We study an unconventional many-body phase, i.e., a chiral odd-Chern-number lattice supersolid (CLSS) state, in a single-component Rydberg-dressed Fermi gas in an optical lattice. We find that in such a CLSS state, contradicting the common sense that typically the density modulation [charge-density-wave (CDW)] and superfluid (SF) orders compete against each other, there is an essentially different interplay between the coexisting two types of orders. The CDW order not only abnormally enhances the coexisting SF, but also provides a scheme to tune the topology of SS through changing the winding of the phase of SF order along the CDW-modified Fermi surface. It thus can support multiple number-tunable chiral Majorana fermions. We develop a scheme of designing the specially spatial dependence of the effective Rydberg-dressed interaction, which turns out to induce an unveiled odd-Chern-number lattice supersolid state confirmed by both mean-field and Monte Carlo calculations. The result may provide an alternative way for manipulating the chiral Majorana fermions, which would be useful in topological quantum computation.

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• Rydberg atoms & molecules