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Physical review. B, Condensed matter

Falicov-Kimball model and the breaking of Fermi-liquid theory in infinite dimensions

Qimiao Si A, Gabriel Kotliar, Antoine Georges

Published Jul 1, 1992• 70 citations
Publisher

Abstract

We reexamine the Brandt-Mielsch solution of the Falicov-Kimball model in infinite dimensions, to elucidate the mechanism for the breakdown of Fermi-liquid theory in this model. We find that this breakdown is due to the pinning of the chemical potential at the localized particle level. When this pinning does not occur, Fermi-liquid theory is restored. Using a Lorentzian density of states, we obtain analytic expressions for spectral densities and local response functions. These expressions give new insights into the formation of upper and lower Hubbard bands, the metal-insulator transition, the softening of the collective modes, and the infrared singularities of this model.

DOI
10.1103/physrevb.46.1261

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We reexamine the Brandt-Mielsch solution of the Falicov-Kimball model in infinite dimensions, to elucidate the mechanism for the breakdown of Fermi-liquid theory in this model. We find that this breakdown is due to the pinning of the chemical potential at the localized particle level. When this pinning does not occur, Fermi-liquid theory is restored. Using a Lorentzian density of states, we obtain analytic expressions for spectral densities and local response functions. These expressions give new insights into the formation of upper and lower Hubbard bands, the metal-insulator transition, the softening of the collective modes, and the infrared singularities of this model.
The diagrammatic formulation of the single-site approximation to the Hubbard model immediately yields the exact solution to the Falikov-Kimball model in infinite dimensions. By an extension of the alloy analogy, we then derive a two-fluid model, dividing self-consistently the electron system among itinerating and relatively localized particles. Our theory coincides with the CPA only the at half-filling and u=∞. We start by studying the mean field approximation to our theory and identify a phase boundary in the particle density-interaction strength (n−u) plane at which the Fermi liquid model breaks down. Then the effect of fluctuations in the Fermi liquid regime is considered and, finally, we construct, at half-filling, a model which interpolates between the Fermi liquid and the Mott insulating phases.
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 Marginal Fermi Liquid (MFL) hypothesis proposed by Varma et al is derived from a consistent treatment of local spinfluctuations in the Falicov-Kimball model (FKM). Within an infinite-dimensional mean-field approach, a marginal Fermi liquid susceptibility and single-particle self-energy are obtained near half-filling, and a Fermi liquid phase is recovered sufficiently away from half-filling, in agreement with indications from experiments which probe the normal state of the cuprates as a function of doping
For pt.I see Physics B, vol.163, p.527 (1990). An exact diagrammatic analysis of the electron self-energy in the Hubbard and related models is given within the local approximation. Contact is made with recent work for dimension d= infinity where the local approximation becomes exact. It is shown that the approximate method of paper I of the series may be fitted into the new framework and yields sensible results for the Anderson impurity model. This lends weight to the prediction of I that for paramagnetic solutions of the Hubbard model Fermi liquid behaviour breaks down in a whole region of the T=0 phase diagram, not just for a half-filled band where the Mott transition occurs.
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].
One of the fundamental questions about high-temperature cuprate superconductors is the size of the Fermi surface underlying the superconducting state. By analyzing the single-particle spectral function for the Fermi-Hubbard model as a function of repulsion UUU and chemical potential $\ensuremath{\mu}$, we find that the Fermi surface in the normal state undergoes a transition from a large Fermi surface matching the Luttinger volume as expected in a Fermi liquid, to a Fermi surface that encloses fewer electrons that we dub the ``Luttinger breaking'' phase, as the Mott insulator is approached. This transition into a non-Fermi-liquid phase that violates the Luttinger count occurs at a critical density in the absence of any other broken symmetry. We obtain the Fermi-surface contour from the spectral weight ${A}_{\mathbf{k}}(\ensuremath{\omega}=0)$ and from an analysis of the singularities of the Green's function ReGk(E=0)\mathrm{Re}\phantom{\rule{0.16em}{0ex}}{G}_{\mathbf{k}}(E=0)ReGk​(E=0), calculated using determinantal quantum Monte Carlo and analytic continuation methods. We discuss our numerical results in connection with experiments on Hall measurements, scanning tunneling spectroscopy, and angle-resolved photoemission spectroscopy.