Symmetry Indicators and Anomalous Surface States of Topological Crystalline Insulators

The rich variety of crystalline symmetries in solids leads to a plethora of topological crystalline insulators (TCIs), featuring distinct physical properties, which are conventionally understood in terms of bulk invariants specialized to the symmetries at hand. While isolated examples of TCI have been identified and studied, the same variety demands a unified theoretical framework. In this work, we show how the surfaces of TCIs can be analyzed within a general surface theory with multiple flavors of Dirac fermions, whose mass terms transform in specific ways under crystalline symmetries. We identify global obstructions to achieving a fully gapped surface, which typically lead to gapless domain walls on suitably chosen surface geometries. We perform this analysis for all 32 point groups, and subsequently for all 230 space groups, for spin-orbit-coupled electrons. We recover all previously discussed TCIs in this symmetry class, including those with “hinge” surface states. Finally, we make connections to the bulk band topology as diagnosed through symmetry-based indicators. We show that spin-orbit-coupled band insulators with nontrivial symmetry indicators are always accompanied by surface states that must be gapless somewhere on suitably chosen surfaces. We provide an explicit mapping between symmetry indicators, which can be readily calculated, and the characteristic surface states of the resulting TCIs.

Popular Summary

In the last two decades, topological materials have garnered a lot of attention because of their unusual conductive properties—their surfaces are metallic, yet their interiors are insulating. In the case of conventional topological insulators, which are protected by internal symmetries, all surfaces are metallic. But topological crystalline insulators (TCIs), protected by the symmetries of the material’s crystal structure, have a variety of surface types, each with different electronic properties. Unfortunately, there is no unified theoretical framework for handling all this variety. Here, we develop such a theory.

At the crux of our analysis is the discovery of a simple algebraic structure emerging from the incorporation of spatial symmetries in the analysis of the surface Dirac equation. Our formulation is inherently physical and leads to concrete predictions regarding the nature of surface states associated with any nontrivial phases. It is also computationally simple, which allows us to obtain exhaustive results for all 230 3D space groups, assuming significant spin-orbit coupling and time-reversal symmetry.

Our theory encapsulates all the known TCIs with stable surface states into a single framework and, in particular, clarifies what combinations of spatial symmetry and sample geometry can lead to higher-order surface states. In addition, we systematically map the symmetry indicators that can diagnose TCIs, which should help in the search for physical realizations in materials.

Figure 1

The parity eigenvalues at high-symmetry momenta. (a) The common choice of coordinates in panels (b)–(s). (b,c) An example of the parity eigenvalues for a strong topological insulator (${\kappa }_1=3$, ${\nu }_0=1$, and ${\nu }_i=0$) and a higher-order topological insulator (${\kappa }_1=6$ and ${\nu }_0={\nu }_i=0$), respectively. Diagram (c) can be realized by stacking two copies of diagram (b). (d)–(k) The parity eigenvalues of the atomic insulators constructed by placing an $s$ orbital at the position specified in each panel in every unit cell. (l)–(s) The same for (d)–(k) but with a $p$ orbital used instead of an $s$ orbital. All atomic insulators have ${\kappa }_1={\nu }_0={\nu }_i=0$ except for diagram (d), ${\kappa }_1=4$, ${\nu }_0={\nu }_i=0$, and diagram (l), ${\kappa }_1=-4$, ${\nu }_0={\nu }_i=0$.

Figure 2

Surface states for the weak topological insulator can be understood as choosing a “$-$” representation for the translation along the “weak” direction.

Figure 3

In the presence of the fourfold rotoinversion $S_4$ only, the equator and the hinge state are deformable into each other since their sum can be trivialized, as shown here.

Figure 4

Graphical illustration of the surface states of the hinge phases, which generate all sTCIs for the 32 crystallographic point groups, on a sphere. For each state, we show the operators that need to be taken in the “$-$” representation as well as the resulting classification. Red circles indicate ${\mathbb{Z}}_2$ modes that would be gapped-out if two copies of the system were stacked together, whereas blue circles indicate $\mathbb{Z}$ modes protected by a mirror Chern number. Rotation axes are shown with black, blue, green, or red for twofold, fourfold, threefold, and sixfold rotations, respectively.

Figure 5

Illustration for the hinge state protected by a fourfold screw (left diagram) and a glide (right diagram) on a cylinder geometry with periodic boundary conditions along the screw direction.

Figure 6

Illustration for the two equivalent surface states corresponding to a fourfold screw. The equivalence of the two can be established by noting that their sum can be deformed to a trivial state while preserving the screw symmetry.

Figure 7

Illustration of the surface dispersion in planes tangent to the sphere whose normal is (a) in a single mirror plane, (b) in two mirror planes, or (c) parallel to a fourfold rotation axis.

Figure 8

Surface hinge mode protected by inversion symmetry on a cubic geometry, assuming no other spatial symmetries are present. The mode is confined to the edges of the cube since the surface mass term cannot change on a flat face.

Figure 9

Hierarchy of crystalline topological insulators with time-reversal symmetry and strong spin-orbit coupling (class AII) in 3D. All nontrivial entries in the symmetry indicator $X_{\mathrm{BS}}$ are compatible with gapped topological band structures with gapless surface states on suitably chosen sample geometries. Excluding the strong TIs from the $X_{\mathrm{BS}}$-nontrivial phases, we found that all the remaining phases are anomalous surface topological crystalline insulators (sTCIs), defined as nontrivial insulators with anomalous surface states, but they can be trivialized upon the breaking of some crystalline symmetries. Note that sTCIs include all gapped topological phases with gapless surface states but are not strong TIs (hatched region). These also include phases not diagnosed by $X_{\mathrm{BS}}$; i.e., the topological nature is not exposed by the symmetry eigenvalues in the bulk.