Geometric Effects in Matter: Topological Insulators, Metals and Superconductors

Topological phases of matter is a new and quickly growing field in condensed matter physics. The fundamental ideas of this field are heavily inspired by high-energy physics and cosmology, and it is concerned with topologically protected properties of materials and devices, which are intrinsically robust against small perturbations. Research in this direction promises us a deeper understanding of the connections between the different fields of physics and also alluring practical applications like a fault-tolerant quantum computer based on non-Abelian Majorana zero modes. In this thesis we address problems from the whole spectrum of topological matter: topological insulators, topological semimetals and topological superconductors. In the first part of the thesis we reflect on the well known topological obstruction for describing the ground state of a topological insulator as a smooth and periodic function of the momentum in the whole Brillouin zone, by a so-called smooth gauge. This problem is mathematically equivalent to finding exponentially localized Wannier functions for a group of energy bands that is topologically nontrivial. We show that the obstruction can be avoided by breaking the symmetries that protect the topology of the system and give a generic procedure for obtaining a smooth gauge based on this principle, given that the total Chern number of the system is zero. In the second part of the thesis we find and describe a new topologically protected fermionic quasiparticle, which consists of the crossings of three bands - the triple point fermion. Unlike the Weyl or Dirac fermion, the triple point fermion has no analogue in the relativistic quantum field theory. We find a number of materials realizing these quasiparticles and discuss their unique physical phenomena like Lifshitz transitions, surface states and transport anomalies. In the last two chapters of the thesis we discuss two problems in the field of Majorana zero modes. The first problem is concerned with finding a material with optimal properties, like particularly a large spin-orbit splitting, for realizing Majorana zero modes. Since both the semiconductors InAs and InSb have been successfully used in several Majorana devices we investigate the ternary alloy InAs$_x$Sb$_{1-x}$. For the disordered alloy we find the optimal composition to maximize the structure-inversion-asymmetry induced spin-orbit splitting. Furthermore, we identify the CuPt-ordered alloy InAs$_{0.5}$Sb$_{0.5}$ to have exceedingly large intrinsic bulk spin-orbit splitting, making it very promising for Majorana devices. In the second problem we find an explanation for the anomalously large Land $g$-factors --- several times larger than the corresponding bulk value --- found in recent Majorana nanowire devices. We show that higher subbands in cylindrical nanowires are subject to a $\mathbf{L}\cdot\mathbf{S}$ coupling that can enhance the $g$-factor by an order of magnitude or more for small effective mass semiconductors. Doing numerical simulations of InAs and InSb wires we find that the effect is robust to some degree to the breaking of cylindrical symmetry