We analyze the impact of the lattice geometry on the thermodynamic transition to magnetically ordered phases in strongly interacting electron systems for various Bravais lattices in three, four, and five dimensions, including both local and nonlocal correlation effects. In a first step we use the dynamical mean-field theory (DMFT), which takes into account purely local correlations, to calculate the magnetic susceptibilities of the Hubbard model on three- (3d-sc), four- (4d-sc), and five- dimensional (5d-sc) simple cubic/hypercubic, as well as on three- dimensional body- (bcc) and face-centered (fcc) cubic lattices, and determine the transition temperature to the corresponding magneticallyordered state. In a second step, we exploit the dynamical vertex approximation ( D Γ A ) , a diagrammatic extension of DMFT, to include the effect of nonlocal correlations which are particularly important in the vicinity of the corresponding phase transition. For the bipartite 3d-sc, 4d-sc, 5d-sc, and bcc lattices nonlocal fluctuations lead to a substantial reduction of the DMFT transition temperature consistent to the overall tendency of mean-field approaches to overestimate the stability of ordered phases. As expected, the magnitude of the difference between the DMFT, being exact in the limit of large connectivity/dimensions, and D Γ A transition temperatures decreases with increasing coordination number. On a more practical perspective, these results also provide a reasonable guidance to evaluate the expected overestimation of the DMFT ordering temperature for different material geometries. For the fcc lattice, on the other hand, the ordered phase observed in DMFT vanishes completely within D Γ A which is consistent with the existence of strong geometric frustration in this lattice.
We analyze the impact of the lattice geometry on the thermodynamic transition to magnetically ordered phases in strongly interacting electron systems for various Bravais lattices in three, four, and five dimensions, including both local and nonlocal correlation effects. In a first step we use the dynamical mean-field theory (DMFT), which takes into account purely local correlations, to calculate the magnetic susceptibilities of the Hubbard model on three- (3d-sc), four- (4d-sc), and five- dimensional (5d-sc) simple cubic/hypercubic, as well as on three- dimensional body- (bcc) and face-centered (fcc) cubic lattices, and determine the transition temperature to the corresponding magneticallyordered state. In a second step, we exploit the dynamical vertex approximation ( D Γ A ) , a diagrammatic extension of DMFT, to include the effect of nonlocal correlations which are particularly important in the vicinity of the corresponding phase transition. For the bipartite 3d-sc, 4d-sc, 5d-sc, and bcc lattices nonlocal fluctuations lead to a substantial reduction of the DMFT transition temperature consistent to the overall tendency of mean-field approaches to overestimate the stability of ordered phases. As expected, the magnitude of the difference between the DMFT, being exact in the limit of large connectivity/dimensions, and D Γ A transition temperatures decreases with increasing coordination number. On a more practical perspective, these results also provide a reasonable guidance to evaluate the expected overestimation of the DMFT ordering temperature for different material geometries. For the fcc lattice, on the other hand, the ordered phase observed in DMFT vanishes completely within D Γ A which is consistent with the existence of strong geometric frustration in this lattice.
We compute the spin susceptibility of the two-dimensional Hubbard model away from half-filling, and analyze the impact of frequency-dependent vertex corrections as obtained from the dynamical mean field theory (DMFT). We find that the local dynamics captured by the DMFT vertex strongly affects nonlocal spin correlations, and thus the momentum dependence of the spin susceptibility. While the widely used random-phase approximation yields commensurate N\'eel-type antiferromagnetism as the dominant instability over a wide doping range, the vertex corrections favor incommensurate ordering wave vectors away from $(\ensuremath{\pi},\ensuremath{\pi})$. Our results indicate that the connection between the magnetic ordering wave vector and the Fermi surface geometry, familiar for weakly interacting systems, can hold in a strongly correlated metal, too.
In this paper, we investigate how nonlocal correlations affect, selectively, the physics of correlated electrons over different energy scales, from the Fermi level to the band edges. This goal is achieved by applying a diagrammatic extension of dynamical mean field theory (DMFT), the dynamical vertex approximation $(\mathrm{D}\mathrm{\ensuremath{\Gamma}}\mathrm{A})$, to study several spectral and thermodynamic properties of the unfrustrated Hubbard model in two and three dimensions. Specifically, we focus first on the low-energy regime by computing the electronic scattering rate and the quasiparticle mass renormalization for decreasing temperatures at a fixed interaction strength. This way, we obtain a precise characterization of the several steps through which the Fermi-liquid physics is progressively destroyed by nonlocal correlations. Our study is then extended to a broader energy range, by analyzing the temperature behavior of the kinetic and potential energy, as well as of the corresponding energy distribution functions. Our findings allow us to identify a smooth but definite evolution of the nature of nonlocal correlations by increasing interaction: They either increase or decrease the kinetic energy w.r.t. DMFT depending on the interaction strength being weak or strong, respectively. This reflects the corresponding evolution of the ground state from a nesting-driven (Slater) to a superexchange-driven (Heisenberg) antiferromagnet (AF), whose fingerprints are, thus, recognizable in the spatial correlations of the paramagnetic phase. Finally, a critical analysis of our numerical results of the potential energy at the largest interaction allows us to identify possible procedures to improve the ladder-based algorithms adopted in the dynamical vertex approximation.
By means of the dynamical vertex approximation (DΓA) we include spatial correlations on all length scales beyond the dynamical mean-field theory (DMFT) for the half-filled Hubbard model in three dimensions. The most relevant changes due to nonlocal fluctuations are (i) a deviation from the mean-field critical behavior with the same critical exponents as for the three dimensional Heisenberg (anti)ferromagnet and (ii) a sizable reduction of the Néel temperature (T(N)) by ~30% for the onset of antiferromagnetic order. Finally, we give a quantitative estimate of the deviation of the spectra between DΓA and DMFT in different regions of the phase diagram.
Almost 50 years after the invention of the Hubbard model [1–3] and despite modern petaflop supercomputers, a precise analysis of the criticality of this most basic model for electronic correlations has not been achieved so far, at least not in three dimensions. Dynamical mean field theory (DMFT) [4–6] was a big step forward to quantitatively calculate the three dimensional Hubbard model since the major contribution of electronic correlations, i.e., the local ones, are well captured within this theory. Local correlations give rise to quasiparticle renormalization, the Mott-Hubbard transition, magnetism, and even more subtle issues such as kinks in a purely electronic models [7]. However, non-local spatial correlations are also naturally generated by a purely local Hubbard interaction, and, as it is well known, they become of essential importance in the vicinity of secondorder phase transitions. As these correlations are neglected in DMFT, this scheme can provide only for a conventional mean-field (MF) description of the critical properties. To overcome this shortcoming cluster extensions to DMFT such as the dynamical cluster approximation (DCA) and cluster-DMFT have been proposed, see Ref. 8 for a review. In these approaches non-local correlations beyond DMFT are taken into account, however only within the range of the cluster size; and due to computational limitations the actual size of the cluster is severely restricted to about 3-5 sites for each direction. Hence, short-range correlations are included by these approaches, whereas long-range ones are not. Nonetheless, Kent et al. [9] were able to extrapolate the cluster size of so-called Betts clusters to infinity, albeit assuming from the very beginning the critical exponents to be those of the Heisenberg model. This way they extrapolated the critical temperature of the paramagnetic-toantiferromagnetic phase transition which was found to be in agreement with earlier quantum Monte Carlo (QMC) simulations of Staudt et al. [10] on finite lattices.
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.
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.