Exact results for the zeros of the partition function of the Potts model on finite lattices

TL;DR

This paper analytically and numerically investigates the distribution of Yang-Lee zeros of the Q-state Potts model across different dimensions, revealing systematic behaviors and exact results for finite lattices.

Contribution

It provides exact analytical results for the zeros of the partition function of the Potts model in one dimension and numerical insights in higher dimensions, enhancing understanding of phase transitions.

Findings

Zeros lie inside or outside the unit circle depending on Q

For Q=2, zeros are exactly on the unit circle

Zeros' behavior helps locate the critical line in external fields

Abstract

The Yang-Lee zeros of the Q-state Potts model are investigated in 1, 2 and 3 dimensions. Analytical results derived from the transfer matrix for the one-dimensional model reveal a systematic behavior of the locus of zeros as a function of Q. For 1<Q<2 the zeros in the complex $x=\exp(\beta H_q)$ plane lie inside the unit circle, while for Q>2 they lie outside the unit circle for finite temperature. In the special case Q=2 the zeros lie exactly on the unit circle as proved by Lee and Yang. In two and three dimensions the zeros are calculated numerically and behave in the same way. Results are also presented for the critical line of the Potts model in an external field as determined from the zeros of the partition function in the complex temperature plane.