Anomalous Transport Gaps of Fractional Quantum Hall Phases in Graphene Landau Levels are Induced by Spin-Valley Entangled Ground States

We evaluate the transport gaps in the most prominent fractional quantum Hall states in the $\mathbf{n}=0$ and $\mathbf{n}=1$ Landau Levels of graphene, accounting for the Coulomb interaction, lattice-scale anisotropies, and one-body terms. We find that the fractional phases in the $\mathbf{n}=0$ Landau level are bond-ordered, while those in the $\mathbf{n}=1$ Landau level are spin-valley entangled. This resolves a long-standing experimental puzzle [Amet, $\mathit{\text{et}}\mathit{\text{al.}}$, Nat. Comm. $\mathbf{6}$, 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.