## 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.

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