Reflection positivity and phase transitions in lattice spin models

TL;DR

This paper reviews reflection positivity in lattice spin models, highlighting its role in proving phase transitions through tools like infrared bounds and chessboard estimates, with historical and recent developments.

Contribution

It summarizes classical and recent advances in reflection positivity and its applications to phase transitions in lattice spin systems.

Findings

Reflection positivity enables proof of phase transitions.

Infrared bounds and chessboard estimates are key tools.

The paper covers both classical and recent results.

Abstract

Reflection positivity (RP) is a property of Gibbs measures exhibited by a class of lattice spin systems that include the Ising, Potts and Heisenberg models. The RP property is useful because of its two basic consequences: infrared bound and chessboard estimates. These are one of basic (and rather efficient) tools for proving phase transitions in many models of physical interest. The notes presented hereby summarize the lectures on reflection positivity and its consequences that the author delivered at the Prague Summer School on Mathematical Statistical Mechanics in September 2006. The text features both the classical material on the subject from the late 1970s as well as some of the more recent developments.