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arXiv:hep-th/9911232 · Journal of Physics A Mathematical and General

Nonstandard coproducts and the Izergin-Korepin open spin chain

Rafael I. Nepomechie

Published Dec 23, 1999• 12 citations
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Abstract

Corresponding to the Izergin-Korepin ( A 2 (2) ) R matrix, there are three diagonal solutions (` K matrices') of the boundary Yang-Baxter equation. Using these R and K matrices, one can construct transfer matrices for open integrable quantum spin chains. The transfer matrix corresponding to the identity matrix K = is known to have U q ( o (3)) symmetry. We argue here that the transfer matrices corresponding to the other two K matrices also have U q ( o (3)) symmetry, but with a nonstandard coproduct. We briefly explore some of the consequences of this symmetry.

DOI
10.1088/0305-4470/33/2/101

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Corresponding to the Izergin-Korepin ( A 2 (2) ) R matrix, there are three diagonal solutions (` K matrices') of the boundary Yang-Baxter equation. Using these R and K matrices, one can construct transfer matrices for open integrable quantum spin chains. The transfer matrix corresponding to the identity matrix K = is known to have U q ( o (3)) symmetry. We argue here that the transfer matrices corresponding to the other two K matrices also have U q ( o (3)) symmetry, but with a nonstandard coproduct. We briefly explore some of the consequences of this symmetry.
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.
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.
The N site open XXZ quantum spin chain with a right non-diagonal boundary and special diagonal left boundary is considered. The boundary non-local charges originally obtained from a field theoretical viewpoint, for the sine Gordon model on the half line, are recovered from the spin chain point of view. Furthermore, the symmetry of the open spin chain is exhibited. More specifically, we show that certain non–local charges commute with the transfer matrix of the open spin chain, depending on the choice of boundary conditions. In addition, we show explicitly that for a special choice of the left boundary, one of the non–local charges, in a particular representation, commutes with each one of the generators of the blob algebra, and hence with the corresponding local Hamiltonian. LAPTH-1029/04 Quantum groups have attracted much interest recently because of their intrinsic mathematical interest [1]–[3], but also because they provide a powerful scheme for deriving solutions of the Yang-Baxter equation [4]–[6] R12(λ1 − λ2) R13(λ1) R23(λ2) = R23(λ2) R13(λ1) R12(λ1 − λ2). (1.1) R acts on V ⊗2, whereas the Yang–Baxter equation on V ⊗3, and as customary R12 = R ⊗ I, R23 = I ⊗ R. The
The Izergin-Korepin model with general non-diagonal boundary terms, a typical integrable model beyond A-type and without U(1)-symmetry, is studied via the offdiagonal Bethe ansatz method. Based on some intrinsic properties of the R-matrix and the K-matrices, certain operator product identities of the transfer matrix are obtained at some special points of the spectral parameter. These identities and the asymptotic behaviors of the transfer matrix together allow us to construct the inhomogeneous T − Q relation and the associated Bethe ansatz equations. In the diagonal boundary limit, the reduced results coincide exactly with those obtained via other methods.