PHYSICAL REVIEW X 6, 021008 (2016)

Topological Insulators from Group Cohomology

A. Alexandradinata,1,2,* Zhijun Wang,1 and B. Andrei Bernevig1 1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA 2Department of Physics, Yale University, New Haven, Connecticut 06520, USA (Received 24 November 2015; revised manuscript received 4 April 2016; published 15 April 2016) We classify insulators by generalized symmetries that combine space-time transformations with quasimomentum translations. Our group-cohomological classification generalizes the nonsymmorphic space groups, which extend point groups by real-space translations; i.e., nonsymmorphic symmetries unavoidably translate the spatial origin by a fraction of the lattice period. Here, we further extend nonsymmorphic groups by reciprocal translations, thus placing real and quasimomentum space on equal footing. We propose that group cohomology provides a symmetry-based classification of quasimomentum manifolds, which in turn determines the band topology. In this sense, cohomology underlies band topology. Our claim is exemplified by the first theory of time-reversal-invariant insulators with nonsymmorphic spatial symmetries. These insulators may be described as “piecewise topological,” in the sense that subtopologies describe the different high-symmetry submanifolds of the Brillouin zone, and the various subtopologies must be pieced together to form a globally consistent topology. The subtopologies that we discover include a glide-symmetric analog of the quantum spin Hall effect, an hourglass-flow topology (exemplified by our recently proposed KHgSb material class), and quantized non-Abelian polarizations. Our cohomological classification results in an atypical bulk-boundary correspondence for our topological insulators.

DOI: 10.1103/PhysRevX.6.021008 Subject Areas: Condensed Matter Physics, Topological Insulators

I. INTRODUCTION method to topologically classify space groups with similar complexity; in contrast, previous classifications Spatial symmetries have enriched the topological classi- – – [1,2,4 7,14] (with one exception by us [8]) have expanded fication of insulators and superconductors [1 10]. A basic the Altland-Zirnbauer symmetry classes [22,23] to include geometric property that distinguishes spatial symmetries only a single point-group generator. For point groups with regards their transformation of the spatial origin: symmor- multiple generators, different submanifolds of the Brillouin phic symmetries preserve the origin, while nonsymmorphic torus are invariant under different symmetries; e.g., mirror symmetries unavoidably translate the origin by a fraction of and glide planes are respectively mapped to themselves the lattice period [11]. This fractional translation is respon- by a symmorphic reflection and a glide reflection, as sible for band topologies that have no analog in symmorphic crystals. Thus far, all experimentally tested topological (a) (b) (c) (d) insulators have relied on symmorphic space groups [12–17]. Here, we propose the first nonsymmorphic theory of time-reversal-invariant insulators, which complements previous theoretical proposals with magnetic, nonsymmor- phic space groups [4,5,18–20]. Motivated by our recently proposed KHgX material class (X ¼ Sb, Bi, As) [21],we present here a complete classification of spin-orbit-coupled 4 insulators with the space group (D6h) of KHgX. The point group (D6h) of KHgX, defined as the quotient FIG. 1. (a) Mirror (red, blue) and glide (green, brown) planes in of its space group by translations, is generated by four the 3D Brillouin torus of KHgX. These planes project to the high- ~ ~ ~ ~ ~ spatial transformations—this typifies the complexity of symmetry line X U Z Γ X in the 2D Brillouin torus of the 010 most space groups. This work describes a systematic surface. (b)–(d) Examples of possible piecewise topologies in the space group of KHgX, as illustrated by their surface band structures along X~ U~ Z~ Γ~ X~ . Panel (b) describes a “quantum glide *[email protected] Hall effect” (along Z~ Γ~ ) and an odd mirror-Chern number (along Γ~ X~ ); these two subtopologies must be pieced together at their Published by the American Physical Society under the terms of Γ~ the Creative Commons Attribution 3.0 License. Further distri- intersection point . Panel (c) describes an hourglass-flow top- ~ ~ ~ ~ ~ ~ bution of this work must maintain attribution to the author(s) and ology (X U Z Γ) and an even mirror-Chern number (Γ X). (d) A the published article’s title, journal citation, and DOI. trivial topology is shown for comparison.

2160-3308=16=6(2)=021008(38) 021008-1 Published by the American Physical Society ALEXANDRADINATA, WANG, and BERNEVIG PHYS. REV. X 6, 021008 (2016)

4 illustrated in Fig. 1(a) for D6h. Wave functions in each correspondence for our topological insulators. This corre- submanifold are characterized by a lower-dimensional spondence describes a mapping between topological topological invariant that depends on the symmetries of numbers that describe bulk wave functions and surface that submanifold; e.g., mirror planes are characterized by a topological numbers [35]—such a mapping exists if the mirror-Chern number [24] and glide planes by a glide- bulk and surface have in common certain “edge sym- symmetric analog [7,21] of the quantum spin Hall (QSH) metries” that form a subgroup of the full bulk symmetry. effect [25] [in short, a quantum glide Hall effect (QGHE)]. This edge subgroup is responsible for quantizing both bulk The various invariants are dependent because wave func- and surface topological numbers; i.e., these numbers are tions must be continuous where the submanifolds overlap; robust against gap- and edge-symmetry-preserving defor- e.g., the intersection of planes in Fig. 1(a) are lines that mations of the Hamiltonian. In our case study, the edge Γ~ ~ ~ ~ project to , X, U, and Z. We refer to such insulators as symmetry is projectively represented in the bulk, where “ ” piecewise topological, in the sense that various subto- quasimomentum provides the parameter space for parallel pologies (topologies defined on different submanifolds) transport; on a surface with reduced translational symmetry, must be pieced together consistently to form a 3D topology. the same symmetry is represented ordinarily. In contrast, all This work addresses two related themes: (i) a group- known symmetry-protected correspondences [27] are one cohomological classification of quasimomentum submani- to one and rely on the identity between bulk and surface folds and (ii) the connection between this cohomological representations; our work explains how a partial corre- classification and the topological classification of band spondence arises where such identity is absent. insulators. In (i), we ask how a mirror plane differs from a glide plane. Are two glide planes in the same Brillouin We summarize our main results in Sec. II, which also 4 serves as a guide to the whole paper. We then preliminarily torus always equal? This equality does not hold for D :in 6h review the tight-binding method in Sec. III A, as well as one glide plane, the symmetries are represented ordinarily, introduce the spatial symmetries of our case study. Next, in while in the other we encounter generalized “symmetries” that combine space-time transformations with quasimo- Sec. IV, we review the Wilson loop and the bulk-boundary mentum translations (W). Specifically, W denotes a dis- correspondence of topological insulators; the notion of a partial correspondence is introduced and exemplified with crete quasimomentum translation in the reciprocal lattice. 4 These symmetries then generate an extension of the point our case study of D6h. We then use the method of Wilson group by W; i.e., W becomes an element in a projective loops to construct and classify a piecewise topological representation of the point group. The various representa- insulator in Sec. V; here, we also introduce the quantum tions (corresponding to different glide planes) are classified glide Hall effect. Our topological classification relies on by group cohomology, and they result in different sub- extending the symmetry group by quasimomentum trans- 4 lations, as we elaborate in Sec. VI; the application of group topologies (e.g., one glide plane in D6h may manifest a quantum glide Hall effect, while the other cannot). In this cohomology in band theory is introduced here. We offer an sense, cohomology underlies band topology. alternative perspective of our main results in Sec. VII, and To determine the possible subtopologies within each end with an outlook. submanifold and then combine them into a 3D topology, we propose a general methodology through Wilson loops II. SUMMARY OF RESULTS of the Berry gauge field [26,27]; these loops represent A topological insulator in d spatial dimensions may quasimomentum transport in the space of filled bands [28]. manifest robust edge states on a (d − 1)-dimensional As exemplified for the space group D4 , our method is 6h boundary. Letting k parametrize the d-dimensional shown to be efficient and geometrically intuitive—piecing Brillouin torus, we then split the quasimomentum coor- together subtopologies reduces to a problem of interpolat- dinate as k ¼ðk⊥; k∥Þ, such that k⊥ corresponds to the ing and matching curves. The novel subtopologies that coordinate orthogonal to the surface, and k∥ is a wave we discover include (i) the quantum glide Hall effect in − 1 Fig. 1(a), (ii) an hourglass-flow topology, as illustrated in vector in a (d )-dimensional surface-Brillouin torus. We Fig. 1(b) and exemplified [21] by KHgX, and (iii) quan- then consider a family of noncontractible circles cðk∥Þ, tized, non-Abelian polarizations that generalize the Abelian where for each circle k∥ is fixed, while k⊥ is varied over a theory of polarization [29]. reciprocal period; e.g., consider the brown line in Fig. 1(a). 4 We propose to classify each quasimomentum circle by the Our topological classification of D6h is the first physical application of group extensions by quasimomentum trans- symmetries that leave that circle invariant. For example, in lations. It generalizes the construction of nonsymmorphic centrosymmetric crystals, spatial inversion is a symmetry − space groups, which extend point groups by real-space of cðk∥Þ for inversion-invariant k∥ satisfying k∥ ¼ k∥ translations [30–34]. Here, we further extend nonsymmor- modulo a surface reciprocal vector. The symmetries of the phic groups by reciprocal translations, thus placing real and circle are classified by the second group cohomology: quasimomentum space on equal footing. A consequence of 2 Z Zd Z this projective representation is an atypical bulk-boundary H ðG∘; 2 × × Þ: ð1Þ

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As further elaborated in Sec. VI and Appendix D, H2 advantage of this analogy is that the Wilson bands classifies the possible group extensions of G∘ by may be interpolated [35,38] to Hamiltonian-energy d Z2 × Z × Z, and each extension describes how the sym- bands in a semi-infinite geometry with a surface metries of the circle are represented. The arguments in H2 orthogonal to k⊥. Some topological properties of the are defined as follows. Hamiltonian and Wilson bands are preserved in this (a) The first argument, G∘, is a magnetic point group [36] interpolation, resulting in a bulk-boundary corre- consisting of those space-time symmetries that (i) pre- spondence that we describe in Sec. IV B. There, we serve a spatial point and (ii) map the circle cðk∥Þ to also introduce two complementary notions of a total 3 and a partial correspondence; the latter is exempli- itself. For d ¼ , the possible magnetic point groups 4 comprise the 32 classical point groups [37] without fied by the space group D6h. time reversal (T), 32 classic point groups with T, and (ii) The symmetries of cðk∥Þ are formally defined as the 58 groups in which T occurs only in combination group of the Wilson loop in Sec. VI; any group of with other operations and not by itself. However, we the Wilson loop corresponds to a group extension consider only subgroups of the 3D magnetic point classified by Eq. (1). That is, our cohomological groups (numbering 32 þ 32 þ 58 ¼ 122) that satisfy classification of quasimomentum circles determines (ii); these subgroups might also include spatial sym- the representation of point-group symmetries that metries that are spoilt by the surface, with the just- constrain the Wilson loop, whether linear or pro- mentioned spatial inversion a case in point. jective. The particular representation determines the (b) The second argument of H2 is the direct product of rules that govern the connectivity of Wilson energies three Abelian groups that we explain in turn. The Z2 (curves), as we elaborate in Sec. VA; we then group is generated by a 2π spin rotation; its inclusion connect the curves in all possible legal ways, as in the second argument implies that we also consider in Sec. VB—distinct connectivities of the Wilson half-integer-spin representations; e.g., at inversion- energies correspond to topologically inequivalent invariant k∥ of fermionic insulators, time reversal is ground states. This program of interpolating and 2 − matching curves, when carried out for the space represented by T ¼ I. 4 (c) The second Abelian group (Zd) is generated by group D6h, produces the classification summarized discrete real-space translations in d dimensions: by in Table I. 4 d extending a magnetic point group (G∘)byZ ,we Beyond D6h, we note that Eq. (1) and the Wilson-loop obtain a magnetic space group; nontrivial extensions method provide a unifying framework to classify chiral are referred to as nonsymmorphic. topological insulators [39], and all topological insulators (d) The final Abelian group (Z) is generated by the with robust edge states protected by space-time sym- discrete quasimomentum translation in the surface- metries. Here, we refer to topological insulators with either normal direction, i.e., a translation along cðk∥Þ and symmorphic [1,2,8] or nonsymmorphic spatial symmetries covering cðk∥Þ once. A nontrivial extension by qua- [4,7,19,40], the time-reversal-invariant quantum spin Hall – simomentum translations is exemplified by one of two phase [25], and magnetic topological insulators [18,41 43]. 4 These case studies are characterized by extensions of G∘ by glide planes in the space group D6h [cf. Sec. VI]. Having classified quasimomentum circles through Eq. (1), we outline a systematic methodology to topologi- TABLE I. Classification of time-reversal-invariant insulators 4 Pη cally classify band insulators. The key observation is that with the space group D6h. A quantized polarization invariant ( Γ~ ) distinguishes between two families of insulators: modulo the quasimomentum translations in the space of filled bands is η e P e=2 0 represented by Wilson loops of the Berry gauge field; the electron charge , Γ~ ¼ (¼ ) characterizes the quantum Pη various group extensions, as classified by Eq. (1), corre- glide Hall effect (its absence). Specifically, Γ~ is the polarization η 1 spond to the various ways in which symmetry may of one of two glide subspaces (as labeled by ¼ ), but time- reversal symmetry ensures there is only one independent polari- constrain the Wilson loop. Studying the Wilson-loop Pþ P− Pη 0 zation: Γ~ ¼ Γ~ modulo e. The Γ~ ¼ family is further spectrum then determines the topological classification. Q ∈ Z subclassified by two non-Abelian polarizations, Γ~ Z~ 2 and A more detailed summary is as follows. Q ∈ Z C X~ U~ 2, and a mirror-Chern number ( e) that is constrained to (i) We consider translations along cðk∥Þ with a certain Q ≠ Q be even; where Γ~ Z~ X~ U~ , the insulator manifests an hour- orientation that we might arbitrarily choose, e.g., the Pη 2 glass-flow topology. The Γ~ ¼ e= family is subclassified by triple arrows in Fig. 1(a). These translations are Q ∈ Z C X~ U~ 2 and odd e. represented by the Wilson loop Wðk∥Þ, and the Q Q C phase (θ) of each Wilson-loop eigenvalue traces out Γ~ Z~ X~ U~ e a “curve” over k∥. In analogy with Hamiltonian- Pη 0 Z Z 2Z Γ~ ¼ 2 2 energy bands, we refer to each curve as the energy of η P ~ ¼ e=2 Z2 2Z þ 1 a Wilson band in a surface-Brillouin torus. The Γ

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d Z2 × Z ; on the other hand, extensions by quasimomentum gap that is finite for all k, such that we can distinguish 4 translations are necessary to describe the space group D6h, occupied from empty bands; the former are projected by but have not been considered in the literature. In particular, Xnocc D4 falls outside the K-theoretic classification of non- 6h PðkÞ¼ ju ihu j symmorphic topological insulators in Ref. [7]. n;k n;k n¼1 Finally, we remark that the method of Wilson loops −1 (synonymous [26] with the method of Wannier centers ¼ VðGÞPðk þ GÞVðGÞ ; ð7Þ [27]) is actively being used in topologically classifying band insulators [26,27,44–46]. The present work advances where the last equality follows directly from Eq. (6). the Wilson-loop methodology by (i) relating it to group cohomology through Eq. (1), (ii) providing a systematic B. Crystal structure and spatial symmetries summary of the method (in this section), and (iii) demon- The crystal structure KHgX is chosen to exemplify the strating how to classify a piecewise-topological insulator 4 spatial symmetries we study. As illustrated in Fig. 2, the Hg for the case study D6h (cf. Sec. V). and X ions form honeycomb layers with AB stacking along ~z; here, x~, y~, ~z denote unit basis vectors for the Cartesian III. PRELIMINARIES coordinate system drawn in the same figure. Between each A. Review of the tight-binding method AB bilayer sits a triangular lattice of K ions. The space 4 ≡ 6 In the tight-binding method, the Hilbert space is reduced group (D6h P 3=mmc) of KHgX includes the following symmetries: (i) an inversion (I) centered around a K ion to a finite number of Löwdin orbitals φR;α, for each unit cell labeled by the Bravais-lattice (BL) vector R [47–49].In (which we henceforth take as our spatial origin), the ¯ 2 ¯ 2 Hamiltonians with discrete translational symmetry, our reflections (ii) Mz ¼ tðc~z= ÞMz, and (iii) Mx ¼ tðc~z= ÞMx, ∈ basis vectors are where Mj inverts the coordinate j fx; y; zg. In (ii) and (iii) and the remainder of the paper, we denote, for any X 1 transformation g, g¯ ¼ tðc~z=2Þg as a product of g with a ffiffiffiffi ik·ðRþrαÞ ϕk;αðrÞ¼p e φR;αðr − R − rαÞ; ð2Þ N translation (t) by half a lattice vector (c~z=2). Among (ii) and R ¯ (iii), only Mx is a glide reflection, wherefore the fractional α 1 … where ¼ ; ;ntot, k is a crystal momentum, N is the translation is unremovable [11] by a different choice number of unit cells, α labels the Löwdin orbital, and rα of origin. While we primarily focus on the symmetries – 4 denotes the position of the orbital α relative to the origin in (i) (iii), they do not generate the full group of D6h; e.g., each unit cell. The tight-binding Hamiltonian is defined as there also exists a sixfold screw symmetry whose impli- Z cations have been explored in our companion paper [21]. d ˆ We are interested in symmetry-protected topologies that HðkÞαβ ¼ d rϕk;αðrÞ Hϕk;βðrÞ; ð3Þ manifest on surfaces. Given a surface termination, we refer where Hˆ is the single-particle Hamiltonian. The energy (a) X Hg (c) eigenstatesP are labeled by a band index n and defined as z ψ ntot α ϕ n;kðrÞ¼ α¼1 un;kð Þ k;αðrÞ, where y c/2 x K k Xntot x HðkÞαβun;kðβÞ¼εn;kun;kðαÞ: ð4Þ a ky β¼1 (b) 010 kz We employ the bra-ket notation: a1 a2 a HðkÞjun;ki¼εn;kjun;ki: ð5Þ y X Hg Because of the spatial embedding of the orbitals, the basis x K vectors ϕk;α are generally not periodic under k → k þ G for a reciprocal vector G. This implies that the tight-binding FIG. 2. (a) 3D view of atomic structure. The Hg (red) and X Hamiltonian satisfies (blue) ions form a honeycomb layers with AB stacking. The K ion (cyan) is located at an inversion center, which we also choose to −1 be our spatial origin. (b) Top-down view of atomic structure that Hðk þ GÞ¼VðGÞ HðkÞVðGÞ; ð6Þ is truncated in the 010 direction; two of three Bravais-lattice vectors are indicated by a~1 and a~2. (c) Center: bulk Brillouin zone where VðGÞ is a unitary matrix with elements ½VðGÞαβ ¼ ¯ (BZ) of KHgX, with two mirror planes of Mz colored red and iG·rα δαβe . We are interested in Hamiltonians with a spectral blue. Top: 100-surface BZ. Right: 010-surface BZ.

021008-4 TOPOLOGICAL INSULATORS FROM GROUP COHOMOLOGY PHYS. REV. X 6, 021008 (2016) to the subset of bulk symmetries that are preserved by that (a) ky (b) ky (c) ky b surface as edge symmetries. The edge symmetries of the 1 k 100 and 001 surfaces are symmorphic, and they have been x previously addressed in the context of KHgX [21].Our b2 a paper instead focuses on the 010 surface, whose edge group / b π b2

2 2 (nonsymmorphic Pma2) is generated by two reflections: k k

a x x /

¯ ¯ π glideless Mz and glide Mx. 4

IV. WILSON LOOPS AND THE BULK-BOUNDARY 2π/ 3a 2π/ 3a CORRESPONDENCE

We review the Wilson loop in Sec. IVA and introduce FIG. 3. (a) A constant-kz slice of quasimomentum space, with ~ ~ the loop geometry that is assumed throughout this paper. two of three reciprocal-lattice vectors indicated by b1 and b2. The relation between Wilson loops and the geometric While each hexagon corresponds to a Wigner-Seitz primitive cell, theory of polarization is summarized in Sec. IV B. it is convenient to pick the rectangular primitive cell that is shaded in cyan. A close-up of this cell is shown in (b). Here, we illustrate There, we also introduce the notion of a partial bulk- pffiffiffi boundary correspondence, which our nonsymmorphic how the glide reflection (M¯ ) maps ðk ; π= 3a; k Þ → pffiffiffi x y z −π 3 2π insulator exemplifies. ðky; =ffiffiffi a; kzÞ (red dot to brown), which connects to ð =a þ p ~ k ; π= 3a; k Þ (blue) through b2. Panel (c) serves two interpre- y z pffiffiffi tations. In the first, TM¯ maps ðk ; π= 3a; k Þ → A. Review of Wilson loops pffiffiffi z y z ð−k ; −π= 3a; k Þ (red dot to brown), which connects to The matrix representation of parallel transport along y pffiffiffiz 2π − π 3 ~ Brillouin-zone loops is known as the Wilson loop of the ð =a ky; = a; kzÞ (blue) through b2. If we interpret 0 Berry gauge field. It may be expressed as the path-ordered (c) as the kz ¼ cross section, the same vectors illustrate the exponential (denoted by exp¯ ) of the Berry-Wilczek-Zee effect of time reversal. ∇ connection [50,51] AðkÞij ¼hui;kj kuj;ki:  Z  B. Bulk-boundary correspondence of topological insulators W½l¼exp¯ − dl · AðkÞ : ð8Þ l The bulk-boundary correspondence describes topo- logical similarities between the Wilson loop and the surface u Here, recall from Eq. (5) that j j;ki is an occupied band structure. To sharpen this analogy, we refer to the eigenstate of the tight-binding Hamiltonian; l denotes a eigenvectors of Wðk∥Þ as forming Wilson bands with loop and A is a matrix with dimension equal to the energies θ . The correspondence may be understood in number (n ) of occupied bands. The gauge-invariant n;k∥ occ two steps. spectrum of W½l is the non-Abelian generalization of (i) The first is a spectral equivalence between the Berry phase factors (Zak phase factors [28])ifl is −i=2π log W k∥ and the projected-position oper- contractible (noncontractible) [26,27]. In this paper, we ð Þ ð Þ ator P⊥ðk∥ÞyPˆ ⊥ðk∥Þ, where consider only a family of loops parametrized by k∥ ¼ pffiffiffi pffiffiffi ðkx ∈ ½−π= 3a; þπ= 3a;kz ∈ ½−π=c; þπ=cÞ, where Z Xnocc π dk for each loop k∥ is fixed while ky is varied over a y ψ ψ P⊥ðk∥Þ¼ j n;k ;k∥ ih n;k ;k∥ jð9Þ −2π 2π −π 2π y y noncontractible circle ½ =a; þ =a [oriented line with n¼1 three arrowheads in Fig. 3(a)]. We then label each Wilson loop as W k∥ and denote its eigenvalues by exp iθ , ð Þ ½ n;k∥ projects to all occupied bands with surface wave 1 … with n ¼ ; ;nocc. Note that k∥ also parametrizes the vector k∥, and ψ n;kðrÞ¼expðik · rÞun;kðrÞ are the 010-surface bands; hence, we refer to k∥ as a surface wave Bloch-wave eigenfunctions of Hˆ . For the position vector. Here and henceforth, we take the unconventional operator yˆ, we have chosen natural units of the lattice pffiffiffi ordering k k ;k ;k k ; k∥ . To simplify the nota- ¼ð y x zÞ¼ð y Þ where 1 ≡ a=2 ¼ a~1 · y~, and a~1=a ¼ − 3x=~ 2 þ tion in the rest of the paper, we reparametrize the rec- y=~ 2 is the lattice vector indicated in Fig. 2(b). tangular primitive cell of Fig. 3 as a cube of dimension 2π; pffiffiffi Denoting the eigenvalues of P⊥ðk∥ÞyPˆ ⊥ðk∥Þ as i.e., k ¼π= 3a → k ¼π, k ¼2π=a → k ¼π, x x y y yn;k∥ , the two spectra are related as yn;k∥ ¼ π → π and kz ¼ =c kz ¼ . The time-reversal-invariant k∥ θ =2π modulo one [26]. Some intuition about ~ ~ ~ n;k∥ are then labeled as Γ ¼ð0; 0Þ, X ¼ðπ; 0Þ, Z ¼ð0; πÞ, and the projected-position operator may be gained U~ ¼ðπ; πÞ. For example, WðΓ~Þ would correspond to a from studying its eigenfunctions; they form a set 0 0 ∈ 1 2 … loop parametrized by ðky; ; Þ. of hybrid functions fjk∥;nijn f ; ; ;noccgg

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that maximally localize in y~ (as a Wannier function) consequently for the entire zigzag topology. The but extend in x~ and ~z [as a Bloch wave with QSH phase thus exemplifies a total bulk-boundary momentum k∥ ¼ðkx;kzÞ]. In this Bloch-Wannier correspondence, where the entire set of boundary topologies (i.e., topologies that are consistent with (BW) representation [27], the eigenvalue (yn;k∥ ) H under P⊥yPˆ ⊥ is merely the center-of-mass the edge symmetries of s) is in one-to-one corre- spondence with the entire set of W topologies (i.e., coordinate of the BW function (jn; k∥i) [26,45]. topologies that are consistent with symmetries of W, Since P⊥ is symmetric under translation [tða~1Þ]by ~ ~ ˆ ~ −1 ˆ − ∈ of which the edge symmetries form a subset). One is a1, while tða1Þytða1Þ ¼ y I, each of [yn;k∥ jn 1 2 … then justified in inferring the topological classifica- f ; ; ;noccg] represents a family of BW functions tion purely from the representation theory of surface related by integer translations. The Abelian polari- — P wave functions this surface-centric methodology zation ( =e) is defined as the net displacement of has been successfully applied to many space BW functions [29,52,53]: groups [4,8,54]. Z While this surface-centric approach is technically easier P 1 Xnocc π k∥ θ i than the representation theory of Wilson loops, it ignores ¼ j;k∥ ¼ TrAyðky; k∥Þdky; ð10Þ e 2π 2π −π the bulk symmetries that are spoiled by the boundary. On j¼1 the other hand, W topologies encode these bulk sym- metries, and are therefore more reliable in a topological where all equalities are defined modulo integers, and P classification. In some cases [26,46,55,56], these bulk TrA ðkÞ¼ nocc hu j∇ u i is the Abelian Berry y i¼1 i;k y i;k symmetries enable W topologies that have no boundary connection. analog. Simply put, some topological phases do not have (ii) The next step is an interpolation [35,38] between robust boundary states, a case in point being the Z topology P⊥yPˆ ⊥ and an open-boundary Hamiltonian (H ) s of 2D inversion-symmetric insulators [26]. In our non- with a boundary termination. Presently, we assume symmorphic case study, it is an out-of-surface translational for simplicity that each of fP⊥; y;ˆ Hsg is invariant symmetry [tða~1Þ] that disables a W topology, and con- under space-time transformations of the edge group. sequently a naive surface-centric approach would over- A simple example is the 2D quantum spin Hall predict the topological classification—this exemplifies a insulator, where time reversal (T) is the sole edge partial bulk-boundary correspondence. As we clarify, the symmetry: by assumption, T is a symmetry of the tða~1Þ symmetry distinguishes between two representations periodic-boundary Hamiltonian (hence also of P⊥); of the same edge symmetries: an ordinary representation furthermore, since T acts locally in space, it is also a with the open-boundary Hamiltonian (H ) and a projective symmetry of yˆ and H . It has been shown in Ref. [35] s s one with the Wilson loop (W). To state the conclusion up that the discrete subset of the Hs spectrum (corre- sponding to edge-localized states) is deformable into front, the projective representation rules out a quantum glide Hall topology that would otherwise be allowed in the a subset of the fully discrete P⊥yPˆ ⊥ spectrum. More ordinary representation. This discussion motivates a careful physically, a subset of the BW functions mutually W and continuously hybridizes into edge-localized determination of the topologies in Sec. V. states when a boundary is slowly introduced, and the edge symmetry is preserved throughout this V. CONSTRUCTING A PIECEWISE- hybridization. Consequently, P⊥yPˆ ⊥ (equivalently, TOPOLOGICAL INSULATOR BY log[W]) and Hs share certain traits that are only well WILSON LOOPS defined in the discrete part of the spectrum, and, We would like to classify time-reversal-invariant insula- moreover, these traits are robust in the continued D4 presence of said symmetries. The trait that identifies tors with the space group 6h; our result should more the QSH phase (in both the Zak phases and the edge- broadly apply to hexagonal crystal systems with the edge 2 ¯ ¯ mode dispersion) is a zigzag connectivity where the symmetry Pma (generated by glide Mx and glideless Mz) spectrum is discrete; here, eigenvalues are well and a bulk spatial-inversion symmetry. Our final result in defined, and they are Kramers degenerate at time- Table I relies on topological invariants that we briefly reversal-invariant momenta but otherwise singly introduce here, deferring a detailed explanation to the degenerate, and, furthermore, all Kramers subspaces sections below. The invariants are (i) the mirror-Chern C 0 are connected in a zigzag pattern [26,44,45]. In the number ( e) in the kz ¼ plane, (ii) the quadruplet Q Q 0 QSH example, it might be taken for granted that the polarization Γ~ Z~ ( X~ U~ ) in the kx ¼ glide plane 2 π Q ≠ Q representation (T ¼ −I) of the edge symmetry is (kx ¼ ), wherefor Γ~ Z~ X~ U~ implies an hourglass flow, Pη 0 2 identical for both Hs and W; the invariance of and (iii) the glide polarization Γ~ ¼ (e= ) indicates the T2 ¼ −I throughout the interpolation accounts for absence (presence) of the quantum glide Hall effect in the the persistence of Kramers degeneracies, and kx ¼ 0 plane.

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Our strategy for classification is simple: we first derive i.e., U maps the Wilson energy as θ → −θ. Here, we the symmetry constraints on the Wilson-loop spectrum, caution that T and U are symmetries of the circle ðky ∈ then enumerate all topologically distinct spectra that are ½−π; π; k∥Þ and preserve the momentum parameter k∥; this consistent with these constraints. Pictorially, this amounts differs from the conventional [23] time-reversal and par- to understanding the rules obeyed by curves (the Wilson ticle-hole symmetries that typically invert momentum. bands) and connecting curves in all possible legal ways; we To precisely state (iii), we first elaborate on how Wilson do these in Secs. VA and VB, respectively. bands may be labeled by mirror eigenvalues, which we ¯ define as λj for the reflection Mj (j ∈ fx; zg). First consider ¯ A. Local rules of the curves the glideless Mz, which is a symmetry of any bulk wave Γ~ ~ 0 ~ ~ π We consider how the bulk symmetries constrain the vector that projects to X (kz ¼ ) and Z U (kz ¼ )iny~. ¯ 2 ¯ 2π Wilson loop Wðk∥Þ, with k∥ lying on the high-symmetry Being glideless, Mz ¼ E ( rotation of a half-integer spin) line Γ~ X~ U~ Z~ Γ~ ; note that Γ~ Z~ and X~ U~ are glide lines that are implies two momentum-independent branches for the ¯ λ ¯ Γ~ ~ ~ ~ eigenvalues of Mz: z ¼i; this eigenvalue is an invariant invariant under Mx, while X and Z U are mirror lines ¯ ¯ of any parallel transport within either Mz-invariant plane. invariant under Mz. The relevant symmetries that constrain That is, if ψ 1 is a mirror eigenstate, any state related to ψ 1 Wðk∥Þ necessarily preserve the circle ðk ∈ ½−π; π; k∥Þ, y by parallel transport must have the same mirror eigenvalue. modulo translation by a reciprocal vector; such symmetries Consequently, the Wilson loop block diagonalizes with comprise the little group of the circle [8]. For example, respect to λ i, and any Wilson band may be labeled I W ¯ ¯ z ¼ (a) T would constrain ðk∥Þ for all k∥, (b) TMx and Mz is by λ . ~ ~ z k∥ Γ X T ¯ constraining only for along , and (c) matters only at A similar story occurs for the glide Mx, which is a ~ ~ ~ ~ the time-reversal-invariant k∥. Along Γ X, we omit dis- symmetry of any bulk wave vector that projects to Γ Z ¯ cussion of other symmetries (e.g., TC2y) in the group of the (kx ¼ 0). The only difference from Mz is that the two circle, because they do not additionally constrain the branches of λx are momentum dependent, which follows ¯ 2 ¯ Wilson-loop spectrum. For each symmetry, only three from Mx ¼ tð~zÞE, with t denoting a lattice translation. ¯ properties influence the connectivity of curves, which Explicitly, the Bloch representation of Mx squares to we first state succinctly. − expð−ik Þ, which implies for the glide eigenval- θ → z (i) Does the symmetry map each Wilson energy as ues λxðkzÞ¼i expð−ikz=2Þ. θ or θ → −θ? Note here we omit the constant To wrap up our discussion of the mirror eigenvalues, we θ ¯ ~ ~ argument of k∥ . consider the subtler effect of M along X U. Despite being a θ → θ x (ii) If the symmetry maps , does it also result in symmetry of any surface wave vector along X~ U~ , Kramers-like degeneracy? By “Kramers-like,” we mean a doublet degeneracy arising from an anti- M¯ ∶ ðπ;k Þ → ð−π;k Þ¼ðπ;k Þ − 2πx;~ ð13Þ unitary symmetry that representatively squares to x z z z −1, much like time-reversal symmetry in half- ¯ with 2πx~ a surface reciprocal vector, Mx is not a symmetry integer-spin representations. ~ ~ (iii) How does the symmetry transform the mirror of any bulk wave vector that projects to X U, but instead eigenvalues of the Wilson bands? Here, we refer relates two bulk momenta that are separated by half a bulk ¯ ¯ reciprocal vector; i.e., to the eigenvalues of mirror Mz and glide Mx along their respective invariant lines. ¯ ∶ π → −π π π − ~ To elaborate, (i) and (ii) are determined by how the Mx ðky; ;kzÞ ðky; ;kzÞ¼ðky þ ; ;kzÞ b2; symmetry constrains the Wilson loop. We say that a T as illustrated in Fig. 3(b). This reference to Fig. 3(b) must symmetry represented by is time-reversal-like at k∥, pffiffiffi π 3 → if for that k∥ be made with our reparametrization (kx ¼ = a kx ¼π, ky ¼2π=a → ky ¼π) in mind. We refer to T W T −1 W −1 such a glide plane as a projective glide plane, to distinguish ðk∥Þ ¼ ðk∥Þ ; it from the ordinary glide plane at k ¼ 0. The absence of T T −1 − T 2 x with i ¼ i; and ¼I: ð11Þ ¯ Mx symmetry at each bulk wave vector implies that the Wilson loop cannot be block diagonalized with respect to T θ → θ Both map the Wilson energy as ,but ¯ the eigenvalues of Mx. However, quantum numbers exist only T − symmetries guarantee a Kramers-like degeneracy. for a generalized symmetry (M¯ ) that combines the glide Similarly, a symmetry represented by U is particle-hole-like x reflection with parallel transport over half a reciprocal at k∥, if for that k∥ period. To be precise, let us define the Wilson line W−π←0 0 π −1 −1 to represent the parallel transport from ð ; ;kzÞ to UWðk∥ÞU ¼ Wðk∥Þ; with UiU ¼ −i; ð12Þ ð−π; π;kzÞ. We demonstrate in Sec. VI that all Wilson

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T U TABLE II. Symmetry constraints of the Wilson bands at generic points along the mirror lines. and are I ¯ ¯ T U possible characterizations of the symmetries (T , TMz, TMx) in the leftmost column: a -type ( -type) symmetry is time-reversal-like (particle-hole-like) symmetry, as defined in Eqs. (11) and (12).Forj ∈ fx; zg, λj is a symmetry eigenvalue that falls into one of two branches: λz ¼i or λxðαÞ¼i expð−iα=2Þ. Along kx ¼ 0, λx is momentum dependent, and along kx ¼ π, it is energy dependent as well; that is, λxðkz þ θÞ is the glide eigenvalue of a Wilson band at momentum kz and energy θ. ~ ~ ~ ~ ~ ~ ~ ~ kx ¼ 0ðΓ ZÞ kx ¼ πðX UÞ kz ¼ 0ðΓ XÞ kz ¼ πðZ UÞ

TIU: λxðkzÞ → −λxðkzÞ U: λxðkz þ θÞ → −λxðkz − θÞ U: λz → −λz U: λz → þλz ¯ T λ → λ T λ θ → λ θ TMz þ: xðkzÞ þ xðkzÞ þ: xðkz þ Þ þ xðkz þ Þ ¯ T λ → λ T λ → −λ TMx þ: z þ z −: z z

bands may be labeled by quantum numbers under simultaneously be labeled by λx. That TI maps λx → −λx ¯ ¯ ¯ I I ¯ Mx ≡ W−π←0Mx, and that these quantum numbers fall then follows from MxT ¼ tð~zÞT Mx, where tð~zÞ origi- into two energy-dependent branches as nates simply from the noncommutivity of I with the ¯ fractional translation [tð~z=2Þ]inMx: λxðθ þ kzÞ¼ηi exp½−iðθ þ kzÞ=2; with η ¼1: ¯ I I ¯ ð14Þ Mx ¼ tð~zÞ Mx: ð15Þ

¯ I λ → −λ That is, λxðθ þ kzÞ is the Mx eigenvalue of a Wilson band To show T : x x in more detail, suppose for a Bloch- at surface momentum ðπ;kzÞ and Wilson energy θ. Wannier function jn; kzi that For the purpose of topological classification, all we need are the existence of these symmetry eigenvalues (ordinary P⊥yPˆ ⊥jn; k i¼y jn; k i and and generalized) that fall into two branches [recall λ z n;kz z z ¼ ¯ ~ ~ Mxjn; kzi¼λxðkzÞjn; kzi; ð16Þ i; λx ¼i expð−ikz=2Þ along Γ Z, and also Eq. (14)], and T U (iii) asks whether the - and -type symmetries preserve λ → λ λ → −λ ( j j) or interchange ( j j) the branch. To clarify a with λxðkzÞ¼i expð−ikz=2Þ and suppression of the label T U 0 I I ˆ 0 possible confusion, both - and -type symmetries are kx ¼ . ½T ;P⊥¼fT ; yg¼ then leads to antiunitary and therefore have no eigenvalues, while the reflections [M¯ ; M¯ (along Γ~ Z~ ) and M¯ (along X~ U~ )] are z x x P⊥yPˆ ⊥TIjn; kzi¼−yn;k jn; kzi and unitary. The answer to (iii) is determined by the commu- z ¯ I I ¯ tation relation between the symmetry in question (whether MxT jn; kzi¼tð~zÞT Mxjn; kzi

T U −ikz or type) and the relevant reflection. To exemplify ¼ e λxTIjn; kzi; ð17Þ (i)–(iii), let us evaluate the effect of TI symmetry along ~ ~ Γ Z. This may be derived in the polarization perspective, 2 with expð−ikzÞλx ¼ −λx following from λx ¼ due to the spectral equivalence of −i=2π log W k∥ ð Þ ð Þ − −ik TI ˆ I expð zÞ. To recapitulate, (a) imposes a particle- and P⊥ðk∥ÞyP⊥ðk∥Þ. Since T inverts all spatial coordi- hole-symmetric spectrum and (b) two states related by TI nates but transforms any momentum to itself (yn;k∥ ¼ ¯ have opposite eigenvalues under Mx. (a) and (b) are θ =2π → −y ), we identify TI as a U-type symmetry n;k∥ n;k∥ summarized by the notation U: λx → −λx in the top left- [cf. Eq. (12)]. Indeed, while TI is known to produce hand entry of Table II. The complete symmetry analysis is Kramers degeneracy in the Hamiltonian spectrum, TI derived in Sec. VI and Appendix B and tabulated in emerges as an unconventional particle-hole-type symmetry Tables II and III. These relations constrain the possible ¯ in the Wilson loop. Since Mx commutes individually with topologies of the Wilson bands, as we show in the next P⊥ð0;kzÞ and yˆ, all eigenstates of P⊥ð0;kzÞyPˆ ⊥ð0;kzÞ may section.

TABLE III. Time-reversal constraint of the Wilson bands at surface wave vectors satisfying k∥ ¼ðkx;kzÞ¼−k∥ modulo a reciprocal T vector. To clarify a possible source of confusion, the actual time-reversal symmetry (T) is, by our definition of in Eq. (11), only “time-reversal-like” at k∥ ¼ −k∥, since T∶k∥ → −k∥ is not a symmetry of the circle ðky ∈ ½−π; π; k∥Þ for generic k∥.

Γ~ ¼ð0; 0Þ X~ ¼ðπ; 0Þ Z~ ¼ð0; πÞ U~ ¼ðπ; πÞ

T T −: λz → −λz, T −: λz → −λz, T −: λz → −λz, T −: λz → −λz, λxðkzÞ → −λxðkzÞ λxðkz þ θÞ → −λxðkz þ θÞ λxðkzÞ → þλxðkzÞ λxðkz þ θÞ → þλxðkz þ θÞ

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(a) (b) (c) (d) (e) B. Connecting curves in all possible legal ways π π π π π Our goal here is to determine the possible topologies of curves (Wilson bands), which are piecewise smooth on the 0 0 0 0 high-symmetry line Γ~ X~ U~ Z~ Γ~ . We first analyze each 0 momentum interval separately, by evaluating the available π π π π subtopologies within each of Γ~ Z~ , Z~ U~ , etc. The various - - - - -π subtopologies are then combined to a full topology, by a (f) (g) (h) (i) (j) program of matching curves at the intersection points (e.g., π π π π π Z~ ) between momentum intervals. Since our program here is to interpolate and match 0 0 0 0 curves (Wilson bands), it is important to establish just how 0 many Wilson bands must be connected. A combination of symmetry, band continuity, and topology dictates this -π -π -π -π -π answer to be a multiple of four. Since the number (nocc) of occupied Hamiltonian bands is also the dimension of the FIG. 4. Possible Wilson spectra along Γ~ Z~ . Wilson loop, it suffices to show that nocc is a multiple of four. Indeed, this follows from our assumption that the ground state is insulating, and a property of connectedness While Figs. 4(c) and 4(d) are not obviously zigzag, they between sets of Hamiltonian bands. For spin systems with are smoothly deformable to Fig. 4(a), which clearly is. A minimally time-reversal and glide-reflection symmetries, unifying property of all five figures [Figs. 4(a)–4(e)]is we prove in Appendix C that Hamiltonian bands divide into spectral flow: the QGHE is characterized by Wilson bands sets of four that are individually connected; i.e., in each set that robustly interpolate across the maximal energy range there are enough contact points to travel continuously of 2π. What distinguishes the QGHE from the usual through all four branches. The lack of gapless excitations quantum spin Hall effect [25]? Despite describing the band in an insulator then implies that a connected quadruplet is topology over all of Γ~ Z~ , the QGHE is solely determined by either completely occupied or unoccupied. Pη Γ~ a polarization invariant ( Γ~ ) at a single point ( ), which we now describe. ~ ~ Pη — ∈ −π π 0 1. Interpolating curves along the glide line Γ Z Definition of Γ~ . Consider the ðky ½ ; Þ; k∥ ¼ Þ 0 Γ~ ~ circle in the 3D Brillouin zone. Each point here has the Along kx ¼ ( Z), the rules are as follows. ¯ (a) There are two flavors of curves (illustrated as solid and glide symmetry Mx, and the Bloch waves divide into two glide subspaces labeled by λx=i ≡ η ¼1. This allows us blue dashed lines in Fig. 4), corresponding to two η to define an Abelian polarization (P ~ =e) as the net branches of the glide eigenvalue λx ¼iexpð−ikz=2Þ. Γ Only crossings between solid and dashed curves are displacement of Bloch-Wannier functions in either η sub- robust, in the sense of being movable but unremovable. space: (b) At any point along Γ~ Z~ , there is an unconventional η 2 I P nXocc= particle-hole symmetry (due to T ) with conjugate Γ~ 1 η θ → −θ ¼ θ mod 1: ð18Þ bands (related by ) belonging in opposite glide e 2π j;Γ~ branches; cf. first column of Table II. Pictorially, [θ, j¼1 ↔ −θ blue solid curve] [ , blue dashed curve]. η λ η (c) At Γ~ , each solid curve is degenerate with a dashed Here, the superscript indicates a restriction to the x ¼ i, occupied subspace, fexpðiθηÞg are the eigenvalues of the curve, while at Z~ the degeneracies are solid-solid and Wη Γ~ dashed-dashed; cf. Table III. These end-point con- Wilson loop ð Þ, and the second equality follows from straints are boundary conditions for the interpolation the spectral equivalence introduced in Sec. IV.Wehave previously determined in this section that n is a multiple along Γ~ Z~ . occ of four, and therefore there is always an even number Given these rules, there are three distinct connectivities 2 η ~ ~ (nocc= ) of Wilson bands in either subspace. Furthermore, along Γ Z, which we describe in turn: (i) a zigzag − Pþ ¼ P modulo e follows from time reversal relating connectivity [Figs. 4(a)–4(e)] defines the quantum glide Γ~ Γ~ θη → θ−η Hall effect, and (ii) two configurations of hourglasses [e.g., Γ~ Γ~ ; cf. Table III. Fig. 4(f) versus Fig. 4(h), and also Fig. 4(g) versus Fig. 4(i)] We claim that the effect of spatial inversion (I) sym- Pη 2 are distinguished by a connected-quadruplet polarization. metry is to quantize Γ~ to 0 and e= , which, respectively, (i) As illustrated in Figs. 4(a)–4(e), the QGHE describes correspond to the absence and presence of the QGHE. a zigzag connectivity over Γ~ Z~, where each cusp of the Restated, the set of occupied Bloch states along a high- zigzag corresponds to a Kramers-degenerate subspace. symmetry line (projecting to Γ~ ) holographically determines

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Γ~ ~ θη θη Pη 2 the topology in a high-symmetry plane (projecting to Z). subspace are defined as 1;k∥ and 2;k∥ .If Γ~ ¼ e= , To demonstrate this, (a) we first relate the Wilson spectrum the boundary conditions are η at Γ~ to the invariant P , then (b) determine the possible Γ~ η η ~ θ ¼ 0; θ ¼π Wilson spectra at Z. (c) These end-point spectra may be 1;Γ~ 2;Γ~ interpreted as boundary conditions for curves interpolating θη θη −θ−η −θ−η and 1 ~ ¼ 2 ~ ¼ 1 ~ ¼ 2 ~ : across Γ~ Z~ —we find there are only two classes of inter- ;Z ;Z ;Z ;Z polating curves that are distinguished by spectral flow. In one of the glide subspaces (say, η), the two Wilson- Pη (a) To prove the quantization of Γ~ , consider how each energy functions are degenerate at Z~ , but are at glide subspace is individually invariant under I. This everywhere else along Γ~ Z~ nondegenerate; particularly invariance follows from Eq. (15), leading to the ¯ at Γ~ , one Wilson energy is fixed to 0 and the other to representative commutivity of I and Mx, where η η π. Consequently, the two energy functions sweep k ¼ 0. We need further that I maps θ → −θ z Γ~ Γ~ out an energy interval that contains at least 0; π 2π ½ mod . This may be deduced from the polarization [see, e.g., Fig. 4(a)], but may contain more [see, e.g., θη 2π perspective, where Γ~ = is an eigenvalue of the Fig. 4(c)]. The particle-hole symmetry (due to TI; position operator yˆ (projected to the occupied sub- cf. Table II) further imposes that the other two energy ~ −η −π 0 — space at surface wave vector Γ and with λx ¼ ηi); our functions (in ) sweep out at least ½ ; the net claim then follows simply from I inverting the result is that the entire energy range is swept; this ˆ θη −θη position operator y. Γ~ and Γ~ may correspond either spectral flow is identified with the QGHE. Pη 0 Γ~ to two distinct Wilson bands (an inversion doublet) or If Γ~ ¼ , the boundary conditions at are instead η to the same Wilson band (an inversion singlet at θ ~ ¼ Γ η −η η −η 0 π θ ¼ θ ¼ −θ ¼ −θ ; ð19Þ or ). Since there are an even number of Wilson 1;Γ~ 1;Γ~ 2;Γ~ 2;Γ~ bands in each η subspace, a 0 singlet is always accompanied by a π singlet—such a singlet pair leading to spectrally isolated quadruplets, e.g., in produces the only nonintegral contribution to Figs. 4(f), 4(h), and 4(i). Since Kramers partners at Γ~ Pη 2 ~ Γ~ ð¼ e= Þ; the absence of singlets corresponds to (Z) belong in opposite η subspaces (the same η subspace), Pη 0 Γ~ ¼ . These two cases correspond to two classes of the interpolation describes an internal partner switching boundary conditions at Γ~ . We remark briefly on fine- within each quadruplet, resulting in an hourglasslike tuned scenarios where an inversion doublet may dispersion. The center of the hourglass is an unavoidable η — accidentally lie at 0 (or π) without affecting the value crossing [57] between opposite- bands this degeneracy η η is movable but unremovable. Finally, we remark that the of P ~ . In complete generality, P ~ ¼ e=2 (0) corre- η Γ Γ interpolations distinguished by P easily generalize sponds to an odd (even) number of bands at both 0 and Γ~ π, in one η subspace. beyond the minimal number of Wilson bands; e.g., com- ▪ (b) What is left is to determine the possible boundary pare Fig. 4(e) to Fig. 4(g). Given that the Abelian polarization depends on the conditions at Z~ . We find here only one class of choice of spatial origin [26], it may seem surprising that boundary conditions, i.e., any one boundary condition η a single polarization invariant (P ) sufficiently indicates may be smoothly deformed into another, indicating the Γ~ absence of a nontrivial topological invariant at Z~ . the QGHE; indeed, each of the inequivalent inversion Indeed, the same nonsymmorphic algebra [Eq. (15)] centers is a reasonable choice for the spatial origin. In contrast, many other topologies are diagnosed by gauge- has different implications where kz ¼ π:nowI relates Wilson bands in opposite glide subspaces; i.e., invariant differences in polarizations of different 1D sub- η −η η manifolds in the same Brillouin zone [26,58]. Unlike I∶θ → θ ¼ −θ . Consequently, the total polariza- η Z~ Z~ Z~ P η generic polarizations, Γ~ is invariant when a different tion (P ~ ) vanishes modulo e, and the analogous P is Z Z~ inversion center is picked as the origin, i.e., this globally well defined but not quantized. With the additional shifts all θ → θ þ π [see, e.g., Figs. 4(a) and 4(b)], which constraint by T (see Table III), any Kramers pair Pη → Pη Z leads to Γ~ =e Γ~ =e modulo , since each glide sub- belongs to the same glide subspace due to the reality of η space is even dimensional. We caution that P will not the glide eigenvalues; on the other hand, each Kramers Γ~ pair at θ is mapped by I to another Kramers pair at −θ, remain quantized if the spatial origin lies away from an and I-related pairs belong to different glide subspaces. inversion center, due to the well-known U(1) ambiguity of (c) Having determined all boundary conditions, we pro- the Wilson loop [26]. Pη ceed to the interpolation. For simplicity, this is first That the QGHE is determined solely by Γ~ makes performed for the minimal number (four) of Wilson diagnosis especially easy: we propose to multiply the bands; the two Wilson-energy functions in each η spatial-inversion (I) eigenvalues of occupied bands in a

021008-10 TOPOLOGICAL INSULATORS FROM GROUP COHOMOLOGY PHYS. REV. X 6, 021008 (2016) single η subspace, at the two inversion-invariant k that continuously travel between the four bands. Such a project to Γ~. This product being þ1 (−1) then implies that property, which we call fourfold connectivity, is illustrated Pη 0 2 in Fig. 5(b) over two spatial unit cells. Here, both Γ~ ¼ (¼ e= ). In contrast, the usual quantum spin Hall effect (without glide symmetry) cannot [26,44,45] be quadruplets fy1;y2;y3;y4g and fy5;y6;y7;y8g are con- formulated as an Abelian polarization, and a diagnosis nected, and their centers of mass differ by unity; on the would require the I eigenvalues at all four inversion- other hand, fy2;y3;y4;y5g is not connected. The following invariant momenta of a two-torus [59]. discussion then hinges on this fourfold connectivity, which η characterizes insulators with glide and time-reversal sym- (ii) With trivial P ~ , the spectrally isolated interpolations Γ metries. Having defined a mod 1 center-of-mass coordinate further subdivide into two distinct classes, which are dis- for one connected quadruplet, we extend our discussion to tinguished by an hourglass centered at θ ¼ π; e.g., contrast insulators with multiple quadruplets per unit cell; i.e., since Figs. 4(h) and 4(i) with Figs. 4(f), 4(g), and 4(j). This there are nocc number of Bloch-Wannier bands, where nocc difference may be formalized by a Z2 topological invariant Q is the number of occupied bands, we now discuss the most ( Γ~ Z~ ), which we introduced in our companion work [21] 4 ≥ 1 Q general case where integral nocc= . Let us define the and presently describe in the polarization perspective. Γ~ Z~ 4 net displacement of all nocc= Pnumber of connected- characterizes a coarse-grained polarization of quadruplets n =4 quadruplet centers: Qðk∥Þ=e ¼ occ Y ðk∥Þ mod 1. A along Γ~ Z~ , as we illustrate in Fig. 5(b). Here, the center-of- j¼1 j I mass position of this quadruplet may tentatively be defined combination of time-reversal (T) and spatial-inversion ( ) Y symmetry quantizes Qðk∥Þ to0ore=2, as we now show. by averagingP four Bloch-Wannier positions: 1ðk∥Þ¼ 4 ~ ~ We have previously described how TI inverts the spatial ð1=4Þ y , with k∥ ∈ Γ Z. Any polarization quantity n¼1 n;k∥ coordinate but leaves momentum untouched; i.e., we have should be well defined modulo 1, which reflects the discrete an unconventional particle-hole symmetry at each k∥: translational symmetry of the crystal. However, we caution TIjk∥;ni¼jk∥;mi, with m ≠ n and y ¼ −y that Y is only well defined mod 1=4 for quadruplet bands n;k∥ m;k∥ I Y → Y −Y without symmetry, due to the integer ambiguity of each mod 1. Consequently, T : jðk∥Þ j0 ðk∥Þ¼ jðk∥Þ Q of fy jn ∈ Zg. To illustrate this ambiguity, consider in mod 1, and the only noninteger contribution to =e n 1 2 Fig. 5(a) the spectrum of P⊥yPˆ ⊥ for an asymmetric insulator (¼ = ) arises if there exists a particle-hole-invariant ¯ Y 1 2 −Y with four occupied bands. Only the spectrum for two spatial quadruplet (j) that is centered at j¯ ¼ = ¼ j¯ mod unit cells (with unit period) are shown, and the discrete 1, as we exemplify in Figs. 4(h) and 4(i); moreover, since − Q translational symmetry ensures yj;k∥ ¼ yjþ4l;k∥ l for j, each yn;k∥ is a continuous function of k∥, k∥ is constant ∈ Z ≡Q Γ~ ~ Q l . Clearly, the centers of mass of fy1;y2;y3;y4g and ( Γ~ Z~ )over Z. Alternatively stated, Γ~ Z~ is a quantized fy2;y3;y4;y5g differ by 1=4 at each k∥, but both choices are polarization invariant that characterizes the entire glide ~ ~ ~ equally natural given level repulsion across Z Γ Z. plane that projects to Γ~ Z~ . However, a natural choice presents itself if the Bloch- Wannier bands may be grouped in sets of four, such that 2. Connecting curves along the glide line X~ U~ within each set there are enough contact points along Γ~ Z~ to As we discuss in Sec. VA, the relevant symmetries that constrain Wðk∥Þ comprise the little group of the circle (a) (b) ~ ~ 1.5 ðk ∈ ½−π; π; k∥Þ [8]. For any k∥ ∈ X U, the corresponding y1 y y ¯ ¯ 1 group has the symmetries Mx, TI, and TMz; these are y2 ~ ~ y2 exactly the same symmetries of the group for k∥ ∈ Γ Z.In 1 1 y ~ ~ 3 y spite of this similarity, the available subtopologies on X U 3 Γ~ ~ y4 and Z differ: while the two hourglass configurations y4 y 0.5 (distinguished by a connected-quadruplet polarization) are y5 available subtopologies on each line, the QGHE is only y5 ~ ~ y6 available along Γ Z. This difference arises because the same y6 — 0 0 symmetries are represented differently on each line the y7 different projective representations are classified by the y7 y8 second cohomology group, as we discuss in Sec. VI.For -0.5 y8 the purpose of topological classification, we need only extract one salient result from that section: any Wilson band π θ ˆ at k∥ ¼ð ;kzÞ and Wilson energy has simultaneously a FIG. 5. Comparison of the spectrum of P⊥yP⊥ for a system “ ” λ θ without any symmetry (a) and one with time-reversal, spatial- glide eigenvalue: xðkz þ Þ in Eq. (14); here, glide refers M¯ ≡ W ¯ inversion, and glide symmetries (b). Only the spectrum for two to the generalized symmetry x −π←0Mx, which ¯ spatial unit cells is shown. combines the ordinary glide reflection (Mx) with parallel

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(a) (b) (c) π π π subtopologies are two hourglass-type interpolations [Figs. 6(a) and 6(b)], which are distinguished by a second Q connected-quadruplet polarization ( X~ U~ ). 0 0 0

3. Connecting curves along the mirror line Γ~ X~ -π -π -π (a) Curves divide into two noninteracting flavors (solid λ ~ ~ and red dashed lines in Fig. 7), corresponding to z ¼ FIG. 6. (a),(b) Possible Wilson spectra along X U. (c) A i subspaces. hypothetical spectrum that is ruled out by a continuity argument. (b) At both boundaries (Γ~ and X~ ), each red solid curve is degenerate with a red dashed curve; cf. Table III. ~ ~ transport (W−π←0). We might ask if η ¼1 in λx labels a (c) At any point along Γ X [θ, red solid] ↔ [−θ, red meaningful division of the Wilson bands; i.e., do we once dashed], due to the TI symmetry of Table II. again have two noninteracting flavors of curves, as we had These rules allow for mirror-Chern [24] subtopol- Γ~ ~ ~ ~ ~ for Z? The answer is affirmative if the Wilson bands are ogies in the torus that projects to X Γ X, where λz ¼i spectrally isolated, i.e., if all nocc Wilson bands lie strictly subspaces have opposite chirality due to time-reversal θ θ θ − θ 2π within an energy interval ½ i; f with j f ij < , for all symmetry; Fig. 7(a) exemplifies a Chern number (Ce) ∈ 0 2π kz ½ ; Þ. For example, the isolated bands of Fig. 6(a) lie of −1 in the λz ¼þi subspace, and Fig. 7(b) exem- within a window of ½−π=2; π=2, whereas no similar plifies Ce ¼ −2. The allowed mirror-Chern numbers window exists in the hypothetical scenario of Fig. 6(c). (C ) depend on our last rule: e ~ ~ If isolated, then at each kz the energy difference (θi − θj) (d) Curves must match continuously at Γ and X. between any two bands is strictly less than 2π—therefore, This last rule imposes a consistency condition with the η ~ ~ ~ ~ there is no ambiguity in labeling each band by from subtopologies at Γ Z and X U: Ce is odd (even) if and only if Eq. (14). Conversely, this potential ambiguity is sufficient Pη 2 Pη 0 – Γ~ ¼ e= [ Γ~ ¼ ], as illustrated in Figs. 8(a) 8(c) to rule out bands with spectral flow, as we now [Figs. 8(d)–8(f)]. demonstrate. To demonstrate our claim, we rely on a single-energy Let us consider a hypothetical scenario with spectral flow criterion to determine the parity of Ce: count the number [Fig. 6(c)], as would describe a QGHE. There is then a (N )ofλ ¼þi states at an arbitrarily chosen energy and smooth interpolation between bands in one energy period þi z along the full circle X~ Γ~ X~ , then apply N mod 2 ¼ C mod to any band in the next, as illustrated by connecting black þi e 2. Supposing we choose θ ¼ π=2 in Fig. 7(a), there is a arrows in Fig. 6(c). As we interpolate θ → θ þ 2π and single intersection (encircled in the figure) with a λz ¼þi kz → kz þ 4π, we of course return to the same eigenvector band, as is consistent with Ce ¼ −1 being odd. For the of W, and therefore the glide eigenvalue must also return to η purpose of connecting C with P , we need a slightly itself. However, the energy dependence leads to λ → −λ . e Γ~ x x ~ ~ More generally, for 4u number of occupied bands, modified counting rule that applies to the half-circle Γ X λ θ → θ þ 2π, while kz → kz þ 4uπ, leading to the same instead of the full circle. Since time reversal relates z ¼i contradiction. We remark that the essential properties that bands at opposite momentum, we would instead count the ¯ jointly lead to a contradiction are that (i) Wilson bands total number of bands in both Mz subspaces, at our chosen come in multiples of four, as we discuss in the introduction energy and along the half-circle; one additional rule regards to Sec. VB, (ii) the Kramers partners at X~ (U~ ) belong to the counting of the Kramers doublet, which comprise time- opposite flavors (the same flavor) (cf. Table III), reversed partners at either Γ~ or X~ . If such a doublet lies at and (iii) bands connect in a zigzag. Certain details of our chosen energy, it counts not as two but as one; every Fig. 6(c) (e.g., that θ is quantized to special values at X~ ) are superfluous to our argument. Besides this argument, we (a) (b) furnish an alternative proof to rule out the QGHE in our π π companion paper [21]. We remark that the QGHE is π/2 π/2 perfectly consistent with the surface symmetries [21], and it is only ruled out by a proper account of the bulk 0 0 symmetries. ▪ Returning to our classification, Tables II and III inform -π -π us of the constraints due to time-reversal and spatial- inversion symmetries. The summary of this symmetry ~ ~ FIG. 7. Illustrating the single-energy criterion to determine the analysis is that our rules for the curves along X U are C C −1 −2 ~ ~ parity of the mirror-Chern number ( e): e ¼ in (a) and in completely identical to that along Γ Z, assuming that bands (b). Red solid (dashed) lines correspond to a Wilson band with are spectrally isolated. We thus conclude that the only mirror eigenvalue λz ¼þi (−i).

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(a) (b) (c) π π π

0 0 0

-π -π -π

(d) (e) (f) π π π

0 0 0

-π -π -π

FIG. 8. Possible Wilson spectra along Γ~ X~ . (a)–(c) With non- FIG. 9. Possible Wilson spectra along Z~ U~ . Pη C trivial glide polarization ( Γ~ ), the mirror-Chern numbers ( e) are, η respectively, −1, þ1, and −3. (d)–(f) With trivial P , C equals 0, Γ~ e These rules are stringent enough to uniquely specify the −2, and −4, respectively. interpolation along Z~ U~ , given the subtopologies at Γ~ Z~ Pη Q ~ ~ Q (specified by Γ~, Γ~ Z~ ) and at X U ( X~ U~ ). Alternatively other singlet state counts as one. With these rules, the parity stated, there are no additional invariants in this already- ~ C of this weighted count (N) equals that of e. Returning to complete classification. To justify our claim, first consider η Fig. 7(a) for illustration, we would still count the single P ¼ e=2, such that doublets at U~ are matched with cusps λ θ π 2 Γ~ z ¼þi crossing (encircled) at ¼ = , but if we instead ~ ~ ~ θ 0 Γ~ of hourglasses (along U X), while doublets at Z connect to pick ¼ , we could count the Kramers doublet at as ~ ~ ~ cusps of a zigzag (along Z Γ). There is then only one type of unity; for both choices of θ, N ¼þ1. In comparison, the η interpolation illustrated in Figs. 9(a)–9(c).IfP ¼ 0,we two θ ¼ 0 states in Fig. 7(b) are both singlets and count Γ~ ~ ~ ~ ~ collectively as two, which is consistent with this figure have hourglasses on both glide lines Γ Z and X U. If on one describing a C ¼ −2 phase. glide line an hourglass is centered at θ ¼ π, while on the e π Q ≠ Q Though our single-energy criterion applies at any θ,itis other line there is no hourglass (i.e., Γ~ Z~ X~ U~ ), the useful to particularize to θ ¼ 0, where in counting N~ we unique interpolation is shown in Figs. 9(d) and 9(e): red would have to determine the number of zero-energy Kramers doublets connect the upper cusp of one hourglass to the doublets at Γ~; e.g., this would be one in Fig. 7(a) and zero in lower cusp of another, in a generalized zigzag pattern with Fig. 7(b). This number may be identified, mod 2, with the spectral flow. A brief remark here is in order: when viewed Γ~ ~ ~ ~ number of zero-energy inversion singlets, which we estab- individually along any straight line (e.g., Z or Z U), bands η η lish in Sec. VB1to be unity if P ¼ e=2 and zero if P ¼ 0. are clearly spectrally isolated; however, when viewed along Γ~ Γ~ Γ~ ~ ~ ~ Γ~ a bent line ( Z U X), the bands exhibit spectral flow. In all Moreover, the parity of zero-energy doublets at may η ~ ~ ~ ~ other cases for P ; Q~ ~ , and Q ~ ~ , bands along Γ Z U X immediately be identified with the parity of N~ (and thus also Γ~ Γ Z X U ~ separate into spectrally isolated quadruplets, as in Fig. 9(f). that of Ce), because every other contribution to N has even parity, as we now show. We first consider the contribution at VI. QUASIMOMENTUM EXTENSIONS AND X~ . Given that the only subtopologies at X~ U~ are hourglasses, GROUP COHOMOLOGY IN BAND INSULATORS there are generically no zero-energy Kramers doublets at X~ [Figs. 8(a), 8(c)–8(f)], though in fine-tuned situations Symmetry operations normally describe space-time [Fig. 8(b)] there might be an even number. Away from transformations; such symmetries and their groups are the end points, any intersection comes in particle-hole- referred to as ordinary. Here, we encounter certain “sym- symmetric pairs [see, e.g., Figs. 8(c), 8(e) and 8(f)]. metries” of the Wilson loop that additionally induce quasimomentum transport in the space of filled bands; ~ ~ we call them W symmetries to distinguish them from the 4. Connecting curves along the mirror line Z U ordinary symmetries. In this section, we identify the ¯ (a) As illustrated in Fig. 9, each red solid curve (Mz ¼þi) relevant W symmetries and show their corresponding group ¯ − is degenerate with a red dashed curve (Mz ¼ i). (Gπ;kz ) to be an extension of the ordinary group (G∘)by Doublet curves cannot cross due to level repulsion, quasimomentum translations, where G∘ corresponds purely and must be symmetric under θ → −θ. to space-time transformations; the inequivalent extensions ~ ~ (b) The curve-matching conditions at Z and U again are classified by the second cohomology group, which we impose consistency requirements. also introduce here. In crystals, G∘ would be a magnetic

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(a) (b) (c) (d) point group [36] for a spinless particle; i.e., G∘ comprise the π space-time transformations (possibly including time rever- π b2 b2 π sal) that preserve at least one real-space point. It is well π known how G∘ may be extended by phase factors to describe half-integer-spin particles, and also by discrete ky ky ky ky spatial translations to describe nonsymmorphic crystals

π k k k k [30–34]. One lesson learned here is that G∘ may be further x x x x 2 extended by quasimomentum translations (as represented by the Wilson loop), thus placing real and quasimomentum space on equal footing. 2π 2π 2π 2π W symmetries are a special type of constraints on the Wilson loop at high-symmetry momenta (k∥). As exem- FIG. 10. Origin of W-glide and W-time-reversal symmetries. ˆ plified in Eqs. (11) and (12), constraints (g) on a Wilson (a)–(d) Constant-kz slices of the bulk Brillouin zone. Panel ¯ loop (W)mapW to itself, up to a reversal in orientation: (a) illustrates how the glide (Mx) maps momenta from the glide ¯ plane kx ¼ π. Under Mx, the Wilson loop is mapped from the red gˆWgˆ−1 ¼ W1; ð20Þ vertical arrow in (a) to the red vertical arrow in (b). Panels (c) and (d) describe the kz ¼ 0 plane and illustrate a similar story for the where W−1 is the inverse of W; all gˆ satisfying this time reversal T. equation are defined as elements in the group (Gk∥ ) of the ¯ Wilson loop. A trivial example of gˆ would be the Wilson The W glide (Mx) squares as loop itself; gˆ may also represent a space-time transforma- ¯ ¯ 2 ¯ −1 tion, as exemplified by a 2π real-space rotation (E). Mx ¼ Etð~zÞW−π; ð23Þ Particularizing to our context, we let ky ∈ ½−π; πÞ para- metrize the noncontractible momentum loop, and choose which may be understood loosely as follows: the glide ¯ the convention that W (W−1) effects parallel transport in component of the W glide squares as a 2π rotation (E) with the positive orientation: þ2πy~ (in the reversed orientation: a lattice translation [tð~zÞ], while the transport component −1 −2πy~), as further elaborated in Appendix B1. squares as a full-period transport (W ); we defer the W symmetries arise as constraints if a space-time trans- detailed derivations of Eqs. (21)–(23) to Appendix B4.For θ formation exists that maps ky → ky þ π. Our first exam- a Wilson band with energy ðkzÞ, Eq. (23) implies that the ple of a W symmetry was introduced in Sec. VA, namely, corresponding W-glide eigenvalue depends on the sum of ¯ → π energy and momentum, as in Eq. (14). Our construction of that the glide reflection (Mx) maps ðky; k∥Þ ðky þ ; k∥Þ ¯ ~ ~ M is a quasimomentum analog of the nonsymmorphic for any k∥ along k ¼ π (X U), as illustrated in Fig. 10(a). x x extension of point groups [30–34]. For example, the glide Consequently, the Wilson loop is mapped as ¯ reflection (Mx) combines a reflection with half a real- ¯ 2 ¯ ¯ −1 lattice translation—M thus squares to a full lattice trans- M W−πðπ;k ÞM ¼ W0ðπ;k Þ; ð21Þ x x z x z lation, which necessitates extending the point group by the ¯ where we indicate the base point of the parameter loop as a group of translations. Here, we further combine Mx with reciprocal-lattice subscript of W; i.e., W ¯ induces parallel transport from half a translation, thus necessitating a ky further extension by Wilson loops. ¯ π ¯ 2π π ðky; ;kzÞ to ðky þ ; ;kzÞ in the positive orientation. Our second example of a W symmetry (T ) combines W This mapping from −π [vertical arrow in Fig. 10(a)]to time reversal (T) with parallel transport over a half-period, W0 [arrow in Fig. 10(b)] is also illustrated. As it stands, and belongs in the groups of WðX~ Þ and WðU~ Þ, which Eq. (21) is not a constraint as defined in Eq. (20). Progress correspond to the two time-reversal-invariant k∥ along k ¼ is made by further parallel transporting the occupied space x π (recall Fig. 2); since both groups are isomorphic, we use a by −πy~, such that we return to the initial momentum: common label: GX~ . Under time reversal, ðky; π;kzÞ. This motivates the definition of a W-glide ¯ symmetry (Mx), which combines the glide reflection ∶ π ¯ → − −π −¯ ¯ T ðky; ; kzÞ ð ky; ; kzÞ (Mx) with parallel transport across half a reciprocal ¯ ~ ¯ ¯ ¼ð−ky þ π; π; kzÞ − b2 − 2kz~z; ð24Þ period—then by our construction, Mx is an element in the group (Gπ;k )ofW−πðπ;kzÞ. To be precise, let us define z for k¯ ∈ f0; πg and 2k¯ ~z a reciprocal vector (possibly zero), the Wilson line W−π←0 to represent a parallel transport z z as illustrated in Fig. 10(c). Consequently, from ð0; π;kzÞ to ð−π; π;kzÞ, then ¯ ¯ ¯ −1 ¯ ¯ −1 kz MxW−πMx ¼ W−π; with Mx ¼ W−π←0Mx: ð22Þ TW−πT ¼Wr;2π; ð25Þ

021008-14 TOPOLOGICAL INSULATORS FROM GROUP COHOMOLOGY PHYS. REV. X 6, 021008 (2016) where Wr;2π denotes the reverse-oriented Wilson loop with further elaborated later in this section), which is inequiva- k¯ lent to G ~ and applies to a different momentum submani- base point 2π [see arrow in Fig. 10(d)], and ¼z indicates that X fold of our crystal; in Sec. VB, we further show that ∈ ~ ~ this equality holds for k∥ fX; Ug. Equation (B62) moti- inequivalent extensions lead to different subtopologies for vates combining T with a half-period transport, such that the Wilson bands. T the combined operation effects From the cohomological perspective, two extensions (of N ¯ ¯ G∘ by ) are equivalent if they correspond to the same −1 kz −1 kz 2 TW−πT ¼W−π; with T ¼W−π←0T: ð26Þ element in the second cohomology group H ðG∘; N Þ. The identity element in this group corresponds to a linear To complete the Wilsonian algebra, we derive in representation of G∘, which we now define. Let the group Appendix B4 that element gi ∈ G∘ be represented by gˆi in the extension of G∘ by N, and further define gij ≡ gigj ∈ G∘ by gˆij. We insist k¯ k¯ 2 z ¯ ¯ ¯ −1 −1 z −1 that fgˆig satisfy the associativity condition: T ¼ E; MxT Mx T ¼W−π: ð27Þ ˆ ˆ ˆ ˆ ˆ ˆ This result, together with Eq. (23), may be compared with ðgigjÞgk ¼ giðgjgkÞ: ð32Þ the ordinary algebra of space-time transformations: In a linear representation, k¯ k¯ M¯ 2 ¼ Et¯ ð~zÞ;T2¼z E;¯ M¯ TM¯ −1T−1¼z I; ð28Þ x x x gˆigˆj ¼ gˆij for all gi;gi;gij ∈ G∘; ð33Þ as would apply to the surface bands at any time-reversal- while in a projective representation, invariant k∥. Both algebras are identical modulo factors of W and its inverse; from here on, Wðk∥Þ−π ¼ W.We gˆigˆj ¼ Ci;jgˆij; where Ci;j ∈ N ; ð34Þ emphasize that the same edge symmetries are represented W— differently in the surface Hamiltonian (Hs) and in this at least one of fCi;jg (defined as the factor system [60])is difference originates from the out-of-surface translational not trivially identity. Equation (23) exemplifies Eq. (34) for ~ symmetry [tða1Þ], which is broken for Hs but not for W; 2 ¯ gi ¼ gj ¼ Mx, satisfying Mx ¼ I, gˆi ¼ Mx, and Ci;j ¼ recall here that a~ 1 is the out-of-surface Bravais-lattice −1 Et¯ ð~zÞW . We say that two representations are equivalent vector drawn in Fig. 2(b). Where tða~1Þ symmetry is if they are related by the transformation preserved, we can distinguish the bulk wave vectors ky from ky þ π, and therefore define W-symmetry operators 0 gˆi → gˆi ¼ Digˆi; with Di ∈ N : ð35Þ that include the Wilson line W−π←0. To further describe this difference group theoretically, let In either representation, the same constraint is imposed on ≅ Z Z us define G∘ 2 × 2 as the symmetry group of a W [cf. Eq. (20)]: spinless particle with glideless-reflection (Mx) and time- reversal (T) symmetries: ˆ 0W ˆ 0−1 W1 ⇔ ˆ W ˆ−1 W1 gi gi ¼ gi gi ¼ ; ð36Þ a b ∈ Z G∘ ¼fMxT ja; b 2g; ð29Þ since any element of N commutes with W. This state of affairs is reminiscent of the U(1) gauge ambiguity in with the algebra representing symmetries of the Hamiltonian (Hˆ ) [61], ˆ ˆ 2 2 0 where if ½g;ˆ H¼0, so would ½gˆ0; H¼0 for any Mx ¼ I; T ¼ I; ½Mx;T¼ : ð30Þ 0 0 gˆ ¼ exp½iϕðgÞgˆ. By this analogy, we also call gˆi and gˆi The algebra of Eq. (28) describes a well-known, non- from Eq. (35) two gauge-equivalent representations of the same element g , though it should be understood in symmorphic extension of G∘ for spinful particles [11];we i this paper that the relevant gauge group is N and not U(1). propose that Eqs. (23) and (27) describe a further extension 2 To recapitulate, each element in H ðG∘; N Þ corresponds to of Eq. (28) by reciprocal translations. That is, GX~ is a 2 an equivalence class of associative representations; in nontrivial extension of G∘ by N, where N ≅ Z × Z2 is an Appendix D2, we further connect our theory to group Abelian group generated by E¯, W, and tð~zÞ: cohomology through the geometrical perspective of ¯ a b c coboundaries and cocycles. N ¼fE tð~zÞ W ja ∈ Z2;b;c∈ Zg: ð31Þ To exemplify an extension or representation that is For an introduction to group extensions and their applica- inequivalent to GX~ , let us consider the group (GΓ~ )of W Γ~ W ~ tion to our problem, we refer the interested reader to ð Þ; GΓ~ is isomorphic to the group of ðZÞ. Recall that ~ ~ Appendix D1. There exists another extension (GΓ~ ,as both Γ and Z are time-reversal-invariant k∥ along kx ¼ 0.

021008-15 ALEXANDRADINATA, WANG, and BERNEVIG PHYS. REV. X 6, 021008 (2016) kx ¼ 0 labels a glide line in the 010-surface BZ, which VII. DISCUSSION AND OUTLOOK guarantees the kx ¼ 0 plane (in the bulk BZ) is mapped to ¯ In the topological classification of band insulators, one itself under the glide Mx; the same could be said for kx ¼ π. π ¯ may sometimes infer the classification purely from the However, unlike kx ¼ , Mx also belongs to the group of representation theory [4,8,54] of surface wave functions. In 0 any bulk wave vector in the kx ¼ plane, and, therefore, our companion work [21], we have identified a criterion on the surface group that characterizes all robust surface ¯ W Γ~ ¯ −1 W Γ~ Mx ð ÞMx ¼ ð Þ; ð37Þ states that are protected by space-time symmetries [1,2,4,7,8,18,19,25,40–43,54]. Our criterion introduces ¯ M¯ with Mx an ordinary space-time symmetry; i.e., unlike x in the notion of connectivity within a submanifold (M)of ¯ Eq. (22), Mx does not encode parallel transport. the surface-Brillouin torus and generalizes the theory of Consequently, this element of GΓ~ satisfies the ordinary elementary band representations [57,62]. To restate the algebra in Eq. (28); by an analogous derivation, the time- criterion briefly, we say there is a D-fold connectivity reversal element in GΓ~ is also ordinary. It is now apparent within M if bands there divide into sets of D, such that M why GX~ and GΓ~ are inequivalent extensions: there exists no within each set there are enough contact points in to gauge transformation, of the form in Eq. (35), that relates their continuously travel through all D branches. If M is a single factor systems. For example, the following elements of G∘, wave vector (k∥), D coincides with the minimal dimension of the irreducible representation at k∥; D generalizes this ≡ ¯ ≡ ¯ −1 −1 ≡ g1 MxT; g2 Mx T ;g1g2 g12 ¼ I; ð38Þ notion of symmetry-enforced degeneracy where M is larger than a wave vector (e.g., a glide line). We are ready may be represented in GX~ by to state our criterion: (a) there exist two separated sub- manifolds M1 and M2, with corresponding D1 ¼ D2 ¼ ˆ ≡ M¯ T ˆ ≡ M¯ −1T −1 ˆ ≡ g1 x ; g2 x ; g12 I; ð39Þ fd (f ≥ 2 and d ≥ 1 are integers), and (b) a third submanifold M3 that connects M1 and M2, with corre- such that the second relation in Eq. (27) translates to sponding D3 ¼ d. This surface-centric criterion is techni- cally simple, and has proven to be predictive of the ˆ ˆ ˆ ≡ W−1 g1g2 ¼ C1;2g12; with C1;2 : ð40Þ topological classification. However, we also found it is sometimes overpredictive [21], in the sense of allowing ¯ ¯ 0 a ¯ Under the gauge transformation, Mx → Mx ¼ W Mx, some surface topologies that are inconsistent with the full T → T 0 ¼ WbT ,anda; b ∈ Z,Eq.(40) transforms as set of bulk symmetries. An alternative and, as far as we know, faithful approach ¯ ¯ −1 −1 −1 MxT Mx T ¼ W would apply our connectivity criterion [21] to the Wilson ¯ ¯ −1 −1 2 −1 “bands,” which properly encode bulk symmetries that are → M0 T 0M0 T 0 ¼ W a ; ð41Þ x x absent on the surface; since Wilson bands also live on the surface-Brillouin torus, we could replace the original which ensures that the factor C1 2 is always an odd product ; meaning of surface bands in the above criterion by of W. This must be compared with the analogous Wilson bands. To determine the possible Wilson “band ¯ → ¯ 0 Wc ¯ algebraic relation in GΓ~ , where with Mx Mx ¼ Mx, structures,” one has to determine how symmetries are → 0 Wd ∈ Z T T ¼ T,andc, d , represented in the Wilson loop; one lesson learned from classifying D4 is that this representation can be projective, ¯ ¯ −1 −1 → ¯ 0 0 ¯ 0−1 0−1 W2c; 6h MxTMx T ¼ I Mx T Mx T ¼ ð42Þ requiring an extension of the point group by the Wilson loop itself. Such an extended group forces us to generalize here, the analogous factor C1;2 is always an even product of the traditional notion of symmetry as a space-time trans- W— there exists no gauge transformation that relates the two formation—we instead encounter symmetry operators that factor systems in GX~ and GΓ~ . We say that the factor system of a combine both space-time transformations and quasimo- projective representation can be lifted if, by some choice of mentum translations, thus putting real and quasimomentum gauge, all of fCi;jg from Eq. (34) may be reduced to the space on equal footing. While our case study involves a N identity element in ;Eq.(41) demonstrates that C1;2 for GX~ nonsymmorphic space group, the nonsymmorphicity (i.e., can never be transformed to identity. GX~ thus exemplifies an nontrivial extensions by spatial translations) is not a intrinsically projective representation, wherefor its nontrivial prerequisite for nontrivial quasimomentum extensions; factor system can never be lifted. e.g., there are projective mirror planes (e.g., in symmorphic Finally, we remark that this section does not exhaust all rocksalt structures) where the reflection also relates Bloch elements in GX~ or GΓ~ ; our treatment here minimally waves separated by half a reciprocal period. The implica- conveys their group structures. A complete treatment of tions are left for future study. GX~ is offered in Appendix B4, where we also derive the To restate our finding from a broader perspective, group above algebraic relations in greater detail. cohomology specifies how symmetries are represented in

021008-16 TOPOLOGICAL INSULATORS FROM GROUP COHOMOLOGY PHYS. REV. X 6, 021008 (2016) the quasimomentum submanifold, which in turn determines We begin by reviewing the effects of spatial symmetries, the band topology. A case in point is time-reversal then generalize our discussion to include time-reversal symmetry (T), which may be extended by 2π-spin rotations symmetry. (which distinguishes half-integer from integer-spin repre- sentations) and also by real-space translations (which 1. Effect of spatial symmetries on the distinguishes paramagnetic and antiferromagnetic insula- tight-binding Hamiltonian tors); only the projective representation (T2 ¼ −I) has a Z Let us denote a spatial transformation by gδ, which well-known 2 topology [18,25]. By our cohomological transforms real-space coordinates as r → D r þ δ, where classification of quasimomentum submanifolds through g D is the orthogonal matrix representation of the point- Eq. (1), we provide a unifying framework to classify chiral g group transformation g in Rd. Nonsymmorphic space topological insulators [39] and topological insulators with δ robust edge states protected by space-time symmetries groups contain symmetry elements where is a rational – fraction [11] of the lattice period; in a symmorphic space [1,2,4,7,8,18,19,25,40 43]. Our framework is also useful δ 0 in classifying some topological insulators without edge group, an origin can be found where ¼ for all symmetry elements. The purpose of this section is to derive the states [26,55,56]; one counterexample that eludes this constraints of gδ on the tight-binding Hamiltonian. First, we framework may nevertheless by classified by bent Wilson clarify how gδ transforms the creation and annihilation loops [46] rather than the straight Wilson loops of this operators. We define the creation operator for a Löwdin work. With the recent emergence of Floquet topological function [47–49] (φα) at Bravais-lattice (BL) vector R as phases, an interesting direction would be to consider further † extending Eq. (1) by discrete time translations [63]. cαðR þ rαÞ. From Eq. (2), the creation operator for a Bloch- wave-transformed Löwdin orbital ϕk;α is X ACKNOWLEDGMENTS † 1 † ffiffiffiffi ik·ðRþrαÞ α 1 … ck;α ¼ p e cαðR þ rαÞ; ¼ ; ;ntot: We thank Ken Shiozaki and Masatoshi Sato for dis- N R cussions of their K-theoretic classification, Mykola ðA1Þ Dedushenko for discussions on group cohomology, and Chen Fang for suggesting a nonsymmorphic model of the A BL that is symmetric under gδ satisfies two conditions. quantum spin Hall insulator. Z. W., A. A., and B. A. B. (i) For any BL vector R, D R is also a BL vector: were supported by NSF CAREER DMR-095242, ONR - g N00014-11-1-0635, ARO MURI on topological insulators, ∀ R ∈ BL;DR ∈ BL: ðA2Þ grant W911NF-12-1-0461, NSF-MRSEC DMR-1420541, g Packard Foundation, Keck grant, “ONR Majorana (ii) If gδ transforms an orbital of type α to another of type β, Fermions” 25812-G0001-10006242-101, and Schmidt then D ðR þ rαÞþδ must be the spatial coordinate of an fund 23800-E2359-FB625. In finishing stages of this work, g orbital of type β. To restate this formally, we define a matrix A. A. was further supported by the Yale Prize Fellowship. Ugδ such that the creation operators transform as

† † gδ APPENDIX A: REVIEW OF SYMMETRIES gδ∶ cαðR þ rαÞ → cβðDgR þ Rβα þ rβÞ½Ugδβα; ðA3Þ IN THE TIGHT-BINDING METHOD gδ ≡ δ − The appendixes are organized as follows: In Appendix A, with Rβα Dgrα þ rβ. Then, we review how space-time symmetries affect the tight- gδ binding Hamiltonian. We introduce notations that are ½Ugδβα ≠ 0 ⇒ Rβα ∈ BL: ðA4Þ employed in the remaining appendixes. In Appendix B, we derive how symmetries of the Wilson loop are repre- Explicitly, the nonzero matrix elements are given by sented and their constraints on the Wilson bands, as X Z summarized in Tables II and III. The first few sections deal d 0 ð1=2Þ −1 ½U δβα ¼ d rφ ðr;sÞ½D 0 φαðD r;sÞ; ðA5Þ with ordinary symmetry representations along X~ Γ~ Z~ U~ , g β g s s g s;s0 while the last section derives the projective representations along X~ U~ . In Appendix C, we prove the fourfold ð1=2Þ where φα is a spinor with spin index s, and Dg represents connectivity of Hamiltonian bands, in spin systems gδ in the spinor representation. with minimally time-reversal and glide-reflection sym- gδ gδ For fixed gδ, α, and β, the mapping T : R → R is metries. This proof is used in the topological classification βα βα of Sec. VB. In Appendix D, we introduce group extensions bijective. Applying Eqs. (A1), (A2), (A4), the orthogon- T gδ by Wilson loops, as well as rederive the extended algebra in ality of Dg, and the bijectivity of βα, the Bloch basis Sec. VI from a group-theoretic perspective. vectors transform as

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X † 1 † δ ∶ → ffiffiffiffi ik·ðRþrαÞ g gδ ck;α p e cβðDgR þ Rβα þ rβÞ½Ugδβα N R X − δ 1 δ † gδ iðDgkÞ· ffiffiffiffi i½Dgk·½DgðRþrαÞþ ¼ e p e cβðDgR þ Rβα þ rβÞ½Ugδβα N R X 1 gδ − δ i D k D R R rβ † gδ iðDgkÞ· ffiffiffiffi ½ g ·½ g þ βαþ ¼ e p e cβðDgR þ Rβα þ rβÞ½Ugδβα N R 1 X − δ 0 † iðDgkÞ· ffiffiffiffi i½Dgk·½R þrβ 0 ¼ e p e cβðR þ rβÞ½Ugδβα N R0 −iðD kÞ·δ † ¼ e g c ½U δβα: ðA6Þ Dgk;β g

¯ This motivates a definition of the operator Mj ¼ IC2j for j ∈ fx; zg. Consequently, Mx ∝ C2x flips Sz → −Sz. In the basis of Bloch waves,

−iðDgkÞ·δ gˆδðkÞ ≡ e Ugδ; ðA7Þ

† − 2 † ¯ ∶ → ikzc= ¯ Mx ck;α e cD k;β½UM βα; ðA10Þ which acts on Bloch wave functions (jun;ki)as x x

gδ∶ u → gˆδ k u : A8 j n;ki ð Þj n;ki ð Þ U ¯ −iτ1σ1 τ3 1 −1 with Mx ¼ . Here, we employ ¼þ ( ) for subcell A (B) and σ3 ¼þ1 for spin-up in ~z. A similar ˆ ¯ The operators fgδðkÞg form a representation of the space- analysis for the other reflection (Mz ∝ C2z ∝ exp½−iSzπ) group algebra [11] in a basis of Bloch-wave-transformed leads to Löwdin orbitals; we call this the Löwdin representation. If the space group is nonsymmorphic, the nontrivial phase (a) factor expð−iDgk · δÞ in gˆδðkÞ encodes the effect of the fractional translation; i.e., the momentum-independent rB rA matrices fUgδg by themselves form a representation of a point group. To exemplify this abstract discussion, we analyze a simple 2D nonsymmorphic crystal in Fig. 11. As delineated c/2 by a square, the unit cell comprises two atoms labeled by subcell coordinates A and B, and the spatial origin is chosen c pffiffiffi z at their midpoint, such that rA ¼ ax=~ 3 − c~z=2 ¼ −rB,as shown in Fig. 11(a). The symmetry group (Pma2) of this ¯ ¯ a/ 3 lattice is generated by the elements Mx and Mz, where in x the former we first reflect across x~ (g ¼ Mx) and then ¯ 3a translate by δ ¼ c~z=2. Similarly, Mz is shorthand for a z → −z reflection followed by a translation by δ ¼ c~z=2. Let us represent these symmetries with spin-doubled (b) (c) (d) s orbitals on each atom. Choosing our basis to diago- nalize Sz,

( † † c ðR þ rAÞ → −ic − ðDxR þ rBÞ ¯ ∶ A;Sz B; Sz Mx † † ðA9Þ c ðR þ r Þ → −ic ðD R þ c~z þ r Þ; B;Sz B A;−Sz x A MX t(cz/2)

t t where Dxðx; zÞ ¼ð−x; zÞ , and in the second mapping, we x;c~z=2 2 − apply RAB ¼ DxrB þ c~z= rA ¼ c~z. It is useful to FIG. 11. (a) Simple example of a 2D nonsymmorphic crystal. recall here that a reflection is the product of an inversion The two sublattices are colored, respectively, dark blue and cyan. with a twofold rotation about the reflection axis: Panels (b)–(d) illustrate the effect of a glide reflection.

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8 † → By assumption of an insulating gap, gˆδðkÞju i belongs in > cA;S ðR þ rAÞ n;k > z <> † the occupied-band subspace for any occupied band jun;ki. −isgn½Szc ðDzR þ c~z þ rAÞ ¯ A;Sz This implies a unitary matrix representation (sometimes Mz∶ ðA11Þ > † called the “sewing matrix”)ofgδ in the occupied-band > c ðR þ rBÞ → > B;Sz : † subspace: −isgn½S c ðD R þ r Þ; z B;Sz z B ˘ ½gδðDgk þ G; kÞmn t t with Dzðx; zÞ ¼ðx; −zÞ , and in the basis of Bloch-wave- ¼hu jVð−GÞgˆδðkÞju i; ðA20Þ transformed Löwdin orbitals, m;DgkþG n;k

† − 2 † 1 … ¯ ∶ → ikzc= with m; n ¼ ; ;nocc. Here, G is any reciprocal vector Mz c α e c β½UM¯ βα; ðA12Þ k; Dzk; z (including zero), and we apply Eq. (7), which may be rewritten as U ¯ −iσ3 gˆδ with Mz ¼ . To recapitulate, we derive f g as

Xnocc Xnocc ˆ −1 ¯ −ikzc=2 MxðkÞ¼−ie τ1σ1 and jun;kihun;kj¼VðGÞ jun;kþGihun;kþGjVðGÞ : n¼1 n¼1 ¯ˆ −ik c=2 M k −ie z σ3; A13 zð Þ¼ ð Þ ðA21Þ which should satisfy the space-group algebra for Pma2, To motivate Eq. (A20), we are often interested in high- namely, that symmetry k which are invariant under gδ; i.e., Dgk þ G ¼ k for some G (possibly zero). At these special momenta, the M¯ 2 ¼ Et¯ ðc~zÞ; M¯ 2 ¼ E¯ and x z sewing matrix is unitarily equivalent to a diagonal matrix, ¯ ¯ ¯ ¯ ¯ MzMx ¼ Etð−c~zÞMxMz; ðA14Þ whose diagonal elements are the gδ eigenvalues of the occupied bands. When we are not at these high-symmetry where E¯ denotes a 2π rotation and tðc~zÞ a translation. momenta, we will sometimes use the shorthand Indeed, when acting on Bloch waves with momentum k, ½g˘δðkÞ ≡ ½g˘δðD k; kÞ ¼hu jgˆδðkÞju i; ðA22Þ ˆ ˆ ˆ ˆ mn g mn m;Dgk n;k ¯ ¯ −ikzc ¯ ¯ MxðDxkÞMxðkÞ¼−e ; MzðDzkÞMzðkÞ¼−I; ¯ˆ ¯ˆ − ¯ˆ ¯ˆ since the second argument is self-evident. We emphasize M ðD kÞM ðkÞ¼−e ikzcM ðD kÞM ðkÞ: ðA15Þ z x x x z z that gˆδ and g˘δ are different matrix representations of the same symmetry element (gδ), and, moreover, the matrix Finally, we verify that the momentum-independent matri- dimensions differ: (i) gˆδ acts on Bloch combinations of ces fUgδg form a representation of the double point group ϕ α 1 … Löwdin orbitals (f k;αj ¼ ; ;ntotg) defined in Eq. (2), C2 , whose algebra is simply v while (ii) g˘δ acts on the occupied eigenfunctions 1 … 2 2 ¯ ¯ (fun;kjn ¼ ; ;noccg)ofHðkÞ. Mx ¼ Mz ¼ E and MzMx ¼ EMxMz: ðA16Þ It is also useful to understand the commutative relation between gˆδðkÞ and the diagonal matrix VðGÞ which A simple exercise leads to encodes the spatial embedding; as defined in Eq. (6), the diagonal elements are ½VðGÞαβ ¼ δαβ expðiG · rαÞ. From 2 2 − 0 U ¯ ¼ U ¯ ¼ I and fUM¯ ;UM¯ g¼ : ðA17Þ Mx Mz x z Eqs. (A2) and (A4),

δ The algebras of C2v and Pma2 differ only in the additional −1 g ½U δαβ ≠ 0 ⇒ D Rαβ ∈ BL elements t c~z , which in the Löwdin representation g g ð Þ −1 −1 iG·ðrβþD δ−D rαÞ (fgˆδðkÞg) is accounted for by the phase factors ⇒ e g g ¼ 1; ðA23Þ expð−ikzc=2Þ. Returning to a general discussion, if the Hamiltonian is for a reciprocal-lattice (RL) vector G. Applying this symmetric under gδ, equation in X † − δ ∶ ˆ → ˆ 0 ≠ ˆ iðDgkÞ· iG·rβ gδ H ¼ ck;αHðkÞαβck;β H; ðA18Þ ½gδðkÞVðGÞαβ ¼ e ½Ugδαβe k −iðDgkÞ·δ iðDgGÞ·ðrα−δÞ ¼ e ½Ugδαβe − δ then Eq. (A6) implies iðDgGÞ· ˆ ¼ e ½VðDgGÞgδðkÞαβ; ˆ ˆ −1 gδðkÞHðkÞgδðkÞ ¼ HðDgkÞ: ðA19Þ we then derive

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− δ ˆ iðDgGÞ· ˆ ˘ ≡ ˘ − gδðkÞVðGÞ¼e VðDgGÞgδðkÞ: ðA24Þ ½TgδðkÞmn ½Tgδð Dgk; kÞmn ˆ ¼hu − jT δðkÞju i: ðA30Þ This equality applies only if the argument of V is a m; Dgk g n;k reciprocal vector. Equations (A25) and (A2) further imply that

2. Effect of space-time symmetry −1 gδ ½UTgδαβ ≠ 0 ⇒ Dg Rαβ ∈ BL on the tight-binding Hamiltonian −1δ− −1 ⇒ eiG·ðrβþDg Dg rαÞ ¼ 1; ðA31Þ Consider a general space-time transformation Tgδ, where now we include the time reversal T. The following which when applied to discussion also applies if gδ is the trivial transformation. ˆ † † Tgδ 0 ≠ ½TgδðkÞVðGÞKαβ Tgδ∶ cαðR þ rαÞ → cβðDgR þ Rβα þ rβÞ½UTgδβα; iðDgkÞ·δ −iG·rβ ¼ e ½UTgδαβe δ − −δ where UTgδ is the matrix representation of Tgδ in the iðDgkÞ· iðDgGÞ·ðrα Þ ¼ e ½UTgδαβe Tgδ Löwdin orbital basis, Rβα ¼ D rα þ δ − rβ, g þiðDgGÞ·δ ˆ ¼ e ½Vð−DgGÞTgδðkÞKαβ ðA32Þ ≠ 0 ⇒ Tgδ ∈ ½UTgδβα Rβα BL; ðA25Þ leads finally to

δ Tg ˆ iðDgGÞ·δ ˆ and the Bravais-lattice mapping of R to DgR þ Rβα is TgδðkÞVðGÞ¼e Vð−DgGÞTgδðkÞ: ðA33Þ bijective. It follows that the Bloch-wave-transformed Löwdin orbitals transform as

† δ † ∶ → iDgk· APPENDIX B: SYMMETRIES Tgδ ck;α e c−D k;β½UTgδβα: ðA26Þ g OF THE WILSON LOOP This motivates the following definition for the Löwdin The goal of this Appendix is to derive how symmetries of representation of Tgδ: the Wilson loop are represented and their implications for the “rules of the curves,” as summarized in Tables II and III. ˆ iðDgkÞ·δ TgδðkÞ ≡ e UTgδK; ðA27Þ After introducing the notations and basic analytic proper- ties of Wilson loops in Appendix B1, we consider in where K implements complex conjugation, such that a Appendix B2 the effect of spatial symmetries, with ˆ ˆ particular emphasis on glide symmetry. We then generalize symmetric Hamiltonian (Tgδ: H → H) satisfies our discussion to space-time symmetries in Appendix B3. ˆ ˆ −1 These first sections apply only to symmetries of the Wilson T δðkÞHðkÞT δðkÞ ¼ Hð−D kÞ: ðA28Þ g g g loops along X~ Γ~ , Γ~ Z~ , and Z~ U~ . These symmetry represen- tations are shown to be ordinary; i.e., they do not encode For a simple illustration, we return to the lattice of Fig. 11, ˆ quasimomentum transport. Their well-known algebra where time-reversal symmetry is represented by TðkÞ¼ includes −iσ2K in a basis where σ3 ¼þ1 corresponds to spin-up in ~z. Observe that time reversal commutes with any spatial 2 2 M¯ ¼ Et¯ ðc~zÞ; ðTM¯ Þ ¼ I; transformation: x z ¯ ¯ ¯ ¯ ¯ 2 2 ¯ MxTMz ¼ Etðc~zÞTMzMx;T¼ðITÞ ¼ E; ∈ ˆ ¯ˆ ¯ˆ − ˆ ¯ ¯ ¯ ¯ for j fx; zg; TðDjkÞMjðkÞ¼Mjð kÞTðkÞ: MxIT ¼ tðc~zÞITMx; MxT ¼ TMx: ðB1Þ

If the Hamiltonian is gapped, there exists an antiunitary In Appendix B4, we move on to derive the projective representation of Tgδ in the occupied-band subspace: representations that apply along X~ U~ .

˘ − ½TgδðG Dgk; kÞmn 1. Notations and analytic properties of the Wilson loop − ˆ ¼hum;G−DgkjVð GÞTgδðkÞjun;ki; ðA29Þ Consider the parallel transport of occupied bands along the noncontractible loops of Sec. IVA. In the Löwdin 1 … where m; n ¼ ; ;nocc, G is any reciprocal vector, and we orbital basis, such transport is represented by the Wilson- apply Eq. (A21). Once again, we introduce the shorthand loop operator [26]:

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πY←−π ˆ The second equality in Eq. (B6) follows from Wðk∥Þ¼Vð2πy~Þ Pðky; k∥Þ: ðB2Þ π←−π ky Y W 2π ji ¼huj;−πjVð y~Þ PðkyÞjui;−πi Recall here our unconventional ordering: k ¼ðky;kx; ky −π←π kzÞ¼ðky; k∥Þ. We discretize the momentum as ky ¼ Y −2π 2πm=Ny for integer m ¼ 1; …;Ny, and ðπ← − πÞ indicates ¼hui;−πj PðkyÞVð y~Þjuj;−πi that the product of projections is path ordered. The role of ky the path-ordered product is to map a state in the occupied −Yπ←π −2π subspace [Hð−π; k∥Þ]atð−π; k∥Þ to one (ju~i) in the ¼hui;πjVð y~Þ PðkyÞjuj;πi k occupied subspace at ðπ; k∥Þ; the effect of Vð2πy~Þ is to y ~ H −π ˆ subsequently map jui back to ð ; k∥Þ, thus closing the ¼hui;πjWrjuj;πi; ðB8Þ parameter loop; cf. Eq. (A21). Equivalently stated, we may represent this same parallel transport in the basis of nocc where we drop the constant argument k∥ for notational occupied bands: simplicity.

W ˆ ½ ðk∥Þij ¼hui;ð−π;k∥ÞjWðk∥Þjuj;ð−π;k∥Þi: ðB3Þ 2. Effect of spatial symmetries of the 010 surface W Let us describe the effect of symmetry on the spectrum of While depends on the choice of gauge for juj;−π;k∥ i, W. First, we consider a generic spatial symmetry gδ, which its eigenspectrum does not. Indeed, under the gauge transforms real-space coordinates as r → Dgr þ δ. From transformation Eq. (A19), we obtain the constraints on the projections as n Xocc −1 → ∈ gˆδðkÞPðkÞgˆδðkÞ ¼ PðDgkÞ: ðB9Þ juj;ð−π;k∥Þi jui;ð−π;k∥ÞiSij; with S UðnoccÞ; i¼1 The constraints on W arise only from a subset of the W → S†WS ¼ S−1WS: ðB4Þ symmetries that either (i) map one loop parametrized by k∥ k∥ The eigenspectrum is also independent of the base point of to another loop at a different or (ii) map a loop to itself at a high-symmetry k∥. We say that two loops are mapped to the loop [26]; our choice of ð−π; k∥Þ as the base point merely renders certain symmetries transparent. In the limit each other even if the mapping reverses the loop orientation [i.e., W → W−1; cf. Eqs. (B5) and (B6)] or translates the of large Ny, W becomes unitary and its full eigenspectrum base point of the loop. For illustration, we consider spatial comprises the unimodular eigenvalues of Wˆ , which we symmetries of the 010 surface that necessarily preserve the label by exp½iθ , with n ¼ 1; …;n . Denoting the n;k∥ occ spatial y coordinate; for these symmetries we add an ˆ eigenvalues of P⊥yP⊥ as yn;k∥ , the two spectra are related additional subscript to gδ ≡ gδ∥, Dg ≡ Dg∥, and δ ≡ δ∥, as y ¼ θ =2π modulo one [26]. n;k∥ n;k∥ such that Dg∥y~ ¼ y~ and δ∥ · y~ ¼ 0. We now demonstrate On occasion, we also need the reverse-oriented Wilson that the Wilson loop is constrained as ˆ loop (Wr), which transports a state from base point π → −π −1 ð ; k∥Þ ð ; k∥Þ: g˘δ∥ð−π; k∥ÞWðk∥Þg˘δ∥ð−π; k∥Þ ¼ WðDg∥k∥Þ; ðB10Þ −Yπ←π where the argument of g˘δ∥ is the base point of Wðk∥Þ. ˆ −2π Wrðk∥Þ¼Vð y~Þ Pðky; k∥Þ: ðB5Þ Proof.—We note that the Löwdin representation ky of gδ only depends on momentum through a multiplicative phase factor: exp½−iðD kÞ · δ½, and for gδ∥ that this same In the occupied-band basis, Wilson loops of opposite g phase factor is independent of k ; hence, the Löwdin orientations are mutual inverses: y representation of gδ may be written as gˆδ∥ðkÞ ≡ gˆδ∥ðk∥Þ. W −1 W ˆ Equation (B9) then particularizes to ½ ðk∥Þ ij ¼½ ðk∥Þji ¼hui;ðπ;k∥ÞjWrðk∥Þjuj;ðπ;k∥Þi; −1 ðB6Þ gˆδ∥ðk∥ÞPðky; k∥Þgˆδ∥ðk∥Þ ¼ Pðky;Dg∥k∥Þ; ðB11Þ with the gauge choice and Eq. (A24) to

−2π 1 … ˆ 2π juj;ðπ;k∥Þi¼Vð y~Þjuj;ð−π;k∥Þi; for j ¼ ; ;nocc: gδ∥ðk∥ÞVð y~Þ

−i2πy~·δ∥ ðB7Þ ¼ e Vð2πy~Þgˆδ∥ðk∥Þ¼Vð2πy~Þgˆδ∥ðk∥Þ: ðB12Þ

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Applying Eqs. (B2), (B11), and (B12), Here, k1 ≡ ð−π; 0;kzÞ; in the second equality, we denote ntot as the total number of bands, and apply that the ˆ −1 ˆ gˆδ∥ðk∥ÞWðk∥Þgˆδ∥ðk∥Þ ¼ WðDg∥k∥Þ: ðB13Þ symmetry representations are block diagonal with respect to the occupied and empty subspaces [i.e., u U ¯ u 0 Into this equation, we then insert complete sets of states h a;k1 j Mx j n;k1 i¼ if the bra (ket) state is occupied [IˆðkÞ]: (empty)]; the completeness relation is used in the third 2 − 2π equality, and ðUM¯ Þ ¼ I represents a rotation. ˆ ˆ ˆ ˆ x Ið−π;Dg∥k∥Þgˆδ∥ðk∥ÞIð−π; k∥ÞWðk∥ÞI −1 ˆ 3. Effect of space-time symmetries × ð−π; k∥Þgˆδ∥ðk∥Þ Ið−π;Dg∥k∥Þ ˆ ˆ ˆ Suppose our Hamiltonian is symmetric under a space- ¼ Ið−π;Dg∥k∥ÞWðDg∥k∥ÞIð−π;Dg∥k∥Þ; ðB14Þ time transformation Tgδ, where gδ is any of the following: ˆ (a) a symmorphic spatial transformation (i.e., the fractional where IðkÞ is resolved by the energy eigenbasis at k: translation δ ¼ 0) that is not necessarily a symmetry of the 010 surface or (b) a nonsymmorphic transformation that is Xn ˆ tot a symmetry of the 010 surface. In the group we study IðkÞ¼ jun;kihun;kj: ðB15Þ 4 n¼1 (space group D6h with time-reversal symmetry), if one considers the subset of space-time symmetries that map the Since all symmetry representations are block diagonal with Wilson loop to itself (recall what “to itself” means in respect to occupied and empty subspaces, Eq. (B14) is Appendix B2), elements of this subset are either of (a) or equivalent to (b) type. ˆ Since the Löwdin representation [recall TgδðkÞ in ˆ Pð−π;D ∥k∥Þgˆδ∥ðk∥ÞPð−π; k∥ÞWðk∥ÞP Eq. (A27)]ofTgδ depends on momentum only through g ˆ −1 the phase factor exp½iðD kÞ · δ, we deduce that T δðkÞ is × ð−π; k∥Þgˆδ∥ðk∥Þ Pð−π;D ∥k∥Þ g g g independent of k . In case (a), this follows trivially from ˆ y ¼ Pð−π;D ∥k∥ÞWðD ∥k∥ÞPð−π;D ∥k∥Þ: g g g δ ¼ 0, while in case (b) we apply that Dgy~ ¼ y~ and ˆ ˆ δ · y~ ¼ 0. Consequently, we write T δðkÞ ≡ T δðk∥Þ with Finally, we apply Eqs. (A20) and (B3) to obtain Eq. (B10), g g possible k∥ dependence through a phase factor. as desired. ▪ Following Eq. (A28), the occupied-band projection is Let us exemplify this discussion with the glide reflection constrained as which transforms spatial coordinates as ðx; y; zÞ → ð−x; y; z þ 1=2Þ. This symmetry acts on Bloch waves as ˆ ˆ −1 ˆ − 2 2 T δ k∥ P k T δ k∥ P −D k : B18 ¯ ikz= − g ð Þ ð Þ g ð Þ ¼ ð g Þ ð Þ MxðkÞ¼e UM¯ [from Eq. (A7)], with U ¯ ¼ I x Mx representing a 2π rotation. Equation (B10) assumes the This, in combination with Eqs. (A33) and (B2), implies form ˆ ˆ ˆ −1 ˘¯ ˘¯ −1 Tgδðk∥ÞWðk∥ÞTgδðk∥Þ Mxð−π;k∥ÞWðkx;kzÞMxð−π;k∥Þ ¼ Wð−kx;kzÞ: ðB16Þ πY←−π ˆ ˆ −1 ¯ ¼ Tgδðk∥ÞVð2πy~Þ PðkÞTgδðk∥Þ Since Mx transforms momentum as k → ðky; −kx;kzÞ,it ky belongs in the little group of any wave vector with kx ¼ 0. ˘ πY←−π ¯ −π 0 W 0 0 ˆ ˆ −1 Indeed, ½Mxð ; ;kzÞ; ð ;kzÞ ¼ , and each Wilson ¼ T δðk∥ÞVð2πy~ÞT δðk∥Þ Pð−D kÞ ˘¯ g g g band may be labeled by an eigenvalue of Mx, which again ky falls into either branch of i expð−ikz=2Þ,aswenow πY←−π iDgð2πy~Þ·δ show: ¼ e Vð−Dgð2πy~ÞÞ Pð−DgkÞ: ðB19Þ ky ˘¯ 2 ½Mxðk1Þmn

Xnocc In the next few sections, we particularize to a few − e ikz u U ¯ u u U ¯ u examples of Tgδ that are relevant to the topology of our ¼ h m;k1 j Mx j a;k1 ih a;k1 j Mx j n;k1 i a¼1 space group.

Xntot − e ikz u U ¯ u u U ¯ u ¼ h m;k1 j Mx j a;k1 ih a;k1 j Mx j n;k1 i a. Effect of space-time inversion symmetry 1 a¼ I − 2 − The space-time inversion symmetry (T ) maps ¼ e ikz hu jðU ¯ Þ ju i¼−e ikz δ : ðB17Þ m;k1 Mx n;k1 mn ðx; y; z; tÞ → −ðx; y; z; tÞ. Let us show how this results in

021008-22 TOPOLOGICAL INSULATORS FROM GROUP COHOMOLOGY PHYS. REV. X 6, 021008 (2016) the eigenvalues of Wðk∥Þ forming complex-conjugate pairs which implies that a Wilson loop at fixed ðkx;kzÞ is mapped at each k∥. If we interpret the phase (θ) of each eigenvalue to a loop at ðkx; −kzÞ, with a reversal in orientation since as an “energy,” then the spectrum has a θ → −θ symmetry; ky → −ky. In more detail, we insert Eq. (B23) into this may be likened to a particle-hole symmetry that Eq. (B19), unconventionally preserves the momentum coordinate (k∥). ˆ ˆ ˆ −1 ˆ Proof.—Inserting D ¼ −I and δ ¼ 0 in Eq. (B19), T ¯ k ;k W k ;k T ¯ k ;k W k ; −k ; g Mx ð x zÞ ð x zÞ Mx ð x zÞ ¼ rð x zÞ

ˆ ˆ ˆ −1 ˆ ˆ TI ðk∥ÞWðk∥ÞTI ðk∥Þ ¼ Wðk∥Þ: ðB20Þ where the reversed Wilson-loop operator (Wr) is defined in Eq. (B5). An equivalent expression in the occupied-band Then by inserting in Eq. (B20) a complete set of states at basis is momentum k ¼ð−π; k∥Þ, and applying the definitions −1 T˘ ¯ −π;k ;k W k ;k T˘ ¯ −π;k ;k Eqs. (B3) and (A29), Mx ð x zÞ ð x zÞ Mx ð x zÞ W − −1 ˘ ˘ −1 ¼ ðkx; kzÞ ; ðB24Þ TI ð−π; k∥ÞWðk∥ÞTI ð−π; k∥Þ ¼ Wðk∥Þ: ðB21Þ where the inverse Wilson loop is defined in Eq. (B6);itis Thus, if an eigensolution exists with eigenvalue ˘ worth clarifying that T ¯ particularizes Eq. (A30) as exp½iθðk∥Þ, there exists a partner solution with the com- Mx plex-conjugate eigenvalue exp½−iθðk∥Þ. These two solu- ˘ ˆ T ¯ −π;k ;k u π − T ¯ k ;k u −π ; tions are always mutually orthogonal, even in cases where Mx ð x zÞmn ¼h m;ð ;kx; kzÞj Mx ð x zÞj n;ð ;kx;kzÞi the eigenvalues are real. The orthogonality follows from with T˘ 2 ¼ −I, as we now show: I −2π jum;ðπ;kx;−kzÞi¼Vð y~Þjum;ð−π;kx;−kzÞi: ðB25Þ

Xnocc ˘ 2 ˆ ˆ ¯ ¯ −¯ 2π ½TI ðkÞ mn ¼ hum;kjTI ðkÞjua;kihua;kjTI ðkÞjun;ki We now focus on kz ¼ kz satisfying kz ¼ kz modulo , a¼1 such that Eq. (B24) particularizes to Xntot ˆ ˆ ¯ ¯ ¯ −1 ¼ hum;kjTI ðkÞjua;kihua;kjTI ðkÞjun;ki T˘ ¯ −π;k ; k W k ; k T˘ ¯ −π;k ; k Mx ð x zÞ ð x zÞ Mx ð x zÞ a¼1 ¯ −1 ¯ −1 ˆ 2 ¼ Wðkx; −kzÞ ¼ Wðkx; kzÞ ; ðB26Þ ¼hum;kj½TI ðkÞ jun;ki ¼hum;kjUITUITjun;ki in the gauge − u u −δ : B22 ¼ h m;kj n;ki¼ mn ð Þ ¯ u ¯ V 2k ~z u ¯ ; B27 j j;ðπ;kx;−kzÞi¼ ð z Þj j;ðπ;kx;kzÞi ð Þ In the second equality, we denote n as the total number of tot ∈ 1 2 … 2¯ bands, and apply that the symmetry representations are for j f ; ; ;noccg, and kz~z a reciprocal vector block diagonal with respect to the occupied and empty (possibly zero). Equation (B26) shows that the symmetry subspaces; the completeness relation is used in the third maps the Wilson loop to itself, with a reversal of − I 2 2π orientation. equality, and UITUIT ¼ I follows because ðT Þ is a rotation. Let us prove that  ¯ 0 b. Effect of time reversal with a spatial glide ˘ ¯ 2 þI; kz ¼ T ¯ ð−π;k ; k Þ ¼ ðB28Þ reflection Mx x z ¯ −I; kz ¼ π; ¯ The symmetry TMx maps space-time coordinates as ðx; y; z; tÞ → ð−x; y; z þ c=2; −tÞ; i.e., from which we may deduce a Kramers-like degeneracy in the spectrum of Wðkx;kz ¼ πÞ but not in Wðkx;kz ¼ 0Þ. −1 1 1 δ 2 DMx ¼ diag½ ; ; and ¼ c~z= : ðB23Þ Employing the shorthand

The momentum coordinates are mapped as −π ¯ t − t π −¯ t k1 ¼ð ;kx; kzÞ; k2 ¼ DMx k1 ¼ð ;kx; kzÞ ; t → − t ðB29Þ ðky;kx;kzÞ DMx ðky;kx;kzÞ − − t ¼ð ky;kx; kzÞ ; and the gauge conditions assumed in Eqs. (B25) and (B27),

021008-23 ALEXANDRADINATA, WANG, and BERNEVIG PHYS. REV. X 6, 021008 (2016)

Xnocc 2 T˘ ¯ k u Tˆ ¯ k u u Tˆ ¯ k u ½ Mx ð 1Þ mn ¼ h m;k2 j Mx ð 1Þj a;k1 ih a;k2 j Mx ð 1Þj n;k1 i a¼1 Xn occ ¯ ˆ ¯ ˆ u V 2πy~ − 2k ~z T ¯ k u u V 2πy~ − 2k ~z T ¯ k u ¼ h m;k1 j ð z Þ Mx ð 1Þj a;k1 ih a;k1 j ð z Þ Mx ð 1Þj n;k1 i a¼1

Xntot ¯ ˆ ¯ ˆ u V 2πy~ − 2k ~z T ¯ k u u V 2πy~ − 2k ~z T ¯ k u ¼ h m;k1 j ð z Þ Mx ð 1Þj a;k1 ih a;k1 j ð z Þ Mx ð 1Þj n;k1 i a¼1 ¯ ˆ ¯ ˆ u V 2πy~ − 2k ~z T ¯ k V 2πy~ − 2k ~z T ¯ k u ¼h m;k1 j ð z Þ Mx ð 1Þ ð z Þ Mx ð 1Þj n;k1 i − ¯ ¯ ¯ ˆ ˆ e ikz u V 2πy~ − 2k ~z V −2πy~ 2k ~z T ¯ k T ¯ k u ¼ h m;k1 j ð z Þ ð þ z Þ Mx ð 1Þ Mx ð 1Þj n;k1 i − ¯ − ¯ − ¯ ikz ikz ikz δ ¼ e hum;k jUTM¯ U ¯ jun;k i¼e hum;k jun;k i¼e mn: ðB30Þ 1 x TMx 1 1 1

W π In the second equality, we apply that the symmetry aim is to derive the algebra of the group (Gπ;kz )of ð ;kzÞ, representations are block diagonal with respect to the which we introduce in Sec. VI. kz ¼ 0 and π mark the time- occupied and empty subspaces; the completeness relation ~ ~ reversal invariant k∥ (namely, X and U). Here, G ~ ≡ Gπ;0 ≅ is used in the third equality, Eq. (A33) in the fourth ¯ X ~ ≡ 2π GU Gπ;π has the elements rotation (E), the lattice equality, and UTM¯ U ¯ ¼þI represents the point-group x 2TMx translation tð~zÞ, the Wilson loop (W), and analogs of time relation that ðTM Þ is just the identity transformation; x reversal (T ), spatial inversion (I), and glide reflection cf. our discussion in Appendix A1. ¯ (Mx) that additionally encode parallel transport; the latter c. Effect of time-reversal symmetry three are referred to as W symmetries. In addition to deriving the algebraic relations in Eqs. (22) and (27),we Let us particularize the discussion in Appendix A2 by also show the following. letting gδ in Tgδ be the trivial transformation. In the Löwdin (a) The combination of time reversal, spatial glide, and representation, Tˆ ¼ U K, where U U ¼ −I corresponds T T T T parallel transport is an element M¯ with the algebra to a 2π rotation of a half-integer spin. We obtain from z Eq. (B19) that T WT −1 W−1 T 2 M¯ ¯ ¼ ; with ¯ ¼ I; z Mz Mz πY←−π ¯ ¯ ¯ ˆ ˆ ˆ −1 and M T ¯ ¼ Etð~zÞT ¯ M W: ðB33Þ T Wðk∥ÞT ¼ Vð−2πy~Þ Pð−kÞ x Mz Mz x

ky −Yπ←π ˆ ¼ Vð−2πy~Þ Pðky; −k∥Þ¼Wrð−k∥Þ; (b) The space-time inversion symmetry acts in the ordi- ky nary manner: ðB31Þ −1 T I WT I ¼ W; with where in the last equality we identify the reverse-oriented 2 ¯ ¯ ¯ T ¼ E; M T I ¼ tð~zÞT I M : ðB34Þ Wilson loop defined in Eq. (B5). Equivalently, in the I x x occupied-band basis,

˘ ˘ −1 −1 Tð−π; k∥ÞWðk∥ÞTð−π; k∥Þ ¼ Wð−k∥Þ ; ðB32Þ The algebra that we derive here extends the ordinary algebra of space-time transformations, which we show in with the inverse Wilson loop defined in Eq. (B6). Time Eq. (B1). For time-reversal-variant k∥ along the same glide θ → θ Gπ k ∉ 0; π G ~ ≡ Gπ 0 reversal thus maps exp½i k∥ exp½i −k∥ . Following an line, ;kz ( z f g) is a subgroup of X ; ,andis ¯ ¯ E t R∥ W M T ¯ T I exercise similar to the previous section (Appendix B3b), instead generated by , ð Þ, , x, Mz ,and . one may derive a Kramers degeneracy where k∥ ¼ −k∥ (up Therefore, Eqs. (22), (B33),and(B34) [but not Eq. (27)] to a reciprocal vector). ∉ 0 π apply to Gπ;kz (kz f ; g). Equations (22), (B33), (B34),and (27) are, respectively, derived in Appendixes B4a–B4d. 4. Extended group algebra of the One motivation for deriving the Wilsonian algebra is that ~ ~ W symmetries along X U it determines the possible topologies of the Wilson bands In Sec. VI we introduce the notion of W symmetries and along X~ U~ . This determination is through “rules of the that section should be read in advance of this section. Our curves” that we summarize in two tables: Table II is derived

021008-24 TOPOLOGICAL INSULATORS FROM GROUP COHOMOLOGY PHYS. REV. X 6, 021008 (2016) Q ~ ~ k2←k1 Gπ k∥ X U from the algebra of ;kz and applies to any along , where k PðkyÞ denotes a path-ordered product of ~ ~ y Table III is derived from GX~ and applies only to X and U. projections sandwiched by Pðk1Þ (rightmost) and Pðk2Þ (leftmost); while k1 and k2 are more generally mod 2π a. Wilsonian glide-reflection symmetry variables due to the periodicity of momentum space, we ¯ always choose a branch that includes both k1 and k2 in our Consider the glide reflection Mx, which transforms definition of W ← , such that if k2 >k1, we parallel spatial coordinates as ðx; y; zÞ → ð−x; y; z þ 1=2Þ.In k2 k1 transport in the direction of increasing k [e.g., Figs. 12(c) Appendix B2, we describe how M¯ constrains Wilson y x and 12(d)], and vice versa [e.g., Fig. 12(e)]. Therefore, this loops in the k ¼ 0 plane, where M¯ belongs in the little x x Wilson line reverts to the familiar Wilson loop if group of each wave vector, and therefore the projections on − 2π W W this plane separate into two mirror representations. This is k2 k1 ¼ , i.e., π←−π ¼ −π of Eq. (B3) with the gauge condition of Eq. (B7). Since parallel transport is no longer true for the k ¼ π plane, since M¯ maps between x x unitary within the occupied subspace [26], two momenta that are separated by half a reciprocal period, ¯ ~ † i.e., M : ðk ; π;k Þ → ðk ; −π;k Þ¼ðk þ π; π;k Þ − b2, W W ⇒ W W x y z y z y z k2←k1 ¼ k1←k2 k2←k1 k1←k2 ¼ I: ðB39Þ as illustrated in Fig. 3(b). Consequently, the Wilson-loop operator is symmetric under a combination of a glide With these definitions in hand, we return to the proof. reflection with parallel transport over half a reciprocal To proceed, we first show that the glide symmetry period, as we now prove. translates the Wilson loop by half a reciprocal period in y~: Proof.—A crucial observation is that the glide symmetry π π π ˘¯ ˘¯ −1 relates projections at ðky; ;kzÞ and ðky þ ; ;kzÞ. That is, Mxð0; −πÞW−πMxð0; −πÞ ¼ W0: ðB40Þ by particularizing Eq. (A19) and (A21), we obtain the symmetry constraint on the projections: This is proven by applying Eqs. (B35) and (A24) to

−1 −1 −1 −1 ~ ¯ π ~ ~ ˆ ~ Vðb2Þ UM Pðky; ;kzÞU ¯ Vðb2Þ Vðb2Þ UM¯ WU ¯ Vðb2Þ x Mx x Mx ¼ Pðk þ π; π;k Þ: ðB35Þ πY←−π y z ~ −1 2π −1 ~ ¼ Vðb2Þ UM¯ Vð y~Þ PðkyÞU ¯ Vðb2Þ x Mx This relation allows us to define a unitary sewing matrix ky π←−π between states at ð0; k∥Þ and ð−π; k∥Þ: Y ~ −1 2π −1 ~ π ¼ Vðb2Þ UM¯ Vð y~ÞU ¯ Vðb2Þ Pðky þ Þ x Mx ˘¯ 0 −π ky ½Mxðð ; k∥Þ; ð ; k∥ÞÞmn 2Yπ←0 −ik =2 ~ e z u 0 V −b2 U ¯ u −π ; B36 2π ¼ h m;ð ;k∥Þj ð Þ Mx j n;ð ;k∥Þi ð Þ ¼ Vð y~Þ PðkyÞ; ðB41Þ ky ~ which particularizes Eq. (A20) with G ¼ b2. Presently, it becomes useful to distinguish the base point of a Wilson loop, and also to define Wilson lines which do (a) (b) (c) (d) (e) (f) (g) not close into a loop. The remaining discussion in this 2π section occurs at fixed k∥ ¼ðπ;kzÞ [for any kz ∈ ½−π; πÞ] and variable ky; subsequently, we suppress the label k∥; π e.g., 0 ≡ ; jum;ky;k∥ i jum;ky i ˘¯ ˘¯ -π Mxðð0; k∥Þ; ð−π; k∥ÞÞ ≡ Mxð0; −πÞ: ðB37Þ

¯ -2π We denote the base point (ky) of a Wilson loop by the subscript in W ¯ and choose an orientation of increasing k ; ky y FIG. 12. To clarify our notations for Wilson loops and lines, we ¯ → ¯ 2π i.e., we parallel transport from ky ky þ , as exempli- draw several examples, all of which occur at fixed k∥ and variable fied by Figs. 12(a) and 12(b). We denote a Wilson line ky (vertical axis). The arrow indicates the orientation for parallel between two distinct momenta by transport. (a) The Wilson loop at base point −π (encircled) is labeled by W−π ≡ Wπ←−π. (b) W−2π ≡ W0←−2π. (c) Wπ←−π=2. ← k2Yk1 (d) Wπ←0. (e) Wπ←2π. W ¯ denotes a Wilson loop with base r;ky ½W ← ¼hu j Pðk Þju i; ðB38Þ ¯ k2 k1 mn m;k2 y n;k1 point ky and oriented in the direction of decreasing ky, e.g., Wr;2π ky in (f) and Wr;π in (g).

021008-25 ALEXANDRADINATA, WANG, and BERNEVIG PHYS. REV. X 6, 021008 (2016) which then implies Eq. (B40) in the occupied-band basis; parallel transport over half a reciprocal period (encoded in cf. Eqs. (B37) and (B50). W−π←0); we call any such symmetry that combines a We now apply the following identity, space-time transformation with parallel transport a Wilsonian symmetry, or just a W symmetry. ▪ 2Yπ←0 ¯ It is worth mentioning that M and W−π always ½W0 ¼hu 0jVð2πy~Þ Pðk Þju 0i x mn m; y n; commute as in Eq. (B43), though the specific representa- ky ¯ tions of M and W−π depend on the gauge of u −π . That 2Yπ←0 0Y←−π −Yπ←0 x j n; i 2π is, in a different gauge labeled by a tilde header, ¼hum;0jVð y~Þ PðryÞ PðqyÞjun;0i ky ry qy 0Y←−π πY←−π Xnocc ~ ≡ ∈ jun;−πi jum;−πiSmn;SUðnoccÞ; ðB44Þ ¼hum;0j PðkyÞVð2πy~Þ PðryÞ m¼1 ky ry −Yπ←0 we would represent the W reflection and the Wilson loop by × PðqyÞjun;0i qy W W W M¯~ −1M¯ W~ −1W ¼½ 0←−π −π −π←0mn; ðB42Þ x ¼ S xS and −π ¼ S −πS; ðB45Þ which is pictorially represented in Fig. 13: Fig. 13(a) represents W0, Fig. 13(b) an intermediate step in which would still commute. Eq. (B42), and Fig. 13(c) the final result. The second Equation (B43) implies that we can simultaneously equality in Eq. (B42) follows from Eq. (B39) and the third diagonalize both Wilson-loop and W-symmetry operators; from Eq. (7). Combining Eqs. (B40) and (B42), for each simultaneous eigenstate, the two eigenvalues are related in the following manner: if exp½iθ is the Wilson- ¯ ¯ ˘¯ loop eigenvalue, then the W-symmetry eigenvalue falls into ½Mx; W−π¼0; with Mx ¼ W−π←0Mxð0; −πÞ: either branch of λxðkz þ θÞ ≡ i exp½−iðkz þ θÞ=2,as ðB43Þ claimed in Eq. (14) and as we proceed to prove. Proof.—Up to a kz-dependent phase factor, this W- To interpret this result, the Wilson-loop operator commutes symmetry operator squares to the reverse-oriented ˘¯ with a combination of glide reflection (encoded in Mx) with Wilson loop:

M¯ 2 W ˘¯ 0 −π W ˘¯ 0 −π ½ xmn ¼½ −π←0Mxð ; Þ −π←0Mxð ; Þmn −Yπ←0 −Yπ←0 −ik ~ −1 ~ −1 e z u −π P k V b2 U ¯ P q V b2 U ¯ u −π ¼ h m; j ð yÞ ð Þ Mx ð yÞ ð Þ Mx j n; i ky qy −Yπ←0 −Yπ←0 −ik ~ −1 ~ −1 e z u −π P k P q π V b2 U ¯ V b2 U ¯ u −π ¼ h m; j ð yÞ ð y þ Þ ð Þ Mx ð Þ Mx j n; i ky qy −Yπ←π −ik ~ −1 ~ 2 e z u −π P k V b2 V b2 − 2πy~ U ¯ u −π ¼ h m; j ð yÞ ð Þ ð Þð Mx Þ j n; i ky −Yπ←π −ikz ¼ −e hum;−πj PðkyÞVð−2πy~Þjun;−πi ky −Yπ←π −ikz ¼ −e hum;πjVð−2πy~Þ PðkyÞjun;πi ky − −1 − ikz W ¼ e ½ð −πÞ mn: ðB46Þ 2 Here, U ¯ ¼ −I represents a 2π rotation, and in the fourth equality we make use of Mx   ~ ~ ~ ~ U ¯ V −b2 iD ¯ b2 ~z=2 V −D ¯ b2 U ¯ V b2 − 2πy~ U ¯ ; Mx ð Þ¼exp Mx · ð Þ ð Mx Þ Mx ¼ ð Þ Mx ðB47Þ

021008-26 TOPOLOGICAL INSULATORS FROM GROUP COHOMOLOGY PHYS. REV. X 6, 021008 (2016)

(a) (b) (c) (d) (e) (f) following similar steps that we use to derive Eq. (B35). 2π V++V This relation allows us to define a unitary sewing matrix between states at ð2π; k∥Þ and ð−π; k∥Þ: π V + ˘ T ¯ 2π; k∥ ; −π; k∥ ½ Mz ðð Þ ð ÞÞmn −1 ~ ˆ ¯ 0 ¼hum;ð2π;k∥ÞjVðb2Þ TM jun;ð−π;k∥Þi; ðB50Þ V--V z ~ -π which particularizes Eq. (A29) with G ¼ −b2. Henceforth, V- suppressing the k∥ labels, and by similar manipulations that FIG. 13. (a)–(c) Pictorial representation of Eq. (B42), with we use to derive Eq. (B40), we are led to solid line indicating a product of projections that is path −1 ordered according to the arrow on the same line, and Vþ ˘ 2π −π W ˘ 2π −π W TM¯ ð ; Þ −πTM¯ ð ; Þ ¼ r;2π; ðB51Þ indicating an insertion of the spatial-embedding matrix Vð2πy~Þ. z z Panels (d)–(f) represent Eq. (B52), with V indicating an insertion þ W of Vð−2πy~Þ. and r;2π denotes the reverse-oriented Wilson loop with base point 2π, as illustrated in Fig. 12(f). ▪ We use the following two identities: as pictorially which follows from Eq. (A24). represented in Figs. 13(d)–13(f), the first identity, The situation is analogous to that of the 010-surface bands along both glide lines (Γ~ Z~ and X~ U~ ), where on each ðiÞ W 2π ¼ W2π←πW πWπ←2π; ðB52Þ line there are also two branches for the glide-mirror r; r; eigenvalues; namely, λzðkzÞ ≡ i expð−ikz=2Þ. A curious difference is that the W-symmetry eigenvalue of a Wilson follows from a generalization of Eq. (B42), and band (only along X~ U~ ) also depends on the energy (θ) ˘ Wπ←2πT ¯ 2π; −π through Eq. (14). ðiiÞ½ Mz ð Þmn We remark on how a W-reflection symmetry more πY←2π ~ −1 ˆ generally arises. While our nonsymmorphic case study is u π P k V b2 T ¯ u −π ¼h m; j ð yÞ ð Þ Mz j n; i a momentum plane with W-glide symmetry, the nonsym- ky morphicity is not a prerequisite for W symmetries. Indeed, −Yπ←0 certain momentum planes in symmorphic crystals (e.g., ~ ˆ ¼hum;−πj PðkyÞVð2πy~ − b2ÞT ¯ jun;−πi rocksalt structures) exhibit a glideless W-reflection sym- Mz ky metry. The rocksalt structure and our case study are each Xn −Yπ←0 characterized by a momentum plane (precisely, a torus occ ¼ hu −πj Pðk Þju 0i within a plane) that is (a) orthogonal to the reflection axis m; y a; a¼1 k and (b) reflected not directly to itself, but to itself translated y ~ ˆ 2 u 0 V 2πy~ − b2 T ¯ u −π by half a reciprocal period (G= ), with G lying parallel to × h a; j ð Þ Mz j n; i the same plane; in our nonsymmorphic case study, the Xnocc ˘ plane is k ¼ π and G ¼ 2πy~. ≡ W−π←0 T ¯ 0; −π : x ½ ma½ Mz ð Þan ðB53Þ a¼1 ¯ b. Wilsonian TMz symmetry ¯ Inserting (i) and (ii) into Eq. (B51), we arrive at TMz transforms space-time coordinates as ðx; y; z; tÞ → ðx; y; −z þ 1=2; −tÞ and momentum coordinates as −1 −1 T ¯ W−πT ¯ W−π ; ~ Mz Mz ¼½ with ðk ; π;k Þ → ð−k ; −π;k Þ¼ðπ − k ; π;k Þ − b2, as illus- y z y z y z ˘ T ¯ W−π←0T ¯ 0; −π : trated in Fig. 3(c). From Eqs. (A27) and (A28), we obtain Mz ¼ Mz ð Þ ðB54Þ its action on Bloch waves, The Wilson-loop operator is thus W symmetric under T ¯ , ˆ − 2 Mz T ¯ k e ikz= U ¯ K; Mz ð Þ¼ TMz ðB48Þ which combines a space-time transformation (encoded in ˘ T ¯ ) with parallel transport over half a reciprocal period and the constraint on the occupied-band projections, Mz (encoded in W−π←0). This constraint does not produce any T 2 ~ −1 ˆ ˆ −1 ~ degeneracy, since (i) ¯ ¼þI and furthermore (ii) the V b2 T ¯ P k ; π;k T ¯ V b2 P π − k ; π;k ; Mz ð Þ Mz ð y zÞ M ð Þ¼ ð y zÞ z M¯ T ¯ eigenvalues of x [cf. Eq. (14)] are preserved under Mz. ðB49Þ The proof of (i) follows as

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−Yπ←0 −Yπ←0 T 2 2π − ~ ˆ 2π − ~ ˆ ½ ¯ ¼hum;−πj PðkyÞVð y~ b2ÞTM¯ PðqyÞVð y~ b2ÞTM¯ jun;−πi Mz mn z z ky qy −Yπ←0 −Yπ←0 ~ ˆ ~ ˆ u −π P k P −π − q V 2πy~ − b2 T ¯ V 2πy~ − b2 T ¯ u −π ¼h m; j ð yÞ ð yÞ ð Þ Mz ð Þ Mz j n; i ky qy −Yπ←0 0Y←−π ~ ~ ˆ 2 u −π P k P q V 2πy~ − b2 V −2πy~ b2 T ¯ u −π ¼h m; j ð yÞ ð yÞ ð Þ ð þ Þð Mz Þ j n; i ky qy −Yπ←0 0Y←−π W W δ ¼hum;−πj PðkyÞ PðqyÞjun;−πi¼½ −π←0 0←−πmn ¼ mn; ðB55Þ ky qy where in the second equality we use Eqs. (7) and (B49), in the third equality, Eq. (A33), and in the fourth, ˆ 2 ðTM¯ Þ ¼ UTM¯ U ¯ ¼ I. To prove (ii), we first demonstrate that UM¯ and UTM¯ K anticommute, which follows from z z TMz x z UM¯ and UTM¯ K forming a symmorphic representation of Mx and TMz [where Mj are glideless reflections; see discussion x z ¯ of Eq. (A16)]. Indeed, MxMz ¼ EMzMx in the half-integer-spin representation, and time reversal commutes with any U ¯ ;U ¯ K 0 spatial transformation, leading to f Mx TMz g¼ . This anticommutation is applied in the fourth equality of

−Yπ←0 −Yπ←0 ¯ −ik ~ ~ M T ¯ e z u −π P k V −b2 U ¯ P q V 2πy~ − b2 U ¯ K u −π ½ x Mz mn ¼ h m; j ð yÞ ð Þ Mx ð yÞ ð Þ TMz j n; i ky qy −Yπ←0 −Yπ←0 −ik ~ ~ e z u −π P k P q π V −b2 U ¯ V 2πy~ − b2 U ¯ K u −π ¼ h m; j ð yÞ ð y þ Þ ð Þ Mx ð Þ TMz j n; i ky qy −Yπ←0 0Y←π −ik ~ ~ e z u −π P k P q V −b2 V b2 U ¯ U ¯ K u −π ¼ h m; j ð yÞ ð yÞ ð Þ ð Þ Mx TMz j n; i ky qy −Yπ←0 0Y←π −ik −e z u −π P k P q U ¯ KU ¯ u −π ¼ h m; j ð yÞ ð yÞ TMz Mx j n; i ky qy −Yπ←0 0Y←π −ik ~ ~ ˆ −ik =2 −e z u −π P k P q V 2πy~ − b2 V −2πy~ b2 T ¯ e z U ¯ u −π ¼ h m; j ð yÞ ð yÞ ð Þ ð þ Þð Mz Þ Mx j n; i ky qy −Yπ←0 0Y←π −ik ~ ˆ −ik =2 ~ −e z u −π P k P q V 2πy~ − b2 T ¯ V 2πy~ e z V −b2 U ¯ u −π ¼ h m; j ð yÞ ð yÞ ð Þ Mz ð Þ ð Þ Mx j n; i ky qy −Yπ←0 0Y←π −ik ~ ˆ −ik =2 ~ −e z u −π P k V 2πy~ − b2 T ¯ V 2πy~ P π − q e z V −b2 U ¯ u −π ¼ h m; j ð yÞ ð Þ Mz ð Þ ð yÞ ð Þ Mx j n; i ky qy −Yπ←0 πY←−π −Yπ←0 −ik ~ ˆ −ik =2 ~ −e z u −π P k V 2πy~ − b2 T ¯ V 2πy~ P q P l e z V −b2 U ¯ u −π ¼ h m; j ð yÞ ð Þ Mz ð Þ ð yÞ ð yÞ ð Þ Mx j n; i ky qy ly −ik ¯ −ik ¯ −e z T ¯ W−πM −e z T ¯ M W−π ; ¼ ½ Mz xmn ¼ ½ Mz x mn ðB56Þ where we also apply Eqs. (B35) and Eq. (A24) in multiple instances. Then we define simultaneous eigenstates of W−π and ¯ Mx such that

iθ iθ iθ ¯ iθ iθ W−πje ; λx; kzi¼e je ; λx; kzi and Mxje ; λx; kzi¼λxje ; λx; kzi: ðB57Þ

Once again, all operators and eigenvalues here depend on kz. Finally,

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¯ iθ −ik ¯ iθ −iðk þθÞ iθ iθ M T ¯ e ; λ ; k −e z T ¯ M W−π e ; λ ; k −e z λ T ¯ e ; λ ; k λ T ¯ e ; λ ; k : x Mz j x zi¼ Mz x j x zi¼ x Mz j x zi¼ x Mz j x zi

2 In the last equality, we apply λx ¼ − exp½−iðθ þ kzÞ from Eq. (14).

c. Effect of space-time inversion symmetry ˘ The first two relations of Eq. (B34) may be carried over from Appendix B3a, if we identify T I ≡ TI. What remains is to ¯ ¯ show MxT I ¼ tð~zÞT I Mx:

−Yπ←0 ¯ −ik =2 ~ ˆ M T I e z u −π P k V −b2 U ¯ TI u −π ½ x mn ¼ h m; j ð yÞ ð Þ Mx j n; i ky −Yπ←0 −ik ˆ −ik =2 ~ e z u −π TI e z P k V −b2 U ¯ u −π ¼ h m; j ð yÞ ð Þ Mx j n; i ky − ikz T M¯ ¼ e ½ I xmn: ðB58Þ

iθ iθ Recalling our definitions in Eq. (B57), we now show that je ; λx; kzi and T I je ; λx; kzi belong in opposite mirror branches. To be precise, since λxðkz þ θÞ ≡ i exp½−iðkz þ θÞ=2 is both momentum and energy dependent, and TI maps ¯ θ → −θ;kz → kz, we would show that two space-time-inverted partners have Mx eigenvalues λxðkz þ θÞ and −λxðkz − θÞ:

¯ iθ −ikz iθ MxT I je ; λx; kzi¼e λxT I je ; λx; kzi

−iðkz−θÞ=2 iθ ¼ ∓ie T I je ; λx; kzi: ðB59Þ

d. Wilsonian time-reversal operator ¯ ¯ In the remaining discussion, we particularize to two Wilson loops at fixed k∥ ¼ðπ; kzÞ, with kz ¼ 0 or π only. Under time ¯ ¯ ¯ ~ ¯ reversal, ðky; π; kzÞ → ð−ky; −π; −kzÞ¼ðπ − ky; π; kzÞ − b2 − 2kz~z, as illustrated in Fig. 3(d). This implies that the occupied-band projections along these lines are constrained as

~ ¯ −1 ˆ ¯ ˆ −1 ~ ¯ ¯ Vðb2 þ 2kz~zÞ TPðky; π; kzÞT Vðb2 þ 2kz~zÞ¼Pðπ − ky; π; kzÞ; ðB60Þ

ˆ where T ¼ UTK is the antiunitary representation of time reversal. This relation allows us to define a unitary sewing matrix between states at ð2π; k∥Þ and ð−π; k∥Þ:

˘ 2π −π −~ − 2¯ ˆ ½Tðð ; k∥Þ; ð ; k∥ÞÞmn ¼hum;ð2π;k∥ÞjVð b2 kz~zÞTjun;ð−π;k∥Þi; ðB61Þ

~ ¯ which particularizes Eq. (A29) with G ¼ −b2 − 2kz~z. Henceforth, suppressing the k∥ labels, we are led to

˘ ˘ −1 Tð2π;−πÞW−πTð2π;−πÞ ¼Wr;2π; where ˘ 2π −π −~ −2¯ ˆ ½Tð ; Þmn ¼hum;2πjVð b2 kz~zÞTjun;−πi; ðB62Þ

and Wr;2π denotes the reverse-oriented Wilson loop with base point 2π, as drawn in Fig. 12(f). Combining this result with Eq. (B52) and the identity

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πY←2π W ˘ 2π −π −~ − 2¯ ˆ ½ π←2πTð ; Þmn ¼hum;πj PðkyÞVð b2 kz~zÞTjun;−πi ky πY←2π ~ ¯ ˆ ¼hum;−πjVð2πy~Þ PðkyÞVð−b2 − 2kz~zÞTjun;−πi ky −Yπ←0 ~ ¯ ˆ ¼hum;−πj PðkyÞVð2πy~ − b2 − 2kz~zÞTjun;−πi ky ≡ W ˘ 0 −π ½ −π←0Tð ; Þmn; ðB63Þ we arrive at

−1 −1 ˘ TW−πT ¼½W−π ; with T ¼ W−π←0Tð0; −πÞ: ðB64Þ

The Wilson-loop operator is thus W symmetric under T , which combines time reversal (encoded in T˘ ) with parallel transport over half a reciprocal period (encoded in W−π←0). While many properties of the ordinary time reversal are well known (e.g., Kramers degeneracy, the commutivity of time reversal with spatial transformations), it is not a priori obvious that these properties are applicable to the W symmetry (T ). We find that T indeed enforces a Kramers degeneracy in the W ¯ spectrum, but it only commutes with the W glide (Mx) modulo a Wilson loop. The Kramers degeneracy follows from (a) T 2 relating two eigenstates of W−π with the same eigenvalue, as follows from Eq. (B64), and (b) T ¼ −I, as we now show:

−Yπ←0 −Yπ←0 T 2 2π − ~ − 2¯ ˆ 2π − ~ − 2¯ ˆ ½ mn ¼hum;−πj PðkyÞVð y~ b2 kz~zÞT PðqyÞVð y~ b2 kz~zÞTjun;−πi ky qy −Yπ←0 0Y←−π ~ ¯ ~ ¯ ˆ 2 ¼hum;−πj PðkyÞ PðqyÞVð2πy~ − b2 − 2kz~zÞVð−2πy~ þ b2 þ 2kz~zÞðTÞ jun;−πi ky qy −Yπ←0 0Y←−π − − W W −δ ¼ hum;−πj PðkyÞ PðqyÞjun;−πi¼ ½ −π←0 0←−πmn ¼ mn; ðB65Þ ky qy where Tˆ 2 ¼ −I represents a 2π rotation. We further investigate if Kramers partners share identical or opposite eigenvalues ¯ under Mx. We find at kz ¼ 0 that T : λx → −λx, while at kz ¼ π, T : λx → λx, as we now prove. Proof.— T;Uˆ ¯ 0 Applying ½ Mx ¼ ,

−Yπ←0 −Yπ←0 ¯ −ik¯ =2 ~ ~ ¯ ˆ M T e z u −π P k V −b2 U ¯ P q V 2πy~ − b2 − 2k ~z T u −π ½ x mn ¼ h m; j ð yÞ ð Þ Mx ð yÞ ð z Þ j n; i ky qy −Yπ←0 0Y←π −ik¯ =2 ~ ik¯ ~ ¯ ˆ e z u −π P k P q V −b2 e z V b2 − 2k ~z TU ¯ u −π ¼ h m; j ð yÞ ð yÞ ð Þ ð z Þ Mx j n; i ky qy −Yπ←0 πY←−π −Yπ←0 ~ ¯ ˆ −ik¯ =2 ~ u −π P k V 2πy~ − b2 − 2k ~z TV 2πy~ P q P l e z V −b2 U ¯ u −π ¼h m; j ð yÞ ð z Þ ð Þ ð yÞ ð yÞ ð Þ Mx j n; i ky qy ly TW M¯ T M¯ W ¼½ −π xmn ¼½ x −πmn: ðB66Þ ¯ This confirms our previous claim that T commutes with Mx modulo a Wilson loop, unlike the algebra of ordinary space- time symmetries. Recalling Eq. (B57),

¯ ¯ iθ ¯ ¯ iθ ¯ −iθ iθ ¯ ikz iθ ¯ MxT je ; λx; kzi¼T MxW−πje ; λx; kzi¼e λxT je ; λx; kzi¼−e λxT je ; λx; kzi: ðB67Þ

2 In the last equality, we apply λxðθ þ kzÞ ¼ − exp½−iðθ þ kzÞ. ▪

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An analog of this result occurs for the surface bands, other spatial symmetries, we show that each quadruplet ¯ where the eigenvalues of Mx are imaginary (real) at kz ¼ 0 splits into a connected hourglass [Fig. 14(b)]. (π), and time reversal pairs up complex-conjugate With vanishing spin-orbit coupling, the system is addi- eigenvalues. tionally symmetric under the spin flip ðFxÞ that rotates spin by π about x~. The double group (G) relations include

2 2 ¯ ¯ 2 ¯ APPENDIX C: CONNECTIVITY OF BULK T ¼ Fx ¼ E; Mx ¼ Etðc~zÞ; HAMILTONIAN BANDS IN SPIN SYSTEMS ½T;F ¼½T;M¯ ¼½F ; M¯ ¼0; ðC1Þ WITH GLIDE AND TIME-REVERSAL x x x x SYMMETRIES with E¯ a 2π rotation and t a lattice translation. It follows In spin systems with minimally time-reversal (T) and from this relations that we can define two operators that act ¯ glide-reflection (Mx) symmetries, we prove that bulk like time reversal and glide reflection in a spinless system: Hamiltonian bands divide into quadruplet sets of hour- ≡ ¯ ≡ ¯ glasses, along the momentum circle parametrized by Tx TFx; mx MxFx; ð0; 0;k Þ. Each quadruplet is connected, in the sense that 2 2 z ½Tx; m¯ x¼0;Tx ¼ I; m¯ x ¼ tðc~zÞ: ðC2Þ there are enough contact points to continuously travel through all four branches. With the addition of other spatial These operations preserve the spin component in ~z—the symmetries in our space group, we further describe how group (G~ ) of a single spin species (aligned in ~z)is degeneracies within each hourglass may be further ¯ generated by Tx, mx, and the lattice translations. It is known enhanced. Our proof of connectivity generalizes a previous from Ref. [62] that the elementary band representation [57] proof [62] for integer-spin representations of nonsymmor- of G~ is two dimensional; i.e., single-spin bands divide into phic space groups. sets of two that cannot be decomposed as direct sums, and The outline of our proof is as follows. We first consider a in each set there are enough contact points (to be found spin system with vanishing spin-orbit coupling, such that it anywhere in the Brillouin zone) to continuously travel has a spin SU(2) symmetry. In this limit, we prove that through both branches. This latter property they call bands divide into doubly degenerate quadruplets with a “connectivity.” We reproduce here their proof of connec- fourfold intersection at ð0; 0; πÞ, as illustrated in Fig. 14(a). tivity by monodromy: Then, by introducing spin-orbit coupling and assuming no Let us consider the single-spin Bloch representation ~ ~ 0 0 ∈ 0 2π (Dkz )ofG along the circle ð ; ;kzÞ; kz ½ ; Þ.For +i exp(-ikz/2) +exp(-ikz/2) kz ≠ π, there are two irreducible representations (labeled by -i exp(-ikz/2) -exp(-ikz/2) ρ) that are each one dimensional: (a) (b) (c) ~ ρ − -i -1 D ðtðc~zÞÞ ¼ e ikz +i +i kz -1,-1 ρ − 2πρ 2 -i +1,+1 +1 -i ⇒ ~ ¯ iðkzþ Þ= ≠ π ρ ∈ 0 1 Dk ðmxÞ¼e ;kz ; f ; g; +i z as follows from Eq. (C2). Making one full turn in this momentum circle (kz → kz þ 2π) effectively permutes the representations as ρ → ρ þ 1. This implies that if we follow continuously an energy function of one of the two FIG. 14. Bulk band structures with glide and time-reversal branches, we would evolve to the next branch after making symmetries, for systems with spin (a),(b) and without (c). one circle, and finally return to the starting point after — Γ ≡ ð0; 0;kz ¼ 0Þ and Z ≡ ð0; 0;kz ¼ πÞ are high-symmetry making two circles both branches form a connected points that are selected because the fractional translation (in the graph. The contact point between the two branches is glide) is parallel to ~z. Panel (a) either has no spin-orbit coupling or determined by the time-reversal symmetry (Tx), which has spin-orbit coupling with an additional spatial-inversion sym- pinches together complex-conjugate representations of m¯ x metry. Panel (b) has spin-orbit coupling but breaks spatial- at kz ¼ π; on the other hand, real representations at kz ¼ 0 inversion symmetry. The crossings between orthogonal mirror are not degenerate, as illustrated in Fig. 14(c). We apply branches (indicated by arrows) are movable along ΓZ but here that ½Tx; m¯ x¼0, and the eigenvalues of m¯ x are unremovable so long as glide and time-reversal symmetries are imaginary (real) at k ¼ π (0). ▪ preserved. The glide eigenvalues are indicated at Γ and Z for one of z Because of the spin SU(2) symmetry, we double all the two hourglasses. Panel (c) could apply to an intrinsically ~ spinless system (e.g., bosonic cold atoms and photonic crystals), irreducible representations of G to obtain representations of and also to an effectively spinless system (e.g., a single-spin G; i.e., in the absence of spin-orbit coupling but accounting species in an electronic system without spin-orbit coupling). for both spin species, bands are everywhere spin

021008-31 ALEXANDRADINATA, WANG, and BERNEVIG PHYS. REV. X 6, 021008 (2016) degenerate, and especially fourfold degenerate at kz ¼ π. representations and group extensions in Appendix D1, For illustration, Fig. 14(a) may be interpreted as a spin- then proceed in Appendix D2 to describe projective doubled copy of Fig. 14(c). Recall that the eigenvalues of representations from the more abstract perspective of ¯ Mx fall into two branches labeled by η ¼1 in cochains, which emphasizes the connection with group λxðkzÞ¼ηi expð−ikz=2Þ. For each spin-degenerate dou- cohomology. Finally, in Appendix D3, we exemplify a ¯ blet, the two spin species fall into opposite branches of Mx, simple calculation of the second group cohomology. as distinguished by solid and dashed curves in Fig. 14(a). 0 This result follows from continuity to kz ¼ , where 1. Connection between projective representations Kramers partners have opposite, imaginary eigenvalues and group extensions under M¯ . x In Sec. VI, we introduce three groups. Without additional spatial symmetries, the effect of spin- ≅ Z Z orbit coupling is to split the spin degeneracy for generic k , (i) As defined in Eq. (29), G∘ 2 × 2 is a symmor- z phic, spinless group generated by time reversal (T) while preserving the Kramers degeneracy at k ¼ 0 and z M π—the final result is the hourglass illustrated in Fig. 14(b). and a glideless reflection ( x), with an algebra summarized in Eq. (30). To demonstrate this, we note at kz ¼ π that each four- (ii) The group (GX~ ) of the Wilson loop is generated by dimensional subspace (without the coupling) splits into two 2π ¯ Kramers subspaces, where Kramers partners have identical, rotation (E), a lattice translation [tðc~zÞ], the Wilson loop (W), and analogs of time reversal real eigenvalues under M¯ ; pictorially, two solid curves x T M¯ emerge from one of the two Kramers subspaces, and for ( ) and glide reflection ( x) additionally encode the other subspace both curves are dashed, as shown in parallel transport. With regard to our KHgX material Fig. 14(b). Furthermore, we know from the previous class, this appendix does not exhaust all elements in GX~ or G∘; our treatment here minimally conveys paragraph that each Kramers pair at kz ¼ 0 combines a solid and dashed curve. These constraints may be inter- their group structures. A more complete treatment of preted as curve boundary conditions at 0 and π, which the material class is described in Appendix B4 for the different purpose of topological classification. impose a solid-dashed crossing between the boundaries, as N ≅ Z2 Z indicated by arrows in Fig. 14(b). There is then an (iii) As defined in Eq. (31), × 2 is an Abelian 2π ¯ “unavoidable degeneracy” [57] that can move along the group generated by rotations (E), momentum W half-circle, but cannot be removed. This contact point, in translations ( ), and real-space translations [tð~zÞ]. N addition to the unmovable Kramers degeneracies at 0 and π, G∘ induces an automorphism on , which we ∈ guarantees that each quadruplet is connected. proceed to define. Letting the group element gi ˆ Let us consider how other spatial symmetries (beyond G∘ be represented in GX~ by gi, we say that gi induces ¯ the automorphism a → σiðaÞ, where Mx) may enhance degeneracies within each hourglass. For illustration, we consider the spatial inversion (I) symmetry, I σ ¯ a bWc ≡ ˆ ¯ a bWc ˆ−1 which applies to the space group of KHgX. Since T iðE tð~zÞ Þ giE tð~zÞ gi belongs in the group of every bulk wave vector, the spin κ ˆ γ ˆ ¯ a ðgiÞb ðgiÞc ˆ ¼ E tð~zÞ W ; with κðgiÞ; degeneracy along ð0; 0;kzÞ does not split, and kz ¼ π remains a point of fourfold degeneracy, as illustrated in γðgˆiÞ ∈ f1g: ðD1Þ Fig. 14(a). One final remark is that the notion of “connectivity” over γðgˆ Þ¼þ1 if gˆ preserves the Wilson loop [e.g., the entire Brillouin zone, as originally formulated in i i gˆ M¯ in Eq. (22)], and γ gˆ −1 if gˆ is Ref. [62], can fruitfully be particularized to “connectivity i ¼ x ð iÞ¼ i orientation reversing [e.g., gˆ T in Eq. (26)]. of a submanifold,” which we introduced in our companion i ¼ Similarly, κ gˆ 1 (−1)ifgˆ preserves (inverts) work [21] as a criterion for topological surface bands; our ð iÞ¼þ i t ~z notion differs from the original formulation in that contact the spatial translation ð Þ; in our example, κ M¯ κ T 1 points must be found only within the submanifold in ð xÞ¼ ð Þ¼þ . Equivalently, we may say N question, rather than the entire Brillouin zone. that is a normal subgroup of GX~ . N To show that GX~ is an extension of G∘ by ,itis sufficient to demonstrate that G∘ is isomorphic to the factor APPENDIX D: PROJECTIVE N group GX~ = [37]. This factor group has as elements the REPRESENTATIONS, GROUP EXTENSIONS, N ∈ N left cosets g , with g GX~ ; by the normality of , AND GROUP COHOMOLOGY gN ¼ N g. This isomorphism respectively maps the ele- N TN M¯ N N Wilsonian symmetries describe the extension of a point ments , , x (in GX~ = ) to identity ðIÞ, T, Mx (in group by quasimomentum translations; such extensions G∘); this is a group isomorphism in the sense that their 2 are also called projective representations. In this appendix, multiplication rules are identical. For example, Mx ¼ I is we elaborate on the connection between projective isomorphic to

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¯ 2 ¯ ¯ ¯ 2 2 ∈ N ðMxN Þ ¼ MxðN MxÞN ¼ MxN To every factor (Ci;j ) of a projective representation, −1 defined again by ¼ Et¯ ðc~zÞW N ¼ N ; ðD2Þ ˆ ˆ ˆ N gigj ¼ Ci;jgij; ðD4Þ where we apply that is a normal subgroup of GX~ , and the third equality relies on Eq. (23). Two extensions are where g ≡ g g and gˆ is the representation of g , there equivalent if there exists a group isomorphism between ij i j i i 2 corresponds a 2-cochain (ν2): them; the second group cohomology [H ðG∘; N Þ,as further elaborated in Appendix D2] classifies the isomor- ν ∈ N N Ci;j ¼ 2ðI;gi;gijÞ ; ðD5Þ phism classes of all extensions of G∘ by , of which GX~ is one example. We also describe an inequivalent extension with the first argument in ν2 set as the identity element in (GΓ~ ) toward the end of Sec. VI, which is nontrivially 3 G∘. ν2 more generally is a map: G∘ → N —besides extended in tð~zÞ (i.e., we are dealing with a glide instead of ¯ informing of the factor system through Eq. (D5), it also a glideless reflection symmetry) and also in E (i.e., this is a encodes how each factor transforms under G∘, through half-integer-spin representation), but not in W. More generally, we could have either an integer-spin or a half- ν ≡ ˆ ν ˆ−1 2ðgig0;gig1;gig2Þ gi 2ðg0;g1;g2Þgi integer-spin representation, with either a glide or glideless ≡ σ ν ∈ N reflection symmetry, and we could be describing a reflec- ið 2ðg0;g1;g2ÞÞ : ðD6Þ tion plane (glide or glideless) in which the reflection either preserves every wave vector or translates each wave vector Presently, we review group cohomology from the perspec- by half a reciprocal period. tive of cochains before establishing its connection with projective representations. Our review closely follows that of Ref. [60], which described only U(1) modules; our 2. Connection between projective representations review demonstrates that the structure of cochains exists for and the second cohomology group more general modules (e.g., N ). Moreover, we are moti- For a group G∘, we define a G∘ module (denoted N )as vated by possible generalizations of our ideas to higher- an Abelian group on which G∘ acts compatibly with the than-two cohomology groups, though presently we do not multiplication operation in N [64]. In our application, G∘ know if such exist. We adopt the convention of Ref. [60] in of Eq. (29) is the point group of a spinless particle, and N defining cochains and the coboundary operator, which they of Eq. (31) is the group generated by real- and quasimo- have shown to be equivalent to standard [64] definitions; mentum-space translations, as well as 2π rotations. Let the the advantage gained is a more compact definition of the ith element (gi)ofG∘ act on a ∈ N by the automorphism: coboundary operator. Generalizing Eq. (D6),ann-cochain → σ ∈ N nþ1 a iðaÞ ; we say this action is compatible if is a map νn: G∘ → N satisfying σ σ σ ∈ N iðabÞ¼ iðaÞ iðbÞ for every a; b : ðD3Þ νnðgig0;gig1;…;gignÞ ≡ σiðνnðg0;g1;…;gnÞÞ ∈ N : ðD7Þ

We show in Eq. (D1) that gi acts on a by conjugation, i.e., Given any νn−1, we can construct a special type of n- σ ˆ ˆ−1 iðaÞ¼giagi , which guarantees that the action is cochain, which we call an n-coboundary, by applying the compatible. coboundary operator dn−1, defined as

Yn ν … ν … … ð−1Þj ½dn−1 n−1ðg0;g1; ;gnÞ¼ n−1ðg0;g1; ;gj−1;gjþ1; ;gnÞ ; j¼0 −1 e:g:; ½d1ν1ðg0;g1;g2Þ¼ν1ðg1;g2Þν1ðg0;g2Þ ν1ðg0;g1Þ: ðD8Þ

n Formally, let C ðN Þ¼fνng be the space of all n-cochains, and the space of all n-coboundaries is an Abelian subgroup of Cn defined by

n n−1 B ¼fνnjνn ¼ dn−1νn−1; νn−1 ∈ C g: ðD9Þ

Applying dn to an n-coboundary always gives the identity element in N :

nYþ1 ν … ν … … ð−1Þj ½dndn−1 n−1ðg0;g1; ;gnþ1Þ¼ ½dn−1 n−1ðg0; ;gj−1;gjþ1; ;gnþ1Þ ¼ I: ðD10Þ j¼0

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Succinctly, a coboundary has no coboundary. Though this conclusion is well known, it might interest the reader how this ν1 result is derived with our nonstandard definition of dn. First, express Eq. (D10) as a product of n−1 by inserting Eq. (D8), e.g.,

−1 −1 −1 ½d2d1ν1ðg0;g1;g2;g3Þ¼fν1ðg2;g3Þν1ðg1;g3Þ ν1ðg1;g2Þgfν1ðg2;g3Þν1ðg0;g3Þ ν1ðg0;g2Þg −1 −1 −1 × fν1ðg1;g3Þν1ðg0;g3Þ ν1ðg0;g1Þgfν1ðg1;g2Þν1ðg0;g2Þ ν1ðg0;g1Þg ¼ I: ðD11Þ

ν1 … 2 Each of n−1 has n distinct arguments drawn from the set fg0;g1; ;gnþ1g of n þ elements. Equivalently, we may label ν1 ∈ 0 1 … 1 2 n−1 by the two elements (gi and gj; i; j f ; ; ;nþ g; i

−1 −1 ½d2d1ν1ðg0;g1;g2;g3Þ ∝ fν1ðg2;g3Þ…gfν1ðg2;g3Þ…g f gf g ; ðD12Þ

−1 where (a) ν1ðg2;g3Þ originates from d2 deleting g0 and d1 deleting g1, while (b) ν1ðg2;g3Þ arises from d2 deleting g1 and d1 deleting g0. These two factors, from (a) and (b), multiply to identity, as is more generally true for any fgi;gjg (recall i

ν … ½dndn−1 n−1ðg0; ;gnþ1Þ ∝ ν … … … ð−1Þið−1Þj−1 ν … … … ð−1Þjð−1Þi n−1ð ;gi−1;giþ1; ;gj−1;gjþ1; Þ n−1ð ;gi−1;giþ1; ;gj−1;gjþ1; Þ ¼ I; ðD13Þ

−1 νð−1Þjð−1Þi νð−1Þið−1Þj where n−1 originates from dn deleting gj, and n−1 from dn deleting gi. We have thus shown that dndn−1νn−1 ¼ I. An n-coboundary is an example of an n-cocycle, which is more generally defined as any n-cochain with a trivial coboundary. The space of all n-cocycles is an Abelian subgroup of Cn defined as

n n Z ¼fνnjdnνn ¼ I;νn ∈ C g: ðD14Þ

The nth cohomology group is defined by the quotient group

n Z G∘; N n N ð Þ ; H ðG∘; Þ¼ n ðD15Þ B ðG∘; N Þ its elements are equivalence classes of n-cocycles, in which we identify any two n-cocycles that differ by an n-coboundary. 2 The rest of this section establishes how H ðG∘; N Þ classifies the different projective representations of G∘, as extended by N . Indeed, we have already identified the factor system of a projective representation with a 2-cochain through Eq. (D5), and the associativity condition on the factor system will shortly be derived as

−1 −1σ Cij;kCi;j iðCj;kÞCi;jk ¼ I; ðD16Þ which translates to a constraint that the 2-cochain is a 2-cocycle. Indeed, from inserting Eqs. (D4)–(D6) into gˆ1ðgˆ2gˆ3Þ¼ðgˆ1gˆ2Þgˆ3, we find that

σ1ðC2;3ÞC1;23gˆ123 ¼ C12;3C1;2gˆ123 ⇒ σ −1 −1 I ¼ 1ðC2;3ÞC12;3C1;23C1;2 −1 −1 ⇒ I ¼ ν2ðg1;g12;g123Þν2ðI;g12;g123Þ ν2ðI;g1;g123Þν2ðI;g1;g12Þ −1 −1 −1 −1 ⇒ I ¼ gˆ0Igˆ0 ¼ gˆ0ν2ðg1;g12;g123Þν2ðI;g12;g123Þ ν2ðI;g1;g123Þν2ðI;g1;g12Þ gˆ0 −1 −1 ⇒ I ¼ ν2ðg01;g012;g0123Þν2ðg0;g012;g0123Þ ν2ðg0;g01;g0123Þν2ðg0;g01;g012Þ −1 −1 ⇒ I ¼ ν2ðg1;g2;g3Þν2ðg0;g2;g3Þ ν2ðg0;g1;g3Þν2ðg0;g1;g2Þ ≡ ½d2ν2ðg0;g1;g2;g3Þ; ðD17Þ

021008-34 TOPOLOGICAL INSULATORS FROM GROUP COHOMOLOGY PHYS. REV. X 6, 021008 (2016) where in the last ⇒ we relabel g01…k → gk and g0 → g0. Furthermore, we recall from Eq. (35) that two projective ˆ ˆ 0 ˆ representations are equivalent if they are related by the gauge transformation gi → gi ¼ Digi, with Di ∈ N . This may be reexpressed as

0 −1 gˆi → gˆi ¼ ν1ðI;giÞ gˆi; ðD18Þ

−1 0 by relabeling Di ≡ ν1ðI;giÞ ∈ N . The motivation for calling gˆi and gˆi gauge equivalent is that both representations induce the same automorphism on the Abelian group N ; i.e.,

ˆ ˆ−1 ≡ σ ∈ N ⇒ ˆ 0 ˆ 0−1 ≡ σ giagi iðaÞ for any a gi agi iðaÞ: ðD19Þ

Let us demonstrate that this gauge-equivalence condition may be expressed as an equivalence of 2-cochains modulo 1- coboundaries. By inserting Eqs. (D18) and (D19) into Eq. (D4), with i ¼ 1 and j ¼ 2,

0 0 0 0 gˆ1gˆ2 ¼ C1;2gˆ12 ⇒ ðν1ðI;g1Þgˆ1Þðν1ðI;g2Þgˆ2Þ¼C1;2ν1ðI;g12Þgˆ12 ≡ ν2ðI;g1;g12Þν1ðI;g12Þgˆ12 0 0 0 ⇒ ν1ðI;g1Þσ1ðν1ðI;g2ÞÞgˆ1gˆ2 ¼ ν2ðI;g1;g12Þν1ðI;g12Þgˆ12 0 0 −1 −1 0 0 0 ⇒ gˆ1gˆ2 ¼ ν2ðI;g1;g12Þν1ðI;g1Þ ν1ðI;g12Þν1ðg1;g12Þ gˆ12 ≡ ν2ðI;g1;g12Þ gˆ12: ðD20Þ

0 To reiterate, the ν2 and ν2 are two gauge-equivalent 2-cochains differing only by multiplication with a 1-coboundary:

0 −1 ν2ð1;g1;g12Þ¼ν2ð1;g1;g12Þ ν1ðI;g1Þν1ðI;g12Þ ν1ðg1;g12Þ −1 0 −1 ⇒ gˆ0ν2ð1;g1;g12Þgˆ0 ¼ ν2ðg0;g01;g012Þ¼ν2ðg0;g01;g012Þ ν1ðg0;g01Þν1ðg0;g012Þ ν1ðg01;g012Þ 0 −1 0 ⇒ ν2ðg0;g1;g2Þ¼ν2ðg0;g1;g2Þ ν1ðg1;g2Þν1ðg0;g2Þ ν1ðg0;g1Þ ≡ ν2ðg0;g1;g2Þ ½d1ν1ðg0;g1;g2Þ; ðD21Þ

where in the last ⇒ we relabel g01…k → gk and g0 → g0. identity according to the multiplication rules of G∘: from We thus demonstrate that different equivalence classes of Eq. (30), these are projective representations correspond to equivalence classes of 2-cocycles, where equivalence is defined modulo M2 ¼ I; T2 ¼ I and M TM−1T−1 ¼ I: ðD24Þ 1-coboundaries; i.e., different projective representations are x x x elements of the second cohomology group [recall 2 A projective representation is obtained by replacing I on H ðG∘; N Þ from Eq. (D15)]. the right-hand side with an element in the G∘ module N:

3. Simple example 2 a 2 b −1 −1 c Mx ¼ W ;T¼ W and MxTMx T ¼ W : For a simple example of H2, consider a reduced problem where we extend G∘ ≅ Z2 × Z2 (as generated by M and T) x That a, b, c are integers does not imply Z3 inequivalent by the group of Wilson loops, extensions; rather, we will see that not all three integers are independent, some integers are only gauge-invariant mod- N ¼fWnjn ∈ Zg ≅ Z; ðD22Þ ulo two, and moreover one of them vanishes so that the representation is associative. Indeed, from the associativity which differs from N in lacking the generators tð~zÞ and E¯ ; of T3, nontrivial extensions by tð~zÞ and E¯ , respectively, describe nonsymmorphic and half-integer-spin representations, and − are already well known [11]. Here, we focus on extensions TWb ¼ TðTTÞ¼ðTTÞT ¼ WbT ¼ TW b purely by momentum translations. G∘ acts on N as ⇒ W2b ¼ I: ðD25Þ

W −1 W−1 W −1 W T T ¼ and Mx Mx ¼ : ðD23Þ Lacking spatial-inversion symmetry, the eigenvalues of W are generically not quantized to any special value, and the Let us follow the procedure outlined in Ref. [65] to only integral solution to W2b ¼ I is b ¼ 0; we comment on determine the possible extensions of G∘. First, we collect the effect of spatial-inversion symmetry at the end of this all nonequivalent products of generators that multiply to example. Similarly, a ¼ c follows from

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a 2 2c 2 2c a 2c−a W T ¼ MxT ¼ W TMx ¼ W TW ¼ W T: to itself by a combination of spatial reflection and quasi- momentum translation across half a reciprocal period. To Moreover, we clarify that only the parity of a labels the clarify a possible confusion, Sec. VI describes a non- 0 inequivalent classes. This follows from Mx and Mx ¼ symmorphic, half-integer-spin representation of a space 2 nx ¯ W Mx (nx ∈ Z) inducing the same automorphism on N: group where Mx is a product of a spatial translation [tð~zÞ] and a 2π rotation (E¯ ), as is relevant to the KHgX material −1 0 0−1 MxWMx ¼ W ⇔ Mx WMx ¼ W: ðD26Þ class [21]; this appendix describes a symmorphic, integer- 2 spin case study where Mx ¼ I. Indeed, we rederive 0 ¯ We say that Mx and Mx are gauge-equivalent representa- Eqs. (23) and (27) in Sec. VI, modulo factors of E and tions; consequently, only the parity of the exponent (a)of tð~zÞ. Despite this difference, all Wilsonian reflections, 2 a W in Mx ¼ W is gauge invariant, as we see from whether glide or glideless, have the same physical origin: some crystal structures host mirror planes (of glide type for 2 a −nx 0 2 a Mx ¼ W ⇒ ðW Mx Þ ¼ W KHgX, but glideless in this appendix) where the group of 2 2 0 any wave vector includes the product of spatial glide or ⇒ M 0 ¼ Waþ nx ≡ Wa : ðD27Þ x reflection with a fractional reciprocal translation. Finally, we address a different example where G∘ One may verify that the relation a ¼ c is gauge invariant, I → 0 2 includes a spatial-inversion ( ) symmetry. We have shown since if a a ¼ a þ nx (as we have just shown), like- W → 0 2 in Ref. [26] that a subset of the eigenvalues may be wise c c ¼ c þ nx (as we now show). To determine 1 I the gauge-transformed c0, we consider two gauge- quantized to depending on the eigenvalues of the occupied bands. Indeed, if we focus only on this quantized equivalent representations of time reversal related by 2b n W I T0 ¼ W T T, with n ∈ Z. By application of Eq. (D23), eigenvalue subset, we might conclude that ¼ [whose T I we derive derivation in Eq. (D25) carries through in the presence of symmetry] could be solved for any b ∈ Z. However, our Wc ¼ M TM−1T−1 perspective is that Wilson-loop extensions classify different x x momentum submanifolds in the Brillouin zone; this clas- − − −1 −1 W nx 0 W nT 0 0 Wnx 0 WnT ¼ð MxÞð T ÞðMx ÞðT Þ; sification should therefore be independent of specific I − − − −1 −1 ⇒ W nx nT nxþnT M0 T0M0 T0 ¼ Wc representations of the occupied bands. Thus, assuming that x x W −1 −1 2 0 a finite subset of eigenvalues are generically not ⇒ 0 0 0 0 Wcþ nx ≡ Wc MxT Mx T ¼ ; ðD28Þ quantized, we conclude that b ¼ 0 even with I symmetry. as desired. We conclude that there are only two elements of H2ðG∘;NÞ. (i) The first is gauge equivalent to a ¼ 0,

2 2 Mx ¼ I; T ¼ I; ½T;Mx¼0; ðD29Þ [1] L. Fu, Topological Crystalline Insulators, Phys. Rev. Lett. 106, 106802 (2011). as expected from the algebra of G∘, and (ii) the second [2] C.-K. Chiu, H. Yao, and S. Ryu, Classification of Topo- logical Insulators and Superconductors in the Presence of element of H2ðG∘;NÞ is gauge equivalent to a ¼ −1, Reflection Symmetry, Phys. Rev. B 88, 075142 (2013). 2 W−1 2 W−1 [3] T. Morimoto and A. Furusaki, Topological Classification Mx ¼ ;T¼ I; TMx ¼ MxT: ðD30Þ with Additional Symmetries from Clifford Algebras, Phys. Rev. B 88, 125129 (2013). The first extension is split (i.e., it is isomorphic to a [4] C.-X. Liu, R.-X. Zhang, and B. K. VanLeeuwen, Topologi- semidirect product of G∘ with N), and corresponds to the cal Nonsymmorphic Crystalline Insulators, Phys. Rev. B 90, identity element of H2ðG∘;NÞ ≅ Z2. Multiplication of two 085304 (2014). elements corresponds to multiplying the factor systems; [5] C. Fang, M. J. Gilbert, and B. A. Bernevig, Entanglement e.g., the two nonsplit elements multiply as Spectrum Classification of Cn-Invariant Noninteracting Topological Insulators in Two Dimensions, Phys. Rev. B 2 −2 2 −2 87, 035119 (2013). M ¼ W ;T¼ I; TM ¼ W M T; ðD31Þ x x x [6] K. Shiozaki and M. Sato, Topology of Crystalline Insulators and Superconductors, Phys. Rev. B 90, 165114 (2014). which is gauge equivalent to Eq. (D29) by the trans- [7] K. Shiozaki, M. Sato, and K. Gomi, Topology of Non- 1 formation of Eq. (D27) with nx ¼ . symmorphic Crystalline Insulators and Superconductors, To realize the nontrivial algebra in Eq. (D30), we need arXiv:1511.01463. that Mx is a Wilsonian symmetry; i.e., it describes not [8] A. Alexandradinata, C. Fang, M. J. Gilbert, and purely a spatial reflection, but also induces parallel trans- B. A. Bernevig, Spin-Orbit-Free Topological Insulators port. As elaborated in Sec. VI, this Wilsonian symmetry is without Time-Reversal Symmetry, Phys. Rev. Lett. 113, realized in mirror planes where any wave vector is mapped 116403 (2014).

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