Generalized T-Q relations and the open XXZ chain

We propose a generalization of the Baxter T-Q relation which involves more than one independent Q(u). We argue that the eigenvalues of the transfer matrix of the open XXZ quantum spin chain are given by such generalized T-Q relations, for the case that at most two of the boundary parameters {\alpha_-, \alpha_+, \beta_-, \beta_+} are nonzero, and the bulk anisotropy parameter has values \eta = i \pi/2, i\pi/4, ...Comment: 14 pages, LaTeX; amssymb, no figure

Exact solution of the open XXZ chain with general integrable boundary terms at roots of unity

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.Comment: 22 page

Boundary energy of the general open XXZ chain at roots of unity

We have recently proposed a Bethe Ansatz solution of the open spin-1/2 XXZ quantum spin chain with general integrable boundary terms (containing six free boundary parameters) at roots of unity. We use this solution, together with an appropriate string hypothesis, to compute the boundary energy of the chain in the thermodynamic limit.Comment: 22 pages, 6 figures; v2: some comments, a reference and a footnote adde

Structure of the two-boundary XXZ model with non-diagonal boundary terms

We study the integrable XXZ model with general non-diagonal boundary terms at both ends. The Hamiltonian is considered in terms of a two boundary extension of the Temperley-Lieb algebra. We use a basis that diagonalizes a conserved charge in the one-boundary case. The action of the second boundary generator on this space is computed. For the L-site chain and generic values of the parameters we have an irreducible space of dimension 2^L. However at certain critical points there exists a smaller irreducible subspace that is invariant under the action of all the bulk and boundary generators. These are precisely the points at which Bethe Ansatz equations have been formulated. We compute the dimension of the invariant subspace at each critical point and show that it agrees with the splitting of eigenvalues, found numerically, between the two Bethe Ansatz equations.Comment: 9 pages Latex. Minor correction

Bethe Ansatz derived from the functional relations of the open XXZ chain for new special cases

The transfer matrix of the general integrable open XXZ quantum spin chain obeys certain functional relations at roots of unity. By exploiting these functional relations, we determine the Bethe Ansatz solution for the transfer matrix eigenvalues for the special cases that all but one of the boundary parameters are zero, and the bulk anisotropy parameter is \eta = i\pi/3, i\pi/5 ,... In an Addendum, these results are extended to the cases that any two of the boundary parameters {\alpha_-, \alpha_+,\beta_-, \beta_+} are arbitrary and the remaining boundary parameters are either \eta or i \pi/2.Comment: 13 pages, LaTeX; amssymb, no figures; v2: published version + Addendum; v3: correct Eq. (3.40

Complete Bethe Ansatz solution of the open spin-s XXZ chain with general integrable boundary terms

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 $(2s+1)^{N}$ eigenvalues of the transfer matrix. For the case s=1, we explicitly determine the Hamiltonian, and find an expression for its eigenvalues in terms of Bethe roots.Comment: 23 pages -- Latex2e; misprints in appendix correcte

Generalized T-Q relations and the open spin-s XXZ chain with nondiagonal boundary terms

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 \eta = i\pi/2, i\pi/4,... We also provide numerical evidence for the completeness of the Bethe-ansatz-type solutions derived, using s = 1 case as an example.Comment: 23 pages. arXiv admin note: substantial text overlap with arXiv:0901.3558; v2: published versio

Equivalences between spin models induced by defects

The spectrum of integrable spin chains are shown to be independent of the ordering of their spins. As an application we introduce defects (local spin inhomogeneities in homogenous chains) in two-boundary spin systems and, by changing their locations, we show the spectral equivalence of different boundary conditions. In particular we relate certain nondiagonal boundary conditions to diagonal ones.Comment: 14 pages, 16 figures, LaTeX, Extended versio

The XXZ model with anti-periodic twisted boundary conditions

We derive functional equations for the eigenvalues of the XXZ model subject to anti-diagonal twisted boundary conditions by means of fusion of transfer matrices and by Sklyanin's method of separation of variables. Our findings coincide with those obtained using Baxter's method and are compared to the recent solution of Galleas. As an application we study the finite size scaling of the ground state energy of the model in the critical regime.Comment: 22 pages and 3 figure

A deformed analogue of Onsager's symmetry in the XXZ open spin chain

The XXZ open spin chain with general integrable boundary conditions is shown to possess a q-deformed analogue of the Onsager's algebra as fundamental non-abelian symmetry which ensures the integrability of the model. This symmetry implies the existence of a finite set of independent mutually commuting nonlocal operators which form an abelian subalgebra. The transfer matrix and local conserved quantities, for instance the Hamiltonian, are expressed in terms of these nonlocal operators. It follows that Onsager's original approach of the planar Ising model can be extended to the XXZ open spin chain.Comment: 12 pages; LaTeX file with amssymb; v2: typos corrected, clarifications in the text; v3: minor changes in references, version to appear in JSTA