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Derivation of the trigonometric Gaudin Hamiltonians

N. Manojlović, Zoltán Nagy, Igor Salom

Published Jan 1, 2015• 3 citations
OpenAlex

Abstract

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.

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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.
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.
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.
The implementation of the algebraic Bethe ansatz for the XXZ Heisenberg spin chain in the case, when both reflection matrices have the upper-triangular form is analyzed. The general form of the Bethe vectors is studied. In the particular form, Bethe vectors admit the recurrent procedure, with an appropriate modification, used previously in the case of the XXX Heisenberg chain. As expected, these Bethe vectors yield the strikingly simple expression for the off-shell action of the transfer matrix of the chain as well as the spectrum of the transfer matrix and the corresponding Bethe equations. As in the XXX case, the so-called quasi-classical limit gives the off-shell action of the generating function of the corresponding trigonometric Gaudin Hamiltonians with boundary terms.