Coupled-layer description of topological crystalline insulators

We introduce a coupled-layer construction to describe three-dimensional topological crystalline insulators protected by reflection symmetry. Our approach uses stacks of weakly coupled two-dimensional Chern insulators to produce topological crystalline insulators in one higher dimension, with tunable number and location of surface Dirac cones. As an application of our formalism, we turn to a simplified model of topological crystalline insulator SnTe, showing that its protected surface states can be described using the coupled-layer construction.

Figure 1

Weakly coupled Chern insulating layers, $H_{\pm }$, stacked in the $z$ direction, with invariants $C_{\pm }=\pm 1$ (a), or $C_{\pm }=\pm 2$ (b). The unit cell (square bracket) is composed of two layers, each of which carries the same number of chiral modes (horizontal arrows). The hopping matrix $T_z$ acts both within a unit cell and between neighboring ones. In both cases, the systems preserve the $\mathcal{S}=\mathrm{\Theta }{T}_{1/2}$ symmetry: $\mathrm{\Theta }$ reverses edge mode chirality, interchanging $H_{+}$ and $H_{-}$, while $T_{1/2}$ translates the system by one layer. This leads to equal coupling between chiral edge modes and a gapless surface in (a), but allows for a fully gapped surface in (b), obtained by coupling different pairs of modes: thick arrow in one layer to thin arrow in the layer above. Imposing reflection symmetry $R$ leads to gapless surfaces in both cases, protected by a $\mathbb{Z}$ valued mirror Chern number $C_M$.

Figure 2

System composed of Chern insulators with alternating Chern numbers, stacked both in the $x$ and in the $z$ directions, shown as gray and black arrows, respectively. The pairs of layers stacked in $z$, with Hamiltonians $H_{+}$ and $H_{-}$, must lie on top of each other in order to preserve the $R_x$ reflection symmetry. The same applies to the layers stacked in the $x$ direction, denoted by ${\tilde{H}}_{+}$ and ${\tilde{H}}_{-}$.

Figure 3

Face-centered cubic structure of rocksalt SnTe. The unit cell (green box) is composed of one Sn (orange) and one Te (dark blue) atom. The principal crystal directions, $x$, $y$, and $z$, as well as the Bravais vectors ${\mathbf{a}}_{\mathbf{1}}$, ${\mathbf{a}}_{\mathbf{2}}$, and ${\mathbf{a}}_{\mathbf{3}}$ are indicated by arrows.

Figure 4

Left: top view of the surface perpendicular to the $z$ direction in an infinite slab geometry. The green box marks our choice of slab unit cell, with ${\mathbf{a}}_{\mathbf{x}}$ and ${\mathbf{a}}_{\mathbf{y}}$ the Bravais vectors. Right: corresponding surface BZ hosting 12 protected Dirac cones, marked with $\times$, located along the projections of the mirror-symmetric planes (dashed lines). The blue Dirac cones are protected by (110) and $\left(1\overline{1}0\right)$ mirror symmetries, while the red and green cones are protected by reflection symmetries about the Sn-Te planes, mapping $x\to -x$ and $y\to -y$, respectively.

Figure 5

Band structure of the model (16) in an infinite slab geometry with hard wall boundaries in the $z$ direction. We use a slab thickness of 80 layers and the tight-binding parameters mentioned in the text. The color scale is given by the eigenstate intensity on the first and last eight layers from the boundaries. The left panel is a cut along the projection of the (110) plane on the surface BZ, showing a pair of surface Dirac cones. The right panel is a cut along (100), on which there are two pairs of surface Dirac cones. The two pairs are distinguished by their different localization lengths, corresponding to red and green colors, respectively.