arXiv:cond-mat/9704086 · Physical review. B, Condensed matter

Critical exponents in Ising spin glasses

L. Bernardi, Ian A. Campbell

Published Sep 1, 1997• 29 citations

Abstract

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.

DOI: 10.1103/physrevb.56.5271

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Critical exponents of the binomial Ising Spin Glass in dimension four; non-universality

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arXiv:cond-mat/9704086 · Physical review. B, Condensed matter

Critical exponents in Ising spin glasses

L. Bernardi, Ian A. Campbell

Published Sep 1, 1997• 29 citations

arXiv HTML PDF

Abstract

\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.

DOI: 10.1103/physrevb.56.5271

Related papers

Critical exponents of the binomial Ising Spin Glass in dimension four; non-universality

P. H. Lundow, Ian A. Campbell

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 $γ$ and $ν$ depend on the form of the interaction distribution. From this observation it can be deduced that critical exponents are not universal in ISGs.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 $γ$ and $ν$ depend on the form of the interaction distribution. From this observation it can be deduced that critical exponents are not universal in ISGs.

arXiv:1310.3484Oct 13, 20131 citation

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.

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Continuous phase transitions are catalogued into universality classes, families of systems having identical values of all the exponents governing the critical behaviour of their different physical properties. Numerical simulations have been carried out on Ising Spin Glasses in dimension three, using a technique where corrections to finite size scaling can be controled. The data show that the critical exponents vary strongly as a function of the kurtosis of the interaction distribution, a parameter which from the standard point of view should not be pertinent. This observation implies that for spin glasses the renormalization group analysis should not be approached in a the same way as in the case of canonical second order transitions; a much richer structure of universality classes would appear to exist for spin glasses.Continuous phase transitions are catalogued into universality classes, families of systems having identical values of all the exponents governing the critical behaviour of their different physical properties. Numerical simulations have been carried out on Ising Spin Glasses in dimension three, using a technique where corrections to finite size scaling can be controled. The data show that the critical exponents vary strongly as a function of the kurtosis of the interaction distribution, a parameter which from the standard point of view should not be pertinent. This observation implies that for spin glasses the renormalization group analysis should not be approached in a the same way as in the case of canonical second order transitions; a much richer structure of universality classes would appear to exist for spin glasses.

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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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Ising spin glasses in dimension five

P. H. Lundow, Ian A. Campbell

Ising spin-glass models with bimodal, Gaussian, uniform, and Laplacian interaction distributions in dimension five are studied through detailed numerical simulations. The data are analyzed in both the finite-size scaling regime and the thermodynamic limit regime. It is shown that the values of critical exponents and of dimensionless observables at criticality are model dependent. Models in a single universality class have identical values for each of these critical parameters, so Ising spin-glass models in dimension five with different interaction distributions each lie in different universality classes. This result confirms conclusions drawn from measurements in dimension four and dimension two.Ising spin-glass models with bimodal, Gaussian, uniform, and Laplacian interaction distributions in dimension five are studied through detailed numerical simulations. The data are analyzed in both the finite-size scaling regime and the thermodynamic limit regime. It is shown that the values of critical exponents and of dimensionless observables at criticality are model dependent. Models in a single universality class have identical values for each of these critical parameters, so Ising spin-glass models in dimension five with different interaction distributions each lie in different universality classes. This result confirms conclusions drawn from measurements in dimension four and dimension two.

arXiv:1606.03108Physical review. EJan 10, 20173 citations

Ordering Temperatures and Critical Exponents in Ising Spin Glasses

L. Bernardi, Sona Prakash, Ian A. Campbell

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.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.

arXiv:cond-mat/9608028Physical Review LettersSep 23, 199645 citations

\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.

γ

and

ν

depend on the form of the interaction distribution. From this observation it can be deduced that critical exponents are not universal in ISGs.

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.

{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.