SciPost Phys. 10, 006 (2021) ·
published 12 January 2021

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We explain how to compute correlation functions at zero temperature within
the framework of the quantum version of the Separation of Variables (SoV) in
the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted
(quasiperiodic) boundary conditions. We first detail all steps of our method
in the case of antiperiodic boundary conditions. The model can be solved in
the SoV framework by introducing inhomogeneity parameters. The action of local
operators on the eigenstates are then naturally expressed in terms of multiple
sums over these inhomogeneity parameters. We explain how to transform these
sums over inhomogeneity parameters into multiple contour integrals. Evaluating
these multiple integrals by the residues of the poles outside the integration
contours, we rewrite this action as a sum involving the roots of the Baxter
polynomial plus a contribution of the poles at infinity. We show that the
contribution of the poles at infinity vanishes in the thermodynamic limit, and
that we recover in this limit for the zerotemperature correlation functions
the multiple integral representation that had been previously obtained through
the study of the periodic case by Bethe Ansatz or through the study of the
infinite volume model by the qvertex operator approach. We finally show that
the method can easily be generalized to the case of a more general nondiagonal
twist: the corresponding weights of the different terms for the correlation
functions in finite volume are then modified, but we recover in the
thermodynamic limit the same multiple integral representation than in the
periodic or antiperiodic case, hence proving the independence of the
thermodynamic limit of the correlation functions with respect to the particular
form of the boundary twist.
Jean Michel Maillet, Giuliano Niccoli, Louis Vignoli
SciPost Phys. 9, 086 (2020) ·
published 11 December 2020

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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 socalled
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
pseudoorthogonal, that is for a given SoV covector, there could be more than
one nonvanishing overlap with the vectors of the chosen right SoV basis. For
simplicity, we describe our method to get these pseudoorthogonality overlaps
in the fundamental representations of the $\mathcal{Y}(gl_3)$ lattice model
with $N$ sites, a case of rank 2. The nonzero couplings between the covector
and vector SoV bases are exactly characterized. While the corresponding
SoVmeasure 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 covector/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.
Jean Michel Maillet, Giuliano Niccoli, Louis Vignoli
SciPost Phys. 9, 060 (2020) ·
published 27 October 2020

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We construct quantum Separation of Variables (SoV) bases for both the
fundamental inhomogeneous $gl_{\mathcal{M}\mathcal{N}}$ supersymmetric
integrable models and for the inhomogeneous Hubbard model both defined with
quasiperiodic twisted boundary conditions given by twist matrices having
simple spectrum. The SoV bases are obtained by using the integrable structure
of these quantum models,i.e. the associated commuting transfer matrices,
following the general scheme introduced in [1]; namely, they are given by set
of states generated by the multiple action of the transfer matrices on a
generic covector. The existence of such SoV bases implies that the
corresponding transfer matrices have non degenerate spectrum and that they are
diagonalizable with simple spectrum if the twist matrices defining the
quasiperiodic boundary conditions have that property. Moreover, in these SoV
bases the resolution of the transfer matrix eigenvalue problem leads to the
resolution of the full spectral problem, i.e. both eigenvalues and
eigenvectors. Indeed, to any eigenvalue is associated the unique (up to a
trivial overall normalization) eigenvector whose wavefunction in the SoV bases
is factorized into products of the corresponding transfer matrix eigenvalue
computed on the spectrum of the separate variables. As an application, we
characterize completely the transfer matrix spectrum in our SoV framework for
the fundamental $gl_{12}$ supersymmetric integrable model associated to a
special class of twist matrices. From these results we also prove the
completeness of the Bethe Ansatz for that case. The complete solution of the
spectral problem for fundamental inhomogeneous $gl_{\mathcal{M}\mathcal{N}}$
supersymmetric integrable models and for the inhomogeneous Hubbard model under
the general twisted boundary conditions will be addressed in a future
publication.
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.
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