## AIMR Math Group On-line Seminar

## #5 FY2021

##

Date :June. 29th (Tue) 2021 15:00–16:00

Venue: Zoom

Speaker: Tomoki Ozawa (WPI-AIMR, Tohoku University)

Title: Chern insulators, quantum metric, and the Kähler geometry

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.

▲TOP

### 【Latest News】

**AIMR Math Group **** Seminar**

Date: Feb. 3rd (Fri) 2023

Speaker: Prof. Makoto Matsumoto

【More Info】

**AIMR Math Group ****On-line Seminar**

※This seminar is organized jointly with (KUAMS) .

Date: Jan. 20th (Fri) 2023

Speaker: Dr. Cavallina Lorenzo

【More Info】

**AIMR Math Group ****Hybrid Seminar**

Date: Dec. 12th (Mon) 2022

Speaker: Shou Yoshikawa

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: Nov. 28th (Mon) 2022

Speaker: Prof. Marco Falconi

【More Info】

The 9th Early Career Researchers Ensemble Workshop

Date：Nov. 22, 2022

Venue：Sakura Hall (Katahira Campus)

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: Oct. 20th (Thu) 2022

Speaker: Shin Hayashi

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: Sep. 21th (Wed) 2022

Speaker: Prof. Dimi Culcer

【More Info】

**Math-Materials: International & Interdisciplinary Workshop Visualization &
Ideal Embeddings of Entangled Structures**

Date: July. 6th(Wed)-7th(Thurs) 2022

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: Feb. 25th (Fri) 2022

Speaker: Hisatoshi Kodani

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: Jan. 13th (Thu) 2022

Speaker: Dr. Yi Huang

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: Dec. 10th (Fri) 2021

Speaker: Jiawei Liu

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: Nov. 12th (Fri) 2021

Speaker: Hiroyasu Ando

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: Oct. 22th (Fri) 2021

Speaker: Dr. Marcello Seri

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: Aug. 27th (Fri) 2021

Speaker: Stefan Junk

【More Info】

"Discussion on promotions of international collaborations" will be held from 15:00 on July 19, 2021 by inviting Mr. Bernard Chenevier, Senior URA of Okayama University. (Application for participation)

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: June. 29th (Tue) 2021

Speaker: Tomoki Ozawa

【More Info】

**0x ****On-line Seminar**

Date: Jun. 19th (Sat) 2021

Speaker: Ryoma Sato/Sho Yokoi

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: May. 28th (Fri) 2021

Speaker: Eriko Shinkawa

【More Info】

**GTF2021**

**The Geometry & Topology Behind Fabrics at Multiple Scales**

Date:May.20th(Thur)-21st(Fri)2021

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: May. 14th (Fri) 2021

Speaker: Koichi Taniguchi

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: April. 30th (Fri) 2021

Speaker: Yuka Fujiki

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: April. 23th (Fri) 2021

Speaker: Jadala Venkata Ramana Reddy

【More Info】

AIMR-IFS-ISM Joint Workshop 2021

Date: April. 21st (Wed) 2021

【More Info】

Open Positions

【Posted on February 12, 2021】

Position: One Assistant Professor

Starting date: June 1, 2021, or earliest possible date

Application deadline: March 31, 2021

【More Info】

**0x ****On-line Seminar**

Date: Oct. 30th (Fri) 2020

Speaker: Yuichi Ike

【More Info】

**0x ****On-line Seminar**

Date: Aug. 29th (Sat) 2020

Speaker: alg-d

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: Aug. 4th (Tue) 2020

Speaker: Kazutoshi Inoue

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: July. 7th (Tue) 2020

Speaker: Chris Bourne

【More Info】

**AIMR Math Group ****On-line Seminar**

Date: April. 24th (Fri) 2020

Speaker: Tomoki Ozawa

【More Info】

Workshop：Jammed matter and its non-Gaussian fluctuations

Date: Mar. 23rd(Mon)-25th(Wed) 2019

【More Info】

**AIMR Math Group Seminar**

Date: Jan. 10th (Fri) 2020

Speaker: Prof. Ayato Mitsuishi

【More Info】

Synchronization phenomena on complex networks 2, from math to experiments - Special workshop for AIMR Advanced Target Projects -

Date: Jan. 8th (Web) 2020

【More Info】

**AIMR Math Group Seminar**

Date: Dec. 11th (Wed) 2019

Speaker: Yves Antonio Brandes Costa Barbosa

【More Info】

**mini course on pseudodifferential operators on non-commutative L^p spaces**

Date:Nov. 25th (Mon)・26th（Tue）

Speaker: Dr. Gihyun Lee

【More Info】

**AIMR Math Group Seminar**

Date: Sep. 6th (Fri) 2019

Speaker: Dr. Xi-Chan Gao

【More Info】

**AIMR Math Group Seminar**

Date: Aug. 9th (Fri) 2019

Speaker: Dr. Clemens Gneiting

【More Info】

**AIMR Math Group Seminar**

Date: July 30th (Tue) 2019

Speaker: Associate Pro. Sungrim Seirin-Lee

【More Info】

Synchronization phenomena on complex networks, from math to experiments – Special workshop for AIMR Advanced Target Projects –

Date: July 29th (Mon) 2019

【More Info】

**AIMR Math Group Seminar**

Date: July 23rd (Tue) 2019

Speaker: Prof. Eric Rowell

【More Info】

**AIMR Math Group Seminar**

Date: July 4th (Thu) 2019

Speaker: Dr. Jordan Hauge

【More Info】

**AIMR Math Group Seminar**

Date: May 17th (Fri) 2019

Speaker: Dr. Yuki Takahashi

【More Info】

**Math discussion for ATP**

Date: April 19th (Fri) 2019

Speaker: Dr. Hirofumi Oka

Speaker: Prof. Tomoteru Fukumura

【More Info】

**AIMR Math Group Seminar**

Date: April 4th (Thu) 2019

Speaker: Dr. Koji Sato

【More Info】