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.
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