Mathematical Theory of Edge Transport in Photonic and Electronic Systems

Abstract

This thesis explores the mathematical theory of edge transport in photonic and electronic systems, with a focus on interface modes and bulk-edge correspondence principles. Motivated by the discovery of physical phenomena such as the valley Hall effect, we investigate transport channels that emerge at interfaces between distinct media, even in systems that lack nontrivial topological characterizations like the Chern number or Z₂ index. The first part of this work focuses on interface and resonant modes in various photonic systems with sharp interfaces and small band gaps, including 2D photonic waveguides, honeycomb lattices, and square-lattice photonic structures. We rigorously prove the existence of interface modes bifurcating from degenerate points in the band structure, such as Dirac points and quadratic degenerate points. The emergence of these modes is tied to mechanisms of band inversion and symmetry-breaking perturbations.

In the second part, we study the edge transport in the non-perturbative regime, with an emphasis on generalizing the existing results on the bulk-edge correspon-dence (BEC) principle. We first establish BEC for photonic structures on finite domains, bridging the gap between theory and experimental setups. Then we study the electronic systems with non-conserved spin charge, and propose a novel BEC correspondence based on spin conductance, incorporating spin drift and spin torque effects near the interface. These findings broaden the scope of BEC, offering insights into interface transport in both topological and non-topological systems. The thesis concludes with discussions on extending these methods to more complex settings, including establishing BEC for systems without internal degrees of freedom or distinct topological classifications.

Date of Award	2025
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology
Supervisor	Hai ZHANG (Supervisor)