TL;DR
This paper establishes a novel relation between Chern number, Berry curvature, and quantum metric in Dirac models, revealing that topological invariants can be derived from quantum geometric properties.
Contribution
It introduces a general relation linking Berry curvature and quantum metric, enabling Chern number characterization via quantum metric and hypersphere surface area.
Findings
Chern number can be encoded in quantum metric.
Quantum metric relates to the hypersphere surface area.
Protocol provided for measuring quantum metric in degenerate systems.
Abstract
Chern number is usually characterized by Berry curvature. Here, by investigating the Dirac model of even-dimensional Chern insulator, we give the general relation between Berry curvature and quantum metric, which indicates that the Chern number can be encoded in quantum metric as well as the surface area of the Brillouin zone on the hypersphere embedded in Euclidean parameter space. We find that there is a corresponding relationship between the quantum metric and the metric on such hypersphere. We show the geometrical property of quantum metric. Besides, we give a protocol to measure the quantum metric in the degenerate system.
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