Constructing local integrals of motion in the many-body localized phase

TL;DR

This paper introduces a physically motivated method to construct local integrals of motion in the many-body localized phase, providing a new way to characterize and analyze MBL systems through numerically constructed quasi-local operators.

Contribution

The authors develop a method to construct LIOMs by time-averaging local operators, offering a physically meaningful and experimentally accessible tool for studying MBL phases.

Findings

Constructed LIOMs are quasi-local with exponential decay.

Localization length can be extracted from LIOM decay.

The method identifies the MBL-ergodic phase transition.

Abstract

Many-body localization provides a generic mechanism of ergodicity breaking in quantum systems. In contrast to conventional ergodic systems, many-body localized (MBL) systems are characterized by extensively many local integrals of motion (LIOM), which underlie the absence of transport and thermalization in these systems. Here we report a physically motivated construction of local integrals of motion in the MBL phase. We show that any local operator (e.g., a local particle number or a spin flip operator), evolved with the system's Hamiltonian and averaged over time, becomes a LIOM in the MBL phase. Such operators have a clear physical meaning, describing the response of the MBL system to a local perturbation. In particular, when a local operator represents a density of some globally conserved quantity, the corresponding LIOM describes how this conserved quantity propagates through the MBL phase. Being uniquely defined and experimentally measurable, these LIOMs provide a natural tool for characterizing the properties of the MBL phase, both in experiments and numerical simulations. We demonstrate the latter by numerically constructing an extensive set of LIOMs in the MBL phase of a disordered spin chain model. We show that the resulting LIOMs are quasi-local, and use their decay to extract the localization length and establish the location of the transition between the MBL and ergodic phases.