Multifractal Analysis of Dirac Fermions Coupled to Disordered Abelian Gauge Field in Curved Space*

A recent paper by Gruzberg et al. [Phys. Rev. B 95, 125414] argued that a certain form of geometric disorder in models for the integer quantum Hall transition may change the universal critical properties of the transition. In the continuum this disorder is described by a fluctuating 2D metric. Motivated by this, we examine a related model of Dirac fermions coupled to random gauge potentials on an arbitrary two-dimensional curved manifold of genus 0 with a fixed (non-fluctuating) metric. We consider the effects of Abelian gauge potential disorder on the averaged generalized inverse participation ratios, and find that their scaling with the system size is described by anomalous multifractal exponents Δ(q) = -(g/π)q(q-1), where g is the strength of the gauge disorder. This scaling does not depend on the metric of the manifold and is the same as that of the Dirac fermions in flat space. The result can be verified using both the replica trick and the supersymmetry method for disorder averaging over the gauge field.