Regular Article

Dynamics of disordered quantum systems using flow equations^★

¹ Institut de Physique Théorique, Université Paris Saclay, CNRS, CEA, 91191 Gif-sur-Yvette, France
² JEIP, USR 3573 CNRS, Collège de France, PSL Research University, 11 Place Marcelin Berthelot, 75321 Paris Cedex 05, France

^a e-mail: steven.thomson@polytechnique.edu
^b Current address: CPHT, CNRS, Institut Polytechnique de Paris, Route de Saclay, 91128 Palaiseau, France.

Received: 30 September 2019
Received in final form: 6 December 2019
Published online: 3 February 2020

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 system, 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 also discuss how the method may be applied to interacting quantum systems, and end with an outlook on 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.