Bethe ansatz for the XXX-S chain with non-diagonal open boundaries

Abstract

We consider the algebraic Bethe ansatz solution of the integrable and isotropic XXX-S Heisenberg chain with non-diagonal open boundaries. We show that the corresponding K-matrices are similar to diagonal matrices with the help of suitable transformations independent of the spectral parameter. When the boundary parameters satisfy certain constraints we are able to formulate the diagonalization of the associated double-row transfer matrix by means of the quantum inverse scattering method. This allows us to derive explicit expressions for the eigenvalues and the corresponding Bethe ansatz equations. We also present evidences that the eigenvectors can be build up in terms of multiparticle states for arbitrary S.

Separation of variables in the open XXX chain

“…In this case Eqs. (5.12) can be obtained by means of the algebraic Bethe ansatz [12] or in the rational limit from the T Q-equation approach for the open XXZ chain [19]. In this trigonometric case the complete set of eigenvalues is obtained from two sets of Bethe equations which both reduce to (5.12).…”

“…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].…”

“…While this establishes the integrability the actual solution of the spectral problem through application of Bethe ansatz methods has been impeded by the absence of a reference state, such as the ferromagnetically polarized state with all spins up for the case of diagonal boundary fields. Interestingly, by imposing certain constraints obeyed by the left and right boundary fields, the eigenvalues of the spin- 1 2 XXZ chain and of the isotropic spin-S model can be obtained by means of the algebraic Bethe ansatz [4,12]. In an alternative approach, Nepomechie et al have been able to derive Bethe type equations whose roots parametrize the eigenvalues of the Hamiltonian for special values of the anisotropy η = iπ/(p + 1) with p a positive integer and where the transfer matrix obeys functional equations of finite order [13].…”

Off-diagonal Bethe ansatz solution of the XXX spin chain with arbitrary boundary conditions

“…Since Yang and Baxter's pioneering works [4,5,1], the quantum Yang-Baxter equation (QYBE), which define the underlying algebraic structure, has become a cornerstone for constructing and solving the integrable models. There are several well-known methods for deriving the Bethe ansatz (BA) solution of integrable models: the coordinate BA [6,1,7,8,9], the T-Q approach [1,10], the algebraic BA [11,12,13], the analytic BA [14], the functional BA [15] and others [16,17,18,19,20,21,22,23,24,25,26,27,28,29].…”

Multiple reference states and complete spectrum of the Belavin model with open boundaries

“…Recently, much progress has been made for the open XXZ spin chain. Bethe Ansatz solutions for non-diagonal boundary terms where the boundary parameters obey some constraints have been proposed by various approaches 1 [9,10,11,12,13,14,15,16]. It has been 1 Solutions with arbitrary boundary parameters were recently proposed by functional Bethe Ansatz [17] and q-Onsager algebra [18].…”