  The critical behavior of quantum Hall transitions in two-dimensional
disordered electronic systems can be described by a class of complicated,
non-unitary conformal field theories with logarithmic correlations. The nature
and the physical origin of these logarithmic correlation functions remain
however mysterious. Using the replica trick and the underlying symmetries of
these quantum critical points, we show here how to construct non-perturbatively
disorder-averaged observables in terms of Green's functions that scale
logarithmically at criticality. In the case of the spin quantum Hall
transition, which may occur in disordered superconductors with spin-rotation
symmetry and broken time reversal invariance, we argue that our results are
compatible with an alternative approach based on supersymmetry. The
generalization to the Integer quantum Hall plateau transition is also
discussed.

Disordered Systems and Neural Networks;Mesoscale and Nanoscale Physics;Statistical Mechanics