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arXiv:2408.15674

Boundary Bethe ansatz in massless AdS3

Daniele Bielli, Vasileios Moustakis, Алессандро Торриелли

Published Aug 28, 2024• 0 citations
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Abstract

We construct the boundary algebraic Bethe Ansatz for the AdS3 X S3 X T4 integrable reflection problem restricted to the massless sector. We derive the double-row monodromy and find the appropriate formulation of the dual equation of Sklyanin's. We perform the algebraic Bethe ansatz and obtain the RTT (more properly the RTRT) relations, from which the spectrum is obtained by repeated action of the B operator on the pseudovacuum. We obtain the Bethe equations by cancelling the unwanted terms, and prove the exact formulae for any number of quantum particles and magnonic excitations.

DOI
10.48550/arxiv.2408.15674

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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.
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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.