Susceptibility amplitude ratios in the two-dimensional Potts model and percolation

TL;DR

This paper investigates the universal susceptibility amplitude ratios in the 2D Potts model and percolation, extending analytic calculations and testing predictions with Monte Carlo simulations, revealing discrepancies and questioning previous assumptions.

Contribution

It extends the analytic calculation of susceptibility amplitude ratios to the low-temperature regime and tests these predictions with simulations, highlighting potential issues with earlier assumptions.

Findings

Analytic prediction for $ ext{Gamma}_T/ ext{Gamma}_L$ agrees with $q=3$ data

Discrepancy found between $ ext{Gamma}/ ext{Gamma}_L$ prediction and $q=3$ data

Large corrections to scaling hinder conclusive results for $q=4$

Abstract

The high-temperature susceptibility of the $q$-state Potts model behaves as $\Gamma|T-T_c|^{-\gamma}$ as $T\to T_c+$, while for $T\to T_c-$ one may define both longitudinal and transverse susceptibilities, with the same power law but different amplitudes $\Gamma_L$ and $\Gamma_T$. We extend a previous analytic calculation of the universal ratio $\Gamma/\Gamma_L$ in two dimensions to the low-temperature ratio $\Gamma_T/\Gamma_L$, and test both predictions with Monte Carlo simulations for $q=3$ and 4. The data for $q=4$ are inconclusive owing to large corrections to scaling, while for $q=3$ they appear consistent with the prediction for $\Gamma_T/\Gamma_L$, but not with that for $\Gamma/\Gamma_L$. A simple extrapolation of our analytic results to $q\to1$ indicates a similar discrepancy with the corresponding measured quantities in percolation. We point out that stronger assumptions were made in the derivation of the ratio $\Gamma/\Gamma_L$, and our work suggests that these may be unjustified.