In 1991, it was proposed that fourfold-degenerate Landau levels formed by a single species of electrons could host a non-Abelian fractional quantum Hall (FQH) state with Fibonacci anyons at filling fraction $\nu =\frac{2}{3}$. In this work, we investigate how such degenerate Landau levels can be realized in rhombohedral-stacked tetralayer graphene. We identify the following key conditions which may stabilize the Fibonacci state: (1) A magnetic field of around 20 T is required if surface and interior carbons have the same energy level. If substrate hybridization raises the surface carbon energy level by ${\mathrm{\Delta }}_2=30\mathrm{meV}$ relative to interior carbon, the required field will have a larger range: 15–20 T. For ${\mathrm{\Delta }}_2=45\mathrm{meV}$, the range reaches a maximum: 7–20 T. (2) The displacement field must be tuned to achieve Landau level degeneracy. $\nu =\frac{3}{5}$ Fibonacci FQH states may also be realized in pentalayer rhombohedral graphene with a magnetic field of 12 T, and $\nu =\frac{1}{2}$ states with Ising anyons may occur in trilayer graphene for magnetic fields of 12–20 T at ${\mathrm{\Delta }}_2=0$ or 5–20 T at ${\mathrm{\Delta }}_2=45\mathrm{meV}$. We also study a simple interaction model to explore spin/valley polarization effects, and we see that the Fibonacci state may occur at $\nu =2/3+$ integer filling fractions, where the integer is 0 and 4 for sufficiently weak interaction, or can shift to 2 and 5 under a stronger interaction. The case ${\mathrm{\Delta }}_2=45\mathrm{meV}$ also produces states at negative filling fraction, e.g., $-\frac{2}{3}, -4\frac{2}{3}$. Here $\nu$ is defined with respect to the Hall conductance, ${\sigma }_{xy}=\nu \frac{e^2}{h}$.

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• Flat bands
• Geometric & topological phases
• Landau levels
• Multilayer graphene