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arXiv:1303.2481 · Reviews in Mathematical Physics

TRIGONOMETRIC sℓ(2) GAUDIN MODEL WITH BOUNDARY TERMS

N. Cirilo António, N. Manojlović, Z. NAGY

Published Nov 1, 2013• 14 citations
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Abstract

We review the derivation of the Gaudin model with integrable boundaries. Starting from the non-symmetric R-matrix of the inhomogeneous spin-½ XXZ chain and generic solutions of the reflection equation and the dual reflection equation, the corresponding Gaudin Hamiltonians with boundary terms are calculated. An alternative derivation based on the so-called classical reflection equation is discussed.

DOI
10.1142/s0129055x13430046

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We review the derivation of the Gaudin model with integrable boundaries. Starting from the non-symmetric R-matrix of the inhomogeneous spin-½ XXZ chain and generic solutions of the reflection equation and the dual reflection equation, the corresponding Gaudin Hamiltonians with boundary terms are calculated. An alternative derivation based on the so-called classical reflection equation is discussed.
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.
We construct a generalized Gaudin system in an external magnetic field corresponding to non-skew-symmetric classical r -matrices r K ( u , v ) associated with the reflection equation algebras. We show that an external magnetic field for which the system stays integrable is constructed with the help of the ‘boundary’ matrix K + ( u , η) satisfying the dual reflection equation, while the corresponding non-skew-symmetric classical r -matrices r K ( u , v ) are constructed with the help of the other ‘boundary matrix’ K ( u ) satisfying the reflection equation.
In the derivation of the generating function of the Gaudin Hamiltonians with boundary terms, we follow the same approach used previously in the rational case, which in turn was based on Sklyanin's method in the periodic case. Our derivation is centered on the quasi-classical expansion of the linear combination of the transfer matrix of the XXZ Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. By defining the appropriate Bethe vectors which yield strikingly simple off-shell action of the generating function, we fully implement the algebraic Bethe ansatz, obtaining the spectrum of the generating function and the corresponding Bethe equations.
Following Sklyanin's proposal in the periodic case, we derive the generating function of the Gaudin Hamiltonians with boundary terms. Our derivation is based on the quasi-classical expansion of the linear combination of the transfer matrix of the XXX Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. By defining the appropriate Bethe vectors which yield strikingly simple off shell action of the generating function, we fully implement the algebraic Bethe ansatz, obtaining the spectrum of the generating function and the corresponding Bethe equations.
Following Sklyanin’s proposal in the rational case, we derive the generating function of the Gaudin Hamiltonians in the trigonometric case. Our derivation is based on the quasi-classical expansion of the linear combination of the transfer matrix of the inhomogeneous XXZ Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians are obtained as the residues of the generating function. PACS: 02.20.Tw, 03.65.Fd, 05.50+q, 75.10.Jm; MSC 2010: 81R12, 82B23.