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arXiv:hep-th/0703106 · Journal of Statistical Mechanics Theory and Experiment

Exact spectrum of the XXZ open spin chain from theq-Onsager algebra representation theory

Pascal Baseilhac, Kozo Koizumi

Published Sep 10, 2007• 131 citations

Abstract

The transfer matrix of the XXZ open spin-½ chain with general integrable boundary conditions and generic anisotropy parameter ( q is not a root of unity and | q | = 1) is diagonalized using the representation theory of the q -Onsager algebra. Similarly to the Ising and superintegrable chiral Potts models, the complete spectrum is expressed in terms of the roots of a characteristic polynomial of degree d = 2 N . The complete family of eigenstates are derived in terms of rational functions defined on a discrete support which satisfy a system of coupled recurrence relations. In the special case of linear relations between left and right boundary parameters for which Bethe-type solutions are known to exist, our analysis provides an alternative derivation of the results of Nepomechie et al and Cao et al . In the latter case the complete family of eigenvalues and eigenstates splits into two sets, each associated with a characteristic polynomial of degree d < 2 N . Numerical checks performed for small values of N support the analysis.

DOI: 10.1088/1742-5468/2007/09/p09006

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The transfer matrix of the XXZ open spin-½ chain with general integrable boundary conditions and generic anisotropy parameter ( q is not a root of unity and | q | = 1) is diagonalized using the representation theory of the q -Onsager algebra. Similarly to the Ising and superintegrable chiral Potts models, the complete spectrum is expressed in terms of the roots of a characteristic polynomial of degree d = 2 N . The complete family of eigenstates are derived in terms of rational functions defined on a discrete support which satisfy a system of coupled recurrence relations. In the special case of linear relations between left and right boundary parameters for which Bethe-type solutions are known to exist, our analysis provides an alternative derivation of the results of Nepomechie et al and Cao et al . In the latter case the complete family of eigenvalues and eigenstates splits into two sets, each associated with a characteristic polynomial of degree d < 2 N . Numerical checks performed for small values of N support the analysis.

In this review, I describe a recent approach based on the representation theory of the

q-

Onsager algebra which is used to derive exact results for the XXZ open spin chain. The complete spectrum and eigenstates are obtained as rational functions of a single variable which discrete values correspond to the roots of a certain characteristic polynomial. Comments and open problems are also presented.

We propose a Bethe-Ansatz-type solution of the open spin-1/2 integrable XXZ quantum spin chain with general integrable boundary terms and bulk anisotropy values i \\pi/(p+1), where p is a positive integer. All six boundary parameters are arbitrary, and need not satisfy any constraint. The solution is in terms of generalized T - Q equations, having more than one Q function. We find numerical evidence that this solution gives the complete set of 2^N transfer matrix eigenvalues, where N is the number of spins.

We consider the open spin- s XXZ quantum spin chain with N sites and general integrable boundary terms for generic values of the bulk anisotropy parameter, and for values of the boundary parameters which satisfy a certain constraint. We derive two sets of Bethe ansatz equations, and find numerical evidence that together they give the complete set of (2 s +1) N eigenvalues of the transfer matrix. For the case of s = 1, we explicitly determine the Hamiltonian, and find an expression for its eigenvalues in terms of Bethe roots.

We propose a Bethe-ansatz-type solution of the open spin-1/2 integrable XXZ quantum spin chain with general integrable boundary terms and bulk anisotropy values iπ/( p +1), where p is a positive integer. All six boundary parameters are arbitrary, and need not satisfy any constraint. The solution is in terms of generalized T – Q equations, having more than one Q function. We find numerical evidence that this solution gives the complete set of 2 N transfer matrix eigenvalues, where N is the number of spins.

We consider the open spin- s XXZ quantum spin chain with nondiagonal boundary terms. By exploiting certain functional relations at roots of unity, we derive a generalized form of T – Q relation involving more than one independent Q ( u ), which we use to propose the Bethe-ansatz-type expressions for the eigenvalues of the transfer matrix. At most two of the boundary parameters are set to be arbitrary and the bulk anisotropy parameter has values η = iπ/2,iπ/4,.... We also provide numerical evidence for the completeness of the Bethe-ansatz-type solutions derived, using the s = 1 case as an example.

We solve the longstanding problem of defining a functional characterization of the spectrum of the transfer matrix associated with the most general spin-1/2 representations of the six-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 variables (SOV) representation, hence proving that it defines a complete characterization of the transfer matrix spectrum. The polynomial form of the Q -function allows us to show that a finite system of generalized Bethe equations can also be used to describe the complete transfer matrix spectrum.