Non-linear $\sigma$-model for long range disorder and quantum chaos

TL;DR

The paper develops a new derivation of a non-linear sigma-model for long-range disorder and quantum billiards, avoiding slow mode separation and saddle-point approximation, and reduces the model to a one-dimensional form on periodic orbits.

Contribution

It introduces a novel derivation method for the ballistic sigma-model applicable to long-range disorder and quantum billiards, including a reduction to a simpler one-dimensional model.

Findings

Derived a sigma-model valid for all distances exceeding the electron wavelength.

Reduced the sigma-model to a one-dimensional form on periodic orbits for quantum billiards.

Resolved the problem of repetitions by solving the reduced model exactly.

Abstract

We suggest a new scheme of derivation of a non-linear ballistic $\sigma$-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 representation 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 $\sigma$-model for all distances exceeding the electron wavelength. 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 $\sigma$-model containing a conventional collision term. For quantum billiards, we demonstrate that, at not very low frequencies, one can reduce the $\sigma$-model to a one-dimensional $\sigma$-model on periodic orbits. Solving the latter model, first approximately and then exactly, we resolve the problem of repetitions.