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

Noise probing of topological band gaps in dispersionless quantum states