Functional renormalization group study of fluctuation effects in fermionic superfluids

Abstract

This thesis is concerned with ground state properties of two-dimensional fermionic superfluids. In such systems, fluctuation effects are particularly strong and lead for example to a renormalization of the order parameter and to infrared singularities. In the first part of this thesis, the fermionic two-particle vertex is analysed and the fermionic renormalization group is used to derive flow equations for a decomposition of the vertex in charge, magnetic and pairing channels. In the second part, the channel-decomposition scheme is applied to various model systems.

In the superfluid state, the fermionic two-particle vertex develops rich and singular dependences on momentum and frequency. After simplifying its structure by exploiting symmetries, a parametrization of the vertex in terms of boson-exchange interactions in the particle-hole and particle-particle channels is formulated, which provides an efficient description of the singular momentum and frequency dependences. Based on this decomposition of the vertex, flow equations for the effective interactions are derived on one- and two-loop level, extending existing channel-decomposition schemes to i) the description of symmetry breaking in the Cooper channel and ii) the inclusion of those two-loop renormalization contributions to the vertex that are neglected in the Katanin scheme.

In the second part, the superfluid ground state of various model systems is studied using the channel-decomposition scheme for the vertex and the flow equations. A reduced model with interactions in the pairing and forward scattering channels is solved exactly, yielding insights into the singularity structure of the vertex. For the attractive Hubbard model at weak coupling, the momentum and frequency dependence of the two-particle vertex and the frequency dependence of the self-energy are determined on one- and two-loop level. Results for the suppression of the superfluid gap by fluctuations are in good agreement with the literature. The two-loop approximation captures the singular infrared behaviour that is expected in a fermionic superfluid at zero temperature. For the repulsive Hubbard model at weak coupling, the momentum dependence of the two-particle vertex and the d-wave superfluid gap are determined on one-loop level as a function of the interaction, the next-nearest neighbour hopping and the fermionic density. The results for the critical scales and superfluid gaps are in qualitative agreement with the literature and suggest the existence of an optimal value of the next-nearest neighbour hopping for pairing.

In dieser Arbeit werden Grundzustandseigenschaften von zweidimensionalen fermionischen Suprafluiden untersucht. In diesen spielen Fluktuationseffekte eine wichtige Rolle und führen beispielsweise zu einer Renormierung des Ordnungsparameters oder zu Infrarotsingularitäten. Im ersten Teil der Arbeit wird der fermionische Zweiteilchenvertex analysiert und die fermionische Renormierungsgruppe wird verwendet um Flussgleichungen für eine Zerlegung des Vertex in Dichte-, Spin- und Paarungskanäle herzuleiten. Im zweiten Teil wird das Kanalzerlegungsschema auf verschiedene Modellsysteme angewendet.