Abstract
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.
Highlights
I IntroductionA model of interacting spins in a chain was first considered by Gaudin [1, 2]
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
Summary
A model of interacting spins in a chain was first considered by Gaudin [1, 2]. Gaudin derived these models as a quasi-classical limit of the quantum chains. Hikami showed how the quasi-classical expansion of the transfer matrix, calculated at the special values of the spectral parameter, yields the Gaudin Hamiltonians in the case of non-periodic boundary conditions [20]. In [41] the off shell action of the generating function of the Gaudin Hamiltonians on the Bethe vectors was obtained through the so-called quasi-classical limit. As one of the main results of the paper, the generating function of the Gaudin Hamiltonians with boundary terms is derived, using the quasi-classical expansion of the linear combination of the transfer matrix of the inhomogeneous XXX spin chain and the so-called Sklyanin determinant. This shows that τ(λ) is the generating function of Gaudin Hamiltonians when the periodic boundary conditions are imposed [3]
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