Stable Pfaffian State in Bilayer Graphene

Here, we show that the incompressible Pfaffian state originally proposed for the $\frac{5}{2}$ fractional quantum Hall states in conventional two-dimensional electron systems can actually be found in a bilayer graphene at one of the Landau levels. The properties and stability of the Pfaffian state at this special Landau level strongly depend on the magnetic field strength. The graphene system shows a transition from the incompressible to a compressible state with increasing magnetic field. At a finite magnetic field of $\sim 10\mathrm{T}$, the Pfaffian state in bilayer graphene becomes more stable than its counterpart in conventional electron systems.

Figure 1

(a) Few lowest LLs of a bilayer graphene, shown for $\Delta U=100\mathrm{meV}$ and $t=400\mathrm{meV}$. The two solid red (light gray) lines belonging to different valleys correspond to the LLs where the $\nu =\frac{1}{2}$ Pfaffian state can be observed. (b) Overlap of the exact many-particle ground state with the Pfaffian function. (c) Excitation gap of the $\nu =\frac{1}{2}$ state. The results are for $N=14$ and $2S=25$ and the zero bias voltage. The black and red (light gray) lines correspond to $t=400\mathrm{meV}$ and 300 meV, respectively. Here, the energy unit is ${\epsilon }_c=e^2/\kappa {\ell }_0$.

Figure 2

(a) The excitation gap (b) and the overlap with the Pfaffian state versus the bias voltage, $\Delta U$. Here, $N=14$ and $2S=25$, and the magnetic field is 10 T.

Figure 3

(a) The excitation gap and (b) the overlap with the Pfaffian state are shown for different number of electrons: $N=8$, 10, and 14. The magnetic field is 10 T, and the bias voltage is zero.

Figure 4

Trajectory of the interelectron interaction with varying magnetic fields, shown by a thick solid red (light gray) line in the plane $(V_1/V_5)-(V_3/V_5)$ for the “special” LL of the bilayer graphene. The initial point of the trajectory (at $B=0$) corresponds to the nonrelativistic system at $n=1$ LL, while the final point (at $B=\infty$) corresponds to the nonrelativistic system at the $n=0$ LL. The shaded region illustrates the compressible $\nu =\frac{1}{2}$ state, while the blank region corresponds to the incompressible $\nu =\frac{1}{2}$ state (Ref. 14). The crossing of the boundary between the compressible and incompressible states occurs at $B\sim 100\mathrm{T}$ for the hopping integral $t=400\mathrm{meV}$. The blue (dark gray) dashed line shows the region of large overlap with the Pfaffian state (Ref. 14).