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

Abstract

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 $p=0,\dots ,n$ and $\epsilon =0,1$. We develop the analytical Bethe ansatz for the previously unexplored case $\epsilon =1$ 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 $0<\gamma <\pi /4$, 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 $\epsilon$: 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 $\epsilon =1$. 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 $\epsilon =0$, the spectrum of conformal weights is purely discrete.

40 pages, typos fixed, appendix for R matrix added

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