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Physical Review Letters

Critical Properties of Spin-Glasses

A. Brooks Harris, T. C. Lubensky, Jing-Huei Chen

Published Feb 23, 1976• 234 citations
Publisher

Abstract

The critical properties of the model of a spin-glass proposed by Edwards and Anderson are studied using the renormalization group. The critical exponents are calculated in $6\ensuremath{-}\ensuremath{\epsilon}$ spatial dimensions. It is argued that a tricritical point can exist where the nonordering field is the skewness of the distribution of $J$.

DOI
10.1103/physrevlett.36.415

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The critical properties of the model of a spin-glass proposed by Edwards and Anderson are studied using the renormalization group. The critical exponents are calculated in 6\ensuremath−\ensuremathϵ6\ensuremath{-}\ensuremath{\epsilon}6\ensuremath−\ensuremathϵ spatial dimensions. It is argued that a tricritical point can exist where the nonordering field is the skewness of the distribution of JJJ.
It is well known that critical phenomena occur in condensed matter under certain conditions, when an abrupt change in its properties occurs. In the vicinity of critical points, various phenomena may arise. The critical region can be described by a set of state parameters (order parameters), which allow one to obtain information about the anomalous behavior of thermodynamic averages, internal processes, and the nature of the objects of study. The abnormal nonlinear behavior of state parameters is described by critical exponents. In this article, we considered spin glass on the example of the Edwards-Anderson model. For the simulation, the replica-exchange Monte-Carlo method was used. Critical exponents were obtained to describe the behavior of the model in the critical region.
By mapping the Edwards-Anderson model onto a nonrandom Ising model we argue that the lower critical dimension d∗{d}^{*}d∗ for the Edwards-Anderson spin-glass is close to 3 and may depend on the probability distribution of the bond strengths. For d>d∗d>{d}^{*}d>d∗, the approach to spin-glass order from the paramagnetic phase is characterized by Ising exponents. At d=d∗d={d}^{*}d=d∗, however, the critical exponents defined for T>TcT>{T}_{c}T>Tc​ are twice the corresponding Ising ones. These results are consistent with recent computer simulations.
A Landau-Ginzburg phenomenological free energy for the Edwards and Anderson spin-glass model when there is competition between spin-glass and ferromagnetic ordering is developed. This free energy obtained with the use of the $n\ensuremath{\rightarrow}0$ replication procedure is analyzed using mean-field theory and the $\ensuremath{\epsilon}$ expansion. Critical exponents for the ferromagnetic-spin-glass multicritical point are calculated in $6\ensuremath{-}\ensuremath{\epsilon}$ dimensions. For Ising systems, $\ensuremath{\nu}=\frac{1}{2}+(\frac{2}{3})\ensuremath{\epsilon}$ and $\ensuremath{\varphi}=1+(\frac{1}{2})\ensuremath{\epsilon}$. For XY\mathrm{XY}XY and Heisenberg systems, these exponents are complex. This result is not fully understood. The Harris-Plischke-Zuckermann model for amorphous magnetism is shown to have an Ising-like spin-glass fixed point in high enough dimension.
The Edwards-Anderson model of a spin glass is studied by position-space renormalization-group techniques, using an inhomogeneous generalization of Migdal's approximate recursion relation. We treat the spin-1/2 Ising model with independently random nearest-neighbor interactions in dimensionalities d=2,3,and4d=2, 3, \mathrm{and} 4d=2,3,and4. The phase diagram, which is in qualitative agreement with mean-field results, exhibits paramagnetic, ferromagnetic, antiferromagnetic, and spin-glass phases. The spin-glass and paramagnetic phases meet along an extended second-order phase boundary, which terminates in two tricritical points. Critical and tricritical exponents are calculated. The spin-glass specific-heat exponent turns out to be large and negative, compatibly with recent experiments which show a rounded specific-heat anomaly.
We present numerical simulations of the four-dimensional Edwards Anderson Ising spin glass with binary couplings. Our results, in the midst of strong finite-size effects, suggest the existence of a spin-glass phase transition. We present a preliminar determination of critical exponents. We discuss spin-glass susceptibilities, cumulants of the overlap and energy overlap probability distributions, finite-size effects, and the behaviour of the disorder-dependent and integrated probability distributions.
Continuous phase transitions are catalogued into universality classes, families of systems having identical values of all the exponents governing the critical behaviour of their different physical properties. Numerical simulations have been carried out on Ising Spin Glasses in dimension three, using a technique where corrections to finite size scaling can be controled. The data show that the critical exponents vary strongly as a function of the kurtosis of the interaction distribution, a parameter which from the standard point of view should not be pertinent. This observation implies that for spin glasses the renormalization group analysis should not be approached in a the same way as in the case of canonical second order transitions; a much richer structure of universality classes would appear to exist for spin glasses.