The six-vertex model eigenvectors as critical limit of the eight-vertex model bethe ansatz

The critical limit of the eight-vertex model eigenvectors obtained by means of the generalized Bethe Ansatz is shown to give the six-vertex eigenvectors as constructed in a previous paper by two of the authors. Furthermore, an explicit mapping is established between these eigenvectors and the usual Bethe Ansatz eigenvectors of the six-vertex model. This allows one to show that the indexv labeling the eight-vertex eigenstates becomes exactly the third component of the total spin in the critical limit.     

Series expansion and CAM study of the nonuniversal behavior of the symmetric 16-vertex model

The phase diagram and the critical indices are investigated for the symmetric 16-vertex model on the square lattice by combining a variational series expansion and the coherent anomaly method. The nonuniversal critical exponents smoothly interpolate between two exactly solvable cases, namely the Baxter eight-vertex model and the Ising model.     

Analytic theory of the eight-vertex model

We observe that the exactly solved eight-vertex solid-on-solid model contains an hitherto unnoticed arbitrary field parameter, similar to the horizontal field in the six-vertex model. The parameter is required to describe a continuous spectrum of the unrestricted solid-on-solid model, which has an infinite-dimensional space of states even for a finite lattice. The introduction of the continuous field parameter allows us to completely review the theory of functional relations in the eight-vertex/SOS-model from a uniform analytic point of view. We also present a number of analytic and numerical techniques for the analysis of the Bethe ansatz equations. It turns out that different solutions of these equations can be obtained from each other by analytic continuation. In particular, for small lattices we explicitly demonstrate that the largest and smallest eigenvalues of the transfer matrix of the eight-vertex model are just different branches of the same multivalued function of the field parameter.     

Domain wall partition function of the eight-vertex model with a non-diagonal reflecting end

With the help of the Drinfeld twist or factorizing F-matrix for the eight-vertex SOS model, we derive the recursion relations of the partition function for the eight-vertex model with a generic non-diagonal reflecting end and domain wall boundary condition. Solving the recursion relations, we obtain the explicit determinant expression of the partition function. Our result shows that, contrary to the eight-vertex model without a reflecting end, the partition function can be expressed as a single determinant.     

Renormalization connection between eight-vertex operators and Gaussian operators

The renormalization connection between the eight-vertex model and the Gaussian fixed line that was obtained previously is further investigated. We obtain the critical indices of a large class of eight-vertex operators via an identification after renormalization with appropriate Gaussian operators. We reproduce all the operator identifications previously obtained by Kadanoff and Brown with a different method. In addition, we conjecture some new identifications involving vector operators. As an application of the method, we discuss the extended scaling relation for the critical exponent of the Potts model.     

Symmetry relations in the sixteen-vertex model

We explore the consequences of the existence of a group of linear symmetry transformations for the sixteen-vertex model. An arbitrary sixteen-vertex model is shown tobe equivalent to a standard model, described in terms of ten parameters having an invariant significance. The general eight-vertex model is obtained when four of the invariant parameters vanish. Two further subciasses of the sixteen-vertex model are found to be equivalent to the eight-vertex model. The invariant conditions leading to the solved cases (i.e the general six-vertex, the free-fermion and the symmetric eight-vertex models) are given.     

Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas

Many two-dimensional spin models can be transformed into Coulomb-gas systems in which charges interact via logarithmic potentials. For some models, such as the eight-vertex model and the Ashkin-Teller model, the Coulomb-gas representation has added significantly to the insight in the phase transitions. For other models, notably theXY model and the clock models, the equivalence has been instrumental for almost our entire understanding of the critical behavior. Recently it was shown that theq-state Potts model and then-vector model are equivalent to a Coulomb gas with an asymmetry between positive and negative charges. Fieldlike operators in these spin models transform noninteger charges and magnetic monopoles. With the aid of exactly solved models the Coulombgas representation allows analytic calculation of some critical indices.     

order and disorder lines in systems with competing interactions: II. The IRF model

Systems with competing interactions can be often exactly solved on a restricted subspace of the parameter space, called an order or disorder trajectory. A simple method introduced within the transfer matrix formalism allows for the calculation of the free energy and spin-spin correlation functions along the order and disorder lines of the Ising model with all possible interactions around a face of the square lattice (IRF model). The general eight-vertex model is thoroughly examined and shows full analogy with the quantum spin chain results of the previous paper.On leave from and address after September 1, 1982: Institute for Theoretical Physics, Eötvös University, 1088 Budapest, Puskin U. 5–7, Hungary.     

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.     

Renormalization connection between the eight-vertex model and the gaussian model

A generalized version of the Villain model is defined which contains the eight-vertex model as a special case. A renormalization pattern is formulated in terms of this Villain model by which critical line(s) in the eight-vertex model are connected with the Gaussian fixed line. Gaussian equivalents are found for various operators along the Baxter line. The relation between the present results and those recently obtained by Kadanoff and Brown is discussed.     

Bethe ansatz calculations for the eight-vertex model on a finite strip

Bethe ansatz equations for the eigenvalues of the transfer matrix of the eight-vertex model are solved numerically to yield mass gap data on infinitely long strips of up to 512 sites in width. The finite-size corrections, at criticality, to the free energy per site and polarization gap are found to be in agreement with recent studies of theXXZ spin chain. The leading corrections to the finite-size scaling estimates of the critical line and thermal exponent are also found, providing an explanation of the poor convergence seen in earlier studies. Away from criticality, the linear scaling fields are derived exactly in the full parameter space of the spin system, allowing a thorough test of a recently proposed method of extracting linear scaling fields and related exponents from finite lattice data.     

The Critical Behavior of the Ising Model on Union Jack Lattice

In this paper, we have investigated the critical behavior of the ferromagnetic Ising model on union jack lattice. The model is equivalent to the eight-vertex model which can be solved as a free fermion model with the free fermion approximation. The critical exponents have been obtained as aα = α = 0, β = β'= 0.125, γ = γ' = 0.875 and δ = δ' = 8.     

Lattice gases and exactly solvable models

We detail the construction of a family of lattice gas automata based on a model of 't Hooft, proceeding by use of symmetry principles to define first the kinematics of the model and then the dynamics. A spurious conserved quantity appears; we use it to effect a radical transformation of the model into one whose spacetime configurations are equivalent to the two-dimensional states of an exactly solvable statistical mechanics model, the symmetric eight-vertex model with parameters restricted to a disorder variety. We comment on the implications of this identification for the original lattice gas.     

Kagome晶格上混合自旋模型的自由费密近似解

本文考虑Kagome晶格上的混合自旋模型,运用decimation和对偶变换,证明它等价于一个八顶角模型。由八顶角模型的自由费密近似解给出了该模型的自由能及临界条件的近似表达式。 关键词：     

Auxiliary Vertices Method for Kagomé-Lattice Eight-Vertex Model

We investigate a class of eight-vertex models on a Kagomé lattice. With the help of auxiliary vertices, the Kagomé-lattice eight-vertex model (KEVM) is related to an inhomogeneous system which leads to a one-parameter family of commuting transfer matrices. Using an equation for commuting transfer matrices, we determine their eigenvalues. From calculated eigenvalues the correlation length of the KEVM is derived with its full anisotropy. There are two cases: In the first case the anisotropic correlation length (ACL) is the same as that of the triangular/honeycomb-lattice Ising model. By the use of an algebraic curve, it is shown that the Kagomé-lattice Ising model, the diced-lattice Ising model, and the hard-hexagon model also have (essentially) the same ACL as the KEVM. In the second case we find that the ACL displays 12fold rotational symmetry.     

Drinfeld twist and the domain wall partition function of the eight-vertex model

With the help of the F-basis provided by the Drinfeld twist or factorising F-matrix for the spatial optical soliton model associated with the eight-vertex model, we calculate the partition function for the eight-vertex model on an N × N square lattice with domain wall boundary condition.     

Free field construction for correlation functions of the eight-vertex model

A free field representation for the type I vertex operators and the corner transfer matrices of the eight-vertex model is proposed. The construction uses the vertex-face correspondence, which makes it possible to express correlation functions of the eight-vertex model in terms of correlation functions of the SOS model with a non-local insertion. This new non-local insertion admits a free field representation in terms of Lukyanov's screening operator. The spectrum of the corner transfer matrix and the Baxter-Kelland formula for the average staggered polarization have been reproduced.     

Solution of Reflection Equation

We give some general results concerning solutions of reflection equation associated with the Z_n symmetry Belavin model. The complete solution for the eight-vertex model is obtained.     

Eight-vertex model: Anisotropic interfacial tension and equilibrium crystal shape

The anisotropic interfacial tension of the eight-vertex model is found by a new method, which introduces two inhomogeneous systems. As the width of the system becomes large, a doublet of the largest eigenvalues of the row-row transfer matrix is asymptotically degenerate. The anisotropic interfacial tension is calculated from their finite-size correction terms in this limit. By the use of the anisotropic interfacial tension, the equilibrium crystal shape of the eight-vertex model is derived via Wulff's construction. The equilibrium crystal shape is represented as a simple algebraic curve. We discuss the close relation between the algebraic curve and the form of an elliptic function appearing in the expression of the interfacial tension.     

Wu Algorithm and Solutions of Yang-Baxter Equation for Eight-Vertex Model

In this note, we claim that we have obtained all nondegenerate solutions of Yang-Baxter equation with spectral parameter for square-lattice eight-vertex model without "zero-field" conditions by several solution transformations and six basic solutiofis with the help of Wu algorithm.