Quantum Spin Systems

TL;DR

This paper reviews the fundamental concepts, techniques, and recent progress in understanding gapped quantum spin systems, emphasizing spectral gaps, locality properties, and their implications for quantum phases.

Contribution

It provides a comprehensive overview of methods for proving spectral gaps, recent advances on conjectures, and the role of Lieb-Robinson bounds in quantum spin systems.

Findings

Lieb-Robinson bounds establish exponential decay of correlations.

Progress on spectral gap conjectures for quantum spin models.

Quasi-adiabatic continuation aids in analyzing gapped phases.

Abstract

This work provides an overview of gapped quantum spin systems, including concepts, techniques, properties, and results. The basic framework and objects of interest for quantum spin systems are introduced, and the main ideas behind methods for proving spectral gaps for frustration-free models are outlined. After reviewing recent progress on several spectral gap conjectures, we discuss quasi-locality of the Heisenberg dynamics and its utility in proving properties of gapped quantum spin systems. Lieb-Robinson bounds have played a central role in establishing exponential decay of ground state correlations, an area law for one-dimensional systems, a many-body adiabatic theorem, and spectral gap stability. They also aided in the development of the quasi-adiabatic continuation, which is a useful for investigating gapped ground state phases, both of which are also discussed.