Ground State of the Falicov-Kimball Model

Here we wish to consider the ground state of the spinless Falicov-Kimball model, which represents one of the few mathematical models that describe strong electron-electron correlations and is exactly solvable (in the infinite dimensional limit). The model itself describes the order- disorder transitions of annealed binary alloys wherein itinerant electrons interact locally with static ions. A Coupled Cluster Method approach will be used to evaluate the ground state properties of the system. Here the wave function for the many particle interacting system is given by $\left\vert \Psi\right\rangle =e^{s}\left\vert \Psi_{0}\right\rangle $ where the operator $S$ represents all one particle, two particle, \ldots, etc.~interactions. A set of non linear equations is generated from the matrix elements $E_{0} =\left\langle \Psi _{0}\right\vert H\left\vert \Psi_{0}\right\rangle $ and $\left\langle \Psi _{0}\right\vert H\left\vert \Psi_{n}\right\rangle =0$ from which the ground state energy $E_{0}$ may be computed.