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.     

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.     

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.     

Free field construction for the eight-vertex model: representation for form factors

The free field realization of the eight-vertex model is extended to form factors. It is achieved by constructing off-diagonal with respect to the ground state sectors matrix elements of the Λ operator which establishes a relation between corner transfer matrices of the eight-vertex model and of the SOS model. As an example, the two-particle form factor of the σ^z operator is evaluated explicitly.     

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.     

The critical line of an asymmetric eight vertex model

Phenomenological renormalization of finite lattice data is used to locate the critical line of an asymmetric eight-vertex model. This model reduces at a special point to a loop gas on the square lattice. The critical fugacity of the loop gas is estimated to be 2.373±0.001, whereas the correlation exponent appears to be Ising-like.     

The phase diagram of the 2D Ising model with anisotropic next-nearest-neighbor and four-spin interactions

In the present paper, the two-dimensional Ising model with anisotropic nearest-neighbor, next-nearest-neighbor and four-spin interactions has been studied. The ground states and energy of the model have been obtained. The model is equivalent to an eight-vertex model on its dual lattice. In some special cases, the model can be solved exactly as a zero-field eight-vertex model or a free-fermion model. Explicit phase diagrams are obtained exactly.     

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.     

On relations between the XY-, Villain and σ-model in one dimension

The correlation functions of the one-dimensional O (2) σ-model are obtained exactly by taking the continuum limit of the XY-model. They are equivalent to those of the Villain model in one dimension, and the McBryan-Spencer upper bound is precise in the one-dimensional Villain model.     

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.     

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.     

On critical properties of the Ashkin-Teller model

The critical properties of the Ashkin-Teller model in the special case where two pair energies are the same are shown to be the same as those of the eight-vertex model in the weak-coupling limit as a consequence of scaling hypothesis.     

具有混合作用的U(1)模型的广义Villain分析

我们引进了广义Villain公式,考察了双周期紧致U(1)理论的电磁涡旋拓扑结构,得到了该模型在其耦合参数空间中的全部相变结构,其中包括部分禁闭相和二光子相。最后,利用Migdal-Kadanoff重整化群方法的数值结果,得到了这个模型在(N₁,N₂)=(1,5),(1,6),(1,7)等情况下的相图,从而证实了我们的周期高斯近似下的理论分析。 关键词：     

Duality in the U(1) Higgs model with an external field

An external electromagnetic field is coupled to the lattice U(1) Higgs model in a Villain form. Duality transformations are then used to express the partition function in terms of an effective lagrangian of topological excitations and their couplings to the external field. Consequences for the phase diagram are derived.     

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.     

Finite-size scaling corrections for the eight-vertex model

Finite-size scaling corrections are calculated analytically for two of the maximal eigenvalues of the transfer matrix in the isotropic eight-vertex model. The valuec=1 for the conformal anomaly of the Virasoro algebra is confirmed.     

Algebraic Bethe ansatz for the eight-vertex model with general open boundary conditions

By using the intertwiner and face-vertex correspondence relation, we obtain the Bethe ansatz equation of the eight-vertex model with open boundary conditions in the framework of algebraic Bethe ansatz method. The open boundary condition under consideration is the general solution of the reflection equation for the eight-vertex model with only one restriction on the free parameters of the right side reflecting boundary matrix. The reflecting boundary matrices used in this paper thus may have off-diagonal elements. Our construction can also be used for the Bethe ansatz of SOS model with reflection boundaries.     

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.     

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.