1,092 research outputs found
Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from SOV
We solve the longstanding problem to define a functional characterization of
the spectrum of the transfer matrix associated to the most general spin-1/2
representations of the 6-vertex reflection algebra for general inhomogeneous
chains. The corresponding homogeneous limit reproduces the spectrum of the
Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most
general integrable boundaries. The spectrum is characterized by a second order
finite difference functional equation of Baxter type with an inhomogeneous term
which vanishes only for some special but yet interesting non-diagonal boundary
conditions. This functional equation is shown to be equivalent to the known
separation of variable (SOV) representation hence proving that it defines a
complete characterization of the transfer matrix spectrum. The polynomial
character of the Q-function allows us then to show that a finite system of
equations of generalized Bethe type can be similarly used to describe the
complete transfer matrix spectrum.Comment: 28 page
On determinant representations of scalar products and form factors in the SoV approach: the XXX case
In the present article we study the form factors of quantum integrable
lattice models solvable by the separation of variables (SoV) method. It was
recently shown that these models admit universal determinant representations
for the scalar products of the so-called separate states (a class which
includes in particular all the eigenstates of the transfer matrix). These
results permit to obtain simple expressions for the matrix elements of local
operators (form factors). However, these representations have been obtained up
to now only for the completely inhomogeneous versions of the lattice models
considered. In this article we give a simple algebraic procedure to rewrite the
scalar products (and hence the form factors) for the SoV related models as
Izergin or Slavnov type determinants. This new form leads to simple expressions
for the form factors in the homogeneous and thermodynamic limits. To make the
presentation of our method clear, we have chosen to explain it first for the
simple case of the Heisenberg chain with anti-periodic boundary
conditions. We would nevertheless like to stress that the approach presented in
this article applies as well to a wide range of models solved in the SoV
framework.Comment: 46 page
The open XXX spin chain in the SoV framework: scalar product of separate states
We consider the XXX open spin-1/2 chain with the most general non-diagonal
boundary terms, that we solve by means of the quantum separation of variables
(SoV) approach. We compute the scalar products of separate states, a class of
states which notably contains all the eigenstates of the model. As usual for
models solved by SoV, these scalar products can be expressed as some
determinants with a non-trivial dependance in terms of the inhomogeneity
parameters that have to be introduced for the method to be applicable. We show
that these determinants can be transformed into alternative ones in which the
homogeneous limit can easily be taken. These new representations can be
considered as generalizations of the well-known determinant representation for
the scalar products of the Bethe states of the periodic chain. In the
particular case where a constraint is applied on the boundary parameters, such
that the transfer matrix spectrum and eigenstates can be characterized in terms
of polynomial solutions of a usual T-Q equation, the scalar product that we
compute here corresponds to the scalar product between two off-shell Bethe-type
states. If in addition one of the states is an eigenstate, the determinant
representation can be simplified, hence leading in this boundary case to direct
analogues of algebraic Bethe ansatz determinant representations of the scalar
products for the periodic chain.Comment: 39 page
Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and matrix elements of some quasi-local operators
The integrable quantum models, associated to the transfer matrices of the
6-vertex reflection algebra for spin 1/2 representations, are studied in this
paper. In the framework of Sklyanin's quantum separation of variables (SOV), we
provide the complete characterization of the eigenvalues and eigenstates of the
transfer matrix and the proof of the simplicity of the transfer matrix
spectrum. Moreover, we use these integrable quantum models as further key
examples for which to develop a method in the SOV framework to compute matrix
elements of local operators. This method has been introduced first in [1] and
then used also in [2], it is based on the resolution of the quantum inverse
problem (i.e. the reconstruction of all local operators in terms of the quantum
separate variables) plus the computation of the action of separate covectors on
separate vectors. In particular, for these integrable quantum models, which in
the homogeneous limit reproduce the open spin-1/2 XXZ quantum chains with
non-diagonal boundary conditions, we have obtained the SOV-reconstructions for
a class of quasi-local operators and determinant formulae for the
covector-vector actions. As consequence of these findings we provide one
determinant formulae for the matrix elements of this class of reconstructed
quasi-local operators on transfer matrix eigenstates.Comment: 40 pages. Minor modifications in the text and some notations and some
more reference adde
Form factors of descendant operators in the massive Lee-Yang model
The form factors of the descendant operators in the massive Lee-Yang model
are determined up to level 7. This is first done by exploiting the conserved
quantities of the integrable theory to generate the solutions for the
descendants starting from the lowest non-trivial solutions in each operator
family. We then show that the operator space generated in this way, which is
isomorphic to the conformal one, coincides, level by level, with that implied
by the -matrix through the form factor bootstrap. The solutions we determine
satisfy asymptotic conditions carrying the information about the level that we
conjecture to hold for all the operators of the model.Comment: 23 page
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