Localization in the random XXZ quantum spin chain

TL;DR

This paper proves many-body localization in the random XXZ quantum spin chain at low energies, extending single-particle localization techniques to the many-body context and identifying a parameter region for localization.

Contribution

It introduces a novel approach reformulating the localization problem as quasi-locality, extending the fractional moment method to many-body systems.

Findings

Localization occurs at the spectrum's bottom in a parameter region including weak interactions and strong disorder.

Localization region is independent of system size and depends only on the energy interval.

The method extends single-particle fractional moment techniques to many-body quantum systems.

Abstract

We study the many-body localization (MBL) properties of the Heisenberg XXZ spin-$\frac12$ chain in a random magnetic field. We prove that the system exhibits localization in any given energy interval at the bottom of the spectrum in a nontrivial region of the parameter space. This region, which includes weak interaction and strong disorder regimes, is independent of the size of the system and depends only on the energy interval. Our approach is based on the reformulation of the localization problem as an expression of quasi-locality for functions of the random many-body XXZ Hamiltonian. This allows us to extend the fractional moment method for proving localization, previously derived in a single-particle localization context, to the many-body setting.