Vertex functions are a crucial ingredient of several forefront many-body algorithms in condensed matter physics. However, the full treatment of their frequency and momentum dependence severely restricts numerical calculations. A significant advancement requires an efficient treatment of the high-frequency asymptotic behavior of the vertex functions. In this work, we first provide a detailed diagrammatic analysis of the high-frequency structures and their physical interpretation. Based on these insights, we propose a parametrization scheme, which captures the whole high-frequency domain for arbitrary values of the Coulomb interaction and electronic density, and we discuss the details of its algorithmic implementation in many-body solvers based on parquet equations as well as functional renormalization group schemes. Finally, we assess its validity by comparing our results for a single impurity Anderson model with exact diagonalization calculations. The proposed parametrization is pivotal for the algorithmic development of all quantum many-body methods based on vertex functions arising from both local and nonlocal static microscopic interactions as well as effective dynamic interactions which uniformly approach a static value for large frequencies. In this way, our present technique can substantially improve vertex-based diagrammatic approaches including spatial correlations beyond dynamical mean-field theory.
Vertex functions are a crucial ingredient of several forefront many-body algorithms in condensed matter physics. However, the full treatment of their frequency and momentum dependence severely restricts numerical calculations. A significant advancement requires an efficient treatment of the high-frequency asymptotic behavior of the vertex functions. In this work, we first provide a detailed diagrammatic analysis of the high-frequency structures and their physical interpretation. Based on these insights, we propose a parametrization scheme, which captures the whole high-frequency domain for arbitrary values of the Coulomb interaction and electronic density, and we discuss the details of its algorithmic implementation in many-body solvers based on parquet equations as well as functional renormalization group schemes. Finally, we assess its validity by comparing our results for a single impurity Anderson model with exact diagonalization calculations. The proposed parametrization is pivotal for the algorithmic development of all quantum many-body methods based on vertex functions arising from both local and nonlocal static microscopic interactions as well as effective dynamic interactions which uniformly approach a static value for large frequencies. In this way, our present technique can substantially improve vertex-based diagrammatic approaches including spatial correlations beyond dynamical mean-field theory.
The multipoint numerical renormalization group (mpNRG) is a powerful impurity solver that provides accurate spectral data useful for computing local, dynamic correlation functions in imaginary or real frequencies nonperturbatively across a wide range of interactions and temperatures. It gives access to a local, nonperturbative four-point vertex in imaginary and real frequencies, which can be used as input for subsequent computations such as diagrammatic extensions of dynamical mean-field theory. However, computing and manipulating the real-frequency four-point vertex on large, dense grids quickly becomes numerically challenging when the density and/or the extent of the frequency grid is increased. In this paper, we compute four-point vertices in a strongly compressed quantics tensor train format using quantics tensor cross interpolation, starting from discrete partial spectral functions obtained from mpNRG. This enables evaluations of the vertex on frequency grids with resolutions far beyond the reach of previous implementations. We benchmark this approach on the four-point vertex of the single-impurity Anderson model across a wide range of physical parameters, both in its full form and in its asymptotic decomposition. For imaginary frequencies, the full vertex can be represented to an accuracy on the order of 2 × 10 − 3 with maximum bond dimensions not exceeding 120. The more complex full real-frequency vertex requires maximum bond dimensions not exceeding 170 for an accuracy of ≲ 2 % . Our work marks another step toward tensor-train-based diagrammatic calculations for correlated electronic lattice models starting from a local, nonperturbative mpNRG vertex.
This thesis aims at developing new schemes for the treatment of correlation effects in condensed matter systems using quantum field theoretical approaches. In particular, our goal is to extend the description of correlation physics at the two-particle level. This is necessary for an unbiased treatment of condensed matter systems that exhibit electronic correlations and competing ordering tendencies. In this respect, the functional renormalization group (fRG) approaches have surely contributed substantially over the last years, as they account for all scattering channels and their mutual feedback effects in an unbiased way. In spite of its flexibility, the application of the fRG is limited by its inherent perturbative nature. To go beyond the conventional weak-coupling implementations, we discuss the general idea to extend fRG based computational schemes by using an exactly solvable interacting reference problem as starting point for the RG flow. The systematic expansion around this solution accounts for a non-perturbative inclusion of correlations at both, the one-particle (self-energy) and two-particle (vertex functions) level. The full treatment of the two-particle vertex functions, however, poses a huge limitation to the numerical performance, not only in the fRG, but in several forefront many-body algorithms. In this perspective, we provide a detailed diagrammatic analysis of the frequency and momentum structures of the vertex functions, together with their physical interpretation. This constitutes the basis for sophisticated parametrization schemes. We then explain the technical details necessary for cutting-edge numerical implementations, and further benchmark our ideas using refined implementations of both, the fRG and the parquet approximation (PA).
Estimating the local two-particle vertex functions, which are crucial for capturing the spatial fluctuation of the effective field beyond the single-site dynamical mean field theory (DMFT), is still challenging. In our previous work [R. Mizuno et al., J. Phys. Soc. Jpn. 91, 034002 (2022)], we developed a computationally efficient method for estimating the local full vertex in DMFT, where we can obtain the local two-particle full vertex from the one-particle self-energy. In this study, we further enhance our method by refining its formulation to be more faithful to the diagrammatic structure of the full vertex. With this improvement, we can qualitatively reproduce the characteristic frequency structures of the full vertex obtained by the numerically exact methods. In particular, the improved version of the simplified full vertex captures a sharp value change in the cross structure.
We derive the equations for calculating the high-frequency asymptotics of the local two-particle vertex function for a multiorbital impurity model. These relate the asymptotics for a general local interaction to equal-time two-particle Green's functions, which we sample using continuous-time quantum Monte Carlo simulations with a worm algorithm. As specific examples we study the single-orbital Hubbard model and the three t2g orbitals of SrVO3 within dynamical mean-field theory (DMFT). We demonstrate how the knowledge of the high-frequency asymptotics reduces the statistical uncertainties of the vertex and further eliminates finite-box-size effects. The proposed method benefits the calculation of nonlocal susceptibilities in DMFT and diagrammatic extensions of DMFT.
We present here two alternative schemes designed to correct the high-frequency truncation errors in the numerical treatment of the Bethe-Salpeter equations. The schemes are applicable to all Bethe-Salpeter calculations with a local two-particle irreducible vertex, which is relevant, e.g., for the dynamical mean-field theory (DMFT) and its diagrammatic extensions. In particular, within a purely diagrammatic framework, we could extend existing algorithms for treating the static case in the particle-hole sector to more general procedures applicable to all bosonic frequencies and all channels. After illustrating the derivation and the theoretical interrelation of the two proposed schemes, these have been applied to the Bethe-Salpeter equations for the auxiliary Anderson impurity models of selected DMFT calculations, where results can be compared against a numerically ``exact'' solution. The successful performance of the proposed schemes suggests that their implementation can significantly improve the accuracy of DMFT calculations at the two-particle level, in particular for more realistic multiorbital calculations where the large number of degrees of freedom substantially restricts the actual frequency range for numerical calculations, as well as---on a broader perspective---of the diagrammatic extensions of DMFT.
Strongly correlated electron systems host a plethora of fascinating physical phenomena and pose formidable challenges in their theoretical analysis. The challenges originate from the inherent complexity of the quantum many-body problem - no classical computer will ever be able to fully simulate these systems - and the lack of an effective single-particle picture, as the strong mutual interactions make it impossible to regard the electrons as independent from each other. As a consequence, most systems of correlated electrons can only be tackled approximately and numerically. In this thesis, we develop a set of numerical methods for strongly correlated electrons, which are inspired by the renormalization group (RG) idea of including degrees of freedom successively from high to low energies. This enables an efficient organization of the diverse fluctuations and is key for an accurate treatment of interacting quantum systems, where collective behavior and composite objects emerge at energy scales far below those of the microscopic constituents. In a first part, we consider the functional renormalization group (fRG), a versatile framework to study the flow of correlation functions upon modulating the underlying action. Though widely used, it has often acted more as a qualitative rather than quantitative method, due a nontransparent approximation induced by truncating the hierarchy of flow equations. We develop an iterative multiloop fRG (mfRG) scheme, which ameliorates this approximation and eliminates many of the drawbacks of fRG experienced hitherto. In particular, it restores the independence of results on the choice of RG regulator and establishes a rigorous relation to the parquet formalism. Furthermore, we show how to derive the flow equations directly from self-consistent many-body relations. This establishes a form of diagrammatic resummations at the two-particle level which circumvents ill-behaved two-particle-irreducible vertices. An application to the prototypical two-dimensional Hubbard model illustrates how our multiloop scheme elevates the fRG approach to correlated electron systems to a quantitative level. Secondly, we employ the numerical renormalization group (NRG), based on the iterative diagonalization of impurity Hamiltonians, in conjunction with the dynamical mean-field theory (DMFT) to describe local correlations in multiorbital systems. Having access to arbitrarily low temperatures and energies, NRG is a unique, real-frequency impurity solver for DMFT. It has been pivotal to the understanding of Hund metals, where strong correlations arise from Hund's rules even at moderate Coulomb repulsion. Building on recent methodological advances, we extend the range of application of DMFT+NRG from orbital-degenerate models to more realistic setups: We first study orbital differentiation in a three-orbital Hund-metal model and unravel key effects of the orbital-selective Mott transition. In a real-materials setting, we then incorporate the bandstructure from density functional theory (DFT) and analyze the archetypal Hund-metal material Sr2RuO4. We particularly follow its RG flow to the Fermi-liquid regime at previously inaccessible low temperatures and generally present DFT+DMFT+NRG as a new computational paradigm for strongly correlated materials. As a side project of our fRG work, we develop an algorithm to count Feynman diagrams from closed many-body relations, which reveals the surprising outcome that totally irreducible contributions are responsible for the factorial growth in the number of diagrams. Additionally, we use NRG to study transport through three-level quantum dots and provide benchmark data for other RG methods, which aim at further describing these systems in nonequilibrium.