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.

arXiv comment40 pages, typos fixed, appendix for R matrix added

References in corpus (35)

• Boundary S-Matrix and Boundary State in Two-Dimensional Integrable Quantum Field Theory
• Addendum to ``Integrability of Open Spin Chains with Quantum Algebra Symmetry''
• Branes in the 2D black hole
• The Partition Function of the Two-Dimensional Black Hole Conformal Field Theory
• An integrable spin chain for the SL(2,R)/U(1) black hole sigma model
• Non-compact String Backgrounds and Non-rational CFT
• Solutions of the reflection equation for face and vertex models associated with $A_n^{(1)},B_n^{(1)},C_n^{(1)},D_n^{(1)}$ and $A_n^{(2)}$
• Non compact conformal field theory and the a_2^{(2)} (Izergin-Korepin) model in regime III
• The Staggered Six-Vertex Model: Conformal Invariance and Corrections to Scaling
• Phase Diagram of an Integrable Alternating $U_q[sl(2|1)]$ Superspin Chain
• Scaling limit of the ${\mathcal{Z}}_2$ invariant inhomogeneous six-vertex model
• Non-Linear Integral Equations for the SL(2,R)/U(1) black hole sigma model
• The open XXZ-chain: Bosonisation, Bethe ansatz and logarithmic corrections
• Integrability of the $D_n^2$ vertex models with open boundary
• Finite size properties of staggered $U_q[sl(2|1)]$ superspin chains
• On the scaling behaviour of the alternating spin chain
• Integrable boundary conditions in the antiferromagnetic Potts model
• Lattice regularisation of a non-compact boundary conformal field theory
• Surveying the quantum group symmetries of integrable open spin chains
• Finite size spectrum of the staggered six-vertex model with $U_q(\mathfrak{sl}(2))$-invariant boundary conditions
• Towards the solution of an integrable $D_2^{(2)}$ spin chain
• The FZZ duality with boundary
• New D_{n+1}^(2) K-matrices with quantum group symmetry
• Integrable boundary conditions for staggered vertex models
• The spectrum of quantum-group-invariant transfer matrices
• Spectral flow for an integrable staggered superspin chain
• Elaborating the phase diagram of spin-1 anyonic chains
• On the critical behaviour of the integrable $q$-deformed $OSp(3|2)$ superspin chain
• Spectrum of the quantum integrable $D_2^{(2)}$ spin chain with generic boundary fields
• Finite-size corrections to the energy spectra of gapless one-dimensional systems in the presence of boundaries
• Scaling limit of the staggered six-vertex model with $U_q(\mathfrak{sl}(2))$ invariant boundary conditions
• Exact solution of the quantum integrable model associated with the twisted $D_3^{(2)}$ algebra
• The $D_3^{(2)}$ spin chain and its finite-size spectrum
• Finite-size spectrum of the staggered six-vertex model with antidiagonal boundary conditions
• Exact physical quantities of the $D_2^{(2)}$ spin chain model with generic open boundary conditions