Topological superfluidity of lattice fermions inside a Bose-Einstein condensate

We calculate the phase diagram of identical fermions in a 2-dimensional (2D) lattice immersed in a 3D Bose-Einstein condensate (BEC). The fermions exchange density fluctuations in the BEC, which gives rise to an attractive induced interaction. The resulting zero temperature phase diagram exhibits topological $p_x+ip_y$ superfluid phases as well as a phase separation region. We show how to use the flexibility of the Bose-Fermi mixture to tune the induced interaction, so that it maximises the pairing between nearest neighbour sites, whereas phase separation originating from long range interactions is suppressed. Finally, we calculate the Berezinskii-Kosterlitz-Thouless (BKT) critical temperature of the topological superfluid in the lattice and discuss experimental realisations.