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Le Journal de Physique Colloques

Phase transition in the Falicov-Kimball model

G. Spronken, M. Avignon

Published May 1, 1979• 0 citations
Publisher

Abstract

Electronic phase transitions between localized and Bloch-like states are studied in the Falicov-Kimball model with and without hybridization between these states. Different decoupling schemes of the Green functions beyond the Hartree-Fock approximation are considered. We then analyze how the nature of the transitions (continuous or discontinuous) and the nature of the phases involved (pure or mixed valence states) are affected by these different decoupling procedures. Results are compared with limiting cases i) single site approximation (independent impurities approximation), ii) zero band width limit of the Bloch-states similar to the atomic limit of the Hubbard model. In the many site problem, effect of intersite correlations will also be discussed.

DOI
10.1051/jphyscol:19795136

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AcPPB•Oct 1, 2001•2 citations
Electronic phase transitions between localized and Bloch-like states are studied in the Falicov-Kimball model with and without hybridization between these states. Different decoupling schemes of the Green functions beyond the Hartree-Fock approximation are considered. We then analyze how the nature of the transitions (continuous or discontinuous) and the nature of the phases involved (pure or mixed valence states) are affected by these different decoupling procedures. Results are compared with limiting cases i) single site approximation (independent impurities approximation), ii) zero band width limit of the Bloch-states similar to the atomic limit of the Hubbard model. In the many site problem, effect of intersite correlations will also be discussed.
In this work, we study the extended Falicov-Kimball model at half-filling within the Hartree-Fock approach (HFA) (for various crystal lattices) and compare the results obtained with the rigorous ones derived within the dynamical mean field theory (DMFT). The model describes a system, where electrons with spin-↓ are itinerant (with hopping amplitude t), whereas those with spin-↑ are localized. The particles interact via on-site U and intersite V density-density Coulomb interactions. We show that the HFA description of the ground state properties of the model is equivalent to the exact DMFT solution and provides a qualitatively correct picture also for a range of small temperatures. It does capture the discontinuous transition between ordered phases at U = 2V for small temperatures as well as correct features of the continuous order-disorder transition. However, the HFA predicts that the discontinuous boundary ends at the isolated-critical point (of the liquid-gas type) and it does not merge with the continuous boundary. This approach cannot also describe properly a change of order of the continuous transition for large V as well as various metal-insulator transitions found within the DMFT.
The extended Falicov-Kimball model, with both an on-site hybridization potential and dispersive narrow band, is examined within the saddle-point approximation to the Kotliar-Ruckenstein [Phys. Rev. Lett. 57, 1362 (1986)] slave-boson theory. We first set the hybridization potential to zero and find that the phase diagram depends strongly on the orbital structure: for degenerate orbitals, a correlated-insulating state is found at sufficiently strong interaction strengths, whereas a finite orbital energy difference can lead to discontinuous valence transitions. The obtained phase diagram is very sensitive to the presence of a finite hybridization potential. As in Hartree-Fock theory, we find an enhancement of the hybridization by the interorbital Coulomb repulsion. The more precise treatment of correlation effects, however, leads to large deviations from the Hartree-Fock results. In the limit of vanishing hybridization, an excitonic insulator state is only found when the orbitals are degenerate, which restricts this phase to a much smaller parameter space than in other available mean-field theories.
The extended Falicov-Kimball model is analyzed exactly in the ground state at half filling in the limit of large dimensions. In the model the on-site and the intersite density-density interactions between all particles are included. We determined the model's phase diagram and found a discontinuous transition between two different charge-ordered phases. Our analytical calculations show that the ground state of the system is insulating for any nonzero values of the interaction couplings. We also show that the dynamical mean-field theory and the static broken-symmetry Hartree-Fock mean-field approximation give the same results for the model at zero temperature. In addition, we prove using analytical expressions that at infinitesimally small, but finite, temperatures the system can be metallic.
The asymmetric Hubbard model is used in investigating the lattice gas of the moving particles of two types.The model is considered within the dynamical mean-field method.The effective single-site problem is formulated in terms of the auxiliary Fermi-field.To solve the problem an approximate analytical method based on the irreducible Green's function technique is used.This approach is tested on the Falicov-Kimball limit (when the mobility of ions of either type is infinitesimally small) of the infinite-U case of the model considered.The dependence of chemical potentials on concentration is calculated using the one-particle Green's functions, and different approximations are compared with the exact results obtained thermodynamically.The densities of states of localized particles are obtained for different temperatures and particle concentrations.The phase transitions are investigated for the case of the Falicov-Kimball limit in different thermodynamic regimes.
The extrapolation of small-cluster exact-diagonalization calculations is used to examine the influence of correlated hopping on valence and metal-insulator transitions in the one-dimensional Falicov-Kimball model. It is shown that in the half-filled band case the ground-state phase diagram as well as the picture of valence and metal-insulator transitions found for the conventional Falicov-Kimball model (without correlated hopping) are strongly changed when the correlated hopping term is added. The effect of correlated hopping is so strong that it can induce the insulator-metal transition. Outside half-filling correlated hopping stabilizes the segregated phase in the ground-state, however, the nature of the ground-state remains qualitatively unchanged.
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: