Scrutinizing the Mori memory function for diffusion in periodic quantum systems

TL;DR

This paper challenges the assumption that converging current autocorrelation functions imply diffusion in quantum systems, introduces a method using Mori memory formalism, and demonstrates its effectiveness with examples.

Contribution

It presents a counterexample to the diffusion assumption and proposes a practical approach to identify diffusion using Lanczos coefficients within the Mori formalism.

Findings

Counterexample showing non-diffusive dynamics despite converging autocorrelation

A method to identify diffusion using a finite number of Lanczos coefficients

Practical applicability of the method in quantum systems

Abstract

Diffusion is an ubiquitous phenomenon. It is a widespread belief that as long as the area under a current autocorrelation function converges in time, the corresponding spatiotemporal density dynamics should be diffusive. This may be viewed as a result of the combination of linear response theory with the Einstein relation. However, attempts to derive this statement from first principles are notoriously challenging. We first present a counterexample by constructing a correlation functions of some density wave, such that the area under the corresponding current autocorrelation function converges, but the dynamics do not obey a diffusion equation. Then we will introduce a method based on the recursion method and the Mori memory formalism, that may help to actually identify diffusion. For a decisive answer, one would have to know infinitely many so called Lanczos coefficients, which is unattainable in most cases. However, in the examples examined in this paper, we find that the practically computable number of Lanczos coefficients suffices for a strong guess.