Knotted Nodal Band Structures

Abstract: It is well known that in conventional three dimensional (3D) Hermitian two band models, the intersections between the energy bands are generically given by points. The typical example are Weyl semimetals, where these singular points can be effectively described as Weyl fermions in the low energy regime. By explicitly imposing discrete symmetries or fine-tuning, the intersection can form higher- dimensional nodal structures, e.g. nodal lines. By instead considering dissipative contributions to such a system, the degeneracies will generically take the form of closed 1D curves, consisting of exceptional points, i.e. points where the Hamiltonian becomes defective. By constructing the Hamiltonian in a particular way, the 1D exceptional curves can host non-trivial topology, i.e. they can form links or knots in the Brillouin zone. In stark contrast to line nodes occurring in Hermitian systems, which inevitably rely on discrete symmetries or fine tuning, the exceptional knots are generically stable towards any small perturbation. In further contrast to point singularities and unknotted circles, the topology of knots cannot be characterized by usual integer valued invariants. Instead, the complexity of the knottedness is captured by polynomial type invariants, making the physical classification and interpretation of these system challenging. To this end, the study of knotted nodal band structures naturally brings two different aspects of topology together – mathematical knot theory on the one hand, and the physical theory of topological phases on the other hand.This licentiate thesis focuses on providing the necessary theoretical background to understand the two accompanying publications entitled Knotted non-Hermitian metals, written by Johan Carlström, together with the author of this thesis, Jan Carl Budich and Emil J. Bergholtz, published in Physical Review B on April 24 2019, and Hyperbolic nodal band structures and knot invariants, written by the author of this thesis, together with Lukas Rødland, Gregory Arone, Jan Carl Budich and Emil J. Bergholtz, published in SciPost Physics August 8 2019. An introduction to gapless topological phases in the Hermitian regime, focusing on Weyl semimetals, their classification and surface states, is provided. Then, the light is brought to non-Hermitian operators and the differences from their conventional Hermitian counterpart, such as the two different set of eigenvectors bi-orthogonal to each other, exceptional eigenvalue degeneracies and some of their consequences, are explained. Afterwards, these operators are applied to dissipative physical system, and some of the striking differences from the conventional Hermitian systems are highlighted, the main focus being the possibly non-trivial topology of the 1D exceptional eigenvalue degeneracies. In order to be somewhat self contained, a brief conceptual introduction to the utilized concepts of knot theory is given, and lastly, further research directions and possible experimental realization of the considered systems are discussed.