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

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