106 research outputs found
Generalized coordinate Bethe ansatz for non diagonal boundaries
We compute the spectrum and the eigenstates of the open XXX model with
non-diagonal (triangular) boundary matrices. Since the boundary matrices are
not diagonal, the usual coordinate Bethe ansatz does not work anymore, and we
use a generalization of it to solve the problem.Comment: 11 pages; References added and misprints correcte
Classification of three-state Hamiltonians solvable by Coordinate Bethe Ansatz
We classify all Hamiltonians with rank 1 symmetry, acting on a periodic
three-state spin chain, and solvable through (generalisation of) the coordinate
Bethe ansatz (CBA). We obtain in this way four multi-parametric extensions of
the known 19-vertex Hamiltonians (such as Zamolodchikov-Fateev,
Izergin-Korepin, Bariev Hamiltonians). Apart from the 19-vertex Hamiltonians,
there exists 17-vertex and 14-vertex Hamiltonians that cannot be viewed as
subcases of the 19-vertex ones. In the case of 17-vertex Hamiltonian, we get a
generalization of the genus 5 special branch found by Martins, plus three new
ones. We get also two 14-vertex Hamiltonians.
We solve all these Hamiltonians using CBA, and provide their spectrum,
eigenfunctions and Bethe equations. A special attention is made to provide the
specifications of our multi-parametric Hamiltonians that give back known
Hamiltonians.Comment: 30 pages; web page: http://www.coulomb.univ-montp2.fr/3Ha
Algebraic Bethe ansatz for open XXX model with triangular boundary matrices
We consider open XXX spins chain with two general boundary matrices submitted
to one constraint, which is equivalent to the possibility to put the two
matrices in a triangular form. We construct Bethe vectors from a generalized
algebraic Bethe ansatz. As usual, the method also provides Bethe equations and
transfer matrix eigenvalues.Comment: 10 pge
Integrable approach to simple exclusion processes with boundaries. Review and progress
We study the matrix ansatz in the quantum group framework, applying
integrable systems techniques to statistical physics models. We start by
reviewing the two approaches, and then show how one can use the former to get
new insight on the latter. We illustrate our method by solving a model of
reaction-diffusion. An eigenvector for the transfer matrix for the XXZ spin
chain with non-diagonal boundary is also obtained using a matrix ansatz.Comment: 44 page
A Calabi-Yau algebra with symmetry and the Clebsch-Gordan series of
Building on classical invariant theory, it is observed that the polarised
traces generate the centraliser of the diagonal embedding of
in . The paper then focuses on and the
case . A Calabi--Yau algebra with three generators is
introduced and explicitly shown to possess a PBW basis and a certain central
element. It is seen that is isomorphic to a quotient of the
algebra by a single explicit relation fixing the value of the
central element. Upon concentrating on three highest weight representations
occurring in the Clebsch--Gordan series of , a specialisation of
arises, involving the pairs of numbers characterising the three
highest weights. In this realisation in , the
coefficients in the defining relations and the value of the central element
have degrees that correspond to the fundamental degrees of the Weyl group of
type . With the correct association between the six parameters of the
representations and some roots of , the symmetry under the full Weyl group
of type is made manifest. The coefficients of the relations and the value
of the central element in the realisation in are
expressed in terms of the fundamental invariant polynomials associated to
. It is also shown that the relations of the algebra can be
realised with Heun type operators in the Racah or Hahn algebra.Comment: 24 page
Set-theoretical reflection equation: Classification of reflection maps
The set-theoretical reflection equation and its solutions, the reflection maps, recently introduced by two of the authors, is presented in general and then applied in the context of quadrirational Yang-Baxter maps. We provide a method for constructing reflection maps and we obtain a classification of solutions associated to all the families of quadrirational Yang-Baxter maps that have been classified recently
3-state Hamiltonians associated to solvable 33-vertex models
Using the nested coordinate Bethe ansatz, we study 33-vertex models, where
only one global charge with degenerate eigenvalues exists and each site
possesses three internal degrees of freedom. In the context of Markovian
processes, they correspond to diffusing particles with two possible internal
states which may be exchanged during the diffusion (transmutation). The first
step of the nested coordinate Bethe ansatz is performed providing the
eigenvalues in terms of rapidities. We give the constraints ensuring the
consistency of the computations. These rapidities also satisfy Bethe equations
involving R-matrices, solutions of the Yang--Baxter equation which
implies new constraints on the models. We solve them allowing us to list all
the solvable 33-vertex models.Comment: 14 pages; title changed according to referee request; an appendix
added to describe explicitely the Hamiltonia
Inhomogeneous discrete-time exclusion processes
We study discrete time Markov processes with periodic or open boundary
conditions and with inhomogeneous rates in the bulk. The Markov matrices are
given by the inhomogeneous transfer matrices introduced previously to prove the
integrability of quantum spin chains. We show that these processes have a
simple graphical interpretation and correspond to a sequential update. We
compute their stationary state using a matrix ansatz and express their
normalization factors as Schur polynomials. A connection between Bethe roots
and Lee-Yang zeros is also pointed out.Comment: 30 pages, 10 figures; a short paragraph at the end to justify the
form of the sequential update has been added; the justification of the
transfer matrix degree is detaile
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