Abstract
In this paper we deal with the trigonometric Gaudin model, generalized using a nontrivial triangular reflection matrix (corresponding to non-periodic boundary conditions in the case of anisotropic XXZ Heisenberg spin-chain). In order to obtain the generating function of the Gaudin Hamiltonians with boundary terms we follow an approach based on Sklyanin’s derivation in the periodic case. Once we have the generating function, we obtain the corresponding Gaudin Hamiltonians with boundary terms by taking its residues at the poles. As the main result, we find the generic form of the Bethe vectors such that the off-shell action of the generating function becomes exceedingly compact and simple. In this way—by obtaining Bethe equations and the spectrum of the generating function—we fully implement the algebraic Bethe ansatz for the generalized trigonometric Gaudin model.
Highlights
The so-called rational s(2) Gaudin model was first introduced in [1] as a model of “long-range”interacting spins in a chain
While in [53] we considered expansion of the XXZ spin-chain expressions to obtain Bethe vectors for the Gaudin model, the open trigonometric Gaudin model can be treated in its own right, by fully implementing the algebraic Bethe ansatz for this case
The Sklyanin linear bracket (7) obeys the Jacobi identity and is anti-symmetric. From here it follows that the entries of the Lax matrix (1) generate a Lie algebra (Gaudin algebra), which in this case corresponds to the trigonometric Gaudin model with periodic boundary conditions [20]
Summary
The so-called rational s(2) Gaudin model was first introduced in [1] as a model of “long-range”. One particular approach to the relation between Heisenberg spin-chains and Gaudin models was due to Hikami, Kulish and Wadati, who showed that the Gaudin Hamiltonians can be obtained by making the so-called quasi-classical expansion of the transfer matrix of the periodic chain [23,24] This was soon demonstrated for cases with non-periodic boundary conditions [25]. While in [53] we considered expansion of the XXZ spin-chain expressions to obtain Bethe vectors for the Gaudin model (i.e., by exploiting the mathematical relation between the two models), the open trigonometric Gaudin model can be treated in its own right, by fully implementing the algebraic Bethe ansatz for this case This is the essential goal of the present paper. In Appendix A we provide proof for the essential commutativity property of the generating function of the Gaudin Hamiltonians, while the Appendix B contains some explicit formulas regarding the Bethe vector φ3 (μ1 , μ2 , μ3 )
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