Antiferromagnetic Potts models on the square lattice: A high-precision Monte Carlo study

Sabino José Ferreira; Alan D. Sokal

doi:10.1023/a:1004599121565

Antiferromagnetic Potts models on the square lattice: A high-precision Monte Carlo study

Sabino José Ferreira, Alan D. Sokal

Research output: Contribution to journal › Article › peer-review

Abstract

We study the antiferromagnetic q-state Potts model on the square lattice for q = 3 and q = 4, using the Wang-Swendsen-Kotecký (WSK) Monte Carlo algorithm and a powerful finite-size-scaling extrapolation method. For q = 3 we obtain good control up to correlation length ξ ∼ 5000; the data are consistent with ξ(β) = Ae^2ββ^p(1 + a₁e^-β + ⋯) as β → ∝, with p ≈.1. The staggered susceptibility behaves as χ_stagg ∼ ξ^5/3. For q = 4 the model is disordered (ξ ≲ 2) even at zero temperature. In appendices we prove a correlation inequality for Potts antiferromagnets on a bipartite lattice, and we prove ergodicity of the WSK algorithm at zero temperature for Potts antiferromagnets on a bipartite lattice.

Original language	English (US)
Pages (from-to)	461-530
Number of pages	70
Journal	Journal of Statistical Physics
Volume	96
Issue number	3-4
DOIs	https://doi.org/10.1023/a:1004599121565
State	Published - Aug 1999

Keywords

Algorithm
Antiferromagnet
Cluster algorithm
Finite-size scaling
Monte Carlo
Phase transition, zero-temperature critical point
Potts model
Square lattice
Swendsen Wang algorithm, Wang Swendsen-Kotecký

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

Access to Document

10.1023/a:1004599121565

Cite this

@article{29cca15c6c884cc49fafe5a5166a0e71,

title = "Antiferromagnetic Potts models on the square lattice: A high-precision Monte Carlo study",

abstract = "We study the antiferromagnetic q-state Potts model on the square lattice for q = 3 and q = 4, using the Wang-Swendsen-Koteck{\'y} (WSK) Monte Carlo algorithm and a powerful finite-size-scaling extrapolation method. For q = 3 we obtain good control up to correlation length ξ ∼ 5000; the data are consistent with ξ(β) = Ae2ββp(1 + a1e-β + ⋯) as β → ∝, with p ≈.1. The staggered susceptibility behaves as χstagg ∼ ξ5/3. For q = 4 the model is disordered (ξ ≲ 2) even at zero temperature. In appendices we prove a correlation inequality for Potts antiferromagnets on a bipartite lattice, and we prove ergodicity of the WSK algorithm at zero temperature for Potts antiferromagnets on a bipartite lattice.",

keywords = "Algorithm, Antiferromagnet, Cluster algorithm, Finite-size scaling, Monte Carlo, Phase transition, zero-temperature critical point, Potts model, Square lattice, Swendsen Wang algorithm, Wang Swendsen-Koteck{\'y}",

author = "Ferreira, {Sabino Jos{\'e}} and Sokal, {Alan D.}",

year = "1999",

month = aug,

doi = "10.1023/a:1004599121565",

language = "English (US)",

volume = "96",

pages = "461--530",

journal = "Journal of Statistical Physics",

issn = "0022-4715",

publisher = "Springer",

number = "3-4",

}

TY - JOUR

T1 - Antiferromagnetic Potts models on the square lattice

T2 - A high-precision Monte Carlo study

AU - Ferreira, Sabino José

AU - Sokal, Alan D.

PY - 1999/8

Y1 - 1999/8

N2 - We study the antiferromagnetic q-state Potts model on the square lattice for q = 3 and q = 4, using the Wang-Swendsen-Kotecký (WSK) Monte Carlo algorithm and a powerful finite-size-scaling extrapolation method. For q = 3 we obtain good control up to correlation length ξ ∼ 5000; the data are consistent with ξ(β) = Ae2ββp(1 + a1e-β + ⋯) as β → ∝, with p ≈.1. The staggered susceptibility behaves as χstagg ∼ ξ5/3. For q = 4 the model is disordered (ξ ≲ 2) even at zero temperature. In appendices we prove a correlation inequality for Potts antiferromagnets on a bipartite lattice, and we prove ergodicity of the WSK algorithm at zero temperature for Potts antiferromagnets on a bipartite lattice.

AB - We study the antiferromagnetic q-state Potts model on the square lattice for q = 3 and q = 4, using the Wang-Swendsen-Kotecký (WSK) Monte Carlo algorithm and a powerful finite-size-scaling extrapolation method. For q = 3 we obtain good control up to correlation length ξ ∼ 5000; the data are consistent with ξ(β) = Ae2ββp(1 + a1e-β + ⋯) as β → ∝, with p ≈.1. The staggered susceptibility behaves as χstagg ∼ ξ5/3. For q = 4 the model is disordered (ξ ≲ 2) even at zero temperature. In appendices we prove a correlation inequality for Potts antiferromagnets on a bipartite lattice, and we prove ergodicity of the WSK algorithm at zero temperature for Potts antiferromagnets on a bipartite lattice.

KW - Algorithm

KW - Antiferromagnet

KW - Cluster algorithm

KW - Finite-size scaling

KW - Monte Carlo

KW - Phase transition, zero-temperature critical point

KW - Potts model

KW - Square lattice

KW - Swendsen Wang algorithm, Wang Swendsen-Kotecký

UR - http://www.scopus.com/inward/record.url?scp=0033245970&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033245970&partnerID=8YFLogxK

U2 - 10.1023/a:1004599121565

DO - 10.1023/a:1004599121565

M3 - Article

AN - SCOPUS:0033245970

SN - 0022-4715

VL - 96

SP - 461

EP - 530

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 3-4

ER -

Antiferromagnetic Potts models on the square lattice: A high-precision Monte Carlo study

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this