Nonlinear σ model for long-range disorder and quantum chaos

We suggest a derivation of a nonlinear ballistic σ model for long-range disorder and quantum billiards. The derivation is based on writing equations for quasiclassical Green functions for a fixed long-range potential and an exact represention of their solutions in terms of functional integrals over supermatrices Q with the constraint Q²=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. Neither singling out slow modes nor a saddle-point approximation are used in the derivation. Carrying out a coarse-graining procedure that allows us to get rid of scales in the Lyapunov 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.