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We propose a single-site mean-field description, an analogue of Weissmean-field theory, suitable for narrow-band systems with correlation-inducedhybridisation at finite temperatures. Presently this approach, based on thenotion of a fluctuating on-site density matrix (OSDM), is developed for thecase of extended Falicov-Kimball model (EFKM). In an EFKM, an excitonicinsulator phase can be stabilised at zero temperature. With increasingtemperature, the excitonic order parameter (interaction-induced hybridisationon-site, characterised by the absolute value and phase) eventually becomesdisordered, which involves fluctuations of both its phase and (at higher T) itsabsolute value. In order to build an adequate finite-temperature description,it is important to clarify the nature of degrees of freedom associated with thephase and absolute value of the induced hybridisation, and correctly accountfor the corresponding phase-space volume. We show that the OSDM-based treatmentof the local fluctuations indeed provides an intuitive and concise description(including the phase-space integration measure). This allows to describe boththe lower-temperature regime where phase fluctuations destroy the long-rangeorder, and the higher temperature crossover corresponding to a decrease of theabsolute value of hybridisation. In spite of the rapid progress in the studiesof excitonic insulators, a unified picture of this kind has not been availableto date. We briefly discuss recent experiments on ${\rm Ta_2 Ni Se_5 }$ andalso address the amplitude mode of collective excitations in relation to themeasurements reported for 1T--${\rm TiSe_2}$. Both the overall scenario and thetheoretical framework are also expected to be relevant in other contexts,including the Kondo lattice model.

An On-Site Density Matrix Description of the Extended Falicov--Kimball Model at Finite Temperatures