This review summarizes recent developments in the theory of spin glasses, as well as pertinent experimental data. The most characteristic properties of spin glass systems are described, and related phenomena in other glassy systems (dielectric and orientational glasses) are mentioned. The Edwards-Anderson model of spin glasses and its treatment within the replica method and mean-field theory are outlined, and concepts such as "frustration," "broken replica symmetry," "broken ergodicity," etc., are discussed. The dynamic approach to describing the spin glass transition is emphasized. Monte Carlo simulations of spin glasses and the insight gained by them are described. Other topics discussed include site-disorder models, phenomenological theories for the frozen phase and its excitations, phase diagrams in which spin glass order and ferromagnetism or antiferromagnetism compete, the Ne\'el model of superparamagnetism and related approaches, and possible connections between spin glasses and other topics in the theory of disordered condensed-matter systems.

Spin glasses: Experimental facts, theoretical concepts, and open questions

We introduce a new method, which does not use replicas, from which we recover all the results of the replica symmetry-breaking solution of the Sherrington-Kirkpatrick model.

https://robots.iopscience.iop.org/article/10.1209/0295-5075/1/2/006/pdf

SK Model: The Replica Solution without Replicas

The generalised random energy model (GREM) is a spin-glass model which can be solved exactly. One can impose arbitrary pair correlations between the energies of configurations. For several examples (the Sherrington-Kirkpatrick model, the p spin-glass model, the Potts glass, spin-glass models on finite-dimensional lattices) the authors calculate the pair correlation between energies and solve the corresponding GREM. In all cases, the free energy of the GREM corresponding to a spin-glass model on a given lattice, has a simple expression in terms of the specific heat of the pure ferromagnetic model on the same lattice. Lastly they compare the correlations between three energy levels in the GREM and in spin-glass models.

https://iopscience.iop.org/article/10.1088/0022-3719/19/13/015/pdf

Solution of the generalised random energy model

The Curie temperature of ferroelectric films described by the transverse Ising model was studied under the mean-field theory. The film layer number, the surface interaction and the surface layer number dependence of the Curie temperature were obtained. There is a critical surface interaction strength, which is JSC=1.25 J for single-surface-layer films, and JSC=1.078 J for multiple-surface-layer films. If the surface interaction strength exceeds the critical value, there exists an optimum film thickness which possesses the maximum Curie temperature; then the surface interaction strength is weaker than the critical value, the Curie temperature decreases monotonically with increasing film thickness, and there exist critical sizes or critical thicknesses at which the ferroelectricity will disappear if the surface interaction is weak enough.

https://iopscience.iop.org/article/10.1088/0953-8984/4/19/014/pdf

The Curie temperature of ultra-thin ferroelectric films

It has been well established that spatially extended, bistable systems that are driven by an oscillating field exhibit a nonequilibrium dynamic phase transition (DPT). The DPT occurs when the field frequency is of the order of the inverse of an intrinsic lifetime associated with the transitions between the two stable states in a static field of the same magnitude as the amplitude of the oscillating field. The DPT is continuous and belongs to the same universality class as the equilibrium phase transition of the Ising model in zero field [G. Korniss et al., Phys. Rev. E 63, 016120 (2001); H. Fujisaka et al., Phys. Rev. E 63, 036109 (2001)]. However, it has previously been claimed that the DPT becomes discontinuous at temperatures below a tricritical point [M. Acharyya, Phys. Rev. E 59, 218 (1999)]. This claim was based on observations in dynamic Monte Carlo simulations of a multipeaked probability density for the dynamic order parameter and negative values of the fourth-order cumulant ratio. Both phenomena can be characteristic of discontinuous phase transitions. Here we use classical nucleation theory for the decay of metastable phases, together with data from large-scale dynamic Monte Carlo simulations of a two-dimensional kinetic Ising ferromagnet, to show that these observations in this case are merely finite-size effects. For sufficiently small systems and low temperatures, the continuous DPT is replaced, not by a discontinuous phase transition, but by a crossover to stochastic resonance. In the infinite-system limit, the stochastic-resonance regime vanishes, and the continuous DPT should persist for all nonzero temperatures.

https://arxiv.org/pdf/cond-mat/0207275.pdf

Absence of first-order transition and tricritical point in the dynamic phase diagram of a spatially extended bistable system in an oscillating field.

We formulate a theory of single-spin-flip dynamics for the infinite-rangeq-state Potts model. We derive a Fokker-Planck equation, without diffusive term, from a phenomenological master equation. It describes the approach to equilibrium of the time-dependent probability density and thus generalizes Griffiths' (1966) result for the Ising model. We particularly compare the dynamic evolutions ofq=2 andq=3 systems when sinusoidal external fields are applied. In the caseq=2 we find evidence of a nonequilibrium phase transition and forq=3 period doubling bifurcations are observed, yielding a good estimate of Feigenbaum's universal exponent.

Dynamics of the infinite-ranged Potts model

The s-state Potts model with a gaussian distribution of pair couplings is found to possess, in the infinite-range limit, a replica symmetric (RS) solution that exhibits a spin glass transition which appears to be second order for s 2, the stability of the PM phase breaks down above the transition temperature indicated by the RS solution suggesting a first-order transition which, however, cannot be located by the RS solution.

The infinite-ranged Potts spin glass model

The stability conditions for replica-generated solutions of Ising spin glass models with general spin magnitude (S), anisotropy energy and gaussian distributed exchange are derived and discussed. The phase diagrams for integer S and half-integer S are argued to be qualitatively different when the anisotropy constant is negative. The spin glass phase (without symmetry breaking) of the S=1 model is shown to be everywhere unstable and the authors argue that the symmetric solution yields wrong first-order transition lines.

http://iopscience.iop.org/article/10.1088/0022-3719/15/33/003/pdf

Stability conditions of generalised Ising spin glass models

An effective-medium theory is presented for the substitutionally disordered transverse Ising model above its transition. The equations of motion for the relevant Green functions are decoupled in RPA. The disorder then occurs through a local parameter and can be treated in the spirit of the single-site coherent potential approximation. Fully self-consistent results are computed at the transition for the transition temperature, densities of states and spectral function. Simple features of the results can be understood using a generalized virtual-crystal approximation which accounts for damping due to disorder. Extensions of the theory to include intrinsic damping, more than one spin per unit cell and coupling to phonons are discussed.

https://iopscience.iop.org/article/10.1088/0022-3719/9/17/022/pdf

Transverse Ising model with substitutional disorder: an effective-medium theory

An extension of the usual diagrammatic coherent-potential approximation is considered to account for the presence of disorder both in off-diagonal and inhomogeneous terms. The method is applied to the quenched site-disordered Ising model (S=1/2) above the transition, where the fundamental equation for the correlation function is a generalized random-phase approximation. The results for the initial slope of the critical temperature and the critical concentration are presented for the simple cubic lattice with nearest-neighbour interactions, and they are in reasonable agreement with the values derived by other methods. The differences are studied and a discussion is presented to show that, near the critical concentration, terms not considered in the coherent-potential approximation are as important as the terms included. The application of this method to other models is also briefly discussed and, in the particular case of the Heisenberg model, it is shown that the method reproduces the basic results derived by Harris et al. (1974) using an effective-medium approach.

https://iopscience.iop.org/article/10.1088/0022-3719/10/2/013/pdf

E J S Lage

Papers

Dynamics of the infinite-ranged Potts model

The infinite-ranged Potts spin glass model

Stability conditions of generalised Ising spin glass models

Transverse Ising model with substitutional disorder: an effective-medium theory

A generalized coherent-potential approximation for site-disordered spin systems