Massive Modes for Quantum Graphs

TL;DR

This paper investigates the role of massive modes in the spectral statistics of chaotic quantum graphs, demonstrating their negligible contribution under certain conditions, thus supporting the universality of spectral correlations.

Contribution

It provides a proof that massive modes in the supersymmetry approach vanish in the large graph limit, under the assumption of a finite spectral gap.

Findings

Massive modes are shown to be negligible in the large graph limit.

The proof relies on the assumption of a finite spectral gap of the Perron-Frobenius operator.

Supports the universality of spectral two-point functions in chaotic quantum graphs.

Abstract

The spectral two-point function of chaotic quantum graphs is expected to be universal. Within the supersymmetry approach, a proof of that assertion amounts to showing that the contribution of non-universal (or massive) modes vanishes in the limit of infinite graph size. Here we pay particular attention to the fact that the massive modes are defined in a coset space. Using the assumption that the spectral gap of the Perron-Frobenius operator remains finite in the limit, we then argue that the massive modes are indeed negligible.

Equations

$${\cal B}_{bd,b^{\prime}d^{\prime}}=\Big{(}\sigma^{D}_{1}\Sigma^{(B)}\Big{)}_{bd,b^{\prime}d^{\prime}}\ .$$

$${\mathcal{F}}_{bd,b^{\prime }{d}^{\prime }}=|{\mathcal{B}}_{bd,b^{\prime }{d}^{\prime }}{|}^2$$

$${\mathcal{A}}_{\mathrm{eff}}=-\mathrm{STr}\ln (1-Z\tilde{Z})+\mathrm{STr}\ln (1-w_{+}{\mathcal{B}}_{+}Z{\mathcal{B}}_{-}^{†}{w}_{-}\tilde{Z}).$$

$${\mathcal{B}}_{\pm }=(1+j_{\pm }\frac{i\pi }{B}{\sigma }_3^s)\mathcal{B},$$

$$\left(\matrix{0&Z\cr\tilde{Z}&0\cr}\right).$$

$$Z=\left(\matrix{Z_{BB}&Z_{BF}\cr Z_{FB}&Z_{FF}\cr}\right),\quad\tilde{Z}=\left(\matrix{\tilde{Z}_{BB}&\tilde{Z}_{BF}\cr\tilde{Z}_{FB}&\tilde{Z}_{FF}\cr}\right),$$

$$Z_{BB}={\tilde{Z}}_{BB}^{\ast },Z_{FF}=-{\tilde{Z}}_{FF}^{\ast },|Z_{BB}|<1.$$

$$\mathcal{A}=-\mathrm{STr}\ln (1-Z\tilde{Z})+\mathrm{STr}\ln (1-w_{+}\mathcal{B}Z{\mathcal{B}}^{†}{w}_{-}\tilde{Z})$$

$$\displaystyle\frac{\pi^{2}}{B^{2}}\Big{(}{\rm STr}\big{[}\sigma^{s}_{3}(1-Z\tilde{Z})^{-1}Z\tilde{Z}\big{]}\,{\rm STr}\big{[}\sigma^{s}_{3}(1-\tilde{Z}Z)^{-1}\tilde{Z}Z\big{]}$$

$$\displaystyle\hskip 36.98866pt+{\rm STr}\big{[}\sigma^{s}_{3}Z(1-\tilde{Z}Z)^{-1}\sigma^{s}_{3}\tilde{Z}(1-Z\tilde{Z})^{-1}\big{]}\Big{)}\,.$$

$${\mathcal{A}}_{\mathrm{bare}}(\tilde{Z},Z)=-\mathrm{STr}\ln (1-Z\tilde{Z})+\mathrm{STr}\ln (1-\mathcal{B}Z{\mathcal{B}}^{†}\tilde{Z}).$$

$$\Lambda=\left(\matrix{1&0\cr 0&-1\cr}\right),\quad M=\left(\matrix{{\cal B}&0\cr 0&{\cal B}^{\dagger}\cr}\right).$$

$${\cal A}_{\rm bare}=-{\rm STr}\ln\frac{2}{1+Q\Lambda}+{\rm STr}\ln\bigg{(}1-M\frac{Q\Lambda-1}{Q\Lambda+1}\bigg{)}$$

$$Q$$

$$g(Z)=\left(\matrix{(1-Z\tilde{Z})^{-1/2}&Z(1-\tilde{Z}Z)^{-1/2}\cr\tilde{Z}(1-Z\tilde{Z})^{-1/2}&(1-\tilde{Z}Z)^{-1/2}\cr}\right).$$

$$g=\left(\matrix{A&B\cr C&D\cr}\right)$$

$$Z(Q)=B(g)(D(g){)}^{-1}=B(gk)(D(gk){)}^{-1},$$

$$\tilde{Z}(Q)=C(g)(A(g){)}^{-1}=C(gk)(A(gk){)}^{-1}$$

$$g_{0}=\left(\matrix{A_{0}&B_{0}\cr C_{0}&D_{0}\cr}\right)\ {\rm and}\ g_{0}g=\left(\matrix{A_{0}A+B_{0}C&A_{0}B+B_{0}D\cr D_{0}C+C_{0}A&D_{0}D+C_{0}B\cr}\right)$$

$$g_0\cdot Z(Q)=(A_{0}Z(Q)+B_0)(D_0+C_{0}Z(Q){)}^{-1},$$

$$g_0\cdot \tilde{Z}(Q)=(D_0\tilde{Z}(Q)+C_0)(A_0+B_0\tilde{Z}(Q){)}^{-1}.$$

$$Y=g_0\cdot 0=B_0{D}_0^{-1},\tilde{Y}=g_0\cdot \tilde{0}=C_0{A}_0^{-1}.$$

$$Z_{bd}=g_0\cdot {\zeta }_{bd},{\tilde{Z}}_{bd}=g_0\cdot {\tilde{\zeta }}_{bd}.$$

$$\xi =A_0\zeta D_0^{-1},\tilde{\xi }=D_0\tilde{\zeta }{A}_0^{-1}.$$

$$Z=g_0\cdot \zeta =(A_0\zeta +B_0)(D_0+C_0\zeta {)}^{-1}=(Y+\xi )(1+\tilde{Y}\xi {)}^{-1},$$

$$\tilde{Z}=g_0\cdot \tilde{\zeta }=(D_0\tilde{\zeta }+C_0)(A_0+B_0\tilde{\zeta }{)}^{-1}=(\tilde{Y}+\tilde{\xi })(1+Y\tilde{\xi }{)}^{-1}.$$

$$\sum _{bd}{\xi }_{bd}=0=\sum _{bd}{\tilde{\xi }}_{bd}.$$

$$\mathrm{STr}\ln (1-(g_0\cdot Z)(g_0\cdot \tilde{Z}))-\mathrm{STr}\ln (1-\mathcal{B}(g_0\cdot Z){\mathcal{B}}^{†}(g_0\cdot \tilde{Z}))$$

$$=\mathrm{STr}\ln (1-Z\tilde{Z})-\mathrm{STr}\ln (1-\mathcal{B}Z{\mathcal{B}}^{†}\tilde{Z}).$$

$${\mathcal{A}}_{\mathrm{bare}}(g_0\cdot \zeta ,g_0\cdot \tilde{\zeta })={\mathcal{A}}_{\mathrm{bare}}(\zeta ,\tilde{\zeta }).$$

$${\mathcal{A}}_{\mathrm{bare}}(\xi ,\tilde{\xi })=-\mathrm{STr}\ln (1-\xi \tilde{\xi })+\mathrm{STr}\ln (1-\mathcal{B}\xi {\mathcal{B}}^{†}\tilde{\xi }).$$

$$\mathcal{A}\approx {\mathcal{A}}_{\mathrm{bare}}(\xi ,\tilde{\xi })-\frac{2i\pi \kappa }{\Delta }{\mathrm{STr}}_s\frac{1}{1-Y\tilde{Y}},$$

$$\displaystyle{\rm STr}_{s}\big{[}\sigma^{s}_{3}(1-Z\tilde{Z})^{-1}Z\tilde{Z}\big{]}=-1+{\rm STr}_{s}\big{[}\Sigma(Y,\tilde{Y})Q(\xi,\tilde{\xi})\Lambda\big{]}\,,$$

Full text

1 Motivation

According to the Bohigas-Giannoni-Schmit (BGS) conjecture [1], the spectral fluctuation properties of Hamiltonian systems that are chaotic in the classical limit, coincide with those of the random-matrix ensemble in the same symmetry class (unitary, orthogonal, symplectic). Numerical simulations (see, f.i., Ref. [2]) strongly suggest that the BGS conjecture holds likewise for chaotic quantum graphs. Analytical arguments in support of the BGS conjecture for chaotic quantum graphs have been presented in several papers [3, 4, 5, 6]. All these approaches use the supersymmetry formalism and a separation of the modes of the system into the universal (or massless or zero) mode and a number of massive modes. An essential part of the argument then consists in showing that the contribution of massive modes to all correlation functions vanishes in the limit of infinite graph size (number $B$ of bonds to $\infty$). That leaves only the contribution of the zero mode, and universality of all correlation functions follows. The zero mode and the massive modes range in a non-linear space of cosets. That fact has not been addressed explicitly in previous work [3, 4, 5, 6]. The problem has not gone unnoticed. Indeed, in Ref. [7] mathematical aspects of the non-linearity of the coset space were discussed in detail without, however, establishing control of the contribution from the massive modes.

In the present paper we aim at filling that gap. We introduce the zero mode and the massive modes in a manner that is consistent with the non-linear coset structure. We do so using the strong assumption that in the limit $B\to\infty$, the spectrum of the Perron-Frobenius operator (defined in Eq. (2) below) has a finite gap that separates the eigenvalue $\lambda_{1}=1$ from the rest. In contrast, Refs. [3] and [4] pose only the weaker condition that the gap closes (for $B\to\infty$) no faster than $B^{-\alpha}$ with $0\leq\alpha<1/2$. That is perhaps a realistic requirement: Numerical simulations [2] suggest that chaotic graphs obeying Kirchhoff boundary conditions at each vertex possess universal spectral correlations even though the gap closes for large $B$. However, we doubt that, in the framework of a perturbative treatment of massive modes, there exists an easy way to turn the reasoning of Refs. [3, 4] into a convincing argument showing that the contribution of the massive modes vanishes in all orders. This is why we settle for the stronger assumption of a gap that remains finite in the limit of infinite graph size $B$. That assumption prevents us from keeping the local structure of the quantum graph fixed in the limit $B\to\infty$: To prevent the gap from closing, the graph connectivity must increase as the graph size is taken to infinity.

In Section 2 we recall some basic facts on chaotic graphs [2]. Section 3 forms the central piece of the paper. We introduce the universal mode and the massive modes in a manner that is consistent with the coset structure, and we express the effective action in terms of these variables. We use the fact that the spectrum of the Perron-Frobenius operator possesses a gap. In Section 4 we address the superintegrals over massive modes, and we argue that the contribution of these modes to the two-point correlation function vanishes in the limit of infinite graph size. We do so under rather restrictive assumptions concerning the matrix ${\cal B}$ that describes amplitude propagation on the quantum graph, see Section 2. We assume that the elements of ${\cal B}$ are all of order $1/\sqrt{2B}$, see Eq. (47). That implies that the fluctuations of the matrix elements about their mean values (defined by the unitarity of ${\cal B}$) are small (of order $1/\sqrt{2B}$). For a general proof of the BGS conjecture it would be necessary to lift that assumption.

We confine ourselves to the unitary case, to closed graphs, and to the two-point function (the correlation function of the retarded and the advanced Green’s function).

2 Two-Point Function

To set the stage, we briefly summarize previous work [2, 3, 4, 5, 6] on chaotic quantum graphs. We consider connected simple graphs with $V$ vertices and $B$ bonds. Each bond has two directions $d=\pm$. The $2B$ directed bonds are labeled $(bd)$. On every directed bond the Schrödinger wave carries the same wave number $k$ and a direction-dependent magnetic phase $\phi_{bd}$ that breaks time-reversal invariance. Hermitean boundary conditions at vertex $\alpha$ (with $\alpha=1,2,\ldots,V$) cause incoming and outgoing waves on the bonds linked to $\alpha$ to be related by a unitary vertex scattering matrix $\sigma^{(\alpha)}$. When arranged in directed-bond representation, the $V$ scattering matrices $\sigma^{(\alpha)}$ form the $(2B)$-dimensional unitary bond scattering matrix $\Sigma^{(B)}$ with elements $\Sigma^{(B)}_{bd,b^{\prime}d^{\prime}}$. Amplitude propagation within the graph depends upon the unitary matrix

[TABLE]

Here $\sigma^{D}_{1}$ is the first Pauli spin matrix in two-dimensional directional space. It flips the direction of bonds, $(\sigma^{D}_{1}\Sigma^{(B)})_{bd,b^{\prime}d^{\prime}}=\Sigma^{(B)}_{b-d,b^{\prime}d^{\prime}}$. To see how $\sigma^{D}_{1}$ arises (see Refs. [5, 6]) we consider a bond connecting vertices $\alpha$ and $\beta$. For vertex $\alpha$ the bond is denoted by $(\alpha,\beta)=(bd)$, for vertex $\beta$ the bond is denoted by $(\beta,\alpha)=(b,-d)$. The two bond directions differ. To correctly describe amplitude propagation through the graph, the bond directions must match. That is achieved by multiplying $\Sigma^{(B)}$ with $\sigma^{D}_{1}$. The graph is classically chaotic if in the limit $B\to\infty$ of infinite graph size the spectrum of the Perron-Frobenius operator, i.e., of the matrix

[TABLE]

possesses a finite gap separating the leading eigenvalue $\lambda_{1}=1$ from all other eigenvalues $\lambda_{i}$ (so that $|\lambda_{i}|\leq(1-a)$ with $a>0$). That is assumed throughout.

Unitary symmetry is realized by averaging separately and independently over the phases $\phi_{bd}$ ranging in the interval $[0,2\pi]$. The averages are carried out using the supersymmetry method and the color-flavor transformation [8]. As a result, the two-point function is written as the derivative of a generating function, an integral in superspace. That function carries in the exponent the effective action ${\cal A}_{\rm eff}$ given here in the form of Ref. [6] (see also Refs. [3, 4]),

[TABLE]

Here $w_{+}=w^{*}_{-}=\exp\{i\kappa{\cal L}\}$ where ${\cal L}$ is the diagonal matrix of bond lengths $L_{b}$, and where $\kappa$ ($-\kappa$) is the wave number increment in the retarded sector (the advanced sector, respectively). The matrices ${\cal B}_{\pm}$ are defined as

[TABLE]

with $\sigma^{s}_{3}$ the third Pauli spin matrix in superspace. The average two-point function is obtained by differentiation of the generating function with respect to $j_{+}$ and $j_{-}$ at $j_{+}=0=j_{-}$.

The supermatrices $Z=\{\delta_{bb^{\prime}}\delta_{dd^{\prime}}Z_{bd;ss^{\prime}}\}$ and $\tilde{Z}=\{\delta_{bb^{\prime}}\delta_{dd^{\prime}}\tilde{Z}_{bd;ss^{\prime}}\}$ are diagonal in directed-bond space. For fixed indices $(bd)$ the matrices $Z$ and $\tilde{Z}$ each have dimension two and form part of a supermatrix of dimension four,

[TABLE]

In Boson-Fermion block notation we have

[TABLE]

where

[TABLE]

The variable transformation $(1+j_{+}\frac{i\pi}{B}\sigma^{s}_{3})Z\to Z$, $(1+j_{-}\frac{i\pi}{B}\sigma^{s}_{3})\tilde{Z}\to\tilde{Z}$ with Berezinian unity is used to simplify the source terms and, after differentiation with respect to $j_{+}$ and $j_{-}$, yields for the effective action

[TABLE]

and for the source terms

[TABLE]

The terms (9) multiply $\exp\{-{\cal A}\}$ (Eq. (8)) under the superintegral over $(Z,\tilde{Z})$ with flat measure. That superintegral constitutes an exact representation of the average two-point function.

3 Effective Action

We introduce the universal mode and the massive modes. We express the effective action (8) and the source terms  (9) as functions of these modes.

3.1 Coset Space

We first focus attention on the bare effective action, obtained from Eq. (8) by omission of $\kappa$, i.e., by putting $w_{+}=1=w_{-}$,

[TABLE]

Using an argument of Ref. [7] we show that ${\cal A}_{\rm bare}(\tilde{Z},Z)$ is defined in a coset space. In retarded-advanced notation we define the matrices

[TABLE]

We expand the logarithms in Eq. (10), resum and obtain

[TABLE]

where

[TABLE]

and

[TABLE]

The supermatrices $Z$ and $\tilde{Z}$ are diagonal in directed bond space and so is, therefore, $Q$.

We consider a specific set of bond indices $(bd)$, omit these and work in four-dimensional superspace only. The matrix $Q$ and the bare action ${\cal A}_{\rm bare}$ remain unchanged under the transformation $g\to gk$ if $k$ commutes with $\Lambda$. Therefore, $Q$ and ${\cal A}_{\rm bare}$ are defined in the coset superspace $G/K$ where $g\in G=U(1,1|2)$ and $k\in K=U(1|1)\times U(1|1)$ with fundamental form $Q=g\Lambda g^{-1}=gk\Lambda k^{-1}g^{-1}$. Writing

[TABLE]

we find from Eq. (14)

[TABLE]

for the local coordinates $(Z,\tilde{Z})$ of $Q$ used in Eqs. (13). As briefly explained in the paragraph following Eq. (22), these and the other local coordinates introduced below are, however, not globally defined.

A group element $g_{0}\in G$ acts on $Q$ by $Q\to g_{0}Qg^{-1}_{0}$ and on $g\in G$ by left multiplication. With

[TABLE]

we obtain from Eqs. (16) for the local coordinates of $g_{0}Qg^{-1}_{0}$

[TABLE]

3.2 Zero Mode plus Fluctuations

Let $g_{0}\in G$ in Eq. (17) be independent of the directed-bond indices $(bd)$, and $Q_{0}=g_{0}\Lambda g^{-1}_{0}$. Following Eqs. (18) we define the zero mode (or universal mode) $(Y,\tilde{Y})$ by the local coordinates of $Q_{0}$,

[TABLE]

Fluctuations about the zero mode are written in the sense of Eq. (18) as

[TABLE]

Eqs. (20) transform the local variables $(Z,\tilde{Z})$ to the local variables $(Y,\tilde{Y})$ and $(\zeta,\tilde{\zeta})$. The latter are not gauge invariant. Therefore, we replace the supermatrices $(\zeta,\tilde{\zeta})$ by the gauge-invariant supermatrices $(\xi,\tilde{\xi})$ defined by

[TABLE]

Suppressing the directed-bond indices we use Eqs. (18) (with $\zeta$ for $Z(Q)$ and $\tilde{\zeta}$ for $\tilde{Z}(Q)$) to express $Z$ and $\tilde{Z}$ as given by Eqs. (20) as

[TABLE]

The transformation (22) from the variables $(Z,\tilde{Z})$ to the variables $(Y,\tilde{Y})$ and $(\xi,\tilde{\xi})$ is somewhat analogous to the transformation from independent coordinates to center-of-mass and relative coordinates. Actually, the transformation (22) is more complicated than that because it has to respect the coset structure. We very briefly remark on the ensuing difficulties. The variables for our coset space $G/K$ cannot be introduced in a globally well-defined way without using the mathematical machinery of an atlas of coordinate charts, transition functions, etc. Therefore, the gauge-independent variables $(Z,\tilde{Z})$, $(Y,\tilde{Y})$ and $(\xi,\tilde{\xi})$ are not globally defined but serve as good local coordinates. A simplification arises because we confine ourselves to small fluctuations of $(Z,\tilde{Z})$ about the “center-of-mass” coordinates $(Y,\tilde{Y})$. That is done by linearizing the transformation (20) in the new variables $(\zeta,\tilde{\zeta})$. Even in the linear regime we have to respect the coset structure, however. We are greatly helped by our assumption that the spectrum of the Perron-Frobenius operator has a sufficiently large gap. By that assumption, the fluctuations of $(Z,\tilde{Z})$ about the “center-of-mass” coset $g_{0}K$ with coordinates $(g_{0}\cdot 0,g_{0}\cdot\tilde{0})$ are small in a quantifiable sense, and the smallness allows us to treat the relative variables $(\zeta,\tilde{\zeta})$ approximately as vectors in the tangent space of $G/K$ at $g_{0}K$. Then the variables $(\xi,\tilde{\xi})$ also lie in a vector space. That is used in what follows.

Constraints are needed because only $(2B-1)$ of the variables $\zeta$ and of the variables $\tilde{\zeta}$ appearing in Eqs. (20) (or of the variables $\xi$ and $\tilde{\xi}$ appearing in Eqs. (21)) are independent. Using the assumption that $(\xi,\tilde{\xi})$ lie in a vector space we impose the constraints

[TABLE]

The transformation (22) with the constraints (23) introduces as new integration variables the coordinates $(Y,\tilde{Y})$ of the zero mode and the independent ones among the variables $(\xi,\tilde{\xi})$. The Berezinian of the transformation is unity.

3.3 Implementation of the New Variables

In order to express the effective action of Eq. (8) and the source terms (9) in terms of the modes $(Y,\tilde{Y})$ and $(\xi,\tilde{\xi})$, we derive an invariance property of the bare effective action, adapting to the present case an argument developed in Ref. [9] for a network model of the Integer Quantum Hall Effect. With the help of the definitions and results of Section 3.1 we show by explicit calculation that

[TABLE]

The invariance property (24) holds provided that $g_{0}\cdot({\cal B}Z{\cal B}^{\dagger})={\cal B}(g_{0}\cdot Z){\cal B}^{\dagger}$. We have not used any specific properties of the matrices $(Z,\tilde{Z})$. Therefore, Eq. (24) holds also for the matrices $(\zeta,\tilde{\zeta})$. We use Eq. (24) for the bare effective action of Eq. (10). We replace the variables $(Z,\tilde{Z})$ by the transformation (20) and apply the invariance property (24) to the resulting expression. That gives

[TABLE]

The matrices $A_{0}$ and $D_{0}$ in Eqs. (21) commute with ${\cal B}$. Therefore, ${\cal A}_{\rm bare}(\zeta,\tilde{\zeta})={\cal A}_{\rm bare}(\xi,\tilde{\xi})$, and we obtain

[TABLE]

We return to the full effective action (8). We are interested in values of $\kappa$ that are of the order of the mean level spacing $\Delta=\pi/\sum_{b}L_{b}$. Therefore, we expand ${\cal A}$ up to terms of first order in $\kappa$. In the first-order terms we neglect the fluctuations by putting all $\xi=0=\tilde{\xi}$. Since ${\cal B}$ commutes with the zero-mode variables $Y$ and $\tilde{Y}$ that gives

[TABLE]

with ${\cal A}_{\rm bare}$ given by Eq. (26). The index $s$ indicates that the supertrace extends only over superspace. Being proportional to ${\rm STr}[g(Y)\Lambda(g(Y))^{-1}\Lambda]={\rm STr}[Q\Lambda]$, the last term in Eq. (27) has the classical form for the symmetry-breaking term.

We turn to the source terms (9). The supermatrices $(Z,\tilde{Z})$ are diagonal in directed bond space; we consider a fixed value of $(bd)$ and omit these indices. Then both $Z$ and $\tilde{Z}$ have dimension two. We define the four-dimensional matrix $Q(\xi,\tilde{\xi})$ as in Eq. (13) but with the replacements $Z\to\xi$, $\tilde{Z}\to\tilde{\xi}$. With $\Lambda$ defined in Eqs. (11), the source terms are written as

[TABLE]

The index $s$ indicates that the supertraces extend over superspace only. The supermatrices $\Sigma$ and $\Sigma^{\prime}$ are defined as

[TABLE]

It is a very convenient feature of expression (28) that the contributions from the universal mode $(Y,\tilde{Y})$ and from the massive modes $(\xi,\tilde{\xi})$ factorize. That separation can be carried a step further. We use the decomposition $Q(\xi_{bd},\tilde{\xi}_{bd})\Lambda=[Q(\xi_{bd},\tilde{\xi}_{bd})\Lambda-1]+1$. Supersymmetry must be broken for the same integration variable in both the advanced and the retarded sector to obtain a non-vanishing result. Therefore, terms linear in $(Q\Lambda-1)$ vanish upon integration, and the contribution of the massive modes to the source terms is

[TABLE]

It is our task to show that the integrals over the terms (30) with weight factor $\exp\{-{\cal A}_{\rm bare}(\xi,\tilde{\xi})\}$ and carried out for all $(\xi,\tilde{\xi})$ vanish for $B\to\infty$.

4 Evaluation

To argue in that direction we proceed as follows. We simplify the source terms (30) by a suitable variable transformation. We expand the bare effective action in powers of the new integration variables. Terms up to second order define Gaussian superintegrals. The exponential containing terms of higher order is expanded in a Taylor series. We perform the Gaussian superintegrals. We give an approximate estimate of the dependence on $B$ of the terms so generated and on that basis argue that they vanish for $B\to\infty$.

We display the procedure for a single contribution to the source terms. The procedure applies likewise to the remaining terms without any additional difficulties and is not given here. We consider the last term in expression (30). We introduce block notation, writing

[TABLE]

and correspondingly for $\Sigma^{\prime}$ and for $Q(\xi_{bd},\tilde{\xi}_{bd})$. It is convenient to write $\mu$ for $(bd)$ and $\nu$ for $(b^{\prime}d^{\prime})$. For the pair $\Sigma_{++}$ $\Sigma^{\prime}_{--}$ the relevant contribution is

[TABLE]

The angular brackets denote the superintegration over all $(\xi,\tilde{\xi})$ with weight factor $\exp\{-{\cal A}_{\rm bare}\}$. According to Eqs. (13) we have $[Q(\xi,\tilde{\xi})\Lambda-1]_{++}=2\xi\tilde{\xi}(1-\xi\tilde{\xi})^{-1}$ and $[Q(\xi,\tilde{\xi})\Lambda-1]_{--}=2\tilde{\xi}\xi(1-\tilde{\xi}\xi)^{-1}$.

4.1 Variable Transformation

We simplify the form of the source term in expression (32) by defining for each set of directed bond indices the variable transformation

[TABLE]

with inverse transformation

[TABLE]

Calculation shows that the Berezinian of the variable transformation (33) is unity. Instead of the constraints (23) we impose

[TABLE]

To justify Eqs. (35) we observe that Eqs. (23) were introduced in an ad-hoc fashion to guarantee that only $(2B-1)$ of the variables $(\zeta_{\mu},\tilde{\zeta}_{\mu})$ and $(\xi_{\mu},\tilde{\xi}_{\mu})$ are independent. Eqs. (35) serve that same purpose.

From Eqs. (33) we have $[Q(\xi,\tilde{\xi})\Lambda-1]_{++}=2\psi\tilde{\psi}$ and $[Q(\xi,\tilde{\xi})\Lambda-1]_{--}=2\tilde{\psi}\psi$. Expression (32) becomes

[TABLE]

For the bare effective action of Eq. (26), the variable transformation (33) leads to

[TABLE]

where the dots indicate terms of higher order in $\psi$ and $\tilde{\psi}$.

4.2 Gaussian Superintegrals

In the expansion of ${\cal A}_{\rm bare}$ in Eq. (37), we retain in the exponent only terms up to second order in $(\psi,\tilde{\psi})$ (last line of Eq. (37)). With ${\cal F}$ defined in Eq. (2) these can be written as

[TABLE]

Eq. (38) defines the Gaussian part ${\cal A}_{0}$ of the bare effective action. The Perron-Frobenius operator ${\cal F}$ is expanded in terms of its complex eigenvalues $\lambda_{k}$ and left and right eigenvectors $\langle w_{k}|$ and $|u_{k}\rangle$ as

[TABLE]

These (non-real) eigenvectors satisfy the relations $\langle w_{k}|u_{l}\rangle=\delta_{kl}$. The matrix ${\cal F}$ is bistochastic, its elements are positive or zero. The graph is connected. It follows from the Perron-Frobenius theorem that there exists a non-degenerate eigenvalue $\lambda_{1}=+1$. The associated left and right eigenvectors $\langle w_{1}|$ and $|u_{1}\rangle$ have the components $(1/\sqrt{2B})\{1,1,\ldots,1\}$. All other eigenvalues $\lambda_{i}$ with $i\geq 2$ lie within or on the unit circle in the complex plane. As stated below Eq. (2), we assume that all other eigenvalues $\lambda_{i}$ with $2\leq i\leq 2B$ obey $|\lambda_{i}|\leq(1-a)$ with $a>0$ even in the limit $B\to\infty$. The matrix $|u_{1}\rangle\langle w_{1}|$ is an orthogonal projector. Eqs. (35) guarantee that $|u_{1}\rangle\langle w_{1}|$ does not contribute to the sum on the right-hand side of expression (38), confirming that the zero mode has been eliminated. We emphasize that fact by defining the complementary projector ${\cal P}=1-|u_{1}\rangle\langle w_{1}|$, and by writing expression (38) as

[TABLE]

The bilinear form ${\cal A}_{0}$ defines the propagator of the theory. The factor $\exp\{-{\cal A}_{0}\}$ defines Gaussian superintegrals. The fundamental integral is

[TABLE]

We have written $\mu=(bd)$ as before. The range of the superindices $(s,t)$ is $(0,1)$ or, equivalently, $(B,F)$.

The Taylor expansion of the exponential containing the dotted terms in expression (37) generates products of supertraces each containing powers of $\psi$ and $\tilde{\psi}$. For the Gaussian integral over the product of these with the source term in expression (36) we use the general result

[TABLE]

The sum is over all permutations of $(1,2,\ldots,n)$. The expressions generated by the Gaussian integrals (42) contain products of factors $\langle\mu|{\cal P}(1-{\cal F})^{-1}{\cal P}|\nu\rangle$ and may become very lengthy. We use the abbreviations

[TABLE]

4.3 Qualitative Estimation

The integrals (42) generate products of matrix elements $W_{\mu\nu}$. Progress hinges on our ability to estimate the dependence of these matrix elements and of sums of their products on the dimension $(2B)$ of directed-bond space for $B\to\infty$. Postponing a strict treatment to future work, we here settle for the simple approximation of using averages based upon the completeness relation.

For the diagonal elements we use Eq. (39), the relation $\langle w_{k}|u_{l}\rangle=\delta_{kl}$, and the completeness relation and find

[TABLE]

Since ${\cal F}$ is real, the eigenvalues $\lambda_{k}$ are either real or come in complex conjugate pairs. Therefore, the sum on the right-hand side is real. By assumption, the eigenvalues obey $|\lambda_{k}|\leq(1-a)$ with $a>0$. Therefore, the expression on the right-hand side is positive and for all $B$ bounded from above by $1/a$. We accordingly estimate $W_{\mu\mu}=\langle\mu|{\cal P}(1-{\cal F})^{-1}{\cal P}|\mu\rangle$ by $1/a$. (Here and in what follows we use the word “estimate” in the non-technical sense of order-of-magnitude estimate). For terms with higher inverse powers of $(1-{\cal F})$ we find correspondingly $W^{(n)}_{\mu\mu}\approx 1/a^{n}$. The factor $1/a^{n}$ stems from the sum $\sum_{k}1/(1-\lambda_{k})^{n}$. In the limit $B\to\infty$ that sum exists for all $n$ only if the gap in the spectrum of the Perron-Frobenius operator ${\cal F}$ does not close. That condition is also used in the estimates given below.

For the non-diagonal elements we have

[TABLE]

Since $\sum_{\mu}\langle\mu|{\cal P}=\sqrt{2B}\langle w_{1}|{\cal P}=0$ and $\sum_{\mu}{\cal P}|\mu\rangle=\sqrt{2B}{\cal P}|u_{1}\rangle=0$, the first term of $(1-\delta_{\mu\nu})$ gives a vanishing contribution. For the second term we use Eq. (44) and find that the typical non-diagonal element $W_{\mu\nu}$ with $\mu\neq\nu$ is of order $1/((2B)a)$. For $W^{(n)}_{\mu\nu}$ with $\mu\neq\nu$ and $n>1$ we correspondingly have $W^{(n)}_{\mu\nu}\approx 1/((2B)a^{n})$.

After integration and use of the order-of-magnitude estimate (44), the remaining terms in the expansion may carry a product of matrix elements of the form $\prod_{i=1}^{n}\langle\mu_{i}|{\cal P}(1-{\cal F})^{-1}{\cal P}|\mu_{i+1}\rangle$. In addition to the steps taken in Eqs. (44) and (45) we use the approximation $|\mu_{i}\rangle\langle\mu_{i}|\approx(1/(2B))\sum_{\mu}|\mu\rangle\langle\mu|$ for the intermediate projectors $|\mu_{i}\rangle\langle\mu_{i}|$ with $2\leq i\leq n$. That gives

[TABLE]

The relation (46) holds likewise (with appropriate changes of the power of $1/a$) in cases where one or several of the denominators $(1-{\cal F})$ carry powers larger than unity.

For an estimate of the large-$B$ dependence of expressions involving ${\cal B}$ or ${\cal B}^{\dagger}$, we use the unitarity relation $\sum_{\sigma=1}^{2B}{\cal B}^{\dagger}_{\rho\sigma}{\cal B}_{\sigma\rho}=1$. It implies that in the ergodic limit we have

[TABLE]

The estimates (42), (46) and (47) provide us with the tools needed to give an order-of-magnitude estimate of the $B$-dependence of the source terms and the terms generated by the Taylor expansion of the higher-order terms in ${\cal A}_{\rm bare}$.

4.4 Source Term

In expression (36) we first disregard contributions due to the Taylor expansion of higher-order terms in ${\cal A}_{\rm bare}$. Eq. (42) implies that the Gaussian integrals over products of factors $(\psi,\tilde{\psi})$ lead to pairwise contractions. In the source term (36) that results in the sum of two terms,

[TABLE]

According to Eq. (42) the first term vanishes, and the second term equals $W_{\mu\nu}W_{\nu\mu}\delta_{s_{1}s^{\prime}_{1}}\delta_{t_{2}t^{\prime}_{2}}$. We have $|\sum_{\nu}W_{\mu\nu}W_{\nu\mu}|\leq||WW||_{\rm op}\leq||W||^{2}_{\rm op}\leq a^{-2}$. The double bars denote the operator norm. Hence $|\sum_{\nu}W_{\mu\nu}W_{\nu\mu}|\leq(2B)/a^{2}$. The term (36) carries the factor $1/B^{2}$. For $B\to\infty$ it vanishes as $1/B$.

4.5 Higher-Order Contributions

Among the terms in expression (36) that arise from the Taylor expansion of higher-order terms in ${\cal A}_{\rm bare}$ in Eq. (37), we first address terms that originate from ${\rm STr}\ln(1+\psi\tilde{\psi})$. Except for numerical factors the general term in the Taylor expansion has the form

[TABLE]

with integer $m\geq 1,n_{i}\geq 2$. Gaussian integration as in Eq. (42) leads to pairwise contraction of all factors $\psi$ and $\tilde{\psi}$ and, thus, to $k=2+\sum_{i}n_{i}$ factors $W$. A non-vanishing result is obtained only if supersymmetry is violated in every supertrace in expression (49). Therefore, all supertraces in expression (49) must be linked by pairwise contractions. That, incidentally, is the reason why in Eq. (48) the first term on the right-hand side vanishes.

We consider two examples. For $m=1,n_{1}=2$, the nonvanishing links yield two terms,

[TABLE]

In the first expression we use $W_{\rho\rho}\approx 1/a$, $W^{(3)}_{\rho\rho}\approx 1/a^{3}$. The sum over $\rho$ yields a factor $(2B)$. Altogether the term is estimated as $1/((2B)a^{4})$ and vanishes for $B\to\infty$. In the second expression we use $W^{(2)}_{\rho\rho}\approx 1/a^{2}$ and find the same result. For $m=2,n_{1}=2=n_{2}$ we correspondingly obtain the terms

[TABLE]

We use the estimates (44, 46). The terms of leading order are the ones where two factors $W^{(n)}$ with $n=1$ or $n=2$ or $n=3$ each carry identical indices. These are the terms number 1, 3, 5, 6. They vanish asymptotically as $1/(2B)$.

From these examples we deduce the following rules. (i) In expression (49), contraction of the four factors $(\psi_{\mu},\tilde{\psi}_{\mu})$ and $(\psi_{\nu},\tilde{\psi}_{\nu})$ with one another and with corresponding factors in the product over traces yields a nonvanishing result only if it generates either two factors $W^{(2)}$ or a single factor $W^{(3)}$. These carry indices of the factors in the product over traces. For every pair $(\rho,\sigma)$, the factors $W^{(2)}_{\rho\sigma}$ and $W^{(3)}_{\rho\sigma}$ are, both for $\rho=\sigma$ and for $\rho\neq\sigma$, of the same order in $1/(2B)$ as the factor $W_{\rho\sigma}$. For purposes of counting powers of $(2B)$ it suffices, therefore, to disregard the four factors $(\psi_{\mu},\tilde{\psi}_{\mu})$ and $(\psi_{\nu},\tilde{\psi}_{\nu})$ together with the summation over $\mu$ and $\nu$ and to consider only contractions within the product of supertraces. (ii) The terms of leading order in $1/(2B)$ are obtained by contracting, in every supertrace ${\rm STr}(\psi\tilde{\psi})^{n_{i}}$, $(n_{i}-1)$ pairs $(\psi_{\rho},\tilde{\psi}_{\rho})$ with one another. That is because contraction of pairs of $\psi$’s in the same (in different) supertraces generates factors $W_{\rho\rho}\propto 1/a$ ($W_{\rho\sigma}\propto 1/(2Ba)$ with $\rho\neq\sigma$, respectively). Contraction of pairs within the same supertrace generates $(n_{i}-1)$ factors $W_{\rho\rho}$ and reduces each supertrace to $(1/a^{n_{i}-1}){\rm STr}(\psi\tilde{\psi})$. These rules imply that the only remaining contractions in expression (49) are over the term

[TABLE]

To the extent that our approximations (of replacing fluctuating quantities by their averages) capture the qualitative aspects of the problem, we may conclude that expression (49) is of order $1/(2B)$ and vanishes for $B\to\infty$.

The general term in the Taylor expansion contains, in addition to the terms in expression (49), also products of supertraces that contain the elements of ${\cal B}$. Each such supertrace has the form ${\rm STr}[{\cal B}\psi(\tilde{\psi}\psi)^{m}{\cal B}^{\dagger}\tilde{\psi}(\psi\tilde{\psi})^{n}]^{l}$ with $l\geq 1$ and $(m,n)$ zero or positive integer, see Eq. (37). In contracting factors $\psi$ and $\tilde{\psi}$ we apply rule (ii). The terms of leading order are obtained by replacing in each supertrace $(\tilde{\psi}\psi)^{m}$ by $1/a^{m}$ and $(\psi\tilde{\psi})^{n}$ by $1/a^{n}$. We proceed likewise for the product of supertraces in expression (49). That reduces the general expression to

[TABLE]

Here $m\geq 0$ and $l_{i}\geq 1$ for $i=1,\ldots,m$. We require $k\geq 1$ as the case $k=0$ has been considered above. We apply rule (i) and disregard the difference between factors $W^{(n)}$ and $W$ in estimating the dependence of expression (53) on $1/(2B)$. In other words, we confine ourselves to contractions involving elements $(\psi,\tilde{\psi})$ in the curly brackets in expression (53). For $m=0$, all supertraces carrying elements of ${\cal B}$ must be connected by pairwise contractions. That generates $\sum_{i}l_{i}$ factors $W$. Each such factor carries a pair of summation indices. No two summation indices in any of these factors are the same. Therefore, each factor $W$ is of order $1/(2B)$. There are $\sum_{i}l_{i}$ elements of ${\cal B}$ and of ${\cal B}^{\dagger}$, each of order $1/\sqrt{2B}$. There are $2\sum_{i}l_{i}$ independent summations over directed-bond space. Together with the prefactor $1/B^{2}$, we therefore expect expression (53) to be of order $1/(2B)^{2}$ for $m=0$.

That expectation remains unaltered for $m>0$. We show that first for $m=1$, writing ${\rm STr}(\psi\tilde{\psi})=\sum_{\rho}{\rm STr}_{s}(\psi_{\rho}\tilde{\psi}_{\rho})$. Contraction links the factor $\psi_{\rho}$ (the factor $\tilde{\psi}_{\rho}$) with some factor $\tilde{\psi}_{\sigma}$ (with some factor $\psi_{\tau}$, respectively). Both $\tilde{\psi}_{\sigma}$ and $\psi_{\tau}$ occur in the product over $i$ in expression (53). The result is $\sum_{\rho}W_{\rho\sigma}W_{\tau\rho}=W^{(2)}_{\tau\sigma}$ while for $m=0$ contracting $\tilde{\psi}_{\sigma}$ and $\psi_{\tau}$ gives $W_{\tau\sigma}$. Since $W_{\tau\sigma}$ and $W^{(2)}_{\tau\sigma}$ are of the same order, the expressions (53) with $m=0$ and with $m=1$ are of the same order, too. The argument can straightforwardly be extended to $m\geq 2$. This concludes our heuristic reasoning that the general term containing matrix elements of ${\cal B}$ vanishes for $B\to\infty$ as $1/(2B)^{2}$.

5 Summary and Discussion

We have given a brief account of an approach to chaotic quantum graphs that aims at demonstrating the BGS conjecture using supersymmetry and the color-flavor transformation. We have paid particular attention to the treatment of the massive modes as these are defined in a coset space. We have used the assumption that the spectrum of the Perron-Frobenius operator possesses a finite gap even for infinite graph size. We have shown that the effective action and the source terms are given by Eqs. (26) and (27) and by Eq. (30), respectively.

To evaluate the resulting generating function, we have defined Gaussian superintegrals by expanding the effective action up to terms of second order. The remaining terms are expanded in a Taylor series. We have carried out the Gaussian superintegrals over the product of that series with the source terms. We have given order-of-magnitude estimates of the resulting expressions using averages based upon completeness and unitarity. Assuming that these approximations are valid, we are led to the conclusion that the contribution of massive modes to the two-point function vanishes for large graph size. Therefore, that function attains universal form.

The rough estimates in Section 4.3 are based on averages and require small fluctuations. That may be unsatisfactory. We are working on strict estimates. We hope to be able soon to report on these, combining that with a more detailed account of conceptual and technical aspects of the steps taken in Section 3 that were treated here only cursorily.

The author is much indebted to M. R. Zirnbauer. Without his numerous useful suggestions, especially concerning the developments in Section 3, this contribution would not have come into existence.