Stochastic Integral Representation for the Dynamics of Disordered Systems

TL;DR

This paper introduces an exact stochastic integral representation for the dynamics of disordered quantum systems, enabling analytic approximations and systematic corrections, with applications to key physical quantities.

Contribution

It provides a novel stochastic integral framework for disordered quantum dynamics, compatible with tensor networks and including diffusive corrections.

Findings

Derived expressions for density of states and spectral form factor.

First approximation includes all diffusive corrections and is completely positive.

Presented systematic corrections through a convergent series.

Abstract

The dynamics of interacting quantum systems in the presence of disorder is studied and an exact representation for disorder-averaged quantities via Ito stochastic calculus is obtained. The stochastic integral representation affords many advantages, including amenability to analytic approximation, applicability to interacting systems, and compatibility with existing tensor network methods. The integral may be expanded to produce a series of approximations, the first of which already includes all diffusive corrections and, further, is manifestly completely positive. The addition of fluctuations leads to a convergent series of systematic corrections. As examples, expressions for the density of states, spectral form factor, and out-of-time-order correlators for the Anderson model are obtained.