Kugler, Fabian (2019): Renormalization group approaches to strongly correlated electron systems. Dissertation, LMU München: Faculty of Physics 

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Abstract
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 manybody problem  no classical computer will ever be able to fully simulate these systems  and the lack of an effective singleparticle 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 selfconsistent manybody relations. This establishes a form of diagrammatic resummations at the twoparticle level which circumvents illbehaved twoparticleirreducible vertices. An application to the prototypical twodimensional 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 meanfield theory (DMFT) to describe local correlations in multiorbital systems. Having access to arbitrarily low temperatures and energies, NRG is a unique, realfrequency 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 orbitaldegenerate models to more realistic setups: We first study orbital differentiation in a threeorbital Hundmetal model and unravel key effects of the orbitalselective Mott transition. In a realmaterials setting, we then incorporate the bandstructure from density functional theory (DFT) and analyze the archetypal Hundmetal material Sr2RuO4. We particularly follow its RG flow to the Fermiliquid 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 manybody 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 threelevel quantum dots and provide benchmark data for other RG methods, which aim at further describing these systems in nonequilibrium.
Item Type:  Theses (Dissertation, LMU Munich) 

Keywords:  Strongly correlated electrons, Functional renormalization group, Numerical renormalization group, Dynamical meanfield theory, Parquet approximation, Feynman diagrams, Hubbard model, Hund metals 
Subjects:  500 Natural sciences and mathematics 500 Natural sciences and mathematics > 530 Physics 
Faculties:  Faculty of Physics 
Language:  English 
Date of oral examination:  29. October 2019 
1. Referee:  Delft, Jan von 
MD5 Checksum of the PDFfile:  7b82a19824278de6d334d7527cbe4abb 
Signature of the printed copy:  0001/UMC 26807 
ID Code:  25359 
Deposited On:  08. Jan 2020 12:23 
Last Modified:  08. Jan 2020 12:23 