Fisher Zeroes and Singular Behaviour of the Two Dimensional Potts Model in the Thermodynamic Limit

TL;DR

This paper investigates the Fisher zeroes in the two-dimensional Potts model with q>4, revealing their locus as a circle and connecting duality, zeroes distribution, and singular thermodynamic behavior in the thermodynamic limit.

Contribution

It introduces a duality-based approach to determine the Fisher zeroes locus and links zeroes distribution to singular thermodynamic functions in the Potts model.

Findings

Fisher zeroes locus is a circle in the complex plane.

Duality relates zeroes distribution to critical thermodynamic discontinuities.

Zeroes density determines the singular behavior of thermodynamic functions.

Abstract

The duality transformation is applied to the Fisher zeroes near the ferromagnetic critical point in the q>4 state two dimensional Potts model. A requirement that the locus of the duals of the zeroes be identical to the dual of the locus of zeroes in the thermodynamic limit (i) recovers the ratio of specific heat to internal energy discontinuity at criticality and the relationships between the discontinuities of higher cumulants and (ii) identifies duality with complex conjugation. Conjecturing that all zeroes governing ferromagnetic singular behaviour satisfy the latter requirement gives the full locus of such Fisher zeroes to be a circle. This locus, together with the density of zeroes is then shown to be sufficient to recover the singular form of the thermodynamic functions in the thermodynamic limit.