  We study the ground states of lattice Hamiltonians that are invariant under
permutations, in the limit where the number of lattice sites, N -> \infty. For
spin systems, these are product states, a fact that follows directly from the
quantum de Finetti theorem. For fermionic systems, however, the problem is very
different, since mode operators acting on different sites do not commute, but
anti-commute. We construct a family of fermionic states, \cal{F}, from which
such ground states can be easily computed. They are characterized by few
parameters whose number only depends on M, the number of modes per lattice
site. We also give an explicit construction for M=1,2. In the first case,
\cal{F} is contained in the set of Gaussian states, whereas in the second it is
not. Inspired by that constructions, we build a set of fermionic variational
wave functions, and apply it to the Fermi-Hubbard model in two spatial
dimensions, obtaining results that go beyond the generalized Hartree-Fock
theory.
