Non-Abelian Fibonacci quantum Hall states in 4-layer rhombohedral stacked graphene

It is known that $n$-degenerate Landau levels with the same spin-valley quantum number can be realized by $n$-layer graphene with rhombohedral stacking under magnetic field $B$. We find that the wave functions of degenerate Landau levels are concentrated at the surface layers of the multi-layer graphene if the dimensionless ratio $\eta = \gamma_1/(v_F\sqrt{2e\hbar B/c}) \approx 9/\sqrt{B[\text{Tesla}]} \gg 1$, where $\gamma_1$ is the interlayer hopping energy and $v_F$ the Fermi velocity of single-layer graphene. This allows us to suggest that: 1) filling fraction $\nu=\frac12$ (or $\nu_n = 5\frac12$) non-Abelian state with Ising anyon can be realized in three-layer graphene for magnetic field $B \in [ 2 , 9]$ Tesla; 2) filling fraction $\nu=\frac23$ (or $\nu_n = 7\frac13$) non-Abelian state with Fibonacci anyon can be realized in four-layer graphene for magnetic field $B \in [ 5 , 9]$ Tesla. Here, $\nu$ is the total filling fraction in the degenerate Landau levels, and $\nu_n$ is the filling fraction measured from charge neutrality point which determines the measured Hall conductance. We have assumed the following conditions to obtain the above results: the exchange effect of Coulomb interaction polarizes the $SU(4)$ spin-valley quantum number in the degenerate Landau levels and effective dielectric constant $\epsilon \gtrsim 10$ to reduce the Coulomb interaction. The high density of states of multi-layer graphene helps to reduce the Coulomb interaction via screening.