Unified theory of local integrals of motion

TL;DR

This paper introduces a general framework for constructing local integrals of motion in many-body localized systems, connecting localization, spin-glass models, and optimization problems to unify existing theories.

Contribution

It provides a method to explicitly construct LIOMs based on input quantum numbers, framing the problem as an optimization task applicable to various localization scenarios.

Findings

Framework unifies previous results on LIOMs and localization.

Connects LIOM construction to classical spin-glass ground state problems.

Demonstrates the approach with Anderson localization and spin chain models.

Abstract

Many-body localization (MBL) is understood theoretically through the existence of an extensive number of local integrals of motion (LIOMs). These conserved quantities are related to the microscopic quantum degrees of freedom that are spatially localized. Here, we present a general framework for constructing exact LIOMs with the desired locality and quantum numbers supplied as input rather than arising as emergent properties. We show that one can express the task of finding LIOMs as an optimization problem. In simple cases, solving this problem amounts to matrix diagonalization, while in more complex settings, it connects to the question of finding classical ground states of spin-glass models. We illustrate our theory using paradigmatic examples of single-particle Anderson localization and MBL in interacting spin chains. These developments unify previous results and reveal intriguing connections among many-body localization, spin-glass physics and constrained optimization problems.