Research Article

Wallpaper fermions and the nonsymmorphic Dirac insulator

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Science  20 Jul 2018:
Vol. 361, Issue 6399, pp. 246-251
DOI: 10.1126/science.aan2802

Exotic topology on the surface

Analyzing the spatial symmetries of three-dimensional (3D) crystal structures has led to the discovery of exotic types of quasiparticles and topologically nontrivial materials. Wieder et al. focus on the symmetry groups of 2D surfaces of 3D materials—the so-called wallpaper groups—and find that some of them allow for an additional topological class. This class hosts a single fourfold-degenerate Dirac fermion on the surface of the material and, on the basis of the authors' calculations, is expected to occur in the compound Sr2Pb3.

Science, this issue p. 246

Abstract

Materials whose gapless surface states are protected by crystal symmetries include mirror topological crystalline insulators and nonsymmorphic hourglass insulators. There exists only a very limited set of possible surface crystal symmetries, captured by the 17 “wallpaper groups.” Here we show that a consideration of symmetry-allowed band degeneracies in the wallpaper groups can be used to understand previously described topological crystalline insulators and to predict phenomenologically distinct examples. In particular, the two wallpaper groups with multiple glide lines, pgg and p4g, allow for a topological insulating phase whose surface spectrum consists of only a single, fourfold-degenerate, true Dirac fermion, representing an exception to a symmetry-enhanced fermion-doubling theorem. We theoretically predict the presence of this phase in Sr2Pb3 in space group 127 (P4/mbm).

Topological phases stabilized by crystal symmetries have a number of distinct realizations. The first class that was theoretically proposed is composed of rotation and mirror topological crystalline insulators (TCIs) that host surface fermions protected by the projection of a bulk mirror plane or rotation axis onto a particular surface (1–3). These fermions have been observed in SnTe (4, 5) and related compounds (6, 7). Recent efforts to expand this analysis to nonsymmorphic systems with surface glide mirrors—symmetries composed of a mirror and a half-lattice translation—have yielded additional exotic free-fermion topological phases, which can exhibit so-called surface gapless “hourglass fermions” and the glide spin Hall effect (8–12). The theoretical proposal of (9, 10) has recently also received incipient experimental support (13).

In addition, topological insulators (TIs), crystalline or otherwise, provide exceptions to fermion-doubling theorems. These theorems impose fundamental bounds on phenomena in condensed matter physics. For example, in two dimensions, a single Kramers degeneracy in momentum space must always have another, partner Kramers crossing elsewhere in the Brillouin zone (BZ); otherwise, the Berry phase of a loop enclosing the degeneracy suffers from ambiguity (14). The discovery of the TI provided the first exception to this theorem: In these systems, Kramers pairs are allowed to be isolated on a single two-dimensional (2D) surface because they are connected to a 3D topological insulating bulk and have their partners on the opposite surface (15, 16).

Bulk fermions with higher-fold degeneracy, such as Dirac points, which are stabilized by crystal symmetry, may come with their own fermion-doubling theorems (17–20). As noted in (21), a single fourfold-degenerate Dirac fermion cannot be stabilized by 2D crystal symmetries as the only nodal feature at a given energy; it must always have at least one partner or accompanying hourglass nodal points. This is because a single Dirac point in two dimensions represents the quantum critical point (QCP) separating a trivial insulator (NI) from a TI. As shown in more detail in section 1B of (22), stabilizing just one of these Dirac points with crystal symmetries would therefore force the broken-symmetry NI and TI phases to be related by just a unitary transformation, violating their ℤ2 topological inequivalence. Here we report a class of symmetry-protected topological materials that, like the TIs before them, circumvent this restriction by placing a single, stable Dirac point on the surface of a 3D material.

The crucial requirement is that the surface preserves multiple nonsymmorphic crystal symmetries. Until now, most attention has been paid to crystal systems with surfaces that preserve only a single glide mirror. However, of the 17 2D surface symmetry groups, called wallpaper groups, there are two that host two intersecting glide lines (23). As we show in section 2 of (22), the algebra of the two glides requires that bands appear with fourfold degeneracy at a single time-reversal–invariant momentum (TRIM) at the edge of the BZ.

We study the noninteracting topological phases allowed in bulk crystals with surfaces that are invariant under the symmetries of the two wallpaper groups mentioned above, pgg and p4g. We show that, in addition to generalizations of the hourglass fermions introduced in (9), they host a topological phase characterized by a single, symmetry-enforced fourfold Dirac surface fermion—i.e., twice the degeneracy of a traditional TI surface state. This Dirac fermion is pinned by nonsymmorphic symmetry to the QCP between a TI and a NI, allowing for controllable topological phase transitions of the 2D surface under spin-independent glide-breaking strain.

Wallpaper groups pgg and p4g

The surface of a nonmagnetic crystal is itself a lower-dimensional crystal that preserves a subset of the bulk crystal symmetries; all 2D nonmagnetic surfaces are geometrically constrained to be invariant under the action of the 17 wallpaper groups. The set of spatial wallpaper group symmetries is restricted to those 3D space group symmetries that preserve the surface normal vector: rotations about that vector and in-plane lattice translations, mirror reflections, and glide reflections. Surface band features will therefore be constrained by the irreducible co-representations of these symmetries and their momentum-space compatibility relations (24, 25). Focusing on the 17 wallpaper groups that describe the surfaces of 3D crystals with strong spin-orbit coupling (SOC) and time-reversal symmetry [section 2 of (22)], we designate all topological 2D surface nodes formed from symmetry-enforced band degeneracies “wallpaper fermions.” In the language of group theory, wallpaper fermions are therefore fully captured by the irreducible co-representations of the wallpaper groups, so that, for example, the fourfold rotation–protected quadratic topological node introduced in (2) is a “spinless wallpaper fermion”; in contrast, the magnetic twofold surface degeneracy in (26), for which both bands have the same co-representations and are instead prevented from gapping by an additional rotation- and time-reversal–enforced (C2×T) 1D loop invariant, is not a wallpaper fermion.

Although other works have focused on the mathematical classification of topological phases protected by wallpaper group symmetries (27, 28), here we detail a further topological classification, as well as a search for topological insulating phases with phenomenologically distinct surface states. For the symmorphic wallpaper groups, topological rearrangements of the minimal surface band connectivities, recently enumerated for 3D bulk systems in (25), allow for only quantum spin Hall (QSH) phases and rotational variants of the mirror TCI phases, which exhibit twofold-degenerate free fermions along high-symmetry lines in the surface BZ (4–7).

For the four wallpaper groups with nonsymmorphic glide lines (pg, pmg, pgg, and p4g), this picture is enriched. Even in two dimensions, glide symmetries require that groups of four or more bands be connected, an effect that frequently manifests itself in hourglass-like band structures (21, 29, 30). For the wallpaper groups with only a single glide line, pg and pmg, surface bands can be connected in topologically nontrivial patterns of interlocking hourglasses (9, 10) or in “Möbius strip” connectivities if time-reversal symmetry is relaxed (26, 31), both of which exhibit twofold-degenerate surface fermions along some of the momentum-space glide lines. In the remaining two wallpaper groups with multiple glide symmetries, pgg and p4g, higher-degenerate wallpaper fermions are uniquely allowed.

We consider a z-normal surface with glides gx,y≡{mx,y|12120}—i.e., a mirror reflection mx,y through the x or y axis followed by a translation of half a lattice vector in the x^ and y^ directions (Fig. 1). When SOC is present, gx,y2=−eiky,x, where kx,y are the in-plane surface crystal momenta. At the corner point M¯ of the surface BZ (kx = ky = π), gx2=gy2=+1 and {gx,gy}=0. When combined with time reversal T2=−1, this symmetry algebra requires that all states at M¯ are fourfold-degenerate. Furthermore, wallpaper groups with two glides are the only nonmagnetic surface groups that admit this algebra and, therefore, the only surface groups that can host protected fourfold degeneracies on strong-SOC crystals (24). The examination of symmetry-allowed terms reveals that fourfold points in these wallpaper groups are linearly dispersing [section 2 of (22)], rendering them true surface Dirac fermions, more closely related by symmetry algebra and quantum criticality to the bulk nodes in nonsymmorphic 3D Dirac semimetals (15, 17, 18) than to the surface states of conventional TIs. In section 2 of (22), we provide proofs relating this algebra to Dirac degeneracy and dispersion.

Fig. 1 Wallpaper groups pgg and p4g.

Shown are the unit cells and Brillouin zone (BZ) for their two-site realizations. The A and B sites are characterized by T-symmetric internal degrees of freedom (blue arrows) that are transformed under crystal symmetry operations. These correspond physically to nonmagnetic properties that transform as vectors, such as atom displacements or local electric dipole moments. Glide lines (green) exchange the sublattices through fractional lattice translations. In p4g, there is an extra C4 symmetry (⊗) about the surface normal with axes located on the sites. The combination of this C4 symmetry and the glides produces additional diagonal symmorphic mirror lines (red) in p4g. In the BZ of p4g, C4 relates Y¯ to X¯.

For bulk insulators, the glide-preserving bulk topological phase and, consequently, z-normal surface states can be determined without imposing a surface by classifying the allowed connectivities of the z-projection Wilson loop holonomy matrix (10, 32–38), a bulk quantity defined by[W(kx,ky,kz0)]ij≡〈ui(kx,ky,kz0+2π)|Π^(kx,ky,kz0)|uj(kx,ky,kz0)〉(1)where we have defined the product of projectorsΠ^(kx,ky,kz)≡P^(kx,ky,kz+2π)P^(kx,ky,kz+2π(N−1)N)⋯P^(kx,ky,kz+2πN)(2)P^(k) is the projector onto the occupied bands at k, and N is the number of discretized kz points. The rows and columns of W correspond to filled bands, where |uj(k)〉 is the cell-periodic part of the Bloch wave function at momentum k with band index j. The eigenvalues of W are gauge-invariant and of the form eiθ(kx,ky), where θ(kx, ky) is the Wilson phase. As detailed in sections 3 and 4 of (22), the Wilson bands inherit the symmetries of the z-normal wallpaper group and, therefore, must also exhibit the required degeneracy multiplets of wallpaper groups pgg and p4g. In particular, both the surface and Wilson bands are twofold-degenerate along X¯M¯ (Y¯M¯) by the combination of time reversal and gy (gx) and meet linearly in fourfold degeneracies at M¯.

By generalizing the ℤ4 invariant defined in (11) for the single-glide wallpaper groups (9, 39), we define topological invariants for double-glide systems using the (001)-directed Wilson loop eigenvalues. For gy in a surface BZ in wallpaper group pgg, the quantized invariant χy is defined in (11) by integrating the Wilson phases, θj(kx,ky), along the path M¯Y¯Γ¯X¯χy≡1π∑j=1nocc/2[θj+(M¯)−θj+(X¯)+∫M¯Y¯dθj++∫Γ¯X¯dθj+]+∑j=1nocc  12π∫Y¯Γ¯ dθj mod 4(3)where nocc is the number of occupied bands, the superscript + indicates the positive glide sector, and the absence of a superscript indicates the line where gy is not a symmetry and the sum is over all bands. In the presence of an additional glide, gx, one can obtain χx by the transformation x↔y, X¯↔Y¯ in Eq. 3. Although Eq. 3 appears complicated, (χx, χy) can be easily evaluated by considering the bands within each glide sector that cross an arbitrary horizontal line in the Wilson spectrum [section 5A of (22)]. Wallpaper group p4g also has C4z symmetry, which requires χx = χy and implies the existence of the symmorphic mirrors{m110|12120}and {m11¯0|12120}. These mirrors yield ℤ mirror Chern numbers n110 and n11¯0, respectively, with values constrained by the glide invariant χx, (−1)n110=(−1)n11¯0=(−1)χx [section 5D of (22)].

To enumerate the allowed topological phases shown in Fig. 2, we consider possible restrictions on (χx, χy). Although χx and χy can individually take on values 0, 1, 2, and 3, only pairs that satisfy χx + χy mod 2 = 0 are permitted in bulk insulators. This can be understood as follows: If χx + χy is odd, the 2D surface consisting of the four partial planes defined by 0 ≤ kx ≤ π ky = 0, π and kx = 0, π 0 ≤ ky ≤ π possesses an overall Chern number, which implies the existence of a bulk gapless point (40, 41), contradicting our original assumption that the system is insulating. We present a rigorous proof in section 5B of (22) and show that the remaining collection of eight insulating phases is indexed by the group ℤ4×ℤ2.

Fig. 2 Double-glide Wilson band connectivities.

Shown are the eight topologically distinct Wilson band connectivities for bulk insulators with crystal surfaces that preserve 2D glide reflections on the projections of two orthogonal bulk glide planes. Each band structure is labeled by its two ℤ4 indices, (χx, χy), subject to the constraint that χx + χy = 0 mod 2. Under the imposition of C4z symmetry in wallpaper group p4g, connectivities are excluded for which χx ≠ χy. Solid black (dotted blue) lines in the regions X¯M¯ and Γ¯Y¯ indicate bands with gx eigenvalue +ieiky/2 (−ieiky/2), whereas the solid (dotted) lines in the regions Y¯M¯ and Γ¯X¯ indicate bands with gy eigenvalue +ieikx/2(−ieikx/2). When bulk inversion symmetry is additionally present, the Wilson spectrum becomes particle-hole–symmetric. Bands along X¯M¯Y¯ are doubly degenerate and meet at M¯ in a fourfold-degenerate point, and bands along Y¯Γ¯X¯ display either the hourglass (left column) or “double-glide spin Hall” (right column) flows. The (2, 0) and (0, 2) phases are relatives of the hourglass topologies proposed in (9, 10). The (2, 2) nonsymmorphic Dirac insulating phase can host a surface state consisting of a single, fourfold-degenerate Dirac fermion.

For χx,y = 1, 3, the system is a strong topological insulator (STI). The four “double-glide spin Hall” STI phases possess the usual twofold-degenerate Kramers pairs at Γ¯, X¯, and Y¯, as well as a fourfold-degenerate Dirac point at M¯. The four STI phases are topologically distinct but will appear indistinguishable in glide-unpolarized angle-resolved photoemission spectroscopy (ARPES) experiments. However, if two double-glide spin Hall systems with differing χx and χy are coupled together, the resulting surface modes will distinguish between χx,y = 1, 3 (11) [section 5B of (22)].

When χx,y = 0, 2, the system is in a topological crystalline phase. For (χx, χy) = (0, 2) or (2, 0), which is only permitted in a C4z-broken surface pgg, a variant of the hourglass insulating phase (9) is present on the surface. For example, when (χx, χy) = (0, 2), either time-reversed partners of twofold-degenerate free fermions live along Γ¯X¯ or both twofold-degenerate fermions live along Γ¯Y¯ and a fourfold-degenerate Dirac fermion exists at M¯.

Lastly, but most interestingly, for χx = χy = 2, we find that the system exists in a previously uncharacterized “nonsymmorphic Dirac insulating” phase, capable of hosting just a single fourfold-degenerate Dirac surface fermion at M¯.

Materials realizations

We applied density functional theory (DFT), including the effects of SOC, to predict topological phases stabilized by wallpaper groups pgg and p4g in previously synthesized, stable materials. The details of these large-scale calculations are provided in section 6 of (22). We find double-glide topological phases on the (001) surface (wallpaper group p4g) of three members of the space group 127 (P4/mbm) A2B3 family of materials: Sr2Pb3 (42, 43), Au2Y3 (44), and Hg2Sr3 (45, 46). We find that Sr2Pb3 has a gap at the Fermi energy at each crystal momentum, in spite of the presence of electron and hole pockets (Fig. 3). A Wilson loop calculation of the bands up to this gap (Fig. 3D) indicates that this material possesses the bulk topology of a (2, 2) nonsymmorphic Dirac insulator [section 6B of (22)]. Calculating the surface spectrum through surface Green’s functions, as described in section 6B of (22) (Fig. 4), we find that the (001) surface of Sr2Pb3, although displaying an overall metallic character, develops a gap of 45 meV at the Fermi energy at M¯. Inside this gap, we observe a single, well-isolated, fourfold-degenerate surface Dirac fermion.

Fig. 3 The crystal and electronic structures of Sr2Pb3.

This compound is in space group 127 (P4/mbm). (A) The unit cell of Sr2Pb3. Mz is a symmorphic mirror reflection through the z axis. (B) The bulk BZ and the (001)-surface BZ (wallpaper group p4g). (C) The electronic bands obtained using DFT. The Fermi level is set to 0 eV. At each point in the BZ, there is a band gap near the Fermi energy, indicated by the dashed red line; insets show magnified images of the boxed regions. (D) The (001)-directed Wilson bands; red (blue) points indicate Wilson bands with positive (negative) surface glide eigenvalues, λx,y+(−). By counting the Wilson bands within each glide sector that cross the dashed green line, we find that Sr2Pb3 has the bulk topology of a (2, 2) nonsymmorphic Dirac insulator [section 6B of (22)].

Fig. 4 The topological fourfold surface Dirac fermion.

Shown is the (001)-surface band structure of Sr2Pb3 (wallpaper group p4g) calculated using surface Green’s functions [section 6B of (22)]. The color bar indicates the relative degree of surface localization. The Fermi level is set to zero. The fourfold surface Dirac fermion appears at M¯ in the region indicated by the red rectangle and is shown in more detail in the inset. Unlike in graphene, the cones of this Dirac point are nondegenerate, except along X¯M¯. The four dark blue surface bands dispersing from M¯ confirm that the red surface-localized point is fourfold-degenerate.

Unlike Sr2Pb3, Au2Y3 and Hg2Sr3 in space group 127 (P4/mbm) are gapless, with bulk C4z-protected Dirac nodes (47) present near the Fermi energy. In section 6B of (22), we show that under weak (100) strain, these Dirac nodes can be gapped to induce the (0, 2) topological hourglass phase in these two materials.

We additionally find that the (001) surface (wallpaper group pgg) of the narrow-gap insulator Ba5In2Sb6 in space group 55 (Pbam) (48) hosts a double-glide topological hourglass fermion. We find that Ba5In2Sb6 develops an indirect band gap of 5 meV (direct band gap, 17 meV) (Fig. 5). The Wilson loop spectrum obtained from the occupied bands (Fig. 5D) demonstrates that this material is a (2, 0) double-glide topological hourglass insulator [section 6A of (22)]. We find that the (001) surface of Ba5In2Sb6 has a projected insulating bulk gap that is spanned along Y¯Γ¯ by the top and bottom bands of two different topological hourglass fermions, which themselves are degenerate with states in the bulk spectrum (Fig. 6). However, these fermions are topologically connected to a clearly distinguishable hourglass fermion along Γ¯X¯ and a fourfold-degenerate surface Dirac fermion at M¯, both of which could in principle be observed using ARPES.

Fig. 5 The crystal and electronic structures of Ba5In2Sb6.

This compound is in space group 55 (Pbam). (A) The unit cell of Ba5In2Sb6. (B) The bulk BZ and the (001)-surface BZ (wallpaper group pgg). (C) The electronic bands obtained using DFT. The Fermi level is set to 0 eV. There is an insulating gap at the Fermi energy, indicated by the dashed black line. (D) The (001)-surface Wilson bands; red (blue) points indicate Wilson bands with positive (negative) surface glide eigenvalues, λx,y+(−). Counting the Wilson bands crossing the dashed green line, we find that the bulk displays a (2, 0) topological hourglass connectivity [section 6A of (22)].

Fig. 6 Double-glide hourglass fermions.

Shown is the (001)-surface band structure of Ba5In2Sb6 (wallpaper group pgg) calculated using surface Green’s functions [section 6A of (22)]. The Fermi level is set to zero. Surface bands from the top of one hourglass fermion and the bottom of another connect the valence and conduction manifolds along Y¯Γ¯; the hourglass fermions themselves are buried in the bulk manifolds along this line. The surface bands also display other signatures of the (2, 0) topological hourglass connectivity, including a fourfold Dirac fermion at M¯ connected to a clearly distinguishable hourglass fermion along Γ¯X¯. The maximally localized Wannier functions obtained numerically from ab initio calculations are only approximately symmetric under the surface wallpaper group, and therefore bands along X¯M¯ appear weakly split.

Discussion

We have demonstrated theoretically the existence of a nonsymmorphic Dirac insulator—a topological crystalline material with a single fourfold-degenerate surface Dirac point stabilized by two perpendicular glides. After an exhaustive study of the 17 time-reversal–symmetric, strong-SOC wallpaper groups, only pgg and p4g are shown to be capable of supporting this fourfold fermion. This phase is one of eight topologically distinct phases that can exist in insulating orthorhombic crystals with surfaces that preserve two perpendicular glides; we have classified all eight phases by topological indices (χx, χy) that characterize the connectivity of the z-projection Wilson loop spectrum. We report the prediction of the nonsymmorphic Dirac insulating phase in Sr2Pb3 and of related double-glide topological hourglass phases in Ba5In2Sb6, as well as in (100)-strained Au2Y3 and Hg2Sr3. We also report the theoretical prediction of a set of previously unknown double-glide spin Hall phases. Although their surface Kramers pairs and fourfold Dirac fermions should be distinctive in ARPES experiments, characterization of transport in the double-glide spin Hall phases remains an open question.

We also find that a simple intuition exists for the topological crystalline phases χx,y = 0, 2. In section 7A of (22), we present an eight-band tight-binding model that, when half-filled, can be tuned to realize all ℤ4×ℤ2 double-glide insulating phases. In a particular regime of parameter space, in which SOC is absent at the X¯ and Y¯ points and bulk inversion symmetry is imposed, the Wilson loop eigenvalues at the edge TRIMs are pinned to ±1 [θ(M¯/X¯/Y¯)=0,π], and each TRIM represents the end of a doubly degenerate Su-Schrieffer-Heeger (SSH) model (49). In this limit, when the product of parity eigenvalues at Γ¯ satisfies ξ(Γ¯)=+1, the bulk topology is fully characterized by the relative SSH polarizations, χx,y=2{[θ(M¯)−θ(Y¯,X¯)] mod 2}.

Lastly, because the (2, 2) topological surface Dirac point is symmetry-pinned to the QCP between a 2D TI and a NI, we examine its potential for hosting strain-engineered topological physics. Consider the two-site surface unit cell in wallpaper group pgg from Fig. 1. In the (2, 2) nonsymmorphic Dirac insulating phase, the surface Dirac fermion can be captured by the k · p Hamiltonian near M¯HM¯=τx(vxσxkx+vyσyky)(4)where τ is a sublattice degree of freedom, vx(y) is the Fermi velocity in the x(y) direction, σ is a T-odd orbital degree of freedom, and gx,y = τyσx,y (gx,y2=+1). There exists a single, T-even mass term, Vm = mτz, which satisfies {HM¯,Vm}=0 and is therefore guaranteed to fully gap HM¯. Therefore, surface regions with differing signs of m will be in topologically distinct gapped phases and must be separated by 1D topological QSH surface domain walls, protected only by time-reversal symmetry (50). Because pg has point group 2mm, and {Vm, gx,y} = 0, Vm can be considered an xy A2 distortion (51), which could be achieved by strain in the x + y direction and compression in the x – y direction. These domain walls would appear qualitatively similar to those proposed in bilayer graphene (52–54). However, whereas those domain walls are protected by sublattice symmetry and are therefore sensitive to disorder, domain walls originating from nonsymmorphic Dirac insulators are protected by only time-reversal symmetry and therefore should be robust against surface disorder. Under the right interacting conditions or chemical modifications, a nonsymmorphic Dirac insulator surface may also reconstruct and self-induce regions of randomly distributed ±m, separated by a network of 1D QSH domain walls. These domain walls are closely related to the 1D helical hinge modes of “second-order” topological insulators (55–57). Furthermore, in the presence of electron-electron interactions, these domain walls form helical Luttinger liquids (58). Although Sr2Pb3 is insufficiently insulating to experimentally isolate these effects, future bulk-insulating nonsymmorphic Dirac insulators could provide an experimental platform for probing and manipulating the physics of time-reversal–invariant topological Luttinger liquids.

Supplementary Materials

www.sciencemag.org/content/361/6399/246/suppl/DC1

Supplementary Text

Figs. S1 to S13

Tables S1 to S3

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Acknowledgments: We thank A. Alexandradinata for a discussion about the invariant in Eq. 3 and E. Mele for fruitful discussions. Funding: B.J.W. and C.L.K. acknowledge support through a Simons Investigator grant from the Simons Foundation to C.L.K. and through Nordita under ERC DM 321031. Z.W. and B.A.B. acknowledge support from the Department of Energy (grant DE-SC0016239), the National Science Foundation (EAGER grant DMR-1643312), Simons Investigator grants (ONR-N00014-14-1-0330, ARO MURI W911NF-12-1-0461, and NSF-MRSEC DMR-1420541), the Packard Foundation, the Schmidt Fund for Innovative Research, and the National Natural Science Foundation of China (grant 11504117). Y.K. and A.M.R. thank the National Science Foundation MRSEC program for support under DMR-1120901 and acknowledge the HPCMO of the U.S. DOD and the NERSC of the U.S. DOE for computational support. B.A.B. wishes to thank Ecole Normale Supérieure, UPMC Paris, and Donostia International Physics Center for their generous sabbatical hosting during some of the stages of this work. The development of Wilson loop–based topological invariants and first-principles characterization of the hourglass insulating phase in Ba5In2Sb6 were supported by Department of Energy grant DE-SC00016239. Author contributions: B.J.W. and B.B. developed the application of wallpaper group co-representations to the prediction of 3D topological phases and related the Dirac insulating phase to symmetry-enhanced fermion doubling. Z.W. and J.C. developed the Wilson loop–based topological invariants; Z.W. discovered the hourglass insulating phase in Ba5In2Sb6 and performed DFT investigations of additional materials. Y.K. and H.-S.D.K. discovered the Dirac and hourglass insulating phases in the A2B3 family and performed DFT investigations of additional materials. A.M.R. was the principal investigator for DFT. C.L.K. and B.A.B. were the principal investigators for crystalline symmetry, fermion doubling, and topological invariants. Competing interests: The authors declare that they have no competing interests. Data and materials availability: For our DFT calculations, we used QUANTUM ESPRESSO, an open-source package (www.quantum-espresso.org/), and VASP, a commercial software package (www.vasp.at/). We used WANNIER90, an open-source software package (www.wannier.org/download.html), to construct the Wannier Hamiltonians for surface Green’s functions. All DFT examinations used the experimental lattice parameters and relaxation schemes detailed in section 6 of (22). The numerical data for all band structure, Wilson band, and surface state calculations are publicly available from the Harvard Dataverse (59).

Correction (27 March 2020): The Funding section of the Acknowledgments has been updated to specify which particular part of the work was supported by the U.S. Department of Energy.

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