# Optimized broad-histogram simulations for strong first-order phase transitions: Droplet transitions in the large-Q Potts model

## TL;DR

This paper introduces a feedback algorithm to optimize broad-histogram ensembles, significantly improving simulation efficiency for strong first-order phase transitions in the large-Q Potts model, though some slowing down persists.

## Contribution

The study develops a feedback-based method to optimize broad-histogram ensembles, enhancing equilibration in simulations of first-order phase transitions, especially in large-Q Potts models.

## Key findings

- Optimized histograms develop multipeak structures resolving entropic barriers.
- Round-trip times scale polynomially for small Q, but exponentially for large Q.
- Ensemble optimization reduces, but does not eliminate, supercritical slowing down.

## Abstract

The numerical simulation of strongly first-order phase transitions has remained a notoriously difficult problem even for classical systems due to the exponentially suppressed (thermal) equilibration in the vicinity of such a transition. In the absence of efficient update techniques, a common approach to improve equilibration in Monte Carlo simulations is to broaden the sampled statistical ensemble beyond the bimodal distribution of the canonical ensemble. Here we show how a recently developed feedback algorithm can systematically optimize such broad-histogram ensembles and significantly speed up equilibration in comparison with other extended ensemble techniques such as flat-histogram, multicanonical or Wang-Landau sampling. As a prototypical example of a strong first-order transition we simulate the two-dimensional Potts model with up to Q=250 different states on large systems. The optimized histogram develops a distinct multipeak structure, thereby resolving entropic barriers and their associated phase transitions in the phase coexistence region such as droplet nucleation and annihilation or droplet-strip transitions for systems with periodic boundary conditions. We characterize the efficiency of the optimized histogram sampling by measuring round-trip times tau(N,Q) across the phase transition for samples of size N spins. While we find power-law scaling of tau vs. N for small Q \lesssim 50 and N \lesssim 40^2, we observe a crossover to exponential scaling for larger Q. These results demonstrate that despite the ensemble optimization broad-histogram simulations cannot fully eliminate the supercritical slowing down at strongly first-order transitions.