Does $mathbb{Z}_2$-flux Disorder Freeze Dirac Fermions?

Motivated by many contemporary problems in condensed matter physics where matter particles experience random gauge fields, we investigate the physics of fermions on a square lattice with $pi$-flux (that realizes Dirac fermions at low energies) subjected to flux disorder arising from a random $mathbb{Z}_2$ gauge field. At half-filling where the system possesses BDI symmetry, we show that a critical phase is realized with the states at the chemical potential (zero energy) showing a multifractal character. The multifractal properties depend on the concentration $c$ of the $pi$-flux defects and are characterized by the singularity spectrum, Lyapunov exponents, and transport properties. For any concentration of flux defects, we find that the multi-fractal spectrum shows termination, but {em not freezing}. Away from half-filling (at finite energies), we show that the fermionic states are localized, for any value of $c$. We formulate a field theory that can describe this physics, and point to the intrinsically non-perturbative nature of the flux-perturbations. These results can throw light on a class of problems where fermions experience random gauge fields.