Connecting higher-order topological insulators to lower-dimensional topological insulators

In recent years, the role of crystal symmetries in enriching the variety of TIs has been actively investigated. Higher-order TIs are a new type of topological crystalline insulators that exhibit gapless boundary states whose dimensionality is lower than those on the surface of conventional TIs. In this paper, relying on a concrete tight-binding model, we show that higher-order TIs can be smoothly connected to conventional TIs in a lower dimension without the bulk-gap closing or symmetry breaking. Our result supports the understanding of higher-order TIs as a stacking of lower-dimensional TIs in a way respecting all the crystalline symmetry.

Figure 1

A schematic illustration of (a) a 3D HOTI, (b) a 2D Chern insulator, (c) a staggered stacking of Chern insulators, (d) a 2D HOTI, (e) a 1D TI in class AIII, and (f) a staggered stacking of 1D TI in class AIII. $O$ is the inversion center. Red (blue) circles represent a right-moving (left-moving) chiral edge mode, while red (blue) dots represent a zero-energy edge mode corresponding to the index $+1(-1)$.

Figure 2

(a) The parity eigenvalue of the valence bands of the model (1) for $m=2$ and $t=c=1$. (b),(c) The top and bottom surfaces host gapless Dirac states protected by the time-reversal symmetry. (d),(e) A uniform magnetic field $\vec{B}=(0,0,\frac{1}{2})$ opens an excitation gap. In panels (c) and (e), the number of layers in $z$ is set to be $L_z=25$, and 30 different values of $k_y$ are overlaid.

Figure 3

3D TI under a uniform magnetic field $\vec{B}=\frac{1}{2}(0,-sin\theta ,cos\theta )$ for $\theta =\frac{\pi }{4}$, $\frac{\pi }{8}$, 0, $-\frac{\pi }{8}$, and $-\frac{\pi }{4}$. For $\left|\theta \right|<\frac{\pi }{4}$, an additional surface Zeeman field [defined in Eqs. (2) and (3)] is applied as illustrated in (a). The boundary condition in $x$ is PBC and that for $y$ and $z$ is OBC. The red and green color in (a) indicate the sign of the mass gap. The panel (b) is the density plot of the weight of the zero-energy states in the band structure (c).

Figure 4

The band dispersion computed under the OBC in $y$ and $z$ and PBC in $x$ for different numbers of layers: $L_z=25$, 15, 9, 5, 3, and 1.

Figure 5

The parity eigenvalue of the valence bands of the model in Eq. (1) for five layers (a), three layers (b), and one layer (c).

Figure 6

(a) The parity eigenvalue of the valence bands of the model in Eq. (1) for $t=c=m=1$. (b),(c) Two edges at $y=\pm l_y$ host a helical edge state protected by the time-reversal symmetry. (d),(e) The uniform magnetic field $\vec{B}=(0,\frac{1}{2},0)$ opens an excitation gap. In panels (c) and (e), the number of layers in $y$ is set to be $L_y=2l_y+1=25$ and 120 different values of $k_x$ are overlaid.

Figure 7

2D TI with chiral symmetry under a uniform magnetic field $\vec{B}=B(-sin\theta ,cos\theta ,0)$ for $\theta =\frac{\pi }{4}$, $\frac{\pi }{8}$, 0, $-\frac{\pi }{8}$, and $-\frac{\pi }{4}$. An additional Zeeman field [defined in Eqs. (9) and (10)] is applied as illustrated in (a). The boundary condition is OBC in both the $x$ and $y$ directions. Panel (b) is the density plot of the weight of the zero-energy modes in panel (c).

Figure 8

The density of states under the OBC in $x$ and $y$ for different numbers of layers: $L_z=25$, 15, 9, 5, 3, and 1.