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Correlated Hopping in the 1D Falicov--Kimball Model

Z. Gajek, R. Lemański

Published Oct 1, 2001• 2 citations

Abstract

Both canonical and grand canonical phase diagrams of the one dimensional Falicov-Kimball model (FKM) with correlated hopping are presented for several values of the model parameters. Using the method of restricted phase diagrams the system has been studied in the thermodynamic limit. As compare to the conventional FKM, the diagrams exhibit a loss of the particle-hole symmetry. PACS D5.30.Fk, 71.30+h As the simplest, still non-trivial model of highly correlated electron systems, the Falicov-Kimball model (FKM) attracts growing attention among solid state physicists. It can describe variety of the most intriguing cooperative phenomena, as metal-insulator transition, mixed-valence phenomenon etc. (see e.g. the review [1]). The model deals with itinerant particles (electrons) that can hope between nearest-neighbor sites. Some of lattice sites are occupied by non-movable particles, playing a role of ions or localized electrons; we call them ”ions”. The only interaction in the system is the on-site, Coulomb-type interaction between the electrons and the ions. The interaction generates long-range correlations between the ions. The model has been investigated thoroughly in nineties. Numerous approximate results supplemented with some exact and rigorous statements reported up to now provide a good basis for further extensions of the model towards more realistic physical situations. These include the discussed here model with correlated hopping, according to which the electron hopping rate depends on occupations of relevant sites. The Hamiltonian of the FKM with correlated hopping reads:

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Both canonical and grand canonical phase diagrams of the one dimensional Falicov-Kimball model (FKM) with correlated hopping are presented for several values of the model parameters. Using the method of restricted phase diagrams the system has been studied in the thermodynamic limit. As compare to the conventional FKM, the diagrams exhibit a loss of the particle-hole symmetry. PACS D5.30.Fk, 71.30+h As the simplest, still non-trivial model of highly correlated electron systems, the Falicov-Kimball model (FKM) attracts growing attention among solid state physicists. It can describe variety of the most intriguing cooperative phenomena, as metal-insulator transition, mixed-valence phenomenon etc. (see e.g. the review [1]). The model deals with itinerant particles (electrons) that can hope between nearest-neighbor sites. Some of lattice sites are occupied by non-movable particles, playing a role of ions or localized electrons; we call them ”ions”. The only interaction in the system is the on-site, Coulomb-type interaction between the electrons and the ions. The interaction generates long-range correlations between the ions. The model has been investigated thoroughly in nineties. Numerous approximate results supplemented with some exact and rigorous statements reported up to now provide a good basis for further extensions of the model towards more realistic physical situations. These include the discussed here model with correlated hopping, according to which the electron hopping rate depends on occupations of relevant sites. The Hamiltonian of the FKM with correlated hopping reads:

Two-dimensional spinless Falicov-Kimball model (FKM) with correlated hopping is studied perturbatively in the limit of large on-site Coulomb interaction U. In the half filled case (i.e., ${\ensuremath{\rho}}_{i}+{\ensuremath{\rho}}_{e}=1,$ where ${\ensuremath{\rho}}_{i},{\ensuremath{\rho}}_{e}$ are densities of ions and electrons, respectively) the effective Hamiltonian in spin variables is derived up to terms proportional to ${U}^{\ensuremath{-}3}.$ Unlike the simplest FKM case, it contains odd parity terms (resulting from the correlated hopping) in addition to even parity ones. The ground-state phase diagram of the effective Hamiltonian is examined in the

(a/t,h)

plane, where

a/t

is a parameter characterizing strength of the correlated hopping and h is a difference of chemical potentials of two sorts of particles present in the system. It appears to be asymmetric with respect to the change $\stackrel{\ensuremath{\rightarrow}}{h}\ensuremath{-}h$ and an additional ordered phase is found for a certain interval of

a/t.

A new numerical method, recently developed to study ground states of the Falicov-Kimball model (FKM), is used to examine the effects of correlated hopping on the ground-state properties of this model in two dimensions. It is shown that the ground-state phase diagram as well as the picture of metal-insulator transitions found for the conventional FKM (without correlated hopping) are substantially changed when the correlated hopping term is added. The effect of correlated hopping is so strong that it can induce the insulator-metal transition, even in the strong-coupling limit, where the ground states of the conventional FKM are insulating for all f-electron densities.

The two-dimensional Falicov-Kimball (FK) model is analyzed using the Monte Carlo method. In the case of concentrations of both itinerant and localized particles equal to 0.5 we determine temperature dependence of specific heat, charge density wave susceptibility, and density-density correlation function. In the weak interaction regime we find a first order transition to the ordered state and anomalous temperature dependence of the correlation function. We construct the phase diagram of half-filled FK model. Also, the role of next-nearest-neighbor hopping on the phase diagram is analyzed. Lastly, we discuss the density of states and the spectral functions for the mobile particles in weak and strong interaction regime.

The two-dimensional Falicov-Kimball (FK) model is analyzed using Monte Carlo method. In the case of concentrations of both itinerant and localized particles equal to 0.5 we determine temperature dependence of specific heat, charge density wave susceptibility and density-density correlation function. In the weak interaction regime we find a first order transition to the ordered state and anomalous temperature dependence of the correlation function. We construct the phase diagram of half-filled FK model. Also, the role of next-nearest-neighbor hopping on the phase diagram is analyzed. Lastly, we discuss the density of states and the spectral functions for the mobile particles in weak and strong interaction regime.

The two-state spinless Falicov-Kimball model on a one-dimensional lattice is studied by means of well-controlled numerical procedures. Restricted phase diagrams in the grand-canonical ensemble and at zero temperature are constructed. The evolution of these phase diagrams, as the interaction parameter U is varied, along with the band structures corresponding to configurations of localized particles (ions) and densities of mobile particles (electrons), is monitored. The changes observed enable us to draw conclusions for the complete phase diagram and to make a qualitative plot of such a diagram.