Inhomogeneous states in two-dimensional frustrated phase separation

We derive the phase diagram of a paradigmatic model of Coulomb frustrated phase separation in two-dimensional systems with negative short-range electronic compressibility. We consider the system subject either to the truly three-dimensional long-range Coulomb interaction (LRC) and to a two-dimensional LRC with logarithmic-like behavior. In both cases we find that the transition from the homogeneous phase to the inhomogeneous phase is generically first-order except for a critical point. Close to the critical point, inhomogeneities arrange in a triangular lattice with a subsequent first-order topological transition to stripe-like objects by lowering the Coulomb frustration. A proliferation of inhomogeneities which have inside smaller inhomogeneities is expected near all the transition lines in systems embedded in the three-dimensional LRC alone.