We apply the functional renormalization group method (FRG) to the problem of interactions in disordered electron systems. To achieve this we develop an approximation that allows us to simplify the FRG equations in inhomogeneous systems in order to make computations feasible. For a model system we demonstrate the efficacy of our method by comparison with exact results for small system sizes. We utilize our method to investigate the phase diagram of this model system.
We apply the functional renormalization group method (FRG) to the problem of interactions in disordered electron systems. To achieve this we develop an approximation that allows us to simplify the FRG equations in inhomogeneous systems in order to make computations feasible. For a model system we demonstrate the efficacy of our method by comparison with exact results for small system sizes. We utilize our method to investigate the phase diagram of this model system.
The functional renormalization group (fRG) approach has the property that, in general, the flow equation for the two-particle vertex generates O(N4) independent variables, where N is the number of interacting states (e.g., sites of a real-space discretization). In order to include the flow equation for the two-particle vertex, one needs to make further approximations if N becomes too large. We present such an approximation scheme, called the coupled-ladder approximation, for the special case of an onsite interaction. Like the generic third-order-truncated fRG, the coupled-ladder approximation is exact to second order and is closely related to a simultaneous treatment of the random phase approximation in all channels, i.e., summing up parquet-type diagrams. The scheme is applied to a one-dimensional model describing a quantum point contact.
Good progress has been made in electronic structure calculations for real materials as well as for model Hamiltonians. However with an eye on $a\phantom{\rule{0}{0ex}}b$ $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o$ calculations, it is still very challenging to accurately calculate, e.g., the dynamical response of small molecules or atom clusters. At present the method of choice is the GW-BSE method. Its notorious drawback is that solving a computationally demanding self-consistency equation is required. Thus motivated, the author explore an alternative idea replacing the self-consistency problem by the functional renormalization group (FRG). The traditional FRG was formulated for systems with translational symmetry. We investigate an alternative ($\ensuremath{\epsilon}$FRG) that is also applicable to inhomogeneous matter, such as disordered metals or molecules. It accounts for Fermi-liquid corrections to quasiparticle energies and particle-hole excitations. In principle it can go beyond the state of the art approximation, GW-BSE, by solving the Bethe-Salpeter equation (BSE) self-consistently We present an efficient implementation of $\ensuremath{\epsilon}$FRG together with performance tests on the spinless disordered Hubbard model.
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.
During the last two decades, functional renormalization group (FRG) emerged as a versatile class of methodologies which can be used in a variety of cases, ranging from statistical mechanics models and interacting field theories to condensed matter strongly interacting systems. The development of FRG techniques from one hand had a methodological side – aimed at testing different ways to handle with approximations of the functional equations – and from the other concurred to define FRG as a method to be seriously considered to face new and poorly/not yet understood problems. Both directions, i.e. improve and test known techniques and proposed refinements against well known problems and concretely deal with non-testbed problems, have motivated the work done during the PhD. This thesis aims at present, provide details and put in perspective results obtained during my PhD in the last three years. Many of such results have been published, in a more synthetic way, in peer review journals, others are going to appear online soon. Some others will probably remain only in this thesis, but together with the published material will be instrumental to delineate a path leading to a deeper understanding and a (hopefully) clearer presentation of potentialities and issues of the FRG method.
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.
In this thesis, the functional renormalization group (FRG) method is applied to quantum spin chains. FRG is a well-known method in the treatment of correlated fermion systems, but the method has been applied to spin systems only recently[1]. The method has been used, however, only as a qualitative tool to find phase boundaries for frustrated spin systems with multiple competing interactions. The main goal of this work is to critically assess the general accuracy of the method by studying toy models, for which quantitative comparison with exact results is possible. After a full derivation of the flow equations for spin systems, the numerical solution of the resulting non-linear system of differential equations is presented. It is shown that in the antiferromagnetic two-site Heisenberg model, FRG can reproduce some known properties of the singlet state but the ground state energy, for example, disagrees with the exact result by 36 %. In addition, the quasiparticle self-energy and dynamic spin-spin correlation functions are of an incorrect form. Unphysical fluctuations in the particle number are shown to be large. In the antiferromagnetic Heisenberg chain, it is shown that whereas the use of properly truncated flow equations regularizes the unphysical antiferromagnetic instability, the resulting ground state is disordered and not critical as expected. [1] J. Reuther and P. Wölfle, Phys. Rev. B 81, 144410 (2010).