  The Hubbard model is used to study an electronic system at half filling.
Starting from a functional integral representation the spin-up Grassmann field
is integrated out. It is shown that the resulting spinless fermion theory has
an instantaneous cluster interaction, and that the spinless fermions are
coupled to thermally fluctuating Ising spins. The coupling parameter of the
spinless fermion interaction is a product of the Hubbard interaction and the
hopping rate. As an example the strongly metallic as well as the strongly
insulating regime are investigated in terms of the effective Ising statistics.
