Topological superconductors and category theory

Abstract

We give a pedagogical introduction to topologically ordered states of matter, with the aim of familiarizing the reader with their axiomatic topological quantum field theory description. We introduce basic noninteracting topological phases of matter protected by symmetries, including the Su-Schrieffer-Heeger model and the one-dimensional p-wave superconductor. The defining properties of topologically ordered states are illustrated explicitly using the toric code and -on a more abstract level -Kitaev's 16-fold classification of two-dimensional topological superconductors. Subsequently, we present a short review of category theory as an axiomatic description of topological order in two-dimensions. Equipped with this structure, we revisit Kitaev's 16-fold way.These lectures were in parts held at:

On 2-group global symmetries and their anomalies

“…We now describe some aspects of 3d TQFTs with global symmetry following [22] (see also [41,47]). Our first task is to identify the intrinsic global symmetry of such a theory as discussed in Section 4.…”

Majorana fermion quantum mechanics for higher rank tensors

“…This is typically done by mapping our model to a random matrix theory ensemble. There are general rules for choosing the associated ensemble based on the various symmetries of the model [34]. The set of possible ensembles we consider is known as the Andreev-Altland-Zirnbaur (AAZ) ten-fold classification.…”

Quantum simulation of strongly correlated condensed matter systems

“…Alternatively, intrinsic topological order may emerge in a many-body system, which is then characterized by topological ground state degeneracy, fractional excitations and topological entanglement entropy [209,210]. Fractional (anyonic) excitations in systems with intrinsic topological order, for example a fractional quantum Hall state, could find a powerful application in topological quantum computation [211].…”