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Physical Review Letters

Three-dimensional random-bond Ising model: Phase diagram and critical properties

Joseph D. Reger, Annette Zippelius

Published Dec 22, 1986• 24 citations
Publisher

Abstract

A dynamic high-temperature series expansion is used to study the three-dimensional random-bond Ising model as a function of temperature and concentration of ferromagnetic bonds. The phase boundaries for the paramagnetic, the ferromagnetic, and the spin-glass phases are determined. We discuss the static and dynamic critical behavior and verify the dynamic scaling hypothesis at the spin-glass transition.

DOI
10.1103/physrevlett.57.3225

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Journal of Applied Physics•Apr 15, 1985•21 citations
A dynamic high-temperature series expansion is used to study the three-dimensional random-bond Ising model as a function of temperature and concentration of ferromagnetic bonds. The phase boundaries for the paramagnetic, the ferromagnetic, and the spin-glass phases are determined. We discuss the static and dynamic critical behavior and verify the dynamic scaling hypothesis at the spin-glass transition.
The four-dimensional $\ifmmode\pm\else\textpm\fi{}J$ random-bond Ising model is studied using ground-state calculations. System sizes up to N=64{N=6}^{4}N=64 spins are considered. Here it is found that the ferromagnetic--spin-glass transition occurs at a critical concentration pc=0.28(1){p}_{c}=0.28(1)pc​=0.28(1) of the antiferromagnetic bonds, which is comparable to values found previously by high-temperature series expansions. The transition is characterized by a correlation-length exponent $\ensuremath{\nu}=1.0(1)$ and an order-parameter exponent $\ensuremath{\beta}=0.4(1).$ Thus, this transition is in a different universality class from four-dimensional bond percolation, where $\ensuremath{\nu}=0.678(50)$ and $\ensuremath{\beta}=0.639(20).$
We investigate the ferromagnetic-glassy transitions that separate the low-temperature ferromagnetic and spin-glass phases in the temperature-disorder phase diagram of three-dimensional Ising spin-glass models. For this purpose, we consider the cubic-lattice $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model with bond distribution $P(J)=p\ensuremath{\delta}(J\ensuremath{-}1)+(1\ensuremath{-}p)\ensuremath{\delta}(J+1)$, and present a numerical Monte Carlo study of the critical behavior along the line that marks the onset of ferromagnetism. The finite-size scaling analysis of the Monte Carlo data shows that the ferromagnetic-glassy transition line is slightly reentrant. As a consequence, for an interval of the disorder parameter ppp, around $p\ensuremath{\approx}0.77$, the system presents a low-temperature glassy phase, an intermediate ferromagnetic phase, and a high-temperature paramagnetic phase. Along the ferromagnetic-glassy transition line, magnetic correlations show a universal critical behavior with critical exponents $\ensuremath{\nu}=0.96(2)$ and $\ensuremath{\eta}=\ensuremath{-}0.39(2)$. The hyperscaling relation $\ensuremath{\beta}/\ensuremath{\nu}=(1+\ensuremath{\eta})/2$ is satisfied at the transitions, so that $\ensuremath{\beta}/\ensuremath{\nu}=0.305(10)$. This magnetic critical behavior represents a new universality class for ferromagnetic transitions in Ising-type disordered systems. Overlap correlations are apparently not critical and show a smooth behavior across the transition.
The phase diagram of modified random-bond Ising model has been investigated. The model is based on random-bond Ising model in which the distribution of magnetic interaction is Gaussian rather than the uniform distribution. The center of the distribution is positive so that most of the interactions are ferromagnetic. However the width of the distribution can be varied. The increasing of the width (J) induces disorder to this system. It is expected that when the tail of the distribution reaches the negative interaction region, the system at low temperature shows the cross over between ferromagnetic phase and spin glass phase. By using magnetization and spin glass magnetization, three ordered phases appear in the J-T phase diagram: (1) the ferromagnetic (FM) phase; (2) the spin glass (SG) phase; (3) the paramagnetic (PM) phase. At very high temperature the system is in paramagnetic phase. At low temperature and small distribution the system is in ferromagnetic phase. The cross over between ferromagnetic phase and spin glass phase is visible as the width of the interaction distribution is increase at low temperature. The critical temperature has been obtained by using Binder cumulant of spin glass magnetization.
An Ising model including field randomness and $\ifmmode\pm\else\textpm\fi{}J$ bond randomness is studied by renormalization-group theory in spatial dimensions d=2d=2d=2 and 3. In d=3,d=3,d=3, with field randomness but no $\ifmmode\pm\else\textpm\fi{}J$ randomness, ferromagnetic and paramagnetic phases occur with no intervening spin-glass phase. Also in d=3,d=3,d=3, at sufficient $\ifmmode\pm\else\textpm\fi{}J$ randomness, a spin-glass phase occurs, but is replaced by the paramagnetic phase for any nonzero random field, also implying its disappearance for any nonzero uniform field. In d=2,d=2,d=2, no ferromagnetic phase under random fields and no spin-glass phase occur. Global phase diagrams uniting the random-field and spin-glass problems are evaluated. The strong violation of critical phenomena universality, previously found adjoining random-bond tricriticality in d=3,d=3,d=3, is seen here in d=2.d=2.d=2.
Large numbers of ground states of the three-dimensional $\ifmmode\pm\else\textpm\fi{}J$ random-bond Ising model are calculated for sizes up to 143{14}^{3}143 using a combination of a genetic algorithm and cluster-exact approximation. Several quantities are calculated as functions of the concentration p of the antiferromagnetic bonds. The critical concentration where the ferromagnetic order disappears is determined using the Binder cumulant of the magnetization. A value of ${p}_{c}=0.222\ifmmode\pm\else\textpm\fi{}0.005$ is obtained. From the finite-size behavior of the Binder cumulant and the magnetization critical exponents \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.3 and \ensuremath{\beta}=0.2\ifmmode\pm\else\textpm\fi{}0.1 are calculated. The behavior of the distribution of overlaps P(q)P(q)P(q) is used to investigate how the spin-glass phase evolves with increasing concentration p.p.p. The spin-glass order is characterized by a broad distribution of overlaps which extends down to q=0.q=0.q=0.
First results of massive Monte Carlo simulations of the d=3 Ising spin glass with ± J bond distribution, performed on a fast special purpose computer, are discussed. Equilibrium static and dynamic correlation functions have been recorded for temperatures T/J=1.10 to 2.00. A qualitative change in the behavior of the system, and the best fits for the spin glass correlation length and relaxation time favor a continuous phase transition at Tc/J≊1.20±0.05. We find no evidence for the Tc=0 freezing picture.