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arXiv:1612.03011 · Physical review. B./Physical review. B

Local correlation functions of arbitrary order for the Falicov-Kimball model

T. Ribic, G. Rohringer, Karsten Held

Published Apr 18, 2017• 13 citations
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Abstract

Local $n$-particle vertex functions represent the crucial ingredient for diagrammatic extensions of dynamical mean field theory (DMFT). Hitherto their application has been restricted---with a few exceptions---to the $n=2$-particle vertex while higher-order vertices have been neglected. In this paper we derive a general analytical expression for the local $n$-particle (one-particle-reducible) vertex of the Falicov-Kimball model (FKM). We observe that the magnitude of such vertex functions itself strongly increases with the number of particles $n$. On the other hand, their effect on generic Feynman diagrams remains rather moderate due to the damping effect of the Green's functions present in such diagrams. Nevertheless, they yield important contributions to the self-energy corrections calculated in diagrammatic extensions of DMFT as we explicitly demonstrate using the example of dual-fermion calculations for the two-dimensional FKM at quarter filling of the stationary $f$ electrons. Here corrections to the self-energy obtained from the three-particle vertex are indeed comparable in magnitude to corresponding corrections stemming from the two-particle vertex.

DOI
10.1103/physrevb.95.155130

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Local nnn-particle vertex functions represent the crucial ingredient for diagrammatic extensions of dynamical mean field theory (DMFT). Hitherto their application has been restricted---with a few exceptions---to the n=2n=2n=2-particle vertex while higher-order vertices have been neglected. In this paper we derive a general analytical expression for the local nnn-particle (one-particle-reducible) vertex of the Falicov-Kimball model (FKM). We observe that the magnitude of such vertex functions itself strongly increases with the number of particles nnn. On the other hand, their effect on generic Feynman diagrams remains rather moderate due to the damping effect of the Green's functions present in such diagrams. Nevertheless, they yield important contributions to the self-energy corrections calculated in diagrammatic extensions of DMFT as we explicitly demonstrate using the example of dual-fermion calculations for the two-dimensional FKM at quarter filling of the stationary fff electrons. Here corrections to the self-energy obtained from the three-particle vertex are indeed comparable in magnitude to corresponding corrections stemming from the two-particle vertex.
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.
Electronic-correlated systems are often well described by dynamical mean field theory (DMFT). While DMFT studies have mainly focused hitherto on one-particle properties, valuable information is also enclosed into local two-particle Green's functions and vertices. They represent the main ingredient to compute momentum-dependent response functions at the DMFT level and to treat nonlocal spatial correlations at all length scales by means of diagrammatic extensions of DMFT. The aim of this paper is to present a DMFT analysis of the local reducible and irreducible two-particle vertex functions for the Hubbard model in the context of a unified diagrammatic formalism. An interpretation of the observed frequency structures is also given in terms of perturbation theory, of the comparison with the atomic limit, and of the mapping onto the attractive Hubbard model.
We present an approach which is based on the one-particle irreducible (1PI) generating functional formalism and includes electronic correlations on all length scales beyond the local correlations of dynamical mean-field theory (DMFT). This formalism allows us to unify aspects of the dynamical vertex approximation (D$\ensuremath{\Gamma}$A) and the dual fermion (DF) scheme, yielding a consistent formulation of nonlocal correlations at the one- and two-particle level beyond DMFT within the functional integral formalism. In particular, the considered approach includes one-particle reducible contributions from the three- and more-particle vertices in the dual fermion approach, as well as some diagrams not included in the ladder version of D$\ensuremath{\Gamma}$A. To demonstrate the applicability and physical content of the 1PI approach, we compare the diagrammatics of 1PI, DF, and D$\ensuremath{\Gamma}$A, as well as the numerical results of these approaches for the half-filled Hubbard model in two dimensions.
We investigate the influence of self-energy diagrams beyond the two-particle vertex level within dual fermion theory. Specifically, we calculate the local three-particle vertex and construct from it selected dual fermion self-energy corrections to dynamical mean field theory. For the two-dimensional Hubbard model, the thus obtained self-energy corrections are small in the parameter space where dual fermion corrections based on the two-particle vertex only are small. However, in other parts of the parameter space, they are of a similar magnitude and qualitatively different from standard dual fermion theory. The high-frequency behavior of the self-energy correction is, surprisingly, even dominated by corrections stemming from the three-particle vertex.
In the study of correlated systems, approximations based on the dynamical mean-field theory (DMFT) provide a practical way to take local vertex corrections into account, which capture, respectively, particle-particle screening at weak coupling and the formation of the local moment at strong coupling. We show that in both limits the local vertex corrections can be efficiently parameterized in terms of single-boson exchange, such that the two-particle physics described by DMFT and its diagrammatic extensions is recovered to good approximation and at a reduced computational cost. Our investigation highlights the importance of the frequency-dependent fermion-boson coupling (Hedin vertex) for local vertex corrections. Namely, at weak coupling the fermion-spin-boson coupling suppresses the N\'eel temperature of the DMFT approximation compared to the static mean-field, whereas for large interaction it facilitates a huge enhancement of local spin-fluctuation exchange, giving rise to the effective-exchange energy scale 4t2/U4t^2/U4t2/U. We find that parameterizations of the vertex which neglect the nontrivial part of the fermion-boson coupling fail qualitatively at strong coupling.
In the study of correlated systems, approximations based on the dynamical mean-field theory (DMFT) provide a practical way to take local vertex corrections into account, which capture, respectively, particle-particle screening at weak coupling and the formation of the local moment at strong coupling. We show that in both limits the local vertex corrections can be efficiently parametrized in terms of single-boson exchange, such that the two-particle physics described by DMFT and its diagrammatic extensions is recovered to good approximation and at a reduced computational cost. Our investigation highlights the importance of the frequency-dependent fermion-boson coupling (Hedin vertex) for local vertex corrections. Namely, at weak coupling the fermion--spin-boson coupling suppresses the N\'eel temperature of the DMFT approximation compared with the static mean field, whereas for large interaction it facilitates a huge enhancement of local spin-fluctuation exchange, giving rise to the effective-exchange energy scale 4t2/U4{t}^{2}/U4t2/U. We find that parametrizations of the vertex which neglect the nontrivial part of the fermion-boson coupling fail qualitatively at strong coupling.