The XY Model and the Berezinskii-Kosterlitz-Thouless Phase Transition

TL;DR

This paper reviews the XY model's Berezinskii-Kosterlitz-Thouless phase transition, emphasizing the importance of subtle effects like logarithmic corrections and recent advances in understanding topological defect-mediated transitions.

Contribution

It provides a comprehensive review of analytical and numerical methods, highlighting the role of vortices and subtle effects in the XY model's phase transition.

Findings

Logarithmic corrections are crucial for understanding the transition.

Vortices play a central role in the phase transition.

Recent research has advanced the understanding of topological defect effects.

Abstract

In statistical physics, the XY model in two dimensions provides the paradigmatic example of phase transitions mediated by topological defects (vortices). Over the years, a variety of analytical and numerical methods have been deployed in an attempt to fully understand the nature of its transition, which is of the Berezinskii-Kosterlitz-Thouless type. These met with only limited success until it was realized that subtle effects (logarithmic corrections) that modify leading behaviour must be taken into account. This realization prompted renewed activity in the field and significant progress has been made. This paper contains a review of the importance of such subtleties, the role played by vortices and of recent and current research in this area. Directions for desirable future research endeavours are outlined.