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arXiv:cond-mat/0103619 · Physical review. B, Condensed matter

Ordering temperature and critical exponents of the binomial Ising spin glass in dimension 3

Pierre‐Olivier Mari, Ian A. Campbell

Published Apr 16, 2002• 28 citations
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Abstract

We compare numerical estimates from different sources for the ordering temperature ${T}_{g}$ and the critical exponents of the Ising spin glass in dimension 3 with binomial $(\ifmmode\pm\else\textpm\fi{}J)$ interactions. Corrections to finite-size scaling turn out to be important especially for parameters such as the Binder cumulant. For nonequilibrium parameters it is easier to approach the large-size limit and to allow for corrections to scaling. Relying principally on such data, a crossing point defines the freezing temperature ${T}_{g};$ the possibility that the ordering temperature is zero can definitively be excluded. We estimate an ordering temperature ${T}_{g}=1.195(15),$ with associated estimates of the critical exponents for which corrections to finite-size scaling are well under control. Among the parameters evaluated is the leading dynamic correction to scaling exponent w.

DOI
10.1103/physrevb.65.184409

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We compare numerical estimates from different sources for the ordering temperature Tg{T}_{g}Tg​ and the critical exponents of the Ising spin glass in dimension 3 with binomial (\ifmmode±\else\textpm\fiJ)(\ifmmode\pm\else\textpm\fi{}J)(\ifmmode±\else\textpm\fiJ) interactions. Corrections to finite-size scaling turn out to be important especially for parameters such as the Binder cumulant. For nonequilibrium parameters it is easier to approach the large-size limit and to allow for corrections to scaling. Relying principally on such data, a crossing point defines the freezing temperature Tg;{T}_{g};Tg​; the possibility that the ordering temperature is zero can definitively be excluded. We estimate an ordering temperature Tg=1.195(15),{T}_{g}=1.195(15),Tg​=1.195(15), with associated estimates of the critical exponents for which corrections to finite-size scaling are well under control. Among the parameters evaluated is the leading dynamic correction to scaling exponent w.
We compare simulation data from different sources on two canonical three-dimensional Ising spin glasses (ISGs): the binomial $\ifmmode\pm\else\textpm\fi{}J$ near-neighbor interaction ISG and the Gaussian interaction ISG. We allow for the possibility of corrections to finite size scaling and estimate the correction exponent w. Consistent estimates for the critical temperatures Tg{T}_{g}Tg​ and for the critical exponents for each system are obtained. The data strongly indicate that critical exponents in the two systems are significantly different from each other. These results thus confirm a breakdown of standard universality rules in Ising spin glasses.
Extensive simulations are made on the bimodal Ising Spin Glass (ISG) in dimension four. The transition temperature is established using a combination of standard finite size scaling and of thermodynamic derivative peak data. Measurements in the thermodynamic limit regime are analysed so as to estimate critical exponents and confluent correction terms. Comparisons with results on other 4d ISGs show that the susceptibility and correlation length critical exponents γγγ and ννν depend on the form of the interaction distribution. From this observation it can be deduced that critical exponents are not universal in ISGs.
We determine accurate values of ordering temperatures and critical exponents for Ising-spin-glass transitions in four dimensions, using a combination of finite-size scaling and nonequilibrium scaling techniques. We find that the exponents $\ensuremath{\eta}$ and zzz vary with the form of the interaction distribution, indicating nonuniversality at Ising-spin-glass transitions. These results confirm conclusions drawn from numerical data for three dimensions.
Extensive simulations are made on Ising Spin Glasses (ISG) with Gaussian, Laplacian and bimodal interaction distributions in dimension four. Standard finite size scaling analyses near and at criticality provide estimates of the critical inverse temperatures βcβ_cβc​, critical exponents, and critical values of a number of dimensionless parameters. Independent estimates are obtained for βcβ_cβc​ and the exponent ννν from thermodynamic derivative peak data. A detailed explanation is given of scaling in the thermodynamic limit with the ISG scaling variable τ=1−β2/βc2τ= 1-β^2/β_c^2τ=1−β2/βc2​ and the appropriate scaling expressions. Data over the entire paramagnetic range of temperatures are analysed in order to obtain further estimates of the critical exponents together with correction to scaling terms. The Privman-Fisher ansatz then leads to compact scaling expressions for the whole paramagnetic regime and for all sample sizes L. Comparisons between the 4d ISG models show that the critical dimensionless parameters characteristic of a universality class, and the susceptibility and correlation length critical exponents γγγ and ννν, depend on the form of the interaction distribution. From these observations it can be deduced that critical exponents are not universal in ISGs, at least in dimension four.
We report a high-precision finite-size scaling study of the critical behavior of the three-dimensional Ising Edwards-Anderson model (the Ising spin glass). We have thermalized lattices up to L=40L=40L=40 using the Janus dedicated computer. Our analysis takes into account leading-order corrections to scaling. We obtain Tc=1.1019(29){T}_{\mathrm{c}}=1.1019(29)Tc​=1.1019(29) for the critical temperature, $\ensuremath{\nu}=2.562(42)$ for the thermal exponent, $\ensuremath{\eta}=\ensuremath{-}0.3900(36)$ for the anomalous dimension, and $\ensuremath{\omega}=1.12(10)$ for the exponent of the leading corrections to scaling. Standard (hyper)scaling relations yield $\ensuremath{\alpha}=\ensuremath{-}5.69(13)$, $\ensuremath{\beta}=0.782(10)$, and $\ensuremath{\gamma}=6.13(11)$. We also compute several universal quantities at Tc{T}_{\mathrm{c}}Tc​.
We perform high-statistics Monte Carlo simulations of three three-dimensional Ising spin glass models: the ± J Ising model for two values of the disorder parameter p , p = 1/2 and 0.7, and the bond-diluted ± J model for bond-occupation probability p b = 0.45. A finite-size scaling analysis of the quartic cumulants at the critical point shows conclusively that these models belong to the same universality class and allows us to estimate the scaling-correction exponent ω related to the leading irrelevant operator, ω = 1.0(1). We also determine the critical exponents ν and η. Taking into account the scaling corrections, we obtain ν = 2.53(8) and η = −0.384(9).