J. D. Reger

Theory of orientational glasses models, concepts, simulations

Abstract This review describes the various attempts to develop a theoretical understanding for ordering and dynamics of randomly diluted molecular crystals, where quadrupole moments freeze in random orientations upon lowering the temperature, as a result of randomness and competing interactions. While some theories attempt to model this freezing into a phase with randomly oriented quadrupole moments in terms of a bond-disorder concept analogous to the Edwards-Anderson model of spin glasses, other theories attribute the freezing to random field-like terms in the Hamiltonian. While models of the latter type have been studied primarily by microscopic molecular field-type treatments, the former models have been treated both in the Sherrington-Kirkpatrick-Parisi infinite-range limit, and in the short-range case. Among the surprising findings of these treatments we emphasize the first-order glass transition (though lacking a latent heat) of the infinite-range Potts glass, the suggestion that the short-range Pot...

Finite size effects at thermally-driven first order phase transitions: A phenomenological theory of the order parameter distribution

We consider the rounding and shifting of a firstorder transition in a finited-dimensional hypercubicLd geometry,L being the linear dimension of the system, and surface effects are avoided by periodic boundary conditions. We assume that upon lowering the temperature the system discontinuously goes to one ofq ordered states, such as it e.g. happens for the Potts model ind=3 forq≧3, with the correlation length ξ of order parameter fluctuation staying finite at the transition. We then describe each of theseq ordered phases and the disordered phase forL≫ξ by a properly weighted Gaussian. From this phenomenological ansatz for the total distribution of the order parameter, all moments of interest are calculated straight-forwardly. In particular, it is shown that forL exceeding a characteristic minimum sizeLmin the forthorder cumulantgL(T) exhibits a minimum atTmin>Tc, withTmin−Tc∝L−d and the value of the cumulant and the minimum (g(Tmin)) behaving asg(Tmin)∝L−d. All cumulantsgL(T) forL≫ξ approximately intersect at a common crossing pointTcross∝L−2d, with a universal valueg(Tcross)=1−n/2q, wheren is the order parameter dimensionality. By searching for such a behavior in numerical simulation data, the first order character of a phase transition can be asserted. The usefulness of this approach is shown using data for theq=3,d=3 Potts ferromagnet.

Monte Carlo Methods For First Order Phase Transitions: Some Recent Progress

This brief review discusses methods to locate and characterize first order phase transitions, paying particular attention to finite size effects. In the first part, the order parameter probability distribution and its fourth-order cumulant is discussed for thermally driven first-order transitions (the 3-state Potts model in d=3 dimensions is treated as an example). First-order transitions are characterized by a minimum of the cumulant, which gets very deep for large enough systems. In the second part, we discuss how to locate first order phase boundaries ending in a critical point in a large parameter space. As an example, the study of the unmixing transition of asymmetric polymer mixtures by a combination of histogram techniques and finite size scaling is described.As a final problem, we discuss the shift of the gas-liquid condensation in thin-film geometry confined between two parallel plates due to boundary fields (“capillary condensation”). Being interested in temperatures far below bulk criticality (e. g. near the wetting transition), special thermodynamic integration techniques are the method of choice, rather than the use of finite sizes scaling to map out the (asymmetric) phase diagram.

Dynamical Critical Exponent of the Two-Dimensional Ising Model

We derive twenty nontrivial terms of the high-temperature series expansion for the linear relaxation time τ of the time-displaced correlation function C(t) = m(0) m(t) of the magnetization m(t) in the two-dimensional nearest-neighbour ferromagnetic Ising model on the square lattice. We study the dynamics introduced by Glauber and compute the longest (characteristic) relaxation time of C(t). We analyse the series by using unbiased and biased methods, such as the ratio method, Pade approximants and generalized differential approximants. It is reassuring that all the methods yield compatible results providing the estimate for the dynamical critical exponent: z = 2.183 ± 0.005.

Monte Carlo study of a vortex glass model

We describe results of Monte Carlo simulations on a model that seems to have the necessary ingredients to describe a disordered type-II superconductor in a magnetic field. We compute the free energy cost to twist the direction of the phase of the condensate and analyse the results by finite-size scaling. The results show convincingly that the model has different behaviour as a function of dimension: in d=4 the model clearly has a finite transition temperature; Tc, while for d=2 only there is only a transition at T=0.

Critical behavior of short range Potts glasses

We study by means of Monte Carlo simulations and the numerical transfer matrix technique the critical behavior of the short rangep=3 state Potts glass model in dimensionsd=2,3,4 with both Gaussian and bimodal (±J) nearest neighbor interactions on hypercubic lattices employing finite size scaling ideas. Ind=2 in addition the degeneracy of the glass ground state is computed as a function of the number of Potts states forp=3, 4, 5 and compared to that of the antiferromagnetic ground state. Our data indicate a transition into a glass phase atT=0 ind=2 with an algebraic singularity, aT=0 transition ind=3 with an essential singularity of the form χ∼exp(const.T−2), and an algebraic singularity atT≈0.25 ind=4. We conclude that the lower critical dimension of the present model isdc=3 or very close to it. Some of the critical exponents are estimated and their respective values discussed.

Evidence for a Disordered Ground State in the Potts Glass with Gaussian Couplings

Numerical transfer-matrix calculations are presented for the two-dimensional p-state Potts glass with Gaussian couplings, for p = 3 and p = 4. From these results we estimate the ground-state entropy per spin in the thermodynamic limit to be S0 = 0.0127 (p = 3) and S0 = 0.1318 (p = 4). This is to be contrasted with the Ising spin glass case (p = 2) for which S0 = 0 for any continuous distribution of the couplings. At T = 0 the glass correlation function decays to zero with an inverse power of the distance.

Dynamics of Ising spin glasses far below the lower critical dimension: The one-dimensional case and small clusters

The Glauber model is studied for symmetric distributions (±J and gaussian) of the nearest-neighbour interactionJ, including a magnetic field. For small clusters of spins (closed rings ofN bonds, withN≦7) the complex magnetic susceptibility χ(ω) and the time-dependent remanent magnetizationm(t) are found exactly for given bond configurations {Jij} by diagonalization of the Liouville operator; apart from the ±J model, the average over {Jij} must be done numerically by simple random sampling Monte Carlo. Nevertheless our accuracy is much better than corresponding dynamic Monte Carlo simulations, even if one considers the extrapolation toN→∞.We analyze the results along the lines of corresponding experimental work, studying the frequency-dependence of the peak in χ(ω), theT lnt-scaling ofm(t) at low temperaturesT, and the decomposition of χ(ω) into a spectrum of relaxation times. Many results are strikingly similar to experimental data for systems such as the Holmium-Borate “spinglass” or the superparamagnet Eu0.05Sr0.95S, for instance. Frequency-dependent critical fieldsHc(ω) in theH-T plane are also identified but do not have the familiar Almeida-Thouless shape, however.

Static magnetic response of the three-state Potts glass☆

Abstract The three-state Potts glass in a three-dimensional cubic lattice in a homogeneous external field was investigated. The temperature and field dependence of the magnetization, my, and the first and second derivative, χn, with respect to the field, h, were measured. For χ1, the data of a Monte Carlo simulation are in good agreement with a high temperature series. Contrary to the Ising spin glass, χ2 does not vanish.

Critical behavior of diluted Heisenberg ferromagnets with competing interactions

The pure and the site-diluted classical Heisenberg model on the face centered cubic (fcc) lattice with ferromagnetic exchange Jnn between nearest neighbors and antiferromagnetic exchange Jnn = −Jnn/2 between next nearest neighbors is studied by Monte Carlo simulation. Data are generated by the heat bath algorithm for lattice sizes L = 4, 8, 12, 16, 20 and 24, using histogram reweighting techniques and sampling up to several hundred configurations of the random site disorder. From a finite size scaling analysis both the critical temperature and the critical exponents are estimated. For the pure system, the data are in very good agreement with the critical exponent estimates 1/v ≈ 1.42, β/v ≈ 0.51 obtained from other methods (as a check of the accuracy of our approach, we also study the nearest neighbor model — where Jnn ≡ 0− and again obtain very good agreement with the known behavior). However, for the diluted systems evidence for a new universality class is found. While for concentration c = 0.875 of occupied sites strong crossover phenomena preclude us from giving exponent estimates, for c = 0.75 we find 1/v ≈ 1.2 and β/v ≈ 0.45. Possible reasons why the Harris criterion may not apply for this system are discussed. The application of this study to experiments on EuxSr1−xS is briefly mentioned.