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arXiv:cond-mat/0701433 · Physical Review B

Static and dynamic properties of the spinless Falicov-Kimball model

K. W. Becker, Steffen Sykora, V. Zlatić

Published Feb 5, 2007• 6 citations
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Abstract

The spinless Falicov-Kimball model is studied by the use of a recently developed projector-based renormalization method for many-particle Hamiltonians. The method is used to evaluate static and dynamic quantities of the one-dimensional model at half-filling. These include the quasiparticle excitation energy ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{k}$, the momentum distribution ${n}_{k}$ of the conduction electrons, and the spatial correlation functions of the localized electrons. One of the most remarkable results is the appearance of a gap in ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{k}$ at the Fermi level of the order of the Coulomb repulsion $U$, which is accompanied by a smooth behavior for ${n}_{k}$. The density of states for the conduction electrons and the one-particle spectral functions for the localized electrons are also discussed. In both quantities, a gap opens with increasing $U$.

DOI
10.1103/physrevb.75.075101

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The spinless Falicov-Kimball model is studied by the use of a recently developed projector-based renormalization method for many-particle Hamiltonians. The method is used to evaluate static and dynamic quantities of the one-dimensional model at half-filling. These include the quasiparticle excitation energy \ensuremathϵ\ifmmode~\else˜\fik{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{k}\ensuremathϵ\ifmmode~\else˜\fi​k​, the momentum distribution nk{n}_{k}nk​ of the conduction electrons, and the spatial correlation functions of the localized electrons. One of the most remarkable results is the appearance of a gap in \ensuremathϵ\ifmmode~\else˜\fik{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{k}\ensuremathϵ\ifmmode~\else˜\fi​k​ at the Fermi level of the order of the Coulomb repulsion UUU, which is accompanied by a smooth behavior for nk{n}_{k}nk​. The density of states for the conduction electrons and the one-particle spectral functions for the localized electrons are also discussed. In both quantities, a gap opens with increasing UUU.
In this work, we study the Anderson-Falicov-Kimball model within the dynamical mean field theory for the Bethe lattice, restricting our analysis to the nonmagnetic case. The one-particle density of states is obtained by both arithmetic and geometric averages over disorder, since only the latter can detect localization in the absence of an energy gap. Varying the strengths of Coulomb interaction and disorder at zero temperature, we construct phase diagrams for this model, where we distinguish spectral regions with localized states, with extended states, or with a correlation-induced gap. With this, we identify metal-insulator transitions driven by correlation and disorder, as well as the competition between these effects. This is done for various band fillings, since our main interest here is to study how the variation of the electron density affects the phase diagrams previously obtained for half-filling. The picture revealed by the density of states is further checked by evaluating the static and dynamic conductivities, including temperature effects.
The Falicov-Kimball model was introduced in 1969 as a statistical model for metal-insulator transitions; it includes itinerant and localized electrons that mutually interact with a local Coulomb interaction and is the simplest model of electron correlations. It can be solved exactly with dynamical mean-field theory in the limit of large spatial dimensions, which provides an interesting benchmark for the physics of locally correlated systems. In this review, the authors develop the formalism for solving the Falicov-Kimball model from a path-integral perspective and provide a number of expressions for single- and two-particle properties. Many important theoretical results are examined that show the absence of Fermi-liquid features and provide a detailed description of the static and dynamic correlation functions and of transport properties. The parameter space is rich and one finds a variety of many-body features like metal-insulator transitions, classical valence fluctuating transitions, metamagnetic transitions, charge-density-wave order-disorder transitions, and phase separation. At the same time, a number of experimental systems have been discovered that show anomalies related to Falicov-Kimball physics [including YbInCu4,{\mathrm{YbInCu}}_{4},YbInCu4​, ${\mathrm{EuNi}}_{2}({\mathrm{Si}}_{1\ensuremath{-}x}{\mathrm{Ge}}_{x}{)}_{2},$ NiI2,{\mathrm{NiI}}_{2},NiI2​, and TaxN].{\mathrm{Ta}}_{x}\mathrm{N}].Tax​N].
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 fff-electron spectral function of the Falicov-Kimball model is calculated within the dynamical mean-field theory using the numerical renormalization group method as the impurity solver. Both the Bethe lattice and the hypercubic lattice are considered at half filling. For small UUU we obtain a single-peaked fff-electron spectral function, which---for zero temperature---exhibits an algebraic (x-ray) singularity $({\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}}^{\ensuremath{-}\ensuremath{\alpha}})$ for $\ensuremath{\omega}\ensuremath{\rightarrow}0$. The characteristic exponent $\ensuremath{\alpha}$ depends on the Coulomb (Hubbard) correlation UUU. This x-ray singularity cannot be observed when using alternative (Keldysh-based) many-body approaches. With increasing $U,\ensuremath{\alpha}$ decreases and vanishes for sufficiently large UUU when the fff-electron spectral function develops a gap and a two-peak structure (metal-insulator transition).
The momentum distribution nk of itinerant electrons in the one-dimensional Falicov-Kimball model is calculated for various ground-state phases. In particular, we examine the periodic phases with period two, three and four (that are ground-states for all Coulomb interactions) as well as the phase separated states (that are ground states for small Coulomb interactions). For all periodic phases examined the momentum distribution is a smooth function of k with no sign of any discontinuity or singular behavior at the Fermi surface k = kF. An unusual behavior of nk (a local maximum) is found at k = 3kF for electron concentrations outside half-filling. For the phase separated ground states the momentum distribution nk exhibits discontinuity at k = k0 < kF. This behavior is interpreted in terms of a Fermi liquid. PACS nrs.: 71.27.+a, 71.28.+d, 71.30.+h
The density matrix renormalization group and small cluster exact diagonalization calculations are used to examine ground-state properties of the Falicov-Kimball model in one and two dimensions. Particular attention is paid to hybridization effects on insulator-metal transitions. The nature of the ground state is analyzed through the behavior of the single-particle excitation energy and the f and d electron density of states. A number of remarkable results are found. (i) At zero hybridization the gaps in the f and d electron density of states do not coincide, and almost all f electron spectral weight is located outside the d electron subbands. (ii) For nonzero hybridization the f and d electron gaps coincide, for both the symmetric and unsymmetric case. (iii) No insulator-metal transitions driven by hybridization are found in one as well as in two dimensions.