Dynamics of disordered quantum systems using flow equations

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. PACS. 72.15.Rn Localization effects

Flow equations for disordered Floquet systems

“…where H 0 contains the diagonal components of the Hamiltonian and V = H − H 0 contains the off-diagonal terms. Other choices are possible [58,64], however the canonical generator tends to be a robust choice that can be stably numerically integrated. For reference, in Appendix A we sketch the application of this method to a system of non-interacting fermions where it can be applied straightforwardly and exactly.…”

“…In the static case, the choice of unitary transform is not unique, and one has a lot of freedom to choose a generator with properties which suit the problem. For example, it is possible to choose a generator which preserves the sparsity of the Hamiltonian [58], although this turns out to be numerically difficult to integrate [64]. In the case of driven systems, however, there is an even richer variety of possible choices.…”

“…A detailed introduction to the flow equation method as applied to a system of non-interacting fermions may be found in Ref. [64].…”

“…In this Appendix, we sketch the application of the flow equation technique to a static, noninteracting Hamiltonian to illustrate the exact application of the method for readers unfamiliar with the technique. Additionally, we show for the first time how to obtain the eigenstates of the non-interacting system using flow equation method, something we did not include in previous work on the topic [64].…”

“…In this work, we set out and demonstrate the use of the flow equation approach to study periodically driven and disordered many-body quantum systems. Flow equations have been used in recent years to study a wide variety of systems with time-independent Hamiltonians, including Kondo and impurity models [55,56], quenches in the Hubbard model [57], and more recently have gained a lot of attention in the context of quantum localization [58][59][60][61][62][63][64][65]. The method has also been successfully employed in the study of dissipative systems, either by decoupling the system from its environment (e.g.…”