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arXiv:cond-mat/0411721 · Physical Review B

X-ray singularities in the f -electron spectral function of the Falicov-Kimball model

Frithjof B. Anders, G. Czycholl

Published Mar 1, 2005• 14 citations
arXivHTMLPDF

Abstract

The $f$-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 $U$ we obtain a single-peaked $f$-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 $U$. 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 $U$ when the $f$-electron spectral function develops a gap and a two-peak structure (metal-insulator transition).

DOI
10.1103/physrevb.71.125101

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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∣\ensuremathω\ensuremath∣\ensuremath−\ensuremathα)({\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}}^{\ensuremath{-}\ensuremath{\alpha}})(\ensuremath∣\ensuremathω\ensuremath∣\ensuremath−\ensuremathα) for \ensuremathω\ensuremath→0\ensuremath{\omega}\ensuremath{\rightarrow}0\ensuremathω\ensuremath→0. The characteristic exponent \ensuremathα\ensuremath{\alpha}\ensuremathα 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,\ensuremathαU,\ensuremath{\alpha}U,\ensuremathα 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 fff-electron spectral function of the Falicov-Kimball model is calculated via a Keldysh-based many-body formalism originally developed by Brandt and Urbanek. We provide results for both the Bethe lattice and the hypercubic lattice at half filling. Since the numerical computations are quite sensitive to the discretization along the Kadanoff-Baym contour and to the maximum cutoff in time that is employed, we analyze the accuracy of the results using a variety of different moment sum rules and spectral formulas. We find that the fff-electron spectral function has interesting temperature dependence, becoming a narrow single-peaked function for small UUU and developing a gap, with two broader peaks for large 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 zero-temperature core-level photoemission spectrum of a Hubbard system is studied across the metal to Mott insulator transition using dynamical mean-field theory and Wilson's numerical renormalization group [Rev. Mod. Phys. 47, 773 (1975)]. An asymmetric power-law divergence is obtained in the metallic phase with an exponent $\ensuremath{\alpha}(U,Q)\ensuremath{-}1$, which depends on the strength of both the Hubbard interaction UUU and the core-hole potential QQQ. For $Q\ensuremath{\lesssim}{U}_{c}∕2$, $\ensuremath{\alpha}$ decreases with increasing UUU and vanishes at the transition $(U\ensuremath{\rightarrow}{U}_{c})$ leading to a symmetric peak in the insulating phase. For $Q\ensuremath{\gtrsim}{U}_{c}∕2$, $\ensuremath{\alpha}$ remains finite, close to the transition, but the integrated intensity of the power law vanishes and there is no associated peak in the insulator. The weight and position of the remaining peaks in the spectra can be understood within a molecular orbital approach.
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 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 ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{k}$ 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.
Using the dynamical mean-field theory (DMFT) with the numerical renormalization-group impurity solver we study the paramagnetic phase of the Hubbard model with the density of states (DOS) corresponding to the three-dimensional (3D) cubic lattice and the two-dimensional (2D) square lattice, as well as a DOS with inverse square-root singularity. We show that the electron correlations rapidly smooth out the square-root van Hove singularities (kinks) in the spectral function for the 3D lattice and that the Mott metal-insulator transition (MIT) as well as the magnetic-field-induced MIT differ only little from the well-known results for the Bethe lattice. The consequences of the logarithmic singularity in the DOS for the 2D lattice are more dramatic. At half filling, the divergence pinned at the Fermi level is not washed out, only its integrated weight decreases as the interaction is increased. While the Mott transition is still of the usual kind, the magnetic-field-induced MIT falls into a different universality class as there is no field-induced localization of quasiparticles. In the case of a power-law singularity in the DOS at the Fermi level, the power-law singularity persists in the presence of interaction, albeit with a different exponent, and the effective impurity model in the DMFT turns out to be a pseudogap Anderson impurity model with a hybridization function which vanishes at the Fermi level. The system is then a generalized Fermi liquid. At finite doping, regular Fermi-liquid behavior is recovered.
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].