Statistics of energy level and eigenfunctions in disordered and chaotic systems: Supersymmetry approach (Q2760016)

The paper gives a detailed account of applications of the supersymmetric approach to the calculation of statistical properties of spectra and eigenfunctions of disordered quantum systems. The supersymmetric approach utilizes anticommuting (Grassmann) variables (in the physical literature often named ``fermionic variables'') to calculate Green's functions for systems with random Hamiltonians. The author starts his tutorial exposition with elementary examples of calculating statistical properties (the density of states and level correlation functions) for the standard ensembles of random matrices -- the Gaussian ensembles of hermitian matrices (i.e. ensembles of hermitian matrices with independently Gauss-distributed matrix elements), as well as for some modified examples: sparse and banded hermitian matrices. The author proceeds with applications of the presented formalism to the problem of the motion of a quantum particle in a random potential and derives a field-theoretical model (the so called nonlinear \(\sigma\)-model) as a low frequency approximation to the original problem. A large part of the paper is devoted to the study of statistical properties of eigenfunctions for various disordered systems also in terms of the corresponding \(\sigma\)-models. The last chapter outlines applications of the formalism to problems of, so called, quantum billiards in which a quantum particle moves freely in a prescribed domain with hard walls -- the paradigmatic models of ``quantum chaos'', i.e. quantum system the classical limits of which are non-integrable.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00065].