We evaluate the transport gaps in the most prominent fractional quantum Hall states in the $n=0$ and $n=1$ Landau levels of graphene, accounting for the Coulomb interaction, lattice-scale anisotropies, and one-body terms. We find that the fractional phases in the $n=0$ Landau level are bond ordered, while those in the $n=1$ Landau level are spin-valley entangled. This resolves a long-standing experimental puzzle [F. Amet et al., Nat. Commun. 6, 5838 (2015)] of the contrasting Zeeman dependence of the transport gaps in the two Landau levels. The spin-valley entangled phases host gapless Goldstone modes that can be probed via bulk thermal transport measurements. As a byproduct of our computations, we place strong constraints on the values of the microscopic anisotropic couplings such that these are consistent with all known experimental results.

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• Quantum Hall effect
• Transport phenomena
• Variational approach