1,059 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

### Current Algebra of Classical Non-Linear Sigma Models

The current algebra of classical non-linear sigma models on arbitrary
Riemannian manifolds is analyzed. It is found that introducing, in addition to
the Noether current $j_\mu$ associated with the global symmetry of the theory,
a composite scalar field $j$, the algebra closes under Poisson brackets.Comment: 6 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

### 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 $XXX$ 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

### Long-distance asymptotic behaviour of multi-point correlation functions in massless quantum models

We provide a microscopic model setting that allows us to readily access to
the large-distance asymptotic behaviour of multi-point correlation functions in
massless, one-dimensional, quantum models. The method of analysis we propose is
based on the form factor expansion of the correlation functions and does not
build on any field theory reasonings. It constitutes an extension of the
restricted sum techniques leading to the large-distance asymptotic behaviour of
two-point correlation functions obtained previously.Comment: 25 page

### The universal R-matrix and its associated quantum algebra as functionals of the classical r-matrix: the $sl_{2}$ case

Using a geometrical approach to the quantum Yang-Baxter equation, the quantum algebra ${\cal U}_{\hbar}(sl_{2})$ and its universal quantum $R$-matrix are explicitely constructed as functionals of the associated classical $r$-matrix. In this framework, the quantum algebra ${\cal U}_{\hbar}(sl_{2})$ is naturally imbedded in the universal envelopping algebra of the $sl_{2}$ current algebra

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