The complete system of algebraic invariants for the sixteen-vertex model: I. Derivation via the three-dimensional orthogonal group

In a previous paper, the partition function of the 16-vertex model was shown to be invariant under a group of linear transformations in the space of the vertex weights. According to a theorem by Hilbert, every algebraic invariant such as the partition function for a finite lattice can be expressed algebraically in terms of a finite set of basic algebraic invariants, which are sums of products of the vertex weights. We construct this set by analysing the structural properties of the transformation group (the direct product of two three-dimensional orthogonal groups). The basic set is found to consist of 21 invariants, ranging from a linear invariant up to invariants of the ninth degree. In particular cases, notably the (general or the symmetric) eight-vertex model, the six-vertex model and the free-fermion model, several invariants vanish and a number of additional algebraic relations between the basic invariants are obtained.     

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.     

Nonintersecting string model and graphical approach: Equivalence with a Potts model

Using a graphical method we establish the exact equivalence of the partition function of aq-state nonintersecting string (NIS) model on an arbitrary planar, even-valenced, lattice with that of a q²-state Potts model on a related lattice. The NIS model considered in this paper is one in which the vertex weights are expressible as sums of those of basic vertex types, and the resulting Potts model generally has multispin interactions. For the square and Kagomé lattices this leads to the equivalence of a staggered NIS model with Potts models with anisotropic pair interactions, indicating that these NIS models have a first-order transition forq > 2. For the triangular lattice the NIS model turns out to be the five-vertex model of Wu and Lin and it relates to a Potts model with two- and three-site interactions. The most general model we discuss is an oriented NIS model which contains the six-vertex model and the NIS models of Stroganov and Schultz as special cases.     

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.     

Fusion of theq-vertex operators and its application to solvable vertex models

We diagonalize the transfer matrix of the inhomogeneous vertex models of the 6-vertex type in the anti-ferroelectric regime directly in the infinite lattice. For this purpose we have introduced new types ofq-vertex operators. The special cases of those transfer matrices were used to diagonalize the s-d exchange model [23, 1, 6]. New vertex operators are constructed from the level one vertex operators by the fusion procedure. Using this construction we determine the commutation relations among new vertex operators which play a crucial role for the diagnoalization. In order to clarify the quasi-particle structure of the model we estabish new isomorphisms of crystals. The isomorphisms figure out, representation theoretically, the ground state degenerations.     

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.     

Conformal invariance in incommensurate phases

We study finite-size corrections to the free energy of free-fermion models on a torus with periodic, twisted, and fixed boundary conditions. Inside the critical (striped-incommensurate) phase, the free energy densityf(N, M) on anN×M square lattice with periodic (or twisted) boundary conditions scales asf(N, M)=f _–A(s)/(NM)+.... We derive exactly the finite-size-scaling (FSS) amplitudesA(s) as a function of the aspect ratios=M/N. These amplitudes are universal because they do not depend on details of the free-fermion Hamiltonian. We establish an equivalence between the FSS amplitudes of the free-fermion model and the Coulomb gas system with electric and magnetic defect lines. The twist angle generates magnetic defect lines, while electric defect lines are generated by competition between domain wall separation and system size. The FSS behavior of the free-fermion model is consistent with predictions of the theory of conformal invariance with the conformal chargec=l. For instance, the FSS amplitude on an infinite cylinder with fixed boundary conditions is found to be one-quarter of that with periodic boundary conditions. Finally, we conjecture the exact form of the FSS amplitudes for an interacting-fermion model on a torus. Numerical calculations employing the Bethe Ansatz confirm our conjecture in the infinite-cylinder limit.     

The complete system of algebraic invariants for the sixteen-vertex model: II. Derivation by means of the theory of algebraic invariants

The construction of a complete system of basic invariants for the sixteen-vertex model on an M x N lattice as described in part I is repeated by means of an alternative method based on the theory of algebraic invariants. We use a generalization of a theorem by Cayley and Sylvester to determine the characteristics of the covariants belonging to the basic system. In this way we arrive at the same set of 21 invariants that was found in part I. The present method offers the possibility of a generalization to the three-dimensional 64-vertex model and the vertex model on a triangular lattice.     

Union Jack晶格上伊辛模型的自由费密近似解

运用自由费密近似对Union Jack晶格上具有各向异性二体耦合作用及三体相互作用的伊辛模型进行了求解,得到了模型的自由能、自发磁矩和临界点方程。在耦合常数简化为正方晶格上的伊辛模型时,得到了与Onsager一致的解。     

Unified algebraic Bethe ansatz for two-dimensional lattice models

We develop a unified formulation of the quantum inverse scattering method for lattice vertex models associated to the nonexceptional A(2)(2r), A(2)(2r-1), B(1)(r), C(1)(r), D(1)(r+1), and D(2)(r+1) Lie algebras. We recast the Yang-Baxter algebra in terms of different commutation relations between creation, annihilation, and diagonal fields. The solution of the D(2)(r+1) model is based on an interesting 16-vertex model, which is solvable without recourse to a Bethe ansatz.     

Duality relation for 32-vertex model on the triangular lattice

The duality relation is derived for a vertex model on the triangular lattice. Vertex configurations are limited to the 32 that have an odd number of incoming arrows, and vertex energies are invariant to rotations ofp/3 and reversal of all arrows. Special cases of the model include the triangular Ising model and Baxter's three-spin model, for each of which the duality relation gives the critical temperature.Research supported in part by NSF Grant No. 33535X.     

EXACT RESULTS FOR A SYMMETRIC 27-VERTEX MODEL

Wu study a symmetric 3-state 27-vertex model and summarize three ways con-tributing to obtaining information of a higher-spin (more-vertex) model in terms of lower-spin (less-vertex) models. We find that this model is equivalent to a symmet-ric 2-state 8-vertex model or an arbitrary 3-state spin models on Kagome lattice in different eases. And 21 exact critical lines and a multicritical point (add something that I cannot recognize) are obtained in ten-dimensional weight space.     

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.     

The exact solution and the finite-size behaviour of the Osp(1|2)-invariant spin chain

We have solved exactly the Osp(1|2) spin chain by the Bethe ansatz approach. Our solution is based on an equivalence between the Osp(1|2) chain and a certain special limit of the Izergin-Korepin vertex model. The completeness of the Bethe ansatz equations is discussed for a system with four sites and the appearance of special string structures is noted. The Bethe ansatz presents an important phase factor which distinguishes the even and odd sectors of the theory. The finite-size properties are governed by a conformal field theory with central charge c = 1.     

Quantum phase transition in spin-3/2 systems on the hexagonal lattice — optimum ground state approach

Optimum ground states are constructed in two dimensions by using so called vertex state models. These models are graphical generalizations of the well-known matrix product ground states for spin chains. On the hexagonal lattice we obtain a one-parametric set of ground states for a five-dimensional manifold of S = 3/2 Hamiltonians. Correlation functions within these ground states are calculated using Monte-Carlo simulations. In contrast to the one-dimensional situation, these states exhibit a parameter-induced second order phase transition. In the disordered phase, two-spin correlations decay exponentially, but in the Néel ordered phase alternating long-range correlations are dominant. We also show that ground state properties can be obtained from the exact solution of a corresponding free-fermion model for most values of the parameter.     

Exact solution of a vertex model ind dimensions

We consider a class of vertex models describing directed lines on a lattice in arbitraryd dimensions, and solve the model exactly for the Cartesian lattice and in the case that each loop of lines carries a fugacity - 1. Our analysis, which can be carried out for arbitrary lattices, is based on an equivalence of the vertex model with a dimer problem. The dimer problem is, in turn, solved using the method of Pfaffians. It is found that the system is frozen below a critical temperatureT _cwith the critical exponent = (3 –d)/2.     

一种推广的混合自旋模型的临界温度曲线

本文提出了一种union jack晶格上推广的混合自旋模型。文中分别用平均场近似、自由费密近似及同普适类等标度变换理论对该模型进行了研究,分别得到了相互之间符合较好的临界温度曲线,并对不同处理方法进行了比较。在简化为特定可解模型时,得到与严格解一致的临界点。 关键词：     

New Developments in the Eight Vertex Model II. Chains of Odd Length

We study the transfer matrix of the 8 vertex model with an odd number of lattice sites N. For systems at the root of unity pointsη=mK/L with m odd the transfer matrix is known to satisfy the famous ‘‘TQ’’ equation where Q(υ) is a specifically known matrix. We demonstrate that the location of the zeroes of this Q(υ) matrix is qualitatively different from the case of evenN and in particular they satisfy a previously unknown equation which is more general than what is often called ‘‘Bethe’s equation.’’ For the case of even m where no Q(υ) matrix is known we demonstrate that there are many states which are not obtained from the formalism of the SOS model but which do satisfy the TQ equation. The ground state for the particular case of η=2K/3 and N odd is investigated in detail.     

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????.     

Ground state phase diagram of a spin-2 antiferromagnet on the square lattice

We use the vertex state model approach to construct optimum ground states for a large class of quantum spin-2 antiferromagnets on the square lattice. Optimum ground states are exact ground states of the model which minimize all local interaction operators. The ground state contains two continuous parameters and exhibits a second order phase transition from a disordered phase with exponentially decaying correlation functions to a Néel ordered phase. The behaviour is very similar to that of the corresponding ground state of a quantum spin-3/2 model on the hexagonal lattice, which has been investigated in an earlier paper. Received 8 April 1999