A wide variety of phenomena in condensed matter systems is driven primarily by interactions between electrons in the system. This work is concerned with the application of functional renormalization group (fRG) as a generalized solver for the multi-band Hamiltonians that describe these systems. We consider a decoupled formulation of the fRG equations that is optimized in the frequency and momentum domains and retains the flow of relevant modes in the system. Approximate truncations that extend the scheme to arbitrary multiband systems are addressed. This optimized decoupling is then used to derive the flow equations that describe fluctuations in model Hamiltonians for Cuprate and Pnictide superconductors. We construct a full phase diagram of the systems studied as a function of doping, temperature and coupling. Access to the frequency modes in the system allows us to explore the impact of coupling phonons to these model Hamiltonians. Alterations to the diagram due to electron-phonon coupling is derived. The results of the decoupled formulation is in agreement with results in the literature for many of the models considered. Further the fRG captures the sensitivity of susceptibilities of Cuprate Hamiltonians to band structure, the enhanced role of Hunds coupling in Pnictide systems and the impact of phonons in multiband Hamiltonians.
A wide variety of phenomena in condensed matter systems is driven primarily by interactions between electrons in the system. This work is concerned with the application of functional renormalization group (fRG) as a generalized solver for the multi-band Hamiltonians that describe these systems. We consider a decoupled formulation of the fRG equations that is optimized in the frequency and momentum domains and retains the flow of relevant modes in the system. Approximate truncations that extend the scheme to arbitrary multiband systems are addressed. This optimized decoupling is then used to derive the flow equations that describe fluctuations in model Hamiltonians for Cuprate and Pnictide superconductors. We construct a full phase diagram of the systems studied as a function of doping, temperature and coupling. Access to the frequency modes in the system allows us to explore the impact of coupling phonons to these model Hamiltonians. Alterations to the diagram due to electron-phonon coupling is derived. The results of the decoupled formulation is in agreement with results in the literature for many of the models considered. Further the fRG captures the sensitivity of susceptibilities of Cuprate Hamiltonians to band structure, the enhanced role of Hunds coupling in Pnictide systems and the impact of phonons in multiband Hamiltonians.
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.
This thesis is devoted to purely fermionic functional renormalization-group (fRG) studies of multiband effects and of antiferromagnetically ordered phases. The formal framework of the one-particle irreducible fRG scheme is recapitulated and extended for charge-conserving theories with a collinear spin symmetry. For multiband models, point-group symmetries and their interplay with the phases of Bloch basis states are discussed. Further, an improved truncation of the fRG flow equations is proposed that keeps track of three-particle interactions generated by virtual excitations of bands away from the Fermi surface. The feedback of this three-particle term on the two-particle interaction is first investigated in a g-ology approach to a simple two-band model for cuprate superconductors at weak coupling. The results suggest a potentially strong impact of the three-particle term. Subsequently, the three-band Emery model is analyzed within a channel-decomposition, again at weak coupling. In contrast to earlier work, the approach pursued here does not rely on a form-factor expansion, which allows for the observation of a deformed d-wave pairing gap for increasingly incommensurate ordering tendencies in the magnetic channel. In this more refined study, the three-particle feedback plays a minor role. Despite considerable qualitative agreement with results for the one-band Hubbard model with effective parameters, multi-orbital effects are found to play an important role on a quantitative level. Phases of antiferromagnetic ordering are first studied for a reduced mean-field model. In order to go beyond the mean-field picture, a hierarchy of approximations is devised, which starts from the corresponding channel-decomposed one-loop flow equations. Interaction terms breaking discrete symmetries are thereby successively neglected. In the course of these approximations, also an exchange parametrization is employed. For the spin-density wave phase of a two-pocket model initially proposed by Chubukov et al. in Phys. Rev. B 78:134512, the flow equations are then integrated numerically at the most approximate level of the hierarchy. This yields a gap that is significantly reduced compared to mean-field theory. The resulting violation of the SU(2) Ward identity appears acceptable on a qualitative level, suggesting that the approximations causing this violation are physically meaningful.
We derive an expansion of the functional renormalization (fRG) equations that treats the frequency and momentum dependencies of the vertices in a systematic manner. The scheme extends the channel-decomposed fRG equations to the frequency domain and reformulates them as a series of linear integral equations in the particle-particle, particle-hole and particle-hole exchange channels. We show that the linearity of the equations offers numerous computational advantages and leads to converged, stable solutions for a variety of Hamiltonians. As the expansion is in the coupling between channels, the truncations that are necessary to making the scheme computationally viable still lead to equations that treat contributions from all channels equally. As a first benchmark we apply the two-loop fRG equations to the single impurity Anderson model. We consider the sources of error within the fRG, the computational cost associated with each, and how the choice of regulator affects the flow of the fRG. We then use the optimal truncation scheme to study the Extended Hubbard Hamiltonian in one and two dimensions. We find that in many cases of interest the fRG flow converges to a stable vertex and self-energy from which we can extract the various correlation functions and susceptibilities of interest.
The treatment of strongly interacting two-dimensional Fermi systems constitutes one of the most challenging problems in the field of condensed matter physics. Many theoretical works focused on the strongly coupled Hubbard model, since it is expected to capture the most important physics of cuprate superconductors. Due to our methodological improvements and understanding of the frequency dependence of the two-particle vertex function, the functional renormalization group combined with the dynamical mean field theory can now be used to study competing correlations in the strongly interacting regime. While limited to moderate interaction strengths, the functional renormalization group describes efficiently systems with a hierarchy of different energy scales and competing correlations. For instance, it provides definite evidence for d-wave superconductivity in the two-dimensional Hubbard model at moderate coupling. In a first project, we study the frequency dependence of the one-particle irreducible vertex function generated by the functional renormalization group flow. The frequency dependence, which becomes singular for strong interactions, appears to be important already for moderate couplings, and it cannot be represented by separate channels each depending on a single linear combination of frequencies. For strongly interacting systems, the dynamical mean field theory captures strong local correlation effects nonperturbatively. With this approximation, we study the impact of local correlations on the magnetic susceptibility. The local dynamics strongly affects the spin response function and its momentum dependence. In contrast to the widely used random-phase approximation with self-energy corrections that predicts Néel antiferromagnetic order, the local vertex corrections favor an incommensurate order similar to the ordering instability predicted by the Fermi surface geometry, as for weakly interacting systems. The dynamical mean field theory is also used as starting point for the functional renormalization group flow. We demonstrate that, due to our improvements for the parametrization of the vertex function, this approach is actually able to access the strong coupling physics. Moreover, we derive a flow scheme that conserves the local contributions and reduces the truncation error of the flow equations. In the strongly interacting regime, we capture strong d-wave pairing correlations driven by magnetic fluctuations with a mechanism similar to the one observed in the weakly interacting system.
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).
The functional renormalization group (fRG) is acknowledged as a powerful tool in quantum many-body physics and beyond. On the technical side, conventional implementations of the fRG rely on regulators for bare propagators only. Starting from Schwinger-Dyson and Bethe-Salpeter equations, we develop here an fRG formulation where both bare propagators and bare interactions can be dressed with regulators. The approach thus obtained is an extension of the multiloop fRG recently introduced for many-fermion systems. Using the single-boson exchange decomposition, we show that the underlying flow equations are simply interpreted as adding a regulator to the bosonic propagator and that such an extension scarcely changes the original structure of the flow equations. Overall, we provide a framework for implementing approaches that cannot be realized with conventional fRG methods, such as temperature flows for models with retarded interactions. For concrete applications, we analyze the loop convergence of our formulation against a conventional cutoff scheme for the Anderson impurity model. Finally, we devise a temperature-flow scheme that implements a cutoff in both the propagator and the bare interaction, and demonstrate its validity on a model of an Anderson impurity coupled to a phonon.