The Odd Eight-Vertex Model

We consider a vertex model on the simple-quartic lattice defined by line graphs on the lattice for which there is always an odd number of lines incident at a vertex. This is the odd 8-vertex model which has eight possible vertex configurations. We establish that the odd 8-vertex model is equivalent to a staggered8-vertex model. Using this equivalence we deduce the solution of the odd8-vertex model when the weights satisfy a free-fermion condition. It is found that the free-fermion model exhibits no phase transitions in the regime of positive vertex weights. We also establish the complete equivalence of the free-fermion odd 8-vertex model with the free-fermion 8-vertex model solved by Fan and Wu. Our analysis leads to several Ising model representations of thefree-fermion model with pure 2-spin interactions.     

Integrable vertex and loop models on the square lattice with open boundaries via reflection matrices

The procedure for obtaining integrable vertex models via reflection matrices on the square lattice with open boundaries is reviewed and explicitly carried out for a number of two- and three-state vertex models. These models include the six-vertex model, the 15-vertex A₂⁽¹⁾ model and the 19-vertex models of Izergin-Korepin and Zamolodchikov-Fateev. In each case the eigenspectra is determined by application of either the algebraic or the analytic Bethe ansatz with inhomogeneities. With suitable choices of reflection matrices, these vertex models can be associated with integrable loop models on the same lattice. In general, the required choices do not coincide with those which lead to quantum-group-invariant spin chains. The exact solution of the integrable loop models — including an O(n) model on the square lattice with open boundaries — is of relevance to the surface critical behaviour of two-dimensional polymers.     

Vertex models on Feynman diagrams

We discuss the statistical mechanics of vertex models on both generic (“thin”) and planar (“fat”) random graphs. Such models can be formulated as the N → 1 and N → ∞ limits of N × N complex matrix models, respectively. From the graph theoretic perspective one is using matrix model and field theory inspired methods to count various classes of directed graphs. For the thin random graphs we use saddle point methods to solve the models in the thermodynamic, large number of vertices limit and note that, as in the case of the eight-vertex model on the square lattice, various other models such as the Ising model appear as particular limits. The generic solution of the fat graph model is rather more elusive, but we show that for several choices of the couplings the models can be reduced to eigenvalue integrals and their critical behaviour deduced.     

Interpretation of the symmetry group of the homogeneous 16-vertex model in terms of Lorentz similarity transformations

A new analysis is made of the symmetry group of the general homogeneous 16-vertex model on a square lattice, i.e. the group of transformations in the parameter space of the model leaving invariant its partition function. The set of 16 vertex weights is decomposed in such a way that the ensuing matrix P of 16 composite parameters transforms according to the group of Lorentz similarity transformations. Equivalence classes of models can be characterized by a suitably chosen ‘normal’ matrix P⁽ⁿ⁾, depending on 10 parameters, four having the significance of principle values, and the remaining six (two angles and two 3-dimensional unit vectors) determining a Lorentz transformation. The analysis is applied to the general eight-vertex model as well as to its soluble subclasses, the symmetric eight-vertex model, the general six-vertex model and the free fermion model.     

Some Exact Results on Bond Percolation

We present some exact results on bond percolation. We derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice ?? by ? bonds connecting the same adjacent vertices, thereby yielding the lattice ?? _? . This relation is used to calculate the bond percolation threshold on ?? _? . We show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality d??2 but changes it on a one-dimensional lattice and quasi-one-dimensional infinite-length strips. We also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the N???? limits of several families of N-vertex graphs. Finally, we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as N????.     

The wrong kind of gravity

The KPZ formula [V.G. Knizhnik, A.M. Polyakov, and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819] shows that coupling central charge c≤1 spin models to 2D quantum gravity dresses the conformal weights to get new critical exponents, where the relation between the original and dressed weights depends only on c. At the discrete level the coupling to 2D gravity is effected by putting the spin models on annealed ensembles of Φ³ planar random graphs or their dual triangulations, where the connectivity fluctuates on the same time-scale as the spins.Since the sole determining factor in the dressing is the central charge, one could contemplate putting a spin model on a quenched ensemble of 2D gravity graphs with the “wrong” c value. We might then expect to see the critical exponents appropriate to the c value used in generating the graphs. In such cases the KPZ formula could be interpreted as giving a continuous line of critical exponents which depend on this central charge. We note that rational exponents other than the KPZ values can be generated using this procedure for the Ising, tricritical Ising and 3-state Potts models.     

Exact results for a random frustrated Ising model on the Kagomé lattice

We perform a slight modification of the decoration-decimation transformation which allows us to map the homogeneous Ising model on the honeycomb lattice on an inhomogeneous Ising model on the Kagomé lattice. Then, we obtain exact results for a class of random bond Ising model on the Kagomé lattice with competing interactions and show that the different types of frustration make the critical point of the pure model disappear.     

Chern–Simons theory, 2d Yang–Mills, and Lie algebra wanderers

The dynamically triangulated random surface (DTRS) approach to Euclidean quantum gravity in two dimensions is considered for the case of the elemental building blocks being quadrangles instead of the usually used triangles. The well-known algorithmic tools for treating dynamical triangulations in a Monte Carlo simulation are adapted to the problem of these dynamical quadrangulations. The thus defined ensemble of 4-valent graphs is appropriate for coupling to it the 6- and 8-vertex models of statistical mechanics. Using a series of extensive Monte Carlo simulations and accompanying finite-size scaling analyses, we investigate the critical behaviour of the 6-vertex F model coupled to the ensemble of dynamical quadrangulations and determine the matter related as well as the graph related critical exponents of the model.     

Competition between Ising spins and classical n-vector spins on the honeycomb and the diced lattice

The competition between ordering and disordering is investigated for mixed spin models of Ising spins and classical n-vector spins on the honeycomb and the diced lattice. The critical indices of the specific heat and the spontaneous magnetization for the mixed spin models turn out to be the same as those for the two-dimensional Ising model.     

Fat Fisher zeroes

We show that it is possible to determine the locus of Fisher zeroes in the thermodynamic limit for the Ising model on planar (“fat”) φ⁴ random graphs and their dual quadrangulations by matching up the real part of the high and low temperature branches of the expression for the free energy. The form of this expression for the free energy also means that series expansion results for the zeroes may be obtained with rather less effort than might appear necessary at first sight by simply reverting the series expansion of a function g(z) which appears in the solution and taking a logarithm.Unlike regular 2D lattices where numerous unphysical critical points exist with non-standard exponents, the Ising model on planar φ⁴ graphs displays only the physical transition at c=exp(−2β)=1/4 and a mirror transition at c=−1/4 both with KPZ/DDK exponents (α=−1, β=1/2, γ=2). The relation between the φ⁴ locus and that of the dual quadrangulations is akin to that between the (regular) triangular and honeycomb lattices since there is no self-duality.     

Bifurcation analysis of the sixteen-vertex model

We shall construct a hierarchy of subclasses of the 16-vertex model having qualitatively different symmetry properties. We determine the bifurcation points in the parameter space of the model where new symmetry elements are added to the invariance group of the partition function. In this paper we restrict ourselves to the study of site-dependent transformations converting a homogeneous 16-vertex model into a different homogeneous model. Apart from a trivial transformation, resulting in a change of sign of all vertex weights, such site-dependent transformations exist only for those points in parameter space where particular relations are satisfied. The solution of these relations gives rise to three 6-parameter families of models, two of which are equivalent to the general 8-vertex model, and two families of 4-parameter models. The primary bifurcation models depending on six parameters contain three different types of secondary bifurcation models, depending on 4 parameters, one of which is equivalent to Baxter's symmetric 8-vertex model.     

Can Extra Updates Delay Mixing?

We consider Glauber dynamics (starting from an extremal configuration) in a monotone spin system, and show that interjecting extra updates cannot increase the expected Hamming distance or the total variation distance to the stationary distribution. We deduce that for monotone Markov random fields, when block dynamics contracts a Hamming metric, single-site dynamics mixes in O(n log n) steps on an n-vertex graph. In particular, our result completes work of Kenyon, Mossel and Peres concerning Glauber dynamics for the Ising model on trees. Our approach also shows that on bipartite graphs, alternating updates systematically between odd and even vertices cannot improve the mixing time by more than a factor of log n compared to updates at uniform random locations on an n-vertex graph. Our result is especially effective in comparing block and single-site dynamics; it has already been used in works of Martinelli, Toninelli, Sinclair, Mossel, Sly, Ding, Lubetzky, and Peres in various combinations.     

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.     

Effects of the bond dilution on the phase diagrams of a spin-1 transverse Ising model with crystal field interaction on a honeycomb lattice

A bond diluted spin-1 Ising model with crystal and transverse field interactions is examined for honeycomb lattice by introducing an effective field approximation that takes into account the correlations between different spins that emerge when expanding the identities. The effects of the crystal field as well as the transverse field and dilution on the phase diagrams and order parameter m_z are discussed in detail. A number of interesting and unusual phenomena such as reentrant phenomena and three successive phase transitions originating from the crystal field as well as the transverse field and bond dilution have been found.     

自旋S=1淬灭键稀释蜂窝格子伊辛模型

本文提出一种新的缀饰方法,将交换作用参数处在子空间exp(K)cosh(J)=1的自旋S=1淬灭键稀释伊辛模型映象到混合自旋淬灭座稀释缀饰格子系统。对此混合自旋缀饰系统,应用退火模型近似求得自旋S=1淬灭键稀释蜂窝格子伊辛系统的临界温度、磁化强度与键浓度之间的解析关系。 关键词：     

Random field effects on the phase diagrams of spin-1/2 Ising model on a honeycomb lattice

The Ising model with quenched random magnetic fields is examined for single Gaussian, bimodal and double Gaussian random field distributions by introducing an effective field approximation that takes into account the correlations between different spins that emerge when expanding the identities. Random field distribution shape dependence of the phase diagrams, magnetization and internal energy is investigated for a honeycomb lattice with a coordination number q=3. The conditions for the occurrence of reentrant behavior and tricritical points on the system are also discussed in detail.     

A set of exactly solvable Ising models with half-odd-integer spin

We present a set of exactly solvable Ising models, with half-odd-integer spin-S on a square-type lattice including a quartic interaction term in the Hamiltonian. The particular properties of the mixed lattice, associated with mixed half-odd-integer spin-(S,1/2) and only nearest-neighbor interaction, allow us to map this system either onto a purely spin-1/2 lattice or onto a purely spin-S lattice. By imposing the condition that the mixed half-odd-integer spin-(S,1/2) lattice must have an exact solution, we found a set of exact solutions that satisfy the free fermion condition of the eight vertex model. The number of solutions for a general half-odd-integer spin-S is given by S+1/2. Therefore we conclude that this transformation is equivalent to a simple spin transformation which is independent of the coordination number.     

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.     

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.