Topological Insulators from Group Cohomology

Topological Insulators from Group Cohomology


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 discovered 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.

Spatial symmetries have enriched the topological classification of insulators and superconductors.Fu (2011); Chiu et al. (2013); Morimoto and Furusaki (2013); Liu et al. (2014); Fang et al. (2013a); Shiozaki and Sato (2014); Shiozaki et al. (); Alexandradinata et al. (2014a); Varjas et al. (2015, ) A basic geometric property that distinguishes spatial symmetries regards their transformation of the spatial origin: symmorphic symmetries preserve the origin, while nonsymmorphic symmetries unavoidably translate the origin by a fraction of the lattice period.Lax (1974) This fractional translation is responsible for band topologies that have no analog in symmorphic crystals. Thus far, all experimentally-tested topological insulators have relied on symmorphic space groups.Xia et al. (2009); Hsieh et al. (2009); Hsieh (2012); Xu (2012); Tanaka (2012); CeX () Here, we propose the first nonsymmorphic theory of time-reversal-invariant insulators, which complements previous theoretical proposals with magnetic, nonsymmorphic space groups.Mong et al. (2010); Liu et al. (2014); Fang et al. (2013a); Shiozaki et al. (2015); Lu et al. () Motivated by our recently-proposed KHg material class (Sb,Bi,As),Hou () we present here a complete classification of spin-orbit-coupled insulators with the space group () of KHg.

Figure 1: (a) Mirror (red, blue) and glide (green, brown) planes in the 3D Brillouin torus of KHg. These planes project to the high-symmetry line in the 2D Brillouin torus of the 010 surface. (b-d) Examples of possible piecewise topologies in the space group of KHg, as illustrated by their surface bandstructures along . (b) describes a ‘quantum glide Hall effect’ (along ), and an odd mirror Chern number (along ); these two subtopologies must be pieced together at their intersection point . (c) describes an hourglass-flow topology (), and an even mirror Chern number (). (d) A trivial topology is shown for comparison.

The point group () of KHg, defined as the quotient of its space group by translations, is generated by four spatial transformations – this typifies the complexity of most space groups. This work describes a systematic method to topologically classify space groups with similar complexity; in contrast, previous classificationsFu (2011); Hsieh (2012); Chiu et al. (2013); Liu et al. (2014); Fang et al. (2013a); Shiozaki and Sato (2014); Shiozaki et al. () (with one exception by usAlexandradinata et al. (2014a)) have expanded the Altland-Zirnbauer symmetry classesKitaev (2009); Schnyder et al. (2009) to include only a single point-group generator. For point groups with multiple generators, different submanifolds of the Brillouin torus are invariant under different symmetries, e.g., mirror and glide planes are respectively mapped to themselves by a symmorphic reflection and a glide reflection, as illustrated in Fig. 1(a) for . Wavefunctions in each submanifold are characterized by a lower-dimensional topological invariant which depends on the symmetries of that submanifold, e.g., mirror planes are characterized by a mirror Chern numberTeo et al. (2008) and glide planes by a glide-symmetric analogHou (); Shiozaki et al. () of the quantum spin Hall effectC. L. Kane and E. J. Mele (2005) (in short, a quantum glide Hall effect). The various invariants are dependent because wavefunctions must be continuous where the submanifolds overlap, e.g., the intersection of planes in Fig. 1(a) are lines that project to and . We refer to such insulators as ‘piecewise topological’, in the sense that various subtopologies (topologies defined on different submanifolds) must be pieced together consistently to form a 3D topology.

This work addresses two related themes: (i) a group-cohomological classification of quasimomentum submanifolds, and (ii) the connection between this cohomological classification and the topological classfication of band insulators. In (i), we ask how a mirror plane differs from a glide plane. Are two glide planes in the same Brillouin torus always equal? This equality does not hold for : in one glide plane, the symmetries are represented ordinarily, while in the other we encounter generalized ‘symmetries’ that combine space-time transformations with quasimomentum translations (). Specifically, denotes a discrete quasimomentum translation in the reciprocal lattice. These ‘symmetries’ then generate an extension of the point group by , i.e., becomes an element in a projective representation of the point group. The various representations (corresponding to different glide planes) are classified by group cohomology, and they result in different subtopologies (e.g., one glide plane in may manifest a quantum glide Hall effect, while the other cannot). In this sense, cohomology underlies band topology.

To determine the possible subtopologies within each submanifold and then combine them into a 3D topology, we propose a general methodology through Wilson loops of the Berry gauge field;Alexandradinata et al. (2014b); Taherinejad et al. (2014) these loops represent quasimomentum transport in the space of filled bands.Zak (1989) As exemplified for the space group , our method is shown to be efficient and geometrically intuitive – piecing together subtopologies reduces to a problem of interpolating and matching curves. The novel subtopologies that we discover include: (i) the quantum glide Hall effect in Fig. 1(a), (ii) an hourglass-flow topology, as illustrated in Fig. 1(b) and exemplifiedHou () by KHg, and (iii) quantized, non-abelian polarizations that generalize the abelian theory of polarization.King-Smith and Vanderbilt (1993)

Our topological classification of is the first physical application of group extensions by quasimomentum translations. It generalizes the construction of nonsymmorphic space groups, which extend point groups by real-space translations.Ascher and Janner (1965, 1968); Hiller (1986); Mermin (1992); Rabson and Fisher (2001) Here, we further extend nonsymmorphic groups by reciprocal translations, thus placing real and quasimomentum space on equal footing. A consequence of this projective representation is an atypical bulk-boundary correspondence for our topological insulators. This correspondence describes a mapping between topological numbers that describe bulk wavefunctions and surface topological numbersFidkowski et al. (2011) – such a mapping exists if the bulk and surface have in common certain ‘edge symmetries’ which form a subgroup of the full bulk symmetry; this edge subgroup is responsible for quantizing both bulk and surface topological numbers, i.e., these numbers are robust against gap- and edge-symmetry-preserving deformations of the Hamiltonian. In our case study, the edge symmetry is projectively represented in the bulk, where quasimomentum provides the parameter space for parallel transport; on a surface with reduced translational symmetry, the same symmetry is represented ordinarily. In contrast, all known symmetry-protected correspondencesTaherinejad et al. (2014) are one-to-one and rely on the identity between bulk and surface representations; our work explains how a partial correspondence arises where such identity is absent.

The outline of our paper: we first summarize our main results in Sec. I, which also serves as a guide to the whole paper. We then preliminarily review the tight-binding method in Sec. II.1, as well as introduce the spatial symmetries of our case study. Next in Sec. III, we review the Wilson loop and the bulk-boundary correspondence of topological insulators; the notion of a partial correspondence is introduced, and exemplified with our case study of . The method of Wilson loops is then used to construct and classify a piecewise topological insulator in Sec. IV; here, we also introduce the quantum glide Hall effect. Our topological classification relies on extending the symmetry group by quasimomentum translations, as we elaborate in Sec. V; the application of group cohomology in band theory is introduced here. We offer an alternative perspective of our main results in Sec. VI, and end with an outlook.

I Summary of results

A topological insulator in spatial dimensions may manifest robust edge states on a -dimensional boundary. Letting parametrize the -dimensional Brillouin torus, we then split the quasimomentum coordinate as , such that corresponds to the coordinate orthogonal to the surface, and is a wavevector in a -dimensional surface-Brillouin torus. We then consider a family of noncontractible circles , where for each circle, is fixed, while is varied over a reciprocal period, e.g., consider the brown line in Fig. 1(a). We propose to classify each quasimomentum circle by the symmetries which leave that circle invariant. For example, in centrosymmetric crystals, spatial inversion is a symmetry of for inversion-invariant satisfying modulo a surface reciprocal vector. The symmetries of the circle are classified by the second group cohomology


As further elaborated in Sec. V and App. D, classifies the possible group extensions of by , and each extension describes how the symmetries of the circle are represented. The arguments in are defined as:

(a) The first argument, , is a magnetic point groupBRADLEY and DAVIES (1968) consisting of those space-time symmetries that (i) preserve a spatial point, and (ii) map the circle to itself. For , the possible magnetic point groups comprise the 32 classical point groupsTinkham (2003) without time reversal (), 32 classic point groups with , and 58 groups in which occurs only in combination with other operations and not by itself. However, we would only consider subgroups of the 3D magnetic point groups (numbering ) which satisfy (ii); these subgroups might also include spatial symmetries which are spoilt by the surface, with the just-mentioned spatial inversion a case in point.

(b) The second argument of is the direct product of three abelian groups that we explain in turn. The group is generated by a spin rotation; its inclusion in the second argument implies that we also consider half-integer-spin representations, e.g., at inversion-invariant of fermionic insulators, time reversal is represented by .

(c) The second abelian group () is generated by discrete real-space translations in dimensions; by extending a magnetic point group () by , we obtain a magnetic space group; nontrivial extensions are referred to as nonsymmorphic.

(d) The final abelian group () is generated by the discrete quasimomentum translation in the surface-normal direction, i.e., a translation along and covering once. A nontrivial extension by quasimomentum translations is exemplified by one of two glide planes in the space group [cf. Sec. V].

Having classified quasimomentum circles through Eq. (1), we outline a systematic methodology to topologically classify band insulators. The key observation is that quasimomentum translations in the space of filled bands is represented by Wilson loops of the Berry gauge field; the various group extensions, as classified by Eq. (1), correspond to the various ways in which symmetry may constrain the Wilson loop; studying the Wilson-loop spectrum then determines the topological classification. A more detailed summary is as follows:

(i) We consider translations along with a certain orientation that we might arbitrarily choose, e.g., the triple arrows in Fig. 1(a). These translations are represented by the Wilson loop , and the phase () of each Wilson-loop eigenvalue traces out a ‘curve’ over . In analogy with Hamiltonian-energy bands, we refer to each ‘curve’ as the energy of a Wilson band in a surface-Brillouin torus. The advantage of this analogy is that the Wilson bands may be interpolatedFidkowski et al. (2011); Huang and Arovas (2012) to Hamiltonian-energy bands in a semi-infinite geometry with a surface orthogonal to . Some topological properties of the Hamiltonian and Wilson bands are preserved in this interpolation, resulting in a bulk-boundary correspondence that we describe in Sec. III.2. There, we also introduce two complementary notions of a total and a partial correspondence; the latter is exemplified by the space group .

(ii) The symmetries of are formally defined as the group of the Wilson loop in Sec. V; any group of the Wilson loop corresponds to a group extension classified by Eq. (1). That is, our cohomological classification of quasimomentum circles determines the representation of point-group symmetries that constrain the Wilson loop, whether linear or projective. The particular representation determines the rules that govern the connectivity of Wilson energies (‘curves’), as we elaborate in Sec. IV.1; we then connect the ‘curves’ in all possible legal ways, as in Sec. IV.2 – distinct connectivities of the Wilson energies correspond to topologically inequivalent groundstates. This program of interpolating and matching curves, when carried out for the space group , produces the classification summarized in Tab. 1.

Table 1: Classification of time-reversal-invariant insulators with the space group . A quantized polarization invariant () distinguishes between two families of insulators: modulo the electron charge , () characterizes the ‘quantum glide Hall effect’ (resp., its absence). Specifically, is the polarization of one of two glide subspaces (as labelled by ), but time reversal symmetry ensures there is only one independent polarization: modulo . The family is further sub-classified by two non-Abelian polarizations: and , and a mirror Chern number () that is constrained to be even; where , the insulator manifests an hourglass-flow topology. The family is sub-classified by and odd .

Beyond , we note that Eq. (1) and the Wilson-loop method provide a unifying framework to classify chiral topological insulators,Haldane (1988) and all topological insulators with robust edge states protected by space-time symmetries. Here, we refer to topological insulators with either symmorphicChiu et al. (2013); Fu (2011); Alexandradinata et al. (2014a) or nonsymmorphic spatial symmetriesLiu et al. (2014); Fang and Fu (2015); Shiozaki et al. (2015, ), the time-reversal-invariant quantum spin Hall phase,C. L. Kane and E. J. Mele (2005) and magnetic topological insulators.Mong et al. (2010); Fang et al. (2013b); Liu (2013); Zhang and Liu (2015) These case studies are characterized by extensions of by ; on the other hand, extensions by quasimomentum translations are necessary to describe the space group , but have not been considered in the literature. In particular, falls outside the K-theoretic classification of nonsymmorphic topological insulators in Ref. [Shiozaki et al., ].

Finally, we remark that the method of Wilson loops (synonymousAlexandradinata et al. (2014b) with the method of Wannier centersTaherinejad et al. (2014)) is actively being used in topologically classifying band insulators.Alexandradinata et al. (2014b); Yu et al. (2011); Soluyanov and Vanderbilt (2011); Taherinejad et al. (2014); Alexandradinata and Bernevig () The present work advances the Wilson-loop methodology by: (i) relating it to group cohomology through Eq. (1), (ii) providing a systematic summary of the method (in this Section), and (ii) demonstrating how to classify a piecewise-topological insulator for the case study (cf. Sec. IV).

Ii Preliminaries

ii.1 Review of the tight-binding method

In the tight-binding method, the Hilbert space is reduced to a finite number of Lwdin orbitals , for each unit cell labelled by the Bravais lattice (BL) vector .Slater and Koster (1954); Goringe et al. (1997); Lowdin (1950) In Hamiltonians with discrete translational symmetry, our basis vectors are


where , is a crystal momentum, is the number of unit cells, labels the Lwdin orbital, and denotes the position of the orbital relative to the origin in each unit cell. The tight-binding Hamiltonian is defined as


where is the single-particle Hamiltonian. The energy eigenstates are labelled by a band index , and defined as , where


We employ the braket notation:


Due to the spatial embedding of the orbitals, the basis vectors are generally not periodic under for a reciprocal vector . This implies that the tight-binding Hamiltonian satisfies:


where is a unitary matrix with elements: . We are interested in Hamiltonians with a spectral gap that is finite for all , such that we can distinguish occupied from empty bands; the former are projected by


where the last equality follows directly from Eq. (6).

ii.2 Crystal structure and spatial symmetries

The crystal structure KHg is chosen to exemplify the spatial symmetries we study. As illustrated in Fig. 2, the Hg and ions form honeycomb layers with AB stacking along ; here, denote unit basis vectors for the Cartesian coordinate system drawn in the same figure. Between each AB bilayer sits a triangular lattice of K ions. The space group () of KHg includes the following symmetries: (i) an inversion () centered around a K ion (which we henceforth take as our spatial origin), the reflections (ii) , and (iii) , where inverts the coordinate . In (ii-iii) and the remainder of the paper, we denote, for any transformation , as a product of with a translation () by half a lattice vector (). Among (ii-iii), only is a glide reflection, wherefor the fractional translation is unremovableLax (1974) by a different choice of origin. While we primarily focus on the symmetries (i-iii), they do not generate the full group of , e.g., there exists also a six-fold screw symmetry whose implications have been explored in our companion paper.Hou ()

We are interested in symmetry-protected topologies that manifest on surfaces. Given a surface termination, we refer to the subset of bulk symmetries which are preserved by that surface as edge symmetries. The edge symmetries of the 100 and 001 surfaces are symmorphic, and they have been previously addressed in the context of KHg.Hou () Our paper instead focuses on the 010 surface, whose edge group (nonsymmorphic ) is generated by two reflections: glideless and glide .

Figure 2: (a) 3D view of atomic structure. The Hg (red) and (blue) ions form a honeycomb layers with AB stacking. The K ion (cyan) is located at an inversion center, which we also choose to be our spatial origin. (b) Top-down view of atomic structure that is truncated in the 010 direction; two of three Bravais lattice vectors are indicated by and . (c) Center: bulk Brillouin zone (BZ) of KHg, with two mirror planes of colored red and blue. Top: 100-surface BZ. Right: 010-surface BZ.

Iii Wilson loops and the bulk-boundary correspondence

We review the Wilson loop in Sec. III.1, as well as introduce the loop geometry that is assumed throughout this paper. The relation between Wilson loops and the geometric theory of polarization is summarized in Sec. III.2. There, we also introduce the notion of a partial bulk-boundary correspondence, which our nonsymmorphic insulator exemplifies.

iii.1 Review of Wilson loops

The matrix representation of parallel transport along Brillouin-zone loops is known as the Wilson loop of the Berry gauge field. It may be expressed as the path-ordered exponential (denoted by ) of the Berry-Wilczek-Zee connectionWilczek and Zee (1984); Berry (1984) :


Here, recall from Eq. (5) that is an occupied eigenstate of the tight-binding Hamiltonian; denotes a loop and is a matrix with dimension equal to the number () of occupied bands. The gauge-invariant spectrum of is the non-abelian generalization of the Berry phase factors (Zak phase factorsZak (1989)) if is contractible (resp. non-contractible).Alexandradinata et al. (2014b); Taherinejad et al. (2014) In this paper, we consider only a family of loops parametrized by , where for each loop is fixed while is varied over a non-contractible circle (oriented line with three arrowheads in Fig. 3(a)). We then label each Wilson loop as and denote its eigenvalues by exp with . Note that also parametrizes the 010-surface bands, hence we refer to as a surface wavevector; here and henceforth, we take the unconventional ordering . To simplify notation in the rest of the paper, we reparametrize the rectangular primitive cell of Fig. 3 as a cube of dimension , i.e., , , and . The time-reversal-invariant are then labelled as: , , and . For example, would correspond to a loop parametrized by .

Figure 3: (a) A constant- slice of quasimomentum space, with two of three reciprocal lattice vectors indicated by and . While each hexagon corresponds to a Wigner-Seitz primitive cell, 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 how the glide reflection () maps (red dot to brown) which connects to (blue) through . Figure (c) serves two interpretation. In the first, maps (red dot to brown) which connects to (blue) through . If we interpret Figure (c) as the cross-section, the same vectors illustrate the effect of time reversal.

iii.2 Bulk-boundary correspondence of topological insulators

The bulk-boundary correspondence describes topological similarities between the Wilson loop and the surface bandstructure. To sharpen this analogy, we refer to the eigenvectors of as forming Wilson bands with energies . The correspondence may be understood in two steps:

(i) The first is a spectral equivalence between log and the projected-position operator , where


projects to all occupied bands with surface wavevector , and are the Bloch-wave eigenfunctions of . For the position operator , we have chosen natural units of the lattice where , and is the lattice vector indicated in Fig. 2(b). Denoting the eigenvalues of as , the two spectra are related as modulo one.Alexandradinata et al. (2014b) Some intuition about the projected-position operator may be gained from studying its eigenfunctions; they form a set of hybrid functions which maximally localize in (as a Wannier function) but extend in and (as a Bloch wave with momentum ). In this Bloch-Wannier (BW) representation,Taherinejad et al. (2014) the eigenvalue () under is merely the center-of-mass coordinate of the BW function ().Soluyanov and Vanderbilt (2011); Alexandradinata et al. (2014b) Since is symmetric under translation by , while , each of represents a family of BW functions related by integer translations. The Abelian polarization () is defined as the net displacement of BW functions:King-Smith and Vanderbilt (1993); Vanderbilt and King-Smith (1993); Resta (1994)


where all equalities are defined modulo integers, and Tr is the Abelian Berry connection.

(ii) The next step is an interpolationFidkowski et al. (2011); Huang and Arovas (2012) between and an open-boundary Hamiltonian () with a boundary termination. Presently, we assume for simplicity that each of is invariant under space-time transformations of the edge group. A simple example is the 2D quantum spin Hall (QSH) insulator, where time reversal () is the sole edge symmetry: by assumption is a symmetry of the periodic-boundary Hamiltonian (hence also of ); furthermore, since acts locally in space, it is also a symmetry of and . It has been shown in Ref. Fidkowski et al., 2011 that the discrete subset of the -spectrum (corresponding to edge-localized states) is deformable into a subset of the fully-discrete -spectrum. More physically, a subset of the BW functions mutually and continuously hybridize into edge-localized states when a boundary is slowly introduced, and the edge symmetry is preserved throughout this hybridization. Consequently, (equivalently, log[]) and share certain traits which are only well-defined in the discrete part of the spectrum, and moreover these traits are robust in the continued presence of said symmetries. The trait that identifies the QSH phase (in both the Zak phases and the edge-mode dispersion) is a zigzag connectivity where the spectrum is discrete; here, eigenvalues are well-defined, and they are Kramers-degenerate at time-reversal-invariant momenta but otherwise singly-degenerate, and furthermore all Kramers subspaces are connected in a zigzag pattern.Yu et al. (2011); Soluyanov and Vanderbilt (2011); Alexandradinata et al. (2014b) In the QSH example, it might be taken for granted that the representation () of the edge symmetry is identical for both and ; the invariance of throughout the interpolation accounts for the persistence of Kramers degeneracies, and consequently for the entire zigzag topology. The QSH phase thus exemplifies a total bulk-boundary correspondence, where the entire set of boundary topologies (i.e., topologies that are consistent with the edge symmetries of ) is in one-to-one correspondence with the entire set of -topologies (i.e., topologies which are consistent with symmetries of , of which the edge symmetries form a subset). One is then justified in inferring the topological classification purely from the representation theory of surface wavefunctions – this surface-centric methodology has successfully been applied to many space groups.Alexandradinata et al. (2014a); Liu et al. (2014); Dong and Liu (2016)

While this surface-centric approach is technically easier than the representation theory of Wilson loops, it ignores the bulk symmetries that are spoilt by the boundary. On the other hand, -topologies encode these bulk symmetries, and are therefore more reliable in a topological classication. In some cases,Alexandradinata et al. (2014b); Hughes et al. (2011); Turner et al. (2012); Alexandradinata and Bernevig () these bulk symmetries enable -topologies that have no boundary analog. Simply put, some topological phases do not have robust boundary states, a case in point being the topology of 2D inversion-symmetric insulators.Alexandradinata et al. (2014b) In our nonsymmorphic case study, it is an out-of-surface translational symmetry () that disables a -topology, and consequently a naive surface-centric approach would over-predict the topological classification – this exemplifies a partial bulk-boundary correspondence. As we will clarify, the symmetry distinguishes between two representations of the same edge symmetries: an ordinary representation with the open-boundary Hamiltonian (), and a projective one with the Wilson loop (). To state the conclusion upfront, the projective representation rules out a quantum glide Hall topology that would otherwise be allowed in the ordinary representation. This discussion motivates a careful determination of the -topologies in Sec. IV.

Iv Constructing a piecewise-topological insulator by Wilson loops

We would like to classify time-reversal-invariant insulators with the space group ; our result should more broadly apply to hexagonal crystal systems with the edge symmetry (generated by glide and glideless ) and a bulk spatial-inversion symmetry. Our final result in Tab. 1 relies on topological invariants that we briefly introduce here, deferring a detailed explanation to the sub-sections below. The invariants are: (i) the mirror Chern number () in the plane, (ii) the quadruplet polarization () in the glide plane (resp. ), wherefor implies an hourglass flow, and (iii) the glide polarization () indicates the absence (resp. presence) of the quantum glide Hall effect in the plane.

Our strategy for classification is simple: we first derive the symmetry constraints on the Wilson-loop spectrum, then enumerate all topologically distinct spectra that are consistent with these constraints. Pictorially, this amounts to understanding the rules obeyed by curves (the Wilson bands), and connecting curves in all possible legal ways; we do these in Sec. IV.1 and IV.2 respectively.

iv.1 Local rules of the curves

We consider how the bulk symmetries constrain the Wilson loop , with lying on the high-symmetry line ; note that and are glide lines which are invariant under , while and are mirror lines invariant under . The relevant symmetries that constrain necessarily preserve the circle , modulo translation by a reciprocal vector; such symmetries comprise the little group of the circle.Alexandradinata et al. (2014a) For example, (a) would constrain for all , (b) and is constraining only for along , and (c) matters only at the time-reversal-invariant . Along , we omit discussion of other symmetries (e.g., ) in the group of the circle, because they do not additionally constrain the Wilson-loop spectrum. For each symmetry, only three properties influence the connectivity of curves, which we first state succinctly:

(i) Does the symmetry map each Wilson energy as or ? Note here we have omitted the constant argument of .

(ii) If the symmetry maps , does it also result in Kramers-like degeneracy? By ‘Kramers-like’, we mean a doublet degeneracy arising from an antiunitary symmetry which representatively squares to , much like time-reversal symmetry in half-integer-spin representations.

(iii) How does the symmetry transform the mirror eigenvalues of the Wilson bands? Here, we refer to the eigenvalues of mirror and glide along their respective invariant lines.

To elaborate, (i) and (ii) are determined by how the symmetry constrains the Wilson loop. We say that a symmetry represented by is time-reversal-like at , if for that


Both map the Wilson energy as , but only symmetries guarantee a Kramers-like degeneracy. Similarly, a symmetry represented by is particle-hole-like at , if for that


i.e., maps the Wilson energy as . Here, we caution that and are symmetries of the circle and preserve the momentum parameter ; this differs from the conventionalSchnyder et al. (2009) time-reversal and particle-hole symmetries which typically invert momentum.

To precisely state (iii), we first elaborate on how Wilson bands may be labelled by mirror eigenvalues, which we define as for the reflection (). First consider the glideless , which is a symmetry of any bulk wavevector which projects to () and () in . Being glideless, ( rotation of a half-integer spin) implies two momentum-independent branches for the eigenvalues of : ; this eigenvalue is an invariant of any parallel transport within either -invariant plane. That is, if is a mirror eigenstate, any state related to by parallel transport must have the same mirror eigenvalue. Consequently, the Wilson loop block-diagonalizes with respect to , and any Wilson band may be labelled by .

A similar story occurs for the glide , which is a symmetry of any bulk wavevector that projects to (). The only difference from is that the two branches of are momentum-dependent, which follows from , with denoting a lattice translation. Explicitly, the Bloch representation of squares to exp, which implies for the glide eigenvalues: exp.

To wrap up our discussion of the mirror eigenvalues, we consider the subtler effect of along . Despite being a symmetry of any surface wavevector along :


with a surface reciprocal vector, is not a symmetry of any bulk wavevector that projects to , but instead relates two bulk momenta which are separated by half a bulk reciprocal vector, i.e.,

as illustrated in Fig. 3(b). This reference to Fig. 3(b) must be made with our reparametrization (, ) in mind. We refer to such a glide plane as a projective glide plane, to distinguish it from the ordinary glide plane at . The absence of symmetry at each bulk wavevector implies that the Wilson loop cannot be block-diagonalized with respect to the eigenvalues of . However, quantum numbers exist for a generalized symmetry () that combines the glide reflection with parallel transport over half a reciprocal period. To be precise, let us define the Wilson line to represent the parallel transport from to . We demonstrate in Sec. V that all Wilson bands may be labelled by quantum numbers under , and that these quantum numbers fall into two energy-dependent branches as:


That is, is the -eigenvalue of a Wilson band at surface momentum and Wilson energy .

For the purpose of topological classification, all we need are the existence of these symmetry eigenvalues (ordinary and generalized) that fall into two branches (recall along , and also Eq. (14) ), and (iii) asks whether the - and -type symmetries preserve () or interchange () the branch. To clarify a possible confusion, both - and -type symmetries are antiunitary and therefore have no eigenvalues, while the reflections ( (along ), and (along ) ) are unitary. The answer to (iii) is determined by the commutation relation between the symmetry in question (whether - or -type) and the relevant reflection. To exemplify (i-iii), let us evaluate the effect of symmetry along . This may be derived in the polarization perspective, due to the spectral equivalence of log and . Since inverts all spatial coordinates but transforms any momentum to itself (), we identify as a -type symmetry (cf. Eq. (12)). Indeed, while is known to produce Kramers degeneracy in the Hamiltonian spectrum, emerges as an unconventional particle-hole-type symmetry in the Wilson loop. Since commutes individually with and , all eigenstates of may simultaneously be labelled by . That maps then follows from , where originates simply from the noncommutivity of with the fractional translation () in :


To show in more detail, suppose for a Bloch-Wannier function that


with exp( and suppression of the label . then leads to


with exp following from exp. To recapitulate, (a) imposes a particle-hole-symmetric spectrum, and (b) two states related by have opposite eigenvalues under . (a-b) is summarized by the notation in the top left entry of Tab. 2. The complete symmetry analysis is derived in Sec. V and App. B, and tabulated in Tab. 2 and 3. These relations constrain the possible topologies of the Wilson bands, as we show in the next section.

- -
- -
Table 2: Symmetry constraints of the Wilson bands at generic points along the mirror lines. and are possible characterizations of the symmetries () in the left-most column: a -type (-type) symmetry is time-reversal-like (resp. particle-hole-like) symmetry, as defined in Eq. (IV.1) and (12). For , is a symmetry eigenvalue which falls into one of two branches: , and exp. Along , is momentum-dependent; along , it is energy-dependent as well, that is, is the glide eigenvalue of a Wilson band at momentum and energy .
Table 3: Time-reversal constraint of the Wilson bands at surface wavevectors satisfying modulo a reciprocal vector. To clarify a possible source of confusion, the actual time-reversal symmetry () is, by our definition of in Eq. (IV.1), only ‘time-reversal-like’ at , since is not a symmetry of the circle for generic .

iv.2 Connecting curves in all possible legal ways

Our goal here is to determine the possible topologies of curves (Wilson bands), which are piecewise smooth on the high-symmetry line . We first analyze each momentum interval separately, by evaluating the available subtopologies within each of , , etc. The various subtopologies are then combined to a full topology, by a program of matching curves at the intersection points (e.g.,) between momentum intervals.

Since our program here is to interpolate and match curves (Wilson bands), it is important to establish just how many Wilson bands must be connected. A combination of symmetry, band continuity and topology dictates this answer to be a multiple of four. Since the number () of occupied Hamiltonian bands is also the dimension of the Wilson loop, it suffices to show that is a multiple of four. Indeed, this follows from our assumption that the groundstate is insulating, and a property of connectedness between sets of Hamiltonian bands. For spin systems with minimally time-reversal and glide-reflection symmetries, we prove in App. C that Hamiltonian bands divide into sets of four which are individually connected, i.e., in each set there are enough contact points to travel continuously through all four branches. The lack of gapless excitations in an insulator then implies that a connected quadruplet is either completely occupied, or unoccupied.

Interpolating curves along the glide line

Along (), the rules are:

(a) There are two flavors of curves (illustrated as blue solid and blue dashed lines in Fig. 4), corresponding to two branches of the glide eigenvalue . Only crossings between solid and dashed curves are robust, in the sense of being movable but unremovable.

(b) At any point along , there is an uncoventional particle-hole symmetry (due to ) with conjugate bands (related by ) belonging in opposite glide branches; cf. first column of Tab. 2. Pictorially, [, blue solid] [, blue dashed].

(c) At , each solid curve is degenerate with a dashed curve, while at the degeneracies are solid-solid and dashed-dashed; cf. Tab. 3. These end-point constraints are boundary conditions for the interpolation along .

Given these rules, there are three distinct connectivities along , which we describe in turn: (i) a zigzag connectivity (Fig. 4(a-e)) defines the quantum glide Hall effect (QGHE), and (ii) two configurations of hourglasses (e.g., Fig. 4(f) vs 4(h), and also 4(g) vs 4(i) ) are distinguished by a connected-quadruplet polarization.

Figure 4: Possible Wilson spectra along .

(i) As illustrated in Fig. 4(a-e), the QGHE describes a zigzag connectivity over , where each cusp of the zigzag corresponds to a Kramers-degenerate subspace. While Fig. 4(c-d) is not obviously zigzag, they are smoothly deformable to Fig. 4(a) which clearly is. A unifying property of all five figures (a-e) is spectral flow: the QGHE is characterized by Wilson bands which robustly interpolate across the maximal energy range of . What distinguishes the QGHE from the usual quantum spin Hall effect:C. L. Kane and E. J. Mele (2005) despite describing the band topology over all of , the QGHE is solely determined by a polarization invariant () at a single point (), which we now describe.

Definition of : Consider the circle in the 3D Brillouin zone. Each point here has the glide symmetry , and the Bloch waves divide into two glide subspaces labelled by . This allows us to define an Abelian polarization () as the net displacement of Bloch-Wannier functions in either subspace:


Here, the superscript indicates a restriction to the , occupied subspace; are the eigenvalues of the Wilson loop , and the second equality follows from the spectral equivalence introduced in Sec. III. We have previously determined in this Section that is a multiple of four, and therefore there is always an even number () of Wilson bands in either subspace. Furthemore,