Ground-state correlations of itinerant electrons in the spinless Falicov-Kimball chain and related tight-binding systems

We consider the one-dimensional spinless Falicov-Kimball model of itinerant fermionic particles (``spinless electrons''), which can hop between nearest-neighbour sites only, and of immobile particles (``classical ions''), with an on-site attraction. Extensive studies of the ground-state phase diagram of this system and its higher dimensional counterparts, carried out up to now, concentrated on determining ground-state arrangements of ions on the underlying lattice, while the properties of electrons were typically ignored. We report studies of short- and long-range correlations between electrons, and between ions and electrons, and of the spatial decay of electron correlations (decay of single-particle density matrix), in the ground state. The studies have been carried out analytically and by means of well-controlled numerical procedures. In the case of period 2 ground state, the single-particle density matrix has been expressed in terms of a hypergeometric function, and its spatial decay has been extracted. Numerical calculations have been done for open chains of various lengths (up to a few thousand sites), in order to control the chain-size dependence of correlations and to extrapolate the results to the limit of infinite chain. A part of the obtained results refers to tight-binding electrons subjected to a periodic external potential due to the ions, which constitute simple models of metals and insulators.