Effective-Geometry Rescaling and Universal Critical Behavior in the Anisotropic Three-State Potts Model on the Square Lattice
Abstract
We study the two-dimensional anisotropic three-state Potts ferromagnet on the square lattice using Wolff single-cluster Monte Carlo simulations and finite-size scaling. For coupling ratios λ=Jy/Jx=0.5, $0.75$, and $1$, finite-size scaling of the correlation ratio yields critical behavior consistent with the two-dimensional three-state Potts universality class. The scaling of the leading Fisher zeros gives a correlation-length exponent consistent with ν=5/6, while their cumulative density is consistent with the expected specific-heat exponent α=1/3. We further characterize the anisotropy at criticality using directional correlation ratios Rx and Ry together with directional FK wrapping probabilities. For λ=0.5 on a physically square lattice, Rx and Ry approach distinct critical values while yielding a common correlation-length exponent. From the wrapping probabilities, we independently determine an effective aspect ratio ρe□=0.6413(5), in close agreement with the theoretical value ρe□,th≃0.64150030 obtained from the isoradial representation. Using the theoretical value to set the physical aspect ratio restores directional equivalence, Rx≃ Ry, and brings the overall correlation ratio toward the isotropic-square reference. The results show that spatial anisotropy changes the effective critical geometry without altering the bulk three-state Potts universality class.
Related
- Effective kinetic theory description of the magnetic field-induced anisotropic gluon pressure during pre-equilibrium in heavy-ion collisions
- Interaction dependence of the critical correlation length amplitude ratio in anisotropic scalar lattice ϕ4 theory
- Effective scatterings and universal clusters of heteronuclear ultracold mixtures in quasi-low dimensions
- A Machine-Learning Based Approach to the Evaluation of the Critical Scaling Behavior of Anisotropic Spin Systems
- Geometry-Controlled Magnetic and Electronic Landscapes in Anisotropic van der Waals Materials
BibTeX
@misc{yang2026effectivegeometry,
title = {{Effective-Geometry Rescaling and Universal Critical Behavior in the Anisotropic Three-State Potts Model on the Square Lattice}},
author = {Fan Yang and Jian Gao and Lu Liu and Yuhai Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10467},
eprint = {2610.10467},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10467},
}
RIS
TY - UNPB
TI - Effective-Geometry Rescaling and Universal Critical Behavior in the Anisotropic Three-State Potts Model on the Square Lattice
AU - Fan Yang
AU - Jian Gao
AU - Lu Liu
AU - Yuhai Liu
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10467
UR - https://arxiv.org/abs/2610.10467
AB - We study the two-dimensional anisotropic three-state Potts ferromagnet on the square lattice using Wolff single-cluster Monte Carlo simulations and finite-size scaling. For coupling ratios λ=J_y/J_x=0.5, $0.75$, and $1$, finite-size scaling of the correlation ratio yields critical behavior consistent with the two-dimensional three-state Potts universality class. The scaling of the leading Fisher zeros gives a correlation-length exponent consistent with ν=5/6, while their cumulative density is consistent with the expected specific-heat exponent α=1/3. We further characterize the anisotropy at criticality using directional correlation ratios R_x and R_y together with directional FK wrapping probabilities. For λ=0.5 on a physically square lattice, R_x and R_y approach distinct critical values while yielding a common correlation-length exponent. From the wrapping probabilities, we independently determine an effective aspect ratio ρ_e^□=0.6413(5), in close agreement with the theoretical value ρ_e^□,th≃0.64150030 obtained from the isoradial representation. Using the theoretical value to set the physical aspect ratio restores directional equivalence, R_x≃ R_y, and brings the overall correlation ratio toward the isotropic-square reference. The results show that spatial anisotropy changes the effective critical geometry without altering the bulk three-state Potts universality class.
ER -