SciPost Phys. 6, 071 (2019) ·
published 21 June 2019

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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 gl(n)invariant Rmatrices in the fundamental
representations. We consider lattices with N sites and quasiperiodic boundary
conditions associated to an arbitrary twist 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, i.e., 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 socalled quantum
spectral curve equation that we obtain as a difference functional equation of
order n. Namely, any eigenvalue satisfies this equation and any solution of
this equation having prescribed properties leads to an eigenvalue. We construct
the associated eigenvector, unique up to normalization, by computing its
decomposition on the SoV basis that is of a factorized form written in terms of
the powers of the corresponding eigenvalues. If the twist matrix K is
diagonalizable with simple spectrum then the transfer matrix is also
diagonalizable with simple spectrum. In that case, we give a construction of
the Baxter Qoperator satisfying a TQ equation of order n, the quantum
spectral curve equation, involving the hierarchy of the fused transfer
matrices.
SciPost Phys. 5, 026 (2018) ·
published 24 September 2018

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This article is a direct continuation of [1] where we begun the study of the
transfer matrix spectral problem for the cyclic representations of the
trigonometric 6vertex reflection algebra associated to the BazhanovStroganov
Lax operator. There we addressed this problem for the case where one of the
Kmatrices describing the boundary conditions is triangular. In the present
article we consider the most general integrable boundary conditions, namely the
most general boundary Kmatrices satisfying the reflection equation. The
spectral analysis is developed by implementing the method of Separation of
Variables (SoV). We first design a suitable gauge transformation that enable us
to put into correspondence the spectral problem for the most general boundary
conditions with another one having one boundary Kmatrix in a triangular form.
In these settings the SoV resolution can be obtained along an extension of the
method described in [1]. The transfer matrix spectrum is then completely
characterized in terms of the set of solutions to a discrete system of
polynomial equations in a given class of functions and equivalently as the set
of solutions to an analogue of Baxter's TQ functional equation. We further
describe scalar product properties of the separate states including eigenstates
of the transfer matrix.
Jean Michel Maillet, Giuliano Niccoli, Baptiste Pezelier
SciPost Phys. 2, 009 (2017) ·
published 28 February 2017

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We study the transfer matrix spectral problem for the cyclic representations
of the trigonometric 6vertex reflection algebra associated to the
BazhanovStroganov Lax operator. The results apply as well to the spectral
analysis of the lattice sineGordon model with integrable open boundary
conditions. This spectral analysis is developed by implementing the method of
separation of variables (SoV). The transfer matrix spectrum (both eigenvalues
and eigenstates) is completely characterized in terms of the set of solutions
to a discrete system of polynomial equations in a given class of functions.
Moreover, we prove an equivalent characterization as the set of solutions to a
Baxter's like TQ functional equation and rewrite the transfer matrix
eigenstates in an algebraic Bethe ansatz form. In order to explain our method
in a simple case, the present paper is restricted to representations containing
one constraint on the boundary parameters and on the parameters of the
BazhanovStroganov Lax operator. In a next article, some more technical tools
(like Baxter's gauge transformations) will be introduced to extend our approach
to general integrable boundary conditions.