We apply our new approach of quantum Separation of Variables (SoV) to
the complete characterization of the transfer matrix spectrum of quantum
integrable lattice models associated to \bm{gl_n}𝐠𝐥𝐧-invariant
\bm{R}𝐑-matrices
in the fundamental representations. We consider lattices with
\bm{N}𝐍-sites
and general quasi-periodic boundary conditions associated to an
arbitrary twist matrix \bm{K}𝐊
having simple spectrum (but not necessarily diagonalizable). In our
approach the SoV basis is constructed in an universal manner starting
from the direct use of the conserved charges of the models, e.g. from
the commuting family of transfer matrices. Using the integrable
structure of the models, incarnated in the hierarchy of transfer
matrices fusion relations, we prove that our SoV basis indeed separates
the spectrum of the corresponding transfer matrices. Moreover, the
combined use of the fusion rules, of the known analytic properties of
the transfer matrices and of the SoV basis allows us to obtain the
complete characterization of the transfer matrix spectrum and to prove
its simplicity. Any transfer matrix eigenvalue is completely
characterized as a solution of a so-called quantum spectral curve
equation that we obtain as a difference functional equation of order
\bm{n}𝐧.
Namely, any eigenvalue satisfies this equation and any solution of this
equation having prescribed properties that we give leads to an
eigenvalue. We construct the associated eigenvector, unique up to
normalization, of the transfer matrices by computing its decomposition
on the SoV basis that is of a factorized form written in terms of the
powers of the corresponding eigenvalues. Finally, if the twist matrix
\bm{K}𝐊
is diagonalizable with simple spectrum we prove that the transfer matrix
is also diagonalizable with simple spectrum. In that case, we give a
construction of the Baxter \bm{Q}𝐐-operator
and show that it satisfies a \bm{T}𝐓-\bm{Q}𝐐
equation of order \bm{n}𝐧,
the quantum spectral curve equation, involving the hierarchy of the
fused transfer matrices.
Conjectures on hidden Onsager algebra symmetries in interacting quantum lattice models
