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arXiv:1408.5200

Algebraic integrability of the classical XXZ spin chain with reflecting boundary conditions

Gus Schrader

Published Aug 22, 2014• 1 citation
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Abstract

In this paper we analyze the classical XXZ spin chain with reflecting boundaries. We exhibit a system of log-canonical coordinates on the phase space generalizing Sklyanin's separation of variables for the periodic XXZ chain, and use these coordinates to construct action-angle variables for the system. We also integrate the flows of the reflection Hamiltonians explicitly in terms of Riemann theta functions. Central to our analysis is the algebraic integrability of the model.

DOI
10.48550/arxiv.1408.5200

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We propose a non-standard separation of variables for the classical integrable XXX and XXZ spin chains with degenerate twist matrix.We show that for the case of such twist matrices one can interchange the role of classical separating functions A(u) and B(u) and construct a new full set of separated variables, satisfying simpler equation of separation and simpler Abel equations in comparison with the standard separated variables of Sklyanin.We show that for certain cases of the twist matrices the constructed separated variables can be directly identified with action-angle coordinates.
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.
Subject of this work are integrable spin chains with general boundary conditions.In the framework of the Quantum Inverse Scattering Method Sklyanin has shown how to construct a family of commuting operators (transfer matrix) containing the hamiltonian of the XXX or XXZ spin chain with general boundary fields.Key ingredient is the underlying algebraic structure which is a combination of the Yang-Baxter algebra, using the known R-matrix representations, and a so-called Reflection algebra.The latter includes fields of arbitrary strength and direction acting on the first and last position of the chain.This setup is solvable via algebraic Bethe ansatz in the case of diagonal boundaries, i.e. the fields are parallel to each other and in the case of the XXZ model parallel to the distinguished direction.Kitanine et al. have managed to express local operators in terms of the non-local elements of the underlying algebraic structure in the case of half-infinite chain length hence establishing a possible approach to evaluate expectation values of physical observables.Their results will be picked up in this work and generalized to spin chains of arbitrary (including finite) lengths using non-linear integral equations for the lowest lying state with zero magnetization.In the case of non-diagonal boundary fields the lack of a reference state or pseudo vacuum prohibits the solution by algebraic Bethe ansatz.The method of separation of variables proposed by Sklyanin is not constrained in that sense and will be applied to this situation.In this approach for the XXX spin chain no restrictions to the boundary parameters are needed.The result is a TQ-equation on a finite discrete set of points and the eigenvalues of the transfer matrix are obtainable from this finite difference equation.As the underlying algebraic structure is independent of the representation, the analysis for the XXX spin chain can be extended to a spin-boson model.Using a known representation of the algebra for the bosonic degree of freedom again a TQ-equation is obtained but in this case being defined on an infinite discrete set of points.Finally a different approach to the case of non-diagonal boundary conditions using non-linear integral equations is studied.By construction of higher dimensional representations of the Yang Baxter algebra a hierarchy of transfer matrices is obtained.In this construction two R-matrices are fused together yielding the name of fusion hierarchy.From such hierarchies it is possible to derive non-linear integral equations for a certain eigenvalue of the transfer matrix using Fourier transform techniques.The equations are valid for arbitrary system sizes and a range of boundary parameters.By careful analysis of the zero and pole structure it is possible to extract information about the eigenvalue of a specific state.
The implementation of the algebraic Bethe ansatz for the XXZ Heisenberg spin chain in the case, when both reflection matrices have the upper-triangular form is analyzed. The general form of the Bethe vectors is studied. In the particular form, Bethe vectors admit the recurrent procedure, with an appropriate modification, used previously in the case of the XXX Heisenberg chain. As expected, these Bethe vectors yield the strikingly simple expression for the off-shell action of the transfer matrix of the chain as well as the spectrum of the transfer matrix and the corresponding Bethe equations. As in the XXX case, the so-called quasi-classical limit gives the off-shell action of the generating function of the corresponding trigonometric Gaudin Hamiltonians with boundary terms.
In this paper we deal with the trigonometric Gaudin model, generalized using a nontrivial triangular reflection matrix (corresponding to non-periodic boundary conditions in the case of anisotropic XXZ Heisenberg spin-chain). In order to obtain the generating function of the Gaudin Hamiltonians with boundary terms we follow an approach based on Sklyanin’s derivation in the periodic case. Once we have the generating function, we obtain the corresponding Gaudin Hamiltonians with boundary terms by taking its residues at the poles. As the main result, we find the generic form of the Bethe vectors such that the off-shell action of the generating function becomes exceedingly compact and simple. In this way—by obtaining Bethe equations and the spectrum of the generating function—we fully implement the algebraic Bethe ansatz for the generalized trigonometric Gaudin model.