949 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
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 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
On algebraic structures in supersymmetric principal chiral model
Using the Poisson current algebra of the supersymmetric principal chiral
model, we develop the algebraic canonical structure of the model by evaluating
the fundamental Poisson bracket of the Lax matrices that fits into the rs
matrix formalism of non-ultralocal integrable models. The fundamental Poisson
bracket has been used to compute the Poisson bracket algebra of the monodromy
matrix that gives the conserved quantities in involution
Resolution of the Nested Hierarchy for Rational sl(n) Models
We construct Drinfel'd twists for the rational sl(n) XXX-model giving rise to
a completely symmetric representation of the monodromy matrix. We obtain a
polarization free representation of the pseudoparticle creation operators
figuring in the construction of the Bethe vectors within the framework of the
quantum inverse scattering method. This representation enables us to resolve
the hierarchy of the nested Bethe ansatz for the sl(n) invariant rational
Heisenberg model. Our results generalize the findings of Maillet and Sanchez de
Santos for sl(2) models.Comment: 25 pages, no figure
Domain wall partition functions and KP
We observe that the partition function of the six vertex model on a finite
square lattice with domain wall boundary conditions is (a restriction of) a KP
tau function and express it as an expectation value of charged free fermions
(up to an overall normalization).Comment: 16 pages, LaTeX2
On factorizing -matrices in and spin chains
We consider quantum spin chains arising from -fold tensor products of the
fundamental evaluation representations of and .
Using the partial -matrix formalism from the seminal work of Maillet and
Sanchez de Santos, we derive a completely factorized expression for the
-matrix of such models and prove its equivalence to the expression obtained
by Albert, Boos, Flume and Ruhlig. A new relation between the -matrices and
the Bethe eigenvectors of these spin chains is given.Comment: 30 page
Petrology and geochemistry of the central North Fiji basin spreading centre (Southwest Pacific) betwenn 16°S and 22°S
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