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arXiv:2001.10760 · Journal of Physics A Mathematical and Theoretical

A Q-operator for open spin chains I. Baxter’s TQ relation

Bart Vlaar, Robert Weston

Published Apr 9, 2020• 13 citations
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Abstract

We construct a Q -operator for the open XXZ Heisenberg quantum spin chain with diagonal boundary conditions and give a rigorous derivation of Baxter’s TQ relation. Key roles in the theory are played by a particular infinite-dimensional solution of the reflection equation and by short exact sequences of intertwiners of the standard Borel subalgebras of U q ( s l ^ 2 ) . The resulting Bethe equations are the same as those arising from Sklyanin’s algebraic Bethe ansatz.

DOI
10.1088/1751-8121/ab8854

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We construct a Q -operator for the open XXZ Heisenberg quantum spin chain with diagonal boundary conditions and give a rigorous derivation of Baxter’s TQ relation. Key roles in the theory are played by a particular infinite-dimensional solution of the reflection equation and by short exact sequences of intertwiners of the standard Borel subalgebras of U q ( s l ^ 2 ) . The resulting Bethe equations are the same as those arising from Sklyanin’s algebraic Bethe ansatz.
We construct Q-operators for the open spin-12 XXX Heisenberg spin chain with diagonal boundary matrices. The Q-operators are defined as traces over an infinite-dimensional auxiliary space involving novel types of reflection operators derived from the boundary Yang–Baxter equation. We argue that the Q-operators defined in this way are polynomials in the spectral parameter and show that they commute with transfer matrix. Finally, we prove that the Q-operators satisfy Baxter's TQ-equation and derive the explicit form of their eigenvalues in terms of the Bethe roots.
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.
Abstract One of the features of Baxter’s Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang–Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang–Baxter equation) associated to these representations. In the case of quantum affine sl2\mathfrak {sl}_2sl2​ sl 2 and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.