Geometry and Topology Tango in Ordered and Amorphous Chiral Matter
GGeometry and Topology Tango in Chiral Materials
Marcelo Guzm´an, Denis Bartolo, David Carpentier
Univ. Lyon, ENS de Lyon, Univ. Claude Bernard,CNRS, Laboratoire de Physique, F-69342, Lyon. (Dated: February 10, 2020)Among all symmetries constraining topological band spectra, chiral symmetry is very distinguish-able. It is naturally realized as a sublattice symmetry in a host of physical systems as diverse aselectrons in solids, photons in metamaterials and phonons in mechanical networks, without rely-ing on sophisticated material designs or crystalographic properties. In this article, we introduce ageneric theoretical framework to generalize and elucidate the concept of topologically protected zeromodes in chiral matter. We first demonstrate that, in the bulk, the algebraic number of zero-energystates of chiral Hamiltonians is solely determined by the real-space topology of their underlyingframe: the chiral charge. In insulators this charge vanishes, however the boundary can support zeromodes which emerge in materials bearing a finite chiral polarization. We establish that this chiralpolarization naturally distinguishes chiral gapped phases from one another by measuring the spatialimbalance of the bulk excitations on each sub-lattice. Crucially, the chiral polarization is not setby the sole topology of Bloch Hamiltonians, but reflects its intimate relation with the underlyingframe geometry. We close our article explaining how the essential interplay between interfacialchiral charge and bulk chiral polarization resolves long-standing ambiguities in the definition oftopologically-protected edge states, and redefines the very concept of bulk-boundary correspondencein chiral matter.
I. INTRODUCTION
A century after the foundations of band theory of solidsby F´elix Bloch [1], our understanding of the topologicalstructure of Bloch theory has lead to the discovery ofnew electronic states of matter ranging from topologi-cal insulators to topological superconductors [2–7]. Thisrevolution built on two cornerstones: the abstract clas-sification of the topology of Bloch eigenspace based onHamiltonian symetries [8–15], and the practical corre-spondence between bulk topology and the existence ofboundary states [2–6, 16]. During the past decade, thesetwo generic principles spread frantically across fields asdiverse as photonics, acoustics, or mechanics, leading todesign principles and practical realization of maximallyrobust waveguides [17–19].Among the number of symmetries constraining wavetopology, chiral symmetry has a special status. Out ofthe three fundamental symmetries of the overarching ten-fold classification, it is the only one naturally realizedwith all quantum and classical waves. It generically takesthe form of a sub-lattice symmetry when waves propa-gate in frames composed of two connected lattices A and B , with couplings only between A and B sites, see e.g.Fig. 1a. In electronic systems, the archetypal example ofa chiral insulator is provided by the Su-Schrieffer-Hegger(SSH) model of polyacetylene [20]. In mechanics, theHamiltonian description of bead-and-spring networks isintrinsically chiral [21–23]: the A sites correspond to thebeads, and the B sites to the springs. In topological pho-tonics and cold atoms chiral wave guides are among thesimplest realizations of topological phases.In this article, we establish that the zero-mode con-tent of chiral insulators stems from an intimate tangobetween Bloch-Hamiltonian topology and the real-space frame structure. We show how this interplay rectifies theparadigmatic bulk-boundary correspondence built on asole spectral picture.In bulk, we show that the chiral charge, which countsthe imbalance between the number of sites on the sublat-tices A and B , predicts the number of zero-energy modesof all Hamiltonians defined on a given chiral frame. Todistinguish chiral insulators, where the chiral charge van-ishes, we use their chiral polarization defined as the spa-tial imbalance of the bulk waves on the two sublattices.This essential quantity although akin to the time-reversalpolarization of Z insulators [24] is not merely set bythe Bloch Hamiltonian topology but also by the under-lying frame geometry. At boundaries, we show how thechiral polarization prescribes the surface chiral charge,and therefore the full edge content of chiral matter. Theintimate relation between the surface chiral charge andthe bulk polarization redefines the very concept of bulk-boundary correspondence in chiral materials from con-densed matter, to mechanics and photonics. II. FROM CHIRAL CHARGE TO CHIRALPOLARIZATION AND ZAK PHASES
Introducing the concepts of chiral charge and polar-ization, we demonstrate that bulk properties of chiralmatter are determined by an intimate interplay betweenthe frame topology, the frame geometry and the chiralZak phases of Bloch Hamiltonians.
A. Chiral charge and chiral polarization
We consider the propagation of waves in chiral ma-terial associated to d -dimensional frames including two a r X i v : . [ c ond - m a t . s t r- e l ] F e b sublattices A and B . The wave dynamics is defined bya Hamiltonian H . By definition, the chiral symmetrytranslates in the anti-commutation of H with the chiralunitary operator C = P A − P B , where P A and P B are thetwo orthogonal projectors on the sub-frames A and B .Simply put, in the chiral basis where C is diagonal, H isblock off-diagonal. a.b. FIG. 1.
Lattices with a finite chiral charge . a. The Lieb(left) and dice (right) frames are both characterized by animbalance between the number N A and N B of sites. In bothcases the chiral charge per unit cell equals 1. Any Hamilto-nian defined on these frames possesses a flat energy band. b. Illustration of two band spectra associated to chiral Hamilto-nians defined on the Lieb (left) and dice (right) frames. Thetwo band spectra are computed for tight-binding Hamiltoni-ans with nearest neighbour coupling and a hopping parameterset to 1, see e.g. [25].
In order to determine the relative weight of the wavefunctions of H on the two sub-lattices, we introduce thechiral charge M = (cid:104) C (cid:105) , (1)where the average is taken over the complete Hilbertspace. Using the basis of fully localized states, we read-ily find that M is fully prescribed by the frame topology:the chiral charge counts the imbalance between the num-ber of A and B sites: M = N A − N B . We can howeveralso evaluate Eq. (1) in the eigenbasis of H . Indexingby n the energy bands of H , the eigenstates of the chiralHamiltonian come by pairs of opposite energies relatedby |− n (cid:105) = C | n (cid:105) . Chirality therefore implies that the chi-ral charge is solely determined by the zero modes of H as M = (cid:80) n (cid:104) n | C | n (cid:105) = (cid:104) | C | (cid:105) . Noting that the | (cid:105) statesare eigenstates of the chiral operator with eigenvalue +1 when localized on the A sites and − M also is an algebraic countthe zero modes of H : M = N A − N B = ν A − ν B . (2)This equality is the classical result established byMaxwell and Calladine in the context of structural me-chanics [26, 27] and independently discussed by Suther-land in the context of electron’s localization [28]. It im-plies that the spectral properties of H are constrainedby the frame topology. In particular, frames with a non-vanishing chiral charge impose all chiral Hamiltonian topossess flat bands. This simple prediction is illustratedin Fig. 1 where we show the Lieb and the dice lattices,which are both characterized by a chiral charge per unitcell equal to one. All Hamiltonians defined on these clas-sical lattices are bound to support at least one flat band,Fig. 1b. No chiral insulators exist on the Lieb and dicelattices.By contrast, in chiral insulators, no zero-energy bulkmodes exist and M must vanish. To probe the relativeweight of the wave functions on the two sub-lattices, wetherefore introduce the chiral polarization vector Π j = (cid:104) C x j (cid:105) . As, in chiral systems, the |± n (cid:105) states contributeequally to Π, we henceforth use the definitionΠ j = (cid:104) C x j (cid:105) E< , (3)with j = 1 , . . . , d and where E < j corresponds to the algebraic distance betweenthe charge centers associated to the A and B atoms, seeFig. 2 where we plot the Wannier functions associatedto the two sub-lattices of a SSH model. In mechanicalnetworks, Π j is the vector connecting the stress-weightedand displacement-weighted positions. A vanishing polar-ization indicates that the average locations of the stressand displacement coincide. Conversely, a finite chiral po-larization reveals an asymmetric mechanical response dis-cussed in [29, 30]. B. Chiral polarization: an interplay between Zakphases and frame geometry
In electronic insulators there exists a direct relationbetween the electric polarization and the so-called Zakphase picked up by the Bloch states as they are trans-ported across the Brillouin zone [31–34]. More generallyholonomy proved to be an effective tool to character-ize the topology imposed by lattice symmetries [10–15].Building on this body of work, we relate the chiral po-larization of a material to the two Zak phases of wavesprojected on sub-lattices A and B .To do so, we first choose a unit cell and consider the ba-sis of Bloch states | k , α (cid:105) = (cid:80) R e i k · R | R + r α (cid:105) , where R is a Bravais lattice vector, α labels the atoms in the unitcell and k is the momentum in the Brillouin Zone (BZ).We henceforth use a convention where the Bloch Hamil-tonian H ( k ) is periodic in the BZ, see [32, 35] and Ap-pendix A 1. More quantitatively, considering first Hamil-tonians with no band crossing [36], we define the A sub-lattice Zak phase of the n th energy band along direction j as γ Aj ( n ) = i (cid:90) C j d k (cid:104) u n | P A ∂ k P A | u n (cid:105) , (4)where the | u n ( k ) (cid:105) are the eigenstates of H k , and C j thenon-contractible loops over the Brillouin zone definedalong the d crystallographic axes. γ Bj ( n ) is defined analo-gously on the B sublattice. The (intercellular) Zak phaseis given by the sum of γ Aj ( n ) and γ Bj ( n ) [37]. In Ap-pendix B, we show how to decompose the chiral polar-ization into a spectral and a frame contributions:Π j = a π ( γ Bj − γ Aj ) + 12 p j , (5)where a is the lattice spacing (assumed identical inall directions), γ Aj and γ Bj are the sublattice Zak phasesdefined by γ Aj = (cid:88) n< γ Aj ( n ) . (6)In Eq. (5) the p j are the components of the geometrical-polarization vector connecting the centers of mass of the A and B sites in the unit-cell: p = (cid:88) α ∈ A r α − (cid:88) α ∈ B r α . (7)In crystals, Eqs. 5 quantifies the difference between thepolarity of the ground-state wave function Π and thegeometric polarization of the frame p . This difference isfinite only when the two sublattice Zak phases differ. III. TOPOLOGY OF CHIRAL INSULATORS
We now elucidate the intimate relation between thechiral polarization and the band topology of chiralgapped phases. We outline the demonstrations of ourcentral results below and detail them in Appendix C.
A. Sublattice Zak phases and winding numbers
Computing the Wilson loop of the non-Abelian con-nection A n,m ( k ) = (cid:104) u n ( k ) | ∂ k | u m ( k ) (cid:105) along C j , we showthat chirality relates the d Zak phases γ Aj + γ Bj to thewindings of the Bloch Hamiltonian as γ Aj + γ Bj = πw j + 2 π Z , (8)where w j = i/ (4 π ) (cid:82) C j d k · Tr (cid:2) ∂ k H C H − (cid:3) ∈ Z . The to-tal Zak phase is quantized but the arbitrary choice of W ann i e r a m p li t ude a.b. FIG. 2.
Chiral polarization and Wannier functions. a.
Square of the Wannier amplitude projected into the A (red)and B (blue) sublattices for the ground state configurationof the two-band SSH model as defined in [38], with hoppingratio t /t = 0 .
79. The chiral polarization Π = (cid:10) x A (cid:11) − (cid:10) x B (cid:11) isnegative: the chain is left polarized regardless of the choice ofunit cell. b. The Bloch Hamiltonian winding number encodesthe chiral polarization relative to a particular unit cell. Itdepends on the choice of unit cell and can take any integervalue for a given ground state. The magnitude of the windingnumber varies linearly with the distance between the A andB sites defining the unit cell. the origin of space implies that both γ A and γ B are onlydefined up to an integer. As a matter of fact, a mere U (1) gauge transformation | u n (cid:105) → e iα n ( k ) | u n (cid:105) arbitrarilymodifies γ Aj ( n ) and γ Bj ( n ) by the same quantized value: γ Aj ( n ) → γ Aj ( n )+ πm , γ Bj ( n ) → γ Bj ( n )+ πm , with m ∈ Z .By contrast, the difference between the two sublatticeZak phases is left unchanged by the same gauge trans-formation which echoes its independence on the spaceorigin. Evaluating the winding of H ( k ) using the Blocheigenstates in Appendix C 1, we readily establish the es-sential relation γ Bj − γ Aj = πw j ∈ π Z . (9)Chirality quantizes the sublattice Zak phases of chiralinsulators, even in the absence of inversion or any otherspecific crystal symmetry. γ Aj and γ Bj are however notindependent. Combining Eqs. (8) and (9) we can alwaysdefine the origin of space so that γ Aj = 0 and γ Bj = πw j .The d winding numbers of Eq. (9) characterize thetopology of H ( k ). In particular, if for a given Wigner-Seitz cell the corresponding H ( k ) is associated to a fi-nite winding ( w j (cid:54) = 0), then it cannot be smoothly de-formed into the atomic limit defined over the same unitcell. The set of winding numbers is however poorly in-formative about the spatial distribution of the charges in a. b. FIG. 3.
Inferring the band topology from frame geometry . a. The two-sites Wigner-Seitz cell on a 1D chiral framehave different geometrical polarizations; their difference is given by one Bravais vector. Consequently, we can always define theunit cell so that the Bloch Hamiltonian has a finite winding. b. All the Wigner-Seitz unit cells on the checkerboard latticeshare the same (vanishing) chiral polarization. Therefore a single winding number w characterizes the Hamiltonians on thisframe in virtue of Eq. (11). Evaluating the winding using the Wigner-Seitz cell compatible with the atomic limit of H yields w = 0, by definition. electronic systems, or about the stress and displacementdistributions in mechanical structures. The values of w j are defined only up to the arbitrary choice of unit cellrequired to construct the Bloch theory. A well knownexample of this limitation is given by the SSH model,where the winding of H k can either take the values 0or ± A or B sublattice, see Fig. 2a and Ap-pendix D. We show in the next section, how the chiralpolarization alleviates this limitation. B. Disentangling Hamiltonian topology from framegeometry
Equations (5) and (9) provide a clear geometrical in-terpretation of the winding number w j as the quantizeddifference between the geometrical and the chiral polar-ization: Π j = 12 ( a j w j + p j ) . (10)We can now use this relation to illuminate the very defi-nition of a chiral topological insulator. The chiral po-larization Π j = (cid:104) C x j (cid:105) E< is a physical quantity thatdoes not depend on the specifics of the Bloch representa-tion. Therefore computing Π j for two unit cells (1) and(2), we find that the windings of the two correspondingBloch Hamiltonians H (1) ( k ) and H (2) ( k ) are related viaEq. (10) as w (2) j − w (1) j = 1 a j (cid:16) p (1) j − p (2) j (cid:17) . (11)This essential relation implies that one can always con-struct a Bloch representation of H where H ( k ) is topolog- ically trivial, at the expense of a suitable choice of a unitcell geometry. As a matter of fact, a suitable redefinitionof the unit cell can increase, or reduce the geometricalpolarization, and therefore the winding numbers, by anarbitrary large multiple of a j as illustrated in Fig. 2b.For instance in the case of Hamiltonians with near-est neighbor couplings ( | w j | ≤ H , as different geometrical polar-izations in the Wigner-Seitz cells. This number providesa direct count of the chiral ’atomic limits’ of H .Defining the topology of a chiral material thereforerequires characterizing both the winding of its BlochHamiltonian, and the frame geometry. Remarkably, thisinterplay provides an insight on topological band prop-erties from the sole inspection of the frame structure. C. Inferring band topology from frame geometry
There exists no trivial chiral phase in one dimension:one can always choose a Wigner-Seitz cell such that theBloch representation of H has a non-vanishing winding.As a matter of fact, the geometrical polarization of theWigner-Seitz cells can only take two finite values of op-posite sign depending on whether the leftmost site in aunit cell is of the A or B type, see Fig. 3a. Equation (11)therefore implies that, in 1 D , there always exists, at least,two topologically distinct gapped phases smoothly con-nected to two atomic limits. The two gapped phases arecharacterized by two distinct pairs of winding numbersdefined by two inequivalent choices of unit cells. In otherwords all SSH Hamiltonians are topological.Similarly, in d > a. b.d.c. FIG. 4.
Bulk-boundary correspondance a . A chiral crystal defined on a honeycomb frame is terminated by an irregularboundary (shaded region). The dashed rectangles indicate the Wigner-Seitz cells compatible with the atomic limit of theHamiltonian. b . Same physical system. The periodic bulk is represented using a different unit cell, therebry implying aredefinition of the crystal boundary (Shaded region) c . Two connected SSH chains. The Wigner-Seitz cell in the two materialsare compatible with their atomic limits. The interface B separating the two materials is one-site wide. d . Redefiningthe Wigner-Seitz cell on the right hand side of the interface requires wideding the boundary region and makes the unit cellincompatible with the atomic limit. The winding of the Bloch Hamiltonian in I R takes a finite value. polarization invariant upon redefinition of the Wigner-Seitz cell can support topologically trivial Hamiltonians.Equation (11) indeed implies that a topologically trivialHamiltonian H constrains the frame geometry to obey p (1) j = p (2) j for all pairs of unit cells and in all directions j . We show a concrete example of such a frame in Fig. 3b.In the next section, we discuss the crucial role of theframe topology and geometry on the bulk-boundary cor-respondance of chiral phases. IV. BULK-BOUNDARY CORRESPONDANCE.A. Topological chiral charge of surfaces andinterfaces
We now establish a bulk-boundary correspondance re-lating the chiral polarization to the number of zero modessupported by the free surface of a chiral insulator. Morespecifically, we consider a crystalline insulator I termi-nated by an arbitrarily complex chiral surface B as il-lustrated in Fig. 4a. We start with a simple situation bysmoothly deforming the Hamiltonian of I into an atomiclimit. In the proper Wigner-Seitz cell, the chiral andgeometrical polarizations are equal ( Π = p ) as the winding vector vanishes ( w = 0). The Wigner-Seitz cellthen prescribes the spatial extent of the free surface: B is the region of space that cannot be tiled by the unit cellcompatible with the atomic limit, Fig. 4a. Noting that I is a gapped phase, all the zero modes, if any, must belocalized in B . Their algebraic count is then given bythe chiral charge of the surface region V = M B .As the atomic limit was taken without closing the gap,this count holds for the initial Hamiltonian as well. Letus show how this seemingly geometrical count relates tothe topology of the Bloch Hamiltonians. To do so, wechange the definition of the unit cell in the bulk, whichrequires a redefinition of the boundary region: B → B ,see Fig. 4b. The chiral charge of this new interface isthen given by the change in the geometrical polarizationof the unit cell: M B − M B = a ( p − p ) · ˆ n , whereˆ n is the outward normal to ∂ I . Importantly, the in-variance of the chiral polarization formalized by Eq. (11)relates the surface chiral charge to the windings of thebulk Hamiltonian as w I = (1 /a )( p − p ). These tworelations demonstrate how the frame topology, the framegeometry and the band topology of Bloch Hamiltoniansdefine a bulk-boundary correspondence generic to all chi-ral insulators: V = M B + w I · ˆ n . (12)This bulk boundary correspondence indicates how thenaive Maxwell count of the zero-energy modes providedby the surface chiral charge M B is corrected by the wind-ing of the Bloch Hamiltonian in the bulk. In addition,it illuminates the geometrical implication of a nonzerowinding. A finite w I indeed echoes the impossibility totile defect-free frames with unit cells compatible with theHamiltonian atomic limit.Two comments are in order. Firstly, we stress thatEq. (12) is readily generalized to interfaces separating twochiral insulators I L and I R : V = M B + w I L · ˆ n L + w I R · ˆ n R ,see e.g. Fig. 4c. Secondly, the formula given by Eq. (12)coincides with the Kane-Lubensky index introduced intheir seminal work to count the zero-energy modes local-ized within isostatic mechanical networks [23] We haveshown here that this index defines a bulk-boundary cor-respondence generic to all chiral insulators. B. Resolving the SSH ambiguity
Given the historical and pedagogical importance of theSSH model illustrated in Fig. 4c, it is worth discussingthe consequences of Eq. (12) for this one-dimensionalsystem. Within the ten-fold classification of topologi-cal gapped phases, 1D chiral insulators are character-ized by the winding of their Bloch Hamiltonian. Follow-ing the standard bulk-boundary-correspondence princi-ple one would then expect the number of topologicallyprotected edge states to be given by the difference ofthe winding numbers across an interface. Eq. (12) re-veals that the number of zero-energy edge states is notsolely determined by the Bloch Hamiltonian topology. V can either be fully determined by the frame, or the BlochHamiltonian topology depending on the specific choice ofthe unit cell underlying the Bloch representation. Thisresult therefore settles a long-standing ambiguity in thetopological characterization of the simplest possible ex-ample of a topological insulator [37, 39–44]. Neither thetopology of the SSH Bloch Hamiltonian, nor its bound-ary correspondence rely on inversion symmetry, both arehowever defined only relative to a unit-cell prescription.This results suggests questioning the practical relevanceof the paradigmatic bulk-boundary correspondence be-yond the specifics of chiral symmetry. V. CONCLUSION
We have established that the frame topology and theframe geometry conspire with Bloch Hamiltonian topol-ogy to determine the bulk and surface properties of chiralmatter. In the bulk, the frame topology fully determinesthe algebraic number of zero-energy modes counted bythe chiral charge M . Chiral insulators, however, are dis-tinguished one another via their chiral polarization Π set both by the frame geometry and Bloch-Hamiltoniantopology. At their surface, the number of zero-energy states is prescribed by the interplay between the BlochHamiltonian topology and the frame geometry in thebulk on one hand, and by the frame topology of theboundary on the other hand. This subtle tango goes be-yond the bulk-boundary-correspondence principles solelybased on Hamiltonian topology. We stress that chiralsymmetry, expressed as a sublattice symmetry, translatesreal-space into spectral properties without relying on anycrystalline symmetry, and therefore complement the clas-sification of topological quantum chemistry [45–48].We expect our framework to extend beyond Hamilto-nian dynamics when dissipative processes obey the chiralsymmetry [49]. We therefore conjecture that real-spacetopology, geometry and non-Hermitian operator topol-ogy should cooperate in chiral dissipative materials asdiverse as cold atoms to photonics and mechanics. ACKNOWLEDGMENTS
We acknowledge support from WTF ANR grant, andToRe IdexLyon breakthrough program. We also thankA. Bernevig, K. Gawedzki, A. Grushin, Y. Hatsugai, A.Po, A. Schnyder and A. Vishwanath for insightful discus-sions.
Appendix A: Bloch theory and Wannier functions1. Conventions for the Bloch decomposition
For the sake of clarity, we first introduce the mainquantities used throughout all the manuscript to de-scribe waves in periodic lattices. We note | Ψ n, k (cid:105) theBloch eigenstates. They correspond to wavefunctions (cid:104) x | Ψ n, k (cid:105) = ϕ n, k ( x ) e i k · x , where k is the momentum inthe Brillouin Zone (BZ), and where the normalized func-tion ϕ n, k has a periodicity of one unit cell [32]. In thisarticle, we use the following convention to express theBloch states as a superposition of plane waves: | Ψ n, k (cid:105) = (cid:88) α u n,α ( k ) | k , α (cid:105) , (A1)where α labels the different atoms in the crystal, and | k , α (cid:105) represents the Fourier transform of the real-spaceposition basis: | k , α (cid:105) = (cid:80) R exp( i k · R ) | R + r α (cid:105) , R be-ing a Bravais lattice vector and r α a site position withinthe unit cell. We stress that here the components u n,α ( k )are periodic functions of k over the BZ. It is worth noting,however, that there exists multiple conventions to decom-pose the Bloch states as discussed e.g in the context ofgraphene-like systems in [50–52]. A common alternativeuses nonperiodic components over the BZ which carryan additional phase encoding the position of each atomwithin the unit cell: | Ψ n, k (cid:105) = (cid:80) α ˜ u n, k ,α e i k · r α | k , α (cid:105) . Wewill comment on the translation of our results from oneconvention to the other in the next Appendix section.
2. Wannier functions
By definition the Wannier function associated to aBloch eigenstate is given by the inverse Fourier trans-form (up to a phase): | W n, R (cid:105) = (cid:90) k e − i k · R | Ψ n, k (cid:105) , (A2)where (cid:82) k · ≡ ( a/ π ) d (cid:82) BZ d d k · . Note that for sake ofclarity, we here and henceforth assume that the spec-trum does not include band crossings. The technicalgeneralization of our results to degenerated spectra isstraightforward but involves some rather heavy algebra,see e.g. [32]. Appendix B: Relating the chiral polarization to thechiral Wannier center
We detail below how to derive the relations betweenthe average projected positions, the chiral polarizationand the corresponding sublattice Zak phases.
1. Projected position operator and sublattice Zakphases
Ignoring the distinction between the A and B sites, wecan first compute the action of the position operator onthe Wannier states following [32]: (cid:104) x | (cid:98) X | W n, R (cid:105) = (cid:90) k x e i k · ( x − R ) ϕ n, k ( x )= (cid:90) k (cid:16) − i∂ k e i k · ( x − R ) + R e i k · ( x − R ) (cid:17) ϕ n, k ( x )= (cid:90) k e − i k · R (cid:2) e i k · x ( R + i∂ k ) (cid:3) ϕ n, k ( x ) , (B1)where in the last step we applied an integration by parts,using that | Ψ n, k (cid:105) = | Ψ n, k + G (cid:105) with G a primitive recipro-cal vector. The generalization of Eq. (B1) to the positionoperator projected on the sublattice a = A, B is straight-forward: (cid:104) x | (cid:98) X P a | W n, R (cid:105) = (cid:90) k e − i k · R (cid:2) e i k · x ( R + i∂ k ) (cid:3) P a ϕ n, k ( x ) , (B2)which allows us to define the average positions (cid:104) x a (cid:105) n, R restricted to the site a = A, B and to the n th band exci-tations: (cid:104) x a (cid:105) n, R ≡ (cid:104) W n, R | P a ˆ X P a | W n, R (cid:105) = R (cid:90) k (cid:104) ϕ n, k | P a | ϕ n, k (cid:105) + 1Ω Γ a Zak ( n ) , (B3) where Ω is the volume of the BZ, | ϕ n, k (cid:105) = e − i k · ˆ X | Ψ n, k (cid:105) ,and Γ a Zak ( n ) is the vector composed of the d sublatticeZak phases associated to the n -th band: Γ a Zak ( n ) = i Ω (cid:90) k (cid:104) ϕ n, k | P a ∂ k P a | ϕ n, k (cid:105) . (B4)We can further simplify Eq. (B3) notingthat the orthonormality of the | ϕ n, k (cid:105) implies (cid:104) ϕ n, k | P A + P B | ϕ n, k (cid:105) = 1 and (cid:104) ϕ n, k | P A − P B | ϕ n, k (cid:105) = 0,which yields (cid:104) ϕ n, k | P a | ϕ n, k (cid:105) = 1 /
2. All in all, we finda simple relation between the average of the positionoperator and the Zak phase of the Bloch eigenstatesover the BZ: (cid:104) x a (cid:105) n, R = R Γ a Zak ( n ) . (B5)
2. Chiral polarization and sublattice Zak phases
We are now equipped to compute the chiral polar-ization, defined as the difference between the expectedvalue of the projected position operators over the occu-pied eigenstates ( n < Π corresponds to the difference of the sublattice Zakphases: Π ≡ (cid:88) n< (cid:10) x A (cid:11) n, R − (cid:10) x B (cid:11) n, R = 1Ω (cid:88) n< Γ A Zak ( n ) − Γ B Zak ( n ) . (B6)Two comments are in order. Firstly, the sum could havebeen also taken over the unoccupied states ( n > C = I , the sublattice phase picked up by | ϕ n, k (cid:105) is indeedthe same as that of its chiral partner | ϕ − n, k (cid:105) = C | ϕ n, k (cid:105) .Secondly, we stress that Eq. (B6) does not depend onthe specific convention of the Bloch representation. Thisrelation, however does not disentangle the respective con-tributions of the frame geometry and of the Hamiltonianon the chiral polarization. To single out the two contribu-tions, we now use the specific Bloch representation (A1).Given this choice, the sublattice Zak phase is naturallydivided into two contributions leading to Γ a Zak ( n ) =Ω (cid:90) k (cid:88) α ∈ a (cid:0) u ∗ n,α u n,α r α + iu ∗ n,α ∂ k u n,α (cid:1) . (B7)The first term on the r.h.s. is the intracellular contribu-tion to the Zak phase while the second is proportionalto the sublattice intercellular Zak phase following to thedefinitions of [37] γ aj ( n ) ≡ i (cid:90) d k j (cid:88) α ∈ a u ∗ n,α ( k ) ∂ k j u n,α ( k ) . (B8)Summing Eq.(B7) over all occupied bands, and using theorthogonality of the chiral component u n,α we then re-cover our central result: Π = 1Ω /d (cid:0) γ A − γ B (cid:1) + 12 p , (B9)where p = (cid:0)(cid:80) α ∈ A r α − (cid:80) α ∈ B r α (cid:1) is the geometricalpolarization of the corresponding unit-cell and γ a = (cid:80) n< γ a ( n ). The chiral polarization is the sum of onecontribution coming only from the frame geometry andone contribution characterizating the geometrical phaseof the Bloch eigenstates.
3. Chiral polarization in different Blochconventions
Although the physical content of the chiral polariza-tion does not depend on the choice of the Bloch con-vention, it is worth explaining how to derive its func-tional form for the other usual representation where | Ψ n, k (cid:105) = (cid:80) α ˜ u n,α ( k ) e i k · r α | k , α (cid:105) . Within this conven-tion the total Zak phase takes the form Γ a Zak ( n ) = i (cid:90) BZ d d k (cid:88) α ∈ a ˜ u ∗ n,α ∂ k ˜ u n,α , (B10)which does not allow the distinction between the geo-metrical and the Hamiltonian contributions to Π whenperforming the sum over the occupied band in Eq. (B6).This observation further justifies our choice for the Blochrepresentation. Appendix C: Topological indices of chiral insulators1. Quantization of the intercellular Zak-phase inchiral insulators
In order to establish the quantization of γ j = γ Aj + γ Bj ,we resort to the Wilson loop formalism reviewed e.g. inRef. [53].Let us first recall the definition of the non-AbelianBerry-Wilczek-Zee connection along the Brillouin zonefor a set of smooth vectors | u n ( k ) (cid:105) , n = 1 , ...M : A nm ( k ) = (cid:104) u n ( k ) | ∂ k | u m ( k ) (cid:105) . (C1)The associated Wilson loop operator defined along thepath C j through the Brillouin zone is given by the orderedexponential W j = exp (cid:32) − (cid:90) C j d k · A ( k ) . (cid:33) (C2)The topological properties of a generic gapped chiralHamiltonian are conveniently captured by smooth defor-mations yielding a flat spectrum E = ±
1. The corre-sponding Bloch Hamiltonian is then given by H = (cid:18) Q ( k ) Q † ( k ) 0 (cid:19) (C3)where Q ( k ) is a nonsingular unitary matrix. Withoutloss of generality, we write the corresponding eigenstates as | u ± n ( k ) (cid:105) = 1 √ (cid:18) ± Q ( k ) (cid:12)(cid:12) e Bn (cid:11)(cid:12)(cid:12) e Bn (cid:11) (cid:19) (C4)where the sign ± identifies the sign of the eigenvalue E = ± (cid:12)(cid:12) e Bn (cid:11) form a basis of theHilbert space of Q † . The non-Abelian connection (C1)for the negative (resp. positive) energy states then takesthe simple form A − nm ( k ) = 12 (cid:10) e Bn (cid:12)(cid:12) Q † ( k ) ∂ k Q ( k ) (cid:12)(cid:12) e Bm (cid:11) (C5)= A + nm ( k ) (C6)It follows from the definition of the Wilson-loop operator(Eq. (C2)) that the intercellular Zak phase for the nega-tive energy bands γ = γ A + γ B is defined in terms of theWilson loops for the non-Abelian connection A − ( k ) as γ j = − i ln det W − j (C7)The quantization of all d intercellular Zakk phases thenfollows from Eqs (C2) and (C5): γ j = − i tr ln (cid:34) exp (cid:32) − (cid:90) C j d k · ∂ k ln Q ( k ) (cid:33)(cid:35) (C8)= πw j mod (2 π ) (C9)where the mod (2 π ) indetermination stems from thechoice of the branch cut of the complex ln function, andwhere w j is the standard winding of the chiral Hamilto-nian (C3): w j = i π (cid:90) C j d k · tr (cid:2) ∂ k H C H − (cid:3) ∈ Z , (C10)= − π (cid:90) C j d k · tr (cid:2) Q − ∂ k Q (cid:3) . (C11)We therefore conclude that the d Zak phases are topo-logical phases defined modulo 2 π .
2. Relating the sublattice Zak phases to theWinding of the Bloch Hamiltonian
We here demonstrate the essential relation gven byEq. (9). To do so, we relate the winding w j tothe sublattice Zak phases by evaluating the trace inEq. (C10) using the eigenstate basis. Noting that (cid:104) u n | ∂ k H ( k ) C H − ( k ) | u n (cid:105) = − (cid:104) u n | C ∂ k | u n (cid:105) , the wind-ing takes the simple form w j = − i π (cid:90) C j d k (cid:88) n (cid:104) u n | C ∂ k | u n (cid:105) . (C12)Decomposing the chiral operator on the two sublatticeprojectors C = P A − P B , yields πw j = (cid:0) γ Bj − γ Aj (cid:1) ∈ π Z . (C13) FIG. 5.
Unit cell transformation.
We illustrate the defi-nition of the R α vectors using the simple example of a SSHchain. For the first atom (empty symbol) R = a ˆ x while R = 0 for the second atom (solid symbol).
3. Quantization of the sublattice Zak phases
Eqs. (C9) and (C13) shows that both the sum and thedifference of the sublattice Zak phases are quantized: γ Aj + γ Bj = πw j + 2 πm, m ∈ π Z ,γ Bj − γ Aj = πw j . (C14)It then follows that both sublattice phases γ Aj and γ Bj are integer multiples of π . Appendix D: How the winding number of a chiralBloch Hamiltonian change upon unit cell redefinition
Starting from a chiral Hamiltonian H , we demonstratebelow the relation between the winding numbers associ-ated to the Bloch Hamiltonians constructed from differ-ent choices of unit cells, Eq. (11).The definition of Bloch waves and Bloch Hamiltoniansrequire prescribing a unit cell. Starting with a first choiceof a unit cell geometry, say unit cell (1), we can write H (1) ( k ) in the chiral basis as H (1) ( k ) = (cid:32) Q (1) Q † (1) (cid:33) , (D1) Let us now opt for a second choice of unit cell, say choice(2). The Bloch Hamiltonians H (1) and H (2) are thenrelated by a unitary transformation H (2) = U † H (1) U, (D2)where the components of the unitary matrix are given by U αβ = exp (cid:16) i k · R (12) α (cid:17) δ αβ , (D3)where the R α are the Bravais vectors connecting theposition of the atoms in the two unit-cell conventions, seeFig. 5 for a simple illustration. We note that, we haveimplicitly ignored the trivial redefinitions of the unit cellthat reduce to permutations of the site indices. We canthen express the winding of H (2) using Eq. (D2) in thedefinition of Eq. (C10), which yields w (2) j = i π (cid:90) C j d k tr (cid:104) ∂ k ( U H (1) U † ) C ( U H (1) U † ) − (cid:105) . (D4)Expanding the gradient, using the trace cyclic propertyand noting that [ C , U ] = 0, we find w (2) j = w (1) j − i π (cid:90) C j d k tr (cid:2) ∂ k U C U − (cid:3) . (D5)This equation relates the winding numbers of the twoBloch Hamiltonians to the winding number of the trans-formation matrix U , which is by definition a geometricalquantity independent of H . Using Eq. (D3) leads to theremarkable relation which relates the spectral propertiesof the Hamiltonian to the unit-cell geometry w (1) j − w (2) j = i π (cid:90) C j d k tr (cid:2) ∂ k U C U − (cid:3) = 1 a j (cid:32) (cid:88) α ∈ A R α − (cid:88) α ∈ B R α (cid:33) . (D6) [1] F. Bloch, “ ¨Uber die quantenmechanik der elektronen inkristallgittern,” Zeitschrift fur Physik , 555 (1929).[2] Qi X.L., Hughes T.L., and Zhang S.C., “Topological fieldtheory of time-reversal invariant insulators,” Phys.Rev.B (2008).[3] M. Z. Hasan and C. L. Kane, “Colloquium: Topologicalinsulators,” Rev. Mod. Phys. , 3045–3067 (2010).[4] B.A.Bernevig and T.L.Hughes, Topological insulatorsand topological superconductors (Princeton UniversityPress, 2013).[5] M. Franz and L. Molenkamp, eds.,