We consider the quantum integrable spin chain models associated with the Jimbo R-matrix based on the quantum affine algebra \( {D}_{n+1}^{(2)} \) , 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 \( {D}_3^{(2)} \) . 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 \( {D}_2^{(2)} \) 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 \( {D}_2^{(2)} \) 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.
