Density-Functional Theory of the Fractional Quantum Hall Effect

A conceptual difficulty in formulating the density-functional theory of the fractional quantum Hall effect is that while in the standard approach the Kohn-Sham orbitals are either fully occupied or unoccupied, the physics of the fractional quantum Hall effect calls for fractionally occupied Kohn-Sham orbitals. This has necessitated averaging over an ensemble of Slater determinants to obtain meaningful results. We develop an alternative approach in which we express and minimize the grand canonical potential in terms of the composite fermion variables. This provides a natural resolution of the fractional-occupation problem because the fully occupied orbitals of composite fermions automatically correspond to fractionally occupied orbitals of electrons. We demonstrate the quantitative validity of our approach by evaluating the density profile of fractional Hall edge as a function of temperature and the distance from the delta dopant layer and showing that it reproduces edge reconstruction in the expected parameter region.

Figure 1

Exchange-correlation energy $E_{\mathrm{xc}}(\nu )$ [Eq. (5)] and potential $V_{\mathrm{xc}}(\nu )=E_{\mathrm{xc}}+\nu \partial E_{\mathrm{xc}}/\partial \nu$ as a function of the filling factor $\nu$ for several temperatures.

Figure 2

(a) Evolution of the $3/7$ edge as a function of the setback distance $d$ for a small temperature $k_{B}T=0.003$. The $\nu (\mathit{r})$ for successive $d$ are vertically displaced for clarity. Edge reconstruction is seen to occur at $d\approx 1.7$. (b),(c) Evolution of the $3/7$ edge as a function of temperature for two values of $d$. For $d=2.2$ the edge structure melts at $k_{B}T\approx 0.017$. $k_{B}T$ is quoted in units of $e^2/\epsilon l$.

Figure 3

Change in the density of the CF Fermi sea due to the presence of a WC of lattice constant $c$ in a nearby layer at a distance $d$. The three panels, from left to right, have $(d,c)=(3,15)$, (3, 20), (6, 30). The color represents $\mathrm{\Delta }\nu (\mathit{r})$ according to scale shown on top. All lengths are in units of $l$.