949 research outputs found

    Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from SOV

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    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

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    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

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    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 XXXXXX 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

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    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

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    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

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    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 FF-matrices in Y(sln)Y(sl_n) and Uq(sln^)U_q(\hat{sl_n}) spin chains

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    We consider quantum spin chains arising from NN-fold tensor products of the fundamental evaluation representations of Y(sln)Y(sl_n) and Uq(sln^)U_q(\hat{sl_n}). Using the partial FF-matrix formalism from the seminal work of Maillet and Sanchez de Santos, we derive a completely factorized expression for the FF-matrix of such models and prove its equivalence to the expression obtained by Albert, Boos, Flume and Ruhlig. A new relation between the FF-matrices and the Bethe eigenvectors of these spin chains is given.Comment: 30 page
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