Kohn-Sham Theory of the Fractional Quantum Hall Effect

The Kohn-Sham (KS) formulation of the density functional theory (DFT) fails for FQHE because the FQHE state cannot be mapped into a system of non-interacting electrons. We instead map the FQHE system into a system of non-interacting composite fermions, which experience an effective magnetic field that depends on density. We generalize the Hohenberg-Kohn theorem to systems with a density-dependent kinetic energy operator, and formulate the KS equations for the FQHE. We also show the equivalence of our formulation with the generalized Kohn-Sham scheme. Numerical results show that our method produces densities that are good approximations to those obtained from many-body wave functions for states with general occupation configurations. As two applications of our method, we (i) obtain the density profile at the edge of fractional quantum Hall states, finding edge reconstruction as the confinement potential becomes softer; and (ii) show how an external charge produces a screening cloud with a fractionally quantized charge accumulation. We speculate on the implications of our method for general strongly correlated systems.