Nonlinearσmodel for long-range disorder and quantum chaos

Abstract

We suggest a new scheme of derivation of a non-linear ballistic σ-model for a long range disorder and quantum billiards. The derivation is based on writing equations for quasiclassical Green functions for a fixed long range potential and exact represention of their solutions in terms of functional integrals over supermatrices Q with the constraint Q 2 = 1. Averaging over the long range disorder or energy we are able to write a ballistic σ-model for all distances exceeding the electron wavelength (Eq. 3.13). Neither singling out slow modes nor a saddle-point approximation are used in the derivation. Carrying out a course graining procedure that allows us to get rid off scales in the Lapunov region we come to a reduced σ-model containing a conventional collision term. For quantum billiards, we demonstrate that, at not very low frequencies, one can reduce the σ-model to a one-dimensional σ-model on periodic orbits. Solving the latter model, first approximately and then exactly, we resolve the problem of repetitions.PACS: 05.45. Mt, 73.23.Ad

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Ballistic electron motion in a random magnetic field

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“…In the present work we follow the method of derivation of the σ-model suggested in our previous work 27 . In order to make the presentation self-contained we repeat the main steps of the derivation.…”

“…Following Ref. 27 we write the solution of Eq. (2.14) in terms of a functional integral over supermatrices Q n (r)…”