Large-q expansion for the second moment correlation length in the two-dimensional q-state Potts model

TL;DR

This paper derives a large-q expansion for the second moment correlation length in the 2D q-state Potts model at the phase transition, revealing spectral properties of the transfer matrix across different phases.

Contribution

It provides a detailed large-q expansion up to order 21 and analyzes the spectral structure of the transfer matrix in the Potts model.

Findings

The second moment correlation lengths in ordered and disordered phases coincide up to third order in the expansion.

The ratio of correlation lengths in the two phases is close to unity for all q>4.

The spectrum of the transfer matrix suggests a continuum of eigenvalues beyond large-q regions.

Abstract

We calculate the large-q expansion of the second moment correlation length at the first order phase transition point of the q-state Potts model in two dimensions both in the ordered and disordered phases to order 21 in $1/\sqrt{q}$. They coincide with each other to the third term of the series but differ a little in higher orders. Numerically the ratio of the second moment correlation length in the two phases is not far from unity in all region of q>4. The ratio of the second moment correlation length to the standard correlation length in the disordered phase is far from unity, which suggests that the second largest and smaller eigenvalues of the transfer matrix form a continuum spectrum not only in the large-q region but also in all the region of q>4.