Detecting signals of weakly first-order phase transitions in two-dimensional Potts models

TL;DR

This study examines weakly first-order phase transitions in 2D Potts models using Monte Carlo simulations, revealing that standard analysis methods can be misleading for small system sizes and emphasizing the importance of careful diagnostics.

Contribution

It demonstrates that traditional data-collapse methods can produce deceptive results for weakly first-order transitions and introduces systematic analysis techniques to accurately identify transition nature.

Findings

Pseudo-critical exponents drift with system size.

The crossover length scale is much smaller than the correlation length.

Proper analysis is crucial for small system sizes.

Abstract

We investigate the first-order phase transitions of the $q$-state Potts models with $q = 5, 6, 7$, and $8$ on the two-dimensional square lattice, using Monte Carlo simulations. At the very weakly first-order transition of the $q=5$ system, the standard data-collapse procedure for the order parameter, carried out with results for a broad range of system sizes, works deceptively well and produces non-trivial critical exponents different from the trivial values expected for first-order transitions. However, a more systematic study reveals significant drifts in the `pseudo-critical' exponents as a function of the system size. For this purpose, we employ two methods of analysis: the data-collapse procedure with narrow range of the system size, and the Binder-cumulant crossing technique for pairs of system sizes. In both methods, the estimates start to drift toward the trivial values as the system size used in the analysis exceeds a certain `cross-over' length scale. This length scale is remarkably smaller than the correlation length at the transition point for weakly first-order transitions, e.g., less than one tenth for $q=5$, in contrast to the naive expectation that the system size has to be comparable to or larger than the correlation length to observe the correct behavior. The results overall show that proper care is indispensable to diagnose the nature of a phase transition with limited system sizes.