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

The Staggered Six-Vertex Model: Conformal Invariance and Corrections to Scaling