数理科学セミナー：Towards A Rigorous Proof of Haldane’s Photonic Bulk-Edge Correspondence

開催日時

2019年7月2日 16時 30分 ～ 2019年7月2日 17時 30分

場所

理学部３号館３－２１０

講演者

Max Lein (AIMR)

概要：

In 2005 Raghu and Haldane predicted an analog of the Integer Quantum Hall Effect in quasi-2d gyrotropic photonic crystals, thereby conjecturing that topological effects are wave rather than quantum phenomena. These unidirectional, backscattering-free edge modes were indeed observed in a number of beautiful experiments with microwaves, light at optical frequencies, coupled oscillators and other classical waves. Haldane proposed to explain this phenomenon via a photonic bulk-edge correspondence: the net number of edge modes that traverse a photonic bulk band gap are given by the bulk Chern number associated to the corresponding “Fermi projection”.

Proving Haldane’s conjecture turns out to be significantly harder than expected, because of a number of conceptual and technical difficulties. This talk will explore four of them: (1) Establishing rigorous quantum-wave analogies that allow us to systematically adapt tools from quantum mechanics to electromagnetism. (2) Identify the physically relevant symmetries and understand their nature. (3) Classify periodic operators with “half a conical crossing”. (4) Further develop tools to prove bulk-boundary correspondences for operators on the continuum.

The first two problems have been understood, whereas the last two problems are the subjects of active research.

This talk is based on a joint work with Giuseppe De Nittis