Evaluating Chern Topology in Discretized Brillouin Zones Using Bargmann Invariants

Abstract

We develop a finite-state geometric framework based on third-order Bargmann invariants for evaluating Chern numbers and characterizing local quantum-state geometry. Using gauge-invariant state overlaps, the framework evaluates the Chern number directly on discretized parameter spaces while retaining local geometric information through the phase-derived density and the complementary amplitude density, as well as signatures associated with changes in topological character. We apply the construction to the Qi--Wu--Zhang, $1/3$- and $1/4$-flux Hofstadter, SSH, and Rice--Mele models, including a rank-$2$ extension for isolated composite subspaces and applications to closed-loop and parameter-space topology. The resulting framework organizes Chern topology and local geometric information directly from finite-state data, providing a compact state-based description without relying on dynamical evolution.