  We consider asymptotically free nonlinear sigma models in two dimensions
which, due to their internal symmetries, allow for a conserved charge.
Introducing nonzero chemical potential for the SO(2) subgroup of the symmetry
group, we discuss two expected phase transitions, which are related to charge
condensation and percolation of merons, respectively. The latter are
topological objects with half integer charge similar to vortices in the
abelianized \emph{O(2)} theory, that emerge for large chemical potentials due
to the suppression of the complementary field components. We conjecture a
particular ordering of these transitions supported by large N calculations, and
discuss dualities helpful for the understanding of these systems in the
continuum and on the lattice. In conclusion we suggest that similar behavior is
to be expected in QCD at finite density.
