RESEARCH ARTICLE◥

TOPOLOGICAL MATTER

Wallpaper fermions and thenonsymmorphic Dirac insulatorBenjamin J. Wieder1,2,3*†, Barry Bradlyn4*, Zhijun Wang1*, Jennifer Cano4*,Youngkuk Kim5,6, Hyeong-Seok D. Kim3,5, Andrew M. Rappe5,C. L. Kane3†, B. Andrei Bernevig1,7,8†

Materials whose gapless surface states are protected by crystal symmetries includemirror topological crystalline insulators and nonsymmorphic hourglass insulators.There exists only a very limited set of possible surface crystal symmetries, captured by the17 “wallpaper groups.” Here we show that a consideration of symmetry-allowed banddegeneracies in the wallpaper groups can be used to understand previously describedtopological crystalline insulators and to predict phenomenologically distinct examples. Inparticular, the two wallpaper groups with multiple glide lines, pgg and p4g, allow for atopological insulating phase whose surface spectrum consists of only a single, fourfold-degenerate, true Dirac fermion, representing an exception to a symmetry-enhancedfermion-doubling theorem.We theoretically predict the presence of this phase in Sr2Pb3 inspace group 127 (P4/mbm).

Topological phases stabilized by crystal sym-metries have a number of distinct realiza-tions. The first class that was theoreticallyproposed is composed of rotation and mir-ror topological crystalline insulators (TCIs)

that host surface fermions protected by the pro-jection of a bulk mirror plane or rotation axisonto a particular surface (1–3). These fermionshave been observed in SnTe (4, 5) and relatedcompounds (6, 7). Recent efforts to expand thisanalysis to nonsymmorphic systems with surfaceglide mirrors—symmetries composed of a mirrorand a half-lattice translation—have yielded addi-tional exotic free-fermion topological phases, whichcan exhibit so-called surface gapless “hourglassfermions” and the glide spin Hall effect (8–12).The theoretical proposal of (9, 10) has recentlyalso received incipient experimental support (13).In addition, topological insulators (TIs), crys-

talline or otherwise, provide exceptions tofermion-doubling theorems. These theoremsimpose fundamental bounds on phenomenain condensed matter physics. For example, intwo dimensions, a single Kramers degeneracyin momentum space must always have another,partner Kramers crossing elsewhere in the Bril-louin zone (BZ); otherwise, the Berry phase ofa loop enclosing the degeneracy suffers fromambiguity (14). The discovery of the TI providedthe first exception to this theorem: In these sys-

tems, Kramers pairs are allowed to be isolatedon a single two-dimensional (2D) surface be-cause they are connected to a 3D topologicalinsulating bulk and have their partners on theopposite surface (15, 16).Bulk fermions with higher-fold degeneracy,

such as Dirac points, which are stabilized bycrystal symmetry, may come with their ownfermion-doubling theorems (17–20). As noted

in (21), a single fourfold-degenerate Dirac fer-mion cannot be stabilized by 2D crystal sym-metries as the only nodal feature at a givenenergy; it must always have at least one partneror accompanying hourglass nodal points. This isbecause a single Dirac point in two dimensionsrepresents 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 withcrystal symmetries would therefore force thebroken-symmetry NI and TI phases to be re-lated by just a unitary transformation, violatingtheir ℤ2 topological inequivalence. Here we re-port a class of symmetry-protected topologicalmaterials that, like the TIs before them, circum-vent this restriction by placing a single, stableDirac point on the surface of a 3D material.The crucial requirement is that the surface

preserves multiple nonsymmorphic crystal sym-metries. Until now, most attention has been paidto crystal systems with surfaces that preserveonly a single glide mirror. However, of the 172D surface symmetry groups, called wallpapergroups, there are two that host two intersect-ing glide lines (23). As we show in section 2 of(22), the algebra of the two glides requires thatbands appear with fourfold degeneracy at asingle time-reversal–invariant momentum (TRIM)at the edge of the BZ.We study the noninteracting topological

phases allowed in bulk crystals with surfacesthat are invariant under the symmetries of thetwo wallpaper groups mentioned above, pggand p4g. We show that, in addition to gener-alizations of the hourglass fermions introducedin (9), they host a topological phase characterized

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1Department of Physics, Princeton University, Princeton, NJ 08544, USA. 2Nordita, Center for Quantum Materials, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23,SE-106 91 Stockholm, Sweden. 3Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104-6323, USA. 4Princeton Center for Theoretical Science, PrincetonUniversity, Princeton, NJ 08544, USA. 5Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104-6323, USA. 6Department of Physics, Sungkyunkwan University (SKKU),Suwon 16419, Korea. 7Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany. 8Max Planck Institute of MicrostructurePhysics, 06120 Halle, Germany.*These authors contributed equally to this work.†Corresponding author. Email: bwieder@princeton.edu (B.J.W.); kane@physics.upenn.edu (C.L.K.); bernevig@princeton.edu (B.A.B.)

Fig. 1. Wallpaper groups pgg and p4g. Shown are the unit cells and Brillouin zone (BZ) for theirtwo-site realizations. The A and B sites are characterized by T -symmetric internal degrees offreedom (blue arrows) that are transformed under crystal symmetry operations. These correspondphysically to nonmagnetic properties that transform as vectors, such as atom displacements orlocal electric dipole moments. Glide lines (green) exchange the sublattices through fractionallattice translations. In p4g, there is an extra C4 symmetry (�) about the surface normal with axeslocated 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.

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by a single, symmetry-enforced fourfold Diracsurface fermion—i.e., twice the degeneracy of atraditional TI surface state. This Dirac fermion ispinned by nonsymmorphic symmetry to the QCPbetween a TI and a NI, allowing for controllabletopological phase transitions of the 2D surfaceunder spin-independent glide-breaking strain.

Wallpaper groups pgg and p4g

The surface of a nonmagnetic crystal is itselfa lower-dimensional crystal that preserves asubset of the bulk crystal symmetries; all 2Dnonmagnetic surfaces are geometrically con-strained to be invariant under the action of the17 wallpaper groups. The set of spatial wallpapergroup symmetries is restricted to those 3D spacegroup symmetries that preserve the surface nor-mal vector: rotations about that vector and in-plane lattice translations, mirror reflections,and glide reflections. Surface band features willtherefore be constrained by the irreducible co-representations of these symmetries and theirmomentum-space compatibility relations (24, 25).Focusing on the 17 wallpaper groups that de-scribe the surfaces of 3D crystals with strongspin-orbit coupling (SOC) and time-reversalsymmetry [section 2 of (22)], we designate alltopological 2D surface nodes formed fromsymmetry-enforced band degeneracies “wallpaperfermions.” In the language of group theory,wallpaper fermions are therefore fully capturedby the irreducible co-representations of the wall-paper groups, so that, for example, the fourfoldrotation–protected quadratic topological nodeintroduced in (2) is a “spinless wallpaper fer-mion”; in contrast, the magnetic twofold sur-face degeneracy in (26), for which both bandshave the same co-representations and are insteadprevented from gapping by an additional rotation-and time-reversal–enforced ðC2 � T Þ 1D loop in-variant, is not a wallpaper fermion.Although other works have focused on the

mathematical classification of topological phasesprotected by wallpaper group symmetries (27, 28),here we detail a further topological classifica-tion, as well as a search for topological insulat-ing phases with phenomenologically distinctsurface states. For the symmorphic wallpapergroups, topological rearrangements of the mini-mal surface band connectivities, recently enum-erated for 3D bulk systems in (25), allow foronly quantum spin Hall (QSH) phases and ro-tational variants of the mirror TCI phases, whichexhibit twofold-degenerate free fermions alonghigh-symmetry lines in the surface BZ (4–7).For the four wallpaper groups with non-

symmorphic glide lines (pg, pmg, pgg, and p4g),this picture is enriched. Even in two dimensions,glide symmetries require that groups of four ormore bands be connected, an effect that fre-quently manifests itself in hourglass-like bandstructures (21, 29, 30). For the wallpaper groupswith only a single glide line, pg and pmg, surfacebands can be connected in topologically non-trivial patterns of interlocking hourglasses (9, 10)or in “Möbius strip” connectivities if time-reversalsymmetry is relaxed (26, 31), both of which

exhibit twofold-degenerate surface fermions alongsome of the momentum-space glide lines. In theremaining two wallpaper groups with multipleglide symmetries, pgg and p4g, higher-degeneratewallpaper fermions are uniquely allowed.We consider a z-normal surface with glides

gx;y ≡ mx;yj 12 12 0

� �—i.e., a mirror reflection mx,y

through the x or y axis followed by a trans-lation of half a lattice vector in the x and ydirections (Fig. 1). When SOC is present, g2x;y ¼�eiky;x, where kx,y are the in-plane surface crystalmomenta. At the corner point �M of the surfaceBZ (kx = ky = p), g2x ¼ g2y ¼ þ1 and fgx; gyg ¼ 0.When combined with time reversal T 2 ¼ �1 ,this symmetry algebra requires that all statesat �M are fourfold-degenerate. Furthermore, wall-paper groups with two glides are the only non-magnetic surface groups that admit this algebraand, therefore, the only surface groups that canhost protected fourfold degeneracies on strong-SOC crystals (24). The examination of symmetry-allowed terms reveals that fourfold points inthese wallpaper groups are linearly dispersing[section 2 of (22)], rendering them true surfaceDirac fermions, more closely related by symmetryalgebra and quantum criticality to the bulknodes in nonsymmorphic 3D Dirac semimetals(15, 17, 18) than to the surface states of conven-tional TIs. In section 2 of (22), we provide proofs

relating this algebra to Dirac degeneracy anddispersion.For bulk insulators, the glide-preserving bulk

topological phase and, consequently, z-normalsurface states can be determined without impos-ing a surface by classifying the allowed connectiv-ities of the z-projection Wilson loop holonomymatrix (10, 32–38), a bulk quantity defined by

½Wðkx ;ky ;kz0Þ�ij ≡ huiðkx; ky; kz0 þ 2pÞj

Pðkx; ky; kz0Þjujðkx; ky; kz0Þi ð1Þ

where we have defined the product of projectors

Pðkx; ky; kzÞ ≡ P ðkx; ky; kz þ 2pÞ

P kx; ky; kz þ2pðN � 1Þ

N

� �

⋯P kx; ky; kz þ2pN

� �ð2Þ

P ðkÞ is the projector onto the occupied bandsat k, and N is the number of discretized kzpoints. The rows and columns ofW correspondto filled bands, where jujðkÞi is the cell-periodicpart of the Bloch wave function at momentum kwith band index j. The eigenvalues of W aregauge-invariant and of the form eiqðkx ;kyÞ, whereq(kx, ky) is the Wilson phase. As detailed insections 3 and 4 of (22), the Wilson bands in-herit the symmetries of the z-normal wallpaper

Wieder et al., Science 361, 246–251 (2018) 20 July 2018 2 of 6

Fig. 2. Double-glide Wilson band connectivities. Shown are the eight topologically distinct Wilsonband connectivities for bulk insulators with crystal surfaces that preserve 2D glide reflections on theprojections of two orthogonal bulk glide planes. Each band structure is labeled by its two ℤ4 indices,(cx, cy), subject to the constraint that cx + cy = 0 mod 2. Under the imposition of C4z symmetry inwallpaper group p4g, connectivities are excluded for which cx ≠ cy. Solid black (dotted blue) lines in the

regions �X �M and �G�Y indicate bands with gx eigenvalue þieiky=2 ð�ieiky=2Þ, whereas the solid (dotted)

lines in the regions �Y �M and �G�X indicate bands with gy eigenvalueþieikx=2ð�ieikx=2Þ.When bulk inversionsymmetry 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�G�Xdisplay 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.

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group and, therefore, must also exhibit the re-quired degeneracy multiplets of wallpaper groupspgg and p4g. In particular, both the surface andWilson bands are twofold-degenerate along �X �Mð�Y �MÞ by the combination of time reversal andgy (gx) and meet linearly in fourfold degener-acies 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 Wilsonloop eigenvalues. For gy in a surface BZ in wall-paper group pgg, the quantized invariant cy isdefined in (11) by integrating theWilson phases,qj(kx,ky), along the path �M �Y �G�X

cy ≡1

p

Xnocc=2

j¼1

½qþj ð �MÞ � qþj ð�XÞ þ ∫�M �Y

dqþj

þ ∫�G�X

dqþj � þXnocc

j¼1

1

2p∫�Y �G

dqj mod 4 ð3Þ

where nocc is the number of occupied bands, thesuperscript + indicates the positive glide sector,and the absence of a superscript indicates theline where gy is not a symmetry and the sum isover all bands. In the presence of an additionalglide, gx, one can obtain cx by the transformationx↔y; �X↔�Y in Eq. 3. Although Eq. 3 appearscomplicated, (cx, cy) can be easily evaluated byconsidering the bands within each glide sectorthat cross an arbitrary horizontal line in theWilson spectrum [section 5A of (22)]. Wallpapergroup p4g also has C4z symmetry, which requirescx = cy and implies the existence of the sym-morphic mirrors m110j 12 1

2 0� �

and m1�10j 12 12 0

� �.

These mirrors yield ℤ mirror Chern numbersn110 and n1�10, respectively, with values con-strained by the glide invariant cx , ð�1Þn110 ¼ð�1Þn1�10 ¼ ð�1Þcx [section 5D of (22)].To enumerate the allowed topological phases

shown in Fig. 2, we consider possible restric-tions on (cx, cy). Although cx and cy can indi-vidually take on values 0, 1, 2, and 3, only pairs

that satisfy cx + cy mod 2 = 0 are permitted inbulk insulators. This can be understood as fol-lows: If cx + cy is odd, the 2D surface consist-ing of the four partial planes defined by 0 ≤ kx≤ p; ky = 0, p and kx = 0, p; 0 ≤ ky ≤ p possessesan overall Chern number, which implies theexistence of a bulk gapless point (40, 41), contra-dicting our original assumption that the systemis insulating. We present a rigorous proof insection 5B of (22) and show that the remainingcollection of eight insulating phases is indexedby the group ℤ4 � ℤ2.For cx,y = 1, 3, the system is a strong topo-

logical insulator (STI). The four “double-glidespin Hall” STI phases possess the usual twofold-degenerate Kramers pairs at �G, �X , and �Y , aswell as a fourfold-degenerate Dirac point at�M . The four STI phases are topologicallydistinct but will appear indistinguishable inglide-unpolarized angle-resolved photoemissionspectroscopy (ARPES) experiments. However, iftwo double-glide spin Hall systems with differing

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Fig. 3. The crystal and electronic structures ofSr2Pb3. This compound is in space group127 (P4/mbm). (A) The unit cell of Sr2Pb3. Mz is asymmorphic mirror reflection through the z axis.(B) The bulk BZ and the (001)-surface BZ(wallpaper group p4g). (C) The electronic bandsobtained using DFT. The Fermi level is set to0 eV. At each point in the BZ, there is aband gap near the Fermi energy, indicatedby the dashed red line; insets show magnifiedimages of the boxed regions. (D) The(001)-directed Wilson bands; red (blue)points indicate Wilson bands with positive(negative) surface glide eigenvalues, lþð�Þ

x;y . Bycounting the Wilson bands within each glidesector that cross the dashed green line,we find that Sr2Pb3 has the bulk topologyof a (2, 2) nonsymmorphic Dirac insulator[section 6B of (22)].

Fig. 4. The topological fourfold surface Diracfermion. Shown is the (001)-surface band structureof Sr2Pb3 (wallpaper group p4g) calculatedusing surface Green’s functions [section 6B of(22)]. The color bar indicates the relative degree ofsurface localization. The Fermi level is set to zero.The fourfold surface Dirac fermion appears at �M inthe region indicated by the red rectangle and isshown 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 redsurface-localized point is fourfold-degenerate.

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cx and cy are coupled together, the resultingsurface modes will distinguish between cx,y =1, 3 (11) [section 5B of (22)].When cx,y = 0, 2, the system is in a topological

crystalline phase. For (cx, cy) = (0, 2) or (2, 0),which is only permitted in a C4z-broken surfacepgg, a variant of the hourglass insulating phase(9) is present on the surface. For example, when(cx, cy) = (0, 2), either time-reversed partners oftwofold-degenerate free fermions live along �G�Xor both twofold-degenerate fermions live along�G�Y and a fourfold-degenerate Dirac fermionexists at �M .Lastly, but most interestingly, for cx = cy = 2,

we find that the system exists in a previously un-characterized “nonsymmorphic Dirac insulating”phase, capable of hosting just a single fourfold-degenerate Dirac surface fermion at �M .

Materials realizationsWe applied density functional theory (DFT), in-cluding the effects of SOC, to predict topologicalphases stabilized by wallpaper groups pgg andp4g in previously synthesized, stable materials.The details of these large-scale calculations areprovided in section 6 of (22). We find double-glide topological phases on the (001) surface(wallpaper group p4g) of three members of thespace group 127 (P4/mbm) A2B3 family of mate-rials: Sr2Pb3 (42, 43), Au2Y3 (44), and Hg2Sr3(45, 46). We find that Sr2Pb3 has a gap at theFermi energy at each crystal momentum, inspite of the presence of electron and hole pock-ets (Fig. 3). A Wilson loop calculation of thebands up to this gap (Fig. 3D) indicates thatthis material possesses the bulk topology of a(2, 2) nonsymmorphic Dirac insulator [section

6B of (22)]. Calculating the surface spectrumthrough surface Green’s functions, as describedin section 6B of (22) (Fig. 4), we find that the(001) surface of Sr2Pb3, although displaying anoverall metallic character, develops a gap of45 meV at the Fermi energy at �M . Inside thisgap, we observe a single, well-isolated, fourfold-degenerate surface Dirac fermion.Unlike Sr2Pb3, Au2Y3 and Hg2Sr3 in space group

127 (P4/mbm) are gapless, with bulk C4z-protectedDirac nodes (47) present near the Fermi energy.In section 6B of (22), we show that under weak(100) strain, these Dirac nodes can be gappedto induce the (0, 2) topological hourglass phasein these two materials.We additionally find that the (001) surface

(wallpaper group pgg) of the narrow-gap insu-lator Ba5In2Sb6 in space group 55 (Pbam) (48)

Wieder et al., Science 361, 246–251 (2018) 20 July 2018 4 of 6

Fig. 5. The crystal and electronic structuresof Ba5In2Sb6.This compound is in space group55 (Pbam). (A) The unit cell of Ba5In2Sb6.(B) The bulk BZ and the (001)-surface BZ(wallpaper group pgg). (C) The electronicbands obtained using DFT. The Fermi level isset to 0 eV. There is an insulating gap atthe Fermi energy, indicated by the dashed blackline. (D) The (001)-surface Wilson bands;red (blue) points indicate Wilson bands withpositive (negative) surface glide eigenvalues,

lþð�Þx;y . Counting the Wilson bands crossing the

dashed green line, we find that the bulkdisplays a (2, 0) topological hourglassconnectivity [section 6A of (22)].

Fig. 6. Double-glide hourglass fermions. Shown isthe (001)-surface band structure of Ba5In2Sb6(wallpaper group pgg) calculated using surfaceGreen’s functions [section 6A of (22)]. The Fermi levelis set to zero. Surface bands from the top of onehourglass fermion and the bottom of another connect

the valence and conduction manifolds along �Y�G; thehourglass fermions themselves are buried in the bulkmanifolds along this line. The surface bands alsodisplay other signatures of the (2, 0) topologicalhourglass connectivity, including a fourfold Dirac

fermion at �M connected to a clearly distinguishable

hourglass fermion along �G�X. The maximally localizedWannier functions obtained numerically from abinitio calculations are only approximately symmetricunder the surface wallpaper group, and therefore

bands along �X �M appear weakly split.

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hosts a double-glide topological hourglass fer-mion. We find that Ba5In2Sb6 develops an in-direct band gap of 5 meV (direct band gap,17 meV) (Fig. 5). The Wilson loop spectrumobtained from the occupied bands (Fig. 5D)demonstrates that this material is a (2, 0) double-glide topological hourglass insulator [section6A of (22)]. We find that the (001) surface ofBa5In2Sb6 has a projected insulating bulk gapthat is spanned along �Y �G by the top and bot-tom bands of two different topological hourglassfermions, which themselves are degenerate withstates in the bulk spectrum (Fig. 6). However,these fermions are topologically connected to aclearly distinguishable hourglass fermion along�G�X and a fourfold-degenerate surface Dirac fer-mion at �M, both of which could in principle beobserved using ARPES.

Discussion

We have demonstrated theoretically the exis-tence of a nonsymmorphic Dirac insulator—atopological crystalline material with a singlefourfold-degenerate surface Dirac point stabilizedby two perpendicular glides. After an exhaustivestudy of the 17 time-reversal–symmetric, strong-SOC wallpaper groups, only pgg and p4g areshown to be capable of supporting this fourfoldfermion. This phase is one of eight topologicallydistinct phases that can exist in insulating or-thorhombic crystals with surfaces that preservetwo perpendicular glides; we have classified alleight phases by topological indices (cx, cy) thatcharacterize the connectivity of the z-projectionWilson loop spectrum. We report the predictionof the nonsymmorphic Dirac insulating phasein Sr2Pb3 and of related double-glide topologicalhourglass phases in Ba5In2Sb6, as well as in (100)-strained Au2Y3 and Hg2Sr3. We also report thetheoretical prediction of a set of previously un-known double-glide spin Hall phases. Althoughtheir surface Kramers pairs and fourfold Diracfermions should be distinctive in ARPES experi-ments, 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 cx,y = 0, 2. Insection 7A of (22), we present an eight-bandtight-bindingmodel that, when half-filled, can betuned to realize all ℤ4 � ℤ2 double-glide insulat-ing phases. In a particular regime of parameterspace, in which SOC is absent at the �X and �Ypoints and bulk inversion symmetry is imposed,the Wilson loop eigenvalues at the edge TRIMsare pinned to ±1 ½qð �M=�X=�Y Þ ¼ 0; p�, and eachTRIM represents the end of a doubly degenerateSu-Schrieffer-Heeger (SSH) model (49). In thislimit, when the product of parity eigenvalues at�G satisfies xð�GÞ ¼ þ1, the bulk topology is fullycharacterized by the relative SSH polarizations,cx;y ¼ 2f½qð �MÞ � qð�Y ; �XÞ� mod 2g.Lastly, because the (2, 2) topological surface

Dirac point is symmetry-pinned to the QCP be-tween a 2D TI and a NI, we examine its poten-tial for hosting strain-engineered topologicalphysics. Consider the two-site surface unit cellin wallpaper group pgg from Fig. 1. In the (2, 2)

nonsymmorphic Dirac insulating phase, the sur-face Dirac fermion can be captured by the k · pHamiltonian near �M

H �M ¼ txðvxsxkx þ vysykyÞ ð4Þ

where t is a sublattice degree of freedom, vx(y)is the Fermi velocity in the x(y) direction, s isa T -odd orbital degree of freedom, and gx,y =tysx,y ðg2x;y ¼ þ1Þ. There exists a single, T -evenmass term, Vm = mtz, which satisfiesfH �M ;Vmg ¼ 0 and is therefore guaranteed tofully gap H �M . Therefore, surface regions withdiffering signs of m will be in topologically dis-tinct gapped phases and must be separated by 1Dtopological QSH surface domain walls, protectedonly by time-reversal symmetry (50). Because pghas point group 2mm, and {Vm, gx,y} = 0, Vm canbe considered an xy A2 distortion (51), whichcould be achieved by strain in the x + y directionand compression in the x – y direction. Thesedomain walls would appear qualitatively sim-ilar to those proposed in bilayer graphene (52–54).However, whereas those domain walls are pro-tected by sublattice symmetry and are thereforesensitive to disorder, domain walls originatingfrom nonsymmorphic Dirac insulators are pro-tected by only time-reversal symmetry and there-fore should be robust against surface disorder.Under the right interacting conditions or chem-ical modifications, a nonsymmorphic Dirac insu-lator surface may also reconstruct and self-induceregions of randomly distributed ±m, separatedby a network of 1D QSH domain walls. Thesedomain walls are closely related to the 1D heli-cal hinge modes of “second-order” topologicalinsulators (55–57). Furthermore, in the presenceof electron-electron interactions, these domainwalls form helical Luttinger liquids (58). AlthoughSr2Pb3 is insufficiently insulating to experimen-tally isolate these effects, future bulk-insulatingnonsymmorphic Dirac insulators could providean experimental platform for probing and manip-ulating the physics of time-reversal–invarianttopological Luttinger liquids.

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ACKNOWLEDGMENTS

We thank A. Alexandradinata for a discussion about the invariant inEq. 3 and E. Mele for fruitful discussions. Funding: B.J.W. and C.L.K.acknowledge support through a Simons Investigator grant from theSimons Foundation to C.L.K. and through Nordita under ERC DM321031. Z.W. and B.A.B. acknowledge support from the Departmentof 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-MRSECDMR-1420541), the Packard Foundation, the Schmidt Fund forInnovative Research, and the National Natural Science Foundation ofChina (grant 11504117). Y.K. and A.M.R. thank the National ScienceFoundation MRSEC program for support under DMR-1120901 andacknowledge 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 NormaleSupérieure, UPMC Paris, and Donostia International Physics Centerfor their generous sabbatical hosting during some of the stages ofthis work. The development of Wilson loop–based topological invariantsand first-principles characterization of the hourglass insulating phasein Ba5In2Sb6 were supported by Department of Energy grantDE-SC00016239. Author contributions: B.J.W. and B.B. developedthe application of wallpaper group co-representations to the predictionof 3D topological phases and related the Dirac insulating phase to

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symmetry-enhanced fermion doubling. Z.W. and J.C. developed theWilson loop–based topological invariants; Z.W. discovered thehourglass insulating phase in Ba5In2Sb6 and performed DFTinvestigations of additional materials. Y.K. and H.-S.D.K. discoveredthe Dirac and hourglass insulating phases in the A2B3 family andperformed DFT investigations of additional materials. A.M.R. was theprincipal investigator for DFT. C.L.K. and B.A.B. were the principalinvestigators for crystalline symmetry, fermion doubling, andtopological invariants.Competing interests:The authors declare thatthey have no competing interests. Data and materials

availability: For our DFT calculations, we used QUANTUMESPRESSO, 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 WannierHamiltonians for surface Green’s functions. All DFT examinations usedthe experimental lattice parameters and relaxation schemes detailedin section 6 of (22). The numerical data for all band structure, Wilsonband, and surface state calculations are publicly available fromthe Harvard Dataverse (59).

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/361/6399/246/suppl/DC1Supplementary TextFigs. S1 to S13Tables S1 to S3References (60–107)

21 March 2017; accepted 22 May 201810.1126/science.aan2802

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Wallpaper fermions and the nonsymmorphic Dirac insulator

L. Kane and B. Andrei BernevigBenjamin J. Wieder, Barry Bradlyn, Zhijun Wang, Jennifer Cano, Youngkuk Kim, Hyeong-Seok D. Kim, Andrew M. Rappe, C.

DOI: 10.1126/science.aan2802 (6399), 246-251.361Science 

, this issue p. 246Science.3Pb2authors' calculations, is expected to occur in the compound Sr

This class hosts a single fourfold-degenerate Dirac fermion on the surface of the material and, on the basis of the and find that some of them allow for an additional topological class.−−the so-called wallpaper groups−−of 3D materials

focus on the symmetry groups of 2D surfaces et al.types of quasiparticles and topologically nontrivial materials. Wieder Analyzing the spatial symmetries of three-dimensional (3D) crystal structures has led to the discovery of exotic

Exotic topology on the surface

ARTICLE TOOLS http://science.sciencemag.org/content/361/6399/246

MATERIALSSUPPLEMENTARY http://science.sciencemag.org/content/suppl/2018/07/18/361.6399.246.DC1

REFERENCES

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