arXiv:cond-mat/9608028 · Physical Review Letters

Ordering Temperatures and Critical Exponents in Ising Spin Glasses

L. Bernardi, Sona Prakash, Ian A. Campbell

Published Sep 23, 1996• 45 citations

Abstract

We propose a numerical criterion which can be used to obtain accurate and reliable values of the ordering temperatures and critical exponents of spin glasses. Using this method we find a value of the ordering temperature for the $\ifmmode\pm\else\textpm\fi{}J$ Ising spin glass in three dimensions which is definitely nonzero and in good agreement with previous estimates. We show that the critical exponents of three-dimensional Ising spin glasses do not obey the usual universality rules.

DOI: 10.1103/physrevlett.77.2798

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arXiv:cond-mat/9608028 · Physical Review Letters

Ordering Temperatures and Critical Exponents in Ising Spin Glasses

L. Bernardi, Sona Prakash, Ian A. Campbell

Published Sep 23, 1996• 45 citations

arXiv HTML PDF

Abstract

\ifmmode\pm\else\textpm\fi{}J

Ising spin glass in three dimensions which is definitely nonzero and in good agreement with previous estimates. We show that the critical exponents of three-dimensional Ising spin glasses do not obey the usual universality rules.

DOI: 10.1103/physrevlett.77.2798

Related papers

Critical exponents in Ising spin glasses

L. Bernardi, Ian A. Campbell

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 $z$ 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.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 $z$ 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.

arXiv:cond-mat/9704086Physical review. B, Condensed matterSep 1, 199729 citations

Ising spin glasses: Corrections to finite size scaling, freezing temperatures, and critical exponents

Pierre‐Olivier Mari, Ian A. Campbell

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 ${T}_{g}$ 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.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 ${T}_{g}$ 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.

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

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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.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.

arXiv:cond-mat/0103619Physical review. B, Condensed matterApr 16, 200228 citations

Numerical studies of Ising spin glasses in two, three, and four dimensions

R. N. Bhatt, A. P. Young

We present the results of numerical simulations on Ising spin glasses in zero magnetic field with nearest-neighbor interactions on hyper-cubic lattices in two, three, and four dimensions with both Gaussian and \ifmmode\pm\else\textpm\fi{}J bond distributions. Finite-size scaling is used to analyze the results. In two dimensions (d=2) we agree with earlier work that the transition temperature is at ${T}_{c}$ =0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero ${T}_{c}$ but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that ${T}_{c}$ is nonzero with a long-range-ordered phase below ${T}_{c}$ . Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.We present the results of numerical simulations on Ising spin glasses in zero magnetic field with nearest-neighbor interactions on hyper-cubic lattices in two, three, and four dimensions with both Gaussian and \ifmmode\pm\else\textpm\fi{}J bond distributions. Finite-size scaling is used to analyze the results. In two dimensions (d=2) we agree with earlier work that the transition temperature is at ${T}_{c}$=0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero ${T}_{c}$ but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that ${T}_{c}$ is nonzero with a long-range-ordered phase below ${T}_{c}$. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.

Physical review. B, Condensed matterApr 1, 1988301 citations

The Ising Spin Glass in dimension four; non-universality

P. H. Lundow, Ian A. Campbell

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 $β_c$ , critical exponents, and critical values of a number of dimensionless parameters. Independent estimates are obtained for $β_c$ and the exponent $ν$ from thermodynamic derivative peak data. A detailed explanation is given of scaling in the thermodynamic limit with the ISG scaling variable $τ= 1-β^2/β_c^2$ 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 $γ$ and $ν$ , 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.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 $β_c$, critical exponents, and critical values of a number of dimensionless parameters. Independent estimates are obtained for $β_c$ and the exponent $ν$ from thermodynamic derivative peak data. A detailed explanation is given of scaling in the thermodynamic limit with the ISG scaling variable $τ= 1-β^2/β_c^2$ 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 $γ$ and $ν$, 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.

arXiv:1402.1991Feb 9, 20141 citation

\ifmmode\pm\else\textpm\fi{}J

z

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.

{T}_{g}

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.

γ

and of the correlation length

ν

depend on the form of the interaction distribution. This observation implies that quite generally critical exponents are not universal in ISGs.

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.

{T}_{c}

=0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero

{T}_{c}

but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that

{T}_{c}

is nonzero with a long-range-ordered phase below

{T}_{c}

. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.

β_c

, critical exponents, and critical values of a number of dimensionless parameters. Independent estimates are obtained for

β_c

and the exponent

ν

from thermodynamic derivative peak data. A detailed explanation is given of scaling in the thermodynamic limit with the ISG scaling variable

τ= 1-β^2/β_c^2

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

γ

and

ν

, 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.