Topological Quantum Matter

Introduction – The recent discovery of new topological electronic phases in insulating materials with strong spin-orbit coupling has lead to a renewed interest in topological states of matter. These insulators have exotic metallic surface states, which are a consequence of bulk topological invariants (see Fig. 1). The aim of the undergraduate research projects is to theoretically investigate novel topological phenomena of condensed matter systems. A particular emphasis will be on edge and surface properties of real materials with non-trivial topological properties. New theoretical and numerical diagnostics of topological order will be devised and used to search for new topological insulators and superconductors. We plan to explore surface and interface phenomena in systems with topological properties, focusing in particular on the experimental implications of the topologically protected surface states that occur at the boundary of crystalline topological materials (i.e, topological materials that are protected by space group symmetries). Furthermore, we propose to study topological characteristics of unconventional superconductors with strong spin-orbit coupling. While these projects mainly aim at the design and fundamental understanding of new topological materials, the insights gained may have important consequences for applications, such as spintronic, thermoelectric, and superconducting devices, which can be utilized for future information and communication technologies.

1) Antiperovskite Dirac materials.The antiperovskite materials Sr3PbO, Ca3PbO, and Eu3PbO are crystalline topological insulators, which exhibit topological boundary states that are protected by reflection symmetries. The low-energy band structure of these systems is given by an eight-component massive Dirac equation. From mfirst-principles-derived tight-binding Hamiltonians we determine the surface spectrum and the surface density of states of these systems using both analytical methods (T-matrix approximation) and numerical techniques (exact diagonalization). The spin and orbital character of the surface states will be investigated in detail. Of particular interest is the compound Eu3PbO, which is magnetic. We will study the intriguing interplay between localized spins and Dirac electrons of Eu3PbO using Dirac field theory methods. Furthermore, we will aim at explaining the exotic magneto-transport properties in the ultra quantum limit that has been observed in these materials.

2)  Quasiparticle scattering interference. The nontrivial spin and orbital character of topological surface states can be tested experimentally from the absence of backscattering processes in quasiparticle interference measurements. In this project, we will perform a detailed theoretical analysis of the quasiparticle scattering interference on the surface of crystalline topological insulators and semimetals. Magnetic and non-magnetic scattering processes from different impurity atoms will be considered. We will focus in particular on Na3Bi and Cd3As2 , which are two topological materials protected by rotation symmetry. Another interesting system is Ru2Sn3, which exhibits one-dimensional Dirac surface states.

3) Josephson currents. Josephson currents will be studied in junctions between different unconventional and conventional superconductors separated by an insulator, ferromagnet, or half-metal. Depending on the topological properties of the different superconductors involved in such a configuration there will be zero-energy modes localized at the interfaces. As a consequence, the Josephson currents are expected to show unconventional effects. We will explore these questions using direct diagonalization as well as within quasiclassical scattering theory and by making use of the Riccati parametrization.