  Landau theory and symmetry considerations lead us to propose an explanation
for several seemingly paradoxical behaviors of charge ordering (CO) and spin
ordering (SO) in the mixed valence compound LuFe2O4. Both SO and CO are highly
frustrated. We analyze a lattice gas model of CO within mean field theory and
determine the magnitude of several of the phenomenological interactions. We
show that the assumption of a continuous phase transition at which CO or SO
develops implies that both CO and SO are incommensurate. To explain how
ferroelectric fluctuations in the charge disordered phase can be consistent
with an antiferroelectric ordered phase, we invoke an electron-phonon
interaction in which a low energy (20meV) zone-center transverse phonon plays a
key role. The energies of all the zone-center phonons are calculated from first
principles. We give a Landau analysis which explains SO and we discuss a model
of interactions which stabilizes the SO state, if it is assumed commensurate.
However, we suggest a high resolution experimental determination to see whether
this phase is really commensurate, as believed up to now. The applicability of
representation analysis is discussed. A tentative explanation for the
sensitivity of the CO state to an applied magnetic field in field-cooled
experiments is given.
