Quantum geometric localization length and localization criticality in an ideally flat Chern band

Abstract

We propose that the localization length in an isolated, ideally flat Chern band is set by quantum geometry. We explore the corresponding localization transition and its critical scaling by applying transfer matrix calculations in the maximally localized hybrid Wannier basis, whose spatial spread is exactly characterized by a quantum geometric length. Remarkably, upon tuning the quantum metric of the Chern band, we observe a crossover from a universal regime controlled by the Dirac fixed point to a non-universal regime with continuously varying critical exponents. Within the universal regime, the localization length exhibits a pronounced linear dependence on the quantum geometric length, supporting its quantum geometric nature. These findings provide a novel quantum geometric perspective on the localization in quantum Hall systems such as twisted moiré superlattices, and shed new light on the long-standing controversy over the criticality of the integer quantum Hall transition.