Interaction-Driven Intervalley Coherence with Emergent Kekul\'e Orbitons

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 non-interacting low-energy physics. In the weak-coupling regime, the Dirac fermions are gaped 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 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 phase. The classical ground-state manifold in the Mott regime enjoys a continuous symmetry characterized by the intervalley coherent phase. Finally, the quantum fluctuation selects a unique ground state with emergent Kekul\'e orbitons through the order-by-disorder mechanism. Our findings provide insights for a direction of searching for Kekul\'e distortion in correlated multi-orbital systems.