  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.
