430
Z. Li et al.
15.5.2 Corner States and Phase Diagram
Figure 15.19b–d shows the bulk band structures under a variety of lattice parameters.
For d 1 = d 2 = 3.6r (see Fig. 15.19c), it can be found that the highest three bands and
the lowest three bands merged separately, leaving a next-nearest hopping-induced
gap centered at 927 MHz. In this case, the FOTI phase was anticipated [83, 125].
However, the six bands are separated from each other when considering the other
two kinds of parameters (see Fig. 15.19b, d), indicating that the system is in the
insulating state. These insulating phases and the phase transition point can be further
distinguished by calculating Chern number and Z 6 Berry phase.
Figure 15.20a plots the dependence of the Chern number (C) and the Z 6 Berry
phase (θ ) on the parameter d 2 /d 1 . Here the material parameters of Py (Ni 80 Fe 20 ) [88,
130] are used and fix d 1 = 2.5r . In addition, the eigenfrequencies of collective vortex
gyration under different ratios d 2 /d 1 for a parallelogram-shaped (see Fig. 15.20b)
structure are also shown in Fig. 15.20c. By considering the topological invariants and
spectrum simultaneously, one can infer that the system is in the trivial phase when
d 2 /d 1 < 0.9 and 1.08 < d 2 /d 1 < 1.49, in the FOTI phase when 0.9 < d 2 /d 1 < 1.08,
and in the HOTI phase when d 2 /d 1 > 1.49. Haldane model is a well-known example
for breaking the time-reversal symmetry [134]. If we consider the limit case of
honeycomb lattice, i.e., d 1 = d 2 , (15.9) can be exactly mapped to the Haldane model,
as shown in [83] and [125]. The very existence of the chiral edge state is thus naturally
expected. For a breathing honeycomb lattice, our (15.11) represents the generalized
form of the mapping, where the last two terms in the right-hand side are the next-tonearest hopping that breaks the time-reversal symmetry. The complete phase diagram
of system can be obtained by systematically changing d 1 and d 2 , with the results
plotted in Fig. 15.20d: The regions labeled gray, white, and red represent the trivial,
FOTI, and HOTI phases, respectively. Importantly, we find that the boundary for
the phase transition between trivial and FOTI phases depends only weakly on the
choice of the absolute values of d 1 and d 2 but is (almost) solely determined by their
ratio, as indicated by dashed black lines (l 1 : d 2 /d 1 = 0.94 and l 2 : d 2 /d 1 = 1.05)
in the figure. However, the boundary for the phase transition between trivial and
HOTI phases is a linear function l 3 : d 2 = 2.24d 1 − 1.88 (see the dashed red line
in Fig. 15.20d). It is worth noting that the topological charge of the vortex has no
influence on higher-order topology for the reason that the sign of topological charge
just determines the direction (clockwise or anti-clockwise) of gyration, see (15.5).
However, it indeed can affect the chiral edge state (first-order topology). Namely the
chirality of edge state will be reversed if the topological charge changes.
The existence of symmetry-protected states on boundaries is the hallmark of a
topological insulating phase. Figure 15.21b–d shows the energy spectrum of the
ribbon configuration with armchair edges (see Fig. 15.21a) for different choices of
d 1 and d 2 . For d 1 = 3.6r and d 2 = 2.08r (black star in Fig. 15.20d), the system
is in the trivial phase without any topological edge mode (see Fig. 15.21b). For
d 1 = d 2 = 3.6r (blue star in Fig. 15.20d), the lattice considered is identical to a
magnetic texture version of graphene (the perfect honeycomb lattice). In contrast to
Z. Li et al.
15.5.2 Corner States and Phase Diagram
Figure 15.19b–d shows the bulk band structures under a variety of lattice parameters.
For d 1 = d 2 = 3.6r (see Fig. 15.19c), it can be found that the highest three bands and
the lowest three bands merged separately, leaving a next-nearest hopping-induced
gap centered at 927 MHz. In this case, the FOTI phase was anticipated [83, 125].
However, the six bands are separated from each other when considering the other
two kinds of parameters (see Fig. 15.19b, d), indicating that the system is in the
insulating state. These insulating phases and the phase transition point can be further
distinguished by calculating Chern number and Z 6 Berry phase.
Figure 15.20a plots the dependence of the Chern number (C) and the Z 6 Berry
phase (θ ) on the parameter d 2 /d 1 . Here the material parameters of Py (Ni 80 Fe 20 ) [88,
130] are used and fix d 1 = 2.5r . In addition, the eigenfrequencies of collective vortex
gyration under different ratios d 2 /d 1 for a parallelogram-shaped (see Fig. 15.20b)
structure are also shown in Fig. 15.20c. By considering the topological invariants and
spectrum simultaneously, one can infer that the system is in the trivial phase when
d 2 /d 1 < 0.9 and 1.08 < d 2 /d 1 < 1.49, in the FOTI phase when 0.9 < d 2 /d 1 < 1.08,
and in the HOTI phase when d 2 /d 1 > 1.49. Haldane model is a well-known example
for breaking the time-reversal symmetry [134]. If we consider the limit case of
honeycomb lattice, i.e., d 1 = d 2 , (15.9) can be exactly mapped to the Haldane model,
as shown in [83] and [125]. The very existence of the chiral edge state is thus naturally
expected. For a breathing honeycomb lattice, our (15.11) represents the generalized
form of the mapping, where the last two terms in the right-hand side are the next-tonearest hopping that breaks the time-reversal symmetry. The complete phase diagram
of system can be obtained by systematically changing d 1 and d 2 , with the results
plotted in Fig. 15.20d: The regions labeled gray, white, and red represent the trivial,
FOTI, and HOTI phases, respectively. Importantly, we find that the boundary for
the phase transition between trivial and FOTI phases depends only weakly on the
choice of the absolute values of d 1 and d 2 but is (almost) solely determined by their
ratio, as indicated by dashed black lines (l 1 : d 2 /d 1 = 0.94 and l 2 : d 2 /d 1 = 1.05)
in the figure. However, the boundary for the phase transition between trivial and
HOTI phases is a linear function l 3 : d 2 = 2.24d 1 − 1.88 (see the dashed red line
in Fig. 15.20d). It is worth noting that the topological charge of the vortex has no
influence on higher-order topology for the reason that the sign of topological charge
just determines the direction (clockwise or anti-clockwise) of gyration, see (15.5).
However, it indeed can affect the chiral edge state (first-order topology). Namely the
chirality of edge state will be reversed if the topological charge changes.
The existence of symmetry-protected states on boundaries is the hallmark of a
topological insulating phase. Figure 15.21b–d shows the energy spectrum of the
ribbon configuration with armchair edges (see Fig. 15.21a) for different choices of
d 1 and d 2 . For d 1 = 3.6r and d 2 = 2.08r (black star in Fig. 15.20d), the system
is in the trivial phase without any topological edge mode (see Fig. 15.21b). For
d 1 = d 2 = 3.6r (blue star in Fig. 15.20d), the lattice considered is identical to a
magnetic texture version of graphene (the perfect honeycomb lattice). In contrast to