| Summary: | Using the framework of the quantum separation of variables (SoV) for higher
rank quantum integrable lattice models [1], we introduce some foundations to go
beyond the obtained complete transfer matrix spectrum description, and open the
way to the computation of matrix elements of local operators. This first
amounts to obtain simple expressions for scalar products of the so-called
separate states (transfer matrix eigenstates or some simple generalization of
them). In the higher rank case, left and right SoV bases are expected to be
pseudo-orthogonal, that is for a given SoV co-vector, there could be more than
one non-vanishing overlap with the vectors of the chosen right SoV basis. For
simplicity, we describe our method to get these pseudo-orthogonality overlaps
in the fundamental representations of the $\mathcal{Y}(gl_3)$ lattice model
with $N$ sites, a case of rank 2. The non-zero couplings between the co-vector
and vector SoV bases are exactly characterized. While the corresponding
SoV-measure stays reasonably simple and of possible practical use, we address
the problem of constructing left and right SoV bases which do satisfy standard
orthogonality. In our approach, the SoV bases are constructed by using families
of conserved charges. This gives us a large freedom in the SoV bases
construction, and allows us to look for the choice of a family of conserved
charges which leads to orthogonal co-vector/vector SoV bases. We first define
such a choice in the case of twist matrices having simple spectrum and zero
determinant. Then, we generalize the associated family of conserved charges and
orthogonal SoV bases to generic simple spectrum and invertible twist matrices.
Under this choice of conserved charges, and of the associated orthogonal SoV
bases, the scalar products of separate states simplify considerably and take a
form similar to the $\mathcal{Y}(gl_2)$ rank one case.
|
|---|
