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arXiv:cond-mat/9605057 · Physical Review Letters

Two-Dimensional Ising Spin Glasses with Nonzero Ordering Temperatures

Ney Lemke, Ian A. Campbell

Published Jun 10, 1996• 18 citations
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Abstract

We demonstrate numerically that, for Ising spins on square lattices with ferromagnetic second neighbor interactions and random near neighbor interactions, two-dimensional Ising spin glass order with a nonzero freezing temperature can occur. We compare some of the physical properties of these spin glasses with those of standard spin glasses in higher dimensions.

DOI
10.1103/physrevlett.76.4616

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We demonstrate numerically that, for Ising spins on square lattices with ferromagnetic second neighbor interactions and random near neighbor interactions, two-dimensional Ising spin glass order with a nonzero freezing temperature can occur. We compare some of the physical properties of these spin glasses with those of standard spin glasses in higher dimensions.
We present the results of numerical simulations on Ising spin glasses in zero magnetic field with nearest-neighbor interactions on hyper-cubic lattices in two, three, and four dimensions with both Gaussian and \ifmmode\pm\else\textpm\fi{}J bond distributions. Finite-size scaling is used to analyze the results. In two dimensions (d=2) we agree with earlier work that the transition temperature is at Tc{T}_{c}Tc​=0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero Tc{T}_{c}Tc​ but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that Tc{T}_{c}Tc​ is nonzero with a long-range-ordered phase below Tc{T}_{c}Tc​. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.
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 Tg{T}_{g}Tg​ 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 investigate the order-parameter distribution P(q) of the Ising spin glass with nearest-neighbor interactions in four dimensions using Monte Carlo simulations on lattices of linear dimension up to L=6. We find that, below the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​, the weight at small q seems to saturate to a nonzero value as the size increases, similar to the infinite-range Sherrington-Kirkpatrick model. We discuss our results in the light of recent theoretical predictions for the nature of the spin-glass phase.
Scaling arguments and precise simulations are used to study the square lattice ±J Ising spin glass, a prototypical model for glassy systems. Droplet theory explains, and our numerical results show, entropically stabilized long-range spin-glass order at zero temperature, which resembles the energetic stabilization of long-range order in higher-dimensional models at finite temperature. At low temperature, a temperature-dependent crossover length scale is used to predict the power-law dependence on temperature of the heat capacity and clarify the importance of disorder distributions.
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.