Noise probing of topological band gaps in dispersionless quantum states

We uncover a useful connection between the integrated current noise $S(\omega )$ and the topological band gap in dispersionless quantum states, $\int d\omega [{\mathcal{S}}_{xx}^{\text{flat}}+{\mathcal{S}}_{yy}^{\text{flat}}]=C{e}^2{\mathrm{\Delta }}^2$ (in units $\mathrm{\hslash }$$=$$1$), where $C$ is the Chern number, $e$ is electric charge, and $\mathrm{\Delta }$ is the topological band gap. This relationship may serve as a working principle for a new experimental probe of topological band gaps in flat band materials. Possible applications include moiré systems, such as twisted bilayer graphene and twisted transition metal dichalcogenides, where a band gap measurement in meV regime presents an experimental challenge.