Quantum Hall liquid on a noncommutative superplane

Supersymmetric quantum Hall liquids are constructed on a noncommutative superplane. We explore a supersymmetric formalism of the Landau problem. In the lowest Landau level, there appear spinless bosonic states and spin-1/2 down fermionic states, which exhibit a superchiral property. It is shown that the Laughlin wave function and topological excitations have their superpartners. Similarities between supersymmetric quantum Hall systems and bilayer quantum Hall systems are discussed.

Figure 1

The left sector about the vertical dashed axis is a “bosonic sector” for the dynamical supersymmetry, and the right sector is a “fermionic sector.” The curved solid arrows represent the nondynamical supersymmetric transformation generated by $(\tilde{Q},{\tilde{Q}}^{†})$, while the curved dashed arrows represent the dynamical supersymmetry transformation generated by $(Q,Q^{†})$. In each of the higher LLs, there are spinless, spin-1/2 up, and spin-1/2 down particles due to the existence of the $\mathcal{N}=2$ supersymmetry, while, in the LLL, the system possesses only $\mathcal{N}=1$ nondynamical supersymmetry, and there appear only spinless and spin-1/2 down particles.

Figure 2

There are spinless bosons and spin-1/2 down fermions in the LLL. They are on the radially symmetric orbits, and rotate around the origin with the same frequency.

Figure 3

In the left figure, the black blobs represent the spinless particles described by the Laughlin state for $\nu =1$. Because of the flux penetration, the spinless particles are pushed outwards by $\delta R=\sqrt{2}{\ell }_B$, and a quasihole is generated at the origin. The penetration of the bosonic flux keeps the particles spinless, while the penetration of the fermionic flux changes the spin of each particle from 0 to $-1/2$.