Amorphous Kane-Mele model in disordered hyperuniform two-dimensional networks*

As a prototype system of the quantum spin Hall effect, the Kane-Mele model which was proposed initially in graphene promotes the search for two-dimensional topological materials of hexagonal lattices. Here we generalize the Kane-Mele model to exotic amorphous systems which possess a remarkable structural property called hyperuniformity. We show that, in general, the Quantum spin Hall state still survives in disordered hyperuniform lattices that are constructed by structural transformation involving Stone-Wales defects. However, compared to that in the honeycomb lattice, the gapped topological region in the phase diagram shrinks and the size of the corresponding topological gap decreases remarkably in disordered hyperuniform lattices. By introducing random vacancies in either perfectly ordered or disordered hyperuniform lattices, we further show that the degradation or destruction of hyperuniformity is detrimental to the existence of gapped topological states. We therefore propose that the hyperuniform metric of an amorphous lattice, which quantifies the extent of disordered hyperuniformity, reflects its ability to preserve topological states. Our finding not only establishes the possible underlying link between electronic topology and disordered hyperuniformity but also provides useful guidance for the seeking of topological materials in amorphous states of matter.