Multi-point local height probabilities of the CSOS model within the algebraic Bethe Ansatz framework

We study the local height probabilities of the exactly solvable cyclic solid-on-solid model within the algebraic Bethe Ansatz framework. We more specifically consider multi-point local height probabilities at adjacent sites on the lattice. We derive multiple integral representations for these quantities at the thermodynamic limit, starting from finite-size expressions for the corresponding multi-point matrix elements in the Bethe basis as sums of determinants of elliptic functions.Comment: 39 page

Antiperiodic dynamical 6-vertex model by separation of variables II: Functional equations and form factors

We pursue our study of the antiperiodic dynamical 6-vertex model using Sklyanin's separation of variables approach, allowing in the model new possible global shifts of the dynamical parameter. We show in particular that the spectrum and eigenstates of the antiperiodic transfer matrix are completely characterized by a system of discrete equations. We prove the existence of different reformulations of this characterization in terms of functional equations of Baxter's type. We notably consider the homogeneous functional $T$-$Q$ equation which is the continuous analog of the aforementioned discrete system and show, in the case of a model with an even number of sites, that the complete spectrum and eigenstates of the antiperiodic transfer matrix can equivalently be described in terms of a particular class of its $Q$-solutions, hence leading to a complete system of Bethe equations. Finally, we compute the form factors of local operators for which we obtain determinant representations in finite volume.Comment: 52 page

Antiperiodic dynamical 6-vertex model I: Complete spectrum by SOV, matrix elements of the identity on separate states and connections to the periodic 8-vertex model

The spin-1/2 highest weight representations of the dynamical 6-vertex and the standard 8-vertex Yang-Baxter algebra on a finite chain are considered in this paper. For the antiperiodic dynamical 6-vertex transfer matrix defined on chains with an odd number of sites, we adapt the Sklyanin's quantum separation of variable (SOV) method and explicitly construct SOV representations from the original space of representations. We provide the complete characterization of eigenvalues and eigenstates proving also the simplicity of its spectrum. Moreover, we characterize the matrix elements of the identity on separated states by determinant formulae. The matrices entering in these determinants have elements given by sums over the SOV spectrum of the product of the coefficients of separate states. This SOV analysis is not reduced to the case of the elliptic roots of unit and the results here derived define the required setup to extend to the dynamical 6-vertex model the approach recently developed in [1]-[5] to compute the form factors of the local operators in the SOV framework, these results will be presented in a future publication. For the periodic 8-vertex transfer matrix, we prove that its eigenvalues have to satisfy a fixed system of equations. In the case of a chain with an odd number of sites, this system of equations is the same entering in the SOV characterization of the antiperiodic dynamical 6-vertex transfer matrix spectrum. This implies that the set of the periodic 8-vertex eigenvalues is contained in the set of the antiperiodic dynamical 6-vertex eigenvalues. A criterion is introduced to find simultaneous eigenvalues of these two transfer matrices and associate to any of such eigenvalues one nonzero eigenstate of the periodic 8-vertex transfer matrix by using the SOV results. Moreover, a preliminary discussion on the degeneracy of the periodic 8-vertex spectrum is also presented.Comment: 36 pages, main modifications in section 3 and one appendix added, no result modified for the dynamical 6-vertex transfer matrix spectrum and the matrix elements of identity on separate states for chains with an odd number of site