# Partition function zeros of the p-state clock model in the complex temperature plane

## TL;DR

This study analyzes the zeros of the partition function in the complex temperature plane for the p-state clock model, revealing the nature of phase transitions and limitations in computational methods for larger systems.

## Contribution

It provides new evidence that the upper transition at p=6 is of BKT type and introduces a modified energy representation to accurately compute zeros without artifacts.

## Key findings

- Upper transition at p=6 is BKT type
- Leading zeros for p=6 collapse onto larger p and XY limit
- Finite-size effects limit access to large systems due to small partition function magnitude

## Abstract

We investigate the partition function zeros of the two-dimensional $p$-state clock model in the complex temperature plane by using the Wang-Landau method. For $p=5$, $6$, $8$, and $10$, we propose a modified energy representation to enumerate exact irregular energy levels for the density of states without any binning artifacts. Comparing the leading zeros between different $p$'s, we provide strong evidence that the upper transition at $p=6$ is indeed of the Berezinskii-Kosterlitz-Thouless (BKT) type in contrast to the claim of the previous Fisher zero study [Phys. Rev. E \textbf{80}, 042103 (2009)]. We find that the leading zeros of $p=6$ at the upper transition collapse onto the zero trajectories of the larger $p$'s including the $XY$ limit while the finite-size behavior of $p=5$ differs from the converged behavior of $p \ge 6$ within the system sizes examined. In addition, we argue that the nondivergent specific heat in the BKT transition is responsible for the small partition function magnitude that decreases exponentially with increasing system size near the leading zero, fundamentally limiting access to large systems in search for zeros with an estimator under finite statistical fluctuations.