Using a leading algorithmic implementation of the functional renormalization group (fRG) for interacting fermions on two-dimensional lattices, we provide a detailed analysis of its quantitative reliability for the Hubbard model.In particular, we show that the recently introduced multiloop extension of the fRG flow equations for the self-energy and two-particle vertex allows for a precise match with the parquet approximation also for twodimensional lattice problems.The refinement with respect to previous fRG-based computation schemes relies on an accurate treatment of the frequency and momentum dependences of the two-particle vertex, which combines a proper inclusion of the high-frequency asymptotics with the so-called truncated unity fRG for the momentum dependence.The adoption of the latter scheme requires, as an essential step, a consistent modification of the flow equation of the self-energy.We quantitatively compare our fRG results for the self-energy and momentumdependent susceptibilities and the corresponding solution of the parquet approximation to determinant quantum Monte Carlo data, demonstrating that the fRG is remarkably accurate up to moderate interaction strengths.The presented methodological improvements illustrate how fRG flows can be brought to a quantitative level for two-dimensional problems, providing a solid basis for the application to more general systems.
Using a leading algorithmic implementation of the functional renormalization group (fRG) for interacting fermions on two-dimensional lattices, we provide a detailed analysis of its quantitative reliability for the Hubbard model.In particular, we show that the recently introduced multiloop extension of the fRG flow equations for the self-energy and two-particle vertex allows for a precise match with the parquet approximation also for twodimensional lattice problems.The refinement with respect to previous fRG-based computation schemes relies on an accurate treatment of the frequency and momentum dependences of the two-particle vertex, which combines a proper inclusion of the high-frequency asymptotics with the so-called truncated unity fRG for the momentum dependence.The adoption of the latter scheme requires, as an essential step, a consistent modification of the flow equation of the self-energy.We quantitatively compare our fRG results for the self-energy and momentumdependent susceptibilities and the corresponding solution of the parquet approximation to determinant quantum Monte Carlo data, demonstrating that the fRG is remarkably accurate up to moderate interaction strengths.The presented methodological improvements illustrate how fRG flows can be brought to a quantitative level for two-dimensional problems, providing a solid basis for the application to more general systems.
Using a leading algorithmic implementation of the functional renormalization group (fRG) for interacting fermions on two-dimensional lattices, we provide a detailed analysis of its quantitative reliability for the Hubbard model. In particular, we show that the recently introduced multiloop extension of the fRG flow equations for the self-energy and two-particle vertex allows for a precise match with the parquet approximation also for two-dimensional lattice problems. The refinement with respect to previous fRG-based computation schemes relies on an accurate treatment of the frequency and momentum dependences of the two-particle vertex, which combines a proper inclusion of the high-frequency asymptotics with the so-called truncated unity fRG for the momentum dependence. The adoption of the latter scheme requires, as an essential step, a consistent modification of the flow equation of the self-energy. We quantitatively compare our fRG results for the self-energy and momentum-dependent susceptibilities and the corresponding solution of the parquet approximation to determinant quantum Monte Carlo data, demonstrating that the fRG is remarkably accurate up to moderate interaction strengths. The presented methodological improvements illustrate how fRG flows can be brought to a quantitative level for two-dimensional problems, providing a solid basis for the application to more general systems.
We present a functional renormalization group (fRG) study of the two dimensional Hubbard model, performed with an algorithmic implementation which lifts some of the common approximations made in fRG calculations. In particular, in our fRG flow; (i) we take explicitly into account the momentum and the frequency dependence of the vertex functions; (ii) we include the feedback effect of the self-energy; (iii) we implement the recently introduced multiloop extension which allows us to sum up all the diagrams of the parquet approximation with their exact weight. Due to its iterative structure based on successive one-loop computations, the loop convergence of the fRG results can be obtained with an affordable numerical effort. In particular, focusing on the analysis of the physical response functions, we show that the results become independent from the chosen cutoff scheme and from the way the fRG susceptibilities are computed, i.e., either through flowing couplings to external fields, or through a “post-processing” contraction of the interaction vertex at the end of the flow. The presented substantial refinement of fRG-based computation schemes paves a promising route towards future quantitative fRG analyses of more challenging systems and/or parameter regimes.
The functional renormalization group (fRG) is an established tool in the treatment of correlated electron systems, notably for the description of competing instabilities. In recent years, methodological advancements led to the multiloop extension of the fRG, which systematically includes loop corrections beyond the conventional one-loop truncation and yields a quantitatively accurate description of two-dimensional lattice systems. At the same time, the single-boson exchange (SBE) decomposition of the two-particle vertex has been shown to offer both computational and interpretative advantages paving the way to more affordable approximation schemes. We here apply their combination coined as multiloop SBE fRG to the two-dimensional Hubbard model at weak coupling. After providing a detailed account of the underlying formalism in physical channels, we analyze the results for the frequency- and momentum-dependent vertex functions. We find that the SBE approximation, i.e., neglecting the flow of the multi-boson exchange contributions, accurately reproduces the parquet approximation at loop convergence. The presented algorithmic improvement opens the route for the treatment of more challenging parameter regimes and more realistic models.
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).
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.
Abstract: We illustrate the algorithmic advantages of the recently introduced single-boson exchange (SBE) formulation for the one-loop functional renormalization group (fRG), by applying it to the two-dimensional Hubbard model on a square lattice. We present a detailed analysis of the fermion-boson Yukawa couplings and of the corresponding physical susceptibilities by studying their evolution with temperature and interaction strength, both at half filling and finite doping. The comparison with the conventional fermionic fRG decomposition shows that the rest functions of the SBE algorithm, which describe correlation effects beyond the SBE processes, play a negligible role in the weak-coupling regime above the pseudo-critical temperature, in contrast to the rest functions of the conventional fRG. Remarkably, they remain finite also at the pseudo-critical transition, whereas the corresponding rest functions of the conventional fRG implementation diverge. As a result, the SBE formulation of the fRG flow allows for a substantial reduction of the numerical effort in the treatment of the two-particle vertex function, paving a promising route for future multiboson and multiloop extensions.