Gradient flow and Bogomolny bounds for quantum metric actions

TL;DR

The paper develops a variational framework for Bloch-band projectors based on two SL$(2,sz)$-covariant quantum-metric actions, $S_{\rm t}$ and $S_{\rm d}$, whose gradient flows monotonically reduce geometric complexity while preserving the Chern number $C$. It proves Bogomolny-type bounds $S_{\rm t}\ge \pi|C|$ and $S_{\rm d}\ge \pi|C|$, with saturating configurations given by (anti-)holomorphic projector conditions, and shows $C$ is conserved along the flows. A key result is that within a fixed topological sector, generic band geometries are driven toward canonical, low-complexity representatives that saturate the bounds, providing a constructive route to ideal Chern-band geometries. The Wilson–Dirac model example illustrates sharpening of Berry curvature into a single, nearly isotropic peak while preserving $C$, highlighting potential use as a preconditioner for variational design of topological bands and ideal-band geometries for interaction-driven 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.

Figures (1)

• Figure 1: Berry-flux density $F_{12}(k)\,d^2k$ on the Brillouin zone $0\le k_\mu\le 2\pi$, discretized on a $21^2$ mesh, for the Wilson-Dirac model with $t=b=1$. (a)-(c) show initial distributions for $m=2.2,3.0,3.8$ ($C=+1$). (d) shows the common configuration after the trace-type gradient flow.