A Mapping Relating Complex and Physical Temperatures in the 2D $q$-state Potts Model and Some Applications

TL;DR

This paper establishes an exact equivalence between the free energies of the q-state Potts antiferromagnet and the Potts model on dual lattices across real and complex temperatures, revealing new insights into complex temperature singularities and phase transitions.

Contribution

It introduces a mapping linking physical and complex temperatures in the 2D q-state Potts model, enabling analysis of complex temperature singularities and phase transition properties.

Findings

Identifies two types of complex temperature singularities: generic and special.

Rules out two existing conjectures about the Potts model.

Determines the critical q-value for phase transition absence on the diced lattice.

Abstract

We show an exact equivalence of the free energy of the $q$-state Potts antiferromagnet on a lattice $\Lambda$ for the full temperature interval $0 \le T \le \infty$ and the free energy of the $q$-state Potts model on the dual lattice for a semi-infinite interval of complex temperatures (CT). This implies the existence of two quite different types of CT singularities: the generic kind, which does not obey universality or various scaling relations, and a special kind which does obey such properties and encodes information of direct physical relevance. We apply this observation to characterize CT properties of the Potts model on several lattices, to rule out two existing conjectures, and to determine the critical value of $q$ above which the Potts antiferromagnet on the diced lattice has no phase transition.