Measure of an ultranarrow topological gap via quantum noise

Advent of new-generation materials, with flat topological bands and ultranarrow band gaps (in the order of 1 meV), poses challenges on their precise characterization. We uncover a useful connection between the integrated current noise $S(\ensuremath{\omega})$ and the topological band gap in dispersionless quantum states, $\ensuremath{\int}d\ensuremath{\omega}[{\mathcal{S}}_xx^{\text{flat}}+{\mathcal{S}}_yy^{\text{flat}}]=C{e}²{\mathrm{\ensuremath{\Delta}}}²$ (in units $\ensuremath{\hbar}=1$), where $C$ is the Chern number, $e$ is electric charge, and $\mathrm{\ensuremath{\Delta}}$ is the topological band gap. This relationship may serve as a working principle for experimental probe of topological band gaps in flat band materials. Possible applications include moir\'e systems, such as twisted bilayer graphene and twisted transition metal dichalcogenides, where a band gap measurement in meV regime presents an experimental challenge.