Reduced density-matrix functional theory in quantum Hall systems

We apply reduced density-matrix functional theory to the parabolically confined quantum Hall droplet in the spin-frozen strong magnetic field regime. One-body reduced density-matrix functional method performs remarkably well in obtaining ground states, energies, and observables derivable from the one-body reduced density matrix for a wide range of system sizes. At the strongly correlated regime, the results go well beyond what can be obtained with the density-functional theory. However, some of the detailed properties of the system, such as the edge Green’s function, are not produced correctly unless we use the much heavier two-body reduced density-matrix method.

Figure 1

(Color online) The minimum interaction energy $V_{\text{ee}}$ at each angular momentum $M$ for (a)–(c) four and (d)–(f) six electrons. The exact diagonalization ground states detected by the convex envelope are marked with circles in (a) and (d).

Figure 2

(Color online) The occupation numbers for selection of the four-electron (upper panel) and six-electron (lower panel) ground states indicated in Figs. 1a, 1d, respectively. A bar that fills the assigned height corresponds to occupation number 1. Occupations are ordered according to the increasing single-particle angular momentum $m$ of the lowest Landau-level orbitals starting with 0 at the left. Because most likely location of electron at orbital $m$ is at distance $r=\sqrt{m}$ from the center, one can think the set of orbitals as a radially discretized disk.

Figure 3

(Color online) The occupation numbers at the angular momentum of the 1/3 Laughlin state for (a) $N=10$, (b) $N=20$, and (c) $N=30$ electrons with $3\left(N-1\right)+1$ natural orbitals. MC is the exact Laughlin wave-function result extracted with Monte Carlo. The dashed line marks 1/3 occupation.

Figure 4

(Color online) (a) 30 electron-density profiles corresponding to Fig. 3c shifted vertically by 0.5 unit. Red (dark gray) dashed line is the phenomenological estimate for the density oscillations at the thermodynamic limit. [(b) and (c)] The decay of the edge Green’s function calculated at $r=\sqrt{3\left(N-1\right)}+1$ from the occupations in Figs. 3a, 3b, respectively. The short red lines illustrate slopes $-3$ (theoretical prediction for $\nu =1/3$) and $-1$ (Fermi liquid).

Figure 5

(Color online) The decay of the edge Green’s function corresponding to the inset three-electron occupation numbers. As previously, the short red lines illustrate slopes $-g=-3$ and $-1$, and $G_{\text{E}}$ is evaluated at $r=\sqrt{3\left(N-1\right)}+1$.

Figure 6

The pair-correlation function $g\left(z_1,z_2\right)={\rho }_2\left(z_1,z_2\right)/\rho \left(z_1\right)$ with the first coordinate placed at the density maximum of the negative $y$ axis $z_1=-i\sqrt{3}$ for (a) exact diagonalization, (b) 2-RDMFT, and (c) 1-RDMFT with $\mathrm{P}\text{−}0.675$. Contours are separated by $0.051/\pi l^2$ and start from $0.051/\pi l^2$ in (a) and (b) and from 0 in (c).

Figure 7

(Color online) Illustration of the 2-RDM ${\Gamma }_{\left(kl\right)}^{\left(ij\right)}$ for three electrons in the $\nu =1/3$ state computed with (a) exact diagonalization and (b) 2-RDMFT. The same color bar applies to both figures while zero values are left white. The negative matrix elements are indicated by black edge.