Kennedy Tom
Some rigorous results on the ground states
of the Falicov-Kimball model
(126K, TeX)
ABSTRACT. The Falicov-Kimball model consists of
nuclei which are not allowed to hop and
spinless electrons which can hop between nearest neighbor sites.
There is an on-site interaction between electrons and nuclei.
We consider the model in two dimensions
with a large attractive potential. For the neutral model
with densities between 1/4 and 1/2 we prove that
the configuration of the nuclei in the ground state must consist
of parallel lines of lattice sites which are either completely
occupied by nuclei or completely free of nuclei.
(The angle of the lines with respect to the lattice depends on
the density. Some mild assumptions on the ground state are
needed for this result.)
For densities 1/3, 1/4 and 1/5 we prove that the ground state
configuration of the nuclei is indeed that which had been
conjectured [8].
For the nonneutral model we show that if the model is close to
neutrality in the sense that both the electron and
nuclear densities are close to 1/2, 1/3, 1/4 or 1/5 then
the configuration of the nuclei in the ground state is close
to the nuclear ground state for the neutral model with density
1/2, 1/3, 1/4 or 1/5.