Using zeros of the canonical partition function map to detect signatures of a Berezinskii-Kosterlitz-Thouless transition

TL;DR

This paper demonstrates that analyzing Fisher zeros of the canonical partition function can effectively identify the Berezinskii-Kosterlitz-Thouless transition in the 2D XY model, providing a new diagnostic tool for phase transitions.

Contribution

The study introduces a method using zeros of the partition function to detect BKT transitions, aligning with theoretical predictions and accurately estimating the transition temperature.

Findings

Zeros' behavior matches theoretical BKT transition signatures.

Transition temperature $T_{BKT}$ agrees with previous studies.

Imaginary part of zeros approaches zero below $T_{BKT}$ and remains finite above.

Abstract

Using the two dimensional $XY-(S(O(3))$ model as a test case, we show that analysis of the Fisher zeros of the canonical partition function can provide signatures of a transition in the Berezinskii-Kosterlitz-Thouless ($BKT$) universality class. Studying the internal border of zeros in the complex temperature plane, we found a scenario in complete agreement with theoretical expectations which allow one to uniquely classify a phase transition as in the $BKT$ class of universality. We obtain $T_{BKT}$ in excellent accordance with previous results. A careful analysis of the behavior of the zeros for both regions $\mathfrak{Re}(T) \leq T_{BKT}$ and $\mathfrak{Re}(T) > T_{BKT}$ in the thermodynamic limit show that $\mathfrak{Im}(T)$ goes to zero in the former case and is finite in the last one.