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arXiv:1202.2796 · Physical Review B

Local electronic correlation at the two-particle level

G. Rohringer, Angelo Valli, A. Toschi

Published Sep 10, 2012• 195 citations
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Abstract

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.

DOI
10.1103/physrevb.86.125114

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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.
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.
In principle, the electronic Coulomb interaction among the correlated orbitals is frequency-dependent. Though it is generally believed that the dynamically screened interaction may play a crucial role in understanding the subtle electronic structures of strongly correlated materials, hitherto we know very little about it. In the Letter, we demonstrate that within the framework of single-site dynamical mean-field theory the local two-particle Green's functions χχχ and vertex functions ΓΓΓ are strongly modified by the dynamically screened interaction. Since both χχχ and ΓΓΓ represent the main ingredients to compute momentum-resolved response functions and to treat non-local spatial correlations by means of diagrammatic extensions of dynamical mean-field theory, it is urgent to reexamine previous results by taking the dynamical screening effect into account. The modifications should be very considerable.
In this thesis, we tackle various problems in strongly correlated electron systems, which can be addressed properly via the non-perturbative dynamical mean field theory (DMFT) approach using the continuous time quantum Monte Carlo method as an impurity solver. First, we revisit the old Nagaoka ferromagnetism problem in the U=oo Hubbard model and study the stability of the ferromagnetic state as a function of the temperature, the doping level, and the next-nearest-neighbor hopping t'. We then address the nature of the Mott transition in the two-dimensional Hubbard model at half-filling using cluster DMFT. Cluster DMFT can incorporate the short-range correlations beyond DMFT by extending the spatial range in which correlations are treated exactly to a finite cluster size. The non-local correlations reduce substantially the critical interaction U and modify the shape of the transition lines in the phase diagram. We then concentrate on the calculation of two-particle response functions from the ab initio perspective by means of computing the one-particle excitation spectrum using the combination of the density functional theory (DFT) and DMFT and extracting the two-particle irreducible vertex function from a local two-particle Green's function computed within DMFT. In particular, we derive the equations for calculating the magnetic/charge susceptibility and the pairing susceptibility in superconductivity. This approach is applied to the Hubbard model and the periodic Anderson model and we determine the phase diagram of magnetism and superconductivity in these models. We show that the superconducting phase is indeed stable near the magnetic phase where the pairing interaction mediated by spin fluctuations is dominantly enhanced. The non-local correlation effect to superconductivity is also discussed using the dual fermion approach and the dynamical vertex approximation. We finally apply the vertex function approach within DFT+DMFT to a Fe-based superconductor, BaFe2As2, and compute the dynamical magnetic susceptibility in this material. Our calculation results show a good agreement with the magnetic excitation spectra observed in a neutron scattering experiment. The response function calculation method derived in this thesis can capture both a localized and an itinerant nature of collective excitations in strongly correlated electron systems.
We propose an approach describing correlated electronic systems within the two-particle irreducible functional renormalization-group (2PI-fRG) approach, which uses initial self-energy and the two-particle irreducible vertices, obtained from (extended) dynamical mean-field theory [E(DMFT)]. Using the 2PI-fRG approach allows us to treat both local and nonlocal interactions. In the case of purely local interaction the corresponding equations have similar (although not identical) structure to the earlier developed DMFT+fRG (DMF2RG)({\mathrm{DMF}}^{2}\mathrm{RG})(DMF2RG) approach. In a simplest truncation, neglecting scale dependence of the two-particle irreducible vertices, we reproduce the results for the two-particle vertices/susceptibilities in the ladder approximation of the dual-boson (DB) or the dynamic vertex approximation approach; in more sophisticated truncations the method allows us to consider nonlocal corrections to the self-energy as well as the interplay of charge and spin correlations. The proposed scheme is tested on the two-dimensional standard and extended $U\text{\ensuremath{-}}V$ half-filled Hubbard models. For the standard Hubbard model we obtain nonlocal self-energy, which is in agreement with numerical studies; for the extended Hubbard model we find the boundary of charge instability, which agrees well with the results of the DB approach. We also ascertain that the effect of spin correlations on electron interaction in the charge channel, not considered previously in the DB approach, only slightly reduces critical next-nearest-neighbor interaction of charge instability of the extended Hubbard model at the considered finite low temperature, yielding better agreement with dynamic cluster approximation. The considered method is rather general and can be applied to study various phenomena in strongly correlated electronic systems.
We present a fully self-consistent combined GW and dynamical mean field (DMFT) study of the extended two-dimensional Hubbard model. The inclusion of the local dynamical vertex stemming from the DMFT self-energy and polarization is shown to cure the known problems of self-consistent GW. We calculate momentum-resolved spectral functions, two-particle polarizations, and electron-loss spectra, as well as the effective dynamical interaction induced by nonlocal screening. The momentum-dependence introduced by GW into the extended DMFT description leads to a narrowing of the quasiparticle width and more pronounced Hubbard bands in the metallic regime as one approaches the charge-ordering transition. It further affects the shape of collective modes, giving rise to dispersive plasmon-like long-wavelength and stripe modes.