Majorana Fermions and Topological Superconductivity 1D Topological classification of CII phases

Abstract

In the last decades, topology has established itself as a fundamental principle in condensed matter physics. Not only does topology explain the incredible robustness of certain physical quantities such as the Hall conductivity, but as a result of the bulkboundary correspondence it also accounts for robust exotic fermionic edge states, e.g. Majorana zero modes in superconductors. In this thesis the field of topological quantum matter is reviewed with particular emphasis on one-dimensional non-interacting superconducting systems and symmetry-protected topological phases. By taking off from Kitaev’s model of p-wave superconductivity and the tenfold classification of topological superconductors and insulators, we also investigate the notion of gapless phases of matter addressing the issue what happens at a topological phase transition in one dimension. For the symmetry classes BDI and CII, which both are known to host topological superconductors in 1D, we obtain an N×Z-classification of gapless phases. In addition to the conventional topological Z-invariant, the other classification is provided by conformal field theory, a framework frequently used when studying critical phenomena.