Correlation inequalities for Ising spin lattices

A change of spin representation is used to present expectation inequalities on Ising lattices directly as sums of terms of like sign. The technique is extended to correlation inequalities by introducing replica variables which convert correlations into expectations on a larger space. Second order correlations are analyzed in full from this viewpoint, recovering the FKG set, among others. Third order correlations are examined in some detail, and the sign of the multi-site Ursell correlationsF ₃,F ₄,F ₆ established under appropriate restrictions.     

Characterisations of phase transitions in Ising spin systems

We construct aC*-algebraic formalism designed to provide a framework for the characterisation of phase transitions in a class of Ising spin systems: this class is large enough to include the rectangular lattice models, of arbitrary finite dimensionality, with nearest neighbour interactions. Using an extension of Onsager's transfer matrix formalism, we express properties of a Gibbs state of a system in terms of a contractive linear transformation, ₀, of a certain Hilbert space, the properties of ₀ being governed by the temperature as well as the interactions in the system. We obtain conditions on ₀ under which the system exhibits a phase transition characterised by (A) a thermodynamical singularity, (B) a change in symmetry, associated with theG-ergodic decomposition of Gibbs states, (C) a divergence of a correlation length (appropriately defined) at the critical point, and (D) scaling laws in the critical region. Applying our formalism to the rectangular two-dimensional Ising model with nearest neighbour interactions, we show that its phase transition possesses the properties (B) and (C), as well as (A).     

Ising自旋体系的非平衡动态相变

在平均场理论架构下, 以含时金兹堡-朗道和Glauber动力学这两类动态模型为基础，研究 了受外场和温度共同驱动的Ising自旋体系的非平衡动态相变.确定了界定动态无序(动态顺 磁相P)和动态有序(动态铁磁相F)转变的动态相界.并根据动态序参量Q和Binder参数U随系统 温度t(r₀)、驱动外场频率ω和振幅h₀的变化规律，就上述两类模 型的动态相界上是否存在区分连续动态转变和非连续动态转变的三临界点这一引发争议的问 题做出了进一步分析说明. 关键词： Ising自旋体系 非平衡动态相变 含时金兹堡-朗道模型 Glauber 动力学模型     

On the uniqueness of the equilibrium state for antiferromagnetic Ising spin system in the phase transition region

We show that at low temperature an Ising spin system with antiferromagnetic interaction in a small enough external magnetic field has only one translationally invariant state.     

Nonequilibrium dynamical phase transition of 3D kinetic Ising／Heisenberg spin system

We have studied the nonequilibrium dynamic phase transitions of both three-dimensional (3D) kinetic Ising and Heisenberg spin systems in the presence of a perturbative magnetic field by Monte Carlo simulation. The feature of the phase transition is characterized by studying the distribution of the dynamical order parameter. In the case of anisotropic Ising spin system (ISS), the dynamic transition is discontinuous and continuous under low and high temperatures respectively, which indicates the existence of a tri-critical point (TCP) on the phase boundary separating low-temperature order phase and high-temperature disorder phase. The TCP shifts towards the higher temperature region with the decrease of frequency, i.e. T_{TCP}=1.33×exp(-ω/30.7). In the case of the isotropic Heisenberg spin system (HSS), however, the situation on dynamic phase transition of HSS is quite different from that of ISS in that no stable dynamical phase transition was observed in kinetic HSS after a threshold time. The evolution of magnetization in the HSS driven by a symmetrical external field after a certain duration always tends asymptotically to a disorder state no matter what an initial state the system starts with. The threshold time τ depends upon the amplitude H_{0}, reduced temperature T/T_C and the frequency ω as τ=C·ω^α·H_0^{-β}·(T/T_C)^{-γ}.     

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     

On the uniqueness of the equilibrium state for Ising spin systems

We show that for an Ising spin system of arbitrary spin with a ferromagnetic pair interaction and a periodic external magnetic field there is a unique equilibrium state if and only if the magnetization is continuous with respect to a uniform change in the external field. Hence, if the critical temperatureT _c is defined as the temperature where the spontaneous magnetization (which is a non-increasing function of the temperature) becomes positive, then the equilibrium state is unique forT>T _c and is non-unique forT<T _c (when the external field is zero). This implies that the correlation functions have a cluster property forT>T _c .We also show that for an anti-ferromagnet consisting of two sublattices there is a unique equilibrium state if and only if the staggered magnetization is continuous with respect to a change in the staggered field.Supported in part by U.S.A.F.O.S.R. under contract F 44620-71-C-0013, P001.     

The hubbard identity and the absence of phase transitions in some spin systems

The Hubbard identity is used to show the absence of magnetic phase transitions in Heisenberg spin systems in one and two dimensions, generalizing Mermin and Wagner's result in an alternative way as Thorpe has done.     

Absence of modulated phase transition and spin density wave phase transition in one- and two-dimensional systems

We prove that the modulated phase transition with any wave vector and the spin density wave (SDW) phase transition cannot exist in one- and two-dimensional systems at finite temperatures.     

Cluster analysis and finite-size scaling for Ising spin systems

Based on the connection between the Ising model and a correlated percolation model, we calculate the distribution function for the fraction (c) of lattice sites in percolating clusters in subgraphs with n percolating clusters, f(n)(c), and the distribution function for magnetization (m) in subgraphs with n percolating clusters, p(n)(m). We find that f(n)(c) and p(n)(m) have very good finite-size scaling behavior and that they have universal finite-size scaling functions for the model on square, plane triangular, and honeycomb lattices when aspect ratios of these lattices have the proportions 1:square root[3]/2:square root[3]. The complex structure of the magnetization distribution function p(m) for the system with large aspect ratio could be understood from the independent orientations of two or more percolation clusters in such a system.     

On the phase transition of the Ising gauge model

We study the Ising gauge model in dimensionsD2. In addition to the plaquette coupling of this model we introduce a nearest neighbour ferromagnetic Ising coupling. By that means the local gauge symmetry is removed while the global symmetry of the Ising model is still present. The phase diagram of the model is found by using general arguments as well as Monte-Carlo methods. The gauge phase of the model is characterized by the presence of a topological excitation (defect) which has no influence on any local physical quantity.     

On a class of inequalities and identities for spin correlation functions of general Ising models

A recently obtained result concerning a class of inequalities and identities for spin correlation functions of Ising models with pair interactions is extended to general Ising models.     

Stochastic resonance and nonequilibrium dynamic phase transition of Ising spin system driven by a joint external field

The dynamic response and stochastic resonance of a kinetic Ising spin system （ISS） subject to the joint action of an external field of weak sinusoidal modulation and stochastic white-nolse are studied by solving the mean-field equation of motion based on Glauber dynamics. The periodically driven stochastic ISS shows that the characteristic stochastic resonance as well as nonequilibrium dynamic phase transition （NDPT） occurs when the frequency ω and amplitude h0 of driving field, the temperature t of the system and noise intensity D are all specifically in accordance with each other in quantity. There exist in the system two typical dynamic phases, referred to as dynamic disordered paramagnetic and ordered ferromagnetic phases respectively, corresponding to a zero- and a unit-dynamic order parameter. The NDPT boundary surface of the system which separates the dynamic paramagnetic phase from the dynamic ferromagnetic phase in the 3D parameter space of ho-t-D is also investigated. An interesting dynamical ferromagnetic phase with an intermediate order parameter of 0.66 is revealed for the first time in the ISS subject to the perturbation of a joint determinant and stochastic field. The intermediate order dynamical ferromagnetic phase is dynamically metastable in nature and owns a peculiar characteristic in its stability as well as the response to external driving field as compared with a fully order dynamic ferromagnetic phase.     

On aC*-algebra approach to phase transition in the two-dimensional Ising model

We investigate the state on theC*-algebra of Pauli spins on a one-dimensional lattice (infinitely extended in both directions) which gives rise to the thermodynamic limit of the Gibbs ensemble in the two-dimensional Ising model (with nearest neighbour interaction). It is shown that the representation of the Pauli spin algebra associated with the state is factorial above and at the known critical temperature, while it has a two-dimensional center below the critical temperature. As a technical tool, we derive a general criterion for a state of the Pauli spin algebra corresponding to a Fock state of the Fermion algebra to be primary. We also show that restrictions of two quasifree states of the Fermion algebra to its even part are equivalent if and only if the projection operatorsE ₁ andE ₂ (on the direct sum of two copies of the basic Hilbert space) satisfy the following two conditions: (1)E ₁ ?E ₂ is in the Hilbert-Schmidt class, (2)E ₁ ∧ (1 ?E ₂) has an even dimension, where the even-oddness of dimE ₁ ∧ (1 ?E ₂) is called ?₂-index ofE ₁ andE ₂ and is continuous inE ₁ andE ₂ relative to the norm topology.     

Lack of ultrametricity in the low-temperature phase of three-dimensional Ising spin glasses

We study the low-temperature spin-glass phases of the Sherrington-Kirkpatrick (SK) model and of the 3-dimensional short-range Ising spin-glass (3DISG). By using clustering to focus on the relevant parts of phase space and reduce finite size effects, we found that for the SK model ultrametricity becomes clearer as the system size increases, while for the short-range case our results indicate the opposite, i.e., lack of ultrametricity. Another method, which does not rely on clustering, indicates that the mean-field solution works for the SK model but does not apply in detail to the 3DISG, for which stochastic stability is also violated.