  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.
