  Recent scanning tunneling microscopy experiments on graphene at charge
neutrality under strong magnetic fields have uncovered a ground state
characterized by Kekul\'e distortion (KD). In contrast, non-local spin and
charge transport experiments in double-encapsulated graphene, which has a
higher dielectric constant, have identified an antiferromagnetic (AF) ground
state. We propose a mechanism to reconcile these conflicting observations, by
showing that Landau-level mixing can drive a transition from AF to KD with the
reduction of the dielectric screening. Our conclusion is drawn from studying
the effect of Landau-level mixing on the lattice-scale, valley-dependent
interactions to leading order in graphene's fine structure constant $\kappa =
e^2/(\hbar v_F \epsilon)$. This analysis provides three key insights: 1)
Valley-dependent interactions remain predominantly short-range with the $m=0$
Haldane pseudopotential being at least an order of magnitude greater than the
others, affirming the validity of delta-function approximation for these
interactions. 2) The phase transition between the AF and KD states is driven by
the microscopic process in the double-exchange Feynman diagram. 3) The
magnitudes of the coupling constants are significantly boosted by remote Landau
levels. Our model also provides a theoretical basis for numerical studies of
fractional quantum Hall states in graphene.
