C-symmetric Chern insulators

Chern insulators (CIs) have attracted great interests for the realization of quantum Hall states without external magnetic field. Recently, CIs have been studied on various curved lattices, such as the cone-like lattices and the fullerenes. However, few works were reported how to identify curved-CIs and explore their topological phase transitions (TPTs). In this paper, we systemically investigate the curved-CIs with arbitraryn-fold rotational symmetry on cone-like and saddle-like lattices (also dubbed asC-symmetric CIs), by 'cutting and gluing' unit sectors with a disk geometry. TheseC-symmetric CIs can be identified based on the chiral edge states, the real-space Chern number and the quantized conductance. Here, we propose two ways to calculate the real-space Chern number, the Kitaev's formula and the local Chern marker. Furthermore, the TPTs of curved CIs are explored by tuning staggered flux and on-site mass.