Mirror Chern numbers in the hybrid Wannier representation
MMirror Chern numbers in the hybrid Wannier representation
Tom´aˇs Rauch, Thomas Olsen, David Vanderbilt, and Ivo Souza
4, 5 Friedrich-Schiller-University Jena, 07743 Jena, Germany Computational Atomic-scale Materials Design, Department of Physics,Technical University of Denmark, 2800 Kgs. Lyngby Denmark Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854-8019, USA Centro de F´ısica de Materiales, Universidad del Pa´ıs Vasco, 20018 San Sebasti´an, Spain Ikerbasque Foundation, 48013 Bilbao, Spain
The topology of electronic states in band insulators with mirror symmetry can be classified in twodifferent ways. One is in terms of the mirror Chern number, an integer that counts the number ofprotected Dirac cones in the Brillouin zone of high-symmetry surfaces. The other is via a Z indexthat distinguishes between systems that have a nonzero quantized orbital magnetoelectric coupling(“axion-odd”), and those that do not (“axion-even”); this classification can also be induced byother symmetries in the magnetic point group, including time reversal and inversion. A systematiccharacterization of the axion Z topology has previously been obtained by representing the valencestates in terms of hybrid Wannier functions localized along one chosen crystallographic direction, andinspecting the associated Wannier band structure. Here we focus on mirror symmetry, and extendthat characterization to the mirror Chern number. We choose the direction orthogonal to the mirrorplane as the Wannierization direction, and show that the mirror Chern number can be determinedfrom the winding numbers of the touching points between Wannier bands on mirror-invariant planes,and from the Chern numbers of flat bands pinned to those planes. In this representation, therelation between the mirror Chern number and the axion Z index is readily established. Theformalism is illustrated by means of ab initio calculations for SnTe in the monolayer and bulk forms,complemented by tight-binding calculations for a toy model. I. INTRODUCTION
The band theory of solids has been enriched in recentyears by a vigorous study of its topological aspects. Thateffort resulted in a systematic topological classification ofinsulators on the basis of symmetry, and in the identifi-cation of a large number of topological materials. Afteran initial focus on the role of time-reversal symmetry, itwas realized that crystallographic symmetries could alsoprotect topological behaviors, leading to the notion of“topological crystalline insulators.”The assignment of an insulator to a particular topolog-ical class can be made by evaluating the correspondingtopological invariant. Depending on the protecting sym-metry, that invariant may assume one of two values ( Z classification), or it may assume any integer value ( Z clas-sification). Other types of classifications such as Z alsooccur, but they do not concern us here. When the invari-ant vanishes the system is classified as trivial, and other-wise it is classified as nontrivial or topological. Topolog-ical insulators typically display robust gapless states atthe boundary, which provide an experimental signatureof topological behavior.In some cases, the same symmetry may induce twodifferent topological classifications. This happens for ex-ample with mirror symmetry, where a Z classification interms of the mirror Chern number (MCN) [1, 2] coex-ists with a Z classification based on the quantized axionangle. The two classifications are not independent, andelucidating the relation between them is one goal of thepresent work.The axion Z classification of three-dimensional (3D) insulators is based on the orbital magnetoelectric effect.In brief, the isotropic part of the linear orbital magneto-electric tensor is conveniently expressed in terms of theaxion angle θ , which is only defined modulo 2 π as a bulkproperty. In the presence of “axion-odd” symmetries thatflip its sign, the axion angle can only assume two values: θ = 0 (trivial), and θ = π (topological) [3–8].The axion Z index was originally introduced for time-reversal invariant insulators, where it was shown to beequivalent to the “strong” Z index ν = 0 or 1, thatis, θ = πν . More generally, axion-odd symmetries canbe classified as proper rotations combined with time re-versal (including time reversal itself), and improper ro-tations (including inversion and reflection) not combinedwith time reversal; in both cases, the associated symme-try operation in the magnetic space group may includea fractional translation. This results in a large numberof magnetic space groups that can host axion-odd topo-logical insulators. A recent realization is the MnBi Te family of antiferromagnetic materials [7–9], whose ax-ion topology is protected by the time reversal operationcombined with a half-lattice translation as envisioned inRef. [10].To aid the computational search for axionic topolog-ical insulators, it is useful to devise simple proceduresfor determining the (quantized) axion angle θ . Unfor-tunately, subtle gauge issues make its direct evaluationfrom the valence Bloch states rather challenging in gen-eral [5]. Notable exceptions are centrosymmetric insula-tors, both nonmagnetic and magnetic. For such systems,the axion Z index can be obtained by counting the num-ber of odd-parity states at high-symmetry points in the a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Brillouin zone (BZ) [11, 12].Recently, an alternative procedure was introducedbased on representing the valence states in terms of hy-brid Wannier (HW) functions that are maximally local-ized along a chosen crystallographic direction z . The HWcenters along z , also known as “Wilson-loop eigenvalues,”form a band structure when plotted as a function of k x and k y ; in the presence of one or more axion-odd sym-metries, the quantized θ value can be determined fromthis “Wannier band structure,” often by mere visual in-spection [13].In the HW representation, axion-odd symmetries arenaturally classified as “ z -preserving” or “ z -reversing,”and the rules for deducing the axion Z index are dif-ferent in each case (they also depend on whether ornot the symmetry operation involves a fractional trans-lation along z ) [13]. Time reversal is an example of a z -preserving operation, while inversion is z reversing. Mir-ror operations may be placed in one group or the other,depending on whether the Wannierization direction z liesin the reflection plane (vertical mirror) or is orthogonalto it (horizontal mirror). In this work we make the latterchoice, so that the mirror operation of interest becomes M z : z → − z , (1)which is manifestly z reversing.A simple mirror symmetry without a glide componentprotects not only the axion Z classification, but also a Z or Z × Z classification based on one or two MCNs, de-pending on the type of mirror. This raises the questionof whether the HW representation might also be con-venient for determining the MCNs, and for illuminatingtheir relationship to the quantized axion angle.In this work, we address the above questions by investi-gating in detail the Wannier bands in the presence of M z symmetry. We clarify the generic behaviors that are ex-pected, and discuss the rules for deducing the MCNs. Bycomparing those rules with the ones obtained in Ref. [13]for the axion Z index, we establish the relation betweenthe two classifications.The paper is organized as follows. In Sec. II wefirst distinguish between “type-1” and “type-2” crystal-lographic mirror operations; we then review the defini-tions of Chern invariants and MCNs in terms of theBloch states in the filled bands; finally, we introducemaximally localized HW functions spanning the valencestates, and assign Chern numbers to isolated groups ofWannier bands. This background material sets the stagefor the developments in the remainder of the paper. InSec. III we discuss the generic features of the Wannierband structure in the presence of M z symmetry, and ob-tain a relation between Chern numbers and winding num-bers in groups of bands touching on a mirror plane. Therules for deducing the MCNs from the Chern numbersand winding numbers on the mirror planes are given inSec. IV, where their relation to the quantized axion angleis also established. In Sec. V we describe the numericalmethods that are used in Sec. VI to apply the formalism to several prototypical systems. We summarize and con-clude in Sec. VII, and present in three Appendices somederivations that were left out of the main text. II. PRELIMINARIESA. Two types of crystallographic mirrors
We begin by observing that if a crystal is left invariantunder an M z reflection operation, then its Bravais latticemust contain vectors pointing along z . To construct theshortest such vector a = c ˆ z , we pick the shortest vector (cid:101) a connecting lattice points on adjacent horizontal latticeplanes. If (cid:101) a points along z then we take it as a , andwe say that the mirror is of type 1. Otherwise we choosethe vector a = (cid:101) a − M z (cid:101) a connecting second-neighborlattice planes, and the mirror is of type 2.The two types of crystallographic mirrors are exempli-fied in 2D in Fig. 1, where the mirror lines z = 0 and c/ k z = 0 and k z = π/c are labeled G and X. The samenotation will be used in 3D, where A and B (G and X)become planes in real (reciprocal) space.The distinction between mirror operations that leavepointwise invariant two inequivalent planes in the BZ,and those that leave invariant only one BZ plane, wasmade in Refs. [14, 15]. Since MCNs are defined on suchplanes [1, 2], a 3D insulator with a type-1 mirror is char-acterized by two separate MCNs µ G and µ X , while fora type-2 mirror there is a single MCN µ G . If the crys-tallographic space group contains additional mirror op-erations, there will be additional MCNs associated withthem. B. Chern invariants in band insulators
1. Generic insulators
Before introducing MCNs for insulators with reflectionsymmetry, let us define Chern invariants for generic 2Dand 3D band insulators in terms of the k -space Berrycurvature of the valence states [5].In 2D, the Berry curvature of a Bloch state | ψ n k (cid:105) withcell-periodic part | u n k (cid:105) is a scalar defined asΩ n k = − (cid:104) ∂ k x u n k | ∂ k y u n k (cid:105) (2)where k = ( k x , k y ), and the Chern number is given by C = 12 π (cid:90) J (cid:88) n =1 Ω n k d k (3)where the summation is over the J filled energy bands.Since the Berry curvature has units of length squared, C is a dimensionless number, and for topological reasons it FIG. 1. The upper panel shows schematically a pair of 2D crystals lying on the ( x, z ) plane; each has one atom per primitivecell (black dots), and lattice constant c along z . The crystal on the left has a rectangular lattice and a type-1 horizontalmirror, with inequivalent mirror lines z = 0 mod c (A) and z = c/ c (B), shown as dashed lines; the one on the righthas a centered rectangular lattice and a type-2 mirror, with equivalent mirror lines A and B. The lattice vectors a and (cid:101) a aredefined in the main text. The lower panel shows the reciprocal lattices, with a separation of 2 π/c between horizontal latticelines G. On the left the periodicity along k z is 2 π/c , and hence both k z = 0 mod 2 π/c (G) and k z = π/c mod 2 π/c (X) arepointwise-invariant mirror lines, as indicated by the dashed lines. On the right, where the periodicity along k z is 4 π/c , G is amirror-invariant line but X is not. The associated Brillouin zones are indicated by the shaded green areas. must be an integer. The Chern number is a global prop-erty of the manifold of occupied states, remaining invari-ant under multiband gauge transformations described by J × J unitary matrices at each k , and it vanishes whenthe crystal has time-reversal symmetry. If a 2D magneticcrystal has a nonzero Chern number C , when that crys-tal is terminated at an edge there will be | C | edge modescrossing the bulk gap, whose chirality will depend on thesign of C .3D insulators are characterized by a Chern vector K = 12 π (cid:90) J (cid:88) n =1 Ω n k d k , (4)where now k = ( k x , k y , k z ) and the Berry curvature hasbecome a vector field, Ω n k = − Im (cid:104) ∂ k u n k |×| ∂ k u n k (cid:105) . TheChern vector has units of inverse length, and is quantizedto be a reciprocal-lattice vector. Like the Chern numberin 2D, the Chern vector always vanishes in nonmagneticcrystals.Given a set of lattice vectors a j and dual reciprocal-lattice vectors b j , the expansion K = (cid:80) j C j b j definesa triad of integer Chern indices C j . Let us orient theCartesian axes such that a = c ˆ z . The vectors b and b then lie on the ( x, y ) plane, and the third Chern indexcan be expressed as C = c π (cid:90) π/c C ( k z ) dk z , (5) where C ( k z ) = 12 π (cid:90) J (cid:88) n =1 Ω zn ( k x , k y , k z ) dk x dk y . (6)The integral in Eq. (6) is over a slice of the 3D BZspanned by b and b at fixed k z . By viewing it asan effective 2D BZ and comparing with Eq. (3), it be-comes clear that C ( k z ) is a Chern number; and since ina gapped system its integer value cannot change with thecontinuous parameter k z , Eq. (5) reduces to C = C ( k z )evaluated at any k z . The Chern indices of 3D insulatorscan therefore be evaluated as Chern numbers defined overindividual BZ slices.
2. Mirror-symmetric insulators
We now consider a 3D crystalline insulator with mirrorsymmetry M z , and assume that its Chern vector K van-ishes. A new integer-valued topological index, the MCN,can be defined for such a system as follows [1, 2].On the mirror-invariant BZ planes, G and possibly X,the energy eigenstates are also eigenstates of M z . Theeigenvalues are i F p , where p = ± F = 0 or 1 when the electrons are treated asspinless or spinful particles, respectively. The occupiedBloch states on those planes can therefore be groupedinto “even” ( p = +1) and “odd” ( p = −
1) sectors underreflection about the A plane z = 0, each carrying its ownChern number. The Chern numbers of the two sectorson the G plane k z = 0 are given by C ± G = 12 π (cid:90) J (cid:88) n =1 f ± n k Ω zn ( k x , k y , k z = 0) dk x dk y , (7)where f + n k = 1 − f − n k equals one or zero for a state with p = ±
1, respectively. The MCN is defined as µ G = 12 (cid:0) C +G − C − G (cid:1) , (8)and it is guaranteed to be an integer since C +G + C − G = C vanishes by assumption. If the mirror is of type 1, theplane X carries a second MCN µ X = 12 (cid:0) C +X − C − X (cid:1) , (9)where C ± X is obtained by replacing k z = 0 with k z = π/c in Eq. (7). The MCNs remain invariant under multibandgauge transformations that do not mix the two mirror-parity sectors. When they are nonzero, protected gaplessmodes appear on surfaces that retain the mirror symme-try M z , with | µ G | and | µ X | counting the number of Diraccones on the two M z -invariant lines in the surface BZ [16].In the case of a 2D or quasi-2D insulator with reflectionsymmetry M z about its own plane, the entire 2D BZ isleft invariant under M z . Such a system has a uniqueMCN µ = 12 ( C + − C − ) , (10)where C + and C − are obtained by inserting the 2D Berrycurvature of Eq. (2) in Eq. (7). When the net Chernnumber C = C + + C − vanishes, | µ | becomes an integerthat counts the number of pairs of counterpropagatingchiral edge modes [17].We note in passing that spin-orbit coupling is requiredto obtain non-vanishing MCNs in systems that are eithernon-magnetic or whose magnetic order is collinear. C. The hybrid Wannier representation
1. Hybrid Wannier functions and Wannier bands
HW functions are obtained from the valence Blochstates of a 2D or 3D crystalline insulator by carryingout the Wannier construction along a chosen reciprocal-lattice direction. They are therefore localized along onedirection only, in contrast to ordinary Wannier functionswhich are localized in all spatial directions.Let us momentarily return to a generic 3D insulatingcrystal, not necessarily mirror-symmetric. We denote by z the chosen localization direction and let κ = ( k x , k y ), so that the wavevector in the 3D BZ becomes k = ( κ , k z ).Given a gauge for the Bloch states that is periodic in k z , | ψ n κ ,k z +2 π/c (cid:105) = | ψ n κ k z (cid:105) , the corresponding HW func-tions are defined as | h ln κ (cid:105) = 12 π (cid:90) π/c − π/c e − ik z lc e − i κ · r | ψ n κ k z (cid:105) dk z , (11)where the index l runs over unit cells along z , and n runsover the J HW functions in one unit cell. By factoringout e − i κ · r , we have made the HW functions cell periodicin the in-plane directions, h ln κ ( r + R ) = h ln κ ( r ) for anyin-plane lattice vector R . This will be convenient lateron when we define Berry curvatures and Chern numbersin the HW representation.For each κ in the projected 2D BZ, we choose themultiband gauge for the Bloch states in such a way thatthe HW functions have the smallest possible quadraticspread along z . Such maximally-localized HW functionssatisfy the eigenvalue equation [18] P κ zP κ | h ln κ (cid:105) = z ln κ | h ln κ (cid:105) , (12)where P κ is the projection operator onto the space of va-lence states with in-plane wave vector κ . The eigenvaluesin Eq. (12) are the HW centers z ln κ = (cid:104) h ln κ | z | h ln κ (cid:105) , (13)which form Wannier bands. These are periodic in realspace along z , as well as in the in-plane reciprocal space, z ln κ = z n κ + lc , z ln, κ + G = z ln κ , (14)where G is an in-plane reciprocal lattice vector.A Wannier band structure is said to be gapped if itcontains at least one Wannier band per vertical cell thatis separated from the band below by a finite gap at all κ . When that is the case, we choose the cell contents insuch a way that the first band, n = 1, has a gap below it.
2. Chern numbers of Wannier bands
The Berry curvature of a HW state is defined asΩ ln = − (cid:104) ∂ k x h ln | ∂ k y h ln (cid:105) , (15)and periodicity along z implies that Ω ln = Ω n . (Hereand in the following, we will frequently drop the index κ .)When the Wannier spectrum is gapped, it becomes possi-ble to associate a Chern number with each isolated group a of bands within a vertical cell, C la = 12 π (cid:90) (cid:88) n ∈ a Ω ln d k = C a . (16)From the HW states in a given group, one can constructBloch-like states at any k = ( k x , k y , k z ) by invertingEq. (11). In general these are not energy eigenstates, andtheir band indices label Wannier bands rather than en-ergy bands. Their Berry curvatures along z are given byΩ zn ( k x , k y , k z ) = (cid:88) l e ik z lc Ω n,ln ( k x , k y ) , (17)whereΩ n,ln = i (cid:104) ∂ k x h n | ∂ k y h ln (cid:105) − i (cid:104) ∂ k y h n | ∂ k x h ln (cid:105) (18)is a matrix generalization of Eq. (15) [19]. To evaluatethe net Chern number C a ( k z ) of that group of Bloch-like states on a slice of the 3D BZ, we insert Eq. (17) inEq. (6) and restrict the summation over n to n ∈ a . Thecontributions from the l (cid:54) = 0 terms drop out, yielding C a ( k z ) = C a . (19)Hence the Chern numbers are the same in the Bloch-likeand HW representations, as expected since the two repre-sentations are related by a unitary transformation. Whenthe group a comprises all J Wannier bands in one ver-tical cell, its Chern number becomes equal to the Chernindex C of Eq. (5), which vanishes by assumption. III. MIRROR-SYMMETRIC WANNIER BANDS
With the above background material in hand, we nowreturn to our system of interest – a 3D insulator with M z symmetry – and construct HW functions localized alongthe direction z orthogonal to the mirror plane. We beginthis section by discussing the generic features of Wannierband structures with M z symmetry. A. Flat vs dispersive bands, and the uniformparity assumption If M z is a symmetry of the system, the operator P zP anticommutes with M z . It follows that if a HW func-tion | h ln (cid:105) satisfies Eq. (12) with eigenvalue z ln , M z | h ln (cid:105) satisfies it with eigenvalue − z ln . Since z ln is only de-fined modulo c , two situations may occur. (i) | h ln (cid:105) and M z | h ln (cid:105) are orthogonal, in which case a pair of disper-sive bands appear at ± z ln . (ii) | h ln (cid:105) and M z | h ln (cid:105) are thesame up to a phase, in which case | h ln (cid:105) is an eigenstateof M z , and a single flat band appears at either z = 0(A plane) or z = c/ The expression for C a ( k z ) involves (cid:82) π/a ∂ k x Y n,ln ( k x ) dk x where Y n,ln ( k x ) = (cid:82) π/b A y n,ln ( k x , k y ) dk y , and another simi-lar integral (cid:82) π/b ∂ k y X n,ln ( k y ) dk y . When l (cid:54) = 0 the quantity Y n,ln ( k x ) becomes fully invariant under band-diagonal gaugetransformations of the HW states. Hence its value at k x = 2 π/a must be the same as at k x = 0, and the integral vanishes. even or odd mirror parity at A; flat bands of even or oddmirror parity at B; and dispersive pairs appearing at ± z .If there are several flat bands on a given mirror planeand not all of them have the same parity, those of oppo-site parity will generally have a nonzero P zP matrix ele-ment between them, and will tend to hybridize and splitto form dispersive pairs. Thus, all flat bands pinned atA are expected to have the same parity p A , and all flatbands pinned at B are expected to have the same parity p B . Following Ref. [13], we call this the “uniform parity”assumption. As discussed in Ref. [13], this assumption isclosely related to a well-known theorem on the minimumnumber of zero-energy modes in bipartite lattices [20–22].Under the uniform parity assumption, the numbers N A and N B of flat bands at A and B can be expressed interms of the imbalance between even- and odd-parity va-lence Bloch states at the mirror-invariant plane(s) in theBZ. For a type-1 mirror we have N A = 12 | ∆ N G + ∆ N X | (20)and N B = 12 | ∆ N G − ∆ N X | , (21)where ∆ N G and ∆ N X denote the excess of even overodd valence states at G and X, respectively. Hence if themirror-parity content is balanced at both G and X, flatWannier bands are absent from both A and B; if is it isbalanced only at G but not at X or vice-versa, the samenumber of flat bands is present at A and at B; and if itis unbalanced at both G and X, the number of flat bandsat B can differ from the number at A. The correspondingrelation for a type-2 mirror is N A = N B = 12 | ∆ N G | . (22)Equations (20-22) are derived in Appendix A. B. Types of generic degeneracies
In this section, we consider the types of degeneraciesthat are typical of the Wannier spectra of insulators with M z symmetry. We call a degeneracy generic when itoccurs without the assistance of any symmetries otherthan M z . If in addition the degeneracy is codimensionprotected, we call it accidental .Accidental degeneracies away from the A and B planeshave codimension three, and hence they require fine tun-ing. On the mirror planes, there are two types of genericdegeneracies: multiple flat bands pinned to the sameplane, and accidental touchings, at isolated points in the2D BZ, between one or more pairs of dispersive bands.Other possibilities such as nodal lines are non-genericand will not be considered further. In the following wefocus on the A plane z = 0, but the discussion would beidentical for the B plane z = c/
1. Point nodes between pairs of dispersive bands
If there are no flat bands pinned at z = 0, any bandsnear z = 0 must come in dispersive pairs at ± z . If there isa single pair, we construct from the two HW functions ateach κ a pair of orthogonal states with opposite paritiesabout z = 0. In this basis, the z operator is representedby a matrix of the form (cid:18) f κ f ∗ κ (cid:19) , (23)with eigenvalues z κ = ±| f κ | . The two bands touch at z = 0 when | f κ | = 0, and for that to happen both the realand imaginary parts of f κ must vanish; this means thatsuch degeneracies have codimension two, and hence theyoccur at isolated points in the 2D BZ. (When the bandsdisperse linearly close to the nodal point, the degeneracyis called a “Dirac node.”) If more than one dispersiveband pair is involved, f κ becomes a matrix. The degen-eracy condition det( f κ ) = 0 again leads to point nodeson the z = 0 plane. Generically, these are simple nodeswhere only two bands meet. However, with additionalsymmetries or fine tuning, more than one pair of bandsmay become degenerate at a given node.In summary, pairs of dispersive Wannier bands cantouch accidentally at isolated points on a mirror planefree of flat bands. We note that the same happens,and for the same mathematical reasons, with the energybands of models with sublattice symmetry [22].
2. Flat bands repel point nodes
When one or more flat bands are present at z = 0, theygap out the point nodes. Let us show this for the simplestcase of one flat band surrounded by a dispersive pair.Choosing a basis of M z eigenstates within this three-bandspace, the matrix representation of the z operator takesthe form f κ g κ f ∗ κ g ∗ κ , (24)where we have chosen the first basis state to have the op-posite mirror parity from the other two. The eigenvaluesare z κ = 0 (flat band) and z κ = ± (cid:112) | f κ | + | g κ | (dis-persive pair). An accidental degeneracy between the pairrequires the real and imaginary parts of both f κ and g κ to vanish (codimension four). In general this cannot beachieved by adjusting κ alone; it also requires fine tuningthe parameters f κ and g κ .In conclusion, flat bands and point nodes do not gen-erally coexist on a mirror plane. Although we have onlyshown this for the case of one flat band plus one dispersivepair, the same result is expected to hold when several flatbands and/or dispersive pairs are present. That scenariohas in fact been considered for the analogous problem ofenergy bands in models with sublattice symmetry [22].
3. Spinful time-reversal symmetry excludes flat bands
The presence of flat bands on the mirror planes cansometimes be ruled out on the basis of symmetry. Thisis the case for a crystal that has both M z symmetry andspinful time-reversal symmetry T . Since [ P κ zP κ , T ] = 0,the standard Kramers-degeneracy argument applies tothe Wannier bands: if | h κ (cid:105) is an eigenstate of P κ zP κ with eigenvalue z κ , then | h (cid:48)− κ (cid:105) = T | h κ (cid:105) is an orthogonaleigenstate with the same eigenvalue. Now suppose that | h κ (cid:105) is a flat-band state at A, with M z eigenvalue λ = ± i .Then | h (cid:48)− κ (cid:105) is also a flat-band state, and using [ M z , T ] =0 we find that its mirror eigenvalue is λ ∗ = − λ . Sincethe two flat bands have opposite mirror eigenvalues, theywill generally hybridize to form a dispersive pair.Another example is a crystal that has both M z symme-try, and spinful T combined with inversion I . The com-bined symmetry I ∗T renders the energy bands Kramers-degenerate at every k , and since [ M z , I ∗ T ] = 0 and M z has purely imaginary eigenvalues, Kramers pairs ofHamiltonian eigenstates on the invariant BZ planes haveopposite M z eigenvalues. The mirror-parity content istherefore balanced on those planes, and from Eqs. (20-22) we conclude that both N A and N B vanish. (Notethat while the energy bands are Kramers degenerate inthe presence of I ∗ T symmetry, the Wannier bands arenot. The difference is that
I ∗ T commutes with theHamiltonian, but it anticommutes with
P zP .)In summary, spinful time-reversal symmetry, either byitself or in combination with inversion, rules out the pres-ence of flat Wannier bands on the mirror planes (underthe uniform parity assumption).
C. Chern numbers in gapped band structures
When an M z -symmetric Wannier band structure isgapped, the J bands per cell can be grouped into three in-ternally connected collections [13]: one containing bandsthat are pinned at A (over the entire 2D BZ or at isolated κ points), another containing bands that are pinned atB, and a third containing “unpinned” bands, in the sensethat they do not touch the mirror planes anywhere in the2D BZ. In Ref. [13] these three collections were called origin-centered , boundary-centered , and uncentered , re-spectively.Letting α = A or B, in each vertical cell l there are ingeneral • N α + flat bands at α of even parity, • N α − flat bands at α of odd parity, • (cid:101) N α dispersive bands touching at α in the α -pinned collection, and (cid:101) N UC dispersive bands inthe unpinned collection. (At this stage we do not yet as-sume uniform parity for the flat bands, nor do we invokethe fact that flat bands repel point nodes.) In the homecell l = 0, the dispersive bands in the A-pinned collectioncome in pairs at ± z , and those in the B-pinned collectioncome in pairs at z and c − z . In the case of the unpinnedcollection we have a choice, since the mirror-symmetricpartners never become degenerate; for definiteness, wechoose the contents of the home cell so that the bands inthe unpinned collection come in pairs at ± z .For each of the seven groups listed above, we can addup the Chern numbers in that group to get C α ± , (cid:101) C α ,and (cid:101) C UC , keeping in mind that their sum C vanishes byassumption, C A + C B + (cid:101) C UC = 0 , (25)where C α = C α + + C α − + (cid:101) C α is the net Chern numberof the α -pinned collection. We further decompose eachof the three dispersive band subspaces into even and oddsectors about their centers, and assign separate Chernnumbers to them, (cid:101) C α = (cid:101) C α + + (cid:101) C α − , (26a) (cid:101) C UC = (cid:101) C UC + + (cid:101) C UC − , (26b)as described in Appendix B. There we show that (cid:101) C α + − (cid:101) C α − = W α , (27)where W α is the sum of the winding numbers (definedin Sec. V C 1) of all the nodal points in the projected 2DBZ on the α mirror plane. Hence W α = (cid:101) C α − (cid:101) C α − , (28)so that (cid:101) C α has the same even or odd parity as W α . Sinceband pairs in the unpinned collection do not touch onthe special planes, by applying the same argument inAppendix B that leads to Eq. (27) we obtain (cid:101) C UC + = (cid:101) C UC − , (29)which implies that their sum (cid:101) C UC is always an even num-ber. IV. MIRROR CHERN NUMBERS IN THEHYBRID WANNIER REPRESENTATION
We are finally ready to evaluate the MCNs in the HWrepresentation, and then relate them to the axion Z in-dex. In Sec. IV A we consider the case of a gapped Wan-nier spectrum, and in Sec. IV B we treat the gapless case. The fact that (cid:101) C UC is even can also be seen as follows [13]. Theunpinned collection is formed by two disconnected groups ofbands related by M z symmetry, which imposes the same Berrycurvature at every κ in the two groups, and hence the sameChern number. TABLE I. Parities under a type-1 mirror M z of Bloch-likestates constructed from HW functions that are maximally lo-calized along z . For spinful electrons, the parity is said to be“even” or “odd” when the M z eigenvalue is + i or − i . Bloch representation G + = even about A (and even about B)G − = odd about A (and odd about B)X + = even about A (and odd about B)X − = odd about A (and even about B) Hybrid Wannier representation A + = even about A, generates G + and X + A − = odd about A, generates G − and X − B + = even about B, generates G + and X − B − = odd about B, generates G − and X + pairs C and C (cid:48) , generates G + G − and X + X − A. Gapped Wannier band structure
To recap, a generic gapped Wannier band structurewith M z symmetry consists of seven band collections percell. The four that are flat have well-defined mirror pari-ties, and the three that are dispersive can be decomposedinto even and odd sectors. This yields a total of ten HWgroups with well-defined parities, each carrying its ownChern number.
1. Type-1 mirrors
To evaluate the MCNs µ G and µ X , we construct fromeach of the ten HW groups a group of Bloch-like states byperforming Bloch sums along z , and recall from Eq. (19)that their Chern numbers on any constant- k z BZ slice(and, in particular, at G and X) are the same as theChern numbers of the parent HW groups. The finalneeded ingredient is Table I, which tells the mirror par-ities at G and X of the Bloch groups coming from eachof the HW groups. That table is valid for both spinlessand spinful mirror symmetry M z , and it agrees with theparity rules for inversion symmetry I in 1D [13]; this isconsistent with the fact that M z = I ∗ C z acts along z in the same way as I .To evaluate µ G , we need to split the occupied Blochspace at G into even- and odd-parity sectors about A.According to Table I, their Chern numbers are C ± G = (cid:16) C A ± + (cid:101) C A ± + (cid:101) C ± UC (cid:17) + (cid:16) C B ± + (cid:101) C B ± (cid:17) , (30)where the first and second groups of terms correspondto Wannier groups that are even or odd about A and B,respectively. Inserting this expression into Eq. (8) for µ G and then using Eqs. (27) and (29), we find2 µ G = (cid:0) C A + − C A − (cid:1) + (cid:0) C B + − C B − (cid:1) + W A + W B . (31)Under the uniform parity assumption the first group ofterms becomes p A C A , where C A is the total Chern num-ber of the flat bands at A, all of the same parity p A = ± p B C B . Thus we ar-rive at µ G = 12 (cid:0) p A C A + W A (cid:1) + 12 (cid:0) p B C B + W B (cid:1) , (32)and via similar steps Eq. (9) for µ X turns into µ X = 12 (cid:0) p A C A + W A (cid:1) − (cid:0) p B C B + W B (cid:1) . (33)Out of the three collections in a type-1 disconnected bandstructure, the uncentered collection does not contributeto the MCNs; and the A-centered and B-centered onescontribute as in Eqs. (32) and (33).Equations (32) and (33) are a central result of thiswork, and in the following sections we will extract sev-eral conclusions from them. In practical applications,those equations can often be simplified: since flat bandsand point nodes do not generically coexist on the mirrorplanes, at least one of the two terms inside each paren-thesis will typically vanish.Before proceeding, let us verify that Eq. (32) correctlyyields an integer value for µ G when C = 0. First weeliminate the winding numbers from Eq. (32) with thehelp of Eq. (28), and then we take mod 2 on both sidesof the resulting equation to find2 µ G mod 2 = (cid:16) C A + (cid:101) C A + C B + (cid:101) C B (cid:17) mod 2= − (cid:101) C UC mod 2 , (34)where Eq. (25) was used to go from the first to the secondline. Given that (cid:101) C UC is an even number, we concludethat µ G is an integer. The proof is identical for Eq. (33).We emphasize that the separate contributions from theA- and B-centered collection to Eqs. (32) and (33) are notalways integer-valued. As can be seen from Eq. (36) be-low, those contributions assume half-integer values whenthe axion angle is quantized to θ = π by mirror symme-try; a concrete example where this happens will be givenin Sec. VI C.
2. Relation to the quantized axion coupling
As mentioned in the Introduction, mirror symmetrybelongs to the group of “axion-odd” symmetries that re-verse the sign of the axion angle θ . When one or moresuch symmetries are present in a 3D insulator with a van-ishing Chern vector, θ is restricted to be zero or π mod2 π , becoming a Z topological index. In the case of mirror symmetry, where the band topol-ogy is already characterized by the MCNs, there shouldbe a relation between them and the quantized θ value.Below we derive that relation for an insulator with a type-1 mirror and a gapped Wannier spectrum. To that end,we make use of the formalism of Ref. [13] for expressing θ in the HW representation.First we write µ G + µ X by combining Eqs. (32) and(33), and eliminate the winding numbers using Eq. (28).Then we take mod 2 on both sides to find( µ G + µ X ) mod 2 = C A mod 2 . (35)Comparing with the relation θ/π = C A mod 2 [13], validfor a gapped spectrum in the presence of a z -reversingaxion-odd symmetry such as M z , we conclude that θπ = ( µ G + µ X ) mod 2 . (36)Thus, the system is axion-even ( θ = 0) or axion-odd( θ = π ) depending on whether the sum of the two MCNsassociated with M z is even or odd. Previously, this resulthad been inferred from an argument based on countingDirac cones in the surface BZ [14, 15]. Here, we haveobtained it directly as a formal relation between bulkquantities expressed in the HW representation. As wewill see shortly, the same relation holds when the Wan-nier spectrum is gapless.
3. Type-2 mirrors
In a crystal with a type-2 mirror, where the planes Aand B are equivalent and G is the only mirror-invariantplane in reciprocal space, the unique MCN µ G is obtainedby setting p B = p A , C B = C A , and W B = W A in Eq. (32), µ G = p A C A + W A . (37)If flat bands are present at A, they repel the pointnodes. Hence W A = 0, and therefore | µ G | = | C A | . Inter-estingly, in this case the magnitude of the MCN does notdepend on the parity of the flat-band states; this simpli-fies considerably its numerical evaluation, since one doesnot need to know how the basis orbitals transform un-der M z . Given that only the magnitude (not the sign)of the MCN is needed to establish the bulk-boundarycorrespondence, this is a potentially useful result.Inserting Eq. (28) for W A in Eq. (37), taking mod 2 onboth sides, and again comparing with θ/π = C A mod 2,we conclude that in this case the relation between theaxion Z index and the MCN reads θπ = µ G mod 2 , (38)as stated in Ref. [15].
4. Weakly coupled layered crystals
Consider a crystal composed of weakly coupled iden-tical layers that remain invariant under reflection abouttheir own planes. Following Ref. [23], we assume thatthe layers are stacked exactly vertically. In this case thereflection symmetry about the individual layers becomesa type-1 mirror of the 3D structure, with two separateMCNs µ G and µ X . In the fully decoupled limit wherethere is no k z dependence the G and X reciprocal planesbecome equivalent, so that µ X = µ G ≡ µ where µ isthe MCN of an isolated layer [Eq. (10)]. But since theMCNs are integers, they cannot change if a weak inter-layer coupling is introduced, and from Eqs. (32) and (33)we obtain µ = 12 (cid:0) p A C A + W A (cid:1) (39)for the unique MCN of a weakly-coupled layered crystal.If flat bands are present at A (the plane of a layer),then W A = 0 and the net Chern number of the valencebands becomes C A + (cid:101) C UC ; since the net Chern numbervanishes by assumption and (cid:101) C UC is even, µ = p A C A / | µ | can be determinedwithout knowing the parity of the flat-band states, as inthe case of a type-2 mirror with flat bands.Let us now evaluate the axion Z index. Since µ G + µ X = 2 µ is an even number, Eq. (36) yields θ = 0 mod 2 π . (40)This is consistent with the assertion made in Ref. [23]that weakly-coupled layered topological crystalline in-sulators are analogous to “weak topological insulators”with a vanishing strong Z invariant ν . B. Gapless Wannier band structure
Let us now apply our formalism to a M z -symmetricsystem with a gapless Wannier spectrum. We start outby noting that such a spectrum must have degeneracies atboth A and B. On those special planes the codimensionis two, so point nodes are allowed. Flat bands can beruled out since they would repel any nodes and generatea gap, and we assume that nodal lines are absent as well.We are left with a scenario where there are point nodesat both A and B, and these are connected by Wannierbands. The only way this can happen without the assis-tance of other symmetries is if there are only two Wannierbands, one in each half unit cell, since otherwise there isgenerically a gap somewhere in each half cell (accidentaldegeneracies away from A and B are not protected, sincethe codimension is three). With the assistance of othersymmetries, the gapless spectrum may contain more thantwo bands per cell.To treat the above scenario, we temporarily add asymmetric pair of occupied orbitals at degeneracy-free planes ± z , and initially do not let them hop at all (com-pletely isolated). This will introduce flat bands on thoseplanes. Now let the added orbitals hybridize with otherorbitals. Since accidental degeneracies away from themirror planes are not protected, gaps will generally openup between the new and the old Wannier bands (theonly exceptions to this rule are treated in the next para-graph). And since the added orbitals are topologicallytrivial, they have no effect on the MCNs, which can nowbe evaluated using the formalism of Sec. IV A for gappedspectra. Setting C A = C B = 0 in Eqs. (32) and (33)therein, we obtain µ G = 12 ( W A + W B ) (41)and µ X = 12 ( W A − W B ) . (42)But since W A and W B cannot be affected by orbitalsinserted far from the A and B planes, we conclude thatEqs. (41) and (42) can be directly applied to the originalsystem with a gapless Wannier spectrum.The above argument needs to be refined if the systemis an axion-odd insulator that has, in addition to M z symmetry, one or more axion-odd symmetries that are z preserving and symmorphic (e.g., spinful time rever-sal or vertical mirrors). The Wannier spectrum is thenguaranteed to be gapless, with adjacent bands touchingat an odd number of Dirac nodes [13]. The solution is toweakly break all such symmetries via some low-symmetryperturbation; the band connectivity then becomes “frag-ile,” allowing gaps to open up once the added orbitalshybridize with the original ones [13, 24]. The rest of theargument proceeds as before, again with the conclusionthat Eqs. (41) and (42) can be directly applied to theoriginal system with a gapless spectrum. This scenariois illustrated in Sec. VI C 2, where the orbital insertionitself acts as the symmetry-lowering perturbation.To conclude, let us show that the relation (36) betweenthe MCNs and the axion angle remains valid for gaplessspectra. Equations (41) and (42) give µ G + µ X = W A ,while θ is equal to the sum of Berry phases of vanishinglysmall loops around the nodes at A [13]. Since those Berryphases divided by π are equal to the node winding num-bers modulo 2 [25], Eq. (36) is immediately recovered. V. METHODSA. Tight-binding, ab initio , and Wannier methods
In this work, the formalism for evaluating MCNs inthe HW representation is implemented in the tight-binding (TB) framework, using a modified version of the
PythTB code [26]. Illustrative calculations are carriedout for 2D and 3D models with mirror symmetry; someare simple toy models, while others are obtained from0 ab initio calculations as described below. Each model isspecified by providing the on-site energies, the hoppingamplitudes, and the matrix elements of the position andmirror operators.In the TB literature, it is common to assume that theposition operator is represented by a diagonal matrix inthe TB basis, (cid:104) ϕ R i | r | ϕ R (cid:48) j (cid:105) = ( R + τ i ) δ R , R (cid:48) δ ij (43)where τ i is the location of the i th basis orbital in thehome cell R = . This approximation is problematic forcalculating the Wannier bands of unbuckled monolayers,since it forces all bands to lie flat on the z = 0 plane:when all basis orbitals lie on the z = 0 plane and all off-diagonal matrix elements (cid:104) ϕ R i | z | ϕ R (cid:48) j (cid:105) vanish, the matrix Z κ that is diagonalized to obtain the HW centers [seeEqs. (44) and (45)] is the null matrix.To apply our formalism to flat monolayers, any flatWannier bands that may be present must be robust andsatisfy the uniform parity assumption, while all otherbands must be dispersive. To ensure that this is so, oneshould retain some off-diagonal z matrix elements. Formodels based on ab initio Wannier functions this occursnaturally, since the position matrix elements between theWannier functions are explicitly calculated, and they aregenerally nonzero for nearby Wannier functions. In thecase of toy models, one needs to assign nonzero values tosome of the off-diagonal z matrix elements under reason-able assumptions.The material chosen for the ab initio calculations isSnTe, which we study as a flat monolayer in Sec. VI A andas a bulk phase in Sec. VI B. We first calculate the elec-tronic structure from density-functional theory (DFT)using the GPAW code [27], and then use the
Wannier90 code [28] to construct well-localized Wannier functions.Lastly, TB models are generated by tabulating the ma-trix elements of the Kohn-Sham Hamiltonian and of theposition operator between those Wannier functions.The self-consistent DFT calculations are performedwithout including spin-orbit coupling, which is added af-terwards non-selfconsistently [29]. We use the Perdew-Burke-Ernzerhof exchange-correlation functional [30, 31],and describe the valence-core interaction via the projec-tor augmented wave method [32]. The valence states areexpanded in a plane-wave basis with an energy cutoff of600 eV, and the BZ is sampled on Γ-centered uniformgrids containing 6 × × × × d semicore states of Snin addition to the 5 s and 5 p states of Sn and Te, yieldinga total of 20 valence electrons for each SnTe formula unit(one per cell for the monolayer, and two for the bulk).For each formula unit, we construct 16 spinor Wannierfunctions of s and p character spanning the upper-valenceand low-lying conduction band states. The Sn 4 d states,which give rise to flat bands lying 22 eV below the Fermilevel, are excluded from the Wannier construction. As a first step towards obtaining well-localized Wan-nier functions, we extract from the space of ab initio Bloch eigenstates at each grid point k an N -dimensionalsubspace with the desired orbital character ( N = 16for the monolayer, and N = 32 for the bulk). This isachieved via the “band disentanglement” procedure ofRef. [33], which involves specifying two energy windows,known as the inner and the outer window, and a set oftrial orbitals. The outer window encloses all the valencebands except for the 4 d semicore states, as well as allthe low-lying conduction states of 5 s and 5 p character.To ensure that the valence states are exactly preservedin the disentangled subspace, we “freeze” them inside aninner window. An initial guess for the target subspaceis obtained by projecting atom-centered s and p trial or-bitals onto the outer-window states. This is followed byan iterative procedure that yields an optimally-smoothdisentangled subspace across the BZ [33].Having extracted a suitable Bloch subspace, we pro-ceed to construct well-localized s - and p -like Wannierfunctions spanning that subspace. This is done by pro-jecting onto it the same s and p trial orbitals that wereused in the disentanglement step, and then orthogonal-izing the resulting orbitals via the L¨owdin scheme [18].This one-shot procedure, without additional maximal-localization steps [18], ensures that the Wannier func-tions retain the orbital character of the trial orbitals.To assess the quality of the Wannier basis we calculatethe energy bands from the Hamiltonian matrix elementsin that basis [33], and find that they are in excellentagreement with the ab initio bands obtained using the GPAW code [34].In addition to the Hamiltonian and position matrixelements, we also require the matrix elements of the mir-ror operator M z in the Wannier basis. These are neededto determine the winding numbers of the nodal touch-ings between Wannier bands on the mirror planes (seeSec. V C), as well as the mirror parities p A and p B ofthe flat-band states. To set up the matrix representationof M z , we assume that the Wannier functions transformunder M z in the same way as pure s and p orbitals. Wefind that the eigenstates of the Wannier Hamiltonian onthe mirror-invariant BZ planes are, to a good approxima-tion, eigenstates of this approximate M z operator, whichvalidates that assumption. B. Construction of hybrid Wannier functions andWannier bands
Formally, maximally-localized HW functions satisfythe eigenvalue equation (12). For a 2D or quasi-2D sys-tem extended along x and y , the matrix elements of the z operator appearing in that equation are well defined.It is therefore straightforward to set up the matrix Z mn k = (cid:104) ψ m k | z | ψ n k (cid:105) , (44)1where k = ( k x , k y ) and m and n run over the J occupiedenergy bands, and to diagonalize it, (cid:104) U † k Z k U k (cid:105) mn = z m k δ mn . (45)The eigenvalues are the HW centers, and from the eigen-vectors (the columns of the U k matrix) we can constructthe maximally-localized HW functions according to | h n k (cid:105) = (cid:88) m e − i k · r | ψ m k (cid:105) U mn k , (46)where the phase factor has been included to render themin-plane periodic.For bulk systems, which are extended in all directionsincluding the wannierization direction z , the above proce-dure fails because the matrix elements in Eq. (44) becomeill defined. In such cases, it is still possible to constructmaximally-localized HW functions by working in recip-rocal space. We now write k = ( κ , k z ), and choose auniform grid; for each point κ in the projected 2D BZ,the problem reduces to the construction of 1D maximally-localized Wannier functions along z . The procedure is de-tailed in Refs. [5, 18]. Briefly, the first step is to establisha “twisted parallel transport gauge” for the valence Blochstates along the string of k z points at each κ , obtainingas a byproduct the HW centers z ln κ . The maximally-localized HW functions | h ln κ (cid:105) are then constructed inthis gauge using Eq. (11), with the integral over k z re-placed by a summation over the string of k z points. C. Winding number of a point node on a mirrorplane
1. Definition
Consider a point node κ j where N pairs of dispersiveWannier bands meet on a mirror plane, and let M z κ bethe matrix representation of the M z operator in the basisof the associated HW functions at a nearby point κ , M zmn κ = (cid:104) h m κ | M z | h n κ (cid:105) . (47)Here, m and n run over the 2 N Wannier bands thatmeet at κ j . By diagonalizing M z κ and then transformingthe | h n κ (cid:105) states accordingly [see Eqs. (45) and (46)], weobtain a new set of 2 N states | ˜ h n κ (cid:105) . Like the originalones they are cell-periodic in plane and localized along z , but they have definite mirror parities. We choose thefirst N to be even under M z , and denote them as | ˜ h + l κ (cid:105) ;the remaining N are odd under M z , and we denote themas | ˜ h − l κ (cid:105) . In both cases, l goes from 1 to N .The matrix representation of the z operator in the newbasis takes the form of Eq. (23), where f κ is the N × N matrix with elements f ll (cid:48) κ = (cid:104) ˜ h + l κ | z | ˜ h − l (cid:48) κ (cid:105) . (48) Letting γ κ = arg(det f κ ) , (49)the winding number of node κ j is defined as [35] W j = 12 π (cid:73) c j ∂ κ γ κ · d κ , (50)where the integral is over a small circle around the node.
2. Numerical evaluation
Suppose a single pair of Wannier bands meet at a pointnode κ j . To evaluate the winding number (50), the phase γ κ must be smooth on c j . In practice, we establish asmooth gauge for the states | ˜ h ± κ (cid:105) as follows. We pick arepresentation of the two states at a reference point κ (cid:48) j inthe vicinity of the node. Then at any point κ (cid:48) j + ∆ κ onthe circle c j we choose the gauge by enforcing maximalphase alignment with the states at κ (cid:48) j , i.e., by requir-ing that the overlaps (cid:104) ˜ h + κ (cid:48) j | ˜ h + κ (cid:48) j +∆ κ (cid:105) and (cid:104) ˜ h − κ (cid:48) j | ˜ h − κ (cid:48) j +∆ κ (cid:105) arereal and positive. In other words, we carry out a one-stepparallel transport from κ (cid:48) j to each circumference point.If several pairs of bands meet at a node, the strategy isbasically the same. The only difference is that one mustnow use the multiband version of the parallel-transportprocedure [5, 18]. VI. NUMERICAL RESULTS
In this section, we use our formalism to calculate theMCNs of three different systems. The first is an unbuck-led monolayer of SnTe, a topological crystalline insulatorprotected by reflection symmetry about its plane. Thesecond is rocksalt SnTe, a 3D topological crystalline in-sulator protected by a type-2 mirror. Our last exampleis a 3D toy model based on a modified Dirac equation. Itis both a strong topological insulator protected by time-reversal symmetry, and a topological crystalline insulatorwith a type-1 mirror. In the first example the Wannierspectrum is trivially gapped, while in the other two it isgapless.
A. Unbuckled monolayer of SnTe
The structure we consider is shown in Fig. 2(a). Itconsist of a single unbuckled layer of Sn and Te atomsarranged in a checkerboard pattern, which can be viewedas a single (001) layer of the bulk rocksalt structure.DFT calculations reveal that the system with an opti-mized lattice constant of a = 6 .
16 ˚A is situated 0.4 eVabove the convex hull and is dynamically unstable [36],and that a buckled structure that breaks mirror symme-try is energetically favored [37]. These results imply that2 M FIG. 2. (a) Atomic structure of monolayer SnTe. The blacksquare is the conventional unit cell with lattice constant a ,and the red square is the primitive cell with lattice constant a (cid:48) = a/ √
2. (b) Brillouin zone and high-symmetry points. a flat SnTe monolayer is not likely to be experimentallyrelevant. This system is nevertheless ideally suited forillustrating our methodology, since it has reflection sym-metry about its own plane and the associated MCN isnonzero [38].We carry out calculations using the primitive cell con-taining one formula unit. The Wannier-interpolated en-ergy bands are shown in Fig. 3(a), where all bands aredoubly degenerate due to time-reversal and inversionsymmetry. There is a robust inverted gap (0 . .
17 meV) aroundthe X point; when the lattice expands the indirect gapincreases, and when it shrinks the system turns into aband overlap semimetal [37, 38]. The lowest four valencebands are predominantly s -type, and the remaining six(plotted in red) are predominantly p -type.Figure 3(b) shows the Wannier bands calculated fromthe Bloch states in the p -type upper valence bands. Thespectrum consists of three mirror-symmetric band pairsthat touch on the A plane z = 0 at isolated points inthe 2D BZ. There are no flat bands on that plane, asexpected from the presence of time-reversal symmetry(Sec. III B 3). Equation (39) therefore reduces to µ = 12 W A , (51)and the MCN can be determined by evaluating the wind-ing numbers of the nodal points on the A plane.To locate those nodal points, we plot in Fig. 3(c) the“gap function” g k = − log(∆ z k /c ) , (52)where ∆ z ( k ) is the separation between the central pair ofbands. Regions with a small gap appear in dark gray, andnodal points as dark spots. The positions and windingnumbers of all the nodal points are indicated in the figure,where we have included only one of the periodic imageswhen a node falls on the BZ boundary. At Γ and M thereare nodes where three pairs of Wannier bands touch, withwinding numbers W j = − W j = +1, respectively. All other nodes on the z = 0 plane are simple Diracnodes where only the two central bands meet, and theyhave W j = ±
1. Adding up the winding numbers of the36 nodal points in the BZ we obtain W A = −
4, and fromEq. (51) we conclude that the group of six p -type valencebands has a MCN of − s -type lower va-lence bands, and find that their net winding number van-ishes. The net MCN of the occupied states is therefore µ = −
2, with the nontrivial topology coming from the p states. This result agrees with the value | µ | = 2inferred from a k · p analysis of the simultaneous bandinversions at the two X points in the BZ [17, 38]. B. Bulk SnTe
Bulk SnTe, which crystallizes in the rocksalt struc-ture, is known both from theory [16] and experiment [39]to be a topological crystalline insulator. The symme-try protecting its nontrivial band topology is reflectionabout the { } family of planes. (Instead, the (001)mirror symmetry responsible for the topological state ofthe monolayer is topologically trivial in the bulk crystal.)The lattice is face-centered cubic lattice, so that theshortest lattice vector perpendicular to the (110) planes is a = a ˆ x / a ˆ y /
2. Since its length is twice the separationbetween adjacent planes, the (110) mirror operation is oftype 2, as is typical of centered lattices (see Fig. 1).For our simulations we pick a tetragonal cell subtendedby a = − a ˆ x / a ˆ y / a = a ˆ z , and a , and reorient theaxes such that those vectors point along ˆ x , ˆ y , and ˆ z , re-spectively. In this new frame, the (110) mirror operationof interest becomes M z . The simulation cell with twoformula units is shown in Fig. 4(a), and the associatedBZ in Fig. 4(b).In Fig. 5(a) we present the energy bands calculatedalong the high-symmetry lines of the folded BZ. The non-trivial topology arises from simultaneous band inversionsat the two L points in the unfolded BZ [16], which maponto the two R points in Fig. 4(b). The inverted bandgap at R and the global indirect band gap amount to 0 . . z . The Wannier spectrumis shown in Fig. 5(b). Its periodicity is c/ z , and only one period is shown. Thespectrum is gapless, with two pairs of bands crossing inopposite directions, between X and Γ, the gap centeredat z = c/ M y symmetry (equivalent to M z ), which leaves in-variant the BZ plane containing the Γ, X, R , and Y points. For a discussion of such “in-plane” Wannier flowassociated with a nonzero MCN, see Ref. [40].Since M z is a type-2 mirror, we evaluate its uniqueMCN using Eq. (37). And since the Wannier spectrumis gapless, and hence devoid of flat bands, we set C A = 03 Γ X M Γ − − − − − − E n e r g y ( e V ) (a) Γ X M Γ − . − . − . . . . . z ( ˚ A ) (b) FIG. 3. (a) Energy bands of monolayer SnTe, with the s -type lower valence bands that are exluded from the Wannierizationshown in grey. All bands are doubly degenerate, and the Fermi level is indicated by the dashed line. (b) Wannier bandsobtained from the Bloch states in the six p -type upper valence bands. (c) Heatmap plot of the gap function of Eq. (52) forthe central pair of Wannier bands, where zero-gap points (nodal points) appear as dark spots. Those with winding numbers W j = ± W j = − x yz FIG. 4. (a) Rocksalt structure of bulk SnTe in a tetragonalconventional cell. a is the lattice constant of the conventionalcubic cell, and b = c = a/ √
2. Green planes are equivalentmirror planes. (b) Brillouin zone associated with the tetrag-onal cell, with its high-symmetry points indicated in red andthe unique M z -invariant plane in green. The projected 2DBrillouin zone with its high-symmetry points is shown on top. in that equation to obtain µ G = W A , (53)which says that the MCN equals the sum of the windingnumbers of all the point nodes on the z = 0 plane.As indicated in Fig. 5(d), there are 16 independentpoint nodes in total on that plane, all of them simplenodes where only two bands meet. Seven have windingnumbers +1 and the other nine have winding numbers −
1, yielding µ G = − k · p analysis of the band inversions. Using Eq. (38), weconfirm that the system is axion-trivial. C. Modified Dirac model on a cubic lattice
In this section we study a 3D toy model constructed byfirst modifying the free Dirac equation to enable topologi-cal phases for certain parameter values, and then placingit on a cubic lattice. The 4 × H ( k ) = m − M K ( k ) 0 c sin k z c (sin k x − i sin k y )0 m − M K ( k ) c (sin k x + i sin k y ) − c sin k z c sin k z c (sin k x − i sin k y ) − m + 2 M K ( k ) 0 c (sin k x + i sin k y ) − c sin k z − m + 2 M K ( k ) , (54)where K ( k ) = 3 − cos k x − cos k y − cos k z , and c , m , and M are dimensionless parameters inherited from the originalisotropic modified Dirac equation [41] by setting the restmass m c to be the energy scale of the model [42].The topological phase diagram of the half-filled modelis shown in Fig. 6 for c = 1 .
0. The system is gapped except on the m = 0 , M, M, M lines, where the gapcloses at Γ = (0 , , π, , π, π, π, π, π ), respectively. As shown in Appendix C,those metallic lines separate axion-trivial from axion-oddinsulating phases.The axion angle is quantized by several axion-odd sym-4 Γ Y M Γ X R A X − − − − E n e r g y ( e V ) (a) Γ X M Γ Y0 . . . . . . z / c (b) Γ X M Γ Y − − z / c (c) FIG. 5. (a) Energy bands of bulk SnTe along high-symmetry lines of the folded tetragonal BZ. The Fermi level is indicatedby the dashed line. (b) Wannier band structure obtained from the full set of valence states. (c) Detail of the Wannier bandsaround the z = 0 mirror plane. (d) Heatmap plot of the gap function of Eq. (52) for the central pair of Wannier bands around z = 0, with the nodal points color-coded as in Fig. 3(c). − . − . . . . m − . − . − . . . . . M θ = 0 θ = π FIG. 6. Topological phase diagram of the model of Eq. (54) for c = 1 .
0. Orange and blue regions denote axion-even ( θ = 0)and axion-odd ( θ = π ) phases, respectively. metries. Some are z -reversing (inversion and horizontalmirror M z ), and others are z -preserving (spinful time re-versal and vertical mirrrors). As M z is a type-1 mirror,it protects two MCNs that are related to the axion angleby Eq. (36).
1. Axion-odd phase with protected Wannier flow
For our numerical tests we set c = m = 1 . M =0 . z -preservingaxion-odd symmetries, the connectivity (or “flow”) of theWannier bands is topologically protected [13]. In partic-ular, spinful time reversal symmetry requires that the Γ M X Γ Z A − − E n e r g y (a) Γ M X Γ − X − . − . . . . z / c (b) FIG. 7. (a) Energy bands of the model described by Eq. (54)with c = m = 1 . M = 0 .
5. The bands are doublydegenerate, and the Fermi level (dashed line) has been placedat midgap. (b) Wannier band structure obtained from thevalence states. (c) and (d) Heatmap plots of the gap functionof Eq. (52) about the z = 0 and z = c/ two bands per vertical cell are glued together as follows:one band touches the band above at one of the four time-reversal invariant momenta (TRIM), and it touches theperiodic image below at the other three. As for the z -reversing axion-odd symmetries, the effect of M z is topin the up-touching to one of the mirror planes and thethree down-touchings to the other, while inversion fur-ther constrains the four touchings to occur at TRIM onthose planes, as already mandated by time reversal.The pattern of band touchings described above is con-firmed by Fig. 7(b), where we plot the Wannier bands.They were obtained by placing at the origin the four ba-sis orbitals that belong to the home unit cell, and makingthe diagonal approximation of Eq. (43) for the positionmatrix. There is one band touching at Γ on the B plane,and three more on the A plane: one at M, and the othersat the two X points.5 Γ M X Γ Z A − − E n e r g y (a) Γ M X Γ − X − . − . . . . z / c (b) FIG. 8. (a) Energy bands of the same model as in Fig. 7, afteradding an extra pair of occupied orbitals with E = − . z = ± . c and coupling them to the other orbitals. The bandsare doubly degenerate, and the Fermi level (dashed line) hasbeen placed at midgap. (b) Wannier band structure obtainedfrom the valence states, with small gaps around z = ± . c due to the added orbitals. Since the Wannier spectrum is gapless, the MCNs µ G and µ X are given respectively by the half-sum and thehalf-difference of the net winding numbers on the A andB planes [Eqs. (41) and (42)]. As indicated in the gap-function plots of Figs. 7(c,d), the three nodes at A give W A = − W B = −
1, sothat µ G = − µ X = 0. Note that µ G + µ X is an oddnumber, as required by Eq. (36) for an axion-odd system.
2. Axion-odd phase with fragile Wannier flow
If the z -preserving axion-odd symmetries of the model(time reversal and vertical mirrors) are weakly broken,the system will remain in an axion-odd phase protectedby M z and inversion. But since these are z -reversing op-erations, the Wannier spectrum is no longer topologicallyrequired to be gapless. The Wannier flow is only pro-tected in a “fragile” sense, and it can be destroyed, whilepreserving M z , by adding some weakly-coupled trivialbands to the valence manifold [13, 24]. Below we carryout this procedure in two different ways, and confirm thatthe MCNs remain the same as in the original model. a. Insertion of a symmetric pair of occupied orbitals Here we implement the strategy outlined in Sec. IV B.We insert in the unit cell two more orbitals, denoted as | (cid:105) and | (cid:105) , that have opposite spins and the same on-siteenergy E = − .
0. To break time reversal and the verti-cal mirrors while preserving M z and inversion, we placethe spin-up orbital | (cid:105) at ( x, y, z ) = (0 . , . , . c ), andthe spin-down orbital | (cid:105) at ( x, y, z ) = (0 . , . , − . c ),keeping the original orbitals | (cid:105) to | (cid:105) at the origin. Fi-nally, we couple the new orbitals to the old via the matrixelements (cid:104) | H | (cid:105) = (cid:104) | H | (cid:105) = 0 .
5. The resulting modelretains the M z and inversion symmetries of the originalmodel, and it breaks the time-reversal and vertical mir-ror symmetries in the Z matrix of Eq. (44) (but not inthe Hamiltonian).The energy and Wannier band structures are plottedin Figs. 8(a,b). Because the Hamiltonian has both inver- Γ (0 , , M (0 . , . , X (0 , . , Γ (0 , , Z (0 , , . A (0 . , . , . − − E n e r g y (a) − . . . .
5Γ M X Γ − X − . . . . . . . . . . . . . . . z / c (b) FIG. 9. (a) Energy bands of the same model as in Fig. 7, afteradding an extra occupied orbital at z = 0 and coupling it tothe other orbitals. The Fermi level (dashed line) has beenplaced in the gap. (b) Wannier band structure obtained fromthe valence states. The added orbital generates a flat bandat z = 0, which repels the nodal points on that plane (lowerpanel). sion and time-reveral symmetry, the energy bands remaindoubly degenerate as in Fig. 7(a). The breaking of the z -preserving symmetries in the Z matrix is reflected inthe Wannier spectrum which is no longer connected asin Fig. 7(b), with small gaps opening up near z = ± . c .The node at Γ on the B plane and those at X , X , andM on the A plane remain intact, protected by M z andinversion. Their winding numbers are also unchanged,leading to the same MCNs as in the original model. b. Insertion of a single occupied orbital at z = 0 . An alternative way of opening up a gap in the Wan-nier spectrum is to insert a flat band on a mirror plane.To illustrate this procedure, we add at the origin a singlespin-up orbital | (cid:105) with on-site energy E = − . (cid:104) | H | (cid:105) = (cid:104) | H | (cid:105) = 2 .
0. Because the orbital is spin-polarized, it breaks time reversal; and because the spinpoints in the vertical direction, it also breaks all verticalmirrors while preserving M z . In addition, the couplingterms break inversion symmetry, leaving M z as the onlyaxion-odd symmetry. The energy bands of the modifiedmodel are shown in Fig. 9(a). A new band has appearedbelow the other four, so that there are now three valencebands in total, leading to three Wannier bands.The added orbital, which belongs to the A + class inTable I, generates an extra even-parity state at both Gand X. This creates an imbalance ∆ N G = ∆ N X = 1between even- and odd-parity states on the two mirror-invariant BZ planes, which according to Eq. (20) resultsin a flat band at A. We emphasize that this extra bandremains flat even after the added orbital is coupled to themodel, as long as the coupling terms respect M z symme-try. As already mentioned, those terms are chosen tobreak inversion symmetry. This is needed to ensure thatthe three point nodes on the A plane are repelled by theflat band in the manner described in Sec. III B 2, sinceinversion symmetry would otherwise protect them.The resulting Wannier bands are displayed in the up-per panel of Fig. 9(b); because of the lowered symme-try, the node at z = c/ z = 0, well separated from a pair of dispersive bandswhose three touchings on the z = 0 plane in Fig. 7(c) havebeen gapped out. Under these circumstances, Eqs. (32)and (33) for the MCNs reduce to µ G = ( p A C A + W B ) (55)and µ X = ( p A C A − W B ) . (56)The single node at B has the same winding number W B = − W A = − p A C A of the flat band ( p A = − C A = +1). Overall, the MCNs remain unchanged. VII. SUMMARY
In summary, we have investigated the topological prop-erties of mirror-symmetric insulating crystals from theviewpoint of HW functions localized along the directionorthogonal to the mirror plane. We first clarified thegeneric behaviors of the associated Wannier bands, andthen derived a set of rules for deducing the MCNs. To val-idate and illustrate the formalism, we applied it to SnTein the monolayer and bulk forms, and to a toy model ofan axion-odd insulator.In the HW representation, the MCNs are expressed interms of a set of integer-valued properties of the Wannierbands on the mirror planes: the Chern numbers and mir-ror parities of flat bands lying on those planes, and thewinding numbers of the touching points on those planesbetween symmetric pairs of dispersive bands. One advan-tage of this representation is that it reveals the relationbetween the MCNs and the axion Z index from purelybulk considerations. That relation is far from obvious inthe standard Bloch representation, and previously it hadonly been obtained via an indirect argument involvingsurface states.In some cases the axion Z index can be determinedby visual inspection of the Wannier band structure, e.g.,by counting the number of nodal points between certainbands [13]. We have found that mere visual inspectiondoes not suffice for obtaining the MCNs since it doesnot reveal, for example, the relative signs of the windingnumbers of different nodes.Interestingly, in certain cases where flat Wannier bandsare present the magnitudes of the MCN can be deter-mined without having to divide the occupied manifoldinto two mirror sectors. This follows from the uniform-parity assumption for the flat bands, which has no coun-terpart in the Bloch representation. Since the determina-tion of the mirror parities is the most cumbersome stepin the calculation of MCNs, this feature of the HW for-malism could lead to a more automated algorithm for computing MCNs. Even without such further develop-ments, the formalism has already proven useful for dis-cussing the topological classification of mirror-symmetricinsulators. ACKNOWLEDGMENTS
Work by T.R. was supported by the DeutscheForschungsgemeinschaft Grant No. Ra 3025/1-1 fromthe Deutsche Forschungsgemeinschaft. Work by D.V.was supported by National Science Foundation GrantDMR-1954856. Work by I.S. was supported by GrantNo. FIS2016-77188-P from the Spanish Ministerio deEconom´ıa y Competitividad.
Appendix A: Derivation of Eqs. (20-22)
According to Table I, the numbers of occupied stateswith each mirror parity at G and X are N G ± = N A ± + N B ± + 12 (cid:101) N , (A1a) N X ± = N A ± + N B ∓ + 12 (cid:101) N , (A1b)where (cid:101) N = (cid:101) N A + (cid:101) N B + (cid:101) N UC is the total number of disper-sive Wannier bands per cell. Letting ∆ N G = N G + − N G − and ∆ N A = N A + − N A − , and defining ∆ N X and ∆ N B in the same way, we find∆ N A = 12 (∆ N G + ∆ N X ) , (A2a)∆ N B = 12 (∆ N G − ∆ N X ) . (A2b)Under the uniform parity assumption | ∆ N A | = N A and | ∆ N B | = N B , resulting in Eqs. (20) and (21). In thecase of a type-2 mirror A and B are equivalent, and fromEq. (A1a) ∆ N A +∆ N B = ∆ N G . Hence ∆ N A = ∆ N B =∆ N G /
2, yielding Eq. (22) under the same assumption.
Appendix B: Derivation of Eq. (27)
Let us prove Eq. (27) for the case of a single pair ofdispersive Wannier bands connected by point nodes onthe A plane. In this case the matrix f κ of Eq. (48) reducesto the scalar f κ ≡ (cid:104) (cid:101) h + κ | z | (cid:101) h − κ (cid:105) = | f κ | e iγ κ , (B1)where | (cid:101) h ± κ (cid:105) are states of even or odd mirror parity con-structed from the pair of HW functions as describedin Sec. V C 1. These states are cell-periodic in planeand localized along z , and we also define new states | ψ ± κ (cid:105) = e i κ · r | (cid:101) h ± κ (cid:105) that are Wannier-like along z andBloch-like in plane.7When the Chern numbers (cid:101) C A ± are nonzero, it becomesimpossible to choose a gauge for the states | ψ ± κ (cid:105) that isboth smooth and periodic in the projected 2D BZ [5].We assume a square BZ with k x , k y ∈ [0 , π ], and choosea smooth but nonperiodic gauge for the | ψ − κ (cid:105) states. Tocharacterize the lack of periodicity, let the phase relationsbetween the edges of the BZ be | ψ − R (cid:105) = e − iµ | ψ − L (cid:105) , | ψ − T (cid:105) = e − iν | ψ − B (cid:105) , (B2)where { L,R,T,B } = { left,right,top,bottom } , µ = µ ( k y ),and ν = ν ( k x ). Also let∆ µ = µ (2 π ) − µ (0) , ∆ ν = ν (2 π ) − ν (0) . (B3)When computing the Berry phase around the BZ bound-ary as an integral of the connection A − κ = i (cid:104) (cid:101) h − κ | ∂ κ (cid:101) h − κ (cid:105) , φ − = (cid:73) ∂ BZ A − κ · d κ , (B4)the contribution from the L and R segments cancel exceptfor terms coming from µ , and similarly for the top andbottom segments. It follows that φ − = ∆ µ − ∆ ν . (B5)We assume a smooth but nonperiodic gauge for the | ψ + κ (cid:105) states as well, so that the phase γ κ in Eq. (B1)becomes a smooth function of κ (except at the nodes,where f κ vanishes and γ κ becomes ill defined). Now wephase-align | ψ + κ (cid:105) with | ψ − κ (cid:105) by re-gauging as follows, | ψ + κ (cid:105) (cid:48) = e iγ κ | ψ + κ (cid:105) . (B6)(In this new gauge f (cid:48) κ is real, and γ (cid:48) κ is zero everywhere.)This will make a gauge for | ψ + κ (cid:105) (cid:48) that is also nonperi-odic. For the moment we only assume that this gauge issmooth in a neighborhood extending some small distanceinside the boundary; we ignore what is going on deeperinside. It is not hard to see that the same relations as inEq. (B2), with the same functions µ and ν , apply to the | ψ + κ (cid:105) (cid:48) states, and it follows that φ (cid:48) + = φ − (call it φ ) . (B7)Now, in the case of the | ψ − κ (cid:105) states the interior wassmooth, so by applying Stokes’ theorem to2 π (cid:101) C A − = (cid:90) BZ Ω − κ d k (B8)where Ω − κ = ∂ k x A − κ ,y − ∂ k y A − κ ,x is the Berry curvature ofstate | u − κ (cid:105) , we get 2 π (cid:101) C A − = φ . (B9)If the interior of | ψ + κ (cid:105) (cid:48) were also smooth, we would con-clude that (cid:101) C A + = (cid:101) C A − . Conversely, when the MCN isnonzero there must exist nonanalytic points where thephase of | u + κ (cid:105) (cid:48) changes discontinuously. Those points are Γ M X Γ − . − . . . . z / c M = 0 . (a) Γ M X Γ − . − . . . . z / c M = 0 . (b) Γ M X Γ − . − . . . . z / c M = 0 . (c) Γ M X Γ − . − . . . . z / c M = 0 . (d) FIG. 10. Wannier bands of the modified Dirac model on acubic lattice [Eq. (54)], for m = 1 . M . precisely the nodes of f κ , which we label by j ; they actas vortex singularities of the Berry connection (cid:0) A + κ (cid:1) (cid:48) = A + κ − ∂ κ γ κ , (B10)and we extract their winding numbers W j [Eq. (50)], typ-ically taking values ±
1, according to how the phase γ κ changes going around each node. Let S be the interior ofthe projected BZ with a small circle c j cut around eachnode, and apply Stokes’ theorem over the region S tofind (cid:90) S Ω + κ d k = (cid:90) ∂ BZ (cid:0) A + κ (cid:1) (cid:48) · d κ − (cid:88) j (cid:73) c j (cid:0) A + κ (cid:1) (cid:48) · d κ . (B11)The first term on the right-hand side is equal to φ (cid:48) + = φ = 2 π (cid:101) C A − . In the limit of small circles the left-hand sidebecomes 2 π (cid:101) C A + , and the second term on the right-handside reduces to 2 π (cid:80) j W j (this follows from Eq. (B10) bynoting that A + κ is smooth everywhere). Thus (cid:101) C A + − (cid:101) C A − equals W A = (cid:80) j ∈ A W j , which is what we set out toprove.The same result holds if more than one pair of bandsmeet at some of the point nodes. Their winding numberare still given by Eq. (50), but γ κ is now given by themore general expression in Eq. (49) instead of Eq. (B1). Appendix C: Phase diagram of the modified Diracmodel on a cubic lattice
In this Appendix, we map out the topological phasediagram of the model of Eq. (54) as a function of theparameters m and M , for c = 1 .
0. The band gap closesfor m = 0 , M, M, M at the points Γ, X, M, and A,respectively [43]. Those lines in the phase diagram mark8the topological phase transitions between axion-even andaxion-odd phases.To decide which phases are trivial and which are topo-logical, it is sufficient to inspect the Wannier band struc-tures in Fig. 10, obtained for representative states in eachof the four phases along the m = 1 . Z index. In the follow-ing, we choose to focus on time-reversal symmetry. The Wannier spectrum of an axion-odd phase withspinful time-reversal symmetry must be gapless, witheach band touching the band above at one of the fourTRIM and the band below at the other three (or vice-versa). From this criterion we conclude that Figs. 10(a,c)correspond to axion-trivial phases, and Figs. 10(b,d) toaxion-odd topological phases. Hence the system is topo-logical for 0 < m/M < < m/M <