Comment on `Series expansions from the corner transfer matrix renormalization group method: the hard-squares model'

TL;DR

This paper analyzes an extended series for the hard-squares model's partition function, revealing a confluent singularity at a specific point and confirming a previously observed critical exponent value.

Contribution

It provides a detailed analysis of the series expansion, identifying the nature of the singularity and confirming the exact critical exponent value for the model.

Findings

Series has a confluent singularity of order 2 at the dominant singularity.

Critical exponent θ is confirmed to be exactly 5/6.

Analysis supports previous observations about the model's critical behavior.

Abstract

Earlier this year Chan extended the low-density series for the hard-squares partition function $\kappa(z)$ to 92 terms. Here we analyse this extended series focusing on the behaviour at the dominant singularity $z_d$ which lies on on the negative fugacity axis. We find that the series has a confluent singularity of order 2 at $z_d$ with exponents $\theta=0.83333(2)$ and $\theta'= 1.6676(3)$. We thus confirm that the exponent $\theta$ has the exact value $\frac56$ as observed by Dhar.