  We study the emergence of non-compact degrees of freedom in the low energy
effective theory for a class of $\mathbb{Z}_2$-staggered six-vertex models. In
the finite size spectrum of the vertex model this shows up through the
appearance of a continuum of critical exponents. To analyze this part of the
spectrum we derive a set of coupled nonlinear integral equations from the Bethe
ansatz solution of the vertex model which allow to compute the energies of the
system for a range of anisotropies and of the staggering parameter. The
critical theory is found to be independent of the staggering. Its spectrum and
density of states coincide with the $SL(2,\mathbb{R})/U(1)$ Euclidean black
hole conformal field theory which has been identified previously in the
continuum limit of the vertex model for a particular 'self-dual' choice of the
staggering. We also study the asymptotic behaviour of subleading corrections to
the finite size scaling and discuss our findings in the context of the
conformal field theory.
