Critical temperatures of the three- and four-state Potts models on the kagome lattice

TL;DR

This paper uses transfer-matrix calculations and duality relations to precisely estimate the critical temperatures of three- and four-state Potts models on the kagome lattice, refining previous approximations.

Contribution

It provides exact conjectures for the critical coupling strengths of the q=3 and q=4 Potts models on the kagome lattice using novel transfer-matrix and duality methods.

Findings

Exact critical coupling for q=3: 1.0565094269290

Exact critical coupling for q=4: 1.1493605872292

Values differ from earlier conjectures and are more precise.

Abstract

The value of the internal energy per spin is independent of the strip width for a certain class of spin systems on two dimensional infinite strips. It is verified that the Ising model on the kagome lattice belongs to this class through an exact transfer-matrix calculation of the internal energy for the two smallest widths. More generally, one can suggest an upper bound for the critical coupling strength $K_c(q)$ for the $q$-state Potts model from exact calculations of the internal energy for the two smallest strip widths. Combining this with the corresponding calculation for the dual lattice and using an exact duality relation enables us to conjecture the critical coupling strengths for the three- and four-state Potts models on the kagome lattice. The values are $K_c(q=3)=1.056~509~426~929~0$ and $K_c(q=4) = 1.149~360~587~229~2$, and the values can, in principle, be obtained to an arbitrary precision. We discuss the fact that these values are in the middle of earlier approximate results and furthermore differ from earlier conjectures for the exact values.