  We analyze two-dimensional nonlinear sigma models at nonzero chemical
potentials, which are governed by a complex action. In the spirit of contour
deformations (thimbles) we extend the fields into the complex plane, which
allows to incorporate the chemical potentials mu as twisted boundary
conditions. We write down the equations of motion and find exact BPS-like
solutions in terms of pairs of (anti)holomorphic functions, in particular
generalizations of unit charge and fractional instantons to generic mu. The
decay of these solutions is controled by the imaginary part of mu and a
vanishing imaginary part causes jumps in the action. We analyze how the total
charge is distributed into localized objects and to what extent these are
characterized by topology.
