Spin-1 Dirac dispersion and Chern insulating phases in 2D honeycomb Sierpi\'nski fractal

Graphene-based Sierpi\'nski fractals host a zero-energy chiral mode and spin-1 Dirac dispersions within the nearest-neighbor tight-binding model. However, the presence of complex next-nearest neighbor hopping arising from the local flux and the staggered Semenoff mass terms, modeled within the Haldane Hamiltonian, breaks the time-reversal and spatial inversion symmetries, respectively, and makes these flat bands dispersive. Moreover, they introduce rich topological phases in this class of systems that can be characterized by Chern numbers up to $\pm 3$, i.e., beyond the conventional honeycomb lattice. These observations pave the way for the exploration of 2D periodic fractals beyond graphene, where topological phase transitions can be realized through externally applied fields.