Review: Local Integrals of Motion in Many-Body Localized systems

TL;DR

This review discusses the concept of local integrals of motion in many-body localized systems, covering theoretical foundations, numerical methods, and debates on their existence in various models.

Contribution

It synthesizes the development, mathematical proofs, and current debates regarding local integrals of motion in many-body localized systems.

Findings

Existence of local integrals of motion explains MBL phenomenology.

Numerical algorithms have been proposed for constructing LIOMs.

Debates continue on LIOMs in systems with mobility edges and higher dimensions.

Abstract

We review the current (as of Fall 2016) status of the studies on the emergent integrability in many-body localized models. We start by explaining how the phenomenology of fully many-body localized systems can be recovered if one assumes the existence of a complete set of (quasi)local operators which commute with the Hamiltonian (local integrals of motion, or LIOMs). We describe the evolution of this idea from the initial conjecture, to the perturbative constructions, to the mathematical proof given for a disordered spin chain. We discuss the proposed numerical algorithms for the construction of LIOMs and the status of the debate on the existence and nature of such operators in systems with a many-body mobility edge, and in dimensions larger than one.