Path Integral Approach to the Scattering Theory of Quantum Transport

TL;DR

This paper introduces a path integral method to analyze the statistics of quantum transport in disordered conductors, connecting transfer matrix distributions to random matrix theory and saddle point solutions.

Contribution

It develops a novel path integral framework for the transfer matrix statistics in quantum transport, applicable to arbitrary dimensions and broken time reversal symmetry.

Findings

Path integral formulation for transfer matrix statistics.

Connection to random matrix theory via saddle point analysis.

Application to quasi-one-dimensional wires and DMPK equation.

Abstract

The scattering theory of quantum transport relates transport properties of disordered mesoscopic conductors to their transfer matrix $\bbox{T}$. We introduce a novel approach to the statistics of transport quantities which expresses the probability distribution of $\bbox{T}$ as a path integral. The path integal is derived for a model of conductors with broken time reversal invariance in arbitrary dimensions. It is applied to the Dorokhov-Mello-Pereyra-Kumar (DMPK) equation which describes quasi-one-dimensional wires. We use the equivalent channel model whose probability distribution for the eigenvalues of $\bbox{TT}^{\dagger}$ is equivalent to the DMPK equation independent of the values of the forward scattering mean free paths. We find that infinitely strong forward scattering corresponds to diffusion on the coset space of the transfer matrix group. It is shown that the saddle point of the path integral corresponds to ballistic conductors with large conductances. We solve the saddle point equation and recover random matrix theory from the saddle point approximation to the path integral.