New classes of topological crystalline insulators having surface rotation anomaly

Rotation anomaly

We consider 2D systems of noninteracting electrons with spin-orbit coupling, time-reversal, and n-fold rotational symmetry, where n = 2, 4, or 6. We aim to establish the number of symmetry-protected Dirac cones—or equivalently the number of stable band crossings—that such systems are allowed to have. For this purpose, it suffices to consider systems with an integer number of electrons per unit cell and with either a bandgap or Dirac points at Fermi energy.

Dirac points can be located at either generic momenta or high-symmetry points in the Brillouin zone. As shown in the Supplementary Materials, one can always choose a C_n symmetric superlattice unit cell such that (i) high-symmetry points fold back to Γ in the reduced Brillouin zone and (ii) the number of electrons in the enlarged unit cell is an even integer. The condition of even-integer electron filling then guarantees that Dirac points at Γ originating from Kramers degeneracy are away from Fermi energy. Hence, in the following analysis, we only study Dirac points at generic momenta, whose presence makes systems at certain even-integer electron fillings gapless.

Because of time reversal and C_n-rotation symmetry (n even), the number of Dirac points at generic momenta must be a multiple of n, and each multiplet consists of n/2 pairs of Dirac points at opposite momenta that are related by symmetry, as shown in Fig. 1.

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We now show that the simplest scenario of n Dirac points is anomalous; i.e., it can only be realized on the surface of a 3D TCI. In any 2D lattice system with time-reversal symmetry, the number of Dirac points away from high-symmetry points must be a multiple of 4, 8, or 12, respectively, when two-, four-, or sixfold rotation symmetry is present.

To prove this fermion multiplication theorem, consider the Berry phase along the closed contours in the Brillouin zone plotted in Fig. 1. Because of time reversal and the twofold rotation symmetry, the Berry curvature vanishes everywhere in momentum space; the Berry phase along each contour is quantized to be either 0 or π (9). In the latter case, there must be an odd number of Dirac cones enclosed by the contour.

The Berry phases that associate with the loops in Fig. 1 (A to C) are determined by the rotation eigenvalues at high-symmetry points Γ, X, Y, M, and K in the corresponding Brillouin zone. Following the method of (17), we findeiΘ2=∏i=1,…,Noccζi(Γ)ζi(X)ζi(Y)ζi(M)eiΘ4=(−1)Nocc∏i=1,…,Noccξi(Γ)ξi(M)ζi(X)eiΘ6=(−1)Nocc∏i=1,…,Noccωi(Γ)θi(K)ζi(M)(1)where N_occ is the number of occupied bands (counting spin) and ζ_i, θ_i, ξ_i, and ω_i are the eigenvalues of C₂, C₃, C₄, and C₆ operation at the corresponding invariant points, respectively.

Since we deal with even-integer electron fillings, N_occ is even. Because of time reversal, at Γ, X, Y, and M points, rotation eigenvalues ζ, ξ, and ω appear in complex conjugate pairs, leading to Kramers degenerate bands. It then follows from Eq. 1 that e^iΘ₂ = e^iΘ₄ = 1. For systems with C₆ symmetry, at K point, the allowed C₃ eigenvalues are e^iπ/3, e^−iπ/3, or −1. The first two appear in complex conjugate pairs due to the composite symmetry C₂T. The number of the remaining bands with C₃ eigenvalue θ = −1 must also be even, as the total number of occupied bands is even. This implies e^iΘ₆ = 1. To summarize, we findΘ2,4,6=0 mod 2π(2)

Equation 2 implies that within each contour, there must be an even number of Dirac cones. In other words, all configurations in Fig. 1 are anomalous because they all have one Dirac cone enclosed by the contour. These states can only be realized on the surface of TCIs, to which we turn our attention below.

New classes of TCIs

A defining property of TCIs is the presence of topological surface states protected by crystal symmetries. The 230 space groups describing all possible crystal symmetries allow for many different classes of TCIs. One class of TCI protected by reflection symmetry was found in IV-VI semiconductors SnTe and Pb_1–xSn_x(Se,Te) (18–22). Recently, another class of TCI protected jointly by glide reflection and time-reversal symmetry was predicted in KHgSb (23), and the experimental signature of its “hourglass” surface states has been reported (24). In addition, several other classes of TCIs have been theorized (9, 10, 23, 25–30). Here, we predict a new class of TCIs protected jointly by n-fold rotation (n = 2, 4, and 6) and time-reversal symmetry (with T² = −1). These TCIs exhibit topological surface states consisting of n massless Dirac cones as shown in Fig. 1 on the top and bottom surfaces.

Topology in momentum space

Our new TCIs can be constructed by “adding” two time-reversal invariant strong TIs (denoted by a flavor index τ_z = ±1), each with n-fold rotational symmetry. By addition, we take the occupied bands of the desired TCI to consist of valence bands of both TIs. Each TI has a single flavor of Dirac fermion on the surface, and without loss of generality, the two Dirac cones are both centered at the center of the surface Brillouin zone. By considering the symmetry-allowed hybridization between the surface Dirac fermions of the two TIs, we obtain the desired surface states of TCIs as shown in Fig. 1.

We explicitly show how this construction works for n = 2 and summarize the results for n = 4 and 6 while leaving details to the Supplementary Materials. With twofold rotation and time-reversal symmetry, the surface state Hamiltonian of a strong TI can be written in the following general formh±(k)=kxσx±kyσy(3)where σ_{x, y, z} are the Pauli matrices and they act on the spin space. Here, we have chosen a basis for the Kramers doublet at k = 0 such that the twofold rotation and the time-reversal symmetry are represented by C₂ = iσ_z and T = (iσ_y)K. When these symmetries are present, the Hamiltonian h₊ and h₋ cannot be deformed into each other, describing two types of surface states. The Dirac fermions described by h₊ and h₋ have vortex-like spin textures in momentum space, with left- and right-handed chirality, respectively (see Fig. 2). h₊ has an emergent rotation invariance that includes twofold rotation as a subgroup, whereas h₋ does not. To be precise, for h₊(k), one can define a continuous rotation operator U₊(θ) = exp (iσ_zθ/2) such that (i) it commutes with time-reversal operator, (ii) R(π) = C₂, and (iii) R(2π) = −1 for 12-spin. This means that the C₂ rotation symmetry in h₊(k) can be promoted to an O(2) rotation symmetry. On the other hand, for h_, one can define a rotation operator U₋(θ) = exp (−iσ_zθ/2), which satisfies (i) and (iii), but U₋(π) ≠ C₂ = iσ_z. Therefore h₋ does not have continuous rotation symmetry. Here, the reason we require that C₂ = iσ_z is that, otherwise if C₂ = −iσ_z, a unitary transform U = iσ_x on the second block sends h₋ to h₊ and, at the same time, −iσ_z to +iσ_z, such that the two blocks are identical, having the same h₊ and the same choice of C₂. The presence and the absence of continuous rotation symmetry can be explicitly seen by looking at the pseudospin vector pattern on some equal energy contours of h₊ and h₋, respectively: The pattern in Fig. 2A is invariant under arbitrary rotation, and the one in Fig. 2B is not, but only invariant under a π-rotation.

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The addition of h₊(k) and h₋(k) in our construction gives rise to two flavors of Dirac fermions on the surface, described by the HamiltonianH(k)=h+(k)⊕h−(k)=kxτ0σx+kyτzσy(4)where τ_i’s are Pauli matrices acting in the flavor space. One can check that no mass term preserving C₂ and T symmetry can be added to gap the spectrum of H(k) (here and below, we define “mass” terms as momentum-independent terms in the Hamiltonian that gaps out the entire energy spectrum). The only T-invariant mass term is proportional to τ_yσ_y, which breaks C₂. Symmetry-preserving terms, on the other hand, can only split the two Dirac cones sitting on top of each other at k = 0 to a pair of Dirac cones centered at opposite momenta, which is precisely the desired TCI surface state shown in Fig. 1A. For example, if one adds mτ_yσ_z into H(k), one finds that (i) it commutes with k_yτ_zσ_y and hence cannot open a full gap and (ii) it splits the two Dirac points along the k_y axis by a separation proportional to ∣m∣. This shows that TCIs protected by C₂ and T symmetry can be realized by adding two TIs whose surface-state spin textures have opposite chirality.

In contrast, adding two surface states with the same helicity leads to a surface spectrum that can be gapped by a symmetry-preserving mass term τ_yσ_z. From these considerations, we conclude that band insulators with twofold rotation and time-reversal symmetry are classified by Z₂ × Z₂ topological invariant [this is consistent with the classification in (31) using K-theory]. The generator of each Z₂ group corresponds to a TI whose surface states have left- and right-handed chiral spin textures. For the TCI surface states shown in Fig. 1A, this topological index takes the value (11).

In Table 1, we summarize how to construct the TCI surface states with 2-, 4-, and 6-fold symmetry by adding two TIs. The proof of these results is found in the Supplementary Materials.

Topology in real space

Having established the TCI surface state band structure, in this subsection, we provide an alternative understanding of their topological nature from the perspective of real space, following the approach introduced in (32, 33). The essence of this real-space approach is to add symmetry-allowed perturbations to break translational symmetry and gap the massless Dirac fermions on the surface as much as possible and study what could be left. This reduction gives a “theoretical minimum” realization of the nontrivial TCI surface states and enables us to demonstrate their robustness under electron interactions. Throughout this section, we assume that the bulk state is described by the double-TI model in the previous section, and the surface-state Hamiltonian is the direct sum of h₊ and h₋ for C₂ and similarly for the case of C_4,6 (see Table 1). We emphasize that the double-TI Hamiltonian represents a rotation-TCI phase, the Hamiltonian of which may be deformed to this model without breaking rotation symmetry, time-reversal symmetry, or closing the gap. Note that specific materials in the rotation-TCI phase may have Hamiltonians different from, and presumably more complicated than, the double-TI model. However, since they have surface states on the top surface that are topologically equivalent to those shown in Fig. 1, by merit of bulk-edge correspondence, we know that they belong to the same topological phase.

We start from considering a double-TI model of a C₂-TCI placed in a cylinder, the size of which is much greater than the correlation length and the surface of which is smooth at atomic scale (the argument for C_4,6-TCI proceeds similarly). Naturally, each surface—top, bottom, or side—has two Dirac cones from the two TIs. On the top surface, as we have argued, because of the presence of rotation symmetry, a mass term is disallowed, but on the side surface, where rotation symmetry is broken, the mass term may be added. The mass term is allowed to be slowly varying with respect to the scale of correlation length so that the massive Dirac Hamiltonian on the side surface becomesH=h+⊕h−+∫dr m(r)ψr†τyσyψr(5)

Since τ_yσ_y anticommutes with h₊ ⊕ h₋, the energy spectrum has a local gap as long as m(r) ≠ 0. However, since C₂ = iσ_z also anticommutes with τ_yσ_y, for the whole system to have twofold rotation symmetry, that is, [C₂, H] = 0, it is required thatm(r)=−m(C2r)(6)

Equation 6 necessitates the presence of domain walls on the side surface where m(r) changes signs. Consider a circumference of the cylinder at any height; then, along this circumference, m(r) must change sign twice, leaving two points far from each other, related by C₂, locally gapless, and since the same holds true for any circumference, these gapless points form two gapless domain walls connecting the top and the bottom surfaces, related to each other under C₂. A domain wall between gapped double-Dirac cones hosts one pair of helical edge modes (18), and the overall surface states of the cylinder consist of 2D gapless modes on the top and the bottom and two helical edge modes on the side surface, shown in Fig. 3A. For C_4,6-TCI placed on a cylinder, the surface states are shown in Fig. 3 (B and C) [also see (16) for the discussion of C₄ case in detail]. We note that the locations of these modes are not pinned to any physical hinges or intersections of crystalline surfaces like in (34–37). Note that while the cylindrical shape may have continuous rotation symmetry, the system described by the Hamiltonian h₊ ⊕ h₋ necessarily breaks it down to discrete rotation symmetry, consistent with the existence of the two 1D gapless lines even on a perfectly atomic-scale smooth cylinder.

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In real materials, samples as grown do not take the shape of the cylinder, and the side surface usually contains several flat planes with different Miller indices. For example, a sample grown along the [001] direction naturally has (100) and (010) for the side surfaces. In that case, we comment that each one of the side surface may be fully gapped individually because of the broken rotation symmetry, but as long as the overall twofold rotation symmetry of the crystallite is preserved, the mass terms on two opposite sides related by C₂ must have opposite sides. Therefore, there must be domain walls along the hinges between surfaces that have opposite masses. In that geometry, we expect vertical helical edge modes along at least two hinges on the side surface that connect the top and the bottom surfaces (18).

The existence of the 1D edge modes hints us to write down the bulk Z₂ topological invariants following (16), and the details are given in the Supplementary Materials. In the Supplementary Materials, we additionally discuss, as contrast, a related TCI protected by inversion and time reversal (38, 39), where surface states do not exist but a single helical edge mode exists on the surface.

We also note that many topological crystalline states can be understood from a dimensional reduction perspective (32, 40–42), where the 3D state can be considered a set of decoupled layers of 2D topological states. From this perspective, all three types of new TCIs here can also be constructed from 2D TIs. This construction helps extend our theory to include strongly interacting symmetry-protected topological states protected by rotation symmetry and any local symmetry, including, but not limited to, time reversal. The discussion is found in the Supplementary Materials.