Excitonic phase transition in the extended three-dimensional Falicov-Kimball model

We study the excitonic phase transition in a system of the conduction band electrons and valence band holes described by the three-dimensional (3D) extended Falicov-Kimball (EFKM) model with the tunable Coulomb interaction $U$ between both species. By lowering the temperature, the electron-hole system may become unstable with respect to the formation of the excitons, i.e, electron-hole pairs at temperature $T=T_{\Delta}$, exhibiting a gap $\Delta$ in the particle excitation spectrum. To this end we implement the functional integral formulation of the EFKM, where the Coulomb interaction term is expressed in terms of U(1) phase variables conjugate to the local particle number, providing a useful representation of strongly correlated system. The effective action formalism allows us to formulate a problem in the phase-only action in the form of the quantum rotor model and to obtain analytical formula for the critical lines and other quantities of physical interest like charge gap, chemical potential and the correlation length.