  We demonstrate that, for a fermionic lattice system, the ground-state
particle density uniquely determines the external potential except for the
sites corresponding to nodes of the wave function, and the limiting case where
the Pauli exclusion principle completely determines the occupation of all
sites. Our fundamental finding completes, for this general class of systems,
the one-to-one correspondence between ground states, their densities, and the
external potential at the base of the Hohenberg-Kohn theorem. Moreover we
demonstrate that the mapping from wave function to potential is unique not just
for the ground state, but also for excited states. To illustrate our findings,
we develop a practical inversion scheme to determine the external potential
from a given density. Our results hold for a general class of lattice models,
which includes the Hubbard model.
