## 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 _{2}TeI, BiSe, and BiTe are helical higher-order topological insulators and propose a realistic experimental setup to detect the hinge states.

## 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 *n*th-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 *1*, *10*–*12*). We show that helical HOTIs may occur when a system is invariant under time reversal

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 *2*). Finally, on the basis of tight-binding and ab initio calculations, we propose SnTe as a material realization for helical HOTIs. We also propose an explicit experimental setup to cleanly create hinge states in a topological SnTe coaxial cable. In contrast, chiral HOTIs may arise in 3D TI materials that exhibit noncollinear antiferromagnetic order at low temperatures. Our work is complemented by two related articles: The study of Langbehn *et al*. (*14*) provides a general classification of second-order phases with reflection symmetry for all 10 Altland-Zirnbauer symmetry classes, and the study of Benalcazar *et al*. (*15*) establishes a physical interpretation of the topological invariants of higher-order phases in terms of electric multipole moments.

## 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 _{xy} = *e*^{2}/*h* and σ_{xy} = − *e*^{2}/*h* on the (100) surfaces and the (010) surfaces, respectively, which respects

A concrete model of this phase is defined via the four-band Bloch Hamiltonian_{i} and τ_{i}, *i* = *x*, *y*, *z*, are the three Pauli matrices acting on spin and orbital degree of freedoms, respectively (see Methods for a real-space representation of the model). For Δ_{2} = 0, *M*| < 3. Time-reversal symmetry is represented by *T* ≡ τ_{0}σ_{y}*K*, where *K* denotes complex conjugation. For Δ_{2} = 0, Hamiltonian (1) has a

The term proportional to Δ_{2} breaks both _{2} ≠ 0. Because

The phase diagram of Hamiltonian (1) is shown in Fig. 2A. For 1 < |*M*/*t*| < 3 and Δ_{1}, Δ_{2} ≠ 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 Δ_{2} 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 _{2}, the (001) surface of the model remains gapless, because its Dirac cone is protected by the

We turn to the bulk topological invariant that describes the *18*). The very same quantity θ is the topological invariant of chiral HOTIs. What changes is that its quantization to values 0, π is not enforced by

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 *19*, *20*). There, we also provide two further topological characterizations, one based on so-called nested Wilson loop (*4*) and entanglement spectra (*21*–*23*) and one applicable to systems that are in addition invariant under the product *3*).

### 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 *x* = −*y* and the *x* = *y* planes invariant and with it a pair of hinges each (as sketched in Fig. 1B). We consider a hypothetical but realizable electronic structure where gapless degrees of freedom are only found on the hinge. At a given hinge, for instance, one that is invariant under *z* direction and have *i*λ, λ = ± 1, by *N*_{R,λ} (*N*_{L,λ}). We argue that the net number of helical hinge pairs *n* ≡ *N*_{R,+} − *N*_{L,+} (which by time-reversal symmetry is equal to *N*_{L,−} − *N*_{R,−}) is topologically protected. In particular, *n* cannot be changed by any surface or hinge manipulation that respects both *N*_{R,+} and *N*_{L,+} are nonzero (assuming from now on that *N*_{R,+} > *N*_{R,−}), then we can always hybridize *N*_{L,+} right-moving modes with all *N*_{L,+} left-moving modes within the λ = + subspace without breaking any symmetry. Therefore, only the difference *n* is well defined and corresponds to the number of remaining pairs of modes.

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 (*N*_{L,−} + *N*_{L,+}) and (*N*_{R,−} + *N*_{R,+}) 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 *N*_{L,+}, *N*_{L,−}, *N*_{R,+}, and *N*_{R,−} 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

The topological invariant for the *6*, *13*) *C*_{m}/2 on the *C*_{m}). First, observe that if *C*_{m} were odd, then the system would be a strong 3D TI: The *C*_{m} is odd, then there is an odd number of Dirac cones on the (110) surface, and time-reversal symmetry implies that such a system is a strong 3D TI. As the surfaces of a strong 3D TI cannot be gapped out with a time-reversal symmetric perturbation, we cannot construct a helical HOTI from it. We conclude that *C*_{m} is even for all systems of interest to us.

We now discuss the correspondence between the bulk topological invariant *C*_{m}/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 **n**_{±} = (1 ± α, 1 ∓ α, 0) for small α, which are mapped into each other under

A nonzero bulk mirror Chern number *C*_{m} with respect to the *k*_{1} = 0, π in the surface Brillouin zone of the (110) surface, where *k*_{1} is the momentum along the direction with the unit vector *M*_{xy} = *i*σ_{x}, preventing mass terms of the form *m*σ_{y} from appearing. The sign of *v*_{z} is tied to the *i* sgn *v*_{z}) of the eigenstate with a positive group velocity in the *z* direction (at *v*_{z} > 0 (*v*_{z} < 0) by *n*_{+} (*n*_{−}), the bulk-boundary correspondence of a topological crystalline insulator (*5*) implies**n**_{+} and **n**_{−}, which are not invariant under the mirror symmetry but map into each other. Mass terms are allowed, and the Hamiltonians on the surfaces with normal **n**_{±} read*m* and κ real parameters. The two surfaces with normals **n _{+}** and

**n**meet in a hinge (see Fig. 3B). Equation 4 describes a Dirac fermion with a mass of opposite sign on the two surfaces. The hinge therefore forms a domain wall in the Dirac mass from which a single chiral channel connecting valence and conduction bands arises (

_{−}*24*). As we show in the Supplementary Materials, this domain wall either binds an R moving mode with

*M*

_{xy}mirror eigenvalue

*i*λ =

*i*sgn(

*v*

_{z}) or an L moving mode with mirror eigenvalue −

*i*sgn(

*v*

_{z}). The equality

*N*

_{L/R,±}we had introduced before to the mirror-graded numbers of Dirac cones on the (110) surface

**n**. From Eq. 3, we obtain

_{±}*C*

_{m}to the number of protected helical hinge pairs

*n*of the HOTI. Notice that by time-reversal symmetry,

*N*

_{R,+}−

*N*

_{R,−}=

*N*

_{L,−}−

*N*

_{L,+}, so that

*n*in Eq. 5 is an integer.

*C*

_{m}is even as aforementioned.

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 *x*, *y*, *z*) → (*y*, *x*, *z*)] as well as its partners under cubic symmetry (*C*_{m} = 2 with respect to the mirror symmetries on the mirror planes that include the Γ point in momentum space. All other mirror planes have *C*_{m} = 0. As a result, cubic SnTe has mirror-symmetry protected Dirac cones on specific surfaces. We consider the geometry of Fig. 1B with open boundary conditions in the *x* and *y* directions and periodic boundary conditions in the *z* direction. The

(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 *C*_{m} = 2.

(2) If uniaxial strain along the (110) direction is applied to SnTe, then *6*) with (110) strain, demonstrating that there exists one Kramers pair of hinge modes on all four hinges in the geometry of Fig. 1B.

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 Bi_{2}TeI (*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 *Fm*3*m*) to the rhombohedral structure (space group *R*3*m*). 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 *R*3*m*). 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 ,

*i*=

*x*,

*y*,

*z*, with two orbitals

^{1}/

_{2}electrons. It is defined by the tight-binding Hamiltonian

**r**. We denoted by σ

_{0}and σ

_{i},

*i*=

*x*,

*y*,

*z*, respectively, the 2 × 2 identity matrix and the three Pauli matrices acting on the spin

^{1}/

_{2}degree of freedom.

### Chern-Simons topological invariant

The invariant for chiral HOTIs with *u*_{n}〉 are the Bloch eigenstates of the Bloch Hamiltonian and *n*, *n*′ are running over the occupied bands of the insulator. ∂_{a} is the partial derivative with respect to the momentum component *k*_{a}, *a* = *x*, *z*, *y*. The trace is performed with respect to band indices.

### Mirror Chern number

The topological invariant of a 3D helical HOTI is the mirror Chern number *C*_{m}. Because for a spinful system a mirror symmetry *M* has eigenvalues ± *i*. Given a surface ∑ in the Brillouin zone, which is left invariant under the action of *u*_{n}〉 of the Bloch Hamiltonian on ∑ can be decomposed into two groups, {*i*, respectively. Time-reversal symmetry maps one mirror eigenspace into the other; if time-reversal symmetry is present, then the two mirror eigenspaces are of the same dimension. We can define the Chern number in each mirror subspace as*i* mirror subspace, with *C*_{+} = − *C*_{−}, and we define the mirror Chern number

## 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 Δ_{1}/*t* = Δ_{2}/*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

**Acknowledgments:**B.A.B. wishes to thank Ecole Normale Superieure, Université Pierre et Marie Curie, Paris, and Donostia International Physics Center for their sabbatical hosting during some of the stages of this work.

**Funding:**F.S. and T.N. acknowledge support from the Swiss National Science Foundation (grant number: 200021_169061) and from the European Union’s Horizon 2020 research and innovation program (ERC-StG-Neupert-757867-PARATOP). A.M.C. wishes to thank the Aspen Center for Physics, which is supported by NSF grant PHY-1066293, for hosting during some stages of this work. M.G.V. was supported by FIS2016-75862-P national projects of the Spanish Ministry of Economy and Competitiveness. B.A.B. acknowledges support for the analytic work from the Department of Energy (de-sc0016239), Simons Investigator Award, the Packard Foundation, and the Schmidt Fund for Innovative Research. The computational part of the Princeton work was performed under NSF Early-Concept Grants for Exploratory Research grant DMR-1643312, ONR-N00014-14-1-0330, ARO MURI W911NF-12-1-0461, and NSF-MRSEC DMR-1420541.

**Author contributions:**F.S., A.M.C., B.A.B., and T.N. worked out the theoretical results presented here, M.G.V. and Z.W. performed first-principles calculations, and S.S.P.P. contributed the experimental proposal for SnTe nanowires.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

- Copyright © 2018 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).