Quantum dynamics in canonical and micro-canonical ensembles. Part I. Anderson localization of electrons

TL;DR

This paper introduces a novel numerical approach combining molecular dynamics and Monte Carlo methods to study quantum dynamics and Anderson localization of electrons in disordered systems using the Wigner representation.

Contribution

It develops a new numerical procedure for quantum dynamics calculations in the Wigner representation, including an integral Wigner-Liouville equation and its solution method.

Findings

Numerical results for quantum operators in disordered systems

Observation of Anderson localization through zero static conductivity

Behavior of position dispersion indicating localization

Abstract

The new numerical approach for consideration of quantum dynamics and calculations of the average values of quantum operators and time correlation functions in the Wigner representation of quantum statistical mechanics has been developed. The time correlation functions have been presented in the form of the integral of the Weyl's symbol of considered operators and the Fourier transform of the product of matrix elements of the dynamic propagators. For the last function the integral Wigner- Liouville's type equation has been derived. The numerical procedure for solving this equation combining both molecular dynamics and Monte Carlo methods has been developed. For electrons in disordered systems of scatterers the numerical results have been obtained for series of the average values of the quantum operators including position and momentum dispersions, average energy, energy distribution function as well as for the frequency dependencies of tensor of electron conductivity and permittivity according to quantum Kubo formula. Zero or very small value of static conductivity have been considered as the manifestation of Anderson localization of electrons in 1D case. Independent evidence of Anderson localization comes from the behaviour of the calculated time dependence of position dispersion.