2018 Fiscal Year Research-status Report
Photonic Topological Insulators: Dynamical and Topological Effects
| Project/Area Number |
16K17761
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| Research Institution | Tohoku University |
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Principal Investigator |
LEIN MAXIMILIAN 東北大学, 材料科学高等研究所, 准教授 (50769891)
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Keywords | topological insulators / photonic crystals / electromagnetic waves |
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| Outline of Annual Research Achievements |
During FY2018, I have worked in 3 collaborations on various aspects of Projects 1 and 3, which resulted in 1 publication and 2 preprints: (1) With Giuseppe De Nittis I have worked towards a proof of Haldane's bulk-boundary correspondence; we have finished one preprint (under review) on the equivalence of the topological classification of first- and second-order formalisms for classical electromagnetism. (2) Konstantin Bliokh, Daniel Leykam, Franco Nori and I have derived hitherto unknown bulk-boundary correspondences that correctly predict the existence of topologically protected modes at interfaces between homogeneous electromagnetic media; our results were published in Nature Communications. And (3) Koji Sato and I have worked out the Krein-Schroedinger formalism for spin waves; we have finished one preprint that is currently under review. While this work concerns different types of classical waves, we used a mathematical formalism which we will apply to electromagnetism in FY2019.
Moreover, together with Mikio Furuta and Chris Bourne I have organized an interdisciplinary, international workshop on topological insulators in Sendai in August 2018; here, mathematicians and physicists met to discuss the state-of-the-art in the field. Additionally, I gave three invited talks at international conferences in Japan and Switzerland, and four seminar talks in The Netherlands and Germany.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
FY 2019 Project 3: When proving Haldane's photonic bulk-edge correspondence, some progress was made. However, we have learnt that the mathematics of the bulk classification is complicated by the singular behavior of the bands which have linear dispersion near k = 0.
FY 2019 Project 3: The results obtained with Bliokh, Leykam and Nori were very surprising: we not only found genuinely new bulk-boundary correspondences, but a mathematical treatment necessitates the use of so-called Krein spaces. Only one classification result for operators on Krein spaces is known (due to Kiyonori Gomi and Giuseppe De Nittis), and a systematic understanding therefore necessitates the development of a new point of view. This is immediately connected to my work with Koji Sato on spin waves: also here, Krein spaces play a role, albeit in a simpler way. So more work is necessary to obtain a mathematical understanding of these bulk-boundary correspondences.
FY 2018 Project 2 was postponed.
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| Strategy for Future Research Activity |
The unexpected discovery of the previously unknown bulk-boundary correspondences for interfaces between homogeneous electromagnetic media have changed my priorities for FY 2019: I would like to change the focus to non-selfadjoint (non-hermitian) operators on Krein spaces rather than work on generalizing the topological classification from linear to weakly non-linear electromagnetic media. This is very timely because of all the other research activities surrounding the topological classification of non-hermitian systems.
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| Causes of Carryover |
The difference between anticipated and actual expenditure is less than 0.6 %. Thus, my research expenses stayed were well-estimated.
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