Universality for 1d Random Band Matrices: Sigma-Model Approximation

Abstract

The paper continues the development of the rigorous supersymmetric transfer matrix approach to the random band matrices started in (J Stat Phys 164:1233–1260, 2016; Commun Math Phys 351:1009–1044, 2017). We consider random Hermitian block band matrices consisting of $W\times W$ random Gaussian blocks (parametrized by $j,k \in \Lambda =[1,n]^d\cap \mathbb {Z}^d$) with a fixed entry’s variance $J_{jk}=\delta _{j,k}W^{-1}+\beta \Delta _{j,k}W^{-2}$, $\beta >0$ in each block. Taking the limit $W\rightarrow \infty$ with fixed n and $\beta$, we derive the sigma-model approximation of the second correlation function similar to Efetov’s one. Then, considering the limit $\beta , n\rightarrow \infty$, we prove that in the dimension $d=1$ the behaviour of the sigma-model approximation in the bulk of the spectrum, as $\beta \gg n$, is determined by the classical Wigner–Dyson statistics.

Appendix

1.1 Grassmann integration

Let us consider two sets of formal variables $\{\psi _j\}_{j=1}^n,\{\overline{\psi }_j\}_{j=1}^n$, which satisfy the anticommutation conditions

$$\begin{aligned} \psi _j\psi _k+\psi _k\psi _j=\overline{\psi }_j\psi _k+\psi _k\overline{\psi }_j=\overline{\psi }_j\overline{\psi }_k+ \overline{\psi }_k\overline{\psi }_j=0,\quad j,k=1,\ldots ,n. \end{aligned}$$

(7.1)

Note that this definition implies $\psi _j^2=\overline{\psi }_j^2=0$. These two sets of variables $\{\psi _j\}_{j=1}^n$ and $\{\overline{\psi }_j\}_{j=1}^n$ generate the Grassmann algebra $\mathfrak {A}$. Taking into account that $\psi _j^2=0$, we have that all elements of $\mathfrak {A}$ are polynomials of $\{\psi _j\}_{j=1}^n$ and $\{\overline{\psi }_j\}_{j=1}^n$ of degree at most one in each variable. We can also define functions of the Grassmann variables. Let $\chi$ be an element of $\mathfrak {A}$, i.e.

$$\begin{aligned} \chi =a+\sum \limits _{j=1}^n (a_j\psi _j+ b_j\overline{\psi }_j)+\sum \limits _{j\ne k} (a_{j,k}\psi _j\psi _k+ b_{j,k}\psi _j\overline{\psi }_k+ c_{j,k}\overline{\psi }_j\overline{\psi }_k)+\ldots . \end{aligned}$$

(7.2)

For any sufficiently smooth function f we define by $f(\chi )$ the element of $\mathfrak {A}$ obtained by substituting $\chi -a$ in the Taylor series of f at the point a. Since $\chi$ is a polynomial of $\{\psi _j\}_{j=1}^n$, $\{\overline{\psi }_j\}_{j=1}^n$ of the form (7.2), according to (7.1) there exists such l that $(\chi -a)^l=0$, and hence the series terminates after a finite number of terms and so $f(\chi )\in \mathfrak {A}$.

Following Berezin [2], we define the operation of integration with respect to the anticommuting variables in a formal way:

$$\begin{aligned} \displaystyle \int d\,\psi _j=\displaystyle \int d\,\overline{\psi }_j=0,\quad \displaystyle \int \psi _jd\,\psi _j=\displaystyle \int \overline{\psi }_jd\,\overline{\psi }_j=1, \end{aligned}$$

and then extend the definition to the general element of $\mathfrak {A}$ by the linearity. A multiple integral is defined to be a repeated integral. Assume also that the “differentials” $d\,\psi _j$ and $d\,\overline{\psi }_k$ anticommute with each other and with the variables $\psi _j$ and $\overline{\psi }_k$. Thus, according to the definition, if

$$\begin{aligned} f(\psi _1,\ldots ,\psi _k)=p_0+\sum \limits _{j_1=1}^k p_{j_1}\psi _{j_1}+\sum \limits _{j_1<j_2}p_{j_1,j_2}\psi _{j_1}\psi _{j_2}+ \ldots +p_{1,2,\ldots ,k}\psi _1\ldots \psi _k, \end{aligned}$$

then

$$\begin{aligned} \displaystyle \int f(\psi _1,\ldots ,\psi _k)d\,\psi _k\ldots d\,\psi _1=p_{1,2,\ldots ,k}. \end{aligned}$$

Let A be an ordinary Hermitian matrix with positive real part. The following Gaussian integral is well-known

$$\begin{aligned} \displaystyle \int \exp \Big \{-\sum \limits _{j,k=1}^nA_{jk}z_j\overline{z}_k\Big \} \prod \limits _{j=1}^n\dfrac{d\,\mathfrak {R}z_jd\,\mathfrak {I}z_j}{\pi }=\dfrac{1}{\mathrm {det}A}. \end{aligned}$$

(7.3)

One of the important formulas of the Grassmann variables theory is the analog of this formula for the Grassmann algebra (see [2]):

$$\begin{aligned} \int \exp \Big \{-\sum \limits _{j,k=1}^nA_{jk}\overline{\psi }_j\psi _k\Big \} \prod \limits _{j=1}^nd\,\overline{\psi }_jd\,\psi _j=\mathrm {det}A, \end{aligned}$$

(7.4)

where A now is any $n\times n$ matrix.

We will also need the following bosonization formula

Proposition 7.1

(see [13]) Let F be some function that depends only on combinations

$$\begin{aligned} \bar{\phi }\phi :=\Big \{\sum \limits _{\alpha =1}^W \bar{\phi }_{l\alpha }\phi _{s\alpha }\Big \}_{l,s=1}^2, \end{aligned}$$

and set

$$\begin{aligned} d\Phi =\prod \limits _{l=1}^2\prod \limits _{\alpha =1}^W d\mathfrak {R}\phi _{l\alpha } d\mathfrak {I}\phi _{l\alpha }. \end{aligned}$$

Assume also that $W\ge 2$. Then

$$\begin{aligned} \int F\left( \bar{\phi }\phi \right) d\Phi =\dfrac{\pi ^{2W-1}}{(W-1)!(W-2)!}\int F(B)\cdot \mathrm {det}^{W-2} B \,dB, \end{aligned}$$

where B is a $2\times 2$ positive Hermitian matrix, and

$$\begin{aligned} dB&=\mathbf {1}_{B>0}dB_{11}dB_{22}d\mathfrak {R}B_{12} d\mathfrak {I}B_{12}. \end{aligned}$$