Lower bound for the segregation energy in the Falicov-Kimball model

In this work, a lower bound for the ground-state energy of the Falicov-Kimball model for intermediate densities is derived. The explicit derivation is important in the proof of the conjecture of segregation of the two kinds of fermions in the Falicov-Kimball model, for sufficiently large interactions. This bound is given by a bulk term, plus a boundary term of the form alpha(1) (n) \partial derivativeLambda\, where Lambda is the region devoid of classical particles and n is the density of electrons. A detailed proof is presented for n = 1/2, where the coefficient alpha(1) (1/2) = 10(-13) is obtained, for the two-dimensional case., Although clearly not optimal in terms of order of magnitude, this is the largest explicitly calculated coefficient in the range of intermediate densities. With suitable modifications the method can also be used to obtain a coefficient for all densities. That is the topic of the last section, where a sketch of the proof for n < 1/2 is shown.