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arXiv:1211.2006 · Physical Review E

Ising ferromagnet in dimension five: Link and spin overlaps

P. H. Lundow, Ian A. Campbell

Published Feb 1, 2013• 0 citations
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Abstract

In the simple [hyper]cubic five-dimension, near-neighbor-interaction Ising ferromagnet, extensive simulation measurements are made of the link overlap and the spin overlap distributions. These "two replica" measurements are standard in the spin glass context but are not usually recorded in ferromagnet simulations. The moments and moment ratios of these distributions (the variance, the kurtosis, and the skewness) show clear critical behaviors at the known ordering temperature of the ferromagnet. Analogous overlap effects can be expected quite generally in Ising ferromagnets in any dimension. The link overlap results in particular, with peaks at criticality in the kurtosis and the skewness, also have implications for spin glasses.

DOI
10.1103/physreve.87.022102

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In the simple [hyper]cubic five-dimension, near-neighbor-interaction Ising ferromagnet, extensive simulation measurements are made of the link overlap and the spin overlap distributions. These "two replica" measurements are standard in the spin glass context but are not usually recorded in ferromagnet simulations. The moments and moment ratios of these distributions (the variance, the kurtosis, and the skewness) show clear critical behaviors at the known ordering temperature of the ferromagnet. Analogous overlap effects can be expected quite generally in Ising ferromagnets in any dimension. The link overlap results in particular, with peaks at criticality in the kurtosis and the skewness, also have implications for spin glasses.
Extensive simulations are made of the link overlap in five dimensional Ising Spin Glasses (ISGs) through and below the ordering transition. Moments of the mean link overlap distributions (the kurtosis and the skewness) show clear critical maxima at the ISG ordering temperature. These criteria can be used as efficient tools to identify a freezing transition quite generally and in any dimension. In the ISG ordered phase the mean link overlap distribution develops a strong two peak structure, with the link overlap spectra of individual samples becoming very heterogeneous. There is no tendency towards a "trivial" universal single peak distribution in the range of size and temperature covered by the data.
Extensive simulations are made of link and spin overlaps in four and five dimensional Ising Spin Glasses (ISGs). Moments and moment ratios of the mean link overlap distributions (the variance, the kurtosis and the skewness) show clear critical behavior around the ISG ordering temperature. The link overlap measurements can be used to identify the ISG transition accurately; the link overlap is often a more efficient tool in this context than the spin overlap because the link overlap inter-sample variability is much weaker. Once the transition temperature is accurately established, critical exponents can be readily estimated by extrapolating measurements made in the thermodynamic limit regime. The data show that the bimodal and Gaussian spin glass susceptibility exponents γγγ are different from each other, both in dimension 5 and in dimension 4. Hence ISG critical exponents are not universal in a given dimension, but depend on the form of the interaction distribution.
Extensive simulations are made of the spin glass susceptibility and correlation length in five dimension Ising Spin Glasses (ISGs) with Gaussian and bimodal interaction distributions. Once the transition temperature is accurately established using a standard criterion, critical exponents and correction terms can be readily estimated by extrapolating measurements made in the thermodynamic limit regime. The data show that the critical exponents of the susceptibility γγγ and of the correlation length ννν depend on the form of the interaction distribution. This observation implies that quite generally critical exponents are not universal in ISGs.
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.
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.