Realizing Universal Edge Properties in Graphene Fractional Quantum Hall Liquids

Universal chiral Luttinger liquid behavior has been predicted for fractional quantum Hall edge states, but so far has not been observed experimentally in semiconductor-based two-dimensional electron gases. One likely cause of this absence of universality is the generic occurrence of edge reconstruction in such systems, which is the result of a competition between confinement potential and Coulomb repulsion. We show that due to a completely different mechanism of confinement, edge reconstruction can be avoided in graphene, which allows for the observation of the predicted universality.

Figure 1

(a) Illustration of the system. A metallic gate is placed at a distance $d$ away from a single-layer graphene. Each electron in the graphene has a positive image charge at a distance $2d$ below the graphene layer. (b) Regions of the $d\mathrm{\text{−}}{N}_{\mathrm{orb}}$ parameter space in which the global ground state has the same total angular momentum $M$ as the Laughlin state $M_0=3N(N-1)/2$, for $N=5–10$ electrons. (c) Collapse of regions for different system sizes when (b) is replotted against $N_{\mathrm{edge}}=N_{\mathrm{orb}}-(3N-2)$. The Laughlin-like regions for $d\sim l_B$ (roughly $0.5\lesssim d/l_B\lesssim 1.5$, shaded area) are not very sensitive to system size, hence a promising window for the observation of universal edge physics. (d) The overlaps between the ground-state and Laughlin wave function as a function of $d$ in the Laughlin phase (along selected paths).

Figure 2

Low-energy spectrum for seven electrons on (a) a sphere ($N_{\varphi }=18$, $d=1.0l_B$) and (b) a disk ($N_{\mathrm{orb}}=26$ and $d=1.8l_B$). The ground state is spin fully polarized on the sphere and partially polarized on the disk. $\delta E$ is the energy gap between the ground state and the lowest energy spin fully polarized state.

Figure 3

(a) Zeeman energy necessary to fully polarize the ground state of seven (six for inset) spinful electrons in the $d\mathrm{\text{−}}{N}_{\mathrm{orb}}$ parameter space. (b) The Zeeman energy and the ground state total angular momentum for the seven electrons along the (green) dashed line in (a) (i.e., $N_{\mathrm{orb}}=26$). (c), (d), (e), and (f) plot the spin-resolved density profiles for four selected ground states in the four different spin regions in (b).