Existence of the thermodynamic limit in nonequilibrium systems

The existence of a thermodynamic limit in nonequilibrium stochastic and quantal systems is proven for finite-range interactions and macrovariables which are bounded in the sense of norm. This condition is easily confirmed to be satisfied for specific models, such as the kinetic Ising model and quantal spin systems.Partially financed by Japanese Scientific Research Fund of the Ministry of Education.     

Microcanonical relation between continuous and discrete spin models

A relation between a class of stationary points of the energy landscape of continuous spin models on a lattice and the configurations of an Ising model defined on the same lattice suggests an approximate expression for the microcanonical density of states. Based on this approximation we conjecture that if a O(n) model with ferromagnetic interactions on a lattice has a phase transition, its critical energy density is equal to that of the n=1 case, i.e., an Ising system with the same interactions. The conjecture holds true in the case of long-range interactions. For nearest-neighbor interactions, numerical results are consistent with the conjecture for n=2 and n=3 in three dimensions. For n=2 in two dimensions (XY model) the conjecture yields a prediction for the critical energy of the Bere?inskij-Kosterlitz-Thouless transition, which would be equal to that of the two-dimensional Ising model. We discuss available numerical data in this respect.     

Kinetic phase transitions and tricritical point in an Ising model with competing dynamics

The nonequilibrium phase diagram of an Ising model in which the evolution of the spins reflects diffusion as well as ordering, is determined via dynamic mean-field theory (pair approximation). There is a first-order transition (in zero field) for sufficiently large diffusion rates.     

Nonequilibrium wetting transition in a nonthermal 2D Ising model

Nonequilibrium wetting transitions are observed in Monte Carlo simulations of a kinetic spin system in the absence of a detailed balance condition with respect to an energy functional. A nonthermal model is proposed starting from a two-dimensional Ising spin lattice at zero temperature with two boundaries subject to opposing surface fields. Local spin excitations are only allowed by absorbing an energy quantum (photon) below a cutoff energy E _c . Local spin relaxation takes place by emitting a photon which leaves the lattice. Using Monte Carlo simulation nonequilibrium critical wetting transitions are observed as well as nonequilibrium first-order wetting phenomena, respectively in the absence or presence of absorbing states of the spin system. The transitions are identified from the behavior of the probability distribution of a suitably chosen order parameter that was proven useful for studying wetting in the (thermal) Ising model.     

The Ising--Sherrington-Kirpatrick Model in a Magnetic Field at High Temperature

We study a spin system on a large box with both Ising interaction and Sherrington-Kirpatrick couplings, in the presence of an external field. Our results are: (i) existence of the pressure in the limit of an infinite box. When both Ising and Sherrington-Kirpatrick temperatures are high enough, we prove that: (ii) the value of the pressure is given by a suitable replica symmetric solution, and (iii) the fluctuations of the pressure are of order of the inverse of the square of the volume with a normal distribution in the limit. In this regime, the pressure can be expressed in terms of random field Ising models.     

Hysteresis behavior of the random-field Ising model with 2-spin-flip dynamics: exact results on a Bethe lattice

We present an exact treatment of the hysteresis behavior of the zero-temperature random-field Ising model on a Bethe lattice when it is driven by an external field and evolved according to a 2-spin-flip dynamics. We focus on lattice connectivities z=2 (the one-dimensional chain) and z=3. For the latter case, we demonstrate the existence of an out-of-equilibrium phase transition, in contrast with the situation found with the standard 1-spin-flip dynamics. We discuss the influence of the degree of cooperativity of the (local) spin dynamics of the nonequilibrium response on the system.     

Completeness of the classical 2D Ising model and universal quantum computation

We prove that the 2D Ising model is complete in the sense that the partition function of any classical q-state spin model (on an arbitrary graph) can be expressed as a special instance of the partition function of a 2D Ising model with complex inhomogeneous couplings and external fields. In the case where the original model is an Ising or Potts-type model, we find that the corresponding 2D square lattice requires only polynomially more spins with respect to the original one, and we give a constructive method to map such models to the 2D Ising model. For more general models the overhead in system size may be exponential. The results are established by connecting classical spin models with measurement-based quantum computation and invoking the universality of the 2D cluster states.     

High-temperature expansion for nonequilibrium steady states in driven lattice gases

We develop a controlled high-temperature expansion for nonequilibrium steady states of the driven lattice gas, the "Ising model" for nonequilibrium physics. We represent the steady state as P(eta) alpha e(-betaH(eta)-psi(eta)) and evaluate the lowest order contribution to the nonequilibrium effective interaction psi(eta). We see that, in dimensions d > or = 2, all models with nonsingular transition rates yield the same summable psi(eta), suggesting the possibility of describing the state as a Gibbs state similar to equilibrium. The models with the Metropolis rule show exceptional behavior.     

Multicanonical sampling of the space of states of ℋ(2, <Emphasis Type="Italic">n</Emphasis>)-vector models

Problems of temperature behavior of specific heat are solved by the entropy simulation method for Ising models on a simple square lattice and a square spin ice (SSI) lattice with nearest neighbor interaction, models of hexagonal lattices with short-range (SR) dipole interaction, as well as with long-range (LR) dipole interaction and free boundary conditions, and models of spin quasilattices with finite interaction radius. It is established that systems of a finite number of Ising spins with LR dipole interaction can have unusual thermodynamic properties characterized by several specific-heat peaks in the absence of an external magnetic field. For a parallel multicanonical sampling method, optimal schemes are found empirically for partitioning the space of states into energy bands for Ising and SSI models, methods of concatenation and renormalization of histograms are discussed, and a flatness criterion of histograms is proposed. It is established that there is no phase transition in a model with nearest neighbor interaction on a hexagonal lattice, while the temperature behavior of specific heat exhibits singularity in the same model, in case of LR interaction. A spin quasilattice is found that exhibits a nonzero value of residual entropy.     

Phase transitions and universality in nonequilibrium steady states of stochastic Ising models

We present results of direct computer simulations and of Monte Carlo renormalization group (MCRG) studies of the nonequilibrium steady states of a spin system with competing dynamics and of the voter model. The MCRG method, previously used only for equilibrium systems, appears to give useful information also for these nonequilibrium systems. The critical exponents are found to be of Ising type for the competing dynamics model at its second-order phase transitions, and of mean-field type for the voter model (consistent with known results for the latter).     

Effects of Superaging and Percolation Crossover on the Nonequilibrium Critical Behavior of the Two-Dimensional Disordered Ising Model

A Monte Carlo study of the specific features of the nonequilibrium critical behavior has been performed for the two-dimensional “pure” and structurally disordered Ising models in the course of their evolution from the low-temperature initial state at spin concentrations p = 1.0, 0.9, and 0.8. It is shown for the first time that the pinning of domain walls by structural defects leads to the anomalously strong slowing down in the evolution of the autocorrelation function characterized by the superaging effect with exponents μ = 6.25(5) and μ = 6.75(5) for the model with the spin concentrations p = 0.9 and 0.8, respectively. The pure model exhibits the conventional aging with the exponent μ = 1. It is found that the superaging effects in structurally disordered systems lead to vanishing of the limiting fluctuation?dissipation ratio X^∞, whereas X^∞ = 0.751(24) for the pure model.     

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.     

Enhancement of next-nearest-neighbouring entanglement in quantum Ising spin chain

Using the method of the Jordan--Wigner transformation for solving different spin--spin correlation functions, we have investigated the generation of next-nearest-neighbouring entanglement in a one-dimensional quantum Ising spin chain with the Gaussian distribution impurities of exchange couplings and external magnetic fields taken into account. The maximal value of entanglement between the next-nearest-neighbouring qubits in the transverse Ising model was analysed in detail by varying the effectively controlled parameters such as interchange coupling, magnetic field and the system impurity. For such systems, where both exchange couplings and external magnetic field disorder appear, we show that it is possible to achieve next-nearest-neighbouring entanglement better than the previously discussed pure Ising spin chain case. We also show that the Gaussian distribution impurity can induce next-nearest-neighbouring entanglement, which can be used as a means to characterize quantum phase transition.     

动态外场作用下Ising自旋体系的非平衡动态相变

系统地考察了Ising自旋体系的动力学方程对三种不同性质的驱动外场（正弦波 、方波和锯齿波）的动态响应及其相应的非平衡动态相变特征.在正弦波和方波的驱动场 作用下，体系存在分别对应于低温对称破缺的铁磁有序态和高温对称顺磁无序态的动态非平衡转变，相应的动态转变相界上存在区分连续转变和非连续转变的三临界点；而锯齿波驱动 场情形下体系始终维持对称性破缺的有序态.体系动态转变表现出的上述差异与作用外场的驱动特征有关.确定了表征相应动态相变相界的临界驱动外场振幅h0C和频率 ωc、体系的温度tc, 并给予了分析讨论 关键词： Ising自旋体系 非平衡动态相变 对称性 平均场     

Precursor phenomena in frustrated systems

To understand the origin of the dynamical transition, between high-temperature exponential relaxation and low-temperature nonexponential relaxation, that occurs well above the static transition in glassy systems, a frustrated spin model, with and without disorder, is considered. The model has two phase transitions, the lower being a standard spin glass transition (in the presence of disorder) or fully frustrated Ising (in the absence of disorder), and the higher being a Potts transition. Monte Carlo results clarify that in the model with (or without) disorder the precursor phenomena are related to the Griffiths (or Potts) transition. The Griffiths transition is a vanishing transition which occurs above the Potts transition and is present only when disorder is present, while the Potts transition which signals the effect due to frustration is always present. These results suggest that precursor phenomena in frustrated systems are due either to disorder and/or to frustration, giving a consistent interpretation also for the limiting cases of Ising spin glass and of Ising fully frustrated model, where also the Potts transition is vanishing. This interpretation could play a relevant role in glassy systems beyond the spin systems case.     

General formulation of Griffiths' inequalities

We present a general framework in which Griffiths inequalities on the correlations of ferromagnetic spin systems appear as natural consequences of general assumptions. We give a method for the construction of a large class of models satisfying the basic assumptions. Special cases include the Ising model with arbitrary spins, and the plane rotator model. The general theory extends in a straightforward way to the non-commutative (quantum) case, but non-commutative examples satisfying all the assumptions are lacking at the moment.     

Two-Point Correlations and Critical Line of the Driven Ising Lattice Gas in a High-Temperature Expansion

Based on a high-temperature expansion, we compute the two-point correlation function and the critical line of an Ising lattice gas driven into a nonequilibrium steady state by a uniform bias E. The lowest nontrivial order already reproduces the key features, i.e., the discontinuity singularity of the structure factor and the (qualitative) E dependence of the critical line. Our approach is easily generalized to other nonequilibrium lattice models and provides a simple analytic tool for the study of the high-temperature phase and its boundaries.     

Localizability and coupling constants

Let us consider for free elementary systems the postulates: (i) localizability of systems would not favor in a physical sense some inertial frames and that (ii) standard quantum mechanics (or, at least, a skeleton of it) applies for localizability. It is known that, at least for the lower values of spin, (i) and (ii) imply a unique solution of the localization problem for the no-interaction case. Extrapolating the analysis to the case when interactions are present, we offer arguments in favor of theconjecture that if elementary systems under interaction are localizable, then (i) and (ii) imply restrictions on the coupling constants, and probably their single-valuedness.