arXiv:0803.3339 · Physical Review B

Universality and universal finite-size scaling functions in four-dimensional Ising spin glasses

Thomas Jörg, Helmut G. Katzgraber

Published Jun 20, 2008• 20 citations

Abstract

We study the four-dimensional Ising spin glass with Gaussian and bond-diluted bimodal distributed interactions via large-scale Monte Carlo simulations and show via an extensive finite-size scaling analysis that four-dimensional Ising spin glasses obey universality.

DOI: 10.1103/physrevb.77.214426

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arXiv:0803.3339 · Physical Review B

Universality and universal finite-size scaling functions in four-dimensional Ising spin glasses

Thomas Jörg, Helmut G. Katzgraber

Published Jun 20, 2008• 20 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevb.77.214426

Related papers

Universality in three-dimensional Ising spin glasses: A Monte Carlo study

Helmut G. Katzgraber, Mathias Körner, A. P. Young

arXiv:cond-mat/0602212Physical Review BJun 23, 2006216 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.

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Martin Hasenbusch B, Andrea Pelissetto C, Ettore Vicari

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arXiv:0809.3329Physical Review BDec 17, 2008119 citations

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.

We perform high-statistics Monte Carlo simulations of three-dimensional Ising spin glass models on cubic lattices of size

L

: the $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model for two values of the disorder parameter

p

p=0.5

and

p=0.7

(up to

L=28

and

L=20

, respectively), and the bond-diluted bimodal model for bond-occupation probability

{p}_{b}=0.45

(up to

L=16

). The finite-size behavior of the quartic cumulants at the critical point allows us to check very accurately that these models belong to the same universality class. Moreover, it allows us to estimate the scaling-correction exponent $\ensuremath{\omega}$ related to the leading irrelevant operator: $\ensuremath{\omega}=1.0(1)$. Shorter Monte Carlo simulations of the bond-diluted bimodal models at

{p}_{b}=0.7

and

{p}_{b}=0.35

(up to

L=10

) and of the Ising spin glass model with Gaussian bond distribution (up to

L=8

) also support the existence of a unique Ising spin glass universality class. A careful finite-size analysis of the Monte Carlo data which takes into account the analytic and the nonanalytic corrections to scaling allows us to obtain precise and reliable estimates of the critical exponents. We obtain $\ensuremath{\nu}=2.45(15)$ and $\ensuremath{\eta}=\ensuremath{-}0.375(10)$.