Title: Quantum fluctuation and SU(4) symmetry breaking in graphene

Recent scanning tunneling microscopy (STM) experiments have found Kekule-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 $\epsilon$. This discrepancy indicates the sensitivity of the magnetic anisotropic energy to variations in the dielectric screening constant and magnetic field strength, influenced by the high density of states in the $\pi$ bands at large energies. To address these complex interaction effects, we employed a two-step approach: initially, we derived the bare parameters of the theory from microscopic calculations and solved a set of renormalization group equations to account for leading logarithmic divergences due to quantum fluctuations with the Dirac sea. Subsequently, we used these renormalized parameters to perform non-perturbative, self-consistent Hartree-Fock calculations.Our results demonstrate that the ground state transitions from a AF state to a KD state when dielectric screening and magnetic fields become small, consistent with experimental observations.Additionally, 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 $\epsilon$ is small. Our work provide insights into the complex interaction effects observed in STM and transport experiments on multilayer graphene electron gases, even in the absence of a magnetic field.