$U_q\mathfrak{sl}_2$-invariant non-compact boundary conditions for the XXZ spin chain

Understand

We introduce new $U_q\mathfrak{sl}_2$-invariant boundary conditions for the open XXZ spin chain.

• For generic values of $q$ we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for $q=e^{\frac{i\pi}{p}}$ a $2p$-th root of unity $ - $ to its $p$-dimensional analogue.
• Both cases are parametrised by a continuous "spin" $\alpha\in\mathbb{C}$.
• To motivate our construction, we first specialise to $q=i$, where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions.