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.
Highlights
I IntroductionThe quantum inverse scattering method (QISM) is an approach to construct and solve quantum integrable systems [1, 2, 3]
The Bethe vectors ΨM(μ1, μ2, . . . , μM) we define here are such that they make the off shell action of the transfer matrix strikingly simple since it almost coincides with the corresponding action in the case when the two boundary matrices are diagonal
As we will show below, the quasi-classical expansion of the Bethe vectors we have defined for he XXX Heisenberg spin chain yields the Bethe vectors of the corresponding Gaudin model
Summary
The quantum inverse scattering method (QISM) is an approach to construct and solve quantum integrable systems [1, 2, 3]. The off shell action yields the spectrum of the transfer matrix and the corresponding Bethe equations To explore further these results we use the so-called quasi-classical limit and obtain the off shell action of the generating function of the Gaudin Hamiltonians, with boundary terms, on the corresponding Bethe vectors. Our derivation is based on the quasi-classical expansion of the linear combination of the transfer matrix of the XXX chain and the central element, the so-called Sklyanin determinant We use this result with the objective to derive the off shell action of the generating function of the Gaudin Hamiltonians. As we will show below, the quasi-classical expansion of the Bethe vectors we have defined for he XXX Heisenberg spin chain yields the Bethe vectors of the corresponding Gaudin model The significance of these Bethe vectors is in the striking simplicity of the formulae of the off shell action of the generating function of the Gaudin Hamiltonians.
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