Revealing Chern number from quantum metric

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.

5pages,2figures

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• Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers
• Nontrivial quantum geometry of degenerate flat bands
• Quantum geometric effect on Fulde-Ferrell-Larkin-Ovchinnikov superconductivity
• From Classical to Quantum Information Geometry: A Guide for Physicists
• Nested-sphere description of the N-level Chern number and the generalized Bloch hypersphere
• Extracting non-Abelian quantum metric tensor and its related Chern numbers
• Quantum metric on the Brillouin Zone in correlated electron systems and its relation to topology for Chern insulators
• Non-Abelian quantum geometric tensor in degenerate topological semimetals
• Adiabatic charge transport in extended SSH models
• Evaluation of real-space second Chern number using the kernel polynomial method
• Magneto-nonlinear Hall effect in time-reversal breaking system
• Numerical calculation of the $k$-space second Chern number in four dimensions
• Quantum theory of nonlinear electromagnetic response