Bose condensation in flat bands

We derive effective Hamiltonians for lattice bosons with strong geometrical frustration of the kinetic energy by projecting the interactions on the flat lowest Bloch band. Specifically, we consider the Bose Hubbard model on the one dimensional sawtooth lattice and the two dimensional kagome lattice. Starting from a strictly local interaction the projection gives rise to effective long-range terms stabilizing a supersolid phase at densities above nu_c=1/9 of the kagome lattice. In the sawtooth lattice on the other hand we show that the solid order, which exists at the magic filling nu_c=1/4, is unstable to further doping. The universal low-energy properties at filling 1/4+delta nu are described by the well known commensurate-incommensurate transition. We support the analytic results by detailed numerical calculations using the Density Matrix Renormalization Group and exact diagonalization. Finally, we discuss possible realizations of the models using ultracold atoms as well as frustrated quantum magnets in high magnetic fields. We compute the momentum distribution and the noise correlations, that can be extracted from time of flight experiments or neutron scattering, and point to signatures of the unique supersolid phase of the kagome lattice.