Fokker-Planck description of the transfer matrix limiting distribution in the scattering approach to quantum transport

TL;DR

This paper derives a Fokker-Planck equation for the distribution of the transfer matrix in quantum transport, showing that the eigenvalues follow a multivariate Gaussian distribution in the insulating regime, connecting with random matrix theory.

Contribution

It introduces a Fokker-Planck framework for the transfer matrix distribution in disordered conductors, linking it to known random matrix results and the DMPK equation.

Findings

Eigenvalues of $box{TT}^{}$ are Gaussian distributed in the limit.

Derived a complete Fokker-Planck equation for the transfer matrix.

Connected the distribution parameters to stationary eigenvector averages.

Abstract

The scattering approach to quantum transport through a disordered quasi-one-dimensional conductor in the insulating regime is discussed in terms of its transfer matrix $\bbox{T}$. A model of $N$ one-dimensional wires which are coupled by random hopping matrix elements is compared with the transfer matrix model of Mello and Tomsovic. We derive and discuss the complete Fokker-Planck equation which describes the evolution of the probability distribution of $\bbox{TT}^{\dagger}$ with system length in the insulating regime. It is demonstrated that the eigenvalues of $\ln\bbox{TT}^{\dagger}$ have a multivariate Gaussian limiting probability distribution. The parameters of the distribution are expressed in terms of averages over the stationary distribution of the eigenvectors of $\bbox{TT}^{\dagger}$. We compare the general form of the limiting distribution with results of random matrix theory and the Dorokhov-Mello-Pereyra-Kumar equation.