h1

TY  - THES
AU  - Diekmann, Jan
TI  - Construction of controlled renormalization-group approximations for low-dimensional quantum systems
PB  - RWTH Aachen University
VL  - Dissertation
CY  - Aachen
M1  - RWTH-2025-07907
SP  - 1 Online-Ressource : Illustrationen
PY  - 2025
N1  - Veröffentlicht auf dem Publikationsserver der RWTH Aachen University
N1  - Dissertation, RWTH Aachen University, 2025
AB  - For low-dimensional condensed-matter systems of interacting fermions, logarithmic divergences are a common problem of perturbation theory. They can appear for example in the perturbative expansion of a power law whose exponent depends on the expansion parameter. Classifying and resumming the leading logarithms allows for approximations with a priori controlled accuracies. For the class of systems where the divergences appear in multiple two-particle channels with one logarithm per diagrammatic bubble, the leading-logarithmic parquet approximation is a well-established tool, specifically designed for this problem. A prototypical system of this class is the X-ray-edge model, which describes the X-ray-absorption singularity in metals. We devise a one-loop functional renormalization group (FRG) approach to the X-ray-edge model that is fully equivalent, step by step, to the leading-logarithmic parquet approximation. Thus, the one-loop FRG can achieve leading-logarithmic accuracy. This capability is well-known for many lowest-order scaling and RG approaches but has been put into question for the one-loop FRG by recent literature on the X-ray-edge model. In our particular case, the FRG and parquet frameworks can even be interpreted as different perspectives on the exact same approximation. Our study shows a way of controlling the accuracy of FRG truncations in regimes not accessible by perturbation theory, in contrast to merely qualitative justifications that are usually invoked there. We also investigate the single-impurity Anderson model (SIAM) that describes a quantum dot connecting two reservoirs. From the point of view of an unusual perturbative expansion in both the interaction on the dot and the coupling between the dot and the reservoirs, the SIAM belongs to the same class as the X-ray-edge model, but it is more complicated. We develop an approximation scheme aimed at resumming the leading logarithms of the SIAM in a similar way. We express this scheme in terms of effective diagrammatic rules, which show similarities to the Kondo model. However, our approach needs to be extended before it can properly capture the Kondo physics of the SIAM. Throughout this thesis we use the zero-temperature formalism. In the context of the X-ray-edge model, it allows for the direct comparison between our FRG approach and the existing leading-logarithmic parquet approximation. It is simpler but more restrictive than the Keldysh and Matsubara formalisms. Since the FRG has never before been used with the zero-temperature formalism, we comprehensively set up the required framework for a general model of interacting fermions in terms of the one-particle irreducible vertex functions.
LB  - PUB:(DE-HGF)11
DO  - DOI:10.18154/RWTH-2025-07907
UR  - https://publications.rwth-aachen.de/record/1018634
ER  -