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 $S\left(\omega \right)$ 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 $\hslash =1$), 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 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.

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• Flat bands
• Quantum fluctuations & noise
• Quantum transport
• Narrow band gap systems
• Topological materials
• Twisted heterostructures