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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.
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.