1207.1928 (G. Niccoli)

Antiperiodic dynamical 6-vertex and periodic 8-vertex models I: Complete
spectrum by SOV and matrix elements of the identity on separate states    [PDF]

G. Niccoli

The spin-1/2 highest weight representations of the dynamical 6-vertex and the standard 8-vertex Yang-Baxter algebra on a finite lattice are considered in this paper. The integrable quantum models associated to the corresponding transfer matrices under antiperiodic boundary conditions for the dynamical 6-vertex and periodic boundary conditions for the 8-vertex case are analyzed by adapting to them Sklyanin's quantum separation of variables (SOV). For these boundary conditions, gauge transformations are defined which allows to use the antiperiodic dynamical 6-vertex transfer matrix as a tool to solve the spectral problem for the periodic 8-vertex transfer matrix. Here, we explicitly construct the SOV representations from the original space of the representations and provide the complete characterization of the eigenvalues and the eigenstates proving also the simplicity of both the transfermatrix spectra. Moreover, for the antiperiodic dynamical 6-vertex model, the matrix elements of the identity on separated states are computed and characterized by determinant formulae of matrices whose elements are given by sums over the SOV spectrum of the product of the coefficients of the separate states. The results here presented define the required setup to extend to both the dynamical 6-vertex and 8-vertex models an approach recently developed in [1]-[5] to compute the formfactors of the local operators in the SOV framework, these results will be presented in a future publication. Finally, let us point out that our SOV analysis is done without any need to be reduced to the case of chains with an even number of quantum sites or to fix the coupling constant values to the so-called elliptic roots of unit.

View original: http://arxiv.org/abs/1207.1928