Ground-state Energies of Spinless Free Fermions and Hard-core Bosons

We compare the groundstate energies of bosons and fermions with the same form of the Hamiltonian. If both are noninteracting, the groundstate energy of bosons is always lower, owing to Bose-Einstein Condensation. However, the comparison is nontrivial when bosons do interact. We first prove that, when the hopping is unfrustrated (all the hopping amplitudes are non-negative), hard-core bosons still must have a lower groundstate energy than the fermions. When the hopping is frustrated, bosons can have higher groundstate energy than fermions. We prove rigorously that this inversion indeed occurs in several examples, using various techniques.