Homotopy invariant in time-reversal and twofold rotation symmetric systems
AA new homotopy invariant in time-reversal and twofold rotation symmetric systems
Haoshu Li ∗ and Shaolong Wan † Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
The primary goal of this paper is to study topological invariants in two dimensional twofoldrotation and time-reversal symmetric spinful systems. In this paper, firstly we build a new homotopyinvariant based on the lifting of the Wilson loop to the universal covering group of the specialorthogonal group. And we prove the invariant we built agrees with the K theory invariant. We gobeyond the previous understanding of the Wilson loop unwinds in more than two occupied bandsby finding an obstruction of such unwinding. Then, within this formalism, we classify four occupiedbands cases into two categories which may have the same Wilson loop spectrum but in differenttopological classes. Our theory implies that even when the Fu-Kane-Mele invariant vanishes, theexistence of a pair of gapless edge modes is topological protected by this new topological invariant.Finally, we present a tight binding model realizing the non-trivial phase.
I. INTRODUCTION
Over the past few decades, there are many studies ontopological phases beyond the Landau paradigm [1–4] inparticular on the subject of topological insulators andtopological superconductors [5–10]. Symmetries play animportant role in researches. The internal symmetriesimplemented in fermionic systems are summarized bythe ten Altland-Zirnbauer(AZ) symmetry classes [11, 12].The classification of non-interacting topological phasesin ten AZ classes was systematically achieved by K the-ory method [13, 14]. In the past few years, topologi-cal insulators and superconductors protected not onlyby internal symmetries but also crystalline symmetrieshave been studied intensively. Topological insulators pro-tected by crystalline symmetries are called topologicalcrystalline insulators(TCI) [15]. TCIs give rise to in-teresting new features, such as the presence of gaplesssurface Dirac cones pinned to mirror planes [16, 17], andhigh-order topological insulators(HOTI) featuring cornercharges, corner states or hinge states [18, 19]. In thestudy of TCIs, one important goal is finding topologi-cal invariants protected by symmetries. Many methodsother than Berry curvature method have been discoveredto formulate topological invariants, for instance, sym-metry indicators at high symmetry points [19–22], Wil-son loop methods [16, 18, 23–26] and elementary bandrepresentations[22, 24].The focus of this paper is to survey two dimensionaltime-reversal(TR) and twofold rotation symmetric spin-ful systems. The K theory classification of this systemhave been finished [27] and the result of the classifica-tion implies the existence of a new Z topological invari-ant. This topological invariant is suggested describingby vortices configuration at high symmetry point in [28].However, to generate a vortices configuration, a smoothgauge is needed which is usually hard to implement. Thissystem was also studied in Ref.[26]. But the topologi- ∗ [email protected] † [email protected] cal invariant described in that paper is more focused onWannier band topology of the system and its obstructedatomic insulator nature. And Wilson loop spectrum con-sidered there is gapped. While in this paper, we will showa phase with non-trivial new topological invariant hav-ing gapless Wilson loop spectrum instead. Until now,there is still a gap between the formulation of the newtopological invariant and the K theory classification.In this paper, we develop a new topological invariantusing homotopy theory and prove it agrees with the Ktheory invariant. We build this new homotopy invariantbased on the lifting of the Wilson loop to the universalcovering group. We will show in our formulation that thetopology origin of the new Z topological invariant is dis-connectedness of some fixed point set, hence its meaningis transparent. We further show while nontrivial Wilsonloop winding in two occupied bands space is protectedby symmetries and can be unwinded when embedded inhigher-dimensional band space as discussed in [29, 30],there exists a further obstruction in certain four occupiedbands case whose existence forbids unwinding of Wilsonloop spectrum. And it is characterized by the new Z topological invariant. Furthermore, the existence of apair of gapless edge mode is topologically protected bythis new invariant.The paper is organized as follows: in Sec.II, we reviewthe topological classification of two dimensional twofoldrotation and TR-symmetric systems and three knowntopological invariants of these systems. A picture onwhat the new Z topological invariant is describing isgiven in this section as well. In Sec.III, we build our the-ory on two and four occupied bands cases. In Sec.III B 1,a new homotopy invariant is formulated and a proof ofthis homotopy invariant agrees with the K theory invari-ant is given. Then we show in Sec.III B 2 that four oc-cupied bands cases can be classified into two categorieswhich may have the same Wilson loop spectrum but indifferent topological classes. In Sec.IV, we introduce aHamiltonian which analytically implements Wilson loopsin previous sections, present some numerical results onits tight binding model, and discuss the physical mean-ing of the new topological invariant. Finally, we give ourconclusions in Sec.V. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b (a) (b) (c) FIG. 1. (a) Two dimensional skeleton X is showed, whichis the whole Brillouin zone. (b) One dimensional skeleton X is showed in red color. When ν Γ X = ν Γ Y = ν FKM = 0,the Hamiltonian of the system can be continuously deformto a Hamiltonian which is constant on X . (c) The effectivehalf Brillouin zone is showed in gray color, which can furthercollapse to a sphere(the boundary of the effective half Bril-louin zone is collapsed to a point) corresponding to one of twocomponents of the space X /X . II. TOPOLOGICAL CLASSIFICATION OF T AND C SYMMETRIC SYSTEM
In this paper, we focus on twofold rotation symmetrictwo dimensional systems in AZ class AII. The classifica-tion of these systems has been obtained in [27], since allsymmetries here are order two symmetries. The systemhas two Z strong indices and two Z weak indices. Thetwo Z weak indices are partial polarizations introducedby Fu and Kane along x and y directions respectively[9, 26], which are expressed as ν Γ X = 1 π (cid:20)(cid:90) π dk x Tr A x ( k x ,
0) + i log P f [ w ( π, P f [ w (0 , (cid:21) mod 2 ,ν Γ Y = 1 π (cid:20)(cid:90) π dk y Tr A y (0 , k y ) + i log P f [ w (0 , π )] P f [ w (0 , (cid:21) mod 2 , (1)where w mn ( k x , k y ) = (cid:104) u m ( − k x , − k y ) | T | u n ( k x , k y (cid:105) isthe sewing matrix of the TR-symmetry, u m ( k x , k y ) isthe periodic part of the Bloch wave function, and A x ( k x , k y )( A y ( k x , k y )) are Berry connections along x(y)-direction. Note these two partial polarizations are quan-tized by the twofold rotation symmetry. As for the two Z strong indices, one of the two Z strong indices is Fu-Kane-Mele invariant in time-reversal invariant insulators[9, 31, 32]. Another one has suggested characterized byvortices at high symmetry points [28] and topological in-variant defined at each C -symmetric channel [33]. How-ever, topological invariants defined in these way have notbeen shown directly related to K theory classification andtheir Z topological nature is not transparent. We adopta homotopic method to study this new Z strong index inthis paper. For convenience, we simply denote this new Z strong topological invariant by ν new .Now, we introduce the concept and notation in [34, 35].For a system with space-group action G , the space-group action on Hamiltonian is introduced as a ”twist” ( τ, c ) ofthat on the base space, where τ is the factor system of thesymmetry group and c ( g ) = 1( −
1) indicates the symme-try element g is a symmetry(anti-symmetry). And antiu-nitary symmetries are specified by a Z -valued function φ for group elements. Then an abelian group φ K ( τ,c ) G ( X )which characterizes the classification of the system canbe introduced. The K group φ K ( τ,c ) G ( T ) for the Bril-louin zone torus T provides topological classification oftwo dimensional crystalline insulators subject to symme-try group G .Make a cell decomposition with respect to symmetriesas in Figure 1. We give in appendix A that the remainingnew Z strong index ν new corresponds to the relative Kgroup φ K ( τ,c ) G ( X , X ) in the framework of twisted equiv-ariant K theory [34–36]. The classification K group of thesystem is φ K ( τ,c ) G ( T ) ∼ = Z ⊕ Z ⊕ φ K ( τ,c ) G ( X , X ) , (2)where the first Z summand is characterized the numberof occupied bands of the system(an even integer), thesecond three Z summands are characterized by threetopological invariants ν Γ X , ν Γ Y , ν FKM as we have men-tioned. Hence topological invariants ν Γ X , ν Γ Y , ν FKM aretopological obstruction invariants without which the re-duced K group is only expressed in terms of the relativeK group φ K ( τ,c ) G ( X , X ).We further point here that since φ K ( τ,c ) G ( X , X ) ∼ = φ ˜ K ( τ,c ) Z TC ( X /X ) ∼ = (cid:103) KO ( S ) ∼ = Z , when ν Γ X = ν Γ Y = ν FKM = 0, the Hamiltonian of the system can be continu-ously deformed [37] to a Hamiltonian whose value is con-stant on the boundary of the effective half Brillouin zone BZ . Then we can view the whole system as two Hamil-tonians over two disjoint two dimensional spheres whichare time reversal related. The other three topological in-variants ν Γ X , ν Γ Y , ν FKM are topological obstructions ofsuch deformation. The new Z strong index is character-ized by the second Stiefel-Whitney number of Hamilto-nian on the effective half Brillouin zone BZ since the Kgroup element in φ ˜ K ( τ,c ) Z TC ( X /X ) ∼ = (cid:103) KO ( S ) is capturedby the second Stiefel-Whitney number on sphere [25, 38],which can be read from Wilson loop spectrum [25]. Wecall the second Stiefel-Whitney number over the effectivehalf Brillouin zone the K theory invariant. However wecan not compute this new topological invariant ν new insuch a way in general as the deformation of the Hamil-tonian is not easy to find. In this paper, we develop ageneral Wilson loop method to compute ν new and proveit agrees with the K theory invariant.Now, we review how to read other three topologicalinvariants ν Γ X , ν Γ Y , ν FKM from the Wilson loop spec-trum. We first make a notation convention which is usedthroughout this paper. Denote the Wilson loop operator W ( k x ,k y +2 π ) ← ( k x ,k y ) used in Ref.[18] by W e ,k y ( k x ). Thepath is a straight line from the start point ( k x , k y ) to theend point ( k x , k y + 2 π ) along k y -direction. The Wanniercenters v yj ( k x ) and v xj ( k y ), where k x , k y ∈ [ − π, π ], canbe read from the Wilson loop spectrum with any chosenbranches, say ( − , ]. Then, ν Γ X = (cid:88) j v xj ( k y = 0) mod 2 ,ν Γ Y = (cid:88) j v yj ( k x = 0) mod 2 ,ν FKM = (cid:88) j [ v xj ( k y = 0) + v xj ( k y = π )] mod 2= (cid:88) j [ v yj ( k x = 0) + v yj ( k x = π )] mod 2 . (3)The Wannier centers of a T, C -symmetric system hassymmetry constraints { v j ( k x ) } T = { v j ( − k x ) }{ v j ( k x ) } T C = {− v j ( k x ) } . (4)Since the system is also T C -symmetric, we can com-pute the second Stiefel-Whitney number w over thewhole Brillouin zone [25]. However, it is not an indepen-dent topological invariant since it is actually equal to theFu-Kane-Mele invariant ν FKM . To prove this fact, notethe second Stiefel-Whitney number w is equal to theparity of the totally number of crossing at v j = whichis nothing but (cid:80) j [ v yj ( k x = 0) + v yj ( k x = π )] mod 2 [39].The symmetry constraint { v j ( k x ) } T = { v j ( − k x ) } has beenused to show crossing at v j = at momentum k x otherthan k x = 0 and k x = π come into pairs, i.e. v yj ( k x ) = implies v yj (cid:48) ( − k x ) = for some Wannier center index j (cid:48) . III. GENERAL THEORY
In this section, we present a general theory to com-pute the new topological invariant ν new under the as-sumption that three obstruction topological invariants ν Γ X , ν Γ Y , ν FKM vanish. This assumption can be weakenin our homotopic treatment, however, we will keep thisassumption to prove that our topological invariant agreeswith the cohomology element in K group.Now we point that considering the case that threeobstruction topological invariants vanish is sufficientfor the goal of distinguish different symmetry pro-tected topological phases. After building the the-ory of computing the new topological invariant ν new ,we can determine whether or not two Hamiltonian H and H are topological equivalent even if theirthree obstruction topological invariants not vanish.We denote their three obstruction topological invari-ants by ν X , ν Y , ν ( ν X , ν Y , ν ) for Hamiltonian H ( H ). For H being equivalent to H , we must first require ν X = ν X ,ν Y = ν Y ,ν = ν . (5)To compare their ν new , we can add a Hamiltonian H which has the same three obstruction topological invari-ants as H and H to Hamiltonian H and H respec-tively. Then the three obstruction topological invariantsof Hamiltonian H ⊕ H and H ⊕ H all vanish. Hence wecan directly compare the new topological invariant ν new of H ⊕ H and H ⊕ H . If their ν new are same, then H is topological equivalent to H in K theory stable sense,since [ H ⊕ H ] = [ H ⊕ H ] ⇐⇒ [ H ] = [ H ] (6)as elements in K group. And in the above process, theHamiltonian H can be simply chosen to be H .In the following two subsections, we discuss the case oftwo occupied bands and four occupied bands. We distin-guish these two cases since their K theory classificationis same but their homotopy classification is not. A. The case of two occupied bands
In this case, the Wilson loop matrix belong to O (2) dueto T C composite symmetry, its first homotopy group is π ( O (2)) ∼ = Z . Hence the homotopy classification is Z instead of Z in K theory classification. The new Z topo-logical invariant is the Euler number of the system. TheEuler number can be written in term of Berry curvaturein real gauge[25] e [ BZ ] = 12 π (cid:90) BZ F dk x dk y , (7)where e is the Euler class, [ BZ ] is the fundamental classof the base manifold BZ in homology, and their cap prod-uct e [ BZ ] is an integer called Euler number. In theWilson loop spectrum viewpoint, the Euler number isequal to a protected non-trivial winding in Wilson loopspectrum as discussed in Refs.[29, 30]. This non-trivialwinding number can not be changed under a symmetryprotected deformation of the Hamiltonian without clos-ing the energy gap. However, when one puts the twooccupied bands system into system having four or moreoccupied bands, the winding number in Wilson loop spec-trum can be changed as discussed in Ref.[30]. Howeverthere is still a Z topological number can not be changedin four occupied bands case. We will discuss this topicin the next subsection.Now, we want to associate the Z homotopy invariant e [ BZ ] to the Z K theory invariant ν new when embeddingthe system into a more occupied bands system. Since thethree obstruction invariants vanish, we can deform theHamiltonian in a symmetric way such that the deformedHamiltonian is constant on the boundary of the effectivehalf Brillouin zone. During this deformation, the Eulernumber e [ BZ ] does not change. After the deformation,the Hamiltonian can be viewed as two Hamiltonians onthe two effective half Brillouin zones which are related byTR-symmetry. The Euler number is then expressed as e [ BZ ] = 12 π (cid:90) BZ F dk x dk y = 12 π (cid:90) BZ (1)12 F dk x dk y + 12 π (cid:90) BZ (2)12 F dk x dk y = e [ BZ (1) ] + e [ BZ (2) ] , (8)where BZ (1) and BZ (2) are the two effective half Bril-louin zone. Since Hamiltonians on BZ (1) and BZ (2) arerelated by TR-symmetry, it is easy to check e [ BZ (1) ] = e [ BZ (2) ] . (9)The K theory topological invariant ν new is the secondStiefel-Whitney number over the effective half Brillouinzone BZ which is mod 2 version of the Euler number[38], that is ν new = w [ BZ (1) ]= e [ BZ (1) ] mod 2= e [ BZ ]2 mod 2 . (10)It should be noted that e [ BZ (1) ] is always an inte-ger, in other word, the Euler number e [ BZ ] is alwaysan even number. Since we have made the assumptionthat ν FKM = 0, which implies the second Stiefel-Whitneynumber over the whole Brillouin zone vanishes and theEuler number over the whole Brillouin zone is an evennumber. Hence ν new is well-defined.This ”proof” of the correspondence from the 2 Z -valuedEuler number e [ BZ ] to the Z -valued topological invari-ant ν new is not rigorous since the deformation of theHamiltonian is only in stable sense. But we will give arigorous proof of this correspondence in the next subsec-tion which discusses the four occupied bands case. Sincethis Z -valued topological invariant ν new is only usefulwhen the two occupied bands space is embedded in ahigher dimensional occupied bands space. B. The case of four occupied bands
1. Homotopy invariant
We present a new homotopy invariant for the case offour occupied bands in this section. The Wilson loopmatrices in four occupied bands case belong to SO (4) group due to TR-symmetry and twofold rotation sym-metry. In this system, we have a loop of Wilson loopmatrices k x (cid:55)→ W e ,π ( k x ) : S → SO (4). The topologyinformation of the system is entirely encoded by this loopsince Wilson loop matrix actually represents the tran-sition function of the system. And we give a proof inappendix D that a homotopy between two Wilson loopsinduce a homotopy between two Hamiltonians with re-spect to symmetry. To study the homotopy between twoWilson loops, we should lift loops in SO (4) to loops inthe universal covering group of SO (4) [40].There is a two-to-one covering map from Sp (1) × Sp (1)group to SO (4) group, where the Sp (1) group is the onedimensional symplectic group, that is the set of quater-nion numbers of modulus one. This covering map is p : Sp (1) × Sp (1) −→ SO (4)( g, h ) (cid:55)−→ ( x (cid:55)→ g − · x · h ) , (11)where x in the expression is a quaternion number, and x (cid:55)→ g − · x · h is a R -linear map from R to R . It is alinear map preserving the norm and having determinantone. Hence it belongs to SO (4). Furthermore, this mapis a group homomorphism p (( g , h ) · ( g , h ))= p (( g · g , h · h ))= x (cid:55)→ g − · g − · x · h · h = p (( g , h )) ◦ p (( g , h )) . (12)The following theorem [41, 42] will be needed in ourfurther discussion. Theorem 1.
Given two maps f , f : [0 , π ] → SO (4) ,such that they are homotoptic, i.e. f (cid:39) f rel ∂ [0 , π ] .And let ˜ f , ˜ f : [0 , π ] → Sp (1) × Sp (1) be liftings of f and f such that ˜ f (0) = ˜ f (0) . Then ˜ f (2 π ) = ˜ f (2 π ) and ˜ f (cid:39) ˜ f rel ∂ [0 , π ] . The rel ∂ [0 , π ] in the homotopy notation used in theabove theorem means the start and end point of thepath should be fixed, which in our case we simply fix f (0) = f (0) = f (2 π ) = f (2 π ) = I × . This theo-rem reduce the problem of finding a homotopy betweentwo Wilson loops to the problem of finding a homotopybetween liftings of the two Wilson loops.In this case, we have made the assumption that ν FKM = w [ BZ ] = 0. This assumption implies [ k (cid:55)→ W e ,π ( k )] = 0 ∈ π ( SO (4)), which implies the lift-ing of the Wilson loop is a loop. That is ˜ W e ,π (0) =˜ W e ,π (2 π ) = (1 , ∈ Sp (1) × Sp (1). Otherwise, if ν FKM = 1, ˜ W e ,π (2 π ) = − ˜ W e ,π (0) = ( − , − T C symmetry, which requires the Wilsonloop matrices to be real. The constraint imposed byTR-symmetry should be considered as well. The TR-symmetry impose that wW e ,π ( k x ) w − = W − e ,π ( − k x ) , (13)where w is the sewing matrix of the TR-symmetry, whichcan be taken to be a constant matrix in general, see equa-tion (35) in the next section. The homotopy between twoWilson loops should respect this constraint as well. De-note it by W ( k, t ), it should satisfy wW ( k, t ) w − = W − ( − k, t ) . (14)Now we lift this constraint to the covering group Sp (1) × Sp (1), p − ( wW ( k, t ) w − ) = p − ( W − ( − k, t )) ⇔ ˜ w − · ˜ W ( k, t ) · ˜ w = ˜ W ( − k, t ) , (15)where the bar on the right side of the equality means theconjugation of quaternion numbers, since for a modulusone quaternion number a we have a − = ¯ a . We denotethe time-reversal operator act on Sp (1) × Sp (1) by T , T : Sp (1) × Sp (1) −→ Sp (1) × Sp (1)( u, v ) (cid:55)−→ ˜ w − · ( u, v ) · ˜ w. (16)Then equation (15) can be written in term of this as T ( ˜ W ( k, t )) = ˜ W ( − k, t ) . (17)Note that for k = 0 or k = π , ˜ W ( k, t ) should be fixedby the time-reversal operator T for any t ∈ [0 , T ( ˜ W (0 , t )) = ˜ W (0 , t ) and T ( ˜ W ( π, t )) = ˜ W ( π, t ). Hencewe study the fixed point set of the time-reversal operator T .Change the basis if necessary, the sewing matrix of TR-symmetry can be assumed to be w = (cid:18) − I × I × (cid:19) ∈ SO (4). It maps a quaternion number x + x i + x j + x k to − x − x i + x j + x k . Since − x − x i + x j + x k =1 − · ( x + x i + x j + x k ) · j , the lifting ˜ w of w is ± (1 , j ) ∈ Sp (1) × Sp (1). The sign of ˜ w does not affect the form ofthe operator T , hence we simply take ˜ w = (1 , j ). For apair ( u, v ) = ( x + x i + x j + x k, x (cid:48) + x (cid:48) i + x (cid:48) j + x (cid:48) k ) ∈ Sp (1) × Sp (1). The image of this point under operator T is T ( u, v ) = ˜ w − · ( u, v ) · ˜ w = ( x + x i + x j + x k, x (cid:48) − x (cid:48) i + x (cid:48) j − x (cid:48) k )= ( x − x i − x j − x k, x (cid:48) + x (cid:48) i − x (cid:48) j + x (cid:48) k ) . (18)And the solution of T ( u, v ) = ( u, v ) is x = x = x = x (cid:48) = 0. Hence the fixed point set of the map T in Sp (1) × Sp (1) is( Sp (1) × Sp (1)) T = { (1 , x (cid:48) + x (cid:48) i + x (cid:48) k ) | ( x (cid:48) ) + ( x (cid:48) ) + ( x (cid:48) ) = 1 } (cid:71) { ( − , x (cid:48) + x (cid:48) i + x (cid:48) k ) | ( x (cid:48) ) + ( x (cid:48) ) + ( x (cid:48) ) = 1 } = X (cid:71) Y, (19) where we denote X = { (1 , x (cid:48) + x (cid:48) i + x (cid:48) k ) | ( x (cid:48) ) + ( x (cid:48) ) +( x (cid:48) ) = 1 } and Y = { ( − , x (cid:48) + x (cid:48) i + x (cid:48) k ) | ( x (cid:48) ) + ( x (cid:48) ) +( x (cid:48) ) = 1 } . We note that X and Y are disconnected in Sp (1) × Sp (1) and X, Y each is a two dimensional sphere.The simplest trivial phase has a constant Wilson loop k (cid:55)→ I × . The lifting of this Wilson loop is also a con-stant loop which is k (cid:55)→ (1 , k = π , it belongto fixed point subset X . We can conclude the followingtheorem: Theorem 2.
The TR-symmetric and twofold rota-tion symmetric Hamiltonian with four occupied bandswith vanishing ν FKM is topologically trivial if the lift-ing ˜ W e ,π ( k x ) of its Wilson loop W e ,π ( k x ) satisfies ˜ W e ,π ( k x = π ) ∈ X . Otherwise if ˜ W e ,π ( k x = π ) ∈ Y , itis non-trivial. We present a proof in appendix B. The basic idea be-hind this is the disconnectedness of the fixed point setis an obstruction to deform the Wilson loop to a trivialloop. It implies that even in four occupied bands case,the spectrum of some non-trivial Wilson loop can notunwind though its winding number can change. Hencethe first component in tuple ˜ W e ,π ( k x = π ) is a Z ho-motopy invariant distinguishing the trivial and the non-trivial phases.
2. Examples
In the following, we will prove the above homotopyclassification agrees with the K theory classification. Andwe show it by two typical examples. As we will show, ex-amples can be classified into two categories. Each systemin the first category has two TR-symmetry related
T C -symmetric channels. It can be expressed as E = E I ⊕ E II T C E I , II ⊆ E I , II T E I ⊆ E II T E II ⊆ E I , (20)where E is the space of occupied bands, E I and E II aretwo channels. While each system in the second categoryis characterized by E = E I ⊕ E II T C E I , II ⊆ E I , II T E I ⊆ E I T E II ⊆ E II . (21)Examples from these two categories may have thesame Wilson loop spectrum, but their topologicalclasses(trivial or non-trivial) is different.We first consider the following example, the Wilsonloop is W e ,k y = π ( k x )= cos nk x sin nk x − sin nk x cos nk x nk x sin nk x − sin nk x cos nk x , (22)where n is an integer. And the sewing matrix of theTR-symmetry is w = (cid:18) − I × I × (cid:19) . (23)Since the Wilson loop is block diagonal and the sewingmatrix w is block off-diagonal, the Wannier bands canbe decomposed to two channels such that they are TR-related and each is T C -symmetric. It is shown inappendix C we can use the Wilson loop operator todecompose the occupied bands space into two chan-nels which are TR-related and each channel is T C -symmetric, which implies this example is in the first cat-egory.We calculate the lifting of the Wilson loop˜ W e ,k y = π ( k x ) = (cos nk x + i sin nk x , W e ,k y = π ( k x = π ) is equal to (( − n ,
1) which be-longs to X for even n and belongs to Y for odd n . Thenby theorem 2, for even n the system is topologically triv-ial and for odd n the system is topologically non-trivial.Without loss of generality, we prove the equivalenceof this homotopy topological invariant to the K theorytopological invariant ν new for n = 1 case. The Wilsonloop spectrum is shown in Figure 2(a). It can be seenthe spectrum is not periodic over the effective half Bril-louin zone. However, the Hamiltonian can be deformto a Hamiltonian which is constant on the boundary ofthe effective half Brillouin zone since three obstructioninvariants is zero. We present a deformation in the Wil-son loop spectrum level such that the winding numberof each Wannier band is not changed. We present thedeformed Wilson spectrum in Figure 2(b). The deforma-tion respects the symmetries of the system. And notethat if we denote black Wannier bands space as channelI and red Wannier bands space as channel II, then thesetwo channels are TR-related and each channel is T C -symmetric. Note that on the left effective half Brillouinzone( k x ∈ [ − π, ν new = w [ BZ ] over the effective half Brillouinzone is 1, which implies this phase is non-trivial. Hencewe conclude the homotopy invariant we obtained aboveagrees with the K theory invariant ν new .Now, we elaborate another example which behaves dif- k x v - π π - (a) k x v - π π - (b) FIG. 2. (a) The original Wilson loop spectrum. There are fourWannier bands label by black, red, dashed black and dashedred. The black band coincides with the red band and theblack dashed band coincides with the red dashed band. (b)The deformed Wilson loop spectrum. The winding numberof each Wannier band is not changed. And symmetries ofWannier bands are preserved. ferent from the first example. The Wilson loop is W e ,k y = π ( k x )= cos mk x sin mk x − sin mk x cos mk x nk x sin nk x − sin nk x cos nk x , (24)where m and n are integers, and m + n is even by therequirement of vanishing of the Fu-Kane-Mele invariant.And the sewing matrix of the TR-symmetry is w = − −
10 0 1 0 . (25)Since both W e ,k y = π ( k x ) and w are block diagonal, theoccupied bands space can be decomposed into two T, C -symmetric subspaces with two occupied bands, which im-plies this example belongs to the second category. Weapply a basis transformation on the sewing matrix to theprevious standard form w = (cid:18) − I × I × (cid:19) . And underthis basis transformation, the Wilson loop becomes W e ,k y = π ( k x )= cos mk x mk x
00 cos nk x nk x − sin mk x mk x − cos nk x nk x . (26)And the lifting of this Wilson loop is˜ W e ,k y = π ( k x )= (cid:18) cos mk x + cos nk x m + n k x ) + j sin mk x − sin nk x m + n k x ) , cos( m + n k x ) − j sin( m + n ) k x m + n k x ) (cid:19) . (27)The midpoint of the lifting is ˜ W e ,k y = π ( k x = π ) = (cid:18) ( − m +( − n − m + n , ∗ (cid:19) = (cid:18) ( − m [1+( − n − m ]2( − m + n , ∗ (cid:19) =(( − m ( − − m − n , ∗ ) = (( − m − n , ∗ ). Hence the systemis in a trivial phase if and only if m − n is even. Note m − n is ensured to be an integer since m + n is even by the re-quirement of vanishing of the Fu-Kane-Mele invariant.Now consider the m = n = 1 case, the Wilson loopspectrum is identical to the first example with n = 1.But their topological classes are different, the first exam-ple is non-trivial and this example is trivial. The reasonis these two examples belong to different categories de-scribed above, and this example can reduce to the prob-lem of two subsystems with two occupied bands and thefirst example can not.Since in this example, the system can be decomposedto two subsystems with two occupied bands. We can rig-orously prove the conclusion ν new = e [ BZ ]2 for the case oftwo occupied bands. Denote occupied spaces of two sub-system by E and E . Let m, n be even so each subsystemhas vanishing Fu-Kane-Mele invariant. The Euler num-bers of two subsystem is given by e ( E )[ BZ ] = m and e ( E )[ BZ ] = n respectively. Adding m by two chang-ing the Euler number of the subsystem E by one, andthe homotopy invariant of the whole four occupied sys-tem changed by one as well. That is 1 = ∆ ν new =∆( e ( E )[ BZ ]2 ) = ∆( e ( E )[ BZ ]2 ) + ∆( e ( E )[ BZ ]2 ), which im-plies ν new = e ( E )[ BZ ]2 + e ( E )[ BZ ]2 . By linearity, one canconclude ν new = e [ BZ ]2 for the case of two occupied bands.Now, we prove the equivalence of this homotopy in-variant to the K theory invariant ν new in this example.Consider a non-trivial phase, say m = 1 , n = 3. It can beseen the Wilson loop spectrum is not periodic over theeffective half Brillouin zone. And we can not deform theWilson loop spectrum with respect to symmetries with-out alternating winding number of each Wannier band.However, we have proved the homotopy class of this ex-ample only depends on m − n . Thus, this phase is equiva-lent to a phase with m = 2+2 j, n = 4+2 j, j ∈ Z . Wilsonloop with m = 2 + 2 j, n = 4 + 2 j has periodic spectrumover the effective half Brillouin zone. And the K the-ory invariant is ν new = w [ BZ ] = m + n mod 2 = 1mod 2 which implies this phase is non-trivial. Therefore,the homotopy invariant pr ( ˜ W e ,π ( k x = π )) agrees withthe K theory invariant ν new , where pr in this expressiontakes the first component of the tuple.From above, all examples can be deformed to the abovetwo typical examples without changing the Wilson loopspectrum winding. And examples belongs to the first cat-egories are characterized by a single Wilson loop windingnumber n , the new topological invariant is expressed by ν new = n mod 2. Meanwhile, examples belongs to thesecond categories are characterized by two Wilson loopwinding numbers m and n , where each winding num-ber is the Euler number of a T, C -symmetric subsys-tem [43]. The new topological invariant is expressed by ν new = m − n mod 2. θ = θ = θ = θ = π θ =- π k y = k y = π k y =- π k y = π k y =- π FIG. 3. The map from the value of k y to the value of θ .The left side of the square(Brillouin zone) shows the valueof k y , and the right side of the square shows the value of θ . The value of θ between two labeled values of θ is linearinterpolated. The value of q ( k x ) in equation (28) is equal to q ( k x ) in gray region, and q ( k x ) in white region. IV. AN EXAMPLE OF HAMILTONIAN
In this section, we give an example of Hamiltonian onmomentum space whose Wilson loop operator can be an-alytically computed. The basic idea of this example isthat Berry connection of this system is flat(vanishes),hence the only contribution to the Wilson loop operatoris the transition function between two patches. The eightbands Hamiltonian is given in the form of the image ofdimension raising isomorphism, H ( k x , k y ) = cos θ (cid:18) q ( k x ) q T ( k x ) 0 (cid:19) + sin θ (cid:18) I × − I × (cid:19) , (28)where θ ∈ [ − π/ , π/ q ( k x ) in the expressionequals q ( k x ) for k y ∈ [ − π , π ] and equals q ( k x ) for k y ∈ [ − π, − π ] ∪ [ π , π ]. The map from the value of k y to the value of θ is shown in Figure 3. For k y ∈ [ − π , π ](gray region in Figure 3), q ( k x ) = q ( k x ) = nk x − sin nk x nk x cos nk x cos nk x − sin nk x nk x cos nk x , (29)where n is an integer. As for k y ∈ [ − π, − π ] ∪ [ π , π ] (whiteregion in Figure 3), q ( k x ) = q ( k x ) = . (30)This Hamiltonian is continuous and periodic on the Bril-louin zone. This system has TR-symmetry T and twofoldrotation symmetry C , which are T = (cid:18) − σ z ⊗ σ σ z ⊗ σ (cid:19) K,C = (cid:18) σ z ⊗ σ − σ z ⊗ σ (cid:19) , (31)where K is the complex conjugation operator. And thecomposite symmetry T C is equal to K .Assume this system is half-filling. The Wilson loopoperator can be analytically computed in this system,we give the computation in appendix D. The result isexpressed in term of q ( k x ) and q ( k x ) W e ,k y = π ( k x )= q T ( k x ) q T ( k x ) − = cos nk x sin nk x − sin nk x cos nk x nk x sin nk x − sin nk x cos nk x . (32)The TR-symmetry of the system induce the followingrelation on Wilson loop operator [18] w ( k x , k y ) W ∗ e ,k y ( k x ) w ( k x , k y ) − = W − e , − k y ( − k x ) , (33)since the Wilson loop operator in this system is real dueto the composite T C symmetry, W ∗ in above equationcan be taken to be W . Take the start point of theWilson loop be ( k x , k y ) = ( k x , π ), the sewing matrix w at the k y = π line can be shown to be independent of k x as in appendix E, w ( k x , π ) = q ( k x ) σ z ⊗ σ = (cid:18) − I × I × (cid:19) . (34)The equation (33) becomes wW e ,π ( k x ) w − = W − e ,π ( − k x ) , (35)where w = (cid:18) − I × I × (cid:19) . Note the Wilson loop W e ,k y = π ( k x ) (32) and the sewing matrix w of TR-symmetry in this example have been analysed in the pre-vious section. And the new topological invariant ν new is equal to n mod 2. For odd n , we have shown in theprevious section that Wannier bands in its Wilson loopspectrum can not unwind. Since the edge mode spectrumis a continuous deformation of the Wilson loop spectrum[44], it can be concluded that at least one pair of gap-less edge states is topological protected even when theFu-Kane-Mele invariant vanishes.We apply a cutoff on the Fourier transformation of theHamiltonian (28) in momentum space to obtain a tight binding model. The hopping matrices are taken to be t = π
00 0 0 0 0 0 0 π π π π π π π ,t = − i − − i − − i − − i − − i − i − i − i ,t = π
00 0 0 0 0 0 0 π π π π π π π ,t = π i π − i π π π i π − i π π π − i π i π π π − i π i π π ,t = i − i i − i − i i − i i ,t = π i π − i π π π i π − i π π π − i π i π π π − i π i π π , (36) - - - k x - - (a) - - - k x - (b) (c) FIG. 4. (a) The Wilson loop spectrum of the tight bindingmodel. (b) The energy spectrum under the periodic boundarycondition along x direction and the open boundary conditionalong y direction. (c) The density function of one of gap-less edge modes. The number of lattice is taken to be 20 innumerical computation. hopping matrices in the inverse direction are taken tobe conjugate transpositions of these tabulated hoppingmatrices, and any other hopping matrices are zero. TheWilson loop spectrum and the edge mode spectrum areplotted in figure 4. The existence of gapless modes pair istopologically protected when ν new = 1, which is a phys-ical implication of this new topological invariant ν new . V. CONCLUSIONS
In this work we presented a new homotopy invariantin twofold rotation symmetric system in AZ class AII.And we proved it agrees with the K theory topologicalinvariant. The main idea is lifting the Wilson loop tothe universal covering group, and the topology origin ofthis invariant is the disconnectedness of the fixed pointset of TR-symmetry acting on the covering group. Weshowed the four occupied case can be classified into twocategories which may have the same Wilson loop but withdifferent topological classes(one is trivial and the otheris non-trivial).In addition, we have shown in four occupied bandscase even when other three topological invariants(partialpolarizations and the Fu-Kane-Mele invariant) vanishes,there is a further obstruction whose existence forbids Wil-son loop spectrum to unwind. This extend the result inthe Refs. [29, 30], in which the conservation of Wilson α ′ α (a) aa ′ cc c ′ c ′ bbb ′ b ′ (b) Γ XY M (c)
FIG. 5. Cell decomposition of the Brillouin zone T . (a) Twodimensional cells α, α (cid:48) are showed. (b) One dimensional cells a, b, c, a (cid:48) , b (cid:48) , c (cid:48) are showed as red arrows. (c) Zero dimensionalcells Γ , X, Y, M are showed as blue points. loop spectrum winding number in two occupied bandscase and the unwinding of Wilson loop spectrum in moreoccupied bands are studied. We have further concludedwhen the new topological invariant is non-trivial, the ex-istence of a pair of gapless edge modes is topologicallyprotected. ACKNOWLEDGMENTS
The authors thank Yongxu Fu, Shuxuan Wang fordiscussions. This work was supported by NSFC GrantNo.11275180.
Appendix A: Computation of the K group
In this section, we present the calculation of the twistedequivariant K group of the system φ K ( τ,c ) G ( X ) using thespectral sequence method [45]. Another approach usingdimension raising isomorphism also works [14, 27, 35].Here we prefer the spectral sequence method for the con-venience of relating topological invariants to pages in aspectral sequence. We just simply give the calculationhere, the detail and explanation of this method can bereferred to [45]. The first step of this method is process-ing an equivariant cell decomposition which is shown asin Figure 5.The symmetry group of each zero dimensional cell is G = Z C × Z T . The little group G splits into the disjointunion of left cosets as G = G (cid:116) T G , (A1)where G = { g ∈ G | φ ( g ) = c ( g ) = 1 } = Z C is the sub-group of unitary symmetries. Time-reversal symmetry T ∈ G is a magnetic symmetry. There are two twistedirreducible representations of G = Z C which are theone dimensional representation with C = i and the onedimensional representation with C = − i . For such twoirreducible representations, Wigner test can be applied,0 E p, − n { Γ , X, Y, M } { a, b, c } { α } n=0 Z + Z + Z + Z Z + Z + Z Z n=1 0 Z + Z + Z Z n=2 Z + Z + Z + Z Z + Z + Z Z n=3 0 0 0TABLE I. The E page of the spectral sequence. which calculate the following formula W Tα = 1 | G | (cid:88) g ∈ G z ag,ag χ α (( ag ) ) , (A2)where α denote the corresponding irreducible represen-tation, the z is the factor system of G and χ α is thecharacter of the irreducible representation α . For theabove two irreducible representations, we obtain W TC = i = 0 W TC = − i = 0 , (A3)this implies that these zero dimensional cells belongto AZ class A, and these two irreducible representa-tions are tied by the magnetic symmetry T . We cando the same process for one dimensional and two di-mensional cells. Each cell has symmetry group G k = { e, T C } , the subgroup of unitary symmetries G k = { e } and the magnetic symmetry a = T C . The result ofWigner test is W T trivial = 1 which implies they belongto AZ class AI. This can be easily understood since T C is the magnetic symmetry satisfies ( T C ) = 1.Using the above facts, the first page of the spectralsequence E p, − n = (cid:81) j ∈ I porb φ K ( τ,c ) , − ( n − p ) G ( X p , X p − ) = (cid:81) j ∈ I porb φ | Dpj K ( τ,c ) | Dpj , − nG Dpj ( D pj ) are listed in Table I.The only non-vanishing differential for n ∈ { , , , } is d , : E , → E , , which is the set of compatible re-lations of symmetry indicators at high symmetry points.The value of d , : E , → E , is listed in Table II.One can calculate the E page of the spectral sequencefrom the above data via definition E p, − n = ker d p, − n / im d p − , − n . (A4)The E page is listed in Table III. One can check all thesecond order differentials d p, − n : E p, − n → E p +2 , − ( n +1)2 for n < E of the spectralsequence which is also the infinite page E ∞ is listed inTable IV.From these pages of the spectral sequence, we conclude E , ∞ = E , ⊆ φ | X K ( τ,c ) | X G ( X ) E , − ∞ = E , − = φ | X K ( τ,c ) | X G ( X , X ) E , − ∞ = E , − = φ K ( τ,c ) G ( X , X ) , (A5) where the first term E , ∼ = Z is characterized by thefilling number of the system, the second term E , − ∞ = φ | X K ( τ,c ) | X G ( X , X ) ∼ = Z + Z + Z is characterized bythree Z topological invariants which are ν Γ X , ν Γ Y and ν FKM , and the third term E , − ∞ = φ K ( τ,c ) G ( X , X ) ∼ = Z is characterized by a new Z topological invariant ν new asdiscussed in the main text. The K group of the system φ K ( τ,c ) G ( X ) relate to the infinite pages of the spectralsequence by the following two short exact sequence,0 → F , − → φ K ( τ,c ) G ( X ) → E , ∞ → , → E , − ∞ → F , − → E , − ∞ → . (A6)Since E , ∞ ∼ = Z is a free Z -module, the first short exactsequence splits and we obtain φ K ( τ,c ) G ( X ) ∼ = E , ∞ ⊕ F , − ∼ = Z ⊕ F , − . (A7)By the dimension raising isomorphism argument [27],we already know the reduced version of K group φ ˜ K ( τ,c ) G ( X ) ∼ = Z , which requires the second short exactsequence also splits, F , − ∼ = E , − ∞ ⊕ E , − ∞ ∼ = Z . (A8)In conclusion, φ K ( τ,c ) G ( X ) ∼ = E , ⊕ φ | X K ( τ,c ) | X G ( X , X ) ⊕ φ K ( τ,c ) G ( X , X ) , (A9)where the first component E , ∼ = Z is characterized bythe filling number of the system, the second component φ | X K ( τ,c ) | X G ( X , X ) is characterized by three Z topo-logical invariants which are ν Γ X , ν Γ Y and ν FKM , and thethird component φ K ( τ,c ) G ( X , X ) is characterized by anew Z topological invariant ν new which is concerned inthis paper. Appendix B: Proof of theorem 2
Proof.
Firstly, we prove if ˜ W e ,π ( k x = π ) ∈ Y , thesystem is non-trivial. Assume a homotopy between k x (cid:55)→ ˜ W e ,π ( k x ) and k x (cid:55)→ I × exists, and denote itby W ( k x , t ). We have a continuous path W ( π, t ) ∈ X (cid:70) Y . Its start point is W ( π, ∈ Y , and its endpoint is W ( π, ∈ X . Since X and Y is disconnectedin Sp (1) × Sp (1), it leads to a contradiction.Secondly, we prove if ˜ W e ,π ( k x = π ) ∈ X , the systemis trivial. It is sufficient to consider the half path of thelifting k x (cid:55)→ ˜ W e ,π ( k x ) , k x ∈ [0 , π ]. The other half can beconstructed via the time-reversal operator T ,˜ W e ,π (2 π − k x ) = T ( ˜ W e ,π ( k x )) . (B1)The topological classification of the half path of the lift-ing k x (cid:55)→ ˜ W e ,π ( k x ) , k x ∈ [0 , π ] is given by the relative1 Γ C i − i X C i − i Y C i − i M C i − i a b c TABLE II. The differential d , : E , → E , . The subscript C = (cid:32) i − i (cid:33) under each high symmetry point symbol meansthe irreducible representations C = i and C = − i come into pair due to time-reversal symmetry. E p, − n p = 0 p = 1 p = 2n=0 Z Z + Z + Z Z n=1 0 Z + Z + Z Z n=2 Z + Z + Z + Z Z + Z + Z Z n=3 0 0 0TABLE III. The E page of the spectral sequence. E p, − n p = 0 p = 1 p = 2n=0 Z Z + Z + Z Z n=1 0 Z + Z + Z Z n=2 Z + Z + Z + Z Z + Z + Z Z TABLE IV. The E = E ∞ page of the spectral sequence. homotopy group π ( Sp (1) × Sp (1) , X ), which is trivial asseen in the following exact sequence,( π ( X ) = π ( S ) ∼ = 0) → ( π ( Sp (1) × Sp (1)) ∼ = 0) → π ( Sp (1) × Sp (1) , X ) → ( π ( X ) ∼ = 0) . (B2)Hence any lifting path k x (cid:55)→ ˜ W e ,π ( k x ) with ˜ W e ,π ( k x = π ) ∈ X is homotopic to a constant path k x (cid:55)→ (1 ,
1) whichis trivial. It follows that k x (cid:55)→ W e ,π ( k x ) is homotopic toa constant loop k x (cid:55)→ I × with respect to symmetries. Appendix C: Decomposition of bands into twotime-reversal related channels
In this section we present a method of decomposition ofoccupied bands into two time-reversal related channels.Such an algorithm has already been investigated in [46].However, since in this article we apply a Wilson loopapproach method, and we do not require each occupiedbands to be smooth, an alternative method will be usedinstead. Following, we first simply review the concept ofWilson loop operator, then illustrate our method.Following the definition in [21], a discrete version of Wilson loop operator is( W k k ) mn = (cid:88) a,b... (cid:104) u m ( k ) | u a ( k (cid:48) ) (cid:105) (cid:104) u a ( k (cid:48) ) | u b ( k (cid:48) ) (cid:105) (cid:104) u b ( k (cid:48) ) | . . . | u n ( k ) (cid:105) , (C1)where k (cid:48) , k (cid:48) , . . . form a path connecting k and k . And m, n, a and b are indices of occupied bands.This is by definition an N occ × N occ matrix. We canalso define an operator ˆ W k k (a ( N occ + N unocc ) × ( N occ + N unocc ) matrix) acting on spin-orbital space associatedwith this matrix:ˆ W k k = (cid:88) i,j ∈ occ ( W k k ) ij | u i ( k ) (cid:105) (cid:104) u j ( k ) | = | u i ( k ) (cid:105) (cid:104) u i ( k ) | u a ( k (cid:48) ) (cid:105) . . . | u j ( k ) (cid:105) (cid:104) u j ( k ) | = P exp (cid:32) i (cid:90) k k ˆ P k ∂ ˆ P k · d k (cid:33) , (C2)where ˆ P k = (cid:80) i ∈ occ | u i ( k ) (cid:105) (cid:104) u i ( k ) | is the projector ontothe occupied subspace at k , and P in the front of expmeans ”path ordered”. We further denoteˆ W e ,k y ( k x ) = ˆ W ( k x ,k y +2 π ) , ( k x ,k y ) , (C3)where the path between the start point ( k x , k y ) and theend point ( k x , k y +2 π ) is a straight line, and e means thepath is along the k y direction. The eigenvalues of this op-erator restrict on occupied bands subspace have modulusone, which can be written as { e iv j ( k x ) | j = 1 , , . . . N occ } for each fixed k x and are independent of k y [18]. We de-note { v j ( k x ) | j = 1 , , . . . N occ } k x the Wilson loop spec-trum. Furthermore, each v j ( k x ) is a Wannier center (cen-ter of the Wannier function) of the system [18].We illustrate our decomposition algorithm by a quan-tum spin Hall insulator model. This insulator has corre-sponding Hamiltonian h ( k ) = sin( k x )(Γ zx + Γ xx ) + sin( k y )(Γ yx + Γ y )+ [2 − m − cos( k x ) − cos( k y )]Γ z , (C4)where Γ ij = σ i ⊗ τ j , and σ i ( τ i ) are Pauli matrices cor-responding to the spin (orbital) degrees of freedom. We2 k x FIG. 6. Wilson loop spectrum of Hamiltonian (C4) with m =3, two wannier bands are plotted by different colors, i.e. redand green, respectively. The green wannier band is labeled asband I, and the red wannier band is labeled as band II. plot its Wilson loop spectrum in Figure 6. In [18], theconcept of Wannier bands is defined as the set of Wanniercenters along x as a function of k y , v x ( k y ), or, vice versa,as the set of Wannier centers along y as a function of k x , v y ( k x ). We simply denote the green Wannier band andthe red Wannier band in Figure 6 by band I and band II,respectively. This system is time-reversal invariant, theWilson loop operator W e ,k y ( k x ) satisfies [18] T ˆ W e ,k y ( k x ) T − = ˆ W − e , − k y ( − k x ) = ˆ W † e , − k y ( − k x ) , (C5)where T is the time-reversal operator, which is anti-unitary. Hence the set of Wannier centers satisfies thefollowing constraint { v j ( k x ) } T = { v j ( − k x ) } , (C6)which in our case implies v I ( k x ) = v II ( − k x ) . (C7)In our case, the Wilson loop operator ˆ W e ,k y ( k x ) ateach ( k x , k y ) has a spectral decompositionˆ W e ,k y ( k x ) = e i πv I ( k x ) P I ( k x , k y ) + e i πv II ( k x ) P II ( k x , k y )+ 0 · P unocc ( k x , k y ) , (C8)where P I ( k x , k y ) is the projection operator of e i πv I ( k x ) eigenspace, P II ( k x , k y ) is the projection operator of e i πv II ( k x ) eigenspace and P unocc ( k x , k y ) is the projection - - - - - - (a) - - (b) FIG. 7. The entry P I ( k x , k y ) , along two circles in Bril-louin zone. (a) The trajectory is chosen to be a straightline between (0 . π, − π ) to (0 . π, π ) which is actually a cir-cle. (b) The trajectory is chosen to be a straight line between( − π, . π ) to ( π, . π ) which is also a circle. operator of the subspace of unoccupied bands. Hence, T ˆ W e ,k y ( k x ) T − = e − i πv I ( k x ) T P I ( k x , k y ) T − + e − i πv II ( k x ) T P II ( k x , k y ) T − = e − i πv II ( − k x ) T P I ( k x , k y ) T − + e − i πv I ( − k x ) T P II ( k x , k y ) T − , (C9)where in second equality we have made use of (C7). Onthe other hand, T ˆ W e ,k y ( k x ) T − = ˆ W † e , − k y ( k x )= e − i πv I ( − k x ) P I ( − k x , − k y ) + e − i πv II ( − k x ) P II ( − k x , − k y ) . (C10)Compare the above two equations, a relation on projec-tion operators can be obtained T P I ( k x , k y ) T − = P II ( − k x , − k y ) T P II ( k x , k y ) T − = P I ( − k x , − k y ) , (C11)which means two subbundles Ran ( P I ( k x , k y )) and Ran ( P II ( k x , k y )) are related by time-reversal symmetry.We further show the two projection operators P I ( k x , k y ) and P II ( k x , k y ) are continuous. Then Ran ( P I ( k x , k y )) and Ran ( P II ( k x , k y )) become two well-defined vector bundles. The two operators are continuoussince the Wilson loop operator W e ,k y ( k x ) is continuousand two Wannier bands are chosen in a continuous way.We check this statement by numerical computing of thetrajectory of P I ( k x , k y ) along some circles in Brillouinzone which is shown in Figure 7. Appendix D: Computation of the Wilson loop
In this section, we show two facts used in the maintext. Firstly, we compute the Wilson loop matrix of theexample mentioned in the main text. We show it canbe analytically computed. Furthermore, it provide anexplicit inverse map of the dimension raising map. Hencewe show the second fact that the classification problem of3
ABA B θ = θ = θ = θ = π θ =- π k y = k y = π k y =- π k y = π k y =- π FIG. 8. Four patches in Brillouin zone. White color patchis denoted by B , orange color patch is denoted by B , yellowcolor patch is denoted by A , and gray color patch is denotedby A . the Hamiltonian with respect to symmetries is equivalentto the classification problem of the path of Wilson loopmatrices with respect symmetries.We recall the Hamiltonian in the main text H ( k x , k y ) = cos θ (cid:32) q ( k x ) q T ( k x ) 0 (cid:33) + sin θ (cid:32) I × OO − I × (cid:33) , (D1)where θ ∈ [ − π/ , π/ q ( k x ) in the expressionequals q ( k x ) for k y ∈ [ − π , π ] and equals q ( k x ) for k y ∈ [ − π, − π ] ∪ [ π , π ]. We divide the Brillouin zoneinto four patches as shown in Figure 8, on each patch theperiodic part of the occupied bands wave function can beanalytically solved. On patch A , | u α ( k x , k y ) (cid:105) = 1 √ θ (cid:32) cos θ · q ( k x ) | v α (cid:105)− (1 + sin θ ) | v α (cid:105) (cid:33) . (D2)On patch B , | u α ( k x , k y ) (cid:105) = 1 √ − θ (cid:32) − (1 − sin θ ) | v α (cid:105) cos θ · q T ( k x ) | v α (cid:105) (cid:33) . (D3)On patch A , | u α ( k x , k y ) (cid:105) = 1 √ θ (cid:32) cos θ · q ( k x ) | v α (cid:105)− (1 + sin θ ) | v α (cid:105) (cid:33) . (D4)On patch B , | u α ( k x , k y ) (cid:105) = 1 √ − θ (cid:32) − (1 − sin θ ) | v α (cid:105) cos θ · q T ( k x ) | v α (cid:105) (cid:33) . (D5)The | v α (cid:105) , α ∈ { , , , } in above equations is the canon- ical orthonormal basis of C , | v (cid:105) = , | v (cid:105) = , | v (cid:105) = , | v (cid:105) = . (D6)We show this basis is parallel transported along k y oneach patch, in other word, the Berry connection vanishesalong k y . On patch A , the element of the Berry connec-tion is expressed as A αβ ( k x , k y )= (cid:104) u α ( k x , k y ) | d | u β ( k x , k y ) (cid:105) = 1 √ θ (cid:16) (cid:104) v α | q T ( k x ) cos θ − (cid:104) v α | (1 + sin θ ) (cid:17) (cid:20) − cos θ (2 + 2 sin θ ) (cid:32) cos θ · q ( k x ) | v β (cid:105)− (1 + sin θ ) | v β (cid:105) (cid:33) + 1 √ θ (cid:32) − sin θ · q ( k x ) | v β (cid:105)− cos θ | v β (cid:105) (cid:33) + 1 √ θ (cid:32) cos θ · dq ( k x ) | v β (cid:105) (cid:33) (cid:21) = − cos θ θ δ αβ + cos θ θ δ αβ +cos θ θ (cid:104) v α | q T ( k x ) · dq ( k x ) | v β (cid:105) = cos θ θ (cid:104) v α | q T ( k x ) · dq ( k x ) | v β (cid:105) , (D7)and in matrix form, A ( k x , k y ) = cos θ θ q T ( k x ) · dq ( k x ) . (D8)Hence the Berry connection vanishes along θ ( k y )-direction(there is no dθ term in equation (D8)). Thisconclusion applies on all patches.The transition function on the intersection of twopatches can be computed as well. t AB ( k x ) αβ = (cid:104) u αA ( k x , | u βB ( k x , (cid:105) = − (cid:104) v α | q T ( k x ) | v β (cid:105) , (D9)which in matrix form is t AB ( k x ) = − q T ( k x ) . (D10)Similarly, all the other transition functions are t A B ( k x ) = − q T ( k x ) t A A ( k x ) = I × t BB ( k x ) = I × . (D11)4The Wilson loop matrix(holonomy) is defined by in-tegral of parallel transports on patches and transitionfunctions. Let { U i } i =1 ,...,N be a cover including the loop l . Divide l to N pieces so that l i ⊆ U i . Let p i be junctionpoints of l i , namely, ∂l i = p i +1 − p i . The Wilson loop isdefined by [25, 35] W l = t ,N ( p ) · P e (cid:82) lN A N · t N,N − ( p N ) . . . P e (cid:82) l A , (D12)where A i is the Berry connection on U i and t i,j is thetransition function on U i ∩ U j . In this example, sincethe connection vanishes along k y -direction, all P e (cid:82) li A i terms are equal to I × . Hence the Wilson loop matrix(holonomy)is expressed by a formula only in terms ontransition functions W e ,π ( k x ) = t A A ( k x ) t AB ( k x ) t BB ( k x ) t B A ( k x )= q T ( k x ) q T ( k x ) − , (D13)which is the result used in the main text.Furthermore, this process provide an explicit inversemap of the dimension raising map. In general, any T, C -symmetric Hamiltonian can be expressed in the formof equation (D1), which suggests the following proce-dure. Given a Hamiltonian of a T, C -symmetric sys-tem, we calculate its Wilson loop matrix W e ,π ( k x ), itcorresponds to an one dimensional chiral system whoseHamiltonian is h chiral ( k x ) = (cid:32) W e ,π ( k x ) W T e ,π ( k x ) 0 (cid:33) , (D14)and chiral symmetry isΓ = (cid:32) I × − I × (cid:33) . (D15)This one dimensional chiral-symmetric system is still T, C -symmetric, and T = (cid:32) ww (cid:33) , C = (cid:32) BB (cid:33) , (D16)where w is the sewing matrix of TR-symmetry as in equa-tion (34), and B is the sewing matrix of twofold rotationsymmetry. This fact implies that the topological equiva-lent class of the Hamiltonian with respect to symmetriesis entirely encoded by the topological equivalent class ofthe Wilson loop matrix with respect to symmetries. Inthe following, we explicit show a homotopy of Wilsonloop with respect to symmetries gives a deformation ofthe Hamiltonian without breaking any symmetries.Assume that we have found a homotopy between twopaths of Wilson loop matrices with respect to symme-tries, that is a map W : [0 , π ] × [0 , → O ( N ) suchthat wW ( k, t ) w − = W − ( − k, t ) , W ( k, t = 0) = W e ,π ( k ) . (D17) We keep q ( k x , t ) invariant during the deformation, and W ( k, t ) and q ( k x , t ) are related by W ( k, t ) = q T ( k, t ) q T ( k ) − , (D18)which in other word says, q T ( k, t ) = W ( k, t ) q T ( k ) . (D19)Use the expression of the time-reversal operator T = (cid:32) − σ z ⊗ σ σ z ⊗ σ (cid:33) K . The TR-invariance of theHamiltonian (28) requires( σ z ⊗ σ ) q T ( k, t ) = − q ( − k, t )( σ z ⊗ σ ) . (D20)We check this requirement on q ( k, t ),( σ z ⊗ σ ) q ( k, t ) T ( σ z ⊗ σ )=( σ z ⊗ σ ) W ( k, t ) q ( k ) T ( σ z ⊗ σ )=( σ z ⊗ σ ) W ( k, t )( σ z ⊗ σ )( σ z ⊗ σ ) q ( k ) T ( σ z ⊗ σ )=( σ z ⊗ σ ) W ( k, t )( σ z ⊗ σ )( − q ( − k ))= − ( σ z ⊗ σ ) W ( k, t ) w − = − ( σ z ⊗ σ ) w − W T ( − k, t )= − ( σ z ⊗ σ )( σ z ⊗ σ ) q W T ( − k, t )= − q W T ( − k, t )= − q ( − k, t ) , (D21)where in the fourth equality w = q ( k )( σ z ⊗ σ ) = q ( σ z ⊗ σ ) has been used.Hence the problem of the classification of the Hamil-tonian with respect to symmetries is equivalent to theproblem of the classification of the Wilson loop matrixwith respect to symmetries. By this reason, we focus onthe classification of the Wilson loop matrices in the maintext, as we have shown above, the homotopy between twopaths of Wilson loop matrices induces a deformation ofHamiltonian. Appendix E: Time reversal related channels and thesewing matrix
The sewing matrix of the TR-symmetry of the examplein the main text is w αβ ( k x , k y = π )= (cid:104) u α ( − k x , π ) | T | u β ( k x , π ) (cid:105) = 12 (cid:16) (cid:104) v α | q T ( k x ) − (cid:104) v α | (cid:17) · (cid:32) − σ z ⊗ σ σ z ⊗ σ (cid:33) · (cid:32) q ( k x ) | v β (cid:105)− | v β (cid:105) (cid:33) = (cid:104) v α | q ( k x )( σ z ⊗ σ ) | v β (cid:105) , (E1)5which in matrix form is w ( k x , k y = π ) = q ( k x )( σ z ⊗ σ ) = q · ( σ z ⊗ σ ) . (E2)Hence the sewing matrix at k y = π line is independentof k x .Next, we show the subbundle of occupied bands canbe divide into two channels which are themselves T C -symmetric and are T -related to each other. Let E I ( E II )denote the channel I(II) subbundle. E I = (cid:71) k ∈ T { abcd | a, b, c, d ∈ C } ∩ E occ,k ,E II = (cid:71) k ∈ T { ab cd | a, b, c, d ∈ C } ∩ E occ,k , (E3) where E occ,k is the fiber of the occupied bundle at mo-mentum k . 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