An antiperiodic dynamical six-vertex model: I. Complete spectrum by SOV, matrix elements of the identity on separate states and connections to the periodic eight-vertex model

摘要

The spin-1/2 highest weight representations of the dynamical six-vertex and the standard eight-vertex Yang-Baxter algebra on a finite chain are considered in this paper. In particular, the integrable quantum models associated with the corresponding transfer matrices under antiperiodic boundary conditions for the dynamical six-vertex case and periodic boundary conditions for the eight-vertex case are analyzed here. For the antiperiodic dynamical six-vertex transfer matrix defined on chains with an odd number of sites, we adapt Sklyanin's quantum separation of variable (SOV) method and explicitly construct the SOV representations from the original space of the representations. In this way, we provide the complete characterization of the eigenvalues and the eigenstates proving also the simplicity of its spectrum. Moreover, we characterize the matrix elements of the identity on separated states of this model by determinant formulae. The matrices entering these determinants have elements given by sums over the SOV spectrum of the product of the coefficients of the separate states. This SOV analysis is done without any need to be reduced to the case of the so-called elliptic roots of unit, and the results derived here define the required setup to extend to the dynamical six-vertex model the approach recently developed by the author and collaborators to compute the form factors of the local operators in the SOV framework. For the periodic eight-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 six-vertex transfer matrix spectrum. This implies that the set of the periodic eight-vertex eigenvalues is contained in the set of the antiperiodic dynamical six-vertex eigenvalues. A criterion is introduced to find simultaneous eigenvalues of these two transfer matrices and associate with any of such eigenvalues one nonzero eigenstate of the periodic eight-vertex transfer matrix by using the SOV results. Moreover, a preliminary discussion on the degeneracy occurring for odd chains in the periodic eight-vertex transfer matrix spectrum is also presented.