Four-Dimensional Topological Insulators with Nodal-Line Boundary States
FFour-Dimensional Topological Insulators with Nodal-Line Boundary States
L. B. Shao
1, 2 and Y. X. Zhao
1, 2, ∗ National Laboratory of Solid State Microstructures and department of Physics, Nanjing University, Nanjing, 210093, China Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
Conventional topological insulators and superconductors have topologically protected nodal pointson their boundaries, and the recent interests in nodal-line semimetals only concerned bulk bandstructures. Here, we present a novel four-dimensional topological insulator protected by an anti-unitary reflection symmetry, whose boundary band has a single
P T -symmetric nodal line with doubletopological charges. Inspired by the recent experimental realization of the four-dimensional quantumHall effect, we also propose a cold-atom system which realizes the novel topological insulator withtunable parameters as extra dimensions.
Introduction
Topological insulators (TIs) and super-conductors have been one of the most developed andprosperous fields in condensed matter physics during thepast decade [1, 2]. We now understand that these topo-logical phases are protected by certain symmetries, anda complete classification has been made for ten Altland-Zirnbauer (AZ) symmetry classes [3] and for all dimen-sions, even beyond the physical limit of three dimen-sions [4–6]. Historically the four-dimensional (4D) time-reversal ( T ) invariant TI played a key theoretical rolein exploring topological phases [7], because its dimen-sion reductions give rise to TIs in three and two dimen-sions [8]. However, it has only become promising untilrecently to experimentally explore the topological phasesin dimensions beyond three [9–11]. Such possibilities re-side in two facts. First, the topological invariants are de-fined in a band theory, and therefore can be well appliedto artificial periodic systems other than electronic crys-tals, such as photonic [12] and phononic [13, 14] crystals,and cold atoms in optical lattices [15, 16]. Second, extradimensions parametrized by momenta can be regardedas highly tunable parameters of these artificial systems,namely, that extra dimensions can be, in a sense, synthe-sized [9–11, 17].Another main focus of topological phases is the topo-logical (semi)metals and nodal superconductors [6, 18],which mainly concerns the topological charges of bandcrossings [19, 20]. While Weyl semimetals are of fun-damental interest with band crossings occuring at dis-crete Weyl points in the Brillouin zone (BZ) [19, 21, 22],nodal-line semimetals, which have the band crossingsforming lines in the BZ, are recently a hot topic [23–26, 30]. Belonging to the spacetime-inversion ( P T ) sym-metric classification of topological gapless phases [27], anodal line can appear in generic locations in the 3D BZwith topological stability, provided
P T is preserved with(
P T ) = 1 [28].One of the significant features of TIs is that gaplessmodes can appear on boundaries, corresponding to non-trivial bulk topological invariants. Conventionally, theboundary gapless modes are located at isolated band-crossing points in the boundary BZ, which includes all topological phases in the periodic classification table often AZ symmetry classes [4, 5]. For instance, Weyl pointsof the same chirality appear on the boundary of the afore-mentioned 4D TI [7, 8]. In this Letter, we present a 4D Z TI protected by an anisotropic two-fold anti-unitaryspatial symmetry RT with ( RT ) = 1, where R inversesthe three dimensions and preserves the fourth. In con-trast to boundary nodal points for conventional TIs, thecrystalline TI generically has odd number of nodal-lineson the 3D boundary normal to the fourth dimension. No-tably, the nodal lines have both 1D and 2D topologicalcharges as detailed below, essentially different from or-dinary nodal lines with only the 1D topological charge.We formulate the bulk Z topological invariant as thesecond Chern number over the half BZ, T / , subtractedby the Chern-Simons integrals over the two RT -invariantboundaries of T / . Analogous to the fact that the 4D T -invariant TI generalizes the 2D Chern insulator [7, 8],the proposed 4D crystalline TI can be regarded as a 4Dgeneralization of the well-known 2D T -invariant TI [29].Moreover, we discuss the general principles for realizingthe topological insulator by artificial systems, and pro-pose a cold-atom realization. Nodal line of double topological charges–
Let us beginby introducing the
P T symmetry. The spatial inversion P is a unitary symmetry, which maps x to − x , therefore k to − k in momentum space, while time-reversal T as ananti-unitary symmetry maps k to − k as well. Accord-ingly, the combination P T is an anti-unitary operatorthat leaves any k invariant. For spinless fermions, it sat-isfies ( P T ) = 1. The nodal-line model we would like toemerge on the boundary of a TI is H NL ( k ) = (cid:88) i =1 k i γ i + im z γ γ . (1)The 4 × γ µ with µ = 1 , · · · , { γ µ , γ ν } = 2 δ µν , among which γ i with i = 1 , , γ and γ are purely imag-inary. Thus, if P T symmetry is represented as PT = ˆ K with ˆ K being the complex conjugate, the model of Eq. (1)as a real Hamiltonian is clearly P T invariant. If m z = 0,the Hamiltonian of Eq. (1) presents the real Dirac point, a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y S N L S FIG. 1. The nodal line with double topological charges inthe boundary Brillouin zone. S encloses the nodal line as awhole, while S surrounds the nodal line locally in momentumspace. which can be regarded as a real generalization of thewell-known Weyl points, namely the real counterpart ofmonopoles in the P T symmetric real band theory [30, 31].But its monopole charge is characterized by a Z invari-ant ν C R , which is defined on a sphere S surrounding thecrossing point in the real band theory, in contrast to the Z -valued Chern number in the complex band theory [31].The second term of Eq. (1) is a “partial” mass term,which anti-commutes with γ while commutes with γ , .Hence, when the second term of Eq. (1) is turned on,the four-fold degeneracy at the momentum-space origin,though, is lifted, there still exist two-fold band crossingpoints forming a nodal circle of k x + k y = m z in theplane with k z = 0 [30]. The fact that the spectrum can-not be fully gapped by the P T -symmetric term actuallyreflects the nontrivial topological charge ν C R of the realDirac model. It is remarkable that, while the topologicalcharge ν C R on an S enclosing the whole nodal circle isinherited, the nodal line also has nontrivial Berry phase ν B R along any loop S surrounding it, which is quan-tized by P T symmetry, and therefore a Z topologicalinvariant. Thus, the Hamiltonian of Eq. (1) describes a P T -symmetric nodal-line semimetal of double topologi-cal charges.
The model–
We now proceed to construct a minimalmodel of 4D TIs, such that the nodal line of Eq. (1)can correspond to its low-energy boundary theory. Ifthere is no symmetry [except U (1)], the only TIs infour dimensions are second Chern insulators, for whichWeyl fermions with given chirality exist on the bound-ary. Hence, certain symmetry is required to realize aboundary nodal line. On the other hand, we expect thebulk symmetry projected on the 3D boundary to be P T symmetry, since the nodal line is protected by
P T sym-metry in three dimensions. Considering this condition,the symmetry we propose is an anti-unitary reflectionsymmetry RT . Here, T is time-reversal symmetry, and R is a unitary spatial symmetry operating as R : ( w, x ) (cid:55)→ ( w, − x ) , ( k w , k ) (cid:55)→ ( k w , − k ) . (2) in real and momentum spaces, respectively, where x is thethree Cartesian coordinates for boundary, and w is thecoordinate perpendicularly towards the bulk. Combinedwith time reversal, RT is anti-unitary with { RT, i } = 0,and operates in momentum space as RT : ( − k w , k ) (cid:55)→ ( − k w , k ) , (3)recalling that time reversal inverses all momentum com-ponents. In general, RT can be represented in momen-tum space as RT = U RT ˆ K ˆ I w , where ˆ I w is the inversionof k w , and U RT is a unitary operator. We further requiresthat ( RT ) = 1, or equivalently U TRT = U RT . Explicitly, P T symmetry constrains the momentum-space Hamilto-nian as U RT H ∗ ( k , k w ) U † RT = H ( k , − k w ) . (4)It is clear that RT with ( RT ) = 1 can always be con-verted to be RT = ˆ K ˆ I w by a unitary transformation.Therefore, we assume RT = ˆ K ˆ I w in the model construc-tion.The minimal model we shall construct has eight bands,which can be inferred from the fact that the doublycharged nodal line has four bands. It is conventionalfor TIs that the bulk band number is a double of thatof the boundary, for instance the 4D Chern insulatorhas four bands with the boundary Weyl points beingtwo-band crossings. Hence, we shall use the Cliffordalgebra Cl , which gives rise to a basis for 8 × × × a , namely { Γ a , Γ b } = 2 δ ab , and Γ † a = Γ a , with a, b = 0 , , · · · ,
6. Among them, following the standardconvention, the first four, Γ a with a = 0 , , ,
3, are real,and remaining three are purely imaginary. For conve-nience, let Γ = (Γ , Γ , Γ ), and the components be Γ i with i = 1 , ,
3. An explicit representation of Γ a conve-nient for our purpose can be found in the SupplementalMaterial (SM) [32].A RT -symmetric massive Dirac Hamiltonian may begiven by H C = k · Γ + k w Γ + m Γ in the continuous form.To compactify momentum space, one may replace m by m − λ ( k + k w ) to get the modified Dirac model. This isso far the standard procedure to construct a TI by Diracmatrices, which however is not sufficient in our case of Cl in four dimensions. There are other terms includ-ing i Γ i Γ , with i = 1 , ,
3, which are real and therefore RT -symmetric, but still hermitian. Such terms, i Γ i Γ , ,are “partial mass” terms, for i Γ i Γ , anti-commutes with k i Γ i , while commutes with the rest terms in the Hamilto-nian, and turn out to be essentially important for bound-ary in-gap modes. For simplicity, adding only i Γ Γ intothe continuous Dirac model, we can readily obtain thelattice Dirac model, H = (cid:88) a =1 sin k i Γ i + [ m − ( (cid:88) a =1 cos k i )]Γ + it Γ Γ , (5)which is clearly RT invariant with RT = ˆ K ˆ I w , and servesas a minimal model for the novel 4D TI with nodal-lineboundary. Nodal-line boundary states
We now open a 3D bound-ary for the Hamiltonian, Eq. (5), perpendicular to the w -direction, and discuss the in-gap boundary states ofthe semi-infinite system with Dirichlet boundary condi-tion. Considering that translational symmetry is vio-lated by the boundary in the w -direction, while is stillpreserved for the other dimensions, we introduce theansatz | ψ k (cid:105) = | ξ k (cid:105) ⊗ (cid:80) ∞ i =0 λ i | i (cid:105) with | λ | < i labels the i th site along the w -direction. Solving the Schr¨odinger equation for the semi-infinite system, we find that the boundary states occurin the 4D subspace corresponding to the positive eigen-value of i Γ Γ or equivalently the image of the projec-tor P = (1 + i Γ Γ ), and λ = m − (cid:80) i =1 cos k i (forderivation details, see the SM) [32]. It is noteworthythat i Γ Γ anti-commutes with Γ and Γ , while com-mutes with the other Γ’s, as well as RT operator RT .Hence, the boundary low-energy effective theory is givenby H eff ( k ) = P H P , for the momentum k satisfying | λ | = | m − (cid:80) i =1 cos k i | <
1. Particularly, we can focuson the eight high symmetry momenta K α with K αi = 0or π and α = 1 , · · · ,
8. If | λ | = | m − (cid:80) i =1 cos K αi | < K α is explicitly givenby H αeff ( q ) = (cid:88) i =1 η α,i q i γ i + itγ γ , (6)where k = K α + q , and η α,i are signs equal to − cos( K αi ).Here, we have introduced γ ’s for Cl , which are speci-fied as γ i = − P Γ i P with i = 1 , , γ = P Γ P and γ = P Γ P , noting that Γ and Γ vanishes under theprojection. Recall that RT symmetry is projected to P T in the boundary, but since RT commutes with the pro-jector P , the low-energy effective theory of Eq. (6) isclearly P T invariant.It is found that if | m | >
4, there is no boundary statebecause | λ | < k . If 2 < m < − < m < − K = (0 , , π, π, π )]. If 0 < m < − < m < K = ( π, , , (0 , π, , (0 , , π )[( π, π, , ( π, , π ) , (0 , π, π )]. However, in general a pairof Eq. (6)’s can be gapped by RT -invariant terms, leav-ing only a single gapless low-energy theory for the lastcase. Thus, we find that the boundary effective theory ofEq. (6) in the topological phase corresponds exactly tothe nodal-line model of Eq. (1), as we claimed. The bulk topological invariant–
In momentum space RT symmetry relates ( k , k w ) to ( k , − k w ). According to k w π XY Y τ τ τ τ (cid:15) (cid:15) FIG. 2. Two coordinate charts X and Y covering the halfBrillouin zone. Y covers the whole half BZ, but has a smallsingular region, which is covered by the regular chart X . Theboundary of X consists of two 3D tori, ∂X = τ (cid:48) − τ (cid:48) , andsimilarly for Y , ∂Y = τ − τ . Periodicity is assumed for k i with i = 1 , , Y and Y are subspaces of Y , and cancomplement X in the half Brillouin zone. Eq. (4), we can require, in a neighborhood of ( k , − k w )and its mirror image, that U RT | α, k , k w (cid:105) ∗ = | α, k , − k w (cid:105) , (7)where | α, k , k w (cid:105) are valence eigenstates of H ( k , k w ) with α labeling valence bands. From Eq. (7), it is sufficientto consider only half of the BZ, T / , as illustrated inFig. 2. Particularly in two mirror-symmetric 3D tori, τ and τ , namely the boundary of T / with k w = 0 and π ,equation (7) puts constraints on eigenstates pointwiselyas a boundary condition. In the nontrivial topologicalphase, it is impossible to find a complete basis of va-lence bands, which are globally well-defined in T / , andsatisfy the boundary condition of Eq. (7) as well. Inother words, if | α, k , k w (cid:105) Y , with Y being the coordinatechart covering the whole T / , are periodic for k i with i = 1 , ,
3, and satisfy Eq. (7) in τ and τ , there mustexist singular points in the bulk of T / for | α, k , k w (cid:105) Y which cannot be eliminated without violating the bound-ary condition given by Eq. (7), while in the trivial phase,we can smooth out all singular points. To characterizethis, we first choose another coordinate chart X in thebulk of T / (Fig. 2) covering all singular points of Y , andassume another basis of valence bands | α, k , k w (cid:105) X with-out singular point in X , which is always possible sinceEq. (7) puts no constraint in the bulk of T / . Then,we can derive the transition function t XY from X to Y in the boundary of X , ∂X = τ (cid:48) − τ (cid:48) . As t XY ∈ U ( N )with N the valence-band number, and π [ U ( N )] = Z , thetransition function may has a nontrivial winding num-ber N [ t XY ]. The winding number can be calculated bythe Chern-Simons terms of the Berry connection on twocharts restricted on ∂X , A X/Yαβ,i = X/Y (cid:104) α, k | ∂ k i | β, k (cid:105) X/Y as N = (cid:82) ∂X Q ( A X ) − (cid:82) ∂X Q ( A Y ), where Q ( A ) = − (cid:15) µνλ tr( A µ ∂ ν A λ + A µ A ν A λ ) / (8 π ) d k . It can beshown that (see the SM for detailed derivations [32]) N = (cid:90) T / ch ( F ) − (cid:90) τ Q ( A )+ (cid:90) τ Q ( A ) mod 2 , (8)where the gauge-invariant second Chern character isgiven by ch ( F ) = − (cid:15) µνλσ tr F µν F λσ / (32 π ) d k , and A can be chosen as A Y or any basis for valence bands sat-isfying the boundary condition given by Eq. (7). Theformula (8) is a Z invariant, because a gauge trans-formation in either one of the Chern-Simons terms inEq. (8) can change N by an even integer, noticing that π [ O ( N )] = 2 Z in the eight-fold periodic homotopygroups of classifying spaces in real K theory [27, 33].Detailed discussions on the topological invariant of theminimal model, Eq. (5), can be found in the SM [32].It noteworthy that the topological invariant of Eq. (8)may be regarded as a 4D generalization of the well-knowntopological invariant for 2D topological insulators with a Z pump interpretation [29], where the first Chern char-acter is replaced by the second, and accordingly the Berryphases are replaced by the Chern-Simons terms. Cold atom realization
To simulate a 4D system by aphysical system with dimension ≤
3, certain momentahave to be simulated by tunable parameters of the phys-ical system. For the particular Hamiltonian of Eq. (5),merely the w -dimension has to be faithfully simulatedin real space, so that a boundary can be opened for ex-ploring the bulk-boundary correspondence. Hence, wesimulate the 4D model by a 1D lattice, and identify thethree momenta k with highly tunable parameters of theartificial system. Then, the band structure of the in-gap boundary states can be mapped out by tuning theparameters accordingly. Based on these general consid-erations of simulating a high-dimensional system, themodel of Eq. (5), in principle, can be realized by arti-ficial systems, such as photonic crystals, mechanical sys-tems, and cold atoms. We now present a simulation bycold atoms trapped in a quasi 1D optical lattice as illus-trated in Fig. S1. To realize the eight bands of Eq. (5),cold atoms with pseudo spin are arranged to hop on thequasi 1D tetragonal optical lattice made by four parallelchains. Observing that each term of Eq. (5) is a tensorproduct of three Pauli matrices, we assign them to theleft-right, up-down and quasi-spin spaces, respectively.In such a setup, the model can be well realized by thecutting-edge techniques of ultracold atom experiment.Particularly, all next-nearest-neighbor hoppings markedby colored arrows in Fig. S1 can be realized by photon-assisted tunnelings [34]. The spin-orbit couplings occurmerely in an inner-cell term of Eq. (5) (i.e., the greenhoppings in Fig. S1), which can be induced by a two-photon process when a pair of incident Raman lasers,counter-propagating along the green lines in Fig. S1, areresonant with the field-split internal levels [34]. More-over, the phases of the hoppings can be created by syn-thetic gauge fields [35–37]. The technical details and the w | u i| d i| l i | r i FIG. 3. The quasi 1D tetragonal optical lattice. There arefour sites positioned as a square in each unit cell, which arelabeled as | ur (cid:105) , | dr (cid:105) , | ul (cid:105) and | dl (cid:105) according to their positions,respectively. All next-nearest hoppings are marked with ar-rows. Specifically, only green arrows involve spin-orbital in-teractions, and the phase difference of red and blue arrows is π . realization of all terms of Eq. (5) are fully addressed inthe SM [32]. Conclusion and Discussions
In conclusion, we havepresented a 4D crystalline topological phases with nodal-line boundary states, and formulated its topological in-variant. Furthermore, a minimal model has been con-structed, and a cold-atom realization has been proposed.Finally, we discuss two aspects of the topological phase.First, the 2D topological charge of the nodal line is es-sential for the bulk boundary correspondence, not the1D topological charge. The 2D topological charge im-plies that the boundary band structure with odd numberof nodal lines cannot be realized by an independent 3Dsystem, and therefore has to be connected to a higher-dimensional bulk, which follows from the index theo-rem of the bulk boundary correspondence [38]. Indeed,any 3D
P T symmetric semimetal satisfies the Nielsen-Ninomiya no-go theorem, namely always has even num-ber of such nodal lines [21, 31]. Second, the topologicalphase is related to time-reversal symmetry with T = 1,so that ( RT ) = 1 is satisfied, provided [ R, T ] = 0. Al-though the requirement of T = 1 is violated by elec-tron systems with spin-orbital couplings (with T = − Acknowledgments–
We thank S. L. Zhu for discussionson the cold-atom realization. This work is supported bythe startup funding of Nanjing University, China. ∗ [email protected][1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010).[2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011).[3] A. Altland and M. R. Zirnbauer, Phys. Rev. B , 1142(1997). [4] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W.Ludwig, Phys. Rev. B , 195125 (2008).[5] A. Kitaev, AIP Conf. Proc. , 22 (2009).[6] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu,Rev. Mod. Phys. , 035005 (2016).[7] S.-C. Zhang and J. Hu, Science , 823 (2001).[8] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B , 195424 (2008).[9] H. M. Price, O. Zilberberg, T. Ozawa, I. Carusotto, andN. Goldman, Phys. Rev. Lett. , 195303 (2015).[10] M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, andI. Bloch, Nature , 55 (2018).[11] O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P.Chen, Y. E. Kraus, and M. C. Rechtsman, Nature ,59 (2018).[12] F. Haldane and S. Raghu, Phys. Rev. Lett. , 013904(2008).[13] E. Prodan and C. Prodan, Phys. Rev. Lett. , 248101(2009).[14] C. L. Kane and T. C. Lubensky, Nat. Phys. , 39 (2014).[15] L. B. Shao, S.-L. Zhu, L. Sheng, D. Y. Xing, and Z. D.Wang, Phys. Rev. Lett. , 246810 (2008).[16] R. O. Umucal ılar, H. Zhai, and M. O. Oktel, Phys. Rev.Lett. , 070402 (2008).[17] D. J. Thouless, Phys. Rev. B , 6083 (1983).[18] N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev.Mod. Phys. , 015001 (2018).[19] G. E. Volovik, Universe in a helium droplet (Oxford Uni-versity Press, Oxford UK, 2003).[20] Y. X. Zhao and Z. D. Wang, Phys. Rev. Lett. , 240404(2013).[21] H. Nielsen and M. Ninomiya, Nucl. Phys. B , 20(1981).[22] X. Wan, A. M. Turner, A. Vishwanath, and S. Y.Savrasov, Phys. Rev. B , 205101 (2011). [23] R. Yu, H. Weng, Z. Fang, X. Dai, and X. Hu, Phys. Rev.Lett. , 036807 (2015).[24] Y. Kim, B. J. Wieder, C. L. Kane, and A. M. Rappe,Phys. Rev. Lett. , 036806 (2015).[25] D.-W. Zhang, Y. X. Zhao, R.-B. Liu, Z.-Y. Xue, S.-L.Zhu, and Z. D. Wang, Phys. Rev. A , 043617 (2016).[26] W. B. Rui, Y. X. Zhao, and A. P. Schnyder, ArXiv e-prints (2017), arXiv:1703.05958 [cond-mat.mes-hall].[27] Y. X. Zhao, A. P. Schnyder, and Z. D. Wang, Phys. Rev.Lett. , 156402 (2016).[28] A nodal line on a mirror plane can also be protected byunitary mirror symmetry.[29] L. Fu and C. L. Kane, Phys. Rev. B , 195312 (2006).[30] C. Fang, Y. Chen, H.-Y. Kee, and L. Fu, Phys. Rev. B , 081201 (2015).[31] Y. X. Zhao and Y. Lu, Phys. Rev. Lett. , 056401(2017).[32] Supplemental Material.[33] M. Karoubi, K-Theory: An Introduction (Springer, NewYork, 1978).[34] S. Zhang, W. S. Cole, A. Paramekanti, and N. Trivedi,“Spin-Orbit Coupling in Optical Lattices,” in
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CLIFFORD ALGEBRAS
For the Clifford algebra C n , there are 2 n generators, namely 2 n × n Dirac gamma matrices γ a with a = 1 , , · · · , n ,satisfying { γ a , γ b } = 2 δ ab . (S1)It is clear that there is the (2 n + 1)th gamma matrix anti-commuting with all γ a with a = 1 , · · · , n , which is γ n +1 = ± i n γ γ · · · γ n . (S2)Note that all gamma matrices are hermitian, γ † a = γ a , and γ a = 1, with a = 1 , , · · · , n + 1. In this paper, Cl and Cl appear as building blocks of the models. Since the representation of the Clifford algebra by 2 n × n matrices isunique up to unitary transformations, we adopt the following convention for gamma matrices for our convenience.For Cl , the 8 × = σ ⊗ σ ⊗ σ , Γ = σ ⊗ σ ⊗ σ , Γ = σ ⊗ σ ⊗ σ , Γ = σ ⊗ σ ⊗ σ , Γ = σ ⊗ σ ⊗ σ , Γ = σ ⊗ σ ⊗ σ , Γ = σ ⊗ σ ⊗ σ , (S3)where the first four Γ µ with µ = 0 , , · · · , P = (1 + i Γ Γ ) to get 4 × Cl , γ = σ ⊗ σ , γ = σ ⊗ σ , γ = σ ⊗ σ , γ = σ ⊗ σ , γ = σ ⊗ σ , (S4)where γ i = − P Γ i P with i = 1 , , γ = − P Γ P , and γ = − P Γ P . Note that P Γ , P = 0. THE BOUNDARY EFFECTIVE THEORY
To open a boundary perpendicular to the w -direction, we first apply the inverse Fourier transform for k w to getthe first quantized Hamiltonian with the w -dimension in real space, H = (cid:88) i =1 sin k i Γ i + 12 i ( S − S † )Γ + (cid:34) m − (cid:88) i =1 cos k i −
12 ( S + S † ) (cid:35) Γ + it Γ Γ (S5)where S is the translation operator along the w -direction, and S † is that along the − w -direction. So, S | i (cid:105) = | i + 1 (cid:105) and S † | i (cid:105) = | i − (cid:105) with integer i numbering lattice site of the w -dimension, and accordingly the matrices are S = . . . ... ... ... ... .... . . 0 0 0 0 · · ·· · · · · ·· · · · · ·· · · · · · ... ... ... ... . . . . . . , S † = . . . . . . ... ... ... ... · · · · · ·· · · · · ·· · · · · ·· · · . (S6)If the boundary is opened with the system being on the non-negative part of the w -axis, the corresponding Hamiltonianis given by (cid:98) H = (cid:88) i =1 sin k i Γ i + 12 i ( (cid:98) S − (cid:98) S † )Γ + (cid:34) m − (cid:88) i =1 cos k i −
12 ( (cid:98) S + (cid:98) S † ) (cid:35) Γ + it Γ Γ , (S7)where still (cid:98) S | i (cid:105) = | i + 1 (cid:105) with i = 0 , , , · · · , but (cid:98) S † | (cid:105) = 0 and (cid:98) S † | i (cid:105) = | i − (cid:105) with i ≥
1. Explicitly, (cid:98) S = · · · · · · · · · · · · ... ... ... . . . . . . , (cid:98) S † = · · · · · · · · · . (S8)We now solve the Schr¨odinger equation for Eq. (S7) by substituting the ansatz | ψ k (cid:105) = | ξ k (cid:105) ⊗ (cid:80) ∞ i =0 λ i | i (cid:105) with | λ | < | i ≥ (cid:105) , the Schr¨odinger equation gives (cid:34)(cid:88) i sin k i Γ i + 12 i ( λ − λ − )Γ + (cid:32) m − (cid:88) i cos k i −
12 ( λ + λ − ) (cid:33) Γ + it Γ Γ (cid:35) | ξ (cid:105) = E | ξ (cid:105) , (S9)while at | i = 0 (cid:105) (cid:34)(cid:88) i sin k i Γ i + 12 i λ Γ + (cid:32) m − (cid:88) i cos k i − λ (cid:33) Γ + it Γ Γ (cid:35) | ξ (cid:105) = E | ξ (cid:105) . (S10)The difference of Eqs. (S9) and (S10) gives i Γ Γ | ξ (cid:105) = | ξ (cid:105) . (S11)This equation implies the boundary states occur merely in the 4D subspace with positive eigenvalue 1 of i Γ Γ , whichcorresponds to the projector P = 12 (1 + i Γ Γ ) . (S12)Applying the projector to Eq. (S10), we have (cid:32)(cid:88) i sin k i Γ i + it Γ Γ (cid:33) | ξ (cid:105) = E | ξ (cid:105) , (S13)The difference of Eqs. (S13) and (S10), together with Eq. (S11), gives λ = m − (cid:88) i cos k i . (S14)The effective Hamiltonian for boundary states is just H eff ( k ) = P H ( k ) P (S15)for regions in k space satisfying Eq. (S14). THE TOPOLOGICAL INVARIANT
In this section, we first give a detailed derivation of the topological invariant in the main text, and then explicitlycalculate the topological invariant of the Dirac model.
The formula
For any region with a well-defined Berry connection, the second Chern character is the total derivative of theChern-Simons form, and therefore we can apply Stokes’ theorem to obtain the following identities, (cid:90) X ch ( F ) = (cid:90) ∂X Q ( A X ) , (S16) (cid:90) Y ch ( F ) = (cid:90) τ Q ( A Y ) − (cid:90) τ (cid:48) Q ( A Y ) , (S17) (cid:90) Y ch ( F ) = (cid:90) τ (cid:48) Q ( A Y ) − (cid:90) τ Q ( A Y ) . (S18)With these identities, we proceed that N = (cid:90) ∂X Q ( A X ) − (cid:90) ∂ X Q ( A Y )= (cid:90) X ch ( F ) − (cid:90) τ (cid:48) Q ( A Y ) + (cid:90) τ (cid:48) Q ( A Y )= (cid:90) X ch ( F ) + (cid:90) Y + Y ch ( F ) − (cid:90) τ Q ( A Y ) + (cid:90) τ Q ( A Y )= (cid:90) T / ch ( F ) − (cid:90) τ Q ( A Y ) + (cid:90) τ Q ( A Y )= (cid:90) T / ch ( F ) − (cid:90) τ Q ( A ) + (cid:90) τ Q ( A ) mod 2 . (S19)In the last equality, the Chern-Simons forms can be derived from any valence basis satisfying the boundary conditionon τ and τ imposed by RT symmetry, and we have used the fact that a gauge transformation preserving the boundarycondition can change either Chern-Simons term by an even number, namely, that only the expression in the last lineis gauge invariant. The second Chern number
Since the topological invariant is preserved under adiabatic deformations of the Hamiltonian without closing theenergy gap, let us set t = 0 throughout this section for technical simplification. We first calculate the second Chernnumber in the half BZ, T / . For this purpose, we need to derive a complete set of globally well-defined wave functionsin the BZ for valence bands, of which the existence is clear because RT symmetry makes the Chern number in thewhole BZ vanishing. Introducing R ( k ) = m − (cid:88) i =1 cos k i − cos k w , (S20) R i ( k ) = sin k i , i = 1 , , , (S21) R ( k ) = sin k w , (S22)we write the Hamiltonian in the concise form, H ( k ) = (cid:88) a =0 R a ( k )Γ a . (S23)Applying the unitary transformation, U = e π Γ Γ e πi Γ e − π Γ Γ , (S24)we have ˜Γ µ = U Γ µ U † (S25)˜Γ = (cid:18) i − i (cid:19) , ˜Γ i = (cid:18) γ i ˜ γ i (cid:19) , ˜Γ = (cid:18) γ ˜ γ (cid:19) , (S26)where ˜ γ = σ ⊗ σ , ˜ γ = σ ⊗ σ , ˜ γ = σ ⊗ σ , ˜ γ = σ ⊗ σ , (S27)and thereby U H U † = (cid:18) Q † Q (cid:19) (S28)with Q = − iR + R j ˜ γ j . (S29)We can renormalize Q to be a unitary matrix, ˆ Q = − i ˆ R + ˆ R j ˜ γ j , (S30)where ˆ R µ = R µ /R with R = (cid:113)(cid:80) µ R µ and µ = 0 , , ,
3. It is noteworthy thatˆ Q T ( k , k w ) = ˆ Q ( k , − k w ) , (S31)as required by RT symmetry. Accordingly, the valence eigenstates are given by | α, k (cid:105) = 1 √ U † (cid:18) − χ α ( k )ˆ Q ( k ) χ α ( k ) (cid:19) (S32)where χ α ( k ) is an orthonormal basis of the 4D Hilbert space for valence bands at each k , namely χ † α ( k ) χ β ( k ) = δ αβ . (S33)For convenience, we choose χ = , χ = χ = , χ = . (S34)The globally well-defined wave functions of Eq. (S32) with Eq. (S34) for valence bands give the concise expressionfor the Berry connection, A µ = 12 ˆ Q † ∂ µ ˆ Q. (S35)Substituting Eq. (S35) into the formula for the Berry curvature, F µν = ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ] (S36)we find F µν = 14 ( ∂ µ ˆ Q † ∂ ν ˆ Q − ∂ ν ˆ Q † ∂ µ ˆ Q ) , (S37)and thereby ch = − π (cid:15) µνλσ tr ∂ µ ˆ Q † ∂ ν ˆ Q∂ λ ˆ Q † ∂ σ ˆ Q, (S38)which turns out to be vanished after tedious derivations. Thus, the topological invariant for this model is determinedmerely by the boundary Chern-Simons terms, derived from valence wave functions with the boundary condition givenby RT symmetry. The representation of RT symmetry We now discuss how the RT operator RT = U RT ˆ K ˆ I w is represented in the valence band structure. First, we havethe identities, U RT U † RT = 1 , U RT = U TRT . (S39)The symmetry operator RT = U RT ˆ K ˆ I w gives the constraint on the Hamiltonian, U RT H ∗ ( k , k w ) U † RT = H ( k , − k w ) , (S40)which implies RT symmetry is represented by the wave functions as U RT | α, k , k w (cid:105) ∗ = | β, k , − k w (cid:105)U βα ( k , − k w ) . (S41) U αβ ( k , k w ) U ∗ βγ ( k , − k w ) = δ αγ , U Tαβ ( k , k w ) = U αβ ( k , − k w ) . (S42)The topological obstruction in the topological phase can be essentially stated as that there exist no globally well-defined valence basis that can simultaneously “diagonalize” H ( k ) and RT in the whole BZ. In other words, if werequire that U ( k ,
0) = U RT = U ( k , π ) , (S43)which is just the boundary condition of Eq. (7) in the main text, the valence basis must be singular at some points inthe bulk of the half BZ, T / . On the other hand, if the basis is well-defined in the whole BZ, the boundary conditionof Eq. (S43) must be violated.In particular, the wave functions of Eq. (S32) are clearly well defined in the whole BZ, provided χ α are well-definedperiodic functions, and accordingly give U αβ ( k , k w ) = iχ † α ( k , k w ) ˆ Q † ( k , k w ) χ ∗ β ( k , − k w ) . (S44)In this case, U RT is just the identity matrix, and one can verify that there are no χ α ( k ) that can transform ˆ Q † intoidentity matrix in the two boundaries with k w = 0 and π at the same time in the topological phases. The boundary Chern-Simons terms
The Hamiltonian restricted on the two boundaries τ and τ is H ( k , π/
0) = (cid:88) i =1 sin k i Γ i + (cid:32) m ± − (cid:88) i =1 cos k i (cid:33) Γ , (S45)In principle, there exists a global basis in τ ( τ ) with the boundary condition for valence bands, | α, k (cid:105) with α =1 , , ,
4, for H ( k , π ) ( H ( k , P T symmetry with (
P T ) =1. Note that RT restricted on either τ or τ is just P T . However, to derive the global wave function is quite technicallyinvolved.To warm up, we diagonalize the 1D Hamiltonian, H ( k ) = cos k σ ⊗ σ + sin k σ ⊗ σ . (S46)It is easy to obtain the two sets of valence eigenstates, ψ ( k ) = sin k − − cos k , ψ ( k ) = k − − cos k , k ∈ [ − π , π ψ ( k ) = − cos k − sin k , ˜ ψ ( k ) = − cos k − sin k , k ∈ [ π , π , (S48)where ψ , and ˜ ψ , have the singular points k = π and k = 0, respectively. To achieve a global basis, we shall gluethem at k = − π and π periodically and continuously. Through they are equal at k = π/
2, they are not smoothlyconnected at 3 π/ − π/
2. The following recombination realizes the continuity,( ψ ( k ) + ψ ( k )) / √ , ( ψ ( k ) − ψ ( k )) / √ , k ∈ [ − π/ , π/
2] (S49)cos( k/
2) ˜ ψ ( k ) + sin( k/
2) ˜ ψ ( k ) , sin( k/
2) ˜ ψ ( k ) − cos( k/
2) ˜ ψ ( k ) , k ∈ [ − π/ , π/
2] (S50)After further simplification, we arrive at the globally well-defined valence orthonormal basis, | , k (cid:105) = 12 sin k − cos k − sin k − − cos k , | , k (cid:105) = 12 − cos k − sin k k − sin k (S51)which can also be derived as | , k (cid:105) = 12 ( ψ + ˜ ψ ) , | , k (cid:105) = 12 ( ˜ ψ − ψ ) . (S52)With the insight from the above simple model, we now return to solving the Hamiltonian. First, we flatten theHamiltonian through dividing it by d = R ( k , π/ H = H ( k , π/ /d = ˆ d Γ + ˆ d i Γ i = (cid:18) ˆ d ˆ∆ˆ∆ − ˆ d (cid:19) , (S53)where (cid:80) µ =0 ˆ d µ = 1, and ˆ∆ is a real symmetric matrix,ˆ∆ = ˆ d d ˆ d d ˆ d − ˆ d ˆ d ˆ d − ˆ d d − ˆ d − ˆ d . (S54)Then, ψ ( ξ ) = (cid:18) ˆ∆ ξ − (1 + ˆ d ) ξ (cid:19) , ψ ( ξ ) = (cid:18) (1 − ˆ d ) ξ − ˆ∆ ξ (cid:19) (S55)are two forms of eigenstates. ψ , is singular if and only if ˆ d = ∓
1, and therefore ψ and ψ cannot both be singularat the same k . To construct globally well-defined wave functions, we combine them as ψ + = ψ + ψ = (cid:18) ˆ∆ ξ + (1 − ˆ d ) ξ − ˆ∆ ξ − (1 + ˆ d ) ξ (cid:19) , (S56)and find ψ † + ψ + = 4 + 2( ξ † ˆ∆ ξ + ξ † ˆ∆ ξ ) . (S57)The norm of ψ + is well defined if the matrix elements of ˆ∆ in the above equation is zero. We observe that if ξ = χ i and ξ = χ j in Eq. (S57), ξ † ˆ∆ ξ is just the ( i, j )-entry of ∆. Since ˆ∆ exactly has four zero entries for ˆ∆, we cancertainly construct a complete set of global wave functions, which are given by | , k (cid:105) = 12 [ ψ ( χ ) − ψ ( χ )] , | , k (cid:105) = 12 [ ψ ( χ ) + ψ ( χ )] , | , k (cid:105) = 12 [ − ψ ( χ ) + ψ ( χ )] , | , k (cid:105) = 12 [ − ψ ( χ ) − ψ ( χ )] . (S58)The signs in the combinations above have been carefully chosen, such that the four wave functions are orthogonal. m ˜ m π ˜ m N = N πW − N W m ’s. Explicitly, the orthonormal wave functions are | , k (cid:105) = 12 − d ˆ d ˆ d − ˆ d ˆ d − − ˆ d ˆ d ˆ d , | , k (cid:105) = 12 ˆ d − ˆ d ˆ d ˆ d − − ˆ d − ˆ d − ˆ d ˆ d , | , k (cid:105) = 12 − ˆ d − ˆ d ˆ d − ˆ d − ˆ d ˆ d d ˆ d , | , k (cid:105) = 12 − ˆ d ˆ d − d ˆ d ˆ d ˆ d − ˆ d d . (S59)Actually the wave functions are carefully ordered, and hence the Berry connection can be written in the conciseform, A αβ,j = 12 ( ˆ d α ∂ j ˆ d β − ˆ d β ∂ j ˆ d α − (cid:15) αβγδ ˆ d γ ∂ j ˆ d δ ) . (S60)Substituting the above expression into the formula of the Chern-Simons form, we find Q ( A ) = − π (cid:15) ijk tr( A i ∂ j A k + 23 A i A j A k )= − π (cid:15) ijk (cid:15) αβγδ ˆ d α ∂ i ˆ d β ∂ j ˆ d γ ∂ k ˆ d δ , (S61)which is just the winding number of the unit vector valued in the unit sphere S over the 3D BZ. The topological invariant
Since the second Chern number is zero, the topological invariant is given by the boundary Chern-Simons terms,more clearly the difference of winding numbers N πW and N W of ˆ d µ ’s in the two boundaries, N = N πW − N W mod 2 . (S62)In the particular model, the winding numbers are determined by˜ m π = m + 1 , ˜ m = m − , (S63)for the two boundaries, respectively. The results for typical masses in gapped regions of m are summarized in theTab. I. We find that all gapped regions with | m | < SIMULATION WITH ULTRACOLD ATOMS
For the simplicity of the simulation, we choose another set of Clifford algebra generatorsΓ (cid:48) = σ ⊗ σ ⊗ σ , Γ (cid:48) = σ ⊗ σ ⊗ σ , Γ (cid:48) = σ ⊗ σ ⊗ σ , Γ (cid:48) = σ ⊗ σ ⊗ σ , Γ (cid:48) = σ ⊗ σ ⊗ σ , Γ (cid:48) = σ ⊗ σ ⊗ σ , Γ (cid:48) = σ ⊗ σ ⊗ σ , (S64)which is related to the original ones by a unitary transformation that does not change the symmetry operator RT = ˆ K .According, the Hamiltonian is transformed to be H (cid:48) ( k , k w ) = sin k w Γ (cid:48) − cos k w Γ (cid:48) + ( m − (cid:88) i =1 cos k i )Γ (cid:48) + sin k Γ (cid:48) + sin k Γ (cid:48) + sin k Γ (cid:48) + t ( i Γ (cid:48) Γ (cid:48) ) , (S65)which has been divided into different parts for simulation. D CA ( a ) ( c )( b ) z = F z = F z = - F u d r B l A DB C
FIG. S1. (a) The quasi 1D tetragonal optical lattice. The four parallel chains are denoted as
A, B, C, D . (b) The unit cell. (c)The internal ground levels of F = 1 manifold split by an external magnetic field. We consider ultra-cold atoms with internal pseudospin moving in two ladders of optical lattice consisting of fourchains denoted as
A, B, C, D in Fig. S1(a) to simulate the Hamiltonian in Eq. (S65). The first two Pauli matrices ofΓ (cid:48) matrices act in the spaces spanned by {| l (cid:105) , | r (cid:105)} and {| u (cid:105) , | d (cid:105)} , respectively. Then, | ψ A (cid:105) = | l (cid:105) ⊗ | u (cid:105) , | ψ B (cid:105) = | l (cid:105) ⊗ | d (cid:105) , | ψ C (cid:105) = | r (cid:105) ⊗ | u (cid:105) , | ψ D (cid:105) = | r (cid:105) ⊗ | d (cid:105) in Fig. S1(b).The internal pseudospin is simulated by two levels | F = 1 , F z = − (cid:105) and | F = 1 , F z = 0 (cid:105) among the three levelswith F = 1, which are slightly split by an external magnetic field, as shown in Fig. S1(c). We set the amplitude ofinter-chain nearest-neighbor (NN) hoppings as unit, and all next-nearest-neighbor (NNN) hoppings are introduced bythe photon-assisted tunneling. When two counter-propagating Raman lasers along r are resonant with the internallevels, the two-photon process can lead to the effective spin-orbital interaction, H SO = (cid:32) ˆ p M + δ Ω e ik r Ω e − ik r ˆ p M − δ (cid:33) (S66)in the subspace spanned by {| F = 1 , F z = 0 (cid:105) , | F = 1 , F z = − (cid:105)} . Here, ˆ p is the momentum operator along r , δ is theenergy splitting by the Zeeman effect, k is the wave vector of the lasers along r and Ω is the Rabbi frequency. Toobtain a tight-binding model, the hopping coefficient T RR (cid:48) between sites R and R (cid:48) can be derived as T RR (cid:48) = (cid:104) W R | H SO | W R (cid:48) (cid:105) , (S67)where | W R (cid:105) are the Wannier wave functions of the optical lattice. The general form of T RR (cid:48) is t σ + t σ + t σ + t σ ,with all t i highly tunable. In addition, we note that all pseudospin-independent phases of hoppings can be tuned bythe usual synthetic gauge field in cold-atom experiments. sin k w Γ (cid:48) This term can be translated to the tight-binding form assin k w Γ = sin k w σ ⊗ σ ⊗ σ ⇒ (cid:88) n (cid:88) s = ↑ , ↓ (cid:104) (cid:16) f † A,n,s f C,n +1 ,s − f † A,n +1 ,s f C,n,s (cid:17) + (cid:16) f † B,n,s f D,n +1 ,s − f † B,n +1 ,s f D,n,s (cid:17) (cid:105) + H . c ., (S68) ( c ) - m m DA CB n + 1n ( f ) n + 1n
A CB D ( e ) n + 1n
A CB D ( d ) n + 1n
DA CB n + 1
DCBA n ( b )( a ) n + 1n DA CB
FIG. S2. The simulation of all terms in Eq. (S65). The light-blue and red arrows represent pseudospin-independent hoppings,and the former has an additional phase π . The hoppings depicted by the green arrows involve pseudospin-orbital couplings. where the first term corresponds to the NNN hoppings between two up-chains and the second term is the NNNhopping between two down-chains. Here and below, f i,n,s with i = A, B, C, D is the annihilation operator of site n of the i -chain with pseudospin s . It is depicted in Fig. S2(a), where the light-blue arrows represent pseudospin-independent hoppings with an additional phase π , and the red arrows represent pseudospin-independent hoppingwithout additional phase. − cos k w Γ (cid:48) This term only contains the NN inter-chain hoppings. There is an phase π for hoppings between two left-chains A, B , and no phase for those between two right chains, C and D . We then have − cos k w Γ = − cos k w σ ⊗ σ ⊗ σ ⇒ − (cid:88) n (cid:88) s = ↑ , ↓ (cid:16) f † A,n +1 ,s f A,n,s + f † B,n +1 ,s f B,n,s (cid:17) + 12 (cid:88) n,s (cid:16) f † C,n +1 ,s f C,n,s + f † D,n +1 ,s f D,n,s (cid:17) + H . c ., (S69)which is illustrated in Fig. S2(b). (cid:16) m − (cid:80) j =1 cos k j (cid:17) Γ (cid:48) We regard m = m − (cid:80) j =1 cos k j as an internal parameter of the cold-atom system. Then, m − (cid:88) j =1 cos k j Γ = m σ ⊗ σ ⊗ σ ⇒ m (cid:88) n (cid:88) s = ↑ , ↓ (cid:16) f † A,n,s f A,n,s + f † B,n,s f B,n,s (cid:17) − (cid:16) f † C,n,s f C,n,s + f † D,n,s f D,n,s (cid:17) , (S70)which means the on-site energy of a cold atom at two left-chains A, B is m , and that at two right-chains C, D is − m , as illustrated in Fig. S2(c). This term can be readily simulated by applying a tilted W -potential.0 (cid:80) j =1 , sin k j Γ (cid:48) j Defining t j = sin k j , for j = 1 ,