Gradient flow and Bogomolny bounds for quantum metric actions

2025/11/11 by Toshizumi Fukui, Fukui, T.
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Geometry and complex manifolds #Mesoscale and Nanoscale Physics (cond-mat.mes-hall) #Quantum chaos and dynamical systems #Topological Materials and Phenomena

paper · pdf · doi:10.48550/arxiv.2511.07855

openalex publication_date 2025/11/11 · openalex created_date 2025/11/13 · openalex updated_date 2026/07/29

Abstract

We formulate gradient flow dynamics generated by two natural actions of the quantum metric for an isolated set of Bloch bands. Specializing to two spatial dimensions, we derive Bogomolny-type lower bounds that relate these actions to the Chern number and show that the bounds are saturated by (anti-)holomorphic projector configurations. Along the flows, the actions decrease monotonically while the Chern number is conserved, giving a constructive route to simplify models within a fixed topological phase toward canonical, low-complexity representatives.

Citations

• Wilson-Loop-Ideal Bands and General Idealization
• Quantum geometric bounds for observables: Linear responses, Drude weight, and orbital magnetization
• Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals
• Quantum weight: A fundamental property of quantum many-body systems
• Chern Numbers in Discretized Brillouin Zone: Efficient Method of Computing (Spin) Hall Conductances

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