PHYSICAL REVIEW X 8, 031070 (2018)
Symmetry Indicators and Anomalous Surface States of Topological Crystalline Insulators
Eslam Khalaf,1,2 Hoi Chun Po,2 Ashvin Vishwanath,2 and Haruki Watanabe3 1Max Planck Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany 2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 3Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
(Received 15 February 2018; published 14 September 2018)
The rich variety of crystalline symmetries in solids leads to a plethora of topological crystalline insulators (TCIs), featuring distinct physical properties, which are conventionally understood in terms of bulk invariants specialized to the symmetries at hand. While isolated examples of TCI have been identified and studied, the same variety demands a unified theoretical framework. In this work, we show how the surfaces of TCIs can be analyzed within a general surface theory with multiple flavors of Dirac fermions, whose mass terms transform in specific ways under crystalline symmetries. We identify global obstructions to achieving a fully gapped surface, which typically lead to gapless domain walls on suitably chosen surface geometries. We perform this analysis for all 32 point groups, and subsequently for all 230 space groups, for spin-orbit-coupled electrons. We recover all previously discussed TCIs in this symmetry class, including those with “hinge” surface states. Finally, we make connections to the bulk band topology as diagnosed through symmetry-based indicators. We show that spin-orbit-coupled band insulators with nontrivial symmetry indicators are always accompanied by surface states that must be gapless somewhere on suitably chosen surfaces. We provide an explicit mapping between symmetry indicators, which can be readily calculated, and the characteristic surface states of the resulting TCIs.
DOI: 10.1103/PhysRevX.8.031070 Subject Areas: Condensed Matter Physics, Topological Insulators
I. INTRODUCTION to a plethora of topological distinctions even when the band insulators admit atomic descriptions and are therefore Topological phases of free fermions protected by internal individually trivial [24,25]. For example, consider trivial symmetries (such as time reversal) feature a gapped bulk (atomic) insulators, where electrons are well localized in and symmetry-protected gapless surface states [1]. Such real space and do not support any nontrivial surface states. bulk-boundary correspondence is an important attribute of Suppose two atomic insulators are built from orbitals of such topological phases, and it plays a key role in their different chemical character, which will typically lead to classification, which has been achieved in arbitrary spatial different symmetry representations in the Brillouin zone dimensions [2–4]. Crystals, however, frequently exhibit a (BZ). These representations cannot be modified without a far richer set of spatial symmetries, like lattice translations, bulk gap closing, and therefore, the two atomic insulators rotations, and reflections, which can also protect new are topologically distinct even though they are both derived topological phases of matter [5–23]. from an atomic limit. However, several new subtleties arise when discussing Since the mathematical classification of topological band topological crystalline phases. For one, spatial symmetries structures, using K theory, is formulated in terms of the are often broken at surfaces, and when that happens, the mutual topological distinction between insulators [2, surface could be fully gapped even when the bulk is 26–28], the mentioned distinction between trivial phases topological. Consequentially, the diagnosis of bulk band is automatically incorporated. In fact, they appear on the topology is a more delicate task as, a priori, the existence of same footing as conventional topological indices like Chern anomalous surface states in this setting is only a sufficient, numbers. While for a physical classification one would but not necessary, property of a topological bulk. want to discern between these different flavors of topo- Furthermore, the presence of crystalline symmetries leads logical distinctions, they are typically intricately related. On the one hand, the pattern of symmetry representations is oftentimes intertwined with other topological invariants, as Published by the American Physical Society under the terms of epitomized by the Fu-Kane parity criterion [29], which the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to diagnoses 2D and 3D topological insulators (TIs) through the author(s) and the published article’s title, journal citation, their inversion eigenvalues; on the other hand, the topo- and DOI. logical distinction between atomic insulators may reside
2160-3308=18=8(3)=031070(33) 031070-1 Published by the American Physical Society KHALAF, PO, VISHWANATH, and WATANABE PHYS. REV. X 8, 031070 (2018) fully in their wave-function properties and, hence, is not Ref. [24], and the indicator groups for all 230 space groups reflected in their symmetry eigenvalues [25,30]. (SGs) have been computed. In this work, we sidestep these issues by focusing on Here, we provide a precise physical interpretation for all topological crystalline phases displaying anomalous gap- the phases captured by these indicators in class AII. In less surface states, which, through bulk-boundary corre- particular, we discover that the majority of them can be spondence, are obviously topological. Unlike the strong TI, understood in terms of familiar indices, like the strong and these phases require crystalline symmetries for their pro- weak TI Z2 indices and mirror Chern numbers. However, tection. Typically, their physical signatures are exposed on there exists symmetry-diagnosed band topology that per- special surfaces where the protecting symmetries are sists even when all the conventional indices have been preserved. Such is the case for the 3D phases like weak silenced. The prototypical example discussed in Ref. [24] is TI [1], the mirror Chern insulator [6,7], and the non- a “doubled strong TI” in the presence of inversion sym- symmorphic insulators featuring so-called hourglass metry, which, as pointed out later in Ref. [23], actually [14,15] or wallpaper [17] fermions, which are, respectively, features hingelike surface states. Motivated by these protected by lattice translation, reflection, and glide sym- developments, we relate the symmetry indicators to our metries. While these conventional phases feature 2D surface Dirac theory and prove that, for spin-orbit coupled gapless surface states on appropriate surfaces, we also systems with time-reversal symmetry (TRS), any system allow for more delicate, 1D “hinge” surface states, whose with a nontrivial symmetry indicator is a band insulator existence is globally guaranteed on suitably chosen sample with anomalous surface states on a suitably chosen boun- geometries, although any given crystal facet can be fully dary. These results are achieved by establishing a bulk- gapped [9,18–23,31–33]. We primarily focus on 3D time- boundary correspondence between the indicators and the reversal symmetric band structures with a bulk gap and surface Dirac theory. Parenthetically, we note that a non- significant spin-orbit coupling (class AII). However, our trivial symmetry indicator constitutes a sufficient, but not approach can be readily extended beyond this specific necessary, condition for the presence of band topology, and setting. In particular, in symmetry classes with particle-hole we discuss examples of topological phases that have symmetry, the notion of hinge states can be generalized to gapless surface states predicted by our Dirac approach, protected gapless points on the surface [9]. although the symmetry indicator is trivial. We base our analysis on two general techniques, which Although the techniques we adopted are tailored for are respectively employed to systematically study gapless weakly correlated materials admitting a band-theoretic surface states and identify bulk band structures associated description, many facets of our analysis have immediate with them. First, we describe a surface Dirac theory bearing on the more general study of interacting topological – approach [1,3,4,23,34]. Specifically, we consider a surface crystalline phases [38 42]. A common theme is to embrace “ ” theory with multiple Dirac cones, which may transform a real-space perspective, where topological crystals are differently under the spatial symmetries [23]. Such a theory built by repeating motifs that are, by themselves, topo- can be viewed as the surface of a “stack of strong TIs,” logical phases of some lower dimension and are potentially which arise, for example, when there are multiple band protected by internal symmetries or spatial symmetries that inversions (assuming inversion symmetry) in the BZ. When leave certain regions invariant (say, on a mirror plane) an even number of Dirac cones are present at the same [38,41]. In particular, such a picture allows one to readily surface momentum, the Dirac cones can generically be deduce the interaction stability of the phases we describe locally gapped out, and, naively, one expects that a trivial [43], and we briefly discuss how our Dirac surface theory surface results. However, spatial symmetries can cast global analysis can be reconciled with such general frameworks. constraints that obstruct the full gapping out of the Dirac cones everywhere on the surface [23]. When that happens, II. SYMMETRY INDICATORS AND the surface remains stably gapless along certain domain BULK TOPOLOGY walls, and hence, one can infer that the bulk is topological. In this section, we provide a precise physical meaning for We provide a comprehensive analysis of such obstructions the phases captured by the symmetry-based indicator due to spatial symmetries and catalogue all the possible groups obtained in Ref. [24], and we give explicit expres- patterns of such anomalous surface states in a space group. sions for each of these indicators, assuming TRS and Next, we turn to bulk diagnostics that informs the focusing on the physically most interesting case of strong presence of such surface states. In particular, we focus spin-orbit coupling. on those which, like the Fu-Kane criterion [29], expose We start by considering a band insulator of spinful band topology using only symmetry eigenvalues [29,35– electrons that are symmetric under a SG and TRS 37]. Such symmetry-based indicators are of great practical (class AII). In this setting, we can define topological value since they can readily be evaluated without comput- indices corresponding to weak and strong TI phases ing any wave-function overlaps and integrals. In fact, a protected by TRS. For centrosymmetric SGs, the Fu- general theory of such indicators has been developed in Kane formula [29] allows us to compute these indices
031070-2 SYMMETRY INDICATORS AND ANOMALOUS SURFACE … PHYS. REV. X 8, 031070 (2018) using only the parities, i.e., inversion eigenvalues, of the rotation followed by a fractional translation along the bands. Namely, they can be determined just by multiplying rotation axis) whose axis is orthogonal to the mirror plane, the parities of the occupied bands without performing any one can diagnose the mirror Chern number CM modulo n sort of integrals. This is one particular instance of the by multiplying the rotation eigenvalues [37]. (For simplic- relation between the symmetry representations in momen- ity, in the following, we often leave the direction of a tum space and band topology. rotation or screw symmetry implicit and, whenever neces- Recently, this relation was extended to all space groups sary, label that as the z axis corresponding to the third (i.e., not restricted to inversion alone) and also to a wider momentum coordinate.) Naively, each CM mod n produces Z class of band topology [24]. By comparing the symmetry a n factor in XBS. However, we sometimes find a “ ” Z representations of the bands against those of trivial (atomic) doubled 2n factor in XBS. The mechanism behind this insulators, the symmetry-based indicator group XBS was enhancement is clarified in Secs. II D and II E. computed for all 230 SGs [24] (and even for all 1651 Following the examination of these familiar indices, we “ magnetic SGs [44]). It was found to take the form find that XBS in Eq. (1) can always be factorized into weak factors” and a “strong factor”: Z Z Z XBS ¼ n1 × n2 × × n ; ð1Þ N ðwÞ ðsÞ XBS ¼ XBS × XBS; where N and fn1;n2; …;nNg depend on the assumed SG. XðwÞ ¼ Z × × Z ;XðsÞ ¼ Z : ð2Þ While XBS has been exhaustively computed for all SGs in BS n1 nN−1 BS nN Ref. [24], the concrete physical interpretation for some of ðwÞ the nontrivial classes was left unclear. We attack this Every factor in XBS can be completely characterized either problem in the following by first recasting these topological by the weak TI indices νi (i ¼ 1, 2, 3) or the weak mirror 0 π indices into more explicit forms, akin to those in ðkz¼ Þ ðkz¼ Þ Chern number CM ¼ CM , both of which can be Refs. [29,37], and then clarifying their physical meanings. understood as arising from stacking 2D TIs. On the other ðsÞ hand, for most SGs, the strong factor XBS cannot be fully A. Review of familiar indices understood by these familiar indicators. We thus focus on Here, we first review some well-known symmetry-based the strong factor in the remainder of this section. indicators and discuss their relations with the general Z results in Ref. [24]. Let us start with the inversion B. 4 index for inversion symmetry symmetry. As mentioned above, for a SG containing the Here, we show that the strong TI index ν0 can actually be I inversion symmetry , the inversion eigenvalues are related promoted to a Z4 index in the presence of inversion to the strong TI index ν0 ∈ Z2 and the three weak TI symmetry. The refined index can capture topological indices νi ∈ Z2 (i ¼ 1,2,3)[29]. Each of the three weak TI crystalline insulators with an anomalous 1D edge state, Z indices gives rise to a 2 factor in XBS when no additional as we discuss in detail in Sec. III. þ − constraints are imposed. Curiously, however, the factor in Let n (nK) be the number of occupied bands with even Z Z K XBS corresponding to the strong TI index is 4, not 2 [24]. (odd) parity at each time-reversal invariant momentum Z This 4 indicator will be discussed in Sec. II B. (TRIM) K. Because of Kramers pairing, nK is even. The Next, let us discuss the mirror Chern numbers, which Fu-Kane formula [29] for the strong TI index ν0 ∈ Z2 may can be defined for every mirror symmetric plane in be expressed as momentum space [6]. On any such plane, the single- Y P 1 þ 1 − 1 − particle Hamiltonian H can be block-diagonalized into ν0 n n 2 n ð−1Þ ¼ ðþ1Þ2 K ð−1Þ2 K ¼ð−1Þ K∈TRIMs K : ð3Þ H , defined, respectively, in the sectors corresponding to K∈TRIMs H the mirror eigenvalues M ¼ i. Since is not neces- H Z sarily time-reversal symmetric, can possess a Chern We now introduce a -valued index [10] that is simply the ∈ Z − ≡ number C , which satisfies Cþ ¼ C− ( CM) sum of the inversion parities of occupied bands (up to a because of the TRS of the total Hamiltonian H.Tobe prefactor): more concrete, let us assume the mirror symmetry is about 1 X the xy plane. For primitive lattice systems, both k ¼ 0 and þ − z κ1 ≡ n − n ∈ Z: π 4 ð K KÞ ð4Þ kz ¼ planes are mirror symmetric and can individually K∈TRIMs ðkz¼0Þ ðkz¼πÞ support CM and CM . For body-centered systems and ≡ þ − face-centered systems, however, kz ¼ 0 is the only mirror Using the total number of occupiedP bands n nK þ nK, κ 2 − 1 − symmetric plane, and there is only one mirror Chern one can rewrite 1 as n 2 K∈TRIMsnK. Comparing this 0 ðkz¼ Þ with Eq. (3), we find number CM . When the SG is further endowed with an n-fold rotation ð−1Þν0 ¼ð−1Þκ1 : ð5Þ symmetry Cn (n ¼ 2, 3, 4, 6) or screw symmetry (Cn
031070-3 KHALAF, PO, VISHWANATH, and WATANABE PHYS. REV. X 8, 031070 (2018)
FIG. 1. The parity eigenvalues at high-symmetry momenta. (a) The common choice of coordinates in panels (b)–(s). (b,c) An example of the parity eigenvalues for a strong topological insulator (κ1 ¼ 3, ν0 ¼ 1, and νi ¼ 0) and a higher-order topological insulator (κ1 ¼ 6 and ν0 ¼ νi ¼ 0), respectively. Diagram (c) can be realized by stacking two copies of diagram (b). (d)–(k) The parity eigenvalues of the atomic insulators constructed by placing an s orbital at the position specified in each panel in every unit cell. (l)–(s) The same for (d)–(k) but with a p orbital used instead of an s orbital. All atomic insulators have κ1 ¼ ν0 ¼ νi ¼ 0 except for diagram (d), κ1 ¼ 4, ν0 ¼ νi ¼ 0, and diagram (l), κ1 ¼ −4, ν0 ¼ νi ¼ 0.
Although Eq. (5) suggests that κ1 ¼ ν0 mod 2, κ1 contains crystallographic references [45]). The index κ1 can be ¯ ¯ more information than ν0 as it is “stable” mod 4, in the defined for every supergroup of P1 (SGs containing P1 as a sense that any trivial insulator has κ1 ¼ 4n (n ∈ Z), and subgroup), i.e., for every centrosymmetric SG. adding or subtracting trivial bands does not alter κ1 mod 4 Z as demonstrated in Fig. 1. Weak topological phases may C. New 2 index for fourfold rotoinversion ¯ ¯ realize κ1 ¼ 2 mod 4, but the most interesting case is when The SGs P4 (No. 81) and I4 (No. 82) possess neither κ 2 ν ν ν 1 ¼ mod 4 while ð 1; 2; 3Þ all vanish. One way of inversion nor mirror symmetry. Hence, the indicator XBS ¼ achieving this phase is to stack two copies of a strong TI Z2 found in Ref. [24] cannot by accounted for by the Fu- as illustrated in Figs. 1(b) and 1(c). The surface signature Kane parity formula [29] or the mirror Chern number [6]. of this phase was discussed in Ref. [23] and is reintroduced Here, we propose a new index κ4 in terms of the in Sec. III. eigenvalues of fourfold rotoinversion S4 (the fourfold The space group that contains only inversion I in rotation followed by inversion) and show that the Z2 addition to translations is called P1¯ (No. 2 in the standard nontrivial phase is actually a strong TI.
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To introduce the invariant, we note that there are four Even when all the mirror Chern numbers and the weak momenta in the BZ invariant under S4, which we denote by indices vanish, Δ can still be 4 ≠ 0 mod 8. Depending on K4; they are (0, 0, 0), ðπ; π; 0Þ, ð0; 0; πÞ, and ðπ; π; πÞ for the geometry, this phase may feature “hinge”-type 1D edge primitive lattice systems and (0, 0, 0), ð0; 0; 2πÞ, ðπ; π; −πÞ, states [19,23], which we discuss in Sec. III. and ðπ; π; πÞ for body-centered systems. For spinful elec- iðαπ=4Þ trons, the four possible values of S4 eigenvalues are e E. Combination of κ and C under sixfold rotation, α 1 α 1 M ( ¼ , 3, 5, 7). Let us denote by nK the number of screw, or rotoinversion occupied bands with the eigenvalue eiðαπ=4Þ at momen- The case of SG P6=m (No. 175) is similar to P4=m.It tum K ∈ K4. has a sixfold rotation C6 in addition to inversion I. The The new Z-valued index is (up to a prefactor) the sum 3 product of I and ðC6Þ is the mirror Mz protecting the of the S4 eigenvalues of occupied bands over the momenta ðkz¼0;πÞ in K4: mirror Chern number CM . Although the eigenvalues of the sixfold rotation can only detect mirror Chern numbers 1 X X ðsÞ iðαπ=4Þ α mod 6, X contains a Z12 factor. In this case, the κ4 ≡ pffiffiffi e n ∈ Z: ð6Þ BS K k 0 2 2 ∈ α ð z¼ Þ κ K K4 combination of CM mod 6 and 1 mod 4 can fully characterize this Z12 factor (thanks to the Chinese remain- This quantity is always an integer in the presence of TRS, der theorem). and it is stable modulo 2 against the stacking of trivial bands. When the rotation C6 is replaced by the screw S6 (i.e., κ 3 The stability can be proven in the same way as we did for 1 C6 followed by a half translation in z), the resulting SG is in the previous section, i.e., by listing all S4 eigenvalues of 6 I 3 P 3=m (No. 176). In this case, the product of and ðS63 Þ atomic insulators. As we show in Appendix A, κ4mod 2 is also a mirror symmetry. The corresponding XðsÞ includes agrees with ν0. However, it is still useful to distinguish κ4 BS ν κ ν the same Z12 factor. from 0,as 4 conveys more information than 0 in the 6¯ presence of additional symmetries, as we discuss in the next In contrast, P (No. 174) generated by the sixfold section. rotoinversion S6 ¼ IC6 does not have an inversion sym- metry, and κ1 is not defined. The mirror Chern numbers ðkz¼0;πÞ 3 κ κ CM protected by Mz ¼ðS6Þ can be diagnosed by D. Combination of 1 and 4 2 threefold rotation C3 ¼ðS6Þ modulo 3, and this fully The SGs P4=m (No. 83) and I4=m (No. 87) contain both ðsÞ Z inversion I and fourfold rotation C4, with the inversion explains XBS ¼ 3. 2 center on the rotation axis. The product of I and ðC4Þ is a mirror Mz about the xy plane, and one can define the mirror F. Summary of XBS ðkz¼0Þ Chern number CM . The eigenvalues of the fourfold The eight SGs studied above (SG 2, 81, 82, 83, 87, 174, 0 ðkz¼ Þ 175, and 176) play the role of “key” SGs, in the sense that rotation determine CM mod 4 [37]. However, XBS studied in Ref. [24] contains a Z8 factor, which cannot they provide an anchor for understanding the symmetry- be explained by the (detected) mirror Chern number alone. based indicator XBS of any other SG. Suppose that we are G Here, we argue that the combination of κ1 and κ4 is interested in one of the 230 SGs . To understand the nature G responsible for this enhanced factor. of a nontrivial XBS, one should first identify its maximal Since these SGs have both inversion I and rotoinversion subgroup among the eight key SGs. Let G0 ≤ G be the key S4 ¼ IC4, κ1 and κ4 can be defined separately using SG. Then, we know that (i) the topological indices character- G G Eqs. (4) and (6). Furthermore, the inversion center coin- G G0 ðsÞ; ðsÞ; 0 izing XBS are the same as those for XBS, (ii) XBS and XBS cides with the rotoinversion center. In this case, we find that G XðwÞ; the difference are the same, and (iii) the weak factor BS may be reduced ðwÞ;G0 from XBS because of the constraints imposed by the Δ ¼ κ1 − 2κ4 ð7Þ additional symmetries in GnG0. In Table I, we identify the key SGs for all the 230 SGs. This table reproduces XBS for all is stable modulo 8, not only 4, against the stacking of trivial 230 SGs presented in Ref. [24] but in a way that renders their bands. Hence, Δ mod 8 should be understood as a new bulk physical properties more transparent. invariant, which can be reconciled with the strong factor Physically, the key space groups play the role of the Z κ 8 031070-5 KHALAF, PO, VISHWANATH, and WATANABE PHYS. REV. X 8, 031070 (2018) TABLE I. Summary of symmetry-based indicator of band topology. Space groups are grouped into their parental key space group in the first column. The second column lists the key topological indices characterizing nontrivial classes in XBS. The space groups are indicated by their numbers assigned in Ref. [45], and those highlighted by an underbar are the key SGs. Key space group Key indices ðwÞ ðsÞ Space groups XBS XBS 3 ðZ2Þ Z4 2, 10, 47. 2 ðZ2Þ Z4 11, 12, 13, 49, 51, 65, 67, 69. ¯ P1 ν1;2;3, κ1 Z2 Z4 14, 15, 48, 50, 53, 54, 55, 57, 59, 63, 64, 66, 68, 71, 72, 73, 74, 84, 85, 86, 125, 129, 131, 132, 134, 147, 148, 162, 164, 166, 200, 201, 204, 206, 224. 0 Z4 52, 56, 58, 60, 61, 62, 70, 88, 126, 130, 133, 135, 136, 137, 138 141, 142, 163, 165, 167, 202, 203, 205, 222, 223, 227, 228, 230. ¯ P4 κ4 0 Z2 81, 111–118, 215, 218. ¯ I4 κ4 0 Z2 82, 119–122, 216, 217, 219, 220. 0 π P4=m ν ðkz¼ ; Þ κ − 2κ Z2 × Z4 Z8 83, 123. 1;2;3, CM , 1 4 Z2 Z8 124. Z4 Z8 127, 221. 0 Z8 128. 0 I4=m ν ðkz¼ Þ κ − 2κ Z2 Z8 87, 139, 140, 229. 1;2;3, CM , 1 4 0 Z8 225, 226 ¯ 0 π P6 ðkz¼ ; Þ Z3 Z3 174, 187, 189. CM 0 Z3 188, 190. 0 π P6=m ðkz¼ ; Þ κ Z6 Z12 175, 191. CM , 1 0 Z12 192 0 P63=m ðkz¼ Þ κ 0 Z12 176, 193, 194. CM , 1 I I 2 inversion . The product of and ðS42 Þ gives rise to a We start by considering a sample with a specific mirror symmetry, whose mirror Chern number is diagnosed geometry and boundary conditions that are periodic or ðsÞ Z open along different directions. The surface is a compact mod 4 by XBS ¼ 4. (In this case, since the inversion center manifold [47] described by a surface Hamiltonian hr;k, and the rotoinversion (IS4 ) center do not agree, κ1 − 2κ2 2 which depends on the position on the surface r and the does not enhance Z4 to Z8.) If we break the mirror surface momentum k ¼ −i∇r, a vector in the tangent space symmetry without breaking inversion, the mirror Chern to the surface at point r. The procedure of obtaining a number becomes undefined, but the phases corresponding surface Hamiltonian from a given bulk Hamiltonian is ðsÞ to the nontrivial XBS are still protected by the inversion explained in detail in Appendix B. To summarize, it starts symmetry and are diagnosed by the inversion index κ1. by introducing some space-dependent parameter in the Hamiltonian and changing it across the surface from its III. SURFACE STATES value inside the sample to a different value outside it (i.e., in the vacuum). This process is followed by projecting the We have seen in the previous section that some phases bulk degrees of freedom onto the space of states localized diagnosed by the symmetry indicator XBS do not fall into close to the surface and rewriting the Hamiltonian in terms the standard categories of strong TI, weak TIs, and mirror of these surface degrees of freedom. Chern insulators. This raises the question about the nature We then proceed to define the notion of an “anomalous of the possible surface states possessed by these phases. surface topological crystalline insulator (sTCI)” based on With this question in mind, we devote this section to its surface properties: We say a band insulator is a sTCI if developing a general approach to studying anomalous there exists a geometry with some boundary conditions surface states protected by crystalline symmetries. We such that the surface Hamiltonian argue that our approach captures most, if not all, TCIs (i) is not gapped everywhere, with anomalous surface states [46]. In the next section, we (ii) cannot be realized in an independent lower-dimen- provide a precise relation between the surface analysis in sional system with the same symmetries, and this section and the bulk symmetry indicator discussed in (iii) can be gapped everywhere by breaking the crystalline the previous section. symmetry or going through a bulk phase transition. 031070-6 SYMMETRY INDICATORS AND ANOMALOUS SURFACE … PHYS. REV. X 8, 031070 (2018) This definition includes familiar phases such as weak will be a part of (i) a single domain wall, (ii) multiple TIs (protected by translation) and conventional TCIs, as intersecting domain walls (this could only happen at a high- well as the recently discovered TCI with surface hinge symmetry point since an extra symmetry is required to modes [19–23]. stabilize the crossing), or (iii) the 2D gapless surface of a Given a gapped bulk, the surface region where the gap strong TI. The last case does not correspond to a sTCI since vanishes is necessarily boundaryless since otherwise the it does not require a crystalline symmetry for its stability, spatial spread of the surface states would diverge as we while the former two cases correspond to sTCIs with approach the boundary of the gapless region, necessitating domain walls, which are either movable or are pinned to the existence of gapless bulk states as in Weyl semimetals high-symmetry points or lines. Consequently, we arrive at a [48–50]. This boundaryless region residing on the surface unified description for the surface states of all sTCIs in of the 3D bulk could be a 2D, 1D (collection of closed terms of “globally irremovable, locally stable topological curves), or 0D region (collection of points). We show below defects.” Our approach is reminiscent of the approach of that the latter is impossible for an insulator in class AII, but Ref. [38], which argued that a symmetry-protected topo- first, let us point out that designating a certain dimension- logical (SPT) phase protected by a point-group symmetry ality to a gapless region generally depends not only on the can be understood in terms of an embedded SPT of a lower symmetries at play but also on the geometry of the sample. dimensionality stabilized by the point-group symmetry. For example, the hinge insulators protected by rotation Having reduced our problem to the study of globally [19,23] possess 1D helical gapless modes when placed on a stable configurations of surface domain walls, the next sphere but exhibit gapless 2D Dirac cones when the surface simplification is achieved by observing that a surface is a plane normal to the rotation axis (with periodic domain wall in class AII can be generically constructed boundary conditions along the two in-plane directions), by first stacking together two strong TIs [52] and then as we illustrate in Fig. 7. The relation between 1D and 2D adding a mass term that changes sign at the domain wall. surface states is explored in more detail in Sec. III D. This is also consistent with the analysis of Sec. II and To explore the types of possible surface states, we start by Refs. [19,23], and it means that the surface theory of any defining a (surface) high-symmetry point as a point left sTCI can be constructed by stacking strong TIs and invariant by some crystalline symmetries. That is to say, the studying the symmetry transformation properties of the stabilizer group of this point contains elements apart from possible mass terms. Our basic building block in the the identity. Let us first consider a generic point r (which is following sections will be the doubled strong TI (DSTI), not a high-symmetry point). Since the only symmetry at r which is constructed by stacking two strong TIs. In is TRS, the surface Hamiltonian at r can only be gapless if it particular, we note that every even entry in the strong ðsÞ is locally identical to (i) the surface of a strong TI or (ii) a factor XBS of any space group can be viewed as the stack of (movable) domain wall between two different quantum- two strong TIs (e.g., by rewriting 4 ¼ 1 þ 3). If the order of spin Hall phases. This can be understood in terms of the ðsÞ ðsÞ Z XBS is even (say, XBS ¼ 4), then the DSTIs correspond classification of stable topological defects [51], which precisely to the even subgroup of XðsÞ (Table I); however, if implies that 1D and 2D defects are topologically stable in BS the order is odd, then this identification does not hold, and class AII (with a Z2 classification), whereas 0D defects are any given entry could correspond to either a strong or a not. The latter typically describe vortices in superconductors doubled strong TI. We elaborate further on such identi- and require particle-hole or chiral symmetry to be stable. fication ambiguity in Sec. IV. At a high-symmetry point, one extra complication arises from the fact that the crystalline symmetry acts locally as an on-site discrete unitary symmetry, which can be used to A. Stacked strong TIs block-diagonalize the local Hamiltonian. Each block can As explained in detail in Appendix B, the gapless potentially have less symmetries than the original Hamiltonian on the surface of a strong TI can be written as Hamiltonian. As a result, we need to consider the additional ˆ possibility of class A (unitary-type) defects. Only 1D hr;k ¼ðk × nrÞ · σ: ð8Þ defects, representing domain walls between two phases ˆ with different Chern numbers, are stable in class A [51] and Here, nr is the normal to the surface at point r, and σ ¼ have a Z classification. They will only exist if the ðσx; σy; σzÞ is the vector of Pauli matrices representing the symmetry leaves a line or curve on the surface invariant, spin degrees of freedom. TRS is implemented as which only occurs for mirror symmetry in 3D. Such a line T ¼ iσyK, where K denotes complex conjugation, and it can host an arbitrary number of 1D gapless modes due to protects the gaplessness of hr;k. the Z classification, which can be understood in terms of A DSTI is constructed by stacking two strong TIs, whose mirror Chern number [6]. surface Hamiltonian can be described by two copies of Our argument can be summarized as follows: A gapless Eq. (8). The only possible T -symmetric mass term that can surface Hamiltonian at a certain point without fine-tuning gap out the surface has the form 031070-7 KHALAF, PO, VISHWANATH, and WATANABE PHYS. REV. X 8, 031070 (2018) ˆ σ ⊗ τ τ ð1Þ ð2Þ Mr ¼ mrnr · y; ð9Þ Here, ug ≡ ug ⊕ ug is the signed representation of g acting on the DSTI, where uðiÞ ¼ ηðiÞv for the two strong τ g g g with x;y;z denoting the Pauli matrices in the orbital space of TIs labeled by i ¼ 1,2. the two copies. The invariance of the mass Mr under symmetry action To study the action of spatial symmetries on the DSTI implies that model, we start by considering a generic spatial symmetry τ ∈ 3 g ¼fRgj gg, where Rg Oð Þ is a point-group operation τ † ˆ σ ⊗ τ τ ˆ σ ⊗ τ mg·r ¼ sgmr; ðugÞ ðng·r · yÞug ¼ sgnr · y; ð12Þ and τg denotes the (possibly fractional) translation in g. R θ nˆ Here, g can be parametrized by three pieces of data: g, g, with sg ¼ 1. This means that, for any signed representa- R θ nˆ τ and det g, where g and g are, respectively, the rotation tion ug of the symmetry group G on the DSTI, we can 1 angle and axis, and det Rg ¼ indicates whether Rg is a define its “signature” as a map s∶G ↦ Z2, which assigns a proper or improper rotation. We can construct a natural sign to each symmetry operation g ∈ G according to spin-1=2 representation of the point-group action of g, whether or not the mass changes sign under the action of g ˆ given by vg ≡ expð−iθgng · σ=2Þ. (in the specified signed representation). The only condition The action of a spatial symmetry g on the surface that should be satisfied by sg ¼ 1 is that it is a group Hamiltonian (8), derived in Appendix B starting from its ∈ G homomorphism, i.e., sg1 sg2 ¼ sg1g2 for g1;2 . The sig- action on the bulk Hamiltonian, is given by nature sg is not to be confused with the sign ηg. While the former describes a representation on the DSTI specifying ≡ † the transformation properties of the mass term (9), the latter g · hr;k vghr;kvg ¼ hg · r;Rgk; ð10Þ describes the sign choice of the representation on a single which leaves the surface Hamiltonian (as a whole) strong TI. The two are related by Eq. (13) given below. invariant for any spatial symmetry g. Clearly, the same The properties of the mass term on the surface are fixed by specifying the signatures of the symmetry operations, conclusion holds when we generalize vg ↦ ug ≡ ηgvg for ηð1;2Þ any U(1) phase factor ηg. Time-reversal symmetry further which are in turn completely fixed by the parameters g and det R via the relation restricts this phase to ηg ¼ 1. In addition, we demand g ηg to respect the (projective) group structure of the point ð1Þ ð2Þ group. Specifically, for a pair of symmetry elements sg ¼ det Rgηg ηg : ð13Þ 0 † † † † η g, g , we demand ugg0 ugug0 ¼ vgg0 vgvg0 , which fixes gg0 ¼ The appearance of det Rg in this expression follows from η η 0 [53]. g g ˆ σ Microscopically, different values of η arise naturally the fact that n · is a pseudoscalar. In the following, g we usually use the terminology “a representation for since the O(3) rotational symmetry is reduced to a finite, ” discrete point group in a crystal, and both spin and orbital symmetry g for symmetry representations on the DSTI to angular momenta contribute to the symmetry representa- indicate that the symmetry g has a signature sg ¼ . Let us point out that, although our DSTI model with tion. Yet, when ug is traceless, say, for ug ¼ iηgσz, the sign η can be removed through a basis transformation, and two flavors of Dirac fermions is not sufficient for g implementing all possible sTCIs of interest—say, those hence, η ¼ 1 do not give rise to distinct representations. g with a high mirror Chern number—it is sufficient for However, such a basis transformation will simultaneously constructing the “root states” that generate such states modify the form of the surface Hamiltonian (8), which upon stacking. To see this, consider a system consisting in turn amounts to a redefinition of the helicity of the of n copies of the DSTIs. In this case, there are k surface Dirac cone, as we discuss later. In the following, we n independent T -preserving mass terms mi;r, i ¼ 1; …;kn. say ug is a “signed representation” when we want to The crystalline symmetries act on the vector mr ¼ emphasize the importance of the sign choice ηg ¼ 1, ðm1 ; …;m Þ as orthogonal transformations leaving regardless of whether this choice actually gives rise to ;r kn;r the length of the vector (which gives the magnitude distinct representations. of the gap at a given point) fixed (see Appendix B). Any Spatial symmetries may force the mass term m to vanish r crystalline symmetry apart from the mirror leaves at along some curve, which creates a domain wall hosting a most two points on any given surface invariant. As a propagating 1D helical gapless mode. To see this, consider result, it will only protect anomalous surface states if it a spatial symmetry operation g, which leaves the mass term enforces the existence of a domain wall between a point (9) invariant. The action of the symmetry on the mass term r and its image under symmetry, g · r. Such a domain can generally be written as wall will be irremovable if and only if there is no trajectory connecting m and O m on the ðk − 1Þ sphere, ≡ τ τ † g n g · Mr ugMrðugÞ ¼ Mg · r: ð11Þ establishing a correspondence between the zeroth 031070-8 SYMMETRY INDICATORS AND ANOMALOUS SURFACE … PHYS. REV. X 8, 031070 (2018) homotopy group of the ðkn − 1Þ sphere and the stable B. Crystallographic point groups domain walls in a model with n DSTIs. Since the zeroth This subsection is devoted to the study of crystalline − 1 1 homotopy group of the (k ) sphere is trivial for k> , point groups, whose associated sTCIs can be understood by we deduce that we cannot build stable domain walls considering a spherical geometry with open boundary 1 1 1 whenever n> . For n ¼ , kn ¼ , and only one mass conditions in all directions. We describe a procedure to term is possible. In this case, the fact that the 0 sphere construct all possible sTCIs in a given symmetry group and (just two points) has two disconnected components then use it to obtain the complete classification of sTCIs Z implies the possibility of having 2 domain walls, protected by point-group symmetries. Z thereby establishing a 2 classification for sTCIs not We begin by reviewing the natural action of the indi- protected by mirror symmetry in class AII. Physically, vidual point-group symmetries on the physical degrees of Z the 2 classification here simply descends from that of freedom and by discussing the types of gapless surface 2D TIs in class AII. states they can protect. Next, we provide a general The only exception to the previous analysis is mirror procedure to construct all sTCIs in a given point group symmetry. In this case, we need to consider the mass vector and use it to obtain an exhaustive classification of sTCIs for in the mirror plane. We find that it has to remain invariant the 32 crystallographic point groups. under the action of mirror symmetry at any point in this plane, O m ¼ m, which is only possible if one of the M 1. Crystallographic point-group symmetries eigenvalues of O is þ1. In a representation that does not M — satisfy this condition, e.g., OM ¼ −1, the mass vector m Inversion. Inversion symmetry acts by inverting the will necessarily vanish in the mirror plane regardless of n. position and momentum while keeping the spin unchanged, Nevertheless, the state with n>1 DSTIs can always be built by stacking the “root” state implemented using a I∶ r → −r; k → −k; σ → σ: ð14Þ single mirror-symmetric DSTI. Notice that, up to this point, the analysis is general and According to Eq. (13), inversion can be represented with a applies to any crystalline symmetry. For instance, weak TIs negative signature by taking its action on the DSTI to be τ ð1Þ ð2Þ can be understood within the DSTI model by considering a I ¼ I ⊕ I ¼ I ⊗ τ0 (corresponding to η ¼ η )or τ surface with a periodic boundary along the “weak” direc- with positive signature by acting on the DSTI as I ¼ I ⊕ ð1Þ ð2Þ tion and choosing a “−” representation for translation along ð−IÞ¼I ⊗ τz (corresponding to η ¼ −η ). this direction. This choice will enforce the existence of a In the “−” representation, the mass term changes sign domain wall for every unit lattice translation along this between a point and its image under inversion, leading to a direction, leading to the surface states obtained by stacking “hinge” phase with a helical gapless mode living on some 2D quantum-spin Hall systems (Fig. 2). Alternatively, for a inversion-symmetric curve on the sphere. The curve can be system without any weak index—i.e., all the lattice trans- moved around but cannot be removed without breaking lations are assigned a “þ” signature—some other elements inversion. Such a phase is graphically illustrated in the Ci in the SG could also be in a “−” representation and lead entry in Fig. 4. Two copies of this state can be trivialized by x;z ˆ to other patterns of gapless modes on suitably chosen adding the mass terms mr ðnr · σÞτx;zμy, where μ denotes surfaces. the Pauli matrices in this additional orbital space of copies. Having outlined our general framework, we now apply it These additional mass terms, like Eq. (9), change sign x;z to classify all sTCIs in class AII. We proceed in two steps: under inversion; however, mr can be chosen so that they First, we focus on phases protected solely by point-group do not both vanish at a point where the original mass term, symmetries; second, we discuss how to extend these results mr in Eq. (9), also vanishes. This leads to a completely consistently to cover all 230 space groups. gapped surface. As a result, gapless modes protected by inversion will have a Z2 classification, as anticipated in the general discussion of the previous subsection. Cn rotation.—An n-fold rotation acts by rotating the position and momentum as vectors and rotating the spin as a spinor. An n-fold rotation about the z axis is implemented as −iðπ=nÞσz iðπ=nÞσz Cnz∶ r→Cnzr; k→Cnzk; σ→e σe : ð15Þ A positive signature representation is obtained by taking τ ⊕ ⊗ τ ηð1Þ ηð2Þ FIG. 2. Surface states for the weak topological insulator can be Cn ¼ Cn Cn ¼ Cn 0 (corresponding to ¼ ), τ ⊕ − ⊗ τ understood as choosing a “−” representation for the translation whereas the choice Cn ¼ Cn ð CnÞ¼Cn z (corre- along the “weak” direction. sponding to ηð1Þ ¼ −ηð2Þ) leads to a negative signature [54]. 031070-9 KHALAF, PO, VISHWANATH, and WATANABE PHYS. REV. X 8, 031070 (2018) Notice that, here, the “þ” and “−” representations are 2. Classification of sTCIs in the 32 crystallographic opposite to the inversion case since det I ¼ −1 but point groups 1 det Cn ¼ [cf. Eq. (13)]. We are now in a position to perform a systematic “−” We note that the representation is only possible for investigation of sTCIs in the 32 crystallographic point n −1 even n since it violates the condition Cn ¼ whenever n groups. As we argued in the beginning of this section, all 3 “ ” is odd. Thus, n ¼ is only consistent with a þ these states can be built from either the DSTI model 2 representation, whereas n ¼ , 4, 6 can have either sig- considered in Sec. III A or copies thereof. “−” nature. A representation in this case forces the mass to In Sec. III A, we propose that there is a one-to-one 2 vanish at n= (closed) curves related by rotation and correspondence between the signatures of the different intersecting at the two points left invariant by rotation symmetries and the pattern of gapless modes on the surface. (the poles) (as shown in Fig. 4 PGs C2, C4, and C6). We One part of this correspondence is obvious since surface remark that these conclusions have already been drawn in states corresponding to different symmetry signatures Ref. [23] using a slightly different language. cannot be deformed into each other without changing these Similar to the case of inversion, two copies of the signatures. We conjecture that the opposite is also true: Two described DSTI can be gapped out by adding the mass patterns of surface modes can always be deformed into 0 ˆ σ τ ⊗ μ term mrðnr · Þ z y, which does not change sign under each other if they correspond to the same representation rotation and can be chosen to be positive everywhere. This signatures for all the possible symmetries. We have Z means that rotations lead to a 2 classification as well, checked this explicitly for several examples, where seem- consistent with our general discussion. ingly different surface-state patterns corresponding to the — Mirror symmetry. Mirror symmetry acts by inverting same symmetry signatures turn out to be deformable into the position and momentum components perpendicular to each other. the mirror plane and the spin component parallel to it. For For example, consider the point group 4¯, where the only example, the action of mirror symmetry about the xy plane symmetry is a fourfold rotoinversion. In this case, there is implemented as seems to be two distinct surface-state patterns correspond- ing to the “−” representation for S4, given by the “equator” “ ” Mz∶ ðx; y; zÞ → ðx; y; −zÞ; σ → σz σ σz: ð16Þ state and the hinge state (cf. Fig. 3). These two patterns can, nevertheless, be deformed into each other. The reason for this is that, unlike rotation, rotoinversion does not leave In the mirror plane, M flips the sign of nˆ · σ. Thus, in the z r the poles fixed. Therefore, there is no symmetry constraint “−” representation, implemented by taking the mirror to act τ on the mass terms at the poles, implying that the inter- in the DSTI as M ¼ M ⊕ M ¼ M ⊗ τ0 (corresponding z z z z section of gapless modes at the poles is spurious; i.e., it is ηð1Þ ηð2Þ to ¼ ), the mass term mr has to vanish in this plane. unstable against symmetry-allowed perturbations. Thus, we The “þ” representation, on the other hand, is implemented τ ⊕ − ⊗ τ can move the hinges symmetrically away from the poles, in the DSTI as Mz ¼ Mz ð MzÞ¼Mz z (corre- bringing them to the equator. This process can be seen more ηð1Þ −ηð2Þ sponding to ¼ ), which does not impose a clearly by adding the two phases and noting that a mass constraint on the mass term in the mirror plane and leads term can be added to gap out the modes at their intersection to a completely gapped surface. points (this is possible since rotoinversion does not enforce “−” As we noted in the previous subsection, the any local constraints on the mass). The resulting surface representation for the mirror implies that the mirror plane can be deformed into a trivial one, as shown in Fig. 3. remains gapless regardless of the number of DSTIs stacked Alternatively, such correspondence can also be understood together. This can be seen from the fact that, for any in a slightly more general language (i.e., beyond the Dirac number of copies, any mass term will have the form theory analysis), as we elaborate in Sec. V. ˆ σ ⊗ Γ Γ mrðnr · Þ , where denotes matrices in the orbital Using our conjecture that the classification of sTCIs space of copies, which will always vanish in the mirror reduces to classifying distinct signatures of the different Z plane. As discussed previously, this implies a classi- symmetries, we now proceed by constructing these phases fication corresponding to the mirror Chern number in real space [6]. Sn rotoinversion.—As Sn ¼ ICn, its action on a DSTI can be readily understood through the corresponding discussions for I and Cn above. Note that a “twofold rotoinversion” is simply a mirror symmetry, which leaves a plane invariant in 3D and is differentiated from S3;4;6, which only leave the origin invariant. It is worth noting that ¯ ¯ ¯ ¯ FIG. 3. In the presence of the fourfold rotoinversion S4 only, the among the three rotoinversion groups 3, 4, and 6, 4 is the equator and the hinge state are deformable into each other since only one that is not a direct product of smaller groups. their sum can be trivialized, as shown here. 031070-10 SYMMETRY INDICATORS AND ANOMALOUS SURFACE … PHYS. REV. X 8, 031070 (2018) ∈ G S explicitly. We first note that two symmetries g1;2 TABLE II. Classification of point-group sTCIs PG denotes the related by conjugation—i.e., there is an h ∈ G such that classification of point-group sTCIs. The generators are chosen −1 according to the criteria defined in the main text. Here, nˆ 1 and nˆ 2 g1 ¼ h g2h—necessarily have the same signature. In the following, we call two symmetries independent if they are denote two of the four threefold rotation axes in the cubic PGs. These describe systems with cubic symmetry with four threefold not related by conjugation, so their signatures can be axes along the cube body diagonal. assigned independently. For two rotations (mirrors), inde- S pendence means that the rotation axes (mirror planes) are Symbol PG Generators PG not related by any symmetry operation in the symmetry 1 C1 10 G ¯ group . This implies that we can classify all possible 1 Ci I Z2 signatures by specifying the signatures for a minimal set of 2 C2 C2z Z2 Z independent generators of the group (i.e., every group mCs Mz 2=m C2h Mz, C2z Z × Z2 element can be written as a product of powers of the 2 222 D2 C2z;C2x Z2 elements in the set, and the size of the set is as small as 2 mm2 C2v Mx, My Z possible). Notice that the generators in a minimal set can 3 mmm D2 M , M , M Z always be chosen to be independent [55]. Due to the fact h x y z 4 C4 C4z Z2 that mirror and C3 rotation symmetries behave differently ¯ 4 S4 S4z Z2 compared to other symmetries in sTCI classification, we 4=m C4h C4z;Mz Z × Z2 2 need the generator set to satisfy the extra conditions (while 422 D4 C4z;C2x Z2 2 respecting the condition that the generator set is minimal): 4mm C4v Mx, Mxy Z ¯ (1) The number of independent mirror symmetries 42m D2d S4z;Mx Z × Z2 3 included as generators is as large as possible. 4=mmm D4h Mx, Mz, Mxy Z (2) The number of independent C3 rotations included as 3 C3 C3z 0 ¯ generators is as large as possible. 3 S6 S3z Z2 Note that these two conditions are consistent with each 32 D3 C3z;C2x Z2 3 Z other since we generally cannot increase (decrease) the mC3v C3z;Mx 3¯ Z Z number of independent mirror symmetries by decreasing m D3d S3z;Mx × 2 6 C6 C6z Z2 (increasing) the number of C3 symmetries in a generator set ¯ 6 C3 C3 ;M Z [56]. Notice also that a minimal generator set with a h z z 6=m C6h C6z;Mz Z × Z2 maximal number of independent mirror symmetries 2 622 D6 C6z;C2x Z2 actually contains all of them, which can be explicitly 6 pffiffi Z2 mm C6v Mx, M 3xþy=2 verified (cf. Table II). These conditions mean that we ¯ 2 6m2 D3h C3z;Mz;Mx Z should include the maximum number of independent C3 6 pffiffi Z3 =mmm D6h Mz, Mx, M 3xþy=2 rotations in the generator set, as long as there is no other 23 TC3;nˆ 1 ;C3;nˆ 2 0 generator set with smaller size. For example, in the PG T, 3¯ Z m Th C3;nˆ 1 ;Mz which can be generated using either two independent C3 Z 432 OC3;nˆ 1 ;C4;z 2 43¯ Z rotations or a C2 and a C3, we have to choose the former m Td C3;nˆ 1 ;Mnˆ 1 3¯ Z2 since it includes more C3 rotations; in contrast, in the PG m m Oh C3;nˆ 1 ;Mx;Mxy C6, which can be generated using a single C6 or a C3 together with a C6, we have to choose the former since it contains a smaller number of generators (the latter is not Implementing these rules leads to the classification of really a minimal generator set). sTCIs in all crystallographic point groups, which we S Once we have a minimal generator set satisfying these tabulate under PG in Table II. The hinge states generating properties, we can read off the classification of sTCIs as the different Z, Z2 factors by choosing the “−” represen- follows: tation in the corresponding symmetry are graphically (1) To every mirror symmetry in the generator set, we illustrated in Fig. 4. assign a factor of Z, indicating the number of gapless modes in this mirror plane. The phase that generates this factor is obtained by choosing the “−” C. Space groups representation for the corresponding mirror sym- In this subsection, we extend the analysis of the previous metry in the DSTI model. section to space groups, which requires considering sym- (2) To every symmetry generator other than mirror and metries that do not fix any point in space (lattice trans- C3, we assign a factor of Z2. Each of these Z2 phases lations and nonsymmorphic symmetries). We first start by is generated by choosing the “−” representation for discussing the main complication arising from the inclusion the corresponding symmetry generator. of nonsymmorphic symmetries, which requires a certain 031070-11 KHALAF, PO, VISHWANATH, and WATANABE PHYS. REV. X 8, 031070 (2018) FIG. 4. Graphical illustration of the surface states of the hinge phases, which generate all sTCIs for the 32 crystallographic point groups, on a sphere. For each state, we show the operators that need to be taken in the “−” representation as well as the resulting classification. Red circles indicate Z2 modes that would be gapped-out if two copies of the system were stacked together, whereas blue circles indicate Z modes protected by a mirror Chern number. Rotation axes are shown with black, blue, green, or red for twofold, fourfold, threefold, and sixfold rotations, respectively. choice of the sample geometry and boundary conditions. translations; i.e., we systematically study the spatial- We then provide a few examples of sTCIs protected by symmetry constraints on weak indices [16]. Finally, we nonsymmorphic symmetries. Next, we discuss how to provide a complete classification of sTCIs in the 230 SGs. consistently combine the preceding results with lattice We note that the order followed in this section, by first 031070-12 SYMMETRY INDICATORS AND ANOMALOUS SURFACE … PHYS. REV. X 8, 031070 (2018) considering nonsymmorphic symmetries and then trans- lations, may seem a bit incongruous given that nonsym- morphic symmetries are generalizations of translation. This ordering is, however, natural within our present context in SðsÞ which the strong part SG of the classification (which does not include translations) is first generalized from the point- group case to include nonsymmorphic symmetries before SðwÞ presenting a separate discussion for the weak part SG where translation plays a major role. FIG. 5. Illustration for the hinge state protected by a fourfold screw (left diagram) and a glide (right diagram) on a cylinder 1. Nonsymmorphic symmetries geometry with periodic boundary conditions along the screw direction. A nonsymmorphic symmetry arises when a point-group symmetry is combined with an irremovable, fractional translation, which results in a symmetry that leaves no Extending a mirror symmetry to its nonsymmorphic point in space invariant. The extension of the analysis of counterpart—a glide—leads to a more drastic modification Sec. III B to include nonsymmorphic symmetries is, in of the physics. Picking a “−” representation for a glide principle, straightforward. As we elaborate later, to analyze symmetry gives rise to a state with two hinge modes sTCIs protected by nonsymmorphic symmetries, it suffices confined to the glide plane shown in Fig. 5. Despite looking to first assume that all the weak indices of the system are similar to surface states protected by mirror symmetry, this trivial; i.e., all the lattice translations are given a “þ” state differs in an essential aspect, as it becomes trivial representation. Microscopically, this is the case when we when added to itself. The reason for this difference is that, stick with DSTI models with degrees of freedom arising unlike mirror symmetry, a mirror Chern number cannot be from the Γ point in the BZ, such that, in momentum space, defined for glide symmetry since it does not act as an on- the fractional translation associated with a nonsymmorphic site symmetry in the glide plane. One can also understand symmetry becomes irrelevant. This restriction is justified in such modification from a momentum-space perspective: Sec. IV and Appendix C. Unlike a mirror, the band eigenvalues i of a glide are not As we can focus on the Γ point in momentum space, the invariantly defined, as they are interchanged upon the main conceptual difference in understanding sTCIs pro- addition of a reciprocal lattice vector to the crystal tected by point-group and nonsymmorphic symmetries, momentum. Consequentially, one cannot define a Z-valued therefore, lies in real space. Recall that, to expose the Chern number using a glide symmetry. We note here that anomalous surface states of a sTCI, one has to consider a glide symmetry fits more naturally than a mirror within our geometry that respects the protecting spatial symmetries. In signed representation approach. The reason for this is that order to preserve a nonsymmorphic symmetry on the the approach, by itself, only allows for Z2-type hinge surface, we need to consider a sample with periodic modes resulting from the symmetry constraints on the boundary conditions along the directions of the fractional surface mass term. The appearance for Z-type hinge modes translation vector of the symmetry. A nonsymmorphic in the mirror case is an exception due to the fact that it symmetry, which can either be a screw (n-fold rotation leaves invariant an extended (1D) region on the surface followed by a fractional lattice translation along the rotation where the effective symmetry class is reduced from AII to axis) or a glide (mirror reflection followed by a fractional A. Such an unusual property, which is not shared by any lattice translation along a vector in the mirror plane), does other symmetry, is the main reason why an integer invariant not leave any point invariant. As a result, the surface states (mirror Chern number) can be defined only in the mir- protected by nonsymmorphic symmetries will have a Z2 ror case. classification (see the discussion at the beginning of this As an example of surface states protected by a non- section). symmorphic symmetry, let us consider SG P42 (No. 77), The anomalous surface states protected by nonsymmor- where the only symmetry is a fourfold screw 42.Ona phic symmetries can be understood by considering a cylinder geometry with a periodic boundary condition cylinder geometry whose axis is parallel to the fractional taken along the screw axis, a “−” representation for the translation axis, along which the periodic boundary con- screw symmetry leads to the state with four hinges shown dition is taken. An n-fold screw in a “−” representation in Fig. 5. We note that a state with domain walls at every would give rise to a state with n symmetry-related hinge half-lattice translation along the screw axis is also con- states along the sides of the cylinder. This surface state is sistent with the “−” representation for the screw symmetry. very similar to the surface state protected by rotation, in that This state is gapless everywhere on the surface and looks the hinges can be moved around freely as long as they very similar to the surface of a weak TI shown in Fig. 2. preserve the screw or rotation as a set. The main difference in this case is that such a state is 031070-13 KHALAF, PO, VISHWANATH, and WATANABE PHYS. REV. X 8, 031070 (2018) Generally, the presence of additional spatial symmetries leads to a restriction on the possible weak indices. For instance, a cubic system cannot realize a weak index that favors a particular axis, say, ν1 ¼ ν2 ¼ 0 but ν3 ¼ 1.As discussed in Ref. [16], such restrictions are encoded in the SG constraints on the admissible values of G, and they originate either from a point-group or nonsymmorphic FIG. 6. Illustration for the two equivalent surface states symmetry. As we discuss below, the same problem can corresponding to a fourfold screw. The equivalence of the two be analyzed through a complementary, though equivalent, can be established by noting that their sum can be deformed to a perspective by studying the symmetry restrictions on the trivial state while preserving the screw symmetry. signature assignments to the lattice vectors t. Let p be a symmetry in the SG and Tt a lattice translation −1 unstable against being gapped out everywhere except for [57]. We then see that the signatures of Tt and pTtp are the four hinges. In other words, the two surface-state necessarily identical. More generally, a weak index is configurations shown in Fig. 6 both correspond to choosing possible if and only if the corresponding signature assign- the “−” representation for the screw symmetry, and they ment to lattice translations, generated by choosing a “−” can be symmetrically deformed into each other as shown in signature for Tt, is symmetric under the described con- the figure. jugation by any symmetry in the SG. This requirement Another example is SG P63=m (No. 176), which is one captures all the restrictions from the point-group actions. of the key space groups considered in Table I. This space The presence of nonsymmorphic symmetries can further group is generated by a sixfold screw and a mirror about a restrict the possible weak indices. To see this, note that, for plane normal to the screw axis. Choosing a cylinder any nonsymmorphic symmetry g, which is not a threefold geometry with a periodic boundary condition along the screw, gn is a lattice translation for some even integer n. screw axis, we may consider a “−” representation for either Therefore, regardless of the signature chosen for g, the the screw or the mirror. In the former case, we get a phase translation gn always takes a “þ” signature, and hence, any with six hinges, whereas the latter is characterized by weak index demanding a “−” signature for gn is forbidden. gapless modes protected by a mirror Chern number in These restrictions can be illustrated using the following the mirror planes. Notice that, in this case, we have several examples. First, consider the (symmorphic) space group mirror planes related by screws, which all have the 143 whose only symmetry is a threefold rotation, and same mirror Chern number. The signatures of the men- suppose the lattice vectors in the plane perpendicular to the tioned mirror and screw are independent, and consequen- rotation axis are denoted by t1 and t2. The action of − tially, the resulting classification is Z × Z2. threefold rotation sends t1 to t2 and t2 to ðt1 þ t2Þ. This −1 While we have argued quite generally that the sTCI means that the translations along t1;2 satisfy C3 Tt1 C3 ¼ −1 −1 −1 classification for nonsymmorphic symmetries can be Tt2 and C3 Tt2 C3 ¼ Tt1 Tt2 . The first equation implies that understood from the point-group counterpart, our discus- translations along t1 and t2 necessarily have the same sion so far has one caveat: Unlike threefold rotations, a signature, and the second equation implies that this threefold screw with a 1=3 lattice translation along the signature is necessarily þ1, thereby ruling out the pos- “−” rotation axis, S31 , does admit a representation. However, sibility of any weak phases for the in-plane translations. 3 Z such a choice implies ðS31 Þ , a unit lattice translation, is The resulting weak classification is 2 corresponding to also assigned a “−” representation, leading to a nontrivial translation along the rotation axis only. weak TI index, and hence, the described phase falls outside To illustrate the effect of nonsymmorphic symmetries, of our present discussion. This brings us to the last item to we consider space group 4, which has a single twofold consider for our classification of sTCIs: the incorporation of screw rotation. Since this screw squares to a lattice trans- weak phases. lation along its axis, it forces translation along this axis to have þ1 signature, ruling out the possibility of a weak phase along this direction. The weak classification is then 2. Lattice translation Z2 × Z2, corresponding to the two independent in-plane As we have briefly addressed at the end of Sec. III A,in translations. Space groups 76 and 167 provide examples in our framework, a weak TI is characterized by the set of which both restrictions are present simultaneously. The lattice translations taking a “−” signature. More specifi- former contains a single fourfold screw rotation, which, in cally,P consider a weak TI characterized by the vector addition to ruling out a weak phase for translation along the ≡ 1 3 ν G 2 i¼1 iGi, where Gi denotes a reciprocal lattice screw axis, also forces the signatures for the two orthogonal vector and νi ¼ 0, 1 is the associated weak index. The in-plane translations to be the same, leading to a Z2 weak signature of a lattice translation, characterized by the vector classification. The sixfold screw characterizing the latter t, is then given by e−iG·t. places even more restrictions by ruling out any weak phase 031070-14 SYMMETRY INDICATORS AND ANOMALOUS SURFACE … PHYS. REV. X 8, 031070 (2018) either in the plane (similar to the case of SG 143 considered same signature. This induces a consistent signature assign- G above) or along the screw axis, leading to the absence of ment on p. Conversely, for any consistent assignment on G G any weak phases. p, one can define an assignment on by using the G → G Extra restrictions on the weak indices may arise because canonical projection p. This demonstrates the stated of the type of unit cell, e.g., primitive vs body centered. For one-to-one correspondence, up to the modification required example, SG 23, which features three orthogonal intersect- for mirror vs glide [58]. Z ing C2 axes in a body-centered unit cell, has a 2 weak Next, we incorporate weak phases into our discussion. classification due to the fact that all translations can be S Recall that the computation of SG amounts to the expressed in terms of a single body-centered translation and identification of a minimal generating set of the SG, paying the action of C2 rotations on it. We point out here that our special attention to the presence of mirror and threefold analysis is equally applicable to the two special space rotation symmetries (conditions outlined in Sec. III B). groups 24 and 199, which are the only nonsymmorphic Consider a weak index, generated by assigning a “−” space groups not containing any screws or glides. Both SGs signature to a lattice vector t, which satisfies all the have a body-centered unit cell containing three noninter- previously outlined SG constraints. This index implies secting C2 axes and are thus very similar to SG 23, with a the signature of t is not fixed by the indices contained in single independent body-centered translation leading to a s Sð Þ, and therefore, t must be incorporated into the gen- Z2 weak classification. SG erating set for the SG. In addition, if the SG possesses a While we have discussed the constraints on weak phases mirror m about a plane normal to t, the mentioned weak TI utilizing the group structure of the SG, it is instructive to is “promoted” to a weak mirror Chern insulator; i.e., the connect it to the more microscopic DSTI model we classification is modified from Z2 to Z. When this happens, described. To this end, observe that, if the surface Dirac we should append the shifted mirror mT instead of the cone originates from degrees of freedom around a TRIM k, t translation T to our generator set. Following this pro- the signed representation for the lattice translation T is t t cedure, one incorporates all the needed lattice translations given by η ¼ e−ik·t ¼ 1. According to Eq. (13), a DSTI Tt or shifted mirrors into the generator set. Each of such built from strong TIs with effective degrees of freedom additional generators for the SG then correspond to either a ð1Þ ð2Þ coming from the TRIMs k and k would then realize a Z2-valued weak TI index or a Z-valued weak mirror Chern ð2Þ ð1Þ weak index of G ¼ k − k . SðsÞ index, which we append to SG to arrive at the full classification SðsÞ . Implementing this rule leads to 3. Classification of sTCIs in the 230 space groups SG Table IV in Appendix C, which provides the sTCI classi- Having separately described how to extend our point- S fication SG for all 230 SGs. group results to sTCIs protected by a lattice translation or Finally, we comment on the meaning of adding two nonsymmorphic symmetry, which does not fix any point in S phases in SG, which is an Abelian group representing the space, we now tackle the problem of classifying all sTCI equivalence classes of distinct sTCI phases. The subtlety phases built from stacking strong TIs in all 230 space arises if there is no geometry for which both phases exhibit groups. From the discussion on point groups, we see that anomalous surface states. This will never be the case for the desired classification can be reconciled with the differ- SGs containing only symmorphic elements, but it is 1 ent ways to assign the signature to a generating set possible in the presence of nonsymmorphic symmetries. S of the space group, which we denote by SG. However, we have to remember that elements of S are SðsÞ ≤ S SG We first focus on a subgroup SG SG classifying distinct bulk phases despite being physically defined by “ ” sTCIs that are not weak, i.e., those for which the their surface signatures. This means that the possibility of 1 signature sT ¼þ for all lattice translations T.Wenow anomalous states on any given surface is completely fixed SðsÞ argue that the sTCIs described by an element of SG can be by the bulk, as will be discussed in detail in Sec. IV. S G readily studied via PG. Recall that, given a SG ,a Moreover, it is always possible to distinguish two distinct G “ ” corresponding point group p is defined by modding out sTCI phases by considering them on many different G G ≡ G the translation part of , i.e., p =TG, where TG is the geometries. For instance, given two phases 1 and 2 and translation subgroup of G (which is always a normal two geometries G1 and G2, such that 1 (2) exhibits surface subgroup). This procedure reduces screws or glides in G states on G1 (G2) but not on G2 (G1), we can distinguish the G sum of the two phases from either phase by the fact that it to rotations or mirrors in p, The classification of sTCIs in G G exhibits surface states on both geometries G1 and G2. any given SG is the same as the classification of p, G except for the fact that every mirror in p that derives from a D. Surface dispersion at special planes glide in G should be assigned a Z2 rather than Z factor. More concretely, we note that, for any consistent signature Although the 1D surface modes shown in Fig. 4 can, in assignment on G satisfying sT ¼þ1, any two symmetries principle, be detected by means of transport measurements, in G with the same point-group action will be given the the most experimentally accessible tool to investigate such 031070-15 KHALAF, PO, VISHWANATH, and WATANABE PHYS. REV. X 8, 031070 (2018) cones apart by adding the T -symmetric term m1σxτy, which shifts the Dirac cones to k ¼ð0; m1Þ [6]. The glide case can be considered very similarly. A surface glide ik·t symmetry has the form Mxe for some fractional trans- lation t. Thus, close to the Γ point, mirror and glide act in the same way. 2. Rotation axis This case was considered in Ref. [23]. We start with the surface Hamiltonian (17), with twofold, fourfold, or sixfold π σ − π σ ≡ ið =nÞ z τ τ ið =nÞ z rotations acting as Cn · hk e zhk ze ¼ hCnk. The only mass term consistent with Cn and time-reversal symmetry is mkσzτx for n ¼ 2,6ormkσzτy for n ¼ 4, both − satisfying mCnk ¼ mk. This case, in particular, implies that the mass mk vanishes at k ¼ð0; 0Þ, so the Dirac cones FIG. 7. Illustration of the surface dispersion in planes tangent to cannot be gapped. However, we can add the symmetry- the sphere whose normal is (a) in a single mirror plane, (b) in two allowed term m1τz, which will move the Dirac cones away mirror planes, or (c) parallel to a fourfold rotation axis. from k ¼ð0; 0Þ, resulting in n Dirac cones related by the n- fold rotation. To see this, let us consider the Hamiltonian (17) (without any mass term), which just describes two surface states is provided by the angle-resolved photo- copies of a gapless Dirac Hamiltonian. Adding the term emission spectroscopy (ARPES), which probes the energy m1τ shifts the two cones in energy by 2m1, so they dispersion at a given surface. The dispersion measured by z intersect at a ring at zero energy. The Hamiltonian of this ARPES can be readily understood from the surface-state nodal ring is given by pattern given in real space in Fig. 4. In brief, we consider what happens when we attach a tangent plane to the sphere 2 2 h ¼ð2m1 − k − k Þμ ; ð18Þ at a given point, i.e., when we consider a flat, macroscopic k x y z crystal facet with the same surface normal as the considered where μx;y;z denote the Pauli matrices in the space of the point on the sphere. two bands intersecting at zero (the lower band from the We note that the surface modes can generally be freely Dirac cone shifted upwards and the upper band from moved on the surface except when they pass through a the one shifted downwards). Adding the mass term rotation-invariant point or lie in a mirror or glide plane. As a m σ τ for n ¼ 2,6orm σ τ for n ¼ 4 and projecting result, a tangent plane at a point that is not rotation invariant k z x k z y it to the μ basis, we find that it has the form m μ or m μ , or lies in a mirror or glide plane will generically have k x k y which will gap out the nodal ring except at n points, where gapped dispersion. Therefore, the analysis of surfaces it necessarily vanishes because of m ¼ −m . where 2D gapless modes may exist boils down to consid- Cnk k ering the cases where the normal to the surface (i) lies in a single mirror or glide plane, (ii) lies at the intersection of 3. Multiple mirror or glide planes multiple mirror or glide planes, or (iii) is parallel to a If the normal to the tangent plane lies in the intersection rotation axis. These three cases are illustrated in Fig. 7, and of multiple mirror planes, all in the “−” representation, we we elaborate on them below. can repeat from the case of a single mirror for each mirror separately and conclude that any T -symmetric mass term will necessarily vanish at k ¼ð0; 0Þ. Similar to the case of 1. Single mirror or glide plane rotation, this does not necessarily imply that we cannot Here, we investigate the dispersion in a tangent plane move the Dirac cones away from (0, 0) since we can add the whose normal lies in a single mirror plane. The dispersion term m1τz to shift them in energy and get a nodal ring in the plane close to the Γ point can be written as given by the Hamiltonian (18), which can be gapped by adding the mass term mkσzτx or mkσzτy (whose projection h k σ − k σ ; 17 k ¼ y x x y ð Þ in the μ basis is mkμx or mkμy). This mass term vanishes at all mirror-invariant lines because of the condition with reflection acting as M · h ≡ σ h σ ¼ h− . − x kx;ky x kx;ky x kx;ky mM·k ¼ mk; e.g., for two perpendicular mirrors Mx and Reflection symmetry implies that any mass term has the My, we get two Dirac cones at ð0; k1Þ and ð k1; 0Þ. The σ ⊗ Γ 0 form mk z , which necessarily vanishes at kx ¼ since case for multiple glides or a glide and a mirror is very − it satisfies m−kx;ky ¼ mkx;ky . Therefore, it is not possible to similar since the action of a glide at the Γ point is identical gap out the surface, but it is possible to move the two Dirac to the action of a mirror. We note here that our results agree 031070-16 SYMMETRY INDICATORS AND ANOMALOUS SURFACE … PHYS. REV. X 8, 031070 (2018) with those of Ref. [17], which showed that we can only pin We start by considering the action of a spatial two Dirac points in the vicinity of a TRIM other than Γ symmetry g, defined as in Sec. III A, on the bulk Dirac using two perpendicular glides. H ≡ H † H Hamiltonian (19), given by g · k Ug kUg ¼ Rgk. Note that, despite the similarity between hinge phases Here, Ug ≡ ug ⊕ ðdet RgugÞ, with ug defined as in protected by rotation and screw symmetries, without fine- Sec. III A. Next, we connect the bulk characterization with tuning, the latter are not detectable by studying the the surface theory. As detailed in Appendix B, for a surface dispersion at any plane on the surface. In this sense, they ˆ with normal nr, the surface Hamiltonian is given by are similar to hinge states protected by inversion or ˆ hr;k ¼ðk × nrÞ · σ, which transforms under symmetry rotoinversion in that they can only be observed if the † as g · h ≡ u h ug ¼ h . Recall that, in Sec. III, surface is considered as a whole. r;k g r;k g · r;Rgk we showed that when two strong TIs are stacked together, the pattern of surface modes (if any) is determined by the IV. BULK-SURFACE CORRESPONDENCE signature sg of the symmetry representation, which can be In Sec. II, we discussed how the symmetry-based expressed in terms of det Rg and the relative sign of the indicators for time-reversal symmetric systems with symmetry representation matrices across the two copies strong spin-orbit coupling can mostly be reconciled [cf. Eq. (13)]. This means that, by specifying the with either a Z2 TI index or a mirror Chern number. (signed) symmetry representation matrices ug of the two However, as listed in Table I, there are new invariants TIs in the bulk, we can predict the pattern of the surface that indicate that certain stacks of strong TIs remain modes, thereby establishing the anticipated bulk-surface nontrivial despite the lack of a conventional index. correspondence. These unconventional phases are expected to feature Such correspondence, however, utilizes more informa- hingelike anomalous surface states, as is discussed for tion than that available from the symmetry representations specific cases in Refs. [19–23] and analyzed thoroughly alone. To make connection with XBS, the symmetry-based in Sec. III. In this section, we tie the bulk and surface indicators of band topology [24], we have to study what perspectives together and describe a concrete bulk-sur- information is lost when we focus only on the symmetry face correspondence. representations. In other words, generally, the identification This case is achieved by making an explicit connection of the surface states associated with a nontrivial class ½b ∈ between the specification of bulk symmetry representa- XBS is not unique. Such nonuniqueness has two origins: η 1 tions and the surface mass-term analysis. In particular, we Either the choices g ¼ correspond to the same repre- show that all XBS-nontrivial phases have anomalous sentation, or certain nontrivial stacking patterns of strong surface states (with suitably chosen sample geometry). TIs have symmetry representations compatible with an However, the XBS diagnosis and our surface-state classi- atomic insulator and hence evades the XBS diagnosis. fication are not generally in one-to-one correspondence; We illustrate these ideas from two concrete examples. rather, for most SG, different sTCIs become indistin- Consider the space group P2 (SG 3), which is generated by guishable when one focuses only on symmetry represen- lattice translation and a C2 rotation about the z axis. As tations, and therefore, the same XBS class is identified with discussed in Ref. [23], when we stack two strong TIs with multiple patterns of surface states. We discuss the general opposite helicities, described, respectively, by the surface σ − σ structure behind such identification ambiguity, and in Hamiltonians h ¼ ky x kx y and with C2 represented σ Table IV in Appendix C, we provide the corresponding by u ;C2 ¼ i z, the resulting DSTI is nontrivial and features results for all SGs. hinge modes when subjected to the C2 symmetric, spherical ðwÞ As the relation between XBS and the weak TI or mirror open boundary conditions in Sec. III. To reconcile with the Chern indices has already been addressed in Sec. II, in the analysis based on Eq. (19), we can perform a basis rotation ðsÞ ðsÞ following, we focus on the relation between XBS and SSG. h− ↦ σyð−kyσx − kxσyÞσy ¼ kyσx − kxσy To proceed, we make a simplifying assumption: We 20 ↦ σ σ σ − σ ð Þ suppose that the topological properties of the strong TI u−;C2 yði zÞ y ¼ i z; of interest can always be analyzed through a k · p (bulk) Γ η 1 Dirac Hamiltonian about [1,3,4,34]: which implies ;C2 ¼ in our present framework. ⊕ Equation (13) then correctly predicts that the DSTI hþ Hk ¼ Mγz þ k · σγx; ð19Þ h− features anomalous surface states. However, as σ 0 ⊕ trð i zÞ¼ , the distinction between hþ h− and the 0 ⊕ where we assume the mass term changes from m< in the trivial phase hþ hþ cannot be diagnosed from symmetry bulk to m>0 outside the system, leading to a Dirac point representations alone. This is consistent with the result 0 γ 0 2 localized to the surface where m ¼ (Appendix B). Here, XBS ¼ found in Ref. [24] for P (SG 3); see also Table I. is a set of Pauli matrices describing the orbital degrees of As a second example, we consider P4=m (SG 83) freedom. generated by the lattice translations and the point group 031070-17 KHALAF, PO, VISHWANATH, and WATANABE PHYS. REV. X 8, 031070 (2018) C4h. In Ref. [24], it was found that the strong TI generates TABLE III. The bulk-surface correspondence for the key space Z ðsÞ the subgroup 8 031070-18 SYMMETRY INDICATORS AND ANOMALOUS SURFACE … PHYS. REV. X 8, 031070 (2018) Dirac cone, as for strong TIs, or are described by our sTCI surface classification. The set of sTCIs also strictly contains the set of conventional TCI phases with 2D surface Dirac cones [6,7,14–17,23]. Note that sTCIs without such 2D gapless surface states then feature only 1D helical hinge modes protected by inversion, rotoinversion, or screw symmetries, whose existence can only be observed by considering the surface as a whole. FIG. 8. Surface hinge mode protected by inversion symmetry Next, we make connections between our results and on a cubic geometry, assuming no other spatial symmetries are more general ideas concerning crystalline SPT classi- present. The mode is confined to the edges of the cube since the fications. Weak TIs, which can be understood in terms surface mass term cannot change on a flat face. of spatially stacking 2D TIs using lattice translation symmetry, exemplify the notion of topological crystal- symmetry-breaking perturbation is large enough (compared line insulators. Curiously, on more general grounds, it to the scale of the surface energy gap). We also note that, has been suggested that many, if not all, crystalline when placed on a surface with planar faces, e.g., a cube, the symmetry-protected topological phases can be under- hinge modes are expected to be localized to the edges of the stood in terms of a similar construction, upon the sample, as shown in Fig. 8, since the mass term is expected replacement of lattice translation by other crystalline to remain constant on any flat surface. symmetries [16,23,38,39,41,42]. Here, we remark that a It is worth reiterating that our classification results rely similar construction is also applicable to the DSTIs. To on a conjectured correspondence between sTCIs in a given illustrate this, consider a space group with only lattice space group and the signatures of the elements of this translation and a mirror symmetry z ↔ −z.Thez ∈ Z group. The resulting classification captures all known and z ∈ ð1=2 þ ZÞ planes are mirror invariant, and if we sTCIs in addition to predicting many new ones. Our results pin a quantum spin Hall insulator to each of these imply that the set of sTCIs strictly contains the set of phases planes, we will arrive at a band insulator that does not that are XBS nontrivial but not strong TIs (Fig. 9). That is to immediately yield an atomic limit. However, such an say, all the nontrivial XBS phases exhibit anomalous gapless insulator does not possess a weak index; rather, we should surface states, which are either of the form of a surface view it as interlacing two weak TIs built from stacking 2D TIs with mirror Chern number þ1, one from pinning them Stable topological phases to the z ¼ 0 plane and the other at the z ¼ 1=2 planes. Computing the mirror Chern numbers, one finds that the described band insulator has mirror Chern number 2 on the k ¼ 0 planeand0onthek ¼ π plane; i.e., it can also Strong TI z z be understood as a DSTI. Schematically, we may “ 0 1 2 ” sTCI write DSTI ¼ weak- þ weak- = . The above picture is, in fact, quite general: It applies to a large class of DSTIs arising in settings where the real-space deformation of “weak-0” to “weak-1=2” is forbidden by symmetry, as in the case when inversion [24] and/or nonsymmorphic [14–16,42] symmetries are Phases with gapless surface states present. In particular, we note that the glide-protected (on suitably chosen boundaries) TCI showcasing the “hourglass fermion” surface states FIG. 9. Hierarchy of crystalline topological insulators with alsofallsintothiscategory[14,15,42]. time-reversal symmetry and strong spin-orbit coupling (class AII) It is also conceptually revealing to abstract our sTCI surface classification from the Dirac equation approach we in 3D. All nontrivial entries in the symmetry indicator XBS are compatible with gapped topological band structures with gapless have employed. Let P be the group of spatial symmetries surface states on suitably chosen sample geometries. Excluding respected by the open boundary condition, e.g., in the the strong TIs from the XBS-nontrivial phases, we found that all filled-sphere geometry, P is the subgroup of the point group the remaining phases are anomalous surface topological crystal- that leaves the origin invariant. In essence, for every spatial line insulators (sTCIs), defined as nontrivial insulators with symmetry g ∈ P, we assign a signature sg ¼ 1, which anomalous surface states, but they can be trivialized upon the conforms to the group multiplication s 0 ¼ s s 0 . A differ- breaking of some crystalline symmetries. Note that sTCIs include gg g g all gapped topological phases with gapless surface states but are ent assignment of signature can then be viewed simply as → Z not strong TIs (hatched region). These also include phases not different homomorphisms from P 2, which are clas- diagnosed by X ; i.e., the topological nature is not exposed by sified by the first cohomology group of P with Z2 BS 1 the symmetry eigenvalues in the bulk. coefficients, H ðP; Z2Þ. 031070-19 KHALAF, PO, VISHWANATH, and WATANABE PHYS. REV. X 8, 031070 (2018) This interpretation is similar to the dimension-reduction by introducing spin-orbit coupling), and hence, we conclude 2 ∈ Z approach advocated in Refs. [38,41], except that, because that the 4 entry in the XBS diagnosis is enforced to be of the presence of additional internal symmetries (time semimetallic by the symmetry representations. We leave the reversal for the case of DSTI), we do not fully reduce the general question of whether any of the XBS entries for any classification to the symmetry charges at the point fixed space group realize a gapped topological phase in class AI by P; rather, we picture the construction of the 3D phase by (with or without surface states) for future works. interlacing 2D SPTs [23], and therefore, the 2D classifi- 1 cation (Z2 for class AII) enters into the coefficient of H . ACKNOWLEDGMENTS One can imagine a similar analysis for DSTI-like phases The authors would like to acknowledge stimulating in a more general setting, where we replace Z2 by the discussions with P. W. Brouwer, M. Geier, and A. P. X appropriate lower-dimensional classification (an Abelian Schnyder. In particular, we thank R. Quieroz for helpful group) and incorporate a group action of P on X (orienta- discussions on surface states protected by multiple mirrors tion-reversing elements in P reverse the phase in X). and/or glides. We also thank C. Fang for informing us of his However, we caution that this provides only a partial recent works [65,66], which also study the physical understanding on the full classification. For instance, in interpretation of phases with nontrivial symmetry indica- the presence of mirror symmetries, the DSTIs can feature tors. A. V. and H. C. P. were supported by NSF Grant Z-valued mirror Chern modes (neglecting interactions) at 1 No. DMR-1411343. A. V. acknowledges support from a the surface, whereas the H description captures only the Simons Investigator Award and ARO MURI Program parity of the number of modes. Such discrepancy arises No. W911NF-12-1-0461. H. W. acknowledges support because the mirror symmetry is effectively reduced to an from JSPS KAKENHI Grant No. JP17K17678. on-site unitary symmetry on the mirror-invariant plane, which enriches the classification [38,41]. We leave the APPENDIX A: RELATION BETWEEN κ AND ν analysis of the general structure of such classifications as an 4 0 interesting open problem. In order to establish the connection between κ4 and ν0, Lastly, we make a few remarks about the spin-orbit- let us focus on the primitive lattice case (the body-centered coupling-free case (class AI), whose indicators were also case can be discussed in the same way) and introduce the obtained in Ref. [24]. In the absence of spin-orbit coupling, 1D Berry phase for each occupied band: a new possibility that may be mandated by a nontrivial Z 1 π indicator is a gapless (semimetallic) phase (dubbed repre- z ≡ dk u ∂ u ; ðkx;kyÞ z k ;k ;k kz ðkx;ky;kzÞ ðA1Þ sentation-enforced semimetals in Ref. [24]) [60–64], and 2πi −π ð x y zÞ we provide an explicit example of such a scenario in the following. which is only well defined modulo 1. The S4 symmetry imposes the relation z ¼ −z − . This, in particular, An example of a semimetallic phase diagnosed by XBS in ðkx;kyÞ ð ky;kxÞ ¯ 1 2 class AI is provided by SG P1 (No. 2), where the only point- implies that zðkx;kyÞ is quantized to either 0 or = at group symmetry is inversion, whose “strong factor” is Z4. ðkx;kyÞ¼ð0; 0Þ and ðπ; πÞ. If (and only if) the two The generator of such a factor has the same inversion quantized values do not agree, the Berry phase zðkx;kyÞ eigenvalues at the TRIMs as the strong TI, and it can be has an odd winding as the 2D momentum changes along thought of as the limit of a strong TI when the spin-orbit the loop ð0; 0Þ → ðπ; 0Þ → ðπ; πÞ → ð0; πÞ → ð0; 0Þ. This coupling is adiabatically switched off. The corresponding nontrivial winding indicates the strong index ð−1Þν0 ¼ −1 phase is a Dirac nodal ring semimetal with so-called drum- [67]. On the other hand, the quantized value of zðk ;k Þ at head surface states [62–64]. Its low-energy physics is x y (0, 0) and ðπ; πÞ can be diagnosed by the ratio of S4 captured by the effective spinless two-band Hamiltonian eigenvalues at k ¼ 0 and k ¼ π, suggesting that ð−1Þν0 H γ 2 2 − 2 γ z z k ¼ kz y þðkx þ ky k0Þ z, which describes a nodal ring can be basically given by the product of S4 eigenvalues of 0 γ centered at k ¼ with radius k0. Here, denotes the Pauli occupied bands over all high-symmetry momenta in K4. H matrices describing the orbital degree of freedom, and k is However, in this product form, the S4 eigenvalues have to invariant under spinless TRS, represented by complex be restricted to those squaring into þi (or, equivalently, −i) γ conjugation, and inversion symmetry, represented by z. to avoid double counting of TR pairs, just as in the case of A system constructed by stacking two copies of this the original Fu-Kane formula. All in all, we find Hamiltonian can be gapped out in the bulk by a mass term Y Y n α γ τ ν0 iðαπ=4Þ n κ4 of the form mk x y. Inversion, however, would require mk to ð−1Þ ¼ð−1Þ2 ½e K ¼ð−1Þ : ðA2Þ satisfy m−k ¼ −mk, which implies that the mass has to K α¼1;5 vanish at least twice along the nodal ring, leaving behind two P α bulk Dirac nodes at generic, but inversion-related, momenta. Again, n ¼ α¼1;3;5;7nK is the total number of occupied This pair of Dirac nodes cannot be removed without breaking bands. The last equality can be verified from the definition the symmetries [35,36] or changing the symmetry class (say, Eq. (6), in the same way as we did for κ1. 031070-20 SYMMETRY INDICATORS AND ANOMALOUS SURFACE … PHYS. REV. X 8, 031070 (2018) ˆ ˆ APPENDIX B: SURFACE THEORY where we used k × n ¼ kk × n. This result explicitly gives 2 2 ≡ ˆ σ In this appendix, we elaborate on the derivation of the a × surface Dirac-cone Hamiltonian hk ðk × nÞ · . surface theory starting from the bulk Dirac Hamiltonian in Before proceeding, we make a few comments on the Eq. (19), with the aim of explicitly connecting the sym- validity of our approach. The curvature of the space enters into our equation only through the (slow) spatial dependence metry properties of the two. ˆ Recall the Dirac Hamiltonian in Eq. (19), of the surface normal n. This might not be the only effect resulting from the background curvature in the Dirac equation since we generally expect both the local and global H ¼ Mγ þ k · σγ : ðB1Þ k z x curvature to introduce extra terms in the equation [68]. However, we argue below that such terms are not relevant for Note that, in principle, there is a sign ambiguity for the term the physics we are considering. To see why, first note that the ∝ k · σγ . However, one can show that this sign choice does x surface states associated with a DSTI are confined to regions not affect the surface analysis in any way, and therefore, we in space corresponding to the domain walls of the mass term simply set it to 1 in our treatment. This should be þ in the Dirac equation. As the presence of such domain contrasted with the helicity of the surface Dirac cone, walls is a topological property of the bulk, it is unaffected by which, in the presence of certain symmetries, plays a key the surface curvature. Now, we can imagine smoothly role in the existence of anomalous surface states in a deforming any given surface such that all the nontrivial DSTI [23]. curvature is concentrated in isolated regions where the mass To expose a boundary, we let the mass term M be spatially term is nonzero (in a symmetry-respecting manner), such dependent, such that the surface region is defined by that the gapless electronic theory at the domain walls Mð0Þ¼0, and MðλÞ → signðλÞ quickly away from the ˆ resembles that defined assuming vanishing curvature. We boundary region. Let n be the surface normal, and decom- then recover the surface theory described throughout, and ˆ ˆ 0 pose k ¼ kk þ k⊥n such that kk · n ¼ . In the Dirac hence the sTCI classification. Note that such an ability to → − ∂ Hamiltonian, we then replace k⊥ i λ, arriving at decouple the possibly nontrivial curvature from the gapless surface states relies on having gapped regions on the surface H λ M λ γ k σγ − inˆ σγ ∂λ: kk; ¼ ð Þ z þ k · x · x ðB2Þ of a sTCI, which prevents the sTCI surface states from exploring the global topology of the surface; this should be H λ Our goal is to find eigenstates of kk; which are exponen- contrasted with the Dirac theory on the surface of a regular tially localized to the surface region. This can be achieved strong TI, which is gapless everywhere and is hence more through the ansatz susceptible to the effect of a nontrivial global curvature. R Next, we connect the symmetry representations fur- λ − dλ0Mðλ0Þ nished by the bulk and surface degrees of freedom. Let Ψðk ; λÞ¼e 0 ψðk Þ; ðB3Þ k k p ¼ð1; 0ÞT be the 4 × 2-dimensional matrix projecting ˆ into the subspace defined by Pþ, and we generalize n to a which gives ˆ slowly varying function nr, where r denotes a point on the boundary. Then, the surface Hamiltonian can be obtained H λΨ k λ k σ γ M λ γ 1 −nˆ σ γ Ψ k λ ; † kk; ð k; Þ¼ð k · x þ ð Þ zð · yÞÞ ð k; Þ T from the bulk through hr;k ≡ p VrHkVr p. We assume the ðB4Þ surface is symmetric under the (unitary) spatial symmetry g ¼fRgjγgg; i.e., for any r on the surface, g · r is also on the and can be solved by surface, and their surface normals are related by ˆ ˆ ng·r ¼ Rgnr. Recall that in the bulk, g is represented by 1 − ˆ σ γ ψ 0 ð n · yÞ ðkkÞ¼ a unitary matrix Ug ¼ ug ⊕ ðdet RgugÞ, where ug equals σ γ ψ ψ ðB5Þ 1=2 R kk · x ðkkÞ¼Ek ðkkÞ: the standard spin- representation of g up to a sign k η 1 H †H g ¼ , and k ¼ Ug RgkUg. We can then deduce the ψ ψ The first condition implies Pþ ðkkÞ¼ ðkkÞ, where Pþ is transformation of the surface Hamiltonian through ≡1 1 ˆ σγ σ γ 0 the projector Pþ 2ð þn· yÞ satisfying ½Pþ;kk · x ¼ . T H † The two-dimensional surface Hamiltonian can then be hr;k ¼ p Vr kVr p σ γ † † † † found by restricting kk · x to the subspace defined by Pþ. T H ¼ p VrUgðVg · rVg · rÞ RgkðVg · rVg · rÞUgVr p This is most easily achieved by introducing a rotation of † † † π 4 ˆ σ γ pTV U V p h pTV U V p ; B7 basis. Let V ¼ exp ½ið = Þn · x , such that ¼ð r g g · r Þ g · r;Rgkð g · r g r Þ ð Þ 1 † † where we use V U V ;ppT 0, and one can check VP V ¼ ð1 − γzÞ; ½ g · r g r ¼ þ 2 † † that pTV U V p ¼ u . Therefore, g ·h ≡u h u ¼ σγ † ˆ σ g · r g r g r;k g r;k g Vðkk · xÞV ¼ðk × nÞ · ; ðB6Þ hg·r;Rgk; i.e., the surface Dirac cone furnishes the same 031070-21 KHALAF, PO, VISHWANATH, and WATANABE PHYS. REV. X 8, 031070 (2018) symmetry representation as the valence bands, as one understood from the existing discussion on the familiar would expect. topological band invariants [29,37]. The ambiguity can be Next, we study the general mass-term structure of summarized as follows: First, for every Z2 factor (i.e., weak 1 ðwÞ ðwÞ stacked strong TIs. Consider the surface theory for n> TI index) in S , X contains a corresponding Z2 if and 1 SG BS copies of the DSTI, with kn > symmetry-allowed only if the SG is centrosymmetric; second, for every Z independent mass terms. The mass terms can be repre- factor (i.e., weak mirror Chern number) in SðwÞ, XðwÞ sented as M ¼ m Γ , i ¼ 1; …;k , with Γ chosen to SG BS i;r i;r i n i contains a Z factor if and only if the SG contains an n- satisfy Γ2 ¼ 1.By“independent,” we mean that the differ- n i fold rotation about an axis normal to the associated mirror ent mass terms anticommute with each other, thus satisfy- plane. These statements completely map out the surface- ing fΓ ; Γ g¼2δ . In addition, we assume all mass terms i j ij SðwÞ anticommute with the surface Hamiltonian so that the state identification ambiguity for SG . ðsÞ SðsÞ Hamiltonian is gapped when any of the mass terms are It remains to study the case of XBS vs SG. This case … is far more complicated, as we detail below. nonzero. We define the mass vector mr ¼ðm1;r; ;mkn;rÞ whose length gives the values of the gap at point r. The transformation properties of the mass vector can be 1. Justification for the simplifying assumption deduced from the requirement of the invariance of P In this subsection, we first justify the simplifying iMi;r under symmetry, X X assumption of establishing the bulk-boundary correspon- dence for the XðsÞ phases; namely, we consider strong TIs g · Mi;r ¼ Mi;g · r; BS i i with only band inversions at Γ involving (signed) repre- X X − θ ˆ σ 2 η i gng· = Γ † Γ sentations of the form ug ¼ gvg ¼ e . For brev- mi;rUg iUg ¼ mi;g · r i; ity, in the following, we refer to such a representation as iX Xi being “quasistandard,” although this terminology is by no Og m Γ m Γ ; ji i;r j ¼ i;g · r i ðB8Þ means standard outside of our present context. ij i A priori, our simplifying assumption is nontrivial as the ≡ g g which implies that g · mr mg · r ¼ O mr, with Oij an symmetry representations of the nonrelativistic electrons in a orthogonal matrix given by crystal are not “quasistandard” in general, and the band structures are also subjected to global connectivity con- 1 g † straints arising from, say, nonsymmorphic symmetries. O ¼ trðΓiUgΓjUgÞ: ðB9Þ ij 4n Establishing the validity of the assumption requires a more Thus, given a unitary representation acting on the technical discussion, which we undertake below. First, we Hamiltonian for the stacked DSTI, we get an induced restate our main conclusion from this analysis: For any space orthogonal representation acting on the (k − 1) sphere. group in class AII, one can choose a basis such that the strong factor in XBS is generated by a strong TI that satisfies the simplifying assumption behind the Dirac analysis. APPENDIX C: BULK-SURFACE We now proceed to show this in the language of CORRESPONDENCE FOR ALL SPACE GROUPS Ref. [24], which we briefly review below. Insofar as In this appendix, we discuss some subtleties regarding symmetry representations, but not the detailed energetics, bulk-surface correspondence of sTCI phases. In particular, are concerned, any band structures isolated from above and we focus on the relation between the sTCI classification below by band gaps can be represented by a D-dimensional and the information contained in the symmetry indicators. “vector” (more accurately, a collection of D integers) As noted in the main text, since the symmetry indicators formed by the irrep multiplicities at different high-sym- only utilize data encoded in the symmetry representations, metry momenta. The symmetry-respecting “vectors” nat- a given nontrivial indicator is generally compatible with urally form an Abelian group fBSg, with group addition more than one sTCI phases, i.e., knowing the indicator corresponding physically to the stacking of the underlying alone does not uniquely identify the surface signature of the systems. We then denote the subgroup of fBSg, which can nontrivial bulk. We refer to this as the “surface-state arise from atomic insulators by fAIg. The symmetry-based identification ambiguity.” In the following, we describe indicator of band topology is defined as the mismatch how such ambiguity can be systematically mapped out. between fBSg and fAIg, which is mathematically captured ≡ Let us first consider the simpler problem concerning by the quotient XBS fBSg=fAIg. “weak” phases defined by having at least one lattice Quasistandard representations at Γ play a special role in translation taking a nontrivial signature, i.e., weak TIs our Dirac analysis. The multiplicities of these irreps are and their mirror enrichments. As discussed in the main text, encoded in certain components of the “vectors” in the ðwÞ SðwÞ Π the relation between XBS and SG , which concerns sTCIs description above. Let be a projection that projects away with either a weak TI or weak mirror Chern index, is readily these components, and suppose b1; b2 ∈ fBSg, b1 ≠ b2, 031070-22 SYMMETRY INDICATORS AND ANOMALOUS SURFACE … PHYS. REV. X 8, 031070 (2018) such that Πðb1 − b2Þ¼0. Physically, this case means that a1 ¼ð1; 0; 1; 0; 1; 0; 1; 0Þ; while b1 and b2 are distinguishable in terms of symmetry a2 ¼ð0; 1; 0; 1; 0; 1; 0; 1Þ; representations, their only distinction arises solely from irrep exchanges involving only the quasistandard repre- a3 ¼ð1; 0; 0; 1; 1; 0; 0; 1Þ; sentations at Γ. Furthermore, suppose b1 is nontrivial in a4 ¼ð1; 0; 1; 0; 0; 1; 0; 1Þ; XBS but b2 is trivial; then, one can choose a representative ∈ a5 ¼ð1; 0; 0; 1; 0; 1; 1; 0Þ; ðC2Þ for the nontrivial class ½b1 XBS which satisfies the simplifying assumption behind the Dirac analysis, insofar as symmetry representations are concerned. which arises by performing Fourier transform on five out of From the discussion above, we see that the kernel of the the eight AIs that we constructed earlier. Note that we use ’ map Π, ker Π ≡ b ∈ BS ∶Π b 0 , plays a key role only five of them, as the remaining ones give n s that are BS f f g ¼ g “ ” in the analysis. We can similarly define ker Π by linearly dependent on the ones above; i.e., the dimension AI 5 replacing fBSg → fAIg in the definition. The mismatch of fAIg is dAI ¼ , as was computed in Ref. [24]. Π Π The next task is to compute fBSg. More systematically, between kerBS and kerAI then describes band inver- Γ this computation can be done via the use of the Smith sions of quasistandard irreps at , which lead to a XBS- nontrivial band structure. Again, this mismatch can be normal form; here, we simply note that the combination captured by a quotient, which has to be a subgroup of X . 1 BS 2 0 1 1 1 1 1 1 Through an explicit computation, we find that 2 ða1 þ a3 þ a4 þ a5Þ¼ð ; ; ; ; ; ; ; ÞðC3Þ ker Π corresponds to a band insulator, which cannot be atomic. XðsÞ ¼ BS ðC1Þ BS ker Π We further assert that this is the only nontrivial combination AI one needs to consider. As such, we infer a possible basis for holds for all space groups in class AII. This result provides fBSg to be a formal justification of our simplifying assumption. b ¼ a for i ¼ 1; 2; 3; 4; More concretely, the validity of Eq. (C1) for any given i i 1 SG can be established in three steps: First, we compute an b5 ¼ ða1 þ a3 þ a4 þ a5Þ; ðC4Þ explicit basis for fAIg and fBSg following the recipe 2 detailed in Ref. [24]; second, we identify the quasistandard ðsÞ and X ¼ Z2. This concludes the first step of the analysis. representations at Γ and construct the projection Π; third, BS Next, we identify the quasistandard representations we compute the null spaces of Π BS and Π AI and − 2π 4 ˆ σ 2 − πσ 4 ðf gÞ ðf gÞ Γ ið = Þz· = i z= at .ForS4,wehavevS4 ¼ e ¼ e , which evaluate the quotient to check if it agrees with the strong z α 1 ðsÞ corresponds simply to the ¼ representation. 3π 4 ∓ π 4 factor XBS. Furthermore, as e i = ¼ −e i = , we conclude that all Let us sketch out the described procedure for an explicit the representations at Γ are quasistandard. The projection Π 4¯ example. Consider SG 81 (P ), whose point group S4 is can therefore be constructed explicitly as generated by the rotoinversion S4z. There are four maximal- symmetry Wyckoff positions with S4 as the site-symmetry Π∶ n ↦ ðn 1;n 3;n 1;n 3;n 1;n 3Þ: ðC5Þ group. As discussed in Sec. II C of the main text, the M M Z Z A A representations of S4 furnished by spinful electrons can be It remains to compute the null spaces of ΠðfBSgÞ and described by the eigenvalue of S4z, which takes the form ΠðfAIgÞ. One finds eiαπ=4 with α ¼ 1; 3. TRS pairs the representations with α 1, and similarly for 3. On each of the four maximal- ¼ b1 þ b2 − b5 ¼ð−1; 1; 0; 0; 0; 0; 0; 0Þ; symmetry Wyckoff positions, we can construct an AI by having two electrons occupying the two orbitals labeled by a1 þ 2a2 − a3 − a4 − a5 ¼ð−2; 2; 0; 0; 0; 0; 0; 0Þ; ðC6Þ α ¼ 1. Similarly, we can construct another AI by putting Π Π the two electrons into the α ¼ 3 orbitals or by localizing which, respectively, generate kerBS and kerAI . We thus Π Π Z them to the other Wyckoff positions. Upon stacking, these conclude kerBS = kerAI ¼ 2; i.e., Eq. (C1) is verified. AIs generate all the possible AIs in our symmetry setting. While we have provided a detailed example using SG 81, In momentum space, the symmetry representations it is conceptually more revealing to understand why are again given by the multiplicities of the α ¼ 1 Eq. (C1) should hold, in general. We first translate and 3 representations at the four S4-symmetric Eq. (C1) into a more physical language: Every nontrivial ðsÞ momenta: Γ ≡ ð0; 0; 0Þ,M≡ ðπ; π; 0Þ,Z≡ ð0; 0; πÞ, and class in XBS can be represented by a BS that differs from an A ≡ ðπ; π; πÞ. The representation data are encoded in the AI only by the exchange of some quasistandard represen- ≡ 1 3 1 3 1 3 1 3 tations at Γ. To establish Eq. (C1) on physical grounds, it eight integers n ðnΓ ;nΓ ;nM ;nM ;nZ ;nZ ;nA ;nA Þ, where the subscript indicates the value of α. In this notation, suffices to show that this defining property holds for the ðsÞ ∈ one can check that a legitimate choice of basis for fAIg is generator of XBS; once a representative b0 fBSg with the 031070-23 KHALAF, PO, VISHWANATH, and WATANABE PHYS. REV. X 8, 031070 (2018) desired band-inversion interpretation is found for the representations at Γ. Such an AI arises from the generic generator, all the other classes can be represented by Wyckoff position, as was established in a corollary in copies of b0, which will automatically enjoy the desired Ref. [24]. Second, we can always perform a band inversion property. on a0 involving only the quasistandard representations at Γ To this end, first note that, by definition, the generator of and arrive at a strong TI. The only concern here is that one ðsÞ might violate some compatibility relations in performing XBS can be chosen to be a strong TI. Given any SG with a symmetry-diagnosable strong TI (i.e., through the indices the desired band inversion and end up with a semimetal. As defined in Sec. II), one can imagine building a strong TI b0 shown in Sec. II, however, the key indices for the diagnosis by first starting with an AI, and one can then exchange of a strong TI use either inversion or S4 eigenvalues, neither a pair of quasistandard representations at Γ; e.g., in a of which is subjected to any compatibility relations as they centrosymmetric SG, one can start with an AI with a only leave isolated points (but not lines) invariant in momentum space. quasistandard representation at Γ, where uI ¼ 1, and then upon a band inversion with uI ¼ −1, one arrives at a strong TI as detected by the Fu-Kane criterion. 2. Surface-state ambiguity It then remains to show that such a strong TI b0 can be Having established the formal framework, we now constructed for any SG. This can be achieved in two steps: ðsÞ derive the surface-state ambiguity in the XBS diagnosis. First, we note that one can always find an AI a0 whose The main result here is reported in Table IV, where we representation content contains each of the quasistandard chart out such ambiguities for all 230 SGs in class AII. TABLE IV. sTCI classification and symmetry indicators for all space groups. The space groups are grouped by their associated point SðwÞ SðsÞ ðwÞ ðsÞ groups. Here, SG ( SG) denotes the sTCI classification for phases with (without) a weak index. K (K ) denotes the subgroup of SðwÞ SðsÞ SG ( SG) corresponding to a trivial symmetry indicator; i.e., it encodes the surface-state identification ambiguity. In each column of SSG=K [i.e., with superscript (w)or(s)], we write out SSG and K explicitly for most SGs. The exceptions are SGs with a trivial K, for which we list only SSG (say, SG 2); in addition, when SSG is trivial, we denote it by a dash. For simplicity, we denote K by h…i when K ¼ SSG, i.e., when the symmetry indicators cannot detect any phase in SSG. Point group (generators) Space group SðwÞ ðwÞ SðsÞ ðsÞ SG =K SG=K C1 (1) 1 Z2 × Z2 × Z2=h…i Ci (I)2 Z2 × Z2 × Z2 Z2 3 Z2 × Z2 × Z2=h…i Z2=h…i C2 (C2z) 4 Z2 × Z2=h…i Z2=h…i 5 Z2 × Z2=h…i Z2=h…i 6 Z × Z2 × Z2=h…i Z=h…i 7 Z2 × Z2=h…i Z2=h…i Cs (Mz) 8 Z2 × Z2=h…i Z=h…i 9 Z2=h…i Z2=h…i 10 Z × Z2 × Z2=hð2; 0; 0Þi Z × Z2=hð1; 1Þi 11 Z2 × Z2 Z × Z2=hð1; 1Þi 12 Z2 × Z2 Z × Z2=hð1; 1Þi C2h (Mz, C2z) 13 Z2 × Z2 Z2 × Z2=hð1; 1Þi 14 Z2 Z2 × Z2=hð1; 1Þi 15 Z2 Z2 × Z2=hð1; 1Þi 16 Z2 × Z2 × Z2=h…i Z2 × Z2=h…i 17 Z2 × Z2=h…i Z2 × Z2=h…i 18 Z2=h…i Z2 × Z2=h…i 19 Z2 × Z2=h…i D2 (C2z;C2x) 20 Z2=h…i Z2 × Z2=h…i 21 Z2 × Z2=h…i Z2 × Z2=h…i 22 Z2 × Z2=h…i Z2 × Z2=h…i 23 Z2=h…i Z2 × Z2=h…i 24 Z2=h…i Z2 × Z2=h…i (Table continued) 031070-24 SYMMETRY INDICATORS AND ANOMALOUS SURFACE … PHYS. REV. X 8, 031070 (2018) TABLE IV. (Continued) Point group (generators) Space group SðwÞ ðwÞ SðsÞ ðsÞ SG =K SG=K 25 Z × Z × Z2=h…i Z × Z=h…i 26 Z × Z2=h…i Z × Z2=h…i 27 Z2 × Z2=h…i Z2 × Z2=h…i 28 Z2 × Z2=h…i Z × Z2=h…i 29 Z2=h…i Z2 × Z2=h…i 30 Z2=h…i Z2 × Z2=h…i 31 Z2=h…i Z × Z2=h…i 32 Z2=h…i Z2 × Z2=h…i 33 Z2 × Z2=h…i 34 Z2=h…i Z2 × Z2=h…i 35 Z2 × Z2=h…i Z × Z=h…i C2v (Mx, My) 36 Z2=h…i Z × Z2=h…i 37 Z2=h…i Z2 × Z2=h…i 38 Z × Z2=h…i Z × Z=h…i 39 Z2 × Z2=h…i Z × Z2=h…i 40 Z2=h…i Z × Z2=h…i 41 Z2=h…i Z2 × Z2=h…i 42 Z2 × Z2=h…i Z × Z=h…i 43 Z2 × Z2=h…i 44 Z2=h…i Z × Z=h…i 45 Z2=h…i Z2 × Z2=h…i 46 Z2=h…i Z × Z2=h…i 47 Z × Z × Z=hð2; 0; 0Þ; ð0; 2; 0Þ; ð0; 0; 2Þi Z × Z × Z=hð1; 1; 0Þ; ð1; 0; 1Þ; ð−1; 0; 1Þi 48 Z2 Z2 × Z2 × Z2=hð1; 1; 0Þ; ð1; 0; 1Þi 49 Z2 × Z2 Z × Z2 × Z2=hð1; 1; 0Þ; ð0; 1; 1Þi 50 Z2 Z2 × Z2 × Z2=hð1; 1; 0Þ; ð1; 0; 1Þi 51 Z × Z2=hð2; 0Þi Z × Z × Z2=hð1; 0; 1Þ; ð0; 1; 1Þi 52 Z2 × Z2 × Z2=hð1; 1; 0Þ; ð1; 0; 1Þi 53 Z2 Z × Z2 × Z2=hð1; 1; 0Þ; ð0; 1; 1Þi 54 Z2 Z2 × Z2 × Z2=hð1; 1; 0Þ; ð1; 0; 1Þi 55 Z=hð2Þi Z × Z2 × Z2=hð1; 1; 0Þ; ð0; 1; 1Þi 56 Z2 × Z2 × Z2=hð1; 1; 0Þ; ð1; 0; 1Þi 57 Z2 Z × Z2 × Z2=hð1; 1; 0Þ; ð0; 1; 1Þi 58 Z × Z2 × Z2=hð1; 1; 0Þ; ð0; 1; 1Þi 59 Z2 Z × Z × Z2=hð1; 0; 1Þ; ð0; 1; 1Þi 60 Z2 × Z2 × Z2=hð1; 1; 0Þ; ð1; 0; 1Þi D2h (Mx, My, Mz) 61 Z2 × Z2 × Z2=hð1; 1; 0Þ; ð1; 0; 1Þi 62 Z × Z2 × Z2=hð1; 1; 0Þ; ð0; 1; 1Þi 63 Z2 Z × Z × Z2=hð1; 0; 1Þ; ð0; 1; 1Þi 64 Z2 Z × Z2 × Z2=hð1; 1; 0Þ; ð0; 1; 1Þi 65 Z × Z2=hð2; 0Þi Z × Z × Z=hð1; 1; 0Þ; ð1; 0; 1Þ; ð−1; 0; 1Þi 66 Z2 Z × Z2 × Z2=hð1; 1; 0Þ; ð0; 1; 1Þi 67 Z2 × Z2 Z × Z × Z2=hð1; 0; 1Þ; ð0; 1; 1Þi 68 Z2 Z2 × Z2 × Z2=hð1; 1; 0Þ; ð1; 0; 1Þi 69 Z2 × Z2 Z × Z × Z=hð1; 1; 0Þ; ð1; 0; 1Þ; ð−1; 0; 1Þi 70 Z2 × Z2 × Z2=hð1; 1; 0Þ; ð1; 0; 1Þi 71 Z2 Z × Z × Z=hð1; 1; 0Þ; ð1; 0; 1Þ; ð−1; 0; 1Þi 72 Z2 Z × Z2 × Z2=hð1; 1; 0Þ; ð0; 1; 1Þi 73 Z2 Z2 × Z2 × Z2=hð1; 1; 0Þ; ð1; 0; 1Þi 74 Z2 Z × Z × Z2=hð1; 0; 1Þ; ð0; 1; 1Þi (Table continued) 031070-25 KHALAF, PO, VISHWANATH, and WATANABE PHYS. REV. X 8, 031070 (2018) TABLE IV. (Continued) Point group (generators) Space group SðwÞ ðwÞ SðsÞ ðsÞ SG =K SG=K 75 Z2 × Z2=h…i Z2=h…i 76 Z2=h…i Z2=h…i 77 Z2 × Z2=h…i Z2=h…i C4 (C4z) 78 Z2=h…i Z2=h…i 79 Z2=h…i Z2=h…i 80 Z2=h…i Z2=h…i 81 Z2 × Z2=h…i Z2=h…i S4 (S4z) 82 Z2=h…i Z2=h…i 83 Z × Z2=hð4; 0Þi Z × Z2=hð2; 1Þi 84 Z2 Z × Z2=hð0; 1Þ; ð2; 0Þi 85 Z2 Z2 × Z2=hð0; 1Þi C4h (Mz, S4z) 86 Z2 Z2 × Z2=hð0; 1Þi 87 Z2 Z × Z2=hð2; 1Þi 88 Z2 × Z2=hð0; 1Þi 89 Z2 × Z2=h…i Z2 × Z2=h…i 90 Z2=h…i Z2 × Z2=h…i 91 Z2=h…i Z2 × Z2=h…i 92 Z2 × Z2=h…i 93 Z2 × Z2=h…i Z2 × Z2=h…i D4 (C4z;C2x) 94 Z2=h…i Z2 × Z2=h…i 95 Z2=h…i Z2 × Z2=h…i 96 Z2 × Z2=h…i 97 Z2=h…i Z2 × Z2=h…i 98 Z2=h…i Z2 × Z2=h…i 99 Z × Z2=h…i Z × Z=h…i 100 Z2=h…i Z × Z2=h…i 101 Z2=h…i Z × Z2=h…i 102 Z2=h…i Z × Z2=h…i 103 Z2=h…i Z2 × Z2=h…i 104 Z2 × Z2=h…i C4v (Mx, Mxy) 105 Z=h…i Z × Z2=h…i 106 Z2 × Z2=h…i 107 Z2=h…i Z × Z=h…i 108 Z2=h…i Z × Z2=h…i 109 Z × Z2=h…i 110 Z2 × Z2=h…i 111 Z2 × Z2=h…i Z × Z2=h…i 112 Z2=h…i Z2 × Z2=h…i 113 Z2=h…i Z × Z2=h…i 114 Z2 × Z2=h…i 115 Z × Z2=h…i Z × Z2=h…i 116 Z2=h…i Z2 × Z2=h…i D2d (Mx, S4z) 117 Z2=h…i Z2 × Z2=h…i 118 Z2=h…i Z2 × Z2=h…i 119 Z2=h…i Z × Z2=h…i 120 Z2=h…i Z2 × Z2=h…i 121 Z2=h…i Z × Z2=h…i 122 Z2 × Z2=h…i (Table continued) 031070-26 SYMMETRY INDICATORS AND ANOMALOUS SURFACE … PHYS. REV. X 8, 031070 (2018) TABLE IV. (Continued) Point group (generators) Space group SðwÞ ðwÞ SðsÞ ðsÞ SG =K SG=K 123 Z × Z=hð4; 0Þ; ð0; 2Þi Z × Z × Z=hð1; 1; 0Þ; ð1; −1; 0Þ; ð0; 1; 2Þi 124 Z2 Z × Z2 × Z2=hð0; 1; 1Þ; ð2; 1; 0Þi 125 Z2 Z × Z2 × Z2=hð1; 0; 0Þ; ð0; 1; 0Þi 126 Z2 × Z2 × Z2=hð1; 0; 0Þ; ð0; 1; 0Þi 127 Z=hð4Þi Z × Z × Z2=hð1; 0; 1Þ; ð0; 2; 1Þi 128 Z × Z2 × Z2=hð0; 1; 1Þ; ð2; 1; 0Þi 129 Z2 Z × Z × Z2=hð1; 0; 0Þ; ð0; 1; 0Þi 130 Z2 × Z2 × Z2=hð1; 0; 0Þ; ð0; 1; 0Þi 131 Z=hð2Þi Z × Z × Z2=hð1; 0; 0Þ; ð0; 0; 1Þ; ð0; 2; 0Þi 132 Z2 Z × Z × Z2=hð1; 0; 0Þ; ð0; 0; 1Þ; ð0; 2; 0Þi D4h (Mx, Mz, Mxy) 133 Z2 × Z2 × Z2=hð1; 0; 0Þ; ð0; 1; 0Þi 134 Z2 Z × Z2 × Z2=hð1; 0; 0Þ; ð0; 1; 0Þi 135 Z × Z2 × Z2=hð0; 1; 0Þ; ð0; 0; 1Þ; ð2; 0; 0Þi 136 Z × Z × Z2=hð1; 0; 0Þ; ð0; 0; 1Þ; ð0; 2; 0Þi 137 Z × Z2 × Z2=hð1; 0; 0Þ; ð0; 1; 0Þi 138 Z × Z2 × Z2=hð1; 0; 0Þ; ð0; 1; 0Þi 139 Z2 Z × Z × Z=hð1; 1; 0Þ; ð1; −1; 0Þ; ð0; 1; 2Þi 140 Z2 Z × Z × Z2=hð1; 0; 1Þ; ð0; 2; 1Þi 141 Z × Z2 × Z2=hð1; 0; 0Þ; ð0; 1; 0Þi 142 Z2 × Z2 × Z2=hð1; 0; 0Þ; ð0; 1; 0Þi 143 Z2=h…i – 144 Z2=h…i – C3 (C3z) 145 Z2=h…i – 146 Z2=h…i – 147 Z2 Z2 S6 (S3z) 148 Z2 Z2 149 Z2=h…i Z2=h…i 150 Z2=h…i Z2=h…i 151 Z2=h…i Z2=h…i D3 (C2x;C3z) 152 Z2=h…i Z2=h…i 153 Z2=h…i Z2=h…i 154 Z2=h…i Z2=h…i 155 Z2=h…i Z2=h…i 156 Z2=h…i Z=h…i 157 Z2=h…i Z=h…i 158 Z2=h…i C3v (Mx, C3z) 159 Z2=h…i 160 Z2=h…i Z=h…i 161 Z2=h…i 162 Z2 Z × Z2=hð1; 1Þi 163 – Z2 × Z2=hð1; 1Þi 164 Z2 Z × Z2=hð1; 1Þi D3d (Mx, C2x;C3z) 165 Z2 × Z2=hð1; 1Þi 166 Z2 Z × Z2=hð1; 1Þi 167 Z2 × Z2=hð1; 1Þi 168 Z2=h…i Z2=h…i 169 Z2=h…i 170 Z2=h…i C6 (C6z) 171 Z2=h…i Z2=h…i 172 Z2=h…i Z2=h…i 173 Z2=h…i (Table continued) 031070-27 KHALAF, PO, VISHWANATH, and WATANABE PHYS. REV. X 8, 031070 (2018) TABLE IV. (Continued) Point group (generators) Space group SðwÞ ðwÞ SðsÞ ðsÞ SG =K SG=K C3h (Mz, C3z) 174 Z=hð3Þi Z=hð3Þi 175 Z=hð6Þi Z × Z2=hð3; 1Þi C6h (Mz, C6z) 176 Z × Z2=hð3; 1Þi 177 Z2=h…i Z2 × Z2=h…i 178 Z2 × Z2=h…i 179 Z2 × Z2=h…i D6 (C6z;C2x) 180 Z2=h…i Z2 × Z2=h…i 181 Z2=h…i Z2 × Z2=h…i 182 Z2 × Z2=h…i 183 Z2=h…i Z × Z=h…i Z Z … pffiffi 184 2 × 2=h i C6v (Mx, M 3 2) xþy= 185 Z × Z2=h…i 186 Z × Z2=h…i 187 Z=hð3Þi Z × Z=hð1; 0Þ; ð0; 3Þi 188 Z × Z2=hð0; 1Þ; ð3; 0Þi D3 (M , M , C3 ) h z x z 189 Z=hð3Þi Z × Z=hð1; 0Þ; ð0; 3Þi 190 Z × Z2=hð0; 1Þ; ð3; 0Þi 191 Z=hð6Þi Z × Z × Z=hð1; 1; 0Þ; ð1; −1; 0Þ; ð0; 1; 3Þi Z Z Z 0 1 1 3 1 0 pffiffi 192 × 2 × 2=hð ; ; Þ; ð ; ; Þi D6h (Mz, Mx, M 3 2) xþy= 193 Z × Z × Z2=hð1; 0; 1Þ; ð0; 3; 1Þi 194 Z × Z × Z2=hð1; 0; 1Þ; ð0; 3; 1Þi 195 Z2=h…i 196 Z … T (C3;nˆ 1 ;C3;nˆ 2 ) 197 2=h i 198 199 Z2=h…i 200 Z=hð2Þi Z=hð2Þi 201 Z2 Z2 202 Z=hð2Þi Z Th (Mz, C3;nˆ 1 ) 203 2 204 Z2 Z=hð2Þi 205 Z2 206 Z2 Z2 207 Z2=h…i Z2=h…i 208 Z2=h…i Z2=h…i 209 Z2=h…i 210 Z2=h…i O (C4;z;C3;nˆ 1 ) 211 Z2=h…i Z2=h…i 212 Z2=h…i 213 Z2=h…i 214 Z2=h…i Z2=h…i 215 Z2=h…i Z=h…i 216 Z=h…i 217 Z2=h…i Z=h…i Td (Mnˆ 1 ;C3;nˆ 1 ) 218 Z2=h…i 219 Z2=h…i 220 Z2=h…i (Table continued) 031070-28 SYMMETRY INDICATORS AND ANOMALOUS SURFACE … PHYS. REV. X 8, 031070 (2018) TABLE IV. (Continued) Point group (generators) Space group SðwÞ ðwÞ SðsÞ ðsÞ SG =K SG=K 221 Z=hð4Þi Z × Z=hð2; 1Þ; ð0; 2Þi 222 Z2 × Z2=hð1; 0Þi 223 Z × Z2=hð0; 1Þ; ð2; 0Þi 224 Z2 Z × Z2=hð1; 0Þi 225 Z × Z=hð2; 1Þ; ð0; 2Þi Oh (Mx, Mxy;C3;nˆ 1 ) 226 Z × Z2=hð2; 1Þi 227 Z × Z2=hð1; 0Þi 228 Z2 × Z2=hð1; 0Þi 229 Z2 Z × Z=hð2; 1Þ; ð0; 2Þi 230 Z2 × Z2=hð1; 0Þi η η 4 The remainder of the section is devoted to an explanation that can detect I and C4;6 . Such is the case of P =m (SG on how these results are obtained. As shown in Sec. III in 83) discussed in the main text. This observation can also be the main text, the surface states of DSTIs can be viewed as a manifestation of the detection of mirror Chern ν ∈ Z classified by two types of indices: M for every numbers through C4;6-symmetry eigenvalues [37].(ForC2 independent mirror (not glide) plane M, and νl ∈ Z2 for together with I, we can only detect the parity of the mirror every other independent generator l of the point group Chern number, but the mirror Chern parity is always even satisfying the conditions outlined in the main text. in a DSTI.) Suppose we have a band insulator with the quasistandard However, there is a second source of surface-state ðiÞ irreps χ , i ¼ 1; …; 2Nχ, and not necessarily distinct, ambiguity that can be present. This originates from non- exchanged at Γ, where we assume each exchange can be trivial stacks of strong TIs, which, when restricted to modeled using the Dirac approach. Let the symmetry g symmetry irreps alone, look indistinguishable from an ðiÞ ðiÞ ðiÞ be represented by ug ¼ ηg vg in χ . The surface indices atomic insulator. In other words, if we are just given a are then given by band insulator that differs from a reference atomic insulator only by the exchange of some quasistandard irreps at Γ,itis X2Nχ Y2Nχ not automatically justified that the exchange arose from a 1 i i ν ηð Þ; −1 νl l Nχ ηð Þ series of consecutive exchanges, each producing a strong TI M ¼ 2 M ð Þ ¼ðdet Þ l : ðC7Þ i¼1 i¼1 amenable to a Dirac analysis. Such is the case when the irrep exchange pattern can be understood as the difference ðiÞ This can be seen by first grouping the fχ ∶i ¼ 1; …; 2Nχg of two atomic insulators, which is captured precisely by Π Π pairwise, applying the bulk-surface correspondence in kerAI in our framework. Therefore, if an entry in kerAI Sec. IV, and then adding the resulting Nχ sets of surface gives rise to a nontrivial surface index according to indices. The resulting indices, Eq. (C7), are insensitive to the Eq. (C7), it implies, from representation alone, that we initial, arbitrary choice on the pairwise grouping and are cannot tell if we have an atomic insulator or a DSTI with therefore well defined. nontrivial surface states. The evaluation of the surface ðsÞ Π To make connection to XBS, we now restrict ourselves to indices on kerAI then gives a subgroup of the surface information available from symmetry representations classification corresponding to possible surfaces of alone. As described in the main text, this introduces 0 ∈ ðsÞ XBS; i.e., this subgroup captures the surface-state surface-state ambiguity as the symmetry representations identification ambiguity. contain less data than the Dirac description. The first source η 1 While the two mentioned sources of ambiguity have of ambiguity is when the choices g ¼ lead to the same rather different physical origins, in practice they may be irrep. For instance, consider the mirror symmetry about a ˆ intertwined with each other. As an illustrative example, plane with surface normal nM, which can be represented by consider the symmorphic SG P2=m (SG 10), which is ˆ σ the traceless unitary matrix vM ¼ inM · . Usually, this generated by lattice translations, inversion I, and a twofold implies that η ¼ 1 is unconstrained by the specification M rotation C2 . Note that this implies a mirror symmetry ðsÞ z of the symmetry representation, and hence, XBS does not Mz ≡ IC2z. As shown in Table II, the surface states of the ν constrain M, which leads to a surface-state ambiguity. DSTIs in this setting are classified by Z × Z2, where Yet, if the point group contains C4 or C6 rotation together the first factor corresponds to equatorial states pinned to with inversion I, then we may have relations like ηM ¼ the mirror plane and protected by a Z-valued mirror Chern 2 η η η number, and the Z2 entry is generated by a hinge mode, I C4 and hence, M becomes tied to the representations 031070-29 KHALAF, PO, VISHWANATH, and WATANABE PHYS. REV. X 8, 031070 (2018) protected by C2z, which passes through the north and south as discussed in Sec. III C 2 of the main text. Nonetheless, poles (and hence gives rise to 2D gapless surface states on for both cases, the symmetry indicators are unable to detect appropriate faces [23]). The quasistandard representations any of the weak TI indices, and hence, we have ðwÞ for both the C2z and Mz are traceless, and therefore, from K ¼h…i, denoting the full group, for both of them. ðsÞ As a more nontrivial example, consider SG 123, which the discussion above, XBS should be blind to the existence of the mentioned surface states. Yet, the inversion sym- contains multiple mirrors extending the weak TI phases to metry I is not traceless, and using its symmetry eigenval- weak mirror Chern insulators. Besides, the C4 rotation ues, one should be able to detect the total parity of the 1D about (say) the z axis relates the weak index along the x and SðwÞ Z Z Z modes on the surface. This small dilemma is resolved by y directions, which leads to SG ¼ × . Since the - η η η noticing that I ¼ Mz C2z , and therefore, although we valued mirror Chern numbers can, at best, be detected η η mod n using an n-fold rotation [37], the surface-state cannot detect the values of Mz and C2z individually, their product becomes detectable. At a more technical level, in identification ambiguity will encode this modulo structure. deriving the surface-state identification ambiguity of an This result is seen in KðwÞ ¼hð4; 0Þ; ð0; 2Þi in Table IV, XðsÞ-nontrivial phase, one first has to study the “preimage” which denotes the subgroup generated by the elements BS (4,0) and (0,2) in Z × Z. Physically, this means that the first of the consistent assignments on the ηg of the traceless elements (in a quasistandard representation), which implies weak mirror Chern index (for phases deformable to stacks z 4 C4 that each element in ker Π is now enhanced correspond- of 2D phases along ) can be detected mod using AI about z, whereas the second weak mirror Chern index can ingly. One then evaluates the surface indices on this be detected mod 2 using (say) a C2 rotation about the enhanced version of ker Π, which would reveal the AI x axis. subgroup of surface signatures that are consistent with Next, we move on to the strong part of the sTCI 0 ∈ ðsÞ the symmetry representations of XBS. classification. Recall that the strong part is tightly tied to the generators of the point group of the SG. The chosen 3. Table for the sTCI classification and generators of the point groups are listed again in Table IV. surface-state ambiguity Note that, for a couple of SGs, the generating set differs slightly from that in Table II. The factors in SðsÞ are listed in The full results of S and its surface-state identification SG SG correspondence with the generators listed in the leftmost ambiguity in X are tabulated in Table IV. Let us illustrate BS column. An exception is made for threefold rotations, how to read it through a few examples. First, consider the SðsÞ weak phases in SG 1 and 2. Both of the SGs have three which do not lead to any factor in SG. They are always independent weak TIs corresponding to the three indepen- listed at the end of the generating set, and when they are SðwÞ Z present, the number of factors in SðsÞ will be less than the dent lattice translations, and hence, we expect SG ¼ 2 × SG Z2 × Z2 for both of them. However, while the weak TI number of independent generators. For instance, the SGs indices are detected through the Fu-Kane parity criterion in with point group D3, generated by C2x and C3z, all have SðsÞ Z SG 2, they are undetectable using symmetry eigenvalues in SG ¼ 2 with the only factor corresponding to C2x. SG 1. This difference implies that there is no surface-state However, depending on the SG, a mirror in the point SðwÞ group may correspond to a glide in the SG, with the former identification ambiguity for SG in SG 2, whereas the ðwÞ ðwÞ leading to a Z factor and the latter a Z2 factor. To see this entire S is consistent with the identity in X ¼ Z1 for SG BS SðsÞ SG 1. In Table IV, the latter case is denoted by a quotient more concretely, compare the SG of SGs 83 and 85. Their with KðwÞ ¼h…i, while the former is denoted by the point group C4h is generated by a mirror Mz and a fourfold absence of a quotient. rotation C4z. Yet, as shown in the table, in tabulating the While the previous discussion focused on the detect- sTCI classification, we instead choose S4z as a generator in ability of the weak phases using the symmetry indicators, lieu of C4z. First, focus on SG 83. While Mz leads to mirror Z we recall that spatial symmetries can also restrict the Chern insulators with a classification, S4z protects sTCIs Z possible set of weak phases, as discussed in the main text. with a nontrivial 2 index. As Mz and S4z are independent, For example, consider the SGs 3 and 4, which have the they lead to independent factors in the sTCI classification, SðsÞ Z Z same point group generated by C2z, but the former is giving SG ¼ × 2. Note that the indices associated with symmorphic and has C2z as a group element whereas the the other symmetries are fixed by the group structure; e.g., −1 latter is nonsymmorphic as the twofold rotation is extended as C4z ¼ MzðS4zÞ , the Z2 index associated with the phase into a 21 screw. Correspondingly, the two SGs have ð1; 0Þ ∈ Z × Z2 is 1 ∈ Z2, but the index associated with SðwÞ Z Z the phase (1, 1) is 0 ∈ Z2. In contrast—although sharing different weak sTCI classifications, with SG ¼ 2 × 2 × Z Z Z — SðsÞ Z Z 2 for SG 3 but 2 × 2 for SG 4. The reduction of one of the same point group SG 85 has SG ¼ 2 × 2 because the Z2 factors in SG 4 arises as the square of 21 is a lattice the would-be mirror in SG 85 has been extended into a translation, and this translation cannot take a nontrivial sg, glide, hence the replacement Z ↦ Z2. 031070-30 SYMMETRY INDICATORS AND ANOMALOUS SURFACE … PHYS. REV. X 8, 031070 (2018) Finally, we describe how to extract the surface-state Ref. [65]. In particular, a more explicit tabulation of the identification ambiguity from Table IV. Consider SG 83 index association can be found there. again. As explained in the main text, the symmetry indicators cannot differentiate between a phase with a mirror Chern index of 2 from one with the nontrivial C4z index, i.e., ð2; 0Þ ≃ ð0; 1Þ. [Here, (0, 1) is a phase with a [1] Topological Insulators, Contemporary Concepts of I I 2 nontrivial S4z index but a trivial index for ,as ¼ MzS4z, Condensed Matter Science, edited by M. Franz and L. Molenkamp (Elsevier, New York, 2013), Vol. 6. and hence, it has a nontrivial index for C4z ¼ IS4z.] This [2] A. Kitaev, Periodic Table for Topological Insulators and idea is represented in Table IV by KðsÞ ¼hð2; 1Þi. For SG Superconductors,inAIP Conference Proceedings Vol. 1134 ðsÞ 0 1 85, however, one finds K ¼hð ; Þi. This result implies (AIP, New York, 2009), pp. 22–30. that the index associated with S4z is undetectable from [3] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, symmetry representations and that a sTCI with a nontrivial Classification of Topological Insulators and Superconduc- 2 ∈ Z indicator of 4 in the strong part of XBS will always tors,inAIP Conference Proceedings Vol. 1134 (AIP, have a nontrivial glide index. New York, 2009), pp. 10–21. As discussed above, we specify KðsÞ by providing its [4] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, generators. In writing down the group elements, we have Topological Insulators and Superconductors: Tenfold Way and Dimensional Hierarchy 12 implicitly assumed a preferred basis. To clarify, we reiterate , New J. Phys. , 065010 (2010). our choice here: Each factor corresponds to a generator of [5] L. Fu, Topological Crystalline Insulators, Phys. Rev. Lett. the associated point group, as tabulated in the leftmost 106, 106802 (2011). column, with the exception of C3 rotations, as they do not [6] T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, and L. Fu, protect nontrivial phases. With this choice in mind, the data Topological Crystalline Insulators in the SnTe Material on KðsÞ can be readily converted into information about the Class, Nat. Commun. 3, 982 (2012). ðsÞ [7] Y. Tanaka, Z. Ren, T. Sato, K. Nakayama, S. Souma, T. possible physical surface states of a given element in XBS. Takahashi, K. Segawa, and Y. Ando, Experimental Reali- Let us illustrate this using a nontrivial example. Consider zation of a Topological Crystalline Insulator in SnTe, Nat. SG 194. As listed in Table IV, its point group is D6h, and Phys. 8, 800 (2012). we choose the generators to be the three independent [8] R.-J. Slager, A. Mesaros, V.Juricic, and J. Zaanen, The Space pffiffi Group Classification of Topological Band-Insulators, Nat. mirrors Mz, Mx, and M 3xþy=2. Furthermore, we have Phys. 9, 98 (2013). SðsÞ ðsÞ Z Z Z 1 0 1 0 3 1 SG=K ¼ × × 2=hð ; ; Þ; ð ; ; Þi. The first and [9] W. A. Benalcazar, J. C. Y. Teo, and T. L. Hughes, Z SðsÞ “ ” Classification of Two-Dimensional Topological Crystalline second factors in SG correspond to strong mirror Superconductors and Majorana Bound States at Disclina- Chern phases protected by Mz and Mx, respectively, ffiffi tions, Phys. Rev. B 89, 224503 (2014). whereas the last Z2 factor implies the mirror Mp 3xþy=2 [10] Y.-M. Lu and D.-H. Lee, Inversion Symmetry has been extended into a glide in this SG. The next step is to Protected Topological Insulators and Superconductors, SðsÞ ðsÞ identify the generator of the group SG=K , which can be arXiv:1403.5558. chosen to be one of the three basis vectors (1, 0, 0), (0, 1, 0), [11] D. Varjas, F. de Juan, and Y.-M. Lu, Bulk Invariants and Topological Response in Insulators and Superconductors or (0, 0, 1). Given that KðsÞ ¼hð1; 0; 1Þ; ð0; 3; 1Þi and that with Nonsymmorphic Symmetries, Phys. Rev. B 92, 195116 the generators should have maximal order, we can choose (2015). the generator to be (0, 1, 0), which has order 6. This case [12] K. Shiozaki, M. Sato, and K. Gomi, Z2 Topology in SðsÞ ðsÞ Z Nonsymmorphic Crystalline Insulators: Möbius Twist in gives SG=K ¼ 6, which, as explained in the main text, ðsÞ Surface States, Phys. Rev. B 91, 155120 (2015). corresponds to the even subgroup of X Z12. BS ¼ [13] X.-Y. Dong and C.-X. Liu, Classification of Topological The preceding analysis also informs the possible surface ðsÞ Crystalline Insulators Based on Representation Theory, states of any given entry of XBS. For instance, suppose a Phys. Rev. B 93, 045429 (2016). material has the symmetry indicator of 2 ∈ Z12. Then, we [14] Z. Wang, A. Alexandradinata, R. J. Cava, and B. A. Bernevig, know that it could be a sTCI classified by ð0; 1; 0Þ ∈ Hourglass Fermions, Nature (London) 532, 189 (2016). Z × Z × Z2; i.e., it has Mx-protected surface states but not [15] M. Ezawa, Hourglass Fermion Surface States in Stacked Topological Insulators with Nonsymmorphic Symmetry, those protected by Mz or the glide. 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