Localization Properties of the XY Spin Chain. A review of mathematical results with an eye toward Many-Body Localization

TL;DR

This review discusses mathematical results on localization phenomena in disordered XY spin chains, highlighting properties like eigenstate localization, correlation decay, and entanglement behavior, with new proofs on energy transport and Fock space localization.

Contribution

It provides a comprehensive review of localization properties in XY chains and introduces new rigorous results on energy transport and Fock space localization.

Findings

Localization of energy eigenstates and thermal states

Decay of correlation functions and absence of particle transport

Bounded entanglement growth and localization in Fock space

Abstract

We review several aspects of Many-Body Localization-like properties exhibited by the disordered XY chains: localization properties of the energy eigenstates and thermal states, propagation bounds of Lieb-Robinson type, decay of correlation functions, absence of particle transport, bounds on the bipartite entanglement, and bounded entanglement growth under the dynamics. We also prove new results on the absence of energy transport and Fock space localization. All these properties are made accessible to mathematical analysis due to the exact mapping of the XY chain to a system of quasi-free fermions given by the Jordan-Wigner transformation. Motivated by these results we discuss conjectured properties of more general disordered quantum spin and other systems as possible directions for future mathematical research.