Geometric study of some higher-order topological invariants related to corners

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K14545
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Tohoku University (2021-2022); National Institute of Advanced Industrial Science and Technology (2019-2020)
• Principal Investigator: Hayashi Shin 東北大学, 材料科学高等研究所, 助教 (70807833)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 指数理論 / K理論 / 四半面テープリッツ作用素 / 高次トポロジカル絶縁体

Outline of Final Research Achievements

We conducted research to elucidate index theory for operators on a discrete quarter-plane (quarter-plane Toeplitz operators). Based on a well-expected relation between index theory for quarter-plane Toeplitz operators and higher-order topological insulators in condensed matter physics, we aimed to clarify the geometric viewpoint for indices of quarter-plane Toeplitz operators as some higher invariants. As a result, we derived an index formula that reveals the underlying geometric picture and the role of analyticity. Additionally, we found explicit contact with higher-order topological insulators and investigated foundational theories for quarter-plane Toeplitz operators for applications. We also conducted some theoretical proposals for the study of topological insulators in collaboration with condensed matter physicists.

Free Research Field

位相幾何学

Academic Significance and Societal Importance of the Research Achievements

ある種の離散的な角と関連したトポロジーの背後にある幾何的描像を明らかにし, 特に解析性の役割を見出したことは, さらなる展開の手がかりとなる可能性がある. また, 物性物理学のトピックである高次トポロジカル絶縁体と指数理論の関連を明確にするとともに, 実際にいくつかの応用を行った. この意味で本研究は高次トポロジカル絶縁体のトポロジーを取り扱う理論基盤の開拓に向けた数学的取り組みとしての意義もある.