A Spin Chain Primer

This is a very elementary introduction to the Heisenberg (XXX) quantum spin chain, the Yang-Baxter equation, and the algebraic Bethe Ansatz

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

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

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

Equivalent T-Q relations and exact results for the open TASEP

Starting from the Bethe ansatz solution for the open Totally Asymmetric Simple Exclusion Process (TASEP), we compute the largest eigenvalue of the deformed Markovian matrix, in exact agreement with results obtained by the matrix ansatz. We also compute the eigenvalues of the higher conserved charges. The key step is to find a simpler equivalent T-Q relation, which is similar to the one for the TASEP with periodic boundary conditions

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

Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms

A Bethe Ansatz solution of the open spin-1/2 XXZ quantum spin chain with nondiagonal boundary terms has recently been proposed. Using a numerical procedure developed by McCoy et al., we find significant evidence that this solution can yield the complete set of eigenvalues for generic values of the bulk and boundary parameters satisfying one linear relation. Moreover, our results suggest that this solution is practical for investigating the ground state of this model in the thermodynamic limit.Comment: 15 pages, LaTeX; amssymb, amsmath, no figures, 5 tables; v2 contains an additional footnote and a "Note Added"; v3 contains an Addendu

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

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