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arXiv:2509.00610 · Journal of High Energy Physics

Quantum-group-invariant $$ {D}_{n+1}^{(2)} $$ models: Bethe ansatz and finite-size spectrum

Holger Frahm, Sascha Gehrmann, Rafael I. Nepomechie, Ana L. Retore

Published Dec 16, 2025• 0 citations
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Abstract

A bstract We consider the quantum integrable spin chain models associated with the Jimbo R-matrix based on the quantum affine algebra $$ {D}_{n+1}^{(2)} $$ D n + 1 2 , subject to quantum-group-invariant boundary conditions parameterized by two discrete variables p = 0, . . . , n and ε = 0, 1. We develop the analytical Bethe ansatz for the previously unexplored case ε = 1 with any n , and use it to investigate the effects of different boundary conditions on the finite-size spectrum of the quantum spin chain based on the rank-2 algebra $$ {D}_3^{(2)} $$ D 3 2 . Previous work on this model with periodic boundary conditions has shown that it is critical for the range of anisotropy parameters 0 < γ < π /4, where its scaling limit is described by a non-compact CFT with continuous degrees of freedom related to two copies of the 2D black hole sigma model. The scaling limit of the model with quantum-group-invariant boundary conditions depends on the parameter ε : similarly as in the rank-1 $$ {D}_2^{(2)} $$ D 2 2 chain, we find that the symmetry of the lattice model is spontaneously broken, and the spectrum of conformal weights has both discrete and continuous components, for ε = 1. For p = 1, the latter coincides with that of the $$ {D}_2^{(2)} $$ D 2 2 chain, which should correspond to a non-compact brane related to one black hole CFT in the presence of boundaries. For ε = 0, the spectrum of conformal weights is purely discrete.

DOI
10.1007/jhep12(2025)117

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