Critical nonequilibrium relaxation in the Swendsen-Wang algorithm in the Berezinsky-Kosterlitz-Thouless and weak first-order phase transitions

TL;DR

This paper analyzes the nonequilibrium relaxation behaviors in the Swendsen-Wang algorithm across various phase transitions, revealing distinct relaxation patterns and proposing a criterion for the weakness of first-order transitions.

Contribution

It extends the analysis of relaxation behaviors in the Swendsen-Wang algorithm to BKT and weak first-order transitions, identifying characteristic relaxation patterns for different transition types.

Findings

BKT transition exhibits exponential relaxation.

Weak first-order transition shows power-law relaxation.

Phase transition types can be distinguished by relaxation behavior.

Abstract

Recently we showed that the critical nonequilibrium relaxation in the Swendsen-Wang algorithm is widely described by the stretched-exponential relaxation of physical quantities in the Ising or Heisenberg models. Here we make a similar analysis in the Berezinsky-Kosterlitz-Thouless phase transition in the two-dimensional (2D) XY model and in the first-order phase transition in the 2D $q=5$ Potts model, and find that these phase transitions are described by the simple exponential relaxation and power-law relaxation of physical quantities, respectively. We compare the relaxation behaviors of these phase transitions with those of the second-order phase transition in the 3D and 4D XY models and in the 2D $q$-state Potts models for $2 \le q \le 4$, and show that the species of phase transitions can be clearly characterized by the present analysis. We also compare the size dependence of relaxation behaviors of the first-order phase transition in the 2D $q=5$ and $6$ Potts models, and propose a quantitative criterion on "weakness" of the first-order phase transition.