The full frequency dependence of electronic interactions is central to dynamical correlations, but its numerical cost restricts multiorbital calculations. We formulate the multiloop functional renormalization group directly in the quantics tensor train representation, retaining all three independent vertex frequencies over broad ranges without an explicit high-frequency parametrization. Analytical tensor operators perform frequency transformations and internal summations without dense vertex intermediates. Multiloop corrections improve agreement of vertices and responses with independent references and reproduce the low-energy numerical renormalization group self-energy. Lower-temperature calculations resolve a quadratic frequency contribution to the analytically continued scattering rate. A degenerate two-orbital impurity exhibits distinct spin, orbital, and charge responses at fixed Hund coupling. In a compensated altermagnetic environment, an orbital-level splitting induces a local spin response despite zero magnetization of the unperturbed impurity. Increasing Hund coupling suppresses the orbital response while increasing its spin polarization. Renormalized one-particle propagation alone does not reproduce this strong polarization dependence. These results extend compressed multiloop calculations to dynamical correlations in multiorbital environments.
The full frequency dependence of electronic interactions is central to dynamical correlations, but its numerical cost restricts multiorbital calculations. We formulate the multiloop functional renormalization group directly in the quantics tensor train representation, retaining all three independent vertex frequencies over broad ranges without an explicit high-frequency parametrization. Analytical tensor operators perform frequency transformations and internal summations without dense vertex intermediates. Multiloop corrections improve agreement of vertices and responses with independent references and reproduce the low-energy numerical renormalization group self-energy. Lower-temperature calculations resolve a quadratic frequency contribution to the analytically continued scattering rate. A degenerate two-orbital impurity exhibits distinct spin, orbital, and charge responses at fixed Hund coupling. In a compensated altermagnetic environment, an orbital-level splitting induces a local spin response despite zero magnetization of the unperturbed impurity. Increasing Hund coupling suppresses the orbital response while increasing its spin polarization. Renormalized one-particle propagation alone does not reproduce this strong polarization dependence. These results extend compressed multiloop calculations to dynamical correlations in multiorbital environments.
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.
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.
We present a tree tensor-network impurity solver suited for general multiorbital systems. The network is constructed to efficiently capture the entanglement structure and symmetry of an impurity problem. The solver works directly on the real-time/frequency axis and generates spectral functions with energy-independent resolution of the order of one percent of the correlated bandwidth. Combined with an optimized representation of the impurity bath, it efficiently solves self-consistent dynamical mean-field equations and calculates various dynamical correlation functions for systems with off-diagonal Green's functions. For the archetypal correlated Hund's metal Sr2RuO4, we show that both the low-energy quasiparticle spectra related to the van Hove singularity and the high-energy atomic multiplet excitations can be faithfully resolved. In particular, we show that while the spin-orbit coupling has only minor effects on the orbital-diagonal one-particle spectral functions, it has a more profound impact on the low-energy spin and orbital response functions.
We investigate several fundamental characteristics of the multiloop functional renormalization group (mfRG) flow by hands of its application to a prototypical many-electron system: the Anderson impurity model (AIM). We first analyze the convergence of the algorithm in the different parameter regions of the AIM. As no additional approximation is made, the multiloop series for the local self-energy and response functions converge perfectly to the corresponding results of the parquet approximation (PA) in the weak- to intermediate-coupling regime. Small oscillations of the mfRG solution as a function of the loop order gradually increase with the interaction, hindering a full convergence to the PA in the strong-coupling regime, where perturbative resummation schemes are no longer reliable. By exploiting the converged results, we inspect the fulfillment of (i) sum rules associated to the Pauli principle and (ii) Ward identities related to conservation laws. For the Pauli principle, we observe a systematic improvement by increasing the loop order and including the multiloop corrections to the self-energy. This is consistent with the preservation of crossing symmetries and two-particle self-consistency in the PA. For the Ward identities, we numerically confirm a visible improvement by means of the Katanin substitution. At weak coupling, violations of the Ward identity are further reduced by increasing the loop order in mfRG. In this regime, we also determine the precise scaling of the deviations of the Ward identity as a function of the electronic interaction. For larger interaction values, the overall behavior becomes more complex, and the benefits of the higher-loop terms are mostly present in the contributions at large frequencies.
We provide a detailed exposition of our computational framework designed for the accurate calculation of real-frequency dynamical correlation functions of the single-impurity Anderson model in the regime of weak to intermediate coupling. Using quantum field theory within the Keldysh formalism to directly access the self-energy and dynamical susceptibilities in real frequencies, as detailed in our recent publication [Ge et al., Phys. Rev. B 109, 115128 (2024)], the primary computational challenge is the full three-dimensional real-frequency dependence of the four-point vertex. Our codebase provides a fully MPI+OpenMP parallelized implementation of the functional renormalization group (fRG) and the self-consistent parquet equations within the parquet approximation. It leverages vectorization to handle the additional complexity imposed by the Keldysh formalism, using optimized data structures and highly performant integration routines. Going beyond the results shown in the previous publication, the code includes functionality to perform fRG calculations in the multiloop framework, up to arbitrary loop order, including self-consistent self-energy iterations. Moreover, implementations of various regulators, such as hybridization, interaction, frequency, and temperature, are supplied.
Including higher order diagrammatic corrections to the strong-coupling expansion is mainly limited by the evaluation of high-dimensional, time-ordered integrals. In this work we present and compare four different parametrizations of the integrands in order to obtain a low-rank (quantics) tensor-train representation using tensor cross interpolation. Particular emphasis is placed on a quantics time-difference formulation in which the required retarded convolutions are performed directly in quantics tensor-train form. Using controlled Gaussian benchmarks, we analyze the accuracy, bond dimensions, and computational scaling of the different approaches. We then validate the most promising formulations in self-consistent equilibrium and nonequilibrium DMFT calculations and demonstrate calculations up to the third order in the strong-coupling expansion. Finally, we extend the solver to impurity models with retarded density-density interactions and apply it within nonequilibrium extended DMFT. Our results show that tensor cross interpolation substantially reduces the cost of evaluating higher-order diagrams and provides a controlled, systematically improvable framework for nonequilibrium quantum impurity calculations.