Abstract
We uncover a useful connection between the integrated current noise S(ω) and the topological band gap in dispersionless quantum states, ∫ d ω [ S^flat_xx + S^flat_yy ] = C e² Δ² (in units ℏ=1), where C is the Chern number, e is electric charge, and Δ 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 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.