The electron-electron interaction in the Landau levels of bilayer graphene is markedly different from that of conventional semiconductors such as GaAs. We show that in the zeroth Landau level of bilayer graphene, in the orbital which is dominated by the nonrelativistic second Landau level wave function, by tuning the magnetic field, a topological quantum phase transition from an Abelian composite fermion to a non-Abelian parton fractional quantum Hall state can be induced at filling factors $1/2$, 2/5, and $3/7$. The parton states host exotic anyons that can potentially be utilized to store and process quantum information. Intriguingly, some of these transitions may have been observed in a recent experiment (K. Huang et al., arXiv:2105.07058).

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• Anyons
• Composite fermions
• Fractional quantum Hall effect
• Bilayer graphene