  It was previously established that the critical staggered XXZ spin chain
provides a lattice regularization of the black hole CFT. We reconsider the
continuum limit of this spin chain with the exact method of non-linear integral
equations (NLIEs), paying particular attention to the effects of a singular
integration kernel. With the help of the NLIEs, we rederive the continuous
black hole spectrum, but also numerically match the density of states of the
spin chain with that of the CFT, which is a new result. Finally, we briefly
discuss the integrable structure of the black hole CFT and the identification
of its massive integrable perturbation on the lattice.
