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Physical Review Letters

Monte Carlo study of the order-parameter distribution in the four-dimensional Ising spin glass

Joseph D. Reger, R. N. Bhatt, A. P. Young

Published Apr 16, 1990• 86 citations
Publisher

Abstract

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 ${\mathit{T}}_{\mathit{c}}$, 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.

DOI
10.1103/physrevlett.64.1859

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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.
We describe results of Monte Carlo simulation studies on the Ising spin glass in four dimensions on a hypercubic lattice with nearest neighbor bonds. Studies of the equilibrium static properties show that the system undergoes a genuine phase transition to an ordered spin glass phase. Critical dynamical behavior is analyzed to obtain the dynamic exponent. Finally, we describe results on the spin glass phase, in particular the finite size scaling of the order parameter distribution function, and compare it with existing models of the spin glass phase, namely the droplet model and the Parisi solution for the low temperature phase of the infinite range spin glass.
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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 discuss the results of numerical simulations of two spin-glass models, the infinite range Sherrington–Kirkpatrick (SK) model, and a nearest–neighbor model in three dimensions. The probability distribution PN(q) for the spin glass order parameter q has been evaluated for the SK model below Tc with N, the number of spins, up to 192. It shows the same basic features as the Parisi function dx/dq as expected, but with sizeable finite size corrections. Results for the short-range model show evidence for a finite temperature transition but further work is needed to confirm this.
The infinite-range Ising spin-glass in the presence of a transverse field is studied by Monte Carlo simulations. The overlap probability distribution P ( q ) is defined and calculated for the present quantum case. The P ( q ) in the spin-glass phase has a long tail extended down to q = 0 qualitatively similar to the classical Sherrington-Kirkpatrick model. The phase boundary of the glass transition temperature vs. the transverse field is obtained. Our results for the phase diagram agree with our recent replica symmetry-breaking solution but disagree with the results of Yokata and of Ray et al.
We perform Monte Carlo simulations of the Ising spin glass at low temperature in three dimensions with a $\ifmmode\pm\else\textpm\fi{}J$ distribution of couplings. Our results display crossover scaling between T=0T=0T=0 behavior, where the order parameter distribution P(q)P(q)P(q) becomes trivial for $\stackrel{\ensuremath{\rightarrow}}{L}\ensuremath{\infty},$ and finite-T behavior, where the nontrivial part of P(q)P(q)P(q) has a much weaker dependence on L, and is possibly size independent.