From topological nodal planes and multifold crossings in 3D to strong correlations in 2D

Abstract

Topological semimetals are hosts of numerous exotic phenomena, like Fermi arc surface states, the anomalous Hall effect, the quantized circular photogalvanic effect, the chiral anomaly, the anomalous Nernst effect and the anomalous thermal Hall effect. These phenomena are directly dependent on the presence and the properties of topological band crossings in the band structures of these compounds. Therefore, this thesis deals with classifying topological crossings in semimetals, deriving constraints on their topological charges, studying the properties and finding material realizations of more exotic band degeneracies, like nodal planes (NPs) and multifold degeneracies. The main results of this work includes an algorithm for fully classifying topological band structures of materials by DFT. This algorithm is used throughout this thesis for many different semimetals. The most detailed topological classification is presented for the B20 compound CoSi, shown to host topologically enforced NPs when including spin-orbit coupling (SOC). On the other hand, in ferromagnetic MnSi NPs are enforced by the space-group symmetries to be topological regardless if SOC is present or not, which we confirmed by a direct computation of symmetry eigenvalues. Tuning the magnetic moment, it is further possible to turn the presence of NPs on and off, leading to divergent Berry curvatures. By performing a complete topological classification of all multifold crossings in all space groups using an algorithm able to derive analytical low-energy Hamiltonians automatically, a 4-fold crossing was found to host an unusually high topological charge of +/-5. A material candidate is presented to feature this new topological phase at the Gamma point of BaAsPt. In two hexagonal space groups, SG 182 and SG 173, the concept of representation-enforced topological NPs have been introduced and material candidates were presented and classified using the above mentioned algorithms. One of these materials, NaCu5S3, exhibits a strikingly simple material band structure and surface DOS, resembling the ones produced by a tight-binding toy model. In chapter 7, we switch to physics of strongly correlated matter and examine the repulsive Hubbard model in 2D at large U. A unitary transformation is presented to map the interaction part of the Hubbard model to a single particle term. Applying this transformation to the whole Hubbard model results in a Hamiltonian of unconstrained fermions, which can then be mapped to a system of fermions interacting with spins. Integrating out the fermions using variational perturbation theory and replacing spins with classical spins reveals a phase transition at non-zero chemical potential to a phase with a finite d-wave order parameter.