Thermal Fluctuations in the Phase Structure of the Excitonic Insulator Charge Density Wave State in the Extended Falicov–Kimball Model
Excitonic insulator (EI) instability accompanied by a charge density wave (CDW) state in the two-dimensional extended Falicov–Kimball model including electron–phonon interaction is investigated by applying the unrestricted Hartree–Fock approximation. Treating both the Coulomb attraction and the electron–phonon coupling on an equal footing, we have derived a set of self-consistent equations, which allows us to determine both the excitonic insulator order parameter and the lattice displacement. The phase structures show us that both the Coulomb interaction and the electron–phonon coupling act together in establishing the EI-CDW state, which typifies either a phononic-CDW or an excitonic-CDW state depending on the competition of the interactions. At a given low temperature, the EI-CDW state is found between two critical values of the Coulomb interaction. The window of the condensation state grows when increasing the electron–phonon coupling. Depending on the temperature and the Coulomb interaction the Bardeen–Cooper–Schrieffer–Bose–Einstein condensation (BCS–BEC) crossover of the EI-CDW state in the systems has been established. Under the effects of phonons, the BCS–BEC crossover deviates to a larger Coulomb interaction while the semimetal–semiconductor transition in the normal state remains.