Symmetry-based Indicators of Band Topology in the 230 Space Groups
SSymmetry-based Indicators of Band Topology in the 230 Space Groups
Hoi Chun Po,
1, 2
Ashvin Vishwanath,
1, 2, ∗ and Haruki Watanabe Department of Physics, University of California, Berkeley, CA 94720, USA Department of Physics, Harvard University, Cambridge MA 02138 Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
The interplay between symmetry and topology leads to a rich variety of electronic topologicalphases, protecting states such as the topological insulators and Dirac semimetals. Previous results,like the Fu-Kane parity criterion for inversion-symmetric topological insulators, demonstrate thatsymmetry labels can sometimes unambiguously indicate underlying band topology. Here we developa systematic approach to expose all such symmetry-based indicators of band topology in all the 230space groups. This is achieved by first developing an efficient way to represent band structures interms of elementary basis states, and then isolating the topological ones by removing the subsetof atomic insulators, defined by the existence of localized symmetric Wannier functions. Asidefrom encompassing all earlier results on such indicators, including in particular the notion of filling-enforced quantum band insulators, our theory identifies symmetry settings with previously hiddenforms of band topology, and can be applied to the search for topological materials.
I. INTRODUCTION
The discovery of topological insulators (TIs) has rein-vigorated the well established theory of electronic bandstructures [1, 2]. Exploration along this new dimensionhas led to an ever-growing arsenal of topological materi-als, which include, for instance, topological (crystalline)insulators [3–5], quantum anomalous Hall insulators [6],and Weyl and Dirac semimetals [7]. Such materials pos-sess unprecedented physical properties, like quantized re-sponse and gapless surface states, that are robust againstall symmetry-preserving perturbations as long as a bandpicture remains valid [1, 2, 7].Soon after these new developments, it was realized thatsymmetries of energy bands, a thoroughly studied aspectof band theory, is also profoundly intertwined with topol-ogy. This is exemplified by the celebrated Fu-Kane cri-terion for inversion-symmetric materials, which demar-cates TIs from trivial insulators using only their parityeigenvalues [8]. This criterion, when applicable, greatlysimplifies the topological analysis of real materials, andunderpins the theoretical prediction and subsequent ex-perimental verification of many TIs [8–12].It is of fundamental interest to obtain results akin tothe Fu-Kane criterion in other symmetry settings. Earlygeneralizations in systems with broken time-reversal(TR) symmetry in 2D constrained the Chern number( C ). The eigenvalues of an n -fold rotation was foundto determine C modulo n [13–15]. This is characteristicof a symmetry-based indicator of topology–when the in-dicator is nonvanishing, band topology is guaranteed, butcertain topological phases (i.e C a multiple of n in thiscontext) may be invisible to the indicator. In 3D, it wasalso recognized that spatial inversion alone can protectnontrivial phases. A new feature here is that these phasesdo not host protected surface states, since inversion sym- ∗ Corresponding author: [email protected] metry is broken at the surface, but they do representdistinct phases of matter. For example they possess non-trivial Berry phase structure in the Brillouin Zone whichleads to robust entanglement signatures [13, 14, 16, 17]and, in some cases, quantized responses [13, 14, 18]. In-terestingly, in the absence of TR invariance the inversioneigenvalues (i.e., parities) can also protect Weyl semimet-als [13, 14], which informed early work on materials can-didates [19]. Hence, these symmetry-based indicatorsare relevant both to the search for nontrivial insulatingphases, and also to the study of topological semimet-als. It is also important to note that the goal here isdistinct from the classification of topological phases, butis instead to identify signatures of band topology in thesymmetry transformations of the state.An important open problem is to extend these power-ful symmetry indicators for band topology to all spacegroups (SGs). Earlier studies have emphasized the topo-logical perspective, which typically rely on constructionsthat are specifically tailored to particular band topologyof interest [8, 15, 20, 21]. While some general mathemat-ical frameworks have been developed [22–24], obtaininga full list of concrete results from such an approach facesan inherent challenge stemming from the sheer multi-tude of physically relevant symmetry settings–there are230 SGs in 3D, and each of them is further enriched bythe presence or absence of both spin-orbit coupling andTR symmetry.A complementary, symmetry-focused perspectiveleverages the existing exhaustive results on band symme-tries [25, 26] to simplify the analysis. Previous work alongthese lines has covered restricted cases [13, 14, 27, 28].For instance, in Ref. [28], which focuses on systems inthe wallpaper groups without any additional symmetry,such an approach was adopted to help develop a morephysical understanding of the mathematical treatment ofRef. [22]. However, the notion of nontriviality is a relativeconcept in these approaches. While such formulation iswell-suited for the study of phase transitions between dif-ferent systems in the same symmetry setting, it does not a r X i v : . [ c ond - m a t . s t r- e l ] J un always indicate the presence of underlying band topology.As an extreme example, such classifications generally re-gard atomic insulators with different electron fillings asdistinct phases, although all the underlying band struc-tures are topologically trivial.Here, we adopt a symmetry-based approach that fo-cuses on probing the underlying band topology. At thecrux of our analysis is the observation that topologicalband structures arise whenever there is a mismatch be-tween momentum-space and real-space solutions to sym-metry constraints [29, 30]. To quantitatively expose suchmismatches, we first develop a mathematical frameworkto efficiently analyze all possible band structures consis-tent with any symmetry setting, and then discuss howto identify the subset of band structures arising fromatomic insulators, which are formed by localizing elec-trons to definite orbitals in real space. The mentionedmismatch then follows naturally as the quotient betweenthe allowed band structures and those arising from real-space specification. We compute this quotient for all 230SGs with or without spin-orbit coupling and/ or time-reversal symmetry. Using these results, we highlight sym-metry settings suitable for finding topological materials,including both insulators and semimetals. In particular,we will point out that, in the presence of inversion sym-metry, stacking two strong 3D TIs will not simply resultin a trivial phase, despite all the Z indices have beentrivialized. Instead, it is shown to produce a quantumband insulator [30] which can be diagnosed through itsrobust gapless entanglement spectrum. II. OVERVIEW OF STRATEGY
Our major goal is to systematically quantify the mis-match between momentum-space and real-space solu-tions to symmetry constraints in free-electron problems[30]. While atomic insulators, which by definition pos-sess localized symmetric Wannier orbitals, can be under-stood from a real-space picture with electrons occupyingdefinite positions as if they were classical particles, topo-logical band structures (that are intrinsic to dimensionsgreater than one) do not admit such a description. When-ever there is an obstruction to such a real-space reinter-pretation, despite the presence of a band gap, the insulat-ing state can only be described through the quantum in-terference of electrons, and we refer in general to such sys-tems as quantum band insulators (QBIs). While all topo-logical phases such as Chern insulators, weak and strong Z topological insulators and topological crystalline in-sulators with protected surface states in d > III. BAND STRUCTURES FORM AN ABELIANGROUP
We will first argue that the possible set of band struc-tures symmetric under an SG G can be naturally iden-tified as the group Z d BS ≡ Z × Z × · · · × Z , where d BS is a positive integer that depends on both G and thespin of the particles (Fig. 1). We will first set aside TRsymmetry, and later discuss how it can be easily incor-porated into the same framework. The discussion in thissection follows immediately from well established resultsconcerning band symmetries [25], and the same set of re-sults was recently utilized in Ref. [28] to discuss an alter-native way to understand the more formal classificationin Ref. [22]. Although there is some overlap betweenthe discussion here and that in Ref. [28], we will focuson a different aspect of the narration: Instead of beingsolely concerned with the values of d BS , we will be moreconcerned with utilizing this framework to extract otherphysical information about the systems. As an interme-diate step we also perform the first computation of d BS for the 230 space groups. The results for TR invariantsystems are summarized in Tables I, and those withoutTR symmetry are tabulated in Appendix H. Atomic InsulatorsBand Structures SymmetryLabels
AtomicTopological (b)(a)
FIG. 1.
Symmetry-based indicators of band topology. (a) Symmetry labelling of bands in a 1D inversion-symmetricexample. k = 0 , π are high-symmetry momenta, where thebands are either even (+) or odd ( − ) under inversion sym-metry (orange diamonds). From a symmetry perspective, atarget set of bands (purple and boxed) separated from all oth-ers by band gaps can be labeled by the multiplicities of thetwo possible symmetry representations, which we denote bythe integers n ± k . Note that such labelling is insensitive tothe detailed energetics within the set. In addition, the setis also characterized by the number of bands involved, whichwe denote by ν . Altogether, the set is characterized by fiveintegers, but are not independent as they are subjected to theconstraints ν = n +0 + n − = n + π + n − π . (b) Symmetry labelslike those described in (a) can be similarly defined for systemssymmetric under any of the 230 space groups in three dimen-sions. Using such labels, one can reinterpret the set of bandstructures as an abelian group. This is schematically demon-strated through the two labels ν and n α , which organize theset of all possible band structures into a two-dimensional lat-tice. Note that the dimension of this lattice is given by thenumber of independent symmetry labels, and is a property ofthe symmetry setting at hand. Organized this way, the bandstructures corresponding to atomic insulators, which are triv-ial by our definition, will generally occupy a sublattice. Anyband structure that does not fall within this sublattice neces-sarily possesses nontrivial band topology. We begin by reviewing some basic notions using a sim-ple example. Consider free electrons in a 1D, inversion-symmetric crystal. The energy bands E m ( k ) are nat-urally labeled by the band index m and the crystalmomentum k ∈ ( − π, π ]. Since inversion P flips k ↔− k , the Bloch Hamiltonian H ( k ) is symmetric under P H ( k ) P − = H ( − k ), which implies E m ( k ) = E m ( − k ),and the wavefunctions are similarly related. The two mo-menta k = 0 and π are special as they satisfy P ( k ) = k (up to a reciprocal lattice vector). As such, the symme-try constraint imposed by P becomes a local constraintat k , which implies the wavefunctions ψ m ( k ) (gener-ically) furnish irreducible representations (irreps) of P : ψ † m ( k ) P ψ m ( k ) = ζ m ( k ) with ζ m ( k ) = ± ζ m ( k ) = ± E m ( k ), and such labels can be readily lifted to a globalone assigned to any set of bands separated from othersby a band gap. We will refer to such sets of bands as‘band structures’ (BSs), though, as we will explain, cau-tion has to be taken when this notion is used in higherdimensions. Insofar as symmetries are concerned, we canlabel the BS by its filling, ν , together with the four non-negative integers, n ± k , corresponding to the multiplicityof the irrep ± at the momenta k = 0 or π (Fig. 1a).Generally, such labels are not independent, since the as-sumption of a band gap, together with the continuity ofthe energy bands, cast global symmetry constraints onthe symmetry labels. These constraints are known ascompatibility relations. For our 1D problem at hand,there are only two of them, which arise from the fillingcondition: ν = n +0 + n − = n + π + n − π . Consequently the BSis fully specified by three non-negative integers, which wecan choose to be n +0 , n + π and ν .This discussion to this point is similar to that ofRef. [28], but we now depart from the combinatorics pointof view of that work. Instead, similar to Ref. [13] wedevelop a mathematical framework to efficiently charac-terize energy bands in terms of their symmetry trans-formation properties, and then show that it provides apowerful tool for the analyzing of general band struc-tures. To begin, we first note that any BS in this 1D,inversion-symmetry problem can be represented by afive-component ‘vector’ n ≡ ( n +0 , n − , n + π , n − π , ν ) ∈ Z ≥ ,where Z ≥ denotes the set of non-negative integers. Inaddition, n is subjected to the two (independent) com-patibility relations. We can arrange these relations intoa system of linear equations and denote them by a 2 × C . The admissible BSs then satisfy C n = , andhence ker C , the solution space of C , naturally enters thephysical discussion. For the current problem, ker C is 3-dimensional, which echoes with the claim that the BS isspecified by three non-negative integers. At this point,however, it is natural to make a mathematical abstrac-tion and lift the physical condition of non-negativity, andthereby define { BS } ≡ ker C ∩ Z D , (1)where for the 1D problem at hand we have D = 5. Themain advantage of this abstraction is that, unlike Z D ≥ , Z D is an abelian group, which has simple structures thatgreatly simplify our forthcoming analysis. In particular, { BS } so defined can be identified with Z d BS , where d BS =3 is the dimension of the solution space ker C . Physically,the addition in Z d BS corresponds to the stacking of energybands.Next, we generalize the discussion to any SG G in threedimensions. (The general framework applies directly toany spatial dimensions; here we focus on three dimen-sions for physical reasons.) We call a momentum k ahigh-symmetry momentum if there is any g ∈ G otherthan the lattice translations such that g ( k ) = k (up toa reciprocal lattice vector). We define a ‘band structure’as ‘a set of energy bands isolated from all others by bandgaps above and below at all high-symmetry momenta’.Note that in 3D, the phrase ‘all high-symmetry momenta’includes all high-symmetry points, lines and planes. Thediscussion for the 1D example carries through, exceptthat one has to consider a much larger zoo of irreps andcompatibility relations [25]. For brevity, we only sum-marize the key generalizations below, and relegate a de-tailed discussion to Appendices B and C.Similar to the 1D example, in the general 3D setting acollection of integers, corresponding to the multiplicitiesof the irreps in the BS, is assigned to each high-symmetrymomentum. By the gap condition, these integers are in-variant along high-symmetry lines. In addition, any pairof symmetry-related momenta will share the same labels.Altogether, we see that the symmetry content of a BS,together with the number of bands ν , is similarly speci-fied by a finite number of integers, which forms the group Z D under stacking. These integers are again subjected tothe compatibility relations, which arise whenever a high-symmetry momentum is continuously connected to an-other with a lower symmetry. By continuity, the symme-try content of the BS at the lower-symmetry momentumis fully specified by that of the higher-symmetry one, giv-ing rise to linear constraints we denote collectively by thematrix C . The group { BS } is then defined as in Eq. (1),and again we find { BS } ≡ ker C ∩ Z D (cid:39) Z d BS , (2)where as before d BS = dim ker C . Note that this resulthas a simple geometric interpretation: From the defini-tion Eq. (1), we can picture ker C as a d BS -dimensionalhyperplane slicing through the hypercubic lattice Z D em-bedded in R D (Appendix D). This gives rise to the sub-lattice Z d BS (Fig. 1b).While Eq. (2) follows readily from definitions, it hasinteresting physical implications. As a group, Z d BS isgenerated by d BS independent generators. In the additivenotation, natural for an abelian group, we can write thegenerators as { b i : i = 1 , . . . , d BS } , and for any givenBS we can expand it similar to elements in a vector spaceBS = d BS (cid:88) i =1 m i b i , (3)where m i ∈ Z are uniquely determined once the basis isfixed. Therefore, full knowledge of { BS } is obtained oncethe d BS generators b i are specified. Having shown that { BS } is a well-defined mathemati-cal entity and identified its general structure, it remainsto connect it to the study of physical band structures.Recall that in motivating the definition Eq. (1), we havelifted the physical condition that all irreps must appeara non-negative number of times. This implies any phys-ical band structure must correspond to elements in thesubset { BS } P ≡ ker C ∩ Z D ≥ ⊂ { BS } . Nonetheless, anyelement of { BS } P enjoys the properties of any general ele-ment of { BS } , and in particular the expansion Eq. (3) ap-plies. Another important observation is that all elementsin { BS } P corresponds to some physical band structures.This is shown in Appendix D, where we argue that thegap condition imposed in the definition of BSs ensuressuch correspondence.So far, we have not addressed the effect of TR sym-metry, which, being anti-unitary, does not lead to newirreps when it is incorporated [25]. Instead, TR sym-metry could force certain irreps to become paired witheither itself or another (Appendix B). When under TRan irrep α at k is paired with a different irrep β at k (cid:48) ,where k (cid:48) = k or k (cid:48) = − k , we simply add to C an addi-tional compatibility relation n α k = n β k (cid:48) ; when α is pairedwith itself, we demand α to be an even integer, whichcan be achieved by redefining ˜ n α k ≡ n α k / C in terms of ˜ n . Note also that TRis not included in our definition of high-symmetry mo-menta, although we will always take Kramers degeneracyin spin-orbit-coupled systems into account. In brief, allTR constraints can be incorporated in a simple mannerin the described framework, and therefore we will treatthem on equal footings with SG constraints from now on. IV. ATOMIC INSULATORS AND MISMATCHCLASSIFICATION
While we have provided a systematic framework toprobe the structure of { BS } , much insight can be gleanedfrom a study of atomic insulators (AIs). AIs correspondto band insulators constructed by first specifying a sym-metric set of lattice points in real space, and then fullyoccupying a set of local orbitals on each of the latticesites. The possible set of AIs can be easily read off fromtabulated data of SGs [26, 33] as we explain in (AppendixC). In addition, once the real-space degrees of freedomare specified one can readily compute the correspondingelement in { BS } . As stacking two AIs lead to anotherAI, we see that { AI } ≤ { BS } as groups. Any subgroupof Z d BS is again a free, finitely-generated abelian group,and therefore we conclude { AI } (cid:39) Z d AI ≡ (cid:40) d AI (cid:88) i =1 m i a i : m i ∈ Z (cid:41) , (4)where we denote by { a i } a complete set of basis for { AI } .Once { BS } and { AI } are separately computed, itis straightforward to evaluate the quotient group (Ap-pendix D) X BS ≡ { BS }{ AI } . (5)Physically, an entry in X BS corresponds to an infiniteclass of BSs that, while distinct as elements of { BS } , onlydiffer from each other by the stacking of an AI. By defi-nition, the entire subgroup { AI } collapses into the trivialelement of X BS . Conversely, any nontrivial element of X BS corresponds to BSs that cannot be be interpretedas AIs, and therefore X BS serves as a symmetry indica-tor of topological BSs.Following the described recipe, we compute { AI } , { BS } , and X BS for all 230 SGs in the four symmetry set-tings mentioned. X BS for spinful fermions with TR sym-metry, relevant for real materials with or without spin-orbit coupling and no magnetic order, are tabulated inTables III and IV. The results for the non-TR-symmetricsettings are presented in Appendix H. We also note thatthe corresponding results for quasi-1D and 2D systems,described respectively by rod and layer group symme-tries, can be readily obtained [26, 34]. The results arealso presented in Appendix H. In particular, we found X BS = Z , the trivial group, for all quasi-1D systems.This is consistent with the picture that topological bandstructures in 1D can be understood as frozen polarizationstates, which are AIs and hence trivial in our definition.An interesting observation from this exhaustive com-putation is the following: for all the symmetry settingsconsidered, we found d BS = d AI (Tables I and II), andtherefore X BS is always a finite abelian group. Equiva-lently, when only symmetry labels are used in the diag-nosis, a BS is nontrivial precisely when it can only beunderstood as a fraction of an AI. In addition, d BS = d AI implies that a complete set of basis for { BS } can be foundby studying combinations of AIs, similar to Eq. (4) butwith a generalization of the expansion coefficients m i ∈ Z to q i ∈ Q , subjected to the constraint that the sum re-mains integer-valued. Although the full set of compati-bility relations is needed in our computation establishing d BS = d AI , using our results { BS } can be readily com-puted directly from { AI } . Since { BS } can be easily foundthis way, we will refrain from providing its complete list.To illustrate the ideas more concretely, we discuss asimple example concerning spinless fermions without TRinvariance, but symmetric under SG 106. In this set-ting, d BS = d AI = 3, and a , one of the three gener-ators of { AI } , has the property that all irreps appearan even number of times, while the other two genera-tors contain some odd entries. Now consider b ≡ a / b satisfies all symmetry con-strains, and therefore b ∈ { BS } . However, b (cid:54)∈ { AI } ≡{ (cid:80) i =1 m a : m i ∈ Z } , and therefore b corresponds toa quantum BS, and indeed it is a representative for thenontrivial element of X BS = Z . In addition, if we con-sider a tight-binding model with representation contentcorresponding to a , the decomposition a = b + b implies it is possible to open a band gap at all high-symmetry momenta at half filling, and thereby realizingthe quantum BS b . It turns out that, in fact, b cor-responds to a filling-enforced QBI (feQBI) [30]. We willelaborate further on this point in Appendix F.Before we move on to concrete applications of our re-sults, we pause to clarify some subtleties in the exposi-tion. Recall that the notion of BS is defined using thepresence of band gaps at all high-symmetry momenta.Generally, however, there can be gaplessness in the in-terior of the Brillouin zone that coexist with our defini-tion of BS. While in some cases such gaplessness is acci-dental in nature, in the sense that it can be annihilatedwithout affecting the BS, in some more interesting casesit is enforced by the specification of the symmetry con-tent. This was pointed out in Refs. [13, 14] for inversion-symmetric systems without TR symmetry, where certainassignment of the parity eigenvalues ensures the presenceof Weyl points at some generic momenta. When a non-trivial element in X BS can be insulating, we refer to itas as a representation-enforced QBI (reQBI); when it isnecessarily gapless, we call it a representation-enforcedsemimetals (reSM). We caution that X BS will naturallyinclude both reQBIs and reSMs, although some symme-try settings naturally forbid the notion of reSMs. In fact,one can show that their individual diagnoses are relatedby X SM = X BS /X BI (Appendix G). Hence, given an en-try of X BS one has to further decide whether it corre-sponds to a reSM or a reQBI. In Appendix G, we providegeneral arguments on the existence of reSMs for systemswith significant spin-orbit coupling and TR symmetry.In addition, we also note that while every BS belongingto a nontrivial class of X BS is necessarily nontrivial, somesystems in the trivial class can also be topological. Bydefinition, an element of BS in the trivial class of X BS can be written as a stacking of atomic insulators. If thestacking necessarily involves at least one negative integercoefficient, the BS element is still topologically nontrivial.Some of the feQBIs discussed in Ref. [30] also fall into thiscategory.Alternatively, when the topological nature of the bandstructure is undetectable using only symmetry labels, sayfor the tenfold-way phases in the absence of any spatialsymmetries, the system belongs to the trivial element of X BS despite it is topological. As an example, considera two-dimensional system with only lattice translationsymmetries. For such systems, the K-theory classifica-tion of band insulators in Refs. [22, 24, 28] gives Z , wherethe two factors correspond respectively to the electronfilling (i.e., number of bands) and the Chern number. Incontrast, within our approach we find { BS } = { AI } = Z ,as in this setting the only symmetry label is the filling,which cannot detect the Chern number of the bands. Fur-thermore, as there exists AI for any filling ν , we find X BS = Z , the trivial group.Finally, we note that all nontrivial entries in X BS havephysical representatives, i.e., any nontrivial class in X BS can be represented by some physical band structures.The proof of this claim will require a small technical corollary concerning the properties of AIs, and we rel-egate the detailed discussion to Appendix D. TABLE I.
Characterization of band structures for systems with time-reversal symmetry and significant spin-orbit coupling. d Space groups1 1, 3, 4, 5, 6, 7, 8, 9, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 3435, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 76, 77, 78, 80, 91, 92, 93, 94, 95, 96, 98, 101, 102, 105, 106,109, 110, 144, 145, 151, 152, 153, 154, 169, 170, 171, 172, 178, 179, 180, 1812 79, 90, 97, 100, 104, 107, 108, 146, 155, 160, 161, 195, 196, 197, 198, 199, 208, 210, 212, 213, 2143 48, 50, 52, 54, 56, 57, 59, 60, 61, 62, 68, 70, 73, 75, 89, 99, 103, 112, 113, 114, 116, 117, 118, 120, 122, 133, 142150, 157, 159, 173, 182, 185, 186, 209, 2114 63, 64, 72, 121, 126, 130, 135, 137, 138, 143, 149, 156, 158, 168, 177, 183, 184, 207, 218, 219, 2205 11, 13, 14, 15, 49, 51, 53, 55, 58, 66, 67, 74, 81, 82, 86, 88, 111, 115, 119, 134, 136, 141, 167, 217, 228, 2306 69, 71, 85, 125, 129, 132, 163, 165, 190, 201, 203, 205, 206, 215, 216, 2227 12, 65, 84, 128, 131, 140, 188, 189, 202, 204, 2238 124, 127, 148, 166, 193, 200, 224, 226, 2279 2, 10, 47, 87, 139, 147, 162, 164, 176, 192, 19410 174, 18711 225, 22913 83, 12314 175, 191, 221 d : the rank of the abelian group formed by the set of band structures. V. QUANTUM BAND INSULATORS INCONVENTIONAL SETTINGS
Having derived the general theory for findingsymmetry-based indicators of band topology, we nowturn to applications of the results. As a first applica-tion, we utilize the results in Table III to look for reQBIsthat are not diagnosed by previously available topologicalinvariants. In particular, we will focus on an interestingresult concerning one of the most well studied symme-try setting: materials with significant spin-orbit couplingsymmetric under TR, lattice translations and inversion(SG 2).As shown in Table III, X BS = ( Z ) × Z for this set-ting. Using the Fu-Kane criterion [8], one can readilyverify the strong and weak TIs respectively serve as thegenerators of the Z and Z factors. This identification,however, fails to account for the nontrivial nature of thedoubled strong TI, which being a nontrivial element in Z corresponds to a reQBI. This reQBI has no protected sur-face states and a trivial magnetoelectric response ( θ = 0),and it cannot be directly explained using earlier works fo-cusing on inversion-symmetric insulators [13, 14, 16].Nonetheless, similar to the argument in Refs. [13, 14,16], the nontrivial nature of the reQBI can be seen fromits entanglement spectrum, which exhibits protected gap- lessness related to the parity eigenvalues of the filledbands (Fig. 2a). In the present context, we define the en-tanglement spectrum as the collection of single-particleentanglement energies arising from a spatial cut, whichcontains an inversion center and is perpendicular to acrystalline axis. Refs. [13, 14, 16] showed that theentanglement spectrum of TR and inversion symmet-ric insulators generally have protected Dirac cones atthe TR invariant momenta of the surface Brillouin zone.These Dirac cones carry effective integer charges underinversion symmetry, and as a result they are symmetry-protected. This is observed for the doubled strong TIphase (Fig. 2b), which has twice the number of Diraccones as the regular strong TI.Yet, one must use caution in interpreting the nontrivialnature of such entanglement, since inversion-symmetricAIs often have protected entanglement surface states.They arise whenever the center of mass of an electronicwavefunction is pinned to the entanglement cut, and suchstates were called frozen-polarization states. The pres-ence of these entanglement surface modes is dependenton the arbitrary choice of the location of the cut, andtherefore are not as robust as the other topological char-acterizations. In contrast, since we have already quotientout all AIs in the definition of X BS , the reQBI at handmust have a more topological origin. This is verified from TABLE II.
Characterization of band structures for systems with time-reversal symmetry and negligible spin-orbit coupling. d Space groups1 1, 4, 7, 9, 19, 29, 33, 76, 78, 144, 145, 169, 1702 8, 31, 36, 41, 43, 80, 92, 96, 110, 146, 161, 1983 5, 6, 18, 20, 26, 30, 32, 34, 40, 45, 46, 61, 106, 109, 151, 152, 153, 154, 159, 160, 171, 172, 173, 178, 179, 199212, 2134 24, 28, 37, 39, 60, 62, 77, 79, 91, 95, 102, 104, 143, 155, 157, 158, 185, 186, 196, 197, 2105 3, 14, 17, 27, 42, 44, 52, 56, 57, 94, 98, 100, 101, 108, 114, 122, 150, 156, 182, 214, 2206 11, 15, 35, 38, 54, 70, 73, 75, 88, 90, 103, 105, 107, 113, 142, 149, 167, 168, 184, 195, 205, 2197 13, 22, 23, 59, 64, 68, 82, 86, 117, 118, 120, 130, 163, 165, 180, 181, 203, 206, 208, 209, 211, 218, 228, 2308 21, 58, 63, 81, 85, 97, 116, 133, 135, 137, 148, 183, 190, 201, 2179 2, 25, 48, 50, 53, 55, 72, 99, 121, 126, 138, 141, 147, 188, 207, 216, 22210 12, 74, 93, 112, 119, 176, 177, 202, 204, 21511 66, 84, 128, 136, 166, 22712 51, 87, 89, 115, 129, 134, 162, 164, 174, 189, 193, 223, 22613 16, 67, 111, 125, 194, 22414 49, 140, 192, 20015 10, 69, 71, 124, 127, 132, 18717 225, 22918 65, 83, 131, 139, 17522 22124 19127 47, 123 d : the rank of the abelian group formed by the set of band structures. the pictorial argument in Figs. 2a-c, where we contrastthe entanglement spectrum of the doubled strong TI withthose that can arise from AIs. Importantly, we see thatthe total Dirac-cone charge of an AI is always 0 mod 4,whereas the doubled strong TI has a charge of 2 mod 4.This immediately implies that the entanglement gapless-ness cannot be reconciled with that arising from any AI,and in fact shows that, in this symmetry setting, thebulk computation of X BS can be reproduced by consid-ering the entanglement spectrum. Note that if TR isbroken, Kramers paring will be lifted and the irrep con-tent of this reQBI becomes achievable with an AI. Thissuggests that the reQBI at hand is protected by the com-bination of TR and inversion symmetry, and in particularit can be interpreted as a TR-symmetric BS mimickinga TR-broken AI through momentum space quantum in-terference. It is an interesting open question to studywhether or not this reQBI has any associated quantizedphysical response [13].Since the strong TI is compatible with any additionalspatial symmetry, the argument above is applicable toany centrosymmetric SGs. Indeed, as can be seen fromTable III, all of them have | X BS | ≥
4, consistent withour claim. From the same table, we can also observethat, very often, a general weak TI becomes incompati-ble with a higher degree of spatial symmetries. This isrelated to the number of factors in X BS , i.e. the number of independent generators N t . For any centrosymmetricSG, the Fu-Kane criterion dictates that all strong andweak TIs are diagnosable using symmetry labels. As onesuch factor is reserved for the strong TI, the SG is com-patible with at most N t − X BS is found for SG 2 in allthe other symmetry settings, their physical interpreta-tions are very different. In particular, the generators of Z < X BS corresponds to a reSM in the other settings. VI. LATTICE-ENFORCED SEMIMETALS
As another application of our results, we demonstratehow the structure of { BS } exposes constraints on thepossible phases of a system arising from the specificationof the microscopic degrees. We will in particular focuson the study of semimetals, but a similar analysis can beperformed in the study of, say, reQBIs.As a warm-up, recall the physics of (spinless) graphene,where specifying the honeycomb lattice dictates that theirrep at the K point is necessarily two-dimensional, andtherefore the system is guaranteed to be gapless at halffilling. Using the structure of { BS } we described, this line TABLE III.
Symmetry-based indicators of band topology for systems with time-reversal symmetry and signif-icant spin-orbit coupling. X BS Space groups Z
81, 82, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 215, 216, 217, 218, 219, 220 Z Z
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 Z Z Z × Z
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 Z × Z
87, 124, 139, 140, 229 Z × Z Z × Z Z × Z Z × Z × Z
11, 12, 13, 49, 51, 65, 67, 69 Z × Z × Z
83, 123 Z × Z × Z × Z
2, 10, 47 X BS : the quotient group between the group of band structures and atomic insulators.TABLE IV. Symmetry-based indicators of band topology for systems with time-reversal symmetry and negli-gible spin-orbit coupling. X BS Space groups Z
3, 11, 14, 27, 37, 48, 49, 50, 52, 53, 54, 56, 58, 60, 66, 68, 70, 75, 77, 82, 85, 86,88, 103, 124, 128, 130, 162, 163, 164, 165, 166, 167, 168, 171, 172, 176, 184, 192, 201, 203 Z × Z
12, 13, 15, 81, 84, 87 Z × Z Z × Z × Z
10, 83, 175 Z × Z × Z × Z X BS : the quotient group between the group of band structures and atomic insulators. of reasoning can be efficiently generalized to an arbitrarysymmetry setting: Any specification of the lattice degreesof freedom corresponds to an element A ∈ { AI } , and onesimply asks if it is possible to write A = B v + B c , where B v,c ∈ { BS } satisfies the physical non-negative condi-tion, such that B v corresponds to a BS with a specifiedfilling ν . Whenever the answer is no, the system is guar-anteed to be (semi-)metallic. We refer to any such sys-tem as a lattice-enforced semimetal (leSM). Note thata stronger form of symmetry-enforced gaplessness canoriginate simply from the electron filling, and such sys-tems were dubbed as filling-enforced semimetal (feSMs)[36, 37]. We will exclude feSMs from the definition ofleSMs, i.e. we only call a system a leSM if the filling ν iscompatible with some band insulators in the same sym-metry setting, but is nonetheless gapless because of theadditional lattice constraints.A preliminary analysis reveals that leSMs abound, es-pecially for spinless systems with TR symmetry. Thisis in fact anticipated from the earlier discussions inRefs. [38–40]. Instead, we will turn our attention to TR- invariant systems with significant spin-orbit coupling,which lies beyond the scope of these earlier studies andoftentimes leads to interesting new physics [1, 7, 30]. Asystematic survey of them will be the focus of anotherstudy. Here, we present a proof-of-concept leSM exam-ple we found, which arises in TR-symmetric systems inSG 219 ( F ¯43 c ) with significant spin-orbit coupling. Wewill only sketch the key features of the model, and theinterested readers are referred to Appendix E for detailsof the analysis.We consider a lattice with 2 sites in each primitiveunit cell, and that each site has a local environment cor-responding to the cubic point group T (Fig. 2d). Wesuppose the relevant on-site degrees of freedom trans-form under the 4D irreducible co-representation of T un-der TR symmetry [25], and that the system is at halffilling, i.e., the filling is ν = 4 electrons per primitiveunit cell. Although the local orbitals are partially filled,generically a band gap becomes permissible once electronhopping is incorporated. Naively, for the present prob-lem this may appear to be the likely scenario, since the Local energylevels and filling(d) (e)(a) (b) (c)
Atomic insulators
Entanglement surface states
FIG. 2.
Examples of topological band structures (a-c) A representation-enforced quantum band insulator ofspinful electrons with time-reversal and inversion symmetries,dubbed the ‘doubled strong topological insulator’. (a) Usingthe Fu-Kane parity criterion [8], the strong and weak Z topo-logical insulator (TI) indices can be computed from the theparities of the occupied bands, which we indicate by ± at theeight time-reversal invariant momenta. Shown are the pari-ties of one state from each Kramers pair for a doubled strongTI with four filled bands. (b) The entanglement spectrumat a spatial cut, parallel to the x - y plane and contains aninversion center, features two Dirac cones at Γ [13, 14, 16].Such Dirac cones are known to possess integer-valued chargesunder the inversion symmetry, and we denote the positively-and negatively-charged cones respectively by blue and red.(c) Inversion-symmetric atomic insulators feature entangle-ment surface Dirac cones in general, but their presence de-pends on the arbitrary choice of the cut. We find that thepossible Dirac-cone arrangement arising from atomic insula-tors can only be a linear combination of four basic config-urations, illustrated as a sum with the integral weights m i .The arrangement in (b) cannot be reconciled with those in(c), confirming the nontriviality of the doubled strong TI. (d-e) Example of a lattice-enforced semimetal for spinful elec-trons with time-reversal symmetry. (d) We consider a site(red sphere) under a local environment (beige) symmetric un-der the point group T , and suppose the relevant local energylevels form the four-dimensional irreducible representation,which is half-filled (boxed). (e) When the red site sits at thehighest-symmetry position of space group 219, the specifiedlocal energy levels and filling gives rise to a half-filled eight-band model (each band shown is doubly degenerate). Such(semi-)metallic behavior is dictated by the specification of themicroscopic degrees of freedom in this model. momentum-space irreps all have dimensions ≤ ν = 4, whichimplies there will be irremovable lattice-enforced gapless-ness at some high-symmetry line. This is indeed verifiedin Fig. 2e, where we plot the band structure obtainedfrom an example tight-binding model (Appendix E). VII. DISCUSSION
In this work, we present a simple mathematical frame-work for efficiently analyzing band structures as entitiesdefined globally over the Brillouin zone. We further uti-lize this result to systematically quantify the mismatchbetween the momentum- and real-space descriptions offree electron phases, obtaining a plethora of symmetrysettings for which topological materials are possible.Our results concern a fundamental aspect of the ubiq-uitous band theory. For electronic problems, we demon-strated the power of our approach by discussing threeparticular applications, predicting quantum band insu-lators and semimetals in both conventional and uncon-ventional symmetry settings. We highlight four inter-esting future directions below: first, to incorporate thetenfold-way classification into our symmetry-based diag-nosis of topological materials [28]; second, to discoverquantized physical responses unique to the phases we pre-dicted [13, 14, 18]; third, to extend the results to mag-netic space groups by including the extra compatibilityrelations [25]; and lastly, to screen materials databasefor topological materials relying on fast diagnosis invok-ing only symmetry labels [41]. More broadly, we expectour analysis to shed light on any other fields of studies,most notably photonics and phononics, where the inter-play between topology, symmetry and band structures isof interest.Note added: Recently, Ref. [42] appeared, which hassome overlap with the present work, in that it also identi-fies topological band insulators by contrasting them withatomic insulators. However, the present work differs fromRef. [42] in important ways in the formulation of theproblem and the mathematical approach adopted.
ACKNOWLEDGMENTS
We thank C.-M. Jian, A. Turner and M. Zaletelfor insightful discussions and collaborations on earlierworks. We also thank C. Fang for useful discussions.AV and HCP were supported by NSF DMR-1411343.AV acknowledges support from a Simons InvestigatorAward. H.W. acknowledges support from JSPS KAK-ENHI Grant Number JP17K17678.
Appendix A: Glossary of abbreviations
For brevity, we have introduced several abbreviationsin the text. For the readers’ convenience, we provide aglossary of the less-standard ones here.‘AI’ (atomic insulator): band insulators possessing lo-calized symmetric Wannier functions.‘BS’ (band structure): a set of energy bands separatedfrom all others by band gaps above and below at all high-symmetry momenta.0‘fe’ (filling-enforced): referring to attributes that followfrom the electron filling of the system.‘le’ (lattice-enforced): referring to attributes that fol-low from the specification of the microscopic degrees offreedom in the lattice.‘QBI’ (quantum band insulators): band insulators,with or without protected surface states, that do not ad-mit any atomic limit provided the protecting symmetriesare preserved.‘re’ (representation-enforced): referring to attributesthat follow from knowledge on the symmetry representa-tions of the energy bands.‘SG’ (space group): any one of the 230 spatial symme-try groups of crystals in three dimensions.‘SM’ (semimetals): filled bands with gap closings thatare stable to infinitesimal perturbations.‘TI’ (topological insulator): note that we use thisphrase in a restricted sense here, always referring to the Z topological insulators in two or three dimensions forspin-orbit-coupled system with time-reversal symmetry. Appendix B: Review of Symmetries in BandStructures
Here, we briefly review some notions and results con-cerning the consequences of symmetries on band struc-ture. The discussion here will closely mirror that of the1D example given in the main text, but we will assumea general 3D setting from the outset.
1. Little group and its representation
Consider a system of noninteracting fermions symmet-ric under a purely spatial symmetry group G with thelattice translation subgroup T . An element g ∈ G thatmaps a point x ∈ R to g ( x ) = p g x + t g ∈ R may becharacterized by an orthogonal matrix p g and a vector t g . Thanks to the lattice translation T , it is natural tolabel the eigenenergy E m ( k ) by the band index m andthe wavevector k ∈ BZ. The notion of energy bands,however, becomes ambiguous whenever they cross, sincethe underlying wave function of a single band will gen-erally become discontinuous. Instead, it is more naturalto consider a set of entangled bands, separated from allothers by band gaps above and below, as a single entity.Whether a given set of energy bands can be isolatedfrom the others dictate whether or not the system can beinsulating, and therefore is a fundamental question in thestudy of electronic band structures. High-symmetry mo-menta play a crucial role in this analysis. The subgroupof G that leaves a momentum k ∈ BZ invariant (up to areciprocal lattice vector G ) is known as the little groupof k , which is commonly denoted by G k [25]. As a set,we have G k = { g ∈ G : p g k = k + ∃ G } . (B1) By definition, lattice translations T automatically form asubgroup of G k , and we say k is a high-symmetry momen-tum whenever G k contains any element aside from latticetranslations. Note that when we say ‘all high-symmetrymomenta’, we include all high-symmetry points, lines andplanes in the BZ.Generally, a spatial symmetry g demands E m ( p g k ) = E m ( k ), and the corresponding wave functions are simi-larly related. When g ∈ G k , the constraint becomes alocal condition in momentum space. This implies thewave functions furnish a representation of G k [25]. Suchrepresentations encode the transformation properties ofthe BS at k , and can be generally decomposed into a di-rect sum of irreducible representations (irreps) u α k , whichhave been exhaustively tabulated in Ref. [25] for bothspinless and spinful fermions. Insofar as symmetry prop-erties are concerned, we can label a BS at k by the setof non-negative integers { n α k : α = 1 , . . . , D k } , (B2)where n α k denotes the number of times the α -th irrepappears in the BS, and D k denotes the number of irreps.Note that such symmetry labeling makes no reference tothe detailed energetics of the system, i.e. the labels areinsensitive to the energetic arrangement of the individualbands within the BS.
2. Types of momenta and compatibility relations
A priori, a BS will carry independent symmetry labels { n α k } and { n β k (cid:48) } at distinct high-symmetry momenta k and k (cid:48) . However, symmetries and continuity can cast ex-tra constraints on such assignment [25]. To see this con-cretely, let us classify all points in BZ into a finite numberof ‘types’: we say k and k are of the same type if either k and k are symmetry-related, i.e., k = p g k + ∃ G ,or each point of the line connecting k and k has thesame G k as G k = G k . For example, imagine a line in-variant under a n -fold rotation (or screw) C n . Any pointon this line (except for possibly the end points) has theidentical little group generated by C n and hence is of thesame type. Similar situation arises for a plane invariantunder a mirror (or a glide). By assumption, the existenceof band gaps imply the representation content of a BS isinvariant within such high-symmetry lines or planes, andtherefore it suffices to specify the representations at onearbitrarily chosen k point for each type of momenta in theBZ. Therefore, a full set of symmetry labels of a BS canbe obtained by specifying only those arising from a finitenumber of representatives. In addition, the BS is alsolabeled by the total number of bands in the BS, whichwe will denote by ν . Note that physically ν is simply theelectron filling of the system when BS is interpreted asthe set of filled bands in the full system. Aggregating allthe labels { n α k } and ν into a single quantity, we denotethe representation content of a BS by n , a set of D non-negative integers where D = 1 + (cid:80) k D k . Here, we have1chosen one arbitrary k for each type of the k vectors inthe BZ.Next, we note that a general n may not be compatiblewith the continuity of the bands and the gap condition weimposed on BS. To find the set of admissible labels { n } ,we introduce the notion of compatibility relations. [25]For any pair of infinitesimally close momenta k and k + δ k , their little groups satisfy G k + δ k ≤ G k (assuming aproper choice of labeling for the two momenta). Sincethe irreps at the higher-symmetry momentum k can bedecomposed into those of the lower-symmetry ones at k + δ k , the symmetry labels { n α k + δ k } are fully constrainedby { n β k } . This is captured by the compatibility relations,one for each value of α , n α k + δ k = (cid:88) β c k ,δ k αβ n β k , (B3)where the coefficients c k ,δ k αβ are non-negative integers.There is also a similar compatibility on the filling: ν = (cid:80) α dim[ u α k ] n α k , where dim[ u α k ] denotes the dimension ofthe irrep α (i.e. the number of bands involved).As our target is to study BSs as global entities, it isinstructive to collect all compatibility relations into a sys-tem of linear equations: C n = 0 , (B4)where C is an integer-valued matrix with coefficients de-termined by those in Eq. (B3). By definition, any BS canbe identified with an n satisfying Eq. (B4). Conversely,any set of D non-negative integers n ∈ Z D ≥ satisfyingEq. (B4) can be identified with a physical band structure(D).
3. Construction of irreps of G Given an irrep u α k of G k , one can construct an irrep of G , which s called the induced representation. It is knownthat every irrep of G can be constructed in this way [25].Here we review the construction in detail for the casewhere u α k is a projective representation of G k due to thespin degrees of freedom.Let {| φ ri, k (cid:105)} dim[ u α k ] i =1 be the basis of an irrep u r x of G x .Namely, | φ ri, k (cid:105) transform under h, h (cid:48) ∈ G k asˆ h | φ αi, k (cid:105) = (cid:88) j | φ αj, k (cid:105) [ u α k ( h )] ji , (B5) u α k ( h ) u α k ( h (cid:48) ) = z h,h (cid:48) u α k ( hh (cid:48) ) , (B6)ˆ h (ˆ h (cid:48) | φ αi, k (cid:105) ) = z h,h (cid:48) ˆ( hh (cid:48) ) | φ αi, k (cid:105) , (B7)where z g,g (cid:48) = ± z g,g (cid:48) should beset to be 1.Since elements g / ∈ G k change k to an inequivalentmomentum p g k by definition, ˆ g | φ ri, k (cid:105) cannot, in general, be expanded by {| φ ri, k (cid:105)} i . The symmetry orbit of k , { k σ } |G / G k | σ =1 = { p g k : g ∈ G} , is called the star of k .We arbitrarily choose a complete set of representatives { g σ } |G / G k | σ =1 ( g σ =1 = e ) of G / G k that satisfies k σ ≡ p g σ k and define | φ αi, k σ (cid:105) ≡ ˆ g σ | φ αi, k (cid:105) . The set {| φ ri, k σ (cid:105)} i,σ servesas the basis of the representation of G . To see how | φ αi, k σ (cid:105) transforms under a general element g ∈ G , note that gg σ ∈ G can be uniquely decomposed into a product of g σ (cid:48) ∈ G / G k and h ∈ G k . Therefore, step-by-step, we haveˆ g | φ αi, k σ (cid:105) = ˆ g (ˆ g σ | φ αi, k (cid:105) )= z g,g σ ˆ( gg σ ) | φ αi, k (cid:105) = z g,g σ ˆ( g σ (cid:48) h ) | φ αi, k (cid:105) = z g,g σ z g σ (cid:48) ,h (cid:88) i (cid:48) ˆ g σ (cid:48) | φ αi (cid:48) , k (cid:105) [ u α k ( h )] i (cid:48) i = z g,g σ z g σ (cid:48) ,h (cid:88) i (cid:48) ˆ g σ (cid:48) | φ αi (cid:48) , k (cid:105) [ u α k ( h )] i (cid:48) i = (cid:88) σ (cid:48) ,i (cid:48) | φ αi (cid:48) , k σ (cid:48) (cid:105) [ U α ( g )] σ (cid:48) i (cid:48) ,σi , (B8)where[ U α ( g )] σ (cid:48) j,σi = δ (cid:48)(cid:48) k σ (cid:48) ,p g k σ z g,g σ z g σ (cid:48) ,g − σ (cid:48) gg σ [ u α k ( g − σ (cid:48) gg σ )] ji (B9)and δ (cid:48)(cid:48) k , k is 1 only when k = k modulo a reciprocallattice vector. This is the induced representation of G constructed from u α k of G k . It has a nice property that U α is irreducible whenever u α k is.
4. Time-reversal symmetry
Here we will follow the discussion in Ref. [25] on theconsequence of the TR symmetry T commuting with ev-ery element of G . We will write ˆ T = ( η T ) ˆ N where ˆ N isthe number of fermions and η T = − η T = +1) for thespinful (spinless) case. a. General case Let us start with a finite group G in general. Supposethat {| i (cid:105)} di =1 ( d ≡ dim[ u ]) is a basis of an irrep u of G :ˆ g | i (cid:105) = | j (cid:105) u ji ( g ) . (B10)Then { ˆ T | i (cid:105)} di =1 is a basis of the conjugate representation u ∗ ( g ): ˆ g ( ˆ T | i (cid:105) ) = ( ˆ T | j (cid:105) ) u ∗ ji ( g ) . (B11)When u and u ∗ are different irreps, the group G + T G issimply represented by D ( g ) = (cid:32) u u ∗ (cid:33) , D ( T ) = (cid:32) η T (cid:33) . (B12)The situation is different when u and u ∗ are the sameirrep, i.e, u ∗ ( g ) = v † u ( g ) v for a unitary matrix v . Using2the definition of v twice, we have ( vv ∗ ) u ( g ) = u ( g )( vv ∗ )for every g . Since u is irreducible, we see vv ∗ = ξη T d for ξ = ±
1. In other words, v T = ( v ∗ ) † = ξη T v, ξ, η T = ± . (B13)It is easy to see that the combination | ¯ i (cid:105) ≡ ( ˆ T | j (cid:105) ) v † ji transforms in the same way as | i (cid:105) under G :ˆ g | ¯ i (cid:105) = | ¯ j (cid:105) u ji ( g ) . (B14)Hence the question is if | i (cid:105) and | ¯ i (cid:105) are the same. To seethis, let us evaluate the inner-product G ij ≡ ( | i (cid:105) , | ¯ j (cid:105) ).Since ˆ g is unitary, we have G ij = ( | i (cid:105) , | ¯ j (cid:105) ) = (ˆ g | i (cid:105) , ˆ g | ¯ j (cid:105) ) = [ u ( g ) † Gu ( g )] ij , (B15)i.e., G and u ( g ) commute for every g and we must have G = c d . To compute c , note thatˆ T | i (cid:105) = | ¯ j (cid:105) v ji , (B16)ˆ T | ¯ i (cid:105) = η T | j (cid:105) ( v T ) ji = ξ | j (cid:105) v ji . (B17)The second line follows from the first by applying ˆ T andusing Eq. (B13). Hence c = 1 d d (cid:88) i =1 ( | i (cid:105) , | ¯ i (cid:105) ) = 1 d d (cid:88) i =1 ( ˆ T | ¯ i (cid:105) , ˆ T | i (cid:105) )= ξd ( vv † ) j,j (cid:48) ( | j (cid:48) (cid:105) , | ¯ j (cid:105) ) = ξc. (B18)Namely, when ξ = − c vanishes and {| i (cid:105)} di =1 and { ˆ T | i (cid:105)} di =1 are orthogonal. On the other hand, when ξ = 1, c is finite and {| i (cid:105)} di =1 and { ˆ T | i (cid:105)} di =1 are the samestate.To compute ξ from the character χ ( g ) = tr[ u ( g )], weuse the following identity z g,g u ( g ) = u ( g ) u ( g ) = [ u ∗ ( g )] ∗ u ( g )= [ v † u ( g ) v ] ∗ u ( g ) = ξη T vu ∗ ( g ) v ∗ u ( g ) . (B19)We used Eq. (B13) in the last line. Therefore, η T | G | (cid:88) g z g,g χ ( g ) = ξ | G | (cid:88) g v ij u ∗ jk ( g ) v ∗ kl u li ( g )= ξ d ( vv † ) i,i = ξ. (B20)where, in the second last line we used the orthogonalityof irreps (cid:80) g u ( α ) ij ( g ) u ( β ) kl ( g ) ∗ = ( | G | /d ) δ ik δ jl δ αβ . Fromthis orthogonality, it is also clear that (cid:80) g z g,g χ ( g ) = 0when u and u ∗ are different irreps.To summarize, to see if the TR symmetry changes ofdegeneracy for the irrep u of G , one should compute η T | G | (cid:88) g ∈ G z g,g χ ( g )= +1 : Degeneracy is unchanged. − u ’s are paired under TR.0 : u and u ∗ are different and are paired . (B21)This is called Wigner’s test in the literature [25]. b. TR pairing of u αk Let us apply this result to irreps u α k of G k . If there is noelement g ∈ G such that p g k = − k modulo a reciprocallattice vector G , then the TR symmetry just implies anadditional degeneracy between u α k and u α − k and this caseis easily handled. Thus, suppose that there is at leastone g ∈ G such that p g k = − k + ∃ G .As explained earlier, an irrep u α k of G k induced an irrep U α of G , see Eq. (B9). Its character can be expressed astr[ U α ( g )] = (cid:88) σ δ (cid:48)(cid:48) k σ ,p g k σ z g,g σ z g σ ,g − σ gg σ tr[ u α k ( g − σ gg σ )]= (cid:88) σ (cid:88) h ∈G k δ g,g σ hg − σ z g,g σ z g σ ,h tr[ u α k ( h )]= (cid:88) σ (cid:88) h ∈G k σ δ g,h tr[ u α k σ ( h )]= (cid:88) σ (cid:88) h ∈G k σ δ g,h χ α k σ ( h ) , (B22)where u α k σ ( h ) ≡ z h,gσ z gσ,g − σ ghgσ u α k ( g − σ hg σ ) and χ α k σ ( h ) =tr[ u α k σ ( h )]. Hence,1 |G| (cid:88) g ∈G z g,g tr[ U α ( g )]= 1 |G| (cid:88) g ∈G (cid:88) σ (cid:88) h ∈G k σ δ g ,h z g,g χ α k σ ( h )= 1 |G k | (cid:88) g ∈G z g,g χ α k ( g )= 1 |G k /T | (cid:88) g ∈G /T δ (cid:48)(cid:48) p g k , − k z g,g χ α k ( g ) (B23)To go to the third line, we used the fact that contributionsfrom each star of k are identical and replaced (cid:80) σ by thefactor |G| / |G k | . To go to the last line, we performed thesum over the translation subgroup T , which results in theconstraints (cid:80) t R ∈ T e − i k · ( p g t R + t R ) = | T | δ (cid:48)(cid:48) p g k , − k . There-fore, the TR index can be computed given by [25] η T |G k /T | (cid:88) g ∈G /T δ (cid:48)(cid:48) p g k , − k z g,g χ α k ( g )= +1 : Degeneracy is unchanged. − u α k ’s are paired under TR.0 : u α k is paired with another irrep u β k . (B24)As explained in the main text, when an irrep u α k is pairedwith a different irrep u β k under TR, we simply add to C an additional compatibility relation n α k = n β k ; when u α k ispaired with itself, we demand n α k to be an even integer,which can be achieved by redefining ˜ n α k ≡ n α k / C in terms of ˜ n .3 Appendix C: Atomic Insulators as Band Structures
In B, we characterized each BS by n , the set of integers n α k that specifies the representation contents of the BS.In this section, we derive a general formula that gives n for each AI.
1. Wyckoff position and site symmetryrepresentations
An AI is specified by the location of the sites on whichatomic orbitals sit, and the type of orbitals on eachsite. Mathematically, these two inputs correspond to thechoice of a Wyckoff position and the representation of thesite symmetry group of a site in the Wyckoff position.Just as we defined the little group of k , let us definethe little group of x as the subgroup of G that leaves x invariant. We call it the site symmetry group G x of x .As a set, we have G x = { h ∈ G | h ( x ) ≡ p h x + t h = x } . (C1)Points in real space are classified based on their littlegroup. Namely, two points x and x belong to the sameWyckoff position iff there exists g ∈ G such that G x = g G x g − . The full list of different Wyckoff positions (inthe real space) is available in Ref. [33].Let us pick a site x in the unit cell UC. By definition,elements of G not belonging to G x will move x . The crys-tallographic orbit { g ( x ) | g ∈ G} defines a G -symmetriclattice L x . Let { x σ } σ =1 , ,... ( x ≡ x ) be the lattice pointsin UC. We choose { g σ } σ =1 , ,... from G in such a waythat g σ =1 = e ( e ∈ G is the identity) and g σ ( x ) = x σ for σ = 2 , , . . . . Namely, { g σ } σ =1 , ,... is a complete set ofrepresentatives of W x ≡ ( G / G x ) /T .We want to introduce an orbital on every site of L x ina symmetric manner. To that end, let us first put states {| φ r x ,i, k (cid:105)} dim[ u r x ] i =1 on x that obey an irrep u r x of G x :ˆ h | φ r x ,i, k (cid:105) = (cid:88) j | φ r x ,j,p h k (cid:105) [ u r x ( h )] ji , (C2) u r x ( h ) u r x ( h (cid:48) ) = z h,h (cid:48) u r x ( hh (cid:48) ) , (C3)ˆ h (ˆ h (cid:48) | φ r x ,i, k (cid:105) ) = z h,h (cid:48) ˆ( hh (cid:48) ) | φ r x ,i, k (cid:105) (C4)for h, h (cid:48) ∈ G x andˆ t R | φ r x ,i, k (cid:105) = | φ r x ,i, k (cid:105) e − i k · R (C5)for t R ∈ T . The orbitals at other sites of L x are thendefined by | φ r x σ ,i, k (cid:105) ≡ ˆ g σ | φ r x ,i,p − σ k (cid:105) . As before, z g,g (cid:48) = ± x to start with and thechoice of an irrep u r x of the site symmetry group G x willspecify an AI and its representation contents as we willsee now.
2. Representation contents of an AI
To determine the transformation of {| φ r x σ ,i, k (cid:105)} σ,i, k un-der G , note that any element g ∈ G can be uniquelydecomposed as g = t R g σ h where t R ∈ T and h ∈ G x .In particular, we can decompose gg σ as t R g σ (cid:48) h with R = g ( x σ ) − x σ (cid:48) . Therefore,ˆ g | φ r x σ ,i, k (cid:105) = z g,g σ ˆ( gg σ ) | φ r x ,i,p − σ k (cid:105) = z g,g σ ˆ( t R g σ (cid:48) h ) | φ r x ,i,p − σ k (cid:105) = z g,g σ z g σ (cid:48) ,h ˆ t R ˆ g σ (cid:48) (ˆ h | φ r x ,i,p − σ k (cid:105) )= z g,g σ z g σ (cid:48) ,h (cid:88) i (cid:48) (ˆ t R | φ r x σ (cid:48) ,i (cid:48) ,p g k (cid:105) )[ u r x ( h )] i (cid:48) i = (cid:88) σ (cid:48) ,i (cid:48) , k (cid:48) | φ r x σ (cid:48) ,i (cid:48) , k (cid:48) (cid:105) [ U r x ( g )] σ (cid:48) i (cid:48) k (cid:48) ,σi k , (C6)where[ U r x ( g )] σ (cid:48) i (cid:48) k (cid:48) ,σi k (C7)= δ (cid:48) x σ (cid:48) ,g ( x σ ) δ (cid:48)(cid:48) k (cid:48) ,p g k e − i k (cid:48) · ( g ( x σ ) − x σ (cid:48) ) z g,g σ z g σ (cid:48) ,h gσ (cid:48) ,σ [ u r x ( h gσ (cid:48) ,σ )] i (cid:48) i ,δ (cid:48) x , x = 1 only when x = x modulo a lattice vector,and h gσ (cid:48) ,σ ≡ g − σ (cid:48) t x σ (cid:48) − g ( x σ ) gg σ . (C8)Note that h gσ (cid:48) ,σ ∈ G x when δ (cid:48) x σ (cid:48) ,g ( x σ ) = 1. An AI con-structed from an irrep u r x of G x has a representation U r x of G . Although the discussion here is similar to the deriva-tion of the induced representation U α of G starting froman irrep u α k of G k in B, there is an important difference.That is, the representation U r x is in general reducible,unlike U α which is always irreducible whenever u α k is ir-reducible.Let us focus on a particular k in BZ. The AI’s repre-sentation of G k is immediately given by U r x by restricting G to G k . In particular, its character is χ r x , k ( g ) ≡ tr[ U r x ( g )]= |W x | (cid:88) σ =1 δ (cid:48) x σ ,g ( x σ ) e − i k · ( g ( x σ ) − x σ ) z g,g σ z g σ ,h gσ,σ χ r x ( h gσ,σ ) , (C9)where χ r x ( h ) ≡ tr[ u r x ( h )]. Note that this result does notdepend on the choice of x σ ; even if one chose x (cid:48) σ = x σ + R σ instead, the character is unchanged.Let χ α k ( g ) = tr[ u α k ( g )] be the character of an irrep u α k of G k . Then, the irrep u α k appears in U r x ( g ) n α k = (cid:88) g ∈G k /T |G k /T | [ χ α k ( g )] ∗ χ r x , k ( g ) ∈ Z ≥ (C10)times. This is the formula that gives n for a AI in general.4
3. The general position
As a special case of the above general discussion, let usassume that the site symmetry group G x is trivial, i.e., G x = { e } . In other words, let us assume that x belongsto the general Wyckoff position. There is only one trivialrepresentation u r =1 x ( e ) = 1 for such a generic position.In this case, Eq. (C9) reduces to χ r =1 x , k ( g ) = |G /T | δ g,e . (C11)Therefore, using χ α k ( e ) = tr[ u α k ( e )] = dim[ u α k ], we get n α k | generic position = |G /T ||G k /T | dim[ u α k ] ≥ . (C12)Namely, the AI constructed from the trivial orbital on ageneric position x contains every irrep u α k at least onceat each k .
4. The special position
Next, consider a position x with a nontrivial G x > e .In other words, x belongs to a special Wyckoff position.In this case, there are several irreps u r x . We have a sum-rule separately for each x : n α k | generic position = (cid:88) r :all irreps on x dim[ u r x ] n α k | irrep u r x on x . (C13)
5. TR invariant AIs
To construct a TR invariant AI, we have to determineif the time-reversal (TR) symmetry T enhances the de-generacy of an irrep u r x . This can be easily done by themethod reviewed in B. Namely, one should compute thefollowing sum, which can be either +1, 0, or − η T |G x | (cid:88) h ∈G x z h,h χ r x ( h )= +1 : u r x by itself is TR invariant. − u r x ’s are paired under TR.0 : u r x and ( u r x ) ∗ are different and are paired . (C14)Here, χ r x ( h ) ≡ tr[ u r x ( h )] and η T = − η T = +1) forthe spinful (spinless) fermions. If the sum is either − u r x on the site x alone is not TR symmetric, and one has to make a properstacking with its TR pair.
6. Uniform basis
So far we have presented two constructions of a rep-resentation of G : In B we constructed one from an irrep u α k of G k , and the present note gives another one from anirrep u r x ( h ) of G x . There is yet another construction of arepresentation of G . Suppose that we know a representa-tion v ( p g ) of the point group G /T . Then a representationof G is given by[ U ( g )] σ (cid:48) i (cid:48) k (cid:48) ,σi k = δ (cid:48) x σ (cid:48) ,g ( x σ ) δ (cid:48)(cid:48) k (cid:48) ,p g k e − i k (cid:48) · ( g ( x σ ) − x σ (cid:48) ) [ v ( p g )] i (cid:48) i . (C15)The advantage of this representation is that we have thesame representation v ( p g ) on every site. As a result, theeffect of spin-orbit coupling, for example, can be inter-preted much easily than U r x . Our leSM example discussedin the main text is formulated using this representation.The drawback is that this construction generally requiresas input particular representations of G x , in contrast tothe earlier construction which applies to any representa-tion, in particular to the irreducible ones. Appendix D: Structure of { BS } and Computation of X BS
1. Mathematical details
Here, we discuss some relatively formal aspects of ourmathematical framework. We will start with the claim { BS } ≡ ker C ∩ Z D ⇒ { BS } (cid:39) Z d BS , (D1)where d BS ≡ dim ker C . This claim can be interpreted ge-ometrically: Embedded in R D , Z D is a hypercubic latticeand ker C defines a d BS -dimensional hyperplane. { BS } isthen the collection of lattice sites sliced by ker C , whichdefines a Bravais lattice in d BS dimensions.Alternatively, this can also be understood alge-braically, as we now discuss in details. First observe { BS } ≤ Z D . Since any subgroup of a finitely generatedabelian group is again finitely generated, and that no el-ement in Z D has finite order, we see that { BS } (cid:39) Z d for some d ≤ D . Next, note that as C is a matrixof integer coefficients, its solution space can be identi-fied as a vector subspace of Q D (instead of R D ). Let { q i : i = 1 , . . . , d BS } be any complete basis for ker C (cid:39) Q d BS . Since only a finite number of rational numbers areinvolved, we can always multiply the basis by the least-common multiple of all the denominators to arrive at aninteger-valued basis. This implies { BS } has at least d BS linearly-independent elements, i.e. d ≥ d BS . Now also ob-serve that as { BS } ≤ ker C , ker C has at least d linearly-independent vectors, which implies d BS ≤ d ≤ d BS .Next we discuss the computation of X BS ≡{ BS } / { AI } , where { AI } (cid:39) Z d AI ≤ { BS } denotes thesubgroup of BS arising from AIs. The first step in theanalysis is to compare their ranks. d BS , a property of C ,is determined once all compatibility relations are found. d AI can be computed as follows: Any AI can be under-stood as a stack of those arising from fully occupying5an irrep of the site-symmetry group of a Wyckoff posi-tion. Hence, by focusing on the finite number of AIsarising from these irreps, we can find a (generally over-complete) basis for { AI } using the formalism developedin B and C. Finally we simply extract the number of lin-early independent combinations among them, which isby definition d AI .Generally, we have d AI ≤ d BS , and the general struc-ture of the quotient group is given by X BS = Z d BS − d AI × Z s × Z s × · · · × Z s d AI . (D2)It remains to compute the integers s i ≥
1. In more phys-ical terms, any element of X BS corresponds to a classof BSs that cannot be obtained from integer combina-tions (i.e. stacking) of entries in { AI } . Now consider a b ∈ { BS } with its equivalence class [ b ] being the genera-tor of a factor Z s i in X BS . From definition, s i [ b ] = [ s i b ]is the trivial element of X BS , i.e. s i b is an AI. Runningthe argument in reverse, the torsion (i.e. finite-order) el-ements of X BS correspond to (the classes of) fractions ofAIs that are nonetheless in { BS } , and hence to computethe s i ’s in Eq. (D2) one simply studies the possible set ofcoefficients for which (cid:80) d AI i =1 q i a i , q i ∈ Q , is integer-valued.This can be readily computed using the Smith normalform, which is, loosely speaking, an integer-valued ver-sion of the singular-value decomposition (in our context).An interesting observation is that, for all the 230 × d BS = d AI . This implies a basis for { BS } can be obtained by a suitable combination of the basisof { AI } using rational coefficients, which are found inthe computation of s i described above. Equivalently, wefound that a BS can always be expanded asBS = (cid:88) x (cid:88) r q x ,r n x ,r , (D3)where n x ,r denotes the n of the AI arising from fullyoccupying a site-symmetry group irrep u r x of the posi-tion x (it is sufficient to choose one representative of x for each Wyckoff position), and q x ,r ∈ Q are rationalnumbers. This is very similar to the decomposition ofenergy bands at high-symmetry momenta into irreps ofthe little group, except that this is now performed in aglobal manner over the BZ. An interesting open questionis whether there is any symmetry setting, say when onestudies the remaining magnetic SGs, for which d BS > d AI and therefore leads to an infinite X BS (i.e. some BSs re-main non-atomic no matter how many copies we take).Alternatively, if such equality always holds for any sym-metry settings, there should be a more elegant methodto prove that X BS is always finite.In closing, we comment that the expansion in Eq. (D3)is similar in spirit to the notion of elementary energybands developed in a series of work by Refs. [38–40]. Inour language, these earlier results focus on such a decom-position between the AIs, and whether or not the build- ing blocks of such decomposition, dubbed elementary en-ergy bands, can be split into energy bands of lower fill-ings. These earlier works approached the problem froma real-space perspective, which in our language is aboutthe structure of { AI } . In contrast, our formalism, cen-tered on the structure of { BS } , automatically capturesall the momentum-space constraints and is more suitedfor studying topological band structures: our notion ofa global decomposition allows for quantum interferencein momentum space, which is more general than thatin Refs. [38–40]. Finally, we also remark that the resultsconcerning energy band connectivity in Refs. [38–40]. hasinteresting implications on finding leSMs (specifically, forspinless fermions with TR symmetry).(Note: From a more mathematical perspective, theearlier results in Refs. [38–40] are concerned with a de-composition in terms of the direct sum ⊕ , whereas wehave first made a generalization ⊕ → +, akin to how rep-resentation rings are constructed, and then further assertthat the decomposition is physically meaningful with ra-tional coefficients, as long as the resulting (formal) sumlies in { BS } .)
2. Physical relevance of X BS Having discussed the mathematical aspect of the for-malism, we now comment on why such notions are rele-vant to physical band structures. Recall we have moti-vated the definition of { BS } by asserting that, as long asonly symmetry properties are concerned, a set of energybands can be labelled simply by a count of the multi-plicities of each irrep at the high-symmetry momenta. Apriori, it is insensible to say that an irrep appears a neg-ative number of times. As mentioned in the main text,this leads to an additional condition in connecting the en-tries of { BS } to physical band structures–for n ∈ { BS } tobe physical, all components of n must be non-negative.Equivalently, one can define a physical subset { BS } P ≡ { BS } ∩ Z D ≥ ⊂ { BS } . (D4)As we have explained, all elements in { BS } P enjoy theproperties of a general element in { BS } , say the decompo-sition in Eq. (D3). Therefore, the inclusion of unphysicalentires, crucial for the group properties of { BS } , shouldbe viewed merely as a mathematical way to simplify theanalysis of the physical problem. (If one insists, one canidentify { BS } P as a commutative monoid enjoying sim-ilar properties as the group { BS } . However, we do notfind this perspective particularly useful in our discussion,and would rather stick with { BS } , a simpler mathemat-ical gadget.)However, there are still two points we have to establishin order in order to quantify the relevance of our math-ematical framework to the study of real, physical bandstructures. First, we argue that all entries of { BS } P cor-respond to physically realizable band structures. To see6this, consider an arbitrary element of { BS } P , and let aset of physical bands possess the desired irrep contentat all high-symmetry momentum points. (If only high-symmetry lines or planes are present, we choose arbi-trary, isolated representatives.) This is always possibleby a suitable arrangement of the energies of the irrepsat these isolated momentum points. By symmetries andcontinuity, all compatibility relations are locally satis-fied near these points. Suppose there are still certainband crossings obstructing us from identifying this set ofbands as a BS. Since any pair of bands carrying the samesymmetry representation will generically anti-cross, theseband crossings must correspond to an exchange of irrepsbetween our target set of bands and the others. As allcompatibility relations are globally satisfied by assump-tion, such exchange must be accidental in nature, i.e. itis possible to perturb the Hamiltonian in a symmetricfashion to get rid of all the band crossings. This thenleads to a BS corresponding to the specified element in { BS } P .Note that, however, there is a subtlety in the statementon generic anti-crossing: Topological band degeneracies,like Weyl points, can only be pushed away but not lifted,unless their topological charges are neutralized by theirpartners. This does not concern us, since by our defi-nition of BS we will only be interested in band gaps athigh-symmetry momenta, and therefore as long as thesedegeneracies can be moved away they do not affect ourdiscussion. This also explains why the notion of reSM,which cannot be insulating due to the specified repre-sentation content, is still consistent with our notion ofBS.Second, we show that all nontrivial classes in X BS havephysical representatives. Let b be a representative of anontrivial class in X BS which is not physical, i.e. cer-tain entries in b are negative. Now, we consider a smallcorollary from C: all irreps appear at least once in theAI corresponding to the generic Wyckoff position [seeEq. (C12)]. Therefore, we can always stack b with asufficiently large number of copies of the generic AI andrectify the representation content. By definition, suchstacking leads to a physical BS belonging to the samenontrivial class as b , and hence all classes of X BS havephysical representatives.
3. Lower dimensional systems
Here we discuss X BS for (quasi-)1D and 2D systems.As far as spinful electrons are concerned, the fact thatlower dimensional systems in reality are embedded in the3D space cannot be neglected, since the electronic spindegree of freedom is a projective representation of O (3),the rotation of the 3D space. For brevity we will focuson 2D systems but the 1D case can be discussed in thesame way.The symmetry groups for 2D lattices lying in the 3Dspace are called layer groups. An element h of a layer group L maps ( x, y, z ) to ( x (cid:48) , y (cid:48) , z (cid:48) ), where ( x (cid:48) , y (cid:48) ) = q h ( x, y ) + s h ( q h is a O (2) matrix and s h is a two com-ponent vector) and z (cid:48) = ξ h z ( ξ h = ± T of L is a group of lattice translations inthe 2D plane, giving rise to the 2D crystal momentum( k x , k y ). (At this moment k z is not defined.) One can fol-low the same steps as in 3D to define { BS L } and { AI L } ,and X L BS = { BS L } / { AI L } for a layer group L .To make use of our results established for space groups,let us consider the space group G corresponding to a layergroup L , which is simply the layer group L endowed witha lattice translation T z along z . More precisely, G is givenby the semi-direct product of L and T z , whose element g = ( h, t ) ∈ G ( h ∈ L and t ∈ T z ) maps ( x, y, z ) to( x (cid:48) , y (cid:48) , z (cid:48) ), where ( x (cid:48) , y (cid:48) ) = q h ( x, y ) + s h and z (cid:48) = ξ h z + t .The product of g = ( h, t ) ∈ G and g (cid:48) = ( h (cid:48) , t (cid:48) ) ∈ G isdefined as gg (cid:48) = ( hh (cid:48) , t + ξ h t (cid:48) ). We list the correspond-ing space group G for each layer group L in Table V.Note that there is one different entry when our tableis compared to the one provided in Ref. [43]: we foundthat the correct correspondence for layer group 35 shouldbe space group 38. To avoid confusion, we use the no-tations { BS G } , { AI G } , and X G BS = { BS G } / { AI G } for aspace group G in this section.Given a layer group L and the corresponding spacegroup G , we expect X G BS to encapsulate that of X L BS :stacking lower-dimensional nontrivial phases by transla-tion symmetries will naturally give rise to weak topolog-ical phases, which are nontrivial as long as the transla-tion symmetries remain intact, i.e., a nontrivial BS of L will never become trivial when lifted to G upon stack-ing. Generally, we also expect X G BS to be richer than X L BS , since certain strong phases should become possi-ble. These observations can be summarized by assertingthe subgroup relation X L BS ≤ X G BS (we will explain shortlythe precise meaning of this symbolic relation). In thefollowing, we formalize these observations and provide a(technical) proof for this relation. Before we dwell intothe technical details, we remark that, given this natu-ral subgroup relation, the finiteness of X G BS implies thatof X L BS , and therefore X L BS can be readily computed us-ing only the data on AI without finding and solving thecompatibility relations.Let us introduce a group homomorphism f : { BS L } →{ BS G } through stacking of layers. Namely, starting froma given b ∈ { BS L } of a layer group L , we can get a bandstructure of G by stacking infinite copies of b in the z direction. To make this idea more concrete, let us takea tight-binding model ˆ H L symmetric under L such thatthe lowest ν bands of ˆ H L are isolated from other bandsby a band gap at all high-symmetry points of the 2DBZ, and the combination of irreps of the lowest ν bandsprecisely agrees with b . Given ˆ H L , one can generate a G symmetric tight-binding model byˆ H G = (cid:88) t ∈ T z ˆ t ˆ H L ˆ t − . (D5)Since there is no inter-layer hopping, the band structure7of ˆ H G is completely flat as a function of k z , and as a re-sult, the lowest ν bands remain isolated from other bandsby a band gap at all high-symmetry points of the 3DBZ. This band structure of ˆ H G defines f ( b ) ∈ { BS G } .By construction, the homeomorphism f ( b + b ) = f ( b )+ f ( b ) is obvious. Furthermore, f maps a ∈ { AI L } to f ( a ) ∈ { AI G } . Therefore, f defines a group homomor-phism f : X L BS → X G BS (D6)Below we show that this f is injective. If this is the case,˜ X L BS ≡ Im f is isomorphic to X L BS and is a subgroup of X G BS .To this end, let us introduce a projection p that de-fines a homomorphism from { BS G } to { BS L } . Given B ∈ { BS G } , one can project out all entries of B asso-ciated with high-symmetry points with k z (cid:54) = 0, keepingonly entries associated with high-symmetry points with k z = 0. By definition p ( B ) satisfies all compatibility con-ditions imposed on the irreps of L and hence is an elementof { BS L } . The projection p in fact acts as an inverse of f : p ( f ( b )) = b . Furthermore, p maps A ∈ { AI G } to p ( A ) ∈ { AI L } . This can be seen by the fact that, forevery Wyckoff position of G , there exists a Wyckoff po-sition of L that is either identical, or differs only by thevalue of z . The difference of z does not affect p ( A ) sincethe projection p sets k z = 0. Given these properties of p ,it is easy to prove that ker f = { e } . If not, there must bea b ∈ { BS L } belonging to a nontrivial class of X L BS whichis mapped to A ∈ { AI G } by f . Then b = p ( A ) ∈ { AI L } is an AI, contradicting with the assumption that b isnontrivial. Hence the proof. Appendix E: Example of Lattice-enforcedSemimetals
Here, we provide details on the leSM example dis-cussed in the main text. We consider a TR-symmetricsystem in SG 219 with significant spin-orbit coupling.We will establish that for a particular lattice specifica-tion, a semimetallic behavior is unavoidable at a filling ν = 4 although band insulators are generally possible atthis filling for the present symmetry setting [36, 37]. Thisarises from the fact that, given the available symmetry ir-reps specified by the lattice, corresponding to an element A ∈ { AI } , there is no way to satisfy all the compatibilityrelations at the filling ν = 4, i.e. A (cid:54) = B v + B c for anynon-zero B v , B c ∈ { BS } satisfying the physical conditionof non-negativity.We consider a lattice in Wyckoff position a , which con-tains two sites at r ≡ { , , } and r ≡ { / , , } inthe unit cell. The two sites are related by a glide symme-try, and the site-symmetry group for each site is givenby the point group T (i.e., the orientation-preservingsymmetries of a tetrahedron, also known as the chiraltetrahedral symmetry group). We suppose the physicallyrelevant degrees of freedom arise from the three p x,y,z TABLE V.
Correspondence between space groups andlayer groups.
LG SG LG SG LG SG LG SG1 1 21 18 41 51 61 1232 2 22 21 42 53 62 1253 3 23 25 43 54 63 1274 6 24 28 44 55 64 1295 7 25 32 45 57 65 1436 10 26 35 46 59 66 1477 13 27 25 47 65 67 1498 3 28 26 48 67 68 1509 4 29 26 49 75 69 15610 5 30 27 50 81 70 15711 6 31 28 51 83 71 16212 7 32 31 52 85 72 16413 8 33 29 53 89 73 16814 10 34 30 54 90 74 17415 11 35 38 55 99 75 17516 13 36 39 56 100 76 17717 14 37 47 57 111 77 18318 12 38 49 58 113 78 18719 16 39 50 59 115 79 18920 17 40 51 60 117 80 191LG: Layer Group; SG: Space Group. orbitals on each site, which together with electron spinleads to a 6-dimensional local Hilbert space. We will let L and S respectively denote the orbital and spin angularmomentum operators in the single-particle basis.As described in the main text, we consider a TR-symmetric system with a strong crystal-field splitting: H ∆ = ∆ (cid:88) r :all sites c † r ( L · S ) c r , (E1)where c r represent the 6-dimensional (row) vector cor-responding to the internal degrees of freedom. One canverify that when ∆ > H ∆ splits the local energy levelsto a total spin-1/2 doublet lying below the total spin-3/2multiplet. While we have chosen H ∆ to conserve thetotal spin L + S for convenience, such conservation is notrequired by the local symmetry, which is described bythe point group T <
SO(3). Therefore, the total spinquantum numbers are not a priori good quantum num-bers for the problem at hand. However, one can verifythat the spinful, time-reversal symmetric irreps of T co-incide with the total spin decomposition described above[25], and hence insofar as symmetries are concerned H ∆ is a sufficiently generic crystal-field Hamiltonian. We alsonote that, if time-reversal symmetry is broken, the four-fold degenerate states originating from the total spin-3/2states can be further split.As discussed in the main text, we are interested in the8system arising from half filling the four-fold degeneratelocal energy levels. To this end, we assume ∆ is thedominant energy scale in the problem, which implies thelow-lying doubly degenerate states can be decoupled fromthe description of the system as long as they are fully-filled. This leaves behind the four-fold degenerate energylevels, which we assume are half-filled. As there are twosymmetry-related sites in each unit cell, these consider-ations altogether imply that the band structure aroundthe Fermi-energy is described by an effective eight-bandtight-binding model at filling ν = 4.Next we consider a nearest-neighbor hopping term H t,λ = (cid:88) g ∈G g (cid:0) c † r ( t + λ ˆ x · ( L × S )) c r (cid:1) g † + h . c ., (E2)where h . c . denotes Hermitian conjugate, and the nota-tion (cid:80) g ∈G g ( . . . ) g † denotes all the terms generated bytransforming the bond in the parenthesis by the symme-try elements of the SG G .The band structure of the full Hamiltonian H = H ∆ + H t,λ is shown in Fig. 2e, with parameters ( t/ ∆ , λ/ ∆) =(0 . , . O (∆). As our computation dictates,the lattice specification gives rise to energy bands thatare necessarily gapless along the high-symmetry lines atfilling ν = 4. Interestingly, note that the lattice-enforcedgaplessness is of a more subtle flavor: Unlike spinlessgraphene, where the the gaplessness is enforced by thedimensions of the irreps involved, here all the irreps havedimensions ≤
4, and therefore the impossibility of find-ing a BS at ν = 4 is reflected in the connectivity of theenergy bands.In closing, we remark that the notion of leSM is not asrobust as the other notions we introduced in this work,say feSM or reQBI. Specifically, the (semi-)metallicbehavior of the system is protected by the assumedmicroscopic degrees of freedom, which is only sensibleassuming a certain knowledge about the energetics of theproblem. Under stacking of a trivial phase, say when weincorporate into the description a set of fully filled bandscorresponding to an atomic insulator, the enforced gap-lessness may become unstable as these apparently inertdegrees of freedom can also supply the representationsneeded to open a gap at the targeted filling. This can bereadily seen from the example above: If we switch thesign of ∆, the same electron filling will now correspondto the full filling of the four-fold degenerate multipleton each site, which leads to an atomic insulator. Suchinstability should be contrasted with, say, the notion ofreQBIs, which by definition remains nontrivial as longas the extra degrees of freedom we introduce are in thetrivial class, i.e. correspond to AIs. Appendix F: Filling-enforced Quantum BandInsulators
In this note we provides details on the feQBIs found inthis work, which we briefly mentioned in the main text.A band insulator, invariant under a set of symmetries(including an SG G ), is called a feQBI if the number ofthe occupied bands (i.e., the filling) is different from thatof any AIs with the same symmetry. Thus feQBIs area special case of reQBIs. As all possible TR-symmetricfeQBIs have been discussed in Ref. [30], we will mainlyconsider systems without TR invariance.
1. TR-breaking spinless filling-enforced quantumband insulators
Let us start with feQBIs in the system of spinlessfermions, which necessarily break the TR symmetry.This is because of the following reason: If there existeda TR-symmetric feQBI for spinless fermions, we couldimmediately construct a TR-symmetric feQBI for spin-ful fermions with the spin SU(2) symmetry. However, weknow that the latter does not exist according to Ref. [30].As a strategy to find feQBIs, one can focus on the elec-tron fillings among elements of { BS } and { AI } . When-ever there is a mismatch between them, it indicates thatcertain nontrivial element of X BS can be diagnosed sim-ply using the electron filling. This question can again betackled efficiently using the vector-space like structure of { BS } and { AI } . We indeed found a mismatch for severalSGs for spinless fermions without TR symmetry.The mismatch can possibly occur only in those 12 SGslisted in Table VI. To see this, we should focus on (themaximal) fixed-point-free subgroups Γ of G (not to beconfused with the Γ point in BZ). For a fixed-point-freeSG Γ, it is easy to show that a certain number of bandsmust cross with each other somewhere at high-symmetrypoints or on high-symmetry lines, and that the BI fillingsare integer multiples of ν Γ >
1. Hence, if G contains Γas a subgroup, we know that any G -symmetric BI shouldhave a filling ν Γ n . For all 218 SGs not listed in Table VI,there exists a G -symmetric AI with the filling ν Γ . Hence,there cannot be any mismatch of the filling between { BS } and { AI } for them. On the other hand, for 8 SGs out ofthe 12 SGs in Table VI, we found that the filling of theelements of { BS } is 2 Z , while that of { AI } is 4 Z . Hencewe can expect a feQBI at filling ν = 4 n + 2.The case of SG 220 is more nontrivial. In Table VI,we show the set of fillings S AI G ( S BI G ) that correspond toat least one AI (BI). This set was computed by the fol-lowing way. Let ν x ,r be the filling of the AI arising fromfully occupying a site-symmetry group irrep u r x of the po-sition x . Then we take superpositions with non-negativeinteger coefficients: S AI G = { (cid:80) x ,r m x ,r ν x ,r : m x ,r ∈ Z ≥ } , which is, in principle, different from the fillings for { AI } ∩ Z D ≥ . On the other hand, S BI G is simply the fillingsfor { BI } ∩ Z D ≥ . In the case of G = 220, although the9filling for { AI } and { BS } are both 2 Z , S AI G and S BI G donot agree with each other and there is a feQBI at filling ν = 4: S AI G = 2 Z ≥ \ { , , } = { , , , , , , . . . } , (F1) S BI G = 2 Z ≥ \ { } = { , , , , , , , , . . . } . (F2)Table VII summarizes TR breaking spinless feQBIs in8+1 = 9 SGs identified above. Note that these new feQBIexamples are, in a sense, more intriguing than the ones wediscussed in Ref. [30]. In our previous examples assumingTR-symmetric spinful electrons, the filling condition wasenriched by Kramers degeneracy, and feQBIs were discov-ered at odd-integer site fillings. In these new examples wediscovered, however, the site filling is fractional for anychoice of SG-symmetric lattices. Superficially, this mightappear to be at odds with the conventional wisdom thatgapped phases at fractional fillings are associated with ei-ther discrete symmetry breaking or intrinsic topologicalorder–both only possible in the presence of interactions.Rather, these examples highlight the fact that spatialsymmetries can lead to intrigue constraints between fill-ing and phases, as was discussed in Ref. [36], which, infact, correctly predicted that symmetry-protected topo-logical phases might be possible for these systems at suchfractional fillings.
2. A tight-binding model
As we have cautioned before, a nontrivial BS can gen-erally be either a reSM or a reQBI. To establish thatthe filling mismatch indeed leads to feQBIs, one has tofurther assert that a band gap is possible at all genericmomenta in the BZ. To achieve this goal we explicitlyconstructed a tight-binding model for each of feQBIs inTable VII. As a concrete example, let us discuss the caseof SG 106, which hosts feQBI at filling ν = 2.Consider a tight-binding model (4 band model) definedon W b with one orbital per site. Here, W b means theWyckoff position with the Wyckoff letter b in Ref. [33]:(0 , , z ) , ( , , z + ) , ( , , z ) , (0 , , z + ) (F3)The site symmetry of this Wyckoff position is the π -rotation about the z -axis. We use a p orbital that flipssign under the rotation. These informations specify therepresentation [ U k ( g )] σ (cid:48) j,σi ≡ [ U r x ( g )] σ (cid:48) j k (cid:48) ,σi k of G = 106in the tight-binding model as discussed in C. One canconstruct U k ( g ) using Eq. (C7). (In this particular ex-ample, i, j label can be dropped since the orbital is a 1Drepresentation). In this notation, U k ( g ) satisfies multi-plication rule U p g k ( g ) U k ( g ) = U k ( g g ).We construct a G -symmetric Hamiltonian H k in twosteps: First, we choose an arbitrary 4 by 4 Hermitianmatrix h k ; next, we symmetrize h k by performing thesummation: H k = (cid:88) g ∈G /T U k ( g ) † h p g k U k ( g ) . (F4) After the summation, H k automatically fulfills the sym-metry requirement, i.e., H p g k U k ( g ) = U k ( g ) H k .To realize an example of feQBI for 106, the followingchoice of h k works h k = ∆ (cos k x − cos k y ) 1 0 − i − ii i , (F5)where ∆ sets the energy scale of the problem. Thesymmetrized Hamiltonian H k has a very large gap. Ateach k , let δE k be the band gap between the secondand the third band and W k be the band width (the en-ergy difference between the lowest and the highest band).Then we found δE = 0 . W where δE = min k δE k and W = max k W k . An example band structure is plotted inFig. 3, which demonstrates that even at a filling of half anelectron per site, a band gap is nonetheless possible. Wealso found that the Chern numbers of the tight-bindingmodel is zero. (a) (b) Local energylevels and filling
Bulk gap
FIG. 3.
Filling-enforced quantum band insulators inunconventional symmetry settings.
Here, we focus onone such example found among spinless systems in the tetrag-onal space group 106 without time-reversal symmetry. (a)For this space group, a maximal-symmetry site (red sphere)is invariant under a C rotation, and symmetries require thatthere are at least four such sites in the unit cell. We considera filling of ν = 2, corresponding to a site filling of 1 /
2. Notethat, given the space group symmetries, any other choice oflattices correspond to site fillings ≤ /
2. (b) Nonetheless, aband insulator is possible at such filling, as shown in the plot-ted band structure. Each band shown is doubly-degenerate,but such degeneracy originates from nonessential additionalsymmetries in the simple tight-binding model we constructed,and therefore can be lifted.
In Table VIII, we list some possible symmetry settingsfor feQBIs in other SGs.
3. TR-symmetric filling-enforced quantum bandinsulators
Finally, we briefly comment on the implications of thepresent work on the feQBIs we introduced in Ref. [30],which arise in systems with TR symmetry and significantspin-orbit coupling. There, the focus of study are SGs199, 214, 220 and 230, which have an intriguing propertyin the ratios between the Wyckoff position multiplicities:They were dubbed Wyckoff-mismatched as some Wyck-off multiplicities are not integer multiples of the smallest0
TABLE VI.
Band insulator fillings for some special space groups
Space group
G S AI G S BI G S Γ G Z ≥ Z ≥ Z ≥ Z ≥ Z ≥ \ { } Z ≥
135 4 Z ≥ Z ≥ Z ≥ Z ≥ \ { } Z ≥ \ { } Z ≥
220 2 Z ≥ \ { , , } Z ≥ \ { } Z ≥
230 4 Z ≥ \ { } Z ≥ \ { , } Z ≥ AI: Atomic Insulator; BI: Band Insulator; S AI G : the set of fillings for AI; S BI G : the set of fillings for BI; S Γ G : is the set of BIfillings for fixed-point free subgroup Γ of G .Note–By definition, we have S AI G ≤ S BI G ≤ S Γ G , and in fact S AI G = S BI G = S Γ G for all 218 SGs not listed here.TABLE VII. Summary of results for the 12 spacegroups listed in Table VI .Space group G X BS Generator of X BS Z feQBI at filling 273,133,142, 206, 228 Z feQBI at filling 6135 1 —199, 214 1 —220 Z feQBI at filling 4230 Z feQBI at filling 6, 10 X BS : the quotient group between the group of bandstructures and atomic insulators. one. In these systems, it was realized that QBIs are pos-sible at non-atomic fillings, and hence the name feQBIs.However, in the analysis of Ref. [30] the mismatch be-tween BSs and AIs is discussed in terms of their corre-sponding physical fillings, which as we have explained isa stronger condition than that exposed using only theabelian group structures of { BS } and { AI } . As a result,these feQBIs could be trivial in X BS . This is indeed thecase for some of them: both 199 and 214 have no non-trivial BSs in X BS = Z . (By our convention, they areomitted from Table III of the main text.)It remains to study 220 and 230, which have X BS = Z and Z respectively. We found that for both cases, somefeQBIs are again in the trivial class, i.e. they can be un-derstood as integer combinations of AIs (but with neg-ative coefficients, resulting in unrealizable AI upon en-forcing the physical conditions). However, the nontriv-ial class for 220 can be represented by a feQBI filling ν = 20. More interestingly, the entry 2 ∈ Z for 230can be represented by a feQBI at filling ν = 8. Since230 is centrosymmetric, as we have argued in the maintext the generator 1 ∈ Z has to be identified with thestrong TI. This implies some ν = 8 feQBIs in 230 re-alize the doubled strong TI phase we discussed in the main text, which has inversion-protected nontrivial en-tanglement signature, albeit no physical surface state isexpected.All in all, we found that generally the TR-symmetricfeQBIs do not have any simple relationship with X BS ,although there are indeed examples which are nontrivialfrom both perspectives. Appendix G: Representation-enforced QuantumBand Insulators and Semimetals1. General relation
As we have discussed in the main text, our notion of aBS is compatible with both band insulators and semimet-als, as long as a continuous band gap is sustained at allhigh-symmetry momenta. In particular, the nontrivialentries in X BS can sometimes correspond to reSM, whichare guaranteed to be semimetallic due to the specificationof the symmetry content. These systems are exemplifiedby the 3D systems with inversion but not TR symmetries[13, 14].Since reSMs are, by definition, also diagnosable usingthe representation content, one can systematically iso-late them from { BS } . However, it is important to real-ize that reSMs do not form a subgroup of { BS } , sincestacking two of them (say) may lead to a band insula-tor [13, 14]. In contrast, it is guaranteed that stackingtwo band insulators will lead to yet another band insula-tor. This suggests that we should identify the subgroup { BI } ≤ { BS } . Further observe { AI } ≤ { BI } , it is naturalto define the following further diagnosis of the elementsin { BS } : X SM ≡ { BS }{ BI } ; X BI ≡ { BI }{ AI } . (G1)The nontrivial entries in X SM and X BI respectively cor-respond to reSMs and reQBIs. As we have alluded to, thenontrivial elements in X BS are either reSMs or reQBIs,1 TABLE VIII.
Possible symmetry setting for the filling-enforced quantum band insulators in Table VII.
Space group Wyckoff position orbital m tot ν b p ∗ a p ∗ a + e s
12 6220 c s > d s
12 6230( > a s + p ∗∗
16 6,10feQBI: filling-enforced quantum band insulators; m tot : the total number of bands in this setting; ν : electron fillings perprimitive unit cell for which feQBIs are possible; ∗ The site symmetry group for the Wyckoff position b of 106 and the Wyckoff position a of 110 are an order 2 group, and the p orbital refers to the representation with − ∗∗ The site symmetry group for the Wyckoff position a of 230 is ¯3 group, generated by the improper three-fold rotation IC .The p orbital refers to the one that has e i π eigenvalue of IC . and hence, unsurprisingly, the three objects X BS , X SM and X BI are not independent. From definitions, one cancheck that X SM = X BS X BI . (G2)Or in a more formal language, X BS can be viewed asa central extension of X SM by X BI . Curiously, this ex-tension is generally nontrivial. As a concrete example,consider the Z factor in X BS = ( Z ) × Z for SG2 (inversion only) assuming no TR symmetry. FromRefs. [13, 14], we see that the generator of this factoris a reSM (with inversion related Weyl points), and thetwice of that, corresponding to 2 ∈ Z , is a reQBI with aquantized magnetoelectric response of θ = π . This showsthat for this particular example, X BI = ( Z ) ; X SM = Z , (G3)and the Z factor in X BS originates from the nontrivialextension of Z by Z .
2. Representation-enforced quantum bandinsulators
As we will discuss in the following, given any symmetrysetting one can systematically study all reSMs using sym-metry arguments together with knowledge on the genericstability of Femri surfaces. For instance, for centrosym-metric systems with TR symmetry and significant spin-orbit coupling, stable band degeneracy must happen ata high-symmetry momentum, and therefore we can ruleout the possibility of reSMs in these problems, i.e. forsuch settings we have X SM = Z and hence X BI = X BS .However, even after X BI is obtained our approach is stillbased on symmetry-labels in nature, and therefore doesnot necessarily detect all nontrivial phases. An impor-tant future direction is to incorporate the tenfold way classification into our current framework [22], akin to thearguments given in Ref. [28].
3. Representation-enforced semimetals
For a tight-binding model with inversion but not TRsymmetry, some combinations of the parity eigenvaluesat TRIMs predict the existence of Weyl points somewherein the interior of the BZ [13, 14]. The semimetallic behav-ior of such systems is enforced by the specification of therepresentations, and we refer to such systems as reSMs.In this section, we ask whether similar phenomena occurfor other SG, i.e., given an SG and a set of integers n forirreps at high-symmetry momenta, we ask if there areenforced gap closing somewhere at non-high-symmetrypoints in the BZ. For simplicity, in the following we an-swer this question for three-dimensional system withoutTR symmetry, such that the topologically-protected gapclosing at non-high-symmetry momenta corresponds toWeyl points.Let us consider a unit sphere S around the Γ pointof the BZ. The radius of the sphere is set to be muchsmaller than any reciprocal lattice vectors. If there areWeyl points at generic momenta in the BZ (i.e. not high-symmetry), we should be able to move them to the sur-face of this sphere without breaking any symmetry orchanging the value of n . Note that, in the following‘sphere’ always refer to S , which does not include theinterior of the sphere.An SG element g ∈ G moves a point k on S to p g k on S . For this transformation, the translation part t g of g does not show up at all and hence what is reallyimportant is the point group P (cid:39) G /T of the SG, whoseelements are the orthogonal matrices p g ( g ∈ G ). Thereare only 32 crystallographic point groups in 3D, and wewill systematically discuss them in the following. Onecan introduce the notion of the little group, the symmetry2orbit, and Wyckoff positions to this S in the same wayas before. Given k on S , the little group G k is definedas G k = { p g ∈ P : p g k = k } . (G4)The symmetry orbit of k (aka the star of k ) is defined as { p g k : p g ∈ P} . Also, two points k and k belong tothe same (momentum-space) Wyckoff position iff thereexists g ∈ G such that G k = p g G k p − g . Finally, let usdefine the irreducible part (or the fundamental domain) F of the sphere. The fundamental domain F tessellatesthe S under the action of P . Namely, any point on thesphere can be uniquely represented as p g k , where k ∈ F and p g ∈ P .With this preparation, let us discuss the stability of theWeyl points against symmetry preserving deformations.Suppose that the fundamental domain F contains a Weylpoint with the chirality +1. (By assumption, genericallythis Weyl point does not sit at the boundary of F .) Thenthere must be at least |P| Weyl points on the sphere intotal. The domain p g F will contain a Weyl point withthe chirality det[ p g ]. Suppose first that det[ p g ] = +1 forall p g ∈ P . This is the case for 11 out of 32 point groupsin 3D. In this case, since the net chirality in the entire BZmust vanish, there must be another Weyl point in F withthe chirality − −
11 = 21 pointgroups, which always have half of the elements havingdet[ p g ] = −
1. Again the number of symmetry-relatedWeyl points is |P| , but now |P| / − S and merge pairs of them with oppositechirality without breaking the SG symmetry or changingthe number of irreps n . We found that 18 of the 21remaining point groups contain at least one mirror re-flection symmetry and there exists a Wyckoff position on the sphere which is symmetric only under the mirror.For these point groups, two Weyl points with the oppo-site chirality should be able to annihilate at this Wyckoffposition. Provided this is true, this again rules out reSMsin these settings. In the following, we will first assumethe validity of this claim and study the remaining pointgroups; a more thorough analysis is left for future works.The remaining three point groups are ¯1, ¯3, and ¯4 inthe Hermann-Mauguin notation. Among them, ¯1 is thepoint group generated by the inversion symmetry, andwe know that Weyl points can be protected by n in thiscase [13, 14]. ¯3 contains ¯1 as a subgroup, and there-fore also contains the rotation C . Due to the relationbetween the C eigenvalues and the Chern number (forany momentum plane perpendicular to the rotation axis)[15], one can show that the only reSMs for ¯3 are thosethat are diagnosed by the ¯1 subgroup. As such, the onlypossibly nontrivial candidate is thus the point group ¯4,which is the point group of the SG symmetry 81 and 82.We looked at 81 for the spinful but TR breaking setting,and indeed found an reSM. X BS for 81 is ( Z ) × Z .The reSM is the generator of one of the two Z factor.In fact, this reSM in SG 81 can also be understood fromthe Chern number arguments in Ref. [15].In closing, we remark that the analysis of reSM hasto be modified in other symmetry setting, since depend-ing on symmetries topologically stable gaplessness maynot be of the Weyl type. For instance, nodal lines arestable in time-reversal and inversion symmetric systemswith negligible spin-orbit coupling, and therefore the ar-gument above has to be modified from the motion ofpoints to nodal rings. In addition, similar to the point-group symmetries we discussed, TR will also constrainthe possible distribution of the gap closing, and thereforehave to be taken into account in the analysis of reSMs inTR-symmetric system. Appendix H: Supplementary tables
Here, we provide tables for the computed values of d ≡ d BS = d AI and X BS for 3D systems without TRsymmetries, and for quasi-2D and 1D systems describedrespectively by layer group and rod group symmetries.3 TABLE IX.
Characterization of band structures for systems with significant spin-orbit coupling and no time-reversal symmetry. d Space groups1 1, 4, 7, 9, 16, 19, 22, 23, 25, 26, 27, 29, 33, 36, 38, 39, 42, 44, 45, 76, 78, 93, 101, 105, 109, 110144, 145, 169, 170, 180, 1812 8, 21, 31, 35, 37, 41, 43, 46, 80, 92, 94, 96, 97, 98, 102, 106, 107, 108, 161, 208, 2143 5, 6, 18, 20, 30, 32, 34, 40, 48, 50, 56, 59, 61, 62, 68, 70, 73, 89, 99, 103, 146, 151, 152, 153, 154, 160171, 172, 178, 179, 185, 186, 195, 196, 197, 198, 209, 210, 211, 212, 2134 24, 28, 54, 57, 60, 72, 77, 90, 91, 95, 100, 104, 133, 137, 142, 155, 158, 159, 177, 183, 184, 199, 2075 3, 14, 17, 52, 63, 64, 67, 79, 111, 112, 115, 116, 119, 120, 121, 126, 130, 134, 138, 156, 157, 173, 1826 11, 15, 49, 69, 71, 113, 114, 117, 118, 125, 129, 132, 135, 141, 149, 150, 215, 216, 217, 218, 2197 13, 51, 55, 66, 74, 122, 131, 136, 143, 167, 220, 228, 2308 58, 65, 75, 88, 140, 163, 165, 222, 223, 2249 2, 47, 53, 86, 139, 168, 201, 203, 205, 206, 22710 12, 187, 189, 193, 194, 202, 204, 22611 82, 85, 124, 148, 166, 200, 225, 22912 81, 127, 128, 162, 164, 188, 19013 84, 123, 147, 19214 191, 22115 1016 87, 17621 17424 8327 175 d : the rank of the abelian group formed by the set of band structures. TABLE X.
Characterization of band structures for systems of spinless fermions without time-reversal symmetry. d Space groups1 1, 4, 7, 9, 19, 29, 33, 76, 78, 144, 145, 169, 1702 8, 31, 36, 41, 43, 80, 92, 96, 110, 1613 5, 6, 18, 20, 26, 30, 32, 34, 40, 45, 46, 61, 106, 109, 146, 151, 152, 153, 154, 160, 171, 172, 178, 179, 198, 2122134 24, 28, 37, 39, 60, 62, 77, 91, 95, 102, 155, 158, 159, 185, 186, 199, 2105 3, 14, 17, 27, 42, 44, 52, 56, 57, 79, 94, 98, 101, 104, 108, 156, 157, 173, 182, 196, 197, 2146 11, 15, 35, 38, 54, 70, 73, 100, 103, 105, 107, 149, 150, 1847 13, 22, 23, 59, 64, 68, 90, 114, 122, 142, 143, 167, 180, 181, 195, 208, 209, 211, 2208 21, 58, 63, 75, 88, 97, 113, 130, 137, 163, 165, 183, 2199 2, 25, 48, 50, 53, 55, 72, 86, 99, 117, 118, 120, 133, 135, 141, 168, 205, 207, 216, 217, 218, 228, 23010 12, 74, 93, 116, 119, 121, 126, 138, 177, 203, 206, 21511 66, 82, 85, 148, 166, 201, 222, 22712 51, 81, 89, 112, 115, 129, 134, 136, 162, 164, 188, 19013 16, 67, 84, 111, 125, 147, 193, 194, 202, 204, 223, 22414 49, 128, 22615 10, 69, 71, 132, 140, 187, 18916 87, 17617 124, 192, 200, 225, 22918 65, 127, 131, 13921 17422 22124 83, 19127 47, 123, 175 d : the rank of the abelian group formed by the set of band structures. TABLE XI.
Symmetry-based indicators of band topology for systems with significant spin-orbit coupling andno time-reversal symmetry. X BS Space groups Z
3, 11, 14, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 70,72, 73, 74, 77, 79, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 125, 126, 129, 130, 133,134, 137, 138, 141, 142, 162, 163, 164, 165, 166, 167, 171, 172, 201, 203, 205, 206, 215, 216, 217, 218, 219,220, 222, 224, 227, 228, 230 Z Z
69, 71, 75, 124, 128, 132, 135, 136, 140, 202, 204, 223, 226 Z Z Z × Z
12, 13, 15, 51, 55, 86, 88 Z × Z
65, 84, 85, 131, 148, 200 Z × Z Z × Z Z × Z Z × Z
87, 127 Z × Z Z × Z Z × Z × Z
10, 82 Z × Z × Z Z × Z × Z Z × Z × Z Z × Z × Z Z × Z × Z Z × Z × Z × Z
2, 47 X BS : the quotient group between the group of band structures and atomic insulators.TABLE XII. Symmetry-based indicators of band topology for systems of spinless fermions without time-reversalsymmetry. X BS Space groups Z
3, 11, 14, 27, 37, 45, 48, 49, 50, 52, 53, 54, 56, 58, 60, 61, 66, 68, 70, 73, 77, 79,103, 104, 106, 110, 112, 114, 116, 117, 118, 120, 122, 126, 130, 133, 142, 162, 163, 164, 165, 166, 167, 171,172, 184, 201, 203, 205, 206, 218, 219, 220, 222, 228, 230 Z Z
75, 124, 128 Z Z × Z
12, 13, 15, 86, 88 Z × Z
84, 85, 148 Z × Z Z × Z Z × Z Z × Z × Z
10, 82 Z × Z × Z Z × Z × Z Z × Z × Z Z × Z × Z Z × Z × Z × Z X BS : the quotient group between the group of band structures and atomic insulators. TABLE XIII.
Characterization of band structures for quasi-two-dimensional systems with significant spin-orbitcoupling and time-reversal symmetry. d Layer groups1 1, 3, 4, 5, 8, 9, 10, 11, 12, 13, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 3435, 362 39, 43, 45, 46, 54, 56, 58, 603 7, 15, 16, 17, 38, 40, 41, 42, 44, 48, 49, 50, 53, 55, 57, 59, 68, 704 18, 47, 52, 62, 64, 65, 67, 69, 73, 76, 775 2, 6, 14, 37, 63, 796 66, 71, 727 74, 788 51, 619 75, 80 d : the rank of the abelian group formed by the set of band structures.TABLE XIV. Characterization of band structures for quasi-two-dimensional systems with negligible spin-orbitcoupling and time-reversal symmetry. d Layer groups1 1, 5, 9, 12, 332 4, 10, 13, 29, 32, 343 8, 11, 17, 21, 25, 28, 30, 31, 364 15, 16, 20, 24, 35, 43, 45, 65, 68, 705 2, 3, 7, 54, 56, 58, 60, 67, 696 18, 22, 26, 27, 39, 42, 44, 46, 49, 50, 52, 66, 738 40, 71, 72, 74, 76, 77, 799 14, 19, 23, 38, 41, 48, 53, 55, 57, 59, 62, 6410 6, 63, 7812 47, 51, 7516 8018 37, 61 d : the rank of the abelian group formed by the set of band structures.TABLE XV. Characterization of band structures for quasi-two-dimensional systems with significant spin-orbitcoupling and no time-reversal symmetry. d Layer groups1 1, 5, 9, 12, 19, 23, 27, 28, 29, 30, 33, 35, 362 4, 10, 13, 22, 26, 32, 34, 39, 463 8, 11, 17, 21, 25, 31, 43, 45, 48, 53, 55, 57, 594 15, 16, 20, 24, 38, 41, 54, 56, 58, 60, 62, 64, 76, 775 2, 3, 7, 37, 40, 44, 47, 67, 68, 69, 706 18, 427 65, 78, 798 49, 50, 52, 61, 63, 71, 729 14, 66, 73, 8010 614 7416 5118 75 d : the rank of the abelian group formed by the set of band structures. TABLE XVI.
Characterization of band structures for quasi-two-dimensional systems of spinless femrions withouttime-reversal symmetry. d Layer groups1 1, 5, 9, 12, 332 4, 10, 13, 29, 32, 343 8, 11, 17, 21, 25, 28, 30, 31, 364 15, 16, 20, 24, 35, 43, 455 2, 3, 7, 67, 68, 69, 706 18, 22, 26, 27, 39, 42, 44, 46, 54, 56, 58, 607 658 40, 49, 50, 52, 71, 72, 76, 779 14, 19, 23, 38, 41, 48, 53, 55, 57, 59, 62, 64, 66, 7310 6, 78, 7912 47, 6314 7416 51, 8018 37, 61, 75 d : the rank of the abelian group formed by the set of band structures.TABLE XVII. Characterization of band structures for quasi-one-dimensional systems with significant spin-orbitcoupling and time-reversal symmetry. d Rod groups1 1, 3, 4, 5, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 24, 25, 26, 31, 32, 33, 35, 43, 44, 47, 48, 5455, 57, 58, 63, 64, 66, 672 7, 12, 21, 22, 23, 30, 34, 36, 38, 42, 46, 49, 50, 56, 65, 703 2, 6, 11, 20, 27, 29, 37, 41, 53, 62, 68, 69, 724 40, 52, 59, 715 61, 756 28, 39, 45, 51, 749 60, 73 d : the rank of the abelian group formed by the set of band structures.TABLE XVIII. Characterization of band structures for quasi-one-dimensional systems with negligible spin-orbitcoupling and time-reversal symmetry. d Rod groups1 1, 5, 9, 24, 26, 43, 44, 54, 582 4, 8, 16, 17, 25, 42, 50, 55, 56, 573 2, 3, 7, 10, 12, 14, 19, 23, 31, 33, 35, 36, 47, 48, 49, 63, 67, 704 15, 27, 46, 53, 65, 695 29, 34, 38, 52, 726 6, 11, 13, 18, 21, 22, 32, 45, 59, 61, 64, 66, 687 30, 378 40, 629 28, 41, 51, 71, 7510 7412 20, 6015 3918 73 d : the rank of the abelian group formed by the set of band structures. TABLE XIX.
Characterization of band structures for quasi-one-dimensional systems with significant spin-orbitcoupling and no time-reversal symmetry. d Rod groups1 1, 5, 9, 13, 15, 16, 17, 18, 24, 26, 32, 35, 43, 44, 54, 58, 64, 662 4, 8, 25, 30, 34, 36, 50, 55, 57, 703 2, 3, 7, 10, 12, 14, 19, 20, 21, 22, 31, 33, 37, 38, 41, 42, 47, 48, 49, 56, 62, 63, 67, 68, 694 23, 46, 65, 716 6, 11, 27, 29, 39, 40, 52, 53, 72, 759 45, 51, 59, 61, 73, 7412 2818 60 d : the rank of the abelian group formed by the set of band structures.TABLE XX. Characterization of band structures for quasi-one-dimensional systems of spinless fermions withouttime-reversal symmetry. d Rod groups1 1, 5, 9, 24, 26, 43, 44, 54, 582 4, 8, 16, 17, 25, 50, 55, 573 2, 3, 7, 10, 12, 14, 19, 31, 33, 35, 36, 42, 47, 48, 49, 56, 63, 67, 704 15, 23, 46, 65, 695 346 6, 11, 13, 18, 21, 22, 27, 29, 32, 38, 52, 53, 64, 66, 68, 727 30, 378 629 40, 41, 45, 51, 59, 61, 71, 7512 20, 28, 7415 3918 60, 73 d : the rank of the abelian group formed by the set of band structures.TABLE XXI. Symmetry-based indicators of band topology for quasi-two-dimensional systems with significantspin-orbit coupling and time-reversal symmetry. X BS Layer groups Z
2, 6, 7, 14, 15, 16, 17, 18, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 52, 62,64, 66, 71, 72 Z
74, 78, 79 Z
51, 61, 63 Z
75, 80 X BS : the quotient group between the group of band structures and atomic insulators.TABLE XXII. Symmetry-based indicators of band topology for quasi-two-dimensional systems with negligiblespin-orbit coupling and time-reversal symmetry. X BS Layer groups Z
2, 3, 7, 49, 50, 52, 66, 73 Z × Z
6, 51, 75 X BS : the quotient group between the group of band structures and atomic insulators. TABLE XXIII.
Symmetry-based indicators of band topology for quasi-two-dimensional systems with significantspin-orbit coupling and no time-reversal symmetry. X BS Layer groups Z
2, 3, 7, 37, 40, 44, 47 Z
65, 78, 79 Z
49, 50, 52, 61, 63 Z
66, 73, 80 Z × Z Z × Z Z × Z Z × Z X BS : the quotient group between the group of band structures and atomic insulators.TABLE XXIV. Symmetry-based indicators of band topology for quasi-two-dimensional systems of spinlessfermions without time-reversal symmetry. X BS Layer groups Z
2, 3, 7 Z Z
49, 50, 52 Z
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