Model anisotropic quantum Hall states

Model quantum Hall states including Laughlin, Moore-Read, and Read-Rezayi states are generalized into appropriate anisotropic form. The generalized states are exact zero-energy eigenstates of corresponding anisotropic two- or multibody Hamiltonians, and explicitly illustrate the existence of geometric degree of freedom in the fractional quantum Hall effect. These generalized model quantum Hall states can provide a good description of the quantum Hall system with anisotropic interactions. Some numeric results of these anisotropic quantum Hall states are also presented.

Figure 1

The density profiles of the generalized LLL wave functions ${\psi }_{0m}(\gamma =1/2,\mathit{r})$ with $m=0$ (a), $m=1$ (b), $m=2$ (c), and $m=3$ (d).

Figure 2

The density profiles of ${\psi }_{03}(\gamma ,\mathit{r})$ with $\gamma =e^{i\phi }/2$ for $\phi =0$ (a), $\phi =\pi /2$ (b), $\phi =\pi$ (c) and $\phi =3\pi /2$ (d). Note that the zeros are localized at $z=0$ and $\pm i\sqrt{3}{z}_0$.

Figure 3

The density profiles [in units of $1/\left(2\pi {\ell }^2\right)$] of the $\nu =1/3$ anisotropic Laughlin state with $\gamma =1/2$ (a) and $\gamma =0$ (b) for an $N=10$ system. Note the yellow bulk whose value approximates $1/3$.

Figure 4

The pair correlation function for $\nu =1/3$ Laughlin state with $\gamma =1/2$ (a) and $\gamma =0$ (b) of an $N=10$ system.

Figure 5

The pair correlation function $g(\gamma ,0,y)$, $g(\gamma ,x,0)$ with $\gamma =1/2$ and $g(0,x,0)$ of an $N=10$ anisotropic Laughlin state. The inset (a) shows the asymptotic (green solid) line of $g(0,x,0)$, which is proportional to $x^6$. The inset (b) shows the asymptotic (green solid) line of $g(\gamma ,x,0)$ [$g(\gamma ,0,y)$], which is proportional to $x^2$ ($y^2$).

Figure 6

The hard-core interaction energy per particle ${\mathcal{E}}_{\mathrm{hc}}^{\left(N\right)}\left(\gamma \right)$ with respect to the distort parameter $\gamma$ for $N=3–8$ system. Here, we also plot ${\gamma }^2$ (solid line) for comparison.

Figure 7

The dipole-dipole interaction energy per particle ${\mathcal{E}}_{\mathrm{dd}}^{\left(N\right)}(\theta ={30}^{\circ },\gamma )$ with respect to the distort parameter $\gamma$ for $N=3–8$ system. Here, axial oscillator length $d$ is set as $0.01\ell$.