15 Topological Dynamics of Spin Texture Based Metamaterials
431
Fig. 15.20 a Dependence of the topological invariants Chern number and Z 6 Berry phase on the
ratio d 2 /d 1 when d 1 is fixed at 2.5r . b Schematic plot of the parallelogram-shaped vortex lattice with
armchair edges. c Eigenfrequencies of collective vortex gyration under different ratios d 2 /d 1 with
the red segment denoting the corner state phase. d Phase diagram of the system with pentagonal
stars of different colors representing three typical parameters of d 1 and d 2 for different phases
considered in the subsequent calculations and analyses. Images are taken from [137]
the gapless band structure for perfect graphene nanoribbons, the imaginary secondnearest hopping term opens a gap at the Dirac point and supports a topologically
protected first-order chiral edge state [83, 125]. For d 1 = 2.08r and d 2 = 3.6r (red
star in Fig. 15.20d), one can clearly see two distinct edge bands, in addition to bulk
ones, as shown in Fig. 15.21d. These localized modes are actually not topological
because they maintain the bidirectional propagation nature, which is justified by
the fact that the wave group-velocity dω/dk x can be either positive or negative at
different k x points. Below, we will show that higher-order topological corner states
exactly emerge around these edge bands when the system is decreased to be finite in
both dimensions.
431
Fig. 15.20 a Dependence of the topological invariants Chern number and Z 6 Berry phase on the
ratio d 2 /d 1 when d 1 is fixed at 2.5r . b Schematic plot of the parallelogram-shaped vortex lattice with
armchair edges. c Eigenfrequencies of collective vortex gyration under different ratios d 2 /d 1 with
the red segment denoting the corner state phase. d Phase diagram of the system with pentagonal
stars of different colors representing three typical parameters of d 1 and d 2 for different phases
considered in the subsequent calculations and analyses. Images are taken from [137]
the gapless band structure for perfect graphene nanoribbons, the imaginary secondnearest hopping term opens a gap at the Dirac point and supports a topologically
protected first-order chiral edge state [83, 125]. For d 1 = 2.08r and d 2 = 3.6r (red
star in Fig. 15.20d), one can clearly see two distinct edge bands, in addition to bulk
ones, as shown in Fig. 15.21d. These localized modes are actually not topological
because they maintain the bidirectional propagation nature, which is justified by
the fact that the wave group-velocity dω/dk x can be either positive or negative at
different k x points. Below, we will show that higher-order topological corner states
exactly emerge around these edge bands when the system is decreased to be finite in
both dimensions.