Knots and Transport in Topological Matter

Sammanfattning: Topology has manifestations in physics ranging from the field of condensed matter to photonics. This dissertation provides a two-fold study on the impact of topology in Hermitian and non-Hermitian band structures. Salient examples include the notion of topological invariants and knots, which are both used to describe characteristics of eigenvalue intersections. The first part focuses on Hermitian topological phases of matter, where general methods predicting transport properties in both gapped and gapless phases are presented. The second part turns to non-Hermitian phases and revolves around the topological properties of their exceptional eigenvalue degeneracies. Through a generic construction originating in knot theory, it is shown that such degeneracies take the form of knots, which furthermore bound open Fermi surfaces coinciding with the respective Seifert surfaces. This construction is then extended and applied in a similar fashion to parity-time-symmetric systems, where the exceptional points form surfaces and curves of any topology, as well as points. These theoretical descriptions constitute a fruitful platform to study dissipative systems—in particular in optics where parity-time symmetry implies a balance between gain and loss in photonic crystals—but also give rise to interesting connections to gravity in the context of analogue black holes.