Fig. 1. Lattice model and bulk-edge correspondence for satellite Dirac cones.

a Schematic of the graphene lattice with nearest-neighbor couplings $t_1$ and TNN couplings $t_3$. b Evolution of the band structure as $t_3$ increases. The red dashed box highlights the appearance of satellite Dirac cones with $t_3=3t_1$. c Distribution of Dirac nodes (black dots) in reciprocal space, with the colormap denoting $\theta$. The dashed hexagon indicates the first BZ. d, e Projected band spectrum and the corresponding winding numbers along $k_x$, respectively. f Evolution of single-band pseudospins $\widehat{\mathbf{n}}$ over a periodic cycle in $k_y$ (toroidal direction). The left panel corresponds to $k_x=0.25$ with the winding number $W=2$, while the right panel corresponds to $k_x=0.36$ with $W=1$. g Eigen spectra for $k_x=0.25$ and $0.36$, obtained using a strip geometry with 40 unit cells. (h) Distributions of the corresponding zero-energy edge states.