Wronskian-type formula for inhomogeneous TQ-equations

The transfer-matrix eigenvalues of the isotropic open Heisenberg quantum spin-1/2 chain with non-diagonal boundary magnetic fields are known to satisfy a TQ-equation with an inhomogeneous term. We derive here a discrete Wronskian-type formula relating a solution of this inhomogeneous TQ-equation to the corresponding solution of a dual inhomogeneous TQ-equation.Comment: 6 pages; to appear in Theor. Math. Phys. as part of the Proceedings of the CQIS-2019 workshop in St. Petersbur

Inhomogeneous T-Q equation for the open XXX chain with general boundary terms: completeness and arbitrary spin

An inhomogeneous T-Q equation has recently been proposed by Cao, Yang, Shi and Wang for the open spin-1/2 XXX chain with general (nondiagonal) boundary terms. We argue that a simplified version of this equation describes all the eigenvalues of the transfer matrix of this model. We also propose a generating function for the inhomogeneous T-Q equations of arbitrary spin.Comment: 8 pages; v2: minor improvement

Supersymmetry in the boundary tricritical Ising field theory

We argue that it is possible to maintain both supersymmetry and integrability in the boundary tricritical Ising field theory. Indeed, we find two sets of boundary conditions and corresponding boundary perturbations which are both supersymmetric and integrable. The first set corresponds to a ``direct sum'' of two non-supersymmetric theories studied earlier by Chim. The second set corresponds to a one-parameter deformation of another theory studied by Chim. For both cases, the conserved supersymmetry charges are linear combinations of Q, \bar Q and the spin-reversal operator \Gamma.Comment: 19 pages, LaTeX; amssymb, no figures; v2 one paragraph and one reference added; v3 Erratum adde

Integrability + Supersymmetry + Boundary: Life on the edge is not so dull after all!

After a brief review of integrability, first in the absence and then in the presence of a boundary, I outline the construction of actions for the N=1 and N=2 boundary sine-Gordon models. The key point is to introduce Fermionic boundary degrees of freedom in the boundary actions.Comment: 10 pages, LaTeX; requires ws-procs9x6.cls and rotating_pr.sty (World Scientific proceedings style, 9 x 6 inch trim size); presented at "Deserfest: A celebration of the life and works of Stanley Deser" in Ann Arbor, Michigan, 3-5 April 2004, and to appear in the Proceedings; v2 and v3 refs adde

Bethe Ansatz for the open XXZ chain from functional relations at roots of unity

We briefly review Bethe Ansatz solutions of the integrable open spin-1/2 XXZ quantum spin chain derived from functional relations obeyed by the transfer matrix at roots of unity.Comment: 10 pages, LaTeX; includes ws-procs9x6.cls and rotating_pr.sty (World Scientific proceedings style, 9 x 6 inch trim size); presented at the 23rd International Conference of Differential Geometric Methods in Theoretical Physics (DGMTP) at the Nankai Institute of Mathematics in Tianjin, China, 20-26 August 2005, and to appear in the Proceeding

Bethe Ansatz solution of the open XX spin chain with nondiagonal boundary terms

We consider the integrable open XX quantum spin chain with nondiagonal boundary terms. We derive an exact inversion identity, using which we obtain the eigenvalues of the transfer matrix and the Bethe Ansatz equations. For generic values of the boundary parameters, the Bethe Ansatz solution is formulated in terms of Jacobian elliptic functions.Comment: 14 pages, LaTeX; amssymb, no figure

Twisting singular solutions of Bethe's equations

The Bethe equations for the periodic XXX and XXZ spin chains admit singular solutions, for which the corresponding eigenvalues and eigenvectors are ill-defined. We use a twist regularization to derive conditions for such singular solutions to be physical, in which case they correspond to genuine eigenvalues and eigenvectors of the Hamiltonian.Comment: 10 pages; v2: references added; v3: introduction expanded, and more references adde

Algebraic Bethe ansatz for singular solutions

The Bethe equations for the isotropic periodic spin-1/2 Heisenberg chain with N sites have solutions containing i/2, -i/2 that are singular: both the corresponding energy and the algebraic Bethe ansatz vector are divergent. Such solutions must be carefully regularized. We consider a regularization involving a parameter that can be determined using a generalization of the Bethe equations. These generalized Bethe equations provide a practical way of determining which singular solutions correspond to eigenvectors of the model.Comment: 10 pages; v2: refs added; v3: new section on general singular solutions, and more reference