Integrals of motion in the Many-Body localized phase

TL;DR

This paper constructs a complete set of quasi-local integrals of motion for the many-body localized phase, providing insights into the transition between localized and delocalized phases in disordered interacting fermion systems.

Contribution

It introduces a method to explicitly construct integrals of motion in the MBL phase using a convergent series in interaction strength, linking to the localization transition.

Findings

Constructed quasi-local integrals of motion with binary spectrum.

Mapped the problem to a non-Hermitian hopping model in operator space.

Estimated the localization-delocalization transition point.

Abstract

We construct a complete set of quasi-local integrals of motion for the many-body localized phase of interacting fermions in a disordered potential. The integrals of motion can be chosen to have binary spectrum $\{0,1\}$, thus constituting exact quasiparticle occupation number operators for the Fermi insulator. We map the problem onto a non-Hermitian hopping problem on a lattice in operator space. We show how the integrals of motion can be built, under certain approximations, as a convergent series in the interaction strength. An estimate of its radius of convergence is given, which also provides an estimate for the many-body localization-delocalization transition. Finally, we discuss how the properties of the operator expansion for the integrals of motion imply the presence or absence of a finite temperature transition.