Physical Review Letters

Correlation length of the three-state Potts model in three dimensions

M. Fukugita, Masanori Okawa

Published Jul 3, 1989• 53 citations

Abstract

A Monte Carlo study is carried out for the three-state Potts model in three dimensions. The correlation length is shown to remain finite and to be discontinuous at the transition point. It is also shown that the transition exhibits genuine first-order characteristics in all aspects.

DOI: 10.1103/physrevlett.63.13

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Physical Review Letters

Correlation length of the three-state Potts model in three dimensions

M. Fukugita, Masanori Okawa

Published Jul 3, 1989• 53 citations

Publisher

Abstract

DOI: 10.1103/physrevlett.63.13

Related papers

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Svend J. Knak Jensen, Ole G. Mouritsen

Physical Review LettersDec 3, 1979101 citations

{\mathit{q}}_{\mathit{c}}

=2.45\ifmmode\pm\else\textpm\fi{}0.1. We also find that the free-energy barrier \ensuremath{\Delta}F(q,L)\ensuremath{\sim}(q-

{\mathit{q}}_{\mathit{c}}

{)}^{2}

{\mathit{L}}^{2.3}

and conclude that there is no essential singularity in the correlation length or latent heat in contrast to two dimensions.

We study the critical behavior of the short-range

p

-state Potts spin glass in three and four dimensions using Monte Carlo simulations. In three dimensions, for

p=3

, a finite-size scaling analysis of the correlation length shows clear evidence of a transition to a spin-glass phase at ${T}_{\mathrm{c}}\ensuremath{\simeq}0.273$ for a Gaussian distribution of interactions and ${T}_{\mathrm{c}}\ensuremath{\simeq}0.377$ for a $\ifmmode\pm\else\textpm\fi{}J$ (bimodal) distribution. These results indicate that the lower critical dimension of the three-state Potts glass is below 3. By contrast, the correlation length of the ten-state

(p=10)

Potts glass in three dimensions remains small even at very low temperatures and thus shows no sign of a transition. In four dimensions we find that the

p=3

Potts glass with Gaussian interactions has a spin-glass transition at ${T}_{\mathrm{c}}\ensuremath{\simeq}0.536$.

{\mathit{L}}^{3}

block lattices (L up to size 36) and obtain high-precision estimates for the leading partition-function zeros. The finite-size-scaling analysis of the first zero exhibits the expected convergence of the critical exponent \ensuremath{\nu} toward 1/D for large L; in particular, we find \ensuremath{\nu}=2.955(26) from our two largest lattices. Analysis of our specific-heat

{\mathit{C}}_{\mathit{v}}

data yields l=0.080 31(26) for the latent heat. Along another line of approach, we calculate the mass gap m=1/\ensuremath{\xi} (\ensuremath{\xi} is the correlation length) for cylindrical

{\mathit{L}}^{2}

{\mathit{L}}_{\mathit{z}}

lattices (L up to 24 and

{\mathit{L}}_{\mathit{z}}

=256). The finite size-scaling analysis of these results is also consistent with the convergence of \ensuremath{\nu} toward 1/D, but that the limiting value is 1/D is not yet conclusively established. Some theoretical arguments favor \ensuremath{\nu}\ensuremath{\rightarrow}0 in case of a first-order transition in a cylindrical ${\mathit{L}}^{\mathit{D}\mathrm{\ensuremath{-}}1}$\ensuremath{\infty} geometry. Therefore, we also applied our approach to the 2D ten-state Potts model, which is known to have a strong first-order transition. In this case we find unambiguous evidence in favor of 1/D as the limiting value.