Critical behavior at edge singularities in one dimensional spin models

TL;DR

This paper investigates the universal behavior of partition function zeros in one-dimensional spin models, revealing a new critical exponent associated with eigenvalue degeneracy.

Contribution

It demonstrates that a triple degeneracy of transfer matrix eigenvalues in 1D spin models leads to a new critical exponent at edge singularities.

Findings

Universal density of zeros near edge singularities with specific exponents

Identification of a new critical exponent for models with eigenvalue degeneracy

Extension of edge singularity analysis to one-dimensional spin models

Abstract

In ferromagnetic spin models above the critical temperature ($T > T_{cr}$) the partition function zeros accumulate at complex values of the magnetic field ($H_E$) with a universal behavior for the density of zeros $\rho (H) \sim | H - H_E |^{\sg}$. The critical exponent $\sg$ is believed to be universal at each space dimension and it is related to the magnetic scaling exponent $y_h$ via $\sg = (d-y_h)/y_h$. In two dimensions we have $y_h=12/5 (\sg = -1/6)$ while $y_h=2 (\sg=-1/2)$ in $d=1$. For the one dimensional Blume-Capel and Blume-Emery-Griffiths models we show here, for different temperatures, that a new value $y_h=3 (\sg =-2/3)$ can emerge if we have a triple degeneracy of the transfer matrix eigenvalues.