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Reviews in Mathematical Physics

LOW TEMPERATURE STATES IN THE FALICOV-KIMBALL MODEL

Alain Messager, Salvador Miracle-Solé

Published Feb 1, 1996• 24 citations
Publisher

Abstract

The ground states and low temperature states of the Falicov-Kimball model are studied in the plane of chemical potentials of the two types of particles involved, for large values of the coupling constant, near the half-filled band. For this study, a cluster expansion is introduced which can be used to investigate the effective interaction energy between the static particles.

DOI
10.1142/s0129055x9600010x

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THE FALICOV-KIMBALL MODEL, EFFECTIVE INTERACTION

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The effective interaction between the static particles in the Falicov-Kimball model is studied, for large values of the coupling constant, using a convergent cluster expansion.

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THE SPINLESS FALICOV–KIMBALL MODEL AWAY FROM HALF-FILLING

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The one-dimensional spinless Falicov–Kimball (FKM) model is studied in the strong coupling limit away from half-filling. The small-cluster exact-diagonalization calculations are employed to obtain valence transitions in the ground state and for finite temperatures. It is found that in the ground state, the model describes continuous valence transitions for small band-fillings and both continuous and discontinuous transitions near half-filling. At finite temperatures, only continuous valence transitions are obtained. The valence is integer at low temperatures for small band-fillings and in other cases, it takes only intermediate values.

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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.

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Thermodynamics of the one-dimensional half-filled-band Falicov-Kimball model

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In order to contribute to understand well the finite-temperature properties of the Falicov-Kimball model, we study the temperature dependence of the specific heat, internal energy, entropy, and mean square fluctuation of the difference between the ion and electron numbers, and of the correlation functions of the one-dimensional half-filled-band Falicov-Kimball model. For this purpose we used the method of small-cluster exact-diagonalization calculations with the application of the grand canonical ensemble and of extrapolation techniques to the infinite chain. Our results show that the examined Falicov-Kimball model exhibits behavior of alternate electron-ion short-range ordering at low temperature, and gradual metal-insulator transition at high temperature, but, does not exhibit the crossing point of the specific heat curves, characteristic of the Hubbard model.

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Ground states of the Falicov-Kimball model with correlated hopping

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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 $a/t.$

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Energy spectrum analysis and finite temperature properties of the Falicov-Kimball model with Hund coupling at half filling

Jakub Wrzodak, R. Lemański

The Falicov-Kimball model supplemented with the Ising-type Hund coupling describes charge and magnetic order induced by on-site, charge-, and spin-dependent interactions between itinerant electrons and localized ions. Motivated by a discovery of the rich structure of ground-state phase diagrams containing various charge and magnetic superstructures, we analyze the energy spectrum and determine numerically finite temperature properties of the model on the square $4\ifmmode\times\else\texttimes\fi{}4$ cluster at half filling. For the density of magnetic ions equal to 1 and 1/2 and not too small coupling constants we show that many-electron collective excitations derived from changes in positions of the ions or inversions their spins have much lower energy than single-electron excitations. From an analysis of the excitation spectrum structure we derive effective interactions between the ions. When the coupling constants are large enough the interactions reduce to two-body forces between ions occupying the nearest-neighboring sites. Finite-temperature characteristics and temperatures of transformation from charge or magnetically ordered to disordered phases are determined in the wide range of the interaction couplings.

Physical Review B•Nov 12, 2010•4 citations
The ground states and low temperature states of the Falicov-Kimball model are studied in the plane of chemical potentials of the two types of particles involved, for large values of the coupling constant, near the half-filled band. For this study, a cluster expansion is introduced which can be used to investigate the effective interaction energy between the static particles.
The one-dimensional spinless Falicov–Kimball (FKM) model is studied in the strong coupling limit away from half-filling. The small-cluster exact-diagonalization calculations are employed to obtain valence transitions in the ground state and for finite temperatures. It is found that in the ground state, the model describes continuous valence transitions for small band-fillings and both continuous and discontinuous transitions near half-filling. At finite temperatures, only continuous valence transitions are obtained. The valence is integer at low temperatures for small band-fillings and in other cases, it takes only intermediate values.
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.
In order to contribute to understand well the finite-temperature properties of the Falicov-Kimball model, we study the temperature dependence of the specific heat, internal energy, entropy, and mean square fluctuation of the difference between the ion and electron numbers, and of the correlation functions of the one-dimensional half-filled-band Falicov-Kimball model. For this purpose we used the method of small-cluster exact-diagonalization calculations with the application of the grand canonical ensemble and of extrapolation techniques to the infinite chain. Our results show that the examined Falicov-Kimball model exhibits behavior of alternate electron-ion short-range ordering at low temperature, and gradual metal-insulator transition at high temperature, but, does not exhibit the crossing point of the specific heat curves, characteristic of the Hubbard model.
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)(a/t,h)(a/t,h) plane, where a/ta/ta/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/t.a/t.
The Falicov-Kimball model supplemented with the Ising-type Hund coupling describes charge and magnetic order induced by on-site, charge-, and spin-dependent interactions between itinerant electrons and localized ions. Motivated by a discovery of the rich structure of ground-state phase diagrams containing various charge and magnetic superstructures, we analyze the energy spectrum and determine numerically finite temperature properties of the model on the square $4\ifmmode\times\else\texttimes\fi{}4$ cluster at half filling. For the density of magnetic ions equal to 1 and 1/2 and not too small coupling constants we show that many-electron collective excitations derived from changes in positions of the ions or inversions their spins have much lower energy than single-electron excitations. From an analysis of the excitation spectrum structure we derive effective interactions between the ions. When the coupling constants are large enough the interactions reduce to two-body forces between ions occupying the nearest-neighboring sites. Finite-temperature characteristics and temperatures of transformation from charge or magnetically ordered to disordered phases are determined in the wide range of the interaction couplings.