Abstract
Three-dimensional topological (crystalline) insulators are materials with an insulating bulk but conducting surface states that are topologically protected by time-reversal (or spatial) symmetries. We extend the notion of three-dimensional topological insulators to systems that host no gapless surface states but exhibit topologically protected gapless hinge states. Their topological character is protected by spatiotemporal symmetries of which we present two cases: (i) Chiral higher-order topological insulators protected by the combination of time-reversal and a fourfold rotation symmetry. Their hinge states are chiral modes, and the bulk topology is
INTRODUCTION
The bulk-boundary correspondence is often taken as a defining property of topological insulators (TIs) (1–3): If a d-dimensional system with given symmetry is insulating in the bulk but supports gapless boundary excitations that cannot be removed by local boundary perturbations without breaking the symmetry, then the system is called TI. The electric multipole insulators in the study of Benalcazar et al. (4) generalize this bulk-boundary correspondence: In two and three dimensions, these insulators exhibit no edge or surface states, respectively, but feature gapless, topological corner excitations corresponding to quantized higher electric multipole moments. Here, we introduce a new class of three-dimensional (3D) topological phases to which the usual form of the bulk-boundary correspondence also does not apply. The topology of the bulk protects gapless states on the hinges, while the surfaces are gapped. Both systems, with gapless corner and hinge states, respectively, can be subsumed under the notion of higher-order TIs (HOTI): An nth-order TI has protected gapless modes at a boundary of the system of codimension n. Following this terminology, we introduce second-order 3D TIs in this work, while the study of Benalcazar et al. (4) has introduced second-order 2D TIs and third-order 3D TIs. The important aspect of 3D HOTIs is that they exhibit protected hinge states with (spectral) flow between the valence and conduction bands, whereas the corner states have no spectral flow.
The topological properties of HOTIs are protected by symmetries that involve spatial transformations, possibly augmented by time reversal. They thus generalize topological crystalline insulators (5, 6), which have been encompassed in a recent exhaustive classification of TIs in the study of Bradlyn et al. (7). Here, we propose two cases: (i) chiral HOTIs with hinge modes that propagate unidirectionally, akin to the edge states of a 2D quantum Hall effect (8), or Chern insulator (9). We show that chiral HOTIs may be protected by the product
For both cases, we show the topological bulk-surface-hinge correspondence, provide concrete lattice-model realizations, and provide expressions for the bulk topological invariants. The latter are given by the magnetoelectric polarizability and mirror Chern numbers (6, 13), for chiral and helical HOTIs, respectively. For the case where a chiral HOTI also respects the product of inversion times time-reversal symmetry
RESULTS
Chiral HOTI
We first give an intuitive argument for the topological nature of a chiral 3D HOTI. We consider a hypothetical but realizable electronic structure where gapless degrees of freedom are only found on the hinge. For concreteness, let us consider a system with a square cross section, periodic boundary conditions in z direction, and
(A) Time-reversal breaking model with chiral hinge currents running along the corners of a
A concrete model of this phase is defined via the four-band Bloch Hamiltonian
The term proportional to Δ2 breaks both
The phase diagram of Hamiltonian (1) is shown in Fig. 2A. For 1 < |M/t| < 3 and Δ1, Δ2 ≠ 0, the system is a chiral 3D HOTI. The spectrum in the case of open boundary conditions in x and y directions is presented in Fig. 2C, where the chiral hinge modes (each twofold degenerate) are seen to traverse the bulk gap. Physically, the term multiplied by Δ2 corresponds to orbital currents that break time-reversal symmetry oppositely in the x and y directions. When infinitesimally small, its main effect is thus to open gaps with alternating signs for the surface Dirac electrons of the 3D TI on the (100) and (010) surfaces. The four hinges are then domain walls at which the Dirac mass changes sign. It is well known (16, 17) that such a domain wall on the surface of a 3D TI binds a gapless chiral mode, which, in the case at hand, is reinterpreted as the hinge mode of the HOTI. Another physical mechanism that breaks time-reversal symmetry and preserves
(A) Schematic phase diagram for model (1), where NI stands for normal insulator. (B) A unit cell of noncollinear magnetic order with
We turn to the bulk topological invariant that describes the
The explicit evaluation of θ is impractical for ab initio computations in generic insulators. This motivates the discussion of alternative forms of the topological invariant. The Pfaffian invariant (1) used to define first-order 3D TIs rests on the group relation
Helical HOTI
Helical HOTIs feature Kramers pairs of counterpropagating hinge modes. They are protected by time-reversal symmetry and a spatial symmetry. For concreteness, let us consider a system with a square (or rhombic) cross section, periodic boundary conditions in the z direction, and two mirror symmetries
The argument for their topological protection proceeds similar to the chiral HOTI case by considering a minimal symmetry-preserving surface perturbation. It consists of a layer of a 2D time-reversal symmetric TI and its mirror-conjugated partner added to surfaces that border the hinge under consideration. Each of the TIs contributes a single Kramers pair of boundary modes to the hinge so that (NL,− + NL,+) and (NR,− + NR,+) each increase by 2 (see Fig. 3A). Because mirror symmetry maps the right-moving modes of the two Kramers pairs onto one another (and the same for the two left-moving modes), we can form a “bonding” and “antibonding” superposition with mirror eigenvalues + i and − i out of each pair. Thus, each of NL,+, NL,−, NR,+, and NR,− increases by 1 because of this minimal surface manipulation. This leaves n invariant, suggesting a Z classification of the helical HOTI for each pair of mirror-invariant hinges. The case depicted in Fig. 1B with two mirror symmetries is then
(A) Additional hinge modes obtained by decorating the surfaces with 2D time-reversal symmetric TIs in a mirror-symmetric fashion. They can always be combined in bonding and antibonding pairs {R1 + R2, L1 + L2} and {R1 − R2, L1 − L2}, with mirror eigenvalues + i and − i, respectively. Therefore, they do not change the net mirror chirality of the hinge. (B) Mirror-symmetry protected Dirac cones on a (110) surface. (C) Slightly tilting the surface normal out of the mirror plane gaps the Dirac cones and forms a Kramers pair of domain wall states between two surfaces with opposite tilting. The mirror eigenvalues of the hinge modes are tied to those of the Dirac cones, which, in turn, are related to a bulk topological invariant, the mirror Chern number Cm. (D) Further deforming the surface to the (100) and (010) orientation in a mirror symmetry–preserving manner does not change this correspondence.
The topological invariant for the
We now discuss the correspondence between the bulk topological invariant Cm/2 and the existence of Kramers-paired hinge modes. For this, we first consider the electronic structure of the (110) surface, which is invariant under
A nonzero bulk mirror Chern number Cm with respect to the
Note that the above deformation of the surfaces can be extended to nonperturbative angles α, until, for example, the (100) and (010) surface orientations are reached. The surfaces on either side of the hinge may undergo gap-closing transitions as α is increased. But as we argued at the beginning of the section, surface transitions of this kind may not change the net number of helical hinge states with a given mirror eigenvalue, if they occur in a mirror-symmetric manner.
We remark that an equation similar to Eq. 5 also holds in the absence of time-reversal symmetry for each mirror subspace. Then, the Chern number in each mirror subspace is an independent topological invariant, which gives rise to a
Material candidates and experimental setup
We propose that SnTe realizes a helical HOTI. In its cubic rocksalt structure, SnTe is known to be a topological crystalline insulator (5, 6). This crystal structure has mirror symmetries
(1) At about 98 K, SnTe undergoes a structural distortion into a low-temperature rhombohedral phase via a relative displacement of the two sublattices along the (111) direction (25, 26). This breaks the mirror symmetries
(2) If uniaxial strain along the (110) direction is applied to SnTe, then
(A) Rocksalt lattice structure of SnTe. Uniaxial strain along the (110) direction breaks the mirror symmetries represented by dotted lines but preserves the ones represented by dashed lines. (B) Circles indicate the location of Dirac cones in the surface Brillouin zone of pristine SnTe for various surface terminations. Those crossed by dotted mirror symmetries are gapped in SnTe with uniaxial strain while the others are retained. The two red Dirac cones are enforced by a mirror Chern number Cm = 2, corresponding to one helical pair of hinge modes. k1 is the momentum along the direction with unit vector
We propose to physically realize the (110) uniaxial strain in SnTe with a topological coaxial cable geometry, which would enable the use of its protected hinge states as quasi-1D dissipationless conduction channels (see Fig. 4F). The starting point is an insulating nanowire substrate made from Si or SiO, with a slightly rhombohedral cross section imprinted by anisotropic etching. SnTe is grown in layers on the surfaces by using molecular beam epitaxy, with a thickness of about 10 layers. SnTe will experience the uniaxial strain to gap out its surfaces and protect the helical HOTI phase. The hinge states can be studied by scanning tunneling microscopy and transport experiments with contacts applied through electronic-beam lithography. Note that in the process of growth, regions with step edges are likely to form on the surfaces and should be avoided in measurements, as they may carry their own gapless modes (27). Alternatively, we propose to use a superconducting substrate to study proximity-induced superconductivity on the helical hinge states.
In addition to the topological crystalline insulator SnTe, we propose weak TIs with nonvanishing mirror Chern numbers as possible avenues to realize helical HOTIs. We computed the relevant mirror Chern numbers for the weak TIs Bi2TeI (28), BiSe (29), and BiTe (30), which all turn out to be 2. These materials are therefore dual TIs, in the sense that they carry nontrivial weak and crystalline topological invariants. Their surface Dirac cones are protected by a nontrivial weak index, that is, by time reversal together with translation symmetry. To gap them, it is necessary to break at least one of these symmetries, which is possible by inducing magnetic or charge density wave order.
DISCUSSION
We have introduced 3D HOTIs, which have gapped surfaces but gapless hinge modes, as intrinsically 3D topological phases of matter. Both time-reversal symmetry breaking and time-reversal symmetric systems were explored, which support hinge states akin to those of the integer quantum Hall effect and 2D time-reversal symmetric TIs, respectively. The former may be realized in magnetically ordered TIs; we propose the naturally occurring rhombohedral or a uniaxially distorted phase of SnTe as a material realization for the latter. Despite their global topological characterization based on spatial symmetries, the hinge states are as robust against local perturbations as quantum (spin) Hall edge modes. The concepts introduced here can be extended to define novel topological superconductors with chiral and helical Majorana modes at their hinges and may further be transferred to strongly interacting, possibly topologically ordered, states of matter and to mechanical (31), electrical (32), and photonic analogs of Bloch Hamiltonians.
METHODS
First-principles calculations
We used DFT as implemented in the Vienna Ab initio Simulation Package (33). The exchange correlation term was described according to the Perdew-Burke-Ernzerhof prescription together with projected augmented-wave pseudopotentials (34). For the autoconsistent calculations, we used a 12 × 12 × 12 k-point mesh for the bulk and 7 × 7 × 1 for the slab calculations.
For the electronic structure of SnTe with (110) distortion, the kinetic energy cutoff was set to 400 eV. We calculated the surface states by using a slab geometry along the (100) direction. Because of the smallness of the bandgap induced by strain, we needed to achieve a negligible interaction between the surface states from both sides of the slab (to avoid a spurious gap opened by the creation of bonding and antibonding states from the top and bottom surface states). To reduce the overlap between top and bottom surface states, we considered a slab of 45 layers and 1 nm vacuum thickness and artificially localized the states on one of the surfaces. The latter was performed by adding one layer of hydrogen to one of the surfaces.
To obtain the electronic structure of bulk SnTe with (111) ferroelectric distortion, we set the cutoff energy for wave-function expansion to 500 eV. We used the parameter λ introduced in the study of Plekhanov et al. (35) to parameterize a path linearly connecting the cubic structure (space group Fm3m) to the rhombohedral structure (space group R3m). Our calculations are focused on the λ = 0.1 structure. Then, to obtain the hinge electronic structure, we first constructed the maximally localized WFs from the bulk ab initio calculations. These WFs were used in a Green’s function calculation for a system finite in a direction, semi-infinite in b direction, and periodic in c direction (a, b, and c are the conventional lattice vectors in the space group R3m). The hinge state spectrum was obtained by projecting on the atoms at the corner, which preserve the mirror symmetry
Chiral HOTI tight-binding model
We considered a model on a simple cubic lattice spanned by the basis vectors
Chern-Simons topological invariant
The invariant for chiral HOTIs with
Mirror Chern number
The topological invariant of a 3D helical HOTI is the mirror Chern number Cm. Because for a spinful system a mirror symmetry
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/6/eaat0346/DC1
Supplementary Text
fig. S1. Nested entanglement and Wilson loop spectra for the second-order 3D chiral TI model defined in Eq. 1 in the main text with M/t = 2 and Δ1/t = Δ2/t = 1.
fig. S2. Real-space hopping picture for the optical lattice model with
fig. S3. Real-space structure for a chiral HOTI.
fig. S4. Constraints on mirror-symmetric domain wall modes in two dimensions.
fig. S5. Wilson loop characterization of helical HOTIs.
fig. S6. Band structure of the surface Dirac cones of the topological crystalline insulator SnTe calculated in a slab geometry.
fig. S7. High-symmetry points in the BZ of SnTe.
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REFERENCES AND NOTES
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