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arXiv:cond-mat/0605010 · Physical Review B

Critical behavior of the three- and ten-state short-range Potts glass: A Monte Carlo study

L. W. Lee, Helmut G. Katzgraber, A. P. Young

Published Sep 26, 2006• 15 citations
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Abstract

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 ${T}_{\mathrm{c}}\ensuremath{\simeq}0.536$.

DOI
10.1103/physrevb.74.104416

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We study the critical behavior of the short-range ppp-state Potts spin glass in three and four dimensions using Monte Carlo simulations. In three dimensions, for p=3p=3p=3, a finite-size scaling analysis of the correlation length shows clear evidence of a transition to a spin-glass phase at Tc\ensuremath≃0.273{T}_{\mathrm{c}}\ensuremath{\simeq}0.273Tc​\ensuremath≃0.273 for a Gaussian distribution of interactions and Tc\ensuremath≃0.377{T}_{\mathrm{c}}\ensuremath{\simeq}0.377Tc​\ensuremath≃0.377 for a \ifmmode±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ (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)(p=10)(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=3p=3p=3 Potts glass with Gaussian interactions has a spin-glass transition at Tc\ensuremath≃0.536{T}_{\mathrm{c}}\ensuremath{\simeq}0.536Tc​\ensuremath≃0.536.
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We have studied the three-dimensional Ising spin glass with a $\ifmmode\pm\else\textpm\fi{}J$ distribution by Monte Carlo simulations. Using larger sizes and much better statistics than in earlier work, a finite-size scaling analysis shows quite strong evidence for a finite transition temperature Tc{T}_{c}Tc​ with ordering below Tc{T}_{c}Tc​. Our estimate of the transition temperature is rather lower than in earlier work, and the value of the correlation length exponent $\ensuremath{\nu}$ is somewhat higher. Because there may be (unknown) corrections to finite-size scaling, we do not completely rule out the possibility that Tc=0{T}_{c}=0Tc​=0 or that Tc{T}_{c}Tc​ is finite but with no order below Tc{T}_{c}Tc​. However, from our data, these possibilities seem less likely.
We investigate the balanced M=4M=4M=4, p=4p=4p=4 spin-glass model for a one-dimensional long-range proxy for the finite dimensional short-range ppp-spin glass model to examine the nature of the glass transition beyond mean-field theory. We perform large-scale Monte Carlo equilibrated simulations for both fully connected and power-law diluted versions of the model. The critical temperatures extracted from the finite-size scaling (FSS) analysis of spin-glass susceptibility are in good agreement with theoretical predictions for σ=0,0.25σ= 0, 0.25σ=0,0.25, and 0.55. For these values of the long-range exponent σσσ (which is the power of the decrease of the interactions between the spins with their separation), one might have expected that mean-field theory would provide a good description of the system. However, the spin-overlap distribution and the value of the λλλ-parameter do not provide numerical evidence for a one-step replica symmetry breaking (1RSB) phase transition. Instead, our results indicate a direct transition from the paramagnetic state to a full replica symmetry broken phase, with a renormalized value of λ≡ω2/ω1<1λ\equiv ω_2/ω_1 < 1λ≡ω2​/ω1​<1 suggesting a continuous FRSB transition, despite this ratio being equal to 2 at mean-field level. A value of λ>1λ> 1λ>1 is required for the discontinuous 1RSB transition. We argue that strong finite-size effects and closely spaced transition temperatures remove the expected 1RSB transition for the system sizes which we can study. For values of the exponent σ=0.85σ= 0.85σ=0.85, which roughly corresponds to a three dimensional system, we find that the renormalized value of λλλ is again less than 1, with no signs of either the 1RSB transition or the continuous FRSB transition, suggesting that the Kauzmann temperature TKT_KTK​ in three dimensions might be zero and the complete absence of phase transitions in structural glasses.
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