A new eight vertex model and higher dimensional, multiparameter generalizations

The Vertex-Face Correspondence and Correlation Functions of the Fusion Eight-Vertex Model I: The General Formalism

Based on the vertex-face correspondence, we give an algebraic analysis formulation of correlation functions of the $k\times k$ fusion eight-vertex model in terms of the corresponding fusion SOS model. Here $k\in Z_{>0}$. A general formula for correlation functions is derived as a trace over the space of states of lattice operators such as the corner transfer matrices, the half transfer matrices (vertex operators) and the tail operator. We give a realization of these lattice operators as well as the space of states as objects in the level $k$ representation theory of the elliptic algebra $U_{q,…

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 nonlocal insertion. This new nonlocal insertion admits of 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.

Bethe Ansatz for higher spin eight vertex models

A generalization of the eight vertex model by means of higher spin representations of the Sklyanin algebra is investigated by the quantum inverse scattering method and the algebraic Bethe Ansatz. Under the well-known string hypothesis low-lying excited states are considered and scattering phase shifts of two physical particles are calculated. The $S$ matrix of two particle states is shown to be proportional to the Baxter's elliptic $R$ matrix with a different elliptic modulus from the original one.

Form factors of spin 1 analogue of the eight-vertex model

The twenty-one-vertex model, the spin $1$ analogue of the eight-vertex model is considered on the basis of free field representations of vertex operators in the $2\times 2$-fold fusion SOS model and vertex-face transformation. The tail operators, which translate corner transfer matrices of the twenty-one-vertex model into those of the fusion SOS model, are constructed by using free bosons and fermions for both diagonal and off-diagonal matrix elements with respect to the ground state sectors. Form factors of any local operators are therefore obtained in terms of multiple integral formulae, in …

Traces on the Sklyanin algebra and correlation functions of the eight-vertex model

We propose a conjectural formula for correlation functions of the Z-invariant (inhomogeneous) eight-vertex model. We refer to this conjecture as Ansatz. It states that correlation functions are linear combinations of products of three transcendental functions, with theta functions and derivatives as coefficients. The transcendental functions are essentially logarithmic derivatives of the partition function per site. The coefficients are given in terms of a linear functional on the Sklyanin algebra, which interpolates the usual trace on finite dimensional representations. We establish the exist…

Classification of eight-vertex solutions of the colored Yang-Baxter equation

In this paper all eight-vertex type solutions of the colored Yang-Baxter equation dependent on spectral as well as color parameter are given. It is proved that they are composed of three groups of basic solutions, three groups of their degenerate forms and two groups of trivial solutions up to five solution transformations. Moreover, all non-trivial solutions can be classified into two types called Baxter and Free-Fermion type.

三维八顶点模型中的组合相关函数

Combinatorial correlation functions in three-dimensional eight-vertex models

提出三维立方晶格上自相似自旋变换的新版本，使等概率“组合”模型中非平凡自旋相关的计算成为可能。

Free field constructions for the elliptic algebra ${\cal A}_{q,p}(\hat{sl}_2)$ and Baxter's eight-vertex model

Three examples of free field constructions for the vertex operators of the elliptic quantum group ${\cal A}_{q,p}(\hat{sl}_2)$ are obtained. Two of these (for $p^{1/2}=\pm q^{3/2},p^{1/2}=-q^2$) are based on representation theories of the deformed Virasoro algebra, which correspond to the level 4 and level 2 $Z$-algebra of Lepowsky and Wilson. The third one ($p^{1/2}=q^{3}$) is constructed over a tensor product of a bosonic and a fermionic Fock spaces. The algebraic structure at $p^{1/2}=q^{3}$, however, is not related to the deformed Virasoro algebra. Using these free field constructions, an …

Q-operators in the six-vertex model

In this paper we continue the study of $Q$-operators in the six-vertex model and its higher spin generalizations. In [1] we derived a new expression for the higher spin $R$-matrix associated with the affine quantum algebra $U_q(\widehat{sl(2)})$. Taking a special limit in this $R$-matrix we obtained new formulas for the $Q$-operators acting in the tensor product of representation spaces with arbitrary complex spin. Here we use a different strategy and construct $Q$-operators as integral operators with factorized kernels based on the original Baxter's method used in the solution of the eight-ve…