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arXiv:cond-mat/0507138 · Physical Review B

Probing the Almeida-Thouless line away from the mean-field model

Helmut G. Katzgraber, A. P. Young

Published Nov 15, 2005• 94 citations
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Abstract

Results of Monte Carlo simulations of the one-dimensional long-range Ising spin glass with power-law interactions in the presence of a (random) field are presented. By tuning the exponent of the power-law interactions, we are able to scan the full range of possible behaviors from the infinite-range (Sherrington-Kirkpatrick) model to the short-range model. A finite-size scaling analysis of the correlation length indicates that the Almeida-Thouless line does not occur in the region with non-mean-field critical behavior in zero field. However, there is evidence that an Almeida-Thouless line does occur in the mean-field region.

DOI
10.1103/physrevb.72.184416

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de Almeida–Thouless instability in short-range Ising spin glasses

Rajiv R. P. Singh, A. P. Young

We use high-temperature series expansions to study the ±J Ising spin glass in a magnetic field in d-dimensional hypercubic lattices for d=5-8 and in the infinite-range Sherrington-Kirkpatrick (SK) model. The expansions are obtained in the variable w=tanh^{2}J/T for arbitrary values of u=tanh^{2}h/T complete to order w^{10}. We find that the scaling dimension Δ associated with the ordering-field h^{2} equals 2 in the SK model and for d≥6. However, in agreement with the work of Fisher and Sompolinsky [Phys. Rev. Lett. 54, 1063 (1985)PRLTAO0031-900710.1103/PhysRevLett.54.1063], there is a violation of scaling in a finite field, leading to an anomalous h-T dependence of the de Almeida-Thouless (AT) [J. Phys. A 11, 983 (1978)JPHAC50305-447010.1088/0305-4470/11/5/028] line in high dimensions, whereas scaling is restored as d→6. Within the convergence of our series analysis, we present evidence supporting an AT line in d≥6. In d=5, the exponents γ and Δ are substantially larger than mean-field values, but we do not see clear evidence for the AT line in d=5.

arXiv:1705.01164•Physical review. E•Jul 13, 2017•8 citations
Results of Monte Carlo simulations of the one-dimensional long-range Ising spin glass with power-law interactions in the presence of a (random) field are presented. By tuning the exponent of the power-law interactions, we are able to scan the full range of possible behaviors from the infinite-range (Sherrington-Kirkpatrick) model to the short-range model. A finite-size scaling analysis of the correlation length indicates that the Almeida-Thouless line does not occur in the region with non-mean-field critical behavior in zero field. However, there is evidence that an Almeida-Thouless line does occur in the mean-field region.
We study in Ising spin glasses the finite-size effects near the spin-glass transition in zero field and at the de Almeida-Thouless transition in a field by Monte Carlo methods and by analytical approximations. In zero field, the finite-size scaling function associated with the spin-glass susceptibility of the Sherrington-Kirkpatrick mean-field spin-glass model is of the same form as that of one-dimensional spin-glass models with power-law long-range interactions in the regime where they can be a proxy for the Edwards-Anderson short-range spin-glass model above the upper critical dimension. We also calculate a simple analytical approximation for the spin-glass susceptibility crossover function. The behavior of the spin-glass susceptibility near the de Almeida-Thouless transition line has also been studied, but here we have only been able to obtain analytically its behavior in the asymptotic limit above and below the transition. We have also simulated the one-dimensional system in a field in the non-mean-field regime to illustrate that when the Imry-Ma droplet length scale exceeds the system size one can then be erroneously lead to conclude that there is a de Almeida-Thouless transition even though it is absent.
We study the de Almeida-Thouless (AT) line in the one-dimensional power-law diluted XY spin-glass model, in which the probability that two spins separated by a distance r interact with each other, decays as 1/r^{2σ}. Tuning the exponent σ is equivalent to changing the space dimension of a short-range model. We develop a heat bath algorithm to equilibrate XY spins; using this in conjunction with the standard parallel tempering and overrelaxation sweeps, we carry out large-scale Monte Carlo simulations. For σ=0.6, which is in the mean-field regime above six dimensions-it is similar to being in 10 dimensions-we find clear evidence for an AT line. For σ=0.75 and σ=0.85, which are in the non-mean-field regime and similar to four and three dimensions, respectively, our data is like that found in previous studies of the Ising and Heisenberg spin glasses when reducing the temperature at fixed field. For σ=0.75, there is evidence from finite-size-scaling studies for an AT transition but for σ=0.85, the evidence for a transition is nonexistent. We have also studied these systems at fixed temperature varying the field and discovered that at both σ=0.75 and at σ=0.85 there is evidence of an AT transition! Confusingly, the correlation length and spin-glass susceptibility as a function of the field are both entirely consistent with the predictions of the droplet picture and hence the nonexistence of an AT line. In the usual finite-size critical point scaling studies used to provide evidence for an AT transition, there is seemingly good evidence for an AT line at σ=0.75 for small values of the system size N, which is strengthening as N is increased, but for N>2048 the trend changes and the evidence then weakens as N is further increased. We have also studied with fewer bond realizations the system at σ=0.70, which is the analog of a system with short-range interactions just below six dimensions, and found that it is similar in its behavior to the system at σ=0.75 but with larger finite-size corrections. The evidence from our simulations points to the complete absence of the AT line in dimensions outside the mean-field region and to the correctness of the droplet picture. Previous simulations which suggested there was an AT line can be attributed to the consequences of studying systems which are just too small. The collapse of our data to the droplet scaling form is poor for σ=0.75 and to some extent also for σ=0.85, when the correlation length becomes of the order of the length of the system, due to the existence of excitations which only cost a free energy of O(1), just as envisaged in the TNT picture of the ordered state of spin glasses. However, for the case of σ=0.85 we can provide evidence that for larger system sizes, droplet scaling will prevail even when the correlation length is comparable to the system size.
We test for the presence or absence of the de Almeida--Thouless (AT) line using one-dimensional power-law diluted Heisenberg spin-glass model, in which the rms strength of the interactions decays with distance, rrr as $1/{r}^{\ensuremath{\sigma}}$. It is argued that varying the power $\ensuremath{\sigma}$ is analogous to varying the space dimension ddd in a short-range model. For $\ensuremath{\sigma}=0.6$, which is in the mean-field regime regime, we find clear evidence for an AT line. For $\ensuremath{\sigma}=0.85$, which is in the non-mean-field regime and corresponds to a space dimension of close to 3, we find no AT line, though we cannot rule one out for very small fields. Finally for $\ensuremath{\sigma}=0.75$, which is in the non-mean-field regime but closer to the mean-field boundary, the evidence suggests that there is an AT line, though the possibility that even larger sizes are needed to see the asymptotic behavior can not be ruled out.
We use high-temperature series expansions to study the ±J Ising spin glass in a magnetic field in d-dimensional hypercubic lattices for d=5-8 and in the infinite-range Sherrington-Kirkpatrick (SK) model. The expansions are obtained in the variable w=tanh^{2}J/T for arbitrary values of u=tanh^{2}h/T complete to order w^{10}. We find that the scaling dimension Δ associated with the ordering-field h^{2} equals 2 in the SK model and for d≥6. However, in agreement with the work of Fisher and Sompolinsky [Phys. Rev. Lett. 54, 1063 (1985)PRLTAO0031-900710.1103/PhysRevLett.54.1063], there is a violation of scaling in a finite field, leading to an anomalous h-T dependence of the de Almeida-Thouless (AT) [J. Phys. A 11, 983 (1978)JPHAC50305-447010.1088/0305-4470/11/5/028] line in high dimensions, whereas scaling is restored as d→6. Within the convergence of our series analysis, we present evidence supporting an AT line in d≥6. In d=5, the exponents γ and Δ are substantially larger than mean-field values, but we do not see clear evidence for the AT line in d=5.