James K. Freericks, Elliott H. Lieb, Daniel Ueltschi
Segregation in the Falicov-Kimball model
(107K, Latex2e)
ABSTRACT. The Falicov-Kimball model is a simple quantum lattice
model that describes light and heavy electrons interacting with an
on-site repulsion; alternatively, it is a model of itinerant electrons
and fixed nuclei. It can be seen as a simplification of the Hubbard
model; by neglecting the kinetic (hopping) energy of the spin up
particles, one gets the Falicov-Kimball model.
We show that away from half-filling, i.e. if the sum of the densities
of both kinds of particles is less than 1, the particles segregate at
zero temperature and for large enough repulsion. In the language of the
Hubbard model, this means creating two regions with a positive and a
negative magnetization.
Our key mathematical results are lower and upper bounds for the sum
of the lowest eigenvalues of the discrete Laplace operator in an
arbitrary domain, with Dirichlet boundary conditions. The lower bound
consists of a bulk term, independent of the shape of the domain, and of
a term proportional to the boundary. Therefore, one lowers the kinetic
energy of the itinerant particles by choosing a domain with a small
boundary. For the Falicov-Kimball model, this corresponds to having a
single `compact' domain that has no heavy particles.