Locality in Quantum Systems

TL;DR

This paper discusses the role of locality, especially Lieb-Robinson bounds, in understanding various properties of quantum many-body systems, emphasizing intuitive explanations over rigorous proofs.

Contribution

It provides an accessible overview of how locality concepts apply to quantum systems, including correlation decay and topological order, with simplified proofs and intuitive insights.

Findings

Correlation decay in quantum systems explained

Application of Lieb-Robinson bounds to topological order

Insights into higher-dimensional Lieb-Schultz-Mattis theorem

Abstract

These lecture notes focus on the application of ideas of locality, in particular Lieb-Robinson bounds, to quantum many-body systems. We consider applications including correlation decay, topological order, a higher dimensional Lieb-Schultz-Mattis theorem, and a nonrelativistic Goldstone theorem. The emphasis is on trying to show the ideas behind the calculations. As a result, the proofs are only sketched with an emphasis on the intuitive ideas behind them, and in some cases we use techniques that give very slightly weaker bounds for simplicity. This is a preliminary version of the lecture notes, with the goal of getting the notes out close to the end of the school. Comments welcome.