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arXiv:1701.07718 · Physical review. B./Physical review. B

Influence of Fock exchange in combined many-body perturbation and dynamical mean field theory

Thomas Ayral, Silke Biermann, Philipp Werner, Lewin Boehnke

Published Jun 23, 2017• 60 citations
arXivHTMLPDF

Abstract

In electronic systems with long-range Coulomb interaction, the nonlocal Fock-exchange term has a band-widening effect. While this effect is included in combined many-body perturbation theory and dynamical mean field theory (DMFT) schemes, it is not taken into account in standard extended DMFT (EDMFT) calculations. Here, we include this instantaneous term in both approaches and investigate its effect on the phase diagram and dynamically screened interaction. We show that the largest deviations between previously presented EDMFT and $GW$+EDMFT results originate from the nonlocal Fock term, and that the quantitative differences are especially large in the strong-coupling limit. Furthermore, we show that the charge-ordering phase diagram obtained in $GW$+EDMFT methods for moderate interaction values is very similar to the one predicted by dual-boson methods that include the fermion-boson or four-point vertex.

DOI
10.1103/physrevb.95.245130

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In electronic systems with long-range Coulomb interaction, the nonlocal Fock-exchange term has a band-widening effect. While this effect is included in combined many-body perturbation theory and dynamical mean field theory (DMFT) schemes, it is not taken into account in standard extended DMFT (EDMFT) calculations. Here, we include this instantaneous term in both approaches and investigate its effect on the phase diagram and dynamically screened interaction. We show that the largest deviations between previously presented EDMFT and GWGWGW+EDMFT results originate from the nonlocal Fock term, and that the quantitative differences are especially large in the strong-coupling limit. Furthermore, we show that the charge-ordering phase diagram obtained in GWGWGW+EDMFT methods for moderate interaction values is very similar to the one predicted by dual-boson methods that include the fermion-boson or four-point vertex.
We discuss the phase diagram of the extended Hubbard model with both attractive and repulsive local and nonlocal interactions. The extended dynamical mean-field theory (EDMFT) and the dual boson method (DB) are compared. The latter contains additional nonlocal correlation effects that are not incorporated in EDMFT. We find that EDMFT and DB give almost identical results in the attractive V regime, where phase separation occurs. This is quite a difference with the previously studied repulsive V regime, where EDMFT and DB give very different phase boundaries for the checkerboard order phase, especially at small U .
In strongly correlated materials, quantum fluctuations taking place at the level of atomic sites play a central role. These local processes are captured nonperturbatively in dynamical mean field theory (DMFT). Nonlocal mechanisms, however, are also suspected to be essential for the proper description of some of these materials, including the well-known high-temperature cuprate superconductors and the more exotic systems of atoms adsorbed on semiconducting surfaces.In this thesis, the effect of nonlocal, potentially long-range fluctuations beyond the DMFT framework is investigated from two angles.First, long-ranged interactions in adatom systems, computed from first principles within the constrained random-phase approximation (cRPA), are treated within the extended DMFT (EDMFT) framework and its combination with the diagrammatic GW method (GW +EDMFT). They are shown to explain some material trends observed in these compounds. In addition, thanks to the spatial and temporal information delivered by GW+EDMFT, they solve existing puzzles raised by different experimental conclusions drawn from different probes.Second, a new approach to the Hubbard model including nonlocal correlation effects is proposed. Formally based on the local expansion of the triply-irreducible functional, this approach, dubbed TRILEX, approximates the renormalized three-leg electron-boson vertex by a local but frequency-dependent vertex computed from an impurity model with retarded density-density and spin-spin interactions. This local vertex is used to construct frequency- and momentum-dependent self-energies capturing nonlocal correlation effects. By construction, this method reconciles two theoretical pictures of cuprate superconductors, namely the spin fluctuation theory at weak interaction strengths and the doped-Mott-insulator picture at strong coupling.This novel approach is applied to the single-band Hubbard model. It is shown to reduce to DMFT at strong coupling and high temperatures, and to fluctuation-exchange theories at weak coupling, while substantial deviations from both limits are observed at intermediate coupling. There, upon doping, antiferromagnetic spin fluctuations are observed to lead to a momentum-differentiated spectral intensity on the Fermi surface, which is reminiscent of the formation of Fermi arcs in photoemission measurements on cuprate superconductors.These two methodological developments are intrinsically linked to algorithmic aspects which are reviewed in the last part of this work. In particular, an open-source library dedicated to massively parallel computations in the field of quantum many-body physics, TRIQS, is presented, with a focus on error estimation through the binning and jackknife methods. The hybridization-expansion continuous-time quantum Monte-Carlo algorithm in the segment picture -- central to the TRILEX method -- and its state-of-the-art implementation are explained. Finally, as a further illustration, a new lightweight method relying on this algorithm and designed for realistic systems, the combined screened-exchange and dynamical DMFT method (SEx+DDMFT), is presented together with an application to the BaCo2As2 compound.
Extended dynamical mean-field theory (EDMFT) is insufficient to describe nonlocal effects in strongly correlated systems, since corrections to the mean-field solution are generally large. We present an efficient scheme for the construction of diagrammatic extensions of EDMFT that avoids the usual double-counting problem by using an exact change of variables (the Dual Boson formalism) to distinguish the correlations included in the dynamical mean-field solution and those beyond. With a computational efficiency comparable to the EDMFT+GW\mathrm{EDMFT}+\mathrm{GW}EDMFT+GW approach, our scheme significantly improves on the charge order transition phase boundary in the extended Hubbard model.
Extended dynamical mean-field theory (EDMFT) is insufficient to describe non-local effects in strongly correlated systems, since corrections to the mean-field solution are generally large. We present an efficient scheme for the construction of diagrammatic extensions of EDMFT that avoids usual double counting problem by using an exact change of variables (the dual boson formalism) to distinguish the correlations included in the mean-field solution and those beyond. With a computational efficiency comparable to EDMFT+GW approach, our scheme significantly improves on the charge order transition phase boundary in the extended Hubbard model.
We develop the extended dynamical mean-field theory (E-DMFT) with a view towards realistic applications. (1) We introduce an intuitive derivation of the E-DMFT formalism. By identifying the Hartree contributions before the E-DMFT treatment, it allows us to handle systems in symmetry-breaking phases within a simple formalism. (2) We make an implementation of E-DMFT through a real Hubbard--Stratonovich transformation to decouple the nonlocal two-particle interactions. We apply it to a three-dimensional $U\ensuremath{-}V$ model, with U the on-site and V the nearest-neighbor interactions, and investigate the behavior of the various Green's functions, especially the density susceptibility, as the density instability is approached. We obtain the phase diagram at a finite temperature. (3) We present a formalism incorporating E-DMFT with cellular DMFT. (4) We suggest an improvement of the E-DMFT approach by combining it with a generalized GW method. The method combines the local self-energy from E-DMFT and the nonlocal ones from the perturbative calculation of GW.\mathrm{GW}.GW. We apply the method to a one-dimensional $U\ensuremath{-}V$ model with two sublattices carrying different chemical potentials. By comparing with those from density matrix renormalization group calculations, we show that the results are shifted in the correct direction due to the GW\mathrm{GW}GW contributions. (5) In order to handle the generic Coulomb repulsion within E-DMFT, we describe a method to tailor E-DMFT so that the proper momentum dependence can be kept in general response functions.
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.