The functional renormalization group provides a nonperturbative and systematically improvable route to constructing density functionals for quantum many-body systems from microscopic Hamiltonians. Here we advance this program by benchmarking functional-renormalization-group density functional theory (FRG-DFT) against the exact thermodynamics of the single-site Bose-Hubbard model. This model provides an ideal testing ground because it is analytically solvable, yet remains subtle in the imaginary-time coherent-state path integral, where a naive continuum treatment generates a spurious self-interaction. We show that a careful Hubbard-Stratonovich derivation identifies the self-interaction correction term that must be included in the FRG-DFT flow to recover the exact thermodynamics. We then systematically compare several closures of the resulting hierarchy of flow equations for the free energy, chemical potential, and connected density correlators over broad ranges of density, temperature, and interaction strength. The benchmark shows that the free energy is comparatively robust, whereas the chemical potential and fluctuation observables provide much sharper diagnostics of the hierarchy closure. A maximum-entropy closure gives the most accurate overall description and reproduces even the low-temperature oscillatory structure of the connected two-density correlator. These results identify two general requirements for functional approaches to quantum thermodynamics: the renormalization group flow equation must retain the equal-time contact subtraction to avoid spurious self-interactions, and any closure of the hierarchy must preserve the statistical consistency of density correlators. This work provides a controlled foundation for deriving ab initio density functionals for quantum many-body systems across condensed-matter, ultracold-atom, and nuclear physics, as well as quantum chemistry.
The functional renormalization group provides a nonperturbative and systematically improvable route to constructing density functionals for quantum many-body systems from microscopic Hamiltonians. Here we advance this program by benchmarking functional-renormalization-group density functional theory (FRG-DFT) against the exact thermodynamics of the single-site Bose-Hubbard model. This model provides an ideal testing ground because it is analytically solvable, yet remains subtle in the imaginary-time coherent-state path integral, where a naive continuum treatment generates a spurious self-interaction. We show that a careful Hubbard-Stratonovich derivation identifies the self-interaction correction term that must be included in the FRG-DFT flow to recover the exact thermodynamics. We then systematically compare several closures of the resulting hierarchy of flow equations for the free energy, chemical potential, and connected density correlators over broad ranges of density, temperature, and interaction strength. The benchmark shows that the free energy is comparatively robust, whereas the chemical potential and fluctuation observables provide much sharper diagnostics of the hierarchy closure. A maximum-entropy closure gives the most accurate overall description and reproduces even the low-temperature oscillatory structure of the connected two-density correlator. These results identify two general requirements for functional approaches to quantum thermodynamics: the renormalization group flow equation must retain the equal-time contact subtraction to avoid spurious self-interactions, and any closure of the hierarchy must preserve the statistical consistency of density correlators. This work provides a controlled foundation for deriving ab initio density functionals for quantum many-body systems across condensed-matter, ultracold-atom, and nuclear physics, as well as quantum chemistry.
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 use the functional renormalization group (FRG) to derive analytical expressions for thermodynamic observables (density, pressure, entropy, and compressibility) as well as for single-particle properties (wave-function renormalization and effective mass) of interacting bosons in two dimensions as a function of temperature T and chemical potential $\ensuremath{\mu}$. We focus on the quantum disordered and the quantum critical regime close to the dilute Bose gas quantum critical point. Our approach is based on a truncated vertex expansion of the hierarchy of FRG flow equations and the decoupling of the two-body contact interaction in the particle-particle channel using a suitable Hubbard-Stratonovich transformation. Our analytic FRG results extend previous analytical renormalization-group calculations for thermodynamic observables at $\ensuremath{\mu}=0$ to finite values of $\ensuremath{\mu}$. To confirm the validity of our FRG approach, we have also performed quantum Monte Carlo simulations to obtain the magnetization, susceptibility, and correlation length of the two-dimensional spin-1/2 quantum XY model with coupling J in a regime where its quantum critical behavior is controlled by the dilute Bose gas quantum critical point. We find that our analytical results describe the Monte Carlo data for $\ensuremath{\mu}\ensuremath{\le}0$ rather accurately up to relatively high temperatures $T\ensuremath{\lesssim}0.1J$.
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).
We investigate the phase diagram of a one-dimensional dissipative Bose-Hubbard model using the nonperturbative functional renormalization group (FRG). Each lattice site is coupled to an independent bath, generating long-range temporal interactions that encode non-Markovian dissipation. For a broad class of bath spectra - ohmic, sub-ohmic, and super-ohmic - we identify two competing low-energy regimes: a Luttinger-liquid line of fixed points and a dissipative fixed point characterized by finite compressibility, vanishing superfluid stiffness, and universal scaling exponents, separated by a Berezinskii-Kosterlitz-Thouless transition. The FRG framework is essential here, as it provides access to the complete renormalization group flow and all fixed points from a single microscopic action, beyond the reach of perturbative or variational methods. This work establishes a unified and systematically improvable framework for describing dissipative quantum phases in one dimension.
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.
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.