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arXiv:cond-mat/9802176

Non-local effects in the fermion Dynamical mean field framework. Application to the 2D Falicov-Kimball model

M. S. Laad, Mathias van den Bossche

Published Feb 17, 1998• 0 citations
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Abstract

We propose a new, controlled approximation scheme that explicitly includes the effects of non-local correlations on the $D=\infty$ solution. In contrast to usual $D=\infty$, the selfenergy is selfconsistently coupled to two-particle correlation functions. The formalism is general, and is applied to the two-dimensional Falicov-Kimball model. Our approach possesses all the strengths of the large-D solution, and allows one to undertake a systematic study of the effects of inclusion of k-dependent effects on the $D=\infty$ picture. Results for the density of states $ρ(ω)$, and the single particle spectral density for the 2D Falicov-Kimball model always yield positive definite $ρ(ω)$, and the spectral function shows striking new features inaccessible in $D=\infty$. Our results are in good agreement with the exact results known on the 2D Falikov-Kimball model.

DOI
10.48550/arxiv.cond-mat/9802176

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We propose a new, controlled approximation scheme that explicitly includes the effects of non-local correlations on the D=∞D=\inftyD=∞ solution. In contrast to usual D=∞D=\inftyD=∞, the selfenergy is selfconsistently coupled to two-particle correlation functions. The formalism is general, and is applied to the two-dimensional Falicov-Kimball model. Our approach possesses all the strengths of the large-D solution, and allows one to undertake a systematic study of the effects of inclusion of k-dependent effects on the D=∞D=\inftyD=∞ picture. Results for the density of states ρ(ω)ρ(ω)ρ(ω), and the single particle spectral density for the 2D Falicov-Kimball model always yield positive definite ρ(ω)ρ(ω)ρ(ω), and the spectral function shows striking new features inaccessible in D=∞D=\inftyD=∞. Our results are in good agreement with the exact results known on the 2D Falikov-Kimball model.
We propose a new, controlled approximation scheme that explicitly includes the effects of non-local correlations on the D = solution. In contrast to the usual D = case, the self-energy is self-consistently coupled to two-particle correlation functions. The formalism is general, and is applied to the two-dimensional Falicov-Kimball model. Our approach possesses all the strengths of the large- D solution, and allows one to undertake a systematic study of the effects of inclusion of k -dependent effects on the D = picture. Results for the density of states ( ), and the single-particle spectral density for the 2D Falicov-Kimball model always yield positive definite ( ), and the spectral function shows striking new features inaccessible for D = . Our results are in good agreement with the exact results known from the 2D Falicov-Kimball model.
An extension of the dynamical mean-field approximation (DMFA) that allows for inclusion of nonlocal correlations is presented. This method is based on the projection technique and the DMFA, in which both local and nonlocal contributions to the dynamics of systems are evaluated exactly within a relevant subspace of Liouville space, while the remaining subspace is kept by local contributions. We examine the method by studying the spinless Falicov-Kimball model. It is found that the sum rules are preserved, the spectral function is positive definite, and nonlocal features of the spectral function appear, especially near the Fermi surface.
We study the effects of correlated hopping in the two-dimensional (2D) Falicov-Kimball model by means of an extension of the dynamical mean-field approximation (DMFA). The extension is based on the projection technique and the DMFA, in which nonlocal correlations are taken into account through static quantities of a relevant subspace. The effect of the correlated hopping is to introduce nonlocal self-energy components which remain even at D → ∞ . We show that the sum rules of the spectral function and its moments are preserved. The spectral function obtained reveals significant nonlocal contributions which are absent in the DMFA.
We derive an analytical expression for the local two-particle vertex of the Falicov-Kimball model, including its dependence on all three frequencies, the full vertex, and all reducible vertices. This allows us to calculate the self-energy in diagrammatic extensions of dynamical mean field theory, specifically in the dual fermion and the one-particle irreducible approach. Nonlocal correlations are thence included and originate here from charge-density wave fluctuations. At low temperatures and in two dimensions, they lead to a larger self-energy contribution at low frequencies and a more insulating spectrum.
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].
The Falicov-Kimball model with a correlated-hopping interaction is solved using an extended dynamical mean-field theory that becomes exact in the limit of large dimensions. The effect of correlated hopping is to introduce nonlocal self-energy components that retain full dynamics as $\stackrel{\ensuremath{\rightarrow}}{D}\ensuremath{\infty},$ thus introducing an explicit k dependence to the single-particle self-energy. An explicit solution for the homogeneous phase at D=2D=2D=2 reveals significant nonlocal dynamical contributions in the physically relevant regime of a moderately large correlated-hopping amplitude, indicating that important nonlocal correlations are omitted in Hubbard-like models upon neglecting the correlated-hopping interaction.