Critical and multicritical behavior in the Ising–Heisenberg universality class

Critical behavior of three-dimensional classical frustrated antiferromagnets with a collinear spin ordering and with an additional twofold degeneracy of the ground state is studied. We consider two lattice models, whose continuous limit describes a single phase transition with a symmetry class differing from the class of non-frustrated magnets as well as from the classes of magnets with non-collinear spin ordering. A symmetry breaking is described by a pair of independent order parameters, which are similar to order parameters of the Ising and O(N) models correspondingly. Using the renormalization group method, it is shown that a transition is of first order for non-Ising spins. For Ising spins, a second order phase transition from the universality class of the O(2) model may be observed. The lattice models are considered by Monte Carlo simulations based on the Wang–Landau algorithm. The models are a ferromagnet on a body-centered cubic lattice with the additional antiferromagnetic exchange interaction between next-nearest-neighbor spins and an antiferromagnet on a simple cubic lattice with the additional interaction in layers. We consider the cases N = 1, 2, 3 and in all of them find a first-order transition. For the N = 1 case we exclude possibilities of the second order or pseudo-first order of a transition. An almost second order transition for large N is also discussed.     

Dynamic critical behavior of the worm algorithm for the Ising model

We study the dynamic critical behavior of the worm algorithm for the two- and three-dimensional Ising models, by Monte Carlo simulation. The autocorrelation functions exhibit an unusual three-time-scale behavior. As a practical matter, the worm algorithm is slightly more efficient than the Swendsen-Wang algorithm for simulating the two-point function of the three-dimensional Ising model.     

Optimized energy calculation in lattice systems with long-range interactions

We discuss an efficient approach to the calculation of the internal energy in numerical simulations of spin systems with long-range interactions. Although, since the introduction of the Luijten-Blote algorithm, Monte Carlo simulations of these systems no longer pose a fundamental problem, the energy calculation is still an O(N2) problem for systems of size N. We show how this can be reduced to an O(N log N) problem, with a break-even point that is already reached for very small systems. This allows the study of a variety of, until now hardly accessible, physical aspects of these systems. In particular, we combine the optimized energy calculation with histogram interpolation methods to investigate the specific heat of the Ising model and the first-order regime of the three-state Potts model with long-range interactions.     

Inference from matrix products: a heuristic spin-glass algorithm

We present an algorithm for finding ground states of two-dimensional spin-glass systems based on ideas from matrix product states in quantum information theory. The algorithm works directly at zero temperature and defines an approximation to the energy whose accuracy depends on a parameter k. We test the algorithm against exact methods on random field and random bond Ising models, and we find that accurate results require a k which scales roughly polynomially with the system size. The algorithm also performs well when tested on small systems with arbitrary interactions, where no fast, exact algorithms exist. The time required is significantly less than Monte Carlo schemes.     

Long-term fluctuations in globally coupled phase oscillators with general coupling: finite size effects

We investigate the diffusion coefficient of the time integral of the Kuramoto order parameter in globally coupled nonidentical phase oscillators. This coefficient represents the deviation of the time integral of the order parameter from its mean value on the sample average. In other words, this coefficient characterizes long-term fluctuations of the order parameter. For a system of N coupled oscillators, we introduce a statistical quantity D, which denotes the product of N and the diffusion coefficient. We study the scaling law of D with respect to the system size N. In other well-known models such as the Ising model, the scaling property of D is D～O(1) for both coherent and incoherent regimes except for the transition point. In contrast, in the globally coupled phase oscillators, the scaling law of D is different for the coherent and incoherent regimes: D～O(1/N(a)) with a certain constant a>0 in the coherent regime and D～O(1) in the incoherent regime. We demonstrate that these scaling laws hold for several representative coupling schemes.     

A systematical improvement of the migdal recursion formula

We construct a new perturbative expansion whose zeroth-order approximation gives the Migdal-Kadanoff recursion equations. By this expansion it is therefore possible to improve systematically the Migdal-Kadanoff results. The first-order corrections are computed for the Ising and bond percolation models. The second-order corrections are computed only for the two-dimensional Ising model. Our method can be easily extended to other systems like the non-linear σ model or gauge theories.     

Wang-Landau study of the 3D Ising model with bond disorder

We implement a two-stage approach of the Wang-Landau algorithm to investigate the critical properties of the 3D Ising model with quenched bond randomness. In particular, we consider the case where disorder couples to the nearest-neighbor ferromagnetic interaction, in terms of a bimodal distribution of strong versus weak bonds. Our simulations are carried out for large ensembles of disorder realizations and lattices with linear sizes L in the range L=8-64L=8{-}64. We apply well-established finite-size scaling techniques and concepts from the scaling theory of disordered systems to describe the nature of the phase transition of the disordered model, departing gradually from the fixed point of the pure system. Our analysis (based on the determination of the critical exponents) shows that the 3D random-bond Ising model belongs to the same universality class with the site- and bond-dilution models, providing a single universality class for the 3D Ising model with these three types of quenched uncorrelated disorder.     

Kinetic lattice models of disorder

We study a class of stochastic Ising (or interacting particle) systems that exhibit a spatial distribution of impurities that change with time. It may model, for instance, steady nonequilibrium conditions of the kind that may be induced by diffusion in some disordered materials. Different assumptions for the degree of coupling between the spin and the impurity configurations are considered. Two interesting well-defined limits for impurities that behave autonomously are (i) the standard (i.e., quenched) bond-diluted, random-field, random-exchange, and spin-glass Ising models, and (ii) kinetic variations of these standard cases in which conflicting kinetics simulate fast and random diffusion of impurities. A generalization of the Mattis model with disorder that describes a crossover from the equilibrium case (i) to the nonequilibrium case (ii) and the microscopic structure of a generalized heat bath are explicitly worked out as specific realizations of our class of models. We sketch a simple classification of transition rates for the time evolution of the spin configuration based on the critical behavior that is exhibited by the models in case (ii). The latter are shown to have an exact solution for any lattice dimension for some special choice of rates.     

Symmetric vertex models on planar random graphs

We solve a 4-(bond)-vertex model on an ensemble of 3-regular (Φ³) planar random graphs, which has the effect of coupling the vertex model to 2D quantum gravity. The method of solution, by mapping onto an Ising model in field, is inspired by the solution by Wu et.al. of the regular lattice equivalent – a symmetric 8-vertex model on the honeycomb lattice, and also applies to higher valency bond vertex models on random graphs when the vertex weights depend only on bond numbers and not cyclic ordering (the so-called symmetric vertex models).The relations between the vertex weights and Ising model parameters in the 4-vertex model on Φ³ graphs turn out to be identical to those of the honeycomb lattice model, as is the form of the equation of the Ising critical locus for the vertex weights. A symmetry of the partition function under transformations of the vertex weights, which is fundamental to the solution in both cases, can be understood in the random graph case as a change of integration variable in the matrix integral used to define the model.Finally, we note that vertex models, such as that discussed in this paper, may have a role to play in the discretisation of Lorentzian metric quantum gravity in two dimensions.     

Boundary between long-range and short-range critical behavior in systems with algebraic interactions

We investigate phase transitions of two-dimensional Ising models with power-law interactions, using an efficient Monte Carlo algorithm. For slow decay, the transition is of the mean-field type; for fast decay, it belongs to the short-range Ising universality class. We focus on the intermediate range, where the critical exponents depend continuously on the power law. We find that the boundary with short-range critical behavior occurs for interactions depending on distance r as r(-15/4). This answers a long-standing controversy between mutually conflicting renormalization-group analyses.     

Stationary State Solutions of a Bond Diluted Kinetic Ising Model: An Effective-Field Theory Analysis

We examined the stationary state solutions of a bond diluted kinetic Ising model under a time dependent oscillating magnetic field within the effective-field theory (EFT) for a honeycomb lattice (q=3). The effects of the Hamiltonian parameters on the dynamic phase diagrams have been discussed in detail. Bond dilution process on the kinetic Ising model causes a number of interesting and unusual phenomena such as reentrant phenomena and has a tendency to destruct the first-order transitions and the dynamic tricritical point. Moreover, we have investigated the variation of the bond percolation threshold as functions of the amplitude and frequency of the oscillating field.     

Renormalization group study of random quantum magnets

We have developed a very efficient numerical algorithm of the strong disorder renormalization group method to study the critical behaviour of the random transverse field Ising model, which is a prototype of random quantum magnets. With this algorithm we can renormalize an N-site cluster within a time NlogN, independently of the topology of the graph, and we went up to N ～ 4 × 10(6). We have studied regular lattices with dimension D ≤ 4 as well as Erd?s-Rényi random graphs, which are infinite dimensional objects. In all cases the quantum critical behaviour is found to be controlled by an infinite disorder fixed point, in which disorder plays a dominant role over quantum fluctuations. As a consequence the renormalization procedure as well as the obtained critical properties are asymptotically exact for large systems. We have also studied Griffiths singularities in the paramagnetic and ferromagnetic phases and generalized the numerical algorithm for other random quantum systems.     

Spin dynamics in hole-doped two-dimensional S = 1/2 Heisenberg antiferromagnets: 63Cu NQR relaxation in La2-xSrxCuO4 for x ≤ 0.04

A cluster algorithm formulated in continuous (imaginary) time is presented for Ising models in a transverse field. It works directly with an infinite number of time-slices in the imaginary time direction, avoiding the necessity to take this limit explicitly. The algorithm is tested at the zero-temperature critical point of the pure two-dimensional (2d) transverse Ising model. Then it is applied to the 2d Ising ferromagnet with random bonds and transverse fields, for which the phase diagram is determined. Finite size scaling at the quantum critical point as well as the study of the quantum Griffiths-McCoy phase indicate that the dynamical critical exponent is infinite as in 1d. Received 6 November 1998     

Nucleation and metastability in three-dimensional Ising models

We present Monte Carlo experiments on nucleation theory in the nearest-neighbor three-dimensional Ising model and in Ising models with long-range interactions. For the nearest-neighbor model, our results are compatible with the classical nucleation theory (CNT) for low temperatures, while for the long-range model a breakdown of the CNT was observed near the mean-field spinodal. A new droplet model and a zeroth-order theory of droplet growth are also presented.Supported in part by grants from ARO, ONR, and NSF.     

Theory for phase transitions in insulating V2O3.

We show that the recently proposed S = 2 bond model with orbital degrees of freedom for insulating V2O3 not only explains the anomalous magnetic ordering but also other mysteries of the magnetic phase transition. The model contains an additional orbital degree of freedom that exhibits a zero temperature quantum phase transition in the Ising universality class.     

Spin glasses on thin graphs

In a recent paper [C. Baillie, D.A. Johnston and J.-P. Kownacki, Nucl. Phys. B 432 (1994) 551] we found strong evidence from simulations that the Ising antiferromagnet on “thin” random graphs — Feynman diagrams — displayed a mean-field spin-glass transition. The intrinsic interest of considering such random graphs is that they give mean-field theory results without long-range interactions or the drawbacks, arising from boundary problems, of the Bethe lattice. In this paper we reprise the saddle-point calculations for the Ising and Potts ferromagnet, antiferromagnet and spin glass on Feynman diagrams. We use standard results from bifurcation theory that enable us to treat an arbitrary number of replicas and any quenched bond distribution. We note the agreement between the ferromagnetic and spin-glass transition temperatures thus calculated and those derived by analogy with the Bethe lattice or in previous replica calculations.

We then investigate numerically spin glasses with a ±J bond distribution for the Ising and Q = 3, 4, 10, 50 state Potts models, paying particular attention to the independence of the spin-glass transition from the fraction of positive and negative bonds in the Ising case and the qualitative form of the overlap distribution P(q) for all of the models. The parallels with infinite-range spin-glass models in both the analytical calculations and simulations are pointed out.     

Universality aspects of the 2d random-bond Ising and 3d Blume-Capel models

We report on large-scale Wang-Landau Monte Carlo simulations of the critical behavior of two spin models in two- (2d) and three-dimensions (3d), namely the 2d random-bond Ising model and the pure 3d Blume-Capel model at zero crystal-field coupling. The numerical data we obtain and the relevant finite-size scaling analysis provide clear answers regarding the universality aspects of both models. In particular, for the random-bond case of the 2d Ising model the theoretically predicted strong universality’s hypothesis is verified, whereas for the second-order regime of the Blume-Capel model, the expected d = 3 Ising universality is verified. Our study is facilitated by the combined use of the Wang-Landau algorithm and the critical energy subspace scheme, indicating that the proposed scheme is able to provide accurate results on the critical behavior of complex spin systems.     

Rounding by disorder of first-order quantum phase transitions: emergence of quantum critical points

We give a heuristic argument for disorder rounding of a first-order quantum phase transition into a continuous phase transition. From both weak and strong disorder analysis of the N-color quantum Ashkin-Teller model in one spatial dimension, we find that, for N > or =3, the first-order transition is rounded to a continuous transition and the physical picture is the same as the random transverse field Ising model for a limited parameter regime. The results are strikingly different from the corresponding classical problem in two dimensions where the fate of the renormalization group flows is a fixed point corresponding to N-decoupled pure Ising models.     

Damage Spreading in the Ising Model with a Special Metropolis Dynamics Approach

The time evolution of the Hamming distance (damage spreading) for the S=1/2 and S=1 Ising models on the square lattice is performed with a special metropolis dynamics algorithm. Two distinct regimes are observed according to the temperature range for both models: a low-temperature one where the distance in the long-time limit is finite and seems not to depend on the initial distance and the system size; a high-temperature one where the distance vanishes in the long-time limit. Using the finite size scaling method, the dynamical phase transition (damage spreading transition) temperature is obtained as T_c≌1.675±0.025 for the S=1 Ising model.