Dynamics of Disordered Quantum Systems Using Flow Equations

TL;DR

This paper demonstrates how flow equation methods can be applied to study localization and non-equilibrium dynamics in disordered quantum systems, providing a versatile framework for analyzing operator spreading and correlations.

Contribution

It introduces a formalism based on flow equations for disordered quantum systems and applies it to the power-law random banded matrix model, highlighting its potential for future research.

Findings

Flow equations effectively analyze localization-delocalization transitions.

The method enables computation of quench dynamics of observables.

Provides a framework for understanding operator spreading and correlations.

Abstract

In this manuscript, we show how flow equation methods can be used to study localisation in disordered quantum systems, and particularly how to use this approach to obtain the non-equilibrium dynamical evolution of observables. We review the formalism, based on continuous unitary transforms, and apply it to a non-interacting yet non trivial one dimensional disordered quantum systems, the power-law random banded matrix model whose dynamics is studied across the localisation-delocalisation transition. We show how this method can be used to compute quench dynamics of simple observables, demonstrate how this formalism provides a natural framework to understand operator spreading and show how to construct complex objects such as correlation functions. We end with an outlook of unsolved problems and ways in which the method can be further developed in the future. Our goal is to motivate further adoption of the flow equation method, and to equip and encourage others to build on this technique as a means to study localisation phenomena in disordered quantum systems.