Hard-square lattice gas

We have studied the hard-square lattice gas, using corner transfer matrices. In particular, we have obtained the first 24 terms of the high-density series for the order parameter ₂– ₁. From these we estimate the critical activity to be 3.7962±0.0001. This is in excellent agreement with the earlier work of Gaunt and Fisher. It conflicts with the value 4.0 given by Müller-Hartmann and Zittartz's formula for the critical point of the antiferromagnetic Ising model in a field, so we conclude that this formula, while a good approximation, is not exact.     

Corner transfer matrices of the eight-vertex model. I. Low-temperature expansions and conjectured properties

A corner transfer matrix (CTM) is defined for the zero-field, eight-vertex model on the square lattice. Its logarithm and its diagonal form are obtained to second order in a perturbation expansion of low-temperature type. They turn out to have a very simple form, apart from certain remainder contributions that can be ignored in the limit of a large lattice. It is conjectured that in this limit the operators have these simple forms for all temperatures less than the critical temperatureT _c. The spontaneous magnetization can then easily be obtained, and agrees with the expression previously proposed. It is intended to prove some of the conjectures in subsequent papers.     

Corner transfer matrices of the triangular Ising model

Recently, a new technique for investigating the zero-field, eight-vertex model on the square lattice using corner transfer matrices was suggested by Baxter. In this paper these ideas are applied to the anisotropic, ferromagnetic, triangular Ising lattice in zero field below its critical temperature. The diagonal form of the corner transfer matrix for the triangular lattice shows essentially the same structure as that for the square Ising lattice. The spontaneous magnetization can be obtained easily and agrees with that previously derived.     

Corner transfer matrices of the chiral Potts model. II. The triangular lattice

We consider a two-dimensional edge-interaction model satisfying the star-triangle relations. For the triangular lattice, the corner transfer matrices are functions of three rapidities: we show that they possess various factorization properties and satisfy certain equations. We indicate how these equations can be solved for the Ising model. We then consider the three-state chiral Potts model and obtain low-temperature solutions to the equations. The conjectured formula for the order parameter (the spontaneous magnetization) is verified to one more order in a series expansion.     

Corner transfer matrices of the eight-vertex model. II. The Ising model case

In a previous paper certain corner transfer matrices were defined. It was conjectured that for the zero-field, eight-vertex model these matrices have a very simple eigenvalue spectrum. In this paper these conjectures are verified for the case when the eight-vertex model reduces to two independent and identical square-lattice Ising models. The Onsager-Yang expression for the magnetization follows immediately.     

Corner transfer matrices of the chiral Potts model

We present some symmetry and factorization relations satisfied by the corner transfer matrices (CTMs) of the chiral Potts model. We show how the single-spin expectation values can be expressed in terms of the CTMs, and in terms of the related boost operator. Low-temperature calculations lead naturally to the variables that uniformize the Boltzmann weights of the model.     

Variational approximations for square lattice models in statistical mechanics

This paper concerns a square lattice, Ising-type model with interactions between the four spins at the corners of each face. These may include nearest and next-nearest-neighbor interactions, and interactions with a magnetic field. Provided the Hamiltonian is symmetric with respect to both row reversal and column reversal, a rapidly convergent sequence of variational approximations is obtained, giving the free energy and other thermodynamic properties. For the usual Ising model, the lowest such approximations are those of Bethe and of Kramers and Wannier. The method provides a new definition of corner transfer matrices.     

Magnetization at corners in two-dimensional Ising models

We investigate the corner spin magnetization of two-dimensional ferromagnetic Ising models in various wedge geometries. Results are obtained for triangular and square lattices by numerical studies on finite wedges using the star-triangle transformation, as well as analytic calculations using corner transfer matrices and a fermionic representation of the row-to-row transfer matrix. The corner magnetizations vanish at the bulk critical temperature T_c with an exponent _c differing from the bulk exponent. For isotropic systems with free edges we find that _c is given simply by _c =/2, where is the angle at the corner. For apex magnetizations of conical lattices we obtain the strikingly similar result _a=/4. These formulas apply equally well to anisotropic lattices if the angle is interpreted as an effective angle, ^eff, determined by the relative strengths of the interactions.     

Spontaneous magnetization of the Ising model on a layered square lattice

We have studied the Ising model on a layered square lattice with four different coupling constants and two different magnetic moments. The partition function at zero magnetic field is derived exactly. We propose a formula for the spontaneous magnetization which agrees with the exact low-temperature series expansion up to the 16th order and reduces to the exact result of Au-Yang and McCoy in a special case.     

A comment on Baxter condition for commutativity of transfer matrices

LetT _N andT _N be the transfer matrices of two vertex models corresponding to two sets of Boltzmann weights. The Baxter condition on Boltzmann weights was known to be sufficient for commutativity ofT _N andT _N for allN. We show that generically it is also necessary.Partially supported by NSF Grant DMS-8403238.     

Phase diagram of the Ising square lattice with competing interactions

We restudy the phase diagram of the 2D-Ising model with competing interactions J₁ on nearest neighbour and J₂ on next-nearest neighbour bonds via Monte-Carlo simulations. We present the finite temperature phase diagram and introduce computational methods which allow us to calculate transition temperatures close to the criticalpoint at J₂ = J₁/2. Further on we investigate the character of the different phase boundariesand find that the transition is weakly first order formoderate J₂ > J₁/2.     

Spontaneous magnetization of the Ising model on a 4–8 lattice

The spontaneous magnetization of the Ising model on a 4–8 lattice with six different coupling constants and two different magnetic moments is studied. A formula for the spontaneous magnetization is proposed. The result agrees with the exact low-temperature series expansions up to the 12th order.     

The Ising model on the layered J1-J2 square lattice

We investigate the phase transitions in the Ising model on a layered square lattice with first-($J_1$) and second-($J_2$) neighbor intralayer interactions and interlayer couplings (J). The thermodynamics of the system is evaluated within a cluster mean-field approximation, which allows us to identify the nature of the thermally driven phase transitions hosted by the model. As a result, we find that interlayer couplings reduce the region of first-order phase transitions between paramagnetic and superantiferromagnetic states. We also find that the interlayer couplings reduce the frustration effects by reducing the entropy content of the low-temperature phases. Our results suggest that tricriticality is present in the special case $J=J_1$, which is in qualitative agreement with recent Monte Carlo simulations for the model.     

The Baxter Revolution

I review the revolutionary impact Rodney Baxter has had on statistical mechanics beginning with his solution of the 8 vertex model in 1971 and the invention of corner transfer matrices in 1976 to the creation of the RSOS models in 1984 and his continuing current work on the chiral Potts model.     

Ground state properties of a spin-3/2 model on a decorated square lattice

We present the construction of an optimum ground state for a quantum spin-3/2 antiferromagnet. The spins reside on a decorated square lattice, in which the basis consists of a plaquette of four sites. By using the vertex state model approach we generate the ground state from the same vertices as those used for the corresponding ground state on the hexagonal lattice. The properties of these two ground states are very similar. Particularly there is also a parameter-controlled phase transition from a disordered to a Néel ordered phase. In the regime of this transition, ground state properties can be obtained from an integrable classical vertex model. Received 28 June 1999     

计算二维声腔传递矩阵的正方形线声源模型

在比较现有二维矩形声腔的声源计算模型特点的基础上,提出了一种新的正方形线声源计算模型,用于计算基于模态级数叠加法的压力响应函数.分析表明,此模型不仅可以克服点声源模型的奇异性,而且在合理选择模型的几何尺寸的前提下能得到比面源模型更为均匀的压力分布,提高了传递函数计算的准确率.数值试验考察了此模型的实际应用效果,表明新的线声源数学模型较圆形线声源更为简便,可以提高计算效率. 关键词： 传递矩阵 正方形线声源模型 二维声腔 模态级数叠加法     

Low-temperature series for renormalized operators: The ferromagnetic square-lattice Ising model

A method for computing low-temperature series for renormalized operators in the two-dimensional Ising model is proposed. These series are applied to the study of the properties of the truncated renormalized Hamiltonians when we start at very low temperature and zero field. The truncated Hamiltonians for majority rule, Kadanoff transformation, and decimation for 2×2 blocks depend on the how we approach the first-order phase-transition line. The renormalization group transformations are multivalued and discontinuous at this first-order transition line when restricted to some finite-dimensional interaction space.     

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.     

Computer simulation study of a two-dimensional nematogenic lattice model based on a mapping from elastic free-energy density

Over the last few years, it has been recognized that on can construct, in different ways, a nematogenic lattice model with pairwise additive interactions, which approximately reproduce the elastic free energy density, and where the parameters defining the pair potential are expressed in terms of elastic constants. An anisotropic nematogenic pair interaction of this kind, originally proposed by Gruhn and Hess [Z. Naturforsch. A 51 (1996) 1] has been investigated by Monte Carlo simulation, for particle centers of mass associated with both a three- and a two-dimensional lattice. Another approximate procedure for the mapping had also been proposed, and studied by simulation on a three-dimensional lattice (Luckhurst and Romano [Liq. Cryst. 26 (1999) 871]) continuing along this line, we investigate here the 2-dimensional lattice counterpart, by means of Mean Field theory and Monte Carlo simulations. In 2 dimensions, the anisotropic character of these potential models does not preclude the existence of orientational order at finite temperature. The model produces a ground state where particles are aligned in the lattice plane; both Mean Field (MF) predictions and simulation results for the second-rank ordering tensor show a low-temperature régime where the system becomes biaxial, with the main director aligned along a lattice axis; at higher temperature there is a transition to uniaxial order with negative order parameter, and director orthogonal to the lattice plane; this orientational order survives up to temperatures higher than the transition temperature of the 3-dimensional counterpart, possibly at all finite temperatures. MF predictions and simulation results appear to agree qualitatively, but in quantitative terms the MF prediction for the transition temperature is some 56% too high.     

The one-dimensional kinetic Ising model: A series expansion study

Exact power series expansions (through eight terms) in the time are derived for relaxation in the one-dimensional Ising model with nearest-neighbor interactions for a general rate parameter where the activation energy is a variable fraction of the energy required to break nearest-neighbor bonds. It is found that the qualitative nature of the relaxation is very dependent on this parameter, varying from nearly simple exponential decay (as with Glauber dynamics) for an intermediate value of this parameter, to an initial rate of change that is either much slower or faster than a simple exponential at the extremes of the range of variation of the parameter. The rate equations for the limit of rapid internal diffusion (internal equilibration) are integrated for several special values of the rate parameter. In general the internal equilibration approximation is not a good representation of the relaxation except when the relaxation is similar to Glauber dynamics.