Aspects of the Mathematical Theory of Disordered Quantum Spin Chains

TL;DR

This paper introduces the mathematical theory of many-body localization in disordered quantum spin chains, discussing key properties, models, results, and future research directions in the field.

Contribution

It provides an overview of mathematical results on MBL in spin chains, including proofs, models, and methods, highlighting recent advances and open problems.

Findings

Verification of MBL properties in XY and XXZ models

Demonstration of zero-velocity Lieb-Robinson bounds

Exponential clustering and low entanglement in eigenstates

Abstract

We give an introduction into some aspects of the emerging mathematical theory of many-body localization (MBL) for disordered quantum spin chains. In particular, we discuss manifestations of MBL such as zero-velocity Lieb-Robinson bounds, quasi-locality of the time evolution of local observables, as well as exponential clustering and low entanglement of eigenstates. Explicit models where such properties have recently been verified are the XY and XXZ spin chain, in each case with disorder introduced in the form of a random exterior field. We introduce these models, state many of the available results and try to provide some general context. We discuss methods and ideas which enter the proofs and, in a few illustrative examples, include more detailed arguments. Finally, we also mention some directions for future mathematical work on MBL.