Analytic theory of the eight-vertex model

Abstract

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.

58 pages, 12 figures, minor corrections

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• Complete Generalized Gibbs Ensemble in an interacting Theory
• Quantum integrability and supersymmetric vacua
• A Shortcut to the Q-Operator
• On the Yang-Baxter equation for the six-vertex model
• The Baxter's Q-operator for the W-algebra $W_N$
• Complete Bethe Ansatz solution of the open spin-s XXZ chain with general integrable boundary terms
• Refined functional relations for the elliptic SOS model
• The 8-vertex model with quasi-periodic boundary conditions
• A new Q-matrix in the Eight-Vertex Model
• Spin-1/2 Ising-Heisenberg model with the pair XYZ Heisenberg interaction and quartic Ising interactions as the exactly soluble zero-field eight-vertex model
• Scalar products of Bethe vectors in the 8-vertex model
• The TQ equation of the 8 vertex model for complex elliptic roots of unity
• New Q matrices and their functional equations for the eight vertex model at elliptic roots of unity
• The Q-operator and Functional Relations of the Eight-vertex Model at Root-of-unity $\eta =\frac{2mK}{N}$ for odd N
• Generalized Gibbs Ensemble and string-charge relations in nested Bethe Ansatz
• Extended T-systems, Q matrices and T-Q relations for $s\ell (2)$ models at roots of unity
• Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder
• Free Energy of the Eight Vertex Model with an Odd Number of Lattice Sites
• Fusion hierarchies, $T$-systems and $Y$-systems for the dilute $A_2^{(2)}$ loop models on a strip
• The Eight Vertex Model.New results