Superbosonization in disorder and chaos: The role of anomalies

TL;DR

This paper develops a rigorous supermatrix approach for calculating fermionic integrals using supersymmetry, with applications to disordered systems and localization phenomena.

Contribution

It introduces a supermatrix representation of superfield integrals and specifies contours, extending superbosonization to systems with anomalies and disorder.

Findings

Derived a supermatrix representation for superfield integrals.

Successfully calculated correlation functions in random matrix models.

Argued applicability to nonperturbative analysis of disordered systems.

Abstract

Superbosonization formula aims at rigorously calculating fermionic integrals via employing supersymmetry. We derive such a supermatrix representation of superfield integrals and specify integration contours for the supermatrices. The derivation is essentially based on the supersymmetric generalization of the Itzikson-Zuber integral in the presence of anomalies in the Berezinian and shows how an integral over supervectors is eventually reduced to an integral over commuting variables. The approach is tested by calculating both one and two point correlation functions in a class of random matrix models. It is argued that the approach is capable of producing nonperturbative results in various systems with disorder, including physics of many-body localization, and other situations hosting localization phenomena.

Equations

$$Q^2(\mathbf{r})=1.$$

$$F\left[Q\right]=\frac{\pi \nu }{8}\int Str\left[D{\left(\nabla Q\right)}^2+2i\left(\omega +i\delta \right)Q\left(\mathbf{r}\right)\right]d\mathbf{r}$$

$$\int D\psi D\bar{\psi }F(\psi \otimes \bar{\psi })={\int }_{{\mathbb{H}}_n}D\mathcal{A}\text{Sdet}{\mathcal{A}}^{-1}F(\mathcal{A}).$$

$$\Gamma \left[\{{\tilde{b}}_j,b_j,\}\mid \{{\tilde{\lambda }}_j,\lambda {}_j\}\right]$$

$$\Gamma \left[\{{\tilde{b}}_j,b,\}\mid \{{\tilde{\lambda }}_j,{\lambda }_j\}\right]=\left[1-\eta \left(\{{\tilde{\lambda }}_i,{\lambda }_i\}\right)\right]$$

$$\displaystyle\times\;\frac{\prod_{i}\delta({\tilde{b}_{i}})\delta(b_{i})}{\Delta^{2}\bigl{(}\{{\tilde{b}_{j}}^{2},b_{j}^{2}\}\bigr{)}}+\Gamma_{0}\left[\{{\tilde{b}_{j},}b_{j}\}\mid\{{\tilde{\lambda}_{j},}\lambda_{j}\}\right].$$

$${\Gamma }_0\left[\{{\tilde{b}}_j,b_j\}\mid \{{\tilde{\lambda }}_j,{\lambda }_j\}\right]$$

$$=\frac{1}{2^{2n^2}(n!{)}^2}\frac{\det J_0[{\tilde{b}}_p{\tilde{\lambda }}_q{]}_{p,q=1\dots n}\det {J_0[b_l{\lambda }_m]}_{l,m=1\dots n}}{\Delta \left(\{{{\tilde{b}}_j}^2,b_j^2\}\right)\Delta \left(\{{{\tilde{\lambda }}_j}^2,{\lambda }_j^2\}\right)},$$

$$\Delta \left(\{{{\tilde{b}}_j}^2,b_j^2\}\right)$$

$$\eta \left(\{{\tilde{\lambda }}_i,{\lambda }_i\}\right)$$

$$\int DB\exp \left\{\frac{1}{2t}\text{Str}[(B-Q_d{)}^2]\right\}=$$

$$I_F=\int D\psi D\bar{\psi }F(\psi \otimes \bar{\psi })$$

$$I_F={\int }_{{\mathbb{H}}_n}D\mathcal{A}F(\mathcal{A})\int D\psi D\bar{\psi }{\int }_{{\mathbb{H}}_n}DB$$

$$\times \exp \left\{i\text{Str}[\mathcal{A}B]-i\sum _j\text{Str}[{\psi }_j\otimes {\bar{\psi }}_{j}B]-\delta \text{Str}[B^2]\right\},$$

$$\int D\psi D\bar{\psi }\exp \left\{-i\sum _j\text{Str}[{\psi }_j\otimes {\bar{\psi }}_{j}B]\right\}$$

$$I_B={\int }_{{\mathbb{H}}_n}DB\text{Sdet}[iB]e^{i\text{Str}[\mathcal{A}B]},$$

$$\text{Sdet}B=\frac{\text{Sdet}{B}^{\prime }}{\text{Sdet}\mathcal{A}},$$

$$I_B$$

$$I_F={\int }_{{\mathbb{H}}_n}D\mathcal{A}\text{Sdet}{\mathcal{A}}^{-1}F(\mathcal{A}).$$

$$\mathcal{P}(H)=\prod _{i,j=1}^N{P}_{ij}(H_{ij}),$$

$$P(H_{ij})=\frac{1}{2\pi A_{ij}}\exp \left\{-\frac{H_{ij}{H}_{ij}^{\ast }}{2A_{ij}}\right\}.$$

$$Z(J_1\dots J_n)$$

$${\mathcal{M}}_{i,j}^J$$

$${\mathcal{H}}_j^J$$

$$L$$

$$\hat{s}$$

$$⟨Z(J_1\dots J_n)⟩=\int D\psi D\bar{\psi }$$

$$\times \exp \left\{i\sum _i{\bar{\psi }}_i{\mathcal{H}}_i^J{\psi }_i-\frac{1}{2}\sum _{i,j}{A}_{i,j}({\bar{\psi }}_i{\psi }_j)({\bar{\psi }}_j{\psi }_i)\right\},$$

$${\bar{\psi }}_i{\mathcal{H}}_i^J{\psi }_i=\text{Str}\left[{\psi }_i\otimes {\bar{\psi }}_i\left(-E+\frac{\omega +i0}{2}\Lambda -J_i\right)\right],$$

$$({\bar{\psi }}_j{\psi }_i)({\bar{\psi }}_i{\psi }_j)=-\text{Str}({\psi }_i\otimes {\bar{\psi }}_i{\psi }_j\otimes {\bar{\psi }}_j),$$

$$⟨Z(J_1\dots J_n)⟩={\int }_{{\mathbb{H}}_n}\prod _{i}D{Q}_i\text{Sdet}[Q_i{]}^{-1}$$

$$\times \exp \left\{i\sum _i\text{Str}[Q_i{\mathcal{H}}_i^J]-\frac{1}{2}\sum _{i,j}{A}_{i,j}\text{Str}[Q_i{Q}_j]\right\},$$

$$G^{R/A}(E_1\dots E_n)=\frac{1}{{\pi }^n}⟨\prod _{i=1}^n\text{Tr}\left[\frac{1}{E_i-H\pm i0}\right]⟩$$

$$⟨Z_0(J_1\dots J_n)⟩={\int }_{{\mathbb{H}}_n}\prod _{i}D{Q}_i\text{Sdet}[Q_i{]}^{-1}$$

Full text

I Introduction

Supersymmetryberezin1 ; berezin2 deals with Grassmann numbers, that were originally invented in mathematics and later used in quantum field theory as the classical analogues of anticommuting operators. This mathematical construction is proven to be a very useful tool for studies in various fields of physics and in particular in models of quantum chaos, involving random matrix theory and various models of disorder.Efetov1 ; Efetov2 ; VWZ

One of the prominent methods employing supersymmetry is the non-linear supersymmetric $\sigma$-modelEfetov1 ; Efetov2 description of disordered metallic conductors. According to this standard formalism, effective field theory is described by an action with coordinate dependent supermatrix field, $Q(\mathbf{r})$, obeying the constraint,

[TABLE]

This method has a broad range of applications including study of Anderson localization, mesoscopic fluctuations, levels statistics in a limited volume, quantum chaos. A general form of the free energy functional $F$ is rather simple

[TABLE]

containing the classical diffusion coefficient $D$, the one particle density of states $\nu$ and frequency $\omega$. Although the free energy $F\left[Q\right],$ Eq. (2), is written in the limit of a weak disorder, it can be used for a strongly disordered samples replacing the gradient by finite differences. At very low energies the effective free energy functional is dominated in a finite volume by the zero spatial mode, $Q(\mathbf{r})=Q_{0}$, which is independent of $\mathbf{r}$. In this limit the model is especially simple containing only the second term in Eq. (2). The spectral properties of the theory are universal and coincide with those of Wigner-Dyson random matrix ensembles with corresponding symmetriesEfetov1 ; Efetov2 ; VWZ ; ABG ; CB .

The derivation of the $\sigma$-model, Eq. (2), from microscopic models is not exact and is based on a saddle point method applicable at weak disorder or the large size of the matrices in the Wigner-Dyson ensembles. At the same time, the “diffusive” $\sigma-$model, Eq. (2) is not applicable for, e.g. description of electron motion in ballistic regime, where characteristic spatial scales are much smaller than the mean free path. Another important problem, that is known to be out of the reach of nonlinear $\sigma$-model, is random matrix models with finite range correlations between the matrix elements that are do not belong the Wigner-Dyson ensembles. One of the examples are models of weakly non-diagonal matrices.OK1 ; OK2 . Of course, there are many other models that cannot be reduced to the $\sigma$-model, Eq. (2).

In many of those models correlation functions of interest can be expressed from the beginning in terms of integrals over supervectors and the problem arise due to absence of a possibility of using the saddle-point approximation leading to Eqs. (1) and (2). Therefore, it is natural to try to generalize the $\sigma$-model, Eq. (2) to a model containing supermatrices but without the constraint, Eq. (1). In such a model, the generating functional $Z(J)$ would be expressed in terms of an integral over unconstrained supermatrices, and having calculated this integral, one would be able to compute correlation functions of interest. It should be noticed here that usually many-level or many point correlation functions are really interesting. One-level or one-point averages (average density of states) usually do not bring an interesting information about the systems (in the problem of Anderson localization, the average density of states cannot help to distinguish between the metal and insulator).

In principle, some dual representations of a generating functional, initially given by an integral over $N\times N$ Hermitian matrices (color space), are known as color-flavor transformationsZirnC_F . They transform the original integral in “color space” to an integral over certain super-manifolds, which are acting in the dual space (flavor space). However, being interesting on its own, this transformation has not yet evolved into a new computational tool.

Trying to find a new method of studying non-standard problems of the supersymmetry method a kind of bosonization procedure to the original fermionic functional $Z(J)$ has been suggested some time ago E ; EK . As a result, the partition function has been represented in a supermatrix action formulation without any constraint what soever; this approach was claimed to be applicable to the physics of electron motion at all scales. It seemed that the limitations due to the non-linearity of the conventional $\sigma$-model representation were overcame. Nevertheless the formula of superbosonization was not well understood from the practical point of view, namely the integration method was not specified.

More precisely, in Ref. E, a new superbosonization formula that allowed the field-integral over supervectors be expressed through a supermatrix integral has been derived,

[TABLE]

where ${\mathbb{H}_{n}}$ is the linear space of $n$-dimensional complex supermatrices, $\psi\in U(n,1|n,1)$ and ${\bar{\psi}}\in U(n,1|n,1)$ are supervectors, and $F:{\mathbb{H}}_{n}\rightarrow{\mathbb{G}}$ is a formal map with ${\mathbb{G}}$ representing a superspace.footnote Importantly, the right-hand-side of Eq. (3) could be evaluated under general conditions, without reducing it to any mean-field manifold. For this reason, it was suggested that Eq. (3) could be capable of producing non-perturbative results in various models of disorder. One can imagine that Eq. (3) can represent a promising approach for non-perturbative studies in physics of many-body localizationMBL ; Altland and other situations where disorder plays an important role.general

OriginallyE , Eq. (3) has been derived rather schematically without discussing contours of integration over the commuting elements of the supermatrix ${\mathcal{A}}.$ An attempt to specify contours of integration has been undertaken in Ref. YEK, . Roughly, speaking it was suggested to integrate over the eigenvalues of the boson-block from $-\infty$ to $\infty,$ while the integration over the eigenvalues of the fermion-fermion block has to be performed over a compact domain (a circle in the simplest case). Surprisingly, it turned out that such an integration was well defined only in rather uninteresting cases. In particular, it worked perfectly well for correlation functions that required a sufficiently small number $q\leq n$ of the bosonic components, where $n$ was a number of artificial “orbitals”. In other words, one could use Eqs. (3) for calculation of the density of states in case of the unitary ensemble, while one encountered a singularity of the type $\infty\times 0,$ when trying to calculate a two-level correlation functions. The situation for, e.g. orthogonal ensemble was even worse and one could not calculate even the density of states in this case. The situation was better when using a sufficiently large number of the “orbitals” $n$ but this could be efficiently closer to results obtained using the standard saddle-point method and therefore less interesting.

These findings have been confirmed rigorously in Ref. Littelmann, but the case $q>n$ was not resolved and it was even concluded that the superbosonization formula, Eq. (3), was not correct for this case. This was a serious obstacle in using the superbosonization for applications to interesting unsolved problems.

In this paper we resolve this long standing problem of the integration in Eq. (3) for the case of hermitian matrices with an arbitrary correlation between the matrix elements. Of course, the suggested approach can be used for disordered systems with a broken time-reversal invariance. We do it integrating over the eigenvalues of the fermion-fermion block along the imaginary axis from $-i\infty$ to $i\infty$ instead of the integration along the circle adopted in Refs. YEK, ; Littelmann, . This does not make a difference in the results for $q\leq n$ but it makes the integral, Eq. (3), well defined for $q>n$ and computation of many point correlation functions feasible, thus establishing a new method of calculations for interesting problems.

The paper is organized as follows. In Section II we set the basis for the subsequent analysis of the bosonization procedure of Ref. E, by calculation of a supersymmetric generalization of Itzykson-Zuber (IZ) integral. In Section III we show how the formulated supermatrix representation of integrals over supervectors (the so called bosonized representation) can be evaluated. In particular, we derive the domains of integration, for which the bosonization formula is exact. It is remarkable that this regularized scheme leads to an effective reduction of dimensionality of the domain of integration, which is noncompact.

The proof is essentially based on the results discussed in Section II: the supersymmetric generalization of the Itzykson-Zuber (IZ) integral, IZ ; G ; S ; AM ; GW ; TS in situations when a boundary term is crucial due to the presence of singularities in the Berezinian. Emergence of this boundary term in the IZ integral ensures that both representations of the generating functional coincide.

In Section IV we apply the regularized superbosonization formula to calculation of correlation functions in random matrix models. We derive both one and two point correlation functions for Hermitian diagonal random matrices with continuously distributed components and correction to the density of states for weakly non-diagonal random matrices.OK1 ; OK2 Technical details of some of the derivations are presented in Appendices A, B, and C.

II Anomaly in supersymmetric Itzykson-Zuber integral

II.1 Supersymmetric Itzykson-Zuber integral

In this section we present useful formulae, that will be applied in subsequent sections. Let us note that in all future considerations the integration over the linear space of complex supermatrices, $\int_{\mathbb{H}_{n}}D{\mathcal{A}}$, with flat Berezin measureberezin1 ; berezin2 is always performed first by diagonalizing the matrix ${\mathcal{A}}$ and then by integrating over the eigenvalues. We distinguish between “fermion-fermion (FF)” and “boson-boson (BB)” blocks of the matrix ${\mathcal{A}}$ corresponding respectively to products $\psi_{F}\otimes{\bar{\psi}}_{F}$ and $\psi_{B}\otimes{\bar{\psi}}_{B}$ of anticommuting and commuting components of the supervectors. After the diagonalization of the supermatrix ${\mathcal{A}}$ one half of the eigenvalues will be in the FF-block, and the other part will be in the BB-block. We will call these eigenvalues FF- and BB-eigenvalues respectively.

We will demonstrate that the integration over the BB-eigenvalues should be performed in the infinite interval ${{\mathbb{R}}}\equiv\{-\infty,\infty\}$, while the integration over the FF-eigenvalues should be performed in the infinite interval $\{-i\infty,i\infty\}$. This contrasts the integration rules of Refs. YEK, ; Littelmann, , where the integration over the FF-eigenvalues was performed along the unit circle. Note, that any complex $2n\times 2n$ supermatrix, ${\mathcal{A}}$, can be diagonalized as ${\mathcal{A}}=U{\mathcal{A}}_{d}{\bar{V}}$, where $U\in U(n\mid n)$, ${\bar{V}}\in{U(n\mid n)}/{U^{2n}(1)}$ are diagonalization matrices restricted correspondingly to the unitary supergroup and its subspace with removed phases.

Here we are interested in Itzykson-Zuber integral of the type

[TABLE]

where $B_{d}={\text{diag}}\{\tilde{b}_{i},b_{i}\}$ and $Q_{d}={\text{diag}}\{\tilde{\lambda},\lambda,\}$ are the FF and BB eigenvalues of supermatrices $B$ and $Q$ respectively and $U\in U(n\mid n)$, ${\bar{V}}\in{U(n\mid n)}/{U^{2n}(1)}$. Then, the result of integration reads G ; AM ; GW

[TABLE]

Here $\Gamma_{0}\left[\{{\tilde{b}_{j},}b_{j}\}\mid\{{\tilde{\lambda}_{j},}\lambda_{j}\}\right]$ is the result of the bulk integration without accounting for the singularity in the Berezinian (if there is such). It has the form

[TABLE]

where

[TABLE]

is the supersymmetric Vandermonde determinant and $J_{0}[b_{p}\lambda_{q}]$ is the zero-order Bessel function. The term $\eta\left(\{{\tilde{\lambda}_{i},}\lambda_{i}\}\right)$ is the boundary term arising from the singularities of the Berezinian (This type of the boundary term in the integrals over supermatrices has been found in Refs. level-level, ; Efetov1, and is sometimes called Efetov-Wegner boundary termVerbaarschot1 ; Kieburg ). It originates from the regularization of the anomaly in the Berezinian and is given by GW

[TABLE]

One can easily check, that the expression (7) for IZ integral boundary terms, $\Gamma_{0}\left[\{{\tilde{b}_{j},}b_{j}\}\mid\{{\tilde{\lambda}_{j},}\lambda_{j}\}\right]$, does not fulfill Eq. (II.2) (see below). The singularity of the Berezinian $\Delta^{2}(\{{\tilde{b}_{j}}^{2},b_{j}^{2}\})$ in (II.2) gives rise to the appearance of the boundary term in \Gamma\Bigl{[}\{{\tilde{b}_{j},}b_{j}\}\mid\{{\tilde{\lambda}_{j},}\lambda_{j}\}\Bigr{]}.

II.2 Origin of the boundary term in the Itzykson-Zuber integral

Our aim in this section is to underline the origin of the anomaly of the Berezinian and the implication for the supersymmetric Itzykson-Zuber integral. For this purpose for any given diagonal complex supermatrix $Q_{d}$ consider the Gaussian integral,

[TABLE]

where we have the diagonalized complex supermatrix $B\rightarrow B_{d}=\text{diag}\{b_{i},\bar{b}_{i}\}$. In Eq. (II.2) $\Delta^{2}(\{{\tilde{b}_{j}}^{2},b_{j}^{2}\})$ is the Berezinian of the transformation, $B=UB_{d}{\bar{V}}$, where $U\in U(n\mid n)$ and ${\bar{V}}\in{U(n\mid n)}/{U^{2n}(1)}$.

The integral, Eq. (13), is originally gaussian and integrating separately over all matrix elements of the supermatrix $B$ gives unity. It is clear that changing the variables of the integration cannot modify this result and one must obtain unity also integrating over the eigenvalues. However, one can easily check, that the ”naive” expression for Itzykson-Zuber integral $\Gamma_{0}\left[\{{\tilde{b}_{j},}b_{j}\}\mid\{{\tilde{\lambda}_{j},}\lambda_{j}\}\right]$, Eq. (6, 7), is not equal to unity. It is the singularity of the Berezinian $\Delta^{2}(\{{\tilde{b}_{j}}^{2},b_{j}^{2}\})$ in Eq. (II.2), that gives rise to the appearance of boundary term $\eta\left(\{{\tilde{\lambda}_{i},}\lambda_{i}\}\right)$ Eq. (II.1), in $\Gamma\left[\{{\tilde{b}_{j},}b_{j}\}\mid\{{\tilde{\lambda}_{j},}\lambda_{j}\}\right]$ , that was found in Ref. GW, . Existence of this boundary term ensures the condition that the integral Eq. (II.2) is unity. Hence, the correct answer for supersymmetric Itzykson-Zuber integral $\Gamma$ has the form Eq. (5).

The following remark is in order. The result, $\Gamma_{0}\left[\{{\tilde{b}_{j},}b_{j}\}\mid\{{\tilde{\lambda}_{j},}\lambda_{j}\}\right]$, of the evaluation of the supersymmetric IZ integral in the absence of singularities was derived by solving the supersymmetric heat equationG ; AM ; technique, that was developed in Ref. IZ, for conventional matrices. It is straightforward to check that the boundary term $\propto(1-\eta)$ in Eq. (5) also satisfies the heat equation.

III Superbosonization: proof and integration contours

In this section we present a derivation of the superbosonization formula and, in particular, of the bosonized $\sigma$-model for random matrices. The derivation is similar to the procedure developed in Refs. Efetov1, ; Efetov2, ; G, ; VWZ, , but here instead of the Hubbard-Stratonovich transformation, which in the standard scheme follows the averaging over random matrices, we use the identities from the above section. Actually, the scheme of the derivation is very close to that of Ref. E, but is more rigorous. It is useful to recall, that formal sums of formal products $\Psi\otimes{\bar{\Psi}}$, where $\Psi\in U(n,1|n,1)$ and ${\bar{\Psi}}\in U(n,1|n,1)$ are supervectors, constitute a vector space. This vector space is defined, up to isomorphism, by the condition that every antisymmetric, bilinear map $f:U(n,1|n,1)\times{\bar{U}}(n,1|n,1)\rightarrow{\mathbb{G}}$ determines a unique linear map $g:U(n,1|n,1)\otimes{\bar{U}}(n,1|n,1)\rightarrow{\mathbb{G}}$ with $f(\Psi,{\bar{\Psi}})=g(\Psi\otimes{\bar{\Psi}})$. This implies that if we consider a map, $F:{\mathbb{H}}_{n}\rightarrow{\mathbb{G}}$, then the integral

[TABLE]

is now well defined. From now on we will restrict ourselves to the case of maps, $F$, such that the integral $I_{F}$ in Eq. (10) is convergent.

As the first step we make use of the identity derived in Appendix A to rewrite the field integral in the left-hand-side of Eq. (3) as

[TABLE]

where $\delta$ is an infinitely small variable that ensures the convergence of the integral over the variable $B$ in Eq.(11); it can be dropped once the integral over $B$ is convergent. Now, due to the convergence of the integral in Eq. (11) and the presence of $\delta$, we are free to change the order of the integration over the supermatrix $B$ and the supermatrices $\psi_{i}\otimes{\bar{\psi}_{i}}$. The integration over the supervectors $\psi,\bar{\psi}$ leads to

[TABLE]

Then, the integral over $B$ acquires the form

[TABLE]

where we dropped $\delta$ due to the convergence of the integral Eq. (13). The integral Eq. (13) can be calculated by changing the integration variable $B$ to $B^{\prime}={\mathcal{A}}B$. Supermatrix $B^{\prime}$ is not necessarily Hermitian, however it obeys the constraint ${\text{Str}}[B^{\prime}]={\text{Str}}\left[\left(B^{\prime}\right)^{\dagger}\right]$. By definition $B^{\prime}$ is an element of the vector space $\Lambda^{2}({\mathbb{H}}_{n})$ (for definition see Appendox B). Taking into account the fact that due to the supersymmetry the Berezinian of the transformation $B^{\prime}={\mathcal{A}}B$ is unity and

[TABLE]

we obtain

[TABLE]

The coefficient $C_{n}$ is calculated in Appendix B, yielding $C_{n}=1$. As a result, one arrives at the bosonized representation for the integral Eq. (11)

[TABLE]

To finalize this section we remind the reader that in Eq. (16) the integration over the linear space of Hermitian supermatrices, $\int_{\mathbb{H}_{n}}D{\mathcal{A}}$, with Berezin measure is understood here as follows: (i) First we diagonalize the matrix ${\mathcal{A}}$ and then integrate over the eigenvalues. (ii) Integration over “boson-boson” eigenvalues is performed in the infinite interval $\{-\infty,\infty\}$, whereas the integration over the “fermion-fermion” eigenvalues is performed (in contrast to Refs. YEK, ; Littelmann, ) in the non-compact interval $\{-i\infty,i\infty\}$. In this way, the integral in Eq. (3) over supervectors is reduced to an integral over commuting variables. It is worth mentioning that the presence of $\text{Sdet}{\mathcal{A}}^{-1}$ in Eq. (3) leads to a singular product $\prod_{i}\tilde{\lambda}_{i}^{-1}$, which make the integral very sensitive to the contour of the integration over the FF-eigenvalues $\tilde{\lambda}_{i}$.

Representing the integral over supervectors in terms of an integral over the supermatrices is more than just changing the variables of the integration. Usually, the term bosonization is used for the procedure of a replacement of an electron model by a model describing collective bosonic excitations. For example, the traditional $\sigma$-model describes so called diffusion modes instead of electrons in a random potential. As our transformation is exact and is based on the supersymmetry, we find it proper to use the word “superbosonization” for the transformation, Eq. (3), complemented by the rules of the integration over the eigenvalues of the supermatrices.

IV Superbosonization of random matrices: Correlation functions

In this section we develop a technique for calculation of various correlation functions in Random Matrix Theory (RMT), such as the averaged density of states, level-level correlations, eigenfunction correlations and higher order correlation functions. For this purpose, without loss of generality, we consider an ensemble of $N$-dimensional Hermitian matrices, $H=\left\{H_{ij}\right\}$, with continuously distributed components. For simplicity let us concentrate on the Gaussian probability density function,

[TABLE]

with the distribution functions $P_{ij}$ ($i,j=1\dots N$) equal to

[TABLE]

Then Eqs. (17), (18) unambiguously define statistical properties of the matrix entries as $\langle H_{ij}\rangle=0$, $\langle H_{ii}^{2}\rangle=A_{0}$, and $\langle H_{ij}^{2}\rangle=A_{ij}$ for $i\neq j$. The Wigner-Dyson unitary ensemble is obtained putting $A_{ij}=const$ independent on $i,j.$

IV.1 Correlation functions in the superbosonized representation:

General framework

We begin with the generating functional for n-point correlation functions

[TABLE]

with the matrix ${\mathcal{M}}_{i,j}^{J}$ is defined as

[TABLE]

In Eq. (LABEL:GF) $\psi_{i}$ are supervectors with $n$ bosonic and $n$ fermionic components, the source terms , $J_{i}$ ($i=1\ldots n$), in Eq. (20) are real parameters multiplied by diagonal $2n\times 2n$ matrices, $\hat{s}$, which break the fermion-boson (FB) symmetry. Parameter $E$ stands for the energy and $\omega$ is the frequency. The $2n$-dimensional supermatrices $L$, $\Lambda$ and $\hat{s}$ are defined as

[TABLE]

with $n$-dimensional unity matrix, $id_{n}$, and $n$-dimensional diagonal matrix, $\hat{k}=diag(1,-1)$. Purpose of introducing the matrix, $\hat{k}$, is that it distinguishes between the advanced and the retarded (A/R) Green functions.

To derive the supersymmetric action for RMT, one has to perform averaging in the generating functional, $Z(J_{1}\ldots J_{n})$, over realizations of the entries of the random matrix, $H$. Carrying out such an averaging with the probability distribution defined in Eqs. (17, 18), one obtains

[TABLE]

where we have defined ${\bar{\psi}_{i}}=\psi_{i}^{+}L$. At this point we note that for the constituent terms of the action (expressions in exponent), Eq. (29), the following identities hold

[TABLE]

The crucial step towards calculation of the correlation functions in Gaussian random matrix theory under consideration, is the evaluation of the super-integrals in Eq. (30) from their superbosonized representation Eq. (3). With the help of the superbosonization formula Eq. (3) we can represent the generating functional, $\langle Z(J)\rangle$, in the form

[TABLE]

where each of the integrals over the linear space of complex Hermitian supermatrices, ${\mathbb{H}_{n}}$, should be performed first diagonalizing matrices ${Q}_{i}$ and then integrating over their eigenvalues. As was mentioned in Inroduction, integration over BB-eigenvalues is performed along the real axis, $(-\infty,\infty)$, whereas integration over FF-eigenvalues is performed along the imaginary axis, $(-i\infty,i\infty)$.

In conclusion of this subsection we note that the derivatives of the averaged generating functional, $\left\langle Z(J_{1}\ldots J_{n})\right\rangle$, taken at zero source, $J=0$, define the advanced and retarded Green functions in RMT VWZ ; G . The $n$-point Green functions can be expressed via the derivatives of $\left\langle Z(J_{1}\ldots J_{n})\right\rangle$ functional in a standard way,

[TABLE]

which define the universal characteristics of RMT. As usual, the sign “$+$” in the denominator corresponds to the retarded Green function $G^{R}$, while the sign “$-$” to the advanced one $G^{A}.$

IV.2 Correlation functions for diagonal random matrices.

Let us first show how the method developed here works for diagonal random matrices. Although this case is not the most interesting one, it allows one to understand how the method works. We remind the reader that the conventional non-linear $\sigma$-model Efetov1 ; Efetov2 ; VWZ is not applicable in this case.

For diagonal random matrices we have $A_{ij}=0$ for $i\neq j$, and thus the averaged generating functional Eq. (31) acquires the form

[TABLE]

where supermatrices ${\mathcal{H}}_{i}^{J}$ are given by Eq. (20), $\hat{s}$ is given by Eq. (25) and $Q_{i}$ are Hermitian supermatrices with $n$ bosonic and $n$ fermionic entries.

Calculation of $\left\langle Z_{0}(J_{1}\ldots J_{n})\right\rangle$ can be performed in a similar way, as the calculation of $C_{n}$ in Appendix B. Namely, first we diagonalize the supermatrices $Q_{i}$, and afterwards perform IZ-type integration. Since the supermatrices $Q_{i}$ in Eq. (33) are Hermitian, they can be diagonalized upon the rotation by the elements of the unitary supergroup, $SU(n\mid n)$. Substituting transformation $Q=UQ_{d}U^{+}$, where $U\in SU(n\mid n)$, into Eq. (33) we arrive to the following form of the generating functional

[TABLE]

where the integration over bosonic eigenvalues of $Q_{i}$, $\lambda_{i,\alpha}$ $(\alpha=1,\cdots n)$, should be carried out along the real axis, $(-\infty,\infty)$, and the integration over fermionic eigenvalues, $\tilde{\lambda}_{i,\alpha}$ $(\alpha=1,\cdots n)$, should be carried out along the imaginary axis, $(-i\infty,i\infty)$. The infinitesimally small terms $\pm i0$ in Eq. (34) arise after removing $\pm i0$ from $H_{i}^{J}$ in Eq. (33) by shifting the variable of the integration $Q$. Berezinian, $\Delta^{2}\left(\{{\tilde{\lambda}_{j,\alpha},}\lambda_{j,\alpha}\}\right)$, is the Jacobian of the diagonalization given by

[TABLE]

It is transparent that the zero order generating functional Eq. (34) has a factorized form and can be represented as

[TABLE]

where

[TABLE]

We see that the calculation of the generating functional for diagonal random matrices reduces to the calculation of the IZ integral. This integral can be calculated employing the result of Section II for the unitary supergroup, G ; GW $U\in SU(n,n)$:

[TABLE]

where the components $h_{\alpha}$ and ${\tilde{h}}_{\alpha},\;(\alpha=1\cdots n)$ are BB and FF eigenvalues of ${\mathcal{H}}_{d}^{J},$ respectively, and the $\Delta$-functions are defined by Eq. (35). The boundary term, $\eta$, is given by Eq. (II.1) and reads

[TABLE]

Here, the matrix \mu_{\alpha\beta}=\mu\bigl{(}{\tilde{h}_{\beta},}h_{\alpha}\bigr{)} is given by

[TABLE]

Now, with the help of the IZ integral, Eq. (IV.2), we can perform integration over $U$ and $U^{+}$, namely the parameter space of the unitary supergroup, in the expression for $\mathcal{Z}_{0}(J)$, Eq. (37). Then, taking into account the determinant form of the super-Vandermonde determinant Eq. (35), we obtain

[TABLE]

With this expression for $\mathcal{Z}_{0}(J)$ we are ready to calculate one and two point (both level-level and eigenfunction-eigenfunction) correlation functions for Gaussian ensemble of unitary diagonal random matrices. These calculations are presented in the next two subsections.

IV.2.1 Density of states for diagonal random matrices

The averaged density of states is expressed in terms of the imaginary part of the one-point Green function, $G^{A}(E)$, as follows

[TABLE]

The function $G^{A}(E)$ is related to the averaged generating functional via Eq. (32). Employing the factorization property Eq. (36) for one point Green function, one is led to evaluate the integral in Eq. (IV.2) for $n=1$, which means that all the supermatrices are two dimensional and thus have one bosonic and one fermionic eigenvalue. Then the Bosonic eigenvalue of the supermatrix ${\mathcal{H}}_{d}^{J}$ will have the form $h=E+\omega+J$, while the fermionic eigenvalue will have the form ${\tilde{h}}=E+\omega-J$. Without loose of generality we can set $\omega=0$.

For one point Green function one has to take a derivative of the generating functional, $G_{0}^{A}(E)=(1/2\pi)\partial\left\langle Z_{0}(J)\right\rangle/\partial J|_{J=0}$, which, as follows from Eqs. (32)-(33), can be equivalently represented as $G_{0}^{A}(E)=i\langle{\text{Str}}[\hat{s}Q]\rangle$. For $n=1$ the supersymmetric Vandermonde determinant simplifies and acquires the form $\Delta(h,{\tilde{h}})=1/(h-{\tilde{h}})={1}/(2J)$. From here one can easily realize, that only ${1}/{\Delta({\tilde{h},}h)}$ term in Eq. (IV.2) contributes into the derivative in the $J=0$ limit. Therefore, the expression for one point Green function takes the form

[TABLE]

Evaluation of the integral in Eq. (47), presented in Appendix C, leads to

[TABLE]

where erfi$\left(x\right)$ is the imaginary error function

[TABLE]

Eq. (48) exactly reproduces the averaged advanced Green function of the Gaussian unitary ensemble of diagonal random matrices (see for example Refs. OK1, ; OK2, ). Substituting Eq. (48) into Eq. (46) we find the density of states $\rho_{0}(E)$ for diagonal random matrices,

[TABLE]

IV.2.2 Two point correlation function for diagonal random matrices

In this subsection we show how the superbosonization formula with the flat integration measure, as defined above, works for four-dimensional supermatrices. Namely, we employ the developed technique of superbosonized generating functional Eqs. (31), (32) for calculation of a two-level correlation function of diagonal random matrices. For simplicity we will concentrate on a level-level correlation function having the following form

[TABLE]

where $\mathbf{N}$ is the size of the matrices. We are aware of the fact that the correlation function $K_{0}^{A}(E_{1},E_{2})$ containing the product of two advanced Green functions is not the most interesting function characterizing the level correlations. However, the computation of this function presented here serves merely as a demonstration of how the method works. We emphasize that the method of integration adopted in Refs. YEK, ; Littelmann, does not work when applied to this problem.

For the case of diagonal random matrices the two-point function Eq. (51) can be derived upon evaluating Eqs. (31), (32). This can be done making use of the factorization property Eq. (36) with $\mathcal{Z}_{0}(J)$ given by (IV.2). The calculation is straightforward. Since the Vandermonde determinant, \Delta\bigl{(}\{{\tilde{h}_{\beta},}h_{\alpha}{\}\bigr{)}}, in Eqs. (39) and (IV.2) is always inverse proportional to the source terms, $J_{1}$ and $J_{2}$, it is easy to see, that only the \left[\Delta\bigl{(}\{h_{\alpha},{\tilde{h}_{\beta}\}\bigr{)}}\right]^{-1} term will contribute to double derivative in Eq. (32) taken at $J_{1}=J_{2}=0$. The double derivative of the Vandermonde determinant is equal to

[TABLE]

Therefore, we have for the function $K_{0}^{A}(E_{1},E_{2})$

[TABLE]

where \mu\bigl{(}\{{\tilde{h}_{\beta},}h_{\alpha}\}\bigr{)} is defined by Eq. (IV.2) and $h_{1,2}={\tilde{h}}_{1,2}=E_{1,2}$.

Analysis of the integral over $\lambda$ and $\tilde{\lambda}$ under the second determinant in Eq. (53) is presented in Appendix C. Result of the integration can be represented as the sum, \tilde{\eta}\bigl{(}\{h_{\alpha},{\tilde{h}_{\beta}}\}\bigr{)}+G_{0}(\{h_{\alpha},{\tilde{h}_{\beta}}\}), where

[TABLE]

Substituting now Eqs. (39) and (54) into the determinants in Eq. (53), we come to the result

[TABLE]

where the function $G_{0}^{A}(E)$ is given by Eq. (48). The result for the level-level correlation function for diagonal random matrices, Eq. (55), coinciding with Eq. (51), together with Eq. (48) agrees with the one found in Ref. Bovier, .

IV.3 Non-diagonal contributions to the density of states for almost

diagonal matrices

In order to show how the superbosonization technique works for less trivial random matrix theories, we calculate in this section a correction to the density of states in the model of almost diagonal matricesOK1 ; OK2 up to the second order in the bandwidth, $b$. By definition, statistical properties of non-diagonal matrices are described by a single, always positive function, $\mathcal{F}(r)$, as $A_{ij}=b^{2}\mathcal{F}\left(|i-j|\right),\;\;i\neq j$. Function $\mathcal{F}(r)$ can adopt any form provided that it has a maximum at the center of the band, $r=0$, and decays with the bandwidth, $b$, as $r$ becomes large. For small $b$ we have the ensemble of almost diagonal random matrices, while for large $b$ we approach the Wigner-Dyson Gaussian Unitary Ensemble (GUE).

We consider the case of $b\ll 1$ when the standard non-linear $\sigma$-model is not applicable. Then, expanding the exponent in Eq. (31) in $b$, we have

[TABLE]

where zero order in $b^{2}$ contribution, $\langle Z_{0}(J)\rangle$, corresponds to diagonal random matrices considered in the previous subsection. Technically, calculation of the correction, $b^{2}\langle Z_{1}(J)\rangle$, is similar to that of $\langle Z_{0}(J)\rangle$. It is determined by the form of $A_{ij}$ for almost diagonal matrices as follows

[TABLE]

where, as usual, integration goes over the linear space ${{\mathbb{H}_{n}}}$ with the flat measure. Then, the correction to the advanced Green function, $b^{2}G_{1}^{A}(E)$, is expressed in terms of the correction, $b^{2}\langle Z_{1}(J)\rangle$, to the averaged generating functional,

[TABLE]

In Eq. (IV.3) the averaging, $\langle\ldots\rangle$, is defined as

[TABLE]

Averaging in the right hand side of Eq. (IV.3) can be performed using the identity, $\langle Q\rangle=\left\langle\frac{1}{2}{\text{Str}}[{\hat{s}}Q]\right\rangle{id}_{2n}$. Next, we make use of this identity to represent the average in Eq. (IV.3) as

[TABLE]

where, according to Eq. (59), we have

[TABLE]

As described above, now again, one has to diagonalize the supermatrix $Q$ and reduce the expression Eq. (57) to IZ integral. For that purpose, we first notice that the only difference between the expressions for $\langle{\text{Str}}[Q]\rangle^{J}$ and $Z_{0}(J)$ is the presence of the term ${\text{Str}}[Q]$ under integral, which, after diagonalization for the one-point Green functions ($n=1$ case), produces an additional $\lambda-{\tilde{\lambda}}$ term under the integral in Eq. (IV.2). Secondly, the boundary $1-\eta$ term does not contribute here, because the presence of $\delta$ functions in Eq. (IV.2) together with $\lambda-{\tilde{\lambda}}$ in the integral makes it zero. Repeating now the calculation for $G_{0}(E)$ and keeping in mind the two observations above, one finds

[TABLE]

Substituting now Eqs. (IV.3) and (IV.3) into the expression for the first order correction to the Green function $G_{1}(E)$, Eq. (IV.3), we obtain

[TABLE]

where $\mathcal{R}_{N}[\mathcal{F}]\equiv 2\sum_{l=1}^{N}\mathcal{F}(l)$. Then, for the first order in $b^{2}$ correction to density of states, $\rho_{1}(E)=\pi^{-1}\text{Im}G_{1}^{A}(E)$, one easily finds

[TABLE]

which exactly reproduces the results first obtained with the help of the virial expansionOK1 ; OK2 .

V Outlook

We have presented a new scheme of computations using the superbosonization formula, Eq. (3), first proposed in Ref. E, . We have proven that this formula is exact and have given a precise recepy for the performing integration for many point correlation functions for the unitary ensemble. In contrast to a previous study YEK ; Littelmann the integration over the eigenvalues in the fermion-fermion block of the supermatrices is performed from $-i\infty$ to $i\infty$ and not along a circle. This way of the integration has allowed us to obtain regular integrals and calculate them in several cases.

The proof of our approach and proposed method of computation of the integrals is heavily based on the supersymmetric extension of the Itzykson-Zuber integral. This integral in known only for systems with broken time-reversal symmetry (unitary ensemble) and this why we consider here only such systems. At the same time, the proposed method of the integration over the eigenvalues of the supermatrices when one integrates over the eigenvalues in the boson-boson block from $-\infty$ to $\infty$ and over the eigenvalues in the fermion-fermion block from $-i\infty$ to $i\infty$ looks very general. This encourages us to make a guess that this way of the integration can also be used for time reversal invariant ensembles. Of course, such a guess must be checked and proven in the future.

We have demonstrated that the application of the bosonization formula to random band matrix (RBM)Fyodorov1 ; MPhysRep ; FM ; FM1 ; Cuevas ; YO models with small bandwidth $b$ reproduces the perturbative expansion of DOS obtained by virial expansion OK1 . We have also computed the simplest two-point correlation function containing a product of two advanced Green functions for the ensemble of diagonal matrices. Of course, calculating an average product of both retarded and advanced Green functions would be a more interesting task but we leave it for future study. It is important at the moment that our method allows us to calculate many-point correlations functions for cases where the way of the integration developed in Refs. YEK ; Littelmann does not work. We have made comparison with the known results only for checking our approach and demonstration of details of the computation.

Eq. (3) complemented by our recipe of the integration is exact and most general representation of the integrals over supervectors in terms of integrals over supermatrices. The traditional non-linear $\sigma$-model, Eqs. (1) and (2), can be obtained using the saddle-point approximation for calculation of the integral over the supermatrix $Q$ is less general. Taking into account a success of the latter in solving numerous problems (see, e.g. Ref. Efetov1, ) we believe that its generalization can also bring new interesting results.

Finally, we would like to mention that to this point disordered systems have been actually successfully studied using supersymmetric $\sigma$-model (including statistical properties of the energy levels in small metallic disordered grains), and we mostly focused here on a field theory for random matrix ensembles and non-perturbative effects therein. Another field of great interest of course is the non-perturbative study of various correlation functions in strongly interacting systems. Examples of such systems that potentially can be studied non-perturbatively using superbosonization include among others (i) the field theory of many-body localization in random spin chainsAltland ; (ii) quantum phase transitions at the boundary of topological superconductors in two and three dimensions, which have been argued to support supersymmetry at long distances and timesgrover .

Acknowledgements.

T.A.S. acknowledges startup funds from UMass Amherst. K.B.E. gratefully acknowledges the financial support of the Ministry of Education and Science of the Russian Federation in the framework of Increase Competitiveness Program of NUST “MISiS” (Nr. K2-2014-015).

Appendix A Integral over ${\mathbb{H}}_{n}$

Let ${\mathbb{H}}_{n}$ is the linear space of Hermitian $2n\times 2n$ supermatrices. Then for all ${\mathcal{A}}\in{\mathbb{H}}_{n}$ the convergent integral,

[TABLE]

taken over ${\mathbb{H}}_{n}$ with Berezin measure satisfies the condition

[TABLE]

Moreover, for any map, ${\mathcal{F}}:{\mathbb{H}}_{n}\rightarrow{\mathbb{G}}$, that converges exponentially (or faster), the identity

[TABLE]

always holds.

To derive identities (65) and (66) for ${\hat{\Upsilon}}({\mathcal{A}})$, one can first formally perform integration in the definition Eq. (64) of ${\hat{\Upsilon}}({\mathcal{A}})$, under the limit. This integration yields

[TABLE]

where the limit is well defined. With the help of Eq. (67), the right hand side of Eq. (66) can be represented as follows

[TABLE]

where we have Tailor expanded the function ${\mathcal{F}}({Q+\mathcal{A}})$ around $\mathcal{A}=0$. We note, that such an expansion exists due to the specific constraints on the function $\mathcal{F}$, outlined in Section III. To finalize our proof, it is left to show that

[TABLE]

Eq. (69) is proven by introducing the generating functional,

[TABLE]

and observing that

[TABLE]

Appendix B Calculation of $C_{n}$

Let the formal sums of Hermitian super-bivectors (product of two supermatrices, each of them being from ${\mathbb{H}}_{n}$) constitute a vector space $\Lambda^{2}({\mathbb{H}}_{n})$ called the second exterior power of ${\mathbb{H}}_{n}$. Then the integral

[TABLE]

over the vector space $\Lambda^{2}({\mathbb{H}}_{n})$ is unity, $C_{n}=1$.

According to the proposed prescription, one evaluates integral Eq. (72) first by diagonalizing the supermatrix $B^{\prime}$. As already stated, any given complex $2n\times 2n$ supermatrix $B^{\prime}$ can be diagonalized by the transformation $B^{\prime}=UB_{d}^{\prime}{\bar{V}}$, where $U\in U(n\mid n)$, ${\bar{V}}\in{U(n\mid n)}/{U^{2n}(1)}$. Substituting this transformation into Eq. (13) we obtain an integral over the eigenvalues, $B_{d}^{\prime}$, and diagonalization ”angles” $U$ and $\bar{V}$; the latter integral is nothing but supersymmetric Itzikson-Zuber IZ integral.

In order to evaluate integral in Eq. (III), we consider the following generalized integral

[TABLE]

which coincides with Eq. (B) in the case when $Q_{d}$ is an identity matrix. Before setting $Q_{d}=id$, first let us note that for any complex $2n\times 2n$ supermatrix of the following diagonal form:

[TABLE]

where $id_{n}$ is the $n\times n$ identity matrix, the $\eta$ term has the form

[TABLE]

The term, $\Gamma_{0}\left[\{b_{j},{\tilde{b}_{j}}\}\mid x,y\right],$ [see Eq. (5)] corresponding to the matrix $\Lambda$ vanishes. This is because the Vandermonde determinant, $\Delta_{\Lambda}(x,y)$, in the denominator will cancel one of determinants involving Bessel function in the nominator. However the next determinant, which is equal to zero, remains. Thus, we see that if our $2n$ dimensional complex supermatrix $Q_{d}=id$ (which means $x=y$ above), then the corresponding $\eta$ term $\eta_{id}(1,1)=\left(1-e^{-{\frac{1-1}{2t}}}\right)^{n}=0$. Therefore, from Eq.(5), we obtain

[TABLE]

Substituting Eq. (78) into Eq. (B), where as $Q_{d}$ a unity matrix is taken with $t=1$, one obtains

[TABLE]

The last equality holds, since our integration contours are shifted by an infinitesimal $\delta$ and $i\delta$ with respect to the imaginary and real axis correspondingly. This completes the computation of $C_{n}$.

Appendix C Evaluation of double-integrals

Here we will evaluate the following integral

[TABLE]

Making use of the decoupling

[TABLE]

we represent the double-integral, ${\mathcal{I}}(h,\tilde{h})$, as the sum

[TABLE]

where

[TABLE]

In the following two subsections we will evaluate integrals ${\mathcal{I}}_{1}(h,\tilde{h})$ and ${\mathcal{I}}_{2}(h,\tilde{h})$ respectively.

C.1 Calculation of ${\mathcal{I}}_{1}$

In order to evaluate ${\mathcal{I}}_{1}(h,\tilde{h})$ we recall that

[TABLE]

where symbol P denotes the principal value of the integral. Then for ${\mathcal{I}}_{1}(h,\tilde{h})$ we have

[TABLE]

with

[TABLE]

The presence of the principle value in Eq. (86) insures the possibility of bringing the integral to the Gaussian form first by taking the derivative over $\tilde{h}$:

[TABLE]

Then the function $\tilde{\mathcal{I}}_{1}$ itself will have the form

[TABLE]

with $C=(A_{0}/2\pi)\exp(h^{2}/2A_{0})\tilde{\mathcal{I}}_{1}(A_{0},h,0)$. On the other hand we have that

[TABLE]

suggesting $C=0$. Substituting Eq. (88) into Eq. (85) we obtain

[TABLE]

C.2 Calculation of ${\mathcal{I}}_{2}$

Introducing new variable, $\lambda^{\prime}=-i\tilde{\lambda}$, we rewrite integral ${\mathcal{I}}_{2}(h,\tilde{h})$ as

[TABLE]

As the next step we shift variables $\lambda$ and $\lambda^{\prime}$ by $ih/A_{0}$ and $\tilde{h}/A_{0}$ respectively. Then Eq. (91) will acquire the form

[TABLE]

It is convenient to evaluate integral Eq. (92) after switching to polar coordinates, $\lambda-i\lambda^{\prime}=ue^{i\theta}$:

[TABLE]

and integrate first over $\theta$ and only then over $u$. Result reads

[TABLE]