Loop effects in the Ising spin glass on the Bethe-like lattices

Equations for the spin glass order in the Ising spin glass model on the Bethe-like lattices with and without small loops are studied. For each lattice, equations are obtained by using and not using the replica method. Within the replica symmetric approximation, equations obtained by the two ways are shown to be identical. To see the effects of the small loops and the replica symmetry breaking, a spin glass order parameter is investigated as a function of the connectivity of the lattices close to the transition temperature. Replica symmetry breaking is enhanced by the existence of small loops.     

The Marginally Stable Bethe Lattice Spin Glass Revisited

Bethe lattice spins glasses are supposed to be marginally stable, i.e. their equilibrium probability distribution changes discontinuously when we add an external perturbation. So far the problem of a spin glass on a Bethe lattice has been studied only using an approximation where marginal stability is not present, which is wrong in the spin glass phase. Because of some technical difficulties, attempts at deriving a marginally stable solution have been confined to some perturbative regimes, high connectivity lattices or temperature close to the critical temperature. Using the cavity method, we propose a general non-perturbative approach to the Bethe lattice spin glass problem using approximations that should be hopefully consistent with marginal stability.     

Replica symmetry breaking for the Ising spin glass within cluster approximations

Within a class of cluster approximations, the Ising spin glass model on a d-dimensional hypercubic lattice is solved near the spin glass transition temperature. Spin glass order parameter function and Almeida-Thouless line are obtained.     

Behavior of Ising spin glasses in a magnetic field

We study the existence of a spin-glass phase in a field using Monte Carlo simulations performed along a nontrivial path in the field-temperature plane that must cross any putative de Almeida-Thouless instability line. The method is first tested on the Ising spin glass on a Bethe lattice where the instability line separating the spin glass from the paramagnetic state is also computed analytically. While the instability line is reproduced by our simulations on the mean-field Bethe lattice, no such instability line can be found numerically for the short-range three-dimensional model.     

Cooperative Behavior of Kinetically Constrained Lattice Gas Models of Glassy Dynamics

Kinetically constrained lattice models of glasses introduced by Kob and Andersen (KA) are analyzed. It is proved that only two behaviors are possible on hypercubic lattices: either ergodicity at all densities or trivial non-ergodicity, depending on the constraint parameter and the dimensionality. But in the ergodic cases, the dynamics is shown to be intrinsically cooperative at high densities giving rise to glassy dynamics as observed in simulations. The cooperativity is characterized by two length scales whose behavior controls finite-size effects: these are essential for interpreting simulations. In contrast to hypercubic lattices, on Bethe lattices KA models undergo a dynamical (jamming) phase transition at a critical density: this is characterized by diverging time and length scales and a discontinuous jump in the long-time limit of the density autocorrelation function. By analyzing generalized Bethe lattices (with loops) that interpolate between hypercubic lattices and standard Bethe lattices, the crossover between the dynamical transition that exists on these lattices and its absence in the hypercubic lattice limit is explored. Contact with earlier results are made via analysis of the related Fredrickson--Andersen models, followed by brief discussions of universality, of other approaches to glass transitions, and of some issues relevant for experiments.     

The Bethe lattice spin glass revisited

So far the problem of a spin glass on a Bethe lattice has been solved only at the replica symmetric level, which is wrong in the spin glass phase. Because of some technical difficulties, attempts at deriving a replica symmetry breaking solution have been confined to some perturbative regimes, high connectivity lattices or temperature close to the critical temperature. Using the cavity method, we propose a general non perturbative solution of the Bethe lattice spin glass problem at a level of approximation which is equivalent to a one step replica symmetry breaking solution. The results compare well with numerical simulations. The method can be used for many finite connectivity problems appearing in combinatorial optimization. Received 27 September 2000     

Cyclic period-3 window in antiferromagnetic potts and Ising models on recursive lattices

The magnetic properties of the antiferromagnetic Potts model with two-site interaction and the antiferromagnetic Ising model with three-site interaction on recursive lattices have been studied. A cyclic period-3 window has been revealed by the recurrence relation method in the antiferromagnetic Q-state Potts model on the Bethe lattice (at Q < 2) and in the antiferromagnetic Ising model with three-site interaction on the Husimi cactus. The Lyapunov exponents have been calculated, modulated phases and a chaotic regime in the cyclic period-3 window have been found for one-dimensional rational mappings determined the properties of these systems.     

A mean field spin glass with short-range interactions

We formulate and study a spin glass model on the Bethe lattice. Appropriate boundary fields replace the traditional self-consistent methods; they give our model well-defined thermodynamic properties. We establish that there is a spin glass transition temperature above which the single-site magnetizations vanish, and below which the Edwards-Anderson order parameter is strictly positive. In a neighborhood below the transition temperature, we use bifurcation theory to establish the existence of a nontrivial distribution of single-site magnetizations. Two properties of this distribution are studied: the leading perturbative correction to the Gaussian scaling form at the transition, and the (nonperturbative) behavior of the tails.Research supported by the NSF under Grant No. DMR-8314625Research supported by the DOE under Grant No. DE-AC02-83ER13044Research supported by the NSF under Grant No. DMR-8503544Research supported by the NSF under Grant No. DMR-8319301     

Alternative Variational Approach to Cactus Lattices

In this paper, I will present an alternative approach to the Bethe or cactus lattice approximation, widely employed in the theory of cooperative phenomena. This approach relies on a variational free energy, which is equivalent to the Bethe free energy in that it has the same stationary points, but allows one to simplify analytical calculations, since it is a function of only single-site probability distributions, in the same way as an ordinary mean-field (Bragg-Williams) free energy. As an application, I shall discuss a derivation of closed-form equations for critical points in Ising-like models. Moreover, I will suggest a rule of thumb to choose the cactus lattice connectivity yielding the best approximation for the corresponding model defined on an ordinary lattice. PACS Numbers: 05.20.-y, 05.50.+q, 05.70.Fh, 64.60.-i, 64.60.Cn     

On the bethe approximation for the random bond Ising model

A random bond Ising model is considered in terms of the pair approximation, which is equivalent to the Bethe approximation, of the cluster variation method. On taking the configurational average over the random distribution of bonds ±J, we take into account the nearest neighbor correlations between effective fields and bonds. We investigate their effects to the phase transition temperature from the paramagnetic phase to the ferro- (or antiferro-) magnetic phase and to the spin glass phase for the Ising model on the square lattice. It turns out that the correlation effects act favorably to the spin glass phase and bend upward the line of transition temperature from the paramagnetic phase to the spin glass phase as the concentration being apart from 0.5. In the appendix, we derive the expression of free energy in the weak interaction limit.     

Slowing down in the three-dimensional three-state Potts glass with nearest neighbor exchange ± J: A Monte Carlo study

,Static and dynamic properties of the Potts model on the simple cubic lattice with nearest neighbor -interaction are obtained from Monte Carlo simulations in a temperature range where full thermal equilibrium still can be achieved (). For a lattice size L = 16, in this range finite size effects are still negligible, but the data for the spin glass susceptibility agree with previous extrapolations based on finite size scaling of very small lattices. While the static properties are compatible with a zero temperature transition, they certainly do not prove it. Unlike the Ising spin glass, the decay of the time-dependent order parameter is compatible with a simple Kohlrausch function, , while a power law prefactor cannot be distinguished. The Kohlrausch exponent y ( T ) decreases from at [0pt] to at [0pt] however. The relaxation time is compatible with the exponential divergence postulated by McMillan for spin glasses at their lower critical dimension, but the exponent that can be extracted still differs significantly from the theoretical value, . Thus the present results support the conclusion that the Potts spin glass in d = 3 dimensions differs qualitatively from the Ising spin glass. Received: 8 October 1997 / Accepted: 27 November 1997     

Variational principle for regular and random Ising models on the cactus tree or on the usual lattice in the “cactus approximation”

The distribution functions and the free energy are expressed in terms of the effective fields for the regular and random Ising models of an arbitrary spin S on the generalized cactus tree. The same expressions apply to systems on the usual lattice in the “cactus approximation” in the cluster variation method. For an ensemble of random Ising models of an arbitrary spin S on the generalized cactus tree, the equation for the probability distribution function of the effective fields is set up and the averaged free energy is expressed in terms of the probability distribution. The same expressions apply to the system on the usual lattice in the “cactus approximation”. We discuss the quantities on the usual lattice when the system or the ensemble of random systems has the translational symmetry. Variational properties of the free energy for a system and of the averaged free energy for an ensemble of random systems are noted. The “cactus approximations” are applicable to the Heisenberg model as well as to the Ising model of an arbitrary spin, and to ensembles of random systems of these models.     

Exact solution of the general non-intersecting string model

We present a thorough analysis of the non-intersecting string (NIS) model and its exact solution. This is an integrable q-states vertex model describing configurations of non-intersecting polygons on the lattice. The exact eigenvalues of the transfer matrix are found by the analytic Bethe ansatz. The Bethe ansatz equations thus found are shown to be equivalent to those for a mixed spin model involving both and infinite spin. This indicates that the NIS model provides a representation of the quantum group corresponding to spins and s = ∞. The partition function and the excitations in the thermodynamic limit are computed.     

Dynamic facilitation picture of a higher-order glass singularity

We show that facilitated spin mixtures with a tunable facilitation reproduce, on a Bethe lattice, the simplest higher-order singularity scenario predicted by the mode-coupling theory (MCT) of liquid-glass transition. Depending on the facilitation strength, they yield either a discontinuous glass transition or a continuous one, with no underlying thermodynamic singularity. Similar results are obtained for facilitated spin models on a diluted Bethe lattice. The mechanism of dynamical arrest in these systems can be interpreted in terms of bootstrap and standard percolation and corresponds to a crossover from a compact to a fractal structure of the incipient spanning cluster of frozen spins. Theoretical and numerical simulation results are fully consistent with MCT predictions.     

Analytic theory of the eight-vertex model

We observe that the exactly solved eight-vertex solid-on-solid model contains an hitherto unnoticed arbitrary field parameter, similar to the horizontal field in the six-vertex model. The parameter is required to describe a continuous spectrum of the unrestricted solid-on-solid model, which has an infinite-dimensional space of states even for a finite lattice. The introduction of the continuous field parameter allows us to completely review the theory of functional relations in the eight-vertex/SOS-model from a uniform analytic point of view. We also present a number of analytic and numerical techniques for the analysis of the Bethe ansatz equations. It turns out that different solutions of these equations can be obtained from each other by analytic continuation. In particular, for small lattices we explicitly demonstrate that the largest and smallest eigenvalues of the transfer matrix of the eight-vertex model are just different branches of the same multivalued function of the field parameter.     

Reentrant phase transitions and multicompensation points in the mixed-spin Ising ferrimagnet on a decorated Bethe lattice

A mixed-spin Ising model on a decorated Bethe lattice is rigorously solved by combining the decoration–iteration transformation with the method of exact recursion relations. Exact results for critical lines, compensation temperatures, total and sublattice magnetizations are obtained from a precise mapping relationship with the corresponding spin-1/2 Ising model on a simple (undecorated) Bethe lattice. The effect of next-nearest-neighbour interaction and single-ion anisotropy on magnetic properties of the ferrimagnetic model is investigated in particular. It is shown that the total magnetization may exhibit multicompensation phenomenon and the critical temperature vs. the single-ion anisotropy dependence basically changes with the coordination number of the underlying Bethe lattice. The possibility of observing reentrant phase transitions is related to a high enough coordination number of the underlying Bethe lattice.     

Bethe ansatz calculations for the eight-vertex model on a finite strip

Bethe ansatz equations for the eigenvalues of the transfer matrix of the eight-vertex model are solved numerically to yield mass gap data on infinitely long strips of up to 512 sites in width. The finite-size corrections, at criticality, to the free energy per site and polarization gap are found to be in agreement with recent studies of theXXZ spin chain. The leading corrections to the finite-size scaling estimates of the critical line and thermal exponent are also found, providing an explanation of the poor convergence seen in earlier studies. Away from criticality, the linear scaling fields are derived exactly in the full parameter space of the spin system, allowing a thorough test of a recently proposed method of extracting linear scaling fields and related exponents from finite lattice data.     

The homology of defective crystal lattices and the continuum limit

The problem of extending fields that are defined on lattices to fields defined on the continua that they become in the continuum limit is basically one of continuous extension from the 0‐skeleton of a simplicial complex to its higher‐dimensional skeletons. If the lattice in question has defects, as well as the order parameter space of the field, then this process might be obstructed by characteristic cohomology classes on the lattice with values in the homotopy groups of the order parameter space. The examples from solid‐state physics that are discussed are quantum spin fields on planar lattices with point defects or orientable space lattices, vorticial flows or director fields on lattices with dislocations or disclinations, and monopole fields on lattices with point defects.     

Exactly solvable two-dimensional quantum spin models

A method is proposed for constructing an exact ground-state wave function of a two-dimensional model with spin 1/2. The basis of the method is to represent the wave function by a product of fourth-rank spinors associated with the nodes of a lattice and the metric spinors corresponding to bonds between nearest neighbor nodes. The function so constructed is an exact wave function of a 14-parameter model. The special case of this model depending on one parameter is analyzed in detail. The ground state is always a nondegenerate singlet, and the spin correlation functions decay exponentially with distance. The method can be generalized for models with spin 1/2 to other types of lattices. Zh. éksp. Teor. Fiz. 115, 249–267 (January 1999)