Physical implications of multi-band Euler topology

Abstract

Recent years have seen rapid advancements in the field of topological band theory. While early research focused on the characterisation of phases that exist in the stable limit of many bands, it has since become clear that few-band systems can host exotic multi-band phases beyond the well-known K-theoretical classification. One particularly interesting topological invariant that exists for certain PT-symmetric systems is the Euler class, which assigns an integer to a real, rank-2n collection of connected bands in 2n spatial dimensions. Systems with a non-trivial Euler class are notable for the unusual properties of their band nodes, which can undergo (classical) non-Abelian braiding processes. The value of the Euler class is moreover equal to the sum of the local topological charges of nodes within the occupied subspace. While the mathematical properties of the Euler class closely parallel those of the Chern number, which is a topological invariant of complex bands, the ways in which the former invariant manifests itself physically remain unclear: while the Chern number is directly measurable through the quantised Hall conductance, no corresponding integral physical observable is known for the Euler class. The identification of systems hosting a non-trivial Euler invariant therefore presents a challenging theoretical and experimental problem. This thesis investigates the ways in which the multi-band Euler class invariant could be detected in real systems, focussing on three examples. Firstly, it is shown that Andreev reflection at the junction between a normal and a superconducting region of an Euler material leads to distinctly recognisable reflection and differential conductance curves. Secondly, the effects of Euler topology in classical spin liquids are investigated, and the nodal stability and pinch point structure induced by the invariant are discussed. Finally, a three-dimensional material with embedded Euler topology and a bulk Hopf invariant is introduced, and it is shown that this material displays topological nodal lines, delocalised Wannier functions, and a quantised nonlinear optical response.