32 research outputs found
GL(3)-Based Quantum Integrable Composite Models. I. Bethe Vectors
We consider a composite generalized quantum integrable model solvable by the
nested algebraic Bethe ansatz. Using explicit formulas of the action of the
monodromy matrix elements onto Bethe vectors in the GL(3)-based quantum
integrable models we prove a formula for the Bethe vectors of composite model.
We show that this representation is a particular case of general coproduct
property of the weight functions (Bethe vectors) found in the theory of the
deformed Knizhnik-Zamolodchikov equation.Comment: The title has been changed to make clearer the connexion with the
preprint arXiv:1502.0196
GL(3)-Based Quantum Integrable Composite Models. II. Form Factors of Local Operators
We study integrable models solvable by the nested algebraic Bethe ansatz and
possessing the GL(3)-invariant R-matrix. We consider a composite model where
the total monodromy matrix of the model is presented as a product of two
partial monodromy matrices. Assuming that the last ones can be expanded into
series with respect to the inverse spectral parameter we calculate matrix
elements of the local operators in the basis of the transfer matrix
eigenstates. We obtain determinant representations for these matrix elements.
Thus, we solve the inverse scattering problem in a weak sense.Comment: The title has been changed to make clearer the connexion with the
preprint arXiv:1501.0756
Generating Series for Nested Bethe Vectors
We reformulate nested relations between off-shell
Bethe vectors as a certain equation on
generating series of strings of the composed
currents. Using inversion of the generating series we find a new type of
hierarchical relations between universal off-shell Bethe vectors, useful for a
derivation of Bethe equation. As an example of application, we use these
relations for a derivation of analytical Bethe ansatz equations [Arnaudon D. et
al., Ann. Henri Poincar\'e 7 (2006), 1217-1268, math-ph/0512037] for the
parameters of universal Bethe vectors of the algebra
.Comment: This is a contribution to the Special Issue on Kac-Moody Algebras and
Applications, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric -Matrix
We study quantum integrable models with GL(3) trigonometric -matrix and
solvable by the nested algebraic Bethe ansatz. Using the presentation of the
universal Bethe vectors in terms of projections of products of the currents of
the quantum affine algebra onto intersections of
different types of Borel subalgebras, we prove that the set of the nested Bethe
vectors is closed under the action of the elements of the monodromy matrix
Classical elliptic current algebras
In this paper we discuss classical elliptic current algebras and show that
there are two different choices of commutative test function algebras on a
complex torus leading to two different elliptic current algebras. Quantization
of these classical current algebras give rise to two classes of quantized
dynamical quasi-Hopf current algebras studied by Enriquez-Felder-Rubtsov and
Arnaudon-Buffenoir-Ragoucy-Roche-Jimbo-Konno-Odake-Shiraishi. Different
degenerations of the classical elliptic algebras are considered. They yield
different versions of rational and trigonometric current algebras. We also
review the averaging method of Faddeev-Reshetikhin, which allows to restore
elliptic algebras from the trigonometric ones
Quantum relativistic Toda chain at root of unity: isospectrality, modified Q
We investigate an N-state spin model called quantum
relativistic Toda chain and based on the unitary
finite-dimensional representations of the Weyl algebra with q
being Nth primitive root of unity. Parameters of the
finite-dimensional representation of the local Weyl algebra form
the classical discrete integrable system. Nontrivial dynamics of
the classical counterpart corresponds to isospectral
transformations of the spin system. Similarity operators are
constructed with the help of modified Baxter's Q-operators. The
classical counterpart of the modified Q-operator for the initial
homogeneous spin chain is a Bäcklund transformation.
This transformation creates an extra Hirota-type soliton in a
parameterization of the chain structure. Special choice of values
of solitonic amplitudes yields a degeneration of spin
eigenstates, leading to the quantum separation of variables, or
the functional Bethe ansatz. A projector to the separated
eigenstates is constructed explicitly as a product of modified
Q-operators