We describe the extension, beyond fundamental representations of the
Yang-Baxter algebra, of our new construction of separation of variables
bases for quantum integrable lattice models. The key idea underlying our
approach is to use the commuting conserved charges of the quantum
integrable models to generate bases in which their spectral problem is separated,
i.e. in which the wave functions are factorized in terms of specific
solutions of a functional equation. For the so-called “non-fundamental”
models we construct two different types of SoV bases. The first is given
from the fundamental quantum Lax operator having isomorphic auxiliary
and quantum spaces and that can be obtained by fusion of the original
quantum Lax operator. The construction essentially follows the one we
used previously for fundamental models and allows us to derive the
simplicity and diagonalizability of the transfer matrix spectrum. Then,
starting from the original quantum Lax operator and using the full tower
of the fused transfer matrices, we introduce a second type of SoV bases
for which the proof of the separation of the transfer matrix spectrum is
naturally derived. We show that, under some special choice, this second
type of SoV bases coincides with the one associated to Sklyanin’s
approach. Moreover, we derive the finite difference type (quantum
spectral curve) functional equation and the set of its solutions
defining the complete transfer matrix spectrum. This is explicitly
implemented for the integrable quantum models associated to the higher
spin representations of the general quasi-periodic
Y(gl_{2})Y(gl2)
Yang-Baxter algebra. Our SoV approach also leads to the construction of
a QQ-operator
in terms of the fused transfer matrices. Finally, we show that the
QQ-operator
family can be equivalently used as the family of commuting conserved
charges enabling to construct our SoV bases.
FLUCTUATING INTERFACES, SURFACE TENSION, AND CAPILLARY WAVES: AN INTRODUCTION
