N-band Hopf insulator
NN -band Hopf insulator Bastien Lapierre, Titus Neupert, and Luka Trifunovic Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland (Dated: February 15, 2021)We study the generalization of the three-dimensional two-band Hopf insulator to the case of manybands, where all the bands are separated from each other by band gaps. The obtained Z classificationof such a N -band Hopf insulator and the formulation of its bulk-boundary correspondence are ourmain results. We find that the quantized boundary effect can only be measured in a non-equilibriumstate. I. INTRODUCTION
Topological materials exhibit robust boundary effectsthat promise many applications. For example, moreenergy-efficient microelectronics and better catalysts can be designed by making use of backscattering-freeedge modes, i.e., chiral (helical) modes appearing in thequantum (spin) Hall systems. Another promising ap-plication is a fault-tolerant quantum computing basedon Majorana zero-energy states appearing at the ends ofcertain topological superconductors. All above mentioned topological phases of matter canbe realized as band insulators or superconductors. Theirtopological classification goes under the name of tenfold-way (or K -theoretic) classification. The mathematicalrules of the tenfold-way classification state that two givenband structures are topologically equivalent if and onlyif they can be continuously deformed into each otherwithout closing the band gap or violating symmetry con-straints. The band structures with different number ofbands can be topologically equivalent too: the tenfold-way classification allows the addition of “trivial” bandsboth above and below the band gap.Initially, the symmetry constraints considered includedtime-reversal, particle-hole and sublattice (chiral) sym-metries which led to an elegant classification result con-taining ten entries with a periodic structure. Recently,crystalline symmetries have been included to extend thetenfold-way classification which now contains many thou-sands entries.
The extended tenfold-way classificationis listed in catalogues that helped to discover manytopological material candidates. A novel robust effect ofsome of these topological crystalline phases are so-calledhigher-order boundary states: chiral (helical) modes canappear not only on the boundary of a two-dimensionalsystems but also on the hinges of three-dimensional sys-tems. Similarly, Majorana zero-energy states can appearas corner states of either two- or three-dimensional sys-tems. These robust boundary effects are guaranteed bythe bulk-boundary correspondence that holds for thetenfold-way topological classification.While the quest for new topological materials is stillan ongoing effort, some more recent theoretical effortsare concerned with the following question: in which waydoes a modification of the tenfold-way classification rulesalter the established classification results? Such “beyond tenfold-way” classification schemes include delicate andfragile (i.e., unstable ) topological classifications. Fordelicate classification (Fig. 1b), both the number of con-duction and valence bands is fixed; The most well stud-ied representative of delicate topological insulator is two-band Hopf insulator. Recently, many fragile topo-logical insulators were accidentally discovered whilecomparing the classification results of “Topological quan-tum chemistry” and that of “Symmetry-based indica-tors”. “Fragile” topological equivalence allows for theaddition of trivial conduction bands while the number ofvalence bands is fixed. In other words, the fragile clas-sification rules are halfway between that of tenfold-way(i.e., stable) and delicate topological classification. Yetanother possibility of going beyond the tenfold-way is tointroduce additional constraints on the band structure.For example, the boundary-obstructed classification requires that the so-called Wannier gap is maintained.We note that so far, the efforts were mainly focusedon obtaining such modified classifications. Despite ef-forts to formulate a bulk-boundary correspondence,it is still unclear if any (possibly subtle) quantized bound-ary effect can be used to uniquely identify any of the“beyond tenfold-way” phases.In this work we focus our attention on delicate topo-logical phases. The constraint that the number of bandsneeds to be fixed hinders direct application to crystallinematerials. For example, the Hopf insulator has exactlyone conduction and one valence band, whereas crystalshave typically many bands. Although the Hopf insulatorcan be turned into a stable topological phase through ad-ditional symmetry constraints , here we take a differentroute and relax the requirements of the delicate topolog-ical classification to allow for a trivial band (or group ofbands) to be added if separated by the gaps from all theother bands, see Fig. 1d. The idea of multi-gap classi-fication is not a new one. The stable multi-gap classifi-cation was used to classify Floquet insulators, whereasthe delicate multi-gap classification, with exception ofthe one-dimensional systems described by real Hamilto-nians, has been largely unexplored.We consider three-dimensional systems with no addi-tional symmetry constraints, and find that delicate multi-gap topological classification is the same as the clas-sification of the Hopf insulator. The obtained phasesare dubbed N -band Hopf insulators. Furthermore, we a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b k kkk(cid:15) (cid:15)(cid:15)(cid:15) (a) (b)(c) (d) FIG. 1. The band structure of a band insulator, where theexistence of a single band gap is guaranteed (a). The sameas in the panel (a) for the case of two bands (b). The bandinsulator can have N − N disjoint sets (c). The same as in the panel (c) for the casewhen each of N sets contains exactly one band (d). formulate the bulk-boundary correspondence for the N -band Hopf insulator. It states that a finite sample of the N -band Hopf insulator can be seen as the bulk, with thetotal magnetolectric polarizability coefficient (of all thebands) taking an integer value, wrapped into a Cherninsulator sheet with the total Chern number of all theband equal to minus the same integer, see Fig. 2. A fi-nite sample fully filled with electrons does not exhibit anyquantized effect, the quantized response is obtained onlyby driving the system into a non-equilibrium state whereeither only the region close to the boundary or deep intothe bulk is fully filled with electrons.The results of topological classifications apply equallywell to periodically and adiabatically driven crystals.In fact, there is a well known one-to-one correspon-dence between a two-dimensional Quantum Hall sys-tem and a one-dimensional Thouless pump. The quan-tized Hall conductance translates into quantized chargepumped during one period of the adiabatic drive. Anal-ogously, there is one-to-one correspondence between athree-dimensional N -band topological insulator and cer-tain two-dimensional adiabatic pumps: the quantizedmagnetoelectric polarizability coefficient of the three-dimensional bulk translates into the quantized orbitalmagnetization of the two-dimensional bulk of the pump,while the surface Chern number translates into the edgeThouless pump. We explicitly construct one such two-dimensional N -band Hopf pump which happens to alsorepresents an anomalous Floquet insulator (AFI). Un-like Floquet insulators, where the condition of the gapin the quasienergy spectrum is difficult to verify exper-imentally, the requirements of N -band Hopf insulatorsare experimentally accessible. FIG. 2. A finite N -band Hopf insulator has the bulk with thetotal (all the bands combined) magnetoelectric polarizabilitycoefficient equal to N Hopf with the gapped boundary that hastotal Chern number equal to − N Hopf . The remaining of the article is organized as follows. InSec. II we review the definition and the classification ofHopf insulators. Sec. III considers N -band Hopf insu-lators and derives their classification and topological in-variant. The bulk-boundary correspondence for N -bandHopf insulators is formulated in Sec. IV. In Sec. V, weconsider a two-dimensional N -band Hopf pump and dis-cuss its orbital magnetization. Examples of both three-dimensional Hopf insulator and two-dimensional N -bandHopf pump with N = 2 and N = 3 can be found inSec. VI. We conclude in Sec. VII. II. HOPF INSULATOR
Consider a three-dimensional, gapped 2-band BlochHamiltonian h (cid:126)k . Assuming that the two bands are “flat-tened” such that the Bloch eigenvalues become ±
1, wewrite h (cid:126)k = U (cid:126)k σ U † (cid:126)k , (1)where σ is Pauli matrix and U (cid:126)k ∈ SU (2). At each (cid:126)k -point in the Brillouin zone (BZ), h (cid:126)k can be seen asan element of the quotient group SU (2) /U (1), where U (1) ∈ SU (2) describes gauge transformations of thetwo Bloch eigenvectors. The group SU (2) /U (1) is iso-morphic to 2-sphere S , hence h (cid:126)k is seen as mappingfrom BZ (3-torus T ) to S h (cid:126)k : T → S . This mappingis defined by the representation of the Bloch state | u (cid:126)k (cid:105) on the Bloch sphere. It follows that the classificationof three-dimensional 2-band Bloch Hamiltonians is givenby homotopy classification of the maps h (cid:126)k : T → S .The complete classification of such maps was first ob-tained by Pointryagyn . There are three weak topo-logical invariants classifying the maps from T → S with T ⊂ T —these invariants are Chern numbers in( k x , k y )-, ( k y , k z )- and ( k x , k z )-manifolds. If any of theweak invariants is non-zero, the homotopy classificationof h (cid:126)k does not have a group structure. In this article weassume that the weak invariants vanish, in which case the Z classification is obtained, given by the Hopf invariant N Hopf = 2 P , with P Abelian the third Chern-Simonsform (Abelian axion coupling) P n = (cid:90) BZ d k π (cid:126)A n · (cid:126) ∇ × (cid:126)A n , (2)with (cid:126)A n = i (cid:104) u (cid:126)kn | (cid:126) ∇ (cid:126)k | u (cid:126)kn (cid:105) .We now proceed with an alternative derivation of theabove results. Vanishing of the weak invariants impliesthe homotopy classification of maps T → S is given bythe homotopy group π ( S ) that classifies maps S → S .Instead of calculating π ( SU (2) /U (1)) = π ( S ), wecalculate the relative homotopy group π ( SU (2) , U (1))which classifies the maps from the 3-disc D to the group SU (2) with the constraint that the disc’s boundary ismapped to the subgroup U (1), ∂D → U (1). The topo-logical invariants for the group π ( SU (2) , U (1)) can beobtained from the knowledge of homotopy groups for SU (2) and U (1) with help of the following exact se-quence π ( U (1)) i −→ π ( SU (2)) i −→ π ( SU (2) , U (1)) (3) ∂ −→ π ( U (1)) i −→ π ( SU (2)) . The exactness of the above sequence means that the im-age of each homomorphism is equal to the kernel of thesubsequent homomorphism. The homomorphisms i and i are induced by the inclusion U (1) → SU (2), the ho-momorphism i identifies the maps from S → SU (2)as maps D → SU (2) where the boundary ∂D ismapped to the identity element of the group SU (2).Lastly, the boundary homomorphism ∂ restricts the map D → SU (2) to its boundary map ∂D = S → U (1)which is classified by the group π ( U (1)). In this partic-ular case the groups π ( U (1)) and π ( U (1)) are trivial,hence the exactness of the sequence (3) implies π ( SU (2) , U (1)) = π ( SU (2)) = Z . (4)The topological invariant for the homotopy group π ( SU (2)) is the third winding number W [ U (cid:126)k ], U (cid:126)k ∈ SU (2) W [ U (cid:126)k ] = (cid:90) BZ d k π Tr (cid:16) U † (cid:126)k ∂ k x U (cid:126)k [ U † (cid:126)k ∂ k y U (cid:126)k , U † (cid:126)k ∂ k z U (cid:126)k ] − (cid:17) , (5)where [ A, B ] − denotes the commutator. Finally, theisomorphism between the groups π ( SU (2) , U (1)) and π ( SU (2) /U (1)) implies that there is a relation betweenthe winding number (5) and the Hopf invariant (2). In-deed, the following relation holds N Hopf = W [ U (cid:126)k ] = P + P = 2 P = N Hopf , (6)where | u (cid:126)k (cid:105) and | u (cid:126)k (cid:105) are two Bloch eigenvectors thatdefine P , via Eq. (2), and U (cid:126)k : T → SU (2) is definedin Eq. (1). The relation (6) was proved in Ref. 39 for thecase when U (cid:126)k | k z =0 is ( k x , k y )-independent. III. N -BAND HOPF INSULATORS The gap of a band insulator divides the Hilbert spaceinto two mutually orthogonal subspaces, with the projec-tor P (cid:126)k ( Q (cid:126)k ≡ − P (cid:126)k ) defined by occupied (empty) Blocheigenvectors; see Fig. 1a. The topological classificationof band insulators is obtained by classifying the subspace P (cid:126)k , or equivalently Q (cid:126)k . Within the K -theory classifica-tion, the ranks of these two projectors, P (cid:126)k and Q (cid:126)k , canbe varied by an addition of topologically trivial bands.On the other hand, the fragile topological classification allows the rank of Q (cid:126)k to be varied while the rank of theprojector P (cid:126)k is fixed. If the ranks of both P (cid:126)k and Q (cid:126)k are required to take some fixed values, as is the case forthe N = 2 band Hopf insulator in Fig. 1b, one then talksabout delicate topological classification.In this work we modify the classification rules by re-quiring not one but N − N projectors P (cid:126)kn , n = 1 , . . . , N , which are projectors onto the sub-spaces spanned by the Bloch eigenvectors with the eigen-values laying between two neighbouring band gaps.As in the case of a single band-gap classification, forthe ( N −
1) band-gap classification one can apply vari-ous classification rules. The K -theoretic version of theclassification, see Fig. 1c, allows the rank of all projec-tors P (cid:126)kn to be varied by the addition of trivial bands—such classification is directly related to a single band-gapclassification, see Ref. 30. On the other hand, if therank of all the projectors P (cid:126)kn is fixed, we refer to thisclassification as delicate multi-gap classification. In con-trast to K -theoretic classification, the delicate multi-gapclassification is not always related to delicate single-gapclassification .Below we show that the delicate ( N − Z for N ≥ P (cid:126)kn = 1for n = 1 , . . . , N . Since the non-trivial topological insu-lators for N = 2 are called Hopf insulators, we call thenon-trivial insulators for N > N -band Hopf insulators.The complete classification of N -band Hopf insulatorsgoes along the lines of N = 2 classification of Sec. II.Given N -band Bloch Hamiltonian h (cid:126)k is flattened suchthat its eigenvalues are distinct integers [1 , N ], the diag-onalized Hamiltonian is written as h (cid:126)k = U (cid:126)k diag(1 , . . . , N ) U † (cid:126)k , (7)where U (cid:126)k ∈ SU ( N ) is continuous on the BZ. At each (cid:126)k -point in the BZ, h (cid:126)k is seen as an element of thegroup SU ( N ) /U (1) N − , where the subgroup U (1) N − ∈ SU ( N ) is generated by U (1) gauge transformations of in-dividual bands. Under the assumption of vanishing weaktopological invariants, that are defined for each P (cid:126)kn , theBZ can be regarded as 3-sphere S . In other words, thestrong classification of N -band Hopf insulators is givenby the homotopy group π ( SU ( N ) /U (1) N − ). We pro-ceed with help of the following isomorphism π ( SU ( N ) /U (1) N − ) = π ( SU ( N ) , U (1) N − ) , (8)where π ( X, A ) for A ⊆ X denotes the relative homotopygroup introduced in the previous Section. The exact se-quence, analogous to the one in Eq. (3), reads π ( U (1) N − ) i −→ π ( SU ( N )) i −→ π ( SU ( N ) , U (1) N − ) ∂ −→ π ( U (1) N − ) i −→ π ( SU ( N )) , (9)implying that π ( SU ( N ) , U (1) N − ) = π ( SU ( N ))because the homotopy groups π ( U (1) N − ) and π ( U (1) N − ) are trivial. The topological invariant, amember of the group π ( SU ( N )) = Z , is the thirdwinding number, which provides the complete classifi-cation of N -band Hopf insulators. As we show in theAppendix B, the classification approach used abovecan be also applied to the case of real one-dimensional N -band systems which were shown to have non-Abelianclassification . The advantage of our classificationapproach is that it gives the complete set of topologicalinvariants that were previously not known.The above considerations give the topological invariantof N -band Hopf insulators N Hopf = W [ U (cid:126)k ] , (10)i.e., N Hopf is the third winding number of the unitary N × N matrix U (cid:126)k ∈ SU ( N ) in Eq. (7). Although U (cid:126)k explicitly depends on the choice of U (1) gauge for eachBloch eigenvector, such gauge transformations cannotchange the third winding number of U (cid:126)k . (This followsdirectly from the exact sequence (9), since img i is triv-ial.) The following relation is easily obtained N Hopf = P ∈ Z , (11)where P is non-Abelian third Chern-Simons form P = (cid:90) BZ d k π tr (cid:18) (cid:126) ˆ A (cid:126)k · (cid:126) ∇ × (cid:126) ˆ A (cid:126)k + 2 i (cid:126) ˆ A (cid:126)k · (cid:126) ˆ A (cid:126)k × (cid:126) ˆ A (cid:126)k (cid:19) , (12)with ( (cid:126) ˆ A (cid:126)k ) nm = i (cid:104) u (cid:126)kn |∇ (cid:126)k | u (cid:126)km (cid:105) . The above topologicalinvariant differs from the tenfold-way topological invari-ants, which vanish when summed over all the bands. Fur-thermore, for tenfold-way classification, the non-Abelianthird Chern-Simons form (12) has an integer ambiguitywhich is removed by requiring N band gaps to stay open.The obtained topological invariant (11) has a physicalmeaning. It is the magnetoelectric polarizability coeffi-cient α of all the bulk bands combined. Unlike thetenfold-way topological invariants which can be assignedto each band (of group of bands) separately, the abovetopological invariant can only be assigned to the wholeband structure. Indeed, we can express the magnetoelec-tric polarizability coefficient α as N Hopf = α = N (cid:88) n =1 α n , (13) where α n is the magnetoelectric polarizability coefficientof the n th band. There are two contributions to themagnetoelectric coefficient α n = α top n + α nontop n , wherethe topological piece α top n is expressed via the Abelianthird Chern-Simons form (2) that involves only Blocheigenvector of the n th band α top n = P n , (14)while for the non-topological piece α nontop n , the knowl-edge of the whole band structure is required. Wenote that, generally, a non-quantized value of (cid:80) Nn =1 α top n ( (cid:80) Nn =1 α nontop n ) cannot change upon a deformation of theHamiltonian that maintains all N − IV. BULK-BOUNDARY CORRESPONDENCE
To formulate bulk-boundary correspondence for N -band Hopf insulator, we consider slab geometry with ar-bitrary termination along y -direction described by the N N y × N N y slab Hamiltonian h k x k z . We assume thatall weak topological invariants (Chern numbers) vanish,hence, there exist continuous bulk Bloch eigenfunctions | ψ (cid:126)kn (cid:105) , n = 1 , . . . , N , of the Hamiltonian h k x k z . In otherwords, each bulk band can be separately “Wannierized” | w (cid:126)Rn (cid:105) = 1 (cid:112) N x N y N z (cid:88) (cid:126)k e i(cid:126)k · (cid:126)R | ψ (cid:126)kn (cid:105) . (15)For a slab terminated in y -direction, we use hybrid bulkWannier functions (WFs) | w k x R y k z n (cid:105) = 1 √ N x N z (cid:88) R x ,R z e − i ( k x R x + k z R z ) | w (cid:126)Rn (cid:105) . (16)The goal is to divide the slab into the three subsystems:the two surfaces and the bulk, the latter being defined bythe choice of the bulk WFs, see Fig. 3. Using the aboveWFs we perform a Wannier cut on all the bands toobtain the projector P Lk x k z onto the two surfaces by re-moving the hybrid bulk WFs from the middle of the slab P Lk x k z ( (cid:126)x (cid:48) , (cid:126)x ) = δ (cid:126)x (cid:48) (cid:126)x − n = NR y = L (cid:88) n =1 R y = − L w k x R y k z n ( (cid:126)x (cid:48) ) ∗ w k x R y k z n ( (cid:126)x ) , (17)which, for large enough integers N y , L with N y (cid:29) ( N y − L ) and 2 L < N y , defines the projector onto the uppersurface P surf k x k z ( (cid:126)x (cid:48) , (cid:126)x ) ≡P Lk x k z ( (cid:126)x (cid:48) , (cid:126)x ) H ( y ) H ( y (cid:48) ) , (18)where (cid:126)x = ( x, y, z ) indexes the orbitals of the slab super-cell, H ( y ) is the Heaviside theta function, and we assumethat the y = 0 plane passes through the middle of theslab. The integer L should be chosen as large as possible FIG. 3. The Hilbert space, spanned by the orbitals of theslab’s supercell, is divided at each ( k x , k z )-point into the threemutually orthogonal subspaces corresponding to the bulk andthe two surfaces. The bulk hybrid WFs | w k x R y k z n (cid:105) are contin-uous in the ( k x , k z )-space. On the other hand, for a non-trivial N -band Hopf insulator, there is an obstruction in finding con-tinuous surface WFs | w surf k x R y k z n (cid:105) . while requiring that in the region where the bulk WFs | w k x Lk z n (cid:105) have support, the Hamiltonian h k x k z is bulk-like. For the slab’s width much larger than the WFs’size, the operator P surf k x k z is a projector. Hence, the firstChern number of P surf k x k z , denoted by Ch surf , readsCh surf = i (cid:90) BZ dk x dk z π Tr (cid:0) P surf k x k z [ ∂ k x P surf k x k z , ∂ k z P surf k x k z ] − (cid:1) . (19)The bulk-boundary correspondence statesCh surf = − N Hopf . (20)The above correspondence can be proved by noticing thatCh surf cannot be changed by surface decorations sincetheir first Chern number summed over all the bands van-ishes. In the previous section we proved that N Hopf is theunique bulk topological invariant of the N -band Hopf in-sulator, it follows that Ch surf can be expressed in termsof N Hopf . Hence, to prove the relation (20) it is sufficientto show that it holds for a generator of the N -band Hopfinsulator, see Sec. VI. Alternatively, the relation (20) fol-lows from Eq. (11) and “Surface theorem for axion cou-pling” of Ref. 47. For N = 2, the correspondence (20)is a generalization of recently discussed bulk-boundarycorrespondence for the Hopf insulator. The above procedure divides a finite sample of the N -band Hopf insulator into bulk and surface subsystems.It is important to note that such division is not unique.Choosing different bulk WFs or different assignment ofthe bulk WFs to their home unit cell yields different bulkand surface subsystems. Despite this non-uniqueness, afinite sample of the N -band Hopf insulator can be seen toconsist of the bulk, with magnetoelectric polarizability being quantized to α = N Hopf , see Eq. (11), “wrapped”into a sheet of a Chern insulator with the total Chernnumber being equal to − N Hopf , see Fig. 2. Clearly, sucha “wrapping paper” cannot exist as a standalone objectsince the total Chern number (of all the bands) of a two-dimensional system needs to vanish. Recently, the concept of multicellularity for band in-sulators was discussed. A band insulator is said to bemulticellular if it can be Wannierized and if it is not pos-sible to deform the band structure such that all the bulkWFs are localized within a single unit cell. The examplesof multicellular band structures include the N = 2 Hopfinsulator and certain insulators constrained by crystallinesymmetries. The bulk-boundary correspondence (20) im-plies that the N -band Hopf insulator is a multicellularphase: if all the bulk WFs are to be localized within asingle unit cell, the resulting projector onto the uppersurface (18) would be ( k x , k z )-independent and the sur-face Chern number (19) would vanish.If the N -band Hopf insulator, with all the bands oc-cupied, is placed into an external magnetic field, thebulk gets polarized due to the magnetoelectric effect, (cid:126)P = α (cid:126)B = N Hopf (cid:126)B . This polarization does not resultin an excess charge density at the boundary, because theexcess charge is compensated by the surface Chern insu-lator. Hence, we see that the two quantized effects, onein the bulk and the other on the boundary, mutually can-cel. It is easy to understand this cancellation by noticingthat the many-body wavefunction of a fully occupied slabis independent of the Hamiltonian. Hence, the fully oc-cupied slab exhibits no magnetoelectric effect, implyingthat the bulk and the boundary magnetoelectric effectsmutually cancel. In order to measure a quantized effect,one needs to drive the system into a non-equilibrium statewhere either the boundary or the bulk subsystem are fullyfilled with electrons, but not both.
V. ORBITAL MAGNETIZATION
Every three-dimensional Bloch Hamiltonian h (cid:126)k of aband insulator defines a periodic adiabatic pump of atwo-dimensional band structure, and vice versa. Thesubstitution k z → πt/T gives the Hamiltonian of thetwo-dimensional adiabatic pump h k x k y t corresponding tothe three-dimensional Hamiltonian h (cid:126)k . As we discuss be-low, this viewpoint sheds light on the link between the N -band Hopf insulators, introduced in this work, and therecently studied anomalous Floquet insulator; see Ap-pendix A for comparison between N -band Hopf pumpsand Floquet insulators.We start by applying the bulk-boundary correspon-dence (20) to the N -band Hopf pump h k x k y t . Con-sider a ribbon h ribb k x t consisting of N y unit cells in y -direction. Similar to Eq. (18), we divide the ribbon-supercell Hilbert space into the two edge and the bulksubspaces NN y × NN y = P edge k x t + P bulk k x t + P edge (cid:48) k x t , (21)where the right-hand side is the sum of three mutually or-thogonal projectors. Importantly, the P bulk k x t projects ontothe space spanned by the bulk hybrid WFs | w k x R y tn (cid:105) with R y ∈ [ − L, L ], and the spaces onto which P edge k x t and FIG. 4. The adiabatic process of a ribbon corresponding to N -band Hopf pump. The ribbon’s supercell is divided intothe three regions, where the middle (gray) region is spannedby the bulk WFs. Whereas the bulk WFs perform periodicmotion, some WFs of the edge subsystems get shifted to theleft or right. P edge (cid:48) k x t project do not contain the orbitals from the middleof the ribbon. This way, at each ( k x , t )-point the ribbonis divided into the bulk and the two edge subsystems, seeFig. 4. The WFs in the bulk subsystem can be chosen tobe periodic, | w k x R y T n (cid:105) = | w k x R y n (cid:105) , (22)i.e., the bulk WFs return to their initial state after oneperiod. On the other hand, from the bulk-boundarycorrespondence (20), it follows that the upper edge P edge k x t has non-zero Chern number equal to N Hopf . Asa consequence, the edge WF | w edge k x R y m (cid:105) is shifted to | w edge k x R y + N Hopf ; T m (cid:105) for some m ∈ [1 , N ]. The edge sub-system acts as a Thouless pump even after consideringall the bands—such a situation cannot occur for a stan-dalone one-dimensional system.Let us consider a fully occupied ribbon. From the rela-tion (11) and the results of Ref. 51, we have that the bulksubsystem has (geometric) orbital magnetization equalto eN Hopf /T . The orbital magnetization gives rise to anedge current that exactly cancels the current pumped bythe Thouless pump of the edge subsystem. Hence, thebulk and the boundary anomalies mutually cancel simi-lar to the three-dimensional case discussed at the end ofthe previous Section.In order to observe the quantized orbital magnetiza-tion, we need to prepare the system at the time t = 0 suchthat only the region (puddle) inside the bulk is fully filledwith electrons. Such an initial state will generally diffuseunder the time evolution and eventually leak into theedge region, in which case, as discussed above, no quan-tization of the orbital magnetization is expected. Such“leakage” is expected also for time-independent band in-sulators. Thus, the quantized (geometric) orbital mag-netization can be measured in the transient state wherethe filled puddle is far from the edges. The flat bandlimit, see Sec. VI, is a special case since there the filledpuddle does not smear in the course of time.The above conclusions parallel the discussion of the so-called anomalous Floquet insulator (AFI). This is nota coincidence, since in Sec. VI, we show that the N -bandHopf pump can at the same time be an AFI, although it is important to note that not every AFI is a N -bandHopf insulator nor vice-versa. For comparison, in Ap-pendix A, we review the stable multi-gap classificationof two-dimensional Floquet insulators. One importantdifference between Floquet insulator and N -band Hopfpump is that the latter is not stable against translation-symmetry-breaking perturbations. Indeed, as we discussin Appendix C 2, doubling of the unit cell violates thecondition of having a single band between the two neigh-bouring band gaps. VI. EXAMPLES
Below, we first consider the three-dimensional Moore-Ran-Wen model ( N = 2 band Hopf insulator), thatwe use to illustrate the bulk-boundary correspondenceof Sec. IV, which generalizes the approach of Ref. 28.Furthermore, two two-dimensional examples correspond-ing to periodic adiabatic processes are considered, whichclarify the relation between the N -band Hopf insulatorand the AFI. A. Moore-Ran-Wen model of Hopf insulator
Here we present an example of a 2-band three-dimensional Hopf insulator, the Moore-Ran-Wen model.The Bloch Hamiltonian is defined as h k x k y k z = (cid:126)v · (cid:126)σ, (23)with v i = (cid:126)z † σ i (cid:126)z , where (cid:126)z = ( z , z ) T , with z = sin( k x ) + i sin( k y ) and z = sin( k z )+ i [cos( k x )+cos( k y )+cos( k z ) − ]. The above model (23) has N Hopf = 1. In the followingwe apply the procedure described in Sec. IV to obtainthe surface Chern number (19) for a three-dimensionallattice with N x × N y × N z unit cells. The two normalizedeigenvectors of the Bloch Hamiltonian (23) are | u (cid:126)k (cid:105) = | (cid:126)z | − ( z , z ) T , | u (cid:126)k (cid:105) = | (cid:126)z | − ( z ∗ , − z ∗ ) T , (24)which are continuous functions of (cid:126)k . We extend thesetwo Bloch eigenvectors to the whole lattice by defining ψ (cid:126)kn ( (cid:126)x ) = e − i(cid:126)k · (cid:126)x u (cid:126)kn ( (cid:126)x ).The WFs | w (0 , , n (cid:105) , n = 1 ,
2, with the home unitcell at (cid:126)R = (0 , ,
0) are given by Eq. (15) and shownin Fig. 5a. For arbitrary (cid:126)R = ( x, y, z ) T , the WFs areobtained from the components w (0 , , n ( (cid:126)x ) of | w (0 , , n (cid:105) w (cid:126)Rn ( (cid:126)x ) = w (0 , , n ( (cid:126)x − (cid:126)R ) . (25)We use the above choice of the bulk WFs to define thebulk subsystem. To this end, we perform the Fouriertransform in x - and y -directions to obtain the hybridbulk WFs | w k x R y k z n (cid:105) . Considering only the components w k x R y k z n ( (cid:126)x ) of the hybrid bulk WFs with (cid:126)x in a supercell, -5 0 52 0-2-2 0 2 -15015-15 0 1512 − π π P s u r f k z k z R x R z R y (a) (b)(c) FIG. 5. (a) Wannier function | w (0 , , (cid:105) localized at the centerof the lattice, for N y = 11, N x = N z = 5. (b) The absolutevalue of the matrix elements of the projector in Eq. (17), |P Lk x k z ( (cid:126)x, (cid:126)x (cid:48) ) | , for N y = 31 with cutoff L = 8. (c) The surfacepolarization along x -direction P surf k z for k z between 0 and 2 π for N y = 31 with cutoff L = 8. we obtain the 2 N y × N y projector | w k x R y k z n (cid:105) (cid:104) w k x R y k z n | .From Eq. (17) we compute the projector P Lk x k z ontothe two surfaces. As shown in Fig. 5b, after remov-ing the hybrid bulk WFs assigned to the units cells at R y ∈ [ − , P surf k x k z is obtained from the upper-left block of the matrix P Lk x k z .The surface Chern number can be obtained from the k z -dependent surface polarization of all the bands P surf k z = − i π ln det (cid:48) (cid:89) k x P surf k x k z , (26)where det (cid:48) ( X ) denotes the product of the non-zero eigen-values of the matrix X . The surface Chern number Ch surf implies that the surface polarization P surf k z is not continu-ous as functions of k z but jumps by Ch surf . The windingof P surf k z is shown in Fig. 5c, where the surface polariza-tion winds once, implying that Ch surf = − B. Two-dimensional Hopf pumps
Here we present examples of N = 2 and N = 3 Hopfpumps. The adiabatic evolution h k x k y t is piecewise de-fined, where each time-segment describes an adiabatictransfer of an electron between two selected orbitals in-side each unit cell of the two-dimensional square lattice.The models considered in this subsection are most eas-ily specified pictorially. In Fig. 6 we consider adiabaticprocess in a half-filled system with 2-sites per unit cell.At t = 0, the orbitals | (cid:126)R (cid:105) (black dots) have negative en-ergy whereas the orbitals | (cid:126)R (cid:105) (empty dots) have positiveenergy (see Fig. 6). We consider the following “building- (b)(a) FIG. 6. Two different adiabatic processes. The full and emptydots denote | (cid:126)R (cid:105) and | (cid:126)R (cid:105) orbitals. If the orbital | (cid:126)R (cid:105) is oc-cupied, the electron is adiabatically transferred (red arrow) tothe orbital | (cid:126)R (cid:105) , see Eq. (27). At the end of such process theorbital | (cid:126)R (cid:105) is empty while the orbital | (cid:126)R (cid:105) is occupied. Thevery same adiabatic process (a) transfers the electron from theorbital | (cid:126)R (cid:105) to the orbital | (cid:126)R (cid:105) as indicated by blue arrow.The second adiabatic process corresponds to an incompetetransfer between the two orbitals (b). In that case the finaloccupied states is superposition orbitals | (cid:126)R (cid:105) and | (cid:126)R (cid:105) . block” adiabatic process h t = Be − iσ πt/T σ e iσ πt/T , (27)where the Pauli matrices act in the space spanned bythe two orbitals. For the initial state | (cid:126)R (cid:105) , the adiabaticprocess is depicted in Fig. 6a by the red arrow. The evo-lution of the excited state | (cid:126)R (cid:105) is shown in Fig. 6a withthe blue arrow. Lastly, one can stop the above adiabaticprocess at times t < T , in which case the charge trans-fer between the sites | (cid:126)R (cid:105) and | (cid:126)R (cid:105) is incomplete. Thefinal state is then a superposition of | (cid:126)R (cid:105) and | (cid:126)R (cid:105) , aswould be the result of the process shown in Fig 6b. Thepictorial representation of the adiabatic process consistsof oriented line segments. The start (end) point of aline segment corresponds to the initial (final) state. Be-low we consider the adiabatic processes where the endpoints of the line segments lie either at the lattice sitesor on the line segment connecting the two neighbouringlattice sites. In the latter case, the initial (final) stateis the superposition of the two orbitals located at thesetwo neighbouring sites. In the following, the number ofarrows enumerates the time-segments, for example, “ → ”describes the first segment, “ (cid:16) ” the second etc. Thetwo examples that follow consider translationally invari-ant systems, hence the adiabatic process (27) is extendedin a translationally-symmetric manner to the whole two-dimensional lattice. N = 2 band Hopf pump Here, we consider a periodically driven system withtwo states per unit cell, labeled by {| (cid:126)R (cid:105) , | (cid:126)R (cid:105)} , wherethe vector (cid:126)R belongs to the square two-dimensional lat-tice. The driving protocol is of period T and is made of4 steps of equal duration T , see Fig. 7. At each of thosesteps, the Hamiltonian reads h k x k y t = U † k x k y t e − πiσ t/T Bσ e πiσ t/T U k x k y t , (28) (a) (b) FIG. 7. Two level periodic drive made of 4 steps of equalduration. Panel (a) shows adiabatic evolution for the initialstates | (cid:105) . The same as the panel (a) for the initial state | (cid:105) (b). with U k x k y t = σ t ∈ [0 , T ) , diag(1 , e − ik y ) t ∈ [ T , T ) , diag( e ik x , t ∈ [ T , T ) , diag(1 , e − i ( k x − k y ) ) t ∈ [ T , T ) , (29)where the Pauli matrices σ i act on the space spanned bythe two orbitals in the unit cell. This two-band Hamilto-nian can equivalently be written as the Hamiltonian of aspin in a time- and momentum-dependent magnetic field, h k x k y t = (cid:126)B k x k y t · (cid:126)σ . Therefore, the unitary transforma-tion in Eq. (1) is given by U k x k y t = e − πi(cid:126)n kxkyt · (cid:126)σt/T , (30)where ˆ n k x k y t is the unit vector along the (cid:126)B k x k y t × ˆ e z vector. The straightforward calculation gives N Hopf = W [ e − πi(cid:126)n kxkyt · (cid:126)σt/T ] = 1 . (31)In other words, the adiabatic process (28) is non-trivial N = 2 Hopf pump. Using the Bloch eigenvectors | u k x k y tn (cid:105) = U k x k y t | n (cid:105) , (32)with n = 1 ,
2, we find that the Berry connection A n(cid:126)kt = A nt depends only on time and the Chern-Simons 3-formis given by the area enclosed by the electron P n = 12 (cid:90) T dt (cid:126)A nt × ∂ t (cid:126)A nt = 12 . (33)Therefore the two Chern-Simons 3-forms sum up to 1,confirming the validity of the relation (6).We now show that the time-dependent Hamilto-nian (28) is at the same time an AFI, see Appendix A.The time-evolution unitary U F k x k y t during each of the foursegments is readily obtained as U F k x k y t = e − πi ˆ n kxkyt · (cid:126)σ ( t − t ) /T e − i ( BT σ − π ˆ n kxkyt · (cid:126)σ )( t − t ) /T . (34)In the adiabatic limit, BT (cid:29)
1, the solution simpli-fies to U F k x k y t = e − πi ˆ n kxkyt · (cid:126)σ ( t − t ) /T e − iBσ ( t − t ) . Asexpected, the unitary U F k x k y t differs from the one in (a) (b) (c) Re( ξ ~k ) Re( ξ ~k ) Re( ξ ~k ) I m ( ξ ~ k ) FIG. 8. The spectrum ξ (cid:126)k of the 1-cycle unitary operator U F k x k y T defined in Eq. (34) for different values of BT . (a) BT = 3. (b) BT = 5, around which the quasienergy gapcloses. (c) BT = 9, after gap closure. Eq. (30) only by a dynamical phase. Since in the adi-abatic limit, U F k x k y T = e − iBT σ , we conclude that themodel (28) corresponds to Floquet insulator, which re-mains to hold as long as BT (cid:38)
5, see Fig. 8. Choosing BT to be integer multiple of 2 π , the relation U F k x k y T = σ holds, and we find that W [ U F k x k y t ] = N Hopf = 1 , (35)i.e., the time-dependent Hamiltonian (28) describesanomalous Floquet insulator when BT (cid:38)
5. See Ap-pendix C 1 for details on the computation of W [ U F k x k y t ].Lastly, we want to verify the bulk-boundary correspon-dence (20), by computing explicitly the edge Chern num-ber (19). We impose open boundary conditions in the y -direction, and for concreteness take 4 layers in this di-rection, which defines the ribbon supercell of 8 orbitals | n (cid:105) , n = 1 , . . . ,
8; see Fig. 9. We note that we can considersuch a narrow ribbon because in this model the WFs arehighly localized. The WFs | w R x R y tn (cid:105) take the followingform in the bulk | w R x R y t (cid:105) = cos( t ) | (cid:126)R (cid:105) − sin( t ) | (cid:126)R (cid:105) , − cos( t ) | (cid:126)R (cid:105) − sin( t ) | (cid:126)R + (cid:126)a y (cid:105) , − cos( t ) | (cid:126)R + (cid:126)a y (cid:105) + sin( t ) | (cid:126)R + (cid:126)a y − (cid:126)a x (cid:105) , cos( t ) | (cid:126)R(cid:126)a y − (cid:126)a x (cid:105) + sin( t ) | (cid:126)R (cid:105) , (36) | w R x R y t (cid:105) = cos( t ) | (cid:126)R (cid:105) + sin( t ) | (cid:126)R (cid:105) , cos( t ) | (cid:126)R (cid:105) − sin( t ) | (cid:126)R − (cid:126)a y (cid:105) , − cos( t ) | (cid:126)R − (cid:126)a y (cid:105) − sin( t ) | (cid:126)R − (cid:126)a y + (cid:126)a x (cid:105) , − cos( t ) | (cid:126)R − (cid:126)a y + (cid:126)a x (cid:105) + sin( t ) | (cid:126)R (cid:105) , (37)where we used the notation t n = πT ( t − nT ) and the fourexpressions for the bulk WFs correspond to the four time-segments as in Eq. (29). The Fourier transform along the x -direction gives the hybrid bulk WFs | w k x R y tn (cid:105) , thatare used to compute the edge projector P edge k x t given inEq. (18). We perform a Wannier cut by removing fourbulk WFs from the middle of the ribbon, followed by pro-jecting onto the upper half of the ribbon supercell. The (a) (b) FIG. 9. (a) Adiabatic time evolution of the upper edge sub-system P edge k x t . (b) The same as panel (a) for the lower edgesubsystem P edge’ k x t (b). The orbital of the ribbon supercell arelabeled by | n (cid:105) , n = 1 , . . . , only nonzero contributions to the edge Chern numbercome from the time-segments t ∈ [ T , T ). For t ∈ [ T , T ), P edge k x t = diag( A, A ), with A = (cid:32) sin( t ) e − ik x cos( t ) sin( t ) e ik x cos( t ) sin( t ) cos( t ) (cid:33) , (38)where the basis {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)} has been usedto write P edge k x t . In this case, we obtain thatTr (cid:16) P edge k x t [ ∂ k x P edge k x t , ∂ t P edge k x t ] − (cid:17) = πiT sin (cid:0) πtT (cid:1) . Simi-larly, for t ∈ [ T , T ), P edge k x t = t ) − e ik x cos( t ) sin( ) 00 − e − ik x cos( t ) sin( t ) cos( t )
00 0 0 0 , (39)we obtain that Tr (cid:16) P edge k x t [ ∂ k x P edge k x t , ∂ t P edge k x t ] − (cid:17) = πiT sin (cid:0) πtT (cid:1) . Therefore Ch edge = −
1, confirming thebulk-boundary correspondence (20). N = 3 band Hopf pump In this example, we consider a N = 3 band modelthat is obtained from the N = 2 band model, introducedin the previous subsection, after adding an additionalorbital in the unit cell. Furthermore, we introduce aparameter δ in the model, such that δ = 0 correspondsto the previously discussed N = 2 Hopf insulator withthe additional orbital not being involved in the adiabaticprocess. This way, for δ = 0 we have P = P = ,while P = 0. For δ (cid:54) = 0 the model is chosen to have theproperty P (cid:54) = P (and P = 0), as we discuss below.We consider a driven model with three sites per unitcell, labeled by {| (cid:126)R (cid:105) , | (cid:126)R (cid:105) , | (cid:126)R (cid:105)} . The driving protocolhas the period T and is made of 6 time segments of equalduration T , which are illustrated in Fig. 10. FIG. 10. Three level periodic drive made of 6 steps of equalduration. At the first and fourth stages, we consider an in-complete rotation between the orbitals | (cid:126)R (cid:105) and | ( (cid:126)R + (cid:126)a x )3 (cid:105) ,such that the area enclosed by the trajectory of the thirdorbital is zero. This is controlled by the parameter δ ∈ [0 , The eigenvalues of the Hamiltonian h k x k y t are chosento be time-independent and take the values 1 , , h k x k y t = (cid:88) n =1 n | u k x k y tn (cid:105) (cid:104) u k x k y tn | , (40)where the Bloch eigenvectors | u k x k y tn (cid:105) can be read offfrom Fig. 10 and are explicitly given in Appendix C 3.At each of the six time segments, the adiabatic processinvolves only two orbitals. For example, during the firstsegment, see Fig. 10, the orbitals involved are | (cid:126)R (cid:105) and | ( (cid:126)R + (cid:126)a x )3 (cid:105) , where the final state is a δ -dependent super-position of these two states. Explicit calculation gives P = (cid:0) − sin (cid:0) π δ (cid:1)(cid:1) ≤ ,P = (cid:0) (cid:0) π δ (cid:1)(cid:1) ≥ ,P = 0 . (41)The contributions P and P are not anymore quantizedand equal if δ (cid:54) = 0. However the sum turns out to bequantized and equal to 1, just like in the previous ex-ample. The winding number can be computed along thesame lines as in the previous section, and is given by N Hopf = W (cid:2) U (cid:126)kt (cid:3) = 1. Since N Hopf = (cid:80) Nn =1 P n holds,we conclude that the non-topological orbital magnetiza-tion (cid:80) Nn =1 m nontop n vanishes for this example. VII. CONCLUSIONS
In this work we explore “beyond tenfold-way” topo-logical phases that belong to the category of delicatemulti-gap phases. These phases can be band insulatorswith ( N −
1) band gaps, where the number of bandsbetween two successive band gaps is fixed. We obtaina Z classification for three-dimensional band insulatorswithout any symmetry constraints. Furthermore, we for-mulate the bulk-boundary correspondence stating that a0finite sample of the N -band Hopf insulator consists of thebulk, with the total magnetoelectric polarizability of allthe bands quantized to an integer value, wrapped into asheet of a Chern insulator with the total Chern number ofall the bands equal to minus the same integer value (seeFig. 2). The obtained classification and bulk-boundarycorrespondence is the same as that of the Hopf insulator(the case N = 2). Hence, we dub these new phases the N -band Hopf insulators.An ultimate “usefulness test” for the beyond-tenfold-way classifications is whether the obtained topologicalphases are accompanied by a quantized boundary ef-fect. While in the tenfold-way classification the chem-ical potential µ plays a crucial role, since only the bandswith the energy below µ are classified, and the quantizedboundary effect can be observed in equilibrium, the quan-tized boundary effect in the multi-gap topological phasescan only be observed out of equilibrium. This is be-cause the multi-gap classifications schemes, both the sta-ble and the delicate ones, classify the whole band struc-ture and the chemical potential plays no role. Hence,one would naturally attempt to fill all the bands withelectrons in order to observe a quantized effect. We findthat if the whole finite sample is fully filled with electrons,the quantized bulk and boundary effects mutually cancel.Therefore, the quantized boundary effect of the N -bandHopf insulator can be observed in a non-equilibrium statewhere only the states close to the boundary are fully filledwith electrons.Our work discusses not only the three-dimensional N -band Hopf insulators but also the two-dimensional N -band Hopf pumps. The Hopf pump can be seen as a bulkwith the quantized (geometric) orbital magnetization ,with a Thouless pump (of all the bands combined) at theedges. Furthermore, we discuss a particular example ofthe Hopf pump that illustrates its similarities with theanomalous Floquet insulator. On the other hand, thedifference between the N -band Hopf pump, introduced inthis work, and the anomalous Floquet insulator is thatthe latter requires the gap in the quasienergy spectrumwhich is difficult to guarantee experimentally, whereashaving a multi-gap band structure is more physical re-quirement. Furthermore, the stable multi-gap classifi-cation (i.e., Floquet insulators) always results in Abeliangroups, while it has been shown that the delicate multi-gap classification of one-dimensional band insulators withcertain magnetic point-group symmetry is non-Abelian.Thus, extending the program outlined in this work tosystems of different dimensions and with additional sym-metry constraints is highly desirable. Acknowledgement
The authors thank Aris Alexandradinata, Tom´aˇsBzduˇsek, Alexandra Nelson, David Vanderbilt, andHaruki Watanabe for fruitful discussions. BL ac-knowledges funding from the European Research Council (ERC) under the European Union’s Hori-zon 2020 research and innovation program (ERC-StG-Neupert-757867-PARATOP). LT acknowledges finan-cial support from the FNS/SNF Ambizione GrantNo. PZ00P2 179962.
Appendix A: Stable multi-gap classification oftwo-dimensional Floquet insulators
Periodically driven band structure h F k x k y t is exampleof Floquet system. The Floquet system is called insula-tor, if the spectrum (quasienergy spectrum e iε kxky T ) ofthe unitary Floquet operator U F k x k y T = T e − i (cid:82) T h F kxkyt dt has at least one gap on the unit circle in the complexplane. Below we review classification of two-dimensionalFloquet insulators.Topological classification of Floquet insulators, classi-fies the unitary evolution operator U F k x k y t under the con-straint that one or multiple gaps in the quasienergy spec-trum are maintained. In other words, two Floquet insu-lators are said to be topologically equivalent if their uni-tary evolution operator can be brought to the same formwithout closing the gap (gaps) in the quasienergy spec-trum. Mathematically, such constraint divides the totalHilbert space into mutually orthogonal subspaces withcorresponding projectors P F k x k y n spanned by the eigen-vectors of the Floquet operator U F k x k y T with quasiener-gies between the two neighbouring gaps. One typicallyconsiders K -theoretic (i.e. stable) classification wherethe rank of the projectors P F k x k y n can be varied by ad-dition of trivial quasibands. For N gaps in quasienergyspectrum, two-dimensional Floquet insulators have Z N classification. The subgroup Z N − ⊂ Z is generated by N − P F k x k y n ,for say n = 1 , . . . , N −
1. The remaining Z topological in-variant is given by the third winding number W [ U F ,εk x k y t ]of the unitary U F ,εk x k y t which is obtained from the unitaryevolution U F k x k y t by continuously deforming the Floquetoperator U F k x k y T to identity matrix while maintaining thegap around some quasienergy ε ( ε belongs to one of the N gaps in quasienergy spectrum).The Floquet insulators with topological invariantsfrom the subgroup Z N − can all be realized as staticsystems. The remaining generator which is diagnozed bythe third winding number exists only for time-dependentband structures and is called anomalous Floquet insula-tor. Anomalous Floquet insulator was found to obey bulk-boundary correspondence. Consider anomalous Flo-quet insulator that satisfies U k x k y T = N × N . We ap-ply open boundary conditions in y -direction and considerslab geometry with N y layers, with time-dependent bandstructure h F k x t and N N y × N N y unitary evolution oper-1ator U F , slab k x t . The bulk-boundary correspondence states N AFI ≡ W [ U F k x k y t ] (A1)= (cid:90) π dk x π Tr[ U F , slab † k x t ∂ k x U F , slab k x t H ( y ) H ( y (cid:48) )] . In other words, anomalous Floquet insulator with N AFI (cid:54) = 0 induces quantized charge pumping of N AFI electrons along the boundary in steady state.
Appendix B: Delicate multi-gap classification ofone-dimensional real band structures
We consider the delicate multi-gap topological classi-fication of one-dimensional real Hamiltonians using themethod of Sec. III of the main text. The resulting clas-sification group is non-Abelian as first discussed by Wu,Soluyanov, and Bzduˇsek. Here we show that the clas-sification method used in the main text also gives theexpressions for strong topological invariants that werepreviously not known.As we show below, the delicate multi-gap classificationof real one-dimensional band structures depends explic-itly on the number of bands N (i.e. gaps). Hence, unlikethe case of N -band Hopf insulators discussed in the maintext, the group structure is given by concatenation of twoBloch Hamiltonians h (1) k and h (2) k with the same numberof bands h (2) k ◦ h (1) k = (cid:40) h (1)2 k for k ∈ [0 , π ) ,h (2)2 k − π for k ∈ [ π, π ) . (B1)In order for the concatenated Hamiltonian to be contin-uous, we require that h (1)0 = h (2)0 , which can be alwaysachieved by deformation given that there are no weaktopological invariants.The flattened Bloch Hamiltonian is diagonalized h k = O k diag(1 , . . . , N ) O Tk , (B2)where O k ∈ SO ( N ) is assumed continuous for k ∈ [0 , π ).Note that the periodicity of the Bloch Hamiltonian h = h π , does not require O = O π , but rather weaker re-quirement O T O π ∈ O (1) N − ⊂ SO ( N ). In other words,the real Bloch eigenvectors | u kn (cid:105) do not need to be con-tinuous at k = 2 π but | u n (cid:105) = ± | u πn (cid:105) . We now defineauxiliary orthogonal matrix o ( k ) o k = O T O k . (B3)The strong classification of the real Hamiltonians (B2)is obtained by classifying orthogonal matrices o k , sincethe orthogonal matrix O (0) contains only weak invari-ants. We have that o = N × N and o π ∈ O (1) N − ,i.e., o k is classified by the relative homotopy group π ( SO ( N ) , O (1) N − ). The group π ( SO ( N ) , O (1) N − ) can be found using the following exact sequence π ( O (1) N − ) i −→ π ( SO ( N )) i −→ π ( SO ( N ) , O (1) N − ) ∂ −→ π ( O (1) N − ) i −→ π ( SO ( N )) , (B4)The groups π ( O (1) N − ) and π ( SO ( N )) are trivial. Wefirst consider case N > N = 2 case see Sec. B 3),where π ( SO ( N )) = Z and π ( O (1) N − ) = Z N − holds,0 i −→ Z i −→ π ( SO ( N ) , O (1) N − ) ∂ −→ Z N − i −→ . (B5)The above extension problem does not have an uniquesolution, i.e., there is more than one group that sat-isfies the above exact sequence if the homomorphisms i and ∂ are not specified. We show in Sec. B 2 that π ( SO ( N ) , O (1) N − ) is non-Abelian group.To each element of the group π ( SO ( N ) , O (1) N − ),that is represented by some path o k defined by rela-tions (B3) and (B2), we can assign topological invari-ants from the Abelian groups π ( SO ( N )) = Z and π ( O (1) N − ) = Z N − . The Z topological invariants ν i for i = 1 , . . . , N − ν i = sign[( o π ) ii ] . (B6)To assign a Z topological invariant p to an arbitrarypath in π ( SO ( N ) , O (1) N − ), we need a convention thatassigns a loop to the given path, because π ( SO ( N )) isdefined for loops only. Using the vector notation ν , where( ν ) m = ν m for i = 1 , . . . , N −
1, we define reference paths o e m ref ,k for m = 1 , . . . , N − o e m ref ,k = e i ˆ L mN k/ , (B7)where ( ˆ L ij ) mn = − i ( δ im δ nj − δ jm δ in ) are generators of SO ( N ). o e m ref , π is the π rotation in the plane spannedby Bloch eigenvectors | u m (cid:105) and | u N (cid:105) . For an arbitraryvalue of the topological invariants from the righthandside of the exact sequence (B4), ν = e m + e m (cid:48) + . . . , wedefine o ν ref ,k = o e m ref ,k ◦ o e m (cid:48) ref ,k ◦ . . . , (B8)where the concatenation is ordered from the smallest tothe largest index m > m (cid:48) > . . . . Hence, a loop of or-thogonal matrices o Lk can be uniquely assigned to each o k , that has the topological invariants ν , o Lk = o k ◦ ( o ν ref ,k ) − , (B9)where the notation ( o k ) − = o π − k has been used. Toeach o Lk an element p ∈ {− , } from π ( SO ( N )) can beassigned, as we review in Sec. B 1. Therefore, each ele-ment of π ( SO ( N ) , O (1) N − ) can be specified by topo-logical invariants p and ν . In Sec. B 2 using the con-catenation of the matrices o k we show that the groupsstructure of π ( SO ( N ) , O (1) N − ) is non-Abelian.2
1. The topological invariant of π ( SO ( N )) To each loop o Lk ∈ SO ( N ), o L = o L π = , we need toassign (continuously) an element ¯ o k ∈ Spin ( N ). Aftersuch assignment, the Z topological invariant p is givenby ¯ o L π = p . (B10)To obtain ¯ o Lk , we need N Dirac matrices γ m for m =1 , . . . , N (the dimension of the representation is unimpor-tant). The Dirac matrices satisfy the following algebra γ m = 1 ,γ m γ n = − γ n γ m , for m (cid:54) = n. (B11)Consider a grid with M points in the Brillioun zone,where each segment of the grid has length δk , M =2 π/δk . The rotation loop o Lk can be approximated byseries of rotations around the piecewise fixed axes, i.e., o Lmδk = o L, ( m ) δk o L, ( m − δk . . . o L, (1) δk , (B12)for m = 1 , . . . , M , where each orthogonal matrix o L, ( n ) δk represents rotation around the fixed axis o L, ( n ) δk = e i (cid:80) ˆ L ab θ ab . (B13)The axes (labeled by the indices a and b ) and the angles θ ab are found by diagonalizing o L, ( n ) δk . The matrix ¯ o L π isfound by replacing each o L, ( n ) δk in the product (B12) by¯ o L, ( n ) δk ¯ o L, ( n ) δk = e i (cid:80) [ γ a ,γ b ] θ ab / . (B14)For practical purposes, one can more easily compute p as follows. The eigenvalues of o Lk come in pairs e ± iθ kn (otherwise they are real). By plotting the phases ± θ kn between [ − π, π ], we can find p by counting the numberof crossings (on real axis) modulo two.
2. The group structure of π ( SO ( N ) , O (1) N − ) For each o k defined by relations (B3) and (B2), we cancompute N topological invariants ν and p , as discussedabove. We use the convention that ν m ∈ { , } whereas p ∈ {− , } . We map an element of π ( SO ( N ) , O (1) N − )to the following string of Dirac matrices p ( iγ ) ν ( iγ ) ν . . . ( iγ N − ) ν N − . (B15)Below we prove that above map is an isomorphism. Tothis end, we need to show that the concatenation satisfiesthe algebra (B11). Consider first o e m ref ,k which is mappedto iγ m (by the construction (B7) it has p = 1 and ν = e m ). The element o e m ref ,k ◦ o e m ref ,k is a loop (i.e. it is equal tothe identity matrix for k = 2 π ), thus ν = 0. Additionally, we have p = − o e m ref ,k ◦ o e m ref ,k represents rotation by2 π angle in the plane spanned by the Bloch eigenvectors | u m (cid:105) and | u N (cid:105) , see Eq. (B7). We conclude that o e m ref ,k ◦ o e m ref ,k should be mapped to − which is in agreementwith ( iγ m ) = − . Next consider an element o e m ref ,k ◦ o e n ref ,k with m > n , which has topological invariants ν = e n + e m and p = 1 and is mapped to ( iγ n )( iγ m ). Onthe other hand, an element o e n ref ,k ◦ o e m ref ,k has the sametopological invariants ν = e n + e m , and rule (B9) assignsthe following loop to it o Lk = o e n ref ,k ◦ o e m ref ,k ◦ ( o e n ref ,k ) − ◦ ( o e m ref ,k ) − . (B16)The mapping (B14) sends o L π to¯ o Lk = e iσ π e iσ π e − iσ π e − iσ π = − σ , (B17)which implies p = −
1, see Eq. (B10). Thus, o e n ref ,k ◦ o e m ref ,k is mapped to − ( iγ n )( iγ m ) = ( iγ m )( iγ n ), which provesthat the considered map is an isomorphism between π ( SO ( N ) , O (1) N − ) and the algebra (B11).
3. The two-band case N = 2 This case is special because π ( SO (2)) = Z , thus theexact sequence (B4) reads0 → Z → π ( SO (2) , O (1)) → Z → . (B18)In the same way as previously, we can assign two topo-logical invariants to o k ∈ SO (2) that we denote by N ∈ Z and ν . To define N we use o ref ,k (in N = 2 case there isonly one reference rotation) to define the correspondingloop o Lk , to which we can associate the winding num-ber. It is easy to check that in this case, o ref ,k withthe topological invariants N = 0 and ν = 1 generatesthe whole group π ( SO (2) , O (1)): the concatenation of o ref ,k n -times with itself gives an element with the topo-logical invariants N = (cid:98) n/ (cid:99) and ν = n mod 2. Thus π ( SO (2) , O (1)) = Z where the topological invariant isequal to θ π /π , with θ k ∈ R being the angle of rotation associated to o k ∈ SO (2). Appendix C: Two-dimensional N -band Hopf pump Below we give details of the calculations for N = 2 and N = 3 Hopf pump.
1. The N = 2 Hopf pump
We compute the winding number for the 2-bandmodel in the adiabatic limit BT (cid:29)
1. In thislimit, the evolution operator takes the form U F k x k y t =3 e − πi ˆ n kxkyt · (cid:126)σ ( t − t ) /T e − iBσ ( t − t ) . This operator takes theform U F k x k y t = e − πiσ t/T e − iBσ t − ie − πi (sin( k y ) σ +cos( k y ) σ ) t × e − iBT σ t σ e − i BT σ − e − πi (sin( k x ) σ +cos( k x ) σ ) t × e − iBT σ t e ik y σ e − πi (sin( δk ) σ +cos( δk ) σ ) t × e − iBT σ t σ e − i ( δk + BT ) σ (C1)where we introduced the notation δk = k x − k y . Wenotice that only during the third segment of the drivethe unitary U F k x k y t will give a non trivial contribution tothe winding number, as it does not depend independentlyon k x and k y for the other segments of the drive. Onethen finds ( U F k x k y t ) † ∂ k y U F k x k y t = iσ (C2)( U F k x k y t ) † ∂ k x U F k x k y t = − i sin (cid:18) πtT (cid:19) σ − i (cid:18) πtT (cid:19) [cos( a ) σ + sin( a ) σ ] (C3)( U F k x k y t ) † ∂ t U F k x k y t = − i BTT σ − πT [cos( a ) σ − sin( a ) σ ](C4)where we introduced the notation a = k x − k y + ( − tT ) BT . We conclude that the trace gives πT sin (cid:0) πtT (cid:1) ,and therefore we obtain W [ U F k x k y t ] = 1, for any value of BT as long as the adiabatic limit holds.
2. Doubling of the unit cell
We now consider a redefinition of the unit cell for theadiabatic N = 2 Hopf pump of Sec.VI B 1. We double the unit cell in the x -direction, hence, there are 4 orbitalsper unit cell, as shown on Fig. 11. The model has 2bands, both doubly degenerate. Thus the classificationdiscussed in this article cannot apply in this example, asall the bands are not separated by a gap.We first consider the gauge | u k x k y tn (cid:105) , n = 1 , . . . ,
4, thatis obtained by Fourier transform of the following fourWFs (36)-(37): | w R x R y t (cid:105) , | w R x R y t (cid:105) , | w ( R x +1) R y t (cid:105) , and | w ( R x +1) R y t (cid:105) , see Fig. 11. The direct calculation of thethird winding number defined by such gauge choice gives W [ U k x k y t ] = 1.Next, we introduce a change of basis between the twolower degenerate bands: (cid:40) | ˜ u k x k y t (cid:105) = α k x k y t | u k x k y t (cid:105) + β k x k y t | u k x k y t (cid:105)| ˜ u k x k y t (cid:105) = γ k x k y t | u k x k y t (cid:105) + δ k x k y t | u k x k y (cid:105) , (C5)which defines a unitary 2 × V k x k y t = (cid:18) α k x k y t β k x k y t γ k x k y t δ k x k y t (cid:19) . (C6)The above gauge defines a new unitary matrix (cid:101) U k x k y t .We make a choice for V k x k y t such that W [ V k x k y t ] = 1,this can be obtained by taking V k x k y t to be the unitaryof the 2 sites unit cell example studied previously. Usingthis definition for the gauge transformation, we obtain (cid:101) U k x k y t FIG. 11. Redefinition of the unit cell. We consider a unit cellconsisting of 4 orbitals. (cid:101) U k x k y t = cos( t ) sin( t ) − sin( t ) cos( t ) 0 − sin( t ) cos( t ) cos( t ) sin( t ) t ) sin( t ) 0 cos( t ) sin( t ) − sin( t ) − sin( t ) cos( t ) cos( t ) , (C7) (cid:101) U k x k y t = sin( t ) e ik y cos( t ) sin( t ) cos( t ) e ik y t ) sin( t ) e ik y − sin( t ) e − ik y cos( t ) − sin( t ) cos( t ) e ik y t ) cos( t ) − cos( t ) t ) sin( t ) e − ik y − sin( t ) e − ik y , (C8)4 (cid:101) U k x k y t = cos( t ) e ik y − cos( t ) sin( t ) e i (2 k y − k x ) − sin( t ) e − i ( k y − k x ) − sin( t ) e ik x − cos( t ) e − ik y − sin( t ) cos( t ) 0cos( t ) sin( t ) e ik x − sin( t ) e − ik y cos( t ) − sin( t ) cos( t ) e i (2 k y − k x ) t ) e i ( k y − k x ) − cos( t ) e − ik y , (C9) (cid:101) U k x k y t = sin( t ) t ) cos( t ) e i ( k y − k x ) − cos( t ) e − i ( k y − k x ) − cos( t ) e ik x sin( t ) cos( t ) sin( t ) e ik y − sin( t ) cos( t ) e i ( k x − k y ) − cos( t ) e − ik y sin( t ) t ) sin( t ) e i ( k y − k x ) t ) e i ( k y − k x ) sin( t ) , (C10)for the 4 different segments of the drive, where we intro-duced the notation t n = πT ( t − nT ). We obtain that onlythe third and the fourth segments of the drive contributeto the third winding number, both giving an opposite contribution. Therefore W [ (cid:101) U k x k y t ] = 0, demonstratingthat the third winding number is not gauge independentin presence of the band degeneracies.Furthermore, imposing open boundary conditions inthe y -direction, the edge Chern number can be com-puted using the Wannier cut procedure described in themain text. We consider for concreteness 8 layers in the y -direction, which defines the ribbon supercell of 32 or-bitals. We compute the bulk hybrid WFs | ˜ w k x R y tn (cid:105) fromthe Bloch eigenvectors | ˜ u k x k y tn (cid:105) . Using these bulk hy-brid WFs, we obtain the upper edge projector P edge k x t byremoving the 16 WFs from the bulk. Explicit computa-tion leads to Ch edge = 0, since the contribution of thethird and fourth time-segments cancel each other.Lastly, we can define ( N = 4)-band Hopf pump usingthe Bloch eigenstates | ˜ u k x k y tn (cid:105) h k x k y t = (cid:88) n =1 n | ˜ u k x k y tn (cid:105) (cid:104) ˜ u k x k y tn | , (C11)such that the 4 bands are now non-denegerate. Theabelian part of the third Chern-Simons form can be com-puted explicitly, (cid:88) n =1 P n = − . (C12)This Hamiltonian (C11) has N Hopf = 0, since the thirdwinding number of (cid:101) U k x k y t vanishes. For this example N Hopf (cid:54) = (cid:80) n =1 P n holds, thus, the non-topological or-bital magnetization (cid:80) n =1 m nontop n = 1 /
3. The N = 3 Hopf pump
For t ∈ (cid:2) , T (cid:3) , the states evolve as | u k x k y t (cid:105) = cos ( t δ ) | (cid:105) − sin ( t δ ) e ik x | (cid:105) , | u k x k y t (cid:105) = | (cid:105) , | u k x k y t (cid:105) = cos ( t δ ) | (cid:105) + sin ( t δ ) e − ik x | (cid:105) , (C13)where we introduced the notation t n = πT ( t − nT ) and theparameter δ ∈ ]0 ,
1[ which creates the asymmetry betweenthe trajectories of | (cid:105) and | (cid:105) . For t ∈ (cid:0) T , T (cid:3) , the statesevolve as | u k x k y t (cid:105) = cos( t ) | u k x k y T (cid:105) − sin( t ) | (cid:105) , | u k x k y t (cid:105) = sin( t ) | u k x k y T (cid:105) + cos( t ) | (cid:105) , | u k x k y t (cid:105) = | u k x k y T (cid:105) . (C14)For t ∈ (cid:0) T , T (cid:3) , the states evolve as | u k x k y t (cid:105) = − cos( t ) | (cid:105) − sin( t ) e − iK | u k x k y T (cid:105) , | u k x k y t (cid:105) = cos( t ) | u k x k y T (cid:105) − sin( t ) e iK | (cid:105) , | u k x k y t (cid:105) = | u k x k y T (cid:105) , (C15)where the notation K = k x + k y has been introduced.For t ∈ (cid:0) T , T (cid:3) , the states evolve as | u k x k y t (cid:105) = − cos (cid:16) δ ( π − t ) (cid:17) e − iK | (cid:105) + sin (cid:16) δ ( π − t ) (cid:17) e − ik y | (cid:105) , | u k x k y t (cid:105) = | (cid:0) T (cid:1) (cid:105) , | u k x k y t (cid:105) = sin (cid:16) δ ( π − t ) (cid:17) e − ik x | (cid:105) + cos (cid:16) δ ( π − t ) (cid:17) | (cid:105) . (C16)For t ∈ (cid:0) T , T (cid:3) , the states evolve as | u k x k y t (cid:105) = − cos( t ) e − iK | (cid:105) + sin( t ) e − ik y | (cid:105) , | u k x k y t (cid:105) = − cos( t ) e iK | (cid:105) − sin( t ) e ik y | (cid:105) , | u k x k y t (cid:105) = | (cid:105) . (C17)5For t ∈ (cid:0) T , T (cid:1) , the states evolve as | u k x k y t (cid:105) = cos( t ) e − ik y | (cid:105) + sin( t ) | (cid:105) , | u k x k y t (cid:105) = − cos( t ) e ik y | (cid:105) + sin( t ) | (cid:105) , | u k x k y t (cid:105) = | (cid:105) . (C18) H. Chen, W. Zhu, D. Xiao, and Z. Zhang, Phys. Rev. Lett. , 056804 (2011). A. Y. Kitaev, Phys. Usp. , 131 (2001). A. Kitaev, AIP Conference Proceedings , 22 (2009). A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud-wig, AIP Conference Proceedings , 10 (2009). A. M. Turner, Y. Zhang, R. S. K. Mong, and A. Vish-wanath, Phys. Rev. B , 165120 (2012). L. Fu, Phys. Rev. Lett. , 106802 (2011). L. Trifunovic and P. W. Brouwer, Phys. Rev. B , 195109(2017). L. Trifunovic and P. W. Brouwer, Phys. Rev. X , 011012(2019). E. Khalaf, H. C. Po, A. Vishwanath, and H. Watanabe,Phys. Rev. X , 031070 (2018). B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang,C. Felser, M. I. Aroyo, and B. A. Bernevig, Nature ,298 (2017). S.-J. Huang, H. Song, Y.-P. Huang, and M. Hermele, Phys.Rev. B , 205106 (2017). K. Shiozaki and M. Sato, Phys. Rev. B , 165114 (2014). M. Geier, P. W. Brouwer, and L. Trifunovic,arXiv:1910.11271 (2019). S. Ono, H. C. Po, and K. Shiozaki, arXiv e-prints , arXiv:2008.05499 (2020), arXiv:2008.05499 [cond-mat.supr-con]. E. Khalaf, Phys. Rev. B , 205136 (2018). F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang,S. S. P. Parkin, B. A. Bernevig, and T. Neupert, ScienceAdvances (2018), 10.1126/sciadv.aat0346. T. Zhang, Y. Jiang, Z. Song, H. Huang, Y. He, Z. Fang,H. Weng, and C. Fang, Nature , 475 (2019). L. Trifunovic and P. W. Brouwer, phys-ica status solidi (b) , 2000090 (2021),https://onlinelibrary.wiley.com/doi/pdf/10.1002/pssb.202000090. In early studies both delicate and fragile phases arecalled unstable topological phases, in order to distinguishthem from the stable (tenfold-way) topological phases. Inthis work we borrow the terminology of Ref. 49 and callphases delicate is they are unstable but not fragile. J. E. Moore, Y. Ran, and X.-G. Wen, Phys. Rev. Lett. , 186805 (2008). R. Kennedy,
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