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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.
The superconducting properties of complex materials like the recently discovered iron-pnictides or strontium-ruthenate are often governed by multi-orbital effects. In order to unravel the superconductivity of those materials, we develop a multi-orbital implementation of the functional renormalization group and study the pairing states of several characteristic material systems. Starting with the iron-pnictides, we find competing spin-fluctuation channels that become attractive if the superconducting gap changes sign between the nested portions of the Fermi surface. Depending on material details like doping or pnictogen height, these spin fluctuations then give rise to $s_{\pm}$-wave pairing with or without gap nodes and, in some cases, also change the symmetry to $d$-wave. Near the transition from nodal $s_{\pm}$-wave to $d$-wave pairing, we predict the occurrence of a time-reversal symmetry-broken $(s+id)$-pairing state which avoids gap nodes and is therefore energetically favored. We further study the electronic instabilities of doped graphene, another fascinating material which has recently become accessible and which can effectively be regarded as multi-orbital system. Here, the hexagonal lattice structure assures the degeneracy of two $d$-wave pairing channels, and the system then realizes a chiral $(d+id)$-pairing state in a wide doping range around van-Hove filling. In addition, we also find spin-triplet pairing as well as an exotic spin-density wave phase which both become leading if the long-ranged hopping or interaction parameters are slightly modified, for example, by choosing different substrate materials. Finally, we consider the superconducting state of strontium-ruthenate, a possible candidate for chiral spin-triplet pairing with fascinating properties like the existence of half-quantum vortices obeying non-Abelian statistics. Using a microscopic three orbital description including spin-orbit coupling, we demonstrate that ferromagnetic fluctuations are still sufficient to induce this $\bs{\hat{z}}(p_x\pm ip_y)$-pairing state. The resulting superconducting gap reveals strong anisotropies on the $d_{xy}$-dominated Fermi-surface pocket and nearly vanishes on the other remaining two pockets.
Magnetism is a phenomenon ubiquitously found in everyday life. Yet, together with superconductivity and superfluidity, it is among the few macroscopically realized quantum states. Although well-understood on a quasi-classical level, its microscopic description is still far from being solved. The interplay of strong interactions present in magnetic condensed-matter systems and the non-trivial commutator structure governing the underlying spin algebra prevents most conventional approaches in solid-state theory to be applied.
On the other hand, the quantum limit of magnetic systems is fertile land for the development of exotic phases of matter called spin-liquids. In these states, quantum fluctuations inhibit the formation of magnetic long-range order down to the lowest temperatures. From a theoretical point of view, spin-liquids open up the possibility to study their exotic properties, such as fractionalized excitations and emergent gauge fields. However, despite huge theoretical and experimental efforts, no material realizing spin-liquid properties has been unambiguously identified with a three-dimensional crystal structure. The search for such a realization is hindered by the inherent difficulty even for model calculations. As most numerical techniques are not applicable due to the interaction structure and dimensionality of these systems, a methodological gap has to be filled.
In this thesis, to fill this void, we employ the pseudo-fermion functional renormalization group (PFFRG), which provides a scheme to investigate ground state properties of quantum magnetic systems even in three spatial dimensions.
We report the status quo of this established method and extend it by alleviating some of its inherent approximations. To this end, we develop a multi-loop formulation of PFFRG, including hitherto neglected terms in the underlying flow equations consistently, rendering the outcome equivalent to a parquet approximation. As a necessary prerequisite, we also significantly improve the numerical accuracy of our implementation of the method by switching to a formulation respecting the asymptotic behavior of the vertex functions as well as employing state-of-the-art numerical algorithms tailored towards PFFRG. The resulting codebase was made publicly accessible in the open-source code PFFRGSolver.jl.
We subsequently apply the technique to both model systems and real materials. Augmented by a classical analysis of the respective models, we scan the phase diagram of the three-dimensional body-centered cubic lattice up to third-nearest neighbor coupling and the Pyrochlore lattice up to second-nearest neighbor. In both systems, we uncover in addition to the classically ordered phases, an extended parameter regime, where a quantum paramagnetic phase appears, giving rise to the possibility of a quantum spin liquid.
Additionally, we also use the nearest-neighbor antiferromagnet on the Pyrochlore lattice as well as the simple cubic lattice with first- and third-nearest neighbor couplings as a testbed for multi-loop PFFRG, demonstrating, that the inclusion of higher loop orders has quantitative effects in paramagnetic regimes and that the onset of order can be signaled by a lack of loop convergence.
Turning towards material realizations, we investigate the diamond lattice compound MnSc\(_2\)S\(_4\), explaining on grounds of ab initio couplings the emergence of a spiral spin liquid at low temperatures, but above the ordering transition.
In the Pyrochlore compound Lu\(_2\)Mo\(_2\)O\(_5\)N\(_2\), which is known to not magnetically order down to lowest temperatures, we predict a spin liquid state displaying a characteristic gearwheel pattern in the spin structure factor.
A general theory for all classes of unconventional superconductors is still one of the unsolved key issues in condensed-matter physics. Actually, it is not yet fully settled if there is a common underlying pairing mechanism. Instead, it might be possible that several distinct sources for unconventional (not phonon-mediated) superconductivity have to be considered, or an electron-phonon interaction is not negligible. The focus of this thesis is on the most probable mechanism for the formation of Cooper pairs in unconventional superconductors, namely a strictly electronic one where spin fluctuations are the mediators. Studying different superconductors in this thesis, the emphasis is put on material-independent features of the pairing mechanism. In addition, the investigation of the phase diagrams enables a view on the vicinity of superconductivity. Thus, it is possible to clarify which competing quantum fluctuations enhance or weaken the propensity for a superconducting state. The broad range of superconducting materials requires the use of more than one numerical technique to study an appropriate microscopic description. This is not a problem but a big advantage because this facilitates the approach-independent description of common underlying physics. For this evaluation, the strongly correlated cuprates are simulated with the variational cluster approach. Especially the question of a pairing glue is taken into consideration. Furthermore, it is possible to distinguish between retarded and non-retarded contributions to the gap function. The cuprates are confronted with the cobaltate NaCoO and graphene. These weakly correlated materials are investigated with the functional renormalization group (fRG) and reveal a comprehensive phase diagram, including a d+id-wave superconductivity, which breaks time-reversal symmetry. The corresponding gap function is nodeless, but for NaCoO, it features a doping-dependent anisotropy. In addition, some general considerations on the kagome lattice are completing the discussion, where a sublattice interference dramatically affects the Fermi-surface instabilities, suppressing the usual spin-density wave and d+id-wave superconductivity. Thereby, some different fascinating charge and bond orders as well as a nematic are observable. In short, this thesis provides an insight to distinct classes of unconventional superconductors with appropriate simulation techniques. This facilitates to separate the material specific properties from the universal ones.