  We explore unconventional superconductivity of repulsive spinless fermions on
square and honeycomb lattices with staggered sublattice potentials. The two
lattices can exhibit staggered $d$-wave and $f$-wave pairing, respectively, at
low doping stemming from an effective two-valley band structure. At higher
doping, in particular, the square lattice displays a much richer phase diagram
including topological $p+ip$ superconductivity which is induced by a
qualitatively different mechanism compared to the $d$-wave pairing. We
illuminate this from several complementary perspectives: We analytically
perform sublattice projection to analyze the effective continuum low-energy
description and we numerically calculate the binding energies for pair and
larger bound states for few-body doping near half filling. Furthermore, for
finite doping, we present phase diagrams based on extensive functional
renormalization group and and density matrix renormalization group
calculations.
