Kohn-Sham Theory of the Fractional Quantum Hall Effect

We formulate the Kohn-Sham (KS) equations for the fractional quantum Hall effect by mapping the original electron problem into an auxiliary problem of composite fermions that experience a density dependent effective magnetic field. Self-consistent solutions of the KS equations demonstrate that our formulation captures not only configurations with nonuniform densities but also topological properties such as fractional charge and fractional braid statistics for the quasiparticles excitations. This method should enable a realistic modeling of the edge structure, the effect of disorder, spin physics, screening, and of fractional quantum Hall effect in mesoscopic devices.

Figure 1

Density profile for $1/3$ droplets. This figure shows the density of a system of $N$ composite fermions. ${\rho }_0$ is the density for Laughlin’s $1/3$ wave function [6], and ${\rho }_{\mathrm{ED}}$ is obtained from exact diagonalization (ED) of the Coulomb interaction at total angular momentum $L_{\text{total}}=3N(N-1)/2$ [28]. The density ${\rho }_{\mathrm{DFT}}$ is calculated from the solution of the KS equations for composite fermions in an external potential produced by a uniform positively charged disk of radius $R$ so that $\pi R^2{\rho }_b=N$. The total angular momentum of the CF state is $L_{\mathrm{tot}}^{*}$, which is related to the total angular momentum of the electron state by $L_{\mathrm{tot}}=L_{\mathrm{tot}}^{*}+N(N-1)$ [29]. The CF-DFT solution produces $L_{\mathrm{tot}}^{*}=N(N-1)/2$, which is consistent with $L_{\mathrm{tot}}=3N(N-1)/2$. All densities are quoted in units of $(2\pi l_B^2{)}^{-1}$, the density at $\nu =1$. We take ${\rho }_b=1/3$.

Figure 2

Screening and fractional charge. This figure shows how the $1/3$ state screens a charged impurity of strength $Q=\pm e$ located at a perpendicular distance $h$ from the origin. (a)–(e), (k)–(o) Self-consistent density ${\rho }_{\mathrm{DFT}}(\mathit{r})$. Also shown are ${\rho }_{\mathrm{DFT}}^0(\mathit{r})$, the “unperturbed” density (for $Q=0$), and ${\rho }_b$, which is the density of the positively charged background. (f)–(j) Occupation of renormalized $\mathrm{\Lambda }\mathrm{Ls}$ in the vicinity of the origin; each composite fermion is depicted as an electron with two arrows, which represent quantized vortices. (The single particle angular momentum is given by $m=-n,-n+1,\dots$ in the $n$th $\mathrm{\Lambda }\mathrm{L}$.) (p) Evolution of the excess charge $\delta q$ and the total CF angular momentum $L_{\mathrm{tot}}^{*}$ as a function of the impurity potential strength at the origin $V_{\mathrm{imp}}(r=0)=Q/h$. Change in the charge at the origin is associated with a change in $L_{\mathrm{tot}}^{*}$. The system contains a total of $N=50$ composite fermions. For $h=\infty$, we have $L_{\mathrm{tot}}^{*}=1225$ and $\delta q=0$. For one and two quasiholes, we have $L_{\mathrm{tot}}^{*}=1225$ and 1275, whereas for one, two, and three quasiparticles we have $L_{\mathrm{tot}}^{*}=1175$, 1127, and 1078, precisely as expected from the configurations in (f)–(j) [29].

Figure 3

Fractional braid statistics. (a) Electron density for a system with a quasihole in angular momentum $m$ orbital, with $m$ changing from 1 to 20 for the curves from the bottom to the top. (Each successive curve has been shifted up vertically for clarity.) (b) The same in the presence of another quasihole at the origin. For each $m$, we indicate the expected position of the outer quasihole (red cross) as well as the position obtained from the DFT density determined by locating the local minimum (blue circle). (c) Calculated statistics parameter $\alpha \equiv (r_{\mathrm{DFT}}^2-r_{\mathrm{DFT}}^{\prime 2})/6l_B^2$. The calculation has been performed for $N=200$ composite fermions at ${\nu }_0=1/3$.