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It was previously established that the critical staggered XXZ spin chainprovides a lattice regularization of the black hole CFT. We reconsider thecontinuum limit of this spin chain with the exact method of non-linear integralequations (NLIEs), paying particular attention to the effects of a singularintegration kernel. With the help of the NLIEs, we rederive the continuousblack hole spectrum, but also numerically match the density of states of thespin chain with that of the CFT, which is a new result. Finally, we brieflydiscuss the integrable structure of the black hole CFT and the identificationof its massive integrable perturbation on the lattice.

Non-Linear Integral Equations for the SL(2,R)/U(1) black hole sigma model