Topological Aspects of Dirac Fermions in a Kagom\'e Lattice

The Dirac fermion with linear dispersion in the kagom\'e lattice governs the low-energy physics of different valleys at two inequivalent corners of 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 term 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 kagom\'e lattice. It is further demonstrated that the twisting of Dirac mass with vorticity leads to zero Dirac modes at the vortex core, which are intimately related to fractionalization. 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 zero Dirac modes around the vortex core.