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Relativistic effects crucially influence the fundamental properties of many quantum materials. In the accelerated reference frame of an electron, the electric field of the nuclei is transformed into a magnetic field that couples to the electron spin. The resulting interaction between an electron spin and its orbital angular momentum, known as spin-orbit coupling (SOC), is hence fundamental to the physics of many condensed matter phenomena. It is particularly important quantitatively in low-dimensional quantum systems, where its coexistence with inversion symmetry breaking can lead to a splitting of spin degeneracy and spin momentum locking. Using the paradigm of Landau Fermi liquid theory, the physics of SOC can be adequately incorporated in an effective single particle picture. In a weak coupling approach, electronic correlation effects beyond single particle propagator renormalization can trigger Fermi surface instabilities such as itinerant magnetism, electron nematic phases, superconductivity, or other symmetry broken states of matter.
In this thesis, we use a weak coupling-based approach to study the effect of SOC on Fermi surface instabilities and, in particular, superconductivity. This encompasses a weak coupling renormalization group formulation of unconventional superconductivity as well as the random phase approximation. We propose a unified formulation for both of these two-particle Green’s function approaches based on the notion of a generalized susceptibility.
In the half-Heusler semimetal and superconductor LuPtBi, both SOC and electronic correlation
effects are prominent, and thus indispensable for any concise theoretical description. The metallic and weakly dispersive surface states of this material feature spin momentum locked Fermi surfaces, which we propose as a possible domain for the onset of unconventional surface superconductivity. Using our framework for the analysis of Fermi surface instability and combining it with ab-initio density functional theory calculations, we analyse the surface band structure of LuPtBi, and particularly its propensity towards Cooper pair formation. We study how the presence of strong SOC modifies the classification of two-electron wave functions as well as the screening of electron-electron interactions. Assuming an electronic mechanism, we identify a chiral superconducting condensate featuring Majorana edge modes to be energetically favoured over a wide range of model parameters.
Despite its history of more than one hundred years, the phenomenon of
superconductivity has not lost any of its allure. During that time the concept
and perception of the superconducting state - both from an experimental and
theoretical point of view - has evolved in way that has
triggered increasing interest. What was initially believed to simply be the
disappearance of electrical resistivity, turned out to be a universal and
inevitable result of quantum statistics, characterized by many more
aspects apart from its zero resistivity. The insights of
BCS-theory eventually helped to uncover its deep connection to particle physics
and consequently led to the formulation of the Anderson-Higgs-mechanism. The
very core of this theory is the concept of gauge symmetry (breaking). Within the
framework of condensed-matter theory, gauge invariance is only one of several
symmetry groups which are crucial for the description and classification of
superconducting states. \\
In this thesis, we employ time-reversal, inversion, point group and spin
symmetries to investigate and derive possible Hamiltonians featuring spin-orbit
interaction in two and three spatial dimensions.
In particular, this thesis aims at a generalization of existing numerical
concepts to open up the path to spin-orbit coupled (non)centrosymmetric
superconductors in multi-orbital models.
This is done in a two-fold way: On the one hand, we formulate - based on the
Kohn-Luttinger effect - the perturbative renormalization group in the
weak-coupling limit. On the other hand, we define the spinful flow equations of
the effective action in the framework of functional renormalization, which is
valid for finite interaction strength as well. Both perturbative and functional
renormalization groups produce a low-energy effective (spinful) theory that
eventually gives rise to a particular superconducting state, which is investigated
on the level of the irreducible two-particle vertex. The symbiotic relationship
between both perturbative and functional renormalization can be traced back to
the fact that, while the perturbative renormalization at infinitesimal coupling
is only capable of dealing with the Cooper instability, the functional
renormalization can investigate a plethora of instabilities both in the
particle-particle and particle-hole channels. \\
Time-reversal and inversion are the two key symmetries, which are being used to
discriminate between two scenarios. If both time-reversal and inversion symmetry
are present, the Fermi surface will be two-fold degenerate and characterized by a
pseudospin degree of freedom. In contrast, if inversion symmetry is broken, the
Fermi surface will be spin-split and labeled by helicity. In both cases, we
construct the symmetry allowed states in the particle-particle as well as the
particle-hole channel. The methods presented are formally unified and implemented
in a modern object-oriented reusable and extendable C++ code.
This methodological implementation is employed to one member of both families of
pseudospin and helicity characterized systems. For the pseudospin case, we choose
the intriguing matter of strontium ruthenate, which has been heavily
investigated for already twenty-four years, but still keeps puzzling researchers.
Finally, as the helicity based application, we consider the oxide heterostructure
LaAlO$_{3}$/SrTiO$_{3}$, which became famous for its highly mobile two-
dimensional electron gas and is suspected to host topological superconductivity.