The free energy of the Potts model: from the continuous to the first-order transition region

TL;DR

This paper develops a large q expansion for the 2D q-states Potts model's free energy, predicting energy cumulant scaling and accurately reproducing numerical data for q=10, with implications for numerical interface tension measurements.

Contribution

It introduces a novel large q expansion approach that bridges the continuous and first-order transition regions in the Potts model, providing new predictions and data comparisons.

Findings

Energy cumulants scale as (3n/2 - 2) power of correlation length for q>4.

Predicted specific heats are significantly larger than previous finite size scaling estimates.

Accurate reproduction of numerical data for q=10 without interface effects.

Abstract

We present a large $q$ expansion of the 2d $q$-states Potts model free energies up to order 9 in $1/\sqrt{q}$. Its analysis leads us to an ansatz which, in the first-order region, incorporates properties inferred from the known critical regime at $q=4$, and predicts, for $q>4$, the $n^{\rm th}$ energy cumulant scales as the power $(3 n /2-2)$ of the correlation length. The parameter-free energy distributions reproduce accurately, without reference to any interface effect, the numerical data obtained in a simulation for $q=10$ with lattices of linear dimensions up to L=50. The pure phase specific heats are predicted to be much larger, at $q\leq10$, than the values extracted from current finite size scaling analysis of extrema. Implications for safe numerical determinations of interface tensions are discussed.