  What kind of lattice Hamiltonian manifestly has an ordered state with
spontaneous orbital currents? We consider interacting spinless fermions on an
array of square plaquettes, connected by weak hopping; the array geometry may
be a 2 x 2L ladder, a 2 x 2 x 2L "tube", or a 2L x 2L square grid. At half
filling, we derive an effective Hamiltonian in terms of pseudospins, of which
one component represents orbital currents, and find the conditions sufficient
for orbital current long-range order. We consider spinfull variants of the
aforesaid spinless models and make contact with other spinfull models in the
literature purported to possess spontaneous currents.
