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arXiv:1210.2712 · Physical Review B

Screening and nonlocal correlations in the extended Hubbard model from self-consistent combined G W and dynamical mean field theory

Thomas Ayral, Silke Biermann, Philipp Werner

Published Mar 29, 2013• 168 citations
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Abstract

We describe a recent implementation of the combined $GW$ and dynamical mean field method ($GW$+DMFT) for the two-dimensional Hubbard model with onsite and nearest-neighbor repulsion. We clarify the relation of the $GW$+DMFT scheme to alternative approaches in the literature, and discuss the corresponding approximations to the free-energy functional of the model. Furthermore, we describe a numerically exact technique for the solution of the $GW$+DMFT equations, namely, the hybridization expansion continuous-time algorithm for impurity models with retarded interactions. We compute the low-temperature phase diagram of the half-filled extended Hubbard model, addressing the metal-insulator transition at small intersite interactions and the transition to a charge-ordered state for stronger intersite repulsions. $GW$+DMFT introduces a nontrivial momentum dependence into the many-body self-energy and polarization. We find that the charge fluctuations included in the present approach have a larger impact on the latter than on the former. Finally, within the $GW$+DMFT framework, as in extended DMFT, the intersite repulsion translates into a frequency dependence of the local effective interaction. We analyze this dependence and show how it affects the local spectral function.

DOI
10.1103/physrevb.87.125149

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We describe a recent implementation of the combined GWGWGW and dynamical mean field method (GWGWGW+DMFT) for the two-dimensional Hubbard model with onsite and nearest-neighbor repulsion. We clarify the relation of the GWGWGW+DMFT scheme to alternative approaches in the literature, and discuss the corresponding approximations to the free-energy functional of the model. Furthermore, we describe a numerically exact technique for the solution of the GWGWGW+DMFT equations, namely, the hybridization expansion continuous-time algorithm for impurity models with retarded interactions. We compute the low-temperature phase diagram of the half-filled extended Hubbard model, addressing the metal-insulator transition at small intersite interactions and the transition to a charge-ordered state for stronger intersite repulsions. GWGWGW+DMFT introduces a nontrivial momentum dependence into the many-body self-energy and polarization. We find that the charge fluctuations included in the present approach have a larger impact on the latter than on the former. Finally, within the GWGWGW+DMFT framework, as in extended DMFT, the intersite repulsion translates into a frequency dependence of the local effective interaction. We analyze this dependence and show how it affects the local spectral function.
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.
Using extended dynamical mean-field theory and its combination with the GWGWGW approximation, we compute the phase diagrams and local spectral functions of the single-band extended Hubbard model on the square and simple cubic lattices, considering long-range interactions up to the third nearest neighbors. The longer-range interactions shift the boundaries between the metallic, charge-ordered insulating, and Mott insulating phases, and lead to characteristic changes in the screening modes and local spectral functions. Momentum-dependent self-energy contributions enhance the correlation effects and thus compete with the additional screening effect from longer-range Coulomb interactions. Our results suggest that the influence of longer-range intersite interactions is significant, and that these effects deserve attention in realistic studies of correlated materials.
We study the half-filled extended Hubbard model on a two-dimensional square lattice using cluster dynamical mean-field theory on clusters of size 8--20. We show that the model exhibits metallic, Mott-insulating, and charge-ordered phases, and determine the location of the charge-ordering phase-transition line and the properties of the phases as a function of temperature, local interaction, and nearest-neighbor interaction. We find strong nonlocal correlations outside the charge-ordered phase and a pronounced screening effect in the vicinity of the phase transition, where nonlocal interactions push the system towards metallic behavior. In contrast, correlations in the charge-ordered phase are mostly local to the unit cell. Finally, we demonstrate how strong nonlocal electron-electron interactions can increase electron mobility by turning a charge-ordered insulator into a metal. We analyze finite-size effects and the convergence of our data to the thermodynamic limit. Control of all sources of errors allows us to assess the regime of applicability of simpler approximation schemes for systems with nonlocal interactions.
We generalize the dynamical-mean field (DMFT) approximation by including into the DMFT equations some length scale $\ensuremath{\xi}$ via a momentum dependent external self-energy ${\ensuremath{\Sigma}}_{\mathbf{k}}$. This external self-energy describes nonlocal dynamical correlations induced by the short-ranged collective spin density wave--like antiferromagnetic spin (or the charge density wave--like charge) fluctuations. At high enough temperatures these fluctuations can be viewed as a quenched Gaussian random field with a finite correlation length. This generalized $\mathrm{DMFT}+{\ensuremath{\Sigma}}_{\mathbf{k}}$ approach is used for the numerical solution of the weakly doped one-band Hubbard model with repulsive Coulomb interaction on a square lattice with the nearest and the next nearest neighbor hopping. The effective single impurity problem in this generalized $\mathrm{DMFT}+{\ensuremath{\Sigma}}_{\mathbf{k}}$ is solved by the numerical renormalization group. Both types of the strongly correlated metals, namely: (i) The doped Mott insulator and (ii) the case of the bandwidth $W\ensuremath{\lesssim}U$ (UUU---value of the local Coulomb interaction) are considered. The densities of states, the spectral functions, and the angle resolved photoemission spectra calculated within the $\mathrm{DMFT}+{\ensuremath{\Sigma}}_{\mathbf{k}}$ show a pseudogap formation near the Fermi level of the quasiparticle band.
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 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.