We present the multi-band dual triply irreducible local expansion (D-TRILEX) approach to interacting electronic systems and discuss its numerical implementation. This method is designed for a self-consistent description of multi-orbital systems that can also have several atoms in the unit cell. The current implementation of the D-TRILEX approach is able to account for the frequency- and channel-dependent long-ranged electronic interactions. We show that our method is accurate when applied to small multi-band systems such as the Hubbard-Kanamori dimer. Calculations for the extended Hubbard, the two-orbital Hubbard-Kanamori, and the bilayer Hubbard models are also discussed.
We present the multi-band dual triply irreducible local expansion (D-TRILEX) approach to interacting electronic systems and discuss its numerical implementation. This method is designed for a self-consistent description of multi-orbital systems that can also have several atoms in the unit cell. The current implementation of the D-TRILEX approach is able to account for the frequency- and channel-dependent long-ranged electronic interactions. We show that our method is accurate when applied to small multi-band systems such as the Hubbard-Kanamori dimer. Calculations for the extended Hubbard, the two-orbital Hubbard-Kanamori, and the bilayer Hubbard models are also discussed.
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 work, we extend the dual triply irreducible local expansion (D-TRILEX) approach for correlated electronic systems by introducing a cluster reference system for the diagrammatic expansion. This framework allows us to consistently combine the exact treatment of short-range correlation effects within the cluster, with an efficient diagrammatic description of the long-range charge and spin collective fluctuations beyond the cluster. We demonstrate the effectiveness of our approach by applying it to the one-dimensional nano-ring Hubbard model, where the low dimensionality enhances non-local correlations. Our results show that the cluster extension of D-TRILEX accurately reproduces the electronic self-energy at momenta corresponding to the Fermi energy, in good agreement with the reliable Hirsch-Fye quantum Monte Carlo solution of the problem. We further compare this method with the more computationally demanding parquet dynamical vertex approximation and find that, our method yields substantially more accurate results at momenta associated with the Fermi surface. We show that the D-TRILEX diagrammatic extension drastically reduces the periodization ambiguity of cluster quantities when mapping back to the original lattice, compared to cluster dynamical mean-field theory (CDMFT). Furthermore, we identify the CDMFT impurity problem as the main source of the translational-symmetry breaking and propose a computational scheme for improving the starting point for the cluster-diagrammatic expansion.
Accurately describing many-body effects in multi-orbital systems remains a major challenge in theoretical condensed matter physics. At present, there is a significant methodological gap between the numerical tools used in ab initio computational materials science and those developed to study strong electronic correlations. The former can treat realistic, large-scale systems but typically neglect many-body effects, while the latter focus on simplified models with only a few degrees of freedom, as only such models can be solved accurately in the presence of strong interactions. The purpose of this thesis is to bridge these two approaches and establish a systematic theoretical framework for realistic correlated electronic materials. This involves a full-cycle methodology that begins with constructing ab initio interacting models from density-functional theory, solving them using dynamical mean-field theory to capture local correlations, and extending beyond to incorporate non-local collective electronic fluctuations. To this end, we introduce the "dual" approach to strong correlations, which includes the dual fermion, dual boson, and dual triply irreducible local expansion methods. The central idea of the dual theories is to shift the reference point of the conventional Feynman diagrammatic expansion from a non-interacting electronic system to an interacting but exactly solvable one. Integrating out this reference system recasts the expansion in an effective dual space, where all diagrammatic building blocks are renormalized by the corresponding impurity quantities. This procedure transforms a non-perturbative expansion in the original variables into a perturbative expansion in terms of dual fermionic and bosonic fields, exact in both the weak- and strong-coupling limits. In this thesis, we collect and systematize the major developments of dual techniques achieved to date.
Extending many-body numerical techniques which are powerful in the context of simple model calculations to the realm of realistic material simulations can be a challenging task. Realistic systems often involve multiple active orbitals, which increases the complexity and numerical cost because of the large local Hilbert space and the large number of interaction terms or sign-changing off-diagonal Green’s functions. The two-particle self-consistent approach (TPSC) is one such many-body numerical technique, for which multi-orbital extensions have proven to be involved due to the substantially more complex structure of the local interaction tensor. In this paper we extend earlier multi-orbital generalizations of TPSC by setting up two different variants of a fully self-consistent theory for TPSC in multi-orbital systems. We first investigate the strengths and limitations of the approach analytically and then benchmark both variants against dynamical mean-field theory (DMFT) and D-TRILEX results. We find that the exact behavior of the system can be faithfully reproduced in the weak-coupling regime, while at stronger couplings the performance of the two TPSC variants strongly depends on details of the system.
We present and implement a self-consistent DΓA approach for multi-orbital models and ab initio materials calculations. It is applied to the one-band Hubbard model at various interaction strengths with and without doping, to the two-band Hubbard model with two largely different bandwidths, and to SrVO3. The self-energy feedback reduces critical temperatures compared to dynamical mean-field theory, even to zero temperature in two-dimensions. Compared to a one-shot, non-self-consistent calculation the non-local correlations are significantly reduced when they are strong. In case non-local correlations are weak to moderate as for SrVO3, one-shot calculations are sufficient.
Abstract One of the most challenging problems in solid state systems is the microscopic analysis of electronic correlations. A paramount minimal model that encodes correlation effects is the Hubbard Hamiltonian, which—regardless of its simplicity—is exactly solvable only in a few limiting cases and approximate many‐body methods are required for its solution. In this review, an overview on the non‐perturbative two‐particle self‐consistent method (TPSC), which was originally introduced to describe the electronic properties of the single‐band Hubbard model, is presented. A detailed derivation of the multi‐orbital generalization of TPSC is introduced here and particular features of the method on exemplary interacting models in comparison to dynamical mean‐field theory results are discussed.