Theory of anomalous Hall effect from screened vortex charge in a phase disordered superconductor

Motivated by recent experiments showing evidence for chiral superconductivity in an anomalous Hall phase of tetralayer graphene, we study the relation between the normal state anomalous Hall conductivity and that in the phase disordered state above the critical temperature of the superconductor. By a numerical calculation of superconductivity in an anomalous Hall metal, we find that a difference in vortex and antivortex charge is determined by the Fermi surface Berry phase. Combining this with the vortex dynamics in a back-ground supercurrent leads to a Hall response in the phase disordered state of the superconductor that is close to the normal state anomalous Hall response. However, using a gauge-invariant superconducting response framework, we find that while vortex charge is screened by interactions, the screening charge, after a time-delay, reappears in the longitudinal current. Thus, the dc Hall conductivity in this phase, instead of matching the screened vortex charge, matches the ac Hall conductance in the superconducting and normal phase, which are similar.