Composite-fermion antiparticle description of the hole excitation in a maximum-density droplet with a small number of electrons

The maximum-density droplet of quantum dots in a high magnetic field, which is a finite-size realization of the state at filling factor 1, becomes unstable to the creation of a central hole (provided it contains a small number of electrons) as the magnetic field is increased or the strength of the confinement potential reduced. The simplest model for the hole is as a vortex at the center, which, however, is renormalized by edge excitations. We show that an accurate description of the actual hole state is achieved in terms of a “composite-fermion antiparticle,” which is surprising in view of the fact that composite fermions are thought to be relevant only in the fractional Hall regime. We extend these considerations to multiple holes in the maximum-density droplet, and also to the quasihole at $\nu =1∕3$. The effect of Landau-level mixing is also considered through a diffusion Monte Carlo calculation.

Figure 1

Configuration of the vortex state. The quantities $n$ and $m$ denote the Landau-level index and the single-particle angular momentum, respectively. Filled solid circles represent occupied orbitals while unoccupied ones are denoted by open dashed circles. All orbitals beyond the last electron are unoccupied.

Figure 2

Examples of other configurations that have the same total angular momentum as the configuration of Fig. 1.

Figure 3

Densities for the true hole (solid line), the vortex (dashed line), and the CF antiparticle (dots) for one hole of the $\nu =1$ maximum-density droplet of $N=5$. The quantity $\rho$ is the one-electron areal density [$\rho ={\left(2\pi l^2\right)}^{-1}$ for $\nu =1$], $l$ is the (effective) magnetic length, and $r$ is the distance from the origin.

Figure 4

(Color online) The zeroes of (a) the vortex state $\left({\Psi }_{\text{vortex}}\right)$, (b) the true hole $\left({\Psi }_{\text{exact}}\right)$, and (c) the CF antiparticle $\left({\Psi }_{\overline{\mathrm{CF}}}\right)$ for $N=6$ at $L=21$. Five electrons are fixed at some typical positions marked by crosses and the zeroes of the last electron are denoted by open circles. The different shades represent the variation of the phase $\theta$ of each wave function in the interval $-\pi ⩽\theta <\pi$ and the phase changes from $\pi$ to $-\pi$ are denoted by the borders between dark and light shades. The plot is shown in the window $-10l⩽x⩽10l$ and $-10l⩽y⩽10l$. The exact density has a maximum at radius $r\approx 2.45l$.

Figure 5

Schematic diagram of $\Lambda$-level occupation of composite fermions for one hole of the $\nu =1$ maximum-density droplet. The quantity $n^{*}$ is the $\Lambda$-level index.

Figure 6

Density at the origin as a function of $1∕N$ for the true hole $\left[{\Psi }_{\text{exact}}(▵)\right]$ and the CF antiparticle $\left[{\Psi }_{\overline{\mathrm{CF}}}(●)\right]$. The density of the vortex state at the origin is zero for all $N$. The numerical uncertainty for the CF antiparticle is comparable to the symbol size.

Figure 7

Addition energy for $N=2$, 3, …, 8 as a function of magnetic field in the vicinity of MDD-hole excitation. A transition from MDD to hole at a given $N$ leads to cusps in two adjacent curves, which are indicated by vertical dashed lines with arrows. Note that only the MDD-hole transition is considered.

Figure 8

Addition energy at the MDD-hole transitions as a function of magnetic field, obtained from vortex (solid line), CF antiparticle (dotted line), vortex with LL mixing (dot-dashed line), and CF antiparticle with LL mixing (dashed line).

Figure 9

Densities for vortex (dashed line), CF antiparticle (solid circle), vortex with LL mixing (dot-dashed line), and CF antiparticle with LL mixing (solid line). The densities are for one hole of the maximum-density droplet of five electrons with $B=4\mathrm{T}$.

Figure 10

Densities for the true holes, the vortices, and the CF antiparticles for (a) two and (b) three holes of the maximum-density droplet of five electrons.

Figure 11

Densities for the true hole $\left({\Psi }_{\text{exact}}\right)$, the vortex $\left({\Psi }_{\text{vortex}}^{(1∕3)}\right)$, and the CF antiparticle $\left({\Psi }_{\overline{\mathrm{CF}}}^{(1∕3)}\right)$ for quantum Hall droplets at $\nu =1∕3$.

Figure 12

Densities for the exact state (solid line), the CF-antiparticle wave function (dots), and the vortex wave function (dashed line) for two and three quasiholes (upper and lower panels, respectively) at $\nu =1∕3$.