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arXiv:cond-mat/0407031 · Physical Review Letters

Absence of an Almeida-Thouless Line in Three-Dimensional Spin Glasses

A. P. Young, Helmut G. Katzgraber

Published Nov 11, 2004• 121 citations
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Abstract

We present results of Monte Carlo simulations of the three-dimensional Edwards-Anderson Ising spin glass in the presence of a (random) field. A finite-size scaling analysis of the correlation length shows no indication of a transition, in contrast with the zero-field case. This suggests that there is no Almeida-Thouless line for short-range Ising spin glasses.

DOI
10.1103/physrevlett.93.207203

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We present results of Monte Carlo simulations of the three-dimensional Edwards-Anderson Ising spin glass in the presence of a (random) field. A finite-size scaling analysis of the correlation length shows no indication of a transition, in contrast with the zero-field case. This suggests that there is no Almeida-Thouless line for short-range Ising spin glasses.
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.
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.
The nature of spin-glass states in a magnetic field remains a major open problem in statistical physics. The existence of the de Almeida-Thouless (dAT) transition for three-dimensional (3D) spin glasses in a field is still debated. We introduce a computational method to define the spin-glass susceptibility, which is robust against the broad tail in the overlap distribution that undermines conventional analyses. Applying this approach to the Edwards-Anderson spin-glass model in 2D and 3D, and contrasting with the 3D Ising (without disorder) and mean-field spin-glass models, we find a stark difference: the locus of susceptibility maxima bends to the right in the field-temperature plane for the Ising and 2D spin-glass cases, indicating a supercritical crossover line, but bends to the left for the mean field and 3D spin glasses-a signature of the dAT line. Finite-size scaling further suggests that the peak susceptibility diverges with system size in 3D spin glasses under a field, while saturating in 2D. These results provide direct numerical evidence for the dAT transition in 3D, supporting the replica symmetry breaking scenario.
We study the existence of a line of transitions of an Ising spin glass in a magnetic field---known as the de Almeida-Thouless line---using one-dimensional power-law diluted Ising spin-glass models. We choose the power-law exponent to have values that approximately correspond to three- and four-dimensional nearest-neighbor systems and perform a detailed finite-size scaling analysis of the data for large linear system sizes, using both a new approach proposed recently [Phys. Rev. Lett. 103, 267201 (2009)], as well as traditional approaches. Our results for the model corresponding to a three-dimensional system are consistent with there being no de Almeida-Thouless line, although the new finite-size scaling approach does not rule one out. For the model corresponding to four space dimensions, the new and traditional finite-size scaling analyses give conflicting results, indicating the need for a better understanding of finite-size scaling of spin glasses in a magnetic field.
The nature of spin-glass states in a magnetic field remains a major open problem in statistical physics. The existence of the de Almeida-Thouless (dAT) transition for three-dimensional (3D) spin glasses in a field is still debated. We introduce a new computational method to define the spin-glass susceptibility, which is robust against the broad tail in the overlap distribution that undermines conventional analyses. Applying this approach to the Edwards-Anderson spin-glass model in 2D and 3D, and contrasting with the 3D Ising (without disorder) and mean-field spin-glass models, we find a stark difference: the locus of susceptibility maxima bends to the right in the field-temperature plane for the Ising and 2D spin-glass cases, indicating a supercritical crossover line, but bends to the left for the mean-field and 3D spin glasses - a signature of the dAT line. Finite-size scaling further suggests that the peak susceptibility diverges with system size in 3D spin glasses under a field, while saturating in 2D. These results provide direct numerical evidence for the dAT transition in 3D, supporting the replica symmetry breaking scenario.