| 研究課題/領域番号 |
16F16728
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| 研究機関 | 東北大学 |
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研究代表者 |
小谷 元子 東北大学, 理学研究科, 教授 (50230024)
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| 研究分担者 |
BOURNE CHRISTOPHER 東北大学, 理学研究科, 外国人特別研究員
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| 研究期間 (年度) |
2016-11-07 – 2019-03-31
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| キーワード | Topological phases / operator algebras / Kasparov theory / aperiodic media |
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| 研究実績の概要 |
We have begun to study topological properties of materials with minimal assumptions on the structure of the material. In particular, we consider mathematical models of Delone sets, which mimic the atomic-lattice of many condensed matter materials at low temperature. In particular, we do not assume that the atomic-lattice is well-structured and may contain impurities and disorder. We have made progress on establishing a coherent mathematical framework that describes the topological phase of materials modelled by Delone sets as well as their corresponding edge/surface effects, though the work is ongoing.
We also attended the international workshop on ‘KK-Theory, Gauge Theory and Topological Phases’ at the Lorentz Center, Leiden University. The workshop helped clarify where the current gaps in knowledge are with regards to mathematical approaches to topological phases.
Lastly, funds were used to acquire necessary resources to effectively conduct our research. These resources included a computer and relevant reference books.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
The project is still in its early stages having been started in November 2016, but currently we feel like we are on schedule to deliver the desired outcomes. We have identified the mathematical models we wish to consider (Delone sets) and have made progress in extracting topological information of relevance for physics.
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| 今後の研究の推進方策 |
Having identified our model of interest, our plan is to develop the mathematical framework to describe the topological phase of Delone sets. Topological phases are often described using a mathematical construction called K-theory, though the relevant K-theory of Delone sets is still an area in development. Our intention for future work is as follows:
1.Understand how the symmetries of interest in topological insulators are realized and implemented in Delone sets.
2.How the symmetries of Delone sets are related to K-theory.
3.How edge/surface effects arise when we consider such topological phases, the bulk-boundary correspondence and its physical interpretation.
4.Potential applications to a wider class of materials than what often considered in the topological phases literature (e.g. quasicrystals). We consider this because Delone sets are capable of modelling a very large class of materials.
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