In this last chapter, very recent promising developments beyond GW+EDMFT by taking into account nonlocal vertex corrections beyond the RPA are described. The first development is the dual fermion and boson approach in which DMFT is used as a starting point for a perturbation expansion of the nonlocal self-energy. This approach is very general and in its lowest similar to the GW+EDMFT scheme. The second development is the TRILEX approach based on the extension of the GW+EDMFT scheme to include higher-order correlation or vertex functions in the self-consistency condition. Like the GW+EDMFT scheme, this approach maintains the basic concept of DMFT in that local projected quantities are made self-consistent. In this way, nonlocal spin fluctuations missing in the GW+EDMFT scheme are incorporated. The third development is the dynamical vertex approximation method, which can be considered as a generalization of DMFT assuming the locality of the two-particle irreducible vertex. This chapter closes with a summary.
In this last chapter, very recent promising developments beyond GW+EDMFT by taking into account nonlocal vertex corrections beyond the RPA are described. The first development is the dual fermion and boson approach in which DMFT is used as a starting point for a perturbation expansion of the nonlocal self-energy. This approach is very general and in its lowest similar to the GW+EDMFT scheme. The second development is the TRILEX approach based on the extension of the GW+EDMFT scheme to include higher-order correlation or vertex functions in the self-consistency condition. Like the GW+EDMFT scheme, this approach maintains the basic concept of DMFT in that local projected quantities are made self-consistent. In this way, nonlocal spin fluctuations missing in the GW+EDMFT scheme are incorporated. The third development is the dynamical vertex approximation method, which can be considered as a generalization of DMFT assuming the locality of the two-particle irreducible vertex. This chapter closes with a summary.
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.
We explore the combination of the extended dynamical mean field theory (EDMFT) with the GW approximation (GWA); the former sums the local contributions to the self-energies to infinite order in closed form and the latter handles the nonlocal ones to lowest order. We investigate the different levels of self-consistency that can be implemented within this method by comparing to the exact quantum Monte Carlo solution of a finite-size model Hamiltonian. We find that using the EDMFT solution for the local self-energies as input to the GWA for the nonlocal self-energies gives the best result.
Abstract The question of spatial locality of electronic correlations beyond GW approximation is one of the central issues of the famous combination of GW and dynamical mean field theory, GW + DMFT. In this work, the above question is addressed directly (for the first time) by performing calculations with and without assumption of locality of the corresponding diagrams. For this purpose we use sc(GW + G3W2) approach where the higher order part (G3W2) is evaluated with fully momentum dependent Green’s function G and screened interaction W and with ‘local’ variant, where the single site approximation is assumed for both G and W. For all three materials studied in this work (NiO, α -Ce, LiFeAs), we have found the spatial non-locality effects to be strong. For NiO and LiFeAs they, in fact, are decisive for the proper evaluation of vertex corrections. The results of this study have direct impact on our understanding of approximations made in practical implementations of GW + DMFT method, where all diagrams beyond GW (DMFT part) are assumed to be local. Taking into account the fact that the first diagrams beyond GW represent the most important contribution also in GW + DMFT calculations, we conclude that the basic assumption of GW + DMFT, namely the locality of diagrams evaluated in the DMFT part, is not as good as it is believed to be.
In systems of indistinguishable particles, correlations between the quantum degrees of freedom can give rise to collective modes where particles behave coherently together over some correlation length and time, thereby forming quasiparticles. One example of such quasiparticle would be the π-ton [66], describing the binding of two particle-hole pairs via the exchange of momentum corresponding to the antiferromagnetic (AFM) wave vector. These quasiparticles would emerge in the vicinity of an AFM ordering instability [66, 119, 120] and nonlocal correlations are crucial to describe them. Especially, in systems out of equilibrium, nonlocality can give rise to prethermal states wherein the distribution function can be momentum-dependent [92, 119], and nonthermal exotic physics can be observed and manipulated [40, 68, 79, 84, 93, 132, 147, 157–159]. To be able to model and simulate systems out of equilibrium is a challenge. A few numerical methods such as nonequilibrium Dynamical Mean Field Theory (DMFT) [44] have been used extensively [5]. Nonequilibrium DMFT is designed to treat nonperturbatively the local time-dependent single-particle correlations. Since phenomena such as quantum criticality, high-Tc superconductivity or the creation of π-tons via light-matter coupling stem from nonlocal two-particle correlations, one needs to resort to diagrammatic extensions to DMFT [103] and other methods to effectively describe them, such as the post-processing DMFT treatment of vertex corrections, which will be introduced. The Two-Particle Self-Consistent approach (TPSC) and its variants figure among those methods that include nonlocal two-particle and single-particle correlations in a self-consistent way [121, 122]. TPSC and its variants have the ability to capture the effect of the growing spin fluctuations (paramagnons) in the self-energy when close to amagnetic phase transition leading to the breakdown of the Fermi-liquid quasiparticles at the Fermi level: this would mark the onset of a pseudo-gap [145, 152]. It fulfils the Mermin-Wagner theorem in two dimensions, the Pauli principle, and various two-particle local sum-rules [152]. In the works embodied in this thesis, numerical methods have been designed to access nonequilibrium regimes, by changing model parameters across time, allowing for nonlocal correlations to be properly included. With these new algorithms at hand, the π-tons were studied in the optical conductivity and magnetic response [119, 120] as well as spin and charge fluctuations in the 2D and 3D single-band Hubbard model [121, 122]. It is demonstrated by means of those methods that nonlocal two-particle vertex corrections can underpin prethermal states and could generate transient nonthermal states such as low-energy charge excitations associated with nonthermal negative temperatures. It is also shown that states lying close to the Fermi level control the thermalization timescale and that lattice hopping quenches changing the unit-cell dimensionality could allow one to play around with the charge and spin scattering channels.
The main focus of the present thesis lies on the development and application of numerical approaches based on quantum lattice models to describe collective excitations in strongly correlated materials. To this end, an efficient and balanced dual triply irreducible local expansion (D-TRILEX) approach that captures the major effects of spatial correlations in strongly correlated fermionic systems is presented, which is based on partial bosonization of the collective electronic fluctuations. Thereby, a specific choice of the bare interaction in the corresponding channel of instability allows a simultaneous description of collective excitations in different channels without facing the Fierz ambiguity problem which usually arises in methods based on partial bosonization. Compared to more elaborate methods, D-TRILEX does not require the computation of the numerically expensive four-point vertex. Furthermore, taking into account only the leading longitudinal contribution of the bosonic modes to the self-energy reduces the complexity of the diagrammatic structure drastically making the method applicable to more complex realistic multi-orbital systems. As the main result of this thesis, the D-TRILEX approach is introduced and its limit of applicability is studied by applying it to the two-dimensional single-band Hubbard model. Performing a comprehensive diagrammatic analysis by comparing D-TRILEX with methods, which consider different sets of diagrams and different approximations for the four-point vertex, we find that contributions that are not considered by the partially bosonized approximation of the four-point vertex have only a minor effect on electronic degrees of freedom and the longitudinal bosonic modes taking into account in D-TRILEX self-energy are the leading ones in a broad range of control parameters. After investigating the applicability of the D-TRILEX approach we apply the method to study collective electronic effects in the hole-doped InSe monolayers using a realistic electronic-structure model. Thereby, we find that due to the weakly screened long-range Coulomb interaction the system shows a charge density waves phase for the broad range of considered parameters, however, inside this regime we have found a coexisting ferromagnetic phase. D-TRILEX can be considered as an approximation of the more elaborate dual boson (DB) theory, which accounts for local correlations within the dynamical mean-field theory (DMFT) and nonlocal correlations perturbatively. As a further result of this thesis, we investigate the weak coupling regime of the two-dimensional single-band Hubbard model within the ladder DB approximation and compare our results to the numerically exact diagrammatic quantum Monte Carlo (DiagMC) method. We find that the DB method qualitatively captures all the different regimes. Even quantitatively DB shows a good agreement with the exact results for single- and two-particle quantities at high and moderate temperatures. However, the ladder approximation slightly underestimates the strong antiferromagnetic fluctuations that appear in the system at low temperatures. As the final result of this thesis, we perform a parametrization of the local four-point vertex in terms of the (three-point) Hedin vertex and screened interactions in the framework of single-boson exchange (SBE) decomposition. Similar to the D-TRILEX approach the computational costs are reduced by neglecting the interaction-irreducible part of the four-point vertex. The approximated vertex is used to calculate the DMFT susceptibility for different interaction strengths at different temperatures. We find that the approximation shows a good agreement in the limits of weak and strong couplings. Thereby, our study highlights the importance of the Hedin vertex for local vertex corrections.
In this chapter, shortcomings of the GW approximation are analyzed, and different ways are proposed to go beyond. The approaches fall into three broad classes: first, one can iterate Hedin's equations that link the self-energy, the polarizability, the screened Coulomb interaction, and the vertex function with the Dyson equation for the one-body Green's function. Second, one can use other Dyson equations like in the T-matrix approach, and third, one can make an ansatz for the one-body Green's function inspired by physical considerations or insight from models, in particular the idea of electron–boson coupling that leads to the cumulant expansion. Applications are given for illustration. Some more ideas and links to subsequent chapters provide an outlook. The GW approximation is a widely used approach in computational electronic structure. However, the previous chapters show some severe limitations. One example is systems with low-energy excitations, as illustrated by the Hubbard dimer in the dissociation limit, where bonding and antibonding orbitals become degenerate. This can be seen in Sec. 11.7. Another example is satellite series, for example in the photoemission spectrum of silicon in Sec. 13.1. Moreover, even in a situation where the approximation is well justified, it induces some error, though maybe small, and one eventually wants to do better. The full self-energy functional is in principle known: it is the Luttinger–Ward sum of skeleton diagrams derived from Eq. (11.35). However, this is an infinite sum, and no closed form of the result is known. Hence, the task is not to invent new diagrams: it is to make a choice of diagrams or combinations of diagrams, which are evaluated fully or in an approximate way. This is the framework of the present chapter. In some cases we will add diagrams to the GWA ones, which gives rise to vertex corrections ; in other cases some of the GWA diagrams are neglected while different ones are taken into account. An example is the T-matrix approximation, which sums ladder diagrams instead of bubbles. In general one tries to combine this with the GWA, especially in extended systems where screening is important. Therefore the GWA plays a prominent role, and the whole chapter is called beyond GW . The chapter contains advanced material, that is in part exploratory and refers to current research. It is intended to make links to communities outside the GW world, and to other chapters.