Gradient flow and Bogomolny bounds for quantum metric actions

TL;DR

This paper develops gradient flow dynamics for quantum metric actions in Bloch bands, deriving bounds related to topological invariants and providing a method to simplify models within fixed topological phases.

Contribution

It introduces a gradient flow framework for quantum metric actions and establishes Bogomolny bounds linked to the Chern number, with saturation by holomorphic projectors.

Findings

Actions decrease monotonically along flows

Chern number remains conserved during the flow

Flow leads to simplified, canonical models within topological phases

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.