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

TL;DR

The paper develops an analytical Bethe ansatz for quantum-group-invariant open spin chains based on the $D^{(2)}_{n+1}$ R-matrix, including the previously unexplored case $\varepsilon=1$ for general $n$ and a detailed study of the rank-2 instance $D^{(2)}_3$. Focusing on the four boundary configurations $(\varepsilon,p)=(0,0),(0,1),(1,0),(1,1)$, it derives Bethe equations, constructs the corresponding Hamiltonians (notably requiring second derivatives for $(1,1)$), and analyzes the thermodynamic and finite-size spectra. The results reveal boundary-condition–dependent critical behavior: for $\varepsilon=1$ the spectrum generally displays spontaneous symmetry breaking with a continuous component (a non-compact degree of freedom) in two cases, while $\varepsilon=0$ yields a purely discrete conformal spectrum akin to two decoupled antiferromagnetic Potts sectors. These findings connect lattice boundary conditions with non-compact boundary CFTs and brane interpretations, offering a lattice realization of rich boundary conformal structures and posing open problems for higher ranks and alternate K-matrix choices.

Abstract

We consider the quantum integrable spin chain models associated with the Jimbo R-matrix based on the quantum affine algebra $D^{(2)}_{n+1}$, subject to quantum-group-invariant boundary conditions parameterized by two discrete variables $p=0,\dots, n$ and $\varepsilon = 0, 1$. We develop the analytical Bethe ansatz for the previously unexplored case $\varepsilon = 1$ 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 $D^{(2)}_3$. 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 $\varepsilon$: 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 $\varepsilon=1$. 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 $\varepsilon=0$, the spectrum of conformal weights is purely discrete.

Figures (7)

• Figure 1: Subalgebras of $D^{(2)}_{n+1}$ corresponding to removing the $p$-th node from the extended Dynkin diagram. The "affine node" is black and labeled 0.
• Figure 2: Effective conformal weight $24 X_{\text{eff}}=\frac{24N}{\pi v_{\rm F}}(E(N)-N e_\infty-f_\infty)$ for states with different $h_1$ of the $D^{(2)}_3$ chain with boundary conditions $(\varepsilon,p)=(1,0)$. The circles display numerical data obtain from the Bethe ansatz. The dotted lines display the lowest state in the $(h_1,0)$ continuum yielding logarithmic correction flowing to \ref{['njdnjsleme']} for $h_1=1,2,3$ for blue, green and purple. The dashed states are discrete states \ref{['ndmfndmf2']} having power law corrections (red $h_1=0$). Note that the ground state is never in the $h_1=0$ sector.
• Figure 3: Effective conformal weight $X_{\text{eff}}=\frac{N}{\pi v_{\rm F}}(E(N)-N e_\infty-f_\infty)$ for excited states in the sector $h_1=1,h_2=0$ of the $D^{(2)}_3$ chain with boundary conditions $(\varepsilon,p)=(1,0)$. We see strong logarithmic corrections.
• Figure 4: Effective conformal weights $X_{\text{eff}}$ for some of the lowest class A states of the $D^{(2)}_3$ chain with boundary conditions $(\varepsilon,p)=(1,1)$: full lines indicate the lower bounds (\ref{['eq:ceff11-k']}) the continua in sectors $[\mathcal{S}^{(\ell)},\mathcal{S}^{(r)}]$, dashed and dash-dotted lines are the conjectures $X_\text{eff}^*$ and $X_\text{eff}^{**}$ for the discrete levels emerging from these continua as given in (\ref{['eq:xeff11_discrete']}), bullets are extrapolations of the numerical finite size data D23.data.
• Figure 5: Root configurations $\{u^{[a]}_{k}\}$ for the $[\mathcal{S}^{(\ell)},\mathcal{S}^{(r)}]=[0,2]$ levels shown in Fig. \ref{['fig:bc11_tower']}: ground state for $N=16$ (top left), ground state for $N=15$ (top right), first excitation in the continuum for $N=16$ (lower left) and for $N=15$ (lower right). Black (red) dots are first (second) level roots, the dashed lines are at $\Im m(u^{[a]}_{k})=0$, $\frac{\pi}{2}-\gamma$, $\frac{\pi}{2}$ and $\pi$.