Shift Insulators: Rotation-Protected Two-Dimensional Topological Crystalline Insulators

We study a two-dimensional tight-binding model of a topological crystalline insulator (TCI) protected by rotation symmetry. The model is built by stacking two Chern insulators with opposite Chern numbers which transform under conjugate representations of the rotation group, e.g., $p_{\pm }$ orbitals. Despite its apparent similarity to the Kane-Mele model, it does not host stable gapless surface states. Nevertheless, the model exhibits topological responses including the appearance of quantized fractional charge bound to rotational defects (disclinations) and the pumping of angular momentum in response to threading an elementary magnetic flux, which are described by a mutual Chern-Simons coupling between the electromagnetic gauge field and an effective gauge field corresponding to the rotation symmetry. In addition, we show that although the filled bands of the model do not admit a symmetric Wannier representation, this obstruction is removed upon the addition of appropriate atomic orbitals, which implies “fragile” topology. As a result, the response of the model can be derived by representing it as a superposition of atomic orbitals with positive and negative integer coefficients. Following the analysis of the model, which serves as a prototypical example of 2D TCIs protected by rotation, we show that all TCIs protected by point group symmetries which do not have protected surface states are either atomic insulators or fragile phases. Remarkably, this implies that gapless surface states exist in free-electron systems if and only if there is a stable Wannier obstruction. We then use dimensional reduction to map the problem of classifying 2D TCIs protected by rotation to a zero-dimensional problem which is then used to obtain the complete noninteracting classification of such TCIs as well as the reduction of this classification in the presence of interactions.

Popular Summary

In topological materials, electrons are neither free to move as in metals nor tied to localized atomic orbitals as in conventional insulators. This feature is responsible for several exotic properties such as being insulating in the inside but conducting on the surface. In conventional topological insulators, which are protected by internal symmetries, different characterizations of topology coincide, such as the existence of surface states and the absence of localized orbitals. Recent research has focused instead on topological phases protected by crystalline symmetries, where these characterizations of topology may not coincide, making the understanding of topological distinctions very subtle. Here, we resolve these conceptual issues by introducing a simple model of a topological crystalline phase that exhibits some features of topology (absence of localized orbitals and fractional charge) but not others (metallic surface states).

Our model, dubbed “shift insulator,” serves as a prototype for discussing the general relationship between the existence of surface states, localized orbitals, and fractional charges, which we investigate in detail. We show under what conditions the absence of localized orbitals implies the presence of surface states, and we provide a general discussion of fractional charges bound to defects in topological crystalline phases. We also show how such topological distinctions are affected by the presence of interaction between electrons.

Our work clarifies several important conceptual issues that have hindered progress towards a complete theory of topological crystalline phases, which should help theoreticians and experimentalists working in this vibrant field.

Figure 1

Sign convention for ${\nu }_{ij}$. ${\nu }_{ij}=+1$ ($-1$) if the hopping direction from $j$ to $i$ is along (against) the directions indicated in the left plaquette.

Figure 2

Brillouin zone (gray region) of the honeycomb lattice. ${\mathit{k}}_1$, ${\mathit{k}}_2$ are the translation basis of the momentum space, dual to the real space translation basis ${\mathit{r}}_1$, ${\mathit{r}}_2$, as shown in the inset. $K$, $K^{\prime }$ are the two gapless points when $\lambda =0$.

Figure 3

Periodic boundary condition in $y$ direction.

Figure 4

Values of ${\mu }_{ij}$ for different hopping directions from $j$ to $i$.

Figure 5

Comparison between $J=0$ (left) and $J=0.1$ (right) with $t=1$, $\lambda =0.2$ fixed.

Figure 6

(a) Schematic illustration of the construction of disclinations. (b) An example of our setup in numerical calculations.

Figure 7

Quantization of the disclination charge. (a) Twelve pentagon disclinations on a sphere in the shape of a dodecahedron, e.g., the buckyball. (b) Six square disclinations on a sphere in the shape of a cube. (c) Two $N$-gons and $N$ squares on a sphere in the shape of a cylinder. (d) Each of the small triangles in the above three examples contains many plaquettes. The triangle vertices coincide with plaquette centers and the triangle edges are perpendicular to plaquette edges. In (a)–(c), disclinations are all located at face centers, and all corners are regular since there are six surrounding wedges.

Figure 8

(a) An $N=3$ example of the expanded honeycomb lattice tori used in our calculation. (b) The $(x^{\prime },y^{\prime })$ coordinates on the expanded torus.

Figure 9

Torus energy spectrum of the Haldane model at $t=1$, $\lambda =0.2$ with respect to the monopole charge $\mathrm{\Phi }/{\mathrm{\Phi }}_0$. The system size is set to be $N=3$ (81 plaquettes).

Figure 10

Wave function of the zero-energy bound state of the Haldane model disclination with $n_{\mathrm{\Omega }}=0$, $\mathrm{\Phi }=1/2$, $t=1$, and $\lambda =0.2$.

Figure 11

Illustration of the Wyckoff positions $a$, $b$, and $c$ for wallpaper group $p6$ corresponding to the hexagon center, corner, and edge, respectively.

Figure 12

The determinant of the overlap matrix $S(\mathit{k})$ for the basis choices given in Eqs. (75), (78)–(80) (note that the curves for $t=\pm 1$ are identical).

Figure 13

The weight of the Wannier function $|W_{n,0}(\mathit{r}){|}^2$ at different sites for the shift insulator model with $\lambda <0$ after the addition of an atomic orbital at the center of the unit cell ($a$ position). The white, red, and black dots denote the center of the unit cell and $A$ and $B$ sublattices, respectively. The orbital character of the orbitals is not shown and the weight of each position is multiplied by a Gaussian function centered around the corresponding position for clarity. We can see that the three Wannier states are centered at the hexagon edges ($c$ position) and are related to each other by rotation.

Figure 14

The Wannier functions $|W_{n,0}(\mathit{r}){|}^2$ for the shift insulator model with $\lambda >0$ after the addition of two atomic orbitals at the $A$ and $B$ sublattices ($b$ position). As in Fig. 13, the white, red, and black dots denote the center of the unit cell and $A$ and $B$ sublattices, respectively, and we use a Gaussian function centered around the corresponding position for clarity. Here, we have three rotation-related states centered at the hexagon edges ($c$ position) and one rotationally symmetric state centered at the hexagon center ($a$ position).

Figure 15

Schematic illustration of the proof that all TCIs without surface states admit a Wannier representation possible after the addition of some atomic d.o.f. Using the layer construction, we show in the main text that any $d$-dimensional TCI without surface states protected by point group symmetries can be mapped to a Wannier representable phase by first using the dimensional reduction to a 0D atomic insulator then using the dimensional raising or layering procedure. The mapping involves the addition of some atomic orbitals at nonmaximal Wyckoff positions.

Figure 16

Illustration of the dimensional reduction procedure for TCIs protected by sixfold rotation symmetry. The 2D plane is divided into six symmetry-related regions separated by three lines at angle $\pi /3$ from each other. The three lines form a 1D symmetry-invariant region for which the construction can be repeated to reduce the problem to the 0D problem with internal ${\mathbb{Z}}_N$ symmetry at the rotation center.