The $p$-orbital doublet in a honeycomb lattice is concretely studied with interacting spinless fermions at half filling. The Dirac fermions with linear dispersion at $\pm K$ valleys govern the noninteracting low-energy physics. In the weak-coupling regime, the Dirac fermions are gapped due to the spontaneous generation of mass terms through a uniform axial orbital ordering, rendering the system into the quantum anomalous Hall insulator phase with a nonzero Chern number. Surprisingly, the intermediate-strength many-particle interaction produces the intervalley coherence between $\pm K$ valleys by developing complex polar orbital orderings in a tripled Wigner-Seitz cell. This phase is shown to have a deep connection with the low-energy physical behavior described by the orbital exchange model in the Mott insulating regime. The classical ground-state manifold enjoys a continuous symmetry characterized by the intervalley coherent phase. Finally, the quantum fluctuation selects a unique ground state with emergent Kekulé orbitons through the order-by-disorder mechanism. Our findings provide insights for a direction of searching for Kekulé distortion in correlated multiorbital systems.

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• Valley degrees of freedom
• Graphene
• Electron-correlation calculations
• Holstein-Primakoff method
• Mean field theory
• Spin wave theory