In this work, we present a nonlocal expansion scheme to study correlated electron systems aiming at a better description of its spatial fluctuations at all length scales. Taking the nonlocal coupling as a perturbation to the local degrees of freedom, we show that the nonlocality in the self-energy function can be efficiently constructed from the coupling between local fluctuations. It can provide one unified framework to incorporate nonlocality to both ordered and disordered correlated many-body fermion systems. In this application, we prove that the dual-fermion approach can be understood as a special case of this nonlocal expansion scheme. The scheme presented in this work is constructed without introducing any dual variable, in which the interacting nature and the correlated behaviors of the lattice fermions have a clear physics correspondence. Thus, in this special case, the equivalence of the dual-fermion approach to the nonlocal expansion scheme beautifully reveals the physics origin of the dual variables. We show that the noninteracting dual-fermion Green's function corresponds exactly to a nonlocal coupling of the lattice fermion renormalized by the local single-particle charge fluctuations, and the dual-fermion self-energy behaves as the one-particle fully irreducible components of the lattice Green's function. Not only limited to this specific example, the nonlocal expansion scheme presented in this work can also be applied to other problems depending on the choice of the local degrees of freedom.
In this work, we present a nonlocal expansion scheme to study correlated electron systems aiming at a better description of its spatial fluctuations at all length scales. Taking the nonlocal coupling as a perturbation to the local degrees of freedom, we show that the nonlocality in the self-energy function can be efficiently constructed from the coupling between local fluctuations. It can provide one unified framework to incorporate nonlocality to both ordered and disordered correlated many-body fermion systems. In this application, we prove that the dual-fermion approach can be understood as a special case of this nonlocal expansion scheme. The scheme presented in this work is constructed without introducing any dual variable, in which the interacting nature and the correlated behaviors of the lattice fermions have a clear physics correspondence. Thus, in this special case, the equivalence of the dual-fermion approach to the nonlocal expansion scheme beautifully reveals the physics origin of the dual variables. We show that the noninteracting dual-fermion Green's function corresponds exactly to a nonlocal coupling of the lattice fermion renormalized by the local single-particle charge fluctuations, and the dual-fermion self-energy behaves as the one-particle fully irreducible components of the lattice Green's function. Not only limited to this specific example, the nonlocal expansion scheme presented in this work can also be applied to other problems depending on the choice of the local degrees of freedom.
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.
Disorder is ubiquitous in real materials and influences the physical properties like the conductivity to varying degrees. \nIf electron-electron interactions are strong, theoretical and numerical treatment of these systems becomes challenging. In this thesis a numerical approach is developed to address these systems, \ntreating both interactions and disorder on equal footing. \nThe approach is based on the dual fermion approach for interacting systems developed by Rubtsov et al. \nTerletska et al. applied the ideas of the dual fermion approach to disordered non-interacting systems. In this approach, the replica trick is used to integrate out the disorder in favor of an effective electron-electron interaction. \nWe extended the approach from Terletska et al. to treat disordered interacting systems. \n \nDual Fermions allow to take into account non-local fluctuations by means of a perturbative expansion around an impurity problem. The impurity reference system is determined self-consistently, analogously to the dynamical mean-field \ntheory. \nThe perturbative expansion is expected to yield good results for small and large values of interaction strength and disorder. A priori, it is not clear what to expect for intermediate values, \nbut experience shows that oftentimes good results are obtained for this region. \n \nAn advantage of the dual fermion approach is that there is no sign-problem for a single orbital model if quantum Monte Carlo is used to solve the interacting reference system. \nAdditionally, perturbation theory is usually numerically much cheaper than fully solving an interacting lattice or cluster problem. \nThus, the dual fermion approach allows to address regions of parameter space that are not accessible to lattice quantum Monte Carlo calculations or cluster extension of dynamical mean-field theory. \nCluster extensions of the dynamical mean-field theory are for example the dynamical cluster approximation or the cellular dynamical mean-field theory. \n \nThe new approach is benchmarked with respect to the Anderson-Falicov-Kimball model. The Anderson-Hubbard model is the main application. \nFor both models, the algorithm is tested in one dimension for the single particle Green function and compared to the dynamical cluster approximation to estimate the quality of the results. \nThe Anderson-Falicov-Kimball model is studied further in two dimensions and the phase diagram on the disorder-interaction plane is calculated from both the single particle Green function and the dc conductivity. \n \nFor the Anderson-Hubbard model, the antiferromagnetic phase transition, the Mott transition and the Anderson transition are investigated in three dimensions. The dynamical mean-field theory results are \nqualitatively confirmed for the most part. Quantitative corrections are in agreement with dynamical cluster approximation results, thus confirming the validity of the results. \nWe found a qualitative difference between the dual fermion approach and dynamical mean-field theory for the temperature dependence of the hysteresis of the double occupancy. The dual fermion result \nshows a decreasing lower critical interaction strength for decreasing temperature, which is in contrast with the dynamical mean-field result that shows an increase of the lower critical interaction strength. \nThis behavior happens for small temperatures that are not accessible using for example the dynamical cluster approximation, thus highlighting the usefulness of the dual fermion approach. \nWe found this behavior for all values of the disorder that we considered.
Disorder is ubiquitous in real materials and influences the physical properties like the conductivity to varying degrees. If electron-electron interactions are strong, theoretical and numerical treatment of these systems becomes challenging. In this thesis a numerical approach is developed to address these systems, treating both interactions and disorder on equal footing. The approach is based on the dual fermion approach for interacting systems developed by Rubtsov et al. Terletska et al. applied the ideas of the dual fermion approach to disordered non-interacting systems. In this approach, the replica trick is used to integrate out the disorder in favor of an effective electron-electron interaction. We extended the approach from Terletska et al. to treat disordered interacting systems. Dual Fermions allow to take into account non-local fluctuations by means of a perturbative expansion around an impurity problem. The impurity reference system is determined self-consistently, analogously to the dynamical mean-field theory. The perturbative expansion is expected to yield good results for small and large values of interaction strength and disorder. A priori, it is not clear what to expect for intermediate values, but experience shows that oftentimes good results are obtained for this region. An advantage of the dual fermion approach is that there is no sign-problem for a single orbital model if quantum Monte Carlo is used to solve the interacting reference system. Additionally, perturbation theory is usually numerically much cheaper than fully solving an interacting lattice or cluster problem. Thus, the dual fermion approach allows to address regions of parameter space that are not accessible to lattice quantum Monte Carlo calculations or cluster extension of dynamical mean-field theory. Cluster extensions of the dynamical mean-field theory are for example the dynamical cluster approximation or the cellular dynamical mean-field theory. The new approach is benchmarked with respect to the Anderson-Falicov-Kimball model. The Anderson-Hubbard model is the main application. For both models, the algorithm is tested in one dimension for the single particle Green function and compared to the dynamical cluster approximation to estimate the quality of the results. The Anderson-Falicov-Kimball model is studied further in two dimensions and the phase diagram on the disorder-interaction plane is calculated from both the single particle Green function and the dc conductivity. For the Anderson-Hubbard model, the antiferromagnetic phase transition, the Mott transition and the Anderson transition are investigated in three dimensions. The dynamical mean-field theory results are qualitatively confirmed for the most part. Quantitative corrections are in agreement with dynamical cluster approximation results, thus confirming the validity of the results. We found a qualitative difference between the dual fermion approach and dynamical mean-field theory for the temperature dependence of the hysteresis of the double occupancy. The dual fermion result shows a decreasing lower critical interaction strength for decreasing temperature, which is in contrast with the dynamical mean-field result that shows an increase of the lower critical interaction strength. This behavior happens for small temperatures that are not accessible using for example the dynamical cluster approximation, thus highlighting the usefulness of the dual fermion approach. We found this behavior for all values of the disorder that we considered.
We formulate a general cluster Dual Fermion Approach to nonlocal correlations in crystals. The scheme allows the treatment of long-range correlations beyond cluster DMFT and nonlocal effects in realistic calculations of multiorbital systems. We show that the the simplest approximation exactly corresponds to free cluster DMFT. We further consider the relation between the two-particle Green functions in real and dual variables. We apply this approach by calculating the Green function of the Hubbard model in one dimension starting from the two-site cluster DMFT solution. The result agrees well with the Green function obtained from a DMRG calculation.
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.
The contributions from the non-local fluctuations to the dynamical mean field theory (DMFT) were studied using the recently proposed dual fermion approach. Straight forward cluster extensions of DMFT need the solution of a small cluster, where all the short-range correlations are fully taken into account. All the correlations beyond the cluster scope are treated in the mean-field level. In the dual fermion method, only a single impurity problem needs to be solved. Both the short and long-range correlations could be considered on equal footing in this method. The weak-coupling nature of the dual fermion ensures the validity of the finite order diagram expansion. The one and two particle Green's functions calculated from the dual fermion approach agree well with the Quantum Monte Carlo solutions, and the computation time is considerably less than with the latter method. The access of the long-range order allows us to investigate the collective behavior of the electron system, e.g. spin wave excitations.