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arXiv:1411.2938

The scalar product of XXZ spin chain revisited. Application to the ground state at $Δ=-1/2$

Alexandr Garbali

Published Nov 11, 2014• 1 citation
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Abstract

For the scalar product $S_n$ of the XXZ $s=1/2$ spin chain we derive a new determinant expression which is symmetric in the Bethe roots. We consider an application of this formula to the inhomogeneous groundstate of the model with $Δ=-1/2$ with twisted periodic boundary conditions. At this point the ground state eigenvalue $τ_n$ of the transfer matrix is known and has a simple form that does not contain the Bethe roots. We use the knowledge of $τ_n(μ)$ to obtain a closed expression for the scalar product. The result is written in terms of Schur functions. The computations of the normalization of the ground state and the expectation value of $σ^z$ are also presented.

DOI
10.48550/arxiv.1411.2938

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The scalar products, form factors, and correlation functions of the XXZ spin chain with a twisted (or antiperiodic) boundary condition are obtained based on the inhomogeneous T–Q relation and the Bethe states constructed via the off-diagonal Bethe ansatz. It is shown that the scalar product of two off-shell Bethe states, the form factors, and the two-point correlation functions can be expressed as the summation of certain determinants. The corresponding homogeneous limits are studied. The results are also checked by the numerical calculations.
Abstract In our previous paper (Kitanine et al 2017 J. Phys. A: Math. Theor . 50 224001) we have obtained, for the XXX spin-1/2 Heisenberg open chain, new determinant representations for the scalar products of separate states in the quantum separation of variables (SoV) framework. In this article we perform a similar study in a more complicated case: the XXZ open spin-1/2 chain with the most general integrable boundary terms. We first revisit the SoV solution of this model using a matrix version of the Vertex-IRF gauge transformation so as to reduce one of the boundary K -matrices to a diagonal form. As usual within the SoV approach, the scalar products of separate states are computed in terms of dressed Vandermonde determinants having an intricate dependency on the inhomogeneity parameters. We show that these determinants can be transformed into different ones in which the homogeneous limit can be taken straightforwardly. These representations generalize in a non-trivial manner to the trigonometric case the expressions found previously in the rational case. We also show that generically all scalar products can be expressed in a form which is similar to—although more cumbersome than—the well-known Slavnov determinant representation for the scalar products of the Bethe states of the periodic chain. Considering a special choice of the boundary parameters relevant in the thermodynamic limit to describe the half infinite chain with a general boundary, we particularize these representations to the case of one of the two states being an eigenstate. We obtain simplified formulas that should be of direct use to compute the form factors and correlation functions of this model.
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Abstract 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.
Abstract We consider the XXZ spin-1/2 Heisenberg chain with antiperiodic boundary conditions. The inhomogeneous version of this model can be solved by separation of variables, and the eigenstates can be constructed in terms of Q -functions, solution of a Baxter TQ-equation, which have double periodicity compared to the periodic case. We compute in this framework the scalar products of a particular class of separate states which notably includes the eigenstates of the transfer matrix. We also compute the form factors of local spin operators, i.e. their matrix elements between two eigenstates of the transfer matrix. We show that these quantities admit determinant representations with rows and columns labelled by the roots of the Q -functions of the corresponding separate states, as in the periodic case, although the form of the determinant are here slightly different. We also propose alternative types of determinant representations written directly in terms of the transfer matrix eigenvalues.