex arc. The limits $\varepsilon \to 0$ and $L \to 0$ imply that $k
\to 1$ and $\lambda \to 0$ and in this case the incomplete elliptic integral of the first kind admits the asymptotic relation $F \sim F_{1}$ where $$\begin{aligned}
\label{F1}
F_{1}(\lambda,k)=\ln\left(
\sqrt{\frac{1+\lambda}{1-\lambda}}\right)+
\frac{1}{\lambda}\ln\left(
\frac{2}{1+\sqrt{(1-k^{2}\lambda^{2})/(1-\lambda^{2})}}
\right) +\frac{1-k^{2}}{8}\ln\left(
\frac{1+\lambda}{1-\lambda}\right).
\end{aligned}$$ Using this, along with standard differentiation formula for incomplete elliptic integrals of the first kind, provides a first order asymptotic form for the local field given by $$\begin{aligned}
\label{LIE}
\textbf{v}_{\varepsilon}(\textbf{x}) &\sim
-8\kappa x_{2} \left\{\frac{9x_{2}F_{1}(\lambda,k)}{2}-
\frac{x_{2}E(L,k)}{(1-k^{2})k} -
\frac{k\sin(2L)\left[\sqrt{1+k^{2}\sin^{2}(L)}+
\sqrt{1-k^{2}\sin^{2}(L)}\right]}
{2(1-k^{2})\sqrt{1-k^{4}\sin^{4}(L)}}\right\}
\hat{\textbf{b}},
\end{aligned}$$ where $E$ is an incomplete elliptic integral of the second kind.
In words, the first theorem expresses the velocity field generated by vortex arc in terms of incomplete elliptic integrals of the first kind. Moreover, this field can be decomposed into three fields controlling the tangential, circulatory and binormal flows. Of these fields, the binormal contribution is $O(1)$ while the remaining fields are $O(\varepsilon)$. The second theorem considers the remaining field in the limits of $\varepsilon \to 0$ and $L \to 0$. In this limit the incomplete elliptic integral of the first kind admits an asymptotic form and consequently provides a representation for the velocity field local to the vortex arc. This asymptotic form is comparable to LIA in that the Biot-Savart integral has been ‘resolved’ and the binormal flow is represented by elementary functions. This form is only valid for filaments of infinitesimal arclength and consequently idealized. However, using the same asymptotic framework, one can construct expansions valid for arcs of finite length. In fact, the remainder terms of such expansions are known and thus the associated approximation error can be controlled. The necessary asymptotic results are quoted in the appendix.
The rest of this document will be organized as follows. In Section II we define the geometry and derive the Biot-Savart representation of the induced velocity field. In Section III we convert this representation into an elliptic form and prove the generalized induction result (\[V\])-(\[V1\]). In Section IV we reduce this elliptic form into a sum of incomplete elliptic integrals of the first kind. Lastly, using known asymptotic results, we derive an expression for the local velocity field and prove the generalized local induction equation (GLIE) result (\[LIE\]). We conclude with some discussion on adapting this result to vortex filament methods and prospective avenues of future work.
Biot-Savart and Quantized Vortex Rings
======================================
It is well known that a vortex-defect with trivial curvature embedded into an incompressible fluid does not induce autonomous dynamics. This is due to an angular symmetry in the induced velocity field. This symmetry is no longer available for curved vortex elements. Using a vortex ring, it is possible to introduce nontrivial curvature and avoid approximations to the Biot-Savart integral. To be precise, we treat a vortex structure $\bm{\omega}: \mathbb{R}^{3} \to
\mathbb{R}^{3}$ such that $$\begin{aligned}
\bm{\omega}(\textbf{x}) = \left\{
\begin{array}{cc}
1, & \textbf{x}\in \bm{\xi}\\
0, & \textbf{x} \notin \bm{\xi}
\end{array}\right.
\end{aligned}$$ where $\bm{\xi}:D \to \mathbb{R}^{3}$, $D \subset
\mathbb{R}$, is parameterized by the ring $$\begin{aligned}
\bm{\xi} &= R\sin(\theta) \hat{\textbf{i}} +
\left[R-R\cos(\theta)\right]\hat{\textbf{j}},\\
d\bm{\xi} &= \left[R\cos(\theta) \hat{\textbf{i}} +
R\sin(\theta)\hat{\textbf{j}}\right]d\theta
\end{aligned}$$ for $\kappa^{-1}=R\in\mathbb{R}^{+}$, $D_{L}=(-L,L]$ and $L\in[0,\pi]$. Thus, at the point $\textbf{x}=(\tilde{x}_{1},\tilde{x}_{2},\tilde{x}_{3})$ we get an element level description of the velocity field, $$\begin{aligned}
v_{i}(\textbf{x}) = \int_{D_{L}}
\frac{\epsilon_{ijk} (\tilde{x}_{j}-\xi_{j})d\xi_{k}}
{\left[|\textbf{x}|^{2} + |\bm{\xi}|^{2}
-2(\tilde{x}_{1}\xi_{1} +\tilde{x}_{2}\xi_{2}+
\tilde{x}_{3}\xi_{3})
\right]^{3/2}}
\end{aligned}$$ where we have used the Levi-Civita symbol, $\epsilon_{ijk}$, and employed silent-summation over repeated indices. Noting that $|\bm{\xi}|^{2} = 2R^{2}-2R^{2}\cos(\theta)$ provides the formulae $$\begin{aligned}
v_{1}&= -|\textbf{x}|\kappa^{2}x_{3}
\int_{D_{L}}\frac{\sin(\theta)}{D^{3/2}}d\theta,\\
v_{2}&= |\textbf{x}|\kappa^{2}x_{3}
\int_{D_{L}}\frac{\cos(\theta)}{D^{3/2}}d\theta,\\
v_{3}&=\kappa^{2}|\textbf{x}|
\int_{D_{L}}\frac{x_{1}\sin(\theta)-x_{2}\cos(\theta)}
{D^{3/2}}d\theta + \kappa\int_{D_{L}}
\frac{\cos(\theta)-1}{D^{3/2}}d\theta
\end{aligned}$$ where the denominator is given by $$\begin{aligned}
D&=\left[c_{1} + c_{2}\cos(\theta)+c_{3}\sin(\theta)\right] \end{aligned}$$ and whose coefficients are $$\begin{aligned}
R^{2}c_{1} &= |\textbf{x}|^{2}+2R^{2}-2\tilde{x}_{2}R,\\
R^{2}c_{2} &= 2|\textbf{x}|{x}_{2}R-2R^{2},\\
R^{2}c_{3} &= -2|\textbf{x}|{x}_{1}R.
\end{aligned}$$ For future limiting work, we have chosen a radial-representation for $\textbf{x}=|\textbf{x}|({x}_{1},{x}_{2}, {x}_{3} )$ where $x_{i}$ is the $i^{th}$ dimensionless angular component of $\textbf{x}$.
Conversion to Elliptic Form
===========================
The previous integral representations for the velocity field can be cast into elliptic form. To do this, we first reduce each integral into an elliptic integral by taking derivatives with respect to internal parameters. Doing so gives $$\begin{aligned}
v_{1}&= 2\kappa x_{3}\varepsilon
\frac{d}{dc_{3}}\int_{D_{L}}\frac{d\theta}{\sqrt{D}},\\
v_{2}&=-2\kappa x_{3}\varepsilon
\frac{d}{dc_{2}}\int_{D_{L}}\frac{d\theta}{\sqrt{D}},\\\
v_{3}&= \left[2\kappa\varepsilon x_{2}
\frac{d}{dc_{2}}-2\kappa \varepsilon x_{1}
\frac{d}{dc_{3}}+2\kappa
\frac{d}{dc_{1}}-2\kappa
\frac{d}{dc_{2}}\right]\int_{D_{L}}\frac{d\theta}
{\sqrt{D}}
\end{aligned}$$ where the parameter $\varepsilon = |\textbf{x}|/R$ is the ratio of radial distance to the radius of curvature. Application of the chain-rule gives the induced velocity field as $$\begin{aligned}
\label{Vint}
\bm{v}(\textbf{x}) = \textbf{V}_{1}(\varepsilon)
\frac{d}{d\varepsilon} \int_{D_{L}}
\frac{d\theta}{\sqrt{D}}
\end{aligned}$$ where the vector $\textbf{V}_{1}$ is given by $$\begin{aligned}
\textbf{V}_{1}(\varepsilon) &=
2\varepsilon \kappa x_{3}
\frac{dc_{3}}{d\varepsilon} \hat{\textbf{i}}-
2\varepsilon \kappa x_{2}
\frac{dc_{2}}{d \varepsilon}\hat{\textbf{j}}+
\left[2\kappa \varepsilon x_{2}
\frac{dc_{2}}{d\varepsilon}-
2\kappa \varepsilon x_{1} \frac{dc_{3}}{d\varepsilon} +
2\kappa\frac{dc_{1}}{d\varepsilon}-
2\kappa \frac{dc_{2}}{d\varepsilon}\right]
\hat{\textbf{k}}
\end{aligned}$$ implying that the velocity field is determined by the derivative of an incomplete elliptic integral. Moreover, this proves Eq. (\[V1\]) from theorem \[thrm:1\] where $$\begin{aligned}
\beta_{1} &= 2\kappa x_{3}\frac{dc_{3}}{d\varepsilon} ,\\
\beta_{2} &= 2\kappa x_{2}\frac{dc_{2}}{d\varepsilon},\\
\beta_{3} &= -2\kappa x_{1}\frac{dc_{3}}{d\varepsilon},\\
\beta_{4} &= 2\kappa\left(\frac{dc_{1}}{d\varepsilon} -
\frac{dc_{2}}{d\varepsilon}\right).
\end{aligned}$$ For $\varepsilon \ll 1$ we find that the velocity field is dominated by $$\begin{aligned}
\label{Ve2}
\bm{v}_{\varepsilon}(\textbf{x}) =
-8\kappa x_{2} \frac{d}{d\varepsilon}
\int_{D_{L}}
\frac{d\theta}{\sqrt{D}}\hat{\textbf{k}}.
\end{aligned}$$ Figure \[fig:1\] shows the vortex configuration as well as the associated field and vortex coordinate geometry. Using the depicted spherical decomposition of $\textbf{x}$ we find that the previous dimensionless parameter is given by $x_{2}=\sin(\gamma_{1})\sin(\gamma_{2})$. Moreover, we observe that the standard basis vector $\hat{\textbf{k}}$ corresponds to the binormal vector $\hat{\textbf{b}}$. These two facts show that the velocity field, asymptotically close to the vortex arc, is asymmetric in the binormal direction and that this affect is extremized for field-points on the normal-axis, which agrees with standard results of induced binormal flow.
Reduction of Elliptic Form to Canonical Elliptic Integrals
==========================================================
Before we construct the asymptotic representation of the velocity field, the previous integrals are converted into canonical forms. The induced velocity field is controlled by an integral of the form $$\begin{aligned}
\int_{D_{L}} \frac{d\theta}
{\sqrt{c_{1}+c_{2}\cos(\theta)+c_{3}\sin(\theta)}},
\end{aligned}$$ which can be converted to a sum of incomplete integrals of the first kind. First, we introduce a new angle defined by $\tan(\phi) = c_{3}/c_{2}$ and hypotenuse $r^{2}=c_{2}^{2}+c_{3}^{2}$ to get $$\begin{aligned}
\int_{D_{L}} \frac{d\theta}
{\sqrt{c_{1}+c_{2}\cos(\theta)+
c_{3}\sin(\theta)}}
& =
\int_{-L}^{L} \frac{d\theta}
{\sqrt{c_{1}+ r \cos(\phi-\theta)}}.
\end{aligned}$$ Now, introducing a change of variable $2\psi = \phi- \theta$ and the notation $L_{\pm} = (\phi\pm L)/2$ we apply trigonometric formulae to get $$\begin{aligned}
\int_{-L}^{L} \frac{d\theta}{\sqrt{c_{1}+
r \cos(\phi-\theta)}}
=
\frac{2}{\sqrt{c_{1}+r}}\int_{L_{-}}^{L_{+}}
\frac{d\psi}{\sqrt{1-k^{2}\sin^{2}(\psi)}}
\end{aligned}$$ where $k^{2} = 2r/(c_{1}+r)$. Lastly, $$\begin{aligned}
\int_{D_{L}} \frac{d\theta}{\sqrt{c_{1}+
c_{2}\cos(\theta)+c_{3}\sin(\theta)}} =
\frac{2\left[ F(L_{+},k) - F(L_{-},k)\right]}
{\sqrt{c_{1}+r}}
\end{aligned}$$ where $F$ is the standard incomplete elliptic integral of the first kind, $$\begin{aligned}
\label{elliptic}
F(\varphi,k) = \int_{0}^{\varphi}\frac{d\psi}
{\sqrt{1-k^{2}\sin^{2}(\psi)}} =
\int_{0}^{\lambda} \frac{dt}
{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}
\end{aligned}$$ such that $\lambda = \sin(\varphi)$.
Asymptotics for the Incomplete Elliptic Integral of the First Kind
==================================================================
Having reduced the Biot-Savart representation of the velocity field to a canonical form, we can now make use of the known asymptotic formula of Karp and Sitnik,[@2007JCoAM.205..186K] which permits the study of (\[elliptic\]) for all $(\lambda,k) \in [0,1]\times [0,1]$. Specifically, they derive a series representation and remainder term for $F$, which is asymptotic for $k \to 1$. Their complete theorem is quoted in the appendix, but we only require the first-order approximation $$\begin{aligned}
F_{1}(\lambda, k) = \ln\left(
\sqrt{\frac{1+\lambda}{1-\lambda}}\right)+
\frac{1}{\lambda}\ln\left(
\frac{2}{1+\sqrt{(1-k^{2}\lambda^{2})/(1-\lambda^{2})}}
\right) +\frac{1-k^{2}}{8}\ln\left(
\frac{1+\lambda}{1-\lambda}\right).
\end{aligned}$$ which is asymptotic to $F$ for $\lambda \to 0$ and $k \to 1$. This asymptotic formula is not suited to differentiation. [@Erdelyi1956; @springerlink:10.1007/BF02787727; @springerlink:10.1007/BF02937348; @springerlink:10.1007/BF02795344] Thus, we must first apply the differentiation formula, for the incomplete elliptic integral of the first kind, prior to its asymptotic evaluation. Doing so gives lengthy formulae and for these we introduce the constants $$\begin{aligned}
A &= \frac{x_{2}c_{2}-x_{1}c_{3}}{r},\\
A_{1} &= \frac{-4}{(c_{1}+r)^{3/2}}
(\varepsilon - x_{2}+A),\\
A_{2} &= \frac{2}{\sqrt{c_{1}+r}},\\
A_{3} &= -2\frac{x_{2}c_{3} + x_{1}c_{2}}{r^{2}},\\
A_{4} &= \frac{\sqrt{2r}(x_{2}-\varepsilon)}
{(c_{1}+r)^{3/2}} + \sqrt{\frac{2}{r}}
\left(\frac{(c_{1}+r)^{3/2}-r\sqrt{c_{1}+r}}
{(c_{1}+r)^{2}}\right)A.
\end{aligned}$$ Using these constants, find $$\begin{aligned}
\frac{d}{d\varepsilon} \int_{D_{L}}\frac{d\theta}{\sqrt{D}} &=
2A_{1}[F(L_{+},k)-F(L_{-},k)] +A_{2}\left[
\Omega(A_{3},A_{4},A_{5},k,L_{+})-
\Omega(A_{3},A_{4},A_{5},k,L_{-})\right]
\end{aligned}$$ where $$\begin{aligned}
\Omega(A_{3},A_{4},A_{5},k,L) &=
\frac{A_{3}}{\sqrt{1-k^{2}\sin^{2}(L)}}+
\frac{A_{4}E(L,k)}{1(1-k^{2})k} -
\frac{A_{4}F(L,k)}{2k} -\frac{A_{4}2k\sin(2L)}
{4(1-k^{2})\sqrt{1-k^{2}\sin^{2}(L)}}
\end{aligned}$$ is given by differentiation formula for incomplete elliptic integrals of the first kind. Together with Eq. (\[Vint\]), proves Eq. (\[V\]) of our first theorem where $\alpha_{1}=2A_{1}$ and $\alpha_{2}=A_{2}$. From Eq. (\[Ve2\]) we find that the local velocity field is given by $$\begin{aligned}
\bm{v}_{\varepsilon}(\textbf{x}) &= -8\kappa x_{2} \left(
2A_{1}[F(L_{+},k)-F(L_{-},k)] +A_{2}\left[
\Omega(A_{3},A_{4},A_{5},k,L_{+})-
\Omega(A_{3},A_{4},A_{5},k,L_{-})\right]\right).
\end{aligned}$$ At this point, the asymptotic formula (\[ASY\]) can now be applied to $F(\lambda,k)$ where $\lambda=\sin(L_{\pm})$. To compare these results to standard LIA we take $\varepsilon
\to 0$ and $L \to 0$. In this case $c_{1}= -c_{2}= r\sim 2$, $c_{3}\sim 0$ and the constants take the asymptotic forms $$\begin{aligned}
A &= \frac{x_{2}c_{2}-x_{1}c_{3}}{r}\sim -x_{2},\\
A_{1} &= \frac{-4}
{(c_{1}+r)^{3/2}}(\varepsilon - x_{2}+A)\sim x_{2},\\
A_{2} &= \frac{2}{\sqrt{c_{1}+r}} \sim 1,\\
A_{3} &= -2\frac{x_{2}c_{3} + x_{1}c_{2}}
{r^{2}} \sim x_{1},\\
A_{4} &= \frac{\sqrt{2r}(x_{2}-\varepsilon)}
{(c_{1}+r)^{3/2}} + \sqrt{\frac{2}{r}}\left(
\frac{(c_{1}+r)^{3/2}-r\sqrt{c_{1}+r}}
{(c_{1}+r)^{2}}\right)A
\sim -\frac{x_{2}}{2}.
\end{aligned}$$ Together this gives the first-order asymptotic representation for the velocity field, $$\begin{aligned}
\textbf{v}_{\varepsilon}(\textbf{x}) &\sim
-8\kappa x_{2} \left\{\frac{9x_{2}F_{1}(\lambda,k)}{2}-
\frac{x_{2}E(L,k)}{(1-k^{2})k} -
\frac{k\sin(2L)\left[\sqrt{1+k^{2}\sin^{2}(L)}+
\sqrt{1-k^{2}\sin^{2}(L)}\right]}
{2(1-k^{2})\sqrt{1-k^{4}\sin^{4}(L)}}\right\}
\hat{\textbf{b}}
\end{aligned}$$ for the limits $\varepsilon \to 0$, $k \to1$ and $L \to 0$, $\lambda \to 0$. This proves Eq. (\[LIE\]) of our GLIE. It should be noted that the above formula is nonzero even for the extreme case of $L=1-k\to0$. The physical meaning of this statement is that the local field induced by an infinitesimal segment of a vortex line is nonzero and asymmetric in the binormal direction.
Discussion and Conclusions
==========================
We have derived an asymptotic representation for the local velocity field induced by a curved vortex filament. This derivation generalizes the previously known statements of induced binormal flow, which play an important role in two-component superfluid simulation. In such simulations one must calculate the superfluid and normal fluid flows as well as their mutual friction interaction. This mutual friction embodies the scattering of rotons and phonons off of the vortex structures.[@1957RSPSA.240..114V; @1957RSPSA.240..128V; @1957RSPSA.242..493V; @1958RSPSA.243..400V; @2000PhRvB..62.3409I] It is possible to calculate this interaction in a manner self-consistent with Navier-Stokes and fully coupled to both components.[@springerlink:10.1023/A:1004641912850] The normal fluid is approximated through Navier-Stokes simulation techniques while the kinematics of the superfluid make use of LIA. Though our focus is LIA dynamics we offer the following references to the computational fluid dynamics literature, which has been used for coupled two-component superfluid simulations. [@1985JCoPh..59..308K; @1965PhFl....8.2182H; @Wray1986] While there has been progress in these techniques, [@Kennedy2000177; @Ragab1997943] the recent growth of the highly adaptable discontinuous Galerkin methods [@NDG2008] and their application to nonlinear fluid flow and acoustic problems [@2010JCoPh.229.6874L] is especially provocative.
Mathematically, the kinematics of a vortex filament, $\bm{\xi}$, are described by[@2005qvhi.book.....D] $$\begin{aligned}
\frac{d\bm{\xi}}{dt} =
\textbf{V}_{S}+\textbf{V}_{I}+\beta\bm{\xi}'\times
\left(\textbf{V}_{N}-\textbf{V}_{S}-\textbf{V}_{I}\right)
- \beta'\bm{\xi}'\times
\left[\bm{\xi}'\times\left(\textbf{V}_{N}-
\textbf{V}_{S}-\textbf{V}_{I}\right)\right].
\end{aligned}$$ In a *filament method*, it is typical to prescribe the normal-fluid flow $\textbf{V}_{N}$ and neglect the mutual friction terms involving $\beta$ and $\beta'$. [@2001LNP...571...97S] This leaves only a potential flow $\textbf{V}_{S}$ and induced flow $\textbf{V}_{I}$. Of the remaining quantities, the computationally costly induced flow is managed through the LIA, $$\begin{aligned}
\textbf{V}_{I}(\textbf{x})\approx
\textbf{V}_{\mathrm{local}}(\textbf{x})=
\kappa \ln\left(\frac{2\sqrt{L_{+}L_{-}}}
{|\textbf{x}|}\right)\hat{\textbf{b}}
\end{aligned}$$ where $L_{\pm} = (\phi\pm L)/2$ is related to the cutoff length $L$ and angle $\phi$. Vortex filament methods avoid integration by application of this approximation to nodal points of the Lagrangian computational mesh attached to the filament centerline. Alternatively, we could simply replace LIA with GLIE (\[Ve\])-(\[F1\]) and write $\textbf{V}_{I} \approx
\textbf{V}_{\varepsilon}$ and prescribe a field point $\textbf{x}$ and arclength $s=2RL$. However, if the higher-order circulatory and binormal terms are desired, one could use (\[V\])-(\[V1\]) and employ efficient numerical routines for the incomplete elliptic integrals.[@Lemczyk1988747; @ISI:A1997XL86800002; @springerlink:10.1007/BF00692874] Either of these changes will then be applied to node points of a computational mesh modeling the filament structure.
The use of piecewise linear interpolants, while prevalent in numerics, cause spurious effects when applied to a vortex centerline. The interpolants themselves have zero local curvature and their connections form cusps with undefined local curvature. Typically, local induction is applied to higher-order interpolations. While one can use the generalized induction equation or GLIE on this mesh, the natural vortex-arc construct has been adapted to efficient meshing techniques. [@Yang20021037] Consequently, computational cusps are avoided and local curvature is always well-defined when GLIE is applied to such vortex-arc meshes. Lastly, what remains is re-meshing to allow for the experimentally witnessed vortex nucleation.[@Hodby2001; @2001LNP...571...36V]
Meshing is the most difficult aspect of vortex filament implementations. Not only must the mesh adapt to the vortex dynamics, it must be made to reconnect filament elements that are not predicted by the Eurelian theory. [@1997PhRvE..55.1617P] The most elementary reconnection algorithms appeal to nonlinear Schödinger theory and force reconnection of filaments passing within a few core widths of each other.[@Schwarz1982; @1993PhRvL..71.1375K] The current theory of the reconnection process is not satisfactory and efforts to avoid ad hoc simulated reconnection continue.[@2000EJMF...19..361L; @2001LNP...571..177L; @2001PhRvB..64u4516L; @2001EL.....54..774K; @2004Nonli..17.2091G; @2010PhyD..239.1367P]
Superfluid turbulence dominated by quantized vortex flows is an active area of analytic, numerical and experimental research.[@2010PhRvA..82c3616S; @Henn2009a; @Henn2009; @Henn2010; @Abo-Shaeer2001; @Madison2000; @Kobayashi2008; @Kasamatsu2003; @Castin1996; @1956RSPSA.238..204H; @1956RSPSA.238..215H; @1960AdPhy...9...89H] Though local induction techniques will play a part in continued numerical investigations, understanding geometric and topological quantification of a tangled state is as important and still a work in progress. [@Ricca2001; @Barenghi2001; @Poole2003; @Jou2010] Lastly, the vortex line approximation, while useful and appropriate, must eventually be discarded in favor of nontrivial core-structure. It is likely that the methods developed within this paper can be adapted to current arguments used to study fields induced by vortex tubes. [@1965PhFl....8..553A; @1965PhFl....8..553A; @1991JFM...222..369F]
That being said, this work makes it clear that binormal flow proportional to curvature is a general feature of vortex filament dynamics. This means that the well-celebrated transformation of Hasimoto[@Hasimoto1972], which connects the filament’s curvature and torsion variables to a wavefunction controlled by nonlinear Schrödinger evolution, is fundamental to vortex filament dynamics. Consequently, even geometrically complicated filament dynamics are rooted in integrable systems theory. This connection underpins efforts to predict allowed filament geometrics from the associated integrable systems. [@Calini97recentdevelopments; @Grinevich_closedcurves; @springerlink:10.1007/s00332-004-0679-9; @1996RSPSA.452.1531U; @1992PhFl....4..938R]
The authors thank Paul Martin for useful discussions, and acknowledge support of the National Science Foundation under Grant PHY-0547845 as part of the NSF CAREER program.
Asymptotic Representation for Incomplete Elliptic Integrals of the First Kind
=============================================================================
The following theorem is one of the two major results proven in Karp and Sitnik. [@2007JCoAM.205..186K] The second result gives a simpler expression but is not valid on the leftmost edge of the unit square and therefore not used in our calculations.
For all $(\lambda,k)\in[0,1]\times[0,1]$ and an integer $N\geq 1$, the previous elliptic integral admits the representation $$\begin{aligned}
\label{ASY}
F(\lambda,k)&=\frac{1}{2}\ln\left(
\frac{1+\lambda}{1-\lambda}\right)
\sum_{j=0}^{N} \frac{(1/2)_{j}(1/2)_{j}}
{(j!)^{2}}(1-k^{2})^{j}+ \frac{1}{2\lambda}
\sum_{n=0}^{N-1} \left(\frac{1-\lambda^{2}}
{-\lambda^{2}}\right)^{n} \,
s_{n}\left(\frac{(1-k^{2})\lambda^{2}}
{1-\lambda^{2}}\right) + R_{N}(\lambda,k),
\end{aligned}$$ where $s_{n}(\cdot)$ is given by the recurrence formulae $$\begin{aligned}
s_{n+3}&=\frac{a_{n}s_{n+2}(x), +b_{n}s_{n+1}(x)+
c_{n}s_{n}(x)+h_{n}}{4(n+3)^{2}},\\
a_{n}(x) &= 8n^{2}+36n+42-x(2n+5)^{2},\\
b_{n}(x) &= 2x(4n^{2}+14n+13)-(2n+3)^{2},\\
c_{n}(x) &= -4x(n+1)^{2},\\
h_{n}(x) &= \frac{x(2n+5)(2n+4)^{2}+(n+3)(8n^{2}+24n+17)}
{8(n+3)[(n+2)!]^{2}} [(3/2)_{n}]^{2}(-x)^{n+2},\\
s_{0}(x) &= -2 \ln \left(\frac{1+\sqrt{1+x}}{2}\right),\\
s_{1}(x) &= \left(\frac{x}{2}-1\right) \ln \left(
\frac{1+\sqrt{1+x}}{2}\right)-
\frac{1}{2}\sqrt{1+x}+\frac{1}{2}+\frac{x}{2},\\
s_{2}(x) &= \left(-\frac{9}{32}x^{2}+\frac{x}{4}-
\frac{3}{4}\right)
\ln \left(\frac{1+\sqrt{1+x}}{2}\right)+
\left(\frac{9}{32}x- \frac{7}{16}\right)
\sqrt{1+x} +\frac{7}{16} + \frac{1}{8}x-
\frac{21}{64}x^{2}
\end{aligned}$$ and the remainder term is negative and satisfies, $$\begin{aligned}
\frac{[(1/2)_{N+1}]^{2}(1-k^{2})^{N}}
{2[(N+1)!]^{2}}f_{N+1}(\lambda,k)\leq
-R_{N}(\lambda,k) \leq
\frac{[(1/2)_{N+1}]^{2}(1-k^{2})^{N}}
{2[(N+1)!]^{2}}f_{N}(\lambda,k) ,
\end{aligned}$$ where the positive function $$\begin{aligned}
f_{N}(\lambda,k) &= \frac{1}{1-\alpha(1-k^{2})}
\left\{\frac{\ln\left(
\frac{\sqrt{ 1+ (1-\lambda^{2})/
[\alpha \lambda^{2}(1-k^{2})]}+1}
{\sqrt{ 1+ (1-\lambda^{2})/
[\alpha \lambda^{2}(1-k^{2})]}-1} \right)}
{\alpha\lambda\sqrt{1+(1-\lambda^{2})/
[\alpha \lambda^{2}(1-k^{2})]}}- (1-k^{2})\ln\left(\frac{1+\lambda}{1-\lambda}
\right)\right\}_{|\alpha=(N+1/2)^{2}/(N+1)^{2}}
\end{aligned}$$ is bounded on every subset of $E$ of the unit square, where $$\begin{aligned}
\sup_{k,\lambda \in E} \frac{1-k}{1-\lambda}< \infty
\end{aligned}$$ and is monotonically decreasing in $N$.
From this theorem we denote its first-order approximation as $$\begin{aligned}
F_{1}(\lambda, k) &= \ln\left(
\sqrt{\frac{1+\lambda}{1-\lambda}}\right)+
\frac{1}{\lambda}\ln\left(
\frac{2}{1+\sqrt{(1-k^{2}\lambda^{2})/
(1-\lambda^{2})}}\right) +
\frac{1-k^{2}}{8}\ln\left(
\frac{1+\lambda}{1-\lambda}\right),
\end{aligned}$$ and note that this expression is asymptotic in the $\lambda$ variable.
|
---
abstract: 'An extreme dissipation event in the bulk of a closed three-dimensional turbulent convection cell is found to be correlated with a strong reduction of the large-scale circulation flow in the system that happens at the same time as a plume emission event from the bottom plate. The reduction in the large-scale circulation opens the possibility for a nearly frontal collision of down- and upwelling plumes and the generation of a high-amplitude thermal dissipation layer in the bulk. This collision is locally connected to a subsequent high-amplitude energy dissipation event in the form of a strong shear layer. Our analysis illustrates the impact of transitions in the large-scale structures on extreme events at the smallest scales of the turbulence, a direct link that is observed in a flow with boundary layers. We also show that detection of extreme dissipation events which determine the far-tail statistics of the dissipation fields in the bulk requires long-time integrations of the equations of motion over at least hundred convective time units.'
author:
- Jörg Schumacher
- 'Janet D. Scheel'
title: Extreme dissipation event due to plume collision in a turbulent convection cell
---
Introduction
============
The highly nonlinear dynamics of fully developed turbulence generates high-amplitude fluctuations of the flow fields and their spatial derivatives. For the latter, amplitudes can exceed the statistical mean values by several orders of magnitude [@Yeung2015]. From a statistical point of view, extreme events correspond to amplitudes in the far tail of the probability density function of the considered field. Although the events are typically rare, they can appear much more frequently than for a Gaussian distributed field – a manifestation of (small-scale) intermittency in turbulence [@Frisch1994; @Ishihara2009]. From a mathematical perspective, these high-amplitude events are solutions of the underlying dynamical equations which display a very rapid temporal variation with respect to a norm defined for the whole fluid volume [@Doering2009]. Typical quantities which can be probed are the vorticity or (local) enstrophy [@Lu2008; @Donzis2010], local strain [@Schumacher2010] or the magnitude of temperature, and passive scalar derivatives [@Kushnir2006]. Numerical studies of extreme events in turbulence have been performed in cubes with periodic boundaries in all three space dimensions [@Boratav1994; @Donzis2010]. With increasing Reynolds number these extreme events in box turbulence are concentrated in ever finer filaments or layers [@Yeung2015].
Alternatively, extreme dissipation events can be connected to flow structures in wall-bounded flows that have a large spatial coherence and exist longer than the typical eddies or plumes. Such dissipation events are observed for example in connection with ramp-cliff structures of the temperature [@Corrsin1962; @Antonia1979], with superstructures of the velocity [@Marusic2010] in atmospheric boundary layers, or with very-large scale motion in pipe flows [@Hellstroem2015]. High-dissipation events are then detected inside the container as well as at the edge of the boundary layers.
In this work, we demonstrate a direct dynamical link between a transition of the large-scale turbulent fields and the rare high-amplitude events of the spatial derivatives which are sampled at the smallest scales of the turbulent flow far away from the boundary layers. The system is a three-dimensional turbulent Rayleigh-Bénard convection (RBC) flow in a closed cylindrical cell. We show how the formation of a rare high-amplitude dissipation rate event in the bulk of the convection cell can be traced back to a plume emission from the bottom plate coinciding with a strong fluctuation of the large-scale circulation which exists in closed turbulent flows [@Ahlers2009; @Chilla2012]. In the large-scale fluctuation event, the large-scale circulation (LSC) roll is significantly weakened and re-oriented afterwards. In the absence of the large-scale ordering circulation (which would sweep the plumes along with it), a collision between a hot upwelling and cold downwelling plume is triggered which generates strong local gradients. Such extreme dissipation events are very rare in the bulk since most of the viscous and thermal dissipation is inside the boundary layers at the top and bottom plates. This has been shown in several direct numerical simulations (DNS) of convection [@Emran2008; @Kaczorowski2013; @Scheel2013]. In our five high-resolution spectral element simulations at different Rayleigh and Prandtl numbers, we monitored the fourth-order moments of the thermal and kinetic energy dissipation rates in the bulk of the cell far away from the boundary layers. After finding one data point in one run which was much larger than the rest, we reran this full simulation twice in the interval around this extreme event at a monitoring frequency five and fifty times higher in order to analyze the dynamics in detail. Our detected rare event reveals a direct connection between a strong large-scale fluctuation of the velocity and a small-scale extreme dissipation (i.e. velocity derivative) event, thus bridging the whole cascade range of the turbulent flow.
Numerical model
===============
We solve the three-dimensional Boussinesq equations for turbulent RBC in a cylindrical cell of height $H$ and diameter $d$. The equations for the velocity field $u_i(x_j,t)$ and the temperature field $T(x_j,t)$ are given by $$\begin{aligned}
\label{ceq}
\partial_i u_i &=0\,,\\
\label{nseq}
\partial_t u_i +u_j \partial_j u_i &=-\partial_i p+\nu \partial_j^2 u_i+ g \alpha (T-T_0) \delta_{iz}\,,\\
\label{pseq}
\partial_t T +u_j \partial_j T&=\kappa \partial_j^2 T\,,\end{aligned}$$ with $i,j=x,y,z$ and the Einstein summation convention is used. The kinematic pressure field is denoted by $p(x_j,t)$ and the reference temperature by $T_0$. The aspect ratio of the convection cell is $\Gamma=d/H=1$ with $x,y\in [-0.5,0.5]$ and $z\in [0,1]$. The Prandtl number which relates the kinematic viscosity $\nu$ and thermal diffusivity $\kappa$ is given by $$Pr=\frac{\nu}{\kappa}\,.$$ The Rayleigh number is given by $$Ra=\frac{g\alpha\Delta T H^3}{\nu\kappa}\,.$$ Here, the variables $g$ and $\alpha$ denote the acceleration due to gravity and the thermal expansion coefficient, respectively. The temperature difference between the bottom and top plates is $\Delta T$. In a dimensionless form all length scales are expressed in units of $H$, all velocities in units of the free-fall velocity $U_f=\sqrt{g\alpha\Delta T H}$ and all temperatures in units of $\Delta T$. Times are measured in units of the convective time unit, the free fall time $T_f=H/U_f$.
We apply a spectral element method in the present direct numerical simulations (DNS) in order to resolve the gradients of velocity and temperature accurately [@bib:nek5000]. More details on the numerical scheme and the appropriate grid resolutions can be found in Ref. [@Scheel2013], and resolution of higher-order moments of the dissipation rates in [@Schumacher2014]. No-slip boundary conditions are applied for the velocity at all the walls. The top and bottom walls are isothermal and the side wall is thermally insulated.
The cylindrical convection cell is covered by $N_e$ spectral elements. On each element all turbulent fields are expanded by $N$th–order Lagrangian interpolation polynomials with respect to each spatial direction. Table \[Tab1\] summarizes our highest Rayleigh number runs on massively parallel supercomputer simulations which have been carried out on up to 262144 MPI tasks. In the course of these production runs we conducted an analysis in which we searched for extreme dissipation events by means of the fourth-order moments obtained in an inner volume of the closed cylindrical cell.
The sequence around the extreme dissipation event was rerun twice to generate a fine sequence of one hundred snapshots with a separation of 0.143 free fall times $T_f$ and then a very fine sequence of five hundred snapshots with a separation of 0.029 $T_f$.
Run $Ra$ $Pr$ $N_{e}$ $N$
----- ----------- ------- ----------- -----
1 $10^8$ 0.7 256,000 11
2 $10^9$ 0.7 875,520 11
3 $10^{10}$ 0.7 2,374,400 11
4 $10^7$ 0.021 875,520 11
5 $10^8$ 0.021 2,374,400 13
: Parameters of the different spectral element simulations. We show the Rayleigh number $Ra$, the Prandtl number $Pr$, the total number of spectral elements $N_e$, and the polynomial order $N$ of the Lagrangian interpolation polynomials in each of the three space directions.[]{data-label="Tab1"}
![(Color online) Appearance of extreme thermal dissipation events in the bulk for five simulation runs which are listed in Table \[Tab1\]. The normalized fourth-order thermal dissipation rate moments $M_{4,4}(t)/\langle M_{4,4}(t) \rangle_t$ are shown versus the number of statistically independent samples saved in the simulation runs in subvolume $V_4$ which is approximately $V_0/5$.[]{data-label="fig1_app"}](extreme_fig1.jpg)
![(Color online) Monitoring of the evolution of the extreme dissipation event in the bulk by means of the fourth moments of the thermal dissipation, $M_{4,j}$, (top panel) and kinetic energy dissipation, $N_{4,j}$, (mid panel). We display the moments in six different subvolumes $V_1\dots V_5$ and the whole cell $V_0$. The vicinity of the extreme event is marked by the vertical dashed lines and replotted in the bottom panel. These data are taken from the run with the finest temporal resolution.[]{data-label="fig1"}](extreme_fig2.jpg)
![(Color online) Five hundred individual probability density functions (PDFs) of the thermal dissipation rate $\epsilon_T$ in the left column and of the kinetic energy dissipation rate $\epsilon$ in the right column which are obtained from the run with the very fine time resolution. Data are for run 1. The insets replot data from the bottom panel of figure \[fig1\]. The data in the vicinity of the local maxima are always highlighted as dark curves.[]{data-label="fig2_app"}](extreme_fig3.jpg)
![(Color online) Extreme thermal dissipation event in the bulk at time $T_{\ast}=5.73$. Combined plot of thermal dissipation rate (isosurfaces at $26 \langle\epsilon_T\rangle_{V_0,t}$) and kinetic energy dissipation rate (horizontal contour slice) on a logarithmic scale. Contour slice levels are from blue ($\log_{10}\epsilon \le -4$) to red ($0.9 \le \log_{10}\epsilon)$. The inner cylinder stands for subvolume $V_4$ with $r\le 0.3$ and $0.2\le z\le 0.8$.[]{data-label="fig2"}](extreme_fig4.jpg)
Results
=======
Detection by fourth-order moments
---------------------------------
The starting point of the analysis is the time evolution of the fourth-order moments of the kinetic energy dissipation rate, $$\epsilon(x,y,z,t)=2\nu S_{ij}S_{ji}\,,$$ with $S_{ij}=(\partial_i u_j+\partial_j u_i)/2$, and the thermal dissipation rate, $$\epsilon_T(x,y,z,t)=\kappa G_i^2\,,$$ with $G_i=\partial_i T$. The Rayleigh-Bénard flow in the cylindrical cell obeys statistical homogeneity in the azimuthal direction only. All statistics will therefore depend on the size of the sample volume. We have monitored the moments in six successively smaller cylindrical subvolumes which are nested in each other. We define $r_0=0.5>r_1=0.45>\dots>r_5=0.25$ and $h_0=1>h_1>\dots >h_5=0.5$ and $V_j= \{ (r,\phi,z)\, |\,r\le r_j\;,(1-h_j)/2\le z\le (1+h_j)/2 \}$ with $j=0\dots 5$. The volume $V_0$ is the full cell. Fourth-order moments of both dissipation rates are given by $$M_{4,j}(t)=\langle \epsilon_T^4\rangle_{V_j}\quad\mbox{and}\quad N_{4,j}(t)=\langle \epsilon^4\rangle_{V_j}\,.$$
In figure \[fig1\_app\] the normalized moments of the thermal dissipation rate are shown for five different runs which are obtained at the highest Rayleigh numbers and two different Prandtl numbers (see table \[Tab1\]). In the primary production runs we analysed the kinetic energy and thermal dissipation rate in the subvolume $V_4$ that is sufficiently far away from all boundaries. Since the simulation runs have a different number of time step widths and a different number of data output steps, the moments are shown versus the number of samples. It is clearly visible that in all runs the volume averages can go far beyond the means at certain times. However, the strongest outlier is observed for run 1 at $Ra=10^8$ and $Pr=0.7$. Therefore, the discussion in this work is dedicated to run 1.
In figure \[fig1\] we display $M_{4,j}(t)$ (top panel) and $N_{4,j}(t)$ (mid panel) on a semi-logarithmic plot for run 1. Data are obtained over a time interval with an output of one hundred snapshots separated by 0.143 free fall times units (see top and mid panels of the figure). $M_{4,0}(t)$ remains nearly unchanged and $N_{4,0}(t)$ fluctuates more strongly, but there is no large event that stands out. The reason is that a major part of both the thermal variance and of the kinetic energy is dissipated in the boundary layers of the temperature and velocity fields close to the walls, respectively [@Emran2008; @Kaczorowski2013; @Scheel2013]. Only in the successively smaller subvolumes $V_j$, that are nested increasingly deeper in the bulk, is the extreme bulk dissipation event detected by the corresponding fourth order moment. It is seen that $M_{4,4}(t)$ grows by three orders of magnitude within $T_f/2$. The bottom panel of figure \[fig1\] shows that a local, but less strong maximum of $N_{4,4}(t)$ occurs approximately $T_f/2$ after the peak in $M_{4,4}$.
The significance of this event for the small-scale statistics of the temperature and velocity derivatives in the bulk region is demonstrated in figure \[fig2\_app\]. In both cases the fattest tail corresponds with this high-amplitude event as seen in the bottom panels of figure \[fig2\_app\]. We display five hundred individual probability density functions (PDFs), each taken at one instant in time. These data have been obtained in a repetition run at the highest temporal resolution in order to resolve the event better. The vicinity of the high-dissipation event is colored differently in both dissipation rates. It is also seen that the high-dissipation bulk event does not contribute significantly to the far tails of the PDFs when averaged over the whole convection cell including all boundary layers. The resulting extension of the far tail of the time-averaged PDFs in the bulk was already shown in ref. [@Scheel2013].
![(Color online) Time evolution of production terms and magnitudes in the course of the extreme event. All quantities are now volume averages taken for box ${\cal B}$. The maximum of $G^2$ is at $T_{\ast}^{({\cal B})}=5.759$ and is slightly shifted with respect to $T_*$ in $V_4$ because $V_4 \gg {\cal B}$ and the temporal resolution is finer. Left: temperature gradient square and production term $P_G$ (see eq. (\[gradbal\])). Right: local strain and production terms $P_S$, $P_{\omega}$ as well as $P_T$ (see eq. (\[strainbal\])). The peaks of $G^2$ (left) and $S^2$ (right) are indicated by solid vertical lines. The dashed vertical line in the right panel is the maximum of $G^2$. Terms are partly rescaled as indicated in the legend.[]{data-label="fig3"}](extreme_fig5.jpg)
Link between high-amplitude thermal and kinetic energy dissipation events
-------------------------------------------------------------------------
Figure \[fig2\] shows isosurfaces of the thermal dissipation rate at $\epsilon_T= 26 \langle\epsilon_T\rangle_{V_0,t}$ which are mostly found close to the top and bottom plates. The same holds for kinetic energy dissipation, but is not shown. It is the high-thermal-dissipation sheet at $T_{\ast}=5.73$ which is mostly inside $V_4$ that contributes to the local maxima of $M_{4,j}(t)$ for $j>0$ in figure \[fig1\]. It can be also seen that the local maximum of $\epsilon_T(x,y,z,t)$ coincides with a local maximum of $\epsilon(x,y,z,t)$. The temperature front generates a strong shear layer which is manifest as a delayed high-amplitude energy dissipation event.
We refined the analysis, both in space and time. We zoom into the small box ${\cal B}=\{(x,y,z)\in[-0.11,-0.05]\times[-0.34,-0.14]\times[0.15,0.33]\}$ that encloses the high-amplitude thermal dissipation layer. The balance equation for the square of the magnitude of $G_i$ is given by [@Pumir1994; @Brethouwer2003] $$\frac{\mbox{d}G^2}{\mbox{d}t}=-2G_i S_{ij} G_j +2\kappa G_i \frac{\partial^2 G_i}{\partial x_j^2}\,.
\label{gradbal}$$ The first term on the right hand side is the gradient production term, $P_G$. Local shear strength is measured by the square of the magnitude of the rate of strain tensor $S^2=S_{ij} S_{ji}$. The balance equation for $S^2$ (see also [@Holzner2008]) has to be extended by a temperature production term and is given by $$\begin{aligned}
\frac{\mbox{d}S^2}{\mbox{d}t}=&-2S_{ij} S_{jk} S_{ki}-\frac{1}{2} \omega_i S_{ij} \omega_j -2 S_{ij} \frac{\partial^2 p}{\partial x_i \partial x_j} \nonumber\\
&+2\nu S_{ij}\frac{\partial^2 S_{ij}}{\partial x_k^2} + 2g\alpha S_{zi}G_i\,.
\label{strainbal} \end{aligned}$$ We have three production terms: strain production (1st, $P_S$), enstrophy consumption (2nd, $P_{\omega}$) and production due to coupling to the temperature gradient (last, $P_T$).
![(Color online) Isocontour plot of the vertical convective current for times: (a) $T_{\ast}-1.146$, (b) $T_{\ast}-0.296$ and (c) $T_{\ast}$. Blue is for downwelling at $\sqrt{Ra Pr}\,u_z T=-900$, red for upwelling plumes at $\sqrt{Ra Pr}\,u_z T=1000$. The collision region is indicated by a box.[]{data-label="fig4"}](extreme_fig6.jpg)
Figure \[fig3\] displays the time evolution of volume averages over ${\cal B}$ for both gradient magnitudes and the corresponding production terms in eqns. (\[gradbal\]) and (\[strainbal\]), respectively. The maximum of $G^2$ coincides with the one of $P_G$ (see left panel). The same holds for the maximum of $S^2$ and the ones of $P_S$ and $\left|P_{\omega}\right|$, respectively (right panel). We also confirm that $\max \langle S^2\rangle_{\cal B}$ lags behind $\max \langle G^2\rangle_{\cal B}$ (see also figure \[fig1\]), a result which is also robust for different sizes of ${\cal B}$. The time of maximum production by $P_T$, falls right between those for $P_G$ and $P_S+P_{\omega}$. This shows that the temperature gradient occurs first, followed by strong shear generation since the colliding fluid masses have to move around each other.
Formation of colliding plumes
-----------------------------
How is the high-amplitude thermal dissipation layer formed? Figure \[fig4\] plots isosurfaces of the vertical component of the convective heat current vector $j^c_z=\sqrt{Ra Pr}\,u_z T$ at three instants. Since $0\le T\le 1$, a negative isolevel of $j_z^c$ corresponds to a downwelling and a positive one to an upwelling plume. The box in the panels indicates the collision point of two plumes in the bulk at time $T_{\ast}$. This collision is caused by the large hot plume from the bottom and a second extended cold plume that falls down at the side wall and turns into the bulk. The high-amplitude thermal dissipation layer is formed at the collision site. The event is comparable with rapid growth events of enstrophy in box turbulence [@Lu2008]. There colliding vortex rings maximized enstrophy growth. Our nearly frontal plume collision can be considered thus a rare event and appears in three-dimensional convection flow much less frequently than in two-dimensional ones [@Chandra2013].
First we will investigate the rising hot plume. Figure \[fig5\] displays contours of $\partial T/\partial z$ at the bottom plate. Local maxima are indicators for rising plumes [@Bandaru2015]. On top of contours we plot field lines of the skin friction field which is given by $\partial_i u_j|_{z=0}=(\partial_z u_x, \partial_z u_y)$ [@Chong2012]. Locally downwelling fluid impacts the bottom plate and generates unstable node points (UN) of the skin friction field. Skin friction lines, which arise from these nodes, form a strong front which starts to form in panel (a) and is moved “upward” in panel (b) of figure \[fig5\]. Saddle points (SP) or stable nodes (SN) are formed between the unstable nodes. The unstable manifold of a saddle [@Bandaru2015] or a sequence of stable nodes, as being the case here, are the preferred sites of plume formation. It is the persistence and convergence of these critical points for a certain time span which causes the rise of a large plume from the bottom just before $T_{\ast}$, that then collides with the downwelling cold plume at $T_{\ast}$.
![(Color online) Strong plume formation at the bottom plate. Contour plots of $\partial T/\partial z$ at $z=0$ are shown together with field lines of the skin friction field $(\partial u_x/\partial z, \partial u_y/\partial z)$. Times are $T_{\ast}-1.146$ for (a) and $T_{\ast}-0.296$ for (b). For better visibility, we seed the skin friction lines only in a square box around the rising plume. Stable nodes (SN) and unstable nodes (UN) are indicated.[]{data-label="fig5"}](extreme_fig7.jpg)
Plume collision due to transition of large-scale flow
-----------------------------------------------------
This raises the last question, namely does a change in the large-scale dynamics enable such a rare plume collision event? It is well-known that in closed convection cells a large-scale circulation (LSC) exists [@Ahlers2009; @Chilla2012]. In cells with $\Gamma=1$, the LSC consists of one big roll which forces the plumes to move along the top or bottom plate, and then to rise dominantly on one side of the cell and to fall down on the other side. This ordering influence stops when the large-scale circulation decelerates strongly and becomes re-oriented. Such events have been studied statistically in experiments [@Sreenivasan2002; @Brown2006; @Xi2007], numerically in two-dimensional [@Chandra2013; @Petschel2011; @Poel2012] or three-dimensional [@Mishra2011] convection as well as in low-dimensional models [@Brown2008].
We quantified the large-scale dynamics by taking a spatial average with respect to the radial and vertical coordinates. We define $$\overline{u_zT}(\phi,t)=\frac{1}{{\cal V}_r}\int_{r_1}^{r_2}\int_{z_1}^{z_2} u_zT(r,\phi,z,t)\, rdrdz\,,
\label{average}$$ with ${\cal V}_r=\pi(r_2^2-r_1^2)(z_2-z_1)$. The complex three-dimensional structure of the up- and downwelling convective currents in the closed cell is thus reduced to a one-dimensional signal. The locally averaged convective current $\overline{u_zT}(\phi,t)$ is expanded in a Fourier series for each instant $$\overline{u_zT}(\phi,t) = \sum_{m=1}^N a_m(t)\cos(m\phi+\gamma_m(t))\,.
\label{average1}$$ Figure \[fig6\] displays the amplitude of the first three modes, $a_1(t)$ to $a_3(t)$. We have chosen different vertical intervals $[z_1,z_2]$, in the upper and lower sections of the cell as well as in the center. At the beginning of the time window, we find $a_1>a_2>a_3$ in all sections of cell. This indicates that a one-roll circulation pattern dominates the LSC as is supported by the isocontours in Figure \[fig6\](d). In Figures \[fig6\](a)–(c), $a_1$ steadily decreases towards $t=T_{\ast}$ with $T_{\ast}$ being the time of the extreme dissipation event. The ratio of the Fourier coefficients is changed to $a_1\sim a_3 > a_2$ for the lower section of the cell (see Figure \[fig6\](c)), while in the mid and upper sections (see Figures \[fig6\](a,b)), $a_1>a_3>a_2$ is observed. The growth of the $m=3$ mode demonstrates that up- and downwelling convective currents are found now close to each other, in particular in the lower section of the cell, as can be seen by the isocontours in Figure \[fig6\](e). For $t>T_{\ast}$, we observe a re-establishment of the one-roll pattern, as supported by Figure \[fig6\](f). The whole process proceeds within $10 T_f$. Also plotted in Figures \[fig6\](d)–(f) are the isocontours for large $\epsilon_T$, which always are located near the bottom and top plates. However, in Figure \[fig6\]e, one sees a region of large $\epsilon_T$ in between the upwelling hot and downwelling cold plumes as they collide, consistent with the increase in $\epsilon_T$ in the bulk seen in Figure \[fig1\].
![(Color online) Time evolution of the three largest Fourier mode amplitudes obtained for $\overline{u_zT}(\phi,t)$. (a) $z_1=0.65< z< z_2=0.85$. (b) $0.4<z<0.6$. (c) $0.15<z<0.35$. Points $0.4< r < 0.48$ were taken in radial direction. At the bottom we add three snapshots of the convective current $\sqrt{Ra Pr}\,u_z T$ (blue for level of -900 and red for level of 1100) together with isocontours of $\epsilon_T= 0.1 \approx 26\langle\epsilon_T\rangle_{V_0,t}$. (d) $t=1.43$. (e) $t=T_{\ast}=5.73$. (f) $t=9.74$.[]{data-label="fig6"}](extreme_fig8.jpg)
Summary
=======
We have connected a far-tail, extreme dissipation event at the small scales in the bulk of a three-dimensional Rayleigh-Bénard convection flow in a closed cell to a reduction event in the LSC accompanied by a plume emission from the bottom boundary layer. Such an event is very rare. In five different simulations spanning a range of $Ra$ and $Pr$ over long evolution times it was the only very high dissipation event in the bulk away from boundary layers as shown in figure \[fig1\_app\].
The detection was possible by monitoring the well-resolved fourth-order dissipation moments in the bulk of the cell during the simulations. We also have showed how a transition of the large-scale flow structures in the cell can impact the dynamics at the smallest scales, the scales across which the steepest gradients are formed. The two events are thus directly linked and bridge the whole scale range of the turbulent cascade. The large-scale coherent fluid motion is established here due to the presence of walls which enclose the convection cell. It can be expected that it would be absent in box turbulence with periodic boundary conditions.
How frequently does such a high-dissipation event appear? If one takes a typical far-tail amplitude of the PDF of $\epsilon_T$ (see lower left panel of figure \[fig2\_app\]) of $p(\epsilon_T)\sim 10^{-6}$ and multiplies it with the binwidth $\Delta \epsilon_T=0.0004$, one gets an estimate of the probability of the appearance of a high-thermal-dissipation event in the bulk of $w\approx p(\epsilon_T)\Delta\epsilon_T\approx 4\times 10^{-10}$, i.e., one out of 2.5 billion data points. The bulk volume $V_4$ contains about a fifth of the total cell volume and about 10 per cent of the total number of mesh cells which translates to roughly 40 million cells for $V_4$. That means that one picks such high-dissipation events every 60 to 70 $T_f$ if one continues with the same sampling frequency as in the original production run. Our total integration time for run 1 was 104 $T_f$. Consequently, if one wants to have a complete picture of the small-scale statistics of a wall-bounded turbulent flow then these events have to be incorporated. As our estimate shows, this requires very long-time integrations of the fully resolved Boussinesq equations which becomes increasingly expensive as the Rayleigh number grows or the Prandtl number decreases.
Computing resources have been provided by the John von Neumann Institute for Computing at the Jülich Supercomputing Centre by Grant HIL09 on Blue Gene/Q JUQUEEN and by Grant SBDA003 of the Scientific Big Data Analytics (SBDA) Program on the Jülich Exascale Cluster Architecture (JURECA), respectively. We thank F. Janetzko for his support in the SBDA project.
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0.5truecm [**Abstract:\
**]{} [Entanglement is a powerful resource for processing quantum information. In this context pure, maximally entangled states have received considerable attention. In the case of bipartite qubit-systems the four orthonormal Bell-states are of this type. One of these Bell states, the singlet Bell-state, has the additional property of being antisymmetric with respect to particle exchange. In this contribution we discuss possible generalizations of this antisymmetric Bell-state to cases with more than two particles and with single-particle Hilbert spaces involving more than two dimensions. We review basic properties of these totally antisymmetric states. Among possible applications of this class of states we analyze a new quantum key sharing protocol and methods for comparing quantum states.]{} 0.1cm PACS: 03.67.-a, 03.65.Ta
Introduction
============
By now, quantum theory has become a well established part of modern physics. We have become accustomed to its results even if some of the concepts involved appear strange from the point of view of classical physics. However, as long as these peculiarities are restricted to the microscopic domain it is not so difficult for us to get used to them. During the last decade there have been various successful attempts to push characteristic quantum phenomena into the macroscopic domain and to exploit these very phenomena for practical purposes. These attempts may be viewed as first steps of a newly emerging quantum technology. Thus, it was possible to propose new, efficient quantum algorithms, to develop methods for the transfer of quantum states and of secret keys, and to invent new quantum error correction methods which suppress decoherence. Quite a number of these new effects rely on the use of states whose correlations are incompatible with local realistic theories. The singlet state of two distinguishable spin-1/2 particles is a prominent example which has been studied extensively in the past. The main purpose of the subsequent contribution is to point out several possible applications of generalizations of this singlet state to cases which involve more than two distinguishable quantum systems of arbitrary but finite dimensions.
Definition and basic properties of totally antisymmetric quantum states
=======================================================================
Totally antisymmetric quantum states are natural generalizations of the singlet state to many-particle quantum systems. In atomic and molecular physics, for example, they have already been playing an important role as Slater-determinant states. These are defined by the relation $$\vert A_N\rangle = \frac{1}{\sqrt{N!}} \sum_\pi (-1)^{sgn (\pi )} \vert \pi_1\rangle ...\vert\pi_N\rangle \label{defA}
\label{definition}$$ with $\{|i_1\rangle ... \vert i_N\rangle; i_1,...,i_N = 0,...,d-1\}$ denoting an orthonormal basis of the Hilbert space of $N$ $d$-dimensional quantum systems. The sum appearing in Eq.(\[definition\]) runs over all possible permutations $\pi$ of the $N$ elementary quantum systems considered. Due to basic properties of determinants this state exists only in cases in which the number of particles $N$ equals the dimension of the one-particle Hilbert spaces $d$ involved. Thus, in the simple case of three qutrits, for example, the totally antisymmetric quantum state is given by $$\vert A_3\rangle = \frac{1}{\sqrt{6}} \left\{\vert 0\rangle\vert 1\rangle\vert 2\rangle + \vert 1\rangle\vert 2\rangle\vert 0\rangle +
\vert 2\rangle\vert 0\rangle\vert 1\rangle - \vert 0\rangle\vert 2\rangle\vert 1\rangle - \vert 1\rangle\vert 0\rangle\vert 2\rangle - \vert 2\rangle\vert 1\rangle\vert 0\rangle\right\} . \label{A3}$$ Let us summarize briefly some of the most important properties of these totally antisymmetric states:
1. They are invariant under local unitary transformations of the form $U\otimes U\otimes ...\otimes U$, i.e. $$U\otimes U\otimes ...\otimes U \vert\psi\rangle \langle \psi\vert
U^{\dagger}\otimes U^{\dagger}\otimes ...\otimes U^{\dagger}
= \vert\psi\rangle \langle \psi\vert.
\label{invpr}$$
2. Simultaneous measurements of all particles in a commonly chosen measurement basis result in perfect correlations, i.e. $$P(i_1,\dots,i_N)\equiv \mid \langle \pi_1| ...\langle \pi_N|\psi\rangle \mid^2
\frac{1}{N!}|\varepsilon_{i_1,\dots,i_N}|^2 \label{correl}$$ with $\varepsilon_{i_1,\dots,i_N}$ denoting the totally antisymmetric tensor which is non-zero only if all its indices are different.
3. They can be generated in an iterative manner by a sequence of generalized XOR-gates and discrete Fourier transforms. The three-particle state $\vert A_3\rangle$, for example, can be prepared from the antisymmetric two-particle state $\frac{1}{\sqrt{2}} (\vert 2\rangle_1 \vert 1\rangle_2 - \vert 1\rangle_1 \vert 2\rangle_2 )$ by $$\vert A_3\rangle = GR_{31}~ GR_{32}~F_3~\vert 0\rangle_3
\frac{1}{\sqrt{2}} (\vert 2\rangle_1 \vert 1\rangle_2 - \vert 1\rangle_1 \vert 2\rangle_2 ).$$ Thereby, $F_3$ denotes the discrete Fourier transformation applied to the third particle and $GR_{ij}$ represents a generalized XOR-operation applied to particles $i$ and $j$. Applied to the first and second particle, for example, this latter operation is defined by $$GR_{12}~\vert i\rangle_1 \vert j\rangle_2 = \vert i\rangle_1 \vert i\ominus j~\rangle_2$$ with $\ominus$ denoting subtraction $mod (d)$. This construction can be generalized in a straightforward way to more than three particles.
4. In the case of $N$ particles the reduced density matrix $\hat\rho_i$ of subsystem $i$ is given by $$\hat\rho_i = \frac{1}{d}\sum\limits^{d-1}_{j=0} \vert j\rangle\langle j\vert ,$$ i.e., the single-particle reduced density matrix describes a completely depolarized state. Projection of one of the particles onto a particular state, say $\vert j\rangle\langle j\vert$, leaves the rest of the system in the pure antisymmetric state which involves all one-particle states except state $|j\rangle$, i.e. $$\vert \overline{A}_{N-1}\rangle = \frac{1}{\sqrt{(N-1)!}} \sum_\pi (-1)^{sgn (\pi )} \vert \pi_1 \rangle ...\vert \pi_{N-1} \rangle.$$ The index of correlation [@Steve] between a particular particle and the remaining part of the system is given by $$I_{i-r} = S_i + S_r - S = 2 S_i = 2 {\rm log~(d)} . \label{IC}$$ with the von-Neumann entropy of particle $i$ being given by $S_i = - Tr{\hat\rho_i \ln\hat\rho_i}$ and with $S_{i-r}$ denoting the von-Neumann entropy of the remaining part. The entropy $S$ of the whole system equals zero as it is in a pure state.
As exemplified in the subsequent sections totally antisymmetric quantum states can be used for many tasks which are of interest for quantum communication.
A quantum mechanical key sharing protocol
=========================================
The secret distribution of a classical key is one of the main aims of quantum cryptography. Known secure protocols of bipartite key distribution are either based on non-orthogonal two dimensional quantum states [@bennet] or on entangled states [@ekert]. These protocols enable two parties to share a common, secret classical key. Recently, several more general situations have been discussed. One of them involves the distribution of a classical key between several parties in such a way that a subset of the parties has access to the key only if they share the information available. Various multi-partite key sharing protocols of this kind have been proposed which are either based on the use of GHZ-states [@hbb] or on the use of pairs of singlet states [@karlsson].
Here we discuss an alternative multi-partite quantum key sharing protocol which is based on anti-symmetric states of qudit systems. (A qudit system is a $d$ dimensional elementary quantum system.) This protocol enables one to generate, to split and to distribute a classical d-ary key securely. We demonstrate the basic principles of this protocol for quantum key sharing in the simplest nontrivial case of three three-dimensional quantum systems. In this case we base our quantum protocol on the totally antisymmetric state $\vert A_3\rangle$ defined by Eq. (\[A3\]). For this key sharing protocol two basic properties of totally antisymmetric states are important. Firstly, all outcomes of simultaneous measurements performed by the participants in identical bases must be different and secondly, the unitary invariance of A-states guarantees that this is also true for any commonly chosen basis.
Let us consider three parties (Alice, Bob and Charlie). Each of them is endowed with a common set ${\cal U}$ of unitary transformations. The protocol runs as follows:
- Alice prepares three qutrits in the anti-symmetric state $\vert A_3\rangle$. She applies a unitary transformation ($\in {\cal U}$) on qutrit one, measures this qutrit and keeps her choice of the unitary transformation and the measurement result secret. This transformation with the subsequent measurement changes the correlations in the anti-symmetric state.
- Alice sends qutrit two to Bob and qutrit three to Charlie.
- In order to recover the original state and the correlations of the measurements, Bob (Charlie) also chooses a unitary transformation ($\in {\cal U}$) randomly and applies it onto his qutrit. Afterwards Bob (Charlie) measures his qutrit. Alice keeps her choice secret.
- Bob (Charlie) transmits his choice of transformation to Alice but keeps the measurement outcome secret. If all three unitary transformations coincide, Alice declares the outcomes of the measurements to be a valid part of the key. In this case, Bob and Charlie can deduce Alice’s result if they share the outcomes of their measurements.
- In order to study the security of the key generated by this protocol Alice requests from Bob and Charlie a subset of the outcomes of their measurements.
Security of the quantum key sharing protocol
============================================
As a general investigation of security is beyond the scope of this contribution, we restrict our subsequent discussion to a cut-and-resend attack which does not involve coherent measurements. In such an attack an external or internal eavesdropper could try to obtain information about the key by attaching an ancilla state to the three qutrits. Subsequently, measurement of the ancilla could reveal information about the outcomes of measurements performed on the qutrits.
The most general state of a system composed of qutrits and an ancilla is given by $$|E\rangle\equiv\sum_{i_1,i_2,i_3=0}^{2}|i_1\rangle |i_2\rangle |i_3\rangle\otimes
|E_{i_1,i_2,i_3}\rangle .$$ Thereby, the ancilla system is described by the states $|E_{i_1,i_2,i_3}\rangle$. These states need not be mutually orthogonal but they obey a normalization condition, namely $\langle E | E\rangle =1$. If the eavesdropper wants to remain undetected he must design the state $|E\rangle$ in such a way that the probabilities $P(i_1,i_2,i_3)$ remain unchanged. This imposes a set of constraints onto the states $|E_{i_1,i_2,i_3}\rangle$. If we choose ${\cal U}=\{{\bf 1},F\}$ with $F$ denoting the discrete Fourier transform these constraints are given by the equations $$\begin{aligned}
|E_{012}\rangle+|E_{021}\rangle+
|E_{120}\rangle+|E_{102}\rangle+
|E_{201}\rangle+|E_{210}\rangle = 0, & & \nonumber \\
|x|^2(|E_{012}\rangle+|E_{021}\rangle)+
x^*(|E_{102}\rangle+|E_{120}\rangle)+
x(|E_{210}\rangle+|E_{201}\rangle) = 0, & & \nonumber \\
x^*(|E_{012}\rangle+|E_{210}\rangle)+
|x|^2(|E_{102}\rangle+|E_{201}\rangle)+
x(|E_{120}\rangle+|E_{021}\rangle)= 0, & & \nonumber \\
x(|E_{012}\rangle+|E_{102}\rangle)+
x^*(|E_{201}\rangle+|E_{021}\rangle)+
|x|^2(|E_{210}\rangle+|E_{120}\rangle)= 0, & & \nonumber \\
|x|^2(|E_{012}\rangle+|E_{021}\rangle)+
x(|E_{102}\rangle+|E_{120}\rangle)+
x^*(|E_{210}\rangle+|E_{201}\rangle)= 0, & & \nonumber \\
x(|E_{012}\rangle+|E_{210}\rangle)+
|x|^2(|E_{102}\rangle+|E_{201}\rangle)+
x^*(|E_{120}\rangle+|E_{021}\rangle)=0,& & \nonumber \\
x^*(|E_{012}\rangle+|E_{102}\rangle)+
x(|E_{201}\rangle+|E_{021}\rangle)+
|x|^2(|E_{210}\rangle+|E_{120}\rangle) = 0 & & \nonumber \end{aligned}$$ with $x\equiv e^{-i\frac{2\pi}{3}}$. The unique solution of this set of equations is given by $$|E_{i_1,i_2,i_3}\rangle=\varepsilon_{i_1,i_2,i_3}|R\rangle.$$ This result implies that, provided the eavesdropper wants to remain undetected, the state $|E\rangle$ has to have the form $$|E\rangle=\left(\frac{1}{3!}\sum_{i_1,i_2,i_3=0}^2
\varepsilon_{i_1,i_2,i_3}|i_1\rangle |i_2\rangle |i_3\rangle
\right) |R\rangle.$$ Thus, the state of the ancilla factorizes from the qutrit-system so that the eavesdropper cannot obtain any information about the key. If the eavesdropper wants to retrieve information about the key he must perturb the state in such a way that the correlations of the outcomes of the measurements are changed.
A protocol for quantum state sharing
====================================
Totally antisymmetric states are also well suited for distributing $d$-dimensional quantum states between $N=d$ parties. The task of quantum state sharing to be realized may be viewed as a generalization of the well-known bipartite entanglement-assisted teleportation protocol. The aim of the protocol is to send the state $\vert\chi\rangle$ from a source to a particular receiver. However, due to security reasons it should be possible to reconstruct this state only if all participants cooperate. Thus, reconstruction of the state $|\chi\rangle$ by the receiver should be possible only if at least one additional mediator communicates additional classical information properly. In the simplest case of three parties, i.e. $N=d=3$, the protocol implementing this task is characterized by the following identity $$|\chi \rangle _{1} |A_3\rangle _{234}\equiv
\sum_{l,\rho
=0}^{2}\frac{1 }{3}|\Psi _{l,\rho }\rangle _{12}\,
\sum_{k=0}^{2}\frac{1}{\sqrt{3}} e{}^{i\frac{2\pi }{3}k\rho
}F_{3}^{-1}|k\rangle_{3}\,U(l,\rho ,k)|\chi \rangle _{4}.
\label{help1}$$ This identity involves four particles, namely particle one which carries the quantum state $|\chi\rangle$ and particles two, three and four which are distributed between the three parties involved in the protocol. The orthonormal states $|\Psi _{l,\rho }\rangle_{12}$ are defined by $$|\psi _{l ,\rho }\rangle_{12} =
\frac{1}{\sqrt{3}}\sum
\limits_{k=0}^{2}{}e^{i\frac{2\pi }{3}l k}|k\rangle _{1}
|k\ominus \rho \rangle _{2}~~{\rm mod~3} . \label{gbell}$$ These orthonormal states generalize the Bell basis to the case of two qutrits. The unitary transformation $U(l,\rho,k)$ is given by $$U(l,\rho ,k)|m\rangle_{4}\equiv e{}^{-i\frac{2\pi }{3}
lm}\sum_{q,r=1}^{3}{}e^{i\frac{2\pi }{3}kq}\varepsilon_{m-\rho
,q,r}|r\rangle _{4}.$$ $F^{-1}$ denotes the inverse discrete Fourier transform. The identity of Eq.(\[help1\]) suggests the following protocol for quantum state sharing: The sender obtains particle two of the totally antisymmetric quantum state. Particles three and four are sent to the other two parties. The sender who is now holding particles one and two performs a maximal quantum test on these two particles by projecting onto the orthonormal basis of generalized Bell states (\[gbell\]). As a consequence he obtains two measurement results, say $l$ and $\rho$, which specify the Bell state particles one and two have been projected onto. Now, one of the other parties applies a discrete Fourier transformation onto particle three and performs a maximal quantum test on this particle. The result of this measurement yields the label of the quantum state particle three has been projected onto, say $k$. The three classical labels, namely $(l,\rho,k)$ are communicated to the receiver. Only after having received this combined classical information from the other two parties is the receiver able to apply the proper inverse transformation, namely $U^{\dagger}(l,\rho,k)$, onto his particle which enables him to recover the original quantum state $\vert\chi\rangle$.
Comparison of two quantum states I
==================================
Quantum state identification and state comparison constitute two other interesting applications of totally antisymmetric quantum states [@comp]. Thereby one wants to answer the basic question whether [*two given quantum states are identical or different*]{}. The simplest version of this problem can be illustrated in the case of two qubits. We shall comment on two separate cases, namely on the case of two unknown and on the case of two known pure states.
Let us first assume that we are given two completely unknown pure quantum states and that we want to decide with maximum probability whether these states are identical or different. In the case of two unknown states, say $\vert\psi\rangle$ and $\vert\phi\rangle$, we cannot give an affirmative answer to the question whether [*these two states are the same*]{}. We can only determine whether theses states are different or whether the answer is inconclusive. The fact that a positive answer to this question cannot be obtained can be demonstrated in several ways. The most straightforward argument relies on continuity. For any pair of different states the affirmative answer should yield a zero result even in cases in which these states are only infinitessimally different. As a consequence the probability for a non-zero result would have to be discontinuous. This contradicts the fact that quantum mechanical probabilities are continuous functions of projection operators.
In view of this impossibility the natural question arises how to proceed in order to obtain at least a negative and an inconclusive answer. The product state of two qubits $\vert\psi\rangle \vert\phi\rangle$ can be decomposed uniquely into the symmetric states $\vert 0\rangle\vert 0\rangle ,\vert 1\rangle\vert 1\rangle ,
(\vert 0\rangle\vert 1\rangle
+ \vert 1\rangle\vert 0\rangle)$ and into the antisymmetric state $(\vert 1\rangle\vert 0\rangle
- \vert 0\rangle\vert 1\rangle)$. If we find a non-zero projection onto the antisymmetric state, the two states cannot be identical. If the measurement yields an overlap with one of the symmetric states the answer is inconclusive. What can we say about the relative frequency of these two possible outcomes? The overlap between the decomposition components is given by $$P_s - P_a = \vert\langle\psi\vert\phi\rangle\vert^2 \geq 0 ,$$ where $P_s = 1 - P_a$ and $P_a = \vert (\langle 1\vert\langle 0\vert - \langle 0\vert\langle 1\vert)\vert \psi\rangle\vert\phi\rangle\vert^2/2$ is the overlap between the tested product state $\vert \psi\rangle\vert\phi\rangle$ and the antisymmetric state. Thus, the measurement will show the inconclusive result (projection onto the symmetric subspace) more often than a negative one.
A realization of this state comparison using passive optical elements (detection in the Bell basis) seems feasible. We have to distinguish in a reliable way the presence of the antisymmetric state from any element of the symmetric subspace. For this purpose also a simple coincidence measurement could be used. The two states can be sent into a multiport, for example, and at the output the coincidences can be detected. Only if both states are identical certain coincidences are absent.
Procedures which are applicable to more than two copies require a more detailed study of the group structure of the corresponding state spaces. If the number of copies equals the dimension of the one-particle Hilbert spaces, i.e. $N = d$, then a comparison is simple as a totally antisymmetric state $\vert A_N\rangle$ exists. Otherwise we have to use projections onto combinations of the ”most antisymmetric” states available. Let us consider the simple example of $N=2$ and $d > 2$. The two-particle Hilbert space can be decomposed into two subspaces, namely a symmetric one, spanned by the vectors $\vert i\rangle\vert i\rangle$ and $(\vert i\rangle\vert j\rangle + \vert j\rangle\vert i\rangle )$, and an antisymmetric one, spanned by the states $(\vert i\rangle\vert j\rangle - \vert j\rangle\vert i\rangle)$ with $i,j = 0,...,d-1$. Successful projection onto the latter state indicates that the two quantum states are different. Another simple case arises if $N=3$ and $d=2$. The eight dimensional three-particle Hilbert space can be decomposed into two subspaces spanned by the states $\vert 1\rangle\vert 1\rangle\vert 1\rangle ,
\vert 0\rangle\vert 0\rangle\vert 0\rangle ,
(\vert 1\rangle\vert 1\rangle\vert 0\rangle + \vert 1\rangle\vert 0\rangle\vert 1\rangle +
\vert 0\rangle\vert 1\rangle\vert 1\rangle),
(\vert 1\rangle\vert 0\rangle\vert 0\rangle + \vert 0\rangle\vert 0\rangle\vert 1\rangle +
\vert 0\rangle\vert 1\rangle\vert 0\rangle)$ and by the states $(2\vert 1\rangle\vert 1\rangle\vert 0\rangle - \vert 1\rangle\vert 0\rangle\vert 1\rangle - \vert 0\rangle\vert 1\rangle\vert 1\rangle ), (2\vert 0\rangle\vert 0\rangle\vert 1\rangle -
\vert 0\rangle\vert 1\rangle\vert 0\rangle - \vert 1\rangle\vert 0\rangle\vert 0\rangle ),
(\vert 1\rangle\vert 0\rangle\vert 1\rangle - \vert 0\rangle\vert 1\rangle\vert 1\rangle ), (\vert 0\rangle\vert 1\rangle\vert 0\rangle - \vert 1\rangle\vert 0\rangle\vert 0\rangle )$. The latter four dimensional subspace can be used to decide whether three two-level states are different.
Comparison of two quantum states II
===================================
Let us now assume that two qubits are each prepared in one of the known states $$\vert\psi_{1,2}\rangle = \cos\theta\vert +\rangle \pm \sin\theta \vert - \rangle .$$ The problem of comparing these two states can be solved either by the strategy of minimum probability of error or by the strategy of unambiguous state identification (for a review see Ref.[@Tony] and references therein). In the first case the minimum error with which both states can be distinguished is given by $$P^{comp}_e = \frac{1}{2}\cos^2 (2\theta ).$$ In the second case the minimum probability of obtaining an inconclusive answer is given by $$P^{comp}_? = \cos (2\theta ) [2 - \cos (2\theta )].$$ The question is whether these two strategies are optimal. Indeed, the minimum error strategy is optimal [@comp]. In the case of unambiguous state identification strategy we can do better. In this latter case the optimum strategy is the following: First we use the Bell state decomposition $$\begin{aligned}
%{ccl}
\vert\psi_{i}\rangle\vert\psi_j\rangle &=& \cos^2\theta\vert +\rangle \vert +\rangle +
(-1)^{i+j} \sin^2\theta\vert -\rangle \vert -\rangle + \\ \nonumber
& & (-1)^i \cos\theta\sin\theta[-\delta_{ij}
(\vert +\rangle \vert -\rangle + \vert -\rangle \vert +\rangle) + \\ \nonumber
& & (1-\delta_{ij}) (\vert +\rangle \vert -\rangle - \vert -\rangle \vert +\rangle)].\end{aligned}$$ If we project successfully onto the antisymmetric state $(\vert +\rangle \vert -\rangle - \vert -\rangle \vert +\rangle)$, the two states are different. If we project onto the symmetric state $(\vert +\rangle \vert -\rangle + \vert -\rangle \vert +\rangle)$, both states have to be identical. If the state is found neither in the symmetric nor in the antisymmetric subspace, it is in one of the two possible states $$\vert\Phi_{\pm}\rangle = \frac{\cos^2\theta \vert +\rangle\vert +\rangle \pm \sin^2\theta \vert -\rangle\vert -\rangle}
{\sqrt{1 - \frac{1}{2}\sin^22\theta}}$$ which can be discriminated unambiguously. Thus, the overall probability for an inconclusive result reads $$P^{comp1}_? = (1 -\frac{1}{2} \sin^2 2\theta )\vert\langle\Phi_+\vert\Phi_-\rangle\vert = \cos 2\theta$$ and $$P^{comp1}_{?} < P^{comp}_? .$$ These simple considerations illustrate that the unambiguous method of state comparison is not the optimal one. It can be shown, however, that the two-step method proposed is the optimal one. The interesting aspect of our analysis is that the unambiguous state discrimination may be viewed as a two-step state comparison. First we find out whether the two states are identical or not and afterwards we determine the label.
Conclusions
===========
We have demonstrated that totally antisymmetric quantum states are useful for various tasks in quantum information processing. Their special features are particularly useful for implementing multi-partite key-sharing and quantum state sharing protocols and for comparing quantum states. All the applications discussed here rely on the high symmetry and the peculiar correlation properties of these quantum states. It is expected that the future development of multi-partite protocols for quantum information processing will stimulate many more interesting applications of totally antisymmetric quantum states. 3truemm [**Acknowledgments**]{}
This work was supported by the European IST-1999-13021 QUIBITS, DLR (CZE00/023) and GAČR (202/01/0318).
[1]{} (1989) 2404 , [Int. Conf. Computers, Systems & Signal Processing]{}, Bangalore, India, (1984) 175 , [Phys. Rev. Lett.]{} [**67**]{} (1991) 661 , [Phys. Rev. A]{} [**59**]{} (1999) 1829 , [Phys. Rev. A]{} [**59**]{} (1999) 162 , [Contemp. Physics]{} [**41**]{} (2000) 401 , quant-phys/0202087
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---
abstract: 'Recent observations of gravitational waves motivate investigations for the existence of Primordial Black Holes (PBHs). We propose the observation of gravitational microlensing of distant quasars for the range of infrared to the submillimeter wavelengths by sub-lunar PBHs as lenses. The advantage of observations in the longer wavelengths, comparable to the Schwarzschild radius of the lens (i.e. $R_{\rm sch}\simeq \lambda$) is the detection of the wave optics features of the gravitational microlensing. The observation of diffraction pattern in the microlensing light curve of a quasar can break the degeneracy between the lens parameters and determine directly the lens mass as well as the distance of the lens from the observer. We estimate the wave optics optical-depth, also calculate the rate of $\sim 0.1$ to $\sim 0.3$ event per year per a quasar, assuming that hundred percent of dark matter is made of sub-lunar PBHs. Also, we propose a long-term survey of quasars with the cadence of almost one hour to few days to resolve the wave optics features of the light curves to discover PBHs and determine the fraction of dark matter made of sub-lunar PBHs as well as their mass function.'
author:
- 'T. Naderi'
- 'A. Mehrabi'
- 'S. Rahvar'
bibliography:
- 'ref.bib'
title: Primordial black hole detection through diffractive microlensing
---
Introduction {#intro}
============
Observations of type Ia supernova [@Perlmutter:1998np; @Riess:2004nr; @Astier:2005qq; @Jha:2006fm], cosmic microwave background (CMB) radiation [@Spergel:2006hy; @Ade:2015rim] and baryon acoustic oscillation (BAO) [@Seo:2005ys; @Blake:2011en] indicate that around $25\%$ of matter content of the universe is made of dark matter (DM). There are many scenarios to explain the nature of DM and one of the models proposes DM might be composed totally or partially by the primordial black holes (PBHs) [@Blais:2002nd; @Khlopov:2008qy; @Frampton:2010sw].
There are several mechanisms to explain the formation of PBHs including sharp peaks in density fluctuations [@GarciaBellido:1996qt], phase transitions [@Jedamzik:1999am], resonant reheating [@Suyama:2004mz], tachyonic preheating [@Suyama:2006sr] and curvaton scenarios [@Kohri:2012yw; @Kawasaki:2012wr; @Bugaev:2013vba]. PBHs smaller than about $10^{12}$kg should have already evaporated through the Hawking radiation [@2010RAA....10..495K; @2010PhRvD..81j4019C]. However, the massive PBHs, are unaffected by the Hawking radiation might have various cosmological consequences, such as seeds for supermassive black holes [@2002PhRvD..66f3505B], generation of the large-scale structures [@Afshordi:2003zb] and affects on the thermal and ionization history of the universe [@2008ApJ...680..829R].
The observations for searching the Massive Astrophysical Compact Halo Objects (MACHOs) by gravitational microlensing and femtolensing of $\gamma$-ray bursts excluded PBHs in the mass range of $\sim10^{-7}M\odot$ -$1 M\odot$ and $10^{14}$-$10^{17}$kg [@Tisserand:2006zx; @Nemiroff:2001bp]. However, assuming an extended mass function for the compact objects, it seems that various observational data along with the dynamical constraints are consistent with PBHs as the dark matter candidate within the mass range of $10 M_\odot<M<10^3 M_\odot$ and/or $10^{17}~\text{kg} <M<10^{21}~\text{kg}$ [@carr; @green]. The later range for the PBHs is convenient to be written in terms of lunar mass, roughly as $10^{-6} <\bar{M}<10^{-2}$ (assuming the lunar mass of $M_{\rm m} \sim 7\times 10^{23}$ kg) where $\bar{M}={M}/{M_{\rm m}}$.
Gravitational lensing provides an exceptional tool for investigating the astrophysical phenomena including indirect detection of the compact objects [@rahvar:2015]. The light deflection in gravitational lensing depends on the configuration of the lens mass distribution and might produce several images. The term of gravitational microlensing is used when the images from the lensing cannot be resolved by the conventional telescopes. In this case, the result of lensing is the magnification of light receiving from the source star. Taking into account the relative motion of the lens, source and the observer results in a bell shape light curve, so-called Paczynski light curve [@pac86]. In recent years, microlensing has been used for discovering extra-solar planets [@Cottle:1991zza; @Gould:1992aj; @Gould:2008zu; @batista; @muraki], investigating the properties of a distant source stars [@Afonso:2001gh; @Gould:2001bg; @Abe:2003xb; @Fields:2003zx; @sajadian] and studying the structure of the Milky Way galaxy [@moniez]. Moreover, in the cosmological scales, the gravitational microlensing provides a useful method for studying compact objects.
The quasar microlensing in the cosmological scales uses the caustic crossing of an ensemble of lenses [@2001ASPC..237..185W] for studying the distribution of stars and micro-halos around the galaxies . The possibility of detection of PBHs have been studied through observation of Quasars in X-ray [@pbh2017] and they found no-candidate in the mass range of $0.05 M_\odot<M<0.45 M_\odot$. Also, some microlensing observation of Quasars has been done in the survey mode where the aim was the detection of the caustic crossing of the lenses in the halo of the strong lensed galaxies [@meld; @Giannini]. All these observations have been done in the optical and shorter wavelengths. Here, we propose extending observations to the infrared and millimeter wavelengths where effects of wave optics is important in the light curve of sub-lunar mass lenses. The advantage of wave optics is that we can obtain more information about the parameters of lenses compared to the geometric microlensing. From the observational point of view, Spitzer space-based telescope and the Atacama Large Millimeter/Submillimeter Array (ALMA) are ideal tools for studying the light curve of quasars in our desired wavelengths [@2015MNRAS.452...88K; @Venemans:2015hyr; @2016ApJ...824..132N] .
In section (\[geometric\]) we introduce the gravitational microlensing in the geometric optics regime. In section (\[WO\]), we introduce the wave optics feature of gravitational microlensing and calculate the diffraction pattern from the scattering of the electromagnetic wave from a PBH on the observer plane. In section (\[obs\]), we study the observational features of the diffraction of light from a PBH as a lens as well as we calculate the optical depth for the observation of this event. Conclusions are given in (\[sec:conclude\]).
Geometric microlensing {#geometric}
======================
The standard gravitational lensing formalism uses the geometric optics for the limit of $\lambda \ll R_{\rm sch}$, where $\lambda$ is the wavelength of the light and $R_{\rm sch}$ is the Schwarzschild radius of the lens. However, when $ \lambda \approx R_{\rm sch}$, the wave optics features of lensing such as interference of the light from different images produces the interference pattern on the observer plane. The relative motion of the observer with respect to the fringes results in a time variation in the intensity of light where the time-scale and the amplitude of these fringes provide more information compared to that of geometric optics. The diffractive gravitational lensing has been studied for a system with a galaxy as a lens and a point radio source [@ohanian:1983]. Following this work, the caustic-crossing of quasars in the wave optics regime has been investigated, where they put a limit on the size of quasars [@Jaroszynski:1995cd]. The wave optics aspect of microlensing with a sub-stellar mass lens, like a free-floating planet in the galaxy, have been studied in a series of papers [@Heyl:2009av; @Heyl:2011b; @Heyl:2010hm]. Also, @Mehrabi:2012dy investigated the wave optics features for a binary lensing system near the caustic lines.
The gravity of a lens deflects light ray from a distant source and this deflection produces multiple images from a single source. It is more convenient to write the lens equation in terms of angular scales [@Schneider1985]: $$\label{eq:lens}
\bs{\beta}=\bs{\theta}-\bs{\alpha}(\bs{\theta})\;,$$ where $\bs{\beta}$ and $\bs{\theta}$ are the source and the image positions and $\bs{\alpha}(\bs{\theta})$ is the deflection angle. Notice that all angles are normalized to the Einstein-angle $$\label{eq:theta-e}
\theta_{E}=\sqrt{\frac{2R_{\rm sch}}{D_s}\frac{1-x}{x}},$$ where $x={D_l}/{D_s}$ is the ratio of the comoving distance of the lens to the comoving distance of the source and $D(z)$ in $\Lambda$CDM model is given by $$\label{eq:dis-cos}
D(z)=\frac{c}{H_0}\int_0^z \frac{dz}{\sqrt{\Omega_m(1+z)^3+\Omega_{\Lambda}}}\;.$$ In the geometric optics limit, after solving equation (\[eq:lens\]), one can find the corresponding map between the source position to the image positions. The Jacobian of transformation in the equation of $\theta = \theta(\beta)$, provides the ratio of areas in the image space to the source space which is equivalent to the magnification in the gravitational microlensing.
In what follows, we concern the wave optics regime of the gravitational microlensing by considering the interferometry of the light rays. It is convenient to use the Fermat potential for the light ray which is proportional to the time delay between a given trajectory and a straight path. The Fermat potential for a generic lens is given by $$\label{eq:ferma}
\phi(\bs{\theta},\bs{\beta})=\frac{1}{2}(\bs{\theta}-\bs{\beta})^2-\psi(\bs{\theta})\;,$$ where $\psi(\bs{\theta})$ is the gravitational potential in 2D and is defined on the lens plane as $$\label{eq:psi}
\psi(\bs{\theta})=\frac{1}{\pi}\int\frac{\Sigma(\bs{\theta'})}{\Sigma_{\rm cr}}\ln\vert\bs{\theta}-\bs{\theta}^{\prime}\vert d^2\bs{\theta}^{\prime},\;$$ and $\Sigma(\bs{\theta})$ is the surface mass density of the lens and the critical mass density is given by $$\nonumber
\Sigma_{\rm cr}=\frac{c^2}{4\pi GD_{\rm l}(1-x)}.$$ The lens equation gives rise from the Fermat principle, $ \nabla_{\bs{\theta}}\phi(\bs{\theta},\bs{\beta})=0,$ and in terms of the Fermat potential, the deflection angle in equation (\[eq:lens\]) is given by $\bs{\alpha(\bs{\theta})}=\bs{\nabla}\psi(\bs{\theta})$.
Diffractive microlensing {#WO}
========================
In wave optics limit where the time delay between trajectories from a source to observer is less than a period of light, the light rays can be considered temporally coherent and the result is the production of the interference pattern on the observer plane. Under condition where the source and deflector are far from the observer, we can use the Huygens-Fresnel principle for analyzing gravitational lensing. Then, every point on the lens plane can be taken as a secondary source and the amplitude of the electromagnetic wave at each point on the observer plane is the superposition of the light from various sources on the lens plane. This analysis can be done for a point source, however, for an extended realistic source the amplification is calculated by the superposition of the infinitesimal incoherent sources. Finally, multiplying the superposition of the electromagnetic wave by its complex conjugate results in the magnification on the observer plane. The magnification for a point source [@Falco:1992] is given by $$\label{eq:mag-wave}
\mu(\bs{\beta};k)=\frac{f^2}{4\pi^2}\vert\int e^{if\phi(\bs{\theta},\bs{\beta})}d^{2}\theta\vert^2,\;$$ where $f=2kR_{\rm sch}$ and $k$ is the wave-number. $\phi(\bs{\theta},\bs{\beta})$ is the Fermat potential for a single lens and is given by: $$\label{eq:ferma:point}
\phi(\bs{\theta},\bs{\beta})=\frac{1}{2}(\bs{\theta}-\bs{\beta})^2-\ln\vert\bs{\theta}\vert.\;$$
Substituting equation (\[eq:ferma:point\]) in equation (\[eq:mag-wave\]), the magnification is given as follows $$\label{eq:mag-poi-main}
\mu(\beta;f)=\frac{\pi \frac{f}{2}}{\sinh(\pi \frac{f}{2})}e^{\pi \frac{f}{2}}\vert 1F1(1-i\frac{f}{2},1,i\frac{f\beta^2}{2})\vert^2, \;$$ where $1F1(a,b,x)$ is the confluent hypergeometric function [@Falco:1992]. Fig. (\[fig:mag-point\]) presents the magnification in terms of $\beta$ for different values of $f$.
![Magnification of a point source for varies values of $f$. The solid red, dot-dashed blue and dashed green lines represent the magnification for $f=5, 10$ and $f=15$, respectively. []{data-label="fig:mag-point"}](f){width=".45\textwidth"}
The diffraction pattern is observable when $\lambda\approx R_{\rm sch}$ (i.e. $f \simeq \mathcal{O}(1)$). By increasing $f$ (smaller wavelength or massive lens), the fringes shrink and the diffraction pattern converge to the geometric optics magnification.
To simplify equation (\[eq:mag-poi-main\]), we expand the Fermat potential in equation (\[eq:ferma:point\]) around the critical point of ($\bs{\theta}=1$, $\bs{\beta}=0$) where according to Fig. (\[fig:mag-point\]), the light curve has peak around it, as follows: $$\label{eq:fer-expand}
\phi(\bs{\theta},\bs{\beta})=\theta^2 -2\theta -\theta\beta\cos\gamma,\;$$ where polar coordinate $(\theta,\gamma)$ is used on the lens plane. Then, from equation (\[eq:mag-wave\]) the magnification simplifies to $$\label{eq:mag-appr}
\mu(\beta;k)=\pi f J_0^2(f\beta),$$ where $J_0$ is the Bessel function of the first kind. We note that the relative difference of magnification from Eq. (\[eq:mag-appr\]) compare to the exact equation is less than $1\%$ for sources with $\bs{\beta}<0.5$ and this difference decreases rapidly when the source moves toward the lens position (i.e. $\beta \rightarrow 0$). In practice, it is possible to observe the light curve in two different wavelengths say $\lambda_1$ and $\lambda_2$. In this case, the relative magnification is given by $$\label{eq:mag-rel}
\frac{\mu_1}{\mu_2}=\frac{\lambda_2}{\lambda_1}\frac{J_0^2(f_1\beta)}{J_0^2(f_2\beta)},$$ where close to the maximum magnification, using the series of $J_0(x)\approx 1 - \frac{x^2}{4}+\frac{x^4}{64}$, equation (\[eq:mag-rel\]) simplifies as $$\label{eq:mag-rel2}
\frac{\mu_1}{\mu_2}=\frac{\lambda_2}{\lambda_1}\frac{1-\frac{1}{2}(f_1\beta)^2+\frac{3}{32}(f_1\beta)^4}{1-\frac{1}{2}(\frac{\lambda_2}{\lambda_1})^2(f_1\beta)^2+\frac{3}{32}(\frac{\lambda_2}{\lambda_1})^4 (f_1\beta)^4}.$$ In this case the right-hand side of this equation is a function of $f_1\beta(t)$ where from the measurement of $\mu_1$ and $\mu_2$ as a function of time, we can extract $f_1\beta(t)$. This parameter depends on the lensing parameters as follows: $$\label{eq:fy}
f_1\beta(t)= f_1 \left(u_0^2 + (\frac{t}{t_E})^2\right)^{\frac{1}{2}},$$ where $u_0$ is the minimum impact parameter and $t_E$ is the Einstein-crossing time. From the observation of a microlensing event in the regime of geometric optics (i.e. $\lambda\ll R_{\rm sch}$), we can extract $u_0$ and $t_E$. On the other hand, knowing the left-hand side of equation (\[eq:fy\]) from the wave optics and right-hand side from the geometric optics at $t=0$, we determine directly $f_1$ or mass of the lens. We note that unlike to the geometric optics, the mass of lens determine from this method is independent of the distance of lens and source as well as their relative velocities with respect to the observer.
In reality, quasars as the source in the lensing have finite sizes and this effect should be taken into account in the wave-optics calculation. For a given source, the total magnification is calculated by integrating over all individual elements on a source where these elements are independent and incoherent. Then the magnification of an extended source [@Mehrabi:2012dy] is given as $$\label{eq:mag-finite}
\mu(\bs{\beta},\rho;k)=\int_{s<\rho}\frac{I_{\rm w}(\bs{\beta})\mu(\bs{\beta};k)d^2 s}{I_{\rm w}(\bs{\beta})d^2 s},\;$$ where $I_{\rm w}(\bs{\beta})$ is the surface brightness of the source (that might depends on the wavelength) and $\rho={\theta_s}/{\theta_E}$ is the angular size of source normalized to the Einstein angle. Here the integration in equation (\[eq:mag-finite\]) is taken over the source area (i.e. $s<\rho$).
In Fig. (\[fig:mag-finite\]), we depict the magnification in terms of $\beta$ for three sources with different sizes in the wave-optics regime. For the small sources, the magnification resembles a point source in the wave-optics regime and by increasing the source size, the fringes are smeared out and the magnification looks like the geometric optics. The distance between the fringes for a point-lens is $\Delta \beta={2\pi}/{f}$ [@Falco:1992] and for a typical extended source with $\rho>\Delta \beta$, fringes smear out due to integration over a highly oscillating function. Hence, the fringes are observable only for the extended source that satisfies the condition of $\rho<\Delta \beta$. Summarizing this part, in the wave optics regime fringes for an extended source can be produced under the following condition of $$\label{eq:sour-size1}
\theta_{\rm s} <\frac{1}{2}\frac{\lambda}{R_{\rm sch}}\theta_{E},\;$$
![Magnification of a uniform luminous source as a function of $\beta$ for $f=10$. The solid red, dot-dashed blue and dashed green lines show the magnification for $\rho=0.01$, $\rho = 0.08$ and $\rho=0.15$, respectively. By increasing the size of source, the incoherent light of the source results in magnification pattern converge to the geometric optics limit profile.[]{data-label="fig:mag-finite"}](finite){width=".45\textwidth"}
where taking into account the redshift of deflector at $z_d$, this condition can be written as $$\label{eq:sour-size2}
\theta_{E}>2\theta_s\left(1+z_d\right){R_{\rm sch}}/{\lambda_{obs}},\;$$ where $\lambda_{obs}$ is the wavelength of observation. Also, since $\theta_E\propto\sqrt{M}$ and $R_{\rm sch}\propto M$, the detection of fringes is in favour of small mass PBHs.
Now, let us assume the lens mass to be in the range of $10^{-6}\lesssim \bar{M}\lesssim 10^{-2}$. Then we rewrite the wave optics parameter of microlensing $f$, as follows $$\label{eq:f-pbh}
f=4\pi(1+z_{\rm d})\frac{R_{\rm sch}}{\lambda_{obs}}=4\pi(1+z_{\rm d})(\frac{\lambda_{obs}}{0.1\text{mm}})^{-1}{\bar{M}}.$$ In the case of strong lensing of a quasar by a galaxy, it is more likely that PBH resides in the halo of the lensed galaxy, which allows us to measure the redshift of lens and from equation (\[eq:f-pbh\]) directly obtain the mass of PBH. We note that we had also another method of mass measurement from equation (\[eq:mag-rel2\]), if we use at least two different wavelengths for the observation.
There are other observables in the geometric optics that can be used for breaking the degeneracy between the lens parameters. Let us take the finite size effect in the geometric optics which smoothes the peak of a light curve [@2011AJ....141..105P; @Paris:2016xdm; @2017PKAS...32..305S]. Knowing the physical size of a quasar as a source from the astrophysical informations, from the finite-size effect, we can extract the projected Einstein radius on the source plane (i.e. $R_E^{(s)}=\theta_{E}D_s$) as $$\label{eq1}
R_E^{(s)} = 1.65 \text{A.U.}(\frac{D_{\rm s}}{6\text{Gpc}})^{1/2}(\frac{1-x}{x})^{1/2}{\bar M}^{1/2},$$ where in this case we assume a source at redshift $z\approx 3$. We can combine equation (\[eq1\]) with the parameter $f = 2kR_{\rm sch}$ from the wave optics observations to obtain the distance of the lens as well as its mass.
Observational prospect and optical depth {#obs}
========================================
From observational point of view, the cadence between the data points in the light curve of a quasar should be small enough to measure the oscillations due to the diffraction pattern in the light curve. In order to estimate the time scale between the fringes, we use $\Delta \beta$ where in terms of $f$ in equation (\[eq:f-pbh\]) is given by $$\label{eq:mag-osc}
\Delta \beta = \frac{2\pi}{f}=\frac12\frac{\lambda _{obs}}{0.1\text{mm}}\frac{1}{1+z_d}\frac{1}{\bar M}.\;$$ By multiplying $\Delta \beta$ to the Einstein crossing time of lens, $t_{E}$, the time-scale for the transit of fringes can be obtained. The Einstein crossing time for typical parameters of a lens with lunar mass at the cosmological distances is $$\label{eq:te-geo}
t_{E} = 5.7{\text d}~(\frac{D_{\rm s}}{6\text{Gpc}})^{1/2}(\frac{1-x}{x})^{1/2}{\bar M}^{1/2}(\frac{500}{v_t}),$$ where $v_t$ is the relative transverse velocity of the lens-source-observer, which is $\sim 1000$ km/s for a rich cluster and $\sim 200$ km/s in the galactic scales. Hence, the time-scale for transit of fringes (i.e. $\Delta t = t_E \Delta\beta $) is given by
$$\label{eq:dely-time}
\Delta t = 2.9\text{d}~ (\frac{\lambda_{obs}}{0.1\text{mm}})\frac{1}{1+z_d}(\frac{D_{\rm s}}{6\text{Gpc}})^{1/2}(\frac{1-x}{x})^{1/2}(\frac{500}{v_t}){\bar M}^{-1/2}.$$
For a lunar mass PBH located at the distance $z_d\sim1$ and wavelength of $\lambda_{obs}=100\mu m$, we have $\Delta\beta\sim 10^{-1}$ and the time scale of fringe-transit is of the order of $\sim 1.5$ days. According to (\[eq:dely-time\]), the transit time is proportional to the $\lambda$ and decreases for shorter wavelengths. For PBHs in the mass range of $10^{-6} <\bar{M}<10^{-2}$ the time scale of fringe-transit is within the range of $15 \text{d}<\Delta t< 4 \text{yr}$.
One of the important technical issues in the observations of quasars is the filtering of intrinsic variabilities compare to the diffraction signals. For a quasar with the variability time scale shorter or in the same order of fringe transit time-scale, it is difficult to filter out the background signals. Some of the quasars with very rapid variabilities have been detected in the timescales shorter than hours to minutes, so-called micro-variability [@Whiting:2001vj]. One solution is to survey those quasars with the low variabilities. The other possibility is to study the quasars in the strong lensing systems and remove any intrinsic variations in the light curve by shifting the light curves according to the time delay between the images [@2001ASPC..237..185W]. This method in recent years is used for detecting microlensing signals in the geometric optics regime [@Ricci2011; @Ricci; @Giannini].
![The wave optics optical depth as a function of PBH mass for sources with the size of 50 AU (solid blue line), 100AU (dashed red line) and 150AU (dot-dashed green line). Here we consider concordance $\Lambda$CDM with $\Omega_m=0.3$, $h=0.7$ and use $\lambda_{\rm obs}=100 \mu m$. While for a lunar mass PBH the optical depth is negligible, for the smaller PBH the optical depth is larger. In addition, the optical depth increases by decreasing the size of source.[]{data-label="fig:opt-dep"}](optical_depth){width=".48\textwidth"}
![The number of expected events per year as a function of PBH mass. Here the wave length of the observation is adapted $\lambda_{\rm obs}=100 \mu m$ and the rate is plotted for three different size of sources as 50 AU (solid blue line), 100AU (dashed red line) and 150AU (dot-dashed green line). We note that $\tau_{\text{w}}$ is larger for the smaller PBHs as $z_{max}$ is getting larger and the optical depth is larger for the smaller mass from Fig. (\[fig:opt-dep\]). On the other hand $\Delta t$ also is getting larger to the smaller PBHs. So the ratio of these two terms in (\[eq:rate-geo\]) is a function of mass and has a peak for the rate of number of events as depicted in this figure. []{data-label="fig:rate"}](rate){width=".48\textwidth"}
In order to estimate the number of detectable events, we calculate the microlensing optical depth for detection of PBH. The optical depth is defined by $$\label{eq:opt-dep1}
\tau=\int\pi R_E^2n(M,z)c\frac{dt}{dz}dz,\;$$ where $n(M,z)$ is the number density of PBHs and it follows the spatial clustering of the cold dark matter. For the lower redshifts, the optical depth is a function of the direction of line of sight, depending on the cosmic density perturbations that cross the line of sight. However, for quasars at the higher redshifts, we can take almost a uniform number density of PBHs that is proportional to the dark matter density of the universe. This assumption has been carefully investigated in [@Zackrisson], where by considering 6 different models for halos and sub-halos, for quasars at higher redshifts (i.e. $z>0.25$) the optical depth from the clustered and uniform distribution of lenses converge. Here, we assume a uniform distribution for the density of PBHs in the optical depth calculation.
For a uniform distribution of PBH, we define a new optical depth, so-called the wave optics optical depth by $\tau_{\text w}$ where in $\Lambda$CDM, it is given by $$\label{eq:opt-dep2}
\tau_{\text{w}}=\frac{3}{2}\frac{D_H}{D_s}\Omega_{pbh}\int_0^{z_{max}}\frac{(1+z)^2(D_s-D_l)D_ldz}{\sqrt{\Omega_m(1+z)^3+\Omega_{\Lambda}}},\;$$ where $\Omega_{pbh}$ is the density parameter of PBHs and $D_H$ is the present horizon size of universe. The difference between this definition with the conventional optical depth is that in this equation, $z_{max}$ is not assigned to the position of the quasar while that is the largest distance for a lens that satisfies the detection of the wave optics condition ($\rho<\Delta y$). Moreover, in the geometric optics $\tau$ is independent of the mass function of the lenses and it depends on the overall mass density of the lenses. However, for the wave optics regime, the optical depth depends on the mass of lens as well as $z_{max}$. In Fig.(\[fig:opt-dep\]) we plot the optical depth in unite of $\Omega_{pbh}$ for three different values of the source sizes. In this plot, we consider the concordance cosmology model of $\Omega_{m}=0.3~,~h=0.7$ and put the source at the redshift $z=3$. As it is expected, the small mass PBH and small size sources are in favor of wave optics microlensing detection.
For a lunar mass PBH, the optical depth is very small, however it grows rapidly by decreasing the mass of lenses. For the case of $\bar{M}\ll 1$, from equation (\[eq:dely-time\]) and equation (\[eq:te-geo\]), $t_E \ll \Delta t$ and we take $\Delta t$ as the corresponding time-scale for the microlensing events in the wave optics regime instead of $t_E$. Now we define the rate of events in the regime of wave optics microlensing as $$\label{eq:rate-geo}
\Gamma_{\text w}=\frac{2}{\pi}\frac{\tau_{\text{w}}}{\Delta t},\;$$ where both $\tau_{\text{w}}$ and $\Delta t $ depend on the mass of PBH. Assuming Dirac-Delta function for the mass function of PBHs, in Fig. (\[fig:rate\]) the rate of events per year is depicted as a function of $\bar{M}$. In equation (\[eq:rate-geo\]), the optical depth increases for the smaller masses. On the other hand $\Delta t$ also increase for the smaller masses with the factor of ${\bar M}^{-1/2}$, so the ratio of these two terms in (\[eq:rate-geo\]) results in a peak as depicted in Fig. (\[fig:rate\]). For a given quasar, the number of detectable events is $N_{obs}=\Gamma T_{obs}$, where $T_{obs}$ is the duration of observation. Let us take a quasar with the size of $50$ AU, then from Fig. (\[fig:rate\]), we expect to detect $N_{obs} \simeq 0.9\Omega_{\rm pbh}/{\text yr}$ for PBHs with $\bar{M}\sim 10^{-3}$ and $N_{obs} = 0.3\Omega_{\rm pbh}/{\text yr}$ for the sources with the radius $r_s=100AU$. Now if hundred percent of the dark matter is made of PBHs (i.e. $\Omega_{\rm pbh} = 0.3$), all with the mass of $\bar{M}\sim 10^{-3}$, we expect to detect $0.27$ and $0.09$ event per year, respectively. Also for this mass, equation (\[eq:dely-time\]) provides $\Delta t \simeq 46$ d and a cadence in the observation of light curve with one day can reveal the oscillation mode of diffraction pattern.
Let us define the contribution of PBH on the density of dark matter as $f_{pbh} = \Omega_{pbh}/\Omega_{m}$. Then we also define the parameter of $df_{pbh}/d\bar{M}$ which provide the fraction of dark matter in form of PBHs within the range of $(\bar{M}, \bar{M} + d\bar{M})$. This function can be measured by a long-term survey of quasars with cadence of order of $\sim$ days. Fig.(\[fig:sampl-data\]) demonstrates the simulation of data points for a microlensing event with the parameters of $t_{E}=10$ hr, $\rho=0.1$, $u_0=0.05$ and $\bar{M}=0.007$. Here we adapt the wavelength of $\lambda=2\mu m$ which results in the transit time scale of fringes of order of a few hours. We assumed a photometric signal to noise ratio of $S/N=50$ and recover the parameters of light curve, using the Markov chain Monte Carlo method. The best values of parameters with 1-$\sigma$ uncertainty for the light curve in Fig. (\[fig:sampl-data\]) are given in Tab.(\[tab:res\]). The maximum magnification for this light curve is around $\sim 17$ and increases rapidly by decreasing $u_0$ and $\rho$. We note that the ratio of light in the anti-nodes to the nodes (where $A_{node}\rightarrow 0$) of the interference pattern in Fig. (\[fig:sampl-data\]) is a large number, much larger than the intrinsic variations of a quasar which is about $\sim 50\%$ [@Soldi:2008ev]. So, once we have enough photometric accuracy and the cadence shorter than the interference crossing time-scale, we can detect our desired signals.
![Simulated data points and the best fit model for a typical event. Here we use $t_E=10$ hr, $\rho=0.1$, $u_0=0.05$, $\bar{M}=7\times 10^{-3}$ and $\lambda=2\mu m$ to simulate data points. []{data-label="fig:sampl-data"}](light_curve){width=".48\textwidth"}
[ l c]{} Parameter & 68% limits\
& $10.035\pm 0.036 $\
[$u_0 $]{} & $0.0557\pm 0.0033 $\
[$\rho $]{} & $0.09995\pm 0.00070 $\
[$\bar{M} $]{} & $0.007027\pm 0.000025 $\
Conclusion {#sec:conclude}
==========
Summarizing this work, we have proposed a new method for the microlensing observation of quasars from the far infrared to the millimeter wavelengths. For the small mass lenses where the Schwarzschild radius of the lens is of the order of the wavelength of observation, the gravitational lens can produce distortions on the wavefront of the light in the order of one wavelength. Since the lens and the source are far enough from the observer, this situation is similar to the Huygens-Fresnel approximation and the result is the diffraction pattern from the phase-shifted electromagnetic wave on the lens plane. A relative motion of the observer through the diffraction pattern on the observer plane produces a modulation in the light curve of the quasar. One of the problems with this wave-optics microlensing observation would be the intrinsic variations of quasars that might be mixed with our desired signals. In order to solve this problem, we suggested the observation of quasars with the multiple images from the strong lensing. The advantage of using these quasars is that by shifting the time delay between the images, we can remove the intrinsic variations of the quasar and extract our desired signals.
We suggested the observation of wave-optics microlensing in two different wavelengths. This technic enable us to measure directly the mass of a lens. Also with single wavelength observation, from the measuring the redshift of the strong lensing galaxy and the redshift of the source, we can break the degeneracy between the lens parameters and extract the mass of lenses. One of the possible candidates for the dark matter is the PBHs in the mass range smaller than the lunar mass. In this work, we proposed the observation of quasars with suitable cadence and photometric accuracy to observe the transit of the fringes in the diffraction pattern by the observer. The optical depth and the rate of events depend on the fraction of dark matter made of PBHs as well as the mass of the PBHs. Assuming the mass of PBHs in the order of $10^{-3}$ lunar mass and hundred percent of dark matter is made of PBHs, we obtained the rate of event detection per year for a given quasar within the range of $\sim 0.1$ to $\sim 0.3$. The wave-optics quasar microlensing can put a constraint on the fraction of dark matter made of PBHs as well as the mass function of PBHs. A long term survey of quasars by the infrared telescopes such as Spitzer space-based telescope or millimeter and submillimeter wavelength ground-based telescopes such as ALMA was suggested for this project.
We thank David Spergel for the useful discussions. Also we thank anonymous referee for his/her useful comments and suggestions. S. Rahvar was supported by Sharif University of Technology’s Office of Vice President for Research under Grant No. G950214.
\[lastpage\]
|
---
author:
- 'Hsiao-Wen Chen'
bibliography:
- 'bibcodes-clean.bib'
title: Outskirts of Distant Galaxies In Absorption
---
Introduction {#sec:intro}
============
![Example of the wealth of information for intervening gas revealed in the optical and near-infrared spectrum of a QSO at $z=4.13$. In addition to broad emission lines intrinsic to the QSO, such as /NV at $\approx 6200$Å, a forest of $\lambda$1215 absorption lines is observed blueward of 6200Å. These forest lines arise in relatively high gas density regions at $z_{\rm abs}\lesssim z_{\rm QSO
}$ along the line of sight. The absorbers span over 10 decades in neutral hydrogen column densities ($N({{\mbox{H\,{\scriptsize I}}}})$), and include (1) neutral damped absorbers (DLAs), (2) optically thick Lyman limit systems (LLS), (3) partial LLS (pLLS), and (4) highly ionized absorbers (see text for a quantitative definition of these different classes). The DLAs are characterized by pronounced damping wings ([*second panel from the top*]{}), while LLS and pLLS are identified based on the apparent flux discontinuities in QSO spectra ([*top panel*]{}). Many of these strong absorbers are accompanied with metal absorption transitions such as the $\lambda\lambda$1031, 1037 doublet transitions which occur in the forest, and the $\lambda\lambda$1548, 1550 and $\lambda\lambda$2796, 2803 doublets. Together, these metal lines constrain the ionization state and chemical enrichment of the gas[]{data-label="fig:qsospec"}](qsospec.pdf)
Absorption-line spectroscopy complements emission surveys and provides a powerful tool for studying the diffuse, large-scale baryonic structures in the distant Universe (e.g., @Rauch:1998 [@Wolfe:2005; @Prochaska:2009ASSP]). Depending on the physical conditions of the gas (including gas density, temperature, ionization state, and metallicity), a high-density region in the foreground is expected to imprint various absorption transitions of different line strengths in the spectrum of a background QSO. Observing the absorption features imprinted in QSO spectra enables a uniform survey of diffuse gas in and around galaxies, as well as detailed studies of the physical conditions of the gas at redshifts as high as the background sources can be observed.
Figure \[fig:qsospec\] displays an example of optical and near-infrared spectra of a high-redshift QSO. The QSO is at redshift $z_{\rm QSO}=4.13$, and the spectra are retrieved from the XQ-100 archive (@Lopez:2016). At the QSO redshift, multiple broad emission lines are observed, including the /NV emission at $\approx 6200$Å, emission at $\approx 7900$Å, and \] emission at $\approx 9800$Å. Blueward of the emission line are a forest of $\lambda$1215 absorption lines produced by intervening overdense regions at $z_{\rm
abs}\lesssim z_{\rm QSO }$ along the QSO sightline. These overdense regions span a wide range in column density ($N({{\mbox{H\,{\scriptsize I}}}})$), from neutral interstellar gas of $N({{\mbox{H\,{\scriptsize I}}}})\ge 10^{20.3}\,{\mbox{${\rm cm^{-2}}$}}$, to optically opaque Lyman limit systems (LLS) of $N({{\mbox{H\,{\scriptsize I}}}})>10^{17.2}\,{\mbox{${\rm cm^{-2}}$}}$, to optically thin partial LLS (pLLS) with $N({{\mbox{H\,{\scriptsize I}}}})=10^{15-17.2}\,{\mbox{${\rm cm^{-2}}$}}$, and to highly ionized forest lines with $N({{\mbox{H\,{\scriptsize I}}}})=10^{12-15}\,{\mbox{${\rm cm^{-2}}$}}$ (right panel of Fig. \[fig:halomap\]).
The large $N({{\mbox{H\,{\scriptsize I}}}})$ in the neutral medium produces pronounced damping wings in the QSO spectrum. These absorbers are commonly referred to as damped absorbers (DLAs). An example is shown in the second panel from the top in Fig. \[fig:qsospec\]. In this particular case, a simultaneous fit to the QSO continuum and the damping wings (red curve in the second panel from the top) yields a best-fit $\log\,N({\rm {\mbox{H\,{\scriptsize I}}}})=21.45\pm 0.05$ for the DLA. At intermediate $N({\mbox{H\,{\scriptsize I}}})$, LLS and pLLS are identified based on the apparent flux discontinuities in QSO spectra (top panel). A significant fraction of these strong absorbers have been enriched with heavy elements which produce additional absorption features due to heavy ions in the QSO spectra. The most prominent features include the $\lambda\lambda$1031, 1037 doublet transitions which occur in the forest, and the $\lambda\lambda$1548, 1550 and $\lambda\lambda$2796, 2803 doublets, plus a series of low-ionization transitions such as C[II]{}, Si[II]{}, and Fe[II]{}. Together, these ionic transitions constrain the ionization state and chemical compositions of the gas (e.g., @Chen:2000 [@Werk:2014]).
Combining galaxy surveys with absorption-line observations of gas around galaxies enables comprehensive studies of baryon cycles between star-forming regions and low-density gas over cosmic time. At low redshifts, $z\lesssim 0.2$, deep 21cm and CO surveys have revealed exquisite details of the cold gas content ($T\lesssim 1000$K) in nearby galaxies, providing both new clues and puzzles in the overall understanding of galaxy formation and evolution. These include extended disks around blue star-forming galaxies with the extent $\approx 2\times$ what is found for the stellar disk (e.g., @Swaters:2002 [@Walter:2008; @Leroy:2008]), extended and molecular gas in early-type galaxies (e.g., @Oosterloo:2010 [@Serra:2012]) with predominantly old stellar populations and little or no on-going star formation (@Salim:2010), and widespread streams connecting regular-looking galaxies in group environments (e.g., @Verdes:2001 [@Chynoweth:2008]).
![Mapping galaxy outskirts in 21cm and in QSO absorption-line systems. [*Left*]{}: Deep 21cm image of the M81 group, revealing a complex interface between stars and gas in the group. The observed neutral hydrogen column densities range from $N({{\mbox{H\,{\scriptsize I}}}})\sim
10^{18}\,{\mbox{${\rm cm^{-2}}$}}$ in the filamentary structures to $N({{\mbox{H\,{\scriptsize I}}}})>
10^{21}\,{\mbox{${\rm cm^{-2}}$}}$ in the star-forming disks of group members (@Yun:1994 [@Chynoweth:2008]). The 21cm image reveals a diverse array of gaseous structures in this galaxy group, but these observations become extremely challenging beyond redshift $z \approx
0.2$ (e.g., @Verheijen:2007 [@Fernandez:2013]). [*Right*]{}: The column density distribution function of absorbers, $f_{N({\mbox{H\,{\scriptsize I}}})}$, uncovered at $z=1.9-3.2$ along sightlines toward random background QSOs (adapted from @Kim:2013). Quasar absorbers in different categories are mapped onto different structures both seen and missed in the 21cm image in the left panel. Specifically, DLAs probe the star-forming ISM and extended rotating disks, LLS probe the gaseous streams connecting different group members as well as stripped gas and high velocity clouds around galaxies, and pLLS and strong absorbers trace ionized gas that is not observed in 21cm signals. Among the quasar absorbers, absorption transitions are commonly observed in strong absorbers of $N({{\mbox{H\,{\scriptsize I}}}})\gtrsim 10^{15}\,{\mbox{${\rm cm^{-2}}$}}$ (see e.g., @Kim:2013 [@Dodorico:2016]), and absorption transitions are seen in most high-$N({{\mbox{H\,{\scriptsize I}}}})$ absorbers of $N({{\mbox{H\,{\scriptsize I}}}})\gtrsim 10^{16}\,{\mbox{${\rm cm^{-2}}$}}$ (see e.g., @Rigby:2002). These metal-line absorbers trace chemically enriched gas in and around galaxies[]{data-label="fig:halomap"}](halomap)
Figure \[fig:halomap\] (left panel) showcases an example of a deep 21cm image of the M81 group, a poor group of dynamical mass $M_{\rm
dyn}\sim 10^{12}\,M_\odot$ (@Karachentsev:2006). Prominent group members include the grand-design spiral galaxy M81 at the centre, the proto-starburst galaxy M82, and several other lower-mass satellites (@Burbidge:1961). The 21cm image displays a diverse array of gaseous structures in the M81 group, from extended rotating disks, warps, high velocity clouds (HVCs), tidal tails and filaments, to bridges connecting what appear to be optically isolated galaxies. High column density gaseous streams of $N({\mbox{H\,{\scriptsize I}}})\gtrsim 10^{18}\,{\mbox{${\rm cm^{-2}}$}}$ are seen extending beyond 50kpc in projected distance from M81, despite the isolated appearances of M81 and other group members in optical images. These spatially resolved imaging observations of different gaseous components serve as important tests for theoretical models of galaxy formation and evolution (e.g., @Agertz:2009 [@Marasco:2016]). However, 21cm imaging observations are insensitive to warm ionized gas of $T\sim 10^4$K and become extremely challenging for galaxies beyond redshift $z=0.2$ (e.g., @Verheijen:2007 [@Fernandez:2013]).
QSO absorption spectroscopy extends 21cm maps of gaseous structures around galaxies to both lower gas column density and higher redshifts. Based on the characteristic $N({\mbox{H\,{\scriptsize I}}})$, direct analogues can be drawn between different types of QSO absorbers and different gaseous components seen in deep 21cm images of nearby galaxies. For example, DLAs probe the neutral gas in the interstellar medium (ISM) and extended rotating disks, LLS probe optically thick gaseous streams and high velocity clouds in galaxy haloes, and pLLS and strong absorbers of $N({{\mbox{H\,{\scriptsize I}}}})\approx 10^{14-17}\,{\mbox{${\rm cm^{-2}}$}}$ trace ionized halo gas and starburst outflows (e.g., supergalactic winds in M82 @Lehnert:1999) that cannot be reached with 21cm observations.
The right panel of Fig. \[fig:halomap\] displays the column density distribution function, $f_{N({\mbox{H\,{\scriptsize I}}})}$, for all absorbers uncovered at $z=1.9-3.2$ along random QSO sightlines (@Kim:2013). $f_{N({\mbox{H\,{\scriptsize I}}})}$, defined as the number of absorbers per unit absorption pathlength per unit column density interval, is a key statistical measure of the absorber population. It represents a cross-section weighted surface density profile of hydrogen gas in a cosmological volume. With sufficiently high spectral resolution and high signal-to-noise, $S/N\gtrsim 30$, QSO absorption spectra probe tenuous gas with $N({\mbox{H\,{\scriptsize I}}})$ as low as $N({\mbox{H\,{\scriptsize I}}})\sim 10^{12}\,{\mbox{${\rm cm^{-2}}$}}$. The steeply declining $f_{N({\mbox{H\,{\scriptsize I}}})}$ with increasing $N({\mbox{H\,{\scriptsize I}}})$ shows that the occurrence (or areal coverage) of pLLS and strong absorbers of $N({{\mbox{H\,{\scriptsize I}}}})\approx 10^{14-17}\,{\mbox{${\rm cm^{-2}}$}}$ is $\approx 10$ times higher than that of optically thick LLS along a random sightline and $\approx 100$ times higher than the incidence of DLAs. Such a differential frequency distribution is qualitatively consistent with the spatial distribution of gas recorded in local 21cm surveys (e.g., Fig. \[fig:halomap\], left panel), where gaseous disks with $N({\mbox{H\,{\scriptsize I}}})$ comparable to DLAs cover a much smaller area on the sky than streams and HVCs with $N({\mbox{H\,{\scriptsize I}}})$ comparable to LLS. If a substantial fraction of optically thin absorbers originate in galaxy haloes, then their higher incidence implies a gaseous halo of size at least three times what is seen in deep 21cm images.
In addition, many of these strong absorbers exhibit associated transitions due to heavy ions. In particular, absorption transitions are commonly observed in strong absorbers of $N({{\mbox{H\,{\scriptsize I}}}})\gtrsim 10^{15}\,{\mbox{${\rm cm^{-2}}$}}$ (see, e.g., @Kim:2013 [@Dodorico:2016]), and absorption transitions are seen in most high-$N({{\mbox{H\,{\scriptsize I}}}})$ absorbers of $N({{\mbox{H\,{\scriptsize I}}}})\gtrsim 10^{16}$ ${\mbox{${\rm cm^{-2}}$}}$ (e.g., @Rigby:2002). While absorbers are understood to originate in photo-ionized gas of temperature $T\sim
10^4$K (e.g., @Bergeron:1986), absorbers are more commonly seen in complex, multi-phase media (e.g., @Rauch:1996 [@Boksenberg:2015]). These metal-line absorbers therefore offer additional probes of chemically enriched gas in and around galaxies.
This Chapter presents a brief review of the current state of knowledge on the outskirts of distant galaxies from absorption-line studies. The review will first focus on the properties of the neutral gas reservoir probed by DLAs, and then outline the insights into star formation and chemical enrichment in the outskirts of distant galaxies from searches of DLA galaxies. A comprehensive review of DLAs is already available in [@Wolfe:2005]. Therefore, the emphasis here focusses on new findings over the past decade. Finally, a brief discussion will be presented on the empirical properties and physical understandings of the ionized circumgalactic gas as probed by strong and various metal-line absorbers.
Tracking the Neutral Gas Reservoir Over Cosmic Time {#sec:DLAstats}
===================================================
DLAs are historically defined as absorbers with neutral hydrogen column densities exceeding $N({\mbox{H\,{\scriptsize I}}})=2\times 10^{20}\,{\mbox{${\rm cm^{-2}}$}}$ (@Wolfe:2005), corresponding to a surface mass density limit of $\Sigma_{\rm atomic}\approx
2\,{\mbox{$M_\odot$}}\,{\rm pc}^{-2}$ for atomic gas (including helium). The large gas surface mass densities revealed in high-redshift DLAs are comparable to what is seen in 21cm observations of nearby star-forming galaxies (e.g., @Walter:2008 [@Leroy:2008]), making DLAs a promising signpost of young galaxies in the distant Universe (@Wolfe:1986). In addition, the $N({\mbox{H\,{\scriptsize I}}})$ threshold ensures that the gas is neutral under the metagalactic ionizing radiation field (e.g., @Viegas:1995 [@Prochaska:1996; @Prochaska:2002]). Neutral gas provides the seeds necessary for sustaining star formation. Therefore, observations of DLAs not only help establish a census of the cosmic evolution of the neutral gas reservoir (e.g., @Neeleman:2016), but also offer a unique window into star formation physics in distant galaxies (e.g., @Lanzetta:2002 [@Wolfe:2006]).
While the utility of DLAs for probing the young Universe is clear, these objects are relatively rare (see the right panel of Fig. \[fig:halomap\]) and establishing a statistically representative sample of these rare systems requires a large sample of QSO spectra. Over the last decade, significant progress has been made in characterizing the DLA population at $z\gtrsim 2$, owing to the rapidly growing spectroscopic sample of high-redshift QSOs from the Sloan Digital Sky Survey (SDSS; @York:2000). The blue points in the right panel of Fig. \[fig:halomap\] are based on $\sim 1000$ DLAs and $\sim 500$ strong LLS identified at $z\approx 2-5$ in an initial SDSS DLA sample (@Noterdaeme:2009). The sample of known DLAs at $z\gtrsim 2$ has continued to grow, reaching $\sim 10,000$ DLAs found in the SDSS spectroscopic QSO sample (e.g., @Noterdaeme:2012).
The large number of known DLAs has led to an accurate characterization of the neutral gas reservoir at high redshifts. Figure \[fig:dla\]a displays the observed $N({\mbox{H\,{\scriptsize I}}})$ distribution function, $f_{\rm DLA}$, based on $\sim 7000$ DLAs identified at $z\approx 2-5$ (@Noterdaeme:2012). The plot shows that $f_{\rm DLA}$ is well represented by a Schechter function (@Schechter:1976) at $\log\,N({\mbox{H\,{\scriptsize I}}})\lesssim 22$ following $$f_{\rm DLA}\equiv f_{N({\mbox{H\,{\scriptsize I}}})}(\log\,N({\mbox{H\,{\scriptsize I}}})\ge 20.3)\propto\left[\frac{N({\mbox{H\,{\scriptsize I}}})}{N_*({\mbox{H\,{\scriptsize I}}})}\right]^{\alpha}\,\exp[-N({\mbox{H\,{\scriptsize I}}})/N_*({\mbox{H\,{\scriptsize I}}})],
\label{eq:fn}$$ with a shallow power-law index of $\alpha\approx -1.3$ below the characteristic column density $\log\,N_*({\mbox{H\,{\scriptsize I}}})\approx 21.3$ and a steep exponential decline at larger $N({\mbox{H\,{\scriptsize I}}})$ (@Noterdaeme:2009 [@Noterdaeme:2012]). At $\log\,N({\mbox{H\,{\scriptsize I}}})>22$, the observations clearly deviate from the best-fit Schechter function. However, DLAs are also exceedingly rare in this high-$N({\mbox{H\,{\scriptsize I}}})$ regime. Only eight such strong DLAs have been found in this large DLA sample (@Noterdaeme:2012), making measurements of $f_{\rm DLA}$ in the two highest-$N({\mbox{H\,{\scriptsize I}}})$ bins very uncertain. In comparison to $f_{N({\mbox{H\,{\scriptsize I}}})}$ established from 21cm maps of nearby galaxies (@Zwaan:2005), the amplitude of $f_{\rm DLA}$ at $z\gtrsim
2$ is $\approx 2\times$ higher than $f_{N({\mbox{H\,{\scriptsize I}}})}$ at $z\approx 0$ but the overall shapes are remarkably similar at both low- and high-$N({\mbox{H\,{\scriptsize I}}})$ regimes (Fig. \[fig:dla\]a; see also @Sanchez:2016 [@Rafelski:2016]).
At $\log\,N({\mbox{H\,{\scriptsize I}}})>21$, numerical simulations have shown that the predicted shape in $f_{\rm DLA}$ is sensitive to the detailed ISM physics, including the formation of molecules ($H_2$) and different feedback processes (e.g., @Altay:2011 [@Altay:2013; @Bird:2014]). Comparison of the observed and predicted $f_{\rm DLA}$ therefore provides an independent and critical test for the prescriptions of these physical processes in cosmological simulations. However, the constant exponentially declining trend at $N({\mbox{H\,{\scriptsize I}}})\gtrsim 2\times
10^{21}\,{\mbox{${\rm cm^{-2}}$}}$ between low-redshift galaxies and high-redshift DLAs presents a puzzle.
![Summary of known DLA properties: ([*a*]{}) evolving neutral hydrogen column density distribution functions, $f_{N({\mbox{H\,{\scriptsize I}}})}$ from DLAs at $z=2-3$ (@Noterdaeme:2012) to galaxies at $z\approx 0$ (@Zwaan:2005); ([*b*]{}) declining cosmic neutral gas mass density with increasing Universe age (or decreasing redshift) from observations of DLAs (solid points from @Noterdaeme:2012, open circles from @Prochaska:2009, open squares from @Crighton:2015, and open triangle from @Neeleman:2016) following Eq. (2), local galaxies (green shaded box, a compilation from @Neeleman:2016), and molecular gas (blue shaded boxes, @Decarli:2016), in comparison to increasing cosmic stellar mass density in galaxies with increasing Universe age (grey asterisks, a compilation from @Madau:2014); ([*c*]{}) gas-phase metallicity ($Z$) relative to Solar ($Z_\odot$) as a function of redshift in DLAs (grey squares for individual absorbers and blue points for $N({\mbox{H\,{\scriptsize I}}})$-weighted mean from Marc Rafelski, @Rafelski:2012 [@Rafelski:2014]), IGM at $z\gtrsim 2$ (orange circles, @Aguirre:2008 [@Simcoe:2011]), ISM of starburst galaxies at $z\approx 2-4$ (light magenta boxes, @Pettini:2001 [@Pettini:2004; @Erb:2006; @Maiolino:2008; @Mannucci:2009]), intracluster medium in X-ray luminous galaxy clusters at $z\lesssim 1$ (red triangles, @Balestra:2007), -selected galaxies (green box, @Zwaan:2005), and stars at $z=0$ (dark purple box, @Gallazzi:2008); and ([*d*]{}) molecular gas fraction, $f_{\rm H_2}$ versus total surface density of neutral gas scaled by gas metallicity for high-redshift DLAs in triangles (@Noterdaeme:2008 [@Noterdaeme:2016]), $\gamma$-ray burst host ISM in star symbols (e.g., @Noterdaeme:2015), and local ISM in the Milky Way (@Wolfire:2008) and Large and Small Magellanic Clouds (@Tumlinson:2002) in dots, blue circles, and cyan squares, respectively[]{data-label="fig:dla"}](dla_summary)
At $z=0$, the rapidly declining $f_{N({\mbox{H\,{\scriptsize I}}})}$ at $N({\mbox{H\,{\scriptsize I}}})\gtrsim N_*({\mbox{H\,{\scriptsize I}}})$ has been interpreted as due to the conversion of atomic gas to molecular gas (@Zwaan:2006 [@Braun:2012]). As illustrated at the end of this Section and in Fig. \[fig:dla\]d, the column density threshold beyond which the gas transitions from to ${\rm H_2}$ depends strongly on the gas metallicity, and the mean metallicity observed in the atomic gas decreases steadily from $z\approx 0$ to $z>4$ (Fig. \[fig:dla\]c). Therefore, the conversion to molecules in high-redshift DLAs is expected to occur at higher $N({\mbox{H\,{\scriptsize I}}})$, resulting in a higher $N_*({\mbox{H\,{\scriptsize I}}})$ with increasing redshift. However, this is not observed (e.g., @Prochaska:2009 [@Sanchez:2016; @Rafelski:2016]; Fig. \[fig:dla\]a). Based on spatially resolved 21cm maps of nearby galaxies with ISM metallicity spanning over a decade, it has been shown that $f_{N({\mbox{H\,{\scriptsize I}}})}$ established individually for these galaxies does not vary significantly with their ISM metallicity (@Erkal:2012). Together, these findings demonstrate that the exponential decline of $f_{\rm DLA}$ at $N({\mbox{H\,{\scriptsize I}}})\gtrsim N_*({\mbox{H\,{\scriptsize I}}})$ is not due to conversion of to ${\rm H_2}$, but the physical origin remains unknown.
Nevertheless, the observed $f_{\rm DLA}$ immediately leads to two important statistical quantities: (1) the number density of DLAs per unit survey pathlength, obtained by integrating $f_{\rm DLA}$ over all $N({\mbox{H\,{\scriptsize I}}})$ greater than $N_0= 2\times 10^{20}\,{\mbox{${\rm cm^{-2}}$}}$ and (2) the cosmic neutral gas mass density, contained in DLAs, $\Omega_{\rm atomic}$, which is the $N({\mbox{H\,{\scriptsize I}}})$-weighted integral of $f_{\rm DLA}$ following $$\label{omega}
\Omega_{\rm
atomic}\equiv\rho_{\rm gas}/\rho_{\rm
crit}=\int_{N_0}^\infty\,(\mu\,H_0/c/\rho_{\rm
crit})\,N({\mbox{H\,{\scriptsize I}}})\,f_{\rm DLA}\,d\,N({\mbox{H\,{\scriptsize I}}}),$$ where $\mu=1.3$ is the mean atomic weight of the gas particles (accounting for the presence of helium), $H_0$ is the Hubble constant, $c$ is the speed of light, and $\rho_{\rm crit}$ is the critical density of the Universe (e.g., @Lanzetta:1991 [@Wolfe:1995]). The shallow power-law index $\alpha$ in the best-fit $f_{\rm DLA}$, together with a steep exponential decline at high $N({\mbox{H\,{\scriptsize I}}})$ from the Schechter function in Eq. (\[eq:fn\]), indicates that while DLAs of $N({\mbox{H\,{\scriptsize I}}})<N_*({\mbox{H\,{\scriptsize I}}})$ dominate the neutral gas cross-section (and therefore the number density), strong DLAs of $N({\mbox{H\,{\scriptsize I}}})\sim N_*({\mbox{H\,{\scriptsize I}}})$ contribute predominantly to the neutral mass density in the Universe (e.g., @Zwaan:2005). A detailed examination of the differential $\Omega_{\rm atomic}$ distribution as a function of $N({\mbox{H\,{\scriptsize I}}})$ indeed confirms that the bulk of neutral gas is contained in DLAs of $N({\mbox{H\,{\scriptsize I}}})\approx 2\times 10^{21}\,{\mbox{${\rm cm^{-2}}$}}$ (e.g., @Noterdaeme:2012).
The cosmic evolution of $\rho_{\rm gas}$ observed in DLAs, from Eq. (2), is shown in black points in Fig. \[fig:dla\]b. Only measurements based on blind DLA surveys are presented in the plot[^1]. These include an early sample of $\approx
700$ DLAs at $z=2.5-5$ in the SDSS Data Release (DR) 5 (open circles; @Prochaska:2009), an expanded sample of $\approx 7000$ DLAs in the SDSS DR12 (solid points; @Noterdaeme:2012), an expanded high-redshift sample of DLAs at $z=4-5$ (open squares; @Crighton:2015), and a sample of $\approx 14$ DLAs at $z\lesssim
1.6$ from an exhaustive search in the [*Hubble Space Telescope*]{} ([*HST*]{}) UV spectroscopic archive (open triangle; @Neeleman:2016).
A range of mean mass density at $z\approx 0$ has been reported from different 21cm surveys (see @Neeleman:2016 for a recent compilation). These measurements are included in the green box in Fig. \[fig:dla\]b. Despite a relatively large scatter between different 21cm surveys and between DLA surveys, a steady decline in $\Omega_{\rm atomic}$ is observed from $z\approx 4$ to $z\approx 0$. For comparison, the cosmic evolution of the molecular gas mass density obtained from a recent blind CO survey (@Decarli:2016) is also included as blue-shaded boxes in Fig. \[fig:dla\]b, along with the cosmic evolution of stellar mass density measured in different galaxy surveys, shown in grey asterisks (data from @Madau:2014). Figure \[fig:dla\]b shows that the decline in the neutral gas mass density with decreasing redshift is coupled with an increase in the mean stellar mass density in galaxies, which is qualitatively consistent with the expectation that neutral gas is being consumed to form stars. However, it is also clear that atomic gas alone is insufficient to explain the observed order-of-magnitude gain in the total stellar mass density from $z\approx 3$ to $z\approx 0$, which implies the need for replenishing the neutral gas reservoir with accretion from the intergalactic medium (IGM) (e.g., @Keres:2009 [@Prochaska:2009]). At the same time, new blind CO surveys have shown that molecular gas contributes roughly an equal amount of neutral gas mass density as atomic gas observed in DLAs at $z\lesssim 3$ (e.g., @Walter:2014 [@Decarli:2016]), although the uncertainties are still very large. Together with the knowledge of an extremely low molecular gas fraction in DLAs (see the discussion on the next page and Fig. \[fig:dla\]d), these new CO surveys indicate that previous estimates of the total neutral gas mass density based on DLAs alone have been underestimated by as much as a factor of two. An expanded blind CO survey over a cosmological volume is needed to reduce the uncertainties in the observed molecular gas mass densities at different redshifts, which will cast new insights into the connections between star formation, the neutral gas reservoir, and the ionized IGM over cosmic time.
Observations of the chemical compositions of DLAs provide additional clues to the connection between the neutral gas probed by DLAs and star formation (e.g., @Pettini:2004). In particular, because the gas is predominantly neutral, the dominant ionization for most heavy elements (such as Mg, Si, S, Fe, Zn, etc.) are in the singly ionized state and therefore the observed abundances of these low-ionization species place direct and accurate constraints on the elemental abundances of the gas (e.g., @Viegas:1995 [@Prochaska:1996; @Vladilo:2001; @Prochaska:2002]). Additional constraints on the dust content and on the sources that drive the chemical enrichment history in DLAs can be obtained by comparing the relative abundances of different elements. Specifically, comparing the relative abundances between refractory (such as Cr and Fe) and non-refractory elements (such as S and Zn) indicates the presence of dust in the neutral gas, the amount of which increases with metallicity (e.g., @Meyer:1989 [@Pettini:1990; @Savage:1996; @Wolfe:2005]). The relative abundances of $\alpha$- to Fe-peak elements determine whether core-collapse supernovae (SNe) or SNe Ia dominate the chemical enrichment history, and DLAs typically exhibit an $\alpha$-element enhanced abundance pattern (e.g., @Lu:1996 [@Pettini:1999; @Prochaska:1999]).
Figure \[fig:dla\]c presents a summary of gas metallity ($Z$) relative to Solar ($Z_\odot$) measured for $>250$ DLAs at $z\lesssim 5$ (grey squares from @Rafelski:2012 [@Rafelski:2014]). The cosmic mean gas metallicity in DLAs as a function of redshift can be determined based on a $N({\mbox{H\,{\scriptsize I}}})$-weighted average over an ensemble of DLAs in each redshift bin (blue points), which is found to increase steadily with decreasing redshift following a best-fit mean relation of $\langle\,Z/Z_\odot\,\rangle=[-0.20\pm
0.03]\,z-[0.68\pm 0.09]$ (dashed blue line, @Rafelski:2014). For comparison, the figure also includes measurements for stars (dark purple box, @Gallazzi:2008) and -selected galaxies (green box, @Zwaan:2005) at $z=0$, iron abundances in the intracluster medium in X-ray luminous galaxy clusters at $z\lesssim 1$ (red triangles, @Balestra:2007), ISM of starburst galaxies (light magenta boxes) at $z\approx 2-3$ (@Pettini:2001 [@Pettini:2004; @Erb:2006]) and at $z=3-4$ (@Maiolino:2008 [@Mannucci:2009]), and IGM at $z\gtrsim 2$ (orange circles, @Aguirre:2008 [@Simcoe:2011]).
It is immediately clear from Fig. \[fig:dla\]c that there exists a large scatter in the observed metallicity in DLAs at all redshifts. In addition, while the cosmic mean metallicity in DLAs is significantly higher than what is observed in the low-density IGM, it remains lower than what is observed in the star-forming ISM at $z=2-4$ and a factor of $\approx 5$ below the mean values observed in stars at $z=0$. The chemical enrichment level in DLAs is also lower than the iron abundances seen in the intracluster medium at intermediate redshifts. The observed low metallicity relative to the measurements in and around known luminous galaxies raised the question of whether or not the DLAs probe preferentially low-metallicity, gas-rich galaxies and are not representative of more luminous, metal-rich galaxies found in large-scale surveys (e.g., @Pettini:2004).
The large scatter in the observed metallicity in DLAs is found to be explained by a combination of two factors (@Chen:2005): (i) the mass-metallicity (or luminosity-metallicity) relation in which more massive galaxies on average exhibit higher global ISM metallicities (e.g., @Tremonti:2004 [@Erb:2006; @Neeleman:2013; @Christensen:2014]) and (ii) metallicity gradients commonly seen in star-forming disks with lower metallicities at larger distances (e.g., @Zaritsky:1994 [@vanZee:1998; @Sanchez:2014; @Wuyts:2016]). If DLAs sample a representative galaxy population including both low-mass and massive galaxies and probe both inner and outer disks of these galaxies, then a large metallicity spread is expected.
The observed low metallicity in DLAs, relative to star-forming ISM, is also understood as due to a combination of DLAs being a gas cross-section selected sample and the presence of metallicity gradients in disk galaxies (@Chen:2005). A cross-section selected sample contains a higher fraction of absorbers originating in galaxy outskirts than in the inner regions, and the presence of metallicity gradients indicates that galaxy outskirts have lower metallicities than what is observed in inner disks (see Sect. \[sec:DLAgals\] and Fig. \[fig:outskirts\] below for more details). Indeed, including both factors, a gas cross-section weighting scheme and a metallicity gradient, for local galaxies resulted in a mean metallicity comparable to what is observed in DLAs (green box in Fig. \[fig:dla\]c; @Zwaan:2005).
While DLAs exhibit a moderate level of chemical enrichment, searches for molecular gas in DLAs have yielded only a few detections (e.g., @Noterdaeme:2008 [@Jorgenson:2014; @Noterdaeme:2016]). Figure \[fig:dla\]d displays the observed molecular gas fraction, which is defined as $f_{\rm H_2}\equiv 2\,N({\rm H_2})/[N({\mbox{H\,{\scriptsize I}}})+2\,N({\rm
H_2})]$, versus metallicity-scaled total hydrogen column density for $\approx 100$ DLAs at $z\approx 2-4$ (triangles). The DLAs span roughly two decades in $N({\mbox{H\,{\scriptsize I}}})$ from $N({\mbox{H\,{\scriptsize I}}})\approx 2\times
10^{20}\,{\mbox{${\rm cm^{-2}}$}}$ to $N({\mbox{H\,{\scriptsize I}}})\approx 2.5\times 10^{22}\,{\mbox{${\rm cm^{-2}}$}}$. Strong limits have been placed for $f_{\rm H_2}$ for the majority of DLAs at $f_{\rm H_2}\lesssim 10^{-5}$ with only $\approx 10$% displaying the presence of ${\rm H}_2$ and two having $f_{\rm
H_2}>0.1$. In contrast, the ISM of the Milky Way (MW), at comparable $N({\mbox{H\,{\scriptsize I}}})$, displays a much higher $f_{\rm H_2}$ than the DLAs at high redshifts.
The formation of molecules is understood to depend on two competing factors: (i) the ISM radiation field which photo-dissociates molecules and (ii) dust which facilitates molecule formation (e.g., @Elmegreen:1993 [@Cazaux:2004]). Dust is considered a more dominant factor because of its dual roles in both forming molecules and shielding them from the ISM radiation field. In star-forming galaxies, the dust-to-gas mass ratio is observed to correlate strongly with ISM gas-phase metallicity (e.g., @Leroy:2011 [@Remy:2014]). It is therefore expected that the observed molecular gas fraction should correlate with gas metallicity (e.g., @Elmegreen:1989 [@Krumholz:2009; @Gnedin:2009]).
In the MW ISM with metallicity roughly Solar, $Z\approx Z_\odot$, the molecular gas fraction is observed to increase sharply from $f_{\rm
H_2}<10^{-4}$ to $f_{\rm H_2}\gtrsim 0.1$ at $N({\mbox{H\,{\scriptsize I}}})\approx 2\times
10^{20}\,{\mbox{${\rm cm^{-2}}$}}$ (see @Wolfire:2008). The sharp transition from atomic to molecular is also observed in the ISM of the Large and Small Magellanic Clouds (LMC and SMC), but occurs at higher gas column densities of $N({\mbox{H\,{\scriptsize I}}})\approx 10^{21}\,{\mbox{${\rm cm^{-2}}$}}$ for the LMC and $N({\mbox{H\,{\scriptsize I}}})\approx 3\times 10^{21}\,{\mbox{${\rm cm^{-2}}$}}$ for the SMC (see @Tumlinson:2002). The ISM metallicities of LMC and SMC are $Z\approx 0.5\,Z_\odot$ and $Z\approx 0.15\,Z_\odot$, respectively. These observations therefore support a simple metallicity-dependent transitional gas column density illustrated in Fig. \[fig:dla\]d. Following the metallicity-scaling relation, it is clear that despite a high $N({\mbox{H\,{\scriptsize I}}})$, most DLAs do not have sufficiently high metallicity (and therefore dust content) to facilitate the formation of molecules (@Gnedin:2010 [@Gnedin:2014; @Noterdaeme:2015]). This finding also applies to $\gamma$-ray burst (GRB) host galaxies (star symbols in Fig. \[fig:dla\]d). With few exceptions (@Prochaska:2009GRB [@Kruhler:2013; @Friis:2015], the ISM in most GRB hosts displays a combination of very high $N({\mbox{H\,{\scriptsize I}}})$ and low $f_{\rm H_2}$ (e.g., @Tumlinson:2007 [@Ledoux:2009]). The observed absence of ${\rm
H}_2$ in DLAs, together with a large molecular mass density revealed in blind CO surveys (e.g., @Walter:2014 [@Decarli:2016]), shows that a complete census for the cosmic evolution of the neutral gas reservoir requires complementary surveys of molecular gas over a broad redshift range. In addition, as described in Sect. \[sec:sfr\] below, the observed low molecular gas content also has important implications for star formation properties in metal-deficient, high neutral gas surface density environments.
Probing the Neutral Gas Phase in Galaxy Outskirts {#sec:DLAgals}
=================================================
Considerable details have been learned about the physical properties and chemical enrichment in neutral atomic gas from DLA studies. To apply the knowledge of DLAs for a better understanding of distant galaxies, it is necessary to first identify DLA galaxies and compare them with the general galaxy population. Searches for DLA galaxies are challenging, because distant galaxies are faint and because the relatively small extent of high-$N({\mbox{H\,{\scriptsize I}}})$ gas around galaxies places the absorbing galaxies at small angular distances from the bright background QSOs. Based on a well-defined size-mass relation observed in local galaxies (e.g., @Broeils:1997 [@Verheijen:2001; @Swaters:2002]), the characteristic projected separation (accounting for weighting by cross section) between a DLA and an $L_*$ absorbing galaxy is $\approx 16$kpc and smaller for lower-mass galaxies. At $z=1-2$, a projected distance of 16kpc corresponds to an angular separation of $\lesssim 2''$, and greater at lower and higher redshifts.
![Neutral gas kinematics and metallicity revealed by the presence of a DLA in the outskirts of two $L_*$ galaxies (adapted from @Chen:2005). The [*top*]{} row presents a DLA found at $d=7.6$kpc from a disk galaxy at $z=0.101$, which also exhibits widespread CO emission in the disk (@Neeleman:2016CO). The [*bottom*]{} row presents a DLA at $d=38$kpc from an edge-on disk at $z=0.525$. Deep $r$-band images of the galaxies are presented in the [*left*]{} panels, which display spatially resolved disk morphologies and enable accurate measurements of the inclination and orientation of the optical disk. The [*middle*]{} panels present the optical rotation curves deprojected along the disk plane (points in shaded area) based on the inclination angle determined from the optical image of each galaxy (Eq. 3 & 4). If the DLAs occur in extended disks, the corresponding galactocentric distances of the two galaxies from Eq. (3) are $R=13.6$kpc (top) and $R=38$kpc (bottom). The DLA in the [*top*]{} panel is resolved into two components of comparable ionic column densities (@Som:2015) but an order of magnitude difference in $N({\rm
H_2})$ (@Muzahid:2015). The component with a lower $N({\rm
H_2})$ appears to be co-rotating with the optical disk (lower DLA data point), while the component with stronger $N({\rm H_2})$ appears to be counter-rotating, possibly due to a satellite (upper DLA data point). The DLA in the [*bottom*]{} panel displays simpler gas kinematics consistent with an extended rotating disk out to $\approx
40$kpc. The [*right*]{} panels present the metallicity gradient observed in the gaseous disks based on comparisons of ISM gas-phase metallicity and metallicity of the DLA beyond the optical disks. In both cases, the gas metallicity declines with increasing radius according to $\Delta\,Z/\Delta\,R=-0.02$ dexkpc$^{-1}$[]{data-label="fig:outskirts"}](outskirts2)
While fewer DLAs are known at $z\lesssim 1$ (see Sect.\[sec:DLAstats\]), a large number ($\approx 40$) of these low-redshift DLAs have their galaxy counterparts (or candidates) found based on a combination of photometric and spectroscopic techniques (e.g., @Chen:2003 [@Rao:2003; @Rao:2011; @Peroux:2016]). It has been shown based on this low-redshift DLA galaxy sample that DLAs probe a representative galaxy population in luminosity and colour. DLA galaxies are consistent with an cross-section selected sample with a large fraction of DLAs found at projected distance $d\gtrsim 10$ kpc from the absorbing galaxies (e.g., @Chen:2003 [@Rao:2011]). In addition to regular disk galaxies, two DLAs have been found in a group environment (e.g., @Bergeron:1991 [@Chen:2003; @Kacprzak:2010; @Peroux:2011]), suggesting that stripped gas from galaxy interactions could also contribute to the incidence of DLAs. The low-redshift DLA sample is expected to continue to grow dramatically with new discoveries from the SDSS (e.g., @Straka:2015). In contrast, the search for DLA galaxies at $z>2$ has been less successful despite extensive efforts (e.g., @Warren:2001 [@Moller:2002; @Peroux:2012; @Fumagalli:2015]). To date, only $\approx 12$ DLA galaxies have been found at $z>2$ (@Krogager:2012 [@Fumagalli:2015]).
In addition to a general characterization of the DLA galaxy population, individual DLA and galaxy pairs provide a unique opportunity to probe neutral gas in the outskirts of distant galaxies. Figure \[fig:outskirts\] shows two examples of constraining the kinematics and chemical enrichment in the outskirts of neutral disks from combining resolved optical imaging and spectroscopy of the galaxy with an absorption-line analysis of the DLA. In the first example (top row), a DLA of $\log\,N({\mbox{H\,{\scriptsize I}}})=19.7$ is found at $d=7.6$kpc from an $L_*$ galaxy at $z=0.101$, which also exhibits widespread CO emission in the disk (@Neeleman:2016CO). The galaxy disk is resolved in the ground-based $r$-band image (upper-left panel), which enables accurate measurements of the disk inclination and orientation (@Chen:2005). While the observed $N({\mbox{H\,{\scriptsize I}}})$ falls below the nominal threshold of a DLA, the gas is found to be largely neutral (e.g., @Chen:2005 [@Som:2015]). In addition, abundant ${\rm H_2}$ is detected in the absorbing gas (@Muzahid:2015). Optical spectra of the galaxy clearly indicate a strong velocity shear along the disk, suggesting an organized rotation motion (@Chen:2005) which is confirmed by recent CO observations (@Neeleman:2016CO). At the same time, the DLA is resolved into two components of comparable ionic column densities (@Som:2015) but an order of magnitude difference in $N({\rm H_2})$ (@Muzahid:2015). A rotation curve of the gaseous disk extending beyond 10kpc (top-centre panel) can be established based on the observed velocity shear ($v_{\rm obs}$) and deprojection onto the disk plane following $$\frac{R}{d}=\sqrt{1+\sin^2(\phi)\tan^2(i)}$$ and $$v=\frac{v_{\rm obs}}{\cos(\phi)\sin(i)}\sqrt{1+\sin^2(\phi)\tan^2(i)},$$ where $R$ is the galactocentric radius along the disk, $v$ is the deprojected rotation velocity, $i$ is the inclination angle of the disk, and $\phi$ is the azimuthal angle from the major axis of the disk where the DLA is detected (@Chen:2005, see also @Steidel:2002 for an alternative formalism). For the two absorbing components in this DLA, it is found that the component with a lower $N({\rm H_2})$ appears to be co-rotating with the optical disk (lower DLA data point), while the component with stronger $N({\rm
H_2})$ appears to be counter-rotating, possibly due to a satellite (upper DLA data point). Comparing the ISM gas-phase metallicity and the metallicity of the DLA shows a possible gas metallicity gradient of $\Delta\,Z/\Delta\,R=-0.02$ dexkpc$^{-1}$ out to $R\approx 14$ kpc.
The bottom row of Fig. \[fig:outskirts\] presents a DLA at $d=38$ kpc from an edge-on disk at $z=0.525$. A strong velocity shear is also seen along the disk of this $L_*$ galaxy. Because the QSO sightline occurs along the extended edge-on disk, Eq. (3) and (4) directly lead to $R\approx d$ and $v\approx v_{\rm obs}$ for this system. This DLA galaxy presents a second example for galaxies with an extended rotating disk out to $\approx 40$kpc. At the same time, the deep $r$-band image (lower-left panel) from [*HST*]{} suggests that the disk is warped near the QSO sightline, which is also reflected by the presence of a disturbed rotation velocity at $R>5$ kpc (bottom-centre panel). The metallicity measured in the gas phase (bottom-right panel) displays a similar gradient of $\Delta\,Z/\Delta\,R=-0.02$ dexkpc$^{-1}$ to the galaxy at the top, which is also comparable to what is seen in the ISM of nearby disk galaxies (e.g., @Zaritsky:1994 [@vanZee:1998; @Sanchez:2014]). A declining gas-phase metallicity from the inner ISM to neutral gas at larger distances appears to hold for most DLA galaxies at $z\lesssim 1$ and the declining trend continues into ionized halo gas traced by strong LLS of $N({\mbox{H\,{\scriptsize I}}})=10^{19-20}\,{\mbox{${\rm cm^{-2}}$}}$ (e.g., @Peroux:2016).
At $z>2$, spatially resolved observations of ISM gas kinematics become significantly more challenging, because the effective radii of $L_*$ galaxies are typically $r_{\rm e}=1-3$kpc (e.g., @Law:2012), corresponding to $\lesssim 0.3''$, and smaller for fainter or lower-mass objects. Star-forming regions in these distant galaxies are barely resolved in ground-based, seeing-limited observations (e.g., @Law:2007 [@Forster:2009; @Wright:2009]). Beam smearing can result in significant bias in interpreting the observed velocity shear and distributions of heavy elements (e.g., @Davies:2011 [@Wuyts:2016]). However, accurate measurements can be obtained to differentiate ISM metallicities of DLA galaxies from metallicities of neutral gas beyond the star-forming regions. Using the small sample of known DLA galaxies at $z\gtrsim 2$, a metallicity gradient of $\Delta\,Z/\Delta\,R=-0.02$ dexkpc$^{-1}$ is also found in these distant star-forming galaxies (@Christensen:2014 [@Jorgenson:2014M]).
The Star Formation Relation in the Early Universe {#sec:sfr}
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While direct identifications of galaxies giving rise to $z>2$ DLAs have proven extremely challenging, critical insights into the star formation relation in the early Universe can still be gained from comparing the incidence of DLAs with the spatial distribution of star formation rate (SFR) per unit area uncovered in deep imaging data (@Lanzetta:2002 [@Wolfe:2006]). Specifically, the SFR per unit area ($\Sigma_{\rm
SFR}$) is correlated with the surface mass density of neutral gas ($\Sigma_{\rm gas}$), following a Schmidt-Kennicutt relation in nearby galaxies (e.g., @Schmidt:1959 [@Kennicutt:1998s]). The global star formation relation, $\Sigma_{\rm SFR}=2.5\times 10^{-4}\,(\Sigma_{\rm
gas}/1\,{\mbox{$M_\odot$}}\,{\rm pc}^{-2})^{1.4}\,{\mbox{$M_\odot$}}\,{\rm yr}^{-1}\,{\rm
kpc}^{-2}$ (dashed line in Fig. \[fig:sfrarea\]), is established using a sample of local spiral galaxies and nuclear starbursts (solid grey points in Fig. \[fig:sfrarea\]) over a broad range of $\Sigma_{\rm gas}$, from $\Sigma_{\rm gas}\approx 10\,{\mbox{$M_\odot$}}\,{\rm
pc}^{-2}$ to $\Sigma_{\rm gas}\approx 10^4\,{\mbox{$M_\odot$}}\,{\rm pc}^{-2}$.
Empirical constraints for a Schmidt-Kennicutt relation at high redshifts require observations of the neutral gas content in star-forming galaxies. Although observations of individual galaxies in emission remain out of reach, the sample of $z=1-3$ galaxies with resolved CO maps is rapidly growing (e.g., @Baker:2004 [@Genzel:2010; @Tacconi:2013]). The observed $\Sigma_{\rm SFR}$ versus $\Sigma_{\rm molecular}$ for the high-redshift CO detected sample is shown in open squares in Fig. \[fig:sfrarea\], which occur at high surface densities of $\Sigma_{\rm molecular}\gtrsim 100\,{\mbox{$M_\odot$}}\,{\rm
pc}^{-2}$. Considering only $\Sigma_{\rm molecular}$ is appropriate for these galaxies, because locally it has been shown that at this high surface density regime molecular gas dominates (e.g., @Martin:2001 [@Wong:2002; @Bigiel:2008]). In contrast, DLAs probe neutral gas with $N({\mbox{H\,{\scriptsize I}}})$ ranging from $N({\mbox{H\,{\scriptsize I}}})=2\times 10^{20}\,{\mbox{${\rm cm^{-2}}$}}$ to $N({\mbox{H\,{\scriptsize I}}})\approx 5\times 10^{22}\,{\mbox{${\rm cm^{-2}}$}}$. The range in $N({\mbox{H\,{\scriptsize I}}})$ corresponds to a range in surface mass density of atomic gas from $\Sigma_{\rm atomic}\approx 2\,{\mbox{$M_\odot$}}\,{\rm pc}^{-2}$ to $\Sigma_{\rm
atomic}\gtrsim 200\,{\mbox{$M_\odot$}}\,{\rm pc}^{-2}$, which is comparable to the global average of total neutral gas surface mass density in local disk galaxies (e.g., Fig. \[fig:sfrarea\]). Therefore, DLAs offer an important laboratory for investigating the star formation relation in the distant Universe, and direct constraints can be obtained from searches of [*in situ*]{} star formation in DLAs.
![The global star formation relation observed in nearby galaxies and at high redshifts. The correlation between the SFR per unit area ($\Sigma_{\rm SFR}$) and the total surface gas mass density ($\Sigma_{\rm gas}$), combining both atomic () and molecular (H$_2$) for nearby spiral and starburst galaxies are shown in small filled circles (@Kennicutt:1998s [@Gracia:2008; @Leroy:2008]), together with the best-fit Schmidt-Kennicutt relation shown by the dashed line (@Kennicutt:1998s). A reduced star formation efficiency is observed both in low surface brightness galaxies and in the outskirts of normal spirals, which are shown in grey star symbols and open triangles, respectively (@Wyder:2009 [@Bigiel:2010]). CO molecules have been detected in many massive starburst galaxies ($M_{\rm star}>2.5\times
10^{10}\,{\mbox{$M_\odot$}}$) at $z=1-3$ (e.g., @Baker:2004 [@Genzel:2010; @Tacconi:2013]), which occur at the high surface density regime of the global star formation relation (open squares). In contrast, searching for [*in situ*]{} star formation in DLAs has revealed a reduced star formation efficiency in this metal-deficient gas. Specifically, green points and orange shaded area represent the constraints obtained from comparing the sky coverage of low surface brightness emission with the incidence of DLAs (@Wolfe:2006 [@Rafelski:2011; @Rafelski:2016]). Cyan squares and red circles represent the limits inferred from imaging searches of galaxies associated with individual DLAs, and the cyan and red bars represent the limiting $\Sigma_{\rm SFR}$ based on ensemble averages of the two samples (@Fumagalli:2015). The level of star formation observed in high-$N({\mbox{H\,{\scriptsize I}}})$ DLAs (green pentagons and orange shaded area) is comparable to what is seen in nearby low surface brightness galaxies and in the outskirts of normal spirals. See the main text for a detailed discussion[]{data-label="fig:sfrarea"}](sfrarea)
In principle, the Schmidt-Kennicutt relation can be rewritten in terms of $N({\mbox{H\,{\scriptsize I}}})$ for pure atomic gas following $$\Sigma_{\rm SFR}=K\times[N({\mbox{H\,{\scriptsize I}}})/N_0]^\beta\ \ \ {\mbox{$M_\odot$}}\,{\rm yr}^{-1}\,{\rm kpc}^{-2},$$ which is justified for regions probed by DLAs with a low molecular gas content (see Sect. \[sec:DLAstats\] and Fig. \[fig:dla\]d). For reference, the local Schmidt-Kennicutt relation has $K=2.5\times
10^{-4}\,{\mbox{$M_\odot$}}\,{\rm yr}^{-1}\,{\rm kpc}^{-2}$, $\beta=1.4$, and $N_0=1.25\times 10^{20}\,{\mbox{${\rm cm^{-2}}$}}$ for a pure atomic hydrogen gas. Following Eq. (5), the $N({\mbox{H\,{\scriptsize I}}})$ distribution function, $f_{N({\mbox{H\,{\scriptsize I}}})}$ (e.g., Fig. \[fig:dla\]a), can then be expressed in terms of the $\Sigma_{\rm SFR}$ distribution function, $h(\Sigma_{\rm
SFR})$, which is the projected proper area per $d\Sigma_{\rm SFR}$ interval per comoving volume (@Lanzetta:2002). The $\Sigma_{\rm SFR}$ distribution function $h(\Sigma_{\rm SFR})$ is related to $f_{N({\mbox{H\,{\scriptsize I}}})}$ according to $h(\Sigma_{\rm SFR})\,d\Sigma_{\rm SFR}=(H_0/c)\,f_{N({\mbox{H\,{\scriptsize I}}})}\,dN({\mbox{H\,{\scriptsize I}}})$.
This exercise immediately leads to two important observable quantities. First, the sky covering fraction ($C_{\rm A}$) of star-forming regions in the redshift range, $[z_1,z_2]$, with an observed SFR per unit area in the interval of $\Sigma_{\rm SFR}$ and $\Sigma_{\rm SFR}+d\Sigma_{\rm
SFR}$ is determined following $$C_{\rm A}[\Sigma_{\rm SFR}|N({\mbox{H\,{\scriptsize I}}})] = \displaystyle\int_{z_1}^{z_2} \frac{c\,(1+z)^2}{H(z)} h(\Sigma_{\rm SFR})\,d\Sigma_{\rm SFR}\,dz,$$ where $c$ is the speed of light and $H(z)$ is the Hubble expansion rate. Equation (6) is equivalent to $f_{N({\mbox{H\,{\scriptsize I}}})}dN({\mbox{H\,{\scriptsize I}}})dX$, where $dX\equiv (1+z)^2\,H_0/H(z)\,dz$ is the comoving absorption pathlength. In addition, the first moment of $h(\Sigma_{\rm SFR})$ leads to the comoving SFR density (@Lanzetta:2002 [@Hopkins:2005]), $$\dot{\rho}_*(>\Sigma_{\rm SFR}^{\rm min})=\int_{\Sigma_{\rm SFR}^{\rm min}}^{\Sigma_{\rm SFR}^{\rm max}}\Sigma_{\rm SFR}h(\Sigma_{\rm SFR})\,d\Sigma_{\rm SFR}.$$ Constraints on the star formation relation at high redshift, namely $K$ and $\beta$ in Eq. (5), can then be obtained by comparing $f_{N({\mbox{H\,{\scriptsize I}}})}$-inferred $C_{\rm A}$ and $\dot{\rho}_*$ with results from searches of low surface brightness emission in deep galaxy survey data. Furthermore, estimates of missing light in low surface brightness regions can also be obtained using Eq. (7) (e.g., @Lanzetta:2002 [@Rafelski:2011]).
In practice, Eq. (5) is a correct representation only if disks are not well formed and a spherical symmetry applies to the DLAs. For randomly oriented disks, corrections for projection effects are necessary and detailed formalisms are presented in [@Wolfe:2006] and [@Rafelski:2011]. In addition, the inferred surface brightness of [*in situ*]{} star formation in the DLA gas is extremely low after accounting for the cosmological surface brightness dimming. At $z=2-3$, only DLAs at the highest-$N({\mbox{H\,{\scriptsize I}}})$ end of $f_{N({\mbox{H\,{\scriptsize I}}})}$ are expected to be visible in ultra-deep imaging data (cf. @Lanzetta:2002 [@Wolfe:2006]). For example, DLAs of $N({\mbox{H\,{\scriptsize I}}})>1.6\times 10^{21}\,{\mbox{${\rm cm^{-2}}$}}$ at $z\approx 3$ are expected to have $V$-band (corresponding roughly to rest-frame 1500Å at $z=3$) surface brightness $\mu_V\lesssim 28.4$ mag arcsec$^{-2}$, assuming the local Schmidt-Kennicutt relation. The expected low surface brightness of UV photons from young stars in high-redshift DLAs dictates the galaxy survey depth necessary to uncover star formation associated with the DLA gas. At $N({\mbox{H\,{\scriptsize I}}})>1.6\times 10^{21}\,{\mbox{${\rm cm^{-2}}$}}$, roughly 3% of the sky ($C_{\rm A}\approx 0.03$) is expected to be covered by extended low surface brightness emission of $\mu_V\lesssim 28.4$ mag arcsec$^{-2}$. For comparison, the sky covering fraction of luminous starburst galaxies at $z=2-3$ is less than 0.1%.
Available constraints for the star formation efficiency at $z=1-3$ are shown in colour symbols in Fig. \[fig:sfrarea\]. Specifically, the Hubble Ultra Deep Field (HUDF; @Beckwith:2006) $V$-band image offers sufficient depth for detecting objects of $\mu_V\approx 28.4$ mag arcsec$^{-2}$. Under the assumption that DLAs originate in regions distinct from known star-forming galaxies, an exhaustive search for extended low surface brightness emission in the HUDF has uncovered only a small number of these faint objects, far below the expectation from applying the local Schmidt-Kennicutt relation for DLAs of $N({\mbox{H\,{\scriptsize I}}})>1.6\times
10^{21}\,{\mbox{${\rm cm^{-2}}$}}$ following Eq. (6). Consequently, matching the observed limit on $\dot{\rho_*}$ from these faint objects with expectations from Eq. (7) has led to the conclusion that the star formation efficiency in metal-deficient atomic gas is more than $10\times$ lower than expectations from the local Schmidt-Kennicutt relation (@Wolfe:2006; green pentagons in Fig. \[fig:sfrarea\]).
On the other hand, independent observations of DLA galaxies at $z=2-3$ have suggested that these absorbers are associated with typical star-forming galaxies at high redshifts. These include a comparable clustering amplitude of DLAs and these galaxies (e.g., @Cooke:2006), the findings of a few DLA galaxies with mass and SFR comparable to luminous star-forming galaxies found in deep surveys (e.g., @Moller:2002 [@Moller:2004; @Christensen:2007]), and detections of a DLA feature in the ISM of star-forming galaxies (e.g., @Pettini:2002 [@Chen:2009GRB; @Dessauges:2010]). If DLAs originate in neutral gas around known star-forming galaxies, then these luminous star-forming galaxies should be more spatially extended than has been realized. Searches for low surface brightness emission in the outskirts of these galaxies based on stacked images have indeed uncovered extended low surface brightness emission out to more than twice the optical extent of a single image. However, repeating the exercise of computing the cumulative $\dot{\rho}_*$ from Eq. (7) has led to a similar conclusion that the star formation efficiency is more than $10\times$ lower in metal-deficient atomic gas at $z=1-3$ than expectations from the local Schmidt-Kennicutt relation (@Rafelski:2011 [@Rafelski:2016]). The results are shown as the orange shaded area in Fig. \[fig:sfrarea\]). In addition, the amount of missing light in the outskirts of these luminous star-forming galaxies is found to be $\approx 10$% of what is observed in the core (@Rafelski:2011).
At the same time, imaging searches of individual DLA galaxies have been conducted for $\approx 30$ DLAs identified along QSO sightlines that have high-redshift LLS serving as a natural coronograph to block the background QSO glare, improving the imaging depth in areas immediate to the QSO sightline (@Fumagalli:2015). These searches have yielded only null results, leading to upper limits on the underlying surface brightness of the DLA galaxies (cyan squares and red circles in Fig. \[fig:sfrarea\]). While the survey depth is not sufficient for detecting associated star-forming regions in most DLAs in the survey sample of [@Fumagalli:2015] based on the local Schmidt-Kennicutt relation, the ensemble average is beginning to place interesting limits (cyan and red arrows).
The lack of [*in situ*]{} star formation in DLAs may not be surprising given the low molecular gas content. In the local Universe, it is understood that the Schmidt-Kennicutt relation is driven primarily by molecular gas mass ($\Sigma_{\rm molecular}$), while the surface density of atomic gas ($\Sigma_{\rm atomic}$) “saturates” at $\sim
10\,{\mbox{$M_\odot$}}\,{\rm pc}^{-2}$ beyond which the gas transitions into the molecular phase (e.g., @Martin:2001 [@Wong:2002; @Bigiel:2008]). As described in Sect. \[sec:DLAstats\] and Fig. \[fig:dla\]d, the transitional surface density from atomic to molecular is metallicity dependent. Therefore, the low star formation efficiency observed in DLA gas can be understood as a metallicity-dependent Schmidt-Kennicutt relation. This is qualitatively consistent with the observed low $\Sigma_{\rm SFR}$ in nearby low surface brightness galaxies (e.g., @Wyder:2009; star symbols in Fig. \[fig:sfrarea\]) and in the outskirts of normal spirals (e.g., @Bigiel:2010; open triangles in Fig. \[fig:sfrarea\]), where the ISM is found to be metal-poor (e.g., @McGaugh:1994 [@Zaritsky:1994; @Bresolin:2012]). Numerical simulations incorporating a metallicity dependence in the H$_2$ production rate have also confirmed that the observed low star formation efficiency in DLAs can be reproduced in metal-poor gas (e.g., @Gnedin:2010).
A metallicity-dependent Schmidt-Kennicutt relation has wide-ranging implications in extragalactic research, from the physical origin of DLAs at high redshifts, to star formation and chemical enrichment histories in different environments, and to detailed properties of distant galaxies such as morphologies, sizes, and cold gas content. It is clear from Fig. \[fig:sfrarea\] that there exists a significant gap in the gas surface densities, between $\Sigma_{\rm
gas}\approx 10\, {\mbox{$M_\odot$}}\,{\rm pc}^{-2}$ probed by these direct DLA galaxy searches and $\Sigma_{\rm gas}\approx 100\, {\mbox{$M_\odot$}}\,{\rm
pc}^{-2}$ probed by CO observations of high-redshift starburst systems (open squares in Fig. \[fig:sfrarea\]). Continuing efforts targeting high-$N({\mbox{H\,{\scriptsize I}}})$ DLAs (and therefore high $\Sigma_{\rm
gas}$) at sufficient imaging depths are expected to place critical constraints on the star formation relation in low-metallicity environments at high redshifts. Similarly, spatially resolved maps of star formation and neutral gas at $z>1$ to mean surface densities of $\Sigma_{\rm SFR}< 0.1\,{\mbox{$M_\odot$}}\,{\rm yr}^{-1}\,{\rm kpc}^{-2}$ and $\Sigma_{\rm atomic, molecular}\approx 10-100\,{\mbox{$M_\odot$}}\,{\rm pc}^{-2}$ will bridge the gap of existing observations and offer invaluable insights into the star formation relation in different environments.
From Neutral ISM to the Ionized Circumgalactic Medium {#sec:cgm}
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Beyond the neutral ISM, strong absorbers of $N({\mbox{H\,{\scriptsize I}}})\approx
10^{14-20}\,{\mbox{${\rm cm^{-2}}$}}$ and associated metal-line absorbers offer a sensitive probe of the diffuse circumgalactic medium (CGM) to projected distances $d\approx
100-500$kpc (e.g., Fig. \[fig:halomap\]). But because the circumgalactic gas is significantly more ionized in the LLS and lower-$N({\mbox{H\,{\scriptsize I}}})$ regime, measurements of its ionization state and metallicity bear considerable uncertainties and should be interpreted with caution.
Several studies have attempted to constrain the ionization state and metallicity of the CGM by considering the relative abundances of different ions at low- and high-ionization states (e.g., @Savage:2002 [@Stocke:2006]). For example, attributing observed absorbers to cool ($T\sim 10^4$K), photo-ionized gas irradiated by the metagalactic ionizing radiation field, the observed column density ratios between and low-ionization transitions (such as and ) require extremely low gas densities of $n_{\rm H}\sim 10^{-5}\,{\rm cm}^{-3}$. Combining the inferred low gas density with observed $N({{\mbox{O\,{\scriptsize VI}}}})$, which are typically $\gtrsim 10^{14.5}\,{\mbox{${\rm cm^{-2}}$}}$ in galactic haloes (e.g., @Tumlinson:2011), leads to a moderate gas metallicity of $\gtrsim
1/10$ Solar and unphysically large cloud sizes of $l_c\sim 1$ Mpc (e.g., @Tripp:2001 [@Savage:2002; @Stocke:2006])[^2]. Excluding due to possible origins in shocks or turbulent mixing layers (e.g., @Heckman:2002OVI) and considering only relative abundances of low-ionization species increases estimated gas densities to $n_{\rm
H}\sim 10^{-4}-10^{-3}\,{\rm cm}^{-3}$. The inferred cloud sizes remain large with $l_c\sim 10-100$kpc, in tension with what is observed locally for the HVCs. The implied thermal pressures in the cool gas phase are still two orders of magnitude lower than what is expected from pressure equilibrium with a hot ($T\approx 10^6$K) medium (e.g., @Stocke:2013 [@Werk:2014]), indicating that these clouds would be crushed quickly. Considering non-equilibrium conditions (e.g., @Gnat:2007 [@Oppenheimer:2013]) and the presence of local ionizing sources may help alleviate these problems (e.g., @Cantalupo:2010), but the systematic uncertainties are difficult to quantify.
Nevertheless, exquisite details concerning extended halo gas have been learned over the past decade based on various samples of close galaxy and background QSO pairs. Because luminous QSOs are rare, roughly one QSO of $g\lesssim 18$ mag per square degree (e.g., @Richards:2006), absorption-line studies of the CGM against background QSO light have been largely limited to one probe per galaxy. Only in a few cases are multiple QSOs found at $d\lesssim 300$kpc from a foreground galaxy (e.g., @Norman:1996 [@Keeney:2013; @Davis:2015; @Lehner:2015; @Bowen:2016]) for measuring coherence in spatial distribution and kinematics of extended gas around the galaxy. All of these cases are in the local Universe, because the relatively large angular extent of these galaxies on the sky increases the probability of finding more than one background QSO. This local sample has now been complemented with new studies, utilizing multiply lensed QSOs and close projected QSO pairs, which provide spatially resolved CGM absorption properties for a growing sample of galaxies at intermediate redshifts (e.g., @Chen:2014 [@Rubin:2015; @Zahedy:2016]).
![Observed absorption properties of halo gas around galaxies. The [*top*]{} panels display the radial profiles of rest-frame absorption equivalent width ($W_{\rm r}$) versus halo-radius $R_{\rm h}$-normalized projected distance for different absorption transitions. Low-ionization transitions are presented in panel ([*a*]{}) and high-ionization transitions in panel ([*b*]{}). data points are presented in both panels for cross-comparison. The galaxy sample includes 44 galaxies at $z\approx 0.25$ from the COS-Halos project (open squares; @Tumlinson:2011 [@Tumlinson:2013; @Werk:2013]) and $\sim 200$ galaxies at $z\approx
0.04$ from public archives (circles; @Liang:2014), for which high-quality, ultraviolet QSO spectra are available for constraining the presence or absence of multiple ions in individual haloes. Different transitions are colour-coded to highlight the differences in their spatial distributions. For transitions that are not detected, a 2-$\sigma$ upper limit is shown by a downward arrow. No heavy ions are found beyond $d=R_{\rm h}$, while continues to be seen to larger distances. Panel ([*c*]{}) displays the ensemble average of gas covering fraction ($\langle\kappa\rangle$) as a function of absolute $r$-band magnitude ($M_{\rm r}$) for (black symbols), (orange), and (purple). Star-forming galaxies (triangles) on average are fainter and exhibit higher covering fractions of hydrogen and chemically enriched gas probed by both low- and high-ionization species than passive galaxies (circles). Measurements of - and -absorbing gas are based on COS-Halos galaxies for $R_{\rm gas}=R_{\rm h}$. Measurements of -absorbing gas are based on $\approx 260$ star-forming galaxies at $z\approx 0.25$ (@Chen:2010), and $\sim 38000$ passive luminious red galaxies at $z\approx 0.5$ (@Huang:2016) for $R_{\rm gas}=R_{\rm h}/3$. Panel ([*d*]{}) illustrates the apparent constant nature of mass-normalized radial profiles of CGM absorption since $z\approx 3$ (e.g., @Chen:2012 [@Liang:2014]). The high-redshift observations are based on mean absorption in stacked spectra of $\sim 500$ starburst galaxies with a mean stellar mass and dispersion of $\langle\,\log\,M_{\rm star}\,\rangle=9.9\pm
0.5$ (@Steidel:2010), and the low-redshift observations are for $\sim 200$ individual galaxies with $\langle\,\log\,M_{\rm
star}\,\rangle=9.7\pm 1.1$ and modest SFR (@Liang:2014) []{data-label="fig:cgm"}](cgm_summary)
With one QSO probe per halo, a two-dimensional map of CGM absorption properties can be established based on an ensemble average of a large sample of QSO-galaxy pairs ($N_{\rm pair}\sim 100-1000$). Fig.\[fig:cgm\] summarizes some of the observable quantities of the CGM. First, panels (a) and (b) at the top display the radial profiles of rest-frame absorption equivalent width ($W_{\rm r}$) for different absorption transitions, including hydrogen , low-ionization and , intermediate-ionization , , and , and high-ionization absorption transitions, colour-coded in black, red, orange, green, blue, magenta, and dark purple, respectively. For transitions that are not detected, a 2-$\sigma$ upper limit is shown as a downward arrow. Because of the large number of data points, the upper limits are shown in pale colours for clarity. The galaxy sample includes 44 galaxies at $z\approx 0.25$ from the COS-Halos project (open squares; @Tumlinson:2011 [@Tumlinson:2013; @Werk:2013]) and $\sim 200$ galaxies at $z\approx 0.04$ from public archives (circles; @Liang:2014), for which high-quality, ultraviolet QSO spectra are available for constraining the presence or absence of multiple ions in individual haloes. These galaxies span four decades in total stellar mass, from $M_{\rm star}\approx 10^7\,{\mbox{$M_\odot$}}$ to $M_{\rm
star}\approx 10^{11}\,{\mbox{$M_\odot$}}$, and a wide range in SFR, from ${\rm SFR}<0.1\,{\mbox{$M_\odot$}}\,{\rm yr}^{-1}$ to ${\rm
SFR}>10\,{\mbox{$M_\odot$}}\,{\rm yr}^{-1}$. Diffuse gas is observed beyond $d=50$kpc around distant galaxies, extending the detection limit of gas in inner galactic haloes from 21cm observations (e.g., Fig. \[fig:halomap\]) to lower column density gas at larger distances and higher redshifts.
While $W_{\rm r}$ is typically found to decline steadily with increasing $d$ for all transitions (e.g., @Chen:2012 [@Werk:2014]), the scatters are large. Including the possibility that more massive haloes have more spatially extended halo gas, the halo radius $R_{\rm h}$-normalized $W_{\rm r}$-$d$ distribution indeed displays substantially reduced scatters in the radial profiles shown in panels (a) and (b) of Fig. \[fig:cgm\]. A reduced scatter in the $R_{\rm h}$-normalized $W_{\rm r}$-$d$ distribution indicates that galaxy mass plays a dominant role in driving the extent of halo gas. In addition, it also confirms that accurate associations between absorbers and absorbing galaxies have been found for the majority of the systems.
A particularly interesting feature in Fig. \[fig:cgm\] is a complete absence of heavy ions beyond $d=R_{\rm h}$, while detections of continue to larger distances. The absence of heavy ions at $d>R_{\rm h}$, which is observed for a wide range of ionization states, strongly indicates that chemical enrichment is confined within individual galaxy haloes. This finding applies to both low-mass dwarfs and massive galaxies. However, it should also be noted that heavy ions are observed beyond $R_{\rm h}$ for galaxies with close neighbours (e.g., @Borthakur:2013 [@Johnson:2015]), suggesting that environmental effects play a role in distributing heavy elements beyond the enriched gaseous haloes of individual galaxies. Comparing panels (a) and (b) of Fig. \[fig:cgm\] also shows that within individual galaxy haloes, a global ionization gradient is seen with more highly ionized gas detected at larger distances. For instance, the observed $W_{\rm r}$ declines to $<0.1$ Å at $d\approx 0.5\,R_{\rm h}$ for and , while and absorbers of $W_{\rm r}>0.1$Å continue to be found beyond $0.5\,R_{\rm h}$.
The observed $W_{\rm r}$ versus $d$ (or $d/R_{\rm h}$) based on a blind survey of absorption features in the vicinities of known galaxies also enables measurements of gas covering fraction[^3]. The mean gas covering fraction ($\langle\kappa\rangle$) can be measured by a simple accounting of the fraction of galaxies in an annular area displaying associated absorbers with $W_{\rm r}$ exceeding some detection threshold $W_0$, and uncertainties can be estimated based on a binomial distribution function. Dividing the sample into different projected distance bins, it is clear from Fig. \[fig:cgm\]a and b that the fraction of non-detections increases with increasing projected distance, resulting in a declining $\langle\kappa\rangle$ with increasing $d$ for all transitions observed (see also @Chen:2010 [@Werk:2014; @Huang:2016]).
It is also interesting to examine how $\langle\kappa\rangle$ depends on galaxy properties. Figure \[fig:cgm\]c displays $\langle\kappa\rangle$ observed within a fiducial gaseous radius $R_{\rm gas}$ for star-forming (triangles) and passive (circles) galaxies. The measurements are made for (black symbols), (orange), and (purple) with a threshold of $W_0=0.1$Å, and shown in relation to the absolute $r$-band magnitude ($M_{\rm r}$). Error bars represent the 68% confidence interval. The absolute $r$-band magnitude is a direct observable of a galaxy and serves as a proxy for its underlying total stellar mass. Measurements of - and -absorbing gas are based on COS-Halos galaxies for $R_{\rm
gas}=R_{\rm h}$ (see also @Johnson:2015 for a sample compiled from the literature). Measurements of -absorbing gas are based on $\approx 260$ star-forming galaxies at $z\approx 0.25$ (@Chen:2010, and $\sim 38000$ passive luminous red galaxies at $z\approx 0.5$ (@Huang:2016) for $R_{\rm gas}=R_{\rm h}/3$ (e.g., @Chen:2008). The larger sample sizes led to better constrained $\langle\kappa\rangle$ for absorbing gas in galactic haloes. In general, star-forming galaxies on average are fainter, less massive, and exhibit a higher covering fraction of chemically enriched gas than passive galaxies (see also @Johnson:2015). At the same time, the covering fraction of chemically enriched gas is definitely non-zero around massive quiescent galaxies.
Comparing the radial profiles of CGM absorption at different redshifts offers additional insights into the evolution history of the CGM, which in turn helps distinguish between different models for chemical enrichment in galaxy haloes. The radial profiles of the CGM have been found to evolve little since $z\sim 3$ (e.g., @Chen:2012), even though the star-forming properties in galaxies have evolved significantly. Figure \[fig:cgm\]d illustrates the apparent constant nature of mass-normalized radial profiles of absorption in galactic haloes (@Liang:2014). The high-redshift observations are based on stacked spectra of $\sim 500$ starburst galaxies with a mean stellar mass and dispersion of $\langle\,\log\,M_{\rm
star}\,\rangle=9.9\pm 0.5$ (@Steidel:2010) and a mean SFR of $\langle\,{\rm SFR}\,\rangle\approx 30-60\,{\mbox{$M_\odot$}}\,{\rm yr}^{-1}$ (e.g., @Erb:2006SFR [@Reddy:2012]). The low-redshift galaxy sample contains individual measurements of $\sim 200$ galaxies with $\langle\,\log\,M_{\rm star}\,\rangle=9.7\pm 1.1$ and more quiescent star-forming activities of $\langle\,{\rm SFR}\,\rangle\sim
1\,{\mbox{$M_\odot$}}\,{\rm yr}^{-1}$ (@Chen:2012 [@Liang:2014]). The constant mass-normalized CGM radial profiles between galaxies of very different SFR indicate that mass (rather than SFR) is a dominant factor that determines the CGM properties over a cosmic time interval. This is consistent with previous findings that CGM absorption properties depend strongly on galaxy mass but only weakly on SFR (e.g., @Chen:2010M), but at odds with popular models that attribute metal-line absorbers to starburst-driven outflows (e.g., @Steidel:2010 [@Menard:2011]).
![Visual comparisons of the geometric alignment of galaxy major axis relative to the QSO sightline and the observed CGM absorption strength (by Rebecca Pierce). [*Top*]{}: Observed column density, $N({{\mbox{O\,{\scriptsize VI}}}})$, versus $d$ for COS-Halos star-forming (in blue) and passive (in red) galaxies (@Tumlinson:2011). [*Bottom*]{}: Comparisons of $N({{\mbox{O\,{\scriptsize VI}}}})$ and $N({{\mbox{Mg\,{\scriptsize II}}}})$ for the COS-Halos galaxies from [@Werk:2013]. When spatially resolved images are available, the data points are replaced with an image panel of the absorbing galaxy. Each panel is 25 proper kpc on a side, and is oriented such that the QSO sightline occurs on the y-axis at the corresponding column density of the galaxy. Disk alignments cannot be determined for face-on galaxies (minor-to-major axis ratio $>0.7$) and galaxies displaying irregular/asymmetric morphologies, which are labeled “F” and “A”, respectively. Galaxies with the QSO located within $30^\circ$ of the minor axis are labeled ’m’ in the lower-left corner, while galaxies with the QSO located within $30^\circ$ of the major axis are labeled ’M’. Galaxies with the QSO sightline occuring intermediate ($30^\circ-60^\circ$) between the minor and major axis are labeled “45”. Downward arrows indicate 2-$\sigma$ upper limits for non-detections, while upward arrows indicate saturated absorption lines. The COS-Halos galaxy sample provides a unique opportunity to examine low- and high-ionization halo gas for the same galaxies at once. Galaxies surrounded by and absorbing gas clearly exhibit a broad range both in morphology and in disk orientation. In addition, the observed $N({{\mbox{Mg\,{\scriptsize II}}}})$ displays a significantly larger scatter than $N({{\mbox{O\,{\scriptsize VI}}}})$[]{data-label="fig:align"}](coshalos)
A discriminating characteristic of starburst-driven outflows is their distinctly non-spherical distribution in galactic haloes in the presence of a well-formed star-forming disk. Specifically, galactic-scale outflows are expected to travel preferentially along the polar axis where the gas experiences the least resistance (e.g., @Heckman:1990). In contrast, accretion of the IGM is expected to proceed along the disk plane with $\lesssim 10$% covering fraction on the sky (e.g., @Faucher:2011 [@Fumagalli:2011]). Such azimuthal dependence of the spatial distribution of infalling and outflowing gas is fully realized in state-of-the-art cosmological zoom-in simulations (e.g., @Shen:2013 [@Agertz:2015]). Observations of $z\approx 0.7$ galaxies have shown that at $d<50$kpc the mean absorption equivalent width within $45^\circ$ of the minor axis is twice of the mean value found within $45^\circ$ of the major axis, although such azimuthal dependence is not observed at $d>50$kpc (@Bordoloi:2011). The observed azimuthal dependence of the mean absorption strength is qualitatively consistent with the expectation that these heavy ions originate in starburst-driven outflows, and the lack of such azimuthal dependence implies that starburst outflows are confined to the inner halo of $d\lesssim 50$kpc.
Many subsequent studies have generalized this observed azimuthal dependence at $d<50$kpc to larger distances and attributed absorbers detected near the minor axis to starburst-driven outflows and those found near the major axis to accretion (e.g., @Bouche:2012 [@Kacprzak:2015]). However, a causal connection between the observed absorbing gas and either outflows or accretion remains to be established. While gas metallicity may serve as a discriminator with the expectation of starburst outflows being more metal-enriched relative to the low-density IGM, uncertainties arise due to poorly understood chemical mixing and metal transport (e.g., @Tumlinson:2006). Incidentally, a relatively strong absorber has been found at $d\approx 60$kpc along the minor axis of a starburst galaxy but the metallicity of the absorbing gas is 10 times lower than what is observed in the ISM (@Kacprzak:2014), highlighting the caveat of applying gas metallicity as the sole parameter for distinguishing between accretion and outflows.
Figure \[fig:align\] presents visual comparisons of the geometric alignment of galaxy major axis relative to the QSO sightline and the observed CGM absorption strength. The figure at the top displays the observed column density, $N({{\mbox{O\,{\scriptsize VI}}}})$, versus $d$ for COS-Halos galaxies at $z\approx 0.2$ (@Tumlinson:2013). The bottom figure displays comparisons of $N({{\mbox{O\,{\scriptsize VI}}}})$ and $N({{\mbox{Mg\,{\scriptsize II}}}})$ for these galaxies. The absorption-line measurements are adopted from [@Werk:2013]. When spatially resolved images are available, the data points are replaced with an image panel of the absorbing galaxy. Each panel is 25 proper kpc on a side, and is oriented such that the QSO sightline falls on the y-axis at the corresponding $N({{\mbox{O\,{\scriptsize VI}}}})$ of the galaxy. The relative alignment between galaxy major axis and the background QSO sightline cannot be determined, if the galaxies are face-on with a minor-to-major axis ratio $>0.7$ or if the galaxies display irregular/asymmetric morphologies. These galaxies are labeled “F” and “A”, respectively. For galaxies that clearly display a smooth and elongated morphology, the orietation of the major axis can be accurately measured. Galaxies with the QSO located within $30^\circ$ of the minor axis are labeled ’m’, while galaxies with the QSO located within $30^\circ$ of the major axis are labeled ’M’. Galaxies with the QSO sightline occuring intermediate ($30^\circ-60^\circ$) between the minor and major axis are labeled “45”. Star-forming galaxies are colour-coded in blue, and passive galaxies in red. Downward arrows indicate 2-$\sigma$ upper limits for non-detections, while upward arrows indicate saturated absorption lines.
While the COS-Halos sample is small, particularly when restricting to those galaxies displaying a smooth, elongated morphology, it provides a unique opportunity to examine low- and high-ionization halo gas for the same galaxies at once. Two interesting features are immediately clear in Fig. \[fig:align\]. First, galaxies surrounded by and absorbing gas exhibit a broad range both in morphology and in star formation history, from compact quiescent galaxies, to regular star-forming disks, and to interacting pairs. The diverse morphologies in and absorbing galaxies illuminate the challenge and uncertainties in characterizing their relative geometric orientation to the QSO sightline based on azimuthal angle alone. When considering only galaxies with smooth and elongated (minor-to-major axis ratio $<0.7$) morphologies, no clear dependence of $N({{\mbox{O\,{\scriptsize VI}}}})$ or $N({{\mbox{Mg\,{\scriptsize II}}}})$ on galaxy orientation is found. Specifically, nine star-forming galaxies displaying strong absorption at $d<80$kpc ($\log\,N({{\mbox{O\,{\scriptsize VI}}}})$) have spatially resolved images available. Two of these galaxies display disturbed morphologies and four are nearly face-on. The remaining three galaxies have the inclined disks oriented at $0^\circ$, $45^\circ$, and $90^\circ$ each. For passive red galaxies, two have spatially resolved images available and both are elongated and aligned at $\approx 45^\circ$ from the QSO sightline. One displays an associated strong absorber and the other has no corresponding detections. At $d>80$kpc, the morphology distribution is similar to those at smaller distances. No strong dependence is found between the presence or absence of a strong absorber and the galaxy orientation. In addition, while the observed $N({{\mbox{O\,{\scriptsize VI}}}})$ at $d<100\,kpc$ appears to be more uniformly distributed with a mean and scatter of $\log\,N({{\mbox{O\,{\scriptsize VI}}}})=14.5\pm 0.3$ (@Tumlinson:2011), the observed $N({{\mbox{Mg\,{\scriptsize II}}}})$ displays a significantly larger scatter. Specifically, the face-on galaxy at $d\approx 32$kpc with an associated absorber of $\log\,N({{\mbox{O\,{\scriptsize VI}}}})\approx 14.7$ does not have an associated absorber detected to a limit of $\log\,N({{\mbox{Mg\,{\scriptsize II}}}})\approx 12.4$. Two quiescent galaxies at $z\approx 20$ and 90kpc (red panels) exhibit saturated absorption of $\log\,N({{\mbox{Mg\,{\scriptsize II}}}})>13.5$ and similarly strong of $\log\,N({{\mbox{O\,{\scriptsize VI}}}})\approx 14.3$. A small scatter implies a more uniformly distributed medium, while a large scatter implies a more clumpy nature of the absorbing gas or a larger variation between different galaxy haloes. Such distinct spatial distributions between low- and high-ionization gas further highlight the complex nature of the chemically enriched CGM, which depends on more than the geometric alignment of the galaxies. A three-dimensional model of gas kinematics that takes full advantage of the detailed morphologies and star formation history of the galaxies is expected to offer a deeper understanding of the physical origin of chemically enriched gas in galaxy haloes (e.g., @Gauthier:2012 [@Chen:2014; @Diamond:2016]).
Summary {#sec:summary}
=======
QSO absorption spectroscopy provides a sensitive probe of both neutral medium and diffuse ionized gas in the distant Universe. It extends 21cm maps of gaseous structures around low-redshift galaxies both to lower gas column densities and to higher redshifts. Specifically, DLAs of $N({\mbox{H\,{\scriptsize I}}})\gtrsim 2\times 10^{20}\,{\mbox{${\rm cm^{-2}}$}}$ probe neutral gas in the ISM of distant star-forming galaxies, LLS of $N({\mbox{H\,{\scriptsize I}}})>10^{17}\,{\mbox{${\rm cm^{-2}}$}}$ probe optically thick HVCs and gaseous streams in and around galaxies, and strong absorbers of $N({\mbox{H\,{\scriptsize I}}})\approx 10^{14-17}\,{\mbox{${\rm cm^{-2}}$}}$ and associated metal-line absorption transitions, such as , , and , trace chemically enriched, ionized gas and starburst outflows. Over the last decade, an unprecedentedly large number of $\sim 10000$ DLAs have been identified along random QSO sightlines to provide robust statistical characterizations of the incidence and mass density of neutral atomic gas at $z\lesssim 5$. Extensive follow-up studies have yielded accurate measurements of chemical compositions and molecular gas content for this neutral gas cross-section selected sample from $z\approx 5$ to $z\approx 0$ (Sect. \[sec:DLAstats\]). Combining galaxy surveys with absorption-line observations of gas around galaxies has enabled comprehensive studies of baryon cycles between star-forming regions and low-density gas over cosmic time. DLAs, while being rare as a result of a small cross-section of neutral medium in the Universe, have offered a unique window into gas dynamics and chemical enrichment in the outskirts of star-forming disks (Sect. \[sec:DLAgals\]), as well as star formation physics at high redshifts (Sect. \[sec:sfr\]). Observations of strong absorbers and associated ionic transitions around galaxies have also demonstrated that galaxy mass is a dominant factor in driving the extent of chemically enriched halo gas and that chemical enrichment is well confined within galactic haloes for both low-mass dwarfs and massive galaxies (Sect. \[sec:cgm\]).
With new observations carried out using new, multiplex instruments, continuing progress is expected in further advancing our understanding of baryonic cycles in the outskirts of galaxies over the next few years. These include, but are not limited to: (1) direct constraints for the star formation relation in different environments (e.g., @Gnedin:2010), particularly for star-forming galaxies at $z\gtrsim
2$ in low surface density regimes of $\Sigma_{\rm SFR}<
0.1\,{\mbox{$M_\odot$}}\,{\rm yr}^{-1}\,{\rm kpc}^{-2}$ and $\Sigma_{\rm
gas}\approx 10-100\,{\mbox{$M_\odot$}}\,{\rm pc}^{-2}$; (2) an empirical understanding of galaxy environmental effects in distributing heavy elements to large distances based on deep galaxy surveys carried out in a large number of QSO fields (e.g., @Johnson:2015); and (3) a three-dimensional map of gas flows in the circumgalactic space that combines absorption-line kinematics along multiple sightlines with optical morphologies of the absorbing galaxies and emission morphologies of extended gas around the galaxies (e.g., @Rubin:2011 [@Chen:2014; @Zahedy:2016]). Wide-field IFUs on existing large ground-based telescopes substantially increase the efficiency in faint galaxy surveys (e.g., @Bacon:2015) and in revealing extended low surface brightness emission features around high-redshift galaxies (e.g., @Cantalupo:2014 [@Borisova:2016]). The [*James Webb Space Telescope*]{} ([*JWST*]{}), which is scheduled to be launched in October 2018, will expand the sensitivity of detecting faint star-forming galaxies in the early Universe. Combining deep infrared images from [*JWST*]{} and CO (or dust continuum) maps from ALMA will lead to critical constraints for the star formation relation in low surface density regimes.
The author wishes to dedicate this review to the memory of Arthur M. Wolfe for his pioneering and seminal work on the subject of damped absorbers and for inspiring generations of scientists to pursue original and fundamental research. The author thanks Nick Gnedin, Sean Johnson, Rebecca Pierce, Marc Rafelski, and Fakhri Zahedy for providing helpful input and comments. In preparing this review, the author has made use of NASA’s Astrophysics Data System Bibliographic Services.
[^1]: At $z\lesssim 1.6$, DLA surveys require QSO spectroscopy carried out in space and have been limited to the number of UV-bright QSOs available for absorption line searches. Consequently, the number of known DLAs from blind surveys is small, $\approx 15$ (see @Neeleman:2016 for a compilation). To increase substantially the sample of known DLAs at low redshifts, Rao & Turnshek (@Rao:2006) devised a clever space programme to search for new DLAs in known absorbers. Their strategy yielded a substantial gain, tripling the total sample size of $z\lesssim 1.6$ DLAs. However, the -selected DLA sample also includes a survey bias that is not well understood. It has been shown that excluding -selected DLAs reduces the inferred $\Omega_{\rm atomic}$ by more than a factor of four (e.g., @Neeleman:2016). For consistency, only measurements of $\Omega_{\rm atomic}$ based on blind DLA surveys are included in the plot.
[^2]: For comparison, the sizes of extended HVC complexes at $d\sim 10$kpc from the MW disk are a few to 15kpc across (e.g., @Putman:2012). HVCs at larger distances are found to be more compact, $\lesssim 2$kpc (e.g., @Westmeier:2008 [@Lockman:2012; @Giovanelli:2013]).
[^3]: A blind survey of absorption features around known galaxies differs fundamentally from a blind survey of galaxies around known absorbers (e.g., @Kacprzak:2008). By design, a blind galaxy survey around known absorbers excludes transparent sightlines and does not provide the sample necessary for measuring the incidence and covering fraction of absorbing species. In addition, because of limited survey depths, a blind galaxy survey is more likely to find more luminous members at larger $d$ that are correlated with the true absorbing galaxies which are fainter and closer to the QSO sightline, resulting in a significantly larger scatter in the $W_{\rm r}$ versus $d$ distribution (e.g., @Kacprzak:2008 [@Nielsen:2013]).
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abstract: 'The ABSTRACT is to be in fully-justified italicized text, at the top of the left-hand column, below the author and affiliation information. Use the word “Abstract” as the title, in 12-point Times, boldface type, centered relative to the column, initially capitalized. The abstract is to be in 10-point, single-spaced type. Leave two blank lines after the Abstract, then begin the main text. Look at previous ICCV abstracts to get a feel for style and length.'
author:
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First Author\
Institution1\
Institution1 address\
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Institution2\
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bibliography:
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title: LaTeX Author Guidelines for ICCV Proceedings
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Introduction
============
Please follow the steps outlined below when submitting your manuscript to the IEEE Computer Society Press. This style guide now has several important modifications (for example, you are no longer warned against the use of sticky tape to attach your artwork to the paper), so all authors should read this new version.
Language
--------
All manuscripts must be in English.
Dual submission
---------------
Please refer to the author guidelines on the ICCV 2019 web page for a discussion of the policy on dual submissions.
Paper length
------------
Papers, excluding the references section, must be no longer than eight pages in length. The references section will not be included in the page count, and there is no limit on the length of the references section. For example, a paper of eight pages with two pages of references would have a total length of 10 pages.
Overlength papers will simply not be reviewed. This includes papers where the margins and formatting are deemed to have been significantly altered from those laid down by this style guide. Note that this LaTeX guide already sets figure captions and references in a smaller font. The reason such papers will not be reviewed is that there is no provision for supervised revisions of manuscripts. The reviewing process cannot determine the suitability of the paper for presentation in eight pages if it is reviewed in eleven.
The ruler
---------
The LaTeX style defines a printed ruler which should be present in the version submitted for review. The ruler is provided in order that reviewers may comment on particular lines in the paper without circumlocution. If you are preparing a document using a non-LaTeXdocument preparation system, please arrange for an equivalent ruler to appear on the final output pages. The presence or absence of the ruler should not change the appearance of any other content on the page. The camera ready copy should not contain a ruler. (LaTeX users may uncomment the `\iccvfinalcopy` command in the document preamble.) Reviewers: note that the ruler measurements do not align well with lines in the paper — this turns out to be very difficult to do well when the paper contains many figures and equations, and, when done, looks ugly. Just use fractional references (e.g. this line is $095.5$), although in most cases one would expect that the approximate location will be adequate.
Mathematics
-----------
Please number all of your sections and displayed equations. It is important for readers to be able to refer to any particular equation. Just because you didn’t refer to it in the text doesn’t mean some future reader might not need to refer to it. It is cumbersome to have to use circumlocutions like “the equation second from the top of page 3 column 1”. (Note that the ruler will not be present in the final copy, so is not an alternative to equation numbers). All authors will benefit from reading Mermin’s description of how to write mathematics: <http://www.pamitc.org/documents/mermin.pdf>.
Blind review
------------
Many authors misunderstand the concept of anonymizing for blind review. Blind review does not mean that one must remove citations to one’s own work—in fact it is often impossible to review a paper unless the previous citations are known and available.
Blind review means that you do not use the words “my” or “our” when citing previous work. That is all. (But see below for techreports.)
Saying “this builds on the work of Lucy Smith \[1\]” does not say that you are Lucy Smith; it says that you are building on her work. If you are Smith and Jones, do not say “as we show in \[7\]”, say “as Smith and Jones show in \[7\]” and at the end of the paper, include reference 7 as you would any other cited work.
An example of a bad paper just asking to be rejected:
> An analysis of the frobnicatable foo filter.
>
> In this paper we present a performance analysis of our previous paper \[1\], and show it to be inferior to all previously known methods. Why the previous paper was accepted without this analysis is beyond me.
>
> \[1\] Removed for blind review
An example of an acceptable paper:
> An analysis of the frobnicatable foo filter.
>
> In this paper we present a performance analysis of the paper of Smith , and show it to be inferior to all previously known methods. Why the previous paper was accepted without this analysis is beyond me.
>
> \[1\] Smith, L and Jones, C. “The frobnicatable foo filter, a fundamental contribution to human knowledge”. Nature 381(12), 1-213.
If you are making a submission to another conference at the same time, which covers similar or overlapping material, you may need to refer to that submission in order to explain the differences, just as you would if you had previously published related work. In such cases, include the anonymized parallel submission [@Authors14] as additional material and cite it as
> \[1\] Authors. “The frobnicatable foo filter”, F&G 2014 Submission ID 324, Supplied as additional material [fg324.pdf]{}.
Finally, you may feel you need to tell the reader that more details can be found elsewhere, and refer them to a technical report. For conference submissions, the paper must stand on its own, and not [*require*]{} the reviewer to go to a techreport for further details. Thus, you may say in the body of the paper “further details may be found in [@Authors14b]”. Then submit the techreport as additional material. Again, you may not assume the reviewers will read this material.
Sometimes your paper is about a problem which you tested using a tool which is widely known to be restricted to a single institution. For example, let’s say it’s 1969, you have solved a key problem on the Apollo lander, and you believe that the ICCV70 audience would like to hear about your solution. The work is a development of your celebrated 1968 paper entitled “Zero-g frobnication: How being the only people in the world with access to the Apollo lander source code makes us a wow at parties”, by Zeus .
You can handle this paper like any other. Don’t write “We show how to improve our previous work \[Anonymous, 1968\]. This time we tested the algorithm on a lunar lander \[name of lander removed for blind review\]”. That would be silly, and would immediately identify the authors. Instead write the following:
> We describe a system for zero-g frobnication. This system is new because it handles the following cases: A, B. Previous systems \[Zeus et al. 1968\] didn’t handle case B properly. Ours handles it by including a foo term in the bar integral.
>
> ...
>
> The proposed system was integrated with the Apollo lunar lander, and went all the way to the moon, don’t you know. It displayed the following behaviours which show how well we solved cases A and B: ...
As you can see, the above text follows standard scientific convention, reads better than the first version, and does not explicitly name you as the authors. A reviewer might think it likely that the new paper was written by Zeus , but cannot make any decision based on that guess. He or she would have to be sure that no other authors could have been contracted to solve problem B.
FAQ\
[**Q:**]{} Are acknowledgements OK?\
[**A:**]{} No. Leave them for the final copy.\
[**Q:**]{} How do I cite my results reported in open challenges? [**A:**]{} To conform with the double blind review policy, you can report results of other challenge participants together with your results in your paper. For your results, however, you should not identify yourself and should not mention your participation in the challenge. Instead present your results referring to the method proposed in your paper and draw conclusions based on the experimental comparison to other results.\
\[fig:onecol\]
Miscellaneous
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Compare the following:\
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`$conf_a$` $conf_a$
`$\mathit{conf}_a$` $\mathit{conf}_a$
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\
See The TeXbook, p165.
The space after , meaning “for example”, should not be a sentence-ending space. So is correct, [*e.g.*]{} is not. The provided `\eg` macro takes care of this.
When citing a multi-author paper, you may save space by using “et alia”, shortened to “” (not “[*et. al.*]{}” as “[*et*]{}” is a complete word.) However, use it only when there are three or more authors. Thus, the following is correct: “ Frobnication has been trendy lately. It was introduced by Alpher [@Alpher02], and subsequently developed by Alpher and Fotheringham-Smythe [@Alpher03], and Alpher [@Alpher04].”
This is incorrect: “... subsequently developed by Alpher [@Alpher03] ...” because reference [@Alpher03] has just two authors. If you use the `\etal` macro provided, then you need not worry about double periods when used at the end of a sentence as in Alpher .
For this citation style, keep multiple citations in numerical (not chronological) order, so prefer [@Alpher03; @Alpher02; @Authors14] to [@Alpher02; @Alpher03; @Authors14].
Formatting your paper
=====================
All text must be in a two-column format. The total allowable width of the text area is $6\frac78$ inches (17.5 cm) wide by $8\frac78$ inches (22.54 cm) high. Columns are to be $3\frac14$ inches (8.25 cm) wide, with a $\frac{5}{16}$ inch (0.8 cm) space between them. The main title (on the first page) should begin 1.0 inch (2.54 cm) from the top edge of the page. The second and following pages should begin 1.0 inch (2.54 cm) from the top edge. On all pages, the bottom margin should be 1-1/8 inches (2.86 cm) from the bottom edge of the page for $8.5 \times 11$-inch paper; for A4 paper, approximately 1-5/8 inches (4.13 cm) from the bottom edge of the page.
Margins and page numbering
--------------------------
All printed material, including text, illustrations, and charts, must be kept within a print area 6-7/8 inches (17.5 cm) wide by 8-7/8 inches (22.54 cm) high. Page numbers should be in footer with page numbers, centered and .75 inches from the bottom of the page and make it start at the correct page number rather than the 4321 in the example. To do this fine the line (around line 23)
%\ificcvfinal\pagestyle{empty}\fi
\setcounter{page}{4321}
where the number 4321 is your assigned starting page.
Make sure the first page is numbered by commenting out the first page being empty on line 46
%\thispagestyle{empty}
Type-style and fonts
--------------------
Wherever Times is specified, Times Roman may also be used. If neither is available on your word processor, please use the font closest in appearance to Times to which you have access.
MAIN TITLE. Center the title 1-3/8 inches (3.49 cm) from the top edge of the first page. The title should be in Times 14-point, boldface type. Capitalize the first letter of nouns, pronouns, verbs, adjectives, and adverbs; do not capitalize articles, coordinate conjunctions, or prepositions (unless the title begins with such a word). Leave two blank lines after the title.
AUTHOR NAME(s) and AFFILIATION(s) are to be centered beneath the title and printed in Times 12-point, non-boldface type. This information is to be followed by two blank lines.
The ABSTRACT and MAIN TEXT are to be in a two-column format.
MAIN TEXT. Type main text in 10-point Times, single-spaced. Do NOT use double-spacing. All paragraphs should be indented 1 pica (approx. 1/6 inch or 0.422 cm). Make sure your text is fully justified—that is, flush left and flush right. Please do not place any additional blank lines between paragraphs.
Figure and table captions should be 9-point Roman type as in Figures \[fig:onecol\] and \[fig:short\]. Short captions should be centred.
Callouts should be 9-point Helvetica, non-boldface type. Initially capitalize only the first word of section titles and first-, second-, and third-order headings.
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---
abstract: 'We present spectra of the white-dwarf companions of the radio pulsars 0655+64 and 0820+02. For the latter, we find a spectrum showing strong lines of hydrogen, i.e., that of a DA star. From modelling these lines, the mass of a white dwarf can, in principle, be determined accurately, thus leading to constraints on the evolution of the binary and the mass of the neutron star. Our present spectrum is not of sufficient quality to set a strong limit, but it does indicate that the white dwarf most likely has a low mass. This is consistent with the star being a helium white dwarf, as would be expected from considerations of the mass function and the preceding evolution. From similar considerations, the companion of PSR 0655+64 is expected to be a more massive, carbon-oxygen white dwarf. This is confirmed by our spectra, which show the Swan bands of molecular carbon, making it a DQ star. Unlike what is observed in other DQ stars, the strength of the bands changes drastically, by a factor two in about two hours. We suggests this reflects large-scale surface inhomogeneities which are rotated in and out of the observed hemisphere. If so, this would imply that the white dwarf rotates supersynchronously.'
author:
- 'M. H. van Kerkwijk, S. R. Kulkarni'
title: 'Spectroscopy of the white-dwarf companions of PSR 0655+64 and 0820+02[^1]$^{,}$[^2]'
---
Introduction
============
About one tenth of the known radio pulsars reside in a binary system. Two have massive, early-type companions, and have properties rather similar to the average isolated radio pulsar. These presumably have been formed in a supernova explosion of a star in an early-type binary. Most of the others have evolved companions — either white dwarfs or neutron stars — and differ markedly from (most of) their isolated counterparts, showing more rapid spin periods and smaller magnetic fields (for recent reviews, see Bhattacharya & Van den Heuvel [@bhatvdh:91]; Verbunt [@verb:93]; Phinney & Kulkarni [@phink:94]; Kulkarni [@kulk:95]; these references have been used throughout this section). The rapid spin is generally believed to result from a phase of mass transfer in the evolutionary history of the binary, during which a large amount of mass and angular momentum is accreted. The reduction of the magnetic field is thought to occur in this phase as well, but the physical mechanism underlying this (e.g., it being “buried” by the accreted matter) is not understood.
The radio pulsars with evolved companions can be divided into three groups, generally referred to as the low-, intermediate- and high-mass binary pulsars (hereafter, LMBP, IMBP and HMBP, respectively). While these classifications are made on the basis of the inferred mass of the companion (from the mass function), stellar evolutionary scenarios allow us to relate these systems to the descendents of binaries composed of a neutron star and a low-mass ($M\simlt1\,M_\odot$), intermediate-mass ($1\,M_\odot\simlt M\simlt8\,M_\odot$) and high-mass ($M\simgt8\,M_\odot$) secondary, respectively. The expectation is that these secondaries evolve to Helium white dwarfs, carbon-oxygen (C-O) white dwarfs, and neutron stars, respectively. There is a fairly systematic trend of decrease in magnetic field strength as one proceeds from HMBPs to LMBPs. Presumably, this reflects the decrease in speed with which evolution proceeds in these systems, and the corresponding increase of the total amount of matter that is accreted.
Observations of the white-dwarf companions provide a number of diagnostics for these objects. For instance, the surface temperature of the white dwarf can be used to infer a cooling age, which sets a lower limit to the age of the neutron star. Such limits have been derived from broad-band photometry for PSR 0655+64 (Kulkarni [@kulk:86]), 0820+02 (ibid.; Koester, Chanmugan, & Reimers [@koescr:92]), 1855+09 (Callanan et al. ; Kulkarni, Djorgovski, & Klemola [@kulkdk:91]), J0437$-$4715 (Bailyn [@bail:93]; Bell, Bailes, & Bessel [@bellbb:93]; Danziger, Baade, & Della Valle [@danzbdv:93]), and J1012+5307 (Lorimer et al. ), and they provide the strongest evidence so far that magnetic fields of neutron stars do not decay on a time scale of millions of years, as had been thought before. In the absence of good constraints on the masses of the white dwarfs, however, the cooling ages cannot be very accurately determined.
Mass determinations would not only lead to better constraints on the cooling ages, but also allow one to verify, e.g., the orbital-period, white-dwarf mass relation predicted for the LMBP (Refsdal & Weigert [@refsw:71]; Savonije [@savo:87]; Joss, Rappaport, & Lewis [@jossrl:87]; Rappaport et al. ), or to constrain the mass of the neutron star. In addition, they could be used to obtain independent distance estimates to the systems, which can be compared to those derived from the dispersion measure of the pulsar.
Constraints on the white-dwarf mass can be derived from Shapiro delay, but this is possible for only a few binaries with favorable geometries. Another possibility is to use spectroscopy of the white dwarfs. In recent years, much progress has been made in the spectroscopic determination of surface gravities (and hence masses and radii), especially for white dwarfs of spectral type DA, i.e., those with a hydrogen atmosphere (e.g., Bergeron, Saffer, & Liebert [@bergsl:92]). Spectroscopy of the very faint companions of binary pulsars is non-trivial, and has so far only been attempted for the brightest, that of PSR J0437$-$4715 (Danziger et al.[@danzbdv:93]). Unfortunately, that white dwarf is rather cool ($T\simeq4000\,$K), and no line features were detected.
Here, we present spectroscopy of two somewhat fainter, but hotter companions, one of an IMBP, PSR 0655+64, and one of a LMBP, PSR 0820+02. For recent radio studies of these binaries, see Taylor & Dewey ([@tayld:88]) and Jones & Lyne ([@jonel:88]). For previous optical studies, see the references listed above.
Observations
============
Spectra were taken on New Year’s eve of 1995 at the Keck 10m telescope with the Low-Resolution Imaging Spectrometer (LRIS). With the $300\,{\rm{}line\,mm}^{-1}$ grating, the wavelength range of 3750 to 8780Å was covered at at $2.5\,{\rm\AA}\,{\rm{}pix}^{-1}$. The observing conditions were good throughout the night, but a substantial amount of time was lost due to telescope problems.
Three spectra were taken of the companion of PSR 0655+64 ($V=22.2\,$mag), starting at 1 January 1995, 8:54, 9:56 and 10:16 [ut]{}, with integration times of 30, 15 and 45 minutes, respectively. For the first spectrum, the slit was positioned over the companion and a nearby elliptical galaxy (see Kulkarni [@kulk:86]), while for the latter two it was set close to the parallactic angle. A 0.7 wide slit was used, giving a resolution of $\sim\!8\,$Å. The companion of PSR 0820+02 ($V=22.8\,$mag) was observed for 45 minutes starting at 14:26 [ut]{}, using a 1 slit to improve the throughput (leading to a resolution of $\sim\!12\,$Å). For approximate flux calibration, the spectrophotometric standard HD84937 (Oke & Gunn [@okeg:83]) was observed.
The reduction of all spectra was done using MIDAS[^3] and programs running in the MIDAS environment. The frames were bias-corrected, flat-fielded and sky-subtracted using standard procedures. The spectra were extracted using an optimal-extraction method similar to that presented by Horne ([@horn:86]). It turned out that the different exposures of the flux standard were inconsistent in both level and slope of the continuum. Since flux calibration was thus impossible, we normalized the spectra for the representation shown here (Fig. \[fig:spectra\]). This was done by dividing by a continuum defined by one of the flux-standard spectra, and scaling by $\lambda^\beta$, with $\beta$ chosen such that the continuum appeared straight.
Results and Discussion
======================
PSR 0655+64
-----------
The spectra of the companion of PSR 0655+64 (Fig. \[fig:spectra\]) are those of a DQ star, showing strong bands (for a review of white-dwarf spectra, see Wesemael et al. ). The bands are thought to be due to traces of carbon in a helium-rich atmosphere, brought up by convection from a deeper region, which is enriched in carbon due to upwards diffusion of carbon from the core (Pelletier et al. ). Thus, the presence of the Swan bands directly confirms the expectation that the object is a carbon-oxygen white dwarf.
The presence of the Swan bands, combined with the absence of lines from atomic carbon, also indicates that the temperature is between 6000 and 9000 K (inferred from Wegner & Yackovich [@wegny:84] and Wesemael et al. ). This compares well with the range of 5500 to 8000 K derived from photometry of this star (Kulkarni [@kulk:86]), and hence provides independent confirmation of the conclusion of Kulkarni ([@kulk:86]) that the cooling age of the white dwarf is about $2\times10^9\,$yr, comparable to the characteristic age of $P/2\dot{P}=3.6\times10^9\,$yr of the pulsar.
From the three spectra that we have, one can see that the strength of the C$_2$ features is variable. In fact, the total equivalent width changes by a factor of 2 over the two hours spanned by the observations, from about 170 to 330$\,$Å in the range 4430–5650Å. As far as we know, this is unprecedented. Since white dwarfs in general are not particularly variable stars, it is tempting to relate the changes to fixed surface patterns — due to, e.g., abundance or temperature variations, perhaps related to the presence of a magnetic field as in magnetic Ap stars (Borra, Landstreet, & Mestel [@borrlm:82]) — that are rotated in and out of the observed hemisphere. If so, then from the swiftness of the change in the spectrum, the strength of the features, and the integration times, it is easy to see that a possible periodicity cannot be shorter than about 3 hours (otherwise, the changes would be washed out), or longer than about 12 hours (to allow a change from about 30 to about 70% depth of the strongest part of the absorption). This is substantially shorter than the orbital period, and thus an association with orbital variations — such as could be produced by, e.g., heating of one hemisphere — seems unlikely.
If we associate the variations with the rotation of the white dwarf, then its rotation period has to be two to eight times shorter than the orbital one. This might actually not be unexpected: the progenitor of the white dwarf was a helium giant transferring mass to the pulsar (e.g., Iben & Tutukov [@ibent:93]), and if it was rotating synchronously — as does not seem implausible given the extremely circular orbit ($e=7.5\,10^{-6}$; Jones & Lyne [@jonel:88]) — then it would have been spun up due to conservation of angular momentum when it shrunk to form a white dwarf. In fact, we can set a rough upper limit to the mass of the envelope that fell back onto the white dwarf if we assume that the effect due to the shrinking of the core can be neglected. For this case, the moment of inertia in the envelope has to be at least equal that of the core – or equivalently (by assumption) that of the white dwarf – in order to be able to spin it up by at least a factor two. Hence, the mass of the envelope should be $\ga{}M_{\rm{}WD}R_{\rm{}WD}^2/R_{\rm{}env}^2$ (ignoring differences in structure). Since the progenitor was filling its Roche lobe, $R_{\rm{}env}\simeq R_{\rm{}L}
\simeq(GM_{\rm{}WD}/10\Omega_{\rm{}orb}^2)^{1/3} \simeq 2\,R_\odot$ (using Eq. 10.1 of Phinney & Kulkarni [@phink:94]). With $M_{\rm{}WD}\simeq0.8\,M_\odot$ and $R_{\rm{}WD}\simeq0.01\,R_\odot$, one finds a mass of the envelope of a few $10^{-4}\,M_\odot$. Interestingly, this is similar to the mass of the helium envelope that is inferred for DQ stars from the presence and strength of the carbon features (Pelletier et al. ; but see also Weideman & Koester [@weidk:95]; Dehner & Kawaler [@dehnk:95]).
While we realise that our estimate is a rough one – the very existence of a periodicity still needs to be confirmed – we note that, in principle, it might be possible to use rotation periods of white dwarfs in similar systems to constrain the final phases of the evolution. Especially for the systems with somewhat longer orbital periods, where the angular momentum is dominated by the envelope, the final spin rate should depend almost uniquely on the radius of the giant. Since the latter is related to the orbital period, one might expect, e.g., to find a correlation between the spin period and orbital period in such systems.
PSR 0820+02
-----------
In contrast to the companion of PSR 0655+64, this white dwarf shows strong lines of hydrogen (see Fig. \[fig:spectra\]), making it a DA white dwarf. As mentioned in the introduction, for a DA white dwarf it is possible to determine surface temperature and gravity uniquely from the spectrum. From a first comparison of the spectrum with model atmospheres, kindly done for us by Dr. Bergeron, it follows that it is consistent with the temperature being in the range of 14000 to 16500 K found by Koester et al. ([@koescr:92]). The best-fitting surface gravities for that range would indicate a mass of 0.25 to 0.35$M_\odot$, consistent with the idea that it is a helium white dwarf. However, the spectrum is of insufficient quality to set a strong limit: the 95% upper limit to the mass is $\sim\!0.9\,M_\odot$. Thus, it is not yet possible to verify whether the mass is within the range of 0.42 to 0.60$\,M_\odot$ expected from the orbital-period, white-dwarf mass relation (Rappaport et al.).
From photometry, combined with an upper limit of 1.9kpc to the distance, Koester et al. ([@koescr:92]) derived a lower limit of about $0.5\,M_\odot$ to the mass of the companion of PSR 0820+02. Indeed, if the white dwarf were to have a mass of about $0.3\,M_\odot$, it would have $M_V\simeq10.4\,$mag (Bergeron, private communication), and with $V\simeq22.8\,$mag and $A_V\simeq0.1\,$mag (Koester et al. [@koescr:92]), the system would be at a distance of $\sim\!2.8\,$kpc. We note, however, that the distance limit Koester et al. used was derived by taking twice the distance of 0.95kpc indicated by the dispersion measure of $23.6(2)\,{\rm{}cm^{-3}\,pc}$ (Taylor & Dewey [@tayld:88]) combined with the model of the Galactic electron distribution of Lyne, Manchester, & Taylor ([@lynemt:85]). With the more recent model of Taylor & Cordes ([@taylc:93]), one would infer a distance of 1.4kpc. Taylor & Cordes estimate that the mean uncertainty in the new dispersion-measure derived distances is about 25%, but they note that there is a dependence on the position of the pulsar. For PSR 0820+02, at $l^{\rm{}II}=222^\circ$, $b^{\rm{}II}=21^\circ$, we find from their Figures 7 and 8 that the model is substantially more uncertain.
While it is thus at present not possible to constrain the mass unambiguously, we note that for a $0.6\,M_\odot$ white dwarf, Koester et al. ([@koescr:92]) derived that the cooling age was within an “allowed range” of 1.5 to $2.7\,10^8\,$yr, which seems only marginally consistent with the characteristic age of $1.1\,10^8\,$yr of the neutron star (which should be an upper limit to the true age). If the white dwarf were to have a lower mass, the two age estimates would be in better agreement.
Conclusions
===========
We have presented spectra of the white-dwarf companions of the radio pulsars 0655+64 and 0820+02. From these spectra, combined with published temperature determinations, we can classify the two as DQ6–9 and DA3/4, respectively. This confirms the expectation that the former is a carbon-oxygen white dwarf, and is consistent with the latter being a helium white dwarf.
As hoped, it will be possible to derive accurate temperatures and surface gravities for the companion of PSR 0820+02. While this will likely be difficult for the companion of PSR 0655+64, its changing spectrum may instead turn out to provide us with a unique possibility to learn more about its previous evolution, as well as about the formation of spectral features in white dwarfs. It seems clear that further spectral studies of both these and other pulsar companions are warranted.
We are very grateful to Pierre Bergeron for making the model-atmosphere comparisons, and thank him, Brad Hansen, Yanqin Wu, and Peter Goldreich for useful discussions. M.H.v.K. is supported by a NASA Hubble Fellowship and S.R.K. by grants from the US NSF, NASA and the Packard Foundation.
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[^1]: Based on observations obtained at the W. M. Keck Observatory on Mauna Kea, Hawaii, which is operated jointly by the California Institute of Technology and the University of California.
[^2]: This is a preprint of a paper accepted by [*The Astrophysical Journal (Letters)*]{}. No bibliographic reference should be made to this preprint. Permission to cite material in this paper must be received from the authors.
[^3]: The Munich Image Data Analysis System is developed and maintained by the European Southern Observatory.
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---
author:
- Alessandro Sfondrini
- 'and Stijn J. van Tongeren'
title: Lifting asymptotic degeneracies with the Mirror TBA
---
Introduction
============
In the decompactification limit, both the light-cone gauge fixed $\AdS$ superstring and its AdS/CFT dual [@Maldacena:1997re] $\mathcal{N}=4$ super Yang-Mills theory have a description through an asymptotic Bethe ansatz[^1]. This description does not apply to either theory at finite size, where the only current non-perturbative description is through a set of equations known as the mirror thermodynamic Bethe ansatz (TBA) equations for the superstring.
The idea of applying methods from integrable relativistic models at finite size [@Zamolodchikov:1989cf] to the AdS/CFT correspondence was initiated in [@Ambjorn:2005wa] and explored in detail in [@Arutyunov:2007tc]. The main step in deriving the mirror TBA equations is the formulation of the string hypothesis [@Takahashi:19721dHubbard], which was done for the present model in [@Arutyunov:2009zu] by using the mirror version of the Bethe-Yang equations [@Beisert:2005fw] for the $\ads$ superstring. This was followed by a derivation of the canonical [@Arutyunov:2009ur; @Bombardelli:2009ns; @Gromov:2009bc] and simplified [@Arutyunov:2009ux] TBA equations that describe the ground state of the theory[^2]. These equations have been used to analyze the vanishing of the ground state energy of the theory at finite size [@Frolov:2009in]. Importantly, these ground state equations can be used to obtain equations for the excited states, through a contour deformation trick [@Arutyunov:2011uz] inspired by the analytic continuation procedure of [@Dorey:1996re]. Using the contour deformation trick, the mirror TBA equations have been used to reproduce perturbative results found through Lüscher’s approach[^3] [@Bajnok:2009vm; @Arutyunov:2010gb; @Balog:2010xa], and to study certain states in the $\alg{sl}(2)$ sector in considerable detail [@Gromov:2009tq; @Arutyunov:2009ax; @Balog:2010vf], specifically at intermediate coupling in [@Gromov:2009zb; @Frolov:2010wt]. The analytic properties of the Y-functions [@Arutyunov:2009ax; @Cavaglia:2010nm; @Cavaglia:2011kd; @Arutyunov:2011inprogress] are essential in determining these equations, and have proved useful for further understanding of the mirror TBA equations. Following these developments, discontinuity relations can now be used to find the excited state equations directly in the $\alg{sl}(2)$ subsector [@Balog:2011nm], giving results in complete equivalence with the contour deformation trick where applicable. Moreover, the simplified TBA equations have recently been brought to a quasi-local form [@Balog:2011cx], another step in the direction of obtaining a so-called non-linear integral equation (NLIE) description of the spectral problem at finite size. Additional steps in this direction had already been taken in [@Gromov:2010km; @Suzuki:2011dj].
It is well known that the asymptotic Bethe ansatz (ABA) captures the leading $1/J$ corrections to the asymptotic energy spectrum, while it misses the exponential corrections due to the finite system size. Therefore the ABA misses quantitative information on the spectrum, as is for example clearly illustrated by wrapping corrections to scaling dimensions in the gauge theory. Following an observation of [@Arutyunov:2011uz], in this paper we will show that for the $\AdS$ superstring finite size effects are not only quantitative in nature. In fact we will demonstrate that the ABA also misses *qualitative* information on the spectrum, owing to a discrete symmetry enhancement of the model in the asymptotic limit, so that certain states become degenerate asymptotically.
This paper is organized as follows. In the next section we start by discussing the symmetries and degeneracies of the asymptotic Bethe ansatz and explain the asymptotic symmetry enhancement alluded to just above. Also, we will indicate how finite size effects should lift this degeneracy. After painting the general picture we illustrate these ideas by considering two concrete states with degenerate energies in the asymptotic limit. We will show that these states have manifestly different TBA equations and explicitly compute the different finite size corrections they receive, in line with the general discussion.
Extra degeneracy in the asymptotic limit
========================================
In general the energy spectrum of string states is expected to have degeneracies, owing to the superconformal symmetry of the model. Because of this symmetry, string states arrange themselves in superconformal multiplets which each share a common energy. At the level of the (asymptotic) Bethe ansatz these degeneracies are reflected by the fact that solutions to the Bethe-Yang equations only give the highest weight states of the underlying symmetry algebra, familiar from e.g. the Heisenberg [@Faddeev:1996iy] and Hubbard model [@Essler:1991wg]. Completeness of the Bethe ansatz then follows by adding the states which lie in the same multiplet, which then by construction have the same energy. In the case of the asymptotic Bethe ansatz for the $\AdS$ superstring however, there is *additional* degeneracy, degeneracy which arises in the decompactification limit and should not be present in the complete model. This degeneracy occurs in the asymptotic Bethe ansatz due to enhanced symmetry in the asymptotic limit, indicating qualitative features of the model that are not captured by the asymptotic solution.
In the light-cone gauge the superstring has manifest $\su_L(2|2)\oplus \su_R(2|2)$ symmetry, where the subscript $L$ and $R$ distinguish the two $\su(2|2)$ factors, conventionally called left and right. By construction, the model possesses a $\mathbb{Z}_2$ symmetry, which we call left-right symmetry, interchanging the sets of left and right $\su(2|2)$ charges. This means that for every state with a given set of $\su_L(2|2)\oplus \su_R(2|2)$ charges, there exists a state with equal energy, with left-right interchanged $\su(2|2)$ charges. It is this left-right symmetry which is enhanced in the asymptotic limit to a larger discrete group. At the level of the Bethe-Yang equations this enhancement is manifested by the fact that they allow more than just an interchange of complete sets of left and right charges, actually allowing free redistribution of roots between the left and right sectors in certain cases. In the finite size model on the contrary, there is no reason to assume states related by such a redistribution should have the same energy, so we expect that finite size effects lift this asymptotic degeneracy. We will now consider these ideas in more detail.
The Bethe-Yang equations
------------------------
The Bethe-Yang equation for the $\AdS$ superstring in the light-cone gauge is given by [@Beisert:2005fw] $$\label{equ:betheyangmain}
1= e^{ip_k J} \prod_{\textstyle\atopfrac{l=1}{l\neq
k}}^{K^{\mathrm{I}}}S_{\sl(2)}(p_k,p_l)\prod_{\alpha = L,R} \prod_{l=1}^{K^{\mathrm{II}}_{(\alpha)}} \frac{x^-_k-y_l^{(\alpha)}}{x^+_k-y_l^{(\alpha)}}\sqrt{\frac{x^+_k}{x^-_k}}\,,\ \ \ \ \ \ \ \ k=1,\dots K^{\rm I}\;.$$ In addition to the rapidities of fundamental particles, this equation contains $y^{(\a)}$ roots. Together with the $w^{(\a)}$ roots ($\a=L,R$) which enter in the auxiliary Bethe equations just below, these correspond to the $\su_L(2|2)\oplus\su_R(2|2)$ symmetry of the model. The auxiliary Bethe equations consist of two independent sets of two coupled equations for the $y$ and $w$ roots, given by $$\begin{aligned}
\label{equ:betheyangaux1}
1=&\prod_{l=1}^{K^{\mathrm{I}}}\frac{y_{k}^{(\a)}-x^{-}_{l}}{y_{k}^{(\a)}-x^{+}_{l}}\sqrt{\frac{x_l^+}{x_l^-}}
\prod_{l=1}^{K^{\mathrm{III}}_{(\a)}}\frac{\nu_{k}^{(\a)}-w_{l}^{(\a)}+\frac{i}{g}}{\nu_{k}^{(\a)}-w_{l}^{(\a)}-\frac{i}{g}}\;, & k = 1, \ldots, K^{\mathrm{II}}_{(\a)}, \; \, &\a=L,R\;,\\
\label{equ:betheyangaux2}
1=&\prod_{l=1}^{K^{\mathrm{II}}_{(\a)}}\frac{w_{k}^{(\a)}-\nu_{l}^{(\a)}+\frac{i}{g}}{w_{k}^{(\a)}-\nu_{l}^{(\a)}-\frac{i}{g}}
\prod_ {\textstyle\atopfrac{l=1}{l\neq
k}}^{K^{\mathrm{III}}_{(\a)}}\frac{w_{k}^{(\a)}-w_{l}^{(\a)}-\frac{2i}{g}}{w_{k}^{(\a)}-w_{l}^{(\a)}+\frac{2i}{g}}\;, & k = 1, \ldots, K^{\mathrm{III}}_{(\a)}, \; \, &\a=L,R\;,\end{aligned}$$ where $\nu_{k}^{(\a)} = y_{k}^{(\a)} + 1/y_{k}^{(\a)}$. The fact that we have two sets of identical left-right decoupled equations corresponds directly to the left-right symmetry mentioned earlier. At the level of the transfer matrix this is reflected by the fact that we have a transfer matrix for each sector, $T^{(L)}$ and $T^{(R)}$; both are $\su(2|2)$-invariant transfer matrices with eigenvalues parametrized by the auxiliary roots, and these eigenvalues are therefore arranged in $\su(2|2)$ multiplets. As mentioned, solutions of the auxiliary equations (\[equ:betheyangaux1\]-\[equ:betheyangaux2\]) identify highest weight states of the $\su(2)$ subalgebras of this symmetry algebra, labeled by the Dynkin labels $(s_{(\alpha)}, q_{(\alpha)})$. The weights are encoded in the excitations numbers as $$s_{(\alpha)}=K^{\mathrm{I}}-K^{\mathrm{II}}_{(\a)}\,, \ \ \ \ q_{(\a)}=K^{\mathrm{II}}_{(\a)}-2K^{\mathrm{III}}_{(\a)}\;,$$ where the excitation numbers satisfy $$K^{\mathrm{I}}\geq K^{\mathrm{II}}_{(\a)}\geq 2 K^{\mathrm{III}}_{(\a)}\,,\ \ \a=L,R\;.$$
Extra degeneracy
----------------
As indicated, the discrete left-right symmetry of the light-cone gauge fixed model can be enhanced in the asymptotic limit, giving a higher amount of degeneracy in the spectrum. This is the case when $K^{\mathrm{III}}_{(\a)} = 0$, for states with a given total number of $y$ roots; $\sum_\a K^{\mathrm{II}}_{(\a)} = K^{\mathrm{II}}_{\scriptscriptstyle \rm Tot} > 1$. *For any such state*, the auxiliary Bethe equations reduce to $$\label{eq:}
\prod_{l=1}^{K^{\mathrm{I}}}\frac{y_{k}^{(\a)}-x^{-}_{l}}{y_{k}^{(\a)}-x^{+}_{l}}\sqrt{\frac{x_l^+}{x_l^-}} = 1\,, \; \; \; k = 1, \ldots, K^{\mathrm{II}}_{(\a)}\,, \; \a=L,R\,,\;$$ showing that we have *one and the same equation for each of the $y_{k}^{(\a)}$* [@Arutyunov:2011uz]. The number of solutions we can pick for each $y$ depends on the main excitation number $K^{\mathrm{I}}$, and in general we must take care to only allow for regular configurations of roots. Nonetheless, the consequence of this degeneration is immediately clear: provided there is more than one allowed solution for $y_{k}^{(\a)}$ we can freely redistribute any number of different $y$ roots between the left and right sectors *without changing the main Bethe-Yang equation*, because it itself contains a product over the left and right roots. This is the enhancement of the left-right symmetry of the finite size model in the asymptotic limit. Let us note that the corresponding symmetry group acts on regular highest weight states only, and that this action cannot be extended to the other states.
Two states differing by such a redistribution will have the same asymptotic momentum, hence energy, while there is no reason to assume their energies should be identical outside the asymptotic regime. Rather, it would actually be a surprising coincidence if their energies agreed. Stated more strongly, looking at the description of the finite size model through the mirror TBA it should be conceptually clear that this symmetry is only present asymptotically; the presence of $w$ roots generically spoils this symmetry, and while an individual state might have no $w$ roots, in the mirror TBA such a state is described in interaction with a thermal background containing all possible excitations. Note also that two such asymptotically equivalent states have manifestly different Dynkin labels, while they are not in the same superconformal multiplet. Indeed, such states correspond to potentially wildly different operators in $\mathcal{N} = 4$ SYM.
Lifting degeneracies through finite size effects
------------------------------------------------
As just stated, we expect finite size corrections to lift this degeneracy of the asymptotic spectrum, whether it be through the thermodynamic Bethe ansatz, or perturbatively through Lüscher corrections. How this happens is perhaps most immediately seen through the complete formula for the energy of a string state in the mirror TBA approach [@Arutyunov:2009ur] $$\label{eq:energy}
E=\sum_{k=1}^{K^{\mathrm{I}}}\mathcal{E}_k-\frac{1}{2\pi} \sum_{Q=1}^{\infty} \int dv\,\frac{d\tilde{p}^Q}{dv}\,\log(1+Y_Q)\;,$$ where $\mathcal{E}_k=i\tilde{p}(u_{*k})$ gives the asymptotic contribution to the energy, while the second term arises from the finite size of the system. In this formula, the $Y_Q$-functions are determined through the mirror TBA equations, which intricately couple the auxiliary left and right $Y$-functions. Now in general there is no reason to expect that the TBA equations for two of these asymptotically degenerate states should be the same, meaning they should receive different finite size corrections, lifting the degeneracy of the asymptotic spectrum.
Alternately, expanding the energy formula (\[eq:energy\]) to leading order around the asymptotic solution (small $Y_Q$-functions) gives a formula in direct agreement with Lüscher’s approach $$\begin{aligned}
\label{eq:perturbativeenergy}
E_{LO} = -\frac{1}{2\pi}\sum_{Q=1}^{\infty}\int dv \frac{d\tilde{p}}{dv} Y^{\circ}_{Q}(v).\end{aligned}$$ Here $Y^{\circ}_{Q}$, in the above expanded to leading order in the coupling constant, is given by the generalized Lüscher’s formula [@Bajnok:2008bm] $$\begin{aligned}
\label{eq:YQasympt}
Y^{\circ}_Q(v) = e^{-J\tilde{\mathcal{E}}_Q(v)}\;T^{(L)}(v|\vec{u})\;T^{(R)}(v|\vec{u})\,\prod_k S^{Q1_*}_{\alg{sl}(2)}(v,u_k).\end{aligned}$$ In this formula $\tilde{\mathcal{E}}_Q(v)$ is the energy of a mirror $Q$-particle, $S^{Q1_*}_{\alg{sl}(2)}(v,u_k)$ denotes the $\alg{sl}(2)$ $S$-matrix with arguments in the mirror ($v$) and string regions ($u_k$) and finally $T^{(L,R)}$ are the left and right transfer matrices, given in appendix \[ap:transfermatrices\]. Clearly these corrections couple the left and right sectors through the product of transfer matrices. Such transfer matrices, and more importantly their products $T^{(L)}T^{(R)}$, will generically be different for two asymptotically degenerate states, giving different perturbative finite size corrections, showing again that the denegeracy of the asymptotic spectrum is lifted in the finite size theory.
In what follows we will illustrate these ideas concretely for two different four-particle states, both parametrized by two $y$ roots. We will show how the full sets of mirror TBA equations describing these states are manifestly different (though naturally in an elegant symmetric way), and explicitly compute different leading-order corrections to the energy, which hence lift the asymptotic degeneracy. Let us first introduce our states.
Two explicit states
===================
We consider two states that have the same value for $K^{\mathrm{I}}$ and $\sum_\alpha K^{\mathrm{II}}_{(\alpha)}$, but different values for the individual $K^{\mathrm{II}}_{(\alpha)}$. When $K^{\mathrm{I}}=2$ there are no non-trivial level-matched solutions of the auxiliary equations, therefore we consider the states ${\Theta}$ and ${\Psi}$, as presented below in table \[tab:states\].
State $K^{\mathrm{I}}$ $K_{(L)}^{\mathrm{II}}$ $K_{(R)}^{\mathrm{II}}$ $K_{(\a)}^{\mathrm{III}}$ Weights
------------ ------------------ ------------------------- ------------------------- --------------------------- ---------------------
${\Theta}$ 4 2 0 0 $[2,J-1,0]_{(2,4)}$
${\Psi}$ 4 1 1 0 $[1,J-1,1]_{(3,3)}$
: The two asymptotically degenerate states we consider. Note the manifestly different excitation numbers. For the readers’ convenience we have also presented the Dynkin labels of the states, denoted by $[q_L,p,q_R]_{(s_L,s_R)}$.
\[tab:states\]
For either state we have four rapidities $u_i$, level-matched, and two auxiliary roots $y^{(\alpha)}_i$, either both left or one left and one right. In both cases we take the four rapidities to come in pairs: $u_1=-u_2>0$ and $u_3=-u_4>0$.
As discussed above, for both states the auxiliary equation for any $y^{(\alpha)}_i$ is the same. From (\[equ:betheyangaux1\]), and imposing the rapidities to come in pairs, we find $$1=\prod_{i=1}^4 \frac{y-x^-_i}{y-x^+_i},\ \ \ \mathrm{with}\ \ x^\pm_1=-x^\mp_2,\ \ x^\pm_3=-x^\mp_4.$$ This admits two regular roots (in addition to $y=0,\infty$) that are opposite to each other, $y=\pm y_o$, where $$y_o=\sqrt{\frac{x^-_1\,x^+_1(x^-_3 -x^+_3)+x^+_3\,x^-_3(x^-_1-x^+_1)}{x^-_1-x^+_1+x^-_3-x^+_3}}\,.$$ Therefore we take $y^{(L)}_{1,2}=\pm y_o$ for state ${\Theta}$, and $y^{(L)}_{1}=+ y_o$ and $y^{(R)}_{1}=- y_o$ for state ${\Psi}$. We can now solve the two Bethe-Yang equations (\[equ:betheyangmain\]) for $u_1$ and $u_3$ at a given value of $J$, by plugging in the auxiliary roots, recalling that the $\alg{sl}(2)$ $S$-matrix is given by $$S_{\sl(2)}(u_1,u_2)=\sigma^{-2}\,\frac{x^+_1-x^-_2}{x^-_1-x^+_2}\,\frac{1-\frac{1}{x^-_1x^+_2}}{1-\frac{1}{x^+_1x^-_2}}\;,$$ where $\sigma$ is the dressing factor.
Solving the resulting equation analytically is not feasible. Therefore, we first consider the limit $g\to0$, rescaling the rapidities such that they remain finite, $u_i\to \frac{1}{g}u^o_i$. Then the equations for $u_1^o$ and $u_3^o$ decouple, and both take the simple form $$1=\left(\frac{u_k^o+i}{u_k^o-i}\right)^{J+2}\ \ \ \Longrightarrow\ \ \ u_k^o=\cot \frac{n_k\,\pi}{J+2},\ \ n_k=1,\dots J+1\;,$$ where the sum of positive $n_k$ giving the string level of the state [@Arutyunov:2004vx]. In order to have a generic root configuration we focus on the case $J=4$ at string level three, where we can solve (\[equ:betheyangmain\]) numerically for arbitrary values of $g$, requiring that at small coupling $$\label{equ:asymptoticrapidities}
u_1^o=-u_2^o=\sqrt{3}\,,\ \ \ \ \ u_3^o=-u_4^o=\frac{1}{\sqrt{3}}\;.$$
![The rapidities $u_1$ and $u_3$ obtained from Bethe-Yang equation at different values of $g$. Note that they asymptote to two as the coupling is increased.[]{data-label="fig:rapidities"}](plotrapidities.pdf){width="10cm"}
In figure \[fig:rapidities\] the numerical solutions $u_1(g)$ and $u_3(g)$ are shown. Note again that these solutions are the same for both states. In solving (\[equ:betheyangmain\]) numerically, the representation of the dressing phase as presented in [@Frolov:2010wt] is most convenient. It is worth remarking that at finite values of $g$, no simple relation between $u_1$ and $u_3$ holds, despite what we see at weak coupling in (\[equ:asymptoticrapidities\]).
Through the AdS/CFT duality, the ${\Theta}$ state corresponds to an operator schematically of the form $\mbox{Tr}(D^2 \bar{\psi}\bar{\psi} Z^3)$. The correspondent operator of the ${\Psi}$ state is actually a linear combination of two types of operators, namely $\mbox{Tr}(D^2 \bar{\psi} \psi Z^3 )$ and $\mbox{Tr}(D^3 W Z^4)$. In these expressions, all excitations have the highest allowed charges.
TBA equations
-------------
In order to obtain TBA equations for an excited state we use the contour deformation trick, following [@Arutyunov:2009ax]. A clear overview of this whole approach can be found in [@Arutyunov:2011uz]. In short we assume that the ground state and excited state TBA equations differ only by the choice of the integration contours. Upon deforming the integration contours of the excited state TBA equations to the ground state ones, we pick up additional contributions whenever there is a singularity in the physical strip of the rapidity plane. This leads to the appearance of new driving terms in the excited state TBA equations.
In the present case, we do this for the left and right sectors, for both states. We denote the $Y$-functions ${Y^{(L)}_{M|w}},{Y^{(L)}_{\pm}},{Y^{(L)}_{M|vw}}$ and ${Y^{(R)}_{M|w}},{Y^{(R)}_{\pm}},{Y^{(R)}_{M|vw}}$ for a given state in the left and right sectors respectively; the sectors are coupled by the $Y_Q$ functions. Below we discuss the analytic properties and related integration contours for both states in detail, followed by the resulting TBA equations.
### Analytic properties
As discussed in [@Arutyunov:2011uz], we will use the (left and right) asymptotic $Y$-functions to study the analytic properties of the TBA equations. Their asymptotic construction is given in appendix \[ap:transfermatrices\]. Let us stress that all the $Y$-functions but $Y_Q$ are defined in a given sector, i.e. $Y_{M|w}\equiv Y_{M|w}^{(\alpha)}$ etc. Only $Y_Q$ couples the left and right sectors, and indeed asymptotically $Y_Q$ contains the product of left and right transfer matrices, as in (\[eq:YQasympt\]).
As shown in detail for the Konishi state in [@Arutyunov:2009ax], the precise analytic structure of asymptotic $Y$-function will depend on the coupling $g$. In general, when we increase $g$ we can expect to encounter an asymptotic critical value [@Arutyunov:2009ax] where some roots enter the physical strip, so that we must include appropriate driving terms. This would make the discussion more technically involved, but is of little relevance to understanding the lifting of degeneracies. We will therefore restrict our analysis to the small coupling region, below the first critical value of $g$.
For both states, it turns out that roots of $1+Y_{M|w}$, $1+Y_{M|vw}$ and $1-Y_-$, and poles of $Y_+$ play an important role. In order to discuss this, let us fix some notation. Roots related to state ${\Theta}$ are described by script letters: ${\varrho_{\scriptscriptstyle M}}$ for $Y_{M|w}$, ${\varrho_{\scriptscriptstyle 0}}$ for $Y_{-}$ and ${r_{\scriptscriptstyle M}}$ for $Y_{M|vw}$. They are fixed by the conditions $$Y_{M|w}({\varrho_{\scriptscriptstyle M}}-i/g)=-1,\ \ Y_{-}({\varrho_{\scriptscriptstyle 0}}-i/g)=1,\ \ Y_{M|w}({r_{\scriptscriptstyle M}}-i/g)=-1\;.$$ Similarly, we will use sans-serif letters to denote roots for state ${\Psi}$: ${\rho_{\scriptscriptstyle M}}$ for $Y_{M|w}$, ${\rho_{\scriptscriptstyle 0}}$ for $Y_{-}$ and ${{\sf r}_{\scriptscriptstyle M}}$ for $Y_{M|vw}$, fixed by $$Y_{M|w}({\rho_{\scriptscriptstyle M}}-i/g)=-1,\ \ Y_{-}({\rho_{\scriptscriptstyle 0}}-i/g)=1,\ \ Y_{M|w}({{\sf r}_{\scriptscriptstyle M}}-i/g)=-1\;.$$ In addition, for both states in both sectors $Y_+$ asymptotically has poles at the rapidities shifted by $i/g$, $Y_{+}(u_i-i/g)=\infty$, as in the Konishi case [@Arutyunov:2009ax]. The relevant roots are summarized in table \[tab:roots\].
$1+{Y^{(L)}_{M|w}}$ $1+{Y^{(R)}_{M|w}}$ $1-{Y^{(L)}_{-}}$ $1-{Y^{(R)}_{-}}$ $1+{Y^{(L)}_{M|vw}}$ $1+{Y^{(R)}_{M|vw}}$
---------------------- -------------------------------------------- -------------------------------------- -------------------------------------------- -------------------------------------- ------------------------------------------ ----------------------------------------- -- --
[Roots ${\Theta}$]{} $\pm {\varrho_{\scriptscriptstyle M}}-i/g$ – $\pm {\varrho_{\scriptscriptstyle 0}}-i/g$ – – $\pm {r_{\scriptscriptstyle M}}-i/g$
[Roots ${\Psi}$]{} ${\rho_{\scriptscriptstyle M}}-i/g$ $-{\rho_{\scriptscriptstyle M}}-i/g$ $- {\rho_{\scriptscriptstyle 0}}-i/g$ $ {\rho_{\scriptscriptstyle 0}}-i/g$ $ -{{\sf r}_{\scriptscriptstyle M}}-i/g$ $ {{\sf r}_{\scriptscriptstyle M}}-i/g$
: Roots for $Y$-functions in the left and right sectors for states ${\Theta}$ and ${\Psi}$ at small coupling. By definition we consider ${\varrho_{\scriptscriptstyle M}},{\rho_{\scriptscriptstyle M}},{r_{\scriptscriptstyle M}},{{\sf r}_{\scriptscriptstyle M}}>0$. Asymptotically, one observes that ${\varrho_{\scriptscriptstyle M}}\neq{\rho_{\scriptscriptstyle M}}$ and ${r_{\scriptscriptstyle M}}\neq{{\sf r}_{\scriptscriptstyle M}}$.[]{data-label="tab:roots"}
We expect that the roots, and hence the driving terms, distribute differently between the left and right sectors for ${\Theta}$ and ${\Psi}$. This is indeed the case, as can be seen in table \[tab:roots\].
On the real mirror line, the asymptotic $Y$-functions for state ${\Theta}$ are even, while for state ${\Psi}$ the $Y$-functions do not have a definite parity but satisfy $Y^{(L)}(v) = Y^{(R)}(-v)$.
### The simplified TBA equations
We now apply the contour deformation trick to the simplified TBA equations of [@Arutyunov:2009ur; @Arutyunov:2009ux]. In order for the asymptotic solution to be a solution, we take the ground state TBA equations and define the integration contour such that it goes slightly below the line $-i/g$, *i.e.* such that it encloses the poles of $Y_+$ at $u_i-i/g$ as well as the roots of table \[tab:roots\] between itself and the real line. By taking the integration contour back to the real line, we find the appropriate driving terms, denoted ${\mathscr{D}}$, and obtain the TBA equations. Below we list the driving terms that appear for each state. The integration kernels and $S$-matrices which enter in the equations below have been defined and are completely listed in [@Arutyunov:2009ax]. As usual, for any kernel or S-matrix we define $S^\pm(v):=S(v\pm i/g)$.\
$\bullet$ $M|w$-strings; $\ M\ge 1\ $, $Y_{0|w}=0$. The equation has the general form $$\log Y^{(\alpha)}_{M|w} = \log(1 + Y^{(\alpha)}_{M-1|w})(1 +
Y^{(\alpha)}_{M+1|w})\star s
+ \delta_{M1}\, \log{1-{1\ov Y^{(\alpha)}_-}\ov 1-{1\ov Y^{(\alpha)}_+} }\hstar s +{\mathscr{D}}^{(\alpha)}_{M|w},~~~~~$$ where ${\mathscr{D}}^{(\alpha)}_{M|w}$ are the driving terms that differ in the left and right sector for each given state and $\alpha =L,R$. For state ${\Theta}$ we have $$\begin{aligned}
{\mathscr{D}}^{(L)}_{M|w}(v)&=&-\log S^{-}\!(\pm{\varrho_{\scriptscriptstyle M-1}}-v)-\log S^{-}\!(\pm{\varrho_{\scriptscriptstyle M+1}}-v)\;,\\
{\mathscr{D}}^{(R)}_{M|w}(v)&=&0\;,
\nonumber\end{aligned}$$ where the terms containing $\pm \varrho$ indicate the sum of two driving terms for opposite roots. For ${\Psi}$ we have, instead, $$\begin{aligned}
{\mathscr{D}}^{(L)}_{M|w}(v)&=&-\log S^{-}\!({\rho_{\scriptscriptstyle M-1}}-v)-\log S^{-}\!({\rho_{\scriptscriptstyle M+1}}-v)\;,\\
{\mathscr{D}}^{(R)}_{M|w}(v)&=&-\log S^{-}\!(-{\rho_{\scriptscriptstyle M-1}}-v)-\log S^{-}\!(-{\rho_{\scriptscriptstyle M+1}}-v)\;.
\nonumber\end{aligned}$$\
$\bullet$ $M|vw$-strings; $\ M\ge 1\ $, $Y_{0|vw}=0$ $$\begin{aligned}
\log Y^{(\alpha)}_{M|vw}(v) = & - \log(1 + Y_{M+1})\star s +
\log(1 + Y^{(\alpha)}_{M-1|vw} )(1 + Y^{(\alpha)}_{M+1|vw})\star
s\\
& + \delta_{M1} \log{1-Y^{(\alpha)}_-\ov 1-Y^{(\alpha)}_+}\hstar s+{\mathscr{D}}_{M|vw}^{(0)} +{\mathscr{D}}^{(\alpha)}_{M|vw}\,\nonumber.\end{aligned}$$ When $M=1$, we find a driving term that is independent of state and sector, arising from the poles of $Y_+$, that is $${\mathscr{D}}_{M|vw}^{(0)}=-\delta_{M1}\!\sum_{i=1}^4\log S^{-}\!(u_i-v)\;.$$ In addition to that, for state ${\Theta}$ we have $$\begin{aligned}
{\mathscr{D}}^{(L)}_{M|vw}(v)&=&0\;,\\
{\mathscr{D}}^{(R)}_{M|vw}(v)&=&-\log S^{-}\!(\pm {r_{\scriptscriptstyle M-1}}-v)-\log S^{-}\!(\pm {r_{\scriptscriptstyle M+1}}-v)\;,
\nonumber\end{aligned}$$ whereas for ${\Psi}$ we have $$\begin{aligned}
{\mathscr{D}}^{(L)}_{M|vw}(v)&=&-\log S^{-}\!(- {{\sf r}_{\scriptscriptstyle M-1}}-v)-\log S^{-}\!(- {{\sf r}_{\scriptscriptstyle M+1}}-v),\\
{\mathscr{D}}^{(R)}_{M|vw}(v)&=&-\log S^{-}\!({{\sf r}_{\scriptscriptstyle M-1}}-v)-\log S^{-}\!({{\sf r}_{\scriptscriptstyle M+1}}-v)\;.
\nonumber\end{aligned}$$\
$\bullet$ $y$-particles $$\begin{aligned}
\label{equ:tbaYproduct}
\log {Y^{(\alpha)}_+\ov Y^{(\alpha)}_-} = \, & \log(1 + Y_{Q})\star K_{Qy}+{\mathscr{D}}^{(0)}_{ratio}\,,\\
\log {Y^{(\alpha)}_- Y^{(\alpha)}_+}= \, & 2 \log{1 + Y^{(\alpha)}_{1|vw} \ov 1 + Y^{(\alpha)}_{1|w} }\star s - \log\left(1+Y_Q \right)\star K_Q \\
&+ 2 \log(1 +Y_{Q})\star K_{xv}^{Q1} \star s+{\mathscr{D}}^{(0)}_{prod}+{\mathscr{D}}^{(\alpha)}_{prod}\; \nonumber.\end{aligned}$$ We expect both of these equations to pick up contributions from the exact Bethe equation[^4] $Y_1(u_{*i}) = -1$, which will yield driving terms that do not depend on the state or sector. These are $$\begin{aligned}
{\mathscr{D}}^{(0)}_{ratio}(v)&=& -\sum_{i=1}^4 \log S_{1_*y}(u_i ,v)\,,\\
{\mathscr{D}}^{(0)}_{prod}(v)&=& -\sum_{i=1}^4 \log {\big(S_{xv}^{1_*1}\big)^2\ov S_2}\star s( u_i,v)\,.\nonumber\end{aligned}$$ where &&s(u,v) \_[-]{}\^ dt s(t-v) . The contribution follows from the identity $$\log{S_1(u_i-v)} - 2 \log{S_{xv}^{1_* 1}}\star s(u_i,v) = -\log{\frac{(S_{xv}^{1_* 1})^2}{S_2}}\star s(u_i,v) ,$$ valid for real $u_i$.
In addition to the above driving terms, we have state-specific contributions in equation (\[equ:tbaYproduct\]); for ${\Theta}$ we have $$\begin{aligned}
{\mathscr{D}}^{(L)}_{prod}(v)&=& 2\log S^{-}\!(\pm {\varrho_{\scriptscriptstyle 1}}-v)\,,\\
{\mathscr{D}}^{(R)}_{prod}(v)&=&-2\log S^{-}\!(\pm {r_{\scriptscriptstyle 1}}-v)\,,\nonumber\end{aligned}$$ and for ${\Psi}$ we have $$\begin{aligned}
{\mathscr{D}}^{(L)}_{prod}(v)&=&2\log \frac{S^{-}\!({\rho_{\scriptscriptstyle 1}}-v)}{S^{-}\!(- {{\sf r}_{\scriptscriptstyle 1}}-v)}\,,\\
{\mathscr{D}}^{(R)}_{prod}(v)&=&-2\log \frac{S^{-}\!({{\sf r}_{\scriptscriptstyle 1}}-v)}{S^{-}\!(- {\rho_{\scriptscriptstyle 1}}-v)}\,.\nonumber\end{aligned}$$\
$\bullet$ $Q$-particles $$\begin{aligned}
\label{equ:hybrid}
\log Y_Q(v) = & - L_{\scriptscriptstyle T\!B\!A}\, \tH_{Q} +\log \left(1+Y_{Q'} \right) \star \left(K_{\sl(2)}^{Q'Q}+2 s \star K^{Q'-1,Q}_{vwx}\right) +{\mathscr{D}}^{(0)}_Q \\
& +\!\!\!\sum_{\alpha\in\{L,R\}}\!\!\left(\log \(1 + Y^{(\alpha)}_{1|vw}\) \star s \hstar K_{yQ}+ \log \(1 + Y^{(\alpha)}_{Q-1|vw}\) \star s \phantom{\frac 1 1 }\right.\nonumber\\
& \left. \phantom{+\sum_{\alpha\in\{L,R\}}}- \log{1-Y^{(\alpha)}_-\ov 1-Y^{(\alpha)}_+} \hstar s \star K^{1Q}_{vwx}+ \frac{1}{2} \log {1- \frac{1}{Y^{(\alpha)}_-} \ov 1-\frac{1}{Y^{(\alpha)}_+} } \hstar K_{Q} \right.\nonumber\\
& \left. \phantom{+\sum_{\alpha\in\{L,R\}}}+ \frac{1}{2} \log \big(1-\frac{1}{Y^{(\alpha)}_-}\big)\big( 1 - \frac{1}{Y^{(\alpha)}_+} \big) \hstar K_{yQ} +{\mathscr{D}}^{(\alpha)}_Q \right).\nonumber\end{aligned}$$ These are the TBA equations for $Q$-particles in the hybrid form of [@Arutyunov:2009ax]. Summation over repeated indices is understood. As before, we split the driving terms in a part independent of the specific state, that is ${\mathscr{D}}^{(0)}_Q$, and sector dependent parts ${\mathscr{D}}^{(\alpha)}_Q$ which will differ between ${\Theta}$ to ${\Psi}$. We then have $${\mathscr{D}}^{(0)}_Q(v)=\sum_{i=1}^4\left(- \log S_{\sl(2)}^{1_*Q}(u_i,v)+ 2 \log{S}\star K^{1Q}_{vwx} (u_i,v)- \log{S^{1Q}_{vwx}} ( u_i,v)\right)\;,$$ where for any kernel $K$ we define $$\log{S}\star K (u,v)=\lim_{\epsilon\to0^+}\int dt\; \log S\left(u-i/g-i\epsilon-t\right)\;K(t+i\epsilon,v)\;,$$ which is the same type of contribution as for the Konishi state.
The left and right driving terms for ${\Theta}$ are $$\begin{aligned}
\nonumber
{\mathscr{D}}^{(L)}_Q(v)&=&\log S\star K^{1Q}_{vwx}(\pm{\varrho_{\scriptscriptstyle 0}},v)-\frac{1}{2}\log S_Q^-(\pm{\varrho_{\scriptscriptstyle 0}}-v)-\frac{1}{2}\log S_{yQ}(\pm{\varrho_{\scriptscriptstyle 0}}-i/g,v),\\
{\mathscr{D}}^{(R)}_Q(v)&=&-\log S\hstar K_{yQ}(\pm{r_{\scriptscriptstyle 1}},v)-\log S^-(\pm{r_{\scriptscriptstyle Q-1}}-v),\end{aligned}$$ while for ${\Psi}$ we have $$\begin{aligned}
\nonumber
{\mathscr{D}}^{(L)}_Q(v)&=&\log S\star K^{1Q}_{vwx}({\rho_{\scriptscriptstyle 0}},v)-\frac{1}{2}\log S_Q^-({\rho_{\scriptscriptstyle 0}}-v)-\frac{1}{2}\log S_{yQ}({\rho_{\scriptscriptstyle 0}}-i/g,v)\\
& &-\log S\hstar K_{yQ}(-{{\sf r}_{\scriptscriptstyle 1}},v)-\log S^-(-{{\sf r}_{\scriptscriptstyle Q-1}}-v),\\
{\mathscr{D}}^{(R)}_Q(v)&=&\log S\star K^{1Q}_{vwx}(-{\rho_{\scriptscriptstyle 0}},v)-\frac{1}{2}\log S_Q^-({\rho_{\scriptscriptstyle 0}}-v)-\frac{1}{2}\log S_{yQ}(-{\rho_{\scriptscriptstyle 0}}-i/g,v)\nonumber\\
& &-\log S\hstar K_{yQ}({{\sf r}_{\scriptscriptstyle 1}},v)-\log S^-({{\sf r}_{\scriptscriptstyle Q-1}}-v).\nonumber\end{aligned}$$ In the above, $K^{0,Q}_{vwx}=0$, $Y_{0|vw}=0$, meaning that the driving $\log S^+(v-{{\sf r}_{\scriptscriptstyle 0}})$ is not present.
Let us stress that in order to check (\[equ:hybrid\]) on the asymptotic solution, $L_{\scriptscriptstyle T\!B\!A}$ needs to be specified. We find that $$L_{\scriptscriptstyle T\!B\!A} = J+2\;,$$ for both ${\Theta}$ and ${\Psi}$, just as for Konishi [@Arutyunov:2009ax]. As discussed in [@Arutyunov:2011uz], $L_{\scriptscriptstyle T\!B\!A}$ is the maximal $J$ charge occurring in the conformal supermultiplet described by the TBA equations, and for a generic state that has full supersymmetry one indeed expects $L_{\scriptscriptstyle T\!B\!A}=J+2$. Nonetheless, there are examples of deformations of the superstring that break supersymmetry where different relations hold [@Arutyunov:2010gu; @deLeeuw:2011rw].
### The exact Bethe equations
As discussed, the finite-size energies of states ${\Theta}$ and ${\Psi}$ depend on the allowed momenta. In the mirror TBA approach, these are found by analytically continuing the $Q$-particle TBA equations to the string region and imposing the exact Bethe equation $Y_1(u_{*i}) = -1$, which is the finite size quantization condition.
The (logarithm of the) exact Bethe equation for a string rapidity $u_k$ is given by $$\begin{aligned}
\label{eq:ExactBethe}
(2n+1)\pi i = & i L_{\scriptscriptstyle T\!B\!A}\, p_k +\log \left(1+Y_{Q'} \right) \star \left(K_{\sl(2)}^{Q'1_*}+2 s \star K^{Q'-1,1_*}_{vwx}\right) +{\mathscr{D}}^{(0)}_{1_*} \\
& +\!\!\!\sum_{\alpha\in\{L,R\}}\!\!\left(\log \(1 + Y^{(\alpha)}_{1|vw}\) \star\left( s \hstar K_{y1_*}+ s^-\right)- \log{1-Y^{(\alpha)}_-\ov 1-Y^{(\alpha)}_+} \hstar s \star K^{11_*}_{vwx} \right.\nonumber\\
&\left.\ \ \ \ \ \ \ \ \ + \frac{1}{2} \log {1- \frac{1}{Y^{(\alpha)}_-} \ov 1-\frac{1}{Y^{(\alpha)}_+} } \hstar K_{1}+ \frac{1}{2} \log \big(1-\frac{1}{Y^{(\alpha)}_-}\big)\big( 1 - \frac{1}{Y^{(\alpha)}_+} \big) \hstar K_{y1_*} +{\mathscr{D}}^{(\alpha)}_{1_*} \right),\nonumber\end{aligned}$$ where the kernels have been analytically continued appropriately[^5]. As for the driving terms, we get the state independent contribution $$\begin{aligned}
{\mathscr{D}}^{(0)}_{1_*}(u_k)&=&\sum_{i=1}^4\left(- \log S_{\sl(2)}^{1_*1_*}(u_i,u_k)+ 2 \log{\rm Res}(S)\,\star K^{11_*}_{vwx} (u_i,u_k)\phantom{\frac 1 1}\right.\\
& &\phantom{\sum_{i=1}^4}\left.- 2 \log{(u_i - u_k - \tfrac{2i}{g})\,\frac{x_j^- -\tfrac{1}{x_k^-}}{x_j^- - x_k^+}}\right)\;.\nonumber\end{aligned}$$ Coming to the state-dependent terms, for ${\Theta}$ we have $$\begin{aligned}
\nonumber
{\mathscr{D}}^{(L)}_Q(u_k)&=&\log S\star K^{11_*}_{vwx}(\pm{\varrho_{\scriptscriptstyle 0}},u_k)-\frac{1}{2}\log S_1^-(\pm{\varrho_{\scriptscriptstyle 0}}-u_k)-\frac{1}{2}\log S_{y1_*}(\pm{\varrho_{\scriptscriptstyle 0}}-i/g,u_k),\\
{\mathscr{D}}^{(R)}_Q(u_k)&=&-\log S\hstar K_{y1_*}(\pm{r_{\scriptscriptstyle 1}},u_k)-\log S(\pm{r_{\scriptscriptstyle 1}}-v),\end{aligned}$$ while for ${\Psi}$ we have $$\begin{aligned}
\nonumber
{\mathscr{D}}^{(L)}_Q(u_k)&=&\log S\star K^{11_*}_{vwx}({\rho_{\scriptscriptstyle 0}},u_k)-\frac{1}{2}\log S_1^-({\rho_{\scriptscriptstyle 0}}-u_k)-\frac{1}{2}\log S_{y1_*}({\rho_{\scriptscriptstyle 0}}-i/g,u_k),\\
& &-\log S\hstar K_{y1_*}(-{{\sf r}_{\scriptscriptstyle 1}},u_k)-\log S(-{{\sf r}_{\scriptscriptstyle 1}}-v),\\
{\mathscr{D}}^{(R)}_Q(u_k)&=&\log S\star K^{11_*}_{vwx}(-{\rho_{\scriptscriptstyle 0}},u_k)-\frac{1}{2}\log S_1^-(-{\rho_{\scriptscriptstyle 0}}-u_k)-\frac{1}{2}\log S_{y1_*}(-{\rho_{\scriptscriptstyle 0}}-i/g,u_k),\nonumber\\
& &-\log S\hstar K_{y1_*}({{\sf r}_{\scriptscriptstyle 1}},u_k)-\log S({{\sf r}_{\scriptscriptstyle 1}}-v).\nonumber\end{aligned}$$ We used the short-hand $$\log {\rm Res}(S)\star K^{11_*}_{vwx} (u,v) = \int_{-\infty}^{+\infty}{\rm d}t\,\log\Big[S(u-i/g -t)(t-u)\Big] K_{vwx}^{11*}(t,v)\;,$$ and indicated the momentum of the magnon as $p = i\tH_{Q}(z_{*})=-i\log{x_s(u+{i\ov g})\ov x_s(u-{i\ov g})}$.
Expanding the exact Bethe equation about the asymptotic $Y$-functions, we find, modulo $2\pi i$, $$\begin{aligned}
\label{equ:Rk}
\mathcal{R}_k \equiv & \,2i\,p_k+\sum_{i=1}^4\left( 2 \log{\rm Res}(S)\,\star K^{11_*}_{vwx} (u_i,u_k)\phantom{\frac 1 1}- 2 \log{(u_i - u_k - \tfrac{2i}{g})\,\frac{x_j^- -\tfrac{1}{x_k^-}}{x_j^- - x_k^+}}\right)\\
&+\!\!\!\sum_{\alpha\in\{L,R\}}\!\!\left(- \log \mathscr{N}_{*}^{(\alpha)}+\log \(1 + Y^{(\alpha)}_{1|vw}\) \star\left( s \hstar K_{y1_*}+ s^-\right)- \log{1-Y^{(\alpha)}_-\ov 1-Y^{(\alpha)}_+} \hstar s \star K^{11_*}_{vwx} \right.\nonumber\\
&\left.\ \ \ \ \ \ \ \ \ \ \ \ + \frac{1}{2} \log {1- \frac{1}{Y^{(\alpha)}_-} \ov 1-\frac{1}{Y^{(\alpha)}_+} } \hstar K_{1}+ \frac{1}{2} \log \big(1-\frac{1}{Y^{(\alpha)}_-}\big)\big( 1 - \frac{1}{Y^{(\alpha)}_+} \big) \hstar K_{y1_*} +{\mathscr{D}}^{(\alpha)}_{1_*} \right) = 0 \, , \nonumber\end{aligned}$$ where the expression is evaluated at $u_k$[^6]. The terms $\log \mathscr{N}_{*}^{(\alpha)}$ arise from the analytic continuation of $$\label{equ:normalizfactor}
\mathscr{N}^{(\alpha)}(v)=\prod_{i=1}^{K^{\rm{II}}_{(\a)}}{{\frac{y^{(\a)}_i-x^-(v)}{y^{(\a)}_i-x^+(v)}\sqrt{\frac{x^+(v)}{x^-(v)}} \, }},$$ that comes from the Bethe-Yang equations (\[equ:betheyangmain\]), appearing whenever $K_{\a}^{\mathrm{II}}>0$. Equation (\[equ:Rk\]) can be verified numerically to ensure that the analytic continuation has been performed correctly.
Since (\[equ:hybrid\]) contains a sum over the left and right sectors, the form of the resulting exact Bethe equation is the same for ${\Theta}$ and ${\Psi}$. We might wonder whether this gives same momenta for both states, but this is of course not the case because the set of auxiliary $Y$-functions for the two states will be completely different. Indeed, even in the asymptotic case, the numerical value of the two set of roots is different: ${\varrho_{\scriptscriptstyle M}}\neq{\rho_{\scriptscriptstyle M}}$ and ${r_{\scriptscriptstyle M}}\neq{{\sf r}_{\scriptscriptstyle M}}$.
Finally, recall that the energy of each state is given by (\[eq:energy\]). Since we have seen that the two set of TBA equations of ${\Theta}$ and ${\Psi}$ differ, we expect the energies $E^{\Theta}$ and $E^{\Psi}$ to be different as well. We will now show this explicitly by evaluating the first order wrapping corrections to the energy in both cases.
Wrapping corrections
--------------------
As shown above, the TBA equations for the two states we consider are not equivalent. Therefore, the resulting $Y_Q$ functions and hence the energies should be different, thus lifting the degeneracy of the asymptotic Bethe ansatz. We will directly compute the leading order wrapping corrections to the energy to see this explicitly, naturally finding different results for the two states.
The leading order wrapping correction to the energy can be conceptually seen to arise from Lüscher corrections [@Luscher:1985dn], or equivalently by perturbatively expanding the free energy of the mirror model [@Arutyunov:2009ur], depending on your point of view.
Using the asymptotic expression for the $Y_Q$-functions, (\[eq:YQasympt\]), we can compute the leading order wrapping correction. To do so we evaluate our $Y_Q$-functions to lowest order in $g$, which give leading order wrapping interactions at seven loops. As expected the resulting $Y_Q$-functions are manifestly different. The expanded $Y_Q$-functions for either state are given in appendix \[ap:expansions\]. Recall that the leading order wrapping correction to the energy is given by $$\begin{aligned}
\nonumber
E_{LO} = -\frac{1}{2\pi}\sum_{Q=1}^{\infty}\int dv \frac{d\tilde{p}}{dv} Y^{\circ}_{Q}(v).\end{aligned}$$ Integrating and summing the Y-functions for $J=4$ yields the following explicit wrapping correction for our states, $$\begin{aligned}
\label{eq:wrappingJ4}
E^{\Theta}_{LO} & = \, - \left(\tfrac{231}{32} \, \zeta(11) + \tfrac{21}{32} \, \zeta(9)- \tfrac{259}{32} \, \zeta(7)- \tfrac{113}{16} \, \zeta(5) +\tfrac{161}{32} \, \zeta(3) + \tfrac{1887}{1024} \right)\, g^{14}\\
& \approx \, \, -0.2761\, g^{14},\nonumber\\
E^{\Psi}_{LO} & = \, - \left(\tfrac{231}{32} \, \zeta(11) + \tfrac{105}{64}\, \zeta(9)- \tfrac{553}{64} \, \zeta(7)- \tfrac{589}{64} \, \zeta(5) +\tfrac{49}{8} \, \zeta(3) + \tfrac{2269}{1024} \right)\, g^{14} \\
& \approx \, \, -0.1889\, g^{14}\nonumber.\end{aligned}$$ This shows explicitly how the degeneracy present in the asymptotic Bethe ansatz is lifted by finite size (wrapping) corrections, with ${\Theta}$ being the lighter state.
Conclusion
==========
In this paper we have described a symmetry enhancement taking place for the $\AdS$ superstring in the asymptotic limit. Due to this enhancement certain states degenerate in the asymptotic limit, as described through the asymptotic Bethe ansatz. This symmetry is not present in the finite size model, indicating a *qualitative* feature of the model that is not captured by the asymptotic solution. We illustrated these ideas on a set of two asymptotically degenerate states, by showing that they have manifestly different TBA equations, as well as explicitly computing their leading order wrapping corrections, clearly showing lifting of the asymptotic degeneracy. It would be interesting to verify these result on the gauge theory side, where two *unrelated* sets of operators should have identical scaling dimensions exactly and only up to wrapping order.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to Gleb Arutyunov for useful discussions, and to Sergey Frolov, Marius de Leeuw and Ryo Suzuki for useful comments on the manuscript. The work by A.S. is part of the VICI grant 680-47-602 of the Netherlands Organization for Scientific Research (NWO). The work by S.T. is a part of the ERC Advanced Grant research programme No. 246974, [*“Supersymmetry: a window to non-perturbative physics"*]{}.
Transfer matrices and asymptotic $Y$-functions {#ap:transfermatrices}
==============================================
The eigenvalues of the transfer matrix $T^{(\alpha)}_{Q,1}$ in the $\sl(2)$-grading are known from [@Beisert:2006qh; @Arutyunov:2009iq]. The index $\alpha=L,R$ labels the sector; for clarity we suppress it from $T_{Q,1}$ as well as from the auxiliary roots $y,w$ that parametrize the eigenvalues. We have $$\begin{aligned}
\label{eqn;FullEignvalue1}
&&T_{Q,1}(v)=\prod_{i=1}^{K^{\rm{II}}}{\textstyle{\frac{y_i-x^-}
{y_i-x^+}\sqrt{\frac{x^+}{x^-}} \, }}\left[1+
\prod_{i=1}^{K^{\rm{II}}}{\textstyle{
\frac{v-\nu_i+\frac{i}{g}Q}{v-\nu_i-\frac{i}{g}Q}}}\prod_{i=1}^{K^{\rm{I}}}
{\textstyle{\left[\frac{(x^--x^-_i)(1-x^-
x^+_i)}{(x^+-x^-_i)(1-x^+
x^+_i)}\frac{x^+}{x^-} \right]}} \right.\\
&&{\textstyle{+}}
\sum_{k=1}^{Q-1}\prod_{i=1}^{K^{\rm{II}}}{\textstyle{
\frac{v-\nu_i+\frac{i}{g}Q}{v-\nu_i+\frac{i}{g}(Q-2k)}}} \Big[
\prod_{i=1}^{K^{\rm{I}}}{\textstyle{\frac{x(v+(Q-2k)\frac{i}{g})-x_i^-}{x(v+(Q-2k)\frac{i}{g})-x_i^+}}}+
\prod_{i=1}^{K^{\rm{I}}}{\textstyle{\frac{1-x(v+(Q-2k)\frac{i}{g})x_i^-}{1-x(v+(Q-2k)\frac{i}{g})x_i^+}
}}\Big]\prod_{i=1}^{K^{\rm{I}}}{\textstyle{\frac{x^+-x_i^+}{x^+-x_i^-}\frac{v-v_i-(2k+1-Q)\frac{i}{g}}{v-v_i+(Q-1)\frac{i}{g}
}}}\nonumber\\
&& -\sum_{k=0}^{Q-1}\prod_{i=1}^{K^{\rm{II}}} {\textstyle{
\frac{v-\nu_i+\frac{i}{g}Q}{v-\nu_i+\frac{i}{g}(Q-2k)}}}\prod_{i=1}^
{K^{\rm{I}}}{\textstyle{\frac{x^+-x^+_i}{x^+-x^-_i}\sqrt{\frac{x^-_i}{x^
+_i}} \frac{v-v_i-(2k+1-Q)\frac{i}{g}}{v-v_i+(Q-1)\frac{i}{g}
}}}\prod_{i=1}^{K^{\rm{III}}}{\textstyle{\frac{w_i-v
+\frac{i(2k-1-Q)}{g}}{w_i-v+\frac{i(2k+1-Q)}{g}} }} \nonumber\\
&&\left. -\sum_{k=0}^{Q-1}\prod_{i=1}^{K^{\rm{II}}} {\textstyle{
\frac{v-\nu_i+\frac{i}{g}Q}{v-\nu_i+\frac{i
}{g}(Q-2k-2)}}}\prod_{i=1}^
{K^{\rm{I}}}{\textstyle{\frac{x^+-x^+_i}{x^+-x^-_i}\sqrt{\frac{x^-_i}{x^
+_i}} \frac{v-v_i-(2k+1-Q)\frac{i}{g}}{v-v_i+(Q-1)\frac{i}{g}
}}}\prod_{i=1}^{K^{\rm{III}}}{\textstyle{\frac{w_i-v+\frac{i}{g}(2k+3-Q)}{
w_i-v+\frac{i}{g}(2k+1-Q)}}}\right]. \nonumber\end{aligned}$$ The variable $$v=x^++\frac{1}{x^+}-\frac{i}{g}a=x^-+\frac{1}{x^-}+\frac{i}{g}a\,$$ takes values in the mirror theory rapidity plane, so that $x^{\pm}=x(v\pm \frac{i}{g}a)$ where $x(v)$ is the mirror theory $x$-function. Similarly, $x^{\pm}_j=x_s(u_j\pm \frac{i}{g})$, where $x_s$ is the string theory $x$-function. Recall that $$\begin{aligned}
&x(u) = \frac{1}{2}(u-i\sqrt{4-u^2}), && x_s(u) = \frac{u}{2}(1+\sqrt{1-4/u^2}).\end{aligned}$$ Notice that the transfer matrix comes with a prefactor of $\mathscr{N}=\prod_{i=1}^{K^{\rm{II}}}\frac{y_i-x^-}
{y_i-x^+}\sqrt{\frac{x^+}{x^-}}$ encountered already in the Bethe-Yang equations (\[equ:betheyangmain\]) and in (\[equ:normalizfactor\]). As discussed in [@Arutyunov:2011uz], this is consistent with the requirement that $Y_{1*}(u_k)=-1$ on a solution of Bethe-Yang equations.
From the transfer matrix one can construct asymptotic $Y$-functions[^7] as follows $$Y_{M|w}^{(\a)} = \frac{T_{1,M}^{(\a)} T_{1,M+2}^{(\a)}}{T_{2,M+1}^{(\a)}} , \ \ Y_{-}^{(\a)} = -\frac{T_{2,1}^{(\a)}}{T_{1,2}^{(\a)}} , \ \ Y_{+}^{(\a)} = - \frac{T_{2,3}^{(\a)}T_{2,1}^{(\a)}}{T_{1,2}^{(\a)}T_{3,2}^{(\a)}}, \ \ Y_{M|vw}^{(\a)} = \frac{T_{M,1}^{(\a)} T_{M+2,1}^{(\a)}}{T_{M+1,2}^{(\a)}},\label{eq:YmvwinT}$$ in each sector, $\a=L,R$. Recall that the $Y_Q$ functions are given asymptotically by (\[eq:YQasympt\]).
Expansions for $Y^{\circ}_Q$ functions {#ap:expansions}
======================================
Taking the transfer matrix (\[eqn;FullEignvalue1\]) and expanding in the coupling constant yields the following expressions for the left and right transfer matrices of state ${\Theta}$ $$\begin{aligned}
\label{eq:T2}
T^{(L)}_{Q,1}(\vec{u}|v) = &\scriptstyle A_Q [6 Q^6+Q^4 \left(5 u_1^2+5 u_3^2+18 v^2+2\right)+Q^2 \left(u_1^4-2 v^2 \left(u_1^2+u_3^2+10\right)-2 \left(u_1^2+u_3^2+3\right)+u_3^4+18 v^4\right)\\
& \scriptstyle -v^4 \left(7 u_1^2+7 u_3^2+22\right)-\left(u_1^2+1\right) \left(u_3^2+1\right)\left(u_1^2+u_3^2+2\right)+v^2 \left(u_1^4+u_1^2 \left(8 u_3^2+6\right)+u_3^4+6u_3^2+2\right)+6 v^6] \nonumber\\
\label{eq:T0}
T^{(R)}_{Q,1}(\vec{u}|v) = &\scriptstyle \frac{A_Q}{3}[ \left(u_1^2+u_3^2+2\right) \left(Q^4+Q^2 \left(3 u_1^2+3 u_3^2-2 v^2+2\right)+v^2 \left(3 u_1^2+3 u_3^2+2\right)-3 \left(u_1^2+1\right) \left(u_3^2+1\right)-3 v^4\right)]\end{aligned}$$ where $$\nonumber
A_Q = {\textstyle \frac{8Q}{\left(u_1^2+1\right) \left(u_3^2+1\right) \left(Q^2+v^2\right)^2 \left(u_1^2+(Q-i v-1)^2\right) \left(u_3^2+(Q-i v-1)^2\right)}\,} .$$ The expansion of the individual transfer matrices is rather convoluted for the ${\Psi}$ state, so we present only the result for the product, given by $$\begin{aligned}
\label{eq:T1T1}
T^{(L)}_{Q,1}T^{(R)}_{Q,1}(\vec{u}|v) = & \scriptstyle A_Q^2 (Q^2 +v^2) \left(u_1^2+u_3^2+2\right) \big[Q^8 \left(9 \left(u_1^2+u_3^2+2\right)-8 v^2\right)\\
&\scriptstyle \nonumber \quad +Q^6 \left(28 v^2 \left(u_1^2+u_3^2+2\right)+6
\left(\left(u_1^2+u_3^2\right)^2-4\right)-32 v^4\right)\\
&\scriptstyle \nonumber \quad +Q^4 \big(46 v^4 \left(u_1^2+u_3^2+2\right)+2 v^2 \left(u_1^4-6 u_1^2 \left(u_3^2+4\right)+u_3^4-24
u_3^2-44\right)\\
&\scriptstyle \nonumber \quad \quad \quad +\left(u_1^2+u_3^2+2\right) \left(u_1^4-2 u_1^2 \left(2 u_3^2+5\right)+u_3^4-10 u_3^2-2\right)-48 v^6\big)\\
&\scriptstyle \nonumber \quad +2 Q^2 \big(22 v^6 \left(u_1^2+u_3^2+2\right)-\left(u_1^2+1\right) \left(u_3^2+1\right) \left(\left(u_1^2+u_3^2\right)^2-4\right)\\
&\scriptstyle \nonumber \quad \quad \quad-v^4 \left(7 u_1^4+6 u_1^2 \left(5 u_3^2+8\right)+7 u_3^4+48 u_3^2+52\right)\\
&\scriptstyle \nonumber \quad \quad \quad+v^2 \left(u_1^2+u_3^2+2\right) \left(u_1^4+u_1^2 \left(8 u_3^2+2\right)+u_3^4+2 u_3^2+10\right)-16 v^8\big)\\
&\scriptstyle \nonumber \quad \quad \quad+17 v^8 \left(u_1^2+u_3^2+2\right)+\left(u_1^2+1\right)^2 \left(u_3^2+1\right)^2 \left(u_1^2+u_3^2+2\right)\\
&\scriptstyle \nonumber \quad \quad \quad-2 v^6 \left(5 u_1^4+6 u_1^2 \left(3 u_3^2+4\right)+5 u_3^4+24 u_3^2+20\right)\\
&\scriptstyle \nonumber \quad \quad \quad+v^4 \left(u_1^2+u_3^2+2\right) \left(u_1^4+2 u_1^2 \left(10 u_3^2+7\right)+u_3^4+14 u_3^2+22\right)\\
&\scriptstyle \nonumber \quad \quad \quad-2 \left(u_1^2+1\right) \left(u_3^2+1\right) v^2 \left(u_1^4+u_1^2 \left(6 u_3^2+4\right)+u_3^4+4 u_3^2\right)-8 v^{10} \big]\end{aligned}$$ The S-matrix in the string-mirror region $S_{\sl(2)}^{1_*Q}$ is found in [@Arutyunov:2009kf] (see also [@Bajnok:2009vm]) and has the following leading behavior in $g$ $$S_{\sl(2)}^{1_*Q}(u,v)= {\textstyle-\frac{\big[(v-u)^2+(Q+1)^2\big]\big[Q-1 + i (v-u)\big]}{(u-i)^2 \big[Q-1-i( v-u)\big]} + O(g^2) }\, .$$ These expressions are enough to build up the leading term in the weak-coupling expansion of the asymptotic function $Y^o_Q$, which is given by $$\begin{aligned}
Y^{\circ}_Q(v) =\, \frac{g^{2J}}{Q^2+v^2} \frac{T^{(L)}(\vec{u}|v)T^{(R)}(\vec{u}|v)}{\prod_i S_0(v,u_i)}\, ,\end{aligned}$$ where for our specific states we take either the product of (\[eq:T0\]) and (\[eq:T2\]), or (\[eq:T1T1\]).
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[^1]: For a review of integrability in the AdS/CFT correspondence see [@Arutyunov:2009ga; @Beisert:2010jr].
[^2]: The associated Y-system was conjectured in [@Gromov:2009tv].
[^3]: The use of Lüscher’s approach [@Luscher:1985dn] in the AdS/CFT correspondence was first advocated in [@Ambjorn:2005wa].
[^4]: Here and afterwards, $*$ indicates analytic continuation to the string region. Also, $S_{1_*y}(u_{j},v) \equiv S_{1y}(u_{*j},v)$ is shorthand notation for the S-matrix with the first and second arguments in the string and mirror regions, respectively. The same convention is used for other kernels and S-matrices.
[^5]: See the appendix of [@Arutyunov:2009ax] for details.
[^6]: As in the Konishi case [@Arutyunov:2010gb], this equation still holds for small perturbations around the solution of the Bethe-Yang equation, $\{u_k\}$.
[^7]: The general construction of the Y-functions in terms of transfer matrices is based on the underlying symmetry group of the model [@Kuniba:1993cn; @Tsuboi:2001ne]. For the string sigma model asymptotic Y-functions were presented in [@Gromov:2009tv]. In fact, this solution can be directly derived from the Bajnok-Janik formula [@Bajnok:2008bm] and the AdS/CFT Y-system, see [@Arutyunov:2011uz].
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abstract: 'The transitions in disordered substances are discussed briefly: liquid–liquid phase transitions, liquid–glass transition and the transformations of one amorphous form to another amorphous form of the same substances. A description of these transitions in terms of many–particle conditional distribution functions is proposed. The concept of a hidden long range order is proposed, which is connected with the broken symmetry of higher order distribution functions. The appearance of frustration in simple supercooled Lennard–Jones liquid is demonstrated.'
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New Kinds of Phase Transitions: Transformations in disordered Substances {#new-kinds-of-phase-transitions-transformations-in-disordered-substances .unnumbered}
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V.N. Ryzhov$^{1}$ and E.E. Tareyeva$^{1}$
[(1) [*Institute for High Pressure Physics, Russian Academy of Sciences, Troitsk 142190, Moscow region, Russia*]{} ]{}
It is well known for a long time that there exist sharp phase transitions between different aggregate states and sharp polymorphic phase transitions between different crystalline phases of the same substances. During last two decades a lot of experimental data was obtained on complicated phase diagrams of liquids and amorphous solids, too. Some of these results were presented on the first international conference that took place here in Russia in 2001 [@book] and that can be considered as a formal claim of a new direction of physical investigations – transformations in disordered substances: liquid–liquid transitions and transformations of one amorphous form to another amorphous form of the same substances. The liquid–glass transition, although has longer history, has to be considered in the same context. It is now firmly established by different experimental techniques that sharp liquid–liquid transitions under pressure, formally similar to first–order phase transitions, exist as well as reversible transformations between amorphous states involving changes in local order structures and density. Usually a crystal melts with a conservation of the short–range order (SRO) structure type or into denser liquid with SRO structure similar to that of high pressure crystalline phase.
It should be emphasized that the transitions in liquids are true phase transitions mainly determined by thermodynamic relationships, whereas the transitions in amorphous solids take place far away from equilibrium and are governed by the corresponding kinetics.
A useful microscopic theory of these transitions is not developed yet and only empirical models and computer simulations have been used in practice to date. For example, interesting results were obtained by Stanley through molecular dynamics study basing on the taking into account the hydrogen bonding in supercooled water [@stanley1]. It is by the demonstration of a simple analytic way of obtaining Stanley results that we begin the presentation of our own results on this subject (see, e.g., [@book; @ryz79; @ryz1; @pn; @clust] and references therein).
From the intuitive point of view liquid-liquid phase transition between low density and high density phases may be related to the competition between expanded and compact structures. This suggests that the potential should have two equilibrium positions. The most obvious form of such potential is: $$\Phi(r)=\left\{
\begin{array}{ll}
\infty, & r\leq \sigma\\
0, & \sigma<r\leq a\\
-\varepsilon_1, & a<r\leq b\\
0, & b<r\leq c\\
-\varepsilon_2, & c<r\leq d.
\end{array}\right..
\label{1}$$ This two–well potential may be considered as a model for the water potential [@stanley1].
To investigate the possibility of the existence of the second critical point in this case we developed the mean-field (van der Waals–like) theory. Using the well-known Bogoliubov inequality for the free energy we can write $F\leq F_{HS}+<U-U_{HS}>_{HS}$. Here $F_{HS}$ is the free energy of the system of hard spheres with diameter $\sigma$, and we consider the attractive part as a perturbation. Here $U=\frac{1}{2}\sum_{i\neq j}^N\Phi(r_{ij})$ and $U_{HS}=\frac{1}{2}\sum_{i\neq j}^N\Phi_{HS}(r_{ij})$. The average over the hard sphere potential has the form $$<U-U_{HS}>_{HS}=2\pi\rho N\int_0^{\infty}\Phi_{atr}(r)g_{HS}(r)r^2dr,
\label{2}$$ where $\Phi_{atr}(r)=\Phi(r)-\Phi_{HS}(r)$, $g_{HS}(r)$ is the radial distribution function of the hard sphere system which we take in the Percus-Yevick approximation [@henderson] and for $F_{HS}$ we use the approximate Carnahan-Starling equation [@barker2] $$\frac{F_{HS}}{k_BTN}=3\ln\lambda-1+\ln\rho+\frac{4\eta-3\eta^2}{(1-\eta)^2}.
\label{3}$$ Here $\lambda=h/(2\pi mk_BT)^{1/2}$.
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The equation of state is given by $P=\rho^2\partial(F/N)/\partial\rho$. In Fig. 1 the two families of isotherms are shown for the temperatures close to two critical points $\beta_1=\varepsilon_1/k_BT_{c1}$ and $\beta_2=\varepsilon_1/k_BT_{c2}$. Fig. 1 shows that at low temperatures ($\beta=\varepsilon_1/k_BT>\beta_1$) there are two van der Waals–like loops in the equation of state which correspond to two fluid-fluid transitions. In the temperature region $\beta_2<\beta<\beta_1$ there is only one loop which corresponds to the well known gas-liquid transition, $\beta_2$ being the gas-liquid critical point temperature and $\beta_1$ – the liquid-liquid critical temperature.
In this example, as at the ordinary critical point, no symmetry of the correlation functions is broken at the transition. The order parameter is the difference of densities of high and low density phases $\Delta\rho=\rho_{l1}-\rho_{l2}$. However, it is interesting to describe the change of the local structure and the cluster symmetry at the transition.
To describe different kinds of space symmetry breaking we use the formalism of classical many particle conditional distribution functions $$F_{s+1}({\bf r}_1|{\bf r}_1^0 ... {\bf r}_s^0)=
\frac{F_{s+1}({\bf r}_1, {\bf r}_1^0,...,{\bf r}_s^0)}
{F_s({\bf r}_1^0,...,{\bf r}_s^0)}.$$ Here $F_s({\bf r}_1,...,{\bf r}_s)$ is the usual $s$–particle distribution function [@NNB2]. The functions $F_{s+1}({\bf r}_1|{\bf r}_1^0 ...
{\bf r}_s^0)$ satisfy the equation $$\begin{aligned}
\frac{\rho F_{s+1}({\bf r}_1|{\bf r}_1^0 ... {\bf r}_s^0)}{z}& = &\exp
\left\{ -\beta \sum_{k=1}^s \Phi({\bf r}_1-{\bf r}_k^0) +\sum_{k \geq 1}
\frac{\rho^k}{k!} \int \, S_{k+1}({\bf r}_1,...,{\bf r}_{k+1}) \right.
\nonumber \\
& &\left.\times F_{s+1}({\bf r}_|{\bf r}_1^0 ... {\bf r}_s^0)...
F_{s+1}({\bf r}_{k+1}|{\bf r}_1^0 ... {\bf r}_s^0)
d{\bf r}_2... d{\bf r}_{k+1} \right\} \label{main}.\end{aligned}$$ Here $z $ is activity, $\rho$ is the mean number density, $S_{k+1}({\bf
r}_1,...,{\bf r}_{k+1})$ is the irreducible cluster sum of Mayer functions connecting (at least doubly) $k+1$ particles, $\beta=1/k_B T$.
The simplest case is the symmetry breaking of the one–particle function. In the solid phase the local density, proportional to the one-particle distribution function, has the symmetry of a crystal lattice and can be expanded in a Fourier series in reciprocal lattice vectors ${\bf
G}$: $$\rho({\bf r})=\sum_{\bf G}\rho_{\bf G}e^{i{\bf Gr}},
\label{4}$$ where the Fourier coefficients $\rho_{\bf G}$ are the order parameters for the transition.
The Taylor expansion of the corresponding free energy functional around the liquid can be written in the following form: $$\beta \Delta F = \int
d{\bf r} \varrho ({\bf r}) \ln \frac {\varrho ({\bf r})}
{\varrho _0} - \sum_{k \geq 2} {1 \over k!} \int c^{(n)} ({\bf
r}_1,...,{\bf r}_k) \Delta \varrho ({\bf r}_1)...\Delta\varrho
({\bf r}_k) d{\bf r}_1 ... d{\bf r}_k , \label{exfree}$$ where $$\Delta \varrho ({\bf r}) = \varrho ({\bf r}) - \varrho_l$$ is the local density difference between solid and liquid phases. This is the base of the density functional theory of freezing (DFT) [@ryz79]. In the frame of this approach tens of melting curves were calculated (see, e.g., the reviews [@rev]). The full system of equations to be solved in DFT contains the nonlinear integral equation for the function $\rho ({\bf r})$, obtained as the extremum condition for the free energy and the equilibrium conditions for the chemical potential and the pressure written in terms of the same functions as in (\[exfree\]). To proceed constructively in the frame of DFT we have to choose an actual form of the free energy functional – a kind of closure or truncating – and we must make an ansatz for the average density of the crystal. The importance of such an ansatz follows from the fact that we are dealing with a theory which is equivalent to Gibbs distribution and one has to break symmetry following the Bogoliubov concept of quasiaverages [@bogol1].
Now let us consider a state of matter which is characterized by the uniform local density, but the broken symmetry of the two–particle distribution function. Such type of order is called the bond orientations order (BOO), where “bond” is the vector joining a particle with its nearest neighbor. This kind of order is well known in theories of two-dimensional melting (hexatic phase) [@halpnel79; @ryz1]. Near the transition to anisotropic liquid state we have: $$F_2({\bf r}_1|{\bf r}_1^0)=g(|{\bf r}_1-{\bf r}_1^0|)
(1+f({\bf r}_1-{\bf r}_1^0)), \label{F2bo}$$ where $f({\bf r}_1-{\bf r}_1^0)$ has the symmetry of the local environment of the particle at ${\bf r}_1^0$ and may be written in the form $f({\bf r}_1-{\bf r}_1^0)=f(|{\bf r}_1-{\bf r}_1^0|,\Omega),$ $\Omega$ determines the direction of the vector ${\bf r}_1-{\bf r}_1^0$. In the case of three dimensions function $f(r,\Omega)$ may be expanded in a series in spherical harmonics: $$f(r,\Omega)=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}f_{lm}(r)Y_{lm}(\Omega).
\label{3d}$$
The microscopic equations for the order parameters $f_{lm}(r)$ can be obtained from the main equation (\[main\]). The linearized equation determines the instability of isotropic liquid against the formation of the state with BOO and has the form [@ryz1]: $$f_{lm}(r)-\frac{4\pi}{2l+1}\int\,\Gamma_l(r,r')g(r')f_{lm}(r')r'^2\,dr'=0.
\label{inst}$$ Here $\Gamma_l(r,r')$ correspond to the isotropic liquid when $$\begin{aligned}
\Gamma({\bf r}_1,{\bf r}_1^0,{\bf r}_2)&=&\sum_{k\geq 1}\frac{\rho^k}
{(k-1)!}\int\,S_{k+1}({\bf r}_1...{\bf r}_{k+1})\times \nonumber\\
&\times&g(|{\bf r}_3-{\bf r}_1^0|)\cdots g(|{\bf r}_{k+1}-{\bf r}_1^0|)\,
d{\bf r}_3 \cdots d{\bf r}_{k+1}. \label{Gamma}\end{aligned}$$ reduces to $$\Gamma({\bf r}_1,{\bf r}_1^0,{\bf r}_2)=
\Gamma(r,r',\theta), \label{K}$$ $$\Gamma(r,r',\theta)=\sum_{l=0}^{\infty}\frac{4\pi}{2l+1}\Gamma_l(r,r')
\sum_{l=-m}^{l}Y_{lm}(\Omega_1)Y_{lm}^*(\Omega_2), \label{Gexp}$$ The angles $\Omega_1$ and $\Omega_2$ determine the directions of the vectors ${\bf r}$ and ${\bf r'}$ and $r=|{\bf r}_1-{\bf r}_1^0|, r'=|{\bf r}_2-{\bf r}_1^0|, \theta$ is the angle between vectors ${\bf r}$ and ${\bf r'}$. It should be notice that the correlation length of the orientational fluctuations $\xi_{l,m}\rightarrow\infty$ when approaching the instability line given by Eq. (\[inst\]).
To describe liquid–liquid and liquid–glass transitions we must consider isotropic case with rotationally invariant two–particle distribution function. A possible description of these cases can be given in terms of broken symmetry of higher order distribution functions. At high temperature the nearest neighbors of a molecule can take different relative positions and there is no SRO. At lower temperature SRO appears which can be of different kinds at different densities. The rotation and the translation of the clusters of preferred symmetry give rise to the fact that one-particle and two-particle distribution functions remain isotropic. If a kind of BOO appears the clusters are oriented in similar way and the two-particle distribution function becomes to be anisotropic (as in 2D hexatic phase). However, we can imagine another situation – freezing of the symmetry axes of the clusters in different position. The isotropic phase can be considered as analogous to the paramagnetic phase (of cluster symmetry axes), the BOO phase – to the ferromagnetic phase, and the mentioned freezed phase – to a spin glass phase.
Let us consider for simplicity a 2D system. In the vicinity of the transition one can write (in the superposition approximation for the liquid) $$F_3({\bf r}_1| {\bf r}_1^0, {\bf r}_2^0) =
g(|{\bf r}_1-{\bf r}_1^0|) g(|{\bf r}_1-{\bf r}_2^0|)(1+f_3({\bf
r}_1| {\bf r}_1^0, {\bf r}_2^0) \label{gla}$$ In 2D case $f_3({\bf r}_1| {\bf r}_1^0, {\bf r}_2^0)$ depends in fact on two distances and two angles $$f_3({\bf r}_1| {\bf r}_1^0, {\bf r}_2^0) =
f_3(R_0, \phi _0;R_1, \Theta _1 ),
\label{gla1}$$ where $ {\bf R}_0 = {\bf r}_2^0 - {\bf r}_1^0$, $ {\bf R}_1 = {\bf r}_1 - {\bf r}_1^0$, $ {\bf R}_2 = {\bf r}_2 - {\bf r}_1^0$ and $\phi _0$ is the angle of the vector ${\bf R}_0$ with the $z$ axis, $ \Theta _1$ – the angle between ${\bf R}_1$ and ${\bf R}_0$ and $ \Theta _2$ – the angle between ${\bf R}_2$ and ${\bf
R}_0$.
The linearization of (\[main\]) for $s=2$ gives: $$f_3(R_0, \phi _0;R_1, \Theta _1)=\int \,
\Gamma'(R_0, \phi _0;{\bf r}_2; R_1, \Theta _1)
f_3(R_0, \phi _0;R_2, \Theta _2)
g(|{\bf R}_2-{\bf R}_0|) g(R_2) d{\bf r}_2,
\label{gla2}$$ where $$\begin{aligned}
\Gamma'(R_0, \phi _0;{\bf r}_2; R_1, \Theta _1)&=&
\sum_{k \geq 1}
\frac{\rho^{k}}{(k-1)!}\, \int\, S_{k+1}({\bf r}_1,...,{\bf
r}_{k+1})g(|{\bf r}_3-{\bf r}_1^0|)\, \nonumber\\ &\times&
g(|{\bf r}_3-{\bf r}_2^0|)...
g(|{\bf r}_{k+1}-{\bf r}_1^0|)g(|{\bf r}_{k+1}-{\bf r}_2^0|) \,
d{\bf r}_3...d{\bf r}_{k+1}. \label{gla3}\end{aligned}$$ There are two kinds of angles entering the equations and two kinds of order parameters, consequently. One angle ($\phi _0$) fixes the position of one pair of particles of the cluster, and the other ($\Theta _i$) – the position of the third particle in the coordinate frame defined by $\phi _0$. The order parameter connected with $\Theta _i$ is the generalization of intracluster hexatic parameter for the case of different coordinate frames. The order parameter connected with $\phi _0$ is an analogue of magnetic moment and in glass–like phase one can consider an Edwards-Anderson parameter $<\cos \phi _0 (t) \cos \phi _0(0)>$. In such a way we come to the concept of a “conditional” or “hidden” long range order: if we consider two pairs of particles at infinite distance from one another then there exists a preferable possibility for the relative position of the third particle near each pair. The directions of the bonds in the pairs of particles themselves are subjects to spin–glass–like order. In 3D case the rotation of clusters is given by rotation matrices $D_{lm}^{l'm'}(\vec \omega _{0i})$ so that we obtain a kind of orientational multipole glass for the clusters. If the intracluster ordering is established then we can consider the system of clusters. The orientational state of this system is defined by the intercluster interaction for different values of temperature an pressure.
Now let us consider this later situation when the intracluster symmetry is fixed and let us try to estimate the intercluster orientational interaction. If the intercluster interaction had the same sign for all cluster sizes (or all clusters had the same size) one would get the state with simple BOO. However, because of the difference in cluster sizes the orientational interaction for some harmonics may change sign as a function of the cluster size (see Fig. 2). In this case the low temperature state should be amorphous for some harmonics. So the difference of the orientational interaction of the clusters for different cluster sizes may be considered as the reason of some kind of frustration in simple liquids. It should be emphasized that the form of the corresponding component of the orientational interaction is the intrinsic statistical property of the liquid and does not depend on the timescale of the fluctuations in size and symmetry of the cluster. There is no real quenched disorder in the system but only an analog of it which may be treated in a formally same way as the quenched disorder in spin glasses. To analyze qualitatively the orientational freezing in the system we introduce simple lattice model which takes into account the interaction only between clusters with definite symmetry. The model gives the possibility to conclude what harmonics freeze first and what local symmetry prevails immediately below the transition. Let us now describe our results in more detail.
Our starting point is the expression for the free energy of the system as a functional of a pair distribution function $g_2({\bf r}_i,{\bf r}_0)$ which has the form [@ryz1]: $$\begin{aligned}
&&F/k_BT=\int d{\bf r}d{\bf r}_0 \rho g_2({\bf r},{\bf r}_0)\left[\ln\left(\lambda^3
\rho g_2({\bf r},{\bf r}_0)\right)-1\right]- \nonumber\\
&&-\sum_n \frac{\rho^{n+1}}{(n+1)!}\int S_{n+1}({\bf
r}_1...{\bf r}_{n+1})
g_2({\bf r}_1,{\bf r}_0)\cdots g_2({\bf r}_{n+1},{\bf r}_0) \times \nonumber\\
&&\times d{\bf r}_1\cdots d{\bf r}_{n+1}d{\bf r}_0
-\int \Phi({\bf r}-{\bf r}_0)\rho g_2({\bf r},{\bf r}_0) d{\bf r}d{\bf r}_0, \label{1}\end{aligned}$$ where the term with logarithm corresponds to the entropy and the other terms — to the interaction energy. Here $\Phi({\bf r}-{\bf r}_0)$ - interparticle potential (for Lennard-Jones potential, $\Phi_0(r)=4\varepsilon((\sigma/r)^{12}-(\sigma/r)^6)$), $\lambda=h/(2\pi mk_BT)^{1/2}$.
We can estimate the change of the energy due to (\[F2bo\]). Omitting the entropy term in Eq.\[1\], we have up to the second order in $\delta g({\bf r},{\bf r}_0)$. $$\begin{aligned}
\Delta F/k_bT&=&-\frac{1}{2}\int \Gamma({\bf r}_1,{\bf r}_0,{\bf
r}_2)
\delta g({\bf r}_1,{\bf r}_0)\delta g({\bf r}_2,{\bf r}_0)
d{\bf r}_1 d{\bf r}_2.\label{5}\end{aligned}$$ Using the approximation for the radial distribution function $g(r)=\rho^{-1}(n_s/4\pi r_s^2)\delta(r-r_s)$ and the Eq.(\[3d\]) we obtain: $$\begin{aligned}
&&\Delta
F(r_s)/k_BT=-\frac{1}{2}\rho^{-2}\left(\frac{n_s}{4\pi}\right)^2\sum_{l=0}^{\infty}\frac{4\pi}{2l+1}
\Gamma_l(r_s,r_s)\sum_{m=-l}^{l}\int Y_{lm}(\Omega_1)\times\nonumber\\
&&\times Y_{lm}^*(\Omega_2)
f(r_s,\Omega_1)f(r_s,\Omega_2) d\Omega_1d\Omega_2=-\frac{1}{2}\sum_{l=0}^{\infty} J_l(r_s)\sum_{m=-l}^l |f_{lm}|^2.
\label{7}\end{aligned}$$ Here $J_l(r_s)=\rho^{-2}\frac{4\pi}{2l+1}(\frac{n_s}{4\pi})^2\Gamma_l(r_s,r_s)$, $n_s$ is the number of nearest neighbors of a particle and $r_s$ is the size of the cluster, which is of the order of the first coordination shell size.
The function $\Delta F(r_s)$ may be interpreted as the mean-field orientational interaction energy of the system of clusters having the size $r_s$. To get the full energy of the system one should integrate (\[7\]) over the probability of finding the cluster with the size $r_s$ which is given by the function $r_s^2g(r_s)$ in the vicinity of the first maximum.
Using the approximations of [@ryz1] for $\Gamma({\bf r}_1,{\bf r}_1^0,{\bf r}_2)$ we obtain the estimation for $J_l(r_s)$ as a function of $r_s$. Fig.2 represents $J_l(r_s)$ for $l=4$ and $6$ along with $r_s^2g(r_s)$ in the vicinity of its first peak. It is seen that $J_l(r_s)$ changes sign. This result enables us to suppose that there is a kind of frustration (which is analogous to that in spin glasses) appearing as a result of variations in the sizes of clusters according to $g(r)$ and that it is possible to study the transition in the system of interacting clusters on the base of simple model lattice Hamiltonian: $$H=-\frac{1}{2}\sum_{<i\neq
j>}\sum_{l=0}^{\infty}J^l_{ij}\sum_{m=-l}^l
U_{lm}(\Omega_i)U_{lm}^*(\Omega_j). \label{8}$$ The functions $U_{lm}(\Omega_i)$ are the lattice harmonics for the point groups corresponding to the cluster symmetry. This Hamiltonian describes correctly BOO of clusters. In this case the energy calculated from Eq. (\[8\]) in the mean-field approximation (taking into account that $\langle U_{lm}(\Omega_i)\rangle=f_{lm}$) coincides with the intercluster energy (\[7\]) under appropriate choice of $J^l_{ij}$. We will use the Hamiltonian (\[8\]) to study the system of interacting clusters with various sizes.
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To simplify the problem we neglect in Hamiltonian (\[8\]) all the terms except ones corresponding to the unit representation of the point group. Furthermore, we consider only the cases $l=4$ and $l=6$ which represent the cases of cubic and icosahedral symmetries. This Ising-like model may be called a “minimal” model: $$\label{one} H=-\frac{1}{2}\sum_{i\neq j}J_{ij}
\hat{U_i}\hat{U_j}.$$ Functions $\hat U\equiv U(\varphi,\theta)$ are the combinations of spherical harmonics. We will consider separately symmetries of “simple” cube ($l=4, m=0,\pm 4$), cube ($l=6, m=0,\pm
4$) and icosahedron ($l=6, m=0,\pm5$) correspondingly [@hay1; @Bredli]. For example, for $l=4$ one has: $$\begin{aligned}
\hat U\equiv
U(\varphi,\theta)=\sqrt{\frac{7}{12}}\left\{Y_{40}(\varphi,\theta)+
\sqrt{\frac{5}{14}}\left(Y_{44}(\varphi,\theta)+
Y_{44}(-\varphi,\theta)\right)\right\}\end{aligned}$$
The interactions $J_{ij}$ are chosen in such a way that the MF approximation gives exact solution (infinite-range interactions). It is easily seen that in the minimal model (\[one\]) without disorder in the framework of the MF approximation there is a first order phase transition to the state with BOO (compare to [@hay1; @Nelson-book]). From Fig.2 it is clear that, as the first qualitative step, $J^l_{ij}$ may be chosen as random interactions with Gaussian probability distribution $$\label{two}
P(J_{ij})=\frac{1}{\sqrt{2\pi
J}}\exp\left[-\frac{(J_{ij}-J_0)^{2}}{ 2J^{2}}\right]$$ where $ J=\tilde{J}/\sqrt{N}$ , $J_{0}=\tilde{J_0}/N$ can be related to the microscopic parameters. We approximate $r^2 g(r)$ by a gaussian exponential near the position of the first maximum $r_0$. So $r^2g(r)\sim
\exp[-(r-r_0)^2/2\sigma]$. The approximation for the functions $\Gamma_l$ is then linear: $\Gamma_l\approx \alpha+\beta(r-r_0)$. That is: $J_0=\alpha, J=\beta\sqrt\sigma$.
The free energy of the system can be obtained using replica approach (see, e.g., [@SK]). In the replica-symmetric (RS) approximation we have [@4avtora]: $$\begin{aligned}
\label{four}
F=-NkT\biggl\{-\left(\frac{\tilde{J_0}}{kT}\right)\frac{m^2}{2}+
t^2\frac{q^2}{4}-t^2\frac{p^2}{4}+\nonumber\\
\int_{-\infty}^{\infty}\frac{dz}{\sqrt{2\pi}}\exp\left(-\frac{z^2}{2}\right)\ln
{{\rm Tr}}\left[\exp\left(\hat\theta\right)\right]\biggr\},\end{aligned}$$ where the trace in this case is defined as follows: ${{\rm Tr}}(\ldots)\equiv \int_0^{2\pi}d\varphi
\int_0^\pi d\cos(\theta)(\ldots)$. Here $t=\widetilde{J}/k_BT$ and $$\hat{\theta}=\left[zt\sqrt{q}+m\left(\frac{\tilde{J_0}}{kT}\right
)\right]\hat{U}+t^2\frac{p-q}{2}\hat{U}^2.$$
The order parameters are: $ m $ is the regular order parameter (an analog of magnetic moment in spin glasses), $ q$ is the glass order parameter and $p$ is an auxiliary order parameter. The extremum conditions for the free energy (\[four\]) give the following equations for these order parameters: $$m=\overline{\langle \hat U\rangle},\qquad
p=\overline{\langle \hat U^2\rangle},\qquad
q=\overline{\langle \hat U\rangle^2} \label{prs},$$ where $\langle\ldots\rangle={{\rm Tr}}( \ldots e^{\hat\theta})/{{\rm Tr}}e^{\hat\theta}$ and $\overline{(\ldots)}=\int_{-\infty}^{\infty}
\frac{dz}{\sqrt{2\pi}}e^{-z^2/2}[\ldots]$. We find from these equations the temperature dependence of the order parameters. The RS solution is stable unless the replicon mode energy $\lambda_{\rm repl}$ is nonzero [@A-T; @4avtora]. For our model we have $$\lambda_{\rm repl}=1-t^2\overline{\langle\langle \hat U^2
\rangle\rangle^2},$$ where $\langle\langle\ldots\rangle\rangle$ denotes the irreducible correlator. We find the temperature $T_{_{A-T}}$ that corresponds to $\lambda_{\rm repl}=0$. To obtain the actual glass transition temperature one has to study the dynamics of the system. In this paper we limit ourselves by the static approach. As is usually believed [@cugl; @KT] and correctly shown in [@Kurchan] the dynamical $T_g$ can be obtained in the frame of the static approach as the temperature $T_m$ of the marginal instability of the one-step RS breaking solution. We have calculated $T_m$ and found that within the accuracy of calculations $T_m$ and $T_{_{A-T}}$ coincide. We expect that as in spin glasses below $T_{_{A-T}}$ the liquid dynamics is characterized by long relaxation times and other phenomena characteristic to glass transitions. So there is the glass transition in the simple cube case with $T_{_{A-T}}\approx 0.39$; in the other cases, icosahedron and cube, there is no glass transition but just a first order transition to BOO state at temperatures about $0.45, 0.42$ correspondingly at $\rho\sigma^3=0.973$. The last two temperatures of BOO transitions are in agreement with the results of molecular dynamics simulations of Ref.[@st83]. It should be noted that all these temperatures are well below the melting temperature $T=0.703$ at this density [@st83; @Nelson-book].
The work was supported in part by the Russian Foundation for Basic Research (Grant No 02-02-16622 (VNR), Grant No 02-02-16621 (EET) and RFBR-NWO Grant No 04-01-89005 (047.016.001.).
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abstract: 'Constitutive equations are derived for enthalpy recovery in polymeric glasses after thermal jumps. The model is based on the theory of cooperative relaxation in a version of the trapping concept. It is demonstrated that some critical temperature $T_{\rm cr}$ and some critical degree of crystallinity $f_{\rm cr}$ exist in a semicrystalline polymer above which the energy landscape becomes homogeneous and structural relaxation ceases.'
author:
- |
Aleksey D. Drozdov\
Institute for Industrial Mathematics\
4 Hanachtom Street, Beersheba, 84311 Israel
title: Enthalpy recovery in semicrystalline polymers
---
230 mm 165 mm -10 mm 0 mm
Introduction
============
This note is concerned with the kinetics of enthalpy recovery in polymeric glasses after thermal jumps. Physical aging of polymers has been the focus of attention in the past three decades, see monographs [@Str78; @Bra85; @Don92; @Mat92; @DB96] and review articles [@MK89; @MCO91; @EAN96]. Despite significant successes in the study of out-of-equilibrium dynamics in disordered materials achieved in the past decade (see reviews [@Kob97; @BCK98; @Bou99; @HVD99; @KBS00; @TK00]), it is difficult to mention a model that adequately predicts structural relaxation in polymers. Even the mode-coupling theory [@GS92] (the most advanced among molecular models) fails to describe slowing down in the response of amorphous polymers below the glass transition temperature $T_{\rm g}$ [@KBS00; @Dyr95; @CLH97]. As a reason for this conclusion, the neglect of cooperativity in the molecular reorientation is conventionally mentioned [@AG65; @MH97].
Enthalpy relaxation provides one of the most convenient ways to study aging of glassy polymers [@MCO91; @Hod94; @RP95; @HSH96; @BCF97]. The kinetics of enthalpy recovery is in close connections with time-varying changes in static and dynamic moduli [@BF95; @Yos95; @CFH98], specific volume [@CHM98; @Dro99] and yield stress [@BCB82; @CME99; @HSH99; @PGM00], as well as with transition from ductile to brittle fracture [@JL92; @MJ95]. For some polymers (e.g., polyesters), DSC measurements demonstrate a dramatic effect of the waiting time [@AH82], whereas the free volume fraction determined by PALS shows no pronounced decrease induced by physical aging [@MDJ00].
Constitutive models for structural relaxation in glassy polymers are conventionally confined to amorphous media (as exceptions, we would mention Refs. [@CM80; @Str87]). This may be explained by a belief that crystalline lamellas do not affect the process of structural relaxation occurring in amorphous regions [@TW81]. This hypothesis may be accepted provided that only one kind of amorphous domains exists in a semicrystalline polymer. A number of observations demonstrate, however, a more complicated picture, where two different kinds of amorphous regions co-exist in a partially crystallized polymer: amorphous layers between lamellas and amorphous domains between spherulites [@IB63; @GBS76; @LC96; @DCM99]. This results in a question whether the rate of physical aging is a linear function of the degree of crystallinity which vanishes when the degree of crystallization $f$ reaches 100% only (as it is postulated in Fig. 6 of Ref. [@TW81]), or the rates of structural relaxation substantially differ in interlamellar and intralamellar zones which implies that physical aging may decline even in a partially crystallized material.
The objective of this study is to derive a constitutive model for enthalpy recovery which is based on the theory of cooperative relaxation in a version of the trapping concept [@Dyr95; @Bou92; @MB96; @Sol98; @DMO99]. To simplify the analysis, we do not distinguish explicitly between interlamellar and intralamellar regions, but presume that the “averaged” distribution of energies of traps may depend on the degree of crystallinity. An amorphous region is treated as an ensemble of mutually independent cooperatively rearranged regions (CRR). A CRR is thought of as a globule consisting of scores of strands of long chains [@Sol98]. The characteristic length of a CRR in the vicinity of $T_{\rm g}$ amounts to several nanometers [@RN99]. In the phase space, a CRR is treated as a point trapped in its potential well (cage). At random times, the unit hops to higher energy levels as it is thermally agitated. When the energy of thermal fluctuation exceeds the height of a barrier between cages, a CRR can change its trap. Introducing several hypotheses regarding the kinetics of hops from one potential well to another, we develop a nonlinear parabolic equation for the probability density of traps with various potential energies. This equation is applied to the analysis of enthalpy recovery in amorphous and semicrystalline poly(ethylene terephthalate) (PET). The following conclusions are drawn:
1. the degree of crystallinity of a semicrystalline polymer affects the model’s parameters in a similar way as temperature influences those for an amorphous polymer (a crystallinity–temperature principle analogous to the time–temperature principle in linear viscoelasticity [@Fer80]);
2. a critical level of crystallinity $f_{\rm cr}<100$% exists at which the energy landscape becomes homogeneous. This level may be treated as an analog to the critical temperature for structural relaxation in amorphous polymers [@Dro00] predicted by the mode-coupling theory.
The exposition is organized as follows. Constitutive equations for the kinetics of enthalpy relaxation are developed in Section 2. These relations are verified in Section 3 by comparison with experimental data. Some concluding remarks are formulated in Section 4.
A model for enthalpy relaxation
===============================
Denote by $w$ the energy of a potential well with respect to some reference state [@Gol69]. It is assumed that $w>0$ for any trap and $w=0$ for the reference state. At random times, a CRR hops to higher energy levels in its potential well as it is thermally activated. Denote by $q(\omega)d\omega$ the probability to reach (in a hop) the energy level that exceeds the bottom level of its potential well by a value located in the interval $[\omega, \omega+d\omega ]$. Referring to [@BCK98], we set $q(\omega)=A\exp(-A\omega)$, where $A$ is a material constant. The probability for a CRR in a trap with potential energy $w$ to reach the reference state in an arbitrary hop is given by $$Q(w)=\int_{w}^{\infty} q(\omega)d\omega =\exp( -A w).$$ The average rate of hops in a cage $\gamma$ is determined by the current temperature $T$ only, $\gamma=\gamma(T)$. The rate of rearrangement $P$ equals the product of the rate of hops $\gamma$ by the probability $Q$ to reach the reference state in a hop, $$P(w)=\gamma \exp(-Aw).$$ Denote by $X$ the (time-uniform) concentration of traps per unit mass, and by $p(t,w)$ the current probability density of traps with potential energy $w$. The number of relaxing regions (per unit mass) trapped in cages with the energy belonging to the interval $[w, w+dw]$ and rearranged during the interval of time $[t,t+dt]$ is $X P(w)p(t,w)dwdt$. Unlike Refs. [@Rob78; @RSC84], we assume that not all flow units change their traps when they reach the reference energy level and denote by $F(t,w)$ the ratio of the number of relaxing regions returning to their traps to the number of those reaching the reference state. The number of relaxing regions leaving their cages (with the energy located within the interval $[w,w+dw]$) per unit mass and unit time is given by $X (1-F) P p dw$. The exchange of flow units is assumed to occur only between the nearest neighbors on the energy landscape, that is between a trap with the energy $[w,w+dw]$ and traps with the energies $[w-dw,w]$ and $[w+dw,w+2dw]$. The balance law for the number of flow units trapped in cages with the energy belonging to the interval $[w,w+dw]$ reads $$\frac{\partial p}{\partial t}=-(1-F)Pp
+\frac{1}{2}\Bigl [(1-F)Pp\Bigr ]_{+}
+\frac{1}{2}\Bigl [(1-F)Pp\Bigr ]_{-},$$ where the subscript indices “$-$” and “$+$” refer to appropriate quantities for the intervals $[w-dw,w]$ and $[w+dw,w+2dw]$. Expanding the right-hand side of this equality into the Taylor series, using Eq. (1) and introducing the notation $\Gamma=\frac{1}{2}\gamma dw^{2}$, we arrive at the differential equation for diffusion over the energy landscape $$\frac{\partial p}{\partial t}=\Gamma \frac{\partial^{2}}{\partial w^{2}}
\Bigl [(1-F)\exp (-Aw)p\Bigr ].$$ We adopt the Metropolis transition rates [@BB85], $$F(t,w)=\left \{ \begin{array}{ll}
1, & p(t,w)\leq p_{\infty} (w),\\
\exp [-\epsilon(p-p_{\infty})], & p(t,w)>p_{\infty}(w),
\end{array}\right .$$ where $p_{\infty}(w)$ is the equilibrium density of traps and $\epsilon>0$ is a material parameter. An important advantage of Eq. (2) compared to relationships suggested in Refs. [@Fel68; @Dob80] is that under condition (3) it does not impose restrictions on the equilibrium density of trap $p_{\infty}(w)$. Referring to [@Dyr95], we suppose that the inequality $$\int_{-\infty}^{0} p(t,w)dw\ll 1$$ is satisfied for any $t\geq 0$ and describe the initial distribution, $p_{0}(w)$, and the equilibrium distribution, $p_{\infty}(w)$, by the Gaussian formulas $$p_{0}(w) = \frac{1}{\sqrt{2\pi}\Sigma_{0}}\exp \biggl [
-\frac{(w-W)^{2}}{2\Sigma_{0}^{2}}\biggr ],
\qquad
p_{\infty}(w) = \frac{1}{\sqrt{2\pi}\Sigma_{\infty}}\exp \biggl [
-\frac{(w-W)^{2}}{2\Sigma_{\infty}^{2}}\biggr ],$$ where $W$, $\Sigma_{0}$, $\Sigma_{\infty}$ are adjustable parameters. Equations (5) imply that the average equilibrium energies of traps are temperature-independent (the same value $W$ is employed for the initial and equilibrium distribution functions), whereas their variances strongly depend on $T$. The first assertion is fairly well confirmed by experimental data in mechanical tests [@Dro99a; @Dro99b], whereas the other hypothesis is in agreement with the conventional scenario for the growth in the ruggedness of the energy landscape with a decrease in temperature [@BCK98; @DMO99; @RSN99].
The level of disorder in an ensemble of CRRs is characterized by the configurational entropy per rearranging region [@AM88], $$s(t)=-k_{B}\int_{0}^{\infty} p(t,w) \ln p(t,w) dw,$$ where $k_{B}$ is Boltzmann’s constant. The configurational enthalpy per unit cage $h(t)$ is expressed in terms of the configurational entropy $s(t)$ by means of the conventional formula $$\frac{\partial h}{\partial s}=T.$$ This equality is integrated for a standard one-step thermal test, $$T(t)=T_{0} \quad (T<0),
\qquad
T(t)=T \qquad (T>0),$$ and an explicit formula is found for the enthalpy per unit mass $H=X h$. The relaxing enthalpy per unit mass $\Delta H(t)=H(t)-H(0)$ is given by $$\Delta H(t)= \Lambda \int_{0}^{\infty} \Bigl [ p_{0}(w)\ln p_{0}(w)
-p(t,w)\ln p(t,w) \Bigr ] dw$$ with $\Lambda=k_{B}TX$. Introducing the dimensionless variables $\bar{w}=Aw$ and $\bar{t}=t/t_{0}$, where $t_{0}$ is the characteristic time of aging, and setting $\bar{\Gamma}=A^{2}\Gamma t_{0}$, $\bar{W}=AW$ and $\bar{\Sigma}_{k}=A\Sigma_{k}$, we arrive at the constitutive model, Eqs. (2), (3), (5) and (7), with six adjustable parameters $\bar{W}$, $\bar{\Sigma}_{0}$, $\bar{\Sigma}_{\infty}$, $\bar{\Gamma}$, $\epsilon$ and $\Lambda$. These kinetic equations substantially differ from conventional relations for enthalpy recovery in polymers [@RP95; @MEW74; @KAH79; @Hod87], because they do not refer to a (purely phenomenological) concept of internal clock. As an analog of the material time $\tau$ in Eqs. (2), (3), (5) and (7), we may mention the parameter $\epsilon$ that characterizes the rate of changes in the energy landscape. An important difference between $\epsilon$ and $\tau$ is that $\epsilon$ is independent of the current energy landscape (but, in general, temperature-dependent), whereas the parameter $\tau$ is conventionally expressed in terms of the current enthalpy $H$ by means of the Narayanaswamy or the Adam–Gibbs equations.
Comparison with experiments
===========================
It is easy to check that the quantities $\bar{W}$ and $\bar{\Gamma}$ are interrelated: when one of them is fixed, the other may be chosen to characterize the time scale. For convenience of numerical simulation, we fix $\bar{W}$ and determine $\bar{\Gamma}$ by fitting observations. The value $\bar{W}=2.5$ ensures that inequality (4) is satisfied with a high level of accuracy.
We begin with experimental data for two amorphous PETs exposed in Ref. [@JL92]. First, observations are fitted for a homopolymer to determine the parameters $\bar{\Sigma}_{0}$, $\bar{\Sigma}_{\infty}$, $\bar{\Gamma}$, $\epsilon$ and $\Lambda$ that ensure the best approximation of observations. Afterwards, the amounts $\bar{\Sigma}_{0}$, $\bar{\Sigma}_{\infty}$ and $\bar{\Gamma}$ are fixed, and the quantities $\epsilon$ and $\Lambda$ are determined by matching experimental data for a copolymer. Figure 1A demonstrates fair agreement between observations and results of numerical simulation. Using the specific gravity $g=1.34$ g/cm$^{3}$ at $T=65$ $^{\circ}$C [@JL92], we find the volume concentration of relaxing regions in PET, $\Xi=6.39\cdot 10^{26}$ m$^{-3}$, which is rather close to $\Xi=4.1\cdot 10^{26}$ m$^{-3}$ for polycarbonate [@BWK99] and $\Xi=7.7\cdot 10^{26}$ m$^{-3}$ for poly(vinyl acetate) [@DLS00] found by PALS. To ensure that adjustable parameters are rather robust with respect to changes in material properties and conditions of the test (a decrease in $T_{\rm g}$ by 5 K and an increase in $T_{0}$ by 15 K), we repeat this procedure using experimental data presented in Ref. [@Pet74]. Figure 1B demonstrates fair agreement between observations and results of numerical simulation for similar values of $\bar{\Sigma}_{0}$, $\epsilon$ and $\Lambda$. An increase in $\bar{\Gamma}$ depicted in Figure 1B compared to Figure 1A seems quite natural, because this parameter is determined by the difference between the glass transition temperature and the annealing temperature, $\Delta T=T_{\rm g}-T$. A decrease in $\bar{\Sigma}_{\infty}$ may be associated with a drop in $T_{\rm g}$, because it is conventionally accepted that the glass transition temperature grows when molecular mobility becomes more restricted (e.g., because of an increase in the number of crosslinks [@CME99] or in the molecular weight [@AHP76]), which may be associated with the growth of inhomogeneity of the energy landscape.
To study the effect of the annealing temperature $T$ on the kinetics of structural recovery, we approximate experimental data for amorphous PET obtained in Ref. [@AH82]. Figure 2A demonstrates fair agreement between observations and results of numerical simulation in the temperature range from $T_{\rm g}-25$ to $T_{\rm g}-5$ $^{\circ}$C. The standard deviation of energies of traps in thermodynamic equilibrium $\bar{\Sigma}_{\infty}$ is plotted versus the degree of undercooling $\Delta T$ in Figure 2B. This figure demonstrate that the dependence $\bar{\Sigma}_{\infty}(T)$ is fairly well approximated by the linear function $$\bar{\Sigma}_{\infty}=a_{0}+a_{1}\Delta T,$$ where $a_{k}$ are adjustable parameters. In follows from Eq. (8) that some critical temperature $T_{\rm cr}$ exists at which $\bar{\Sigma}_{\infty}$ vanishes and the energy landscape becomes homogeneous. Results of simulation imply that $T_{\rm cr}=T_{\rm g}+34.11$ $^{\circ}$C which is in accord with the values of $T_{\rm cr}$ found by fitting date in mechanical tests [@Dro00]. The quantities $\bar{\Gamma}$ and $\epsilon$ are depicted in Figure 3A. This figure show that the dependences $\bar{\Gamma}(T)$ and $\epsilon(T)$ are correctly approximated by the “linear” functions $$\log \bar{\Gamma}=b_{0}-b_{1}\Delta T,
\qquad
\log \epsilon=c_{0}+c_{1}\Delta T$$ with adjustable parameters $b_{k}$ and $c_{k}$. In the vicinity of the glass transition temperature, the apparent activation energy $\Delta E$ is calculated as [@Str97] $$\Delta E=-R\frac{d\ln \bar{\Gamma}}{d(1/T)} \biggl |_{T=T_{\rm g}},$$ where $R$ is the gas constant. This equality together with Eq. (9) results in the formula $\Delta E=RT_{\rm g}^{2}b_{1}$. According to Figure 3A, $\Delta E=156.1$ kJ/mol, which is quite comparable with $\Delta E=206$ kJ/mol determined for PET in mechanical tests [@CYM99]. An increase in the parameter $\bar{\Gamma}$ with temperature $T$ is in agreement with the theory of thermally activated processes. A surprising result is a decrease in $\epsilon$ with temperature. It follows from Eq. (3) that this decrease provides an additional source for slowing down of the aging process in the close vicinity of $T_{\rm g}$ which has not been accounted for in previous studies.
To assess the effect of crystallinity on the rate of structural relaxation, we approximate experimental data for enthalpy recovery in semicrystalline PET with various degrees of crystallinity $f$. For a detailed description of specimens and the experimental procedure, see Ref. [@DCM99]. First, we approximate observations for a sample with the smallest value of $f$ and determine adjustable parameters of the model using the steepest-descent procedure. Afterwards, we fix the quantities $\bar{\Sigma}_{0}$ and $\Lambda$ and repeat matching experimental data with three adjustable parameters, $\bar{\Sigma}_{\infty}$, $\bar{\Gamma}$ and $\epsilon$. Figure 4A demonstrates fair agreement between observations and predictions of the model. The equilibrium standard deviation of energies of traps $\bar{\Sigma}_{\infty}$ is depicted versus the degree of crystallinity $f$ in Figure 4B. The dependence $\bar{\Sigma}_{\infty}(f)$ is correctly approximated by the linear function $$\bar{\Sigma}_{\infty}=a_{0}-a_{1}f$$ with adjustable parameters $a_{k}$. Comparing Figures 2B and 4B, we conclude that an increase in $f$ for a semicrystalline polymer affects the equilibrium distribution of cages in a way similar to that in which the annealing temperature $T$ influences the energy landscape for an amorphous material. By analogy with an amorphous polymer, one may define the critical degree of crystallinity $f_{\rm cr}$ for a semicrystalline medium as the percentage of crystallites at which the parameter $\bar{\Sigma}_{\infty}$ vanishes and the energy landscape becomes homogeneous. In contrast to the conclusions of Ref. [@TW81], Figure 4B demonstrates that $f_{\rm cr}$ is essentially less than 100%. The parameters $\bar{\Gamma}$ and $\epsilon$ are plotted versus the degree of crystallinity in Figure 3B. This figure shows that the dependences $\bar{\Gamma}(f)$ and $\epsilon(f)$ are correctly approximated by the “linear” functions $$\log \bar{\Gamma}=b_{0}+b_{1}f,
\qquad
\log \epsilon=c_{0}-c_{1}f$$ with adjustable parameters $b_{k}$ and $c_{k}$.
Two kinds of crystallization of polymers are conventionally studied: thermal crystallization, when a specimen is annealed at a fixed temperature in the rubbery region for a given time (this procedure was employed in Ref. [@DCM99]) and mechanically induced crystallization, when a sample is stretched slightly above $T_{\rm g}$ to a given extension ratio $\lambda$. To demonstrate that the effect of the degree of crystallinity on the kinetics of physical aging weakly depends on the crystallization procedure, we approximate experimental data for semicrystalline PET samples where the level of crystallization is established by uniaxial stretching at an elevated temperature. A detailed description of specimens and the experimental procedure can be found in Ref. [@MJ95]. The extension ratio $\lambda$ is transformed into the level of crystallinity using Figure 1 of Ref. [@VLJ98]. Unlike previous sets of experimental data, where $\epsilon$ is treated as a function of the level of crystallinity $f$, we fit observations at various elongations $\lambda$ with the same value of $\epsilon$ found for an undeformed specimen. Figure 5A demonstrates good agreement between experimental data and predictions of the model. The functions $\bar{\Sigma}_{\infty}(f)$ and $\bar{\Gamma}(f)$ are plotted in Figure 5B. This figure demonstrates that Eqs. (10) and (11) adequately describe observations. The parameter $b_{1}$ (that characterizes the influence of crystallinity on the rate of rearrangement) acquires similar values for thermally and mechanically crystallized samples. The same assertion is true for the critical degree of crystallinity $f_{\rm cr}$ which adopts the value 33.9% for the stretched PET versus 42.7% for the PET annealed above $T_{\rm g}$. On the contrary, the parameter $\bar{\Sigma}_{\infty}$ for the PET obtained by thermal crystallization exceeds that for the hot-drawn PET by twice. This may be explain by the fact that stretching a rubbery polymer establishes an additional order caused by partial destruction of a polymeric network and alignment of long chains along the axis of loading [@PGM00].
Concludung remarks
==================
Constitutive equations have been derived for enthalpy relaxation in glassy polymers after thermal jumps. The model is based on the trapping concept which treats a disordered medium as an ensemble of flow units rearranged at random times as they are thermally activated. Adjustable parameters are found by fitting experimental data for amorphous and semicrystalline poly(ethylene terephthalate). The following conclusions are drawn:
1. the constitutive equations correctly describe the kinetics of structural relaxation in the sub–$T_{\rm g}$ region.
2. the model predicts the existence of some critical temperature $T_{\rm cr}$ at which the energy landscape becomes homogeneous.
3. for semicrystalline polymers, the model implies the existence of some critical degree of crystallinity $f_{\rm cr}$ at which the energy landscape becomes homogeneous. The inequality $f_{\rm cr}<100$% may serve as an indirect confirmation that the processes of structural relaxation in interlamellar and intralamellar regions differ from one another.
4. a correspondence may be established between the effects of temperature and the level of crystallinity on the kinetics of enthalpy recovery.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The author is grateful to A.L. Svistkov for fruitful discussions. The work is supported by the Israeli Ministry of Science through grant 1202–1–98.
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(102,89)[1]{} (102,96)[2]{} (87,4)[A]{}
( 1.58, 42.83) ( 20.84, 55.43) ( 32.11, 60.28) ( 46.64, 67.40) ( 56.11, 72.71) ( 76.30, 82.96) ( 88.19, 88.50) ( 91.99, 90.17)
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(100,100) (0,0)[(100,100)]{} (20,0)(20,0)[4]{}[(0,1)[2]{}]{} (0,16.67)(0,16.67)[5]{}[(1,0)[2]{}]{} (0,-10)[$0.0$]{} (91,-10)[2.0]{} (50,-10)[$\log t$]{} (-16,0)[1.0]{} (-16,94)[4.0]{} (-16,70)[$\Delta H$]{} (87,4)[B]{}
( 4.23, 7.31) ( 19.28, 27.98) ( 28.09, 38.24) ( 34.33, 46.88) ( 39.18, 52.13) ( 54.23, 68.37) ( 63.03, 76.21) ( 69.28, 81.41) ( 74.13, 84.39) ( 78.09, 86.66) ( 81.43, 88.59) ( 84.33, 89.29) ( 86.89, 89.82) ( 89.18, 90.52) ( 91.25, 90.87) ( 93.14, 90.87) ( 94.88, 91.04) ( 96.48, 91.22)
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(100,100) (0,0)[(100,100)]{} (20,0)(20,0)[4]{}[(0,1)[2]{}]{} (0,16.67)(0,16.67)[5]{}[(1,0)[2]{}]{} (0,-10)[$0.0$]{} (91,-10)[2.0]{} (50,-10)[$\log t$]{} (-16,0)[0.0]{} (-16,94)[3.0]{} (-16,70)[$\Delta H$]{} (102,56)[1]{} (102,84)[2]{} (102,92)[3]{} (102,76)[4]{} (87,4)[A]{}
( 8.34, 30.63) ( 26.93, 41.19) ( 42.21, 47.71) ( 68.89, 55.42) ( 83.79, 54.72)
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40.51) ( 26.59, 40.63) ( 26.80, 40.74) ( 27.00, 40.85) ( 27.20, 40.96) ( 27.41, 41.07) ( 27.61, 41.17) ( 27.81, 41.28) ( 28.01, 41.39) ( 28.22, 41.49) ( 28.42, 41.59) ( 28.62, 41.70) ( 28.82, 41.80) ( 29.02, 41.90) ( 29.23, 42.00) ( 29.43, 42.10) ( 29.63, 42.20) ( 29.84, 42.29) ( 30.04, 42.39) ( 30.24, 42.49) ( 30.44, 42.59) ( 30.64, 42.71) ( 30.84, 42.82) ( 31.05, 42.94) ( 31.25, 43.06) ( 31.46, 43.18) ( 31.66, 43.30) ( 31.86, 43.41) ( 32.06, 43.53) ( 32.27, 43.65) ( 32.47, 43.76) ( 32.67, 43.87) ( 32.87, 43.98) ( 33.08, 44.09) ( 33.28, 44.20) ( 33.48, 44.31) ( 33.68, 44.41) ( 33.88, 44.52) ( 34.09, 44.62) ( 34.29, 44.72) ( 34.49, 44.81) ( 34.70, 44.91) ( 34.90, 45.01) ( 35.10, 45.10) ( 35.30, 45.19) ( 35.51, 45.28) ( 35.71, 45.37) ( 35.91, 45.46) ( 36.11, 45.55) ( 36.31, 45.63) ( 36.51, 45.72) ( 36.72, 45.80) ( 36.92, 45.88) ( 37.12, 45.96) ( 37.32, 46.04) ( 37.53, 46.11) ( 37.73, 46.19) ( 37.93, 46.26) ( 38.13, 46.34) ( 38.34, 46.41) ( 38.54, 46.48) ( 38.74, 46.55) ( 38.94, 46.62) ( 39.15, 46.70) ( 39.35, 46.77) ( 39.55, 46.85) ( 39.75, 46.93) ( 39.95, 47.01) ( 40.16, 47.09) ( 40.36, 47.17) ( 40.56, 47.25) ( 40.76, 47.33) ( 40.96, 47.41) ( 41.16, 47.49) ( 41.37, 47.57) ( 41.57, 47.65) ( 41.77, 47.73) ( 41.97, 47.81) ( 42.17, 47.89) ( 42.37, 47.97) ( 42.58, 48.05) ( 42.78, 48.12) ( 42.98, 48.20) ( 43.18, 48.27) ( 43.38, 48.35) ( 43.58, 48.42) ( 43.78, 48.50) ( 43.98, 48.58) ( 44.19, 48.67) ( 44.39, 48.75) ( 44.59, 48.84) ( 44.79, 48.93) ( 44.99, 49.02) ( 45.19, 49.10) ( 45.39, 49.19) ( 45.60, 49.27) ( 45.80, 49.35) ( 46.00, 49.44) ( 46.20, 49.51) ( 46.40, 49.59) ( 46.60, 49.67) ( 46.80, 49.75) ( 47.00, 49.82) ( 47.20, 49.90) ( 47.41, 49.97) ( 47.61, 50.04) ( 47.81, 50.11) ( 48.01, 50.18) ( 48.21, 50.25) ( 48.41, 50.31) ( 48.61, 50.38) ( 48.82, 50.44) ( 49.02, 50.50) ( 49.22, 50.57) ( 49.42, 50.63) ( 49.62, 50.69) ( 49.82, 50.74) ( 50.02, 50.80) ( 50.22, 50.86) ( 50.42, 50.91) ( 50.62, 50.97) ( 50.83, 51.02) ( 51.03, 51.07) ( 51.23, 51.13) ( 51.43, 51.18) ( 51.63, 51.22) ( 51.83, 51.27) ( 52.03, 51.32) ( 52.23, 51.37) ( 52.43, 51.41) ( 52.63, 51.46) ( 52.84, 51.50) ( 53.04, 51.55) ( 53.24, 51.59) ( 53.44, 51.63) ( 53.64, 51.68) ( 53.84, 51.72) ( 54.04, 51.77) ( 54.24, 51.81) ( 54.44, 51.86) ( 54.64, 51.91) ( 54.85, 51.95) ( 55.05, 52.00) ( 55.25, 52.05) ( 55.45, 52.10) ( 55.65, 52.14) ( 55.85, 52.19) ( 56.05, 52.24) ( 56.25, 52.29) ( 56.45, 52.34) ( 56.65, 52.39) ( 56.85, 52.45) ( 57.05, 52.50) ( 57.26, 52.56) ( 57.46, 52.61) ( 57.66, 52.67) ( 57.86, 52.72) ( 58.06, 52.77) ( 58.26, 52.82) ( 58.46, 52.87) ( 58.66, 52.92) ( 58.86, 52.97) ( 59.06, 53.02) ( 59.26, 53.07) ( 59.46, 53.12) ( 59.66, 53.16) ( 59.87, 53.21) ( 60.07, 53.25) ( 60.27, 53.30) ( 60.47, 53.34) ( 60.67, 53.38) ( 60.87, 53.42) ( 61.07, 53.46) ( 61.27, 53.50) ( 61.47, 53.54) ( 61.67, 53.58) ( 61.87, 53.62) ( 62.07, 53.65) ( 62.27, 53.69) ( 62.47, 53.73) ( 62.67, 53.76) ( 62.87, 53.79) ( 63.08, 53.83) ( 63.28, 53.86) ( 63.48, 53.89) ( 63.68, 53.93) ( 63.88, 53.96) ( 64.08, 53.99) ( 64.28, 54.02) ( 64.48, 54.05) ( 64.68, 54.07) ( 64.88, 54.10) ( 65.08, 54.13) ( 65.28, 54.16) ( 65.48, 54.18) ( 65.68, 54.21) ( 65.88, 54.24) ( 66.08, 54.26) ( 66.28, 54.29) ( 66.48, 54.31) ( 66.68, 54.33) ( 66.88, 54.36) ( 67.09, 54.38) ( 67.29, 54.40) ( 67.49, 54.42) ( 67.69, 54.45) ( 67.89, 54.47) ( 68.09, 54.49) ( 68.29, 54.51) ( 68.49, 54.54) ( 68.69, 54.56) ( 68.89, 54.59) ( 69.09, 54.62) ( 69.29, 54.65) ( 69.49, 54.67) ( 69.69, 54.70) ( 69.89, 54.73) ( 70.09, 54.76) ( 70.29, 54.79) ( 70.49, 54.81) ( 70.69, 54.84) ( 70.89, 54.87) ( 71.09, 54.89) ( 71.29, 54.92) ( 71.49, 54.94) ( 71.69, 54.97) ( 71.89, 54.99) ( 72.09, 55.02) ( 72.29, 55.04) ( 72.50, 55.07) ( 72.70, 55.09) ( 72.90, 55.11) ( 73.10, 55.14) ( 73.30, 55.16) ( 73.50, 55.18) ( 73.70, 55.20) ( 73.90, 55.22) ( 74.10, 55.24) ( 74.30, 55.26) ( 74.50, 55.28) ( 74.70, 55.30) ( 74.90, 55.32) ( 75.10, 55.34) ( 75.30, 55.36) ( 75.50, 55.38) ( 75.70, 55.40) ( 75.90, 55.41) ( 76.10, 55.43) ( 76.30, 55.45) ( 76.50, 55.47) ( 76.70, 55.48) ( 76.90, 55.50) ( 77.10, 55.51) ( 77.30, 55.53) ( 77.50, 55.54) ( 77.71, 55.56) ( 77.91, 55.57) ( 78.11, 55.59) ( 78.31, 55.60) ( 78.51, 55.61) ( 78.71, 55.63) ( 78.91, 55.64) ( 79.11, 55.65) ( 79.31, 55.67) ( 79.51, 55.68) ( 79.71, 55.69) ( 79.91, 55.70) ( 80.11, 55.71) ( 80.31, 55.73) ( 80.51, 55.74) ( 80.71, 55.75) ( 80.91, 55.76) ( 81.11, 55.77) ( 81.31, 55.78) ( 81.51, 55.80) ( 81.71, 55.81) ( 81.91, 55.82) ( 82.11, 55.83) ( 82.31, 55.84) ( 82.51, 55.85) ( 82.71, 55.86) ( 82.91, 55.87) ( 83.11, 55.88) ( 83.31, 55.90) ( 83.51, 55.91) ( 83.71, 55.92) ( 83.91, 55.93) ( 84.11, 55.94) ( 84.31, 55.95) ( 84.52, 55.96) ( 84.72, 55.98) ( 84.92, 55.99) ( 85.12, 56.00) ( 85.32, 56.01) ( 85.52, 56.02) ( 85.72, 56.03) ( 85.92, 56.04) ( 86.12, 56.05) ( 86.32, 56.06) ( 86.52, 56.07) ( 86.72, 56.08) ( 86.92, 56.09) ( 87.12, 56.10) ( 87.32, 56.11) ( 87.52, 56.12) ( 87.72, 56.13) ( 87.92, 56.14) ( 88.12, 56.15) ( 88.32, 56.16) ( 88.52, 56.16) ( 88.72, 56.17) ( 88.92, 56.18) ( 89.12, 56.19) ( 89.32, 56.20) ( 89.52, 56.21) ( 89.72, 56.21) ( 89.92, 56.22) ( 90.12, 56.23) ( 90.32, 56.24) ( 90.52, 56.24) ( 90.72, 56.25) ( 90.92, 56.26) ( 91.12, 56.27) ( 91.32, 56.27) ( 91.52, 56.28) ( 91.72, 56.28) ( 91.92, 56.29) ( 92.12, 56.30) ( 92.32, 56.30) ( 92.52, 56.31) ( 92.72, 56.32) ( 92.92, 56.32) ( 93.12, 56.33) ( 93.32, 56.33) ( 93.52, 56.34) ( 93.72, 56.34) ( 93.92, 56.35) ( 94.12, 56.35) ( 94.32, 56.36) ( 94.52, 56.36) ( 94.72, 56.37) ( 94.92, 56.38) ( 95.12, 56.38) ( 95.32, 56.39) ( 95.52, 56.39) ( 95.72, 56.40) ( 95.92, 56.40) ( 96.12, 56.41) ( 96.32, 56.41) ( 96.52, 56.42) ( 96.72, 56.42) ( 96.92, 56.42) ( 97.12, 56.43) ( 97.33, 56.43) ( 97.53, 56.44) ( 97.73, 56.44) ( 97.93, 56.45) ( 98.13, 56.45) ( 98.33, 56.45) ( 98.53, 56.46) ( 98.73, 56.46) ( 98.93, 56.47) ( 99.13, 56.47) ( 99.33, 56.47) ( 99.53, 56.48) ( 99.73, 56.48) ( 99.93, 56.49)
( 8.34, 55.72) ( 26.93, 67.86) ( 42.21, 74.40) ( 68.89, 81.81) ( 83.79, 84.35)
( 0.22, 48.80) ( 0.43, 48.99) ( 0.64, 49.19) ( 0.85, 49.39) ( 1.06, 49.60) ( 1.27, 49.81) ( 1.47, 50.01) ( 1.67, 50.21) ( 1.87, 50.41) ( 2.17, 50.70) ( 2.46, 50.98) ( 2.75, 51.25) ( 3.03, 51.51) ( 3.32, 51.76) ( 3.59, 52.01) ( 3.87, 52.24) ( 4.14, 52.47) ( 4.41, 52.69) ( 4.67, 52.91) ( 4.93, 53.11) ( 5.19, 53.31) ( 5.45, 53.50) ( 5.70, 53.69) ( 5.95, 53.87) ( 6.19, 54.05) ( 6.44, 54.21) ( 6.68, 54.38) ( 6.92, 54.54) ( 7.15, 54.69) ( 7.38, 54.84) ( 7.61, 54.98) ( 7.84, 55.12) ( 8.07, 55.26) ( 8.29, 55.39) ( 8.51, 55.51) ( 8.73, 55.64) ( 8.95, 55.76) ( 9.16, 55.87) ( 9.38, 55.99) ( 9.59, 56.10) ( 9.79, 56.22) ( 10.00, 56.35) ( 10.21, 56.48) ( 10.41, 56.61) ( 10.61, 56.74) ( 10.87, 56.92) ( 11.14, 57.10) ( 11.39, 57.28) ( 11.65, 57.46) ( 11.90, 57.63) ( 12.15, 57.81) ( 12.40, 57.98) ( 12.64, 58.15) ( 12.88, 58.32) ( 13.12, 58.52) ( 13.36, 58.74) ( 13.59, 58.96) ( 13.82, 59.18) ( 14.05, 59.40) ( 14.28, 59.61) ( 14.50, 59.81) ( 14.72, 60.01) ( 14.94, 60.21) ( 15.16, 60.40) ( 15.37, 60.59) ( 15.59, 60.77) ( 15.80, 60.94) ( 16.01, 61.11) ( 16.21, 61.28) ( 16.42, 61.44) ( 16.62, 61.60) ( 16.82, 61.75) ( 17.07, 61.93) ( 17.32, 62.11) ( 17.56, 62.29) ( 17.80, 62.45) ( 18.04, 62.62) ( 18.27, 62.77) ( 18.51, 62.93) ( 18.74, 63.07) ( 18.97, 63.22) ( 19.19, 63.36) ( 19.41, 63.49) ( 19.63, 63.62) ( 19.85, 63.75) ( 20.07, 63.87) ( 20.28, 63.99) ( 20.50, 64.10) ( 20.71, 64.21) ( 20.92, 64.32) ( 21.12, 64.43) ( 21.33, 64.53) ( 21.53, 64.63) ( 21.73, 64.73) ( 21.97, 64.84) ( 22.20, 64.95) ( 22.44, 65.05) ( 22.67, 65.15) ( 22.89, 65.25) ( 23.12, 65.36) ( 23.34, 65.48) ( 23.56, 65.59) ( 23.78, 65.71) ( 24.00, 65.83) ( 24.21, 65.97) ( 24.43, 66.13) ( 24.64, 66.30) ( 24.85, 66.46) ( 25.05, 66.63) ( 25.26, 66.79) ( 25.46, 66.95) ( 25.66, 67.11) ( 25.89, 67.29) ( 26.12, 67.46) ( 26.35, 67.63) ( 26.57, 67.79) ( 26.80, 67.95) ( 27.02, 68.11) ( 27.23, 68.26) ( 27.45, 68.41) ( 27.66, 68.55) ( 27.88, 68.69) ( 28.08, 68.83) ( 28.29, 68.96) ( 28.50, 69.09) ( 28.70, 69.21) ( 28.90, 69.33) ( 29.13, 69.47) ( 29.36, 69.60) ( 29.58, 69.72) ( 29.80, 69.85) ( 30.02, 69.96) ( 30.24, 70.08) ( 30.45, 70.19) ( 30.67, 70.30) ( 30.88, 70.41) ( 31.08, 70.51) ( 31.29, 70.61) ( 31.50, 70.70) ( 31.70, 70.80) ( 31.90, 70.89) ( 32.12, 70.99) ( 32.34, 71.08) ( 32.56, 71.18) ( 32.78, 71.27) ( 33.00, 71.35) ( 33.21, 71.44) ( 33.42, 71.52) ( 33.63, 71.60) ( 33.83, 71.68) ( 34.04, 71.75) ( 34.24, 71.83) ( 34.44, 71.90) ( 34.66, 72.01) ( 34.88, 72.11) ( 35.10, 72.22) ( 35.31, 72.34) ( 35.53, 72.46) ( 35.74, 72.57) ( 35.95, 72.69) ( 36.15, 72.81) ( 36.36, 72.93) ( 36.56, 73.04) ( 36.76, 73.15) ( 36.98, 73.27) ( 37.20, 73.39) ( 37.41, 73.51) ( 37.62, 73.63) ( 37.83, 73.74) ( 38.04, 73.85) ( 38.25, 73.95) ( 38.45, 74.06) ( 38.65, 74.16) ( 38.85, 74.26) ( 39.07, 74.36) ( 39.28, 74.46) ( 39.50, 74.56) ( 39.71, 74.66) ( 39.92, 74.75) ( 40.12, 74.85) ( 40.33, 74.93) ( 40.53, 75.02) ( 40.73, 75.10) ( 40.94, 75.19) ( 41.16, 75.28) ( 41.37, 75.36) ( 41.58, 75.44) ( 41.78, 75.52) ( 41.99, 75.60) ( 42.19, 75.67) ( 42.39, 75.74) ( 42.61, 75.82) ( 42.82, 75.89) ( 43.03, 75.96) ( 43.24, 76.03) ( 43.45, 76.10) ( 43.65, 76.16) ( 43.85, 76.22) ( 44.05, 76.29) ( 44.27, 76.37) ( 44.48, 76.45) ( 44.69, 76.53) ( 44.90, 76.61) ( 45.10, 76.68) ( 45.30, 76.76) ( 45.50, 76.83) ( 45.72, 76.90) ( 45.93, 76.97) ( 46.14, 77.04) ( 46.34, 77.11) ( 46.55, 77.18) ( 46.75, 77.25) ( 46.95, 77.32) ( 47.16, 77.39) ( 47.37, 77.47) ( 47.58, 77.54) ( 47.78, 77.62) ( 47.99, 77.69) ( 48.19, 77.76) ( 48.40, 77.84) ( 48.61, 77.91) ( 48.82, 77.98) ( 49.02, 78.06) ( 49.23, 78.13) ( 49.43, 78.19) ( 49.64, 78.26) ( 49.85, 78.33) ( 50.05, 78.40) ( 50.26, 78.47) ( 50.46, 78.53) ( 50.66, 78.59) ( 50.87, 78.65) ( 51.08, 78.72) ( 51.29, 78.78) ( 51.49, 78.84) ( 51.69, 78.89) ( 51.90, 78.95) ( 52.11, 79.01) ( 52.32, 79.06) ( 52.52, 79.11) ( 52.72, 79.16) ( 52.92, 79.21) ( 53.13, 79.27) ( 53.33, 79.32) ( 53.54, 79.37) ( 53.74, 79.43) ( 53.94, 79.49) ( 54.15, 79.54) ( 54.35, 79.60) ( 54.56, 79.66) ( 54.76, 79.71) ( 54.97, 79.76) ( 55.17, 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(100,100) (0,0)[(100,100)]{} (16.67,0)(16.67,0)[5]{}[(0,1)[2]{}]{} (0,20)(0,20)[4]{}[(1,0)[2]{}]{} (0,-10)[0.0]{} (87,-10)[30.0]{} (50,-10)[$\Delta T$]{} (-16,0)[0.0]{} (-16,94)[1.0]{} (-16,70)[$\bar{\Sigma}_{\infty}$]{} (87,4)[B]{}
( 80.00, 72.00) ( 53.33, 65.00) ( 30.00, 57.00) ( 16.67, 48.00)
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(100,100) (0,0)[(100,100)]{} (16.67,0)(16.67,0)[5]{}[(0,1)[2]{}]{} (0,16.67)(0,16.67)[5]{}[(1,0)[2]{}]{} (100,16.67)(0,16.67)[5]{}[(-1,0)[2]{}]{} (0,-10)[0.0]{} (87,-10)[30.0]{} (50,-10)[$\Delta T$]{} (-16,0)[0.0]{} (-16,94)[3.0]{} (-19,70)[$\log \bar{\Gamma}$]{} (102,0)[$-3.0$]{} (102,94)[0.0]{} (102,70)[$\log \epsilon$]{} (14,102)[1]{} (90,90)[2]{} (87,4)[A]{}
( 80.00, 3.11) ( 53.33, 33.33) ( 30.00, 78.72) ( 16.67, 98.47)
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( 80.00, 75.18) ( 53.33, 61.50) ( 30.00, 43.37) ( 16.67, 28.17)
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88.18) ( 94.75, 88.37) ( 95.00, 88.55) ( 95.25, 88.73) ( 95.50, 88.91) ( 95.75, 89.10) ( 96.00, 89.28) ( 96.25, 89.46) ( 96.50, 89.64) ( 96.75, 89.83) ( 97.00, 90.01) ( 97.25, 90.19) ( 97.50, 90.37) ( 97.75, 90.56) ( 98.00, 90.74) ( 98.25, 90.92) ( 98.50, 91.10) ( 98.75, 91.29) ( 99.00, 91.47) ( 99.25, 91.65) ( 99.50, 91.83) ( 99.75, 92.02) ( 100.00, 92.20)
(100,100) (0,0)[(100,100)]{} (20,0)(20,0)[4]{}[(0,1)[2]{}]{} (0,10)(0,10)[9]{}[(1,0)[2]{}]{} (100,16.67)(0,16.67)[5]{}[(-1,0)[2]{}]{} (0,-10)[0.0]{} (87,-10)[50.0]{} (50,-10)[$f$]{} (-16,0)[1.0]{} (-16,94)[2.0]{} (-19,70)[$\log \bar{\Gamma}$]{} (102,0)[$-3.0$]{} (102,94)[0.0]{} (102,70)[$\log\epsilon$]{} (72,102)[1]{} (6,102)[2]{} (90,4)[B]{}
( 28.00, 14.61) ( 42.00, 39.79) ( 54.00, 65.32) ( 64.00, 81.29)
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( 48.75, 45.75)[$\Box$]{} ( 52.62, 47.35)[$\Box$]{} ( 58.05, 49.66)[$\Box$]{} ( 62.70, 52.40)[$\Box$]{} ( 77.57, 55.45)[$\Box$]{}
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( 47.12, 0.97)[$\ast$]{} ( 54.05, 8.29)[$\ast$]{} ( 62.70, 6.79)[$\ast$]{} ( 72.52, 8.53)[$\ast$]{} ( 75.90, 7.56)[$\ast$]{} ( 81.94, 9.36)[$\ast$]{}
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(100,100) (0,0)[(100,100)]{} (16.67,0)(16.67,0)[5]{}[(0,1)[2]{}]{} (0,20)(0,20)[4]{}[(1,0)[2]{}]{} (100,20)(0,20)[4]{}[(-1,0)[2]{}]{} (0,-10)[0.0]{} (87,-10)[30.0]{} (50,-10)[$f$]{} (-16,0)[0.0]{} (-16,94)[1.0]{} (-16,70)[$\bar{\Sigma}_{\infty}$]{} (102,0)[$-1.0$]{} (102,94)[0.0]{} (102,70)[$\log\bar{\Gamma}$]{} (86,18)[1]{} (86,91)[2]{} (7,90)[B]{}
( 0.00, 62.00) ( 15.33, 50.00) ( 62.58, 28.00) ( 90.00, 13.60)
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( 0.00, 9.92) ( 15.33, 27.10) ( 62.58, 66.90) ( 90.00, 84.61)
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39.44) ( 32.25, 39.64) ( 32.50, 39.85) ( 32.75, 40.06) ( 33.00, 40.26) ( 33.25, 40.47) ( 33.50, 40.67) ( 33.75, 40.88) ( 34.00, 41.08) ( 34.25, 41.29) ( 34.50, 41.49) ( 34.75, 41.70) ( 35.00, 41.90) ( 35.25, 42.11) ( 35.50, 42.31) ( 35.75, 42.52) ( 36.00, 42.72) ( 36.25, 42.93) ( 36.50, 43.13) ( 36.75, 43.34) ( 37.00, 43.55) ( 37.25, 43.75) ( 37.50, 43.96) ( 37.75, 44.16) ( 38.00, 44.37) ( 38.25, 44.57) ( 38.50, 44.78) ( 38.75, 44.98) ( 39.00, 45.19) ( 39.25, 45.39) ( 39.50, 45.60) ( 39.75, 45.80) ( 40.00, 46.01) ( 40.25, 46.21) ( 40.50, 46.42) ( 40.75, 46.62) ( 41.00, 46.83) ( 41.25, 47.04) ( 41.50, 47.24) ( 41.75, 47.45) ( 42.00, 47.65) ( 42.25, 47.86) ( 42.50, 48.06) ( 42.75, 48.27) ( 43.00, 48.47) ( 43.25, 48.68) ( 43.50, 48.88) ( 43.75, 49.09) ( 44.00, 49.29) ( 44.25, 49.50) ( 44.50, 49.70) ( 44.75, 49.91) ( 45.00, 50.11) ( 45.25, 50.32) ( 45.50, 50.53) ( 45.75, 50.73) ( 46.00, 50.94) ( 46.25, 51.14) ( 46.50, 51.35) ( 46.75, 51.55) ( 47.00, 51.76) ( 47.25, 51.96) ( 47.50, 52.17) ( 47.75, 52.37) ( 48.00, 52.58) ( 48.25, 52.78) ( 48.50, 52.99) ( 48.75, 53.19) ( 49.00, 53.40) ( 49.25, 53.60) ( 49.50, 53.81) ( 49.75, 54.02) ( 50.00, 54.22) ( 50.25, 54.43) ( 50.50, 54.63) ( 50.75, 54.84) ( 51.00, 55.04) ( 51.25, 55.25) ( 51.50, 55.45) ( 51.75, 55.66) ( 52.00, 55.86) ( 52.25, 56.07) ( 52.50, 56.27) ( 52.75, 56.48) ( 53.00, 56.68) ( 53.25, 56.89) ( 53.50, 57.09) ( 53.75, 57.30) ( 54.00, 57.51) ( 54.25, 57.71) ( 54.50, 57.92) ( 54.75, 58.12) ( 55.00, 58.33) ( 55.25, 58.53) ( 55.50, 58.74) ( 55.75, 58.94) ( 56.00, 59.15) ( 56.25, 59.35) ( 56.50, 59.56) ( 56.75, 59.76) ( 57.00, 59.97) ( 57.25, 60.17) ( 57.50, 60.38) ( 57.75, 60.58) ( 58.00, 60.79) ( 58.25, 61.00) ( 58.50, 61.20) ( 58.75, 61.41) ( 59.00, 61.61) ( 59.25, 61.82) ( 59.50, 62.02) ( 59.75, 62.23) ( 60.00, 62.43) ( 60.25, 62.64) ( 60.50, 62.84) ( 60.75, 63.05) ( 61.00, 63.25) ( 61.25, 63.46) ( 61.50, 63.66) ( 61.75, 63.87) ( 62.00, 64.07) ( 62.25, 64.28) ( 62.50, 64.49) ( 62.75, 64.69) ( 63.00, 64.90) ( 63.25, 65.10) ( 63.50, 65.31) ( 63.75, 65.51) ( 64.00, 65.72) ( 64.25, 65.92) ( 64.50, 66.13) ( 64.75, 66.33) ( 65.00, 66.54) ( 65.25, 66.74) ( 65.50, 66.95) ( 65.75, 67.15) ( 66.00, 67.36) ( 66.25, 67.56) ( 66.50, 67.77) ( 66.75, 67.98) ( 67.00, 68.18) ( 67.25, 68.39) ( 67.50, 68.59) ( 67.75, 68.80) ( 68.00, 69.00) ( 68.25, 69.21) ( 68.50, 69.41) ( 68.75, 69.62) ( 69.00, 69.82) ( 69.25, 70.03) ( 69.50, 70.23) ( 69.75, 70.44) ( 70.00, 70.64) ( 70.25, 70.85) ( 70.50, 71.05) ( 70.75, 71.26) ( 71.00, 71.47) ( 71.25, 71.67) ( 71.50, 71.88) ( 71.75, 72.08) ( 72.00, 72.29) ( 72.25, 72.49) ( 72.50, 72.70) ( 72.75, 72.90) ( 73.00, 73.11) ( 73.25, 73.31) ( 73.50, 73.52) ( 73.75, 73.72) ( 74.00, 73.93) ( 74.25, 74.13) ( 74.50, 74.34) ( 74.75, 74.54) ( 75.00, 74.75) ( 75.25, 74.96) ( 75.50, 75.16) ( 75.75, 75.37) ( 76.00, 75.57) ( 76.25, 75.78) ( 76.50, 75.98) ( 76.75, 76.19) ( 77.00, 76.39) ( 77.25, 76.60) ( 77.50, 76.80) ( 77.75, 77.01) ( 78.00, 77.21) ( 78.25, 77.42) ( 78.50, 77.62) ( 78.75, 77.83) ( 79.00, 78.03) ( 79.25, 78.24) ( 79.50, 78.45) ( 79.75, 78.65) ( 80.00, 78.86) ( 80.25, 79.06) ( 80.50, 79.27) ( 80.75, 79.47) ( 81.00, 79.68) ( 81.25, 79.88) ( 81.50, 80.09) ( 81.75, 80.29) ( 82.00, 80.50) ( 82.25, 80.70) ( 82.50, 80.91) ( 82.75, 81.11) ( 83.00, 81.32) ( 83.25, 81.52) ( 83.50, 81.73) ( 83.75, 81.94) ( 84.00, 82.14) ( 84.25, 82.35) ( 84.50, 82.55) ( 84.75, 82.76) ( 85.00, 82.96) ( 85.25, 83.17) ( 85.50, 83.37) ( 85.75, 83.58) ( 86.00, 83.78) ( 86.25, 83.99) ( 86.50, 84.19) ( 86.75, 84.40) ( 87.00, 84.60) ( 87.25, 84.81) ( 87.50, 85.01) ( 87.75, 85.22) ( 88.00, 85.43) ( 88.25, 85.63) ( 88.50, 85.84) ( 88.75, 86.04) ( 89.00, 86.25) ( 89.25, 86.45) ( 89.50, 86.66) ( 89.75, 86.86) ( 90.00, 87.07) ( 90.25, 87.27) ( 90.50, 87.48) ( 90.75, 87.68) ( 91.00, 87.89) ( 91.25, 88.09) ( 91.50, 88.30) ( 91.75, 88.50) ( 92.00, 88.71) ( 92.25, 88.92) ( 92.50, 89.12) ( 92.75, 89.33) ( 93.00, 89.53) ( 93.25, 89.74) ( 93.50, 89.94) ( 93.75, 90.15) ( 94.00, 90.35) ( 94.25, 90.56) ( 94.50, 90.76) ( 94.75, 90.97) ( 95.00, 91.17) ( 95.25, 91.38) ( 95.50, 91.58) ( 95.75, 91.79) ( 96.00, 91.99) ( 96.25, 92.20) ( 96.50, 92.41) ( 96.75, 92.61) ( 97.00, 92.82) ( 97.25, 93.02) ( 97.50, 93.23) ( 97.75, 93.43) ( 98.00, 93.64) ( 98.25, 93.84) ( 98.50, 94.05) ( 98.75, 94.25) ( 99.00, 94.46) ( 99.25, 94.66) ( 99.50, 94.87) ( 99.75, 95.07) ( 100.00, 95.28)
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abstract: 'We consider a model for deformations of a homogeneous isotropic body, whose shear modulus remains constant, but its bulk modulus can be a highly nonlinear function. We show that for a general class of such models, in an arbitrary space dimension, the respective PDE problem has a unique solution. Moreover, this solution enjoys interior smoothness. This is the first regularity result for elasticity problems that covers the most natural space dimension $3$ and that captures behaviour of many typical elastic materials (considered in the small deformations) like rubber, polymer gels or concrete.'
address:
- 'Mathematical Institute, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic'
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Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland\
OxPDE, Mathematical Institute, University of Oxford, Oxford, UK
author:
- Miroslav Bulíček
- Jan Burczak
bibliography:
- 'references.bib'
title: Existence and smoothness for a class of $n$D models in elasticity theory of small deformations
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nonlinear small strain elasticity,regularity ,nonconstant bulk modulus
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Introduction
============
This note provides existence and, more importantly, interior smoothness of solutions to a PDE system describing a static problem in a linearized yet nonlinear elasticity theory in an arbitrary space dimension. Let us begin with the model description. We consider an isotropic homogeneous elastic body occupying in a reference configuration a Lipschitz domain $\Omega \subset {\mathbb{R}}^d$. The body is affected by external forces of density $\bef:\Omega \to {\mathbb{R}}^d$ and surface forces $\bg:\Gamma_N \to {\mathbb{R}}^d$, where $\Gamma_N \subset \partial \Omega$. In addition, another $\Gamma_D$ part of the boundary $\partial \Omega$ is deformed by a displacement $\bu_0:\Gamma_D \to {\mathbb{R}}^d$. $\Gamma_N$ and $\Gamma_D$ are disjoint open subsets of $\partial \Omega$, whose union has the same $(d-1)$ Hausdorff measure as $\partial \Omega$. Then, the described deformation must satisfy the following balance of forces $$\begin{aligned}
\label{eqa1}
- {{\rm {\bf div \,}}}\bsig& = \bef \quad (= -{{\rm {\bf div \,}}}\bF) &&\textrm{in } \Omega, \\
\bsig \bn&= \bg \quad (=\bF \bn) &&\textrm{on }\Gamma_N,\\
\bu &= \bu_0 &&\textrm{on } \Gamma_D.
\end{aligned}$$ Here, $\bsig:\Omega \to {\mathbb{R}}^{d\times d}_{sym}$ is the Cauchy stress tensor and $\bu:\Omega \to {\mathbb{R}}^d$ is the sought displacement field; $\bn$ is the outer normal unit vector. (For clarity, let us denote tensors by bold letters, vectors by regular letters and scalars by regular Greek letters. Accordingly, we disambiguate operators acting on tensors in bold and those acting on vectors in non-bold.) Already at this point, we assume that the external body and surface forces can be expressed by a given tensor $\bF:\Omega \to {\mathbb{R}}^{d\times d}_{sym}$. Naturally, if the problem is solvable, then any couple $(\bef, \bg)$ can be described (non-uniquely) by a tensor $\bF$. Additionally, in the case of the pure Neumann problem $\Gamma_N = \partial \Omega$, we avoid thereby using the necessary compatibility condition $\int_{\partial \Omega} g + \int_{\Omega} f = 0$, since it is already encoded into the existence of $\bF$.
In order to complete the problem , it remains to prescribe the constitutive relations for the Cauchy stress $\bsig$. The first classical law of linearized elasticity for isotropic homogeneous material is the generalized Hooke law, which has the form $$\label{Hook}
\bsig = 2\mu \, {{\tens {D}}}u + \lambda {\mathop{\text{div}}}\bu \;\bI = 2\mu\, {{\tens {D^d}}}u + (2\mu/d + \lambda) \diver \bu \;\bI,$$ where the constants $\mu$ and $\lambda$ are the so-called Lamé coefficients, the shear and the bulk modulus, respectively. Here, we denoted by ${{\tens {D}}}u$ the linearized strain tensor and by ${{\tens {D^d}}}u$ its deviatoric part, i.e., $${{\tens {D}}}u:= \frac12 ({{\tens {\nabla}}}\bu + ({{\tens {\nabla}}}\bu)^T), \qquad {{\tens {D^d}}}u := {{\tens {D}}}u - \frac{\diver \bu}{d} \; \bI.$$ Notice, that such a setting corresponds to a stored energy of the form $$W(\bu):= \mu |{{\tens {D^d}}}\bu|^2 + \tilde \lambda |\diver \bu|^2,$$ where $ \tilde \lambda := \frac{2\mu+d\lambda}{2d}$. Consequently, natural constraints on the coefficients are $\mu>0$ and ${2\mu+d\lambda} >0$. Furthermore, the problem can be now equivalently restated as $$\min_{\bu: \bu=\bu_0 \textrm{ on } \Gamma_D}\int_{\Omega} W(\bu) - \bF \cdot {{\tens {D}}}\bu \; dx.
\label{min}$$ For suitably chosen data one can always find a unique weak solution $u$. In addition, due to linearity of the problem, it is then standard to show higher regularity properties of $u$ that are restricted only by smoothness of $\bF$, $\bu_0$ and $\partial \Omega$.
However, it is experimentally established that the Hooke law is not valid anymore for a body undergoing a large loading. Recently, many phenomenological laws allowing for more involved constitutive relations between the Cauchy stress and the linearized strain tensor were investigated. Although they are far from being linear, they can be theoretically justified even within the theory of small deformations (i.e., the ‘linearized’ theory), see [@Ra07]. For an isotropic material, the most general constitutive law falling into the framework of the theory developed in [@Ra07] reads $$\bsig = \alpha_1 \bI + \alpha_2 {{\tens {D}}}\bu + \alpha_3 {{\tens {D}}}\bu \, {{\tens {D}}}\bu, \label{Hook-b}$$ where $\alpha_i$ may depend on all invariants of ${{\tens {D}}}\bu$. Decomposing further the dependence on the symmetric displacement gradient into its deviatoric and the trace part as well as requiring further that the material is hyperelastic, i.e., that there exists a potential - the generalized stored energy $W$, one arrives at the relation$$\bsig = 2\mu(|{{\tens {D}}}^d \bu|) {{\tens {D}}}^d \bu + \tilde{\lambda}(\diver \bu) \diver \bu \bI, \label{Hook-c}$$ where now $\mu$ and $\tilde{\lambda}$ are nonnegative scalar functions. The corresponding stored energy is then of the form $$\label{stored-b}
W(\bu):= \psi(|{{\tens {D}}}^d \bu|) + \varphi(\diver \bu), \qquad \psi(s):= \int_{0}^s 2\mu(t)t\; dt, \qquad \varphi(s):=\int_0^s \tilde{\lambda}(t)t\; dt.$$ Both functions $\psi$ and $\varphi$ are assumed to be nonnegative, convex and vanishing at zero. With this notation, solvability of supplemented with the constitutive relation is still equivalent to minimization of with the stored energy . Hence the solvability (in the weak sense) directly follows from the assumed convexity of $\psi$ and $\varphi$, provided reasonable growth and coercivity conditions for $W$ are assumed. Furthermore, an application of the standard difference quotient method enables to show certain regularity properties of the (unique) solution (at most the interior $W^{2,2}$ regularity for strongly coercive potentials). However, any more complex regularity theory, e.g. the $\mathcal{C}$-, $\mathcal{C}^{1}$- or $\mathcal{C}^{\infty}$-everywhere regularity is missing in general. The positive results in this direction for a rather general class of $W$’s in are known only either in the two dimensional setting or for general dimensional case if $\mu$ and $\lambda$ are ‘almost’ constant functions, compare [@FrNe88] or [@KaMaSt99] for incompressible fluids.
In this short paper, we provide further regularity (smoothness) properties of a weak solution to , in a general multidimensional setting, that go much beyond the classical results. Indeed, we are able to cover certain highly nonlinear dependences of the Cauchy stress on the small strain tensor. It is worth noticing that the investigated problem falls into the class of ‘generalized’ elliptic[^1] problems, where one cannot expect the full regularity of a solution, see the counterexamples in [@Ne77; @SvYa02]. The only known structural assumption allowing for the full regularity reads $$W(\bu) \sim \tilde{W}(|\nabla \bu|),$$ due to the classical result by Uhlenbeck [@Uh77]. However, for a nonlinear stored energy of the form , such a result is currently not available, due to ‘non-diagonality’ of the corresponding elliptic operator and nonexistence of a proper substitute to the Uhlenbeck’s method. Nevertheless, we show that for a certain class of models within one can overcome these difficulties and improve the smoothness of the solution significantly. Namely, our main result can be summarised as follows (for the fully rigorous formulation we refer the reader to Theorem \[T:PR\])
\[T:main\] Let $\bu_0, \bF$ be smooth and $\mu$ be a constant. In , let $\psi(s):=\mu s^2$ and $\varphi$ be a smooth convex function, having at most the polynomial growth. Then there exists a unique weak solution $\bu$ to the problem belonging to $\mathcal{C}^{\infty}_{loc}(\Omega)$.
Let us emphasize here, that this is the first regularity result for nonlinear systems od PDEs’ arising in the linearized elasticity theory, with no data-smallness or low-dimensionality restrictions. Despite being clearly far from covering the most natural case of (namely, both $\psi$ and $\varphi$ being reasonably nonlinear), our result can be seen as the first step forward in the regularity theory for such problem. Beyond its theoretical novelty, it is viable applicatively: it covers real-world materials whose shear modulus remains constant, but the bulk modulus may change drastically with respect to the volume changes. Their examples are rubber, certain polymers, concrete etc., see e.g. [@Hi91; @Ho71; @Ma92] and the references therein.
Notation and definitions
========================
Firstly, let us introduce function spaces relevant to the stored energy with a constant $\mu$. Since we want to keep the allowed bulk modulus nonlinearity ($\varphi$) possibly general, we resort to Orlicz growths in our analysis. Nevertheless, let us accentuate at the very beginning that our results are new even for power–law growths. Let us briefly recall the Orlicz framework. Since we intend to keep this part as concise as possible, we refer the interested reader e.g. to [@BurKap15 Appendix] for more details. We shall start with the notion of the ${{\mathcal{N}}}$ function.
\[def:app\_orlicz:nf\] A real function ${{\varphi}}: {\mathbb{R}}\to {\mathbb{R}}_+$ is an *${{\mathcal{N}}}$-function* iff it is even and there exists ${{\varphi}}'\!: {\mathbb{R}}_+ \to {\mathbb{R}}_+$
- \[N1\] that is right-continuous, non-decreasing,
- \[N2\] that satisfies ${{\varphi}}' (0) =0$, $ {{\varphi}}' (t) >0$ for $t>0$ and ${{\varphi}}' (+ \infty^-) =+ \infty$,
such that for all $t>0$ it holds $${{\varphi}}(t) = \int_0^t {{\varphi}}' (s) \, ds.$$
Notice here, that it follows from (N\[N1\]) that ${{\varphi}}$ is convex. Thus, we can also introduce its convex conjugate $\varphi^*$ by the formula $\varphi^*(s):=\sup_{t}(st-\varphi(t))$. Next, let us define the Orlicz class $L^{{{\varphi}}}(\Omega)$ as $$L^{{{\varphi}}}(\Omega):=\left\{u\in L^1(\Omega): \; \int_{\Omega} {{\varphi}}(u) \dx <\infty \right\}.$$ It becomes the Banach space for ${{\varphi}}$ satisfying the so–called $\Delta_2$ condition[^2], i.e., $$\label{Delta2}
{{\varphi}}(2t)\le C({{\varphi}}(t)+1), \quad \textrm{ for some } C>0 \textrm{ and all } t\in \mathbb{R}.$$ The importance of the $\Delta_2$ condition appears also in the regularity theory, since it directly implies[^3] $$\varphi(t)\le t\varphi'(t) \le C(\varphi(t)+1).\label{delta2-better}$$ Let us provide most typical growths that stay within our Orlicz structure and satisfy $\Delta_2$ condition. Let $\kappa \ge 0$, $p > 1$. The classical power-law growths $${{\varphi}}_1 (t) = \int_0^t \! (\kappa + s^{p-2}) \, s \,ds, \qquad {{\varphi}}_2 (t) = \int_0^t \!(\kappa + s^2)^\frac{p-2}{2} \, s \,ds$$ are naturally allowed.
An example of an admissible ${{\mathcal{N}}}$-function related to a non-polynomial growth reads $${{\varphi}}_3 (t) = \int_0^t (\kappa +s^2)^\frac{p-2}{2} s \ln^\beta (e+s) ds, \quad \beta >0.$$
In what follows we will use standard notions of Lebesgue and Sobolev spaces $L^p$, $W^{k,p}$ respectively, where the subscript $_{\Gamma_D}$ will indicate that the considered functions vanish on $\Gamma_D$ (the Dirichlet part of the boundary). Next, let us introduce generalized Sobolev–Orlicz classes of vector–valued functions compatible with our problem setting. Here, $\bu_0:\Omega \to \mathbb{R}^d$ is a given measurable function. $$\begin{aligned}
WD_{\Gamma_D}&:=\left\{\bu\in W^{1,1}_{\Gamma_D}(\Omega; \mathbb{R}^d): \; {{\tens {D}}}^d\bu \in L^2(\Omega; {\mathbb{R}}^{d\times d}), \; \diver \bu \in L^{{{\varphi}}}(\Omega) \right\},\\
WD_{\bu_0}&:=\left\{\bu\in W^{1,1}_{\Gamma_D}(\Omega; \mathbb{R}^d): \; \bu=\bu_0 \textrm{ on } \Gamma_D, \; {{\tens {D}}}^d\bu \in L^2(\Omega; {\mathbb{R}}^{d\times d}), \; \diver \bu \in L^{{{\varphi}}}(\Omega) \right\}.
\end{aligned}$$ Please notice that if ${{\varphi}}$ satisfies the $\Delta_2$ condition, $\Omega$ is Lipschitz and $\bu_0$ has sufficiently highly integrable first order derivatives, then the notion $\bu \in WD_{u_0}$ is equivalent to $(\bu-\bu_0) \in WD_{\Gamma_D}$. Furthermore, in the case of $\Gamma_D = \emptyset$, we shall consider all functions belonging to $WD_{\Gamma_D}$ and $WD_{\bu_0}$ up to a rigid body motions, i.e., modulo all linear functions fulfilling ${{\tens {D}}}\bu \equiv 0$, in order to guarantee uniqueness of the desired displacement.
We shall frequently use a generic constant $C>0$ that depends only on the data of our problem. It may generally vary line to line. If we need to trace any data dependences more precisely, it will be clearly indicated in the text.
To conclude this section, let us introduce a notion of a weak solution. Below, we use the decomposition of (symmetric) $\bF$ into the deviatoric and the traceless part, i.e., $\bF=\bF^d + d^{-1} \tr \bF \, \bI$.
\[D1\] Assume that $\Omega\subset \mathbb{R}^d$ is a Lipschitz domain and $\bu_0 \in W^{1,2}(\Omega, \mathbb{R}^d)$ is such that $\diver \bu_0 \in L^{\varphi}$. Further, let $\bF$ be such that $\bF^d \in L^2(\Omega;\mathbb{R}^{d\times d}_{sym})$ and $\tr \bF \in L^{\varphi^*}(\Omega)$. Let $\varphi$ be an ${{\mathcal{N}}}$ function that satisfies and $\mu$ be a positive constant. We say that $\bu \in WD_{\bu_0}$ is a weak solution to with the constitutive law – iff for all $\bv \in WD_{\Gamma_D}$ there holds $$\label{w-f}
\int_{\Omega}2\mu\, {{\tens {D}}}^d \bu \cdot {{\tens {D}}}\bv+ \varphi'(\diver \bu) \diver \bv = \int_{\Omega}\bF \cdot {{\tens {D}}}\bv.$$
The Neumann part (formally) cancels out thanks to our ‘compatibility condition’, i.e. the use of $\bF$ in . Observe that our weak formulation is meaningful. Indeed, the critical term can be estimated thanks to (N\[N1\]) and as follows $$|\varphi'(\diver \bu) \diver \bv| \le |\varphi'(\diver \bv) \diver \bv| + |\varphi'(\diver \bu) \diver \bu| \le C(1+ \varphi(\diver \bu) + \varphi(\diver \bv)) \in L^1(\Omega).$$
Result {#sec:rel}
======
\[T:PR\] Let all assumptions of Definition \[D1\] be satisfied. Then, there exists a weak solution to such that $$\begin{aligned}
\|{{\tens {D}}}^d\bu\|^2_{L^2(\Omega)} + \int_{\Omega}\varphi(\diver \bu)&\le C(\mu,d)\left(\|\bF^d\|^2_{L^2(\Omega)}+\|{{\tens {D}}}^d\bu_0\|^2_{L^2(\Omega)} + \int_{\Omega}\varphi^*(\tr \bF)+\varphi(\diver \bu_0)\right)=:A, \label{TE1}\end{aligned}$$ This weak solution is unique, provided that $\varphi$ is strictly convex. More surprisingly, it enjoys the following smoothness properties in any compact set $K\subset \Omega$:
$$\begin{aligned}
\left\|\frac{\partial \bu_i}{\partial x_j} - \frac{\partial \bu_j}{\partial x_i}\right\|_{W^{k,p}(K)}+\left\|\frac{2\mu(d-1)}{d}\diver \bu + \varphi'(\diver \bu)\right\|_{W^{k,p}(K)} &\le C(K,\Omega, A,k,p)(\|\bF\|_{W^{k,p}(\Omega)}+1), \label{TE2}\\
\|\nabla \bu\|_{W^{k,p}(K)}&\le C(K, \Omega, A,k,p, \|\varphi\|_{\mathcal{C}^{k+2}}, \|\bF\|_{W^{k,p}(\Omega)}),\label{TE3}\end{aligned}$$
where is valid for all $k\in \mathbb{N}_0$ and all $p\in (1,\infty)$, whereas holds both for $k=0,1$ with all $p\in (1,\infty)$ and for arbitrary $k\in \mathbb{N}$ with any $p\in (d/2, \infty)$.
Proof of Theorem \[T:PR\] {#sec:proof}
=========================
#### Weak existence & uniqueness
We use the direct methods of the calculus of variations to obtain the existence. Indeed, let us mimic the problem and can seek for a minimizer $\bu\in WD_{\bu_0}$ fulfilling for all $\bv \in WD_{\bu_0}$ $$\label{min-g}
\int_{\Omega}\mu|{{\tens {D}}}^d \bu|^2 + \varphi(\diver \bu) - \bF \cdot {{\tens {D}}}\bu \le \int_{\Omega}\mu|{{\tens {D}}}^d \bv|^2 + \varphi(\diver \bv) - \bF \cdot {{\tens {D}}}\bv.$$ The functional is definitely coercive and due to convexity of $\varphi$ and its superlinearity at infinity, we can find a minimizer fulfilling . In addition, the uniform estimate easily follows. Furthermore, since $\varphi$ satisfies the $\Delta_2$ condition, and consequently $\bv:=\bu+\bw$ with an arbitrary $\bw \in WD_{\Gamma_0}$ is an admissible competitor in , we can easily derive the Euler–Lagrange equations for , which is nothing else than the identity . The uniqueness then follows from the strict convexity of $\varphi$.
We are now approaching the crucial part, namely our proof of the interior smoothness. Before passing to details, let us observe that the main idea behind this proof is the following observation: Since $$\diver {{\rm {\bf div \,}}}{{\tens {D^d}}}f = \frac{d-1}{d} \Delta \diver f,$$ then $\diver \eqref{eqa1}$ produces a scalar, well-manageable equation for $\diver u$.
#### Estimate
We start with deriving an elliptic equation for $\diver \bu$. Let $v\in \mathcal{C}^2_0(\Omega)$ be arbitrary. Setting $\bv:=\nabla v$ in we observe that $$\label{MB1}
\int_{\Omega}2\mu {{\tens {D}}}^d \bu \cdot \nabla^2 v + \varphi'(\diver \bu) \Delta v = \int_{\Omega}\bF \cdot \nabla^2 v.$$ Next, let $G$ be the Green function to the Laplace equation in ${\mathbb{R}}^d$ and let us define (in the sense of distribution) $$g:=G*\diver {{\rm {\bf div \,}}}\bF,$$ where we extend $\bF\equiv 0$ outside $\Omega$. Notice that such a $g$ solves the problem $$\label{dfg}
\int_{\Omega} g \Delta v = \int_{\Omega} \bF \cdot \nabla^2 v \qquad \textrm{ for all } v \in \mathcal{C}^{\infty}_{0}(\Omega).$$ In addition for any compact $K \subset \Omega$, all $k\in \mathbb{N}$ and all $p\in (1,\infty)$, we have the estimate $$\|g\|_{W^{k,p}(K)} \le C(K,k,p)\|\bF\|_{W^{k,p}(\Omega)}. \label{g-est}$$
Finally, using integration by parts, it is not difficult to deduce that $$\begin{aligned}
\int_{\Omega}{{\tens {D}}}^d \bu \cdot \nabla^2 v=\int_{\Omega}{{\tens {D}}}^d \bu \cdot (\nabla^2 v - d^{-1}\Delta v \, \bI)=\int_{\Omega}\nabla \bu \cdot (\nabla^2 v - d^{-1}\Delta v \, \bI)=\frac{d-1}{d}\int_{\Omega}\diver \bu \Delta v.
\end{aligned}$$ Consequently, using also and , we obtain $$\int_{\Omega} \left(\frac{2\mu(d-1)}{d}\diver \bu + \varphi'(\diver \bu) -g\right)\Delta v =0,$$ which means that $\frac{2\mu(d-1)}{d}\diver \bu + \varphi'(\diver \bu) -g$ is harmonic in $\Omega$. Therefore, for any compact $K\subset \Omega$ we have that $$\left\|\frac{2\mu(d-1)}{d}\diver \bu + \varphi'(\diver \bu) -g\right\|_{W^{k,p}(K)} \le C(K)\left\|\frac{2\mu(d-1)}{d}\diver \bu + \varphi'(\diver \bu) -g\right\|_{L^1(\Omega)}\le C(A,K),$$ where $A$ is defined in . Thus, using also , we finally deduce that $$\left\|\frac{2\mu(d-1)}{d}\diver \bu + \varphi'(\diver \bu)\right\|_{W^{k,p}(K)} \le C(K,A,p,k)(\|\bF\|_{W^{k,p}(\Omega)}+ \|\diver \bu\|_{L^\varphi (\Omega)}) \label{fin-div}.$$ Hence the second part of the estimate follows from estimating the right hand side of with the help of .
Similarly, using , we deduce that in the distributional sense $$\label{w-f-c}
-\mu \Delta \bu + \mu \nabla \diver \bu - \nabla \left( \frac{2\mu(d-1)}{d}\diver \bu + \varphi'(\diver \bu)\right) = -{{\rm {\bf div \,}}}\bF.$$ Consequently, taking distributionally aa $i,j$ component of $\nabla \wedge$ in yields for any $i,j=1,\ldots,d$ $$\label{curl}
\mu \Delta \left (\frac{\partial \bu_i}{\partial x_j} - \frac{\partial \bu_j}{\partial x_i}\right ) = \sum_{k=1}^d \frac{\partial}{\partial x_k} \left(\frac{\partial \bF_{ik}}{\partial x_j} - \frac{\partial \bF_{jk}}{\partial x_i}\right).$$ Thus, since we already know that $\nabla \bu \in L^{1}$, we can use the singular integral theory to conclude that for all $i,j=1,\ldots,d$, all $k\in \mathbb{N}$, all $p\in (1,\infty)$ and all compact set $K\subset \Omega$ $$\label{est}
\left\|\frac{\partial \bu_i}{\partial x_j} - \frac{\partial \bu_j}{\partial x_i}\right\|_{W^{k,p}(K)}\le C(\mu, k,p,K)(\| \nabla \bu\|_{L^{1}(\Omega)}+\|\bF\|_{W^{k,p}(\Omega)})$$ and follows from and .
#### Estimate
Going back to and using the theory for the Laplace equation, we directly deduce the estimate $$\label{26}
\|\bu\|_{W^{k+1,p}(K')} \le C(k,p,K,K')\left(\|\bu\|_{L^2(\Omega)} +\|\bF\|_{W^{k,p}(\Omega)} + \|\diver \bu\|_{W^{k,p}(K)} + \left\|\frac{2\mu(d-1)}{d}\diver \bu + \varphi'(\diver \bu)\right\|_{W^{k,p}(K)} \right)$$ for arbitrary compact sets $K'$, $K$ such that there is an open set $\Omega'$ fulfilling $K'\subset \Omega'\subset K\subset \Omega$, arbitrary $k\in \mathbb{N}_0$ and arbitrary $p\in (1,\infty)$. Therefore due to the already obtained estimate , we see that it is enough to get the bound on $\diver \bu$ which will be however read again from .
We start with the case $k=0$. Since $\varphi'(s)s\ge 0$ (which follows from the fact that $\varphi$ is even), hence $|s| \le |\varphi'(s)+s|$ and one has $$\|\diver \bu\|_p \le C(d,\mu)\left\|\frac{2\mu(d-1)}{d}\diver \bu + \varphi'(\diver \bu)\right\|_{p}.$$ Consequently, it follows from and that $$\label{27}
\|\bu\|_{W^{1,p}(K')} \le C(k,p,K,K')\left(\|\bu\|_{L^2(\Omega)} +\|\bF\|_{L^{p}(\Omega)} \right)$$ and holds for $k=0$ and $p\in (1,\infty)$. Secondly, take $k=1$. Since $\varphi'$ is nondecreasing (and Lipschitz, recall for $k=1$), it is easy to observe that $$\|\diver \bu\|_{1,p} \le C(d,\mu)\left\|\frac{2\mu(d-1)}{d}\diver \bu + \varphi'(\diver \bu)\right\|_{1,p}$$ for all $p\in (1,\infty)$. Consequently, we again get that (now we set $k=1$ in ) $$\label{28}
\|\bu\|_{W^{2,p}(K')} \le C(k,p,K,K')\left(\|\bu\|_{L^2(\Omega)} +\|\bF\|_{W^{1,p}(\Omega)} \right)$$ and for $k=1$ again follows. Finally, for the remaining range of $k$’s we shall restrict ourselves to the case $p> d/2$. From one obtains $$\left\|\left(\frac{2\mu(d-1)}{d}+\varphi''(\diver \bu)\right) \nabla \diver \bu\right\|_{W^{k,p}(K')} \le C(K')(\|\bF\|_{W^{k+1,p}(\Omega)}+ \|\diver \bu\|_{L^\varphi(\Omega)}). \label{fin-div2}$$ From and using $p> d/2$ $$\label{eq:i1}
\|\diver \bu\|_{L^\infty(K')} \le C(d,p,K',\Omega)\left(\|\bu\|_{L^2(\Omega)} +\|\bF\|_{W^{1,d+\delta}(\Omega)} \right) \le C(d,p,K',\Omega)\left(\|\bu\|_{L^2(\Omega)} +\|\bF\|_{W^{k+1,p}(\Omega)} \right),$$ therefore arguments of $\varphi''$ and its derivatives will remain bounded. Let us now show inductively that for $p> d/2$ there holds $$\left\|\nabla \diver \bu\right\|_{W^{k,p}(K')} \le C(K', \Omega, d, p,\|\varphi\|_{\mathcal{C}^{k+2}}, k, A) (1+ \|\bF\|_{W^{k+1,p}(\Omega)})^{3^k}. \label{fin-div3}$$ This relation is valid for $k=0$ thanks to and . Assuming now that holds for $k-1$, we shall show that it also holds for $k$. One has
$$\label{eq:ind}
\left\|\nabla \diver \bu\right\|_{W^{k,p}(K')} =\left\|\nabla^{k} \nabla \diver \bu\right\|_{L^{p}(K')} + \left\|\nabla \diver \bu\right\|_{W^{k-1,p}(K')} := I + II.$$
Furthermore $$\label{eq:i2}
\begin{aligned}
I &\le \frac{d}{2\mu(d-1)} \left\| \left(\frac{2\mu(d-1)}{d}+\varphi''(\diver \bu)\right) \nabla^k \left( \nabla \diver \bu \right)\right\|_{L^{p}(K')} \\
& \le C(\mu, d) \left\| \nabla^k \left(\left(\frac{2\mu(d-1)}{d}+\varphi''(\diver \bu)\right) \nabla \diver \bu \right)\right\|_{L^{p}(K')} + \sum^k_{i=1} C \left\| \nabla^i ( \varphi''(\diver \bu) )\nabla^{k-i}\nabla \diver \bu \right\|_{L^{p}(K')} \\
& \le C(\mu, d) \left\| \nabla^k \left(\left(\frac{2\mu(d-1)}{d}+\varphi''(\diver \bu)\right) \nabla \diver \bu \right)\right\|_{L^{p}(K')} + C \|\varphi\|_{\mathcal{C}^{k+2}} \|\diver \bu\|_{L^\infty(K')} \|\diver \bu\|^{{2}}_{W^{k,2p}(K')}.
\end{aligned}$$ The last term of , via the inductive assumption for $k-1$, and next by $W^{k+1,p}\hookrightarrow W^{k,2p}$ for $p> d/2$ can be estimated with $$C \left(\|\bu\|_{L^2(\Omega)} +\|\bF\|_{W^{k+1,p}(\Omega)} \right) (\|\bF\|_{W^{k,2p}(\Omega)}+ \|\diver \bu\|_{L^\varphi(\Omega)})^{2 \cdot 3^{k-1}} \le C ( 1+ \|\bF\|_{W^{k+1,p}(\Omega)})^{3^k},$$ where we have used and $C= C(K', \Omega, d, p,\|\varphi\|_{\mathcal{C}^{k+2}}, k, A)$. Since the last but one of is controlled by , it holds $$I \le C ( 1+ \|\bF\|_{W^{k+1,p}(\Omega)})^{3^k},$$ Since the lower-order term $II$ is estimated by the inductive assumption for $k-1$, we have arrived at validity of for $k$. Estimate is hence proven.
[^1]: We are using the word ‘generalized’ here, because the potential $W$ depends only the symmetric gradient.
[^2]: For ${{\varphi}}$ not satisfying $\Delta_2$ condition, the relevant function space related to the corresponding Orlicz class is then defined as a union of such functions $u$ for which there exists $\lambda>0$ such that $\lambda u \in L^{{{\varphi}}}(\Omega)$.
[^3]: Indeed, for $t\ge 0$ we can use the fact that $\varphi'$ is nondecreasing and nonnegative to observe that $$\varphi(t)=\int_0^t \varphi'(\tau) \le t\varphi'(t) \le \int_t^{2t}\varphi'(\tau)\le \varphi(2t) \overset{\eqref{Delta2}}{\le} C(\varphi(t)+1).$$ For more details about the $\Delta_2$ condition and the above so–called good $\varphi'$ property see also [@BurKap15 Appendix], [@DieEtt08] or [@DieKap13].
|
---
abstract: 'The shock response of two-dimensional model high explosive crystals with various arrangements of circular voids is explored. We simulate a piston impact using molecular dynamics simulations with a Reactive Empirical Bond Order (REBO) model potential for a sub-micron, sub-ns exothermic reaction in a diatomic molecular solid. In square lattices of voids (of equal size), reducing the size of the voids or increasing the porosity while holding the other parameter fixed causes the hotspots to consume the material more quickly and detonation to occur sooner and at lower piston velocities. The early time behavior is seen to follow a very simple ignition and growth model. The hotspots are seen to collectively develop a broad pressure wave (a sonic, diffuse deflagration front) that, upon merging with the lead shock, transforms it into a detonation. The reaction yields produced by triangular lattices are not significantly different. With random void arrangements, the mean time to detonation is 15.5% larger than with the square lattice; the standard deviation of detonation delays is just 5.1%.'
author:
- 'S. Davis Herring'
- 'Timothy C. Germann'
- 'Niels Grønbech[-]{}Jensen'
title: Sensitivity effects of void density and arrangement in a REBO high explosive
---
Introduction
============
Heterogeneities such as inclusions, voids, cracks, and other defects enhance the shock sensitivity of high explosives by causing additional shock dissipation that creates small regions of high temperature called hotspots[ [@info:lanl-repo/inspec/1653210]]{}. Chemical reactions initiated in the hotspots emit pressure waves that merge with the lead shock and strengthen it, so that further hotspots are created with more vigor. This positive feedback is the principal mechanism of the shock-to-detonation transition in inhomogeneous explosives[ [@beyond-standard]]{}. While the significance of heterogeneities is well known, which of their characteristics are most important are not. In particular, since the details of the growth of reactions from the hotspots are not well understood, it is not known whether hotspots act separately or if the spatial arrangement of hotspots determines their efficacy.
Spherical voids are an often-studied, common defect in explosives[ [@zukas/walters; @bowden/yoffe; @explosives/propellants; @beyond-standard]]{}. They can become hotspots upon collapsing under shock loading, and may also cause hotspots elsewhere by their partial reflection of the lead shock and by emitting further shocks upon their collapse and explosion. Inert beads produce similar effects; while they do not collapse violently enough to become hotspots, their reflected shocks are not weakened by immediately following rarefactions and so may more easily create hotspots where they collide.
Experimental, theoretical, and numerical studies have sought to explain the sensitivity enhancement caused by voids and inert inclusions. Bourne and Field[ [@info:lanl-repo/eixxml/1991060235317]]{} reported results from shocked two-dimensional samples of gelatin or an emulsion explosive that had large cylindrical voids introduced. They observed that voids could shield their downstream neighbors from the lead shock, but that they could also effect the collapse of their neighbors by emitting shock waves when they collapsed. Dattelbaum [ [@nm-hotspots2]]{} shocked samples of nitromethane with randomly embedded glass beads or microballoons, observing that their presence decreased the run distance to detonation and the pressure dependence of that distance. The balloons were found to have a greater effect than the beads, and small beads were in turn more effective than large beads.
Medvedev [ [@emulsion-microballoons]]{} conducted a theoretical analysis of emulsion explosives with microballoons that explained changes in detonation velocity with microballoon concentration via an ignition and growth model with a constant mass burn rate per hotspot. Bourne and Milne[ [@cavity-collapse]]{} experimentally and computationally considered a hexagonal lattice of cylindrical voids in an emulsion explosive or nitromethane and observed that the reactions at the hotspots accelerated the shock relative to a comparison with water. In a previous report[ [@paper1]]{} we used molecular dynamics (MD) to simulate single circular voids (and their periodic images) in a two-dimensional model solid explosive; the one rank of voids was able to induce a shock-to-detonation transition in the downstream material. In this study, we extend those simulations to samples with structured and unstructured two-dimensional arrangements of voids.
Method
======
We simulate piston impacts on a number of samples with several equal-sized circular voids either randomly placed or in a regular square or triangular lattice. We parameterize the possible lattices by their symmetry, the total porosity $p$ (proportion of molecules removed from the perfect crystal), and the radius $r$ of each void. The spacing between voids in the square lattice is then $$\delta\equiv r\sqrt{\pi/p}.$$ The principal goal is to determine which of these parameters have a significant effect on the sensitivity of the explosive, as measured by the time [$t_D$]{} from piston impact to detonation transition. We also look for nonadditive contributions from the arrangement of voids (rather than their simple number density) and explore the mechanism of the development of detonation.
Model
-----
The Reactive Empirical Bond Order (REBO) “AB” potential (originally developed in [@info:lanl-repo/inspec/4065531; @info:lanl-repo/inspec/4411199; @extreme-dynamics]) describes an exothermic $\mathrm{2AB \rightarrow A_2 + B_2}$ reaction in a diatomic molecular solid and exhibits typical detonation properties but with a sub-micron, sub-ns reaction zone that is amenable to MD space and time scales. Heim modified it to give a more molecular (and less plasmalike) Chapman-Jouguet state[ [@info:lanl-repo/isi/000260573900087]]{}. We utilize the SPaSM (Scalable Parallel Short-range Molecular dynamics) code[ [@info:lanl-repo/eixxml/1994121436059]]{} and the modified REBO potential (“ModelIV”) also utilized in our previous study [@paper1]. The masses of A and B atoms are 12 and 14 amu; a standard leapfrog-Verlet integrator is used with a fixed timestep of 0.509 fs in the NVE ensemble.
Our two-dimensional samples are rectangles of herringbone crystal with two AB molecules in each $6.19\times4.21$ Å unit cell. The shock propagates in the $+z$ direction; the samples are periodic in the transverse $x$ direction. Each circular void is created by removing all dimers whose midpoints lie within a circle of a given radius. The atoms are assigned random velocities corresponding to a temperature of 1 mK and an additional bulk velocity $v_z=-u_p$ directed into an infinite-mass piston formed by three frozen layers of unit cells at the $-z$ end. The temperature is chosen to be small to avoid significant thermal expansion of the crystal but nonzero to avoid spurious effects from a mathematically perfect crystal; the RMS atomic displacement it causes is 2.1 pm.
Each simulation is run until the shock (whether reacting or not) reaches the free end of the sample; the traversal time of the shock (assuming that it does not accelerate) is ${t_t}=Z/u_s(u_p)$, where $u_s(u_p)$ is the shock velocity Hugoniot. It is broadly similar throughout the simulations, so the determination of whether or not a detonation occurs is meaningfully consistent. In particular, in each of the principal studies the sample length $Z$ is held fixed so that ${t_t}$ is constant for each $u_p$. Analysis of the results, including dynamic identification of molecules, location of the lead shock, and determination of detonation transition times, follows [@paper1], except that when finding the detonation transition we keep the shock positions 15 nm apart for greater noise resistance.
Systems
-------
We consider 94 square lattices: 27 with an integer number $n$ of voids (in each periodic image) and 67 with $n$ allowed to merely approximate an integer (so the last void’s distance to the end of the sample differs slightly from the first’s to the piston). For brevity, we will term these two cases [S$_1$]{} and [S$_2$]{} respectively.
In [Case [S$_1$]{}]{}, each combination of $p\in\{1.0,1.778,2.25\}\%$ and $r\in\{3,4,6\}{\text{~nm}}$ is considered, with $Z=1.28$ [m]{} chosen so that $n\equiv Z/\delta$ is always an integer ($n\in[12,36]$), and each choice is simulated at each piston velocity $u_p\in\{{1.96},{2.95},{3.93}\}{\text{~km/s}}$; 694776–2070048 atoms are simulated. In [Case [S$_2$]{}]{}, $Z=845{\text{~nm}}$ and $u_p={2.95}{\text{~km/s}}$ are fixed, and the 67 $p$-$r$ pairs from $\{1.0,1.1,\dotsc,2.0\}\%\times\{15,16,\dotsc,59\}~{\text\AA}$ that yield an $n$ within 0.075 of an integer are simulated ($n\in[8,42]$, 260736–1341296 atoms). The fixed velocity and loosened constraint on $n$ allow this study to explore the $p$-$r$ space more effectively.
In an ancillary study called [Case T]{}, rectangular and triangular lattices of $n=10$ voids are each simulated 27 times with a fixed $u_p={1.96}{\text{~km/s}}$ and every combination of $p\in\{1.0,1.5,2.0\}\%$, $r\in\{3,4,5\}{\text{~nm}}$, and $Z\in\{416,521,627\}{\text{~nm}}$. Here the void spacings are $\delta_z=Z/n$ and $\delta_x=\pi r^2/p\delta_z=n\pi r^2/pZ$; 214060–1206248 atoms are simulated.
In the random case, a fixed sample size of $201{\text{~nm}}\times1015{\text{~nm}}$ is used with three different $(p,r,u_p)$ triples taken from the [Case [S$_1$]{}]{} lattices (but with more voids because of the increased sample area): 1 with $(1\%,6{\text{~nm}},{1.96}{\text{~km/s}})$ and $n=18$, 2 with $(2.25\%,6{\text{~nm}},{1.96}{\text{~km/s}})$ and $n=40$, and 3 with $(2.25\%,4{\text{~nm}},{2.95}{\text{~km/s}})$ and $n=90$. For each set of parameters, 10 simulations are run with different random arrangements of voids chosen by the simple rejection method such that all void centers are at least $2r$ away from either surface of the sample and at least $4r$ away from each other. This largest case involves $\sim$3.1 million atoms.
Results
=======
The transition times for the 53 (of 67) [Case [S$_2$]{}]{} samples that detonated are shown in [[Fig. ]{}]{}[vn2]{}. The plane is a fit to the data; its equation is $t_D(p,r)/\text{ps}=9.718r/\text{nm}-1506p+45.67$. The cases that did not detonate before the shock reached the free surface (${t_t}\approx78.2{\text{~ps}}$) would occupy the rear corner of the plot (smallest $p$ and largest $r$). The proportion of atoms that form product molecules was generally 85% but dropped to 65% in that corner. The half of those 14 simulations closer to the ones which detonated ended with detonation evidently imminent, but they are not counted as having detonated since a transition time could not be identified.
![Detonation transition times [$t_D$]{} from [Case [S$_2$]{}]{}. The grid shows every $p$ and every fourth $r$ value and is a planar fit to the data, which are shown as crosses connected to it by lines to place them in space and indicate the discrepancies. The upper limit of the plotting region is the length of the longest, non-detonating simulations (78.2 ps).[]{data-label="vn2"}](vn2.pdf){width="\figwidth"}
At early times, we observe that the extent of reaction closely follows a very simple ignition and growth model. Suppose that a reacting hotspot is a disk that is created with a finite radius $r_0$ upon collapse of the void and grows at a constant speed $v$ into the surrounding unreacted material until it overlaps its periodic images. If the disk contains a constant areal number density $n_a$ of reacted atoms, the number of reacted atoms as a function of time since initiation has the form $$N(t)=n_aR(t)^2=n_a\pi(r_0+vt)^2.$$ In [[Fig. ]{}]{}[iandg]{} are plotted the counts of reacted atoms from the beginning of the least reactive [Case [S$_2$]{}]{} simulation, and the results of fitting one and two copies of $N(t)$ to them, corresponding to the first and then the second periodic line of voids being ignited. The two copies use the same growth parameters and are merely each shifted in time to match the data. Other [Case [S$_2$]{}]{} simulations have similar behavior, but the growth parameters depend in an unknown fashion on $r$ and $u_p$, so we have no general model for $N(r,u_p,t)$.
![Growth of reaction from the first two voids in a sample with $p=1\%$ and $r=59$ Å. The values from the ignition and growth model applied to the first void and to the first two voids are also shown; the latter curve is almost everywhere indistinguishable from the simulation values.[]{data-label="iandg"}](iandg.pdf){width="\figwidth"}
[[Fig. ]{}]{}[spires]{} shows the temperature distribution history from the other extreme [Case [S$_2$]{}]{} simulation, with the smallest (eligible) $r$ at the largest $p$. Each point in the figure is calculated with respect to the center of mass motion of, and averaged over, a column of computational cells of width $\Delta z\approx0.53$ nm. We note here the development of a broad pressure wave in the shocked material, visible both as a region of strong advection intersecting the detonation transition and as a temperature increase between the last few hotspots created before the transition. It appears that the pressure waves emitted by the first several hotspots merge and the combined wave strengthens itself by encouraging the deflagration at each hotspot it encounters. When this wave overtakes the lead shock, its particle velocity is approximately equal to $u_p={2.95}{\text{~km/s}}$, so the relative velocity in the collisions at the shock doubles and detonation begins immediately.
![[(Color online) ]{}Temperature over space and time in a sample with a square lattice of 42 voids: $p=2\%$, $r=16$ Å, $u_p={2.95}{\text{~km/s}}$. Only the region of space occupied by the sample at its final compression is shown. Each spire at the lower left is a hotspot; the very hot region at large $z$, bounded below by a much faster shock, is the detonation. Note the left-moving shock generated at the transition and the pressure wave (apparent as a temporary strong advection) visible in the shocked material between $z=150$ nm and $z=400$ nm.[]{data-label="spires"}](spires_s.pdf){width="\figwidth"}
All but 3 of the 27 [Case [S$_1$]{}]{} simulations produced a detonation; those that did not used the lowest $u_p$ and (again) occupied the $-p$/$+r$ corner of that slice of the parameter space. The transition times for the rest are shown in [[Fig. ]{}]{}[vn]{}; they follow the pattern of [Case [S$_2$]{}]{} with the unsurprising addition that ${\partial{t_D}/\partial{u_p}}<0$. The middle surface has the same $u_p$ as [Case [S$_2$]{}]{} and shows some non-planarity beyond the range of [[Fig. ]{}]{}[vn2]{}. With appropriate $p$ and $r$, we see detonations even at $u_p<2{\text{~km/s}}$, which is much smaller than any value observed to trigger detonation with merely one periodic rank of voids[ [@paper1]]{}; the feedback is much strengthened by the subsequent hotspots.
![Detonation transition times [$t_D$]{} from [Case [S$_1$]{}]{}. The missing points for the smallest $u_p$ indicate failures to detonate.[]{data-label="vn"}](vn.pdf){width="\figwidth"}
In [Case T]{}, the rectangular and triangular lattices did not differ significantly: they produced similar reaction yields and no detonations (presumably because they were relatively short; the [Case [S$_1$]{}]{} simulations corresponding to the most reactive case treated here detonated after 87–90 ps). We would expect the triangular case to have a greater reactivity than the rectangular because the overlap between hotspots in adjacent columns is delayed by their offset. The difference is never more than 5% of the area, however, and lasts only until the overlap is complete (about 30% of the reaction phase), so it may be difficult to measure. For 3, a consistent transition time of $t_D=59.3{\text{~ps}}\pm5.1\%$ was observed. The mean is 15.5% larger than the [$t_D$]{} from the corresponding [Case [S$_1$]{}]{} simulation; the small standard deviation suggests that the details of the void arrangement are not significant. Furthermore, examination of one such simulation shows that a tight arrangement of 5 voids approximately 30% of the way down the sample produces no extra reactions. Later, a triangle of 6 voids triggers the transition, but spontaneous reactions elsewhere along the shock are also doing so simultaneously, as shown in [[Fig. ]{}]{}[rvrace]{}. None of the other random arrangements detonated. 2 produced yields of $68.0\%\pm3.7\%$ (where the second percentage is relative) whereas its [Case [S$_1$]{}]{} counterpart detonated. 1 produced $52.1\%\pm6.3\%$ as compared to 70.1% for its counterpart. These differences are between normalized quantities, yet are partially due to the fact that the [Case [S$_1$]{}]{} samples were 26.1% longer.
![[(Color online) ]{}Snapshot from a 3 simulation just as the upward-propagating shock is becoming a detonation. The whole width of the sample but only one ninth of its (original) length is shown. Blue atoms are unreacted, green are reacted, and red and purple are intermediate states. The red disk is a fresh hotspot; note the isolated reactions close to the shock everywhere.[]{data-label="rvrace"}](rvrace.pdf){width="\figwidth"}
Discussion
==========
It is to be expected that the detonation transition time [$t_D$]{} decreases with an increase in either the porosity $p$ or piston velocity $u_p$, but that it increases with radius (${\partial{t_D}/\partial{r}}>0$) deserves further consideration. First it should be noted that $({\partial{t_D}/\partial{r}})_\delta<0$: enlarging each of a set of voids without moving them ([, ]{}keeping the separation $\delta$ fixed) does enhance the reactivity. However, when holding $p$ fixed the reduction in number density overwhelms the effect of increasing the void size.
The simple ignition and growth model used earlier provides an explanation. Before any detonation begins, any point in the material is reacted if and only if it is closer to the location of a void collapse than the $R(t)$ associated with that void[ [@info:lanl-repo/inspec/9166224]]{}. Since all voids near a given point will collapse at nearly the same time, what matters is the expected minimum distance ${{\left\langle{d_\text{min}}\right\rangle}}$ to a void (after shock compression). We expect that ${{\left\langle{d_\text{min}}\right\rangle}}\propto\delta\propto r$, so the material will react sooner, on average, with small voids[ — ]{}until, of course, the voids become so small that they no longer reliably produce any reactions at all. (For the smallest $u_p$ considered here, that failure radius is approximately 2 nm[ [@paper1]]{}.)
A caveat is that if $r_0$ or $v$ is a strong function of $r$, the larger average distances from larger voids might be ovewhelmed by more vigorous growth of the hotspots created by larger voids. While we do not have a model for ${r_0(r,u_p)}$ and ${v(r,u_p)}$, they appear to be weak (perhaps sublinear) functions of $r$, in which case the conclusion of more reactivity from smaller voids holds. That $v$ is a function of $r$ at all is interesting; we suppose that the $r$-dependent strength of the reshock emitted when the void collapses and explodes in place may imprint on its surrounds a memory of the void’s size.
The model also explains how disorder in the arrangement of voids increases [$t_D$]{}. Whenever, in the random placement of voids, two or more are placed much closer than $\delta$ to one another, their hotspots will overlap very quickly and the total burn front area will then be reduced; equivalently, ${{\left\langle{d_\text{min}}\right\rangle}}$ is larger for a random arrangement than for a lattice (especially a hexagonal one) at the same $p$ and $r$.
The broad pressure wave created by the lead hotspots seems to be the principal mechanism for the detonation transition in this system. Its development, the identical growth of the first two hotspots, the similarity of the results from rectangular and triangular lattices, the consistency among the results from random void arrangements, and the apparent irrelevance of void clusters all suggest that the development of a detonation is a collective effect that depends on $p$, $r$, and the regularity of the void arrangement but not on the details of that arrangement. This collectivity affords a major simplification in predicting the behavior of collections of voids: a model might need only $r$, $p$, and $\sigma_\delta$.
Finally, we note that the function ${v(r,u_p)}$, since it will likely dominate ${r_0(r,u_p)}$ and appears to depend on both its arguments but not on time, may prove useful as a measure of the strength or activity of a hotspot that might be incorporated into an analytical reaction rate model. For voids of non-uniform size, it might be sufficient to consider the variation of $v$ in calculating a point’s expected burn time.
This report was prepared by Los Alamos National Security under contract no. DE-AC52-06NA25396 with the U.S. Department of Energy. Funding was provided by the Advanced Simulation and Computing (ASC) program, LANL MDI contract 75782-001-09, and the Fannie and John Hertz Foundation.
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abstract: 'A detailed quantum analysis of a ionic reaction with a crucial role in the ISM is carried out to generate ab initio reactive cross sections with a quantum method. From them we obtain the corresponding CH$^+$ depletion rates over a broad range of temperatures. The new rates are further linked to a complex chemical network that shows the evolution in time of the CH$^+$ abundance in photodissociation region (PDR) and molecular cloud (MC) environments. The evolutionary abundances of CH$^+$ are given by numerical solutions of a large set of coupled, first-order kinetics equations by employing the new chemical package <span style="font-variant:small-caps;">krome</span>. The differences found between all existing calculations from low-T experiments are explained via a simple numerical model that links the low-T cross section reductions to collinear approaches where nonadiabatic crossings dominate. The analysis of evolutionary abundance of CH$^+$ reveals that the important region for the depletion reaction of this study is that above 100 K, hence showing that, at least for this reaction, the differences with the existing low-temperature experiments are of essentially no importance within the astrochemical environments. A detailed analysis of the chemical network involving CH$^+$ also shows that a slight decrease in the initial oxygen abundance might lead to higher CH$^+$ abundance since the main chemical carbon ion depletion channel is reduced in efficiency. This simplified observation might provide an alternative starting point to understand the problem of astrochemical models in matching the observed CH$^+$ abundances.'
author:
- |
S. Bovino$^1$, T. Grassi$^2$, M. Tacconi$^3$, and F. A. Gianturco$^{2,4,5}$[^1]\
$^1$Institut für Astrophysik Georg-August-Universität, Friedrich-Hund Platz 1, 37077 Göttingen, Germany\
$^2$Department of Chemistry, Sapienza University of Rome, P.le A. Moro, 00185, Rome, Italy\
$^3$CINECA, via Magnanelli 6/3, 40033 Casalecchio di Reno, Bologna, Italy\
$^4$Institute of Ion Physics, University of Innsbruck, Technikerstrasse 25, 6020, Innsbruck, Austria\
$^5$Scuola Normale Superiore, Piazza de’ Cavalieri, 56125, Pisa, Italy
date: 'Accepted \*\*\*\*\*. Received \*\*\*\*\*; in original form \*\*\*\*\*\*'
title: 'CH$^+$ depletion by atomic hydrogen: accuracy of new rates in photo-dominated and self-shielded environments'
---
\[firstpage\]
Astrochemistry –Molecular processes – ISM: molecules –Methods: numerical
Introduction
============
The methylidine cation CH$^+$ was observed a while ago for the first time [@Douglas1941] in the diffuse interstellar medium (ISM) and was followed by further detections in a variety of interstellar and circumstellar environments. In its earlier detections, its A$^1\Pi\leftarrow X^1\Sigma^+$ electronic band system was observed [@Crane1995; @Weselak2008], while the more recent data from the *Infrared Space Observatory*[^2] [@Kessler1996] and from the *Herschel Space Telescope*[^3] [@Pillbratt2010] have given access to the far infrared (FIR) spectrum of this molecule in several remote star forming regions [@Falgarone2010]. The fairly large number of subsequent detections shows that the presence of CH$^+$ can be considered as confirmed throughout the interstellar matter [@Godard2013].
Despite the large effort made by observers to detect with great accuracy the CH$^+$ bands, this molecule still remains a puzzle in the modern molecular astrophysics. The observed averaged abundances, in fact, are still orders of magnitudes larger compared to the one provided by astrochemical models, even if some attempts to solve this puzzle came out during the last years (see discussion below).
The methylidine has also been known to play a significant role in various steps of the complex chemical network of those molecular processes and reactions which are taken to occur in interstellar and circumstellar regions. For example, its hydrogenation reaction is involved in two important species in that network: the methyledene ion CH$_2^+$ and the methyl cation CH$_3^+$. Although the latter molecules are found to be rapidly destroyed by efficient dissociative recombination with electrons via $$\rm CH_3^+ + e^- \rightarrow CH + H_2$$ $$\rm CH_2^+ + e^- \rightarrow C + H_2\,,$$ they can also react with oxygen and nitrogen atoms forming in addition CO$^+$, CN$^+$, HCO$^+$, and HCN$^+$ plus other cation precursors of CO, HCN, etc. [@Godard2013], thus expected to induce a departure of carbon from its ionisation equilibrium [@Godard2009]. CH$^+$ therefore initiates an extensive chemical chain of processes which evolve into the formation of more complex species.
Since CH$^+$ is chemically a very reactive ion, its reaction path to destruction by hydrogen abstraction has been considered, together with dissociative recombination and the above hydrogenation processes, one of the important ways to its destruction [@McEwan1999; @Larsson2008; @Mitchell1990]: it is thus both important and interesting to assess as accurately as possible the actual efficiency of that destruction path
$$\label{eq:reaction}
\rm CH^+ + H \rightarrow C^+ + H_2\,,$$
so that one may link the above reaction with the additional variety of chemical processes which are important to model the chemistry of CH$^+$ in various ISM environments [@Godard2013].
In the last few years there have been a series of papers which have dealt first with its photon-induced formation: $$\rm{ C^+(^2P_{3/2,1/2}) + H(^2S_{1/2}) \rightarrow CH^+(^1\Sigma^+) + }h\nu$$ thereby producing the relevant radiative association rates [@Barinovs2006], and also with the experimental, low-temperature study of its destruction reaction by Eqn.(\[eq:reaction\]) with slow H atoms [@Plasil2011].
More recently, the formation reaction of CH$^+$ has been analysed with new computational data which start from a vibrationally excited ($\nu$ = 1) H$_2$ partner of C$^+$ and investigate the high-T regimes using a quantum wave-packet method [@Zanchet2013].
The latter formation reaction is thought to be the main path of CH$^+$ formation even if a series of other physical processes have to be invoked to excite the H$_2$ molecules into higher vibrational levels in order to overcome the endothermicity of the reaction (4177 K). For instance, neutral shocks [@Elitzur1978 e.g.], and magnetohydrodynamic (MHD) shocks [@Draine1986 e.g] have been suggested earlier but seem to be both ruled out by observations [@Gredel1993; @Crawford1995]. The presence of Alfvén waves [@Federman1996], and the occurrence of turbulent dissipation [@Godard2009 e.g] still remain viable mechanisms in diffuse clouds. An attempt to provide a model which is able to reproduce the observed abundances has been proposed by @Falgarone2010 exploring turbulent dissipation regions (TDR) in which dissipation of turbulent energy locally triggers a specific warm chemistry. This was found to well reproduce the results for the inner Galaxy conditions.
The analysis of the present destruction reaction has also been carried out by several authors, who employed a variety of potential energy surfaces (PES) and tried to provide realistic estimates of the reaction rates for the destruction path of Eqn.(\[eq:reaction\]) using an adiabatic fit of the various surfaces involved. Some have used the Negative Imaginary Potential (NIP) code within an infinite order sudden (IOS) approximation for the reaction dynamics [@Stoecklin2005]. More recent calculations which employ a newly computed reactive potential energy surface (RPES) were carried out by @Warmbier who used quasi-classical trajectories (QCT) and close-coupling (CC) quantum methods.
We therefore think that it would be useful to employ the adiabatic PES approach once more to the depletion reaction (3), but using an accurate quantum method for angular momentum coupling, linking our findings to the recent study of @Grozdanov2013 and also embed our final rates, after comparisons with the experiments, into a broader chemical network modelling the CH$^+$ evolution in dark clouds. To these ends, we decided to employ the recent RPES of @Warmbier and to use our recently developed NIP approach to quantum reactive dynamics [@Tacconi2011] to study the depletion reaction of Eqn.(\[eq:reaction\]), where the quantum dynamics was treated within the coupled-states (CS) approach. The use of the NIP method, originally suggested by @Baer1990, had been proved already to be quite realistic when dealing with the many channels which are usually dynamically coupled in ionic reactions [@Bovino2011b]. We shall specifically show below that the present results turn out to provide the best overall agreement with the existing experiments over a broad range of temperatures.
The following Section briefly describes the main features of the employed RPES, together with an outline of our computational method for yielding reaction cross sections and rates. Section \[sect:rate\_coefficients\] presents our computed quantities, their low-T behaviour vis-á-vis changes in the RPES features, and further carries out an extensive comparison with a very recent computational modelling of non-adiabatic effects [@Grozdanov2013] together with a detailed analysis in relation to existing experiments. Section \[sect:modelling\_ISM\] describes the effects of our computed rates on realistic evolutionary models which include the CH$^+$ network of reactions, while the present conclusions are given in Section \[sect:conclusions\].
The quantum dynamics
====================
The CH$_2^+$ reactive system is characterised by the presence of a conical intersection between the $^2\Sigma^+$ and the $^2\Pi$ PES [@Stoecklin2005; @Halvick2006; @Warmbier]. The latter becomes avoided crossing for near-linear, bend configurations and suddenly disappears when the molecule is clearly bent. This behaviour has been discussed in great detail in the above studies and will not be repeated here. By performing an adiabatization of the relevant RPES one can obtain a smoother description of the lower, approximately adiabatic RPES which we have employed in the present work, as also done in all the existing previous studies. The resulting angular shape of this single RPES could be seen from the data in Fig.\[figure1\], where different cuts are shown for the CH$^+$ molecule at its equilibrium geometry. We note there that at angles of 0$^{\circ}$ or 180$^{\circ}$ (linear configurations) a small barrier occurs around 5 a.u., obviously depicting the presence of the avoided crossing. Therefore, the fit of the RPES should be carefully checked in those regions. In our calculations we employ the RPES calculated by @Warmbier constructed following a modified ansatz of @Braams [@Braams2008] and @murrell1984. This RPES is better in terms of accuracy than the previous work of @Stoecklin2005 for two main reasons: (i) the number of points used to construct it is 16259 to be compared with 3291 used by the earlier authors, and (ii) the global root-mean-square found here is 15.5 meV to be compared with the 59.4 meV value of @Stoecklin2005. It should be added that, in both cases similar results for the two fragments (CH$^+$ and H$_2$) diatomic curve are provided, also in good agreement with experiments. Furthermore, the RPES of @Warmbier provides a fitting which works better in the complex’s region (CH$_2^+$), the latter being very important in terms of reactive scattering studies involving ionic species since they indeed proceed via the complex formation regions. An additional set of polynomials has been included for each intersection: as stated by the authors this approach is very sensitive to the chosen parametrisation but offers a more accurate description of the conical intersections when compared to the global accuracy of the whole potential fit [@Warmbier].
To describe the dynamics of the reaction in Eqn.(\[eq:reaction\]), due to the features outlined above, is computationally very demanding since it should correctly be solved by taking into account the presence of at least two different PESs. In practice, all published calculations thus far have resorted to using a single adiabatic RPES, due to the generally low-temperature regimes which are indicated for the ISM environments, and therefore considered the lower portion only of the conical intersection region and of both reagents’ and products’ states.
A very recent publication [@Grozdanov2013] has analyzed within a modified statistical model the possible effects of the conical intersections at the collinear orientations and we shall discuss their findings below, showing them to be in line with our own model analysis of such effects, also detailed below. Furthermore, the earlier computational/theoretical data related to this reaction have been obtained by employing methods which use an asymptotic basis or some approximation in the dynamics. For instance the negative imaginary potential (NIP) calculations reported by @Stoecklin2005 include the IOS approximation which is generally known to underestimate the reactive cross sections [@Huarte]. The subsequent work by @Halvick2006 employed a quasi-classical trajectory (QCT) and phase-space theory (PST) methods which are expected to be inaccurate at very low temperatures, where the quantum mechanical effects are dominant. The results from @Warmbier try to use a more accurate RPES and more accurate dynamics. They have been obtained via the close-coupling (CC) ABC code [@Skouteris] and explored the first five rotational levels for the CH$^+$ molecule.
Our approach is based on the NIP method introduced by @Baer1990 and extended in our previous works (@Bovino2011a [@Bovino2011b; @Tacconi2011], see these papers for the mathematical details) which made use of an additional potential term, $V_{NIP}$, aimed at absorbing the reactive flux. Because of the flux-absorbing effects coming from it, the resulting S-matrix is non-unitary and its default to unitarity yields the (state-to-all) reaction probability, as discussed in @Tacconi2011. From the reaction probability one can in turn obtain the reactive cross section $$\label{eq:sigma}
\sigma(E) = \frac{\pi}{(2j + 1)k^2}\sum_J\sum_{\Omega}(2J + 1) P^{J\Omega}\,,$$ evaluated from a given initial state ($\nu$, $j$) and summed over all the final roto-vibrational states of the products. In Eqn.(\[eq:sigma\]) $E$ is the collision energy, $k^2$ is the wave vector, $J$ the total angular momentum, and $\Omega$ the projection of the rotational angular momentum along the Body Fixed (BF) axis. Once the reactive cross sections are obtained, the rate coefficients are computed by averaging the appropriate cross sections over a Boltzmann distribution of relative velocities: $$\alpha(T) = \frac{1}{(k_BT)^2}\sqrt{\frac{8k_BT}{\pi\mu}}
$$ where $T$ is the gas temperature, $k_B$ is the Boltzmann constant and $\mu$ the reduced mass of the system.
It is important to note here that our NIP implementation is different from the one discussed by @Stoecklin2005, where an asymptotic basis for the roto-vibrational CH$^+$ states is used throughout the entire reactive domain. In addition, their calculations make the assumption of the IOS approximation [@Kouri] that treats the rotational motion as “frozen” during the collisions so that the final cross sections are evaluated at fixed Jacobi angles. In our approach an adiabatic basis set is employed instead, thus taking into account the physical effects by which interaction between the incoming atom and the molecule can adiabatically modify the reagents’ rotovibrational states. The reactive scattering calculations are carried out in the body-fixed frame making use of the coupled-states coupling scheme [@Mcguire; @Mcguire2] approximation which has been found to provide accurate results when compared with experiments: see the earlier works by @Bovino2011a [@Bovino2011b].
![Cuts of the PES at different angles for the CH$^+$ molecule at its equilibrium distance. []{data-label="figure1"}](figs/eps/figure1.eps){width=".45\textwidth"}
Rate coefficients and low temperature behaviour {#sect:rate_coefficients}
===============================================
The measured and computed rates
-------------------------------
We carried out calculations for the reaction in Eqn.(\[eq:reaction\]) starting from the initial roto-vibrational level $\nu=0,j=0$ and for energies ranging from 10$^{-5}$ to 1.0 eV. Our results are thus providing final rates between 10-1000 K, which therefore extend the range of temperatures sampled by the earlier, accurate quantum calculations of @Warmbier. All the parameters employed ensured a relative error of the calculated reactive cross sections within 1%. A basis set expansion of 800 functions has been used for the CH$^+$ reagent, leading to an equal number of coupled equations. The molecular basis functions were expanded over a direct product of a Colbert-Miller discrete variable representation (DVR) of 150 points (ranging from 0.35 $a_0$ to 15.0 $a_0$) and a set of 48 spherical harmonics. The convergence over the total angular momentum values ($J$) has also been checked: we used a number of $J$ ranging from 10 to 42 for the highest energy. Following the Baer criteria [@Baer1990] we have also obtained the following stable NIP parameters that have been employed in the calculations: $r_{\rm min} = 6.75\,a_0$, $r_{\rm max} =
10.25\,a_0$, and the NIP order $n = 2$ (see ref. @Tacconi2011 for further details).
The final rates are shown in Fig.\[figure3\] where they are compared with previous calculations and some experimental data. As can be seen from that figure, our results are both in good agreement with the accurate ABC-CC calculations which used the same RPES [@Warmbier] and with the experiment for temperature above 60 K [@Plasil2011]. It is worth noting here that for temperatures below 60 K the experiments by @Plasil2011 are still seen to decrease with temperature faster than any of the available calculations. A possible explanation for such a behavior will be further discussed, while it is worth noting here that an independent set of additional measurements would be a welcomed addition to the available data. Other computed values obtained with the ABC and QCT methods [@Warmbier] employing the same RPES are, like us, in good agreement with the experiments, while the results from @Stoecklin2005 and @Halvick2006 appear to deviate from both set of data. This behaviour may be attributed to both their specific, model quantum approach (involving a series of approximations) as the poor accuracy of the PES involved.
One of the surprising results reported by @Plasil2011 is the sudden drop in the rates at temperature below 60 K seen by their experiments. Since the dynamics of the system has been explored with different computational methods, we thought that it would be interesting to further explore via a computational experiment another possible physical cause for such a feature, since the numerical diabatization artificially introduces a barrier as seen in our Fig.\[figure1\]. More specifically, the presence of the barrier is due to conical intersections occurring for the collinear alignments of the reactants: H-H-C and H-C-H. A very interesting theoretical paper recently published on this subject [@Grozdanov2013] analyses the reactive behavior of such collinear configurations within a modified statistical treatment and finds that, by artificially suppressing the reactivity of the rotational states of CH$^+$ which dominate the reaction at the alignment channels, the final rates drop with temperature more rapidly, although not quite as the experiments. Their conclusions indicate that, if the presence of a barrier after diabatization were to be used to represent reactive flux losses into the other coupled surfaces, then a possible explanation of the existing experiments could be linked to the physical presence of such intersections. To explore this option we have carried out further model calculations in which we arbitrarily modify that barrier by using a disposable artificial factor ($\alpha$) that can increase the barrier height along the crucially important linear configurations. This numerical experiment would reduce tunneling of the reagents into the inner region of the reactive lower surface, thereby mimicking the flow of dynamical flux into the upper surfaces excluded by diabatization. We have therefore carried out different calculations with different factor values to see how much changes in the barrier height at the collinear crossings can affect the final cross sections at low energies. In Fig.\[figure2\] we report the results of our model for different values of $\alpha$ ranging from 0.7 to 3.0. We show in this figure that an increase of the barrier leads to lower values of the cross sections at low temperatures, while for energies larger than 1 meV the differences are not so marked. It should be noted that in this numerical study we are testing only the contribution from the most important partial wave ($J=0$) and that when going to larger energies the higher $J$ contributions should be added. However, since the behaviour at very low energies is largely controlled by the s-wave, this test is a reasonable one and it allows to see the clear presence of a correlation between barrier and reactive probabilities. This fairly simple numerical experiment is therefore telling us that diabatization is particularly important at the lower temperatures since the low-T alignment of the reactants would lead to additional flux losses into the other RPES’s. Since the low temperature regime is exactly the region were the experiments are showing a marked drop in the size of the cross sections and rates, it follows that our numerical tests in that region have the dynamics efficiently modelled by observing instead the flux losses into different channels induced during the collinear encounters of reactants. The results of Fig.\[figure2\] indicate that such effects are shown by our present study to be much less important as the temperature increases. These findings are also relevant for the analysis, discussed in the next Section, of the CH$^+$ evolution within a large network of linked chemical processes in order to establish their bearing on the evolutionary abundances of the CH$^+$ molecule. We shall, in fact, show below that other chemical processes take over at the lower temperatures, thereby making less relevant the role of the present reaction. The latter, however, becomes very important again at the higher temperatures where the agreement between our calculations and the experiments, as shown in figure \[figure2\], is indeed quite good.
![Reaction rates for the destruction channel of Eqn.(\[eq:reaction\]) as a function of the Temperature for different calculations and experiments. Colours online.[]{data-label="figure3"}](figs/eps/figure3.eps){width=".45\textwidth"}
![Reactive cross sections as the function of the collision energy for different values of the $\alpha$ parameter. See text for the details. Colours online.[]{data-label="figure2"}](figs/eps/figure2.eps){width=".45\textwidth"}
The CH$^+$ chemistry in different ISM environments {#sect:modelling_ISM}
==================================================
We have therefore analyzed the effects of the present rates on an ISM model which selects a specifically simplified chemistry based on H, He, C, and O elements, since the above components are the most abundant under the evolutionary conditions that we shall discuss below. Due to the ubiquitous presence of the CH$^+$ molecules we decided to model the evolution of the gas by exploring different environments: from a PDR to a self-shielded molecular cloud (MC). In order to do that, we essentially vary the extinction coefficient $A_v$ and make it range from 0.2 to 10; we further select the temperature according to what is suggested by Sect.9.6 of @Tielens2005. As indicated above, we employ a sub-network of the chemistry given in the UMIST database[^4] [@UMIST2013] which includes 356 reactions and 34 species, the latter chemical species being listed in Tab.\[tab:species\]. We then follow a model of a chemical network made up by simple radicals, and for this reason we do not include any of the more complex molecules such as the larger carbon chains. It is worth noting that some of the rate coefficients in the UMIST database have a limited temperature range of applicability which is mainly focussed into the cold regions ($\lesssim$300 K). We therefore extended such rate coefficients in order to cover a wider temperature range, following (whenever possible) the indications from the works where the rates had been evaluated.
We performed a series of standard one-zone models with different temperatures and different visual extinction coefficient values. In each model we take the temperature and density to be constant quantities during the evolution, while the chemical species are computed within a non-equilibrium scheme by using the chemical package <span style="font-variant:small-caps;">KROME</span>[^5] [@GrassiKROME] that employs a <span style="font-variant:small-caps;">DLSODES</span> solver [@Hindmarsh83; @Hindmarsh2005]. The initial conditions for the selected chemical species are given in Tab.\[tab:specie\_init\] as fractional abundances with respect to the total hydrogen number density $n_\mathrm{Htot}=2\times10^3$ cm$^{-3}$, while for the free electrons we adopt $n_\mathrm{e^-}=n_\mathrm{C^+}$ as initial value [@Cardelli1993; @Meyer1998; @Wakelam2008]. Finally, the cosmic rays ionisation rate is set to $1.3\times10^{-17}$ s$^{-1}$ in all the models, and we do not include any cooling or heating term during the 10$^8$ years evolution that we have sampled, in order to specifically assess the expected role of the chemical reactions included in our present network.
The results of our calculations are shown in Fig.\[fig:Av\_CHp\] as the fractional abundance of the CH$^+$ molecule that varies with the visual extinction $A_v$, the later going from 0.2 to 10. This range of values influences the photoionisation and photodissociation rate coefficients that in the UMIST database have the form $k=\alpha\exp(-\gamma\,A_v)$ in units of s$^{-1}$, where $\alpha$ and $\gamma$ are two parameters that depend on the reaction considered.
The amount of CH$^+$ shown in Fig.\[fig:Av\_CHp\] is markedly higher in the region with lower $A_v$, where the dominant formation path is controlled by the endothermic (4177 K) reaction [@Zanchet2013] $$\label{eqn:main_formation}
\rm H_2 + \rm C^+ \rightarrow \rm{CH}^+ + \rm H\,,$$ that is considered to be more efficient at higher temperatures (lower $A_v$). On the other hand, within the MC-like regions (higher $A_v$) where the temperature becomes lower the rate coefficient for Eqn.(\[eqn:main\_formation\]) is no longer efficient. Therefore, in the region around $A_v\approx2.2$ the profile presents a dip: from an anlysis of our calculations we found that this is due to the sudden increase of the H$_2$O abundance which depletes the C$^+$ abundance and hence reduces the efficiency of the formation by Eqn.(\[eqn:main\_formation\]). In the simplified network employed here this competing process is, in fact, triggered by the following endothermic reaction with oxygen $$\label{eqn:react_O}
\rm H_2 + \rm O \rightarrow \rm{OH} + \rm H\,,$$ that produces the OH molecule which in turn feeds the reaction $$\label{eqn:react_OH}
\rm H_2 + \rm OH \rightarrow \rm{H_2O} + \rm H\,,$$ and finally the H$_2$O produced is additionally further involved in the reactions $$\begin{aligned}
\label{eqn:react_H2O}
\rm H_2O + \rm C^+ &\rightarrow \rm{HCO^+} + \rm H\nonumber\\
&\rightarrow \rm{HOC^+} + \rm H\,,\end{aligned}$$ which consume the C$^+$ ion, the latter being the main source of CH$^+$ formation as indicated by Eqn.(\[eqn:main\_formation\]). To numerically test the above chain of reactions we found, in fact, that when we reduce the initial abundance of the atomic oxygen by a factor of two in our network, the dip is removed and the final amount of CH$^+$ increases by more than four orders of magnitudes, especially in the optically thick region ($A_v\gtrsim3$) corresponding to lower temperatures. This numerical experiments therefore confirms the suggested role of the above reactions within the chemistry of CH$^+$ and of CH$_2^+$.
We provide here two functional forms for our rate coefficent: one considered to be less accurate that follows the classical Kooij form $$\label{eq:ko}
k(T) = \alpha\left(\frac{T}{300 \mathrm{K}}\right)^\beta\exp(-\gamma/T)\qquad \mathrm{cm^{3}\,s^{-1}}\,,$$ with $\alpha = 8.72$ 10$^{-10}$, $\beta = -0.075$, $\gamma = 10.1866$, and $T$ in K, and another, more accurate fit (employed in these calculations) given by: $$\label{eq:koj}
k(T) = a\sqrt{T} + bT + cT^{3/2} + dT^2\qquad \mathrm{cm^{3}\,s^{-1}}\,,$$ where $a = 1.9336$ 10$^{-10}$, $b = -1.4423$ 10$^{-11}$, $c = 4.3965$ 10$^{13}$, $d = -4.8821$ 10$^{-15}$, and $T$ in K. Both fits are valid in the range of temperatures from 10 to 1000 K.
We report in Fig.\[fig:Av\_CHp\] the results from our evolutionary models which are obtained when using the various rate coefficients discussed in Sect.\[sect:rate\_coefficients\] and shown in Fig.\[figure3\], but also by additionally adding the Langevin value $k_{LV}=1.89\times10^{-9}$ cm$^3$ s$^{-1}$ which we obtained by following @Levine1987. Marked differences are found at lower $A_v$ when using either the NIP-IOS calculations or the Langevin rate coefficients. As expected, the latter rate is an upper limit for the reaction coefficient and therefore produces a smaller amount of CH$^+$, while the rate coefficient from @Stoecklin2005, which employs a dynamical approximation for the angular momentum coupling which has been found to underestimate the values of the cross sections, is also seen to generate larger abundances of CH$^+$. This behaviour is even more visible in the inset of Fig.\[fig:Av\_CHp\]. The evolutionary calculations which include our computed rates (labelled *this work* in the figure) are found to be in good agreement with the UMIST, which is based on the experimental values of @Plasil2011 above 60 K (see Sect.\[sect:rate\_coefficients\]). We are also, as expected, very close to the accurate quantum calculations of @Warmbier, labelled *ABC* in the figure.
It is here useful to note that the differences at the lower $A_v$ do not originate here from the discrepancies of the rates at low temperature discussed in Sect.\[sect:rate\_coefficients\], but rather depend on the high temperature conditions at these extinction coefficient values (see Fig.\[figure3\]) where even the PST and the QCT produce reasonable results, although they are both failing at lower temperatures. To better understand this behaviour, we show the evolution of the velocities of the main reactions for the destruction of the CH$^+$ molecule at two different $A_v$ conditions: one representing a PDR-like environment ($A_v=0.5$, $T_\mathrm{gas}\approx800$ K), and another describing a MC-like ($A_v=8$, $T_\mathrm{gas}\approx30$ K) environment. The velocity $\epsilon_i$ of the [$i$th ]{}reaction is defined as $$\epsilon_i=\frac{k_i\,n_{r1i}\,n_{r2i}}{\max\left(k\,n_{r1}\,n_{r2}\right)}\,,$$ where $k_i$ is the rate coefficient, $n_{r1i}$ and $n_{r2i}$ are the abundances of the reactants and the denominator represents the velocity of the fastest destruction rate at a given time. These quantities are reported in Figs.\[fig:destAv05\] and \[fig:destAv80\], for $A_v=0.5$ and $A_v=8$, respectively. In the optically thin regime, where the photodissociation of H$_2$ is more efficient, we note that the reaction $$\label{eqn:R1}
\rm H_2 + \rm{CH}^+ \rightarrow \rm{CH_2^+} + \rm H\,,$$ dominates the CH$^+$ destruction, while, when the hydrogen becomes atomic, the channel of Eqn.(\[eq:reaction\]) starts to be important. The optically thick environment is dominated by the presence of H$_2$ during the whole evolution, and therefore the reaction of Eqn.(\[eqn:R1\]) always plays a key role. These plots clearly show that in the lower temperature region, where the computed rates are markedly different from one another (see Fig.\[figure3\]), the present evolutionary results are seen to be only marginally influenced by such differences since this destruction channel becomes in fact negligible (see Fig.\[fig:destAv80\]). Conversely, at the expected higher temperatures of the PDR-like region, the smaller differences between the rates arise because this new environment is now becoming richer in atomic hydrogen at the later temporal stages (see Fig.\[fig:destAv05\]). In addition, from Figs.\[fig:destAv05\] and \[fig:destAv80\] we note that, given our selected initial conditions and the sub-network employed, the dissociative recombination is not likely to be important for the destruction of the CH$^+$ molecule when compared to the other existing destruction channels.
![Fractional abundances of the CH$^+$ molecule normalised to the total density of the gas as a function of the visual extinction $A_v$. We present different profiles for different choices of the desctruction rates discussed in the present work (see text for details). The inset is an enlargement of the region between $A_v=0.3$ and $A_v=1.3$. Colours online.[]{data-label="fig:Av_CHp"}](figs/eps/figure4.eps){width=".45\textwidth"}
Species
---------- ------------ ------------ ---------
e$^-$ H H$^-$ H$^+$
He He$^+$ C C$^-$
C$^+$ O O$^-$ O$^+$
H$_2$ H$_2^+$ C$_2$ CO
CO$^+$ O$_2$ O$_2^+$ OH
OH$^+$ CH CH$^+$ HCO
HCO$^+$ CH$_2$ CH$_2^+$ CH$_3$
CH$_3^+$ H$_2$O H$_2$O$^+$ HOC$^+$
H$_3^+$ H$_3$O$^+$
: List of the species included in our ISM model. See text for details.
\[tab:species\]
Species Fraction Species Fraction
--------- --------------------- --------- ---------------------
H$_2$ $0.50$ He $9.00\times10^{-2}$
O $2.56\times10^{-4}$ C$^+$ $1.20\times10^{-4}$
e$^-$ see text
: Initial fractions respect to $n_\mathrm{Htot}$.
\[tab:specie\_init\]
![Evolution of the velocities of the main destruction reactions normalized to the most important one for $A_v=0.5$, see colured figure online.[]{data-label="fig:destAv05"}](figs/eps/figure5.eps){width=".48\textwidth"}
![Same as Fig.\[fig:destAv05\], but for $A_v=8$. Colours online.[]{data-label="fig:destAv80"}](figs/eps/figure6.eps){width=".48\textwidth"}
Conclusions {#sect:conclusions}
===========
In the present work we have attempted to address various, separate aspects involving the chemistry of the CH$^+$ in the ISM, combining them together in order to gain a better understanding of its role as a chemical component in many diverse environments:
1. we have carried out a new theoretical evaluation of the chemical path to its destruction via the chemical reaction with H atom discussed in the introduction as Eqn.(\[eq:reaction\]);
2. we have employed the reaction rates given by our calculations within different models of its evolution associated with either a PDR-like, high-temperature environment or with an MC-like model at lower temperatures.
3. we have carried out numerical modelling tests on the features of the employed RPES in the region of the conical intersections in order to show that reactive flux losses into the other coupled RPES’s could be relevant at the very-low temperatures, and therefore could be used to partly explain the experimentally observed drops in rate values as T decreases.
4. the present evolutionary study of a large set of connected chemical reactions clearly shows that the above, low-T regime of the reaction (3) is marginally affecting the CH$^+$ presence in the astrophysical environments that we have modelled, since under those conditions other chemical reactions begin to play the role of destroying that species.
In particular, the new analysis of CH$^+$ depletion channel (Eqn. \[eq:reaction\]) stems from the quantum calculations of the reactive cross section using an accurate RPES already available from the recent literature [@Warmbier] and employing a quantum reactive method, the NIP approach used by us before [@Bovino2011a], with a CS treatment of the dynamical coupling during the evolution from reagents to products. A detailed comparison of our new findings with the available experiments and with all recent calculations of the same reaction (see details in Sect.\[sect:rate\_coefficients\] and Fig.\[figure3\]) turns out to indicate that our calculated rates are in very good agreement with experiments and also with the best available quantum calculations for the same depletion reaction over the range of astrophysically relevant temperatures. Furthermore, given the differences shown by all existing calculations (old and new) with the experiments for the low-T data [@Plasil2011], and in order to check a possible physical cause of the rate values reduction at the low temperatures (see Fig.\[figure3\]), we have also carried out a numerical experiment whereby the barrier in the product region, caused by the avoided crossings at the different orientations of the regents (see Fig.\[figure1\]), is artificially modified in height and the ensuing reactive probabilities analysed. We thus found that any role of such a barrier in driving the destruction reaction is very relevant in the low-T region, while becoming negligible as the temperature is increased. We could therefore suggest from such computations that the low-T behaviour of the reaction rates for this system are indeed sensitive to specific details of the adiabatisation of a multiple-RPES reaction. It will translate, in the experiments, into the possible role of non-adiabatic effects which would be very delicate to measure reliably in that region. Additionally, our detailed modelling of two distinct astrophysical environments, within which the reaction (3) is linked to a broader range of chemical processes, has shown that the low-T regimes of the latter reaction play a marginal role for describing its evolution, which it is instead dominated by the high-T behaviour where our quantum calculations indeed match existing experiments.
The second aspect of the CH$^+$ chemistry that we have analysed in some detail, as described in the previous Sect.\[sect:modelling\_ISM\], has been the monitoring of its evolution with different visual extinction values ($A_v$) in order to model either the high temperature conditions of a PDR environment with intense photon flux (low $A_v$ values) or the low temperature conditions of a photon-thick environment for a MC. In both cases, at each $A_v$ value the temperature has been kept constant and the additional presence of other radical atoms and ions (together with all their interconnected reactions) has been included in our simulations by accessing the database UMIST to assemble a sub-network excluding the presence of less important paths. The evolution has been followed through the publicly available package <span style="font-variant:small-caps;">KROME</span> [@GrassiKROME] for $10^8$ years and the results were collected in Figs.\[fig:Av\_CHp\] through \[fig:destAv80\].
From those data we were able to understand that the role of the present destruction reaction is an important one in the PDR conditions while, as the temperature and the photon flux decrease, other destruction reactions take over and therefore even large changes in the destruction rates for the present reaction play an overall minor role.
In addition, we should consider that our calculated rate coefficient could play a key role at lower temperatures in the H-dominated enviroment like the cold neutral medium (CNM) regions, where the differences in the CH$^+$ evolution is supposed to be led by the destruction reaction studied in this work.
In conclusion, we have shown that using accurate quantum methods for the destruction reaction of CH$^+$ with H does yield very reliable reaction rates, which are in accord with existing experiments and which further appear to be important at high-T values, while becoming less crucial in modelling the CH$^+$ abundances in the photon-poor, low-T ISM regions. The present computational rates are also found to agree well with those included already within the UMIST database since the latter rates are indeed based on the experimental data above 60 K [@Plasil2011].
Acknowledgements {#acknowledgements .unnumbered}
================
S.B. thanks for funding through the DFG priority programme ‘The Physics of the Interstellar Medium’ (project SCHL 1964/1-1). One of us (T.G.) also thanks CINECA consortium of the awarding of financial support while the present research was carried out. We are very grateful to Robert Warmbier for having provided the PES and shared his results and to D.R.G Schleicher for having read this Manuscript and for his useful suggestions on the presentation of our results.
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\[lastpage\]
[^1]: Corresponding author: francesco.gianturco@uniroma1.it
[^2]: http://iso.esac.esa.int
[^3]: http://herschel.esac.esa.int/
[^4]: http://www.udfa.net
[^5]: Available for download from http://www.kromepackage.org
|
---
abstract: 'Nova shells can provide us with important information on their distance, their interactions with the circumstellar and interstellar media, and the evolution in morphology of the ejecta. We have obtained narrow-band images of a sample of five nova shells, namely DQHer, FHSer, TAur, V476Cyg, and V533Her, with ages in the range from 50 to 130 years. These images have been compared with suitable available archival images to derive their angular expansion rates. We find that all the nova shells in our sample are still in the free expansion phase, which can be expected, as the mass of the ejecta is 7-45 times larger than the mass of the swept-up circumstellar medium. The nova shells will keep expanding freely for time periods up to a few hundred years, reducing their time dispersal into the interstellar medium'
author:
- 'E. Santamaría'
- 'M.A. Guerrero'
- 'G. Ramos-Larios'
- 'J.A.Toalá'
- 'L. Sabin'
- 'G. Rubio'
- 'J.A. Quino-Mendoza'
title: 'Angular Expansion of Nova Shells[^1]'
---
Introduction {#sec:intro}
============
Novae are the result of the interaction of stars in close binary systems, where a white dwarf (WD) accretes H-rich material from a companion, typically a giant or sub-giant low-mass star [@2008B]. When the accreted material reaches a critical mass, a thermonuclear runaway (TNR) occurs. Temperatures can reach values of $\sim$1–4$\times$10$^{8}$ K in a few seconds [@2016PASP..128e1001S] and up to $\sim$2$\times$10$^{-4}$ M$_{\odot}$ [@1998PASP..110....3G] of highly processed material is ejected at velocities $\sim$1000 km s$^{-1}$ [@2010AN....331..160B] in a classical nova (CN) event. With time, the nova remnants will expand and mix into the interstellar medium (ISM).
--------- --------------- ----------- ---------- ---------------------- ------ ------------------ ---------------- ---
TAur 177.14$-$1.70 fast 1891 Dec 880$^{+50}_{-35}$ 25 25.4$\times$18.6 655 1
V476Cyg 87.37$+$12.42 very fast 1920 Aug 670$^{+110}_{-50}$ 145 14.6$\times$13.4 725 2
DQHer 73.15$+$26.44 slow 1934 Dec 501$^{+6}_{-6}$ 220 32.0$\times$24.2 370 3
V533Her 69.19$+$24.27 slow 1963 Feb 1200$^{+50}_{-40}$ 495 16.8$\times$15.2 850 4
FHSer 32.91$+$5.79 slow 1970 Feb 1060$^{+110}_{-70}$ 105 12.4$\times$10.6 490$\times$385 4
--------- --------------- ----------- ---------- ---------------------- ------ ------------------ ---------------- ---
The morphology and expansion of a nova remnant depend on the details of the nova event, but also on the interactions of the ejecta with the stellar companion and the pre-existing circumstellar material, which may consists of an accretion disk and a common envelope. A typical CN outburst includes an initial slow (500–2000 km s$^{-1}$) wind followed by a longer phase with a faster (1000–4000 km s$^{-1}$) wind [@BODE1989]. The interaction of these two winds forms a double shock structure, with the fast wind passing through the slow one until it dissipates and cools adiabatically as it expands [@OBRIEN1994].
Additionally, the interaction of the ejecta with the binary companion and material in a common envelope has effects in the asphericity of the nova shell [@LIVIO1990] and dynamics of the ejecta [@SHANKAR1991]. The effects of these interactions vary among novae of different speed class [@LLOYD1997], which are basically associated to the different time-scales of the slow and fast wind phases, providing an interpretation for the larger asphericities of the remnants of slow novae with respect to those of fast novae [@1995MNRAS.276..353S]. Finally, the WD rotation may also feed the ejecta with angular momentum, which can produce noticeable effects on the structure of nova shells [@PORTER1998].
The late expansion of nova shells can help us gain insights into the plasma physics and shock phenomena associated with the blast produced by the interaction of hydrogen-poor, metal-rich ejecta with the circumstellar environment and to investigate the ingestion of this ejecta by the ISM. The complete dynamical evolution of a nova occurs in time scales comparable to that of human life, and thus it provides a first class comparison to investigate the much slower evolution of planetary nebulae (PNe) or the processes involved in the evolution of the much rare supernova remnants (SNR). The time scale for a nova dispersal is an important parameter to assess the duration of the different stages of hibernation between a CN eruption and their parents cataclysmic variables [@SHARA2017].
Very little attention has been paid to the late expansion of nova shells, however. @DUER1987 conducted a heroic investigation of the angular expansion of nova shells using images of limited quality and concluded that they have mean half-time of 75 years, noting that this deceleration is most noticeable for novae with higher expansion velocities. Since then, very few detailed studies of the angular expansion of nova shells have been carried out, including those of GKPer, perhaps the most studied nova shell [@LIIM2012; @TAK2015; @HAR2016], DQHer [@HER1992; @VOR2007; @HARR2013], FHSer [@VAL1997; @ESEN1997], and recently IPHASXJ210204.7$+$471015 [@SAN2019]. The multiple knots in GKPer expand isotropically at an angular velocity of 03-05 yr$^{-1}$, which has been kept unchanged since their ejection a century ago [@Shara2012b; @LIIM2012]. This is somehow surprising, because detailed analyses of individual knots reveal the notable interaction with each other and with the ISM [@HAR2016]. On the other hand, a noticeable deceleration of a bow-shock component of the nova IPHASXJ210204.7$+$471015 has been recently reported [@SAN2019].
The lack of agreement between these results most likely implies that the expansion of a nova shell depends on the details of the nova outburst and the local properties of the ISM. The availability of high-quality archival images of nova shells allows a precise investigation of the expansion of a meaningful sample of sources. Using the sample of images presented by [@1995MNRAS.276..353S], we have selected five nova shells with multi-epoch high-quality images, namely DQHer, FHSer, TAur, V476Cyg, and V533Her, to carry out a pilot study of the investigation of the angular expansion of nova shells. Basic information on these novae, including their Galactic coordinates, type, outburst date, distance [as adapted from @2018MNRAS.481.3033S], height over the Galactic Plane, and expansion velocity derived from spectroscopic observations, is compiled in Table \[tab:nov\]. The latter is provided for the major and minor axes of FHSer.
{width="95.00000%"}
Imaging {#sec:obs}
=======
Contemporary Imaging {#sec:own_img}
--------------------
Present day (2016-2019) images of the nova shells in Table \[tab:nov\] were obtained using the Alhambra Faint Object Spectrograph and Camera (ALFOSC) at the 2.5m Nordic Optical Telescope (NOT) of the Roque de los Muchachos Observatory (ORM) in La Palma, Spain. The E2V 231-42 2k$\times$2k CCD was used with pixel size 15.0 $\mu$m, providing a plate scale of 0211 pix$^{-1}$ and a field of view (FoV) of 72 arcmin. The images used to investigate the angular expansion of these novae were obtained through H$\alpha$ filters with FWHM of 33 Å for the 2016 run of TAur and 13 Å for the others. Total exposure times and spatial resolutions, as determined from the FWHM of field stars, are listed in Table \[tab:obs\].
Images were also acquired in other filters as described in the caption of Figure \[fig:fig1\] to obtain colour-composite pictures of these novae. All images were processed using standard [iraf]{} routines.
Archival Imaging {#sec:arc_img}
----------------
Archival CCD images of the novae in Table \[tab:nov\] have been obtained using different telescopes and instruments as listed in Table \[tab:obs\]. The images were downloaded from the European Southern Observatory (ESO) Science Archive Facility the Isaac Newton Group (ING) data archive and the Mikulski Archive for Space Telescopes (MAST) and Hubble Legacy Archive (HLA) at the Space Telescope Science Institute The ESO images were obtained using SUper Seeing Instrument (SUSI) and ESO Faint Object Spectrograph and Camera 2 (EFOSC2) at the 3.5m New Technology Telescope (NTT) of the ESO’s La Silla Observatory. The ING images were acquired using the Auxiliary-port CAMera (ACAM) of the 4.2m William Herschel Telescope (WHT) and Jacobus Kapteyn Telescope (JKT).
The *HST* images were obtained using the Wide Field and Planetary Camera 2 [WFPC2 Instrument Handbook, @2009wfpc.rept....4B] under programs ID 6770 (PI O’Brien) and 6060 (PI Shara). The filters, exposure times, pixel scales, and spatial resolutions of these images are listed in Table \[tab:obs\].
--------- -------------- ----------------- -------------------- ------------- --------- ---------
TAur 1956 Dec PO & 103aE Red $\dots$ 1.7 $\dots$
1978 Mar KPNO & ISIT H$\alpha$ 1800 $\dots$ $\dots$
1989 Nov 22 POSS2 Red RG610 4800 1.0 $\dots$
1998 Nov 2 *HST* & WFPC2 F656N 5400 0.05 0.2
2016 Nov 28 NOT & ALFOSC NOT \#21 H$\alpha$ 1800 0.21 0.6
2018 Jan 03 NTT & EFOSC2 H$\alpha$ 1440 0.12 0.5
2019 Oct 11 NOT & ALFOSC OSN H01 H$\alpha$ 2700 0.21 0.9
V476Cyg 1944 Jan-Jun MW & 100-inch H$\alpha$ $\dots$ $\dots$ $\dots$
1993 Sep 12 WHT & Aux. Port H$\alpha$ 6569 900 0.25 1.2
2018 Jun 08 NOT & ALFOSC OSN H01 H$\alpha$ 2700 0.21 0.7
DQHer 1977 May 15 BokT & ITT 40mm H$\alpha$ $\dots$ $\dots$ $\dots$
1993 Jul 31 JKT & AGBX H$\alpha$ 7200 0.33 2.0
1995 Sep 04 *HST* & WFPC2 F656N 2000 0.05 0.1
1997 Oct 25 WHT & Aux. Port H$\alpha$ 656 1200 0.11 0.4
2012 Aug 15 WHT & ACAM T6565 40 0.25 0.7
2017 May 27 NOT & ALFOSC OSN H01 H$\alpha$ 2700 0.21 0.8
2018 Jun 05 NOT & ALFOSC OSN H01 H$\alpha$ 2700 0.21 0.8
V533Her 1993 Sep 11 WHT & Aux. Port H$\alpha$ 6569 1800 0.25 1.0
1997 Sep 03 *HST* & WFPC2 F656N 2600 0.05 0.2
2018 Jun 06 NOT & ALFOSC OSN H01 H$\alpha$ 4800 0.21 0.6
FHSer 1996 Mar 18 NTT & SUSI H$\alpha$ 720 0.13 0.9
1997 May 11 *HST* & WFPC2 F656N 4800 0.05 0.1
2017 May 29 NOT & ALFOSC OSN H01 H$\alpha$ 2700 0.21 0.6
2018 Jun 06 NOT & ALFOSC OSN H01 H$\alpha$ 3600 0.21 0.7
--------- -------------- ----------------- -------------------- ------------- --------- ---------
Ancient images were acquired using photographic plates. The 1956 image of TAur was taken by Walter Baade at the Palomar Observatory , whereas that of 1978 was obtained at Kitt Peak National Observatory (KPNO) using the 4m telescope [@1980ApJ...237...55G]. The oldest image of V476 Cyg was acquired in 1944 at Mount Wilson Observatory [for more details, see @1944MWOAR..16....1A; @1970IAUS...39..281B]. The 1977 image of DQHer was obtained with the Steward Observatory 2.3m telescope using an ITT 40 mm tube [@1978ApJ...224..171W].
![ Expansion of the semi-minor axis of nova shells. The angular size of the semi-minor axis measured in the images has been converted into linear size in pc using the *Gaia* DR2 distance to each nova as listed in Table \[tab:nov\], whereas the epoch of each measurement is referred to the time since the nova outburst. The error bars correspond to the dispersion of the individual values obtained for each epoch, which is smaller than the symbol size in a few cases. The expansion of all novae is consistent with free expansion. The slope of the linear fits has been converted to expansion velocity in the common units of km s$^{-1}$. \[fig:fig2\]](fig2.eps){width="48.00000%"}
Results {#sec:res}
=======
We present in the left and middle columns of Figure \[fig:fig1\] archival and present day images of the nova shells in our sample, respectively. These images reveal all nova shells in our sample to have elliptical morphologies with different degree of ellipticity. TAur and DQHer have similar knotty morphologies, with cometary knots showing remarkable tails mainly along the major axis. V476Cyg also seems to have a broken, clumpy morphology, whereas the shells of FHSer and V533Her have smoother appearance. The comparison of present day images (Fig. \[fig:fig1\] middle column) with representative archival images (Fig. \[fig:fig1\] left column) unveils clear expansion patterns. A careful examination of multi-epoch images also discloses small-scale morphological variations, including changes in the size and distribution of clumps. A detailed study is deferred to a subsequent work (Santamaría et al., in preparation).
To investigate and quantify the expansion of these nova shells, radial spatial profiles across individual discrete features have been extracted from the images at the different epochs listed in Table \[tab:obs\]. The distance of these features to the central star has been determined by measuring their position using Gaussian fits. The angular sizes along different directions have then been normalized to the minor axis using an elliptical fit to the shape of the nova shell, and an averaged value for the size of the minor axis and its 1$\sigma$ dispersion derived for each epoch. These are shown in Figure \[fig:fig2\], together with linear fits for all the nova shells in our sample, where the time is computed from the nova outburst date and the angular size of the semi-minor axis has been converted to linear size using the nova distance. The increase of the size of these nova shells with time can be described by linear fits (Figure \[fig:fig2\]) with correlation coefficients $\geq$0.98, implying $t$-test significance probabilities $\geq$98% for V533Her and V476Cyg, $>$99% for FHSer, and $>$99.9% for DQHer and TAur. The slope of these fits correspond to the angular expansion rates along the minor axis of these nova shells (column 2 of Table \[tab:nov\]).
We note that the angular expansion rates derived from these fits are consistent with those previously reported.
[@VOR2007] derived angular expansion rates of 0205$\pm$0014 yr$^{-1}$ and 0165$\pm$0012 yr$^{-1}$ along the major and minor axes of DQHer, respectively, whereas angular expansion rates of 0128 yr$^{-1}$ [@SD1987], 0136 yr$^{-1}$ [@DUER1992], and 0104–0146 yr$^{-1}$ [@VAL1997] have been reported for FHSer. Similarly, @Harvey2018 provides angular expansion rates $\approx$012 yr$^{-1}$ and $\approx$0088 yr$^{-1}$ along the major and minor axes of TAur, respectively, and $\approx$0075 yr$^{-1}$ for V476Cyg.
--------- -------------------------------------------- ---------------------------------- ---------------------- ---------------------- --------------------- ----------------------
TAur (0.097$\pm$0.004)$\times$(0.072$\pm$0.001) (410$\pm$40)$\times$(315$\pm$22) 1.3$\times$10$^{-5}$ 3.6$\times$10$^{-4}$ 2$\times$10$^{-13}$ 4.2$\times$10$^{44}$
V476Cyg (0.073$\pm$0.008)$\times$(0.067$\pm$0.007) (230$\pm$60)$\times$(200$\pm$50) 1.7$\times$10$^{-6}$ 2.2$\times$10$^{-5}$ 1$\times$10$^{-14}$ 1.1$\times$10$^{43}$
DQHer (0.188$\pm$0.008)$\times$(0.139$\pm$0.005) (460$\pm$25)$\times$(325$\pm$16) 5.2$\times$10$^{-6}$ 2.3$\times$10$^{-4}$ 7$\times$10$^{-13}$ 3.3$\times$10$^{44}$
V533Her (0.152$\pm$0.006)$\times$(0.139$\pm$0.007) (850$\pm$70)$\times$(770$\pm$70) 1.3$\times$10$^{-5}$ 9.0$\times$10$^{-5}$ 7$\times$10$^{-15}$ 5.9$\times$10$^{44}$
FHSer (0.125$\pm$0.002)$\times$(0.109$\pm$0.002) (630$\pm$80)$\times$(540$\pm$70) 3.5$\times$10$^{-6}$ 1.4$\times$10$^{-4}$ 9$\times$10$^{-14}$ 4.7$\times$10$^{44}$
--------- -------------------------------------------- ---------------------------------- ---------------------- ---------------------- --------------------- ----------------------
Discussion
==========
The main result from the investigation of the angular expansion of this sample of nova shells is their linear expansion with time (Fig. \[fig:fig2\]). This linear increase of size with time implies a free expansion, where the initial velocity of the ejecta remains the same since the nova event with no sign of deceleration. Thus, we should expect the expansion velocity of a nova shell derived from its angular expansion rate and distance ($\bar{v}_{\rm exp}$, column 3 of Table \[tab:mass\]), which is the averaged expansion velocity of the ejecta since the nova outburst projected on the plane of the sky, to be consistent with the expansion velocity derived from spectroscopic observations ($v_{\rm exp}^{\rm sp}$, column 8 of Table \[tab:nov\]), which is the expansion velocity along the line of sight at the time of the observation[^2]. These two expansion velocities are found to be in excellent agreement for DQHer and V533Her, and within the uncertainties for FHSer, whose $v_{\rm exp}^{\rm sp}$ have been derived from spatiokinematic models. Remarkable discrepancies are found, however, for TAur and V476Cyg, whose spectroscopic observations are of low quality [@1983ApJ...268..689C; @DUER1987].
An orientation of the major axis of these nova shells close to the line of sight cannot explain the much larger spectroscopic velocities of TAur and V476Cyg than the expansion velocities on the plane of the sky along the minor axis. We note that recent high-dispersion spectra of TAur imply expansion velocities similar to those found here [@Harvey2018].
The linear expansion with time of the nova shells in this sample strengthens the idea that the ejecta has kept expanding at its initial velocity since the nova event. This result confirms previous results presented for TAur and V476Cyg, but also for V1500Cyg and V4362Sgr [@Harvey2018]. Apparently, the circumstellar medium around these novae has not been able to slow down their expansion, which can be expected if the mass of the ISM material swept up by the nova shell is much smaller than the mass of the nova ejecta. This can be tested by computing their values. Assuming an ISM density[^3]
$n_{\rm ISM}$=1 cm$^{-3}$, the volume evacuated by the nova shell implies swept-up masses of 10$^{-6}$–10$^{-5}$ $M_\odot$ (column 4 in Table \[tab:mass\]). These can be compared with the nova masses 2$\times$10$^{-5}$–3$\times$10$^{-4}$ $M_\odot$ (column 5 in Table \[tab:mass\]). The latter have been derived following , using the unabsorbed H$\alpha$ fluxes listed in column 6 and assuming a filling factor $\epsilon$=0.1. The H$\alpha$ fluxes are computed from our H$\alpha$ narrow-band images, using intermediate-dispersion flux-calibrated spectra of DQHer to derive a count-to-flux conversion factor and corrected for absorption using the extinction values given by @1995MNRAS.276..353S for TAur and V476Cyg, @SG2013 for DQHer and V533Her, and @GIL200 for FHSer. The masses of the nova ejecta are indeed much greater than the masses of the swept up ISM, by factors from 7 to 45, which is consistent with their free expansion. At their present expansion rates, the free expansion can be expected to last from $\simeq$100 yr for V533Her up to $\simeq$400 yr for TAur until the time when the swept up ISM mass equals that of the nova ejecta. Since the nova ejecta is not slowed down, it reduces the time for dispersal of nova shells into the ISM.
The free expansion is supported by the large kinetic energy ($\frac{1}{2}\,M_{\rm shell}\,v_{\rm exp}^2$) of the nova shells, which have been computed adopting a weighted expansion velocity among the polar and equatorial velocities. The kinetic energies listed in column 7 of Table \[tab:mass\] are in the range of a few times 10$^{44}$ erg, but for V476Cyg, which is 1$\times$10$^{43}$ erg.
The free expansion of the nova shells in our sample is in sharp contrast with the conclusions drawn by @DUER1987, who proposed that the expansion velocity of a nova shell reduces to half every 75 years. This is particularly shocking for DQHer and V476Cyg (this work), and GKPer [@Shara2012b; @LIIM2012], which were proposed to have deceleration half-times of 67, 117, and 58 years, respectively. The free expansion of nova shells applies to different nebular morphologies, from the smooth elliptical morphology of FHSer, V476Cyg, and V533Her, the mildly broken elliptical structure of TAur and DQHer, and the knotty morphology of GKPer. Only the faint bow-shock structural component of IPHASXJ210204.7$+$471015 seems to have experienced a notable braking in its interaction with the ISM [@2018ApJ...857...80G; @SAN2019].
Summary and Conclusions
=======================
The comparison between multi-epoch suitable broadband and narrowband images of the nova shells DQHer, FHSer, TAur, V476Cyg, and V533Her has been used to derive their angular expansion rates. This is found to be unchanged since the nova event, i.e., the nova shells in this sample are still in a free expansion phase. This can be expected, as the mass of the ejecta is 7-45 times larger than the mass of the swept-up circumstellar medium.
The images of the nova shells in our sample cover a time lapse since the nova event from 20 to 130 yrs.
Given the large ratio between the mass of the ejecta and that of the swept-up circumstellar medium, the free expansion phase can be expected to last for a few hundred years, during most (if not all) their whole visible phase.
E.S. G.R. and J.A.Q.M. acknowledges support from CONACyT and Universidad de Guadalajara. M.A.G. and E.S. acknowledge financial support by grants AYA 2014-57280-P and PGC 2018-102184-B-I00, co-funded with FEDER funds. M.A.G. acknowledges support from the State Agency for Research of the Spanish MCIU through the “Center of Excellence Severo Ochoa” award for the Instituto de Astrofísica de Andalucía (SEV-2017-0709). E.S. acknowledges the hospitality of the IAA during a short-term visit. G.R.L. acknowledges support from CONACyT and PRODEP (Mexico). L.S. acknowledges support from UNAM DGAPA PAPIIT project IN101819. M.A.G. and J.A.T. acknowledge support from the UNAM DGAPA PAPIIT project IA 100318. We appreciate the valuable comments of the referee, Dr Nye Evans. Finally, we thank Alessandro Ederoclite for useful discussion and comments. The data presented here were obtained in part with ALFOSC, which is provided by the Instituto de Astrofísica de Andalucía (IAA) under a joint agreement with the University of Copenhagen and NOTSA. This research made use of [iraf]{}, distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. We acknowledge the use of The ESO Science Archive Facility developed in partnership with the Space Telescope European Coordinating Facility (ST-ECF). Also, the ING archive, maintained as part of the CASU Astronomical Data Centre at the Institute of Astronomy, Cambridge and finally, the STScI, operated by the Association of Universities for research in Astronomy, Inc., under NASA contract NAS5-26555.
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[^1]: Released on November, 8th, 2019
[^2]: Spatiokinematic models of nova shells found them to be described as prolate ellipsoid with axial ratios $\leq$1.3 [e.g., FHSer @GIL200]. Since the angular expansion along the minor axis probes the equatorial expansion of such a prolate ellipsoid, the spectroscopic velocity can be expected to be at most 1.3 times larger than the latter in the most favorable case of pole-on ellipsoids.
[^3]: The averaged density of the ISM along the path towards these novae is in the range from 0.1 to 2 cm$^{-3}$ [@HI4PI2016].
|
---
abstract: |
We investigate how efficiently a known underlying causality structure of a simulated multivariate process can be retrieved from the analysis of time-series . Causality is quantified from conditional transfer entropy and the network is constructed by retaining only the statistically validated contributions. We compare results from three methodologies: two commonly used regularization methods, Glasso and ridge, and a newly introduced technique, LoGo, based on the combination of information filtering network and graphical modelling. For these three methodologies we explore the regions of time series lengths and model-parameters where a significant fraction of true causality links is retrieved. We conclude that, when time-series are short, with their lengths shorter than the number of variables, sparse models are better suited to uncover true causality links with LoGo retrieving the true causality network more accurately than Glasso and ridge.\
[Keywords: LoGo, Sparse Modelling, Information Filtering Networks, Graphical Modeling, Machine Learning. ]{}
address: |
$^1$ Department of Computer Science, UCL, London, UK\
$^{2}$ UCL Centre for Blockchain Technologies, UCL, London, UK\
$^{3}$ Systemic Risk Centre, London School of Economics and Political Sciences, London, UK\
$^{4}$ Department of Mathematics King’s College London, London, UK
author:
- 'Tomaso Aste$^{1,2,3}$ and T. Di Matteo$^{1,2,3,4}$'
title: Sparse causality network retrieval from short time series
---
Introduction
============
Establishing causal relations between variables from observation of their behaviour in time is central to scientific investigation and it is at the core of data-science where these causal relations are the basis for the construction of useful models and tools capable of prediction. The capability to predict (future) outcomes from the analytics of (past) input data is crucial in modeling and it should be the main property to take into consideration in model selection, when the validity and meaningfulness of a model is assessed. From an high-level perspective, we can say that the whole scientific method is constructed around a circular procedure consisting in observation, modelling, prediction and testing. In such a procedure, the accuracy of prediction is used as a selection tool between models. In addition, the principle of parsimony favours the simplest model when two models have similar predictive power.
The scientific method is the rational process that, for the last 400 years, has mostly contributed to scientific discoveries, technological progresses and the advancement of human knowledge. Machine learning and data-science are nowadays pursuing the ambition to mechanize this discovery process by feeding machines with data and using different methodologies to build systems able to make models and predictions by themselves. However, the automatisation of this process requires to identify, without the help of human intuition, the relevant variables and the relations between these variables out of a large quantity of data. Predictive models are methodologies, systems or equations which identify and make use of such relations between sets of variables in a way that the knowledge about a set of variables provides information about the values of the other set of variables. This problem is intrinsically high-dimensional with many input and output data. Any model that aims to explain the underlying system will involve a number of elements which must be of the order of magnitude of the number of relevant relations between the system’s variables. In complex systems, such as financial markets or the brain, prediction is probabilistic in nature and modeling concerns inferring the probability of the values of a set of variables given the values of another set. This requires the estimation of the joint probability of all variables in the system and, in complex systems, the number of variables with potential macroscopic effects on the whole system is very large. This poses a great challenge for the model construction/selection and its parameter estimation because the number of relations between variables scales with -at least- the square of the number of variables but, observation window, the amount of information gathered from such variables scales -at most- linearly with the number of variables [@bruckstein2009sparse; @theodoridis2012sparsity].
For instance, a linear model for a system with $p$ variables requires the estimation from observation of $p(p+1)/2$ parameters (the distinct elements of the covariance matrix). In order to estimate $\mathcal O(p^2)$ parameters one needs a comparable number of observations requiring time series of length $q \sim p$ or larger to gather a sufficient information content from a number of observations which scales as $p \times q \sim \mathcal O(p^2)$. However, the number of parameters in the model can be reduced by considering only $\mathcal O(p)$ out of the $\mathcal O(p^2)$ relations between the variables reducing in this way the required time series length to $\mathcal O(p)$. Such models with reduced numbers of parameters are referred in the literature as sparse models. In this paper we consider two instances of linear sparse modelling: Glasso [@tibshirani1996] which penalizes non-zero parameters by introducing a $\ell_1$ norm penalization and LoGo [@LoGo16] which reduces the inference network to an $\mathcal O(p)$ number of links selected by using information filtering networks [@asteetal2005; @tumminelloetal2005; @TMFG]. The results from these two sparse models are compared with the $\ell_2$ norm penalization (non-sparse) ridge model [@tikhonov1963solution; @hoerl1970ridge].
This paper is an exploratory attempt to map the parameter-regions of time series length, number of variables, penalization parameters and kinds of models to define the boundaries where probabilistic models can be reasonably constructed from the analytics of observation data. In particular, we investigate empirically, the true link retrieval performances in the region of short time-series and large number of variables which is the most critical region – and the most interesting – in many practical cases.
Results are reported for artificially generated time series from an autoregressive model of $p=100$ variables and time series lengths $q$ between 10 and 20,000 data points. Robustness of the results has been verified over a wider range of $p$ from 20 to 200 variables. Our results demonstrate that sparse models are superior in retrieving the true causality structure for short time series. Interestingly, this is despite considerable inaccuracies in the inference network of these sparse models. We indeed observe that statistical validation of causality is crucial in identifying the true causal links, and this identification is highly enhanced in sparse models.
The paper is structured as follows. In section \[s.definitions\] we briefly review the basic concepts of mutual information and conditional transfer entropy and their estimation from data that will then be used in the rest of the paper. We also introduce the concepts of sparse inverse covariance, inference network and causality networks. Section \[s.causalNet\] concerns the retrieval of causality network from the computation and statistical validation of conditional transfer entropy. Results are reported in Section \[s.results\] where the retrieval of the true causality network from the analytics of time series from an autoregressive process of $p=100$ variables is discussed. Conclusions and perspectives are given in Section \[s.conclusions\].
Estimation of conditional transfer entropy from data {#s.conditionslSigma}
====================================================
\[s.definitions\]
In this paper causality is quantified by means of statistically-validated transfer entropy. Transfer entropy $T(\mathbf{ Z_i} \rightarrow \mathbf{ Z_j})$ quantifies the amount of uncertainty on a random variable, $\mathbf{ Z_j}$, explained by [the past]{} of another variable, $\mathbf{Z_i}$ conditioned to the knowledge about the past of $\mathbf{ Z_j}$ itself. Conditional transfer entropy, $T(\mathbf{ Z_i} \rightarrow \mathbf{ Z_j} | \mathbf W)$, includes an extra condition also to a set variables $\mathbf W$. These quantities are introduced in details in Appendix \[cTE\] (see also [@shannon2001mathematical; @schreiber2000measuring; @anderson1984multivariate]). Let us here just report the main expression for the conditional transfer entropy that we shall use in this paper: $$\begin{aligned}
\label{General_cTE}
T(\mathbf{ Z_i} \rightarrow \mathbf{ Z_j}|\mathbf{ W}) & {}=
H(\mathbf Z_{\mathbf j,t} | \{\mathbf Z_{\mathbf j,t}^{lag},\mathbf{ W}_{t}\})
-
H(\mathbf Z_{\mathbf j,t} | \{\mathbf Z_{\mathbf i,t}^{lag},\mathbf Z_{\mathbf j,t}^{lag},\mathbf{ W}_{t}\}) \;\;. \end{aligned}$$ Where $H(.|.)$ is the conditional entropy, $\mathbf Z_{\mathbf j,t}$ is a random variable at time $t$, whereas $\mathbf Z_{\mathbf i,t}^{lag} =\{ \mathbf Z_{\mathbf i,t-1},...,\mathbf Z_{\mathbf i,t-\tau}\}$ is the lagged set of random variable ‘$\mathbf i$’ considering previous times $t-1...t-\tau$ and $\mathbf{ W}_{t}$ are all other variables and their lags (see Appendix \[cTE\], Eq.\[General\_cTE1\]).
In this paper we use Shannon entropy and restrict to linear modeling with multivariate normal setting (see Appendix \[Shannon\]). In this context the conditional transfer entropy can be expressed in terms of the determinants of conditional covariances $\mbox{det}( \mathbf \Sigma(.|.))$ (see Eq.\[H2\] in Appendix \[Shannon\]): $$\begin{aligned}
\label{General_cTE_Sigma}
T(\mathbf{ Z_i} \rightarrow \mathbf{ Z_j}|\mathbf{ W}) & {}=
\frac12 \log \mbox{det}\!\left( \mathbf \Sigma(\mathbf Z_{\mathbf j,t} | \{\mathbf Z_{\mathbf j,t}^{lag},\mathbf{ W}_{t}\})\right)
-
\frac12 \log \mbox{det}\!\left( \mathbf \Sigma(\mathbf Z_{\mathbf j,t} | \{\mathbf Z_{\mathbf j,t}^{lag},\mathbf Z_{\mathbf i,t}^{lag},\mathbf{ W}_{t}\})\right) \;\;. \end{aligned}$$
Conditional covariances can be conveniently computed in terms of the inverse covariance of the whole set of variables $\mathbf Z_t = \{ \mathbf Z_{k,t},\mathbf Z_{k,t-1},...\mathbf Z_{k,t-\tau}\}_{k=1}^p \in \mathbb R^{p\times(\tau+1)}$ (see Appendix \[InverseCovJandSigma\]). Such inverse covariance matrix, $\mathbf J$, represents the structure of conditional dependencies among all couples of variables in the system and their lags. Each sub-part of $\mathbf J$ is associated with the conditional covariances of the variables in that part with respect to all others. In terms of $\mathbf J$, the expression for the conditional transfer entropy becomes: $$\begin{aligned}
\label{TE20}
T(\mathbf{ Z_i} \rightarrow \mathbf{ Z_j}|\mathbf{ W}) = - \frac12 \log\mbox{det}\! \left(\mathbf {J_{1,1}} - \mathbf {J_{1,2}} (\mathbf {J_{2,2}})^{-1} \mathbf {J_{2,1}})\right)+ \frac12 \log \mbox{det}\! \left (\mathbf{ J_{1,1} }\right ) \;\;.
\end{aligned}$$ where the indices ‘$\mathbf 1$’ and ‘$\mathbf 2$’ refer to sub-matrices of $\mathbf J$ respectively associated with the variables $ \mathbf Z_{\mathbf j,t}$ and $\mathbf Z_{\mathbf i,t}^{lag}$.
Conditioning eliminates the effect of the other variables retaining only the exclusive contribution from the two variables in consideration. This should provide estimations of transfer entropy that are less affected by spurious effects from other variables. On the other hand, conditioning in itself can introduce spurious effects, indeed two independent variables can become dependent due to conditioning [@anderson1984multivariate]. In this paper we explore two extreme conditioning cases: i) condition to all other variables and their lags; ii) unconditioned.
[ In principle, one would like to identify the maximal value of $T(\mathbf{ Z_i} \rightarrow \mathbf{ Z_j} | \mathbf W)$ over all lags and all possible conditionings $\mathbf W$. However, the use of multiple lags and conditionings increases the dimensionality of the problem making estimation of transfer entropy very hard especially when only a limited amount of measurements is available (i.e. short time-series). This is because the calculation of the conditional covariance requires the estimation of the inverse covariance of the whole set of variables and such an estimation is strongly affected by noise and uncertainty. Therefore, a standard approach is to reduce the number of variables and lags to keep dimensionality low and estimate conditional covariances with appropriate penalizers [@tikhonov1963solution; @hoerl1970ridge; @tibshirani1996; @friedmanetal2008]. An alternative approach is to invert the covariance matrix only locally on low dimensional sub-sets of variables selected by using information filtering networks [@asteetal2005; @tumminelloetal2005; @TMFG] and then reconstruct the global inversion by means of the LoGo approach [@LoGo16]. Let us here briefly account for these two approaches. ]{}
Penalized inversions
--------------------
The estimate of the inverse covariance is a challenging task to which a large body of literature has been dedicated [@friedman2008sparse]. From an intuitive perspective, one can say that the problem lies in the fact that uncertainty is associated with nearly zero eigenvalues of the covariance matrix. Variations in these small eigenvalues have relatively small effects on the entries of the covariance matrix itself but have major effects on the estimation of its inverse. Indeed small fluctuations of small values can yield to unbounded contributions to the inverse. A way to cure such near-singular matrices is by adding finite positive terms to the diagonal which move the eigenvalues away from zero: $\hat {\mathbf J} = \left((1-\gamma) \mathbf S +\gamma \mathbf I_N \right)^{-1}$, where $\mathbf S = \mbox{Cov}(\mathbf Z)$ is the covariance matrix of the set of variables $\mathbf Z \in \mathbb R^N$ estimated from data and $\mathbf I_{N}\in \mathbb R^{N\times N}$ is the identity matrix (where $N=p\times(\tau+1)$, see later). This is what is performed in the so-called ridge regression [@hoerl1970ridge], also known as shrinkage mean-square-error estimator [@gruber1998improving] or Tikhonov regularization [@tikhonov1963solution]. The effect of the additional positive diagonal elements is quivalent to compute the inverse covariance which maximizes the log-likelihood: $\log \mbox{det}( \hat {\mathbf J}) - \mbox{tr}(\mathbf S\hat{\mathbf J}) - \gamma ||\hat {\mathbf J} ||_2$, where the last term penalizes large off-diagonal coefficients in the inverse covariance with a $\ell_2$ norm penalization [@witten2009covariance]. The regularizer parameter $\gamma$ tunes the strength of this penalization. This regularization is very simple and effective. However, with this method insignificant elements in the precision matrix are penalized toward small values but they are never set to zero. By using instead $\ell_1$ norm penalization: $\log \mbox{det}( \hat {\mathbf J}) - \mbox{tr}(\mathbf S\hat{\mathbf J}) - \gamma ||\hat{\mathbf J} ||_1$, insignificant elements are forced to zero leading to a sparse inverse covariance. This is the so-called lasso regularization [@tibshirani1996; @meinshausenbuehlmann2006; @friedmanetal2008]. The advantage of a sparse inverse covariance consists in the provision of a network Indeed, let us recall that zero entries in the inverse covariance are associated with couples of non-conditionally dependent variables.
Information filtering network approach: LoGo
--------------------------------------------
An alternative approach to obtain sparse inverse covariance is by using information filtering networks generated by keeping the elements that contribute most to the covariance by means of a greedy process. of this approach is that inversion is performed at local level on a small subsets of variables and then the global inverse is reconstructed by joining the local parts through the information filtering network. Because of this Local-Global construction this method is named LoGo. It has been shown that LoGo method yields to statistically significant sparse precision matrices that outperform the ones with the same sparsity computed with lasso method [@LoGo16].
Causality network reterival {#s.causalNet}
===========================
Simulated multivariate autoregressive linear process
----------------------------------------------------
In order to be able to test if causality measures can retrieve the true causality network in the underlying process, we generated artificial multivariate normal time series with known sparse causality structure by using the following autoregressive multivariate linear process [@hamilton1994time]: $$\begin{aligned}
\label{Process}
\mathbf Z_t = \sum_{\lambda=1}^\tau \mathbf A_\lambda \mathbf Z_{t-\lambda} + \mathbf U_t \end{aligned}$$ where $\mathbf A_\lambda \in \mathbb R^{p \times p}$ are matrices with random entries drawn from a normal distribution. The matrices are made upper diagonal (diagonal included) by putting to zero all lower diagonal coefficients and made sparse by keeping only a $\mathcal O(p)$ total number of entries different from zero in the upper and diagonal part. $\mathbf U_t \in \mathbb R^p$ are random normally distributed uncorrelated variables. This process produces autocorrelated, cross-correlated and causally dependent time series. We chose it because it is among the simplest processes that can generate this kind of structured datasets. The dependency and causality structure is determined by the non-zero entries of the matrices $\mathbf A_\lambda$. The upper-triangular structure of these matrices simplify the causality structure eliminating causality cycles. Their sparsity reduces dependency and causality interactions among variables. The process is made autoregressive and stationary by keeping the eigenvalues of $\mathbf A_\lambda$ all smaller than one in absolute value. For the tests we used $\tau=5$, $p=100$ and sparsity is enforced to have a number of links approximately equal to $p$. We reconstructed the network from time series of different lengths $q$ between 5 to 20,000 points. To test statistical reliability the process was repeated 100 times with every time a different set of randomly generated matrices $\mathbf A_\lambda$. We verify that the results are robust and consistent by varying sample sizes from $p=20$ to $200$, by changing sparsity with number of links from $0.5 p$ to $5 p$ and for $\tau$ from 1 to 10.
Causality and inference network retrieval
-----------------------------------------
We tested the agreement between the causality structure of the underlying process and the one inferred from the analysis of $p$ time-series of different lengths $q$, $\mathbf Z_t \in \mathbb R^p$ with $t=1..q$, generated by using Eq.\[Process\]. We have $p$ different variables and $\tau$ lags. The dimensionality of the problem is therefore $N = p \times (\tau+1)$ variables at all lags including zero.
To estimate the inference and causality networks we started by computing the inverse covariance, $\mathbf J \in \mathbb R^{ N \times N}$, for all variables at all lags $\mathbf Z \in \mathbb R^{ N \times q}$ by using the following three different estimation methods:
- $\ell_1$ norm penalization (Glasso [@friedmanetal2008]);
- $\ell_2$ norm penalization (ridge [@tikhonov1963solution]);
- information filtering network (LoGo [@LoGo16]).
We retrieved the inference network by looking at all couples of variables, with indices ${\mathbf i} \in[1,..,p]$ and ${\mathbf j} \in[1,..,p]$, which have non-zero entries in the inverse covariance matrix $\mathbf J$ between the lagged set of $\mathbf j$ and the non-lagged $\mathbf i$. Clearly, for the ridge method the result is a complete graph but for the Glasso and LoGo the results are sparse networks with edges corresponding to non-zero conditional transfer entropies between variables $\mathbf i$ and $\mathbf j$. For the LoGo calculation we make use of the regularizer parameter as a local shrinkage factor to improve the local inversion of the covariance of the 4-cliques and triangular separators (see [@LoGo16]).
We then estimated transfer entropy between couples of variables, ${\mathbf i} \rightarrow {\mathbf j}$ conditioned to all other variables in the system. This is obtained by estimating of the inverse covariance matrix (indicated with an ‘hat’ symbol) by using Eq.\[TE20b\] (see Appendix \[s.CTE\]) with: $$\begin{aligned}
\mathbf{Z_1} &= \mathbf{ Z_j}_{,t}\\ \nonumber
\mathbf{Z_2} &=\{\mathbf{ Z_i}_{,t-1}...\mathbf{ Z_i}_{,t-\tau}\}\\ \nonumber
\mathbf{Z_3} &= \{\mathbf{ Z_j}_{,t-1}...\mathbf{ Z_j}_{,t-\tau},\mathbf{ W}\} \;.
\end{aligned}$$ With $\mathbf{ W}$ a conditioning to all variables $\mathbf{Z}$ except $\mathbf{Z_1},\mathbf{Z_2}$ and $\{\mathbf{ Z_j}_{,t-1}...\mathbf{ Z_j}_{,t-\tau}\}$. The result is a $p \times p$ matrix of conditional transfer entropies $T(\mathbf{ Z_i}_{,t} \rightarrow \mathbf{ Z_j}_{,t})$. Finally, to retrieve the causality network we retained the network of statistically validated conditional transfer entropies only. Statistical validation was performed as follows.
Statistical validation of causality
-----------------------------------
[Statistical validation has been performed from likelihood ratio statistical test. Indeed, entropy and likelihood are intimately related: entropy measures uncertainty and likelihood measures the reduction in uncertainty provided by the model. Specifically, the Shannon entropy associated with a set of random variables, $\mathbf{ Z_i}$, with probability distribution $p(\mathbf{ Z_i})$ is $H(\mathbf{ Z_i}) = - \mathbb E [ \log p(\mathbf{ Z_i}) ]$ (Eq.\[entropy\]) whereas the log-likelihood for the model $\hat p(\mathbf{ Z_i})$ associated with a set of independent observations $\mathbf{ \hat Z}_{i,t}$ with $t=1..q$ is $\log \mathcal L(\mathbf{\hat Z}_{\mathbf i}) = \sum_{t=1}^q \log \hat p(\mathbf{\hat Z}_{i,t})$ which can be written as $\log \mathcal L(\mathbf{\hat Z}_{\mathbf i}) = q \mathbb E_{\hat p} [ \log \hat p(\mathbf{ Z_i}) ]$. Note that $q$ is the total available number of observations which, in practice, is the length of the time-series minus the maximum number of lags. It is evident from these expressions that entropy and the log-likelihood are strictly related trough this link might be non-trivial. In the case of linear modeling this connection is quite evident because the entropy estimate is $H = \frac 12 (- \log |\mathbf{\hat J} | + p \log (2\pi) + p)$ and the log-likelihood is $\log \mathcal L = \frac q2 ( \log |\mathbf{\hat J} | - Tr( \mathbf{\hat \Sigma} \mathbf{\hat J} ) - p \log (2\pi))$. For the three models we study in this paper we have $Tr( \mathbf{\hat \Sigma} \mathbf{\hat J} ) =p$ and therefore the log-likelihood is equal to $q$ times the opposite of the entropy estimate. Transfer entropy, or conditional transfer entropy, are differences between two entropies: the one of a set of variables conditioned to their own past minus the one conditioned also to [the past]{} of another variable. This, in turns, is the difference of the unitary log-likelihood of two models and therefore it is the logarithm of a likelihood ratio. As Wilks pointed out [@wilks1938large; @vuong1989likelihood] the null distribution of such model is asymptotically quite universal. Following the likelihood ratio formalism, we have $\lambda=q T$ and the probability of observing a transfer entropy larger than $T$, estimated under null hypothesis, is given by $p_v \sim 1-\chi_c^2(r qT,d)$ with $r\simeq 2$ and $\chi_c^2$ the chi-square the cumulative distribution function with $d$ degrees of freedom which are the difference between the number of parameters in the two models. In our case the two models have respectively $\tau (p_j^2+1)$ and $\tau(p_j^2+1) + \tau (p_j \, p_i)$ parameters. ]{}
Statistical validation of the network
-------------------------------------
The procedures described in the previous two subsections produce the inference network and causality network. Such networks are then compared with the known underlying network of true causalities in the underlying process which is defined by the non-zero elements in the matrices $A_\lambda$ (see Eq.\[Process\]). The overlapping between the retrieved links in the inference or causality networks with the ones in the true network underlying the process is an indication of a discovery of a true causality relation. However some discoveries can be obtained just by chance or some methodologies might discover more links only because they produce denser networks. We therefore tested the hypothesis that the matching links in the retrieved networks are not obtained just by chances by computing the null-hypothesis probability to obtain the same or a larger number of matches randomly. Such probability is given by the conjugate cumulative hypergeometric distribution for a number equal or larger than ${\mathrm{TP}}$ of ‘true positive’ matching causality links between an inferred network of $n$ links and a process network of $K$ true causality links, from a population of $p^2-p$ possible links: $$\begin{aligned}
\label{HyperTest}
P(X \ge {\mathrm{TP}}| n , K , p)= 1 - \sum_{k=0}^{{\mathrm{TP}}-1} {\frac {{\binom {K}{k}}{\binom {p^2-p -K}{n-k}}}{\binom {p^2-p}{n}}} \;\;\;.\end{aligned}$$ Small values of $P$ indicate that the retrieved ${\mathrm{TP}}$ links out of $K$ are unlikely to be found by randomly picking $n$ edges from $p^2-p$ possibilities. Note that in the confusion matrix notation [@swets2014signal] we have with ${\mathrm{TP}}$ number of true positives, ${\mathrm{FP}}$ number of false positives, ${\mathrm{FN}}$ number of false negatives and ${\mathrm{TN}}$ number of true negatives.
Results {#s.results}
=======
Computation and validation of conditional transfer entropies
------------------------------------------------------------
By using Eq.\[Process\] we generated 100 multivariate autoregressive processes with known causality structures. We here report results for $p=100$ but analogous outcomes were observed for dimensionalities between $p=20$ and $200$ variables. Conditional transfer entropies between all couples of variables, conditioned to all other variables in the system, were computed by estimating the inverse covariances by using tree methodologies, ridge, lasso and LoGo and applying Eq.\[TE20\]. Conditional transfer entropies were statistically validated with respect to null hypothesis (no causality) at $p_v=1\%$ p-value. Results for Bonferroni adjusted p-value at 1% (i.e. $p_v = 0.01/(p^2-p) \sim 10^{-6}$ for $p=100$) are reported in Appendix \[BonferroniValidated\]. We also tested other values of $p_v$ from $10^{-8}$ to 0.1 obtaining consistent results. We observe that small $p_v$ reduce the number of validated causality links but increase the chance that these links match with the true network in the process. Conversely large values of $p_v$ increase the numbers of mismatched links but also of the true links discoveries. Let us note that here we use $p_v$ as a thresholding criteria and we are not claiming any evidence of statistical significance of the causality. We assess the goodness of this choice a-posteriori by comparing the resulting causality network with the known causality network of the process.
![ [**Regions in the $p/q$-$\gamma$ space where causality networks for the three models are statistically significant.**]{} The significance regions are all at the left of the corresponding lines. Tick line reports the boundary $P< 0.05$ (Eq.\[HyperTest\]) and dotted lines indicate $P< 10^{-8}$ significance levels ($P$ is averaged over 100 processes). The plots refer to $p=100$ and report the region where the causality network are all significant for 100 processes. []{data-label="f.significance"}](SignificantRetreivals.pdf){width="80.00000%"}
![ [**True positive rate: fraction of retrieved true causality links (${\mathrm{TP}}$) with respect to the total number of links in the process (n).**]{} The three panels refer to ridges, Glasso and LoGo (top, central and bottom). Data are average fractions over 100 processes. []{data-label="f.RegionFractionRetreived"}](FractionRetreivedLinks_Colromap.pdf){width="80.00000%"}
Statistical significance of the recovered causality network.
------------------------------------------------------------
Results for the contour frontiers of significant causality links for the three models are reported in Fig.\[f.significance\] for a range of time series with lengths $q$ between 10 and 20,000 and regularizer parameters $\gamma$ between $10^{-8}$ and $0.5$. Statistical significance is computed by using Eq.\[HyperTest\] and results are reported for both $P < 0.05$ and $P < 10^{-8}$ (continuous and dotted lines respectively). As one can see, the overall behaviours for the three methodologies are little affected by the threshold on $P$. We observe that LoGo significance region extends well beyond the Glasso and ridge regions.
The value of the regularizer parameter $\gamma$ affects the results for the three models in a different way. Glasso has a region in the plane $\gamma-p/q$ where it has best performances (in this case it appears to be around $\gamma \simeq 0.1$ and $p/q \simeq 2.5$). Ridge appears instead to be little affected with mostly constant performances across the range of $\gamma$. LoGo has best performances for small, even infinitesimal, values of $\gamma$. Indeed, differently from Glasso in this case $\gamma$ does not control sparsity but instead acts as local shrinkage parameter. Very small values can be useful in some particular cases to reduce the effect of noise but large values have only the effect to reduce information.
q 10 20 30 50 200 300 1000 20000
------------- ------ -------------------- ----------------------- ------------- ------------- ------------- ------------- ------------- --
ridge TP/n 0.00 0.00 0.00 0.00 0.23$^{**}$ 0.49$^{**}$ 0.76$^{**}$ 0.93$^{**}$
ridge FP/n 0.00 0.00 0.00 0.00 0.00 0.10 0.65 1.06
Glasso TP/n 0.00 0.00 0.00 0.13$^{**}$ 0.48$^{**}$ 0.53$^{**}$ 0.62$^{**}$ 0.74$^{**}$
Glasso FP/n 0.00 0.00 0.00 0.00 0.06 0.10 0.23 0.54
LoGo TP/n 0.00 0.08$^*$ 0.21$^{**}$ 0.37$^{**}$ 0.61$^{**}$ 0.65$^{**}$ 0.75$^{**}$ 0.90$^{**}$
LoGo FP/n 0.00 0.00 0.00 0.01 0.06 0.08 0.15 0.34
$^{*}$ $P < 0.05$; $^{**}$ $P < 10^{-8}$
: [**Causality network validation.**]{} Comparison between fraction of true positive (${TP/n}$) and fraction of false positive ($FP/n$), statistically validated, causality links for the three models and different time-series lengths. The table reports only the case for the parameter $\gamma=0.1$. Statistical validation of conditional transfer entropy is at $p_v=1\%$ p-value. Note that LoGo can perform better than reported in this table for smaller values of $\gamma$ (see Figs.\[f.significance\] and \[f.RegionFractionRetreived\]).
\[tab:fraction\]
Causality links retrieval
-------------------------
Once identified the parameter-regions where the retrieved causality links are statistically significant, we also measured the fraction of true links retrieved. [Indeed, given that the true underlying causality network is sparse, one could do significantly better than random by discovering only a few true positives. Instead, from any practical perspective we aim to discover a significant fraction of the edges. Figure \[f.RegionFractionRetreived\] shows that the fraction of causality links correctly discovered (true positive, ${\mathrm{TP}}$) with respect to the total number of causality links in the process ($n$) is indeed large reaching values above 50%.]{} This is the so-called true positive rate or sensitivity, which takes values between 0 (no links discovered) and 1 (all links discovered). Reported values are averages over 100 processes. We observe that the region with discovering of 10% or more true causality links greatly overlaps with the statistical validity region of Fig.\[f.significance\].
We note that when the observation time becomes long, $p/q \lessapprox 0.25$, ridge discovery rate becomes larger than LoGo. However, statistical significance is still inferior to LoGo, indeed the ridge network becomes dense when $q$ increases and the larger discovery rate of true causality links is also accompanied by a larger rate of false links incorrectly identified (false positive ${\mathrm{FP}}$).
The fraction of false positives with respect to the total number of causality links in the process ($n$) are reported in Table \[tab:fraction\] together with the true positive rate for comparison. This number can reach values larger than one because the process is sparse and there are much more possibilities to randomly chose false links than true links. Consistently with Fig.\[f.significance\] we observe that, for short time series, up to $p/q \sim 0.5$ the sparse models have better capability to identify true causality links and to discard the false ones with LoGo being superior to Glasso. Remarkably, LoGo can identify a significant fraction of causality links already from time-series with lengths of 30 data-points only. P-value significances, reported in the table with one or two stars indicate when all values of $P(X \ge {\mathrm{TP}}| n , K , p)$ from Eq.\[HyperTest\] for all 100 processes have respectively $P<0.05$ or $P<10^{-8}$. Again we observe that LoGo discovery rate region extends well beyond the Glasso and ridge regions.
q 10 20 30 50 200 300 1000 20000
------------- ---------- -------------------- ----------------------- ------------- ------------- ------------- ------------- ------------- --
ridge TP/n 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
ridge FP/n 97.84 97.84 97.84 97.84 97.84 97.84 97.84 97.84
Glasso TP/n 0.61$^*$ 0.74$^*$ 0.79$^*$ 0.85$^*$ 0.87$^{**}$ 0.84$^{**}$ 0.80$^{**}$ 0.80$^{**}$
Glasso FP/n 28.39 38.11 45.79 53.58 40.61 26.60 1.54 0.92
LoGo TP/n 0.31$^*$ 0.50$^{**}$ 0.58$^{**}$ 0.63$^{**}$ 0.75$^{**}$ 0.78$^{**}$ 0.85$^{**}$ 0.93$^{**}$
LoGo FP/n 4.53 4.27 4.18 4.03 3.72 3.63 3.44 3.21
$^{*}$ $P < 0.05$; $^{**}$ $P < 10^{-8}$
: [**Inference network validation: comparison between fraction of true positive (TP/n) and fraction of false positive (FP/n).**]{} Data for ridge are only for comparison because it is a complete graph with all links present. The table reports only the case for the parameter $\gamma=0.1$.
\[tab:fraction\_Inference\]
![ [**ROC values, for each model and each parameter combination.**]{} X-axis false positive rates (${\mathrm{FP}}/m$), y-axis true positive rates (${\mathrm{TP}}/n$). Left and right figures are the same with X-axis expanded on the low values only for the right figure to better visualise the differences between the various models. Large symbols refer to $\gamma = 0.1$ and validation at p-value $p_v=0.01$. Color intensity is proportional to time series length. Inference network results are all outside the range of the plot. Reported values are averages over 100 processes. []{data-label="f.ROC"}](ROC_cond.pdf "fig:"){width="45.00000%"} ![ [**ROC values, for each model and each parameter combination.**]{} X-axis false positive rates (${\mathrm{FP}}/m$), y-axis true positive rates (${\mathrm{TP}}/n$). Left and right figures are the same with X-axis expanded on the low values only for the right figure to better visualise the differences between the various models. Large symbols refer to $\gamma = 0.1$ and validation at p-value $p_v=0.01$. Color intensity is proportional to time series length. Inference network results are all outside the range of the plot. Reported values are averages over 100 processes. []{data-label="f.ROC"}](ROC_All_detail.pdf "fig:"){width="45.00000%"}
Inference network
-----------------
We have so far empirically demonstrated that a significant part of the true causality network can be retrieved from the statistically validated network of conditional transfer entropies. Results depend on the choice of the threshold value of the $p_v$ at which null hypothesis is rejected. We observed that lower $p_v$ are associated with network with fewer true positives but also fewer false positives and conversely larger $p_v$ yield to causality networks with larger true positives but also larger false positives. Let us here report on the extreme case of the inference network which contains all causality channels with no validation. For the ridge model this network is the complete graph with all variables connected to each-other. Instead, for Glasso and LoGo the inference network is sparse.
Results are summarized in Table.\[tab:fraction\_Inference\]. In terms of true positive rate we first notice that they are all larger than the ones in Table.\[tab:fraction\]. Indeed, the network of statistically validated conditional transfer entropies is a sub-network of the inference network. On the other hand we notice that the false positive fraction is much larger than the ones in Table.\[tab:fraction\_Inference\]. Ridge network has a fraction of 1 because, in this case, the inference network is the complete graph.
Galsso also contains a very large number of false positives reaching even 55 times the number of links in the true network and getting to lower fractions only from long time-series with $q >1000$. These numbers also indicate that Galsso networks are not sparse. LoGo has a sparser and more significant inference network with smaller fractions of false positives which stay below $5n$, which is anyway a large number of misclassification. Nonetheless, we observe that, despite such large fractions of ${\mathrm{FP}}$, the discovered true positives are statistically significant.
q 10 20 30 50 200 300 1000 20000
------------- ------ ------------- ----------------------- ------------- ------------- ------------- ------------- -------------
ridge TP/n 0.02 0.39$^{**}$ 0.45$^{**}$ 0.51$^{**}$ 0.65$^{**}$ 0.69$^{**}$ 0.78$^{**}$ 0.92$^{**}$
ridge FP/n 0.07 1.06 0.95 0.85 0.93 0.99 1.20 1.73
Glasso TP/n 0.00 0.24$^{**}$ 0.35$^{**}$ 0.43$^{**}$ 0.57$^{**}$ 0.60$^{**}$ 0.67$^{**}$ 0.77$^{**}$
Glasso FP/n 0.00 0.10 0.20 0.29 0.51 0.56 0.73 1.66
LoGo TP/n 0.11 0.34$^{**}$ 0.41$^{**}$ 0.47$^{**}$ 0.63$^{**}$ 0.66$^{**}$ 0.76$^{**}$ 0.89$^{**}$
LoGo FP/n 0.02 0.16 0.25 0.34 0.59 0.66 0.87 1.49
$^{**}$ $P < 10^{-8}$
: [**Unconditioned transfer entropy network: comparison between fraction of true positive (TP/n) and fraction of false positive (FP/n).**]{} Statistical validation of transfer entropy is at $p_v=1\%$ p-value. The table reports only the case for the parameter $\gamma=0.1$.
\[tab:fraction\_Unconditional\]
Unconditioned transfer entropy network
--------------------------------------
We last tested whether conditioning to [the past]{} of all other variables gives better causality network retrievals than the unconditioned case. Here, transfer entropy, $T(\mathbf{ Z_i} \rightarrow \mathbf{ Z_j})$, is computed by using Eq.\[TE20\] with $\mathbf W=\emptyset$, the empty set. For the ridge case this unconditional transfer entropy depends only from the time-series, $\mathbf{ Z}_{i,t}$, $\{\mathbf{ Z}_{i,t-1},...,\mathbf{ Z}_{i,t-\tau}\}$ and $\{\mathbf{ Z}_{j,t-1},...,\mathbf{ Z}_{j,t-\tau}\}$ (with $\tau = 5$ in this case). Glasso and LoGo cases are instead hybrid because a conditional dependency has been already introduced in the sparse structure of the inverse covariance $\mathbf J$ (the inference network). Results are reported in Tab.\[tab:fraction\_Unconditional\] where we observe that these networks retrieve a larger quantity of true positives than the ones constructed from conditional entropy. However, the fraction of false positive is also larger than the ones in Tab.\[tab:fraction\] although it is smaller than what observed in the inference network in Tab.\[tab:fraction\_Inference\]. [Overall, these results indicate that conditioning is effective in discarding false positives. ]{}
Summary of all results in a single ROC plot
-------------------------------------------
In summary, we have investigated the networks associated with conditional transfer entropy, unconditional transfer entropy and inference for three models under a range of different parameters. In the previous sub-sections we have provided some comparisons between the performances of the three models in different ranges of parameters. Let us here provide a summary of all results within a single ROC plot [@swets2014signal]. Figure \[f.ROC\] reports the ROC values, for each model and each parameter combination, x-axis are false positive rates (${\mathrm{FP}}/m$) and y-axis true positive rates (${\mathrm{TP}}/n$). Each point is an average over 100 processes. Points above the diagonal line are associated to relatively well performing models with the upper left corner representing the point where models correctly discover all true causality links without any false positive. The plot reports with large symbols the cases for $\gamma = 0.1$ and validation at p-value $p_v=0.01$, which can be compared with the data reported in the tables.
Overall from Tables \[tab:fraction\], \[tab:fraction\_Inference\], \[tab:fraction\_Unconditional\] and Fig. \[f.ROC\] we conclude that all models obtain better results for longer time series and that conditional transfer entropy over-performs the unconditional counterparts (see, Tables \[tab:fraction\] and \[tab:fraction\_Unconditional\] and the two separated ROC figures for conditional and unconditional transfer entropies reported in Fig.\[f.ROCCoUc\] in appendix \[A.Co.UC\]). In the range of short time series, when $q\le p$, which is of interest for this paper, LoGo is the best performing model with better performances achieved for small $\gamma \lesssim 10^{-4}$ and validation with small p-values $p_v \lesssim 10^{-4}$. LoGo is consistently the best performing model also for longer time-series up to lengths of $q \sim1000$. Instead, above $q=2000$ ridge begins to provide better results. For long time series, at $q=20,000$, the best performing model is ridge with parameters $\gamma =10^{-5}$, p-value $p_v = 5\;10^{-6}$. LoGo is also performing well when time series are long with best performance obtained at $q=20,000$ for parameters $\gamma =10^{-10}$, p-value $p_v = 5\;10^{-6}$. We note that LoGo instead performs poorly in the region of parameters with $\gamma \le 0.1$ and $p_v \le 0.01$ for short time-series $q\le p/2$.
Conclusions and perspectives {#s.conclusions}
============================
In this paper we have undertaken the challenging task to explore models and parameter regions where the analytics of time series can retrieve significant fractions of true causality links from linear multivariate autoregressive process with known causality structure. Results demonstrate that sparse models with conditional transfer entropy are the ones who achieve best results with significant causality link retrievals already for very short time series even with $q \le p/5 = 20$.
Unexpectedly, we observed that the structure of the inference networks in the two sparse models, Glasso and LoGo, have excessive numbers of false positives yielding to rather poor performances. However, in these models false positive can be efficiently filtered out by imposing statistical significance of the transfer entropies.
Results are affected by the choice of the parameters and the fact that the models depend on various parameters ($q$, $p$, $\gamma$, $p_v$, $P$) make the navigation in this space quite complex. We observed that the choice of p-values, $p_v$, for valid transfer entropies affects results. Within our setting we obtained best results with the smaller p-values especially in the regions of short time-series. We note that the regularizer parameter $\gamma$ also plays an important role and best performances are obtained by combination of the two parameters $\gamma$ and $p_v$. Not surprizingly, longer time-series yield to better results. We observe that conditioning to all other variables or unconditioning is affecting the transfer entropy estimation with better performing causality network retrieval obtained for conditioned transfer entropies. However, qualitatively, results are comparable. Other intermediate cases, such as conditioning to past of all other variables only, have been explored again with qualitatively comparable results. It must be said that in the present system results are expected to be robust to different conditionings because the underlying network of the investigated processes is sparse. For denser inference structures, conditioning could affect more the results.
Consistently with the findings in [@LoGo16] we find that LoGo outperforms the other methods. This is encouraging because the present settings of LoGo is using a simple class of information filtering networks, namely the TMFG [@TMFG], obtained by retaining largest correlations. There are a number of alternative information filtering networks which should be explored. In particular, given the importance of statistical validation emerged from the present work, it would be interesting to explore statistical validation within the process of construction of the information filtering networks themselves.
In this paper we investigate a simple case with a linear autoregressive multivariate normal process analysed by means of linear models. Both LoGo and Glasso can be extended to the non-linear case with LoGo being particularly suitable for non-parametric approaches as well [@LoGo16].
There are Alternative methods to extract causality networks from short time series, in particular Multispatial CCM [@sugihara2012detecting; @clark2015spatial] appears to perform well for short time series. A comparison between different approaches and the application of these methods to real data will be extremely interesting. However this should be the object of future works.
Acknowledgement {#acknowledgement .unnumbered}
---------------
T.A. acknowledges support of the UK Economic and Social Research Council (ESRC) in funding the Systemic Risk Centre (ES/K002309/1). TDM wishes to thank the COST Action TD1210 for partially supporting this work and Complexity Science Hub Vienna.
Conflict of interest disclosure {#conflict-of-interest-disclosure .unnumbered}
-------------------------------
The authors declare that there is no conflict of interest regarding the publication of this paper.
Conditional transfer entropy {#cTE}
============================
\[s.definitions1\] Let us here briefly review two of the most commonly used information theoretic quantities, that we use in this paper, namely, mutual information (quantifying dependency) and transfer entropy (quantifying causality) for the multivariate case [@shannon2001mathematical; @schreiber2000measuring; @anderson1984multivariate].
Mutual information
------------------
Let us first start from the simplest case of two random variables, $X\in \mathbb R^{1}$ and $Y\in \mathbb R^{1}$, where dependence can be quantified by the amount of shared information between the two variables, which is called mutual information: $I(X;Y)=H(X)+H(Y)-H(X,Y)$ where $H(X)$ is the entropy of variable $X$, $H(Y)$ is the entropy of variable $Y$ and $H(X,Y)$ is the joint entropy of variables $X$ and $Y$ [@anderson1984multivariate]. Extending to the multivariate case, the shared information between a set of $n$ random variables $\mathbf X = (X_1,...,X_n)^T\in \mathbb R^{n}$ and another set of $m$ random variables $\mathbf Y = (Y_1,...,Y_m)^T \in \mathbb R^{m}$ is $$\begin{aligned}
\label{I1}
I(\mathbf X;\mathbf Y) & {}
= H(\mathbf X) + H(\mathbf Y) - H(\mathbf X,\mathbf Y) \end{aligned}$$ with $H(\mathbf X)$, $H(\mathbf Y)$, the entropies respectively for the set of variables $\mathbf X$ and $\mathbf Y$ and $H(\mathbf X,\mathbf Y)$ their joint entropy. It must be stressed that this quantity is the mutual information between two sets of multivariate variables and it is not the multivariate mutual information between all variables $\{ \mathbf X,\mathbf Y \}$ which instead measures the intersection of information between all variables. Mutual information in Eq.\[I1\] can also be written as $$\begin{aligned}
\label{I2}
I(\mathbf X;\mathbf Y) & {}
= H(\mathbf Y) - H(\mathbf Y|\mathbf X)=H(\mathbf X) - H(\mathbf X|\mathbf Y)\end{aligned}$$ which makes use of the conditional entropy of $\mathbf Y$ given $\mathbf X$: $H(\mathbf Y|\mathbf X)=H(\mathbf Y,\mathbf X)-H(\mathbf X)=\mathbb E(H(\mathbf Y)|\mathbf X)$.
Conditioning to a third set of variables $\mathbf W$ can also be applied to mutual information itself and its expression is a direct extension of Eq.\[I1\] and it is called conditional mutual information: $$\begin{aligned}
\label{Ic}
I(\mathbf X;\mathbf Y|\mathbf W) & {}
= H(\mathbf X|\mathbf W) + H(\mathbf Y|\mathbf W) - H(\mathbf X,\mathbf Y|\mathbf W) \;\;.\end{aligned}$$ Eq.\[I1\] and Eq.\[Ic\] coincide in the case of an empty set $\mathbf W=\emptyset$. Mutual information and conditional mutual information are symmetric measures with $I(\mathbf X;\mathbf Y|\mathbf W) =I(\mathbf Y;\mathbf X|\mathbf W) $ always. Let us note that symmetry is unavoidable for information measures that quantify the simultaneous effect of a set of variables onto another. Indeed, in a simultaneous interaction cause and effect cannot be distinguished from the exchange of information and direction cannot be established. To quantify causality one must investigate the transmission of information not only between two sets of variables but also trough time.
Conditional transfer entropy {#cTE.s}
----------------------------
Causality between two random variables, $X\in \mathbb R^{1}$ and $Y\in \mathbb R^{1}$, can be quantified by means of the so-called transfer entropy which quantifies the amount of uncertainty on $Y$ explained by [the past]{} of $ X $ given [the past]{} of $Y$. Let us consider a series of observations and denote with $X_{t}$ the random variable $X$ at time $t$ and with $X_{t-\tau}$ the random variable at a previous time, $\tau$ lags before $t$. Using this notation, we can define transfer entropy from variable $X$ to variable $Y$ in terms of the following conditional mutual information: $T(X \rightarrow Y) = I(Y_{t};X_{t-\tau}|Y_{t-\tau})$ [@schreiber2000measuring; @anderson1984multivariate].
For the multivariate case, given two sets of random variables $\mathbf{ X} \in \mathbb R^{n}$ and $\mathbf{ Y} \in \mathbb R^{m}$, the transfer entropy is the conditional mutual information between the set of variables $\mathbf{Y}_t$ at time $t$ and [the past]{} of the other set of variables, $\mathbf{ X}_{t-\tau}$ conditioned to [the past]{} of the first variable $\mathbf{Y}_{t-\tau}$. This is: $T(\mathbf X \rightarrow \mathbf Y) = I(\mathbf{Y}_t;\mathbf{ X}_{t-\tau}|\mathbf{ Y}_{t-\tau}) $ [@anderson1984multivariate]. In general, the influence from [the past]{} can come from more than one lag and we can therefore extend the definition including different sets of lags for the two variables: $\tau_1,...,\tau_k$, $\lambda_1,...,\lambda_h$: $$\begin{aligned}
\label{GeneralTE}
T(\mathbf X \rightarrow \mathbf Y) & {}= I(\mathbf{Y}_t; \{\mathbf{ X}_{t-\tau_1}...\mathbf{ X}_{t-\tau_k}\}|\{\mathbf{ Y}_{t-\lambda_1}...\mathbf{ Y}_{t-\lambda_h} \}) \\& {} = H(\mathbf{Y}_t | \{\mathbf{ Y}_{t-\lambda_1}...\mathbf{ Y}_{t-\lambda_h} \}) - H(\mathbf{Y}_t | \{\mathbf{ X}_{t-\tau_1}...\mathbf{ X}_{t-\tau_k},\mathbf{ Y}_{t-\lambda_1}...\mathbf{ Y}_{t-\lambda_h} \}) \nonumber\end{aligned}$$ a further generalization, which we use in this paper, includes conditioning to any other set of variables $\{\mathbf{ W}_{t-\theta_1}...\mathbf{ W}_{t-\theta_g}\}$ lagged at $\theta_1,...,\theta_g$: $$\begin{aligned}
\label{General_cTE1}
T(\mathbf X \rightarrow \mathbf Y|\mathbf{ W}) & {}= I(\mathbf{Y}_t; \{\mathbf{ X}_{t-\tau_1}...\mathbf{ X}_{t-\tau_k}\}|\{\mathbf{ Y}_{t-\lambda_1}...\mathbf{ Y}_{t-\lambda_h},\mathbf{ W}_{t-\theta_1}...\mathbf{ W}_{t-\theta_g}\}) \;\;.\end{aligned}$$
In this paper we simplify notation using $\mathbf{ X}_{t}^{lag}=\{\mathbf{ X}_{t-\tau_1}...\mathbf{ X}_{t-\tau_k}\}$, $\mathbf{ Y}_{t}^{lag} = \{\mathbf{ Y}_{t-\lambda_1}...\mathbf{ Y}_{t-\lambda_h}\}$ and $\mathbf{ W}_{t} = \{\mathbf{ W}_{t-\theta_1}...\mathbf{ W}_{t-\theta_g}\}$.
In the literature, there are several examples that use adaptations of Eq.\[General\_cTE\] to compute causality and dependency measures [@pearl2009causality]. A notable example is the directed information, introduced by Massey in [@massey1990causality], where $\tau$ spans all lags in a range between $0$ to $s-1$, $\lambda$ spans the lags from $1$ to $s-1$. The directed information is then defined as the sum over transfer entropies from $s=1$ to present: $$\begin{aligned}
I(\{\mathbf X\}_{1}^t \rightarrow \{\mathbf Y\}_{1}^t|\mathbf{ W}) & {}= \sum_{s=1}^t I(\mathbf{Y}_s; \{\mathbf{ X}\}_{1}^s|\{\mathbf{ Y}\}_{1}^{s-1},\mathbf{ W}) \;\;.\end{aligned}$$ where we adopted the notation $\{\mathbf X\}_{1}^t = \{\mathbf{ X}_{1}...\mathbf{ X}_{t}\}$ and $\{\mathbf Y\}_{1}^t = \{\mathbf{ Y}_{1}...\mathbf{ Y}_{t}\}$. Interestingly, this definition includes the conditional synchronous mutual information contributions between $\mathbf{ X}_{s}$ and $\mathbf{ Y}_{s}$. Following Kramer [@kramer1998directed; @amblard2012relation] we observe that for stationary processes $$\begin{aligned}
\lim_{t \to \infty} \frac{1}{t} I(\{\mathbf X\}_{1}^t \rightarrow \{\mathbf Y\}_{1}^t) = \lim_{t \to \infty} I(\{\mathbf X\}_{1}^t; \mathbf Y_t | \{\mathbf Y\}_{1}^{t-1})=T(\{\mathbf X\}_{1}^{t-1} \rightarrow \mathbf Y_t) + I(\{\mathbf X\}_{1}^t ; \{\mathbf Y\}_{1}^t |\{\mathbf X\}_{1}^{t-1})\;\;,\end{aligned}$$ with $T(\{\mathbf X\}_{1}^{t-1} \rightarrow \mathbf Y_t)= I(\mathbf{Y}_t; \{\mathbf{ X}_{1}...\mathbf{ X}_{t-1}\}|\{\mathbf{ Y}_{1}...\mathbf{ Y}_{t-1}\} )$. This identity supports the intuition that the directed information accounts for the transfer entropy plus an instantaneous term.
Shannon-Gibbs entropy {#Shannon}
=====================
The general expression for the transfer entropy reported di in Sec.\[s.definitions1\], Eq.\[General\_cTE\] is independent on the kind of entropy definition. In this paper we use Shannon entropy, which is defined as $$\begin{aligned}
\label{entropy}
H(\mathbf X) & {}= - \mathbb E [ \log p(\mathbf X) ] \\
H(\mathbf Y) & {}= - \mathbb E [ \log p(\mathbf Y) ] \end{aligned}$$ where $p(\mathbf X)$ and $p(\mathbf Y)$ are the probability distribution function for the set of random variables $\mathbf X$ and $\mathbf Y$. Similarly, the joint Shannon entropy for the variables $\mathbf X$ and $\mathbf Y$ is defined as $$\begin{aligned}
H(\mathbf X,\mathbf Y) = - \mathbb E [ \log p(\mathbf X,\mathbf Y) ] \end{aligned}$$ with $p(\mathbf X,\mathbf Y )$ the joint probability distribution function of $\mathbf X$ and $\mathbf Y$. This is the most common definition of entropy. It is a particularly meaningful and suitable entropy for linear modelling, as we focus in the paper.
Multivariate normal modelling {#ss.linearModelling}
-----------------------------
For multivariate normal variables the Shannon-Gibbs entropy is: $$\begin{aligned}
\label{HLoGo}
H(\mathbf X) = \frac12 \log \left(\mbox{det} {\mathbf \Sigma(\mathbf X)}\right) + \frac{n}2 \log\left( 2\pi e \right)\end{aligned}$$ and its conditional counterpart is $$\begin{aligned}
\label{H2}
H(\mathbf X | \mathbf W) = \frac12 \log\left( { \mbox{det} \mathbf \Sigma(\mathbf X | \mathbf W)} \right) + \frac{n}2 \log\left( 2\pi e \right)\end{aligned}$$ with $\mathbf \Sigma$ the covariance matrix and $\mbox{det}( . )$ the matrix determinant. In the paper we use these expressions to compute mutual information and conditional transfer entropy.
![ The inverse of parts the inverse covariance $\mathbf J$ gives the covariance of the variables corresponding to that part conditioned to the other variables. []{data-label="f.inversions"}](Inversions.pdf){width="80.00000%"}
Computing conditional covariances for sub-sets of variables from the inverse covariance {#InverseCovJandSigma}
=========================================================================================
Let us consider three sets of variables $\mathbf{ Z_1} \in \mathbb R^{p_1}$, $\mathbf{ Z_2} \in \mathbb R^{p_2}$ and $\mathbf{ Z_3} \in \mathbb R^{p_3}$ and the associated inverse covariance $\mathbf J \in \mathbb R^{{(p_1+p_2+p_3)}\times{(p_1+p_2+p_3)}}$ for $\{\mathbf{ Z_1 },\mathbf{ Z_2 },\mathbf{ Z_3 }\} \in \mathbb R^{(p_1+p_2+p_3)}$. The conditional covariance of $\mathbf{ Z_1}$ given $\mathbf{ Z_2}$ and $\mathbf{ Z_3}$ is the inverse of the $p_1 \times p_1$ upper left part of $\mathbf J$ with indices in $V_1=(1,...,p_1)$ (see Fig.\[f.inversions\]): $$\begin{aligned}
\label{Cov1_3}
{\mathbf\Sigma}(\mathbf{Z_1}|\mathbf{Z_2},\mathbf{Z_3}) = \left(\mathbf{ J_{1,1}}\right)^{-1} \;\;.\end{aligned}$$
Instead, the conditional covariance of $\mathbf{Z_1}$ given $\mathbf{Z_3}$ is obtained by inverting the larger upper left part $\mathbf{ J_{12,12}}$ with both indices in $\{V_1,V_2\}$ with $V_2=(p_1+1,...,p_1+p_2)$, and then taking the inverse of the part with indices in $V_1$ which, using the Schur complement [@anderson1984multivariate], is: $$\begin{aligned}
\label{Cov2_3}
{\mathbf\Sigma}(\mathbf{Z_1}|\mathbf{Z_3}) = (\mathbf {J_{1,1}} - \mathbf {J_{1,2}} (\mathbf {J_{2,2}})^{-1} \mathbf {J_{2,1}})^{-1} \;\;.\end{aligned}$$
Figure \[f.inversions\] schematically illustrates these inversions and their relations with conditional covariances. Let us note that these conditional covariances can also be expressed directly in terms of sub-covariances by using again the Schur complement: $$\begin{aligned}
\label{Cov1_3COV}
{\mathbf\Sigma}(\mathbf{Z_1}|\mathbf{Z_2},\mathbf{Z_3})
&= {\mathbf{ \Sigma_{1,1}}}- \mathbf{{ \Sigma_{1,23}}}( {\mathbf{ \Sigma_{23,23}}})^{-1} {\mathbf{ \Sigma_{23,1}}}\\
{\mathbf\Sigma}(\mathbf{Z_1}|\mathbf{Z_3})
&= {\mathbf{ \Sigma_{1,1}}}- \mathbf{{ \Sigma_{1,3}}}( {\mathbf{ \Sigma_{3,3}}})^{-1} {\mathbf{ \Sigma_{3,1}}}\;\;.
\label{Cov1_3COV1}\end{aligned}$$ However, when $p_3$ (cardinality of $V_3$) is much larger than $p_1$ and $p_2$ (cardinalities of $V_1$ and $V_2$) then the equivalent expressions, Eqs.\[Cov1\_3\] and \[Cov2\_3\], that use the inverse covariance involve matrices with much smaller dimensions. This can become computationally crucial when very large dimensionalities are involved. Furthermore, if the inverse covariance $\mathbf J$ is estimated by using a sparse modeling tool such as Glasso or LoGo [@friedmanetal2008; @LoGo16] (as we do in this paper), then computations in expressions Eqs.\[Cov1\_3\] and \[Cov2\_3\] have to handle only a few non-zero elements providing great computational advantages over Eqs.\[Cov1\_3COV\] and \[Cov1\_3COV1\].
In the paper we make use of Eqs.\[Cov1\_3\]-\[Cov2\_3\] to compute mutual information and conditional transfer entropy for the system of all variables and their lagged versions.
Mutual information
------------------
Let us consider the mutual information between any two subsets $\mathbf X \in \mathbb R^n$ and $\mathbf Y \in \mathbb R^m$ of variables conditioned to all other variables, which we shall call $\mathbf W\in \mathbb R^{p-n-m}$. For these three sets of variables $\{\mathbf X,\mathbf Y,\mathbf W\} \in \mathbb R^{p}$ the conditional mutual information, $I(\mathbf X,\mathbf Y,\mathbf W)=H(\mathbf X,\mathbf Y|\mathbf W)-H(\mathbf X|\mathbf Y,\mathbf W)$ (Eq.\[Ic\]), can be expressed in terms of the conditional covariances by using Eq.\[H2\]: $$\begin{aligned}
\label{JMI0c}
I(\mathbf X;\mathbf Y|\mathbf W) = \frac12 \log \mbox{det} {\mathbf\Sigma}(\mathbf X|\mathbf W)- \frac12 \log \mbox{det} {\mathbf\Sigma}(\mathbf X | \mathbf Y,\mathbf W) \;\;.\end{aligned}$$ Given the inverse covariance $\mathbf J \in \mathbb R^{p\times p}$, by using Eqs.\[Cov1\_3\] and \[Cov2\_3\] and substituting $$\begin{aligned}
\mathbf{Z_1} &= \mathbf X\\ \nonumber
\mathbf{Z_2} &= \mathbf Y\\ \nonumber
\mathbf{Z_3} &= \mathbf W
\end{aligned}$$ we can express the conditional mutual information, Eq.\[JMI0c\], directly in terms of the parts of $\mathbf J$: $$\begin{aligned}
\label{JMI1c}
I(\mathbf X;\mathbf Y|\mathbf W) = - \frac12 \log \mbox{det}\left( \mathbf {J_{1,1}} - \mathbf {J_{1,2}} (\mathbf {J_{2,2}})^{-1} \mathbf {J_{2,1}})\right)+ \frac12 \log \left (\mbox{det}\mathbf{ J_{1,1} }\right ) \;\;.\end{aligned}$$ Note that, although this is not directly evident, Eq.\[JMI1c\] is symmetric by exchanging $\mathbf 1$ and $\mathbf 2$ (i.e. $\mathbf X$ and $\mathbf Y$).
Conditional transfer entropy {#s.CTE}
-----------------------------
Conditional transfer entropy (Eq.\[General\_cTE\] ) is a conditional mutual information between lagged sets of variables and therefore it can be computed directly from Eq.\[JMI1c\]. In this case we shall name $$\begin{aligned}
\label{TE20b}
\mathbf{Z_1} &= \mathbf{Y}_t\\ \nonumber
\mathbf{Z_2} &=\{\mathbf{ X}_{t-\tau_1}...\mathbf{ X}_{t-\tau_k}\}\\ \nonumber
\mathbf{Z_3} &= \{\mathbf{ Y}_{t-\lambda_1}...\mathbf{ Y}_{t-\lambda_h},\mathbf{ W}_{t-\theta_1}...\mathbf{ W}_{t-\theta_g}\}\\\nonumber
T(\mathbf X \rightarrow \mathbf Y|\mathbf{ W}) &= - \frac12 \log \mbox{det} \left(\mathbf {J_{1,1}} - \mathbf {J_{1,2}} (\mathbf {J_{2,2}})^{-1} \mathbf {J_{2,1}})\right)+ \frac12 \log \mbox{det} \left (\mathbf{ J_{1,1} }\right )
\end{aligned}$$ obtaining an expression which is formally identical to Eq.\[JMI1c\] but with indices $\mathbf{ 1}$ and $ \mathbf 2$ referring to the above sets of variables instead.
Note that the index $\mathbf 3$ does not appear in this expression. Information from variables $\mathbf 3$ ($\mathbf W$) has been used to compute $\mathbf J$ but then only the sub-parts $\mathbf{ 1}$ and $ \mathbf 2$ are required to compute the conditional transfer entropy. The fact that these expressions for conditional mutual information and conditional transfer entropy involve only local parts ($\mathbf{ 1}$ and $ \mathbf 2$) of the inverse covariance can become extremely useful when high dimensional datasets are involved.
![ [**ROC values, for conditional (left) and unconditional (right) transfer entropies.**]{} X-axis false positive rates (${\mathrm{FP}}/m$), y-axis true positive rates (${\mathrm{TP}}/n$). Large symbols refer to $\gamma = 0.1$ and validation at p-value $p_v=0.01$. Color intensity is proportional to time series length. Inference network results are all outside the range of the plot. Reported values are averages over 100 processes. \[f.ROCCoUc\]](ROC_cond.pdf "fig:"){width="45.00000%"} ![ [**ROC values, for conditional (left) and unconditional (right) transfer entropies.**]{} X-axis false positive rates (${\mathrm{FP}}/m$), y-axis true positive rates (${\mathrm{TP}}/n$). Large symbols refer to $\gamma = 0.1$ and validation at p-value $p_v=0.01$. Color intensity is proportional to time series length. Inference network results are all outside the range of the plot. Reported values are averages over 100 processes. \[f.ROCCoUc\]](ROC_uncond.pdf "fig:"){width="45.00000%"}
Comparison between conditional and unconditional transfer entropies {#A.Co.UC}
===================================================================
The two ROC plots for conditional and unconditional transfer entropies are displayed in Fig.\[f.ROCCoUc\]. Form the comparison it is evident that, for the process studied in this paper, conditional transfer entropy provides best results. This is in line with what observed in Tab.s \[tab:fraction\],\[tab:fraction\_Unconditional\],\[tab:fraction\_B\] and \[tab:fraction\_U\_B\].
Causality network results for transfer entropy validation with 1% Bonferroni adjusted p-values {#BonferroniValidated}
==============================================================================================
In tables \[tab:fraction\_B\] and \[tab:fraction\_U\_B\], are reported true positive rates (${{\mathrm{TP}}/n}$) and fraction of false positives (${\mathrm{FP}}/m$) statistically validated, causality links with validation at 1% Bonberroni adjusted p-value (i.e. $p_v \lesssim 10^{-6}$). These tables must be compared with Tab.s \[tab:fraction\] and \[tab:fraction\_Unconditional\], in the main text where causality links are validated at $p_v=1\%$ non-adjusted p-value.
q 10 20 30 50 200 300 1000 20000
------------- ------ ------ ----------------------- ------------- ------------- ------------- ------------- -------------
ridge TP/n 0.00 0.00 0.00 0.00 0.00 0.30$^{**}$ 0.67$^{**}$ 0.89$^{**}$
ridge FP/n 0.00 0.00 0.00 0.00 0.00 0.01 0.18 0.75
Glasso TP/n 0.00 0.00 0.00 0.00 0.35$^{**}$ 0.43$^{**}$ 0.57$^{**}$ 0.71$^{**}$
Glasso FP/n 0.00 0.00 0.00 0.00 0.01 0.03 0.13 0.45
LoGo TP/n 0.00 0.00 0.02 0.17$^{**}$ 0.50$^{**}$ 0.56$^{**}$ 0.69$^{**}$ 0.87$^{**}$
LoGo FP/n 0.00 0.00 0.00 0.00 0.01 0.03 0.09 0.28
$^{**}$ $P < 10^{-8}$
: [**Causality network validation with conditional transfer entropy validation at 1% Bonberroni adjusted p-value.**]{} Fraction of true positive (${TP/n}$) and fraction of false positive ($FP/n$), statistically validated, causality links for the three models and different time-series lengths. The table reports only the case for the parameter $\gamma=0.1$.
\[tab:fraction\_B\]
q 10 20 30 50 200 300 1000 20000
------------- ------ ------ ----------------------- ------------- ------------- ------------- ------------- -------------
ridge TP/n 0.00 0.00 0.22$^{**}$ 0.36$^{**}$ 0.55$^{**}$ 0.59$^{**}$ 0.70$^{**}$ 0.88$^{**}$
ridge FP/n 0.00 0.00 0.09 0.21 0.47 0.55 0.77 1.32
Glasso TP/n 0.00 0.00 0.00 0.27$^{**}$ 0.48$^{**}$ 0.53$^{**}$ 0.62$^{**}$ 0.75$^{**}$
Glasso FP/n 0.00 0.00 0.00 0.11 0.37 0.43 0.61 1.41
LoGo TP/n 0.00 0.00 0.22$^{**}$ 0.35$^{**}$ 0.53$^{**}$ 0.58$^{**}$ 0.69$^{**}$ 0.86$^{**}$
LoGo FP/n 0.00 0.00 0.05 0.16 0.42 0.49 0.71 1.27
$^{**}$ $P < 10^{-8}$
: [**Causality network validation with unconditional transfer entropy validation at 1% Bonberroni adjusted p-value.**]{} Fraction of true positive (${TP/n}$) and fraction of false positive ($FP/n$), statistically validated, causality links for the three models and different time-series lengths. The table reports only the case for the parameter $\gamma=0.1$.
\[tab:fraction\_U\_B\]
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---
abstract: 'Analytical and numerical methods are developed to analyze the quantum nature of the big bang in the setting of loop quantum cosmology. They enable one to explore the effects of quantum geometry both on the gravitational and matter sectors and significantly extend the known results on the resolution of the big bang singularity. Specifically, the following results are established for the homogeneous isotropic model with a massless scalar field: i) the scalar field is shown to serve as an internal clock, thereby providing a detailed realization of the ‘emergent time’ idea; ii) the physical Hilbert space, Dirac observables and semi-classical states are constructed rigorously; iii) the Hamiltonian constraint is solved numerically to show that *the big bang is replaced by a big bounce*. Thanks to the non-perturbative, background independent methods, unlike in other approaches the quantum evolution is *deterministic across the deep Planck regime*. Our constructions also provide a conceptual framework and technical tools which can be used in more general models. In this sense, they provide foundations for analyzing physical issues associated with the Planck regime of loop quantum cosmology as a whole.'
author:
- 'Abhay Ashtekar${}^{1,2,3}$'
- 'Tomasz Pawlowski${}^{1}$'
- 'Parampreet Singh${}^{1,2}$'
title: |
Quantum Nature of the Big Bang:\
An Analytical and Numerical Investigation
---
Introduction {#s1}
============
Loop quantum gravity (LQG) is a background independent, non-perturbative approach to quantum gravity [@alrev; @crbook; @ttbook]. It is therefore well-suited for the analysis of certain long standing questions on the quantum nature of the big bang. Examples of such questions are: Over the years, these and related issues had been generally relegated to the ‘wish list’ of what one would like the future, satisfactory quantum gravity theory to eventually address. However, over the past five years, thanks to the seminal ideas introduced by Bojowald and others, notable progress was made on such questions in the context of symmetry reduced, minisuperspaces [@mbrev]. In particular, it was found that Riemannian quantum geometry, which comes on its own in the Planck regime, has just the right features to resolve the big-bang singularity [@mb1] in a precise manner [@abl]. However, the physical ramifications of this resolution —in particular the answer to what is ‘on the other side’— have not been worked out. It is therefore natural to ask whether one can complete that analysis and systematically address the questions listed above, at least in the limited context of simple cosmological models. It turns out that the answer is in fact in the affirmative. A brief summary of arguments leading to this conclusion appeared in [@aps1]. The purpose of this paper is to provide the detailed constructions and numerical simulations that underlie those results.
Let us begin with a brief summary of the main results of loop quantum cosmology (LQC). (For a comprehensive survey, see, e.g., [@mbrev].) They can be divided in two broad classes:\
i) singularity resolution based on exact quantum equations (see e.g. [@mb1; @bh; @hm1; @abl; @bdv; @mb3; @ab1; @lm]), and,\
ii) phenomenological predictions based on ‘effective’ equations (see e.g. [@mb2; @bd1; @tsm; @jw; @dh1; @dh2; @bms; @hw; @jel; @sd; @gh; @ps1; @sv]).
Results in the first category make a crucial use of the effects of quantum geometry on the *gravitational part* of the Hamiltonian constraint. Because of these effects, the quantum ‘evolution’ is now dictated by a second order *difference equation* rather than the second order differential equation of the Wheeler-DeWitt (WDW) theory. Nonetheless, the intuitive idea of regarding the scale factor $a$ as ‘internal time’ was maintained. The difference is that the ‘evolution’ now occurs in discrete steps. As explained in detail in Sec. \[s2.2\], this discreteness descends directly from the quantum nature of geometry in LQG; in particular, the step size is dictated by the lowest non-zero eigenvalue of the area operator. When the universe is large, the differential equation is an excellent approximation to the ‘more fundamental’ difference equation. However, in the Planck regime, there are major deviations. In particular, while the classical singularity generically persists in the theory without additional inputs, this is not the case in LQG. This difference does *not* arise simply because the discrete ‘evolution’ enables one to ‘jump over’ the classical singularity. Indeed, even when the discrete evolution passes *through* the point $a=0$, the difference equation remains well-defined and enables one to ‘evolve’ *any* initial data across this classically singular point. It is in this sense that the singularity was said to be resolved.
In the second category of results, by and large the focus was on quantum geometry modifications of the *matter part* of the Hamiltonian constraint. The main idea is to work with approximation schemes which encode the idea of semi-classicality and to incorporate quantum geometry effects on the matter Hamiltonian by adding suitable ‘effective’ terms to the classical Hamiltonian constraint. The hope is that dynamics generated by the ‘quantum-corrected’ classical Hamiltonian would have a significant domain of validity and provide a physical understanding of certain aspects of the full quantum evolution. This strategy has led to a number of interesting insights. For example, using effective equations, it was argued that the singularity resolution is generic for all homogeneous models; that there is an early phase of inflation driven by quantum gravity effects; that this phase leads to a reduction of power spectrum in CMB at large angular scales; and that quantum geometry effects suppress the classical chaotic behavior of Bianchi IX models near classical singularity.
Attractive as these results are, important limitations have persisted. Let us begin with the results on singularity resolution. As in most non-trivially constrained systems, solutions to the quantum constraint fail to be normalizable in the kinematical Hilbert space on which the constraint operators are well-defined. Therefore one has to endow physical states (i.e. solutions to the Hamiltonian constraint) with a new, *physical inner product*. Systematic strategies —e.g., the powerful ‘group averaging procedure’ [@dm; @almmt]— typically require that the constraint be represented by a self-adjoint operator on the kinematic Hilbert space while, so far, most of the detailed discussions of singularity resolution are based on Hamiltonian constraints which do not have this property. Consequently, the space of solutions was not endowed with a Hilbert space structure. This in turn meant that one could not introduce Dirac observables nor *physical* semi-classical states. Therefore, as pointed out, e.g. in [@bt], the physical meaning of singularity resolution remained somewhat obscure. In particular, even in simple models, there was no clear-cut answer as to what the universe did ‘before’ the big-bang. Was there a genuine ‘quantum foam’ or was the quantum state peaked at a large classical universe on the ‘other side’? In absence of a physical Hilbert space, the second and the third questions posed in the beginning of this section could only be answered partially and the first and the fourth remained unanswered.
The phenomenological predictions have physically interesting features and serve as valuable guides for future research. However, the current form of this analysis also faces some important limitations. Many of these discussions focus only on the quantum geometry modifications of the matter Hamiltonian. This strategy has provided new insights and does serve as a useful starting point. However, there is no a priori reason to believe that it is consistent to ignore modifications of the gravitational part of the Hamiltonian and retain only the matter modifications. Conclusions drawn from such analysis can be taken as attractive suggestions, calling for more careful investigations, rather than firm predictions of LQC, to be compared with observations. On the conceptual side, a number of semi-classical approximations are made while deriving the effective equations. Many of them are largely violated in the Planck regime. Therefore it is difficult to regard conclusions drawn from effective equations on singularity avoidance, and on the fate of the universe beyond, as reliable. It does happen surprisingly often in physics that approximation schemes turn out to work even outside the regimes for which they were originally intended. But there is no a priori reason to think that this *must* happen. To develop intuition on the validity of approximations, it is essential to make, in at least a few simple models, detailed comparisons of predictions of effective equations with those of quantum equations that are being approximated. To summarize, while LQC has led to significant progress by opening new avenues and by indicating how qualitatively new and physically desirable results can arise, the program has remained incomplete even within the realm of symmetry reduced, mini-superspace models.
The goal of this paper is to complete the program in the simplest of models by using a combination of analytical and numerical methods. The resulting theory will enable us to answer, in the context of these models, the questions raised in the beginning of this section. Specifically, we will show from first principles that: i) a classical space-time continuum is an excellent approximation till very early times; ii) the singularity is resolved in the sense that a complete set of *Dirac observables* on the *physical* Hilbert space remains well-defined throughout the evolution; iii) the big-bang is replaced by a big-bounce in the *quantum* theory; iv) there is a large classical universe on the ‘other side’, and, v) the evolution bridging the two classical branches is deterministic, thanks to the background independence and non-perturbative methods. While the paper is primarily concerned with basic conceptual and computational issues, our constructions also provide some tools for a more systematic analysis of phenomenological questions. Finally, our approach can be used in more general models. In particular, our constructions can be used also for anisotropic models and for models in which the scalar field has a potential, although certain conceptual subtleties have to be handled carefully and, more importantly, the subsequent numerical analysis is likely to be significantly more complicated. Nonetheless, in a rather well defined sense, these constructions provide a foundation from which one can systematically analyze the Planck regime in LQC well beyond the specific model discussed in detail.
The main ideas of our analysis can be summarized as follows. First, our Hamiltonian constraint is self-adjoint on the kinematical (more precisely, auxiliary) Hilbert space. Second, we use the scalar field as ‘internal time’. In the classical theory of $k\!=\! 0$ models with a massless scalar field $\phi$, the scale factor $a$ as well as $\phi$ are monotonic functions of time in any given solution. In the $k\! =\! 1$ case, on the other hand, since the universe recollapses, only $\phi$ has this property. Therefore, it seems more natural to use $\phi$ as ‘internal time’, which does not refer to space-time coordinates or any other auxiliary structure. It turns out that $\phi$ is well-suited to be *‘emergent time’* also in the quantum theory. Indeed, our self-adjoint Hamiltonian constraint is of the form [\^2]{}[\^2]{} = - where ${\Theta}$ does not involve $\phi$ and is a positive, self-adjoint, difference operator on the auxiliary Hilbert space of quantum geometry. Hence, the quantum Hamiltonian constraint can be readily regarded as a means to ‘evolve’ the wave function with respect to $\phi$. Moreover, this interpretation makes the group averaging procedure similar to that used in the quantization of a ‘free’ particle in a static space-time, and therefore conceptually more transparent. Third, we can carry out the group averaging and arrive at the physical inner product. Fourth, we identify complete sets of Dirac observables on the physical Hilbert space. One such observable is provided by the momentum $\h{p}_\phi$ conjugate to the scalar field $\phi$ and a set of them by $\hat{a}|_{\phi_o}$, the scale factor at the ‘instants of emergent time’ $\phi_o$. [^1] Fifth, we construct semi-classical states which are peaked at values of these observables at late times, when the universe is large. Finally we ‘evolve’ them backwards using the Hamiltonian constraint using the adaptive step, 4th order Runge-Kutta method. The numerical tools are adequate to keep track not only of how the peak of the wave function ‘evolves’ but also of fluctuations in the Dirac observables in the course of ‘evolution’. A variety of numerical simulations have been performed and the existence of the bounce is robust.
The paper is organized as follows. In section \[s2\] we summarize the framework of LQC in the homogeneous, isotropic setting, keeping the matter field generic. We first summarize the kinematical structure [@abl], highlighting the origin of the qualitative difference between the theory and LQC already at this level. We then provide a self-contained and systematic derivation of the quantum Hamiltonian constraint, spelling out the underlying assumptions. In section \[s3\] we restrict the matter field to be a massless scalar field, present the detailed theory and show that the singularity is not resolved. The choice of a massless scalar field in the detailed analysis was motivated by two considerations. First, it makes the basic constructions easier to present and the numerical simulations are substantially simpler. More importantly, in the massless case *every* classical solution is singular, whence the singularity resolution by LQC is perhaps the clearest illustration of the effects of quantum geometry. In section \[s4\] we discuss this model in detail within the LQC framework, emphasizing the dynamical differences from the theory. Specifically, we present the general solution to the quantum constraint, construct the physical Hilbert space and introduce operators corresponding to Dirac observables. In section \[s5\] we present the results of our numerical simulations in some detail. Solutions to the Hamiltonian constraint are constructed using two different methods, one using a Fast Fourier Transform and another by ‘evolving’ initial data using the adaptive step, 4th order Runge-Kutta method. To further ensure the robustness of conclusions, the initial data (at late ‘times’ when the universe is large) is specified in three different ways, reflecting three natural choices in the construction of semi-classical states. In all cases, the classical big-bang is replaced by a quantum big-bounce and the two ‘classical branches’ are joined by a deterministic quantum evolution. Section \[s6\] compares and contrasts the main results with those in the literature.
Issues which are closely related to (but are not an integral part of) the main results are discussed in three appendices. Appendix \[a1\] is devoted to certain heuristics on effective equations and uncertainty relations which provide a physical intuition for ‘mechanisms’ underlying certain constructions and results. In Appendix \[a2\] we discuss technical aspects of numerical simulations which are important but whose inclusion in the main text would have broken the flow of the argument. Finally in Appendix \[a3\] we present an alternate physical Hilbert space which can be constructed by exploiting certain special features of LQC which are not found in the general setting of constrained systems. This space is more closely related to the theory and could exist also in more general contexts. Its existence illustrates an alternate way to address semi-classical issues which may well be useful in full LQG.
LQC in the $k\!=\! 0$ homogeneous, isotropic setting {#s2}
====================================================
This section is divided into three parts. To make the paper self-contained, in the first we provide a brief summary of the kinematical framework, emphasizing the conceptual structure that distinguishes LQC from the theory. In the second, we introduce the self-adjoint Hamiltonian constraints and their limits. By spelling out the underlying assumptions clearly, we also pave the way for the construction of a more satisfactory Hamiltonian constraint in [@aps3].[^2] In the third part, we first list issues that must be addressed to extract physics from this general program and then spell out the model considered in the rest of the paper.
Kinematics {#s2.1}
----------
### Classical phase space {#s2.1.1}
In the $k\! = \! 0$ case, the spatial 3-manifold $\M$ is topologically $\R^3$, endowed with the action of the Euclidean group. This action can be used to introduce on $\M$ a fiducial, flat metric $\q_{ab}$ and an associated constant orthonormal triad $\e^a_i$ and co-triad $\w_a^i$. In full general relativity, the gravitational phase space consists of pairs $(A_a^i,\, E^a_i)$ of fields on $\M$, where $A_a^i$ is an $\SU(2)$ connection and $E^a_i$ an orthonormal triad (or moving frame) with density weight 1 [@alrev]. As one would expect, in the present homogeneous, isotropic context, one can fix the gauge and spatial diffeomorphism freedom in such a way that $A_a^i$ has only one independent component, $\tilde{c}$, and $E^a_i$, only one independent component, $\tilde{p}$: \[ss\] A = \^[i]{} \_i, = \_[i]{}\^i ,where $\tilde{c}$ and $\tilde{p}$ are constants and the density weight of ${E}$ has been absorbed in the determinant of the fiducial metric.[^3] Denote by $\Gamma^S_{\rm grav}$ the subspace of the gravitational phase space ${\Gamma_{\rm grav}}$ defined by (\[ss\]). Because $\M$ is non-compact and our fields are spatially homogeneous, various integrals featuring in the Hamiltonian framework of the full theory diverge. However, the presence of spatial homogeneity enables us to bypass this problem in a natural fashion: Fix a ‘cell’ $\V$ adapted to the fiducial triad and restrict all integrations to this cell. (For simplicity, we will assume that this cell is cubical with respect to $\q_{ab}$.) Then the gravitational symplectic structure $\Omega_{\rm grav}$ on $\Gamma_{\rm grav}$ and fundamental Poisson brackets are given by [@abl]: \^S\_[grav]{} = , {, } = [8G]{}[3V\_o]{} where $V_o$ is the volume of $\V$ with respect to the auxiliary metric $\q_{ab}$. Finally, there is a freedom to rescale the fiducial metric $\q_{ab}$ by a constant and the canonical variables $\tilde{c}, \tilde{p}$ fail to be invariant under this rescaling. But one can exploit the availability of the elementary cell $\V$ to eliminate this additional ‘gauge’ freedom. For, c = V\_o\^[[1]{}[3]{}]{} p = V\_o\^[[2]{}[3]{}]{} are independent of the choice of the fiducial metric $\q_{ab}$. Using $(c,\, p)$, the symplectic structure and the fundamental Poisson bracket can be expressed as \[qcsym\] \^S\_[grav]{} = c p {c, [p]{}} = [8G]{}[3]{} . Since these expressions are now independent of the volume $V_o$ of the cell ${\cal V}$ and make no reference to the fiducial metric $\q_{ab}$, it is natural to regard the pair $(c,p)$ *as the basic canonical variables* on $\Gamma^S_{\rm grav}$. In terms of $p$, the physical triad and co-triad are given by: \[e1\] e\^a\_i = ([sgn]{}p) |p|\^[-]{} (V\_o\^ \^a\_i), \_a\^i = ([sgn]{} [p]{}) |p|\^ (V\_o\^[- ]{} \_a\^i) The function ${\rm sgn}\, p$ arises because in connection-dynamics the phase space contains triads with both orientations, and since we have fixed a fiducial triad $\e^a_i$, the orientation of the physical triad $e^a_i$ is captured in the sign of ${p}$. (As in the full theory, we also allow degenerate co-triads which now correspond to ${p}=0$, for which the triad vanishes.)
Finally, note that although we have introduced an elementary cell $\V$ and restricted all integrals to this cell, the spatial topology is still $\R^3$ and not $\mathbb{T}^3$. Had the topology been toroidal, connections with non-trivial holonomy around the three circles would have enlarged the configuration space and the phase space would then have inherited additional components.
### Quantum kinematics {#s2.1.2}
To construct quantum kinematics, one has to select a set of elementary observables which are to have unambiguous operator analogs. In non-relativistic quantum mechanics they are taken to be $x,p$. One might first imagine using $c,p$ in their place. This would be analogous to the procedure adopted in the theory. However, unlike in geometrodynamics, LQG now provides a well-defined kinematical framework for full general relativity, without any symmetry reduction. Therefore, in the passage to the quantum theory, we do not wish to treat the reduced theory as a system in its own right but follow the procedure used in the full theory. There, the elementary variables are: i) holonomies $h_e$ defined by the connection $A_a^i$ along edges $e$, and, ii) the fluxes of triads $E^a_i$ across 2-surfaces $S$. In the present case, this naturally suggests that we use: i) the holonomies $h^{(\mu)}_k$ along straight edges $(\mu\, \e^a_k)$ defined by the connection $A = (c/\sqrt[3]{V_o})\, \w^i\tau_i$ , and ii) the momentum $p$ itself [@abl]. Now, the holonomy along the $k$th edge is given by: h\^[()]{}\_k = [c]{}[2]{} + 2 [c]{}[2]{} \_k \[hol\]where $\mathbb{I}$ is the identity 2X2 matrix. Therefore, the elementary configuration variables can be taken to be $\exp (i\mu
c/2)$ of $c$. These are called *almost periodic* functions of $c$ because $\mu$ is an arbitrary real number (positive if the edge is oriented along the fiducial triad vector $\e^a_i$, and negative if it is oriented in the opposite direction). The theory of these functions was first analyzed by the mathematician Harold Bohr (who was Niels’ brother).
Thus, in LQC one takes $e^{i\mu c/2}$ and $p$ as the elementary classical variables which are to have unambiguous operator analogs. They are closed under the Poisson bracket on $\Gamma^S_{\rm grav}$. Therefore, as in quantum mechanics, one can construct an abstract $\star$-algebra $\a$ generated by $e^{i\mu
c/2}$ and $p$, subject to the canonical commutation relations. The main task in quantum kinematics is to find the appropriate representation of $\a$.
In this task, one again follows the full theory. There, surprisingly powerful theorems are now available: By appealing to the background independence of the theory, one can select an irreducible representation of the holonomy-flux algebra *uniquely* [@lost; @cf]. The unique representation was constructed already in the mid-nineties and is therefore well understood [@al2; @jb1; @mm; @al3; @al4; @almmt]. In this representation, there are well defined operators corresponding to the triad fluxes and holonomies, but the connection itself does not lead to a well-defined operator. Since one follows the full theory in LQC, the resulting representation of $\a$ also has well-defined operators corresponding to $p$ and almost periodic functions of $c$ (and hence, holonomies), but there is no operator corresponding to $c$ itself.
The Hilbert space underlying this representation is $\Hkg =
L^2(\R_{\rm Bohr}, d\mu_{\rm Bohr})$, where $\R_{\rm Bohr}$ is the Bohr compactification of the real line and $\mu_{\rm Bohr}$ is the Haar measure on it. It can be characterized as the Cauchy completion of the space of continuous functions $f(c)$ on the real line with finite norm, defined by: ||f||\^2 = \_[D ]{} [1]{}[2D]{} \_[-D]{}\^D |[f]{} f dc . An orthonormal basis in $\Hkg$ is given by the almost periodic functions of the connection, $N_{(\mu)}(c) := e^{i\mu c/2}$. (The $N_{(\mu)} (c)$ are the LQC analogs of the spin network functions in full LQG [@rs2; @jb2]). They satisfy the relation N\_[()]{}|N\_[(\^)]{} e\^[[ic]{}[2]{}]{}|e\^[[i\^c]{}[2]{}]{} = \_[, \^]{} . Note that, although the basis is of the plane wave type, the right side has a Kronecker delta, rather than the Dirac distribution. Therefore a generic element $\Psi$ of $\Hkg$ can be expanded as a *countable sum* $\Psi (c) = \sum_k\, \alpha_k N_{(\mu_k)}$ where the complex coefficients $\alpha_k$ are subject to $\sum_k
|\alpha_k|^2 < \infty$. Consequently, the intersection between $\Hkg$ and the more familiar Hilbert space $L^2(\R, dc)$ of quantum mechanics (or of the theory) consists only of the zero function. Thus, already at the kinematic level, the LQC Hilbert space is *very* different from that used in the theory.
The action of the fundamental operators, however, is the familiar one. The configuration operator acts by multiplication and the momentum by differentiation: \_[()]{}(c)= [ic]{}[2]{} (c), (c) = -i [8\^2]{}[3]{} [c]{} where, as usual, $\lp^2 =G\hbar$. The first of these provides a 1-parameter family of unitary operators on $\Hkg$ while the second is self-adjoint.
It is often convenient to use the Dirac bra-ket notation and set $e^{i\mu c/2} = \langle c|\mu\rangle$. In this notation, the eigenstates of $\hat{p}$ are simply the basis vectors $|\mu\rangle$: | = [8\^2]{}[6]{} | . Finally, since the operator $\hat{V}$ representing the volume of the elementary cell $\V$ is given by $\hat{V} =
|\hat{p}|^{{\frac}{3}{2}}$, the basis vectors are also eigenstates of $\hat{V}$: | = ([8]{}[6]{} ||)\^[[3]{}[2]{}]{} \^3 | . The algebra $\a$, of course, also admits the familiar representation on $L^2(\R, dc)$. Indeed, as we will see in section \[s3\], this is precisely the ‘Schrödinger representation’ underlying the theory. The LQC representation outlined above is unitarily inequivalent. This may seem surprising at first in the light of the von-Neumann uniqueness theorem of quantum mechanics. The LQG representation evades that theorem because there is no operator $\hat{c}$ corresponding to the connection component $c$ itself. Put differently, the theorem requires that the unitary operators $\hat{N}_{(\mu)}$ be *weakly continuous* in $\mu$, while our operators on $\Hkg$ are not. (For further discussion, see [@afw].)
The Hamiltonian constraint {#s2.2}
--------------------------
### Strategy {#s2.2.1}
Because of spatial flatness, the gravitational part of the Hamiltonian constraint of full general relativity simplifies and assumes the form: C\_[grav]{} = -\^[-2]{}\_[V]{}d\^3x N \_[ijk]{}F\_[ab]{}\^i e\^[-1]{} [E\^[aj]{}E\^[bk]{}]{} where $e:=\sqrt{|\det E|}$, and where we have restricted the integral to our elementary cell $\V$. Because of spatial homogeneity the lapse $N$ is constant and we will set it to one.
To obtain the corresponding constraint operator, we need to first express the integrand in terms of our elementary phase space functions $h_e, p$ and their Poisson brackets. The term involving triad can be treated using the Thiemann strategy [@ttbook; @tt] : \[cotriad\] \_[ijk]{} e\^[-1]{}E\^[aj]{}E\^[bk]{} = \_k [([sgn]{}p)]{}[2G\_o V\_o\^[[1]{}[3]{}]{}]{} \^o\^[abc]{} \^k\_c (h\_k\^[(\_o)]{} {h\_k\^[(\_o)]{}\^[-1]{}, V} \_i) where $h_k^{(\mu_0)}$ is the holonomy along the edge parallel to the $k$th basis vector of length $\mu_o V_o^{1/3}$ with respect to $\q_{ab}$, and $V= |p|^{3/2}$ is the volume function on the phase space. While the right side of (\[cotriad\]) involves $\mu_o$, it provides an exact expression for the left side which is independent of the value of $\mu_o$.
For the field strength $F_{ab}^i$, we use the standard strategy used in gauge theories. Consider a square $\Box_{ij}$ in the $i$-$j$ plane spanned by two of the triad vectors $\e^a_i$, each of whose sides has length $\mu_o V_o^{1/3}$ with respect to the fiducial metric $\q_{ab}$. Then, ‘the $ab$ component’ of the curvature is given by: \[F\] F\_[ab]{}\^k = -2\_[\_o 0]{} ([h\^[(\_o)]{}\_[\_[ij]{}]{}-1 ]{}[\_o\^2V\_o\^[2/3]{}]{} ) \^k \^i\_a \^j\_b where the holonomy $h^{(\mu_o)}_{\Box_{ij}}$ around the square $\Box_{ij}$ is just the product of holonomies (\[hol\]) along the four edges of $\Box_{ij}$: h\^[(\_o)]{}\_[\_[ij]{}]{}=h\_i\^[(\_o)]{} h\_j\^[(\_o)]{} (h\_i\^[(\_o)]{})\^[-1]{} (h\_j\^[(\_o)]{})\^[-1]{} . By adding the two terms and simplifying, $C_{\rm grav}$ can be expressed as: \[reg\] C\_[grav]{} &=& \_[\_o 0]{} C\^[(\_o)]{}\_[grav]{},\
C\^[(\_o)]{}\_[grav]{} &=& - \_[ijk]{} \^[ijk]{} [Tr]{} (h\_i\^[(\_o)]{}h\_j\^[(\_o)]{} h\_i\^[(\_o)-1]{}h\_j\^[(\_o)-1]{} h\_k\^[(\_o)]{}{h\_k\^[(\_o)-1]{},V}) Since $C^{(\mu_o)}_{\rm grav}$ is now expressed entirely in terms of holonomies and $p$ and their Poisson bracket, it is straightforward to write the corresponding quantum operator on $\Hkg$: \[qh1\] \_[grav]{}\^[(\_o)]{} &=& \_[ijk]{}\^[ijk]{}(\_i\^[(\_o)]{} \_j\^[(\_o)]{} \_i\^[(\_o)-1]{}\_j\^[(\_o)-1]{} \_k\^[(\_o)]{}\[\_k\^[(\_o)-1]{},\])\
&=& [24i([sgn]{}p)]{}[8 \^3\_o\^3\^2]{} \^2\_o c ( - ) . However, the limit $\mu_o \rightarrow 0$ of this operator does not exist. This is not accidental; had the limit existed, there would be a well-defined operator directly corresponding to the curvature $F_{ab}^i$ and we know that even in full LQG, while holonomy operators are well defined, there are no operators corresponding to connections and curvatures. This feature is intimately intertwined with the quantum nature of Riemannian geometry of LQG. The viewpoint in LQC is that the failure of the limit to exist is a reminder that there is an underlying *quantum* geometry where eigenvalues of the area operator are *discrete*, whence there is a smallest non-zero eigenvalue, $\Delta$, i.e., an *area gap* [@rs1; @al5]. Thus, quantum geometry is telling us that it is physically incorrect to let $\mu_o$ go to zero because that limit corresponds to shrinking the area enclosed by loops $\Box_{ij}$ to zero. Rather, *the ‘correct’ field strength operator in the quantum theory should be in fact non-local, given by setting $\mu_o$ in (\[F\]) to a non-zero value, appropriately related to $\Delta$.* In quantum theory, we simply can not force locality by shrinking the loops $\Box_{ij}$ to zero area. In the classical limit, however, we are led to ignore quantum geometry, and recover the usual, local field $F_{ab}$ [@abl; @jw].
There are two ways of implementing this strategy. In this paper, we will discuss the one that has been used in the literature [@mbrev; @abl; @ab1]. The second strategy will be discussed in [@aps3]; it has a more direct motivation and is more satisfactory in semi-classical considerations especially in presence of a non-zero cosmological constant.
In the above discussion, $\mu_o$ enters through holonomies $h_k^{(\mu_o)}$. Now, it is straightforward to verify that (every matrix element of) $h_k^{(\mu_o)}$ is an eigenstate of the area operator $\widehat{Ar} = \widehat{|p|}$ (associated with the face of the elementary cell $\V$ orthogonal to the $k$th direction): h\_k\^[(\_o)]{}(c) = ([8\_o]{}[6]{} \^2) h\_k\^[(\_o)]{}(c).In the first strategy, one fixes $\mu_o$ by demanding that this eigenvalue be $\Delta = 2\sqrt{3}\pi \g \lp^2$, the area gap [@alrev; @al5], so that $\mu_o = 3\sqrt{3}/{2}$.
To summarize, by making use of some key physical features of quantum geometry in LQG, we have arrived at a ‘quantization’ of the classical constraint $C_{\rm grav}$: \[qh2\] \_[grav]{} = \_[grav]{}\^[(\_o)]{}\_[\_o = [3]{}[2]{}]{}
There are still factor ordering ambiguities which will be fixed in the section \[s2.2.2\].
We will conclude our summary of the quantization strategy by comparing this construction with that used in full LQG [@tt; @alrev; @ttbook]. As one would expect, the curvature operator there is completely analogous to (\[F\]). However in the subsequent discussion of the Hamiltonian constraint certain differences arise: While the full theory is diffeomorphism invariant, the symmetry reduced theory is not because of gauge fixing carried out in the beginning of section \[s2.1.1\]. (These differences are discussed in detail in sections 4 and 5 of [@abl].) As a result, in the dynamics of the reduced theory, we have to ‘parachute’ the area gap $\Delta$ from the full theory. From physical considerations, this new input seems natural and the strategy is clearly viable since that the constraint operator has the correct classical limit. However, so far there is no systematic procedure which leads us to the dynamics of the symmetry reduced theory from that of the full theory. This is not surprising because the Hamiltonian constraint in the full theory has many ambiguities and there isn’t a canonical candidate that stands out as being the most satisfactory. The viewpoint in LQC is rather that, at this stage, it would be more fruitful to study properties of LQC constraints such as the one introduced in this section and in [@aps3], and use the most successful of them as a guide to narrow down the choices in the full theory.
### Constraint operators and their properties {#s2.2.2}
It is easy to verify that, although the classical constraint function $C_{\rm grav}$ we began with is real on the phase space $\Gamma^S_{\rm grav}$, the operator $\hat{C}_{\rm grav}^{(\mu_o)}$ is not self-adjoint on $\Hkg$. This came about just because of the standard factor ordering ambiguities of quantum mechanics and there are two natural re-orderings that can rectify this situation.
First, we can simply take the self-adjoint part of (\[qh2\]) on $\Hkg$ and use it as the gravitational part of the constraint: \[qh3\] \_[grav]{}\^ = [1]{}[2]{}It is convenient to express its action on states $\Psi(\mu) :=
\langle\Psi|\mu\rangle$ in the $\mu$ —or the triad/geometry— representation. The action is given by \[qh4\] \_[grav]{}\^ () = f\^\_+ (+ 4\_o) + f\^\_o () + f\^\_- (- 4\_o)where the coefficients $f_{\pm}^\prime, f_o^\prime$ are functions of $\mu$: f\^\_o () &=& - | |+\_o|\^-|-\_o|\^ |\
f\^\_+ () &=& - (f\^\_o()+f\^\_o(+4\_o))\
f\^\_- () &=& - (f\^\_o()+f\^\_o(-4\_o)) By construction, this operator is self-adjoint and one can show that it is also bounded above (in particular, $\langle\Psi,\,
\hat{C}_{\rm grav}^{\prime}\, \Psi\rangle < \sqrt{\pi/3}\,\lp\,
(\gamma \mu_o)^{-\frac{3}{2}}(5^{\frac{3}{2}}-3^{\frac{3}{2}})
\langle\Psi|\Psi\rangle$). Next, consider the *‘parity’* operator $\Pi$ defined by $\Pi \Psi(\mu) := \Psi(-\mu)$. It corresponds to the flip of the orientation of the triad and thus represents a large gauge transformation. It will play an important role in section \[s4\]. Here we note that the functions $f^\prime_\pm, f^\prime_o$ are such that the constraint operator $\hat{C}_{\rm grav}^{\prime}$ commutes with $\Pi$: = 0 Finally, for $\mu \gg \mu_o \equiv 3\sqrt{3}/2$, this difference operator can be approximated, in a well-defined sense, by a second order differential operator (a connection-dynamics analog of the operator of geometrodynamics). Since $\hat{C}_{\rm
grav}^{\prime}$ is just the self-adjoint part of the operator used in [@abl], we can directly use results of section 4.2 of that paper. Set $p = 8\pi \g\mu \lp^2/6$ and consider functions $\Psi(p)$ which, together with their first four derivatives are bounded. On these functions, we have \[wdw1\] \_[grav]{}\^ (p) [64\^2]{}[3]{} \^4 \[ + [\^2]{}[p\^2]{} \] (p) =: \_[grav]{}\^[[WDW]{}]{} (p)where $\approx$ stands for equality modulo terms of the order $O(\mu_o)$. That is, had we left $\mu_o$ as a free parameter, the equality would hold in the limit $\mu_o \rightarrow 0$. This is the limit in which the area gap goes to zero, i.e., quantum geometry effects can be neglected.
Let us now turn to the second natural factor ordering. The form of the expression (\[qh1\]) of $\hat{{C}}_{\rm grav}$ suggests [@wk] that we simply ‘re-distribute’ the $\sin^2 \mu_o c$ term in a symmetric fashion. Then this factor ordering leads to the following self-adjoint gravitational constraint: \[qh5\]\_[grav]{} &=& [24i ([sgn]{} p)]{}[8 \^3\_o\^3\^2]{} \_[\_o = [3]{}[2]{}]{}\
& =: & \_[\_o = [3]{}[2]{}]{} *For concreteness in most of this paper we will work with this form of the constraint and, for notational simplicity, unless otherwise stated set $\mu_o = 3\sqrt{3}/2$.* However, numerical simulations have been performed also using the constraint $\hat{C}_{\rm grav}^{\prime}$ of Eq. (\[qh3\]) and the results are robust.
Properties of $\hat{C}_{\rm grav}$ which we will need in this paper can be summarized as follows. First, the eigenbasis $|\mu\rangle$ of $\hat{p}$ diagonalizes the operator $\hat{A}$. Therefore in the $p$-representation, $\hat{A}$ acts simply by multiplication. It is easy to verify that \[A\] () = - 2 [(\_o\^2)\^[[3]{}[2]{}]{}]{} | |+\_o|\^[[3]{}[2]{}]{} - |-\_o|\^[[3]{}[2]{}]{} | ()By inspection, $\hat{A}$ is self-adjoint and negative definite on $\Hkg$. The form of $\hat{C}_{\rm grav}$ now implies that it is also self-adjoint and negative definite. Its action on states $\Psi(\mu)$ is given by \[qh6\] \_[grav]{} () = f\_+ (+ 4\_o) + f\_o () + f\_- (- 4\_o)where the coefficients $f_{\pm}, f_o$ are again functions of $\mu$: f\_+ () &=& [1]{}[2]{} [(\_o\^2)\^[[3]{}[2]{}]{}]{} | |+3\_o|\^[[3]{}[2]{}]{} - |+\_o|\^[[3]{}[2]{}]{} |\
f\_- () &=& f\_+(-4\_o)\
f\_o () &=& -f\_+() - f\_-() . It is clear from Eqs (\[qh5\]) and (\[A\]) that $\hat{C}_{\rm
grav}$ commutes with the parity operator $\Pi$ which flips the orientations of triads: =0 .Finally, the ‘continuum limit’ of the difference operator $\hat{C}_{\rm grav}$ yields a second order differential operator. Let us first set () = f\_+(- 2\_o) \[(+2\_o) - (-2\_o)\] Then, \_[grav]{}() = (+2\_o) - (-2\_o) Therefore, if we again set $p = 8\pi \g\mu \lp^2/6$ and consider functions $\Psi(p)$ which, together with their first four derivatives are bounded we have \[wdw2\] \_[grav]{} (p) [128\^2]{}[3]{} \^4 \[[p]{} [p]{}\] =: \_[grav]{}\^[[WDW]{}]{} (p)where $\approx$ again stands for equality modulo terms of the order $O(\mu_o)$. That is, in the limit in which the area gap goes to zero, i.e., quantum geometry effects can be neglected, the difference operator reduces to a type differential operator. (As one might expect, the limiting operator is independent of the Barbero-Immirzi parameter $\g$.) We will use this operator extensively in section \[s3\].
In much of computational physics, especially in numerical general relativity, the fundamental objects are differential equations and discrete equations are introduced to approximate them. In LQC the situation is just the opposite. The physical fundamental object is now the discrete equation (\[qh6\]) with $\mu_o= 3\sqrt{3}/2$. The differential equation is the approximation. The leading contribution to the difference between the two —i.e., to the error — is of the form $O(\mu_o)^2 \Psi^{''''}$ where the second term depends on the wave function $\Psi$ under consideration. Therefore, the approximation is not uniform. For semi-classical states, $\Psi^{''''}$ can be large, of the order of $10^{16}/\mu^4$ in the examples considered in section \[s5\]. In this case, the continuum approximation can break down already at $\mu \sim 10^4$.
As one might expect the two differential operators, $\hat{C}_{\rm
grav}^{\prime{\rm WDW}}$ and $\hat{C}_{\rm grav}^{{\rm WDW}}$ differ only by a factor ordering: \[wdwdiff\] (\_[grav]{}\^ - \_[grav]{}\^[[WDW]{}]{}) (p) = - [16\^2]{}[3]{}\^4 |p|\^[-[3]{}[2]{}]{} (p) .
**Remark:** As noted in section \[s2.2.1\], the ‘continuum limit’ $\mu_o \rightarrow 0$ of any of the quantum constraint operators of LQC does not exist on $\Hkg$ because of the quantum nature of underlying geometry. To take this limit, one has to work in the setting in which the quantum geometry effects are neglected, i.e., on the type Hilbert space $L^2(\R, dc)$. On this space, operators $\hat{C}_{\rm grav}^{\prime}$, $\hat{C}_{\rm
grav}$, $\hat{C}_{\rm grav}^{\rm WDW}$ and $\hat{C}_{\rm
grav}^{\prime{\rm WDW}}$ are all densely defined and the limit can be taken on a suitable dense domain.
Open issues and the model {#s2.3}
-------------------------
In section \[s2.1\] we recalled the kinematical framework used in LQC and in \[s2.2\] we extended the existing results by analyzing two self-adjoint Hamiltonian constraint operators in some detail. Physical states $\Psi(\mu, \phi)$ can now be constructed as solutions of the Hamiltonian constraint: \[qc1\] (\_[grav]{} + \_[matt]{}) (, ) =0where $\phi$ stands for matter fields. Given any matter model, one could solve this equation numerically. However, generically, the solutions would not be normalizable in the total kinematic Hilbert space $\Hk$ of gravity plus matter. Therefore, although section \[s2.2\] goes beyond the existing literature in LQC, one still can not calculate expectation values, fluctuations and probabilities —i.e., extract physics— knowing only these solutions.
To extract physics, then, we still have to complete the following tasks:
In the classical theory, seek a dynamical variable which is monotonically increasing on all solutions (or at least ‘large’ portions of solutions). Attempt to interpret the Hamiltonian constraint (\[qc1\]) as an ‘evolution equation’ with respect to this ‘internal time’. If successful, this strategy would provide an ‘emergent time’ in the background independent quantum theory. Although it is possible to extract physics under more general conditions, physical interpretations are easier and more direct if one can locate such an emergent time.\
Introduce an inner product on the space of solutions to (\[qc1\]) to obtain the physical Hilbert space $\Hp$. The fact that the orientation reversal induced by $\Pi$ is a large gauge transformation will have to be handled appropriately.\
Isolate suitable Dirac observables in the classical theory and represent them by self-adjoint operators on $\Hp$.\
Use these observables to construct physical states which are semi-classical at ‘late times’, sharply peaked at a point on the classical trajectory representing a large classical universe.\
‘Evolve’ these states using (\[qc1\]). Monitor the mean values and fluctuations of Dirac observables. Do the mean values follow a classical trajectory? Investigate if there is a drastic departure from the classical behavior. If there is, analyze what replaces the big-bang.
In the rest of the paper, we will carry out these tasks in the case when matter consists of a zero rest mass scalar field. In the classical theory, the phase space $\Gamma^S_{\rm grav}$ is now 4-dimensional, coordinatized by $(c,p;\, \phi, p_\phi)$. The basic (non-vanishing) Poisson brackets are given by: \[pbs\] {[c]{}, [p]{}} = [8G]{}[3]{} , {, p\_} = 1 .The symmetry reduction of the classical Hamiltonian constraint is of the form \[cc\] C\_[grav]{} + C\_[matt]{} - [6]{}[\^2]{} c\^2 + 8G [p\_\^2]{}[|p|\^[[3]{}[2]{}]{}]{} = 0 .
![Classical phase space trajectories are plotted in the $\phi, p\sim\mu$ -plane. For $\mu \ge 0$, there is a branch which starts with a big-bang (at $\mu =0$) and expands out and a branch which contracts into a big crunch (at $\mu =0$). Their mirror images appear in the $\mu \le 0$ half plane.[]{data-label="f1"}](symmetric-phi-class1.eps){width="3.5in"}
Using this constraint, one can solve for $c$ in terms of $p$ and $p_\phi$. Furthermore, since $\phi$ does not enter the expression of the constraint, $p_\phi$ is a constant of motion. Therefore, each dynamical trajectory can be specified on the 2-dimensional $(p,\, \phi)$ plane. Typical trajectories are shown in Fig. 1. Because the phase space allows triads with both orientations, the variable $p$ can take both positive and negative values. At $p=0$ the physical volume of the universe goes to zero and, if the point lies on any dynamical trajectory, it is an end-point of that trajectory, depicting a curvature singularity. As the figure shows, for each fixed value of $p_\phi$, there are four types of trajectories, two in the $p \ge 0$ half plane and two in the $p\le
0$ half plane. Analytically they are given by: = [|p|]{}[|p\_|]{} + \_where $p_\star, \phi_\star$ are integration constants. These trajectories are related by a ‘parity transformation’ on the phase space which simply reverses the orientation of the physical triad. As noted before, since the metric and the scalar field are unaffected, it represents a large gauge transformation. Therefore, it suffices to focus just on the portion $p \ge 0$ of figure 1. Then for each fixed value of $p_\phi$, there are two solutions passing through any given point $(\phi_\star, p_\star)$. In one, the universe begins with the big-bang and then expands and in the other the universe contracts into a big crunch. Thus *in this model, every classical solution meets the singularity.*
Finally, we can introduce a natural set of *Dirac observables*. Since $p_\phi$ is a constant of motion, it is obviously one. To introduce others, we note that $\phi$ is a monotonic function on each classical trajectory. Furthermore, in each solution, the space-time metric takes the form $ds^2 = -dt^2
+ V_o^{-2/3}|p|(t) dS_o^2$ and the time dependence of the scalar field is given by [dt]{} = [16G p\_\^]{}[|p\_|\^[[3]{}[2]{}]{}]{} \[\] , where $p_\star,\phi_\star$ and $p_\phi^\star$ are constants. Thus, in every solution $\phi$ is a monotonic function of time and can therefore serve as a good ‘internal clock’. This interpretation suggests the existence of a natural family of Dirac observables: $
p|_{\phi_o}$, the value of $p$ at the ‘instant’ $\phi_o$. The set $(p_\phi,\, p|_{\phi_o})$ constitutes a complete set of Dirac observables since their specification uniquely determines a classical trajectory on the symmetry reduced phase space $\Gamma^{S}$, i.e., a point in the reduced phase space $\tilde\Gamma^{S}$. While the interpretation of $\phi$ as ‘internal time’ motivates this construction and, more generally, makes physics more transparent, it is not essential. One can do all of physics on $\tilde\Gamma^{S}$: Since physical states are represented by points in $\tilde\Gamma^{S}$ and a complete set of observables is given by $(p_\phi, p|_{\phi_o})$, one can work just with this structure.
**Remark:** In the open i.e., $k\! = \! 0$ model now under consideration, since $p$ is also monotonic along any classical trajectory, $\phi_{p_o}$ is also a Dirac observable and $p$ could also be used as an ‘internal time’. However, in LQC the expression of the gravitational part of the constraint operator makes it difficult to regard $p$ as the ‘emergent time’ in quantum theory. Moreover even in the classical theory, since the universe expands and then recollapses in the $k\! =\! 1$ case, $p$ fails to be monotonic along solutions and cannot serve as ‘internal time’ globally.
theory {#s3}
=======
The theory of the model has been analyzed in some detail within geometrodynamics (see especially [@ck]). However, that analysis was primarily in the context of a WKB approximation. More recently, the group averaging technique was used to construct the physical Hilbert space in a general cosmological context [@dm; @hm2], an elementary example of which is provided by the present model. However, to our knowledge a systematic completion of the program outlined in section \[s2.3\] has not appeared in the literature.
In this section we will construct the type quantum theory in the *connection dynamics*. This construction will serve two purposes. First it will enable us to introduce the key notions required for the completion of the program in a familiar and simpler context. Second, we will be able to compare and contrast the results of the theory and LQC in detail, thereby bringing out the role played by quantum geometry in quantum dynamics.
Emergent time and the general solution to the WDW equation {#s3.1}
----------------------------------------------------------
Recall that the phase space of the model is 4-dimensional, coordinatized by $(c,p;\, \phi,p_\phi)$ and the fundamental non-vanishing Poisson brackets are given by (\[pbs\]). The Hamiltonian constraint has the form: C\_[grav]{}+ C\_ - c\^2 + 8G [p\_\^2]{}[|p|\^[[3]{}[2]{}]{}]{} = 0 . To make comparison with the standard geometrodynamical theory, it is most convenient to work in the $p,\phi$ representation. Then, the kinematic Hilbert space is given by $\Hkwdw =L^2(\R^2, \dd p\, \dd\phi)$. Operators $\hat{p},
\hat{\phi}$ operate by multiplication while $\hat{c}$ and $\hat{p}_\phi$ are represented as: = i [8G]{}[3]{} \_= -iNote that, while in geometrodynamics the scale factor is restricted to be non-negative, here $p$ ranges over the entire real line, making the specification of the Hilbert space and operators easier.
To write down the quantum constraint operator, we have to make a choice of factor ordering. Since our primary motivation behind the introduction of the theory is to compare it with LQC, it is most convenient to use the factor ordering that comes from the continuum limit (\[wdw2\]) of the constraint operator of LQC. Then, the equation becomes: \[wdw3\] [2]{}[3]{} (8G)\^2 [p]{}[p]{} = [8G\^2]{} [\^2]{}[\^2]{} =: 8G \^2 (p) [\^2]{}[\^2]{} ,where we have denoted the eigenvalue $|p|^{-{\frac}{3}{2}}$ of $\widehat{|p|^{-{\frac}{3}{2}}}$ by $\ub{B}(p)$ to facilitate later comparison with LQC.[^4] The operator on the left side of this equation is self-adjoint on $L^2(\R, dp)$ and the equation commutes with the orientation reversal operator $\Pi \Psi(p,\phi) = \Psi(-p, \phi)$ representing a large gauge transformation. Thus, if $\Psi$ is a solution to Eq. (\[wdw3\]), so is $\Psi(-p, \phi)$.
For a direct comparison with LQC, it is convenient to replace $p$ with $\mu$ defined by $p = ({8\pi \g G\hbar}/{6})\, \mu$. Then, (\[wdw3\]) becomes: \[wdw4\][\^2 ]{}[\^2]{} &=& [16G]{}[3]{} \[()\]\^[-1]{}\
&=:& - where we have recast the equation in such a way that operators involving only $\phi$ appear on the left side and operators involving only $\mu$ appear on the right. The equation now has the same form as the Klein-Gordon equation in a static space-time, $\phi$ playing the role of time and $\ul{\Theta}$ of the (elliptic operator constructed from the norm of the Killing field and the) spatial Laplacian. Thus, the form of the quantum Hamiltonian constraint is such that $\phi$ can be interpreted as *emergent time* in the quantum theory. In this factor ordering of the constraint operator, which emerged in the continuum limit (\[wdw2\]) of LQC, it is not as convenient to regard $p \sim \mu$ as emergent time.
Physical states will be suitably regular solutions to (\[wdw4\]). Since $\Pi$ is a large gauge transformation, we can divide physical states into eigenspaces of $\hat{P}$. Physical observables will preserve each eigenspace. Since $\Pi^2 =1$, there are only two eigenspaces, one representing the symmetric sector and the other, anti-symmetric. Since the standard theory deals with metrics, it is completely insensitive to the orientation of the triad. Therefore, it is natural to work with the symmetric sector. Thus, *the physical Hilbert space will consist of suitably regular solutions $\Psi(\mu, \phi)$ to (\[wdw4\]) which are symmetric under $\mu \rightarrow -\mu$*.
The mathematical similarity with the Klein Gordon equation in static space-times immediately suggests a strategy to obtain the general solution of (\[wdw4\]). We first note that $(\dd/\dd\mu) \, \sqrt{\mu}\, (\dd/\dd \mu)$ is a negative definite, self-adjoint operator on $L^2_S(\R, d\mu)$, the symmetric sector of $L^2(\R, d\mu)$. Therefore, $\ul{\Theta}$ is a positive definite, self-adjoint operator on $L^2_S(\R, \ub{B}(\mu)
d\mu)$. Its eigenfunctions provide us with an orthonormal basis. It is easy to verify that the eigenvectors $\ub{e}_k(\mu)$ can be labelled by $k\in \R$ and are given by \[eq:ek\] \_k() := [[||]{}\^[[1]{}[4]{}]{}]{}[4]{} e\^[ik||]{} Their eigenvalues are given by: \_k() = \^2 \_k(), \^2 = [16G]{}[3]{}(k\^2 + [1]{}[16]{}) ; (where the factor of $1/16$ is an artifact of the factor ordering choice which we were led to from LQC). The eigenfunctions satisfy the orthonormality relations: \_[-]{}\^d() |\_k() \_[k’]{}() = (k,k’) , (where the right side is the standard Dirac distribution, not the Kronecker symbol as on $L^2(\R_{\rm Bohr}, d\mu_{\rm Bohr})$); and the completeness relation: \_[-]{}\^d() |\_k()() =0 k () =0 for any $\Psi(\mu) \in L^2_S(\R, \ub{B}(\mu) d\mu)$.
With these eigenfunctions at hand, we can now write down a ‘general’ symmetric solution to (\[wdw3\]). Any solution, whose initial data at $\phi=\phi_o$ is such that $\mu^{-1/4}
\Psi(\mu,\phi_o)$ and $\mu^{-1/4} \dot\Psi(\mu,\phi_o)$ are symmetric and lie in the Schwartz space of rapidly decreasing functions, has the form: \[sol1\] (,) = \_[-]{}\^dk \_+(k) \_k() e\^[i]{} + \_-(k) |\_k() e\^[-i]{} ,for some $\t\Psi_\pm (k)$ in the Schwartz space. Following the terminology used in the Klein-Gordon theory, if $\t\Psi_\pm(k)$ have support on the negative $k$-axis, we will say the solution is ‘outgoing’ (or ‘expanding’) while if it has support on the positive $k$ axis, it is ‘incoming’ (or, ‘contracting’). If $\t\Psi_-(k)$ vanishes, the solution will be said to be *positive frequency* and if $\t\Psi_+(k)$ vanishes, it will be said to be of *negative frequency*. Thus, every solution (\[sol1\]) admits a natural decomposition into positive and negative frequency parts. Finally we note that positive (respectively negative) frequency solutions satisfy a first order (in $\phi$) equation which can be regarded as the square-root of (\[wdw4\]): \[sch\] i[\_]{} = \_where $\sqrt{\ul{\Theta}}$ is the positive, self-adjoint operator defined via spectral decomposition of $\ul{\Theta}$ on $L^2(\R,
\ub{B}(\mu)d\mu)$. Regarding $\phi$ as time, this is just a first order Schrödinger equation with a *non-local* Hamiltonian $\sqrt{\ul{\Theta}}$. Therefore, a general ‘initial datum’ $f_\pm
(\mu)$ at $\phi=\phi_o$ can be ‘evolved’ to obtain a solution to (\[sch\]) via: \[sol2\] \_(,) = e\^[i(-\_o)]{} f\_(, \_o)
The physical sector {#s3.2}
-------------------
Solutions (\[sol1\]) are not normalizable in $\Hkwdw$ (because zero is in the continuous part of the spectrum of the operator). Our first task is to endow the space of these physical states with a Hilbert space structure. There are several possible avenues. We will begin with one that is somewhat heuristic but has direct physical motivation. The idea [@aabook; @at] is to introduce operators corresponding to a complete set of Dirac observables and select the required inner product by demanding that they be self-adjoint. In the classical theory, such a set is given by $p_\phi$ and $\mu|_{\phi_o}$. Since $\hat{p}_\phi$ commutes with the operator in (\[wdw4\]), given a (symmetric) solution $\Psi(\mu,\phi)$ to (\[wdw4\]), \[dirac1\]\_(, ) := -i is again a (symmetric) solution. So, we can just retain this definition of $\hat{p}_\phi$ from $\Hkwdw$. The Schrödinger type evolutions (\[sol2\]) enable us to define the other Dirac observable $\widehat{|\mu|_{\phi_o}}$, where the absolute value suffices because the states are symmetric under $\Pi$. Given a (symmetric) solution $\Psi(\mu,\phi)$ to (\[wdw4\]), we can first decompose it into positive and negative frequency parts $\Psi_\pm(\mu,\phi)$, freeze them at $\phi=\phi_o$, multiply this ‘initial datum’ by $|\mu|$ and evolve via (\[sol2\]): \[dirac2\] (,) = e\^[i(-\_o)]{} || \_+(,\_o) + e\^[-i(-\_o)]{} || \_-(,\_o) The result is again a (symmetric) solution to (\[wdw4\]). Now, we see that both these operators have the further property that they preserve the positive and negative frequency subspaces. Since they constitute a complete family of Dirac observables, we have *superselection*. In quantum theory we can restrict ourselves to one superselected sector. In what follows, for definiteness *we will focus on the positive frequency sector and, from now on, drop the suffix $+$*.
We now seek an inner product on positive frequency solutions $\Psi(\mu,\phi)$ to (\[wdw4\]) (invariant under the $\mu$ reflection) which makes $\hat{|p|}_\phi$ and $\hat{|\mu|}_{\phi_o}$ self-adjoint. Each of these solutions is completely determined by its initial datum $\Psi(\mu,\phi_o)$ and the Dirac observables have the following action on the datum: (,\_o) = || (,\_o), \_(,\_o) = (,\_o) . Therefore, it follows that (modulo an overall rescaling,) the unique inner product which will make these operators self-adjoint is just: \[ip1\] \_1|\_2\_[phy]{} = \_[=\_o]{} d() |\_1 \_2 (see e.g. [@aabook; @at]). Note that the inner product is conserved, i.e., is independent of the choice of the ‘instant’ $\phi=\phi_o$. Thus, *the physical Hilbert space $\Hpwdw$ is the space of positive frequency wave functions $\Psi(\mu,\phi)$ which are symmetric under $\mu$ reflection and have a finite norm (\[ip1\]).* The procedure has already provided us with a representation of our complete set of Dirac observables on this $\Hpwdw$: \[dirac3\] (,) = e\^[i(-\_o)]{}|| (,\_o), \_(,) = (,) . We will now show that the same representation of the algebra of Dirac observables can be obtained by the more systematic group averaging method [@dm; @hm2] which also brings out the mathematical inputs that go in this choice. (The two methods have been applied and compared for a non-trivially constrained system in [@lr]). Here, one first notes that the total constraint operator is self-adjoint on an auxiliary Hilbert space $\Hawdw :=
L^2_S(\R^2, \ub{B}(\mu)\dd\mu\,\dd\phi)$, (where, as before the subscript $S$ denotes restriction to functions which are symmetric under $\mu$-reflection). One must then select an appropriate dense subspace $\Phi$ of $\Hawdw$. A natural candidate is the Schwartz space of rapidly decreasing functions. One then ‘averages’ elements of $\Phi$ under the 1-parameter family group generated by the constraint operator $\hat{\ul{C}}= {\partial}_\phi^2 +
{\underline\Theta}$ to produce a solution to (\[wdw4\]): \_f(,) :&=& \_[-]{}\^ e\^[i]{} f(,)\
&=& \_[-]{}\^[k]{}[2||]{} ((k,)\_k()e\^[i]{} +(k,-) \_k()e\^[-i]{}) where to arrive at the second step we expanded $f$ in the eigenbasis of $\ul{\Theta}$ and $\hat{p}_\phi$ with $\tilde{f}$ as the coefficients. Thus, the group averaging procedure reproduces the solution (\[sol1\]) with $\t\Psi_+(k)=
\t{f}(k,\omega)/2|\omega|$ and $\t\Psi_- (k)=
\t{f}(k,-\omega)/2|\omega|$. Solutions $\Psi_f$ are regarded as ‘distributions’ or elements of the dual $\Phi^\star$ of $\Phi$ and the physical norm is given by the action of this distribution, $\Psi_f$, on the ‘test function’ $f$. However, there is some freedom in the specification of this action which generally results in seemingly different but unitarily equivalent representations of the algebra of Dirac observables. For us the most convenient choice is: ||||\^2 := \_f(f) := \_[-]{}\^\_[-]{}\^() |\_f (,) f(,)where $\Psi_f(f)$ is the action of the distribution $\Psi_f$ on the test field $f$. Then, the inner product coincides with (\[ip1\]) and the representation of the Dirac observables is the same as in (\[dirac1\]) and (\[dirac2\]). Had we chosen to drop the factor of $\sqrt{\ul{\Theta}}$ in defining the action of $\Psi_f$ on $f$, we would have obtained a unitarily equivalent representation in which the action of $\hat{\mu}|_{\phi_o}$ is more complicated.
Finally, with the physical Hilbert space and a complete set of Dirac observables at hand, we can now introduce semi-classical states and study their evolution. Let us fix a ‘time’ $\phi=\phi_o$ and construct a semi-classical state which is peaked at $p_\phi = p_\phi^\star$ and $|\mu|_{\phi_o} = \mu^\star$. We would like the peak to be at a point that lies on a large classical universe. This implies that we should choose $\mu^\star
\gg 1$ and (in the natural classical units $c$=$G$=1), $p_\phi^\star \gg \hbar$. In the closed ($k\!=\! 1$) models for example, the second condition is necessary to ensure that the universe expands out to a size much larger than the Planck scale. At ‘time’ $\phi=\phi_o$, consider the state \[sc\] (,\_o) = \_[-]{}\^dk (k) \_k() e\^[i(\_o-\^)]{}, (k) = e\^[-[(k-k\^)\^2]{}[2\^2]{}]{} . where $k^\star = -\sqrt{3/16\pi G\hbar^2}\,\, p_\phi^\star$ and $\phi^\star = -\sqrt{3/16\pi G} (\ln |\mu^\star|) + \phi_o$. It is easy to evaluate the integral in the approximation $\omega =
-\sqrt{(16\pi G/3)}\, k$ (which is justified because $k^\star \ll
-1$) and calculate mean values of the Dirac observables and their fluctuations. One finds that, as required, the state is sharply peaked at values $\mu^\star, p_\phi^\star$. The above construction is closely related to that of coherent states in non-relativistic quantum mechanics. The main difference is that the observables of interest are not $\mu$ and its conjugate momentum but rather $\mu$ and $p_\phi$ —the momentum conjugate to ‘time’, i.e., the analog of the Hamiltonian in non-relativistic quantum mechanics.
We can now ask for the evolution of this state. Does it remain peaked at the classical trajectory defined by $p_\phi =
p_\phi^\star$ passing through $\mu^\star$ at $\phi =\phi_o$? This question is easy to answer because (\[sol2\]) implies that the (positive frequency) solution $\Psi(\mu,\phi)$ to (\[wdw4\]) defined by ‘initial data’ (\[sc\]) is obtained simply by replacing $\phi_o$ by $\phi$ in (\[sc\])! Since the measure of dispersion $\sigma$ in (\[sc\]) does not depend on $\phi$, it follows that the initial state $\Psi(\mu,\phi_o)$ which is the semi-classical, representing a large universe at ‘time’ $\phi_o$ continues to be peaked at a trajectory defined by: = + \_o .This is precisely the classical trajectory with $p_\phi =
p_\phi^\star$, passing through $\mu^\star$ at $\phi =\phi_o$. This is just as one would hope during the epoch in which the universe is large. However, this holds also in the Planck regime and, in the backward evolution, the semi-classical state simply follows the classical trajectory into the big-bang singularity. (Had we worked the positive $k^\star$, we would have obtained a contracting solution and then the forward evolution would have followed the classical trajectory into the big-crunch singularity.) In this sense, the evolution does not resolve the classical singularity.
**Remark:** In the above discussion for simplicity we restricted ourselves to eigenfunctions $\ub{e}_k(\mu)$ which are symmetric under $\mu\,\rightarrow\, -\mu$ from the beginning. Had we dropped this requirement, we would have found that there is a 4-fold (rather than 2-fold) degeneracy in the eigenfunctions of $\ul{\Theta}$. Indeed, if $\theta(\mu)$ is the step function ($\theta(\mu) =0$ if $\mu < 0$ and $\, =1$ if $\mu >0$), then $\theta(\mu)\ub{e}_{|k|},\, \theta(\mu) \ub{e}_{-|k|},\,
\theta(-\mu)\ub{e}_{|k|},\, \theta(-\mu) \ub{e}_{-|k|}$ are all continuous functions of $\mu$ which satisfy the eigenvalue equation (in the distributional sense) with eigenvalue $\omega^2 =
(16\pi G/3)(k^2 + 1/16)$. This fact will be relevant in the next section.
Analytical issues in Loop quantum cosmology {#s4}
===========================================
We will now analyze the model within LQG. We will first observe that the form of the Hamiltonian constraint is such that the scalar field $\phi$ can again be used as emergent time. Since the form of the resulting ‘evolution equation’ is very similar in the theory, we will be able to construct the physical Hilbert space and Dirac observables following the ideas introduced in section \[s3.2\].
Emergent time and the general solution to the LQC Hamiltonian constraint {#s4.1}
------------------------------------------------------------------------
The quantum constraint has the form \_[grav]{} + \_ =0 where $\hat{C}_{\rm grav}$ is given by (\[qh6\]). Since $\hat{C}_{\rm \phi} = (8\pi G)\, (\widehat{1/p^{3/2}})\,
(\hat{p}_\phi^2)$, the constraint becomes: \[eq:main\] 8G \_\^2 (,) = \[B(p)\]\^[-1]{} \_[grav]{} (,) where $\tilde B(p)$ is the eigenvalue of the operator $\widehat{1/|p|^{3/2}}$: \[eq:bp\] B(p) =: ([6]{}[8 \^2]{})\^[3/2]{} B(), [where]{} B() = ([2]{}[3\_o]{})\^6 \^6 .Thus, we now have a separation of variables. Both the classical and the theory suggests that $\phi$ could serve as emergent time. To implement this idea, let us introduce an appropriate kinematical Hilbert space for both geometry and the scalar field: $\Hk := L^2(\R_{\rm Bohr}, B(\mu)\dd\mu_{\rm Bohr}) \otimes
L^2(\R, \dd \phi)$. Since $\phi$ is to be thought of as ‘time’ and $\mu$ as the genuine, physical degree of freedom which evolves with respect to this ‘time’, we chose the standard Schrödinger representation for $\phi$ but the ‘polymer representation’ for $\mu$ which correctly captures the quantum geometry effects. This is a conservative approach in that the results will directly reveal the manifestations of quantum geometry; had we chosen a non-standard representation for the scalar field, these effects would have been mixed with those arising from an unusual representation of ‘time evolution’. Comparison with the theory would also become more complicated. (However, the use of a ‘polymer representation’ for $\phi$ may become necessary to treat inhomogeneities in an adequate fashion.)
On $\Hk$, the constraint takes the form: \[qh7\] [\^2]{}[\^2]{} &=& \[B()\]\^[-1]{} ( C\^+()(+4\_o, ) + C\^o()(, ) + C\^-() (-4\_o, ) )\
&=& - (, ) were the functions $C^{\pm}, C^o$ are given by:[^5] C\^+() &=& [G]{}[9|\_o|\^[3]{}]{} |+3\_o|\^[[3]{}[2]{}]{} - |+\_o|\^[[3]{}[2]{}]{}\
C\^-() &=& C\^+ (-4\_o)\
C\^o() &=& - C\^+() - C\^-() . The form of (\[qh7\]) is the same as that of the constraint (\[wdw4\]), the only difference is that the $\phi$-independent operator $\Theta$ is now a difference operator rather than a differential operator. Thus, the the LQC quantum Hamiltonian constraint can also be regarded as an ‘evolution equation’ which evolves the quantum state in the emergent time $\phi$.
However, since $\Theta$ is a difference operator, an important difference arises from the analysis. For, now the space of physical states, i.e. of appropriate solutions to the constraint equation, is naturally divided into sectors each of which is preserved by the ‘evolution’ and by the action of our Dirac observables. Thus, there is super-selection. Let $\La_{|\epsilon|}$ denote the ‘lattice’ of points $\{|\epsilon|+4n\mu_o,\, n\in \Z\}$ on the $\mu$-axis, $\La_{-|\epsilon|}$ the ‘lattice’ of points $\{-|\epsilon|+4n\mu_o,\, n\in \Z\}$ and let $\La_{\epsilon} =
\La_{|\epsilon|} \cup \La_{-|\epsilon|}$. Let $\H_{|\epsilon|}^{\rm grav},\H_{-|\epsilon|}^{\rm grav}$ and $\H_{\epsilon}^{\rm grav}$ denote the subspaces of $L^2(\R_{\rm
Bohr}, B(\mu)d\mu_{\rm Bohr})$ with states whose support is restricted to lattices $\La_{|\epsilon|}, \La_{-|\epsilon|}$ and $\La_\epsilon$. Each of these three subspaces is mapped to itself by $\Theta$. Since $\hat{C}_{\rm grav}$ is self-adjoint and positive definite on $\Hkg \equiv L^2(\R_{\rm Bohr}, \dd\mu_{\rm
Bohr})$, it follows that $\Theta$ is self-adjoint and positive definite on all three Hilbert spaces.
Note however that since $\H_{|\epsilon|}^{\rm grav}$ and $\H_{-|\epsilon|}^{\rm grav}$ are mapped to each other by the operator $\Pi$, only $\H_\epsilon^{\rm grav}$ is left invariant by $\Pi$. Now, because $\Pi$ reverses the triad orientation, it represents a large gauge transformation. In gauge theories, we have to restrict ourselves to sectors, each consisting of an eigenspace of the group of large gauge transformations. (In QCD in particular this leads to the $\theta$ sectors.) The group generated by $\Pi$ is just $\Z_2$, whence there are only two eigenspaces, with eigenvalues $\pm 1$. Since there are no fermions in our theory, there are no parity violating processes whence we are led to choose the symmetric sector with eigenvalue $+1$. (Also, in the anti-symmetric sector all states are forced to vanish at the ‘singularity’ $\mu=0$ while there is no such a priori restriction in the symmetric sector.) Thus, we are primarily interested in the symmetric subspace of $\H_\epsilon^{\rm grav}$; the other two Hilbert spaces will be useful only in the intermediate stages of our discussion.
Our first task is to explore properties of the operator $\Theta$. Since it is self-adjoint and positive definite, its spectrum is non-negative. Therefore, as in the theory, we will denote its eigenvalues by $\omega^2$. Let us first consider a generic $\epsilon$, i.e., not equal to $0$ or $2\mu_o$. Then, on each of the two Hilbert spaces $\H_{\pm|\epsilon|}^{\rm grav}$, we can solve for the eigenvalue equation $\Theta\, e_\omega(\mu) = \omega^2\,
e_\omega (\mu)$, i.e., \[eq:eigen\] C\^+()e\_(+4\_o) + C\^o()e\_() + C\^-() e\_(-4\_o) = \^2 B() e\_() Since this equation has the form of a recursion relation and since the coefficients $C^\pm(\mu)$ never vanish on the ‘lattices’ under consideration, it follows that we will obtain an eigenfunction by freely specifying, say, $\Psi(\mu^\star)$ and $\Psi(\mu^\star+4\mu_o)$ for any $\mu^\star$ on the ‘lattice’ ${\cal L}_{|\epsilon|}$ or ${\cal L}_{-|\epsilon|}$. Hence the eigenfunctions are 2-fold degenerate on each of $\H_{|\epsilon|}^{\rm grav}$ and $\H_{-|\epsilon|}^{\rm grav}$. On $\H_\epsilon^{\rm grav}$ therefore, the eigenfunctions are 4-fold degenerate as in the theory. Thus, $\H_\epsilon^{\rm
grav}$ admits an orthonormal basis $e_\omega^I$ where the degeneracy index $I$ ranges from $1$ to $4$, such that e\_\^I|e\_[’]{}\^[I’]{} = \_[I,I’]{} (, ’). (The Hilbert space $\H_\epsilon^{\rm grav}$ is separable and the spectrum is equipped with the standard topology of the real line. Therefore we have the Dirac distribution $\delta(\omega,\,\omega')$ rather than the Kronecker delta $\delta_{\omega,\,\omega'}$.) As usual, every element $\Psi(\mu)$ of $\H_\epsilon^{\rm grav}$ can be expanded as: () = \_[[sp]{}]{} d \_I() e\^I\_() \_I() = e\^I\_| , where the integral is over the spectrum of $\Theta$. The numerical analysis of section \[s5\] and comparison with the theory are facilitated by making a convenient choice of this basis in $\H_\epsilon^{\rm grav}$, i.e., by picking specific vectors from each 4 dimensional eigenspace spanned by $e^I_\omega$. To do so, note first that, as one might expect, every eigenvector $e_\omega^I(\mu)$ has the property that it approaches unique eigenvectors $\ub{e}^\pm(\omega)$ of the differential operator $\ul\Theta$ as $\mu \rightarrow \pm \infty$. The precise rate of approach is discussed in section \[s5.1\]. In general, the two eigenfunctions $\ub{e}^\pm(\omega)$ are distinct. Indeed, because of the nature of the operator $\ul\Theta$, its eigenvectors can be chosen to vanish on the entire negative (or positive) $\mu$-axis; their behavior on the two half lines is uncorrelated. (See the remark at the end of section \[s3.2\].) Eigenvectors of the LQC $\Theta$ on the other hand are rigid; their values at any two lattice points determine their values on the entire lattice $\La_{\pm|\epsilon|}$. Second, recall that the spectrum of the operator $\ul\Theta$ is bounded below by $\omega^2 \ge \pi G/3$, whence $\ub{e}_\omega$ with $\omega^2 <
\pi G/3$ does not appear in the spectral decomposition of $\ul\Theta$. Note however that solutions to the eigenvalue equation $\ul\Theta\, \ub{e}_\omega = \omega^2 \ub{e}_\omega$ continue to exist even for $\omega^2 < \pi G/3$. But such eigenfunctions diverge so fast as $\mu \rightarrow \infty$ or as $\mu \rightarrow -\infty$ that $\langle\ub{e}_\omega|\Psi\rangle$ fails to converge for all $\Psi \in L^2(\R, \ub{B}(\mu) d\mu)$, whence they do not belong to the basis. What is the situation with eigenvectors of the LQC $\Theta$? Since eigenvectors $e_\omega$ of $\Theta$ approach those of $\ul\Theta$, $\langle
e_\omega|\Psi\rangle$ again fails to converge for all $\Psi \in
\H_\epsilon^{\rm grav}$ if $\omega^2 < \pi G/3$. Thus the spectrum of $\Theta$ is again bounded below by $\pi
G/3$.[^6] Therefore, to facilitate comparison with the theory we will introduce a variables $k$ via $\omega^2 - \pi G/3 = (16\pi G/3)
k^2$ and use $k$ in place of $\omega$ to label the orthonormal basis. To be specific, let us
> i\) Denote by $e_{-|k|}^\pm(\mu)$ the basis vector in $\H_{\pm|\epsilon|}^{\rm grav}$ with eigenvalue $\omega^2$, which is proportional to the $\ub{e}_{-|k|}$ as $\mu \rightarrow
> \infty$; (i.e., it has only ‘outgoing’ or ‘expanding’ component in this limit);\
> ii) Denote by $e_{|k|}^\pm(\mu)$ the basis vector in $\H_{\pm|\epsilon|}^{\rm grav}$ with eigenvalue $\omega^2$ which is orthogonal to $e_{-|k|}^\pm(\mu)$ (since eigenvectors are 2-fold degenerate in each of $\H_{\pm|\epsilon|}^{\rm grav}$, the vector $e_{|k|}^\pm (\mu)$ is uniquely determined up to a multiplicative phase factor.)
As we will see in section \[s5.1\], this basis is well-suited for numerical analysis.
We thus have an orthonormal basis $e_k^\pm$ in $\H_{\epsilon}^{\rm
grav}$ with $k \in \R$: $\langle e_k^\pm|e_{k'}^\pm\rangle =
\delta(k,k')$, and $\langle e_k^+|e_{k'}^-\rangle = 0$. The four eigenvectors with eigenvalue $\omega^2$ are now $e^+_{|k|},
e^+_{-|k|}$ which have support on the ‘lattice’ $\La_{|\epsilon|}$, and $e^-_{|k|}, e^-_{-|k|}$ which have support on the ‘lattice’ $\La_{-|\epsilon|}$. We will be interested only in the symmetric combinations: \[eq:e-symm\] e\^[(s)]{}\_k () = [1]{}[[2]{}]{} (e\^+\_k() + e\^+\_k(-) + e\^-\_k() + e\^-\_k(-) ) which are invariant under $\Pi$. Finally we note that any symmetric element $\Psi(\mu)$ of $\H_{\epsilon}^{\rm grav}$ can be expanded as \[sym\] () = \_[-]{}\^ dk (k) e\^[(s)]{}\_k()
We can now write down the general symmetric solution to the quantum constraint (\[qh7\]) with initial data in $\H_{\epsilon}^{\rm grav}$ : \[sol3\](,) = \_[-]{}\^ dk \[\_+(k) e\^[(s)]{}\_k() e\^[i]{} + \_- (k) |[e]{}\^[(s)]{}\_k() e\^[-i]{}\]where $\t{\Psi}_\pm(k)$ are in $L^2(\R, dk)$. As $\mu \rightarrow
\pm \infty$, these approach solutions (\[sol1\]) to the equation. However, the approach is not uniform in the Hilbert space but varies from solution to solution. As indicated in section \[s2.2\], the LQC solutions to (\[qh7\]) which are semi-classical at late times can start departing from the solutions for relatively large values of $\mu$, say $\mu \sim 10^4
\mu_o$.
As in the theory, if $\Psi_-(k)$ vanishes, we will say that the solution is of positive frequency and if $\Psi_+(k)$ vanishes we will say it is of negative frequency. Thus, every solution to (\[qh7\]) admits a natural positive and negative frequency decomposition. The positive (respectively negative) frequency solutions satisfy a Schrödinger type first order differential equation in $\phi$: i[\_]{} = \_but with a Hamiltonian $\sqrt{\Theta}$ (which is non-local in $\mu$). Therefore the solutions with initial datum $\Psi(\mu,
\phi_o) = f_\pm(\mu)$ are given by: \_(,) = e\^[i(-\_o)]{} f\_(,) **Remark:** In the above discussion, we considered a generic $\epsilon$. We now summarize the situation in the special cases, $\epsilon= 0$ and $\epsilon =2\mu_o$. In these cases, differences arise because the individual lattices are invariant under the reflection $\mu \rightarrow -\mu$, i.e., the lattices $\La_{|\epsilon|}$ and $\La_{-|\epsilon|}$ coincide. As before, there is a 2-fold degeneracy in the eigenvectors of $\Theta$ on any one lattice. For concreteness, let us label the Hilbert spaces $\H_{|\epsilon|}^{\rm grav}$ and choose the basis vectors $e_k^+(\mu)$, with $k\in \R$ as above. Now, symmetrization can be performed on each of these Hilbert spaces by itself. So, we have: \[sb\] e\_k\^[(s)]{}() = [1]{} (e\^+\_k() + e\^+\_k (-))However, the vector $e_{|k|}^{(s)}(\mu)$ coincides with the vector $e_{-|k|}^{(s)}(\mu)$ so there is only one symmetric eigenvector per eigenvalue. This is not surprising: the original degeneracy was 2-fold (rather than 4-fold) and so there is one symmetric and one anti-symmetric eigenvector per eigenvalue. Nonetheless, it is worth noting that there is a precise sense in which the Hilbert space of symmetric states is only ‘half as big’ in these exceptional cases as they are for a generic $\epsilon$.
For $\epsilon=2\mu_o$, there is a further subtlety because $C^+$ vanishes at $\mu= -2\mu_o$ and $C^-$ vanishes at $\mu= 2\mu_o$. Thus, in this case, as in the theory, there is a decoupling and the knowledge of the eigenfunction $e^+_k(\mu)$ on the positive $\mu$-axis does not suffice to determine it on the negative $\mu$ axis and vice-versa. However, the degeneracy of the eigenvectors does not increase but remains 2-fold because (\[qh7\]) now introduces two new constraints: $C^\pm(2\mu_o)
e^+_k(\pm 6\mu_o) = [\omega^2 B(\pm 2\mu_o) - C^o(\pm 2\mu_o)]
e^+_k(\pm 2\mu_o) =0$. Conceptually, this difference is not significant; there is again a single symmetric eigenfunction for each eigenvalue.
The Physical sector {#s4.2}
-------------------
Results of section \[s4.1\] show that while the LQC operator $\Theta$ differs from the operator $\ul\Theta$ in interesting ways, the structural form of the two Hamiltonian constraint equations is the same. Therefore, apart from the issue of superselection sectors which arises from the fact that $\Theta$ is discrete, introduction of the Dirac observables and determination of the inner product either by demanding that the Dirac observables be self-adjoint or by carrying out group averaging is completely analogous in the two cases. Therefore, we will not repeat the discussion of section \[s3.2\] but only summarize the final structure.
The sector of physical Hilbert space $\Hp^\epsilon$ labelled by $\epsilon \in [0,\, 2 \mu_o]$ consists of positive frequency solutions $\Psi(\mu,\phi)$ to (\[qh7\]) with initial data $\Psi (\mu,
\phi_o)$ in the symmetric sector of $\H^\epsilon_{\rm grav}$. Eq. (\[sol3\]) implies that they have the explicit expression in terms of our eigenvectors $e^{(s)}_k(\mu)$ \[eq:psi-int\] (,) = \_[-]{}\^dk (k) e\^[(s)]{}\_k() e\^[i]{} , where, as before, $\omega^2 = (16\pi G/3)(k^2 + 1/16)$ and $e^{(s)}_k(\mu)$ is given by (\[eq:e-symm\]) and (\[sb\]). By choosing appropriate functions $\t\Psi(k)$, this expression will be evaluated in section \[s5.1\] using Fast Fourier Transforms. The resulting $\Psi(\mu,\phi)$ will provide, numerically, quantum states which are semi-classical for large $\mu$. The physical inner product is given by: \[ip2\] \_1|\_2\_ = \_[{|| + 4n\_o; n}]{} B() |\_1(,\_o) \_2(,\_o) for any $\phi_o$. The action of the Dirac observables is independent of $\epsilon$, and has the same form as in the theory: \[dirac4\] (,) = e\^[i(-\_o)]{} || (,\_o), \_(,) = - i [(,)]{} . The kinematical Hilbert space $\Hk$ is non-separable but, because of super-selection, each physical sector $\Hp^\epsilon$ is separable. Eigenvalues of the Dirac observable $\widehat{|\mu|_{\phi_o}}$ constitute a discrete subset of the real line in each sector. In the kinematic Hilbert space $\Hkg$, the spectrum of $\hat{p}$ is discrete in a subtler sense: while every real value is allowed, the spectrum has discrete topology, reflecting the fact that each eigenvector has a finite norm in $\Hkg$. Thus, the more delicate discreteness of the spectrum of $\hat{p}$ on $\Hkg$ descends to the standard type of discreteness of Dirac observables. Question is often raised whether the kinematic discreteness in LQG will have strong imprints in the physical sector or if they will be washed away in the passage to the physical Hilbert space. A broad answer —illustrated by the area eigenvalues of isolated horizons [@abck; @abk]— is that the discreteness will generically descend to the physical sector at least in cases where one can construct Dirac observables directly from the kinematical geometrical operators [@alrev]. The present discussion provides another illustration of this situation.
Note that the eigenvalues of $\widehat{|\mu|_{\phi_o}}$ in distinct sectors are distinct. Therefore which sector actually occurs is a question that can be in principle answered experimentally, provided one has access to microscopic measurements which can distinguish between values of the scale factor which differ by $\sim 10 \lp$. This will not be feasible in the foreseeable future. Of greater practical interest are the coarse-grained measurements, where the coarse graining occurs at significantly greater scales. For these measurements, different sectors would be indistinguishable and one could work with any one.
The group averaging procedure used in this section is quite general in the sense that it is applicable for a large class of systems, including full LQG if, e.g., its dynamics is formulated using Thiemann’s master constraint program [@ttmc]. In this sense, the physical Hilbert spaces $\Hp^\epsilon$ constructed here are natural. However, using the special structures available in this model, one can also construct an inequivalent representation which is closer to that used in the theory. The main results on the bounce also hold in that representation. Although that construction appears to have an ad-hoc element, it may well admit extensions and be useful in more general models. Therefore, it is presented in Appendix \[a3\].
Numerics in loop quantum cosmology {#s5}
==================================
In this section, we will find physical states of LQG and analyze their properties numerically. This section is divided into three parts. In the first we study eigenfunctions $e^\pm_k (\mu)$ of $\Theta$ and then use them to directly evaluate the right side of (\[eq:psi-int\]), thereby obtaining a ‘general’ physical state. In the second part we solve the initial value problem starting from initial data at $\phi=\phi_o$, thereby obtaining a ‘general’ solution to the difference equation (\[qh7\]). In the third we summarize the main results and compare the outcome of the two methods. Readers who are not interested in the details of simulations can go directly to the third subsection.
A large number of simulations were performed within each of the approaches by varying the parameters in the initial data and working with different lattices ${\cal L}_\epsilon$. They show that the final results are robust. To avoid the making the paper excessively long, we will only show illustrative plots.
Direct evaluation of the integral representation (\[eq:psi-int\]) of solutions {#s5.1}
------------------------------------------------------------------------------
The goal of this sub-section is to evaluate the right side of (\[eq:psi-int\]) using suitable momentum profiles $\t\Psi(k)$. This calculation requires the knowledge of eigenfunctions $e_k^{(s)}(\mu)$ of $\Theta$. Therefore, we will first have to make a somewhat long detour to numerically calculate the basis functions $e^\pm_k(\mu)$ and $e_k^{(s)}(\mu)$ introduced in section \[s4.1\]. The integral in (\[eq:psi-int\]) will be then evaluated using a fast Fourier transform.
### General eigenfunctions of the $\Theta$ operator: asymptotics {#sec:eig-as}
We will first establish properties of the general eigenfunctions $e_\omega(\mu)$ of $\Theta$ that were used in section \[s4.1\].
Let us fix a ‘lattice’, say $\La_{|\epsilon|}$. Since the left side of (\[eq:eigen\]) approaches $\hat{C}_{\text{grav}}^{\text{WDW}} e_{\omega}(\mu)$ as $|\mu|\to\infty$, in this limit one would expect each $e_{\omega}(\mu)$ to converge to an eigenfunction $\ul{e}_{\omega}(\mu)$ of $\ul{\Theta}$ with the same eigenvalue. Numerical simulations have shown that this expectation is correct and have also provided the rate of approach.
![Crosses denote the values of an eigenfunction $e_{\omega}(\mu)$ of $\Theta$ for $\epsilon=0$ and $\omega=20$. The solid curve is the eigenfunction $\ub{e}_\omega(\mu)$ of the $\ul\Theta$ to which $e_\omega(\mu)$ approaches at large positive $\mu$. As $\mu$ increases, the set of points on ${\cal
L}_\epsilon$ becomes denser and fill the solid curve. For visual clarity only some of these points are shown for $\mu >100$.[]{data-label="fig:e-LQC-WdW"}](e-conv.eps){width="5in"}
Recall that each eigenfunction $\ul{e}_{\omega}(\mu)$ is a linear combination of basis functions $\ub{e}_{|k|}(\mu)$, $\ub{e}_{-|k|}(\mu)$ defined in section \[s3.1\]. Therefore, given an $e_{\omega}(\mu)$ it suffices to calculate the coefficients of the decomposition of $\ub{e}_{\omega}(\mu)$ with respect to this basis. The method of finding these coefficients is presented in detail in Appendix \[a2\].[^7] Once the limiting $\ub{e}_{\omega}(\mu)$ were found, they were compared with the original eigenfunction $e_{\omega}(\mu)$ for a variety of values of $\omega$. An illustrative plot comparing ${e}_{\omega}(\mu)$ with its limit is shown in Fig. \[fig:e-LQC-WdW\]. In general, each $e_\omega(\mu)$ approaches distinct eigenfunctions $\ub{e}_{\omega,\,\pm}(\mu)$ of $\ul\Theta$ in the limits $\mu \rightarrow \pm \infty$. The rate of approach is given by: $$\label{eq:e-asympt}
\mu^{-\frac{1}{4}} e_{\omega}(\mu)\ = \begin{cases}
\mu^{-\frac{1}{4}} \underline{e}_{\omega,\, +} (\mu)
+ O\left( \frac{1}{\mu^2} \right) \ ,
& \text{for }\mu>0 \ , \\
\mu^{-\frac{1}{4}} \underline{e}_{\omega,\, -} (\mu)
+ O\left( \frac{1}{\mu^2} \right) \ ,
& \text{for }\mu<0 \ .
\end{cases}$$ Numerical tests were performed up to $|\mu| = 10^6\mu_o$. The quantity $\mu^{7/4}\mid e_\omega -\ub{e}_{\omega,\,\pm} \mid
(\mu)$ was found to be bounded. The bound decreases with $\mu$. For the case $\epsilon=0,\, \omega =20$ depicted in Fig. \[fig:e-LQC-WdW\] the absolute bound in the $\mu$-interval $(10^2\mu_o,\, 10^6\mu_o)$ was less than 90. Because the eigenfunctions $e_\omega(\mu)$ of $\Theta$ are determined on the entire lattice $\La_{|\epsilon|}$ by their values on (at most) two points, the WDW limits for positive and negative $\mu$ are not independent. Thus, if the limits are expressed as
\[eq:asympt-coeffs\]$$\begin{aligned}
e_{\omega}(\mu) &\xrightarrow{\mu\gg 1}
A\,\ub{e}_{|k|}(\mu) + B\,\ub{e}_{-|k|}(\mu), &
e_{\omega}(\mu) &\xrightarrow{\mu\ll -1}
C\,\ub{e}_{|k|}(\mu) + D\, \ub{e}_{-|k|}(\mu)
\tag{\ref{eq:asympt-coeffs}}\end{aligned}$$
the coefficients $C,D$ are uniquely determined by values of $A,B$ (and vice versa). One relation, suggested by analytical considerations involving the physical inner product, was verified in detail numerically: $$\label{eq:e-norm-pres}
|A|^2 - |B|^2\ =\ |C|^2 - |D|^2 \ .$$ It will be useful in the analysis of basis functions in the next two sub-sections.
### Construction of the basis $e^{\pm}_{-|k|}$ {#sec:basis-pm}
In section \[s4.1\] we introduced a specific basis of $\H_{\epsilon}$ which is well adapted for comparison with the theory. We will now use numerical methods to construct this basis and analyze its properties. Our investigation will be restricted to the vectors $e^{\pm}_{-|k|}(\mu)$ because, the physical states $\Psi(\mu,\phi)$ of Eq. (\[eq:psi-int\]) we are interested will have negligible projections on the vectors $e^{\pm}_{|k|}(\mu)$. (Recall that in general $\Psi \in \H_\epsilon$ has support on $\La_{|\epsilon|}\cup \La_{-|\epsilon|}$. $e^+_{-|k|}(\mu)$ has support on $\La_{+|\epsilon|}$ and $e^-_{-|k|}(\mu)$ on $\La_{-|\epsilon|}$.)
Each of the eigenfunctions $e^{\pm}_{-|k|}$ is calculated as follows. To solve (\[eq:eigen\]), we need to specify initial conditions at two points on each of the two lattices. We fix large positive $\mu_{\pm}^\star\in\La_{\pm|\epsilon|}$ and demand that the values of $e^{\pm}_{-|k|}(\mu)$ agree with those of $\ub{e}_{-|k|}(\mu)$ at the points $\mu_{\pm}^\star$ and $\mu_{\pm}^\star +4\mu_o$. Then $e^\pm_{-|k|}$ are evaluated separately on finite domains $\La_{\pm|\epsilon|} \cap
[-\mu_{\pm}^\star,\mu_{\pm}^\star]$ of each of the two lattices. For large negative $\mu$, these eigenfunctions are linear combinations of the basis functions $C^\pm\, \ub{e}_{|k|}+
D^\pm\, \ub{e}_{-|k|}$. The coefficients $C^\pm, D^\pm$ are evaluated using the method specified in Appendix \[a2\].
![The exponential growth of $|e^{\pm}_{-|k|}|(\mu)$ in the ‘genuinely quantum region’ is shown for three different values of $\omega$, where $\omega$ is given by $\Theta\, e^{\pm}_{-|k|} =
\omega^2\, e^{\pm}_{-|k|}$.[]{data-label="fig:ek-log"}](ek-sing.eps){width="5in"}
![The amplification factor $\lambda^{\pm}$ in the ‘genuinely quantum region is shown as a function of the parameter $\epsilon$ labeling the lattice and $\omega$.[]{data-label="fig:ampl-3d"}](ampl-3d.eps){width="5in"}
![The function $a(\epsilon)$ of Eq \[eq:lambda-fit\] is plotted by connecting numerically calculated data points.[]{data-label="fig:ampl-a"}](a_ep.eps){width="5in"}
![The function $b(\epsilon)$ of Eq \[eq:lambda-fit\] is plotted by connecting numerically calculated data points.[]{data-label="fig:ampl-b"}](b_ep.eps){width="5in"}
Eigenfunctions $e^{\pm}_{-|k|}(\mu)$ were calculated for approximately $2\times 10^4$ different $|k|$’s in the range $5\le
\omega \le 10^3$. They revealed the following properties:
1. Each $e^{\pm}_{-|k|}(\mu)$ is well approximated by a eigenfunction until one reaches the ‘genuinely quantum region’. In this region the absolute value $|e^{\pm}_{-|k|}|$ grows very quickly as $\mu$ decreases: $|e^{\pm}_{-|k|}|\,\propto\, e^{-{\rm
sgn}(\mu) \,\alpha\sqrt{\omega\,|\mu|}}$, where $\alpha \equiv
\alpha (\epsilon)$ is a constant on any given lattice. This property is illustrated by Fig. \[fig:ek-log\]. This region of rapid growth is symmetric about $\mu=0$ and its size depends linearly on $\omega$ (the square-root of the eigenvalue of ${\Theta}$); its boundary lies at $\mu\approx 0.5\omega\,\mu_o$). (However, this region excludes the interval $[-4\mu_o,\,4\mu_o]$ where $B(\mu)$ decreases and goes to zero, departing significantly from its analog $\ub{B}(\mu)$.)
2. After leaving this region of growth, the basis function $e^{\pm}_{-|k|}(\mu)$ again approaches some WDW eigenfunction $$e^{\pm}_{-|k|} \xrightarrow{\mu << -1 }
C^{\pm}\, \ub{e}_{|k|} + D^{\pm}\, \ub{e}_{-|k|} \ .$$ where the coefficients $C^{\pm},\, D^{\pm}$ are large. Their absolute values grow exponentially with $\omega$. To investigate this property qualitatively we defined an ‘amplification factor’ $$\lambda^{\pm}\ := |C^{\pm}| + |D^{\pm}|$$ The numerical calculations show that $\lambda^+ = \lambda^-$, so parts of $e^\pm_{-|k|}$ supported on $\La_{+|\epsilon|}$ and $\La_{-|\epsilon|}$ are amplified equally. The dependence of $\lambda^{\pm}$ on $\omega$ and $\epsilon$ is shown in Fig. \[fig:ampl-3d\]. Almost everywhere it can be well approximated by the function $$\label{eq:lambda-fit}
\lambda^{\pm} (\omega, \epsilon) \approx e^{a\omega+b} \ .$$ where $a,b$ are rather complicated functions of $\epsilon$. The fits of $a(\epsilon),b(\epsilon)$ are presented in Fig. \[fig:ampl-a\] and Fig. \[fig:ampl-b\]. The actual simulations were carried out for various values of $p_\phi = \hbar \omega$, up to $p_\phi=10^3$.
3. The general relation (\[eq:e-norm-pres\]) holds in our case with $|B| =0$. Existence of the tremendous amplification now implies that the absolute values of $C$ and $D$ are almost equal (with differences of the order of $1/\lambda^{\pm}$) $$|C^{\pm}| \approx |D^{\pm}| \ .$$ Thus for negative $\mu$, eigenfunctions asymptotically approach eigenfunctions and are almost equally composed of incoming and outgoing waves.
### Basis for the symmetric sector {#sec:basis-sym}
Once the basis functions $e^{\pm}_{-|k|}$ are known one can readily use to construct the basis $e^{(s)}_{-|k|}$ for solutions to (\[eq:eigen\]) which are symmetric under $\mu \rightarrow -\mu$. Due to strong amplification in the region around $\mu=0$ the behavior of $e^{(s)}_{-|k|}$ is dominated by properties of $e^{\pm}_{-|k|}$ for $\mu<0$. The numerical calculations show the following properties:
1. Each symmetric basis eigenfunction is strongly suppressed in the ‘genuinely quantum region’ around to $\mu=0$. The behavior of $e^{\pm}_{-|k|}$ in this region implies that $|e^{(s)}_{-|k|}|\propto\cosh(\alpha\sqrt{\omega\mu})$, where $\alpha$ is a function of $\epsilon$ only.
2. Outside the ‘genuinely quantum region’ $e^{(s)}_{-|k|}$ quickly approaches the eigenfunction almost equally composed of incoming and outgoing ‘plane waves’ ($\ub{e}_{|k|}$ and $\ub{e}_{-|k|}$). In the exceptional cases $\epsilon = 0$ [or]{} $2\mu_o$ the two contributions are exactly equal. To establish this result, note first that the symmetry requirement and the fact that $C^{+}(-2\mu_o) = C^{-}(2\mu_o) = 0$ imply that in both cases the value of $e^{(s)}_{-|k|}$ at one point already determines the complete ‘initial data’ (i.e., values at some $\mu_*$ and $\mu_*+4\mu_o$) and hence the eigenfunction on the entire $\La_{\epsilon}$:
\[eq:02-symm-cond\]$$\begin{aligned}
e^{(s)}_{-|k|}(4\mu_o)\ &=\ e^{(s)}_{-|k|}(-4\mu_o)\
=\ e^{(s)}_{-|k|}(0) \ , &
\text{for }\epsilon &= 0 \ , \\
e^{(s)}_{-|k|}(6\mu_o)\
&=\ \frac{\omega^2B(2\mu_o)-C^o(2\mu_o)}{C^{+}(2\mu_o)}
e^{(s)}_{-|k|}(2\mu_o) \ , &
\text{for }\epsilon &= 2\mu_o \ .
\end{aligned}$$
Because of the reality of coefficients of Eq. (\[eq:eigen\]), this implies that the phase of $e^{(s)}_{-|k|}$ is exactly constant, whence contributions of $\ub{e}_{|k|}$ and $\ub{e}_{-|k|}$ are also exactly equal.
3. On each lattice $\La_{\pm|\epsilon|}$ the incoming and outgoing components are rotated with respect to each other by an angle $\alpha^{\pm}$ $$\label{eq:phases}
e^{(s)}_{-|k|}\mid_{\La_{\pm|\epsilon|}} \xrightarrow{\mu\to\infty}
z^{\pm} ( e^{i\alpha^{\pm}}\ub{e}_{|k|} +
e^{-i\alpha^{\pm}}\ub{e}_{-|k|} ) \ ,$$ where $z^{\pm}$ are some complex *constants* satisfying $|z^+|\approx|z^-|$, while the phases $\alpha^{\pm}$ are functions of $\epsilon$ and $\omega$. In general, for $\epsilon \not= 0$ or $\epsilon \not= 2\mu_o$, $\alpha_{+}$ need not equal $\alpha_{-}$.
![Plot of the wave function $\Psi(\mu,\phi)$ obtained by directly evaluating the right side of (\[eq:psi-int\]). Parameters are $p_{\phi}=500$, $\Delta p_{\phi}/p_{\phi} = 0.05$ and $\epsilon = \mu_o$.[]{data-label="fig:direct-3d"}](direct-3d.eps){width="5in"}
![Expectation values and dispersion of $\widehat{|\mu|_{\phi}}$ for wave function presented in Fig. \[fig:direct-3d\] are compared with classical trajectories.[]{data-label="fig:direct-mu"}](direct-mu.eps){width="5in"}
### Evaluation of the integral in (\[eq:psi-int\]) {#sec:state-direct}
Now that we have the symmetric basis functions $e^{(s)}_{-|k|}$ at our disposal, we can obtain the desired physical states by directly evaluating the integral in (\[eq:psi-int\]).
We wish to construct physical states which are sharply peaked at a phase space point on a classical trajectory of the expanding universe at a late time (e.g., ‘now’). The form of the integrand in (\[eq:psi-int\]), the expression (\[ip2\]) of the physical inner product, the functional form of $\omega(k)$ and standard facts about coherent and squeezed states in quantum mechanics provide a natural strategy to select an appropriate $\t\Psi(k)$. If we set (k) = e\^[-[(k - k\^)\^2]{}[2 \^2]{}]{} e\^[-i\^]{} with suitably small $\sigma$, the final state will be sharply peaked at $p_\phi = p_\phi^\star$ p\_\^= -([16G\^2]{}[3]{} )\^[[1]{}[2]{}]{} k\^and the parameter $\phi^\star$ will determine the value $\mu^\star$ of the Dirac observable $\widehat{|\mu|_{\phi_o}}$ at which the state will be peaked at ‘time’ $\phi_o$. As mentioned in section \[s3.2\], to obtain a state which is semi-classical at a late time, we need a large value of $p_\phi$: $p_\phi^\star
\gg \hbar$ in the classical units, $c$=$G$=1. Therefore, we need $k^\star \ll -1$, whence the functions $\t\Psi(k)$ of interest will be negligibly small for $k>0$. Therefore, without loss of physical content, we can set them to zero on the positive $k$ axis. This is why the explicit form of eigenfunctions $e^{(s)}_{|k|}$ is not required in our analysis.
Thus, the integral we wish to evaluate is: $$\label{eq:state-direct}
\Psi(\mu, \phi) \ =\ \int_0^{\infty} \, d k \, e^{-{\frac}{(k -
k^\star)^2}{2 \sigma^2}} \, e^{(s)}_{k}(\mu)\, e^{i \omega(k)(\phi
- \phi^\star)} \ ,$$ where $\sigma$ is the spread of the Gaussian. The details of the numerical evaluation can be summarized as follows.
- For a generic $\epsilon$, the $e^{(s)}_{-|k|}$ were found numerically following the procedure specified in sections \[sec:basis-pm\] and \[sec:basis-sym\]. For the exceptional cases, $\epsilon= 0,2\mu_o$, in order to avoid loss of precision in the region where $e^{(s)}_{-|k|}$ is very small, we provided ‘initial values’ of $e^{(s)}_{k_j}$ at $\mu=\pm\epsilon$ and $\mu
= \pm\epsilon+4\mu_o$ using . On the $k$ axis we chose a set $\{k_j\}$ of points which are uniformly distributed across the interval $[k^\star-10\sigma,
k^\star+10\sigma]$. In numerical simulations, the number $l$ of points in the set $\{k_j\}$ ranged between $2^{11}$ and $2^{13}$.
- Next, for each $k_j$, we calculated $e_{-|k_j|}^{(s)}(\mu_i)$ for $\mu\in \{\pm\epsilon+4n\mu_o:\, n\in\{-N,\ldots,N\}\}$ where $N$ is a large constant $(\sim\, 50 p_\phi^\star)$.
- Finally, we evaluated using fast Fourier transform. The result was a set of profiles $\Psi(\mu_i,\phi_j)$ where $\phi_j=\sqrt{3\pi/(4G)}(j-l/2)/(k_l-k_1)$ This is a positive frequency solution to the LQC equation (\[qh7\]).
Our next task is to analyze properties of these solution. Given any one $\Psi(\phi_j,\mu_i)$, we chose ‘instants of time’ $\phi$ and calculated the norm and the expectation values of our Dirac observable $|\hat{\mu}|_{\phi}$ and $\hat{p}_{\phi}$ using:
\[eq:expect\]$$\begin{aligned}
||\Psi||^2\ &=\ \sum_{\mu_i\in\La_{\epsilon}} B(\mu_i)
|\Psi(\phi,\mu_i)|^2 \ , \\
\left<\widehat{|\mu|_{\phi}}\right>\
&=\ \frac{1}{||\Psi||^2}
\sum_{\mu_i\in\La_{\epsilon}}
B(\mu_i)\,|\mu_i|\,|\Psi(\phi,\mu_i)|^2 \ , \\
\left<\hat{p}_{\phi}\right>\ &=\ \frac{1}{||\Psi||^2}
\sum_{\mu_i\in\La_{\epsilon}}
B(\mu_i)\,\bar{\Psi}(\phi,\mu_i)\,\,{\frac}{\hbar}{i} \partial_{\phi}
\Psi(\phi,\mu_i)
\, .\end{aligned}$$
Finally, the dispersions were evaluated using their definitions:
\[eq:disp\]$$\begin{aligned}
\left<\Delta \widehat{|\mu|_{\phi}}\right>^2\ &=\
|\left<\widehat{|\mu^2|_{\phi}}\right>
- \left<\widehat{|\mu|_{\phi}}\right>^2| \ , &
\left<\Delta \hat{p}_{\phi}\right>^2\ &=\
|\left<\hat{p}_{\phi}^2\right>
- \left<\hat{p}_{\phi}\right>^2| \ .
\tag{\ref{eq:disp}}\end{aligned}$$
These calculations were performed for 16 different choices of $\epsilon$ and for ten values of $p_\phi$ up to a maximum of $p_\phi= 10^3$, and for 5 different choices of the dispersion parameter $\sigma$. Fig. \[fig:direct-3d\] provides an example of a state constructed via this method. The expectation values of $\left<{\widehat{|\mu|_{\phi}}}\right>$ are shown in Fig. \[fig:direct-mu\].
Results are discussed in section \[s5.3\] below.
Evolution in $\phi$ {#s5.2}
-------------------
We can also regard the quantum constraint (\[qh7\]) as an initial value problem in ‘time’ $\phi$ and solve it by carrying out the $\phi$-evolution. Conceptually, this approach is simpler since it does not depend on the properties of the eigenfunctions of $\Theta$. However, compared to the direct evaluation of the integral (\[eq:psi-int\]), this method is technically more difficult because it entails solving a large number of coupled differential equations. Nonetheless, to demonstrate the robustness of results, we carried out the $\phi$-evolution as well. This sub-section summarizes the procedure.
### Method of integration {#sec:int-method}
At large $|\mu|$ the difference equation is well approximated by the equation which is a hyperbolic partial differential equation (PDE). However since $\Theta$ couples $\mu$ in discrete steps, for a given $\epsilon$-sector is just a system of a countable number of coupled ordinary differential equations (ODEs) of the 2nd order.[^8] Technical limitations require restriction of the domain of integration to a set $|\mu-\epsilon|\leq 4N\mu_o$, where $1 \ll N
\in \mathbb{Z}$. This restriction makes the number of equations finite. However one now needs to introduce appropriate boundary conditions. The fundamental equation on the boundary is $i{\partial}_\phi
\Psi = s\sqrt{\Theta}\Psi$, where $s=+1$ ($-1$) for the forward (backward) evolution in $\phi$. It is difficult to calculate $\sqrt{\Theta}\Psi$ at each time step. Therefore, *just on the boundary itself*, this equation was simplified to: $$\label{eq:boundary}
\partial_{\phi} \Psi(\mu,\phi)\ =\ s\sqrt{\pi G/3}(|\mu|-2\mu_o)
( \Psi(\mu,\phi) - \Psi(\mu-4\operatorname*{sgn}(\mu)\mu_o,\phi) ) \ .$$ This is the discrete approximation of the continuum operator ${\partial}_\phi\Psi = s(\sqrt{16\pi G/3})\mu\,{\partial}_\mu \Psi$ which itself is an excellent approximation to the fundamental equation when the boundary is far. The boundary condition requires the solution to leave the domain of integration. (For, to make the evolution deterministic in the domain of interest, it is important to avoid waves entering the integration domain from the boundary.) The boundary was chosen to lie sufficiently far from the location (in $\mu$) of the peak of the initial wave packet. Its position was determined by requiring that the value of the wave function at the boundary be less than $10^{-n}$ times that at its value of the peak, and $n$ ranged between $9$ and $24$ in different numerical simulations.
Three different methods were used to specify the initial data $\Psi$ and $\partial_{\phi}\Psi$ at $\phi=\phi_o$. These are described in section \[sec:phi-init-data\]. The data were then evolved using the fourth order adaptive Runge-Kutta method (RK4). To estimate the numerical error due to discretization of time evolution, two sup-norms were used: $$\label{eq:psi-psi2_1}
|\Psi_1 -\Psi_2|_{I}\, (\phi) = \frac{\sup_{|\mu_i-\epsilon|\leq
N\mu_o}\,|\Psi_1-\Psi_2|(\phi) } {\sup_{|\mu_i-\epsilon|\leq
N\mu_o} |\Psi_2|(\phi) } \ .$$ and $$\label{eq:psi-psi2_2}
|\Psi_1 -\Psi_2|_{II}\,(\phi) = \frac{\sup_{|\mu_i-\epsilon|\leq
N\mu_o} \left| |\Psi_1| -|\Psi_2|\right|(\phi)}
{\sup_{|\mu_i-\epsilon|\leq N\mu_o} |\Psi_2|(\phi)} \ .$$ Fig. \[fig:conv-test\] shows an example of the results of convergence tests for the solution corresponding to $p_\phi =
10^3$ with initial spread in $p_\phi$ of $3\%$ (used in figures \[fig:rel-dmu\] — \[fig:zoom-symm\]). One can see that the phase of $\Psi$ is more sensitive to numerical errors than its absolute value. Therefore, although the accuracy is high for the solution $\Psi(\mu,\phi)$ itself, it is even higher for the mean values and dispersions of $\widehat{|\mu|_{\phi_o}}$ and $\hat{p}_\phi$.
![Error functions $|\Psi_{(M')} - \Psi_{(M)}|_{I}$ (upper curve) and $|\Psi_{(M')} - \Psi_{(M)}|_{II}$ (lower curve) are plotted as a function of time steps. Here $\Psi_{(M')}$ refers to final profile of wave function for simulation with $M'$ time steps. $\Psi_{(M)}$ refers to the final profile for the finest evolution (approximately $2.88 \times 10^6$ time steps). In both cases, the evolution started at $\phi=0$ and the final profile refers to $\phi = -1.8$.[]{data-label="fig:conv-test"}](conv-test.eps){width="5in"}
### Initial data {#sec:phi-init-data}
Although we are interested in positive frequency solutions, to avoid having to take square-roots of $\Theta$ at each ‘time step’, the second order evolution equation (\[qh7\]) was used. Thus, the initial data, consist of the pair $\Psi$ and its time derivative ${\partial}_{\phi} \Psi$, specified at some ‘time’ $\phi_o$. The positive frequency condition was incorporated by specifying, in the initial data, the time derivative of $\Psi$ in terms of $\Psi$. Since $\Psi(\mu) = \Psi(-\mu)$, we will restrict ourselves to the positive $\mu$ axis. The idea is to choose semi-classical initial data peaked at a point $(p_\phi^\star,\, \mu^\star)$ on a classical trajectory at ‘time’ $\phi=\phi_o$, with $(p_\phi^\star \gg
\sqrt{G\hbar^2},\, \mu^\star\gg 1)$, and evolve them. To avoid philosophical prejudices on what the state should do at or near the big bang, we specify the data on the expanding classical trajectory at ‘late time’ (i.e., ‘now’) and ask it to be semi-classical like the observed universe.
The idea that the data be semi-classical was incorporated in three related but distinct ways.
1. *Method I*: This procedure mimics standard quantum mechanics. Since $(c,\mu)$ are canonically conjugate on the phase space, we chose $\Psi(\mu)$ to be a Gaussian (with respect to the measure defined by the inner product) and peaked at large $\mu^\star$ and the value $c^\star$ of $c$ at the point on the classical trajectory determined by $(\mu^\star,p_{\phi}^\star,\phi_o)$): $$\Psi\mid_{\phi_o} := N \, |\mu|^{{\frac}{3}{4}} \, e^{-\frac{(\mu -
\mu^*)^2}{2\tilde{\sigma}^2}} \, e^{-i {\frac}{c^\star
(\mu - \mu^*)}{2}} \ .$$ where $N$ is a normalization constant. The initial value of $\partial_\phi\Psi|_{\phi_o}$ was calculated using the classical Hamilton’s equations of motion $$\partial_\phi\Psi|_{\phi_o}\
=\ \left( -\operatorname*{sgn}(k^\star)\sqrt{\frac{16\pi G}{3}}\,
\frac{(\mu-\mu^{\star})\mu^{\star}}{\tilde{\sigma}^2}
+ i \, \omega^\star\,{\frac}{\mu}{\mu^\star} \right)
\Psi\mid_{\phi_o} \ ,$$ where $\omega^\star = p^\star_\phi/\hbar$.
2. *Method II*: This procedure takes advantage of the fact that provides a physical state which is semi-classical at late times. The idea is to calculate $\Psi$ and ${\partial}_\phi\Psi$ at $\phi=\phi_o$ and use their restrictions to lattices ${\cal L}_\epsilon$ as the initial data for LQC, setting $\omega = \sqrt{16\pi G/3} \, |k|$ as in the discussion following . Thus the initial data used in the simulations were of the form: $$\begin{aligned}
\Psi\mid_{\phi_o}\
&=&\ \left(\frac{\mu}{\mu^\star}\right)^{\frac{1}{4}}
e^{ -\frac{\sigma^2}{2} \ln^2 \frac{|\mu|}{|\mu^{\star}|}
}\,\,
e^{i k^\star \ln \frac{|\mu|}{|\mu^\star|} } \ , \\
\partial_{\phi}\Psi|_{\phi_o}\
&=&\ \left(-\operatorname*{sgn}(k^\star)\sqrt{\frac{16\pi}{3}}
\,\sigma^2\ln\frac{|\mu|}{|\mu^\star|}
+ i \, \omega(k^\star) \right)
\Psi|_{\phi_o} \\end{aligned}$$ This choice is best suited to comparing the LQC results with those of the theory.
The spreads $\t\sigma$ of *Method I* and ${\sigma}$ of *Method II* are related to the initial spread $\Delta\mu|_{\phi_o}$ as follows $$\tilde{\sigma}\ =\ \frac{\mu^\star}{\sigma}\
=\ \sqrt{2}\Delta\mu|_{\phi_o} \ .$$
3. *Method III*: To facilitate comparison with the direct evaluation of the integral solution described in section \[s5.1\], a variation was made on *Method II*. Specifically, in the expression of $\Psi$, the basis eigenfunctions $\ub{e}_k(\mu)$ were rotated by multiplying them with a $k$ and $\epsilon$ dependent phase factor defined in Eq. $$\label{eq:base-rot}
\ub{e}_{-|k|}\mapsto e^{-i\alpha^{+}} \ub{e}_{-|k|} \ .$$ These phases were first found numerically using the method specified in appendix \[a2\] and then functions of the form $$\alpha^{+}\, =\, A \, \ln(Bk+C)k+D \ ,$$ (where $A,B,C,D$ are real constants) were fitted to the results. After subtraction of the $0$th and the $1$st order terms in the expansion in $k$ around $k^\star$ (which respectively correspond to a constant phase and a shift of origin of $\phi$) the resulting function was used to rotate the basis $\ub{e}_k(\mu)$ appearing in ) via ). The expression on the right side of Eq and its $\phi$-derivative were then integrated numerically at $\phi=\phi_o$.
In the simulations, 15 different values of $p_\phi^\star$ were used ranging between $10^2$ and $10^5$. $\mu^\star$ was always greater than $2.5p_\phi$. The dispersion $\sigma$ was allowed to have five different values. These simulations involved four different, randomly chosen values of $\epsilon$.
![The absolute value of the wave function obtained by evolving an initial data of *Method II*. For clarity of visualization, only the values of $|\Psi|$ greater than $10^{-4}$ are shown. Being a physical state, $\Psi$ is symmetric under $\mu \rightarrow -\mu$. In this simulation, the parameters were: $\epsilon=2 \mu_o$, $p_{\phi}^{\star}=10^4$, and $\Delta p_\phi/p_\phi^{\star} = 7.5 \times 10^{-3}$.[]{data-label="fig:l-3d"}](letter-3d-color.eps){width="5in"}
![The expectation values (and dispersions) of $\widehat{|\mu|_{\phi}}$ are plotted for the wavefunction in Fig. \[fig:l-3d\] and compared with expanding and contracting classical trajectories.[]{data-label="fig:l-mu"}](letter-trajectory.eps){width="5in"}
![Comparisons between the relative dispersions $\Delta\mu/\mu$ as functions of $\phi$ for all three methods specifying initial data and the result of direct construction.[]{data-label="fig:rel-dmu"}](relative-disp-new.eps){width="5in"}
![A zoom on the absolute value of the wave function near the bounce point. Initial data was specified using *Method I*.[]{data-label="fig:zoom-gauss"}](zoom-gauss.eps){width="5in"}
![A zoom on the absolute value of the wave function near the bounce point. Initial data was specified using *Method II*.[]{data-label="fig:zoom-wdw"}](zoom-WdW.eps){width="5in"}
![A zoom on the absolute value of the wave function near the bounce point. Initial data was specified using *Method III*.[]{data-label="fig:zoom-symm"}](zoom-symm.eps){width="5in"}
![The uncertainty product $\Delta\phi \Delta p_\phi$ for three methods of specifying the initial data.[]{data-label="fig:dphi1"}](coherence123.eps){width="5in"}
Results and Comparisons {#s5.3}
-----------------------
We now summarize the results obtained by using the two constructions specified in sections \[s5.1\] and \[s5.2\]. The qualitative results are robust and differences lie in the finer structure. In particular, evolutions of initial data constructed from the three methods described in section \[sec:phi-init-data\] yield physical states $\Psi(\mu,\phi)$ which are virtually identical except for small differences in the behavior of relative dispersions of the Dirac observables. The numerical evaluation of the integral (\[eq:psi-int\]) in section \[sec:state-direct\] yielded results very similar to those obtained in \[sec:phi-init-data\] using initial data of *Method III*. An example of results is presented in Fig. \[fig:l-3d\] and Fig. \[fig:l-mu\].
Highlights of the results can be summarized as follows:
1. The state remains sharply peaked throughout the evolution. However, as shown in Fig. \[fig:dphi1\], while the product $\Delta \phi\,\Delta p_\phi$ is nearly constant for large $\mu$, there is a substantial increase near $\mu=0$.
2. The expectation values of $\hat \mu_\phi$ and $\hat p_\phi$ are in good agreement with the classical trajectories, until the increasing matter density approaches a critical value. Then, the state bounces from the expanding branch to a contracting branch with the same value of $\langle \hat{p}_\phi\rangle$. (See Fig. \[fig:l-mu\]). This phenomena occurs [*universally*]{}, i.e., in every ${\epsilon}$-sector, for all three methods of choosing the initial data and for any choice of $p_\phi\gg \sqrt{G\hbar^2}$. *In this sense the classical big-bang is replaced by a quantum bounce.* Note that this is in striking contrast with the situation with the theory we encountered in section \[s3.2\], even when the initial data is chosen using *Method II* which is tailored to the theory. As indicated in Appendix \[a1\], the existence of the bounce can be heuristically understood from an ‘effective theory’. The detailed numerical work supports that description, thereby providing a justification for the approximation involved.
3. If the state is peaked on the expanding branch $\mu(\phi) =
\mu^\star \exp (\sqrt{16\pi G/3}\, (\phi-\phi_o))$ in the distant future, due to the bounce it is peaked on a contracting branch in the distant past, given by $\mu(\phi) = D(p_\phi^\star)\, \mu^\star
\exp (-\sqrt{16\pi G/3}\, (\phi-\phi_o))$, where $D(p_\phi) =
\mu_o^2p_\phi^2/12\pi G\hbar^2$. Thus, for large $|\mu|$ the solution $\Psi(\mu,\phi)$ exhibits reflection symmetry (about $\phi = \phi_o -{\frac}{1}{2}\ln D(p_\phi^\star)$). However, it is not exactly reflection symmetric (compare [@shw]).
4. As a consistency check, we verified that the norm and the expectation value $\langle \hat{p}_\phi \rangle$ are preserved during the entire evolution. Furthermore, the dispersion also remains small throughout the evolution, although the precise behavior depends on the method of specification of the initial data. Differences arise primarily near the bounce point and manifest themselves through the behavior of the relative dispersion $\Delta \mu/\mu$ as a function of $\phi$. These are illustrated (for all three methods as well as for the direct evaluation of section \[s5.1\]) in Fig. \[fig:rel-dmu\]. Finally, as argued in Appendix \[a1\], since the state is sharply peaked, the value of $\Delta\mu/\mu$ can be related to $\Delta\phi$ via Eq \[dp\_dm\_m\]. One finds that the product $\Delta\phi\Delta p_{\phi}$ is essentially constant in the region away from the bounce but grows significantly near the bounce.
The differences can be summarize as follows:
1. In *Method I* the initial state is a minimum uncertainty state in $(\mu, c)$ but doesn’t minimize the uncertainty in $(\phi, p_\phi)$ at any value of $\phi$. The relative spread in $\mu$ remains approximately constant in the regions where $\langle\hat{\mu}_{\phi}\rangle$ is large and increases near the bounce point monotonically. The wave function interpolates ‘smoothly’ between expanding and contracting branches (see Fig. \[fig:zoom-gauss\]). On the other hand the product $\Delta \phi \Delta p_\phi$ of uncertainties has a value much higher than 1/2, grows quickly near the bounce and settles down to constant value after it. See Fig. \[fig:dphi1\].
2. The state obtained by evolving the initial data constructed from *Method II* has minimal uncertainty in $(\phi, p_\phi)$. $\Delta \mu/\mu$ is approximately constant for large $\langle\widehat{\mu_{\phi}}\rangle$ and it decreases quickly near the bounce point, reaching its minimal value shortly before the bounce point. After the bounce it grows and stabilizes at the value of the relative spread found for data constructed using *Method I* (for the same values of $p_{\phi}^\star$ and initial $\Delta\mu/\mu$). Behavior of the wave function is also different from that in the previous case. Near the bounce point its value grows to form a bulge (see Fig. \[fig:zoom-wdw\]). The product $\Delta\phi\Delta p_{\phi}$ remains almost constant for as long as the results of the numerical measurement are reliable (see Fig. \[fig:dphi1\]) approaching a constant after the bounce, with a somewhat higher value. The heuristic estimate on $\Delta\phi\Delta p_{\phi}$ (see Appendix \[a1\]) agrees with these results. Finally, we also found that the increase of the relative spread depends on $p_{\phi}^\star$ and initial $\Delta\mu/\mu$.
3. The state obtained by evolving the initial data constructed from *Method III* does not have minimum uncertainty in $(\phi, p_\phi)$. The behavior of $\Delta\mu/\mu$ is similar to the previous case except that it becomes equal to $\Delta
p_{\phi}/p_{\phi}$ at the bounce point and its asymptotic value in the contracting branch is same as its starting value. Thus the spread is symmetric with respect to reflection in $\phi$ around the bounce point. The difference between the value of $\Delta\mu/\mu$ for large $\langle\widehat{\mu_{\phi}}\rangle$ and the one corresponding to the minimum uncertainty state (for the same $p_{\phi}^\star$ and $\Delta p_{\phi}$) is a function of $p_{\phi}$ and $\Delta p_{\phi}$. The wave function forms a symmetric bulge near the bounce point and the value $\Delta\phi\Delta p_{\phi}$ remains constant within regime of validity of its estimation (see Fig.\[fig:dphi1\]).
4. The direct construction of section \[s5.1\] yields results similar to those obtained by using the third method to choose the initial data for the $\phi$ evolution.
5. The differences between the relative dispersion $\Delta
\mu/\mu$ resulting from different methods of choosing initial data can also be estimated using the solutions to the effective dynamical equations (for details of the method see Appendix \[a1\]).
We will conclude with two remarks.
(i) Let us return to the comparison between the results of LQC and the theory in light of our numerical results. As remarked in section \[s2.2\], on functions $\Psi(\mu)$ which we used to construct the semi-classical initial data, the leading term in the difference $(\hat{C}_{\rm grav} - \hat{C}_{\rm grav}^{\rm
wdw})\Psi$ between the actions of the two constraint operators goes as $O(\mu_o^2) \Psi''''$. Now, in LQC $\mu_o$ is fixed, ($\mu_o= 3\sqrt{3}/2$), and on semi-classical states $\Psi''''
\sim k^{\star 2}/\mu^4\, \Psi$. Hence the difference is negligible only in the regime $k^{\star 4}/\mu^4 \ll 1$. In our simulations, $k^\star \sim 10^4$ whence the differences are guaranteed to be negligible only for $\mu \gg 10^4$, i.e., well away from the bounce. But let us probe the situation in greater detail. Let us regard $\mu_o$ as a mathematical parameter which can be varied and shrink it. In the limit $\mu_o \rightarrow 0$, $(\hat{C}_{\rm
grav} - C^{\rm wdw}_{\rm grav})\Psi$ should tend to zero. Since there is no bounce in the theory, we are led to ask: Would the LQC bounce continue to exist all the way to $\mu_o=0$ or is there a critical value at which the bounce stops? The answer is that for any finite value of $\mu_o$ there is a bounce. However, if we keep the *physical* initial data the same, we find that as we decrease $\mu_o$ the solution follows the classical trajectory into the past more and more and bounce is pushed further and further in to the past. In the limit as $\mu_o$ goes to zero, the wave function follows the classical trajectory into infinite past, i.e., the bounce never occurs. This is the sense in which the result is recovered in the limit $\mu_o \rightarrow
0$.
(ii) Numerical simulations show that the matter density at the bounce points is inversely proportional to the expectation value $\langle \hat{p}_\phi\rangle \equiv p_\phi^\star$ of the Dirac observable $\hat{p}_\phi$: Given two semi-classical states with $\langle \hat{p}_\phi \rangle = p_\phi^\star$ and $\bar{p}_\phi^\star$, we have $\rho_{\rm crit}/ \bar{\rho}_{\rm
crit} = \bar{p}^\star_\phi/p^\star_\phi$. Therefore, this density can be made small by choosing sufficiently large $p_\phi^\star$. Physically, this is unreasonable because one would not expect departures from the classical theory until matter density becomes comparable to the Planck density. This is a serious weakness of our framework. Essentially every investigations within LQC we are aware of has this —or a similar— drawback but it did not become manifest before because the physics of the singularity resolution had not been analyzed systematically. The origin of this weakness can be traced back to details of the construction of the Hamiltonian constraint operator, specifically the precise manner in which the operator corresponding to the classical field strength $F_{ab}^i$ was introduced. The physical idea that in LQC the operator corresponding to the field strength $F_{ab}^i$ should be defined through holonomies, and that quantum geometry does not allow us to shrink the loop to zero size, seem compelling. However, the precise manner in which the value of $\mu_o$ was determined using the area gap $\Delta$ is not as systematic and represents only a ‘first stab’ at the problem. In [@aps3] we will discuss an alternate and more natural way of implementing this idea. The resulting Hamiltonian constraint has a similar form but also important differences. Because of similarities the qualitative conclusions of this analysis —including the occurrence of the quantum bounce— are retained but the differences are sufficiently important to replace the expression of the critical density by $\rho_{\rm crit}^\prime = (3/8\pi
G\gamma^2 \Delta)$ where, as before, $\Delta = 2\sqrt{3}\pi\gamma
\lp^2$ is the area gap. Since $\rho_{\rm crit}^\prime$ is of Planck scale and is independent of parameters associated with the semi-classical state, such as $p_\phi^\star$, the departures from classical theory now appear only in the Planck regime. This issue is discussed in detail in [@aps3].
Discussion {#s6}
==========
We will first present a brief summary and then compare our analysis with similar constructions and results which have appeared in the literature. However, since this literature is vast, to keep the discussion to a manageable size, these comparisons will be illustrative rather than exhaustive.
In general relativity, gravity is encoded in space-time geometry. A basic premise of LQG is that geometry is fundamentally quantum mechanical and its quantum aspects are central to the understanding of the physics of the Planck regime. In the last three sections, we saw that LQC provides a concrete realization of this paradigm. In our model every classical solution is singular and the singularity persists in the theory. The situation is quite different in LQC. As in full LQG, the kinematical framework of LQC forces us to define curvature in terms of holonomies around closed loops. The underlying quantum geometry of the full theory suggests that it is physically incorrect to shrink the loops to zero size because of the ‘area gap’ $\Delta$. This leads to a replacement of the differential equation by a difference equation whose step size are dictated by $\Delta$. Careful numerical simulations demonstrated in a robust fashion that the classical big-bang is now replaced by a quantum bounce. Thus, with hindsight, one can say that although the theory is quite similar to LQC in its structure, the singularity persists in the theory because it ignores the quantum nature of geometry.[^9]
Emergent time
In this paper we isolated the scalar field $\phi$ as the emergent time and used it to motivate and simplify various constructions. However, we would like to emphasize that this —or indeed any other— choice of emergent time is not *essential* to the final results. In the classical theory (for any given value of the constant of motion $p_\phi$), we can draw dynamical trajectories in the $\mu-\phi$ plane without singling out an internal clock. A complete set of Dirac observables can be taken to be $p_\phi$, and either $|\mu|_{\phi_o}$ *or* $\phi|_{\mu_o}$.[^10] What these observables measure is correlations and their specification singles out a point in the reduced phase space. However, we do not have to single out a time variable to define them. The same is true in quantum theory. To have complete control of physics, we need to construct the physical Hilbert space $\Hp$ and introduce on it a complete set of Dirac observables. Again, both these steps can be carried out without singling out $\phi$ as emergent time. For example, the scalar product can be constructed using group averaging which requires only the knowledge of the full quantum constraint and its properties, and not its decomposition into a ‘time evolution part’ ${\partial}_\phi^2$ and an operator $\Theta$ on the ‘true degrees of freedom’. Once the scalar product is constructed, we can introduce a complete set of Dirac observables consisting of $\hat{p}_\phi$, and $\widehat{|\mu|_{\phi_o}}$ *or* $\widehat{\phi|_{\mu_o}}$. Again, what matters is the correlations. This and related issues have been discussed exhaustively in the quantum gravity literature in relativity circles. In particular, a major part of a conference proceedings [@asbook] was devoted to it in the late eighties and several exhaustive reviews also appeared in the nineties (see in particular [@cji; @kk]).
However, thanks to our knowledge of how quantum theory works in static space-times, singling out $\phi$ as the emergent time turned out to be extremely useful in practice because it provided guidance at several intermediate steps. In particular, it directly motivated our choice of $L^2(\R_{\rm Bohr}, B(\mu)
\dd\mu_{\rm Bohr}) \otimes L^2(\R, \dd\phi)$ as our auxiliary Hilbert space; stream-lined the detailed definition of operators representing the Dirac observables; and facilitated the subsequent selection of the inner product by demanding that these be self-adjoint. More importantly, by enabling us to regard the constraint as an evolution equation, it transformed the ‘frozen formalism’ to a familiar language of ‘evolution’ and enabled us to picture and interpret the bounce and associated physics more easily. Indeed, following the lead of early LQC papers, initially we tried to use $\mu$ as time and ran in to several difficulties: specification of physically interesting data became non-intuitive and cumbersome; one could not immediately recognize the occurrence of the bounce; and the physics of the singularity resolution remained obscure.
Our specific identification and use of emergent time differs in some respects from that introduced earlier in the literature. For example, in the context of the theory, there is extensive work on isolating time in the WKB approximation (see e.g., [@ck; @wkb]). By contrast, a key feature of our emergent time is that it is not restricted to semi-classical regimes: We first isolated the scalar field $\phi$ as a variable which serves as a good internal clock away from the singularity in the classical theory, but then showed that the form of the *quantum* Hamiltonian constraint is such that $\phi$ can be regarded as emergent time in the *full* quantum theory, without a restriction that the states be semi-classical or stay away from the singularity. The idea of identifying an emergent time and exploiting the resulting ‘deparametrization’ to select an inner product on the space of solutions to the Hamiltonian constraint is not new [@asbook; @aabook; @cji; @kk; @dm; @hm2]. However, several of the concrete proposals turn out to have serious deficiencies (for a further discussion see, e.g., [@greensite; @kiefertime]). The idea of using a matter field to define emergent time is rather old. In the framework of geometrodynamics, it was carried out in detail for dust in [@bktime]. A proposal to use a massless scalar field as time was also made in the framework of LQG [@lstime] but its implementation remained somewhat formal. In particular, it is unlikely that the required gauge conditions can be imposed globally in the phase space and the modifications in the construction of the physical scalar product that are necessary to accommodate a more local constructions were not spelled out. More recently, a massless scalar field was used as the internal clock in quantum cosmology in the connection dynamics framework [@kodama]. However, the focus of discussion there is on the Kodama state in inflationary models. Because of the inflationary potential, the quantum constraint has explicit time dependence and the construction of the physical inner product is technically much more subtle. In particular, a viable inner product cannot depend on any auxiliary structure such as the choice of an ‘instant of time’. These issues do not appear to have been fully addressed in [@kodama].
Resolutions of the Big-Bang Singularity
The issue of obtaining singularity free cosmological models has drawn much attention over the years. The discovery of singularity theorems sharpened this discussion and there is a large body of literature on how one may violate one or more assumptions of these theorems, thereby escaping the big bang. Proposals include the use of matter which violate the standard energy conditions, addition of higher derivative corrections to the Einstein-Hilbert action and introduction of higher dimensional scenarios inspired by string theory. To facilitate comparison with the model discussed in this paper, we will restrict ourselves to spatially non-compact situations.
Already in the seventies, Bekenstein investigated a model where the matter source consisted of incoherent radiation and dust, interacting with a conformal massless scalar field (which can have a negative energy density). He showed that Einstein’s equations admit solutions which are free of singularities [@bek]. In the eighties, Narlikar and Padmanabhan found a singularity free solution to Einstein’s equation with radiation and a *negative energy* massless scalar field (called the ‘creation field’) as source, and argued that the resulting model was consistent with the then available observations [@narlikar_paddy]. Such investigations were carried out entirely in the paradigm of classical relativity and the key difference from the standard Friedmann-Robertson-Walker models arose from the use of ‘non-standard’ matter sources. Our analysis, by contrast, uses a standard massless scalar field and *every* solution is singular in the classical theory. The singularity is resolved because of quantum effects.
Another class of investigations starts with actions containing higher derivative terms which are motivated by suitable fundamental considerations. For example, to guide the search for an effective theory of gravity which is viable close to the Planck scale, Mukhanov and Brandenberger proposed an action with higher order curvature terms for which all isotropic cosmological solutions are non-singular, even when coupled to matter [@mukh_bran]. The modifications to Einstein’s equations are thought of as representing quantum corrections. However one continues to work with differential equations formulated in the continuum. By contrast, our investigation is carried out in the framework of a genuine quantum theory with a physical Hilbert space, Dirac observables and detailed calculations of expectation values and fluctuations. Departures from classical general relativity arise directly from the quantum nature of geometry. The final results are also different: While solutions in [@mukh_bran] asymptotically approach de Sitter space, in our analysis the classical big-bang is replaced by a quantum bounce.
Perhaps the most well-known discussions of bounces come from the pre-big-bang cosmology and Ekpyrotic/Cyclic models. The pre-big-bang model uses the string dilaton action and exploits the scale factor duality to postulate the existence of a super-inflating pre-big-bang branch of the Universe, joined to the radiation dominated post-big-bang branch [@pbb1]. However, the work was carried out in the framework of perturbative string theory and the transition from the pre-big-bang to post-big-branch was *postulated*. The initial hope was that non-perturbative stringy effects would enforce such a transition. However, as of now, such mechanisms have not been found [@pbb2]. Although subsequent investigations have shown that a bounce can occur in simplified models [@pbb3] or by using certain effective equations (see, e.g., [@pbb4]), it is not yet clear that this is a consequence of the fundamental theory. The Ekpyrotic and the more recent Cyclic models [@ekp1; @ekp2] are motivated by certain compactifications in string theory and feature a five dimensional bulk space-time with a 4-d branes as boundaries. In the Ekpyrotic model, the collision between a bulk brane with a boundary brane is envisioned as a big bang. A key difficulty is the singularity associated with this collision [@kklt] (which can be avoided but at the cost of violating the null energy condition [@ekp2]). In the Cyclic model, collision occurs between the boundary branes [@ekp2], however it has been shown that the singularity problems persists [@ffkl]. Thus, a common limitation of these models is that the branch on ‘our side’ of the big-bang is not joined deterministically to the branch on the ‘other side’. In LQC by contrast, the quantum evolution is fully deterministic. This is possible because the approach is non-perturbative and does not require a space-time continuum in the background.
Finally, the idea of a bounce has been pursued also in the context of braneworld models. In the original Randall-Sundrum scenario, the Friedmann equation *is* modified by addition of a term on the right side which is quadratic in density: $\dot{a}^2/a^2 =
(8\pi G/3)\, \rho(1 + \rho/(2 \sigma))$ where $\sigma$ is brane tension. However, since $\sigma >0$, the sign of the quadratic term in $\rho$ is positive whence $\dot{a}$ can not vanish and there is no bounce. To obtain a bounce, the correction should be negative, i.e., make a ‘repulsive’ contribution. One way to reverse the sign is to introduce a second time-like dimension in the bulk [@shtanov_sahni]. However, this strategy does not appear to descend from fundamental considerations and the physical meaning of the second time-like direction is also unclear. Another avenue is to consider a bulk with a charged black hole. A non-vanishing charge leads to terms in the modified Friedmann equation which are negative. $\dot{a}$ can now vanish and a bounce can occur [@mp_brane]. However it was shown that transition from contraction to expansion for the brane trajectory occurs in the Cauchy horizon of the bulk which is unstable to small excitations, thus the brane encounters singularity before bouncing [@hov_myers]. In LQC, by contrast, while the Friedmann equation *is* effectively modified, the corrections come from quantum geometry and they are automatically negative.
Extensions
A major limitation of our analysis—shared by all other current investigations in quantum cosmology— is that the theory is not developed by a systematic truncation of full quantum gravity. This is inevitable because we do not have a satisfactory quantum gravity theory which can serve as an unambiguous starting point. The viewpoint is rather that one should use lessons learned from mini and midi superspace analysis to work one’s way up to more general situations, especially to reduce the large number of ambiguities that exist in the dynamics of the full theory.
Even within quantum cosmology, our detailed analysis was restricted to a specific mini-superspace model. In more complicated models, differences are bound to arise. For example, the full solution is not likely to remain so sharply peaked on the classical trajectory till the bounce point and even the existence of a bounce is not a priori guaranteed, especially when inhomogeneities are added. However, the *methods* developed in the paper can be applied to more general situations. First, one could consider anisotropies. Now, the main structural difference is that the operator $\Theta$ will no longer be positive definite. However, a detailed analysis shows that what matters is just the operator $|\Theta|$, obtained by projecting the action of $\Theta$ to the positive eigenspace in its spectral decomposition. Therefore, our analytical considerations should go through without a major modification. The numerical simulations will be more complicated because we have to solve a higher dimensional difference equation (involving 4 variables in place of 2). Another extension will involve the inclusion of non-trivial potentials for the scalar field. Now, generically $\phi$ will no longer be a monotonic function on the classical trajectories and one would not be able to use it as ‘internal time’ globally. In the quantum theory, the operator $\Theta$ becomes ‘time-dependent’ (i.e. depends on $\phi$) and the mathematical analogy between the quantum constraint and the Klein-Gordon equation in a static space-time is no longer valid. Nonetheless, one can still use the group averaging procedure [@dm; @hm2] to construct the physical Hilbert space. For a general potential, a useful notion of time will naturally emerge only in the semi-classical regimes. For specific potentials (such as the quadratic one used in chaotic inflation) one should be able to use methods that have been successfully employed in the quantization of model systems [@at; @lr] (in particular, a pair of harmonic oscillators constrained to have a fixed total energy).
Incorporation of spherical inhomogeneities seems to be within reach since significant amount of technical groundwork has already been laid [@mb-ss]. Incorporation of general inhomogeneities, on the other hand, will be substantially more difficult. Background dependent treatments have suggested that results obtained in the mini-superspace approximation may be qualitatively altered once field theoretical complications are unleashed (see, e.g., [@hh1]). However, already in the anisotropic case, there is a qualitative difference between perturbative and non-perturbative treatments. Specifically, if anisotropies are treated as perturbations of a background isotropic model, the big-bang singularity is not resolved while if one treats the whole problem non-perturbatively, it is [@mb-aniso]. Therefore definitive conclusions can not be reached until detailed calculations have been performed in inhomogeneous models. However, if a quantum bounce does generically replace the big bang singularity, it would be possible to explore the relation between the effective descriptions of LQG and the Hartle-Hawking ‘no boundary’ proposal [@hh2]. For, in the effective description, the extrinsic curvature would vanish at the bounce. Therefore generically it may be possible to attach to the Lorentzian, post-bounce effective solution representing the universe at late times, a *Riemannian* pre-bounce solution without boundary. If so, it would be very interesting to analyze the sense in which this Riemannian solution captures the physics of the pre-bounce branch of the full quantum evolution.
Finally, it is instructive to recall the situation with singularities in classical general relativity. There, singularities first appeared in highly symmetric situations. For a number of years, arguments were advanced that this is an artifact of symmetry reduction and generic, non-symmetric solutions will be qualitatively different. However, singularity theorems by Penrose, Hawking, Geroch and others showed that this is not correct. An astute use of the *differential geometric* Raychaudhuri equation revealed that singularities first discovered in the simple, symmetric solutions are in fact a generic feature of classical general relativity. A fascinating question is whether the singularity resolution due to quantum geometry is also generic in an appropriate sense [@nd]. Is there a general equation in *quantum geometry* which implies that gravity effectively becomes repulsive near generic space-like singularities, thereby halting the classical collapse? If so, one could construct robust arguments, now establishing general ‘singularity resolution theorems’ for broad classes of situations in quantum gravity, without having to analyze models, one at a time.
**Acknowledgments:** We would like to thank Martin Bojowald, Jim Hartle and Pablo Laguna for discussions. This work was supported in part by the NSF grants PHY-0354932 and PHY-0456913, the Alexander von Humboldt Foundation, and the Eberly research funds of Penn State.
Heuristics {#a1}
==========
Quantum corrections to the classical equations can be calculated using ideas from a geometric formulation of quantum mechanics where the Hilbert space is regarded as (an infinite dimensional) phase space, the symplectic structure being given by the imaginary part of the Hermitian inner product (see, e.g., [@as]). This ‘quantum phase space’ has the structure of a bundle with the classical phase space as the base space, and all states with the same expectation values for the canonically conjugate operators $(\hat{q}^i,\, \Hat{p}_i)$ as an (infinite dimensional) fiber. Thus, any horizontal section provides an embedding of the classical phase space into the ‘quantum phase space’. In the case of a harmonic oscillator (or free quantum fields) coherent states constitute horizontal sections which are furthermore preserved by the full quantum dynamics. In the semi-classical sector defined by these coherent states, the effective Hamiltonian coincides with the classical Hamiltonian and there are no quantum corrections to classical dynamics. For more general systems, using suitable semi-classical states one may be able to find horizontal sections which are preserved by the quantum Hamiltonian flow to a desired accuracy (e.g. in an $\hbar$ expansion). The effective Hamiltonian governing this flow —the expectation value of the quantum Hamiltonian operator in the chosen states, calculated to the desired accuracy— is generally different from the classical Hamiltonian. In this case, dynamics generated by the effective Hamiltonian provides systematic quantum corrections to the classical dynamics [@jw] (see also [@dh1; @dh2; @sv]).
This procedure has been explicitly carried out in LQC for various matter sources [@jw; @vt]. For a massless scalar field, the leading order quantum corrections are captured in the following effective Hamiltonian constraint [@vt]: \[heff0\] C\_ = - [6]{}[\^2 \_o\^2]{} |p|\^[[1]{}[2]{}]{}\^2(\_o c) + 8GB(p) [p\_\^2]{} where $B(p)$ is the eigenvalue of $\widehat{1/|p|^{3/2}}$ operator given by (\[eq:bp\]). [^11] For $|\mu| \gg \mu_o$, $B(p)$ can be approximated as B(p) = ([6]{}[8 \^2]{})\^[3/2]{} ||\^[-3/2]{} (1 + [5]{}[96]{} [\_o\^2]{}[\^2]{} + [ O]{}([\_o\^4]{}[\^4]{}) ) . The leading order term is $1/p^{3/2}$, thus $B(p)$ quickly approaches its classical value for $|\mu| \gg\mu_o$, corrections being significant only in the ‘genuinely quantum region’ in the vicinity of $\mu=0$. *From now on, we will ignore the quantum corrections to $B(p)$.*
![Expectation values (and dispersions) of $\widehat{|\mu|_{\phi}}$ are plotted near the bounce point, together with with classical and effective trajectories (fainter and darker dots, respectively). While the classical description fails in this region, the effective description provides an excellent approximation to the exact quantum evolution. In this plot, $p_\phi = 3000$ and $\epsilon = 2
\mu_o$.[]{data-label="fig:mu-traj-zoom"}](new_traj2.eps){width="5in"}
To obtain the equations of motion, we need the effective Hamiltonian ${{\cal H}_{\mathrm{eff}}}$. As usual it is obtained simply by a rescaling of $C_{\rm eff}$ which gives ${{\cal H}_{\mathrm{eff}}}$ the dimensions of energy and ensures that the matter contribution to it is the standard matter Hamiltonian: \[heff1\] [[H]{}\_]{}= [C\_[eff]{}]{}[16G]{} = - [3]{}[8G\^2 \_o\^2]{} |p|\^[1/2]{} \^2(\_o c) + [p\_\^2]{}[2 p\^[3/2]{}]{}. Then, the Hamilton’s equation for $\dot p$ become: \[dotp\] p = {p,[[H]{}\_]{}} = - [8 G]{}[3]{} [c]{} = [(\_o c)]{} [(\_o c)]{} . Further, since the Hamiltonian constraint implies that ${{\cal H}_{\mathrm{eff}}}$ of (\[heff1\]) vanishes, we have: [\^2(\_o c)]{}= p\_\^2 , which, on using Eq.(\[dotp\]), provides the *modified Friedmann equation* for the Hubble parameter $H$: \[mod\_fried\] H\^2 [p\^2]{}[4 p\^2]{} = ( 1 - [[\_]{}]{} ), [\_]{}= ([3]{}[8 G \^2 \_o\^2]{})\^[3/2]{} [ p\_]{} . To obtain the dynamical trajectory, we also need the Hamilton’s equation for $\phi$, \[dotphi\] = {, [[H]{}\_]{}} = [p\_]{}[p\^[3/2]{}]{} . By combining Eq. (\[dotp\]) and (\[dotphi\]) we obtain the effective equation of motion in the $\mu$-$\phi$ plane: \[mod\_dmdf\] [d]{}[d ]{} = (1 - [[\_]{}]{})\^[1/2]{} . The classical Friedmann dynamics results if we set ${\rho_{\mathrm{crit}}}=\infty$. Eqs. (\[mod\_fried\]) and (\[mod\_dmdf\]) suggest that the LQC effects significantly modify the Friedmann dynamics once the matter density reaches a critical value, ${\rho_{\mathrm{crit}}}$. In the classical dynamics, the Hubble parameter $H$ can not vanish (except in the trivial case with $p_\phi =0$). In the modified dynamics, on the other hand, $H$ vanishes at $\rho ={\rho_{\mathrm{crit}}}$. At this point, the Universe bounces. Thus, the bounce predicted by Eq.(\[mod\_fried\]) has its origin in quantum geometry. (The critical value $\mu$ at which this bounce occurs is given by $\mu_{\mathrm{crit}} = (\sqrt{6/8 \pi G \hbar^2})\, p_\phi \, \mu_o$.) As pointed out in the main text, a physical limitation of the present framework is that if $p_\phi$ is chosen to be sufficiently large, the critical density ${\rho_{\mathrm{crit}}}$ can be small.
To obtain the effective equations, several approximations were made [@jw; @vt] which are violated in the deep Planck regime. Nonetheless, the resulting picture of the bounce is consistent with the detailed numerical analysis. In fact, within numerical errors the trajectory \[mu\_phi\] () = [1]{}[2]{} ( ( (- \_o)) + D(p\_) (- (- \_o)) ) obtained by integrating Eq (\[mod\_dmdf\]) approximates the expectation values of $\widehat{|\mu|_\phi}$ quite well. (As in the main text, $D(p_\phi) = \mu_o^2p_\phi^2/12\pi G\hbar^2$). An illustrative plot of this generic behavior is shown in Fig. \[fig:mu-traj-zoom\]. Therefore, in retrospect, this analysis can be taken as a justification for the validity of the approximation throughout the evolutionary history of semi-classical states used in this paper. However, by its very nature, the effective description can not reproduce the interesting features exhibited by quantum states captured in Figs. \[fig:zoom-gauss\]-\[fig:dphi1\].
![The darker ‘lattice’ show the difference between final and initial relative spreads $\Delta\mu/\mu$ for states obtained by evolving initial data of $\emph{Method II}$. The upper, fainter ‘lattice’ shows the heuristic bound $\delta\mu/\mu$ given by Eq (\[est\_wdw\]).[]{data-label="fig:bound-wdw"}](WDW-bound.eps){width="5in"}
![The darker ‘lattice’ show the difference between final and initial relative spreads $\Delta\mu/\mu$ for states obtained by evolving initial data of $\emph{Method III}$. The upper, fainter ‘lattice’ shows the heuristic bound $\delta\mu/\mu$ given by Eq (\[est\_sym\_wdw\]).[]{data-label="fig:bound-symm"}](sym-WDW-bound.eps){width="5in"}
However, the effective description can be used to provide an intuitive understanding of the behavior of various uncertainties discovered through numerical analysis. Let us first note that the position of the bounce point depends linearly on the value of $p_\phi$. Next, consider two near-by solutions with slightly different $p_\phi$ which asymptote to the same solution for the expanding branch in the distant future. We wish to know the way in which $(\delta\mu/\mu)(\phi) :=
((\mu_1-\mu_2)/\mu_2)(\phi)$ changes in the backward evolution as the two wave functions asymptote to solutions in the distant past. This relative difference can be found using Eq (\[mu\_phi\]) and is given by \[est\_wdw\] = 2 [p\_]{}[p\_]{} + ([p\_]{}[p\_]{} )\^2 , where $\delta p_\phi$ is the difference between values of $p_\phi$ of the two classical trajectories. A heuristic estimate on the relative difference in $\mu$ can be compared with the relative dispersion $\Delta \mu/\mu$ obtained from the *Method II* in the $\phi$ evolution of section \[s5.2\]. It turns out that estimate in Eq (\[est\_wdw\]) provides a reasonably good upper bound to the relative dispersion found numerically (see Fig. \[fig:bound-wdw\]). A similar comparison can be made for *Method III* of section \[s5.2\]. In this case the corresponding solution to Eq (\[mod\_dmdf\]) is () = [1]{}[2]{} (D(p\_)\^[-1/2]{} ( (- \_o)) + D(p\_)\^[1/2]{} (- (- \_o)) ) with relative difference \[est\_sym\_wdw\] = [p\_]{}[p\_]{} . A comparison with the relative dispersions in numerical analysis is shown in Fig. \[fig:bound-symm\]. As in the case of construction of coherent states via *Method II*, above estimate serves as an upper bound for relative dispersions computed by numerical analysis.
Finally, in the numerical analysis an important issue concerns with the behavior of dispersions of our Dirac observables $\hat
\mu_\phi$ and $\hat p_\phi$ and the product $\Delta \phi \Delta
p_\phi$. Intuitive understanding of our numerical results of Fig. \[fig:dphi1\] can be gained by casting Eq. (\[mod\_dmdf\]) in the form \[dp\_dm\_m\] = (1 - [[\_]{}]{})\^[-1/2]{} . Since $\Delta \mu/\mu$ can be determined numerically, we can then estimate $\Delta \phi$ throughout the evolution. The factor $\left(1 - {\frac}{\rho}{{\rho_{\mathrm{crit}}}}\right)^{-1/2}$ is approximately equal to unity for $\rho \ll {\rho_{\mathrm{crit}}}$ or equivalently for $\mu \gg
\mu_{\mathrm{crit}}$. However near the bounce point, $\left(1 -
{\frac}{\rho}{{\rho_{\mathrm{crit}}}}\right)^{-1/2} \gg 1$. In *Method II* and *Method III* of constructing initial states described in Sec. \[sec:phi-init-data\], this change compensates the corresponding decrease in $\Delta \mu/\mu$ and leads to a nearly constant value of $\Delta \phi$. However, for *Method I*, since $\Delta
\mu/\mu$ increases monotonically, the fluctuation $\Delta \phi$ increases significantly near the bounce point.
Issues in numerical analysis {#a2}
============================
In this Appendix we will spell out the way in which the limit of the eigenfunctions of $\Theta$ were found.
Consider a general eigenfunction $\ub{e}_{\omega}$ of $\ul{\Theta}$ ( for $\omega^2 \ge \pi G/3$). It is always a linear combination of basis functions $\ub{e}_{|k|},
\ub{e}_{-|k|}$ (where $k^2 = 3/(16\pi G)\omega^2 - 1/16$) defined in Eq. (\[eq:ek\]). For later convenience, let us express the linear combination as: $$\label{eq:e-dec}
\ub{e}_{\omega} = r^+ e^{i(\beta+\alpha)} \ub{e}_{|k|}
+ r^- e^{i(\beta-\alpha)} \ub{e}_{-|k|} \, ,$$ where $r_\pm,\alpha,\beta$ are real numbers. Since each $\ub{e}_{\pm|k|}$ is a product of an ‘amplitude’ $|\mu|^{\frac{1}{4}}/4\pi$ and a ‘phase’ $e^{\pm i|k|\ln|\mu|}$, it is natural to rescale $\ub{e}_{\omega}$: $$\tilde{\ub{e}}_{\omega}(\mu)
:= 4\pi|\mu|^{-\frac{1}{4}}\ub{e}_{\omega} \ .$$ In terms of coefficients defined in , we have $$\tilde{\ub{e}}_{\omega}(\mu)\ =\ e^{i\beta} \left(
(r^{+}+r^{-})\cos(\alpha+|k|\ln|\mu|)
+ i(r^{+}-r^{-})\sin(\alpha+|k|\ln|\mu|) \right) \ .$$ The values of $\t{\ub{e}}_{\omega}(\mu)$ trace out an ellipse on the complex plane, parameterized by $\ln|\mu|$. The length of semi-major and semi-minor axis of this ellipse is equal to, respectively
\[eq:axis\]$$\begin{aligned}
r^{+}+r^{-} &= \sup_{\mu}|\tilde{\ub{e}}_{\omega}| &
|r^{+}-r^{-}| &= \inf_{\mu}|\tilde{\ub{e}}_{\omega}| \ ,
\tag{\ref{eq:axis}}\end{aligned}$$
whereas the phase $\alpha$ is related to positions of maxima of $\tilde{\ub{e}}_{\omega}$ as follows $$\label{eq:phase}
|\ub{e}_{\omega}| = r^{+}+r^{-} \quad \Leftrightarrow \quad
\alpha+|k|\ln|\mu|\ =\ n\pi\ , \quad n\in{\mathbb{Z}}\ .$$ The remaining phase $\beta$ is just the phase of $\tilde{\ub{e}}_{\omega}$ at maximum. The sign of $(r^{+}-r^{-})$ is, on the other hand, determined by direction of the rotation of the curve as $\mu$ increases.
The method specified above allows us to calculate the decomposition in $\ub{e}_k$ basis of a function $\ub{e}_{\omega}$ specified in the form of numerical data (i.e. array of values at sufficiently large domain). The same algorithm can be applied to identify the WDW limit of any eigenfunction of the LQC operator $\Theta$. Indeed given an eigenfunction $e_{\omega}(\mu)$ supported on the lattice $\La_{|\epsilon|}$ (or $\La_{-|\epsilon|}$) one can again define $\tilde{e}_{\omega}$ analogously to $\tilde{\ub{e}}_{\omega}$ and find its (local) extrema for large $\mu$. (For definiteness, we restrict our consideration to finding the limit on the positive $\mu$ side, however this method can be used of course also for the negative $\mu$ domain.) Next, the positions of extremas and values of $\tilde{e}_{\omega}$ at them can be used to calculate coefficients $r^{+}\pm r^{-},\alpha,\beta$ at each extremum independently. If $\tilde{e}_{\omega} \to \tilde{\ub{e}}_{\omega}$ then these coefficients form sequences $(\{r^++r^-\}_i, \{r^+-r^-\}_i,
\{\alpha\}_i, \{\beta\}_i)$ which converge to the analogous coefficients corresponding to $\tilde{\ub{e}}_{\omega}$ as $\mu
\to \infty$. Finding the WDW limit of $\tilde{e}_{\omega}$ reduces then to finding the limit of $(\{r^++r^-\}_i, \{r^+-r^-\}_i,
\{\alpha\}_i, \{\beta\}_i)$.
In actual numerical work the following method was used:
- After given eigenfunction $e_{\omega}$ was calculated using the positions of extrema $\{\mu\}_i$ were found.
- Around the extrema the function $e_{\omega}$ supported on $\La_{|\epsilon|}$ (or $\La_{-|\epsilon|}$) was extended to neighborhoods of $\{\mu\}_i$ via polynomial interpolation. Then the positions and values of extrema were recalculated with use of this extension. This allowed us to construct sequences converging to the WDW limit much more quickly than the ones constructed in the first step. The motivation for this construction is the expectation that for sufficiently large $\mu$ the values of $e_{\omega}$ at $\La_{|\epsilon|}$ should be good estimates of its WDW limit (being regular function defined on entire ${\mathbb{R}}^+$).
- Extrema found in previous step were next used to calculate sequences $(\{r^++r^-\}_i, \{r^+-r^-\}_i, \{\alpha\}_i,
\{\beta\}_i)$ in a way analogous to that specified by Eqs. (\[eq:axis\])and (\[eq:phase\]) and in the description below them.
- Finally the limits of coefficients at ${1}/{\mu} \to 0$ were calculated by polynomial extrapolation.
An alternate physical Hilbert space {#a3}
===================================
In this Appendix we will construct a physical Hilbert space $\Hp'$ in LQC which is qualitatively different from the spaces $\Hp^\epsilon$ constructed in section \[s4.2\]. In its features, it interpolates between these and the Hilbert space $\Hpwdw$ of section \[s3.2\]. For completeness, we will first explain why a new representation of the algebra of Dirac observables can arise, then summarize the results and finally compare and contrast them with those obtained in sections \[s3.2\] and \[s4.2\]. The first part is somewhat technical but we have organized the presentation such that the readers can go directly to the summary without loss of continuity.
Let us begin by recalling the situation for a general system with a single constraint $C$. In the refined version of Dirac quantization [@almmt], one introduces an auxiliary Hilbert space $\Ha$, and represents the constraint by a self-adjoint operator $\hat{C}$ on it. The technically difficult task is to chooses a dense sub-space $\Phi$ of $\Ha$ such that for all $f,g
\in \Phi$, (\_f| := \_[-]{}\^d e\^[-i]{} f|is a well-defined element of $\Phi^\star$, such that the action $(\Psi_f|g\rangle$ of $(\Psi_f|\in \Phi^\star$ on $|g\rangle\in
\Phi$ yields a Hermitian scalar product on the space of solutions $(\Psi_f|$ to the quantum constraint (see, e.g. [@dm; @almmt; @abc]). Results in section \[s4.2\] were obtained using $L^2(\R_{\rm Bohr}, B(\mu)\,\dd\mu_{\rm Bohr})\otimes
L^2(\R,\dd\phi)$ for $\Ha$, and the space of rapidly decreasing functions $f(\mu, \phi)$ in this $\Ha$ for $\Phi$. In the LQC literature, $\Phi$ is sometimes called $\cyl$ and $\Phi^\star$ is taken to be its algebraic dual, denoted by $\cyl^\star$.
The construction given above is rather general. For the model under consideration, we can extend this construction by using an entirely different subspace of $\Phi^\star$ for the auxiliary Hilbert space. This is possible because by duality the action of $\hat{C}$ can be extended to all of $\Phi^\star$. Let us set $\Ha^\prime := L^2(\R^2, B(\mu)\dd\mu\dd\phi)$. This is a subspace of $\Phi^\star$ because each $\Psi \in \Ha^\prime$ defines a linear map from $\Phi$ to $\Comp$: (|f:= \_\_[-]{}\^B() |(,) f(, ) \^where the sum over $\mu$ converges because $f \in \Phi$ has support only on a countable number of points on the $\mu$-axis and a rapid fall-off. The dual action of $\hat{C}$ on $\Ha^\prime$ can now be calculated: Since (|f= \_\_[-]{}\^B() | it follows from the definitions of $C^\pm(\mu)$ that ()(,) &=& [\^2 ]{}[\^2]{} + \[B()\]\^[-1]{} (C\^+() (+4\_o, ) + C\^o() (, ) + C\^[-]{}() (-4\_o))\
&& (,)It is straightforward to verify that $\hat\Theta$ and $\hat{C}$ are self-adjoint on $\Ha^\prime$. Therefore, we can carry out group averaging on $\Ha'$ and obtain a new physical Hilbert space. As in the theory of section \[s3.2\], there are two superselected sectors. We will work with the positive frequency sector and denote it by $\Hp^\prime$. The Dirac observables $\hat{p}_\phi$ and $\widehat{|\mu|_{\phi_o}}$ on $\Phi$ act by duality on $\Ha^\prime$ and descend naturally to $\Hp^\prime$.
The final results can be summarized as follows. The new physical Hilbert space $\Hp^\prime$ is the space of functions $\Psi(\mu,\phi)$ satisfying the positive frequency equation: $-i{\partial}_\phi \Psi = \sqrt{\Theta} \Psi$, with finite norm: \[ip3\]||||\^[ 2]{}\_[phy]{} = \_[=\_o]{} dB() |(,)|\^2 and the action of the Dirac observables is the standard one: \[dirac5\] (,) = e\^[i(-\_o)]{}|| (,\_o), \_(,) = - i [(,)]{} . Note that the final physical theory is different from both the theory of \[s3.2\] and the ‘standard’ LQC theory of \[s4.2\]. Since the inner product (\[ip3\]) involves an integral rather than a sum, states $\Psi(\mu,\phi)$ now have support on continuous intervals of the $\mu$-axis as in the theory, rather than on a countable number of points as in LQC. However, the states satisfy the LQC type positive frequency equation $-i{\partial}_\phi \Psi = \sqrt{\Theta} \Psi$, where the operator on the right is the square-root of a positive, self-adjoint *difference* operator rather than of a differential operator, and the measure determining the inner product also involves $B(\mu)$ from LQC rather than $\ub{B}(\mu)$ from the theory. Thus, the dynamical operator is the same as in LQC. *In particular, as in section \[s5\], the quantum states exhibit a big bounce.* However, since typical states are continuous on the $\mu$-axis, the spectrum of the Dirac observable $\widehat{|\mu|_{\phi_o}}$ is now continuous. In essence, the states $\Psi(\mu,\phi)$ have support on *2-d continuous* regions of the $\mu-\phi$ plane as in the theory but their dynamics is dictated by a *difference* operator as in LQC. When the cosmological constant is non-zero, the analog of this physical Hilbert space appears to be a natural home to analyze the role of the Kodama state in quantum cosmology [@kodama].
In the literature on ‘polymer representations’, non-relativistic quantum mechanics of point particles and the quantum theory of a Maxwell theory have been discussed in some detail[@afw; @almax]. In the first case, the standard Schrödinger Hilbert space $L^2(\R, \dd x)$, and in the second case, the standard Fock space turned out to be subspaces of $\cyl^\star$ which were especially helpful for semi-classical analysis. The present auxiliary Hilbert space $\Ha' \subset \Phi^\star$ is completely analogous to these. Therefore the resulting $\Hp^\prime$ may be more useful for semi-classical considerations. Indeed, since it does not refer to any $\epsilon$, no coarse graining is required to carry out the semi-classical analysis. Therefore the analog of this construction may well be useful in more general contexts in full LQG.
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[^1]: As we will see in section \[s2\], to construct a Hamiltonian framework in the open model, one has to fix a fiducial cell. The scale factor $a$ (and the momentum $p$ conjugate to the gravitational connection introduced later) refers to the volume of this cell. Alternatively, one can avoid the reference to the fiducial cell by fixing a $\phi_o$ and considering the ratios $a|_\phi/a_{\phi_o}$ as Dirac observables. However, for simplicity of presentation we will not follow this route.
[^2]: Our conventions are somewhat different from those in the literature, especially [@abl]. First, we follow the standard quantum gravity convention and set $\lp^2 = G\hbar$ (rather than $8\pi G\hbar$). Second, we follow the general convention in geometry and set the volume element $e$ on $\M$ to be $e :=
\sqrt{|\det E|}$ (rather than $e := \sqrt{|\det E|}\, {\rm sgn}
(\det E)$). This gives rise to some differences in factors of ${\rm sgn} \, p$ in various terms in the expression of the Hamiltonian constraint. Finally, the role of the the minimum non-zero eigenvalue of area is spelled out in detail, and the typographical error in the expression of $\mu_o$ that features in the Hamiltonian constraint is corrected.
[^3]: Our conventions are such that $\tau_i\, \tau_j =
{\frac}{1}{2}\epsilon_{ijk}\tau^k - {\frac}{1}{4} \delta_{ij}$. Thus, $2i
\tau_k = \sigma_k$, where $\sigma_i$ are the Pauli matrices.
[^4]: Here, and in what follows, quantities with an underbar will refer to the theory.
[^5]: Note that this fundamental evolution equation makes no reference to the Barbero-Immirzi parameter $\gamma$. If we set $\t\mu = \mu/\mu_o$ and $\t\Psi(\t\mu, \phi) = \Psi(\mu,\phi)$, the equation satisfied by $\t\Psi(\t\mu,\phi)$ makes no reference to $\mu_o$ either. *This is the equation used in numerical simulations.* To interpret the results in terms of scale factor, however, values of $\gamma$ and $\mu_o$ become relevant.
[^6]: A rigorous version of this argument can be constructed e.g. by using the Gel’fand triplet [@gs] associated with the operator $\Theta$. However, this step has not been carried out.
[^7]: If $\omega^2 < \pi G/3$, then $\omega^2$ is not part of the spectrum of the self-adjoint operator $\ul\Theta$. Nonetheless, by directly solving the eigenvalue equation $\ul\Theta \ub{e}_\omega = \omega^2 \ub{e}_\omega$ one can introduce an analogous decomposition onto fixed eigenfunctions, $\ul{e}_{\pm|k'|} := |\mu|^{\frac{1}{4}\pm k'} \ ,
\quad {k'}^2 = 1/16 - 3/(16\pi G)\omega^2$ and also write the limit of $e_{\omega}$ in terms of coefficients in this ‘basis’.
[^8]: Unfortunately for the $\epsilon=0$ sector the equation is singular at $\mu=0$, so the analysis of this sub-section will not go through. This sector was handled by the direct evaluation of the integral representation of the solution, presented in the last sub-section.
[^9]: Differences arise in two places. The first occurs in the matter part of the constraint and stems from the fact that the functions $B(\mu)$ representing the eigenvalues of the operator $\widehat{1/|p|^{3/2}}$ in LQG is different from the corresponding $\ub{B}(\mu)$ of the theory. The second comes from the role of quantum geometry in the gravitational part of the Hamiltonian constraint, emphasized above. In our model, qualitatively new features of the LQG quantum dynamics can be traced back to the second. In particular, the bounce would have persisted even if we had used $\ub{B}(\mu)$ in place of $B(\mu)$ in the analysis presented in this paper.
[^10]: In the closed models, care is needed to specify the latter because they can not be defined globally. But this issue is well-understood in the literature, especially through Rovelli’s contributions [@crtime]. See also [@dm].
[^11]: In the literature, eigenvalues $B(p)$ often contain a half-integer $j$, a parameter representing a quantization ambiguity. In view of the general consistency arguments advanced in [@ap], we have set its value to its minimum, i.e. $j =
1/2$.
|
---
abstract: 'First results on the longitudinal asymmetry and its effect on the pseudorapidity distributions in collisions at [$\sqrt{s_{\mathrm{NN}}}$]{}= 2.76 TeV at the Large Hadron Collider are obtained with the ALICE detector. The longitudinal asymmetry arises because of an unequal number of participating nucleons from the two colliding nuclei, and is estimated for each event by measuring the energy in the forward neutron-Zero-Degree-Calorimeters (ZNs). The effect of the longitudinal asymmetry is measured on the pseudorapidity distributions of charged particles in the regions $|\eta| < 0.9$, $2.8 < \eta < 5.1$ and $-3.7 < \eta < -1.7 $ by taking the ratio of the pseudorapidity distributions from events corresponding to different regions of asymmetry. The coefficients of a polynomial fit to the ratio characterise the effect of the asymmetry. A Monte Carlo simulation using a Glauber model for the colliding nuclei is tuned to reproduce the spectrum in the ZNs and provides a relation between the measurable longitudinal asymmetry and the shift in the rapidity ($y_{\mathrm{0}}$) of the participant zone formed by the unequal number of participating nucleons. The dependence of the coefficient of the linear term in the polynomial expansion, $c_{\rm 1}$, on the mean value of $y_{\mathrm{0}}$ is investigated.'
bibliography:
- 'biblio.bib'
title: |
Longitudinal asymmetry and its effect on pseudorapidity distributions\
in collisions at [$\mathbf{{\sqrt{\textit{s}_{NN}}}}$]{} = 2.76 TeV
---
Introduction {#sec:intro}
============
In a heavy-ion collision, the number of nucleons participating from each of the two colliding nuclei is finite, and will fluctuate event-by-event. The kinematic centre of mass of the participant zone, defined as the overlap region of the colliding nuclei, in general has a finite momentum in the nucleon-nucleon centre of mass frame because of the unequal number of nucleons participating from the two nuclei. This momentum causes a longitudinal asymmetry in the collision and corresponds to a shift of rapidity of the participant zone with respect to the nucleon-nucleon centre of mass (CM) rapidity, termed the rapidity-shift $y_{\mathrm{0}}$. The value of $y_{\rm 0}$ is indicative of the magnitude of the longitudinal asymmetry of the collision [@Vovchenko:2013viu; @Raniwala:2016ugm]. Assuming the number of nucleons participating from each of the two nuclei is A and B, the longitudinal asymmetry in participants is defined as $\alpha_{\rm {part}} = \frac{A-B}{A+B}$ and the rapidity-shift can be approximated as $y_{\rm 0} \cong
\frac{1}{2}ln\frac{A}{B}$ at LHC energies [@Raniwala:2016ugm].
The shift in the CM frame of the participant zone, which evolves into a state of dense nuclear matter, needs to be explored in heavy-ion collision models. Comparison of model predictions with the observed $\Lambda$-polarisation, possibly due to vorticity from the initial state angular momentum surviving the evolution, requires a precise determination of initial conditions and hence the shift in the CM frame [@STAR:2017ckg; @Becattini:2013vja; @Xie:2017upb]. Such a shift may also affect observations on correlations amongst particles, which eventually provide information about the state of the matter through model comparisons. Further, the resultant decrease in the CM energy may affect various observables including the particle multiplicity. The transverse spectra are known to be affected by the initial geometry of the events, as estimated through techniques of event shape engineering, indicating an interplay between radial and transverse flow [@Adam:2015eta]. The measurement of longitudinal asymmetry will provide a new parameter towards event shape engineering, affecting many other observables.
The simplest of all possible investigations into the effect of longitudinal asymmetry is a search for modification of the kinematic distribution of the particles. The pseudorapidity distribution (${\mathrm{d}N/\mathrm{d}\eta}$) of soft particles, averaged over a large number of events, is symmetric in collisions of identical nuclei. These distributions were observed to be asymmetric in collisions of unequal nuclei such as [@Back:2003hx] and [@ALICE:2012xs; @Adam:2014qja; @Aad:2015zza] and have been explained in terms of the rapidity-shift of the participant zone [@Steinberg:2007fg]. In a heavy-ion collision, the effect of the rapidity-shift of the participant zone should be discernible in the distribution of produced particles. This small effect can be estimated by taking the ratio of pseudorapidity distributions in events corresponding to different longitudinal asymmetries [@Raniwala:2016ugm].
It was suggested that the rapidity distribution of an event, scaled by the average rapidity distribution, can be expanded in terms of Chebyshev polynomials, where the coefficients of expansion are measures of the strength of longitudinal fluctuations and can be determined by measuring the two particle correlation function [@Bzdak:2012tp]. Using the same methodology, the event-by-event pseudorapidity distributions are also expanded in terms of Legendre polynomials [@Jia:2015jga]. The ATLAS collaboration expanded the pseudorapidity distributions in terms of Legendre polynomials and obtained the coefficients by studying pseudorapidity correlations [@Aaboud:2016jnr].
In the present work, the events are classified according to the asymmetry determined from the measurement of energies of neutron spectators on both sides of the collision [@Raniwala:2016ugm]. The effect of asymmetry is investigated by taking the ratio of the measured raw ${\mathrm{d}N/\mathrm{d}\eta}$ distributions for events from different regions of the distribution of measured asymmetry. A major advantage of studying this ratio is the cancellation of (i) systematic uncertainties and (ii) the effects of short range correlations. The first measurements of the effect of asymmetry on the raw ${\mathrm{d}N/\mathrm{d}\eta}$ distributions are reported here. The paper is organised as follows: Sect. \[sec:expdetail\] provides an introduction to the experimental setup and the details of the data sample. Section \[sec:Analysis\] discusses the characterisation of the change in raw ${\mathrm{d}N/\mathrm{d}\eta}$ distributions for events classified in different asymmetry regions. Section \[sec:Simula\] describes the simulations employed to provide a relation between the measured asymmetry and the rapidity-shift $y_{\rm 0}$ of the participant zone. The relation between the parameter characterising the change in raw ${\mathrm{d}N/\mathrm{d}\eta}$ distributions is shown for different centralities in Sect. \[sec:Result\], along with its relation to the estimated values of $y_{\mathrm{0}}$.
Experimental details and data sample {#sec:expdetail}
====================================
The analysis uses data from collision events at [$\sqrt{s_{\mathrm{NN}}}$]{}= 2.76 TeV, recorded in the ALICE experiment in 2010, with a minimum bias trigger [@Aamodt:2010pa; @Aamodt:2010cz]. The data used in the present analysis is recorded in the neutron Zero Degree Calorimeters (ZNs), the V0 detectors, the Time Projection Chamber (TPC) and the Inner Tracking System (ITS). Both ZNs and V0 detectors are on either side of the interaction vertex, those in the direction of positive pseudorapidity axis are referred as V0A and ZNA and those in the opposite direction are referred as V0C and ZNC. A detailed description of the ALICE detectors and their performance can be found elsewhere [@Aamodt:2008zz; @Abelev:2014ffa].
The event asymmetry is estimated using the energy measured in the two ZNs situated 114 metres away from the nominal interaction point (IP) on either side. The ZNs detect only spectator neutrons that are not bound in nuclear fragments, since the latter are bent away by the magnetic field of the LHC separation dipole. The ZN detection probability for neutrons is 97.0% $\pm$ 0.2%(stat) $\pm$3%(syst) [@ALICE:2012aa]. The relative energy resolution of the 1[*n*]{} peak at 1.38 TeV is 21% for the ZNA and 20% for the ZNC [@ALICE:2012aa]. The production of nuclear fragments increases with collision impact parameter degrading the resolution on the number of participating nucleons. The energy in the ZNs is a good measure of the number of spectator neutrons only for the more central collisions [@Abelev:2014ffa]. The analysis is limited to the top 35% most central sample and employs data from $\sim 2.7$ million events.
The raw ${\mathrm{d}N/\mathrm{d}\eta}$ distributions in the region $|\eta| < 0.9$ are obtained by reconstructing the charged particle tracks using the TPC and ITS. The requirements on the reconstructed tracks obtained using the measurements in these detectors are the same as in other earlier analyses [@Aamodt:2010pa]. The measured amplitudes in the V0A ( $+2.8 < \eta < +5.1$) and V0C ($-3.7 < \eta < -1.7$ ) are used to estimate the raw ${\mathrm{d}N/\mathrm{d}\eta}$ distributions of charged particles in the forward regions. Both V0A and V0C are scintillator counters, each with four segments in pseudorapidity and eight segments in azimuth. The raw distributions measured are termed as ${\mathrm{d}N/\mathrm{d}\eta}$ distributions throughout the manuscript. In order to ensure a uniform detector performance, the present analysis uses events with z position (along the beam direction) of the interaction vertex, $V_{\rm z}$, within $\pm$ 5 cm of the IP in ALICE. The centrality of Pb-Pb collisions was estimated by two independent methods. One estimate was based on the charged particle multiplicity reconstructed in the TPC and the other was based on the amplitudes in the V0 detectors [@Abelev:2013qoq].
Analysis and systematic uncertainties {#sec:Analysis}
=====================================
In the present analysis, changes in the raw pseudorapidity distribution of charged particles are investigated for different values of measured asymmetry of the event. The method of measurement of the asymmetry and the parameters characterising the change in ${\mathrm{d}N/\mathrm{d}\eta}$ distributions are discussed in this section.
Analysis
--------
Any event asymmetry due to unequal number of nucleons from the two participating nuclei may manifest itself in the longitudinal distributions, i.e. $\mathrm{d}N/\mathrm{d}y$ (or ${\mathrm{d}N/\mathrm{d}\eta}$) of the produced particles because of a shift in the effective CM. Assuming that the rapidity distributions can be described by a symmetric function about a mean $y_{\rm 0}$ ($y_{\rm 0}$ = 0.0 for symmetric events), the ratio of the distributions for asymmetric and symmetric events may be written as $$\begin{split}
\frac {(dN/dy)_{\rm {asym}}}{(dN/dy)_{\rm {sym}}} = \frac {f
(y-y_{\rm 0})} { f(y)}
\propto \sum_{n=0}^{\infty} c_{\rm n}(y_{\rm 0})y^{\rm n} \\
\end{split}
\label{eq:RatioEqn}$$ For any functional form of the rapidity distribution, this ratio may be expanded in a Taylor series. The coefficients $c_{\rm n}$ of the different terms in the expansion depend on the shape and the parameters of the rapidity distribution [@Raniwala:2016ugm]. In the ALICE experiment, the pseudorapidities of the emitted particles were measured. The effect of a rapidity-shift $y_{\rm 0}$ on the pseudorapidity distribution is discussed in Sect. \[rapshift\].
The unequal number of participating nucleons will yield a non-zero $y_{\rm 0}$ of the participant zone and will cause an asymmetry in the number of spectators. This asymmetry can provide information about the mean values of $y_{\rm 0}$ using the response matrix discussed in Sect. \[sec:Simula\]. The asymmetry of each event is estimated by measuring the energy in the ZNs on both sides of the interaction vertex: $E_{\rm {ZNA}}$ on the side referred to as the A-side ($\eta > 0 $) and $E_{\rm {ZNC}}$ on the side referred to as the C-side ($\eta < 0 $). A small difference in the mean and the relative energy resolution of the 1[*n*]{} peak at 1.38 TeV was observed in the performance of the two ZNs [@ALICE:2012aa]. For each centrality interval, the energy distribution in each ZN is divided by its mean, and the width of the $E_{\rm {ZNC}}/\langle E_{\rm {ZNC}} \rangle$ distribution is scaled to the width of the corresponding distribution using $E_{\rm {ZNA}}$. The asymmetry in ZN is defined as $$\alpha_{\rm {ZN}} = \frac{\epsilon_{\rm {ZNA}}-\epsilon_{\rm
{ZNC}}}{\epsilon_{\rm {ZNA}}+\epsilon_{\rm {ZNC}}}$$ where $\epsilon_{\rm {ZNC(A)}}$ is a dimensionless quantity for each event, obtained after scaling the distributions of $E_{\rm {ZNC(A)}}$ as described above.
For the 15–20% centrality interval, Fig. \[fig:AlphaDisbnData\] shows the distribution of the asymmetry $\alpha_{\rm{ZN}}$.
To investigate the significance of this distribution, the contribution of the resolution of ZNs to the resolution of the asymmetry parameter $\alpha_{\rm {ZN}}$ is evaluated. For each centrality interval, values of $E_{\rm {ZNC}}$ and $E_{\rm {ZNA}}$ are simulated for each event by assuming a normal distribution peaked at the mean value corresponding to the average number of neutrons and the corresponding energy resolution. The average number of neutrons is estimated by dividing the experimental distribution of energy in ZN by 1.38 TeV. These values are used to obtain $\alpha_{\rm{ZN}}$ for each event and its distribution. The width of the distribution corresponds to the intrinsic resolution of the measured parameter $\alpha_{\rm{ZN}}$ and varies from 0.023 to 0.050 from the most peripheral (30–35%) selection to the most central (0–5%) selection. The observed width of 0.13 of the distribution of $\alpha_{\rm {ZN}}$ reported in Fig. \[fig:AlphaDisbnData\] is considerably larger than the resolution of $\alpha_{\rm {ZN}}$ (0.027 for the centrality interval corresponding to the data in the figure) and the increase in width may be attributed to the event-by-event fluctuations in the number of neutrons detected in each ZN. To investigate the effect of $\alpha_{\rm{ZN}}$ on the ${\mathrm{d}N/\mathrm{d}\eta}$ distributions, the events are demarcated into three regions of asymmetry by choosing a cut value $\alpha_{\rm{ZN}}^{\rm{cut}}$. These regions correspond to (i) $\alpha_{\rm {ZN}}
< -\alpha_{\rm {ZN}}^{\rm {cut}} $ (Region 1), (ii) $\alpha_{\rm {ZN}}
\geq \alpha_{\rm {ZN}}^{\rm {cut}} $ (Region 2) and (iii) $-\alpha_{\rm {ZN}}^{\rm {cut}}\leq \alpha_{\rm {ZN}} < \alpha_{\rm {ZN}}^{\rm {cut}}$ (Region 3). Regions 1 and 2 are referred to as the asymmetric regions and Region 3 is referred to as the symmetric region.
The effect of the measured asymmetry $\alpha_{\rm{ZN}}$ on the pseudorapidity distributions is investigated by studying the ratio of ${\mathrm{d}N/\mathrm{d}\eta}$ distribution in events from the asymmetric region to those from the symmetric region. There are small differences in the distributions of centrality and vertex position in events of different regions of asymmetry. It is necessary to ensure that any correlation between the ratio of ${\mathrm{d}N/\mathrm{d}\eta}$ and the asymmetry is not due to a systematic effect of a shift in the interaction vertex. To eliminate any possible systematic bias on the measured distributions, the ${\mathrm{d}N/\mathrm{d}\eta}$ distributions are corrected by weight factors obtained by normalising the number of events in asymmetric and symmetric regions in each 1% centrality interval and each 1 cm range of vertex positions.
For the 15-20% centrality interval, the distributions of these factors in the two cases corresponding to the asymmetry regions 1 and 2 have a mean of 1.0 and an rms of 0.05 and 0.06 respectively. The weight factors do not show any systematic dependence on the position of the vertex. This is expected considering the large distance between the ZNs as compared to variations in the vertex position. The factors show a systematic dependence on 1% centrality bins within each centrality interval. The 1% centrality bin with the greater number of participants tends to have more asymmetric events, presumably to compensate for the decrease in the effective CM energy due to the motion of the participant zone; the weight factor is 1.08 for the most central 15–16% centrality bin and is 0.94 for the 19–20% centrality bin.
The ratio of ${\mathrm{d}N/\mathrm{d}\eta}$ for events corresponding to different regions of asymmetry, as shown in Fig. \[fig:AlphaDisbnData\], is determined. For $|\eta| < 1.0$, the ratio is obtained using ${\mathrm{d}N/\mathrm{d}\eta}$ for tracks. For $|\eta| > 1.0$, the ratio shown in Fig. \[fig:RatiodNdetaV0A\](a) and (b) is obtained from amplitudes measured in V0A and the one shown in Fig. \[fig:RatiodNdetaV0A\](c) and (d) is from amplitudes measured in V0C. The squares in Fig. \[fig:RatiodNdetaV0A\] (a) and (c) represent the ratio of ${\mathrm{d}N/\mathrm{d}\eta}$ in the asymmetry Region 1 to that in Region 3 (R13), and the stars represent the corresponding ratio in Region 2 to Region 3 (R23). The filled circles in Fig. \[fig:RatiodNdetaV0A\] (b) and (d) are obtained by (i) reflecting the data points labelled R23 across $\eta = 0$ and (ii) taking the averages of R13 and reflected-R23 for $|\eta| < 1.0 $. A third order polynomial is fitted to the points and the values of the coefficients $c_n$ along with the $\chi^{\rm 2}$ are shown. The polynomial fit to the ratio of ${\mathrm{d}N/\mathrm{d}\eta}$ distribution has a dominantly linear term. A small residual detector effect is observed when determining $c_{1}$ using data measured in V0A and when using data measured in V0C. In all subsequent discussion, the values of $c_{\rm 1}$ quoted are the mean of values obtained from the measurements in V0A and V0C.
![ The coefficient $c_{\rm 1}$ characterising the change in ${\mathrm{d}N/\mathrm{d}\eta}$ distribution for asymmetric regions is shown for different values of $\alpha_{\rm{ZN}}^{\rm{cut}}$ ($\alpha_{\rm{ZN}}^{\rm{cut}}$ demarcates the asymmetric and symmetric events) for each centrality interval.[]{data-label="fig:c1alphazdc"}](c1vsAsym.pdf){width="60.00000%"}
Considering that the event samples corresponding to different regions of asymmetry are identical in all aspects other than their values of measured $\alpha_{\rm{ZN}}$, the observation of non-zero values of $c_{\rm 1}$ can be attributed to the asymmetry. For a fixed centrality interval, $c_{\rm 1}$ depends on the choice of $\alpha_{\rm{ZN}}^{\rm{cut}}$. The analysis is repeated for different values of $\alpha_{\rm{ZN}}^{\rm{cut}}$ and the dependence of $c_{\rm 1}$ on $\alpha_{\rm{ZN}}^{\rm{cut}}$ is shown in Fig. \[fig:c1alphazdc\], for different centralities. For each centrality interval the coefficient $c_{\rm 1}$ has a linear dependence on $\alpha_{\rm{ZN}}^{\rm{cut}}$ and the slope increases with decreasing centrality; $c_{\rm 1}$ increases for events corresponding to larger values of average event asymmetry. The range of values of $\alpha_{\rm{ZN}}^{\rm{cut}}$ was guided by the resolution and the width of the distribution of $\alpha_{\rm{ZN}}$, as mentioned in reference to Fig. \[fig:AlphaDisbnData\]. Increasing the value of $\alpha_{\rm{ZN}}^{\rm{cut}}$ increases the mean $\langle \alpha_{\rm {ZN}} \rangle$ for events from the asymmetric class (Region 1 or Region 2), and increases the RMS of $\alpha_{\rm{ZN}}$ for events from the symmetric class (Region 3).
Systematic uncertainties
------------------------
The current method of analysis uses the ratio of two ${\mathrm{d}N/\mathrm{d}\eta}$ distributions from events divided on the basis of measurements in ZNs, within a centrality interval. All effects due to limited efficiency, acceptance or contamination would cancel while obtaining the value of the ratio. The contributions to the systematic uncertainties on $c_{\rm 1}$ are estimated due to the following sources:
1. Centrality selection: the ratio of ${\mathrm{d}N/\mathrm{d}\eta}$ is obtained from the measurements of tracks in the ITS+TPC at midrapidity and charge particle signal amplitudes in the V0 at forward rapidities. For the former, the event centrality is determined using the measurements in the V0 and for the latter using the track multiplicity in the TPC. The analysis is repeated by interchanging the centrality criteria and the resultant change in the values of $c_{\rm 1}$ for different centrality intervals is in the range 0.1% to 3.6%.
2. V0A and V0C: the systematic uncertainty on the mean value of $c_{\rm 1}$ is estimated by assuming a uniform probability distribution for the correct value of $c_{\rm 1}$ to lie between the two values obtained using the charged-particle signal amplitudes measured in the V0A and the V0C. The uncertainty is in the range 2.1% to 4.6% and does not depend on the centrality value.
3. Vertex position: the analysis is repeated for the z position of the interaction vertex $|V_{\rm z}| \leq
3.0 $ cm. For the most central interval, the results change by less than 0.1%. For the 15–20% centrality interval, the results change by 3.3% and for all other centrality intervals, the changes are less than 1.3%.
4. Weight factors for normalisation: the analysis is also repeated without the weight factors mentioned in Sect. 3.1 for the centrality and the vertex normalisation in the number of events. The change in the results is 4.9% in the most central class and less than 1% for all other centrality intervals.
The total systematic uncertainty is obtained by adding the four uncertainties in quadrature. The resultant uncertainty is in the range 2.3% to 5.8% and is shown by the band in Fig. \[fig:Meanc1centrality\].
Simulations {#sec:Simula}
===========
The simulation used for obtaining a relation between rapidity-shift $y_{\rm 0}$ and the measurable asymmetry $\alpha_{\rm{ZN}}$ is described in this section. This simulation has three components: (i) a Glauber Monte Carlo to generate number of participants and spectator protons and neutrons, (ii) a function parametrised to fit the average loss of spectator neutrons due to spectator fragmentation (the loss of spectator neutrons in each event is smeared around this average) and (iii) the response of the ZN to single neutrons. The simulation encompassing the above is referred to in the present work as Tuned Glauber Monte Carlo (TGMC), and reproduces the energy distributions in the ZNs. The effect of $y_{\rm 0}$ on the pseudorapidity distributions has been estimated using additional simulations for a Gaussian ${\mathrm{d}N/\mathrm{d}y}$ and are also described in this section.
Asymmetry and rapidity-shift {#subsec:ZNasymtoy0}
----------------------------
The Glauber Monte Carlo model [@Alver:2008aq] used in the present work assumes a nucleon-nucleon interaction cross section of 64 mb at [$\sqrt{s_{\mathrm{NN}}}$]{} = 2.76 TeV. The model yields the number of participating nucleons in the overlap zone from each of the colliding nuclei. The range of impact parameters for each 5% centrality interval is taken from our Pb-Pb centrality paper [@Abelev:2013qoq]. For each centrality interval, 0.4 million events are generated.
![Rapidity-shift $y_{\rm 0}$ as a function of asymmetry in (a) number of participants (b) number of spectators (c) number of spectator neutrons and (d) energy in ZN obtained using TGMC as described in the text. The results in all four panels are shown for the 15–20% centrality interval.[]{data-label="fig:GlauberMCy0"}](Panel4_y0Asym.pdf){width="100.00000%"}
For each generated event, the number of participating protons and neutrons is obtained, enabling a determination of the rapidity-shift $y_{\rm 0}$ and the various longitudinal asymmetries. If A and B are the number of spectators (spectator neutrons) in the two colliding nuclei, the asymmetry is referred to as $\alpha_{\rm spec}$ ($\alpha_{\rm {spec-neut}} $). Figure \[fig:GlauberMCy0\] (a) shows the correspondence between $y_{\rm 0}$ and $\alpha_{\rm {part}}$. Figures \[fig:GlauberMCy0\] (b) and (c) show the relation between $y_{\rm 0}$ and $\alpha_{\rm {spec}}$ and $\alpha_{\rm {spec-neut}}$ respectively. These figures show that the rapidity-shift $y_{\rm 0}$ can be estimated by measuring $\alpha_{\rm {spec}}$ or $\alpha_{\rm {spec-neut}}$ in any experiment that uses Zero Degree Calorimeters. However, the lack of information on the number of participants worsens the precision in determining $y_{\mathrm{0}}$. Figure \[fig:GlauberMCy0\] (d) shows the relation between $y_{\rm 0}$ and $\alpha_{\rm{ZN}}$ obtained in TGMC, as described in the next paragraph.
The Glauber Monte Carlo is tuned to describe the experimental distributions of ZN energy. For each 1% centrality interval, the mean number of spectator neutrons ($N_s$) is obtained in the Glauber Monte Carlo. Folding the ZN response yields the simulated values of mean energy as a function of centrality. The experimentally measured mean energy in the ZN is also determined for each 1% centrality interval. The ratio of the measured value of mean energy to the simulated value of mean energy gives the fractional loss (*f*) of neutrons due to spectator fragments that veer away due to the magnetic field. The value of *f* for the 0-5% centrality interval is 0.19. For all other centralities it varies from 0.40 for 5-10% to 0.55 for 30-35% centrality interval.
A fluctuation proportional to the number of remaining neutrons ($N_s\times (1-f)$) is incorporated to reproduce the experimental distribution of the energy deposited in the ZN shown in Fig. \[fig:ZNDist\] (a). The peak and the RMS of the energy distributions match well. The fractional difference in the position of the peak varies between 3.7% for the 0-5% centrality interval and 0.1% for the 30-35% centrality interval. The fractional difference in RMS for the most central class is 8.6% and is in the range 1.0–2.0% for all other centrality intervals. The distributions of the asymmetry parameter for the TGMC events and the measured data for each centrality interval are shown in Fig. \[fig:ZNDist\] (b). The
. \[fig:Meany0vsAsym\]
TGMC contains information of $y_{\rm 0}$ and $\alpha_{\rm{ZN}}$ for each event. A scatter plot between $y_{\rm 0}$ and $\alpha_{\rm{ZN}}$ is shown in Fig. \[fig:GlauberMCy0\](d) for the 15–20% centrality interval. This constitutes the response matrix. For any measured value of $\alpha_{\rm{ZN}}$, the distribution of $y_{\rm 0}$ can be obtained. Any difference in the experimental and TGMC distributions of $\alpha_{\rm {ZN}}$ can be accounted for by scaling the $y_{\rm 0}$ distribution by the ratio of number of events in data to the number in TGMC as $$f (y_{\rm 0}, \alpha_{\rm {ZN}}^{\rm {Data}}) = f (y_{\rm 0},\alpha_{\rm {ZN}}^{\rm {TGMC}}) \frac{N_{\rm{events}}^{\rm{Data}}}{N_{\rm{events}}^{\rm{TGMC}}},
\label{eq:alphazn}$$ with Data (TGMC) in the superscript of number of events, $N_{\rm{events}}$, denoting the experimental data (TGMC events). For each of the three regions of asymmetry shown in Fig. \[fig:AlphaDisbnData\], corresponding to a chosen value of $\alpha_{\rm {ZN}}^{\rm {cut}} = 0.1$, the distribution of rapidity-shift $y_{\rm 0}$ obtained using the response matrix is shown in Fig. \[fig:rapshiftdisbn\]. It is worth mentioning that the width of the distribution of $y_{\rm 0}$ for events from Region 3, corresponding to $-\alpha_{\rm {ZN}}^{\rm {cut}}\leq \alpha_{\rm {ZN}} < \alpha_{\rm {ZN}}^{\rm {cut}}$, is comparable to the widths of the corresponding distributions from Regions 1 and 2. The effect of difference in the value of the means of the $y_{\rm 0}$ distributions is investigated in the present work.
Effect of rapidity-shift on pseudorapidity distributions {#rapshift}
--------------------------------------------------------
The effect of a shift in the rapidity distribution by $y_{\rm 0}$ on the measurable pseudorapidity distribution (${\mathrm{d}N/\mathrm{d}\eta}$) is investigated using simulations. For each event, the rapidity of charged particles is generated from a Gaussian distribution of a chosen width *$\sigma_y$* [@Abbas:2013bpa]. The pseudorapidity is obtained by using the Blast-Wave model fit to the data for the transverse momentum distributions and the experimentally measured relative yields of pions, kaons and protons [@Abelev:2013vea]. To simulate the effect of different widths of the parent rapidity distribution for different centralities, different $\sigma_{\rm y}$ widths are chosen to reproduce the measured FWHM (Full Width at Half Maximum) of the pseudorapidity distribution [@Adam:2015kda]. For the most central (0–5%) class, a value 3.86 is used for the width of the rapidity distribution, and a value 4.00 is used for the width of the least central class employed in this analysis (30–35%).
The distribution of rapidity-shift $y_{\rm 0}$, similar to the one shown in Fig. \[fig:rapshiftdisbn\], is obtained for each centrality interval and each $\alpha_{\rm{ZN}}^{\rm{cut}}$ using TGMC. Figure \[fig:Meany0vsAsym\] (a) shows the $\langle y_{\rm 0} \rangle$ as a function of $\alpha_{\rm{ZN}}^{\rm{cut}}$ for different centralities. One observes a linear relation between the two quantities, showing that an asymmetry in the ZN measurement, arising from the unequal number of participating nucleons, is related to the mean rapidity-shift $\langle y_{\rm 0} \rangle$. The rapidity distribution of the particles produced in each event is generated assuming a Gaussian form centred about a $y_{\rm 0}$, which is generated randomly from the $y_{\rm 0}$ distribution. Events with a rapidity distribution shifted by *$y_0$* $ \neq 0$ yield an asymmetric pseudorapidity distribution. A third order polynomial function in *$\eta$* is fitted to the ratio of the simulated ${\mathrm{d}N/\mathrm{d}\eta}$ for the asymmetric region to the simulated ${\mathrm{d}N/\mathrm{d}\eta}$ for the symmetric region. The values of the coefficients in the expansion depend upon the rapidity-shift $y_{\rm 0}$ and the parameters characterising the distribution [@Raniwala:2016ugm].
The simulations described above were repeated for different values of $\alpha_{\rm{ZN}}^{\rm{cut}}$ to obtain the pseudorapidity distributions for symmetric and asymmetric regions. Fitting third order polynomial functions to the ratios of the simulated pseudorapidity distributions determines the dependence of ${c_{\rm 1}}$ on $\alpha_{\rm{ZN}}^{\rm{cut}}$. Figure \[fig:Meany0vsAsym\](b) shows that ${c_{\rm 1}}$ has a linear dependence on $\alpha_{\rm{ZN}}^{\rm{cut}}$ for each centrality interval. The difference in the slopes for different centralities is due to differences in the distributions of $y_{\rm 0}$ and to differences in the widths of the rapidity distributions. It is important to note that the parameter ${c_{\rm 1}}$, characterising the asymmetry in the pseudorapidty distribution, shows a linear dependence on the parameter $\alpha_{\rm{ZN}}^{\rm{cut}}$ in the event sample generated using TGMC and simulations for a Gaussian ${\mathrm{d}N/\mathrm{d}y}$, akin to the dependence of the estimated value of rapidity-shift $y_{\rm 0}$ for the same sample of events.
Results {#sec:Result}
=======
The longitudinal asymmetry in a heavy-ion collision has been estimated from the difference in the energy of the spectator neutrons on both sides of the collision vertex. The effect of the longitudinal asymmetry is observed in the ratio of ${\mathrm{d}N/\mathrm{d}\eta}$ distributions corresponding to different asymmetries. The linear term in a polynomial fit to the distribution of the ratio is dominant, and is characterised by its coefficient $c_{\rm1}$. The centrality dependence of the coefficient $c_{\rm 1}$ for $\alpha_{\rm {ZN}}^{\rm {cut}} = 0.1$ is shown in Fig. \[fig:Meanc1centrality\].
![The mean values of the coefficient $c_{\rm 1}$ are shown as filled (red) circles for different centralities. These correspond to the ratio of ${\mathrm{d}N/\mathrm{d}\eta}$ distributions of populations of events demarcated by $\alpha_{\rm{ZN}}^{\rm{cut}}$ = 0.1. The squares show the corresponding values from simulations, and correspond to $\alpha_{\rm{ZN}}^{\rm{cut}}$ = 0.1 in Fig. \[fig:Meany0vsAsym\], for different centralities. The systematic uncertainties are shown as bands.[]{data-label="fig:Meanc1centrality"}](Meanc1vsCentrality.pdf){width="90.00000%"}
It is worth emphasising that the values of $c_{\rm 1}$ and hence its centrality dependence are affected by (i) the distribution of rapidity-shift $y_{\rm 0}$ for each centrality interval, (ii) the chosen value of $\alpha_{\rm{ZN}}^{\rm{cut}}$, as seen in Fig. \[fig:Meany0vsAsym\] and (iii) the shape or the width of the parent rapidity distribution for each centrality. Figure \[fig:Meanc1centrality\] also shows the results obtained using simulations as described in Sec. \[rapshift\] for $\alpha_{\rm {ZN}}^{\rm {cut}} = 0.1$. The systematic uncertainty on the simulated event sample is estimated by (i) varying the resolution of ZNs from 20% to 30%, (ii) assuming all charged particles are pions and (iii) varying the width of the parent rapidity distribution within the range corresponding to the uncertainties on FWHM quoted in Ref. [@Adam:2015kda]. The simulated events show a good agreement with the experimental data providing credence to the assumptions of the simulation, in particular that the asymmetry in the distributions arises from the shift of rapidity of the participant zone.
There are two quantities from independent measurements for each selection of asymmetric events. These are (i) $c_{\rm 1}$, the parameter characterising the effect of asymmetry in the ${\mathrm{d}N/\mathrm{d}\eta}$ distributions and shown in Fig. \[fig:c1alphazdc\] and (ii) the mean rapidity-shift $\langle
y_{\rm 0} \rangle$ obtained from the measured asymmetry, filtered through the corresponding response matrix (Fig. \[fig:GlauberMCy0\] (d)), and shown in Fig. \[fig:Meany0vsAsym\] (a). The relation between $c_{\rm
1}$ and $\langle y_{\rm 0} \rangle$ is shown in Fig. \[fig:c1vsy0\]. The parameter $c_{\rm 1}$ shows a linear dependence on *$\langle y_{\rm 0} \rangle$* for each centrality. The difference in the slopes indicates the sensitivity of the longitudinal asymmetry to the details of the rapidity distribution. For a Gaussian rapidity distribution the corresponding parameter $c_{\rm 1}$ would be related to the rapidity-shift as $c_{\rm 1} = \frac {y_{\rm 0}}{\sigma_{\rm
{y^2}}}$ [@Raniwala:2016ugm], implying that the slope is inversely proportional to the square of the width of the distribution. The observation of an increase in the slope with an increase in the centrality in the present data indicates a decrease in the width of the pseudorapidity distribution with increasing centrality. Such a decrease in the width of the pseudorapidity distribution with increasing centrality has been observed independently by fitting the pseudorapidity distributions in a broad range of pseudorapidity [@Adam:2015kda].
Conclusions
===========
The present analysis demonstrates the existence of a longitudinal asymmetry in the collision of identical nuclei due to fluctuations in the number of participants from each colliding nucleus. This asymmetry has been measured in the ZNs in the ALICE experiment (Fig. \[fig:AlphaDisbnData\]), and affects the pseudorapidity distributions, as demonstrated by taking the ratio of distribution of events from the asymmetric region to the corresponding one from the symmetric region (Fig. \[fig:RatiodNdetaV0A\]). The effect can be characterised by the coefficient of the linear term in the polynomial expansion of the ratio. The coefficients show a linear dependence on $\alpha_{\rm{ZN}}^{\rm{cut}}$, a parameter to classify the events into symmetric and asymmetric regions (Fig. \[fig:c1alphazdc\]). Different values of $\alpha_{\rm{ZN}}^{\rm{cut}}$ correspond to different values of the mean rapidity shift $\langle y_{\rm 0} \rangle$ (Fig. \[fig:Meany0vsAsym\] (a)). The parameter describing the change in the pseudorapidity distributions ($c_{\rm 1}$) has a simple explanation in the rapidity-shift $ \langle y_{\rm 0} \rangle $ of the participant zone (Fig. \[fig:c1vsy0\]). The analysis confirms that the longitudinal distributions are affected by the rapidity-shift of the participant zone with respect to the nucleon-nucleon CM frame. The results provide support to the relevance of number of nucleons affecting the production of charged particles, even at such high energies.
The longitudinal asymmetry is a good variable to classify the events and provides information on the initial state of each event. A systematic study of the effects of longitudinal asymmetry on different observables, e.g. the odd harmonics of anisotropic flow, the forward-backward correlations, the source sizes, in heavy-ion collisions may reveal other characteristics of the initial state and of particle production phenomena.
Acknowledgements
================
The ALICE Collaboration {#app:collab}
=======================
|
---
abstract: 'Particle dynamics in the electron current layer in collisionless magnetic reconnection is investigated by using a particle-in-cell simulation. Electron motion and velocity distribution functions are studied by tracking self-consistent trajectories. New classes of electron orbits are discovered: figure-eight-shaped regular orbits inside the electron jet, noncrossing regular orbits on the jet flanks, noncrossing Speiser orbits, and nongyrotropic electrons in the downstream of the jet termination region. Properties of a super-Alfvénic outflow jet are attributed to an ensemble of electrons traveling through Speiser orbits. Noncrossing orbits are mediated by the polarization electric field near the electron current layer. The noncrossing electrons are found to be non-negligible in number density. The impact of these new orbits to electron mixing, spatial distribution of energetic electrons, and observational signatures, is presented.'
author:
- Seiji Zenitani
- Tsugunobu Nagai
title: Particle dynamics in the electron current layer in collisionless magnetic reconnection
---
Introduction
============
Collisionless magnetic reconnection is a basic plasma process for the abrupt release of magnetic energy. Understanding the process is crucial to discuss planetary magnetospheres, solar corona, the solar wind, laboratory plasmas, and astrophysical plasma environments. Collisionless reconnection is a highly nonlinear, complex process, in which the electromagnetic field and the plasma particle motion interact with each other. The reconnection mechanism has not yet been fully understood, but numerical simulations provide a way to investigate the underlying physics.
Since early research with particle-in-cell (PIC) simulations, it has been recognized that a small-scale electron-physics layer is embedded inside a broader ion-physics layer in collisionless magnetic reconnection. For example, @prit01a presented a narrowly collimated electron jet inside a broader ion outflow. This picture was further extended by successive PIC simulations, [@dau06; @keizo06; @kari07; @shay07; @drake08; @klimas08] which were large enough to separate electron-scale structure from the ion-scale structure. @kari07 and @shay07 demonstrated that the electron-physics layer evolves into the inner core region and the fast elongated jet. Although these results raised a question concerning long-term behavior of magnetic reconnection,[@klimas08] now many scientists agree that the inner core region controls the reconnection rate. The entire layer is often called the electron current layer (ECL) or the electron diffusion layer. Hereafter we call it the electron current layer (ECL).
The inner core is called the dissipation region (DR) or the electron diffusion region (EDR). This is the site of dissipation physics, arising from complex electron motions (See @hesse11 for a review). As of today, the DR is ambiguously defined, and there are many different opinions on its rigorous definition.[@goldman16] Promising signatures to identify the DR are enhanced energy dissipation,[@zeni11c] electron nongyrotropic behavior,[@scudder08; @aunai13c; @swisdak16] electron phase-space hole along the inflow direction,[@hori08; @chen11] and characteristic velocity distribution functions (VDFs).[@ng11; @bessho14]
The jet is popularly referred to as the super-Alfvénic electron jet, because its bulk speed exceeds the Alfvén speed in the inflow region. This fast jet has characteristic features such as violation of the electron ideal condition,[@prit01a; @kari07; @shay07] diamagnetic-type momentum balance,[@hesse08] an electron pressure anisotropy,[@le10a] electron nongyrotropy,[@scudder08; @aunai13c; @swisdak16] bipolar polarization electric fields $E_z$ (the so-called Hall electric fields), and highly structured electron VDFs.[@aunai13c; @bessho14; @shuster15] These issues are usually discussed separately. The number of attempts to comprehensively explain these signatures has been limited.
There have been many observational studies on the ECL during magnetic reconnection in the Earth’s magnetosphere. @chen08 [@chen09] studied a magnetotail reconnection event with Cluster spacecraft. With the help of a map of electron VDFs by PIC simulation, they identified that the satellite crossed an electron-scale thin current layer near the X-line. @nagai11 reported an informative reconnection event in the magnetotail with the Geotail spacecraft. They detected both bi-directional electron flows that outrun ion flows and an energy dissipation site around the X-line.[@zeni12] @nagai13b studied another magnetotail reconnection event with ion-electron decoupling. They reported additional signatures in the ion-electron decoupling region, such as the energetic electron fluxes. @oka16 reported a DR-crossing in the magnetotail with THEMIS. They observed nongyrotropic electron VDFs and perpendicular heating inside the ECL. They also showed that the ECL is a site of electron energetization. In addition, there have been several observations of super-Alfvénic electron jets inside the reconnection outflow exhaust in the magnetosheath,[@phan07] in the magnetotail,[@zhoum14] and in the solar wind.[@xux15] These results support the standard picture of the ECL, the central DR and extended electron jets.
In the above observations, time and spatial resolutions were rather limited to discuss electron physics in great detail. In order to probe electron-scale structures in near-Earth reconnection sites at ultra-high resolutions, NASA recently launched the Magnetospheric MultiScale (MMS) spacecraft in 2015.[@burch16] MMS is planned to move to the Earth’s magnetotail in 2017, where magnetic reconnection often occurs in an anti-parallel configuration. It is expected that MMS will encounter ${\approx}10$ reconnection events.[@kevin14] Once it encounters reconnection events, MMS is expected to resolve the aforementioned structures.
Since spacecraft observe VDFs, it is important to understand the electron motion behind the VDFs in PIC simulations. @hoshino01b was one of the first to discuss electron VDF and associated particle motion. They examined VDFs near the magnetic island in the downstream region. @egedal05 [@egedal08] found that electron VDFs are elongated in the field-aligned direction in the inflow region. As already introduced, @chen08 [@chen09] studied spatial distribution of electron VDFs by PIC simulation to interpret satellite data.
In order to prepare for the MMS observation, a growing number of works are devoted to electron VDFs in PIC simulations. @ng11 visualized the complex structure of the electron VDF in the DR. They reconstruct the VDF at high resolution, by back-tracing particle orbits in the PIC field and by using a Liouville’s theorem. The VDF contains discrete striations in a triangular-shaped envelope. Shuster et al.[@shuster14; @shuster15] presented that electron VDFs contain various discrete components in the outflow exhaust. They further examined the structure of the electron VDF over the ECL, with help from test-particle simulations. @bessho14 examined electron VDFs in the ECL. In the DR, they gave semi-analytic expressions to the fine structure of the VDF. In the downstream, many arcs were found in the VDFs and they were attributed to a gradual remagnetization of electrons. @frankcheng15 inferred particle dynamics in driven reconnection from the spatial distribution of VDFs. They claimed that the field-aligned electron population in the inflow region is injected into the super-Alfvénic electron jet. Most recently, @wang16a studied the electron heating mechanism in the exhaust region in detail. With help from test particle simulations, they found that parallel heating by the curvature drift acceleration and perpendicular heating by the gradient-B drift acceleration account for a highly structured VDF near the magnetic flux pile-up region.
These VDFs are ensembles of electrons, following various complex trajectories in the reconnection system. In addition to a gyration and a parallel motion, several classes of electron particle motions are reported in the previous literature, such as a Speiser motion around the DR[@speiser65; @ng11; @bessho14] and a field-aligned bounce motion in the inflow region.[@egedal05; @egedal08] They are proven to be building blocks of the VDFs. However, it is not clear whether these particle motions and/or their combinations can explain everything about electron VDFs. In fact, as reviewed in this section, the ECL structure is found to be much more complicated than previously expected. It is possible that some electrons travel through new orbits near the reconnection site and that they have an impact on the reconnection dynamics and observational signatures. In order to better interpret the electron VDFs to deeper discuss kinetic reconnection physics, it is important to understand electrons particle orbits and dynamics in a modern reconnection simulation.
The purpose of this paper is to comprehensively discuss electron fluid properties, VDFs, self-consistent trajectories, and relevant particle dynamics around the ECL in magnetic reconnection. The paper is organized as follows. First, we briefly review nongyrotropic particle motions in a curved magnetic field in Section \[sec:theory\]. Next, we describe the numerical setup of a 2D PIC simulation in Section \[sec:setup\]. The simulation results are presented in several ways. Section \[sec:fluid\] presents macroscopic fluid quantities. In Section \[sec:kinetic\], we present electron kinetic signatures such as VDFs and phase-space diagrams. In Section \[sec:traj\], we will discuss self-consistent electron trajectories in detail. In Section \[sec:comp\], we utilize the trajectory datasets to further examine VDFs and spatial distribution of electrons. The relevance to the satellite observation is briefly addressed in Section \[sec:ET\]. Section \[sec:discussion\] contains discussions and summary.
Electron motion in a curved field reversal {#sec:theory}
==========================================
We outline basic properties of particle (electron) trajectories in a highly bent magnetic field.[@chen86; @BZ89] Here we consider a simple parabolic field, $$\begin{aligned}
\label{eq:B}
{\boldsymbol{B}}=B_0(z/L){\boldsymbol{e}}_x+B_n{\boldsymbol{e}}_z, ~~{\boldsymbol{E}}=0,\end{aligned}$$ where $B_0$ is the reference magnetic field, $L$ is the length scale of the current sheet, and $B_n$ is the normal magnetic field. This system resembles the outflow region in magnetic reconnection at the lowest order. The equation of motion, $m_e({d{\boldsymbol{v}}_e}/{dt}) = -e({\boldsymbol{v}}_e\times{\boldsymbol{B}})$, can be rewritten $$\begin{aligned}
~~~~~~~~~~~~~~~
\left\{
\begin{aligned}
\ddot{x} &=& &-\Omega_n \dot{y} & ~~~~~~~~~~~~{\textrm (2a)} \\
\ddot{y} &=& -\omega_b^2(\dot{z}/|v_e|)z &+\Omega_n \dot{x} & {\textrm (2b)} \\
\ddot{z} &=&~~\omega_b^2(\dot{y}/|v_e|)z & & {\textrm (2c)}
\end{aligned}
\right.
\notag\end{aligned}$$ where $\Omega_n=eB_n/m_e$ is the gyrofrequency about $B_n$ and $\omega_b=\sqrt{eB_0|v_e|/m_eL}$ is a characteristic frequency. The electron motion is characterized by the ratio of the two frequencies,[@BZ89] $$\begin{aligned}
\label{eq:kappa}
\kappa
\equiv
\frac{\Omega_n}{\omega_b}
=
\sqrt{\frac{R_{\rm c,min}}{r_{\rm L,max}}}
=
\Big|\frac{B_n}{B_0}\Big|\sqrt{\frac{L}{r_{\rm L,max}}}
,\end{aligned}$$ where $R_{\rm c,min}$ is the minimum curvature radius of magnetic field line and $r_{\rm L,max} = m_e|v_e| / eB_n$ is the electron’s maximum gyroradius. This parameter is known as the curvature parameter. The curvature radius solely depends on the field-line geometry, while the gyroradius $r_{\rm L,max}$ depends on the electron velocity $|v_e|$ or the electron energy $\mathcal{E}\equiv\frac{1}{2}m_ev_e^2$. Note that both $|v_e|$ and $\mathcal{E}$ are constant in this system, because ${\boldsymbol{E}}=0$.
When $\kappa \gg 1$, the gyration about $B_n$ dominates. If the parameter falls below the unity, $\kappa \lesssim 1$, the electron motion becomes nongyrotropic. In particular, the motion becomes highly chaotic for $\kappa \sim 1$ ($\Omega_n \sim \omega_b$), because the two oscillations of different kinds interfere with each other. Several characteristic orbits appear for $\kappa \ll 1$, as will be shown in the next paragraphs.
Figure \[fig:theory\] demonstrates typical electron orbits for $\kappa=0.1$ in the 3D space (Fig. \[fig:theory\]a), the velocity space (Figs. \[fig:theory\]b and \[fig:theory\]c), and the phase-space ($v_z$–$z$; Fig. \[fig:theory\]d). Parameters are chosen to be $|R_c|=1$ and $|v_e|=1$. The blue orbit demonstrates a well-known Speiser orbit.[@speiser65] After entering the midplane from the upper left, it slowly turns its direction from $-x$ to $+x$ due to the gyration about $B_n$. Then it exits in the $+x$ direction. Near the midplane ($z\sim 0$), the electron mainly travels in $-y$, while bouncing in $z$ at the frequency of $\omega_b$ ($\dot{y}\approx -|v_e|$ in Eq. (2c)). This is the so-called meandering motion. In the velocity space, the $B_n$-gyration near the midplane corresponds to a half circle in $v_x$–$v_y$, as can be seen in Figure \[fig:theory\]c. The electron velocity rotates anti-clockwise from $-v_x$ to $+v_x$. The fast $z$-bounce motion is evident in ${\pm}v_z$, as indicated by the arrow in Figure \[fig:theory\]b. Since the meandering motion consists of two opposite gyrations, the electron moves back-and-forth along arcs in ${\pm}v_z$. It finally exhibits a zigzag pattern in the 3D velocity space. The $z$-bounce motion also corresponds to the rotation around the center in the phase-space (Fig. \[fig:theory\]d).
The red orbit belongs to another kind of nongyrotropic orbit. It is called a regular orbit or an integrable orbit.[@chen86] When the $z$-motion fully resonates with the gyration about $B_n$, the electron travels through a figure-eight-shaped orbit, hitting a fixed point at the midplane (See also the left panel of Fig. 4 in Ref. ). In the vicinity of the figure-eight-shaped orbit, electrons keep bouncing in $z$ and do not escape away from the midplane. It was further shown that the electrons are trapped on the surface of a ring-type torus, but the reader is referred to @BZ89 paper for detail. Because of the $z$-bounce motion, we usually see closed circuits in the phase space (Fig. \[fig:theory\]d). Note that the regular-orbit electrons travel in $+v_y$ near the midplane ($z \sim 0$). They appear in the $+v_y$ side in the velocity space near the midplane (Fig. \[fig:theory\]c).
Simulation {#sec:setup}
==========
We use a partially implicit PIC code[@hesse99] to study our reconnection problem. The length, time, and velocity are normalized by the ion inertial length $d_i=c/\omega_{pi}$, the ion cyclotron time $\Omega_{ci}^{-1}=m_i/(eB_0)$, and the ion Alfvén speed $c_{Ai}=B_0/(\mu_0 m_i n_0)^{1/2}$, respectively. Here, $\omega_{pi}=( e^2n_0/\varepsilon_0 m_i)^{1/2}$ is the ion plasma frequency, and $n_0$ is the reference density. We employ a Harris-like configuration, ${\boldsymbol{B}}(z)=B_0 \tanh(z/L) {\boldsymbol{\hat{x}}}$ and $n(z) = n_{0} \cosh^{-2}(z/L) + n_b$, where the half thickness is set to $L=0.5 d_i$ and $n_b = 0.2 n_0$ is the background density. The ion-electron temperature ratio is $T_i/T_e=5$. The mass ratio is $m_i/m_e=100$. The ratio of the electron plasma frequency to the electron cyclotron frequency is $\omega_{pe}/\Omega_{ce}=4$. Our domain size is $x,z \in [0,76.8]\times[-19.2, 19.2]$. It is resolved by $2400 \times 1600$ grid cells. Periodic ($x$) and reflecting wall ($z$) boundaries are employed. $1.7{\times}10^9$ particles are used. Reconnection is triggered by a small flux perturbation, $\delta A_y = - 2L B_1 \exp[-(x^2+z^2)/(2L)^2]$, where $B_1=0.1 B_0$ is the typical amplitude of the perturbed fields. The initial electric current is configured accordingly.
This run was analyzed in our previous paper on ion VDFs and ion particle dynamics.[@zeni13] The parameters and system evolution are similar to those in run 1A in Ref. on the electron-scale structure. Several aspects of this reconnection system were presented in these papers. We explore new aspects of electron VDFs and particle dynamics in this paper.
Fluid quantities {#sec:fluid}
================
The reconnection occurs at the center of the simulation domain. The magnetic flux transfer rate across the X-line grows in time until $t \approx 14.5$. If normalized by quantities at $3 d_i$ upstream of the X-line, the reconnection rate reaches $0.14$, gradually decreases to $0.11$ at $t \approx 25$, and then remains constant after that. An electron jet as well as other electron-scale structures grows in time. They are well developed at the time of our interest, $t=35$. We study this time step, because we studied other aspects in our previous studies,[@zeni11d; @zeni13] and because it is early enough to avoid major effects from the periodic boundary in $x$. In fact, the electron jet continues to evolve until $t \approx 44$,[@zeni11d] but minor boundary effects appear in particle signatures at $t \approx 35$, as will be shown later.
Figure \[fig:snapshot\] shows various fluid and field quantities of our PIC simulation at $t=35$. They are averaged over $\Delta t=0.25$ to remove noises. The X-line is located at $(x,z)=(38.1,0.0)$. Figure \[fig:snapshot\]a shows the $x$ component of the electron bulk velocity ${\boldsymbol{V}}_{e}$. One can see narrow bi-directional electron jets from the X-line. The rightward jet ranges $38.1 < x < 48$. The jet speed is higher than the ion bulk speed $V_{ix}$ and the inflow Alfvén speed $\approx 1.62$ at this time. This is consistent with previous studies.[@prit01a; @shay07; @kari07] It has been known that electrons are unmagnetized in the electron jet region, while they are magnetized again farther downstream. Ref. called the jet front boundary ($x\approx 48$) an “electron shock,” but it would be more appropriate to call it the “remagnetization front.” In the downstream of the remagnetization front, unmagnetized ions form a broad current layer.[@zeni13; @le14] Properties of this ion current layer was studied in our previous work.[@zeni13] In Figure \[fig:snapshot\]b, it is interesting to see electron divergent flows in the vertical (${\pm}z$) directions near the remagnetization front. The maximum jet speed is $|V_{ez}| = 1.67$, also comparable with the inflow Alfvén speed. These divergent flows generate the vertical electric currents $J_z$, which correspond to a step-shaped pattern in the out-of-plane magnetic field $B_y$ (See Figs. 1c and 3 in Ref. ). Obviously the super-Alfvénic electron jet is responsible for the divergent flows. These electron flows further correspond to narrow electron jets in red ($V_{ex}>0$) near the separatrices at $x>47$, as the dashed arrows indicate in Figure \[fig:snapshot\]a. These jets penetrate into a broader distribution of incoming electrons ($V_{ex}<0$). Hereafter we call these jets the “field-aligned electron outflows.” Similar electron jets were reported by recent studies.[@zhoum12; @zeni13]
Figure \[fig:snapshot\]c shows the vertical electric fields $E_z$. They consist of a large-scale X-shaped structure along the separatrices and a small-scale bipolar structure along the electron jet region. They are polarization electric fields, due to a broad ion distribution and a narrow electron distribution. Figure \[fig:snapshot\]d shows the parallel electric field $E_{\parallel}$. To better see a weak background structure, we smooth it with boxcar averaging over ${\sim}0.16$ and then we adjust the range of the color bar. Aside from plasma instabilities along the separatrices, one can recognize a double quadrupole structure.[@prit01b; @chen09] The first inner quadrupole is found near the midplane. It features a strong $E_{\parallel}$ toward the X-line, a projection of the Hall electric field $E_z$. This inner quadrupole plays a crucial role for electron dynamics, as will be shown in Section \[sec:noncrossing\]. The second outer quadrupole features the parallel field $E_{\parallel}$ away from the X-line. Although they are hard to recognize, one can see $E_{\parallel}>0$ in the first quadrant ($x \gtrsim 45, z>0$) and $E_{\parallel}<0$ in the lower half ($x \gtrsim 45, z<0$) inside the exhaust region. The other two quadrants are found outside the displayed domain ($x \lesssim 31$). This quadrupole is a projection of the reconnection electric field $E_y$ to the quadrupole Hall magnetic field $B_y$.[@sonnerup79] The incoming electrons near the separatrices, discussed in the previous paragraph, are weakly accelerated toward the X-line by this $E_{\parallel}$.
Figure \[fig:snapshot\]e shows the $x$ component of the ideal flow vector, ${\boldsymbol{w}} \equiv {\boldsymbol{E}}\times{\boldsymbol{B}}/B^2$. Although it saturates in the close vicinity of the X-line ($37\lesssim x \lesssim 39, z\approx 0$), it exhibits a characteristic picture. It looks bifurcated in the electron jet region: It is super-Alfvénic $w_x = 5$–$6$ on the upper and lower sides of the electron jet. Strangely, it is relatively low $w_x = 2$–$3$ at the midplane $z=0$. It is also enhanced along the separatrices. These structures are largely attributed to the Hall electric field $E_z$ (Fig. \[fig:snapshot\]c). We also note that $w_y$ looks bifurcated in the electron jet region \[not shown\].
Figure \[fig:snapshot\]f shows the electron density $n_e$. The ion density looks similar \[not shown\]. We adjust the color bar $0<n_e<0.3$ for discussion later in this paper, while the density reaches $n_e=0.37$ in the downstream side. At this stage, the reconnection process has flushed out the Harris current sheet into the outflow region. The reconnection region is occupied by the inflow plasmas, whose initial density is $n_{b} = 0.2$. The typical electron density is $n_e \sim 0.1$–$0.2$ around the center. One can see two high-density yellow bands around $30<x<45$. This consists of a plasma distribution over an ion meandering width ($|z| < 2$–$3$) and a density cavity near the midplane ($|z| < 1$) in green. The cavity stretches in $x$ and it covers the electron jet region.
Figure \[fig:snapshot\]g displays a nongyrotropy measure $\sqrt{Q}$, which quantifies the deviation of the electron VDF from gyrotropic one.[@swisdak16] It is defined in the following way, $$\begin{aligned}
\label{eq:sqrtQ}
\sqrt{Q} \equiv
\Big\{
\frac{P_{e12}^2 + P_{e13}^2 + P_{e23}^2}{P_{e\perp}^2 + 2P_{e\perp}P_{e\parallel}}\Big\}^{1/2},\end{aligned}$$ where $\overleftrightarrow{P}_{e}$ is the electron pressure tensor. The subscripts ($\parallel, \perp$) and numeral subscripts indicate the parallel, perpendicular, and three off-diagonal components of $\overleftrightarrow{P}_{e}$ in the field-aligned coordinates. Equation ranges from $0$ in the fully gyrotropic case to $1$ in the nongyrotropic limit. In Figure \[fig:snapshot\]g, the measure highlights the ECL near the midplane. If one takes a closer look, two narrow bands are highlighted along the electron jets. These are consistent with previous studies.[@scudder08; @aunai13c; @shuster15] It also marks small-scale regions near the remagnetization front and the separatrices farther downstream $x > 47$.
Figure \[fig:snapshot\]h shows a frame-independent energy dissipation, $\mathcal{D}_e = \gamma_e [ {\boldsymbol{J}} \cdot ({\boldsymbol{E}}+{\boldsymbol{V}}_e\times{\boldsymbol{B}}) - \rho_c ( {\boldsymbol{V}}_e \cdot {\boldsymbol{E}} ) ],$ where $\gamma_e=[1-(V_e/c)^2]^{-1/2}$ is the Lorentz factor and $\rho_c$ is the charge density.[@zeni11c] This is equivalent to the nonideal energy conversion in the nonrelativistic MHD in a neutral plasma. The measure marks the electron-scale dissipation region around the X-line. In addition, it also marks the vertical flow region, as indicated by the circle.
Let us take a closer look at the electron jet. Panels in Figure \[fig:cut432\] show 1D cuts at $x=43.2$. The black arrow in Figure \[fig:snapshot\]a indicates this $x$-position. The velocity profile (Fig. \[fig:cut432\]a) tells us that the electron perpendicular flow outruns the ideal MHD velocity ($V_{e{\perp}x}\approx V_{ex}>w_x$) around the ECL ($|z| < 0.2$) and that the electron are threaded by the magnetic field (${\boldsymbol{V}}_{e\perp} \simeq {\boldsymbol{w}}$) outside the ECL. The profile of $w_x$ is bifurcated, as discussed in the previous section. The bifurcation of $w_x$ can be seen in Figure 2 in @hesse08 as well. The $y$ components ($V_{e{\perp}y}$ and $w_y$) are similarly bifurcated and $V_{e{\perp}y}$ outruns $w_y$ in the $-y$ direction near the midplane. We note that $V_{ex}<V_{e{\perp}x}$ and $V_{ey}<V_{e{\perp}y}$ outside the ECL, because there is a field-aligned electron outflow in the $(-x,+y)$ direction toward the X-line.
Figure \[fig:cut432\]b displays the variation in the field properties. Both the reconnecting magnetic field $B_x$ and the Hall magnetic field $B_y$ change their polarities across the midplane. The Hall electric field $E_z$ has a large-scale bipolar structure. It is negative in $z>0$ and positive in $z<0$. Its amplitude is eight times stronger than the reconnection electric field $|E_y|=0.15$. This also corresponds to the bipolar ${E}_{\parallel}$ at $|z| \lesssim 0.5$. Note that parallel electric field remains nonzero in any moving frame, because ${\boldsymbol{E}}\cdot{\boldsymbol{B}}$ is invariant. Outside there, since ${E}_{\parallel} \simeq 0$ and since ${\boldsymbol{V}}_{e\perp} \simeq {\boldsymbol{w}}$, the electron ideal condition is recovered, ${\boldsymbol{E}} + {\boldsymbol{V}}_e \times {\boldsymbol{B}} \simeq 0$. In a close vicinity of the midplane, $|z| < 0.1$, one can recognize a reverse bipolar structure in $E_z$ and $E_{\parallel}$. It is positive in $z>0$ and negative in $z<0$. This is an electrostatic field due to the electron meandering motion in $z$.[@chen11]
Kinetic Signatures {#sec:kinetic}
==================
Panels in Figure \[fig:VDF\] show electron velocity distribution functions (VDFs) at various locations at $t=35$. These VDFs are computed in small boxes of $0.5{\times}0.5$. The boxes are indicated in Figure \[fig:snapshot\]a and the figure titles indicate the box center positions. The VDFs are two-dimensional. The electron number is integrated in the third direction.
The top four panels (Figs. \[fig:VDF\]a–d) display VDFs ($v_{x}$–$v_{z}$) around the upper boundary regions. Note that we use lowercase ${\boldsymbol{v}}$ for the particle velocity in order to distinguish it from the bulk velocity ${\boldsymbol{V}}$. Figure \[fig:VDF\]a shows the electron VDF in $v_{x}$–$v_{z}$ just above the X-line. It looks highly anisotropic. Here, the magnetic field is directed in $x$, and so the electrons are heated in the parallel direction. @egedal05 [@egedal08] explained that the electrons are trapped by the parallel potential and that they are fast traveling in the field-aligned directions. Figure \[fig:VDF\]b is a VDF at $(x,z)=(43.2,0.6)$. It looks slightly tilted, because the magnetic field is directed in a slightly upward direction. These two VDFs are gyrotropic. The nongyrotropy measure does not mark these regions (Fig. \[fig:snapshot\]g).
The third panel (Fig. \[fig:VDF\]c) shows a VDF at $(x,z)=(47.7,1.2)$. In addition to the field-aligned component, one can see a hot outgoing component in the right half. A similar VDF was reported by @chen08 (\#10 in Fig. 4; §2.2.2 in Ref. ). The hot component looks partially gyrotropic; the electrons are found in the $v_{ey}<0$ half in $v_y$ \[not shown\]. As a result, the entire VDF is weakly nongyrotropic, because two or more components start to mix with each other here. The $\sqrt{Q}$ measure weakly marks this and nearby regions (Fig. \[fig:snapshot\]g). Interestingly, this hot component suddenly appears here. We do not find it in the separatrix regions closer to the X-line. In Figure \[fig:VDF\]d, one can see two field-aligned components, a cold incoming component and a high-energy outgoing component. The cold electrons are weakly accelerated by $E_{\parallel}$ toward the X-line. We will discuss the outgoing electrons later in this paper. Since both populations are gyrotropic, the VDF is anistropic but gyrotropic.
Panels in the bottom three rows show VDFs at four locations at the midplane. The VDFs in $v_{x}$–$v_{y}$ (the second row), in $v_{x}$–$v_{z}$ (the third), and in $v_{y}$–$v_{z}$ (the bottom row) are presented. Figure \[fig:VDF\]e exhibits typical signatures of a VDF in the DR. Compared with the inflow region (Fig. \[fig:VDF\]a), the VDF is stretched in $-y$, due to the $y$ acceleration by the reconnection electric field $E_y$. The VDF in $v_{x}$–$v_{y}$ looks triangular.[@ng11; @bessho14; @shuster15] Small $v_{x}$ electrons stay longer in the DR and therefore they are more accelerated in $-y$.[@prit05] The structure in $v_z$ is not so clear, because our box size in $z$ ($\Delta z = 0.5$) is larger than the electron meandering width. The electron VDFs in $|z| < 0.1$ are bifurcated in $v_{z}$ \[not shown\], as reported by previous studies.
Figure \[fig:VDF\]f shows the VDF at $(x,z)=(43.2,0)$ in the middle of the electron jet. The overall VDF is shifted in $+v_{x}$ in agreement with the fast bulk flow. The bulk velocity is ${\boldsymbol{V}}_{e} \approx (+7,-3,0)$. The electrons are spread in $v_{x}$–$v_{y}$, while they are confined in $v_{z}$. At the midplane $z=0$, the magnetic field is directed in $z$ and so the electron perpendicular pressure exceeds the parallel pressure. This VDF is highly nongyrotropic $\sqrt{Q}\approx 0.4$, as evident in Figure \[fig:snapshot\]g. In $v_{x}$–$v_{y}$ (Fig. \[fig:VDF\]f1), one can see a narrow ridge in the right, as indicated by the white arrow. This is related to $y$-accelerated electrons in the DR, indicated the white arrow in Figure \[fig:VDF\]e1. As we depart from the X-line, the bottom ridge in Figure \[fig:VDF\]e1 rotates anti-clockwise and then evolves into the right ridge in Figure \[fig:VDF\]f1. In $v_{y}$–$v_{z}$ (Fig. \[fig:VDF\]f3), the VDF is weakly bifurcated in $v_z$ for $v_{ey}<0$. Figure \[fig:VDF\]g shows the VDF at $(x,z)=(47.7,0)$ in the jet termination region. The VDF looks fairly isotropic in $v_{x}$–$v_{y}$. In the bottom two panels, the major outgoing component looks similar to one in Figure \[fig:VDF\]f. In addition, one can recognize a hot low-density component in the $v_x<0$ half. These electrons come from the downstream region and then they start to spread in ${\pm}v_{z}$. This corresponds to the vertical divergent flows in Figure \[fig:snapshot\]b.
Figure \[fig:VDF\]h shows the VDF at $x=51.6$, downstream of the remagnetization front. The electrons are isotropic in all three planes. One can recognize two small peaks in $v_{y}$–$v_{z}$, as indicated by the arrows in Panel h3. Similar components in VDFs were reported by @shuster14. These energetic electrons travel backward from the downstream magnetic island, but it is not clear how they are accelerated. This is one of the earliest signals from the downstream region. To avoid side-effects from the downstream, we limit our attention to the upstream side, $x \lesssim 51.6$.
Figure \[fig:phase432\] provides additional information to electron kinetic physics. Figure \[fig:phase432\]a shows the phase-space diagram in $v_{y}$–$z$ along the inflow line at $x=38.1$. Here, as the electrons travel in ${\pm}z$ from the inflow regions toward the midplane $z=0$, they start to drift in $-y$ due to the polarization electric field $E_z$. Once they enter the ECL, they are accelerated by the reconnection electric field $E_y$ through Speiser motion.[@speiser65] In the $v_{z}$–$z$ diagram (Fig. \[fig:phase432\]b), a circle around the central hole corresponds to a bounce motion during the Speiser motion.[@hori08; @chen11] Interestingly, the electron density is high outside the ECL, $|z|{\gtrsim}0.5$ (Figs. \[fig:phase432\]a and \[fig:phase432\]b). Figure \[fig:phase432\]c is the energy-space diagram in $\mathcal{E}$–$z$, where $\mathcal{E} = \frac{1}{2}m_ev_e^2$ is the electron kinetic energy, normalized by $m_i c_{Ai}^2$. High-energy electrons ($\mathcal{E} > 1.0$) are only found inside the ECL, $|z|<0.25$.
The right Panels show similar diagrams for the electron jet region at $x=43.2$. Figures \[fig:phase432\]d and \[fig:phase432\]e are the $v_{x}$–$z$ and $v_{z}$–$z$ diagrams. The electrons are accelerated in $x$ (Fig. \[fig:phase432\]d) and one can see a phase-space hole in Figure \[fig:phase432\]e. There are high-density regions outside the ECL, $|z|{\gtrsim}0.5$ (Figs. \[fig:phase432\]d and \[fig:phase432\]e). In Figure \[fig:phase432\]f, although medium-energy electrons ($\mathcal{E} = 0.5$–$1.0$) are distributed wider in $z$, high-energy electrons ($\mathcal{E} > 1.0$) are confined around the ECL.
Particle trajectories {#sec:traj}
=====================
In this work, we manage to record as many electron trajectories as possible in our PIC simulation. Using the particle ID number, we select $3\%$ ($1/32$) of electrons without bias. Then we output the selected particle data to a hard drive every two plasma periods $\Delta t = 2\omega_{pe}^{-1}$ during $30<t<36.25$. The time interval is comparable with one ion gyroperiod, $6.25 \approx 2\pi$. The time resolution is sufficient to see electron gyrations, i.e., the typical electron gyroperiod is $2\pi \Omega_{ce}^{-1} \approx 25 \omega_{pe}^{-1} \gg \Delta t$. It is even sufficient for plasma oscillations around the reconnection site, i.e., $2\pi\omega_{pe}^{-1}(n_{b}/n_0)^{-1/2} \approx 14\omega_{pe}^{-1} > \Delta t$. As a result, we obtain $2.3{\times}10^7$ electron trajectories. We introduce characteristic trajectories in our dataset in the following subsections.
Speiser orbits {#sec:Speiser}
--------------
Panels in Figure \[fig:traj\_a\] show the first set of representative trajectories. The electron orbits during the interval are presented in spatial/energy/velocity/phase spaces. The circles indicate their positions at $t=35$. The first orbit in red is a typical Speiser orbit.[@speiser65] In the $x$–$z$ space (Fig. \[fig:traj\_a\]a), this electron comes from the bottom left to enter the DR. Then it is accelerated in the $-y$ direction by the reconnection electric field $E_y$ (Fig. \[fig:traj\_a\]b). As a consequence, it quickly gains energy near the X-line (Fig. \[fig:traj\_a\]c). The electron gradually turns in the $+x$ direction (Fig. \[fig:traj\_a\]b), while staying around the midplane ($z \approx 0$; Fig. \[fig:traj\_a\]a). Finally it escapes from the midplane to the upper right, gyrating about the magnetic field line. In the $v_{x}$–$v_{y}$ space (Fig. \[fig:traj\_a\]d), it initially starts from the center, moves downward in $-v_{y}$ due to the $y$-acceleration near the X-line, rotates anti-clockwise as it turns in $v_{x}$, and then exhibits larger gyration after it exits from the midplane. The $z$-bounce motion around the midplane is also evident in the central circles in the $v_{z}$–$z$ space (Fig. \[fig:traj\_a\]g).
The second orbit in green is another example of the Speiser orbit. This one is much more accelerated around the X-line than the first one (Fig. \[fig:traj\_a\]c). After the Speiser rotation, this electron wanders around the midplane, $|z| \lesssim 1$, in the downstream region ($x \gtrsim 47$). This is interesting, because we expect that the electron escapes along the field line like in the first orbit after the Speiser motion.
The third orbit in blue represents a Speiser motion of different kind. The electron comes from the upper right and reaches the midplane at $x\approx 43$ (Fig. \[fig:traj\_a\]a). There, it slowly gyrates about $B_z$, turns its direction from $-y$ to $+x$ while bouncing in $z$. Instead of passing through the DR, it is locally reflected to the downstream. This is a Speiser orbit of local reflection-type. Following our previous work on ion orbits,[@zeni13] we call this blue electron orbit a “local Speiser orbit,” and the previous red and green orbits “global Speiser orbits.” The blue electron gains less energy than the other electrons through global Speiser orbits. This is because the local magnetic field $B_z$ is stronger, and because the electron turns more quickly than in the DR. During the local reflection phase, the velocity vector rotates anti-clockwise from $-v_{y}$ to $+v_{y}$ (Fig. \[fig:traj\_a\]d). This is because the magnetic field line also turns from $\pm x$ near the X-line to $\pm y$ near $x \approx 47$, where this blue electron escapes from the midplane. Interestingly, the electron still remains near the midplane, $|z| \lesssim 1$, chaotically bouncing in $z$, similar to the second electron in green.
Let us examine the first and third orbits near $x=43.2$ in more detail. Both are located near the midplane ($|z| \lesssim 0.1$; Figs. \[fig:traj\_a\]a, \[fig:traj\_a\]f, and \[fig:traj\_a\]g) at $t=35$. In such close vicinity to the midplane, both $B_x$ and $B_y$ are approximately linear in $z$ (Fig. \[fig:cut432\]b), while $B_z$ is roughly constant, $B_z \approx 0.06$. This configuration is similar to the system in Section \[sec:theory\], (1) if the system is uniform in $x$ and (2) if we switch to an appropriately moving frame in which the electric field vanishes. Even though the two conditions are not exactly met in the PIC simulation, the theory provides insight into electron motions. We fit the magnetic field across the midplane at $x=43.2$ to the parabolic model (Eq. ) to obtain the magnetic curvature radius $R_{\rm c,min}$. A similar procedure is presented in Section III B in Ref. . At $x=43.2$, the field line is so sharply bent that the curvature radius is $R_{\rm c,min} = 0.068$. The electron maximum Larmor radius and the curvature parameter are as follows, $$\begin{aligned}
\frac{r_{\rm L,max}}{d_i}
&=
\Big( \frac{v'_e}{c_{Ai}} \Big)
\Big( \frac{m_e}{m_i} \Big)
\Big( \frac{B_0}{|B_z|} \Big)
\approx
{1.64}
\Big( \frac{v'_e}{10 c_{Ai}} \Big) \\
\kappa
& \approx
{0.2}
\Big( \frac{v'_e}{10 c_{Ai}} \Big)^{-1/2}
=
{0.16}
\Big( \frac{\mathcal{E}'}{m_i c^2_{Ai}} \Big)^{-1/4}
,
\label{eq:kappa432}\end{aligned}$$ where the prime sign $'$ denotes a physical quantity in a rest frame. Here we consider an appropriate rest-frame velocity ${\boldsymbol{U}}$, so that ${\boldsymbol{v}}'_e = {\boldsymbol{v}}_e-{\boldsymbol{U}}$. In this case, since the [**E**]{}$\times$[**B**]{} velocity is non-uniform (Fig. \[fig:cut432\]a) and since $E_{\parallel}$ is finite (Fig. \[fig:cut432\]b), it is impossible to find out ${\boldsymbol{U}}$ that transforms away the electric field. We approximate ${\boldsymbol{U}}_{43.2}\approx (4,-3,0)$ by referring to the [**E**]{}$\times$[**B**]{} velocity at $z = \pm 0.22$–$0.24$, where $|E_z|$ hits its maximum (Fig. \[fig:cut432\]b). In this case, compared with the electron velocities in Figure \[fig:traj\_a\]d, the frame speed $|{\boldsymbol{U}}_{43.2}|$ is relatively small and so one can approximate $\mathcal{E}'\approx \mathcal{E}$. From Equation and Figure \[fig:traj\_a\]c, one can see that $\kappa \sim \mathcal{O}(0.1)$ for the two electrons. This is reasonable, because the Speiser motion appears in the $\kappa \ll 1$ regime. Regardless of whether they follow global or local Speiser orbits, we find that the electrons undergo either of the following two orbits after the Speiser phase. One follows a field-aligned outgoing orbit, like in the first red electron. This corresponds to the field-aligned electron outflow near the separatrix, discussed in Section \[sec:fluid\]. The other follows a chaotic bounce motion around the midplane, as evident in the second green orbit. It is located at $(x,z)=(50.0,-0.1)$ at $t=35$ and so we discuss the electron motion near $x \sim 50$. This region corresponds the middle of a broader current layer of unmagnetized ions.[@zeni13; @le14] The electric current is weaker, and therefore the magnetic curvature radius is larger than in the super-Alfvénic jet. We estimated the curvature radius $R_{\rm c,min}=0.62$ at $x = 50.0$ and the frame velocity ${\boldsymbol{U}}_{50.0}=(1.37,-0.1,0)$. The electric field is excellently transformed away. The electron maximum Larmor radius and the curvature parameter are, $$\begin{aligned}
\frac{r_{\rm L,max}}{d_i}
\approx
0.83
\Big( \frac{\mathcal{E}'}{m_i c^2_{Ai}}\Big)^{1/2},
~~
\kappa
\approx
0.86
\Big( \frac{\mathcal{E}'}{m_i c^2_{Ai}}\Big)^{-1/4}
.\end{aligned}$$ One can approximate $\mathcal{E}'\approx\mathcal{E}$, because the frame speed $|{\boldsymbol{U}}_{50.0}|=1.37$ is negligible. One can see $\kappa \lesssim 1$ from Figure \[fig:traj\_a\]c. Figure \[fig:traj\_a\]a tells us that the typical bounce width $|z|\sim 1$ is comparable with $R_{\rm c,min}=0.62$. In this regime, the particle motion becomes highly variable. Although we do not track the orbit long enough, the third blue electron exhibits a similar nongyrotropic motion around $x\gtrsim 48$.
Noncrossing Speiser orbits {#sec:noncrossing}
--------------------------
In Figure \[fig:traj\_b\], we show a new class of electron orbits in the same format as Figure \[fig:traj\_a\]. These electrons approach the midplane $z=0$ and then all of them exhibit Speiser-like rotations in the $x$–$y$ plane (Fig. \[fig:traj\_b\]b). During the motion, particles bounce in $z$. However, surprisingly, they do not cross the midplane (Fig. \[fig:traj\_b\]a). To guide our eyes, we indicate $z=0$ and $z=0.6$ by the dotted lines in Figures \[fig:traj\_b\]a, \[fig:traj\_b\]f, and \[fig:traj\_b\]g. One can see that all orbits are above the $z=0$ line. Owing to this, we call them “noncrossing Speiser orbits.” Strictly speaking, there is no guarantee that the reconnecting magnetic field changes its polarity at the midplane, $z=0$. However, we confirm that the midplane is fairly identical to the field reversal, because of the perfectly symmetric configuration in $z$ and a large number of particles per cell in our simulation. In panels in Figure \[fig:snapshot\], the dashed line indicates the field reversal, $B_x=0$. One can see that it is located at $z \approx 0$ and that it is sometimes slightly [*below*]{} the midplane ($z<0$) near the remagnetization front. This provides further confidence that these electrons do not cross the field reversal plane.
First, we examine the motion of electron \#1 in red. It starts from the upper inflow region (Fig. \[fig:traj\_b\]a). Since there is a reconnection electric field $E_y$, it drifts in $-z$ at the speed of $-E_y/B_x$ while traveling along the field line. Once it reaches the $0.3 \lesssim z \lesssim 0.8$ region above the DR, it travels in the $-y$ direction. Its energy starts to increase (Fig. \[fig:traj\_b\]c). The $-y$-motion is attributed to the [**E**]{}$\times$[**B**]{} drift by the polarization electric field $E_z$.[@keizo06; @li08] The gyrocenter velocity approaches $(0,E_z/B_x,-E_y/B_x)$, which becomes faster in the closer vicinity of the midplane, because the magnetic field decreases $B_x \rightarrow 0$. This drift motion in $-y$ is also evident in Figure \[fig:phase432\]a. Note that $E_z$ is negative here (Fig. \[fig:snapshot\]c). At $t=35$, the electron is located at $(x,z)=(38.15,0.46)$ with the velocity of ${\boldsymbol{v}}_e = (1.0,-4.3,-0.06)$. Below this position, the electron starts to turn to the outflow direction (Fig. \[fig:traj\_b\]b) above the midplane. Figures \[fig:traj\_b\]d and \[fig:traj\_b\]e present the velocity-space trajectories within $0<z<0.6$. The velocity for the electron \#1 rotates anti-clockwise in $v_x$–$v_y$ (Fig. \[fig:traj\_b\]d), turning to the outflow direction. During this phase, the electron bounces in $z$ (Figs. \[fig:traj\_b\]a and \[fig:traj\_b\]e). These features are similar to those in the Speiser motion.
We further examine the orbit \#1 in the 3D velocity space (Fig. \[fig:orbit247\]a). One can see that the velocity vector keeps rotating in the same direction. This tells us that the electron motion is a combination of a gyration and a guiding-center motion. It is apparently different from the conventional Speiser motion with a meandering motion, which exhibits the zigzag pattern in the velocity space (Fig. \[fig:theory\]b). In Figure \[fig:orbit247\]a, the spiral path further indicates that the electron is continuously accelerated in the parallel direction. We confirm that the electron \#1 is accelerated by a parallel electric field, by reconstructing the electromagnetic field at the particle position. Except for minor noises, the parallel field $E_{\parallel}$ points inward to the X-line, continuously accelerating the electron away from the X-line in $+x$. This is consistent with the spatial profile of $E_{\parallel}$ (Fig. \[fig:snapshot\]d). In the case of the electron \#1, the parallel acceleration is responsible for the most of the energy gain, in particular at the later stage. On a longer time scale, the electron velocity slowly rotates anti-clockwise in $v_x$–$v_y$ (Fig. \[fig:traj\_b\]d). One can also interpret that the electron slowly gyrates about $B_z$, while the $E_z$ field prevents the particle from crossing beyond a certain distance in $z$. Note that a field-aligned component $E_{\parallel}$ is a projection of the polarization electric field $E_z$. Summarizing these results, this orbit is similar to but different from the traditional Speiser orbit, in the sense that it relies on a combination of the drift motion and the parallel acceleration instead of the meandering motion. Hereafter we call the orbit the “[*noncrossing*]{} Speiser orbit.”
The second (green) and third (blue) orbits in Figure \[fig:traj\_b\] are other examples of the noncrossing Speiser orbits. The electron \#2 exhibits multiple reflections in the $z$ direction. After entering the central region at $x \approx 42$, it slowly turns to the $+x$ direction, travels upward at $x \approx 47$, and then comes back to the central channel once again. It travels fast in $x$ and in $-y$ near the midplane due to the [**E**]{}$\times$[**B**]{} drift by the Hall field $E_z$ (Figs. \[fig:snapshot\]c and \[fig:snapshot\]e), while it slowly moves in the pedestal region outside the electron jet. The electron \#3 in blue travels backward along the field lines into the central channel at $x=44$, and then it drifts in the $-y$ direction due to the Hall field $E_z$. The initial energy of this electron is very low. It is accelerated to the [**E**]{}$\times$[**B**]{} speed $\approx |E_z/B|$ in this jet flank region. Then the electron turns round to $+x$ around $44<x<47$. One can also see the spiral in the velocity spaces (Figs. \[fig:traj\_b\]d and \[fig:traj\_b\]e), indicating a parallel acceleration by $E_{\parallel}$. Finally, the electron escapes upward along the field lines.
We verify the forces acting on the noncrossing Speiser-orbit electrons using a conceptual model. To mimic the Hall field $E_z$, we impose ${\boldsymbol{E}} = -|v_{e0}|B_0 \sin(\pi z/L) {\boldsymbol{e}}_z$ near the midplane ($|z|<L$) to the parabolic model in Section \[sec:theory\]. Here, $|v_{e0}|=1$ is the initial electron velocity outside the Hall-field region ($|z|>L$). Corresponding electrostatic potential $\int_0^L |E_z| dz = 2/\pi $ is sufficient to reflect electrons whose normalized energies are $\frac{1}{2}m_e|v_{e0}^2| = 0.5$. Figure \[fig:orbit247\]b displays test-particle orbits in the modified field. The blue orbit (solid line) employs the same initial condition as the Speiser-orbit electron (dashed line) in Figure \[fig:theory\]a. As can be seen, it excellently reproduces qualitative features for noncrossing Speiser orbits. The electron remains on the upper half due to the electric field, turns its direction, and then exits in the $+x$ direction. Figure \[fig:orbit247\]c shows the velocity-space trajectory, when the electron is in the right half ($y<0$) around the midplane ($z<1.5$). It exhibits a similar spiral of the [**E**]{}$\times$[**B**]{} drift and the parallel motion as in Figure \[fig:orbit247\]a.
We note that these noncrossing electrons have lower energy than the Speiser electrons in Section \[sec:Speiser\]. If electrons have enough energies, they will cross the midplane. Among the three, the first one gains the highest energy, probably because $B_z$ is weak near the X-line. Its radius in the $x$–$y$ plane (Fig. \[fig:traj\_b\]b) is the largest. The other two are picked up by the outflow exhaust and then they are locally reflected above the midplane. All these features are similar to the Speiser orbits, even though electrons are always reflected upward by the Hall field $E_z$ or by the parallel electric field $E_{\parallel}$ above the midplane at $z = 0.2$–$0.4$ (Figs. \[fig:snapshot\]c and \[fig:cut432\]b). In analogy with the conventional Speiser orbits, we classify the first one as the noncrossing global Speiser orbit, and the other two as the noncrossing local Speiser orbits.
Regular orbits {#sec:trapped}
--------------
Figure \[fig:traj\_c\] presents electron orbits of another kind in the same format as in Figure \[fig:traj\_a\]. The first one in red originally comes from the bottom right and then undergoes the local Speiser motion. The velocity vector rotates anti-clockwise in $v_{x}$–$v_{y}$ (Fig. \[fig:traj\_c\]d) and then the electron eventually turns in the $-x$ direction. Very interestingly, it starts to bounce in $z$ at $x>45$. This orbit looks stable. We argue that this is a regular orbit in a curved magnetic geometry (Section \[sec:theory\]). The diagonal oscillation in $v_{x}$–$v_{y}$ (Fig. \[fig:traj\_c\]d), the inverse C-shaped oscillation in $v_{x}$–$v_{z}$ (Fig. \[fig:traj\_c\]e), and the V-shaped path in the phase space (Fig. \[fig:traj\_c\]f) suggest a trapped motion in an appropriately moving frame. The diagonal oscillation is transverse to the magnetic fields outside the electron current layer. The characteristic closed circuit in the $v_z$–$z$ space (Fig. \[fig:traj\_c\]g) is consistent with the regular orbit in Figure \[fig:theory\]d (the red orbit). At $x=45.6$, the curvature radius is $R_{\rm c,min}=0.079$ and the normal magnetic field is $B_z=0.069$. The corresponding curvature parameter is $$\begin{aligned}
\label{eq:kappa456}
\kappa
& \approx
{0.23}
\Big( \frac{v'_e}{10 c_{Ai}} \Big)^{-1/2}
=
{0.20}
\Big( \frac{\mathcal{E}'}{m_i c^2_{Ai}} \Big)^{-1/4}
.\end{aligned}$$ In this case, it is difficult to estimate the reference-frame velocity ${\boldsymbol{U}}$, because the ideal velocity ${\boldsymbol{w}}$ has the variation in $z$ (Fig. \[fig:snapshot\]e). We roughly evaluate $v'_e=7.5$–$10$ (Fig. \[fig:traj\_c\]e) and $\mathcal{E}' \approx 0.7$ (Fig. \[fig:traj\_c\]c), and then obtain $\kappa \approx 0.2$.
In the second case, the green electron enters the DR and then undergoes a global Speiser motion. After leaving the midplane at $x=43.7$, it starts gyrating in the lower half. The orbit looks similar to the first regular orbit in red in the phase spaces (Figs. \[fig:traj\_c\]f and \[fig:traj\_c\]g), except that the electron always remains below the midplane. Figures \[fig:snapshot\]c–\[fig:snapshot\]d suggest that the Hall field $E_z$ keeps the electron away from the midplane. Without $E_z$ or $E_{\parallel}$, an electron usually crosses the midplane in such a field configuration, because it is reflected by the mirror force toward the midplane. This electron keeps gyrating around $z \sim -0.5$, because it is also mediated by the Hall field $E_z$. We argue that this is a noncrossing variant of the electron regular orbit. It is detached from the midplane, due to the Hall field $E_z$.
Both the third electron in blue and the last electron in magenta travel through similar stable orbits. The blue one comes from the inflow region and then enters the stable channel after crossing the separatrix. The magenta one directly enters the channel, traveling above the X-line. They keep gyrating on the upper flank of the electron jet region (Figs. \[fig:traj\_c\]a, \[fig:traj\_c\]f, and \[fig:traj\_c\]g). They have lower energies than the previous two cases. One may interpret these electrons as just drifting. This is a good point, but we remark that drift motions have no influence in the parallel motion. These electrons are trapped in the parallel direction, balanced by the mirror force toward the midplane and the parallel electric force $-eE_{\parallel}$ away from the midplane. We verify the forces on these orbits using the test-particle model in Section \[sec:noncrossing\]. A stable orbit is shown in red in Figure \[fig:orbit247\]b. Therefore, it is appropriate to call the orbits the noncrossing regular motions, rather than drift motions.
Theoretically, the figure-eight-shaped regular orbits exist in the field reversal for $\kappa \lesssim 0.53$.[@wang94] However, we find the figure-eight-shaped regular orbits only for $\kappa \lesssim 0.2$. We attribute to this to the Hall electric field $E_z$, which remains finite in the moving frame. Keeping electrons away from the midplane, $E_z$ delays the $z$-bounce motion. Recalling that $\kappa$ is the frequency ratio of the gyration about $B_z$ to the $z$-bounce motion, the Hall field $E_z$ increases an effective $\kappa$ and therefore the threshold is reduced to $\kappa \approx 0.2$. One can also interpret in the following way: While lower-$\kappa$ (higher-energy) electrons are insensitive to $E_z$, higher-$\kappa$ (lower-energy) electrons are sensitive. The Hall field $E_z$ transforms the high-$\kappa$ figure-eight-shaped orbits ($0.2 \lesssim \kappa \lesssim 0.5$) to the noncrossing regular orbits. In Figures \[fig:traj\_c\]c–\[fig:traj\_c\]e, one can see that all the noncrossing regular electrons have lower energy than the crossing electron in red in an appropriate frame. Although $\kappa$ is not so meaningful for the noncrossing electrons, we plug in their typical velocities to Eq. \[eq:kappa456\] to obtain $0.2 \lesssim \kappa \lesssim 0.4$. This further suggests that they are detached variants of the regular orbits.
Composition Analysis {#sec:comp}
====================
The trajectory dataset allows us to explore kinetic signatures in further detail. Figure \[fig:VDF2\] shows the velocity, energy, and phase-space distribution of electrons at $t=35$. Each symbol stands for the electrons in the dataset. Based on their trajectories, we classify the electrons into the following three classes. The green circles indicate electrons that never cross the midplane $z=0$ during the interval ($30<t<36.25$). We call them “noncrossing candidates.” They may cross the midplane before $t=30$ or after $t=36.25$. The red triangles indicate electrons that spent some time in the square region of $x,z \in [35.5,40.5] \times [-0.2, 0.2]$ during the interval. This region approximates the DR, which is indicated in Figure \[fig:snapshot\]h. We call them “DR-crossing electrons.” The blue crosses indicate the other electrons. They have crossed the midplane at least once, but they spent no time in the DR during the interval. We call them “crossing electrons.” Please note that the frontmost symbols sometimes overwrite the background ones in panels in Figure \[fig:VDF2\]. The order is carefully selected to emphasize interesting features.
Panels in the first three rows show the electron VDFs at $t=35$. They are equivalent to those in Figure \[fig:VDF\]. Typical features of the separatrix VDFs are found in Figures \[fig:VDF2\]b–d. In general, one can recognize outgoing red particles from the DR and incoming blue electrons toward the midplane. In Figure \[fig:VDF2\]b, the leftmost electrons turn red, because they are going to enter the DR. Figure \[fig:VDF2\]c contains the outgoing blue population, which crossed the midplane outside the DR. In Figure \[fig:VDF2\]d, the outgoing red population is more prominent in the $v_{ex}>0$ half, even though the outgoing blue population is also hidden behind them. This is reasonable because the electrons from the DR (global Speiser electrons) are more energetic than the local Speiser electrons, and therefore the DR-crossing electrons travel deeper into the exhaust region beyond the remagnetization front.
Surprisingly, we find a substantial amount of noncrossing candidates in the first four VDFs (Figs. \[fig:VDF2\]a–d). In these panels, a number on the bottom-right corner indicates the ratio of the number of noncrossing candidates to the total number. In Figure \[fig:VDF2\]a, right above the DR, 89% of the electrons are noncrossing candidates. The noncrossing candidates are also hidden behind the central red population. For example, the diamond symbol indicates the electron \#1 in Figure \[fig:traj\_b\]. As discussed, it travels through the global noncrossing Speiser orbit. Here this electron is classified as noncrossing electrons in green. Interestingly, this electron also hits the DR of $x,z \in [35.5,40.5] \times [-0.2, 0.2]$, even though it does not cross the midplane. In contrast, only a limited number of electrons are entering the DR and crossing the midplane. The red population is found only around the center $|v_{ex}| \sim 0$, while left-going and right-going populations are noncrossing. In the next domain (Fig. \[fig:VDF2\]b), although some blue crosses are hidden behind the green circles, the noncrossing candidates are majority, accounting for 68% of the total electron number. The purple and magenta circles indicate the electrons \#3 and \#4 in Figure \[fig:traj\_c\]. As discussed in Section \[sec:trapped\], they are trapped on the upper flank of the electron jet, traveling through noncrossing regular orbits. The relevant blue orbit in $v_x$–$v_z$ (Fig. \[fig:traj\_c\]e) is in excellent agreement with the green region in Figure \[fig:VDF2\]b. These results suggest that the green noncrossing population in Figure \[fig:VDF2\]b are likely to travel through the noncrossing regular orbits.
Farther away from the DR, in Figure \[fig:VDF2\]c, 25% of electrons are noncrossing candidates. The two squares indicate the noncrossing Speiser electrons \#2 and \#3 in Figure \[fig:traj\_b\]. They are either reflecting back to the midplane (\#2) or escaping outward (\#3). In this VDF, the noncrossing candidates are found around the center. Their velocity distribution is fairly unchanged from the previous case (Fig. \[fig:VDF2\]b). The hot outgoing population consists of the crossing electrons in either red or blue. This suggests that they are Speiser-accelerated electrons from the midplane. One can see in Figure \[fig:traj\_a\] that the Speiser-accelerated electron in red wraps around the magnetic field line along the separatrix (Fig. \[fig:traj\_a\]a). Its velocity (Fig. \[fig:traj\_a\]e) explains the hot population in Figure \[fig:VDF2\]c very well. The hot population is evident in the (c) region and in further downstream along the separatrix, because Speiser electrons are ejected from the midplane at the end of the ECL, as can be seen in the orbit \#3 (blue) in Figure \[fig:traj\_a\]a (See also Fig. 4(a) in Ref. ). This agrees with the divergent flows in ${\pm}z$ (Fig. \[fig:snapshot\]b; Sec. \[sec:fluid\]) and the vertically spread VDF (Fig. \[fig:VDF\]g; Sec. \[sec:kinetic\]).
In Figure \[fig:VDF2\]d, we recognize green noncrossing candidates in the incoming direction ($v_x<0$). Some are hidden behind the blue crossing electrons. These green candidates may be overemphasized, because the (d) region is far from the DR and the midplane. The field-line through $(x,z)=(52.5,3.0)$ at $t=35$ is convected to $(x,z)=(48.4,0.0)$ at $t=36.25$, and the field-aligned distance to the midplane is $\approx 5 [d_i]$. The electrons at a velocity $|v_e|=4$ will travel $5 [d_i]$ from $t=35$ to $t=36.25$. Therefore, some green electrons could be crossing electrons. On the other hand, our classification will be valid in the outgoing part ($v_x>0$). One can see energetic electrons from the DR in red. They are more pronounced than blue crossing populations. Importantly, the green population retains signatures similar to those in the previous cases. From these four panels, we find a non-negligible amount of noncrossing electrons. This will be further analyzed in this section.
Panels in the second and third rows show the VDFs at the midplane, similar to those in Figure \[fig:VDF\]. The left panels (Fig. \[fig:VDF2\]e) indicate the VDFs around the X-line. We recognize some amount of green noncrossing electrons here, because the VDF is calculated in a thicker box in $z$ than the square region to classify the red population, and because some noncrossing electrons come close to the midplane (e.g., the orbit \#1 in Fig. \[fig:traj\_b\]a). Aside from them, the (e) region is filled with the DR-crossing electrons in red. As we depart from the X-line in the $+x$ direction, the blue population gradually replaces the red population in the VDFs. In the $v_x$–$v_y$ space (Fig. \[fig:VDF\]f1), the blue population appears in the bottom ($v_{ey}<0$). Then they evolve anti-clockwise, as the dashed arrow indicates. The red population rotates anti-clockwise accordingly. Finally, all these electrons are mixed with each other around the remagnetization front (Fig. \[fig:VDF\]g). We see no remarkable separation in color farther downstream.
In Figure \[fig:VDF2\]f1, we argue that the global Speiser motion accounts for the DR-crossing electrons in red and that the local Speiser motion accounts for the other crossing electrons in blue. The two circles in Figure \[fig:VDF2\]f indicate the representative electrons for the global Speiser motion (the orbit \#1 in Fig. \[fig:traj\_a\]) and for the local Speiser motion (\#3), discussed in Section \[sec:Speiser\]. For the local-type Speiser motion, we expect a half-ring distribution function in $v_x$–$v_y$, corresponding to the slow half-gyration about $B_z$. In Figure \[fig:VDF2\]f1, the gray arrow indicates the orientation of the magnetic field at $z = 0.22$–$0.24$. It is tilted by 56 degrees due to the Hall effect. In Section \[sec:Speiser\], we estimated the frame velocity ${\boldsymbol{U}}_{43.2}=(4,-3,0)$. This is indicated by the black cross in Figure \[fig:VDF2\]f1. Keeping these in mind, one can see that the blue electrons are distributed in a semicircle or a half ring surrounding ${\boldsymbol{U}}_{43.2}$ in this velocity space. The semicircle is tilted, similar to the magnetic field outside the ECL (the gray arrow). From Figures \[fig:VDF2\]f1 and \[fig:VDF2\]f2, one can see the typical velocity for the blue electrons ${v}'_e=|{\boldsymbol{v}}_e-{\boldsymbol{U}}_{43.2}| \approx 5$–$15~c_{Ai}$, which corresponds to the Speiser regime of $\kappa < 1$ (Eq. \[eq:kappa432\]). All these features are consistent with the Speiser motion of local-reflection type.
In the (f) region, the green noncrossing candidates are found near ${\boldsymbol{U}}_{43.2}$ in the velocity spaces. Their thermal velocity is smaller than in the upper (b) region, probably because they lose their energy due to the Hall field $E_z$. Some more signatures of the green noncrossing electrons are evident in the energy-space and phase-space diagrams for the electron distribution (Figs. \[fig:VDF2\]fa and \[fig:VDF2\]fb), which correspond to Figs. \[fig:phase432\]f and \[fig:phase432\]e. In contrast to the two crossing populations (blue and red), the green noncrossing candidates are found only outside the midplane $|z| \gtrsim 0.1$. Their energy is low, $\mathcal{E} \lesssim 0.3$, in agreement with small thermal velocity in the VDFs. By definition, particles move downward (upward) in the $v_{ez}<0$ ($v_{ez}>0$) region in the $v_z$–$z$ space (Fig. \[fig:VDF2\]fb). With this in mind, we see that the green electrons are reflected away from the midplane. They rotates anti-clockwise near the $v_{z}=0$ axis, as indicated by the green arrows. These features are consistent with the noncrossing orbits in Section \[sec:traj\] (see Figs. \[fig:traj\_b\]g and \[fig:traj\_c\]g).
Next, we investigate spatial distribution of the noncrossing candidates. Figure \[fig:dist\]a shows the density of the noncrossing candidates, reconstructed from our 3% dataset, in the same color range as in Figure \[fig:snapshot\]f. Around the ECL, there are three layers of (1) the high-density yellow layers in the inflow region, (2) the medium-density green layers near the separatrices, and (3) the blue cavity along the midplane. Comparison between Figures \[fig:snapshot\]f and \[fig:dist\]a tells us that the noncrossing electrons are the majority in the inflow region, in particular in the high-density yellow layer. The noncrossing electron density decreases in the medium-density green layers, but it is non-negligible $\sim 0.1$. The noncrossing electrons disappear in the downstream of the remagnetization front, $x \gtrsim 48$. We note that the noncrossing electron density could be underestimated in the flanks of the ECL at $45 \lesssim x \lesssim 47$. Even if some electrons do not cross the midplane at $t<35$, once they cross the midplane somewhere in the downstream ($x \gtrsim 48$) during $35<t<36.25$, we count them as crossing electrons in our analysis. Despite these concerns, we recognize many noncrossing candidates.
The presence of noncrossing electrons implies that upper-origin and lower-origin electrons may not mix with each other across the ECL. Figure \[fig:dist\]b shows the electron mixing fraction at $t=35$, computed from the full PIC datasets. The fraction is defined in the following way $$\begin{aligned}
\label{eq:mixing}
\mathcal{M} \equiv \frac{N_{\rm up}-N_{\rm low}}{N_{\rm up}+N_{\rm low}}\end{aligned}$$ where $N_{\rm up}$ is the number density of electrons that were in the upper half ($z>0$) at $t=30$ and $N_{\rm low}$ in the lower half ($z<0$). It ranges from $\mathcal{M} \rightarrow +1$ in the upper inflow region to $\mathcal{M} \rightarrow -1$ in the lower region. During $30<t<35$, the magnetic flux across the X-line is transported by 1.7 in ${\pm}z$ and by 11.4 in ${\pm}x$. The latter is comparable with the distance between the X-line and the remagnetization front. Thus, we expect that electrons are fully mixed $\mathcal{M} \approx 0$ in the exhaust region between the separatrices. However, surprisingly, the electrons are mixed only inside the ECL in the upstream of the remagnetization front ($x{\lesssim}48$). They are largely unmixed outside the ECL. Weakly mixed regions around $45 < x <48$ between the ECL and separatrices do not change this picture. The electrons are quickly mixed in the downstream, $x{\gtrsim}48$. Based on these results, we conclude that electron mixing is inefficient in the upstream side of the remagnetization front ($x{\lesssim}48$) and that the electron mixing occurs mainly in the downstream of the remagnetization front.
Crossing electrons are distributed in the outflow region between the separatrices. Many of them follow the Speiser orbits. Through Speiser-type orbits, electrons can be accelerated to higher energies than the noncrossing electrons. Motivated by this, we examine the spatial distribution of energetic electrons. Figure \[fig:dist\]c shows a number density of electrons whose energy exceeds a threshold, $\mathcal{E} > 1.0$. They are localized around the ECL ($x<48$). The localization of the high-energy electrons is also evident in Figures \[fig:phase432\]c and \[fig:phase432\]f. They are crossing populations, as confirmed in Figure \[fig:VDF2\]fa. These electrons follow either the global Speiser orbits from the DR or the local Speiser orbits that turn around inside the ECL. After the Speiser phase, these electrons escape along the separatrices or they chaotically remain around the midplane, as discussed in Section \[sec:Speiser\]. In Figure \[fig:dist\]c, the energetic electrons are located on separatrices in the downstream of the remagnetization front ($47<x<53$). This supports the former (the orbit \#1 in Fig. \[fig:traj\_a\]), while we do not see significant energization near the midplane. We find that some nongyrotropic electrons lose their energy as shown by the orbit \#2 in Figure \[fig:traj\_a\]c. In these regions, @hoshino01a proposed a two-step mechanism of the Speiser acceleration and the $\nabla{B}$ acceleration. Our results are not favorable to the @hoshino01a’s proposal, probably because combinations of acceleration mechanisms vary from case to case. Farther downstream ($x>52$), the energetic electrons are again found near the midplane. The region is equivalent to an outer edge of a long magnetic island, inside which these electrons are confined. We confirm that the energetic electrons are repeatedly accelerated inside the island across the periodic boundary in $x$.[@drake06] Since we focus on electrons from the ECL side, these energetic electrons are out of the scope of this study.
In summary, a substantial amount of noncrossing electrons are found outside the ECL. They have less energy than the high-energy population inside the ECL, because the crossing electrons are accelerated via the Speiser process, once they enter the ECL. The energetic electrons exit from the ECL toward the separatrices.
Observational signatures {#sec:ET}
========================
We discuss potential observational signatures in this section. Figures \[fig:ET\]a and \[fig:ET\]b show electron energy-space spectrograms (E-S diagrams), computed from the PIC simulation at $t=35$. We count the electron particle [*flux*]{} as a function of the logarithmic energy above the ECL at $z=0.5$ (Fig. \[fig:ET\]a) and along the ECL at $z=0.0$ (Fig. \[fig:ET\]b). The spatial resolution is $\Delta x = 0.89, \Delta z =0.5$. The vertical axis is equivalent to the energy spectrum of $\mathcal{E}^{1.5}f(\mathcal{E})d\mathcal{E}$. In Figure \[fig:ET\]b, one can see a two-step profile of the electron count rates. The reconnection site ($18<x<59$) is filled with tenuous plasmas from the inflow region. The ECL ($28<x<48$) is embedded inside the region, as indicated by the dashed arrows. Around the X-line, we recognize many energetic electrons of $\mathcal{E}>1$ (the solid arrow). Since they are absent above the ECL (Fig. \[fig:ET\]a), they are quite probably accelerated near the X-line via the Speiser process. [@speiser65; @zeni01; @prit05] The energy spectrum has a power-law tail $f(\mathcal{E})d\mathcal{E}\propto\mathcal{E}^{-5.8}$ around the X-line. As we depart from the X-line in the ECL, the spectral index gradually decreases. The electron fluxes slightly shift to the higher energies and then shift to the lower energies. Another remarkable signature is the absence of the low-energy electron flux in the ECL. One can see that the electron flux of $10^{-1.5}$–$10^{-1}$ suddenly disappears around the ECL in Figure \[fig:ET\]b. In contrast, the low-energy electron flux remains fairly unchanged above the ECL (Fig. \[fig:ET\]a).
The bottom two panels in Figure \[fig:ET\] show Geotail observation of magnetic reconnection from 0659:18 UT to 0708:16 UT on 5 May 2007. The satellite was located in the magnetotail at (-21.3, 6.9, 1.3 $R_E$) in the geocentric solar magnetospheric (GSM) coordinates at 0700 UT. This event was studied by @nagai13b in detail. Here we briefly introduce key signatures. Figure \[fig:ET\]d presents the ion and electron bulk velocities, obtained from plasma moments. The $x$ components of plasma perpendicular velocities are presented. Both the ion velocity ($V_{{\perp}x}$ in gray color) and the electron velocity ($V_{{\perp}x}$ in black) are initially negative, reverse their signs at 0702:41 UT, and then remain positive until $\sim$0707 UT. This and other signatures[@nagai13b] suggest that the Geotail encountered bidirectional reconnection outflows in the close vicinity of the X-line. In the period 0701:17–0705:29 UT, the electron flow is decoupled from the ion flow. The shadow in Figure \[fig:ET\]d indicates this ion-electron decoupling interval. Figure \[fig:ET\]c shows electron counts per sample time in the energy-time (E-T) diagram. If the structure of the reconnection region is stationary, the E-S diagrams are equivalent to the E-T diagram. One can see that the electron fluxes shift to higher energies during the reconnection event from $\sim$0700 UT to $\sim$0707 UT. The dashed arrows indicated the ion-electron decoupling interval (0701:17 UT to 0705:29 UT). One can see that the electron fluxes shift to even higher energies and that the low-energy electron fluxes disappear.
We argue that the ion-electron decoupling interval corresponds to the ECL surrounding the X-line. The plasma velocities (Fig. \[fig:ET\]d) are consistent with the super-Alfvénic electron jets (Fig. \[fig:snapshot\]a). The profile of the E-T diagram (Fig. \[fig:ET\]c) resembles the two-step profile of the E-S diagram (Fig. \[fig:ET\]b). In particular, one can clearly see the two-step profile in the second half of the event; The inner ion-electron decoupling region (before 0705:29 UT) and the outer region (for example, 0705:29 UT – 0706:29 UT). The step-like transition is more evident in the observation, because we employ an artificially high electron temperature in the PIC simulation. Immediately after the flow reversal (0702:41 UT), the satellite observed the highest-energy electron flux, as indicated by the solid arrow in Figure \[fig:ET\]c. This is consistent with the DR in PIC simulation (Fig. \[fig:ET\]b). In the next few intervals, the highest-energy fluxes temporarily decreased, while the low-energy electron fluxes increased. Probably the electron flux in the pedestal region above the ECL (Fig. \[fig:ET\]a) are contaminated, because the magnetic field was positive ($B_x>0$; not shown) during the interval. In both PIC simulation and the satellite observation, we recognize the two-step profile of electron fluxes around the reconnection site. Both in the ECL and in the ion-electron decoupling interval, high-energy fluxes are found in the absence of the low-energy fluxes. The observation is consistent with our picture of the high-energy Speiser-accelerated electrons in the ECL and the low-energy noncrossing electrons outside the ECL.
Discussion and summary {#sec:discussion}
======================
We have investigated the basic properties of the ECL from various angles: Fluid quantities, VDFs, trajectories, and compositions. This allows us to understand the physics in and around the ECL much more deeply than before. For example, in the electron jet, we have found that the electrons are in the $\kappa \lesssim 1$ regime. They are unmagnetized, and follow Speiser orbits. Our orbit analysis and composition analysis suggest that the electron VDF consists of the following two Speiser populations. One is a global-type Speiser motion traveling through the DR, and the other is a local-type Speiser motion picked-up by the outflow exhaust. Since the Speiser electrons gyrate about $B_z$ for a half gyroperiod near the midplane, the average electron velocity can be faster in $x$ than the [**E**]{}$\times$[**B**]{} velocity, $V_{ex} \approx V_{e{\perp}x} > w_x$, when the reconnecting magnetic fields have out-of-plane ${\pm}y$ components. This results in the violation of the electron ideal condition, ${\boldsymbol{E}} + {\boldsymbol{V}}_e \times {\boldsymbol{B}} \ne 0$ in the electron jet.[@prit01a; @kari07; @shay07] In addition, since electrons travel through chaotic or Speiser orbits, the VDF is no longer gyrotropic, and therefore the jet is marked by the nongyrotropy measures.[@scudder08; @aunai13c; @swisdak16] The $z$-bounce motion during the Speiser motion is responsible for the phase-hole in $v_z$–$z$ (Fig. \[fig:phase432\]e), similar to the DR (Fig. \[fig:phase432\]b).[@hori08; @chen11] Since the $z$-bounce speed is smaller than the rotation speed in the $x$–$y$ plane, one can see an electron pressure anisotropy with a stronger perpendicular pressure.[@le10a]
@hesse08 argued that the super-Alfvénic electron jet speed can be explained by diamagnetic effects. They showed that the force balance is similar to the diamagnetic drift and that the $z$-profile of the off-diagonal component of the pressure tensor can be fitted by a gyrotropic pressure model. During the Speiser motion, electrons exhibit the meandering motion. It is plausible to categorize the meandering motion as the diamagnetic drift, because the diamagnetic drift is not a guiding center drift. However, it may not be the best way to discuss a gyrotropic pressure model inside the ECL,[@hesse08] because most electrons are in the unmagnetized regime of $\kappa<1$. It is more appropriate to say that “nongyrotropic electrons carry the diamagnetic-type electric current” in the ECL.
For ions, similar semicircle or half-ring-type VDFs by Speiser motions were reported by hybrid simulations,[@nakamura98; @lot98] PIC simulations,[@zeni13; @hietala15; @keizo16] and satellite observations.[@hietala15] In particular, Ref. discussed impacts of Speiser VDFs in PIC simulations in detail. They attributed a slow ion flow at a sub-Alfvénic speed and the violation of ion ideal condition to Speiser orbits. In this work, we attribute both a fast electron flow at a super-Alfvénic speed and the violation of electron ideal condition to Speiser motion. Ion physics and electron physics are similar, but some apparent effects are opposite: The ion flow looks slow while the electron flow looks fast. Ref. further argued that some ions travel through figure-eight-shaped regular orbits.[@chen86] We support this discovery by presenting electron regular orbits. The V-shaped path (Fig. \[fig:traj\_c\]f) corresponds to a narrow ion channel in the phase-space diagrams (Figs. 6b and 11b in Ref. ). Therefore, both Speiser orbits and regular orbits appear in the nongyrotropic region in magnetic reconnection, regardless of plasma species.
We have further introduced a new family of electron orbits, the noncrossing orbits. They are attributed to the polarization electric field $E_z$. Particle motions are organized in a conceptual model in Figure \[fig:orbit247\]b. Similar to conventional orbits, there exist the noncrossing Speiser orbits and the noncrossing regular orbits. The noncrossing Speiser orbits can be further classified into noncrossing global Speiser orbits and noncrossing local Speiser orbits (Sec. \[sec:noncrossing\]). As seen in VDFs in Figure \[fig:VDF2\], the noncrossing electrons are confined in a low-energy part of the VDFs. They are the majority in number density (Fig. \[fig:dist\]a). One can order-estimate a typical energy of the noncrossing electrons ($\mathcal{E}_{\rm NC}$) in the following way: Considering that the plasma density is nearly uniform over the reconnection site ($\sim n_b$; Fig. \[fig:snapshot\]f), we obtain $n_b(T_i+T_e) \approx \frac{1}{2\mu_0}B_{0}^2$ from the pressure balance across the ECL. We find that ions sustain most of the perpendicular pressure. Since the ion bounce motion sustains the polarization field, the electrostatic potential energy should be a small fraction ($\delta$) of the ion energy, $\delta \lesssim \mathcal{O}(0.1)$. The noncrossing electron energy satisfies $n_b \mathcal{E}_{\rm NC} \lesssim \delta n_b T_i \approx \delta\frac{1}{2\mu_0}B_{0}^2$. This yields $\mathcal{E}_{\rm NC} \lesssim 2.5 \delta~m_ic_{Ai}^2$, in agreement with $\mathcal{E}_{\rm NC} \lesssim 0.25$ in Figures \[fig:phase432\]c, \[fig:phase432\]f, and \[fig:VDF2\]fa.
From the macroscopic viewpoint, the noncrossing electrons challenge the conventional understanding of (1) electron mixing and (2) electron heating during magnetic reconnection. First, as discussed in Section \[sec:comp\], the high-energy electrons from the two inflow regions mix with each other in the ECL. On the other hand, the noncrossing electrons do not enter the ECL due to the Hall field $E_z$. They start to mix with each other only downstream of the remagnetization front (Fig. \[fig:dist\]b), where the Hall field $E_z$ disappears (Fig. \[fig:snapshot\]c). In Figure \[fig:snapshot\]h, we recognize the enhanced energy dissipation around the DR, where the high-energy global Speiser electrons start to mix with each other, and near the remagnetization front (indicated by a circle), where the noncrossing electrons start to mix with each other. It is interesting to see that the nonideal energy transfer corresponds to these sites of electron mixing. The relevance between the dissipation measure $\mathcal{D}_e$ and electron mixing deserves further research. Second, the electron heating mechanisms have been actively studied in the last few years.[@phan13b; @shay14; @haggerty15] These works reported electrons parallel heating outside the ECL inside the outflow exhaust. They implicitly assume the local-type Speiser motion, while the relevant self-consistent orbits have not been investigated. In fact, @haggerty15 showed in Figure 2(d) of their paper that the electron jet region, flanked by the bipolar $E_{\parallel}$ layers, extends $40 d_i$’s away from the X-line. This is favorable for the noncrossing electrons. We expect that the noncrossing electrons are an integral part of the electron VDFs of the exhaust region. The electron heating mechanism should be investigated in more detail, by considering the noncrossing electrons.
A question is whether these results are applicable to magnetic reconnection in the actual magnetotail, because we have employed artificial parameters in our PIC simulation. The parameters include the mass ratio ($m_i/m_e$), the density ratio ($n_b/n_0$), the ratio of the plasma and gyro frequencies ($\omega_{pe}/\Omega_{ce}$), and plasma beta ($\beta$) in the inflow region. The mass ratio controls the relative size of the ECL within the reconnection site. However, as long as the ECL is well resolved, there is no reason to alter the electron physics. We expect that our results scale to the real mass ratio. Since tenuous inflow plasma occupies the reconnection site, the density ratio ($n_b/n_0$) should control only the build-up time of the ECL structure. The frequency ratio $(\omega_{pe}/\Omega_{ce})$ in the inflow region will be important, because it controls electrostatic properties around the ECL. [@li08; @zeni11d; @chen12; @jara14] We expect that the Hall field $|E_z| \sim c_{Ae,in}B_0 = (\omega_{pe,in}/\Omega_{ce})^{-1} cB_0$ will be important for $(\omega_{pe}/\Omega_{ce})\sim \mathcal{O}(1)$. As we estimated in Ref. , the inflow frequency ratio in our PIC simulation is $$\begin{aligned}
\frac{\omega_{pe,in}}{\Omega_{ce}} = \frac{\omega_{pe}}{\Omega_{ce}} \Big(\frac{n_b}{n_0}\Big)^{1/2} \approx 1.8.\end{aligned}$$ In the tail lobe, we expect $\omega_{pe,in}/\Omega_{ce} = 1.6$–$2.3$ for $B=20$ nT and $n_{b}=0.01$-$0.02~{\rm cm}^{-3}$. Thus, our results will be applicable to the magnetotail reconnection. The plasma beta $\beta$, in particular, the electron beta $\beta_e\equiv 2\mu_0 p_e/B^2$, may be another important factor. In the cold limit of $\beta_e\rightarrow 0$, as in the magnetotail, we expect that fine structures will be more evident in electron VDFs.[@bessho14; @shuster15]
The noncrossing electrons can be more pronounced in magnetic reconnection in different configurations. We address two favorable cases of driven-type reconnection and asymmetric reconnection. In the so-called driven systems, inflow plasmas are continuously injected toward the reconnection site. In such a case, the ions inter-penetrate deeper than in the undriven case and therefore the polarization electric field becomes stronger. Then we expect more noncrossing electrons on both sides of the midplane. In fact, @hori08 presented an electron phase-space diagram across the X-line in a PIC simulation of driven reconnection (Fig. 5 in Ref. ). The diagram shows clear signatures of noncrossing electrons, two high-density electron regions in red inside the ion meandering region. In asymmetric systems, there is often a density gradient across the two inflow regions. Previous PIC simulations[@prit08; @tanaka08] reported a strong normal electric field on the low-density side of the boundary layer. This is a polarization field and this electric field layer overlaps the DR. In such a case, the low-density side of the DR is very favorable for noncrossing electrons. At the Earth’s magnetopause, magnetic reconnection takes place between a high-density magnetosheath plasma and a tenuous magnetospheric plasma, continuously driven by the solar wind. We expect that noncrossing electrons will be a key player for understanding the physics of magnetopause reconnection. Since the sunward polarization electric field will be enhanced, magnetospheric electrons will rarely mix with magnetospheric plasma in the exhaust region.
In addition, the reconnection system may involve an out-of-plane magnetic field (the so-called guide-field). In fact, reconnection events with a guide-field have been observed even in the Earth’s magnetotail,[@eastwood10] where we expect antiparallel magnetic fields. Since a small guide-field alters the ECL structure,[@prit04; @swisdak05; @le13] the noncrossing electron orbits will be modified accordingly. A guide-field tends to magnetize electrons even in the DR, while the polarization electric field persists in the guide-field case. Electron orbits in guide-field reconnection deserve independent research. We have also ignored the variation in the $y$ direction. At this point, it is not clear whether our picture persists in 3D configurations or not. Nevertheless, it is encouraging that the satellite observation is consistent with the simulation with noncrossing electrons (Sec. \[sec:ET\]).
In this work, we have investigated particle dynamics in the electron current layer (ECL) in collisionless magnetic reconnection in an antiparallel magnetic field. By tracking self-consistent trajectories in the two-dimensional PIC simulation, we have found new classes of electron orbits. The electron motion in and around the ECL is much more complicated than we have expected before. The new orbits, as well as previously known orbits, are schematically illustrated in Figure \[fig:diagram\]. In the inflow region, electrons are gyrating and fast-bouncing in the parallel direction, as extensively studied by @egedal05 [@egedal08]. Near the separatrices, some electrons stream along the field lines toward the X-line. Once electrons enter the DR, they undergo the Speiser motions of global type.[@speiser65] The electrons slowly turn around to the outflow directions while bouncing in $z$. Others travel through the Speiser motions of local-reflection type. Inside the ECL, there exists an figure-eight-shaped (crossing) regular orbit.[@chen86; @zeni13] The polarization electric field introduces noncrossing regular orbits on the jet flank and noncrossing Speiser orbits. Similar to the traditional Speiser orbits, the noncrossing Speiser orbits can be categorized as the global type and the local reflection type, although their difference is ambiguous. Downstream of the remagnetization front, some Speiser electrons remain around the center as nongyrotropic electrons, while others travel near the separatrices in field-aligned electron outflows.
Considering particle orbits, we have discussed key properties of the electron jet. The electrons are traveling through Speiser orbits. The fast bulk speed, electron nonidealness, anisotropy, and nongyrotropy are consequences of the electron nongyrotropic motion in the $\kappa \lesssim 1$ regime. The noncrossing orbits are consistent with the electron density profile, the energy-dependent spatial distribution of electrons, and the electron mixing sites with nonideal energy transfer. They correspond to the following observational signatures of the ECL: (1) The super-Alfvénic electron jet will be populated by high-energy nongyrotropic electrons. (2) The electron density is lower than in the jet flank region. (3) The electron energy-time diagram will exhibit the two-step profile. We hope these predictions will be confirmed by the MMS spacecraft[@burch16] in the second science phase targeting the magnetotail.
The authors acknowledge I. Shinohara for comments. This work was supported by Grant-in-Aid for Young Scientists (B) (Grant No. 25871054). The authors acknowledge facilities at Center for Computational Astrophysics, National Astronomical Observatory of Japan.
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abstract: 'We evaluate the octupole in the large-$N_c$ limit in the McLerran-Venugopalan model, and derive a general expression of the 2n-point correlator, which can be applied in analytical studies of the multi-particle production in the scatterings between hard probes and dense targets.'
author:
- Yu Shi
- Cheng Zhang
- Enke Wang
title: Multipole scattering amplitudes in the Color Glass Condensate formalism
---
Introduction
============
Multi-point functions from the dipole amplitude to 2n-point correlators described by Wilson lines play an important role in calculating the cross sections of the multi-particle production processes, which could take place in deep inelastic scattering (DIS) experiments and proton-nucleus (pA) collisions.
In high-energy DIS and pA collisions, high-energy partons multiply scatter with a small-x gluon field inside a large nucleus in an eikonal way [@Kovner:2001vi]. In the Color Glass Condensate (CGC) [@Iancu:2002xk; @arXiv:1002.0333] formalism, the color source in the heavy target is regarded as a classical color field, and the color charges are assumed to obey the Guassian distribution. In the above framework, multipoles written in terms of Guassian average of Wilson lines appear in the multiple scattering cross sections, and each Wilson line in the scattering amplitudes represents a parton traversing the gluon field with a fixed transverse coordinate in the eikonal approximation, and resums the multiple interactions between the parton and the dense target.
Dipoles and quadrupoles correspond to two types of gluon distributions [@Dominguez:2010xd; @Catani:1990eg; @Collins:1991ty] (the Weiszäcker-Williams gluon distribution [@Kovchegov:1998bi; @McLerran:1998nk] and the dipole gluon distribution). The evolution of the dipole gluon distribution obeys the Balitsky-Kovchegov equation [@Balitsky:1995ub+X; @Kovchegov:1999yj], while the Weiszäcker-Williams gluon distribution is governed by the evolution of the quadrupole amplitude, which have been successfully evaluated in Ref. [@Dominguez:2011gc] by using the JIMWLK [@Jalilian-Marian:1997jx+X; @Ferreiro:2001qy] Hamiltonian method. As shown in Ref. [@Dominguez:2010xd], the generalized factorization in high-density QCD can be established in the CGC formalism with only dipole and quadrupole amplitudes. Higher multiple scattering amplitudes, such as 6-point (sextupole) and 8-point (octupole) functions are suppressed in the large-$N_c$ limit in physical processes [@Dominguez:2012ad]. Nevertheless, it is interesting to evaluate the leading $N_c$ contribution of higher-point correlators, which may help us to better understand the underlying dynamics and estimate the size of finite-$N_c$ corrections to multi-parton productions. The complexity naturally arises as we compute the correlators consisting of more than four Wilson lines. Fortunately, the large-$N_c$ limit simplifies the calculation and makes it possible to give the relatively simple analytical results for higher-point correlators.
In this paper, we evaluate the correlator of eight Wilson lines and then conjecture a general 2n-point expression, which can be used as the initial condition for small-$x$ evolutions [@Iancu:2011ns; @Dumitru:2011vk] of any even number point correlators. We also use $n=3$ case as an example to show how the general formula gives the correct expression for the six-point function.
This paper is organised as follows. In section II, we outline the main aspects of the McLerran-Venugopalan (MV) [@McLerran:1993ni] model and briefly review the results of the dipole, quadrupole and sextupole amplitudes. We next proceed to compute the octupole amplitude and conjecture a general expression for 2n-point correlators in section III. Finally section IV is devoted to the conclusion.
Correlators in the McLerran-Venugopalan model
=============================================
We aim to compute the 2n-point correlator, namely
$$\begin{aligned}
\frac{1}{N_c}\left\langle\text{Tr}\left[U({\boldsymbol{x}}_{1\perp})U({\boldsymbol{x}}_{2\perp})^{\dagger}...U({\boldsymbol{x}}_{2n-1\perp})U({\boldsymbol{x}}_{2n\perp})^{\dagger}\right]\right\rangle,\end{aligned}$$
where in the MV model, $U({\boldsymbol{x}}_\perp)$ is a Wilson line in the fundamental representation defined as
$$U({\boldsymbol{x}}_\perp)=\mathcal{P}\exp\Biggl[-{\mathrm{i}}g^2\int_{-\infty}^{+\infty}{\mathrm{d}}x^+{\mathrm{d}}^2 {\boldsymbol{z}}_\perp G_0({\boldsymbol{x}}_\perp-{\boldsymbol{z}}_\perp)\rho_a(x^+,{\boldsymbol{z}}_\perp)t^a\Biggr],$$
where $t^a$ is a color matrix in the fundamental $\textrm{SU}(N_\textrm{c})$ representation, $\rho_a$ is the color source inside a target, and $\mathcal{P}$ represents $x^+$ ordering operator. $G_0$ is the 2-dimensional propagator which satisfies
$$\frac{\partial^2}{\partial{\boldsymbol{x}}_\perp^2} G_0({\boldsymbol{x}}_\perp-{\boldsymbol{z}}_\perp)=\delta({\boldsymbol{x}}_\perp-{\boldsymbol{z}}_\perp).$$
One can solve the above equation and get the explicit form as
$$G_0({\boldsymbol{x}}_\perp-{\boldsymbol{z}}_\perp)=\int\frac{{\mathrm{d}}^2{\boldsymbol{k}}_\perp}{(2\pi)^2}\frac{{\mathrm{e}}^{{\mathrm{i}}{\boldsymbol{k}}_\perp\cdot({\boldsymbol{x}}_\perp-{\boldsymbol{z}}_\perp)}}{{\boldsymbol{k}}^2_\perp}.$$
As mentioned before, our calculation is based on the MV model, which gives the color field average of a given physical quantity as the Guassian weighted functional integral
$$\langle f[\rho]\rangle=\int\mathcal{D}\rho\exp\left\{-\int {\mathrm{d}}^2{\boldsymbol{x}}\,{\mathrm{d}}^2{\boldsymbol{y}}\,{\mathrm{d}}z^+\,\frac{\rho_c(z^+,{\boldsymbol{x}})\rho_c(z^+,{\boldsymbol{y}})}{2\mu^2(z^+)}\right\}f[\rho],$$
where $\mu^2(z^+)$ is the variance of the Guassian distribution of color field representing the color charge density at coordinate $z^+$, and its integration over $z^+$ is proportional to the saturation momentum square $Q_{\rm s}^2$.
With the help of the Wick’s theorem, any correlators can be obtained by the most elementary correlator which reads
$$\langle\rho_a(x^+,{\boldsymbol{x}})\rho_b(y^+,{\boldsymbol{y}})\rangle=\delta_{ab}\delta(x^+-y^+)\delta({\boldsymbol{x}}-{\boldsymbol{y}})\mu^2(x^+).$$
Under the framework above, several studies [@HiroFujii; @Gelis:2001da; @Blaizot:2004wv; @Dominguez:2012ad; @Fukushima:2007dy; @Dominguez:2008aa; @Dominguez:2011wm; @Kovchegov:2008mk; @Marquet:2010cf] have obtained the finite-$N_c$ expressions for the dipole and the quadrupole in the MV model, as well as the large-$N_c$ expressions for the quadrupole and the sextupole. In the following discussion, in order to be self-contained, we summarize the known results in the large-$N_c$ limit first. The dipole amplitude reads [@Gelis:2001da]
$$\begin{aligned}
\frac{1}{N_c}\left\langle\text{Tr}\left[U_{1}U_{2}^{\dagger}\right]\right\rangle\
=e^{-\Gamma_{12}},\end{aligned}$$
where
$$\begin{aligned}
&&\Gamma_{ij}=\frac{C_F}{2}\mu^2(L_{ii}+L_{jj}-2L_{ij}),
\\
&&\mu^2=\int{\mathrm{d}}x^{+}\mu^{2}\left(x^{+}\right),\label{2-point}\end{aligned}$$
with $L_{ij}$ given by the two-dimensional massless propagator $G_0$ as
$$\begin{aligned}
L_{ij}=g^4\int {\mathrm{d}}^2{\boldsymbol{z}}_\perp\;G_0({\boldsymbol{x}}_{i\perp}-{\boldsymbol{z}}_\perp)G_0({\boldsymbol{y}}_{i\perp}-{\boldsymbol{z}}_\perp).\end{aligned}$$
The quadrupole can be written as [@Blaizot:2004wv; @Dominguez:2011wm]
$$\begin{aligned}
&&\frac{1}{N_c}\left\langle\text{Tr}\left[ U_{1}U_{2}^{\dagger }U_{3}U_{4}^{\dagger}\right]\right\rangle\nonumber
\\
&&=e^{-\Gamma_{12}-\Gamma_{34}}-\frac{F_{1234}}{F_{1324}}\left[e^{-\Gamma_{12}-\Gamma_{34}}-e^{-\Gamma_{14}-\Gamma_{32}}\right].\label{4-point}\end{aligned}$$
The sextupole [@Dominguez:2012ad] is given by the following expression in the MV model
$$\begin{aligned}
&&\frac{1}{N_c}\left\langle\text{Tr}\left[ U_{1}U_{2}^{\dagger }U_{3}U_{4}^{\dagger}U_{5}U_{6}^{\dagger }\right]\right\rangle\nonumber
\\
&&=e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}}-\frac{F_{1234}}{F_{1324}}\left[e^{-\Gamma_{12}-\Gamma_{34}}-e^{-\Gamma_{14}-\Gamma_{32}}\right]e^{-\Gamma_{56}}\nonumber
\\
&&-\frac{F_{1256}}{F_{1526}}\left[e^{-\Gamma_{12}-\Gamma_{56}}-e^{-\Gamma_{16}-\Gamma_{52}}\right]e^{-\Gamma_{34}}-\frac{F_{3456}}{F_{3546}}\left[e^{-\Gamma_{34}-\Gamma_{56}}-e^{-\Gamma_{36}-\Gamma_{54}}\right]e^{-\Gamma_{12}}\nonumber
\\
&&+F_{1234}F_{1456}\left[\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}}}{F_{1324}G}-\frac{e^{-\Gamma_{14}-\Gamma_{32}-\Gamma_{56}}}{F_{1324}F_{1546}}+\frac{e^{-\Gamma_{16}-\Gamma_{32}-\Gamma_{54}}}{F_{1546}G}\right]\nonumber \\
&&+F_{1256}F_{2543}\left[\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}}}{F_{1526}G}-\frac{e^{-\Gamma_{16}-\Gamma_{34}-\Gamma_{52}}}{F_{1526}F_{2453}}+\frac{e^{-\Gamma_{16}-\Gamma_{32}-\Gamma_{54}}}{F_{2453}G}\right]\nonumber \\
&&+F_{3456}F_{1236}\left[\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}}}{F_{3546}G}-\frac{e^{-\Gamma_{12}-\Gamma_{36}-\Gamma_{54}}}{F_{3546}F_{1326}}+\frac{e^{-\Gamma_{16}-\Gamma_{32}-\Gamma_{54}}}{F_{1326}G}\right],
\label{6-point}\end{aligned}$$
with
$$\begin{aligned}
&&F_{ijkl}=L_{ik}-L_{jk}+L_{jl}-L_{il},
\\
&&G=L_{12}+L_{34}+L_{56}-L_{16}-L_{32}-L_{54}.\end{aligned}$$
From the above three expressions for the dipole, quadupole, and sextupole amplitudes, we find that higher-point functions have terms with a larger number of factors $F_{jikl}$’s, which equals the number of transitions between color singlet states. For example, the sextupole amplitude contains three types of terms, corresponding to singlet states with no color transitions, one and two times of color transitions, respectively.
The evaluation of 2n-point correlators in the large-$N_c$ limit
===============================================================
Based on the formalism developed in Refs. [@HiroFujii; @Blaizot:2004wv; @Dominguez:2012ad], we calculate the 8-point correlator and conjecture a general expression for 2n-point functions in the MV model. Our calculation is based on the derivation introduced in [@Dominguez:2012ad], in which the result of the 6-point correlator was derived in the large-$N_c$ limit.
The 8-point correlator takes the form
$$\begin{aligned}
&&\frac{1}{N_c}\left\langle\text{Tr}\left[U_{1}U_{2}^{\dagger}U_{3}U_{4}^{\dagger}U_{5}U_{6}^{\dagger}U_{7}U_{8}^{\dagger}\right]\right\rangle\nonumber
\\
&=&\frac{T}{N_c}\sum_{n=0}^\infty\int_{z_1^+<\cdots<z_n^+}\left[N_ca_n(z_1^+,\dots,z_n^+)+N_c^2b_n(z_1^+,\dots,z_n^+)+N_c^3c_n(z_1^+,\dots,z_n^+)+N_c^4d_n(z_1^+,\dots,z_n^+)\right],\label{8}\end{aligned}$$
where $T$ is the so-called tadpole contribution corresponding to the diagrams where each gluon link attaches to a single Wilson line. The tadpole contribution can be evaluated straightforwardly and gives
$$T=e^{-\frac{C_F}{2}\mu^2\sum\limits_{i=1}^{8}L_{ii}}.$$
The rest contributions arising from the states in which every gluon link attaches to two different Wilson lines are presented in the square bracket. Similar to the sextupole calculation, one considers all the possible states contribute in the system of 8 Wilson lines, and finds that there are 24 singlet states in total. In the large-$N_c$ limit [@Dominguez:2012ad], only transitions to states with a higher power of $N_c$, namely, with one more fermion loop, are allowed, which picks out 14 states whose topologies are shown in Fig. \[topo\], with $N_c$, $N_c^2$, $N_c^3$ and $N_c^4$ terms representing the first, the next 6, the later 6 and the last diagrams, respectively. Therefore, $b_n$ and $c_n$ in Eq. (\[8\]) equal the sums of elements in the following column matrices ${\boldsymbol{b}}_{n(6\times 1)}$ and ${\boldsymbol{c}}_{n(6\times 1)}$ respectively, and the corresponding evolution coefficients take the form
$$\left(\begin{matrix}a_n \\ {\boldsymbol{b}}_{n(6\times 1)} \\ {\boldsymbol{c}}_{n(6\times 1)} \\ d_n\end{matrix}\right)=\left[\prod_{i=1}^n\mu^2(z^+_i)\right]M^n\left(\begin{matrix}1\\ \boldsymbol{0}_{(6\times 1)}\\ \boldsymbol{0}_{(6\times 1)}\\ 0\end{matrix}\right),$$
in which the transition matrix $M$ can be divided into blocks as
$$M=\left(\begin{matrix}M_1 & 0 & 0&0\\ M_2 &M_3 & 0&0\\ 0 & M_4 & M_5&0\\0&0& M_6 & M_7\end{matrix}\right).\label{block}$$
The evolution part of our desired correlator before integrating over the longitudinal coordinate is
$$\left(\begin{matrix}
1 & \boldsymbol{N_c}_{(1\times 6)} & \boldsymbol{N_c^2}_{(1\times 6)} & N_c^3
\end{matrix}\right)
M^n
\left(\begin{matrix}
1 \\ \boldsymbol{0}_{(13\times 1)}
\end{matrix}\right),\label{mainpart}$$ which picks out the first column of the $n$th power of this matrix
$$\left(\begin{matrix}M_1^n \\ \sum_{i=0}^{n-1}M_3^iM_2M_1^{n-i-1} \\ \sum_{i=0}^{n-2}\sum_{j=0}^{n-i-2}M_5^iM_4M_3^jM_2M_1^{n-i-j-2} \\ \sum_{i=0}^{n-3}\sum_{j=0}^{n-i-3}\sum_{k=0}^{n-i-j-3}M_7^iM_6M_5^jM_4M_3^kM_2M_1^{n-i-j-k-3}\end{matrix}\right).$$
As a 14x14 matrix, $M$ has the form $$M=\begin{pmatrix}
m_{11} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
m_{21} & m_{22} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
m_{31} & 0 & m_{33} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
m_{41} & 0 & 0 & m_{44} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
m_{51} & 0 & 0 & 0 & m_{55} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
m_{61} & 0 & 0 & 0 & 0 & m_{66} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
m_{71} & 0 & 0 & 0 & 0 & 0 & m_{77} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & m_{82} & m_{83} & 0 & m_{85} & 0 & 0 & m_{88} & 0 & 0 & 0 & 0 & 0 & 0\\
0 & m_{92} & 0 & m_{94} & 0 & m_{96} & 0 & 0 & m_{99} & 0 & 0 & 0 & 0 & 0\\
0 & m_{102} & 0 & 0 & 0 & 0 & m_{107} & 0 & 0 & m_{1010} & 0 & 0 & 0 & 0\\
0 & 0 & m_{113} & m_{114} & 0 & 0 & m_{117} & 0 & 0 & 0 & m_{1111} & 0 & 0 & 0\\
0 & 0 & 0 & m_{124} & m_{125} & 0 & 0 & 0 & 0 & 0 & 0 & m_{1212} & 0 & 0\\
0 & 0 & 0 & 0 & m_{135} & m_{136} & m_{137} & 0 & 0 & 0 & 0 & 0 & m_{1313} & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & m_{148} & m_{149} & m_{1410} & m_{1411} & m_{1412} & m_{1413} & m_{1414}\\
\end{pmatrix},\label{M-matrix}$$ which can be written in forms of block matrices in Eq. (\[block\]) as
$$\begin{aligned}
&&m_{11}=C_FL_{12,34,56,78},
\ m_{1414}=C_FL_{18,23,45,67},
\ M_6=\frac{1}{2}\begin{pmatrix}
F_{1687} & F_{4756} & F_{1485} & F_{2534} & F_{2736} & F_{1283}
\end{pmatrix},
\\
&&M_2=\frac{1}{2}\begin{pmatrix}
F_{1243} \\ F_{1265} \\ F_{1287} \\ F_{3465} \\ F_{3487} \\ F_{5687}
\end{pmatrix},
M_4=\frac{1}{2}\begin{pmatrix}
F_{1465} & F_{2534} & 0 & F_{1263} & 0 & 0
\\
F_{1487} & 0 & F_{2734} & 0 & F_{1283} & 0
\\
F_{5687} & 0 & 0 & 0 & 0 & F_{1243}
\\
0 & F_{1687} & F_{2756} & 0 & 0 & F_{1285}
\\
0 & 0 & F_{3465} & F_{1287} & 0 & 0
\\
0 & 0 & 0 & F_{3687} & F_{4756} & F_{3485}
\end{pmatrix},
\\
&&M_3=C_F\begin{pmatrix}
L_{14,23,56,78} & 0 & 0 & 0 & 0 & 0
\\
0 & L_{16,25,34,78} & 0 & 0 & 0 & 0
\\
0 & 0 & L_{18,27,34,56} & 0 & 0 & 0
\\
0 & 0 & 0 & L_{12,36,45,78} & 0 & 0
\\
0 & 0 & 0 & 0 & L_{12,38,47,56} & 0
\\
0 & 0 & 0 & 0 & 0 & L_{12,34,58,67}
\end{pmatrix},
\\
&&M_5=C_F\begin{pmatrix}
L_{16,23,45,78} & 0 & 0 & 0 & 0 & 0
\\
0 & L_{18,23,47,56} & 0 & 0 & 0 & 0
\\
0 & 0 & L_{14,23,58,67} & 0 & 0 & 0
\\
0 & 0 & 0 & L_{18,25,34,67} & 0 & 0
\\
0 & 0 & 0 & 0 & L_{18,27,36,45} & 0
\\
0 & 0 & 0 & 0 & 0 & L_{12,38,45,67}
\end{pmatrix},\end{aligned}$$
where
$$\begin{aligned}
L_{ab,cd,ij,kl}=L_{ab}+L_{cd}+L_{ij}+L_{kl}.\end{aligned}$$
The first column of the $n$th power of $M$ is found to be
$$\begin{pmatrix}
m_{11}^n\\
m_{21}\sum_{i=0}^{n-1}m_{11}^im_{22}^{n-i-1}\\
m_{31}\sum_{i=0}^{n-1}m_{11}^im_{33}^{n-i-1}\\
m_{41}\sum_{i=0}^{n-1}m_{11}^im_{44}^{n-i-1}\\
m_{51}\sum_{i=0}^{n-1}m_{11}^im_{55}^{n-i-1}\\
m_{61}\sum_{i=0}^{n-1}m_{11}^im_{66}^{n-i-1}\\
m_{71}\sum_{i=0}^{n-1}m_{11}^im_{77}^{n-i-1}\\
\sum_{k=2,3,5}m_{8 k}m_{k1}\left[\sum_{i=0}^{n-2}\sum_{j=0}^{n-i-2}m_{11}^im_{kk}^jm_{8 8 }^{n-i-j-2}\right]\\
\sum_{k=2,4,6}m_{9 k}m_{k1}\left[\sum_{i=0}^{n-2}\sum_{j=0}^{n-i-2}m_{11}^im_{kk}^jm_{9 9 }^{n-i-j-2}\right]\\
\sum_{k=2,7 }m_{10k}m_{k1}\left[\sum_{i=0}^{n-2}\sum_{j=0}^{n-i-2}m_{11}^im_{kk}^jm_{1010}^{n-i-j-2}\right]\\
\sum_{k=3,4,7}m_{11k}m_{k1}\left[\sum_{i=0}^{n-2}\sum_{j=0}^{n-i-2}m_{11}^im_{kk}^jm_{1111}^{n-i-j-2}\right]\\
\sum_{k=4,5 }m_{12k}m_{k1}\left[\sum_{i=0}^{n-2}\sum_{j=0}^{n-i-2}m_{11}^im_{kk}^jm_{1212}^{n-i-j-2}\right]\\
\sum_{k=5,6,7}m_{13k}m_{k1}\left[\sum_{i=0}^{n-2}\sum_{j=0}^{n-i-2}m_{11}^im_{kk}^jm_{1313}^{n-i-j-2}\right]\\
\sum_{p=8}^{13}\sum_{q=2}^{7}m_{14p}m_{pq}m_{q1}\left[\sum_{i=0}^{n-3}\sum_{j=0}^{n-i-3}\sum_{k=0}^{n-i-j-3}m_{11}^im_{qq}^jm_{pp}^km_{1414}^{n-i-j-k-3}\right]\\
\end{pmatrix},$$
which is derived from the following expression
$$\begin{pmatrix}
m_{11}^n\\
\frac{m_{21}}{m_{11}-m_{22}}\left[m_{11}^n-m_{22}^n\right]\\
\frac{m_{31}}{m_{11}-m_{33}}\left[m_{11}^n-m_{33}^n\right]\\
\frac{m_{41}}{m_{11}-m_{44}}\left[m_{11}^n-m_{44}^n\right]\\
\frac{m_{51}}{m_{11}-m_{55}}\left[m_{11}^n-m_{55}^n\right]\\
\frac{m_{61}}{m_{11}-m_{66}}\left[m_{11}^n-m_{66}^n\right]\\
\frac{m_{71}}{m_{11}-m_{77}}\left[m_{11}^n-m_{77}^n\right]\\
\sum_{k=2,3,5}m_{8k}m_{k1}\left[\frac{m_{11}^n}{(m_{11}-m_{kk})(m_{11}-m_{88})}+\frac{m_{kk}^n}{(m_{kk}-m_{11})(m_{kk}-m_{88})}+\frac{m_{88}^n}{(m_{88}-m_{kk})(m_{88}-m_{11})}\right]\\
\sum_{k=2,4,6}m_{9k}m_{k1}\left[\frac{m_{11}^n}{(m_{11}-m_{kk})(m_{11}-m_{99})}+\frac{m_{kk}^n}{(m_{kk}-m_{11})(m_{kk}-m_{99})}+\frac{m_{99}^n}{(m_{99}-m_{kk})(m_{99}-m_{11})}\right]\\
\sum_{k=2,7}m_{10k}m_{k1}\left[\frac{m_{11}^n}{(m_{11}-m_{kk})(m_{11}-m_{1010})}+\frac{m_{kk}^n}{(m_{kk}-m_{11})(m_{kk}-m_{1010})}+\frac{m_{1010}^n}{(m_{1010}-m_{kk})(m_{1010}-m_{11})}\right]\\
\sum_{k=3,4,7}m_{11k}m_{k1}\left[\frac{m_{11}^n}{(m_{11}-m_{kk})(m_{11}-m_{1111})}+\frac{m_{kk}^n}{(m_{kk}-m_{11})(m_{kk}-m_{1111})}+\frac{m_{1111}^n}{(m_{1111}-m_{kk})(m_{1111}-m_{11})}\right]\\
\sum_{k=4,5}m_{12k}m_{k1}\left[\frac{m_{11}^n}{(m_{11}-m_{kk})(m_{11}-m_{1212})}+\frac{m_{kk}^n}{(m_{kk}-m_{11})(m_{kk}-m_{1212})}+\frac{m_{1212}^n}{(m_{1212}-m_{kk})(m_{1212}-m_{11})}\right]\\
\sum_{k=5,6,7}m_{13k}m_{k1}\left[\frac{m_{11}^n}{(m_{11}-m_{kk})(m_{11}-m_{1313})}+\frac{m_{kk}^n}{(m_{kk}-m_{11})(m_{kk}-m_{1313})}+\frac{m_{1313}^n}{(m_{1313}-m_{kk})(m_{1313}-m_{11})}\right]\\
\sum_{p=8}^{13}\sum_{q=2}^{7}m_{14p}m_{pq}m_{q1}[\frac{m_{11}^n}{(m_{11}-m_{qq})(m_{11}-m_{pp})(m_{11}-m_{1414})}+\frac{m_{qq}^n}{(m_{qq}-m_{11})(m_{qq}-m_{pp})(m_{qq}-m_{1414})}\\
+\frac{m_{pp}^n}{(m_{pp}-m_{11})(m_{pp}-m_{qq})(m_{pp}-m_{1414})}+\frac{m_{1414}^n}{(m_{1414}-m_{11})(m_{1414}-m_{qq})(m_{1414}-m_{pp})}]\\
\end{pmatrix}.\label{nth1stcol}$$
Plugging the first column into the above evolution part in Eq. (\[mainpart\]), integrating over the longitudinal coordinate, summing over $n$ with a factor of $\frac{1}{n!}$, and including the tadpole contribution, one obtains
$$\begin{aligned}
&&\frac{1}{N_c}\left\langle\text{Tr}\left[U_{1}U_{2}^{\dagger}U_{3}U_{4}^{\dagger}U_{5}U_{6}^{\dagger}U_{7}U_{8}^{\dagger}\right]\right\rangle\nonumber
\\
&=&e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}
-\frac{F_{1234}}{F_{1324}}\left[e^{-\Gamma_{12}-\Gamma_{34}}-e^{-\Gamma_{14}-\Gamma_{32}}\right]e^{-\Gamma_{56}-\Gamma_{78}}
-\frac{F_{1256}}{F_{1526}}\left[e^{-\Gamma_{12}-\Gamma_{56}}-e^{-\Gamma_{16}-\Gamma_{52}}\right]e^{-\Gamma_{34}-\Gamma_{78}}\nonumber
\\
&&-\frac{F_{1278}}{F_{1728}}\left[e^{-\Gamma_{12}-\Gamma_{78}}-e^{-\Gamma_{18}-\Gamma_{72}}\right]e^{-\Gamma_{34}-\Gamma_{56}}
-\frac{F_{3456}}{F_{3546}}\left[e^{-\Gamma_{34}-\Gamma_{56}}-e^{-\Gamma_{36}-\Gamma_{54}}\right]e^{-\Gamma_{12}-\Gamma_{78}}\nonumber
\\
&&-\frac{F_{3478}}{F_{3748}}\left[e^{-\Gamma_{34}-\Gamma_{78}}-e^{-\Gamma_{38}-\Gamma_{74}}\right]e^{-\Gamma_{12}-\Gamma_{56}}
-\frac{F_{5678}}{F_{5768}}\left[e^{-\Gamma_{56}-\Gamma_{78}}-e^{-\Gamma_{58}-\Gamma_{76}}\right]e^{-\Gamma_{12}-\Gamma_{34}}\nonumber
\\
&&+F_{1243}F_{1465}\left[\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}}}{F_{1324}G_1}-\frac{e^{-\Gamma_{14}-\Gamma_{32}-\Gamma_{56}}}{F_{1324}F_{1546}}
+\frac{e^{-\Gamma_{16}-\Gamma_{32}-\Gamma_{54}}}{F_{1546}G_1}\right]e^{-\Gamma_{78}}\nonumber
\\
&&+F_{1265}F_{2534}\left[\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}}}{F_{1526}G_1}-\frac{e^{-\Gamma_{16}-\Gamma_{34}-\Gamma_{52}}}{F_{1526}F_{2453}}
+\frac{e^{-\Gamma_{16}-\Gamma_{32}-\Gamma_{54}}}{F_{2453}G_1}\right]e^{-\Gamma_{78}}\nonumber
\\
&&+F_{3465}F_{1263}\left[\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}}}{F_{3546}G_1}-\frac{e^{-\Gamma_{12}-\Gamma_{36}-\Gamma_{54}}}{F_{3546}F_{1326}}
+\frac{e^{-\Gamma_{16}-\Gamma_{32}-\Gamma_{54}}}{F_{1326}G_1}\right]e^{-\Gamma_{78}}\nonumber
\\
&&+F_{1243}F_{1487}\left[\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{78}}}{F_{1324}G_2}-\frac{e^{-\Gamma_{14}-\Gamma_{32}-\Gamma_{78}}}{F_{1324}F_{1748}}
+\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{74}}}{F_{1748}G_2}\right]e^{-\Gamma_{56}}\nonumber
\\
&&+F_{1287}F_{2734}\left[\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{78}}}{F_{1728}G_2}-\frac{e^{-\Gamma_{18}-\Gamma_{72}-\Gamma_{34}}}{F_{1728}F_{2473}}
+\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{74}}}{F_{2473}G_2}\right]e^{-\Gamma_{56}}\nonumber
\\
&&+F_{3487}F_{1283}\left[\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{78}}}{F_{3748}G_2}-\frac{e^{-\Gamma_{12}-\Gamma_{38}-\Gamma_{74}}}{F_{3748}F_{1328}}
+\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{74}}}{F_{1328}G_2}\right]e^{-\Gamma_{56}}\nonumber
\\
&&+F_{1243}F_{5687}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{1324}G_3}-\frac{e^{-\Gamma_{14}-\Gamma_{32}-\Gamma_{56}-\Gamma_{78}}}{F_{1324}F_{5768}}
+\frac{e^{-\Gamma_{14}-\Gamma_{32}-\Gamma_{58}-\Gamma_{76}}}{F_{5768}G_3}\right]\nonumber
\\
&&+F_{5687}F_{1243}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{5768}G_3}-\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{58}-\Gamma_{76}}}{F_{5768}F_{1324}}
+\frac{e^{-\Gamma_{14}-\Gamma_{32}-\Gamma_{58}-\Gamma_{76}}}{F_{1324}G_3}\right]\nonumber
\\
&&+F_{1265}F_{1687}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{56}-\Gamma_{78}}}{F_{1526}G_4}-\frac{e^{-\Gamma_{16}-\Gamma_{52}-\Gamma_{78}}}{F_{1526}F_{1768}}
+\frac{e^{-\Gamma_{18}-\Gamma_{52}-\Gamma_{76}}}{F_{1768}G_4}\right]e^{-\Gamma_{34}}\nonumber
\\
&&+F_{1287}F_{2756}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{56}-\Gamma_{78}}}{F_{1728}G_4}-\frac{e^{-\Gamma_{18}-\Gamma_{72}-\Gamma_{56}}}{F_{1728}F_{2675}}
+\frac{e^{-\Gamma_{18}-\Gamma_{52}-\Gamma_{76}}}{F_{2675}G_4}\right]e^{-\Gamma_{34}}\nonumber
\\
&&+F_{5687}F_{1285}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{56}-\Gamma_{78}}}{F_{5768}G_4}-\frac{e^{-\Gamma_{12}-\Gamma_{58}-\Gamma_{76}}}{F_{5768}F_{1528}}
+\frac{e^{-\Gamma_{18}-\Gamma_{52}-\Gamma_{76}}}{F_{1528}G_4}\right]e^{-\Gamma_{34}}\nonumber
\\
&&+F_{1287}F_{3465}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{1728}G_5}-\frac{e^{-\Gamma_{18}-\Gamma_{72}-\Gamma_{34}-\Gamma_{56}}}{F_{1728}F_{3546}}
+\frac{e^{-\Gamma_{18}-\Gamma_{72}-\Gamma_{36}-\Gamma_{54}}}{F_{3546}G_5}\right]\nonumber
\\
&&+F_{3465}F_{1287}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{3546}G_5}-\frac{e^{-\Gamma_{12}-\Gamma_{36}-\Gamma_{54}-\Gamma_{78}}}{F_{3546}F_{1728}}
+\frac{e^{-\Gamma_{18}-\Gamma_{72}-\Gamma_{36}-\Gamma_{54}}}{F_{1728}G_5}\right]\nonumber
\\
&&+F_{3465}F_{3687}\left[
\frac{e^{-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{3546}G_6}-\frac{e^{-\Gamma_{36}-\Gamma_{54}-\Gamma_{78}}}{F_{3546}F_{3768}}
+\frac{e^{-\Gamma_{38}-\Gamma_{54}-\Gamma_{76}}}{F_{3768}G_6}\right]e^{-\Gamma_{12}}\nonumber
\\
&&+F_{3487}F_{4756}\left[
\frac{e^{-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{3748}G_6}-\frac{e^{-\Gamma_{38}-\Gamma_{74}-\Gamma_{56}}}{F_{3748}F_{4675}}
+\frac{e^{-\Gamma_{38}-\Gamma_{54}-\Gamma_{76}}}{F_{4675}G_6}\right]e^{-\Gamma_{12}}\nonumber
\\
&&+F_{5687}F_{3485}\left[
\frac{e^{-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{5768}G_6}-\frac{e^{-\Gamma_{34}-\Gamma_{58}-\Gamma_{76}}}{F_{5768}F_{3548}}
+\frac{e^{-\Gamma_{38}-\Gamma_{54}-\Gamma_{76}}}{F_{3548}G_6}\right]e^{-\Gamma_{12}}\nonumber
\\
&&+F_{1243}F_{1465}F_{1687}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{1324}G_1K}-\frac{e^{-\Gamma_{14}-\Gamma_{32}-\Gamma_{56}-\Gamma_{78}}}{F_{1324}F_{1546}H_1}+
\frac{e^{-\Gamma_{16}-\Gamma_{32}-\Gamma_{54}-\Gamma_{78}}}{G_1F_{1546}F_{1768}}-\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{76}}}{KH_1F_{1768}}\right]\nonumber \\
&&+F_{1243}F_{1487}F_{4756}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{1324}G_2K}-\frac{e^{-\Gamma_{14}-\Gamma_{32}-\Gamma_{56}-\Gamma_{78}}}{F_{1324}F_{1748}H_1}+
\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{74}-\Gamma_{56}}}{G_2F_{1748}F_{4675}}-\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{76}}}{KH_1F_{4675}}\right]\nonumber \\
&&+F_{1243}F_{5687}F_{1485}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{1324}G_3K}-\frac{e^{-\Gamma_{14}-\Gamma_{32}-\Gamma_{56}-\Gamma_{78}}}{F_{1324}F_{5768}H_1}+
\frac{e^{-\Gamma_{14}-\Gamma_{32}-\Gamma_{58}-\Gamma_{76}}}{G_3F_{5768}F_{1548}}-\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{76}}}{KH_1F_{1548}}\right]\nonumber \\
&&+F_{1265}F_{2534}F_{1687}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{1526}G_1K}-\frac{e^{-\Gamma_{16}-\Gamma_{52}-\Gamma_{34}-\Gamma_{78}}}{F_{1526}F_{2453}H_2}+
\frac{e^{-\Gamma_{16}-\Gamma_{32}-\Gamma_{54}-\Gamma_{78}}}{G_1F_{2453}F_{1768}}-\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{76}}}{KH_2F_{1768}}\right]\nonumber \\
&&+F_{1265}F_{1687}F_{2534}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{1526}G_4K}-\frac{e^{-\Gamma_{16}-\Gamma_{52}-\Gamma_{34}-\Gamma_{78}}}{F_{1526}F_{1768}H_2}+
\frac{e^{-\Gamma_{18}-\Gamma_{52}-\Gamma_{34}-\Gamma_{76}}}{G_4F_{1768}F_{2453}}-\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{76}}}{KH_2F_{2453}}\right]\nonumber \\
&&+F_{1287}F_{2734}F_{4756}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{1728}G_2K}-\frac{e^{-\Gamma_{18}-\Gamma_{72}-\Gamma_{34}-\Gamma_{56}}}{F_{1728}F_{2473}H_3}+
\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{56}}}{G_2F_{2473}F_{4675}}-\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{76}}}{KH_3F_{4675}}\right]\nonumber \\
&&+F_{1287}F_{2756}F_{2534}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{1728}G_4K}-\frac{e^{-\Gamma_{18}-\Gamma_{72}-\Gamma_{34}-\Gamma_{56}}}{F_{1728}F_{2675}H_3}+
\frac{e^{-\Gamma_{18}-\Gamma_{52}-\Gamma_{34}-\Gamma_{76}}}{G_4F_{2675}F_{2453}}-\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{76}}}{KH_3F_{2453}}\right]\nonumber \\
&&+F_{1287}F_{3465}F_{2736}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{1728}G_5K}-\frac{e^{-\Gamma_{18}-\Gamma_{72}-\Gamma_{34}-\Gamma_{56}}}{F_{1728}F_{3546}H_3}+
\frac{e^{-\Gamma_{18}-\Gamma_{72}-\Gamma_{36}-\Gamma_{54}}}{G_5F_{3546}F_{2673}}-\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{76}}}{KH_3F_{2673}}\right]\nonumber \\
&&+F_{3465}F_{1263}F_{1687}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{3546}G_1K}-\frac{e^{-\Gamma_{12}-\Gamma_{36}-\Gamma_{54}-\Gamma_{78}}}{F_{3546}F_{1326}H_4}+
\frac{e^{-\Gamma_{16}-\Gamma_{32}-\Gamma_{54}-\Gamma_{78}}}{G_1F_{1326}F_{1768}}-\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{76}}}{KH_4F_{1768}}\right]\nonumber \\
&&+F_{3465}F_{1287}F_{2736}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{3546}G_5K}-\frac{e^{-\Gamma_{12}-\Gamma_{36}-\Gamma_{54}-\Gamma_{78}}}{F_{3546}F_{1728}H_4}+
\frac{e^{-\Gamma_{18}-\Gamma_{72}-\Gamma_{36}-\Gamma_{54}}}{G_5F_{1728}F_{2673}}-\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{76}}}{KH_4F_{2673}}\right]\nonumber \\
&&+F_{3465}F_{3687}F_{1283}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{3546}G_6K}-\frac{e^{-\Gamma_{12}-\Gamma_{36}-\Gamma_{54}-\Gamma_{78}}}{F_{3546}F_{3768}H_4}+
\frac{e^{-\Gamma_{12}-\Gamma_{38}-\Gamma_{54}-\Gamma_{76}}}{G_6F_{3768}F_{1328}}-\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{76}}}{KH_4F_{1328}}\right]\nonumber \\
&&+F_{3487}F_{1283}F_{4756}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{3748}G_2K}-\frac{e^{-\Gamma_{12}-\Gamma_{38}-\Gamma_{74}-\Gamma_{56}}}{F_{3748}F_{1328}H_5}+
\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{74}-\Gamma_{56}}}{G_2F_{1328}F_{4675}}-\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{76}}}{KH_5F_{4675}}\right]\nonumber \\
&&+F_{3487}F_{4756}F_{1283}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{3748}G_6K}-\frac{e^{-\Gamma_{12}-\Gamma_{38}-\Gamma_{74}-\Gamma_{56}}}{F_{3748}F_{4675}H_5}+
\frac{e^{-\Gamma_{12}-\Gamma_{38}-\Gamma_{54}-\Gamma_{76}}}{G_6F_{4675}F_{1328}}-\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{76}}}{KH_5F_{1328}}\right]\nonumber \\
&&+F_{5687}F_{1243}F_{1485}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{5768}G_3K}-\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{58}-\Gamma_{76}}}{F_{5768}F_{1324}H_6}+
\frac{e^{-\Gamma_{14}-\Gamma_{32}-\Gamma_{58}-\Gamma_{76}}}{G_3F_{1324}F_{1548}}-\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{76}}}{KH_6F_{1548}}\right]\nonumber \\
&&+F_{5687}F_{1285}F_{2534}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{5768}G_4K}-\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{58}-\Gamma_{76}}}{F_{5768}F_{1528}H_6}+
\frac{e^{-\Gamma_{18}-\Gamma_{25}-\Gamma_{43}-\Gamma_{76}}}{G_4F_{1528}F_{2453}}-\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{76}}}{KH_6F_{2453}}\right]\nonumber \\
&&+F_{5687}F_{3485}F_{1283}\left[
\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{56}-\Gamma_{78}}}{F_{5768}G_6K}-\frac{e^{-\Gamma_{12}-\Gamma_{34}-\Gamma_{58}-\Gamma_{76}}}{F_{5768}F_{3548}H_6}+
\frac{e^{-\Gamma_{12}-\Gamma_{38}-\Gamma_{54}-\Gamma_{76}}}{G_6F_{3548}F_{1328}}-\frac{e^{-\Gamma_{18}-\Gamma_{32}-\Gamma_{54}-\Gamma_{76}}}{KH_6F_{1328}}\right],
\label{8result}\end{aligned}$$
where
$$\begin{aligned}
&&G_1=L_{12}+L_{34}+L_{56}-L_{16}-L_{32}-L_{54},
\\
&&G_2=L_{12}+L_{34}+L_{78}-L_{18}-L_{32}-L_{74},
\\
&&G_3=L_{12}+L_{34}+L_{56}+L_{78}-L_{14}-L_{32}-L_{58}-L_{76},
\\
&&G_4=L_{12}+L_{56}+L_{78}-L_{18}-L_{52}-L_{76},
\\
&&G_5=L_{12}+L_{34}+L_{56}+L_{78}-L_{18}-L_{72}-L_{36}-L_{54},
\\
&&G_6=L_{34}+L_{56}+L_{78}-L_{38}-L_{54}-L_{76},
\\
&&H_1=L_{14}+L_{56}+L_{78}-L_{18}-L_{54}-L_{76},
\\
&&H_2=L_{16}+L_{52}+L_{34}+L_{78}-L_{18}-L_{32}-L_{54}-L_{76},
\\
&&H_3=L_{72}+L_{34}+L_{56}-L_{32}-L_{54}-L_{76},
\\
&&H_4=L_{12}+L_{36}+L_{78}-L_{18}-L_{32}-L_{76},
\\
&&H_5=L_{12}+L_{38}+L_{74}+L_{56}-L_{18}-L_{32}-L_{54}-L_{76},
\\
&&H_6=L_{12}+L_{34}+L_{58}-L_{18}-L_{32}-L_{54},
\\
&&K=L_{12}+L_{34}+L_{56}+L_{78}-L_{18}-L_{32}-L_{54}-L_{76}.\end{aligned}$$
Along with 2, 4 and 6-point correlators described above and according to patterns that we observe, we can conjecture a general expression of the 2n-point correlator given as
$$\begin{aligned}
&&\frac{1}{N_c}\left\langle\text{Tr}\left[U_{1}U_{2}^{\dagger}...U_{2n-1}U_{2n}^{\dagger}\right]\right\rangle\nonumber
\\
&=&\left[e^{C_F\mu^2A_1} +\sum_{k=2}^n \sum_{I_k}
\left[\prod_{i=1}^{k-1}F_{a_ib_ic_id_i} \sum_{l=1}^k\frac{e^{C_F\mu^2A_l}}{\prod\limits_{j=1;j\neq l}^k(A_l-A_j)}
\right]
\right]
e^{-\frac{C_F}{2}\mu^2\sum\limits_{r=1}^{2n}L_{rr}},\label{2n-point}\end{aligned}$$
where $I_k$ represents summing over all possible permutations $a_1,b_1,c_1,d_1,a_2,b_2,c_2,d_2,...,a_{k-1},b_{k-1},c_{k-1},d_{k-1}$, $A_1,A_2,...,A_k$ that satisfy the following conditions
$$\begin{aligned}
&&a_i<d_i; \ (a_i,b_i),(d_i,c_i)\in U_i,
\\
&&A_i=\sum_{(a,b)\in U_i}L_{ab},\end{aligned}$$
where $U_i$ satisfies
$$\begin{aligned}
&&L(U_{i}+U_0)=i; \ i\geq 1,
\\
&&U_0=\{(2,3),(4,5),...,(2n-2,2n-1),(2n,1)\},\end{aligned}$$
and the recurrence relation between $U_i$ and $U_{i+1}$ is
$$\begin{aligned}
U_{i+1}=U_i-\{(a_i,b_i),(d_i,c_i)\}+\{(a_i,c_i),(d_i,b_i)\}; \ i\geq 1,\end{aligned}$$
with its first term
$$\begin{aligned}
U_1=\{(1,2),(3,4),...,(2n-1,2n)\}.\end{aligned}$$
$U=\{(a,b),(b,c),...\}$ representing that the element $a$ is connected with $b$, $b$ is connected with $c$..., is a set of 2-dimensional row matrices $(i,j)$. $L(U)$ is a function of $U$, which equals the number of loops of the elements in $U$. For example, $U=\{(1,6),(2,3),(3,4),(4,5),(5,2),(6,1)\}$, which consists of two loops $1-6-1$ and $2-3-4-5-2$, therefore $L(U)=2$.
In the above expression, $U_i+U_0$ represents the configuration, which contains $i$ fermion loops, before the $i$th transition, then one can easily see that the sum condition $L(U_{i+1}+U_0)=i+1$ ensures that only transitions to states with one more fermion loop, as well as one more order of $N_c$, are allowed, while the recurrence relation between $U_i$ and $U_{i+1}$ shows how the configurations of these states translate.
One can use the above formula to get any 2n-point correlators, as well as the 6-point one, which can be subsequently derived as an illustration. For $n=3$, one gets
$$\begin{aligned}
&&[e^{C_F\mu^2A_1}
+\sum_{a_1,b_1,c_1,d_1,A_1,A_2}F_{a_1b_1c_1d_1}
\left[\frac{e^{C_F\mu^2A_1}}{(A_1-A_2)}+\frac{e^{C_F\mu^2A_2}}{(A_2-A_1)} \right]
+\sum_{ \substack{a_1,b_1,c_1,d_1,A_1,A_2,\\a_2,b_2,c_2,d_2,A_3} }F_{a_1b_1c_1d_1}F_{a_2b_2c_2d_2}\nonumber
\\
&&\times\left[ \frac{e^{C_F\mu^2A_1}}{(A_1-A_2)(A_1-A_3)}+\frac{e^{C_F\mu^2A_2}}{(A_2-A_1)(A_2-A_3)}+\frac{e^{C_F\mu^2A_3}}{(A_3-A_1)(A_3-A_2)}\right]
]T,\label{6}\end{aligned}$$
where
$$\begin{aligned}
&&U_1=\{(1,2),(3,4),(5,6)\};A_1=L_{12}+L_{34}+L_{56},\end{aligned}$$
for the second term
$$\begin{aligned}
&(a_1,b_1,d_1,c_1,U_2,A_2)=&(1,2,3,4,\{(1,4),(3,2),(5,6)\},L_{14}+L_{32}+L_{56}),\nonumber
\\
&&(1,2,5,6,\{(1,6),(3,4),(5,2)\},L_{16}+L_{34}+L_{52}),\nonumber
\\
&&(3,4,5,6,\{(1,2),(3,6),(5,4)\},L_{12}+L_{36}+L_{54}),\end{aligned}$$
for the third term
$$\begin{aligned}
&&(a_1,b_1,d_1,c_1,U_2,A_2,a_2,b_2,d_2,c_2,U_3,A_3)\nonumber
\\
&=&(1,2,3,4,\{(1,4),(3,2),(5,6)\},L_{14}+L_{32}+L_{56},1,4,5,6,\{(1,6),(3,2),(5,4)\}),\nonumber
\\
&& (1,2,5,6,\{(1,6),(3,4),(5,2)\},L_{16}+L_{34}+L_{52},3,4,5,2,\{(1,6),(3,2),(5,4)\}),\nonumber
\\
&& (3,4,5,6,\{(1,2),(3,6),(5,4)\},L_{12}+L_{36}+L_{54},1,2,3,6,\{(1,6),(3,2),(5,4)\}).\end{aligned}$$
Putting the values of the above sum variables and the tadpole term $T$ into Eq. (\[6\]), one can get the exact 6-point correlator shown in Eq. (\[6-point\]).
When calculating the multiple scattering factors in the multi-particle production processes, one may encounter the case that several coordinates coincide in the multipole amplitude, which can be entirely derived by taking the corresponding limits of coordinates in the above general expression of 2n-point function. We derive the simplest case with an explicit expression that only two different coordinates $x_1$ and $x_2$ are involved in correlators as an example. Taking the limit $x_{3,4},x_{5,6}\rightarrow x_{1,2}$, namely $A_{1,2,3}\rightarrow3L_{1,2}$, in Eq. (\[6\]), which gives
$$\begin{aligned}
\frac{1}{N_c}\left\langle\text{Tr}\left[ U_{1}U_{2}^{\dagger}U_{1}U_{2}^{\dagger}U_{1}U_{2}^{\dagger}\right]\right\rangle=e^{-3\Gamma_{12}}\left[1+3C_F\mu^2F_{1221}+\frac{3}{2}(C_F\mu^2F_{1221})^2\right].\end{aligned}$$
In the same way, one also gets
$$\begin{aligned}
\frac{1}{N_c}\left\langle\text{Tr}\left[ U_{1}U_{2}^{\dagger}U_{1}U_{2}^{\dagger}\right]\right\rangle=e^{-2\Gamma_{12}}\left[1+C_F\mu^2F_{1221}\right],\end{aligned}$$
and
$$\begin{aligned}
\frac{1}{N_c}\left\langle\text{Tr}\left[U_{1}U_{2}^{\dagger}U_{1}U_{2}^{\dagger}U_{1}U_{2}^{\dagger}U_{1}U_{2}^{\dagger}\right]\right\rangle
=e^{-4\Gamma_{12}}\left[1+6C_F\mu^2F_{1221}+\frac{16}{2!}(C_F\mu^2F_{1221})^2+\frac{16}{3!}(C_F\mu^2F_{1221})^3\right].\label{8simp}\end{aligned}$$
And the expression for the 2n-point correlator with two different coordinates involved can be also obtained as
$$\begin{aligned}
\frac{1}{N_c}\left\langle\text{Tr}\left[(U_{1}U_{2}^{\dagger})^{n}\right]\right\rangle
=e^{-n\Gamma_{12}}\sum_{i=0}^{n-1}\left[\frac{n^{i-1}C_n^{i+1}}{i!}(C_F\mu^2F_{1221})^i\right],\end{aligned}$$
where the factor $n^{i-1}C_n^{i+1}$ is the exact number of color transition ways from the initial color structure with one fermion loop to the color singlet state with $i+1$ fermion loops after the $i$th transition. For the octupole case, namely $n=4$, we have $C_n^2=6$, $nC_n^3=n^2C_n^4=16$, which are the exact corresponding coefficients in the second, the third and the fourth term in Eq. (\[8simp\]), and also equal the numbers of the terms with one, two and three factors $F_{abcd}$ in Eq. (\[8result\]), respectively.
The general expression contains different terms corresponding to the number of factors $F_{a_ib_ic_id_i}$, which represents the number of transitions between the color singlet states. One can recognise that the 8-point correlator has one term with no color transitions, and several terms with one, two and three transitions, while it is not difficult to see from Eq. (\[2n-point\]) that the largest number of color transitions allowed in the terms of 2n-point correlators is $n-1$.
Conclusion
==========
In conclusion, we find that, in the large-$N_c$ limit in the MV model, the octupole amplitude as well as general 2n-point correlators can be written in analytical forms, which may help us to outline the underlying dynamics and estimate the size of finite-$N_c$ corrections to multi-particle productions, and provide initial conditions for small-$x$ evolutions.
We thank Dr. Bo-Wen Xiao for discussions and comments. This material is based upon work supported by the Natural Science Foundation of China (NSFC) under Grant No. 11575070, No. 11435004 and by the Ministry of Science and Technology of China under Projects No. 2014CB845404.
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|
---
abstract: 'In this paper I developed a classical model of elementary particle that is associated with a membrane of finite size, surrounded by non-linear electromagnetic field. The form of local interaction which lead to bounded states of finite masses, charges and spins was constructed. To do this I added Kaluza-Klein Lagrangian on the surface, which is associated with extended particle, to quadratic in field potentials term (a la tachyon mass). This ensures that Lagrangian remains an even function of the field, but spontaneous symmetry breaking leads to nontrivial soliton-like solutions. I assumed that the particle has axial symmetry and that the surface has only one degree of freedom, i.e. is a disk with the radius determined from the equations of motion. The solution of system of two non-linear partial differential equations for the field potentials was obtained numerically by different methods. Several solutions with increasing orders of leading field multipoles and disk radius were obtained, and masses, electric charges and spins were calculated. In the framework of this model the ratio of a charge square to a double spin, i.e. the fine structure constant, which do not depend on parameters of the model, was calculated.'
author:
- |
E. P. Likhtman\
All - Russian Institute for Scientific and Technical Information\
(VINITI), Usievicha 20, Moscow 125315, Russia\
e-mail: peisv@viniti.ru
title: Simple classical model of spin membrane particle
---
PACS numbers: 11.10.Lm, 11.27.d, 12.90+b
Introduction
============
Physicists handle the laws of nature in two ways: they either explain them using more fundamental hypotheses or just postulate them. The values of electric charges and spins of elementary particles are still almost postulated in the framework of quantum relativistic field theory, whereas their masses became a subject of non-trivial calculations a long time ago. Wineberg-Salam model predicted not only existence and characteristics of intermediate bosons but allowed to calculate electroweak processes in tree-like approach. However, the calculation of radiation corrections is limited by the absence of experimental data on Higgs bosons’ masses. Also further unifications of interactions such as supersymmetry [@GL] and great unification [@VN] were not supported by clear experimental observations so far, and electrodynamics are still leading in calculation precision of stationary states. Nevertheless electrodynamics can not be considered as an ideal model for the lepton world because of two problems: divergences in both classical and quantum theories, and the absence of calculations of some dimensionless quantities such as fine structure constant and lepton mass ratios.
Non-linear extensions of electrodynamics were first proposed by Kaluza [@KK1], Klein [@KK2] as well as by Born and Infeld [@BI]. Non-linearityleads to non-singular solution in case of point-like particle [@RK], but the particle mass, charge and magnetic moment remain arbitrary. To change this situation one can introduce a fundamental length into the model while preserving the local character of interaction, and construct a non-singular solution with finite size field sources.
In the present paper I propose an extended particle model, which consists of only one real vector electromagnetic field. The model is based on bilinear Maxwell Lagrangian $$L_M=\frac{1}{8\,\pi}\,\int({\it E}^{2}-{\it B}^{2})\,dV, \label{lm}$$ and quadrolinear Kaluza-Klein Lagrangian (self-interaction of vector field) $$L_{K-K}=\frac{1}{8\,\pi \,\beta ^2}\,\int {({\it E\cdot B}})^{2}\,dV,
\label{lkk}$$ where constant $\beta $ has dimension of field strength.
The key assumption of this work is proportionality of the four-current on some surface associated with the particle to the four-potential of the field. Therefore the gauge invariance is broken on the surface. In order to generate this dependence in the form of Lagrange equations I add quadratic in potentials term on the surface to the Lagrangian $$L_{surface}=-\frac{1}{4\,\pi \,l}\,\int ({\it p}^{2}-{\it A}^{2})\,dS,
\label{ls}$$ where $l$ is a fundamental length and $p$ and $A$ are scalar and vector potentials. The Lagrangian remains an even function of the field, and to provoke spontaneous symmetry breaking the sign of the last term must be the same as the sign used in the theory of vector tachyon field.
The solution of the problem was found by using approximation of infinite number of degrees of freedom by the finite number. It appears that non-linearity of Kaluza-Klein interaction on the one hand and the competition of surface and volume energy on the other hand lead to a spectrum of non-trivial stationary states with the finite values of all observables. This result was obtained by assuming an axial symmetry of the whole system, and that the surface is a disk with only one degree of freedom — disk radius. The description of the field by only two degrees of freedom has already led to a set of stationary solutions. Then more field degrees of freedom were added (up to 1000) so that solutions achieved an asymptotic behavior.
An important feature of the presented classical model is that the charges, magnetic moments and spins are no longer parameters of the Lagrangian, but arise due to a spontaneous symmetry breaking. These observables are proportional to some powers of two fundamental parameters of the Lagrangian $\beta$ and $l$. To compare predictions of the model with experiment one can construct several dimensionless parameters, which do not depend on $\beta$ and $l$. Therefore, in the following text I set these two parameters to unity without loss of generality. For each state I calculate Lande coefficient and the ratio of a charge square to a double spin $e^2/(2\,s)$, i.e. the fine structure constant. I also extract the ratios of masses, spins and magnetic moments of different states.
Lagrangian in ellipsoidal coordinates
=====================================
Owing to an axial symmetry and particular choice of calibration the problem was reduced to solution of two non-linear partial differential equations for electric field potential $p$ and the only non-zero component of vector-potential $A=A_{\phi}$, which both depend on two spacial coordinates. I introduced Cartesian coordinates so that $z$ is perpendicular to the disk plane, and then did the following transformation to the ortogonal ellipsoidal coordinates $$x=\rho \,\cos \left( \phi \right),\label{xx}$$ $$y=\rho \,\sin \left( \phi \right),\label{yy}$$ $$z=R\,\frac{\bar v(v)\,\cos \left( \pi/2\,\bar w(w)\right) }{\sin
\left( \pi/2
\,\bar w(w)\right) },\label{zz}$$ $$\rho ={R\,\frac{\sqrt{1-\bar v(v)^{2}}}{\sin \left( \pi/2\,\bar
w(w)\right) }},$$ where $R$ is the disk radius. This transformation reduced an infinite space to a unit square ($w,v$) (note that none of the functions depends on the third angular coordinate $\phi $). I have concentrated on the mirror symmetry case $L(-v)=L(v)$, and therefore evaluated all functions in the interval $0\leq v\leq 1$. The transformation \[xx\]–\[zz\] can be tuned by two monotonically increasing functions $\bar w(w)$ and $\bar v(v)$ ($0\leq w,\bar w,\bar v\leq 1$). The simplest case is $\bar w(w)=w,\ \bar v(v)=v$, which we explore in subsection \[basis\]. The particular choice of functions $\bar w(w)$ and $%
\bar v(v)$ should not affect the values of observables, but could be used to achieve better convergence to the asymptotic solution. I have also introduced contravariant potentials $$\tilde{\it p}=p\,R, \tilde{\it A}=A\,R\,h_{\phi},\label{Ap}$$ where $h_{\phi }=\sqrt{g_{\phi \phi }}/R$ is dimensionless Lame coefficient, and ”field strength” $$E_{w}=-{\frac{d}{dw}}\tilde{{\it p}}, E_{v}=-{\frac{d}{dv}}\tilde{{\it
p}},\label{E}$$ $$B_{w}={\frac{d}{dv}}\tilde{{\it A}}, B_{v}=-{\frac{d}{dw}}\tilde{{\it A}}.
\label{z}$$
The boundary conditions on the axis $v=1$ are $\tilde{{\it A}}=0$ with finite $E_v$, and at infinity $w=0$ are $\tilde{{\it p}}=0$, $\tilde{{\it
A}}%
=0$. The boundary conditions in the disk plane outside the disk itself $v=0$ are $E_v=0$ and $B_w=0$. Now we can write Lagrangian using new variables as $$L=L_p+L_A+L_{K-K}, L_p=L_{pv}+L_{ps}, L_A=L_{Av}+L_{As},\label{L}$$ where subscripts $v$ and $s$ denote volume and surface terms, and $$L_{pv}=-\frac {R}{2}\,\int _{0}^{1}\!\int
_{0}^{1}\!(F^w\,E_w^2+F^v\,E_v^2)dv\,dw,
\label{Lp}$$ $$L_{Av}=\frac {R}{2}\,\int _{0}^{1}\!\int
_{0}^{1}\!(G^w\,B_w^2+G^v\,B_v^2)dv\,dw,
\label{LA}$$ $$L_{K-K}=\frac {1}{2\,R}\,\int _{0}^{1}\!\int _{0}^{1}\!K\,(B\cdot
E)^{2}\,dv\,dw,
\label{LKK}$$ $$L_{ps}=\frac {R^2}{2}\,\int _{0}^{1}\!S^p\,\tilde{{\it p}}^2\,dv,
\label{Lps}$$ $$L_{As}=-\frac {R^2}{2}\,\int _{0}^{1}\!S^A\,\tilde{{\it A}}^2\,dv,
\label{LAs}$$ where $$F^w=h_v\,h_\phi/h_w, F^v=h_w\,h_\phi/h_v,$$ $$G^w=1/F^w, G^v=1/F^v,$$ $$K=1/(h_v\,h_w\,h_\phi), (B\cdot E)=(B_w\,E_w+ E_v\,B_v)$$ $$S^p=h_v\,h_\phi, S^A=h_v/h_\phi.$$
Orbital contribution to the angular momentum is zero due to the axial symmetry $E_{\phi }=0$. The $z$-component of the spin is determined by the integral $$s=R^2\,\int _{0}^{1}\!\int _0^1\!( D_w\,\frac {dw}{d\rho}\,h_w^2+ D_v\,%
\frac {dv}{d\rho}\,h_v^2 )\, \tilde{{\it A}}\,h_\phi\,dv\,dw,
\label{s}$$ where $$D_w= E_w+ B_w\,(B\cdot E)\,K/F^w,D_v= E_v+ B_v\,(B\cdot E)\,K/F^v.$$
Numerical solution
==================
In order to solve the model numerically, I approximated the fields by the finite number of degrees of freedom either by expansion of both potentials in series or by discretization of the unit square. The third method combines both approaches. In all methods the problem was reduced to a solution of the system of many non-linear algebraic equations. The system was solved by the iterative Newton’s tangent method for many variables. I supposed that the approximate asymptotic solution was found if solutions converged and the energy values did not change significantly with increasing number of variables. To choose initial approximation I have used the property of Kaluza-Klein interaction (generally speaking, of any Lagrangian with only quadratic and fourth-order dependence on the fields), that $$L_{A}+L_{K-K}=0,L_{p}+L_{K-K}=0,
\label{LL}$$ on exact solutions. To ensure positive values of energy $H$ (in the following I use $H=-L$) one requires $H_{A}=H_{p}=-H_{K-K}>0$.
Basic functions method {#basis}
----------------------
The simplest method, which allows to find qualitative solutions with only two field degrees of freedom, is basic functions method. In this method one expresses potentials as a finite series of normalized functions with unknown amplitudes, so that Lagrangian becomes a function of these amplitudes. Calculation of bilinear part of Lagrangian requires evaluation of relatively small number of integrals, whereas number of integrals required for quadrolinear terms is proportional to fourth power of basic functions number. Therefore the advantages of the method can be seen only for the small number of correctly chosen functions.
The natural choice of basic functions is solutions of Maxwell equations (fixing the choice of coordinate system so that $\bar w(w)=w$ and $\bar
v(v)=v$) with the first few multipole moments
$$\tilde p_1={\frac {\pi\,w}{2}},
\label{p1}$$
$$\tilde A_2={\frac {\left( -\pi\,w+\sin \left( \pi \,w \right) \right) \left(
1-{v}^{2} \right) }{-1+\cos \left( \pi \,w \right) }},
\label{A2}$$
$$\tilde p_3={\frac {\left( \left( 4+2\,\cos \left( \pi \,w \right) \right)
\pi \,w-6\,\sin \left( \pi \,w \right) \right) \left( 3/2\, {v}^{2}-1/2
\right) }{-1+\cos \left( \pi \,w \right) }},
\label{p3}$$
$$.............,$$ which were normalized at $v=0$ and $w=1$ to be $\pi/2$. The potentials become $$\tilde p(v,w) =\sum _{k=1}\tilde p_{2\,k-1}(v,w)\, a_{2\,k-1},
\label{pvw}$$ $$\tilde A(v,w) =\sum _{k=1}\tilde A_{2\,k }(v,w)\, a_{2\,k },
\label{Avw}$$ where $a_1=e/R$, $a_2=\mu /R^2\,3/2$, $a_3$,... are unknown multipole amplitudes, $e$ – charge, and $\mu$ – magnetic moment.
Integration of bilinear part of energy leads to (for the first six multipoles):
$$H_{pv}=-R/2\,(\pi/2) \,(a_1^2 +16/5\,a_3^2+4096/729\,a_5^2),
\label{Hpv}$$
$$H_{Av}=R/2\,(\pi/2)\,(4/3\,a_2^2+256/63\,a_4^2+16384/2475\,a_6^2),
\label{HAv}$$
$$H_{ps}=R^2/2\,(\pi/2)^2\,(1/2\, a_1^2+1/2\, a_3^2+1/2\, a_5^2
-1/2\,a_1\,a_3 - 1/9\,a_1\,a_5 -13/36\,a_3\,a_5),
\label{Hps}$$
$$H_{As}=-R^2/2\,(\pi/2)^2\,(1/4\,a_2^2+11/24\,a_4^2+29/60\,a_6^2
-1/3\,a_2\,a_4-1/12\,a_2\,a_6-1/3\,a_4\,a_6).
\label{HAs}$$
Integration of quadrolinear part of energy is evaluated analytically over $v$ and numerically over $w$. An approximate values of first $3\times 3=9$ integrals are listed below $$H_{K-K}=1/(2\,R)\,( 2.13\,a_1^2\,a_2^2 +6.19\,a_1^2\,a_2 \,a_4
+15.4\,a_1^2\,a_4^2 +22.2\,a_3^2\,a_2^2$$ $$+16.7\,a_1 \,a_2 \,a_3\,a_4 + 121\,a_3^2\,a_4^2 -12.0\,a_1 \,a_2^2\,a_3
+15.0\,a_3 \,a_4^2\,a_1 -98.5\,a_3^2\,a_4 \,a_2).
\label{HKK}$$
To find a simplest solution with only one electric and one magnetic multipoles, we note that the bilinear parts of the energies $H_{p}$ and $%
H_{A}$ must be positive, which is achieved if $$\begin{aligned}
1.27 &<&R_{1}, R_{2}<3.40 \nonumber \label{r1r10} \\
4.07 &<&R_{3}, R_{4}<5.64 \nonumber \\
7.15 &<&R_{5}, R_{6}<8.72 \\
10.3 &<&R_{7}, R_{8}<11.8 \nonumber \\
13.4 &<&R_{9}, R_{10}<15.0 \nonumber\end{aligned}$$ where the subscripts denote the number of corresponding multipole. I considered values of radii which satisfy each line of eq. \[r1r10\] separately in order to obtain solutions with minimal energy. First pair of multipoles is centrosymmetric electric field and dipolar magnetic moment. Therefore the energy $H_{1-2}$ is a function of three variables: two multipole amplitudes $a_1$, $a_2$ and the radius $R$. From eqs. \[Hpv\]–\[HKK\] one gets:
$$H_{1,2}=-R/2\,(1.57\,a_1^2-2.09\,a_2^2)+ R^2/2\,(1.23\,a_1^2-0.617\,a_2^2)
-2.13\,a_1^2\,a_2^2/(2\,R).
\label{H12}$$
The values of these unknowns are determined from solution of the system of three algebraic equations, which are obtained from requiring that partial derivatives of the energy in respect to these variables are equal to zero. To obtain the next state I chose two other degrees of freedom, which are quadrupole electric and octupole magnetic moments, and then solve analogous equations for $H_{3,4}$, etc.
Then the obtained solutions were used as a starting point for solution of Lagrange equations with 8 degrees of freedom in order to increase the precision. This procedure did not lead to a convergent process for the first pair of multipoles but did improve calculations for all others. The possible reason is that the determinant of the matrix of linearized equations does not have constant sign and can be close to zero, which lead to poor convergence. The physical reason of this problem is the areas with very strong electrical field where equation for magnetic field becomes hyperbolic rather than elliptic.
The results of calculation for the first three pairs of leading multipoles are listed in Table 1 ($m=H$).
------------ ------------ ------ ------- ------
Leading Included $R$ $m$ $e$
multipoles multipoles
1,2 1..2 2.81 3.57 3.25
1,2 1..6 - - -
3,4 3..4 5.03 0.431 -
3,4 1..8 5.49 0.366 1.14
5,6 5..6 8.05 0.337 -
5,6 1..10 8.05 0.923 2.05
------------ ------------ ------ ------- ------
[**Table 1**]{}\
The table shows that as expected from eqs. \[r1r10\] the disk size increases with increasing number of leading multipoles. Further increase of number of multipoles must be accompanied by simultaneous addition of new basic functions with non-Maxwellian $w$-dependence for lower multipoles, which would lead to significant increase of number of integrals and calculation time.
Lattice approach
----------------
An alternative method is to define the potentials on some lattice and to assume linear interpolation between lattice points. Due to local character of interactions, presence of first derivatives only and linear interpolation assumption the energy depends only on the values of potentials at the neighboring sites, so that each algebraic equation contains limited number of variables (not more than 18), which does not increase with increasing number of lattice points. Therefore calculation time increases slower than the cube of number of points.
Linear interpolation and integration by trapezium rule works well only for slowly varying functions. To increase smoothness of potentials I used specific choice of function $\bar v(v)$, which ”stretched” oscillations of the potentials near the disk axis. In particular I used $\bar
v(v)=3/2\,v-1/2\,v^{3}$ and $\bar w(w)=w$. The results of calculation for different number of lattice points are listed in Table 2.
------------ ---------------------- ------ ------ ------
Leading Lattice $R$ $m$ $e$
multipoles size
3,4 $10\times 10$ 6.01 1.40 3.55
3,4 $12\times 12$ 5.98 1.43 3.68
3,4 $15\times 15$ 5.96 1.47 3.81
3,4 $20\times 20$ 5.96 1.52 3.91
3,4 $\infty\times\infty$ 5.96 1.60 4.04
5,6 $12\times 12$ 9.88 1.83 3.92
5,6 $14\times 14$ 9.62 1.58 3.72
5,6 $18\times 18$ 9.38 1.41 3.69
5,6 $24\times 24$ 9.23 1.32 3.68
5,6 $\infty\times\infty$ 9.05 1.22 3.68
7,8 $12\times 12$ 15.0 3.61 5.89
7,8 $14\times 14$ 14.2 2.44 4.40
7,8 $18\times 18$ 13.4 1.70 3.96
7,8 $24\times 24$ 12.8 1.40 3.90
7,8 $\infty\times\infty$ 12.1 1.20 3.88
------------ ---------------------- ------ ------ ------
[**Table 2**]{}\
I used solutions of the previous section as a starting point for the lattice method. At the end of each part of the table extrapolated values for the infinite lattice are listed. For leading multipoles 1,2 solutions was found for lattice 6$\times 6$ only. For leading multipoles 3,4 stable solutions exist for a large range of number of degrees of freedom, and observables quickly achieve their asymptotic values. Convergence gets worse with increasing order of leading multipole because higher order multipoles have more oscillations on the disk surface. In particular potentials of 3,4 multipole have two extremums, 5,6 — 3 extremums and 7,8 — 4 extremums.
Combined method
---------------
To improve precision of calculations and the consistency of two previous methods I divided space into two parts: for large distances from the disk ($%
w=0..3/4$) I used basic functions method with $2\times 5=10$ functions, and in the vicinity of the disk ($w=3/4..1$) I used lattice approach. The results of combined method are listed in Table 3.
------------ ------------------------- ------ ------ ------
Leading Lattice $R$ $m$ $e$
multipoles size
3,4 $ 12 \times 3 $ 5.97 1.41 3.63
3,4 $ 16 \times 4 $ 5.95 1.47 3.77
3,4 $ 20 \times 5 $ 5.96 1.51 3.88
3,4 $ 28 \times 7 $ 5.96 1.55 3.92
3,4 $\infty\times\infty /4$ 5.97 1.57 3.94
3,4 $\infty\times\infty$ 5.96 1.60 4.04
5,6 $ 16\times 4 $ 9.31 1.37 3.70
5,6 $ 20\times 5 $ 9.36 1.49 3.91
5,6 $ 28\times 7 $ 9.21 1.35 3.77
5,6 $ 32\times 8 $ 9.15 1.27 3.65
5,6 $ 40\times 10 $ 9.13 1.25 3.63
5,6 $\infty\times\infty /4$ 9.06 1.19 3.56
5,6 $\infty\times\infty$ 9.05 1.22 3.68
7,8 $ 16\times 4 $ 13.7 1.96 4.13
7,8 $ 20\times 5 $ 13.1 1.55 3.87
7,8 $ 28\times 7 $ 12.6 1.32 3.92
7,8 $ 32\times 8 $ 12.5 1.27 3.91
7,8 $ 40\times 10 $ 12.4 1.21 3.90
7,8 $\infty\times\infty /4$ 12.1 1.10 3.90
7,8 $\infty\times\infty$ 12.1 1.20 3.88
------------ ------------------------- ------ ------ ------
[**Table 3**]{}\
The last lines of each part of this table repeats the asymptotic results of the previous section. The results of lattice and combined methods are in good agreement with each other. However, combined method is 10 times faster and therefore gives more precise results. The results for $\bar v=2\,v-v^2$ and $40\times 10$ lattice differ from Table 3 only by a few percent even for 7,8 multipole.
Conclusions
===========
I proposed a new concept of extended particle that is associated with a membrane of finite size. Charges and currents on this surface are chosen to be proportional to the corresponding potentials of the non-linear Kaluza-Klein electromagnetic field, so that the gauge invariance is broken on this surface. Then I assumed the axial and mirror symmetry of the system and existence of the only one surface degree of freedom: radius of the disk. Number of field degrees of freedom was also chosen to be finite. Described set of assumptions allowed me to develop a technique for iterative calculation of the first three stationary states of the model. Then I demonstrated that increasing number of degrees of freedom (up to 1000) leads to the asymptotic solutions of Lagrange equations. Several states with increasing order of leading electric and magnetic multipoles and disk radii were constructed numerically using this method. Each state can be potentially associated with an elementary particle.
The key result of the paper is that the values of all observables are finite. The masses were found to decrease weakly with increasing order of leading multipoles. The charges, magnetic moments and spins were also calculated from the model rather than postulated. It was not obvious that charges and spins of different states are identical or that their ratios are rational numbers. Calculations have shown that they are equal for the first three states within 10-20%:
------------ ------ ----- ----- ----- ------------ -------------
Leading $R$ $m$ $e$ $s$ $1/\alpha$ $\mu/\mu_B$
multipoles
3,4 6.0 1.6 4.0 12 1.4 1.0
5,6 9.1 1.2 3.6 11 1.5 1.3
7,8 12.1 1.1 3.9 12 1.5 1.1
------------ ------ ----- ----- ----- ------------ -------------
[**Table 4**]{}\
More detailed and extensive calculations are needed to make more precise conclusion. From the other side, the question about equality of charges and spins can be resolved by discovering some hidden symmetry of the described or modified interaction. It is also possible to change non-linearity to Born-Infeld type without adding any new parameters to the model.
Masses, charges, magnetic moments and spins are proportional to some powers of the two fundamental constants of the model with dimensions of length and field strength. However, there are two observables which do not depend on these constants — Lande coefficient ($\mu/\mu_B$) and the ratio of charge square to a double spin, which is equal to the fine structure constant if one assumes that the spin is $\hbar /2$. Lande coefficient was found about 1, and the fine structure constant — more than 100 times larger than expected. Therefore, calculated values of these parameters do not allow to associate discovered stationary states with charged leptons. Another interpretations (quarks) would require development of a calculation technique for bound states probably with non-additive charges and spins.
The presented method allows to modify calculations for different non-linearity and/or boundary conditions on the surface without adding new model parameters. This will preserve the predictive ability of the approach and the existence of finite values of the observables.
Acknowledgment {#acknowledgment .unnumbered}
==============
I thank I.V. Tutin, B.L. Voronov and A.E. Likhtman for continuous support and discussions.
[9]{} Yu.A. Golfand, E.P. Likhtman. [*JETP Letters*]{}, 13, 452 (1971).
M.I. Vysotskii, R.B. Nevzorov. [*Uspekhi Fizicheskikh Nauk*]{}, 171, 919 (2001).
Th. Kaluza. [*Sitzungsber. Preuss. Acad. Wiss.*]{}, K1, 966, (1921).
O. Klein. [*Z. Phys.*]{}, 37, 895, (1926).
M. Born and L. Infeld. [*Nature*]{}, 132, 970, (1932).
R. Kerner, A.L. Barbosa and D.V. Gal’tsov.[*arXiv:hep-th*]{}, 0108026 v2, (2001)
|
---
abstract: |
Traditional thermoelectric Peltier coolers exhibit a cooling limit which is primarily determined by the figure of merit, *zT*. Rather than a fundamental thermodynamic limit, this bound can be traced to the difficulty of maintaining thermoelectric compatibility. Self-compatibility locally maximizes the cooler’s coefficient of performance for a given $zT$ and can be achieved by adjusting the relative ratio of the thermoelectric transport properties that make up $zT$. In this study, we investigate the theoretical performance of thermoelectric coolers that maintain self-compatibility across the device. We find such a device behaves very differently from a Peltier cooler, and term self-compatible coolers “Thomson coolers” when the Fourier heat divergence is dominated by the Thomson, as opposed to the Joule, term. A Thomson cooler requires an exponentially rising Seebeck coefficient with increasing temperature, while traditional Peltier coolers, such as those used commercially, have comparatively minimal change in Seebeck coefficient with temperature. When reasonable material property bounds are placed on the thermoelectric leg, the Thomson cooler is predicted to achieve approximately twice the maximum temperature drop of a traditional Peltier cooler with equivalent figure of merit ($zT$). We anticipate the development of Thomson coolers will ultimately lead to solid state cooling to cryogenic temperatures.\
\[1ex\] PACS numbers: 84.60.Rb, 05.70.Ce, 72.20.Pa, 85.80.Fi
author:
- 'G. Jeffrey Snyder$^{1*}$, Eric S. Toberer$^2$, Raghav Khanna$^1$, Wolfgang Seifert$^3$'
bibliography:
- 'toberer.bib'
title: Improved Thermoelectric Cooling Based on the Thomson Effect
---
Introduction
============
Peltier coolers are the most widely used solid state cooling devices, enabling a wide range of applications from thermal management of optoelectronics and infra-red detector arrays to polymerase chain reaction (PCR) instruments. Thermoelectric coolers have been traditionally understood by means of the Peltier effect, which describes the reversible heat transported by an electric current. This effect is traditionally understood in terms of absorption or release of heat at the junction of two dissimilar materials. The conventional analysis of a Peltier cooler approximates the material properties as independent of temperature (Constant Property Model (CPM)). This results in a maximum cooling temperature difference $\Delta T_{max}$ for a CPM cooler, which dependent on the figure of merit $ZT$ of the device [@GoldsmidBook; @Heikes]. $$\label{Eq_dTmax_CPM}
\Delta T_{max} = \frac{ZT_c^2}{2}$$
For the best commercial materials this leads to a $\Delta T_{max}$ of 65K (single stage) [@Marlow], which translates to a device $ZT$ at 300K of 0.74. In the CPM the device $ZT$ is equal to the material $zT$. Material $zT$ depends on the Seebeck coefficient ($\alpha$), temperature ($T$), electrical resistivity ($\rho$), and thermal conductivity ($\kappa$), $zT = \frac{\alpha^2 T}{\rho \kappa}$. In the CPM, the only way to increase $\Delta T_{max}$ for a single stage is to increase $zT$, leading to the focus of much thermoelectric research on improving $zT$. It is well known that even further cooling to lower temperatures can be achieved using multi-stage Peltier coolers [@GoldsmidBook; @Heikes]. In principle, each stage can produce additional cooling to lower temperatures, regardless of the $zT$ of the thermoelectric material in the stage. In practice, the thermal losses and complications of fabrication limit the performance of such devices. The 6-stage cooler of Marlow achieves a $\Delta T_{max}$ of 133K; this doubling of $\Delta T_{max}$ compared to a single stage cooler is achieved despite using materials with similar $zT$ [@Marlow]. Alternatively, such $\Delta T_{max}$ with a single-stage CPM cooler would require $ZT$ to be 2.5.
The transport properties across a single thermoelectric leg can be manipulated to improve cooling performance, although it has been less effective in reducing $\Delta T_{max}$ than a multi-stage approach. One common strategy is to engineer a change in extrinsic dopant concentration across a thermoelectric element which can significantly alter $\alpha$, $\rho$ and even $ \kappa$. For example, this has been demonstrated for thermoelectric generators in n-type PbTe doped with I [@Gelbstein2005]. Similar efforts have been done with cooling materials, as has been reviewed in ref [@KuznetsovCRC]. The simplest explanation for an improvement is an in increase in the local $zT$ at some temperatures by spatially adjusting the dopant composition within a material [@MahanJAP91].
Early theoretical work by Sherman et al for TEC found that different $\Delta T_{max} $ could be predicted from materials have the same or similar average $zT$ but different temperature dependence of the individual properties $\alpha, \rho, \kappa$ [@Sherman1960]. This demonstrated that optimizing cooler performance is significantly more complex than simply maximizing $zT$. More recently, Müller et al. [@mueller2003; @muellerpssa2006; @mueller06] and Bian et al. [@bian2006; @bian2007] used different numerical approaches to predict substantial gains in cooling to $\Delta T_{max}$ from functionally grading where an average $zT$ remains constant in an effort to determine the best approach to functionally grading.
Different material classes optimized for different temperatures can also be segmented together to improve performance of thermoelectric generators but the current must also be matched [@Moizhes1962]. The analysis of segmentation strikingly demonstrates that increasing the average $zT$ does not always lead to an increase in overall thermoelectric efficiency and so an understanding of the thermoelectric compatibility factor is needed to explain device performance [@snyder2004]. This paper derives the cooling limit for a single stage, fully optimized (self-compatible) TEC that functions as an infinitely staged cooler. The Fourier heat divergence in such an optimized cooler is found to be dominated by the Thomson effect rather than the Joule heating as in traditional Peltier coolers. This new opportunity presents a new challenge for material optimization based on compatibility factor rather than only $zT$.
\
Theory
======
Coolers are characterized by the coefficient of performance ($\phi = Q_c / P $), which relates the rate of heat extraction at the cold end $Q_c$ to the power consumption $P$ in the device [@SnyderCRC]. For simplicity, but without loss of generality, a single thermoelectric element can be considered rather than a complete device. A TEC leg can be treated as an infinite series of infinitesimal coolers, each of which is operating locally with some COP. Scaling this COP to the local Carnot COP ($T/dT$) yields the local reduced coefficient of performance $\phi_r$. [@seifertJMR2011]. This relationship between local performance across the leg and global COP, $\phi$, given in Eq.\[Eq\_COP\] is derived in the Appendix based on Ref.[@Sherman1960][@ZenerEgli1960]. While TECs are traditionally analysed using a global approach, we have previously shown the utility of a local approach [@snyder03; @snyder2004; @SnyderCRC]. This local approach leads to a consideration of material ‘compatibility’, as discussed below. $$\frac{1}{\phi}= \exp{\left(\int^{T_h}_{T_c}\frac{1}{T}\frac{1}{\phi_r(T)}dT\right)}-1
\label{Eq_COP}$$
The compatibility approach to optimizing thermoelectric cooling arises naturally from an analysis of the thermal and electric transport equations. This method has been described in detail for thermoelectric generators [@SnyderCRC] and coolers[@seifert06a] and are reproduced here for TEC. The method has been experimentally verified [@crane2009] and shown to reproduce results using a more traditional finite element results but with less computational complexity. This method has been incorporated into several engineering models such as those used by NASA for Radioisotope Thermoelectric Generators [@NASATechBriefNPO-45252] and Amerigon/BSST for automotive applications [@crane2011; @crane2009]. Consider an infinitesimal section of thermoelectric leg in a temperature gradient and an electric field. The temperature gradient will induce a Fourier heat flux ($\textbf{q}_\kappa = -\kappa \nabla T$) across this segment. The divergence of this heat (Eq.\[heateq\]) is equal to the source terms: irreversible Joule heating ($\rho j^2$) and the reversible Thomson heat ($T \frac{d\alpha}{dT} j \nabla T$), both of which depend on the electric current density ($j$). From these two effects, the governing equation for heat flow in vector notation is
$$\nabla \cdot \textbf{q}_\kappa = \nabla \cdot (-\kappa \nabla T)=\rho j^2 - \tau \,\mathbf{j} \cdot \nabla T
\label{heateq}$$
with Joule heat per volume $\rho j^2$, Thomson coefficient $\tau=T
\frac{d\alpha}{dT}$ and Thomson heat per volume $\tau \,\mathbf{j} \cdot \nabla T$. The Peltier, Seebeck and Thomson effect are all manifestations of the same thermoelectric property characterized by $\alpha$. The Thomson coefficient ($\tau=T\frac{d\alpha}{dT}$) describes the Thomson heat absorbed or released when current flows in the direction of a temperature gradient.
Restricting the problem to one spatial dimension, Eq. is typically examined assuming the heat flux and electric current are parallel [@SnyderCRC]. In the typical CPM model used to analyze Peltier coolers, the Thomson effect is zero because $\alpha$ is constant along the leg ($\frac{d \alpha}{dT} = 0$).
The exact performance of a thermoelectric leg with $\alpha(T)$, $\rho(T)$, and $\kappa(T)$ possessing arbitrary temperature dependence can be straightforwardly computed using the reduced variables: relative current density ($u$) and thermoelectric potential ($\Phi$) [@snyder03]. The relative current density $u$, given in Eq.\[Eq\_u\], is primarily determined by the electrical current density $j$, which is adjusted to achieve maximum global COP. The thermoelectric potential $\Phi$ is a state function which simplifies Eq.\[Eq\_COP\] to Eq.\[phifromPhi\] [@SnyderCRC]. $$\label{Eq_u}
u = \frac{-j^2}{\kappa \nabla T \cdot \textbf{j}}$$
$$\label{Eq_TEpotential}
\Phi = \alpha T + 1/u$$
$$\label{phifromPhi}
\phi = \frac{\Phi(T_c)}{\Phi(T_h)-\Phi(T_c)}$$
Changing variables to $T$ via the monotonic function $x(T)$, Eq.\[heateq\] simplifies to the differential equation in $u(T)$. $$\label{deqrelcurrent} \frac{du}{dT} = u^2 \left( T \frac{d \alpha}{dT} + \frac{\alpha^2}{z} u \right)$$ Using this formalism, the reduced coefficient of performance ($\phi_r$) can be simply defined for any point in the cooler (Eq.\[Eq\_eta\_r\]). Fig.\[fig1\] shows this relationship between $u$ and $\phi_r$. From Eq.\[Eq\_COP\], it can be shown that $\phi$ is largest when $\phi_r$ is maximized for every infinitesimal segment along the cooler. Hence, global maximization can be traced back to local optimization [@seifert2010].
$$\label{Eq_eta_r}
\phi_r = \frac{ u\, \frac{\alpha}{z} + \frac{1}{z\,T}}{u
\frac{\alpha}{z}~(1 - u \,\frac{\alpha}{z})}
= \frac{u \alpha + \frac{1}{T}}{u(\alpha - u \rho \kappa)}$$
The optimum $u$ which maximizes $\phi_r$ ($\frac{d\phi_r}{du}=0$) can be expressed solely in terms of local material properties (Eq.\[Eq\_sc\]). This optimum value of $u$ is defined as the thermoelectric compatibility factor $s_c$ for coolers. $$\label{Eq_sc}
s_c = \frac{-\sqrt{1+zT}-1}{\alpha T}$$ As this paper strictly focuses on coolers, we will refer to $s_c$ as simply $s$.
The maximum local $\phi_r$, denoted $\phi_{r,max}$, occurs when $u = s$. The expression for $\phi_{r,max}$ (Eq.\[Eq\_phirmax\]) is an explicit function of the material $zT$ and is independent of the individual properties $\alpha$, $\rho$, $\kappa$. This maximum allowable local efficiency provides a natural justification for the definition of $zT$ as the material’s figure of merit.
$$\label{Eq_phirmax}
\phi_{r,max}=\frac{\sqrt{1+zT}-1}{\sqrt{1+zT}+1}$$
One thus wishes to construct devices where, locally, each segment has “$u=s$" and thus $\phi_{r,max}$ is obtained. Globally, maximum $\phi$ is found when the entire cooler satisfies $u=s$.
Cooling Performance
===================
To compare the cooling performance of traditional Peltier coolers and $u = s$ coolers, we consider coolers with equivalent $z$. Traditional Peltier coolers have typically been analyzed with the constant property model (CPM), yielding a constant $z$ (where $zT$ is linearly increasing with temperature). We will show that constant $z$, but allowing $\alpha$, $\kappa$, $\rho$ to vary with $T$, can lead to substantial improvement in cooling. At the limit of this variation, we will assume the properties can be varied to satisfy $u = s$.
#### Performance of a CPM cooler {#performance-of-a-cpm-cooler .unnumbered}
CPM coolers have been extensively studied, typically using a global approach to the transport behavior. The $\phi$ for a CPM cooler (operated at optimum $j$) is given by Eq.\[COP\_CPM\] [@heikes1961]. Figure \[fig2\]a shows the $\phi$ of a CPM cooler decreases with increasing $\Delta T$. With increasing cooling, this $\phi$ decreases and reaches zero at $\Delta T_{max}$ (Eq.\[Eq\_dTmax\_CPM\]).
$$\phi^{CPM} = \left( \frac{T_c}{\Delta T} \right) \left( \frac{ \sqrt{1+zT_{avg}}-\frac{T_h}{T_c}}{ \sqrt{1+zT_{avg}} +1} \right)
\label{COP_CPM}$$
To understand what is limiting the CPM cooler at $\Delta T_{max}$, we derive the local reduced coefficient of performance $\phi_r^{CPM}(T)$. To obtain $\phi_r^{CPM}$ we need $u$ as a function of $T$. The solution to differential equation\[deqrelcurrent\] for CPM is $$\frac{1}{u(T)^2}=\frac{1}{u_h^2}+\frac{2\alpha^2}{z}(T_h-T)
\label{CPM_u}$$ where the value of $u$ at $T=T_h$ ($u_h$) serves as an initial condition. This expression allows $u(T)$ to be determined for any CPM cooler, regardless of temperature drop ($\Delta T \leq \Delta T_{max}$) and applied current density (**j**). The global maximum COP ($\phi$) is obtained when the optimum $u_h$ from Eq.\[maxu\_h\] is employed.
$$\frac{1}{ u_h}=\frac{-\alpha}{z}\frac{z T_c^2-2(T_h-T_c)}{T_h+T_c\sqrt{z(\frac{T_h+T_c}{2})+1}}
\label{maxu_h}$$
Consideration of Eq.\[maxu\_h\] reveals that the maximum $T_c$ is obtained when $1/u_h$ approaches zero. Figure\[fig3\] shows $|u|$ becoming infinite at $T_h$ for the CPM cooler. In this limit, Eq.\[maxu\_h\] can be simplified to give Eq.\[Eq\_dTmax\_CPM\] with $Z=z$. Thus, a local approach to transport yields the classic CPM limit typically obtained through an evaluation of global transport behavior.
Combining Eq.\[Eq\_eta\_r\], \[CPM\_u\], and \[maxu\_h\] results in $\phi_r(T)$ at $\Delta T_{max}$ for the CPM Peltier cooler (Eq.\[CPM\_phidtmax\]). This expression reveals $\phi_r$ drops to zero at both ends of the CPM cooler leg, as shown in Figure\[fig2\]b. This prohibits additional cooling and sets $\Delta T_{max}$.
$$\label{CPM_phidtmax}
\phi^{CPM}_{r,\Delta T_{max}}=\frac{\sqrt{2z(T_h-T)}-2\frac{T_h-T}{T}}{1+\sqrt{2z(T_h-T)}}
$$
To achieve cryogenic cooling ($T_c \rightarrow 0$) within the CPM, $zT$ must approach infinity (Eq.\[Eq\_dTmax\_CPM\]). For example, cooling with a single-stage CPM cooler to $10\,K$ would require $zT$ to be over $1000$ if the hot side is $300\,K$. When $\phi$ is negative, the net effect of the thermoelectric device is to supply heat, rather than remove heat, from the cold side. For negative $\phi$ values for the CPM cooler to be obtained requires certain parts of the cooler to locally possess $\phi_r<0$. Such a result may be surprising at first as this $\phi_r<0$ region is made from material possessing positive $zT$. This seems particularly odd when compared to the behavior of staged generators, discussed above. Clearly, single- and multi-staged CPM legs exhibit fundamentally different behavior, despite being composed of exactly the same material. Such behavior can be rationalized using the thermoelectric compatibility concept.
Figure\[fig3\] shows that the compatibility condition ($u=s$) is maintained at only one point in the CPM cooler. Consequently, CPM coolers operate inefficiently ($u \neq s$) at both the hot and cold ends. This is demonstrated in Figure\[fig2\]b, where $\phi_{r} < \phi_{r,max}$ for all but one point. Once $\phi_r$ goes below zero at low temperature, the thermoelectric device is no longer cooling the cold end and $\Delta T_{max}$ is reached (Figure\[fig2\]a).
While real coolers do not possess temperature-independent properties, the qualitative results for CPM translate well to traditional Peltier coolers due to their weak material gradients. Considering a Bi$_2$Te$_3$ leg with temperature-dependent properties described in Ref. [@seifert2002], we find $u$ and $s$ to be quite close to a $z$-matched CPM cooler (Figure\[fig3\]). Like the CPM cooler, $u=s$ at only one temperature along the leg. This leads to similar $\phi_{r}(T)$ for the Bi$_2$Te$_3$ and CPM coolers, shown in Figure\[fig2\]b.
Within the CPM, large $zT$ results in a high upper limit to $\phi_r$ but does not ensure this $\phi_{r,max}$ is achieved. Generaly, commercial cooling materials such as Bi$_2$Te$_3$ and any material that can be described by the CPM model will be operating significantly below the $\phi_r$ predicted by the $zT$ they possess (Eq. \[Eq\_eta\_r\], \[Eq\_phirmax\]).\
\
#### Performance of a $u=s$ cooler {#performance-of-a-us-cooler .unnumbered}
We now consider an idealized cooler which maintains $u=s$ across the entire leg. $\phi_r$ for this cooler is simply given by Eq.\[Eq\_phirmax\]. This $\phi_r$ is found to be positive for all $T$, as $z$ is always a positive real number. Globally, this translates to the analytic maximum for $\phi$ for a cooler where $z$ is defined and limited.
To facilitate comparison with CPM, we consider a constant $z$ model where the individual properties are adjusted to maintain $u=s$. The constant $z$ approach yields vanishing $zT$ at low $T$, consistent with real materials. Evaluating $\phi$ (Eq.\[Eq\_COP\]) for a $u=s$ cooler and the assumption of constant $z$, one obtains Eq.\[COP\_const\_z\], where $M_i = \sqrt{1+zT_i}$ with $T_i = T_h$, $T_c$. $$\label{COP_const_z}
\frac{1}{\phi^{u=s}}=\left(\frac{M_h-1}{M_c-1}\right)^2 \exp{ \left( \frac{2(M_h-M_c)}{(M_h-1)(M_c-1)} \right)}-1$$ Inspection of Eq.\[COP\_const\_z\], where $M_h > M_c > 1$, reveals that $\phi$ is always greater than zero for a $u=s$ cooler.
The difference between CPM and $u = s$ coolers can be visualized in Fig.\[fig2\]a, with the $\phi$ of the Thomson cooler asymptotically approaching zero with increasing $\Delta T$. Figure\[fig2\]b shows that $\phi_r$ for a self-compatible cooler with constant $z$ remains finite and positive throughout the device. In contrast, the CPM cooler is operating inefficiently at both the hot and cold ends, limiting its temperature range.
In principle, if $u = s$ can be maintained, the idealized $u = s$ cooler can achieve an arbitrarily low cold side temperature as long as the all of the materials have a finite $zT$. However, the material requirements to maintain $u = s$ become exceedingly difficult to achieve as the cooling temperature is reduced and the ultimate cooling will be finite, yielding $T_c> 0$.\
Material Requirements
=====================
A CPM cooler has a fixed $z$ and performance which is independent of the ratio of individual properties as long as they are constant with respect to temperature (Eq.\[Eq\_dTmax\_CPM\]). In contrast, a $u = s$ cooler requires dramatic changes in properties with temperature to maintain self-compatibility. Within the constraint of constant $z$, consideration of Eq.\[Eq\_sc\] suggests that the Seebeck coefficient must be varied across the device to maintain $u = s$. Additionally, as ${\alpha(T)} =\sqrt{z \rho(T) \kappa(T)}$ within a constant $z$ model, the product $\rho(T) \kappa(T)$ must also vary across the device.
The Seebeck coefficient profile $\alpha (T)$ for a $u = s$ cooler with constant $z$ can be solved analytically. Combining Eq.\[deqrelcurrent\] and $u=s$ yields the simple differential equation of $\alpha(T)$: $$ \frac{d}{dT} \left( \frac{\alpha T}{1+\sqrt{1 + z\,T}} \right)
= T\, \frac{d \alpha}{dT} - \frac{\alpha}{z} ~\frac{1+\sqrt{1 + z\,T}}{T}~.$$ Solving this equation yields $$\label{alphasoln}
\alpha(T)=\alpha_0 \frac{\sqrt{1+zT}-1}{\sqrt{1+zT}}\exp{ \left( \frac{-2}{\sqrt{1+zT}-1}\right) } ~.$$
With this expression for $\alpha(T)$, it is possible to evaluate $s(T)$ with Eq.\[Eq\_sc\]. Figure\[fig3\] shows the variation in $s$ required for a $u = s$ cooler with constant $z$. The self-compatible cooler modeled in Figure\[fig3\] has $z= 0.002$; 60K of cooling results in a change in $s$ of one order of magnitude.
The approximation for small $zT$ yield a simple expression for $\alpha(T)$, given by Eq.\[alpha\_approx\]. $$\label{alpha_approx}
\frac{d}{dT} \left( \ln \alpha(T) \right) = \frac{4}{zT^2} ~\longrightarrow ~
\alpha(T) \propto \exp{ \left( \frac{-4}{zT} \right)}$$ This reveals that $\alpha$ should be very large at the hot end and must decrease to a low value at the cold end. This exponentially varying $\alpha(T)$ required to maintain $u=s$ for constant $z$ is anticipated to be the limiting factor in real coolers and place bounds on the maximum cooling obtainable. We consider the realistic range of $\alpha$ below.
Large values of $\alpha$ are found in lightly doped semiconductors and insulators with large band gaps ($E_g$) that effectively have only one carrier type, thereby preventing compensated thermopower from two oppositely charged conducting species. Using the relationship between peak $\alpha$ and $E_g$ of *Goldsmid* (Eq.\[GoldsmidSharp\]) allows an estimate for the highest $\alpha (T_h)$ we might expect at the hot end, $\alpha_h$ [@thermalbandgap]. Good thermoelectric materials with band gap of 1eV are common while 3eV should be feasible. For a cooler with an ambient hot side temperature, this would suggest $\alpha_{h}$ should be $\sim$1-5mV/K. Maintaining $zT$ at such large $\alpha$ will require materials with both extremely high electronic mobility and low lattice thermal conductivity. $$\label{GoldsmidSharp}
\alpha_{h}= E_g / (2 eT_{h}) ~$$
A lower bound to $\alpha_c$ also arises from the interconnected nature of the transport properties. We require $zT$ to be finite; thus the electrical conductivity $\sigma$ must be large as $\alpha_c$ tends to zero. In this limit, the electronic component of the thermal conductivity ($\kappa_E$) is much larger than the lattice ($\kappa_L$) contribution and $\kappa \sim \kappa_E$. To satisfy the Wiedemann-Franz law ($\kappa_E = L \sigma T$ where $L = \frac{\pi^2}{3} \frac{k^2}{e^2}$ is the Lorenz factor in the free electron limit), $\alpha_c$ has a lower bound given by Eq.\[alphac\]. For example, a $z = \frac{1}{300}$ K$^{-1}$ and $T_c = 175$K results in a lower bound to $\alpha_c$ of $119\,\mu$V/K.
$$\label{alphac}
\alpha_c^2 = L z T_c = \frac{\pi^2}{3}\frac{k_B^2}{e^2} z T_c ~$$
The maximum cooling temperature $T_c$ can be solved as a function of $z$, $E_g$ and $T_h$ from equations Eq.\[alphasoln\], Eq.\[GoldsmidSharp\] and Eq.\[alphac\]. For small $z$ the approximate solution $$\label{ThomsonApprox}
\Delta T \approx \frac{z}{8} \, T_h^2 ~\ln{ \left( \frac{E_g^2}{\frac{4}{3} \pi^2 k_B^2 \, z \, T_h^3} \right) } ~$$ gives an indication of the important parameters but quickly becomes inaccurate for $zT$ above 0.1.
#### Material limits to performance {#material-limits-to-performance .unnumbered}
With these bounds on material properties, we consider the $\Delta T_{max}$ of a $u=s$ cooler. Figure\[fig2\] suggests that the $\phi$ of a $u=s$ cooler remains positive for all temperature. However, obtaining materials with the required properties limits $\Delta T_{max}$ to a finite value. Fig.\[Fig\_DeltaTlimit\] compares the $\Delta T_{max}$ solution for $u=s$ and CPM coolers with the same $z$. Here, the maximum Seebeck coefficient is set by the band gap ($E_g = 1-3~eV$), per Eq.\[GoldsmidSharp\]. The $u=s$ cooler provides significantly higher $\Delta T_{max}$ than the CPM cooler with the same $zT$, nearly twice the $\Delta T_{max}$ for $E_g = 3~eV$.
#### Spatial dependence of material properties {#spatial-dependence-of-material-properties .unnumbered}
These analytic results are possible because the compatibility approach does not require an exact knowledge of the spatial profile for the material properties. Nevertheless, it is possible solve for the spatial dependence of the $u = s$ cooler, given some material constraints. To determine $x(T)$, we integrate Eq.\[Eq\_u\], recalling we have assumed constant cross-sectional area ($j(x)=const.$), obtaining Eq.\[CRC40\]. $$\label{CRC40}
x(T) = \frac{-1}{j}\int_T^{T_h} u \kappa dT$$ Thus, the natural approach to cooler design within the $u = s$ approach is to determine the temperature dependence of the material properties, and then determine the required spatial dependence from the resulting $u(T)$ and $\kappa(T)$.
Figure\[fig5\]a shows an example of the Seebeck distribution $\alpha(x)$ along the leg that will provide the necessary $\alpha(T)$, where a constant $\kappa_L = 0.5$ W/mK is assumed. The $\alpha$ of Figure\[fig5\]a spans the range permitted by Eq.\[GoldsmidSharp\] and \[alphac\].
In a real device the spatial profile of thermoelectric properties will need to be carefully engineered. If this rapidly changing $\alpha(x)$ is achieved by segmenting different materials, low electrical contact resistance is required between the interfaces. We anticipate such control of semiconductor materials may require thin film methods on active bulk thermoelectric substrates.
\
Cooler phase space
==================
The improved performance of a $u = s$ cooler is not simply an incremental improvement, but rather we find CPM and $u=s$ coolers operate in fundamentally different phase-spaces. Here, by phase space we refer to the class of solutions defined by the sign of the Fourier heat divergence ($\nabla \cdot \textbf{q}_\kappa$ in Equation 3). The Fourier heat divergence in a cooler contains both the Joule $(\rho j^2)$ and Thomson $(\tau \textbf{j} \cdot \nabla T)$ terms.
We begin by considering the Fourier heat divergence in CPM and Bi$_2$Te$_3$ coolers and then compare this behavior to $u = s$ coolers. In the typical CPM model to analyze Peltier coolers, $\tau = 0$ as there is no variation in $\alpha$. In a CPM cooler, $\nabla \cdot \textbf{q}_\kappa$ is thus greater than zero. This can be seen by the downward concavity of the temperature distribution in Figure\[fig5\]b. In a typical Peltier cooler (e.g. Bi$_2$Te$_3$), the concavity is the same as the CPM cooler and thus the divergence is likewise positive. This is because the Thomson term is always less than the Joule term in a conventional thermoelectric cooler.
In contrast, a $u=s$ cooler changes the sign of the Fourier heat divergence such that $\nabla \cdot \textbf{q}_\kappa$ is less than zero. This can be readily visualized in Figure\[fig5\]b, where the concavity of the $u = s$ cooler is opposite the CPM and Bi$_2$Te$_3$ coolers. This difference in concavity must come from the Thomson term being positive and greater than the Joule heating term. The large magnitude of Thomson term is understandable with the exponentially rising Seebeck coefficient seen in Figure\[fig5\]a. The reversibility of the Thomson effect requires that for $\nabla \cdot \textbf{q}_\kappa$ to be less than zero, the hot end must have a high $|\alpha|$ relative to the cold end, and not vice versa. This translates to a requirement for $\tau$ such that $\tau\textbf{j} \cdot \nabla T > \rho j^2$.
We can also express the Fourier heat divergence in terms of reduced variables. $$\nabla \cdot \textbf{q}_\kappa = \textbf{j} \cdot \nabla \frac{1}{u} = \frac{- 1}{u^2} \frac{du}{dT} \textbf{j} \cdot \nabla T$$
Manipulation with Eq.\[Eq\_u\] produces a form where the sign of $u$ and directions of $j$ and $\nabla T$ are irrelevant. $$\nabla \cdot \textbf{q}_\kappa =\frac{j^2}{2 \kappa u^4}\frac{d}{dT}u^2$$ Thus the sign of $\nabla \cdot \textbf{q}_\kappa$ is determined by the sign of $\frac{d|u|}{dT}$, which is valid for both $p$ and $n$-type elements regardless of the sign of $u$.
The Fourier heat divergence criterion is a convenient definition to distinguish these two regions of thermoelectric cooling in experimental data. The Peltier cooling region, defined by $\nabla \cdot \textbf{q}_\kappa>0$, is found in the phase space where $\frac{d|u|}{dT}>0$. Likewise, the Thomson cooling region defined by $\nabla \cdot \textbf{q}_\kappa<0$ is the phase space where $\frac{d|u|}{dT}<0$. The constant relative current $u(T) = const.$ separates the Thomson-type and from the Peltier-type solutions to the differential equation. In Figure\[fig3\], the CPM and $u = s$ cooler have opposite slopes, indicating these coolers exist in separate regions of the cooling phase space. This result is consistent with our discussion above concerning the concavity of $T(x)$ in Figure\[fig5\].
For clarity, we suggest coolers which are predominately in the Thomson phase-space, $\nabla \cdot \textbf{q}_\kappa<0$ but may not have $u = s$ be referred to as “Thomson coolers”. Similarly, “Peltier coolers" should refer to coolers operating in the usual $\nabla \cdot \textbf{q}_\kappa>0$ Fourier heat divergence phase-space where Joule heating dominates.
This understanding of phase space for $u = s$ and CPM coolers enables us to hypothesize that the performance advantages of $u = s$ coolers extends to imperfect Thomson coolers. We expect such coolers possess two primary advantages over traditional Peltier coolers. First, for a given material $zT$, performance ($\Delta T_{max}$ and $\phi$) of the Thomson cooler is greater (Figure\[fig2\]). The $\Delta T_{max}$ solution for the $u=s$ cooler is compared to a Peltier cooler with the same material assumption for $z$ in Fig.\[Fig\_DeltaTlimit\]. Here, the maximum Seebeck coefficient is set by the band gap ($E_g = 1-3~eV$), per Eq.\[GoldsmidSharp\]. The Thomson cooler provides significantly higher $\Delta T_{max}$ than the Peltier cooler with the same $zT$, nearly twice the $\Delta T_{max}$ for $E_g = 3~eV$. Second, in a Thomson cooler, the temperature minimum is not limited by $zT$ explicitly like it is in a traditional Peltier coolers.
Discussion
==========
Efficiency improvements from staging and maintaining $u=s$ also exists for thermoelectric generators, but the improvement is small ($< 10\%$ compared to CPM). This is because the $u$ does not typically vary by more than a factor of two across the device. However, in a TEC the compatibility requirement is much more critical. When operating a TEC to maximum temperature difference, the temperature gradient varies from zero to very high values, which means $u$ will have a much broader range (Figure\[fig3\]) in a TEC than in a generator. Thus, unless compatibility is specifically considered, the poor compatibility will greatly reduce the performance of the thermoelectric cooler, and this results in the $\Delta T_{max}$ limit well known for Peltier coolers.
In real materials, changing material composition also changes $zT$ so the effect of maximizing average $zT$ is difficult to decouple from the effect of compatibility. As such, efforts which are focused on maximizing $zT$ will generally fail to create a material with $u = s$ and may only marginally increase $\Delta T_{max}$. Conversely, focusing on $u = s$ without consideration of $zT$ could rapidly lead to unrealistic materials requirements.
In this new analysis we have focused on the compatibility criterion, $u=s$, with constant $z$ (as opposed to $zT$ [@seifertJMR2011]) to demonstrate the differences between a Thomson and a Peltier cooler typically analyzed with the CPM model. Generally, achieving $u=s$ in a material with finite $zT$, is more important to achieve low temperature cooling than increasing $zT$.
Minor improvements in thermoelectric cooling beyond increasing average $zT$ by increasing the Thomson effect in a functionally graded material were predicted as early as 1960 [@Sherman1960]. Similarly Müller et al. describe modest gains in cooling from functionally grading [@mueller2003; @muellerpssa2006; @mueller06] where material properties are allowed to vary in a constrained way such that the average $zT$ remains constant. Such additional constraints can keep the analysis within the Peltier region, preventing a full optimization to a $u = s$ solution.
Bian et al, [@bian2006; @bian2007] propose a thermoelectric cooler with significantly enhanced $\Delta T_{max}$ using a rapidly changing Seebeck coefficient in at least one region. The method of Bian et al focuses on the redistribution of the Joule heat rather than a consideration of the Thomson heat or the effect of compatibility. Nevertheless, the region of rapidly changing Seebeck coefficient would also create a significant Thomson effect and likely place that segment of the cooler in the Thomson cooler phase space while other segments would function like a CPM Peltier cooler.
In a traditional single-stage (or segmented) thermoelectric device, the current flow and the heat flow are collinear and flow through the same length and cross-sectional area of thermoelement. This leads to the compatibility requirement between the optimal current density and optimal heat flux to achieve optimal efficiency. In a multi-stage (cascaded) device the thermal and electrical circuits become independent and so the compatibility requirement is avoided between stages.
A transverse Peltier cooler also decouples the current and heat flow by having them transport in perpendicular directions. Gudkin showed theoretically that the transverse thermoelectric cooler, could function as an infinite cascade by the appropriate geometrical shaping of the thermoelement [@Gudkin1977]. The different directions of heat and current flow enable, in principle, an adjustment of geometry to keep both heat and electric current flow independently optimized. Cooling of 23 K using a rectangular block was increased to 35 K using a trapezoidal cross-section [@Gudkin1978].
Conclusion
==========
Here, we compare self-compatible coolers with CPM and commercial Bi$_2$Te$_3$ thermoelectric coolers. Significant improvements in cooling efficiency and maximum cooling are achieved for equivalent $z$ when the cooler is self-compatible. Such improvement is most pronounced when the goal is to achieve maximum temperature difference, rather than high coefficient of performance at small temperature difference. Optimum material profiles are derived for self-compatible Thomson coolers and realistic material constraints are used to bound the performance. Self-compatible coolers are found to operate in a fundamentally distinct phase space from traditional Peltier coolers. The Fourier heat divergence of Thomson coolers is dominated by the Thomson effect, while this divergence in Peltier coolers is of the opposite sign, indicating the Joule heating is the dominant effect. This analysis opens a new strategy for solid state cooling and creates new challenges for material optimization.
We thank AFOSR MURI FA9550-10-1-0533 for support. E. S. T. acknowledges support from the U.S. National Science Foundation MRSEC program - REMRSEC Center, Grant No. DMR 0820518.
Appendix
========
The metric for summing the efficiency and coefficient of performance of these thermodynamic processes is not a simple summation because energy is continuously being supplied or removed so that neither the heat nor energy flow is a constant. The derivation, attributed to Zener[@ZenerEgli1960], of the coefficient of the performance Eq.\[Eq\_COP\] and efficiency summation metric for a continuous system in one-dimension given here is based on Zener and similar derivations given in [@Sherman1960][@Freedman1966][@Harman1968Book][@seifert06a].
Consider $n$ heat pumps (or heat engines) connected in series such that the heat entering the $i$th pump, $Q_i$, is the same as the heat exiting the $i-1$ pump, namely $Q_{i-1}$. Then by conservation of energy $$\label{Q_n}
Q_i=Q_{i-1}+P_i$$ where $P_i$ is the power entering the pump $i$. The coefficient of performance $\phi_n$ of the pump $n$ is defined by $$\label{phi_n}
\frac{1}{\phi_i} = \frac{P_i}{Q_{i-1}}$$ This gives the recursive relation $$\label{Q2_n}
Q_i=Q_{i-1}(1+\frac{1}{\phi_i})$$ which can be solved for $Q_n$ as $$\label{Q3_n}
Q_n=Q_{0}\prod_{i=1}^n (1+\frac{1}{\phi_i})$$
The total power added to the system $P$ from the $n$ pumps is $$\label{powereq1}
P = \sum_{i=1}^n P_i$$
The coefficient of performance for the $n$ pumps, $\phi$, with heat $Q_c = Q_0$ entering from the cold side is defined by $$\label{phi_totn}
\frac{1}{\phi} = \frac{P}{Q_0}$$
By conservation of energy (or recursive relation Eq \[Q\_n\]), the heat exiting the $n$ pumps, $Q_n$ is the heat pumped by the first pump plus the total power, Eq \[powereq1\]. $$\label{heateq1}
Q_n = Q_0 + P$$
Combining equations \[phi\_totn\], \[heateq1\], and \[Q3\_n\], it is straightforward to show $$\label{phi_product}
(1+\frac{1}{\phi}) = \prod_{i=1}^n (1+\frac{1}{\phi_{i}})$$ This can be transformed into a summation by use of a natural logarithm $$\label{phi_sum}
ln(1+\frac{1}{\phi}) = \sum_{i=1}^n ln(1+\frac{1}{\phi_{i}})$$
While $1/\phi_i$ should become very small with small $\delta T_i = T_i - T_{i-1}$ the reduced coefficient of performance $\phi_{r,i}$ should remain finite $$\label{phir_n}
\frac{1}{\phi_i} = \frac{1}{\phi_{r,i}} \frac{\delta T_i}{T_i}$$ Assuming a monotonic temperature distribution (for a simple TEC $T_i >T_{i-1}$ for all $i$) the sum in Eq. \[phi\_sum\] can be converted to an integral in the limit that $n\to\infty$, where $ \delta T_i = (T_h-T_c)/n$ and $ln(1+x)\to x$ for small $x$ $$\label{phi_integral}
ln(1+\frac{1}{\phi}) = \int_{T_c}^{T_h}\frac{1}{T\phi_{r}(T)}dT$$ If the temperature distribution is not monotonic but can be divided up into monotonic segments, each of these monotonic segments can be individually transformed into integrals.
For a generator (as opposed to a cooler or heat pump) power is extracted rather than added at each segment in the series. Then the sign of the power in equation \[Q\_n\] is negative for a generator with the efficiency given by $\eta = -P/Q_0 = -1/\phi$ and reduced efficiency $\eta_r = -1/\phi_r$. Thus the above method can be used to derive the analogous equation for generator efficiency: $$\label{eta_integral}
ln(1-{\eta}) = -\int_{T_c}^{T_h}\frac{\eta_{r}}{T}dT$$
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