Bethe ansatz solution of the open XXZ chain with nondiagonal boundary terms

Abstract

We propose a set of conventional Bethe Ansatz equations and a corresponding expression for the eigenvalues of the transfer matrix for the open spin-1 2 XXZ quantum spin chain with nondiagonal boundary terms, provided that the boundary parameters obey a certain linear relation.

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

“…The transfer matrix has the important commutativity property 6) and it "contains" the Hamiltonian (1.1),…”

“…Finally, we remark that our numerical results suggest that the Bethe Ansatz correctly yields 2 N −1 eigenvalues for k = 0 (N odd), and 2 N −1 + 1 2 N N/2 eigenvalues for k = 1 (N even). 6 …”

“…However, the case of nondiagonal boundary terms [3] has resisted solution for many years (see, e.g., [4]). A Bethe Ansatz solution for the latter case has recently been proposed in [5,6] (see also [7]). In terms of the parameters introduced there, the Hamiltonian is given by where σ x , σ y , σ z are the usual Pauli matrices, η is the bulk anisotropy parameter, α ± , β ± , θ ± are boundary parameters, and N is the number of spins.…”

“…1 The energy eigenvalues are given by [6] E = sinh 2 η M j=1 1 sinh u j sinh(u j + η) + 1 2 sinh η (coth α − + tanh β − + coth α + + tanh β + ) + 1 2 (N − 1) cosh η , (1.3) where the Bethe roots {u j } satisfy the Bethe Ansatz equations…”

Separation of variables in the open XXX chain

“…At the same time various problems concerning systems with open boundaries are still not solved completely. Even for the prototype spin-1 2 XXZ chain with general open boundary conditions techniques for the solution of the spectral problem have been developed only recently [1,4,[11][12][13][14]. This model, apart from being the simplest starting point for studies of boundary effects in a correlated system, allows to investigate the approach to a stationary state in one-dimensional diffusion problems for hard-core particles [7,8] and transport through one-dimensional quantum systems [5].…”