Weyl semimetals, named after the physicist Hermann Weyl, were predicted theoretically about ten years ago and were found experimentally only in 2015. The topological properties of Weyl semimetals and their analogy to massless elementary particles make them a field of great interest in condensed matter physics. The key feature of Weyl semimetals is the crossing of two linearly dispersing bands, forming a Weyl cone. Here we study the effects of breaking the rotational symmetry of the Weyl cone by a tilt, which refers in this context to an additive linear term proportional to the unit matrix in the Weyl equation, leaving the topological invariants unaffected. This thesis is built on a work originally published in Physical Review B, which is one of the first works to discuss the generic but previously overlooked tilt in Weyl semimetals and the very first to point out the tilt’s relevance for the transport properties. Based on our theoretical results we can conclude which properties are useful to demonstrate the tilt in Weyl semimetals experimentally. We distinguish between type-I Weyl semimetals, which have moderate tilts and a point-like Fermi surface, and type-II Weyl semimetals, characterized by large tilts leading to finite electron and hole pockets. This thesis begins with a general introduction to transport properties, Weyl semimetals and the methods we use. The following main content consists of two parts. The first part (chapters 2 and 3) addresses the electronic quantum transport properties of type-I Weyl semimetals, including the conductivity and the Fano factor. We discuss the symmetry-breaking tilt in detail and consider the influence of the tilt on the transport properties. We show that the Fano factor is a useful indicator to detect the tilt of a Weyl semimetal, as it is independent of other parameters. Further, we study the phase transition induced by disorder from a quasi-ballistic to a diffusive phase and we determine how the critical disorder strength depends on the tilt. In doing so, we show that the transition from type-I to type-II Weyl semimetals is always preceded by the disorder-induced transition. To find these results, we apply a variety of methods, including scattering matrix theory and the self-consistent Born approximation, and show exemplary lattice models. The second part (chapters 4 and 5) considers type-II Weyl semimetals in a magnetic field including electron-electron interactions. By developing a general mean field approach for type- II Weyl semimetals, we show that the interaction induces a charge density wave (CDW). We compute the critical temperature of the transition to the CDW phase and find a cascade of successive transitions for different Landau levels. We determine the resistivity ρ using the quan- tum mechanic linear response theory and confirm the semi-classical picture of a ρ ∝ B 2 scaling. The semi-classical picture is valid below the quantum limit reached at high magnetic fields. We further quantify the transition to a different scaling above the quantum limit. While the semi- classical picture only considers the non-interacting case, we are able to extend the analysis to the interacting case and show in which regime the ρ ∝ B2 scaling is still present. Additionally, we study the oscillations of ρ(B), finding that the CDW increases their amplitude and leads to an unusual temperature dependence. Altogether this thesis demonstrates the importance of the tilt in Weyl semimetals on the basis of quantum mechanical transport properties.
