Influence of the Dirac sea on phase transitions in monolayer graphene under strong magnetic fields

Recent scanning tunneling microscopy experiments have found Kekul\'e-distorted (KD) ordering in graphene subjected to strong magnetic fields, a departure from the antiferromagnetic (AF) state identified in earlier transport experiments on double-encapsulated devices with larger dielectric screening constant $\ensuremath{\epsilon}$. This variation suggests that the magnetic anisotropic energy is sensitive to the dielectric screening constant. To calculate the magnetic anisotropic energy without resorting to perturbation theory, we adopted a two-step approach. First, we derived the bare valley-sublattice-dependent interaction coupling constants from microscopic calculations and accounted for the leading logarithmic divergences arising from quantum fluctuations by solving renormalization group flow equations in the absence of a magnetic field, from the carbon lattice scale up to the much larger magnetic length. Subsequently, we used these renormalized coupling constants to perform nonperturbative, self-consistent Hartree-Fock calculations. Our results demonstrate that the ground state at neutrality ($\ensuremath{\nu}=0$) transitions from an AF state to a spin-singlet KD state when dielectric screening and magnetic fields become small, consistent with experimental observations. For filling fraction $\ensuremath{\nu}=\ifmmode\pm\else\textpm\fi{}1$, we predict a transition from spin-polarized charge-density wave states to a spin-polarized KD state when dielectric screening and magnetic fields become small. Our self-consistent Hartree-Fock calculations, which encompass a large number of Landau levels, reveal that the magnetic anisotropic energy receives substantial contributions from the Dirac sea when $\ensuremath{\epsilon}$ is small. Our paper provides insights into how the Dirac sea, which contributes one electron per graphene unit cell, affects the small magnetic anisotropic energy in graphene.