(a) The DOS of TI with triangular flux-superlattice: the mid-gapped states are induced by the flux-superlattice with flux-superlattice constant in the case of T = 1.0, T′ = 0.1. (b) The illustration of details of the mid-gap band in panel (a). (c) The edge states of the composite system, of which the parent Haldane model has periodic boundary along both x and y directions, and while the flux-superlattice has periodic boundary along y direction and open boundary condition along x direction. From Ref. [17].