Supersymmetry in Disorder and Chaos

Abstract

Notes by S. Ribault Here ǫ is a function, for instance ǫ(p) = p 2 2m −ǫ F where ǫ F is the Fermi energy. And U r is a random potential. As we are looking for universal results, we may as well assume that U r has a Gaussian law

The supersymmetric technique for random-matrix ensembles with zero eigenvalues

“…The reader may also refer to Ref. [2] for conventions regarding manipulations with supermatrices. The Lie superalgebras of OSp(n|2m) and GL(n|m) are denoted as osp(n|2m) and gl(n|m).…”

“…Then, for the convergence of the integral (32), the matrix Q must satisfy Im Q B > 0 at infinity on the contour Γ [Q B is the bosonic diagonal element, as shown in (25)]. Besides, the contour Γ must be compact in the fermionic and non-compact in the bosonic sector (see, e.g., [1,2,23]). It is shown in [1] that Γ is the Riemannian symmetric superspace SpO(2|2)/GL(1|1) (class CI|DIII).…”

“…In parameterizing the saddle-point manifold we use the usual trick of splitting the rotation of the supermatrix Q i into the two rotations by even and odd generators of the supergroup [2]. Namely, we parameterize…”

“…It has been shown by Zirnbauer that in the large-N limit (N is the matrix dimension) correlation functions in random-matrix ensembles may be represented as integrals over appropriate Riemannian symmetric superspaces (with dimensions independent of N ) [1]. This relation to symmetric superspaces is based on Efetov's supersymmetric technique introducing auxiliary anticommuting (Grassmann) variables in order to directly average correlation functions over the statistical ensemble [2].…”

“…It has been shown by Zirnbauer that in the large-N limit (N is the matrix dimension) correlation functions in random-matrix ensembles may be represented as integrals over appropriate Riemannian symmetric superspaces (with dimensions independent of N ) [1]. This relation to symmetric superspaces is based on Efetov's supersymmetric technique introducing auxiliary anticommuting (Grassmann) variables in order to directly average correlation functions over the statistical ensemble [2].At the same time, the random-matrix ensembles are known to be in one-to-one correspondence with symmetric spaces (Cartan symmetry classes) [3,4]. The classification of Zirnbauer thus establishes a correspondence between large families of symmetric spaces and Riemannian symmetric superspaces [1].…”