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Journal of the Physical Society of Japan

Critical Behavior of the Three-Dimensional ±JModel in a Magnetic Field

Naoki Kawashima, Nobuyasu Ito

Published Feb 15, 1993• 65 citations
Publisher

Abstract

The three-dimensional ± J Ising spin glass model is investigated by means of Monte Carlo simulation. In uniform fields of various strength, the Edwards-Anderson susceptibility χ EA Q is calculated at T =0.9 J which is lower than the transition temperature in the zero-field case. It is found that χ EA decreases monotonically when the field strength is increased and that the data do not suggest a phase transition in a finite field. The best parameters for the scaling function of the form \(\chi _{\rm EA}(H)\simeq l^{\psi}\tilde{\chi}_{\rm EA}((H-H_{\rm c})l^{-\phi})\) are estimated to be H c =0.00(10) J , ψ=2.16(22) and φ=-1.13(6). These results support the hypothesis that the phase transition does not exist in a finite magnetic field. \sloppy

DOI
10.1143/jpsj.62.435

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The three-dimensional ± J Ising spin glass model is investigated by means of Monte Carlo simulation. In uniform fields of various strength, the Edwards-Anderson susceptibility χ EA Q is calculated at T =0.9 J which is lower than the transition temperature in the zero-field case. It is found that χ EA decreases monotonically when the field strength is increased and that the data do not suggest a phase transition in a finite field. The best parameters for the scaling function of the form (\chi {\rm EA}(H)\simeq l^{\psi}\tilde{\chi}{\rm EA}((H-H_{\rm c})l^{-\phi})) are estimated to be H c =0.00(10) J , ψ=2.16(22) and φ=-1.13(6). These results support the hypothesis that the phase transition does not exist in a finite magnetic field. \sloppy
We present results of a Monte Carlo simulation study of the zero-temperature quantum phase transition of a three-dimensional short-range spin-1/2 Ising spin glass, driven by a transverse magnetic field. This quantum transition is equivalent to the finite-temperature transition in the (3+1)-dimensional anisotropic classical random Ising model that is the path integral of the quantum system. The critical exponents are estimated using a finite-size scaling analysis. The uniform linear susceptibility is finite at the transition, while the nonlinear susceptibility diverges. The results are more consistent with conventional, rather than activated, dynamic scaling in the quantum system.
The three-dimensional Ising spin-glass in zero field with nearest-neighbor interactions having \ifmmode\pm\else\textpm\fi{}J distribution is studied by Monte Carlo simulations for samples of linear dimension L with 3\ensuremath{\le}L\ensuremath{\le}20. Results for the probability distribution, PL{\mathrm{P}}_{\mathrm{L}}PL​(q), of the spin-glass order parameter are analyzed by finite-size scaling. Data for T\ensuremath{\ge}1.2 are consistent with a conventional phase transition at Tc{\mathrm{T}}_{\mathrm{c}}Tc​${=1.2}_{\mathrm{\ensuremath{-}}0.2}^{+0.1}$, with exponents \ensuremath{\nu}=1.3\ifmmode\pm\else\textpm\fi{}0.3 and \ensuremath{\eta}=-0.3\ifmmode\pm\else\textpm\fi{}0.2. However, results at lower temperature indicate marginal behavior and suggest that the lower critical dimensional is close to three.
We study spin glass behavior in a random Ising Coulomb antiferromagnet in two and three dimensions using Monte Carlo simulations. In two dimensions, we find a transition at zero temperature with critical exponents consistent with those of the Edwards-Anderson model, though with large uncertainties. In three dimensions, evidence for a finite-temperature transition, as occurs in the Edwards-Anderson model, is rather weak. This may indicate that the sizes are too small to probe the asymptotic critical behavior, or possibly that the universality class is different from that of the Edwards-Anderson model and has a lower critical dimension equal to three.
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 three-dimensional (3D) bond-diluted Edwards-Anderson (EA) model with binary interactions at a bond occupation of 45% by Monte Carlo (MC) simulations. Using an efficient cluster MC algorithm we are able to determine the universal finite-size scaling (FSS) functions and the critical exponents with high statistical accuracy. We observe small corrections to scaling for the measured observables. The critical quantities and the FSS functions indicate clearly that the bond-diluted model for dilutions above the critical dilution p∗{p}^{*}p∗, at which a spin glass (SG) phase appears, lies in the same universality class as the 3D undiluted EA model with binary interactions. A comparison with the FSS functions of the 3D site-diluted EA model with Gaussian interactions at a site occupation of 62.5% gives very strong evidence for the universality of the SG transition in the 3D EA model.
We present a scaling theory for the low-temperature phase transition of the Ising spin glass in a transverse magnetic field (\ensuremath{\Gamma}). The theory provides relations and bounds on some critical exponents, which are supported by results from actual calculations. A renormalization-group analysis of the zero-temperature transition in the presence of a longitudinal symmetry-breaking field enables us to estimate the critical exponents \ensuremath{\beta} and \ensuremath{\gamma}, associated with the Edwards-Anderson order parameter and the nonlinear susceptibility, respectively. We have found no indications of the transition being of first order.