[geometry-ml:04364] AIMR数学セミナーのお知らせ（6月29日小澤知己氏）

[Hiroshi Suito]

応用数学ML、幾何学ML、トポロジーMLの皆様
（重複投稿ご容赦下さい。）

東北大学AIMR数学連携グループが主催するオンライン・セミナーのご案内です。参加ご希望の方はフォーム
https://forms.gle/UKzoWWNbLqAQfd9X8
からご登録下さい。ZoomミーティングIDを開催当日の開始1時間前までにお送りします。

どうぞよろしくお願い致します。

東北大学AIMR
水藤　寛

=========================================

東北大学AIMR数学連携グループセミナー

日時：2021年6月29日（火）15時～16時
Date: June 29th, 2021 (Tue) 15:00-16:00

講演者： 小澤知己（東北大学WPI-AIMR材料物理グループ）
Speaker: Tomoki Ozawa (Materials Physics Group, WPI-AIMR, Tohoku University)
https://tomokiozawa.com

Title: Chern insulators, quantum metric, and the Kähler geometry
（講演は英語で行われます。）

Abstract:
Two-dimensional Chern insulators are topological insulators, which are
modeled by a vector bundle with a base manifold given by momentum
space and fibers are given by quantum states which form a vector
space. The Chern number, a topological invariant characterizing the
vector bundle, is related to local geometrical quantity known as the
Berry curvature; the integral of the Berry curvature over momentum
space gives rise to the Chern number. I discuss how the Chern number
and the Berry curvature are related to another geometrical quantity
known as the quantum metric. The quantum metric is a naturally
equipped Riemannian metric for the Chern insulators; we can consider
the associated volume of momentum space measured with the quantum
metric and call it the quantum volume. Besides momentum space, it is
often useful to consider another parameter space which is the
twist-angle space; the twist-angle space is spanned by the twist in
the boundary condition of the periodic boundary condition. We found an
inequality among the Chern number, the quantum volume of momentum
space, and the quantum volume of the twist-angle space. Using this
relation, we can infer the topology (Chern number) of the system just
by studying the metric in certain limits. Furthermore, by looking at
the Brillouin zone as a complex manifold (Riemann surface), we can
study complex structure of the Brillouin zone. We have found that,
when the map from the Brillouin zone to the space of quantum states
(which is complex Grassmannian), is holomorphic, the above-mentioned
inequality between the Chern number and the quantum volume becomes an
equality. When the inequality is saturated, we found that the
Brillouin zone is a Kähler manifold with the quantum metric as the
Kähler metric and the Berry curvature as the Kähler form.

http://www.wpi-aimr.tohoku.ac.jp/mathematics_unit/english/seminar/20210629.html