Ground state degeneracy of interacting spinless fermions

We propose an eigenoperator scheme to study the lattice model of interacting spinless fermions at half filling and show that this model possesses a hidden form of reflection positivity in its Majorana fermion representation. Based on this observation, we prove rigourously that the ground state of this model is either unique or doubly degenerate if the lattice size $N$ is even, and is always doubly degenerate if $N$ is odd. This proof holds in all dimensions with arbitrary lattice structures.