# Logarithmic finite-size scaling correction to the leading Fisher zeros in the p-state clock model: A higher-order tensor renormalization group study

## TL;DR

This study analyzes the finite-size scaling of Fisher zeros in p-state clock models, deriving and verifying logarithmic corrections using HOTRG, revealing complex transition behaviors beyond simple first or second order.

## Contribution

We derive and numerically verify logarithmic finite-size corrections to Fisher zeros in clock models using HOTRG, highlighting the importance of deterministic methods for accurate zero identification.

## Key findings

- Logarithmic FSS corrections match BKT predictions at both transitions.
- HOTRG provides reliable zero identification unaffected by stochastic noise.
- Clock models exhibit non-ordinary phase transition behaviors.

## Abstract

We investigate the finite-size-scaling (FSS) behavior of the leading Fisher zero of the partition function in the complex temperature plane in the $p$-state clock models of $p=5$ and $6$. We derive the logarithmic finite-size corrections to the scaling of the leading zeros which we numerically verify by performing the higher-order tensor renormalization group (HOTRG) calculations in the square lattices of a size up to $128 \times 128$ sites. The necessity of the deterministic HOTRG method in the clock models is noted by the extreme vulnerability of the numerical leading zero identification against stochastic noises that are hard to be avoided in the Monte-Carlo approaches. We characterize the system-size dependence of the numerical vulnerability of the zero identification by the type of phase transition, suggesting that the two transitions in the clock models are not of an ordinary first- or second-order type. In the direct FSS analysis of the leading zeros in the clock models, we find that their FSS behaviors show excellent agreement with our predictions of the logarithmic corrections to the Berezinskii-Kosterlitz-Thouless ansatz at both of the high- and low-temperature transitions.