The Dirac fermion with linear dispersion in the kagomé lattice governs the low-energy physics of different valleys at two inequivalent corners of the hexagonal Brillouin zone. The effective Hamiltonian based on the cyclic permutation symmetry of sublattices is constructed to show that the topology of Dirac fermions at these two valleys is characterized by opposite winding numbers. For spinless fermions, the many-particle interactions produce intervalley scattering and drive an intervalley coherent state with spontaneous translation symmetry breaking. The Dirac fermions acquire a mass from the simultaneous charge and bond orderings. In this phase, the developed bond texture underlies a hollow-star-of-David pattern in a tripled Wigner-Seitz cell of the kagomé lattice. It is further demonstrated that the twisting of Dirac mass with vorticity leads to Dirac zero modes at the vortex core, which lead to fractionalization and anyon statistics. The hollow-star-of-David phase is shown to have a distinct ${\mathbb{Z}}_6$ Berry phase with its sign-change counterpart of Dirac mass, i.e., the hexagonal phase, shedding light on the topological origin of Dirac zero modes around the vortex core.

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• Charge density waves
• Topological materials
• Topological order
• Topological phases of matter
• Valley degrees of freedom