We introduce a diagrammatic multi-scale approach to the Hubbard model based on the interaction-irreducible (multi-boson) vertex of a small cluster embedded in a self-consistent medium. The vertex captures short-range correlations up to the length scale of the cluster, while long-range correlations are recovered from a set of diagrammatic equations for the Hedin three-leg vertex. By virtue of the crossing symmetry, the Fierz decoupling ambiguity of the Hubbard interaction is resolved exactly. Our benchmarks for the half-filled Hubbard model on the square lattice are in very good agreement with numerically exact diagrammatic Monte Carlo simulations. Published by the American Physical Society 2024
We introduce a diagrammatic multi-scale approach to the Hubbard model based on the interaction-irreducible (multi-boson) vertex of a small cluster embedded in a self-consistent medium. The vertex captures short-range correlations up to the length scale of the cluster, while long-range correlations are recovered from a set of diagrammatic equations for the Hedin three-leg vertex. By virtue of the crossing symmetry, the Fierz decoupling ambiguity of the Hubbard interaction is resolved exactly. Our benchmarks for the half-filled Hubbard model on the square lattice are in very good agreement with numerically exact diagrammatic Monte Carlo simulations. Published by the American Physical Society 2024
A numerically implementable multi-scale many-body approach to strongly correlated electron systems is introduced. An extension to quantum cluster methods, it approximates correlations on any given length-scale commensurate with the strength of the correlations on the respective scale. Short length-scales are treated explicitly, long ones are addressed at a dynamical mean-field level and intermediate length-regime correlations are assumed to be weak and are approximated diagrammatically. To illustrate and test this method, we apply it to the one-dimensional Hubbard model. The resulting multi-scale self-energy provides a very good quantitative agreement with substantially more numerically expensive, explicit quantum Monte Carlo calculations.
Starting with Hedin's equations, simple expressions for the irreducible self-energy are derived. The derivation with vertex effects included in the self-energy results in a number of terms beyond G W such as second-order screened exchange (the term that also gives rise to vertex/excitonic effects in the polarization) and an infinite series describing correlations between the added-particle (removed-hole) and the excited electrons and holes: the screened T -matrix channels. Two-body correlated propagation is considered, with the third propagating freely, however, three-body correlations are discussed and these can be added hierarchically to the method. For electron-hole propagation, the reducible polarization is calculated, which results in an expression for the self-energy that can be solved analytically without the need for widely used expensive numerical methods for frequency integration or having to adopt the plasmon-pole approximation. The method requires diagonalization of the usual particle-hole matrix—derived from the Bethe-Salpter equation—and also a particle-particle + hole -hole matrix that has a similar structure; with the second-order exchange present in all channels. The method goes beyond current widely used methods that adopt the G W approximation, whilst being in nature and requiring relatively modest extensions to existing functionality, e.g., it could be implemented in the sophisticated quasiparticle self-consistent framework. It is anticipated the method will significantly improve on the accuracy of current state-of-the-art calculations, improving our understanding of processes in simple and strongly correlated systems, with the additional contributions demanding a similar cost to methods that include vertex effects in the polarization only. Published by the American Physical Society 2024
We investigate analytically the performance of many-body energy functionals, derived by Klein, Luttinger, and Ward, at different levels of diagrammatic approximations, ranging from second Born, to , to the so-called T matrix, for the calculation of total energies and potential energy surfaces. We benchmark our theoretical results on the extended two-site Hubbard model, which is analytically solvable and for which several exact properties can be calculated. Despite its simplicity, this model displays the physics of strongly correlated electrons: it is prototypical of the H 2 dissociation, a notoriously difficult problem to solve accurately for the majority of mean-field-based approaches. We show that both functionals exhibit good to excellent variational properties, particularly in the case of the Luttinger-Ward one, which is in close agreement with fully self-consistent calculations, and we elucidate the relation between the accuracy of the results and the different input one-body Green's functions. Provided that these are wisely chosen, we show how the Luttinger-Ward functional can be used as a computationally inexpensive alternative to fully self-consistent many-body calculations, without sacrificing the precision of the results obtained. Furthermore, in virtue of this accuracy, we argue that this functional can also be used to rank different many-body approximations at different regimes of electronic correlation, once again bypassing the need for self-consistency. Published by the American Physical Society 2024
In the study of correlated systems, approximations based on the dynamical mean-field theory (DMFT) provide a practical way to take local vertex corrections into account, which capture, respectively, particle-particle screening at weak coupling and the formation of the local moment at strong coupling. We show that in both limits the local vertex corrections can be efficiently parameterized in terms of single-boson exchange, such that the two-particle physics described by DMFT and its diagrammatic extensions is recovered to good approximation and at a reduced computational cost. Our investigation highlights the importance of the frequency-dependent fermion-boson coupling (Hedin vertex) for local vertex corrections. Namely, at weak coupling the fermion-spin-boson coupling suppresses the N\'eel temperature of the DMFT approximation compared to the static mean-field, whereas for large interaction it facilitates a huge enhancement of local spin-fluctuation exchange, giving rise to the effective-exchange energy scale 4t2/U. We find that parameterizations of the vertex which neglect the nontrivial part of the fermion-boson coupling fail qualitatively at strong coupling.
We present an embedded-cluster method, based on the triply irreducible local expansion formalism. It turns the Fierz ambiguity, inherent to approaches based on a bosonic decoupling of local fermionic interactions, into a convergence criterion. It is based on the approximation of the three-leg vertex by a coarse-grained vertex computed from a self-consistently determined cluster impurity model. The computed self-energies are, by construction, continuous functions of momentum. We show that, in three interaction and doping regimes of the two-dimensional Hubbard model, self-energies obtained with clusters of size four only are very close to numerically exact benchmark results. We show that the Fierz parameter, which parametrizes the freedom in the Hubbard-Stratonovich decoupling, can be used as a quality control parameter. By contrast, the GW+extended dynamical mean field theory approximation with four cluster sites is shown to yield good results only in the weak-coupling regime and for a particular decoupling. Finally, we show that the vertex has spatially nonlocal components only at low Matsubara frequencies.
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.