Localization renormalization and quantum Hall systems

The obstruction to constructing localized degrees of freedom is a signature of several interesting condensed matter phases. We introduce a localization renormalization procedure that harnesses this property. To demonstrate our method, we distinguish between topological and trivial phases in quantum Hall and Chern insulators. By iteratively removing a fraction of orthogonal basis states, we find that the maximally-localized length scale supported in the residual Hilbert space exhibits a power-law divergence as the fraction of remaining states approaches zero, with an exponent of ν=0.5. In sharp contrast, the localization length converges to a system-size independent constant in the trivial phase.