  There is growing interest to search the chiral Majorana fermions that could
arise as the quasi-particle edge state of a two-dimensional topological state
of matter. Here we propose a new platform, i.e., a two-dimensional chiral odd
Chern number lattice supersolid state, for supporting multiple number-tunable
chiral Majorana fermions from a single component Rydberg-dressed Fermi gas in
an optical lattice. The attractiveness of our idea rests on the fact that by
introducing the unveiled competition between two distinct length scales, i.e.,
lattice period and the distance of resonant Rydberg-dressing, can provide a new
way to manipulate the spatial dependence of both the strength and sign of the
effective Rydberg-dressed interaction. Such a designed effective interaction
turns out, can induce an unveiled odd Chern number lattice supersolid state,
which is confirmed by both the mean-field and Monte Carlo calculations.
Furthermore, we also find that the spontaneously formed density modulation
resulted from the discrete translational symmetry breaking provides a natural
way of tuning the system's topology arising from the superfluidity induced by
the $U(1)$ symmetry breaking. It thus provide an alternative way for
manipulating the chiral Majorana fermions, which would be useful in topological
quantum computation.
