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We explore the spontaneous formation of an excitonic insulator state at thesemimetal-semiconductor transition of mixed-valence materials in the frameworkof the spinless Falicov-Kimball model with direct $f$-$f$ electron hopping.Adapting the projector-based renormalization method, we obtain a set ofrenormalization differential equations for the extended Falicov-Kimball modelparameters and finally derive analytical expressions for the order parameter,as well as for the renormalized $c$- and $f$-electron dispersions, momentumdistributions, and wave-vector resolved single-particle spectral functions. Ournumerical results proved the valence transition picture, related to theappearance of the excitonic insulator phase, in the case of overlapping $c$ and$f$ bands. Thereby the photoemission spectra show significant differencesbetween the weak-to-intermediate and intermediate-to-strong Coulomb attractionregimes, indicating a BCS-BEC transition of the excitonic condensate.

Spectral signatures of the BCS-BEC crossover in the excitonic insulator phase of the extended Falicov-Kimball model