Importance of interband transitions for the fractional quantum Hall effect in bilayer graphene

Several recent works have proposed that electron-electron interactions in bilayer graphene can be tuned with the help of external parameters, making it possible to stabilize different fractional quantum Hall states. In these prior works, phase diagrams were calculated based on a single Landau level approximation. We go beyond this approximation and investigate the influence of polarization effects and virtual interband transitions on the stability of fractional quantum Hall states in bilayer graphene. We find that for realistic values of the dielectric constant, the phase diagram is strongly modified by these effects. We illustrate this by evaluating the region of stability of the Pfaffian state.

Figure 1

The dependence of the lowest LL energies on (a) the magnetic field $B$ at $U=50\text{meV}$ and (b) the gap parameter $U$ at $B=10\text{T}$. Each level is labeled by a pair of quantum numbers $(n,\xi )$.

Figure 2

Feynman diagrams[18] showing renormalization of the electron-electron interaction due to (a) the vacuum polarization processes, and (b) the simplest processes involving virtual hopping of one or both of the two interacting electrons from the $n$th LL to the $n^{\prime }$th LL.

Figure 3

The region [(a) for the $(1,-1)$ LL; (b) for the $(2,-1)$ LL] of the applicability of perturbative analysis for fixed values of $\kappa =5$, 10, and 15. The size of the region increases with increasing $\kappa$. The thick black line shows where the maximum overlap with the Moore-Read Pfaffian for the bare Coulomb interaction is achieved.

Figure 4

Color plot of the overlap of the ground state with the Moore-Read Pfaffian for 12 particles at $U=50\text{meV}$ as a function of the magnetic field $B$ and the dielectric constant $\kappa$ [(a) for the $(1,-1)$ LL; (b) for the $(2,-1)$ LL]. Contours show the lines of constant overlap. The region where perturbative analysis is not applicable is hatched.