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

Nonlinear sigma models at nonzero chemical potential: breaking up instantons and the phase diagram