Algebraic Bethe ansatz for the XXX chain with triangular boundaries and Gaudin model

Abstract

We implement fully the algebraic Bethe ansatz for the XXX Heisenberg spin chain in the case when both boundary matrices can be brought to the upper-triangular form. We define the Bethe vectors which yield the strikingly simple expression for the off shell action of the transfer matrix, deriving the spectrum and the relevant Bethe equations. We explore further these results by obtaining the off shell action of the generating function of the Gaudin Hamiltonians on the corresponding Bethe vectors through the so-called quasi-classical limit. Moreover, this action is as simple as it could possibly be, yielding the spectrum and the Bethe equations of the Gaudin model.

Slavnov and Gaudin–Korepin formulas for models withoutU(1) symmetry: the XXX chain on the segment

“…In the case of the Heisenberg spin chain on the segment, this is a consequence of the breaking of the U(1) symmetry by off-diagonal boundaries. Many approaches have been developed to handle this problem, including generalizations of the Bethe ansatz to consider special non-diagonal boundaries, see for instance [1,4,9,26,29] and references therein, the SoV method [13-15, 21, 28], the functional method [16], the q-Onsager approach [8] and the non-polynomial solution from the homogeneous Baxter T-Q relation [25].…”

Non-compact Quantum Spin Chains as Integrable Stochastic Particle Processes

“…to coincide with the one of the spin chain with diagonal boundary conditions. The algebraic Bethe ansatz for these types of transfer matrices with triangular K-matrices and for finite-dimensional representations in the quantum space has been studied in [48,49]. We plan to elaborate on the non-compact case, which is relevant here, elsewhere.…”

Thermodynamic limit of the XXZ central spin model with an arbitrary central magnetic field

“…[10][11][12][13][14][15] For the θ = 0 case, the eigenvalues and eigenstates of the Hamiltonian (1) can be exactly solved by the algebraic Bethe ansatz method. [16,17] Based on the exact solutions, on the one hand, an efficient technique has been devised for the numerical solution of the Bethe ansatz equations (BAEs); [18] on the other hand, the thermodynamic limit of the models is a subject of intense research. [19][20][21][22][23] For the θ ̸ = 0 case, the integrability and solutions of the model have been studied when [24,25]  In this paper, we consider the coupling constant…”