Power-law growth of shift current with superlattice period in flat Chern bands

Quantum geometry governs a wide array of physical observables in topological quantum materials. In its ideal limit, quantum geometry yields exact bounds and analytical results for a growing class of observables. However, a comparable framework for the shift current remains elusive because it depends on geometric relations between multiple bands. In this work, we show that the shift current is proportional to the cyclotron shift, the displacement of the cyclotron center induced by Landau-level mixing. This mechanism yields a universal scaling law: the integrated weight and the peak magnitude scale as $l^{i+1-n}$ and $l^{2i+1-n}$ in the magnetic length or moiré period $l$, where $n$ is the momentum order of the symmetry-breaking perturbation and $\mathrm{\Delta }E\propto l^{-i}$ is the level spacing. We verify the scaling law in exactly solvable chiral-$N$ Landau levels under a uniform magnetic field and in Schrödinger and Dirac moiré skyrmion crystals, which capture the flat Chern bands of twisted MoTe${}_2$ and twisted bilayer graphene. For a Schrödinger flat band with a distorted skyrmion texture, the integrated weight grows linearly with the moiré period and the resonant peak grows cubically.