Plateau transitions of spin pump and bulk-edge correspondence
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Plateaux Transitions of Spin Pump and Bulk-Edge Correspondence
Yoshihito Kuno and Yasuhiro Hatsugai Department of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan (Dated: February 19, 2021)Sequential plateaux transitions of quantum spin chains ( S =1,3/2,2 and 3) are demonstrated bya spin pump using dimerization and staggered magnetic field as synthetic dimensions. The bulk ischaracterized by the Chern number associated with the boundary twist and the pump protocol asa time. It counts the number of critical points in the loop that is specified by the Z Berry phases.With open boundary condition, discontinuity of the spin weighted center of mass due to emergenteffective edge spins also characterizes the pump as the bulk-edge correspondence. It requires extralevel crossings in the pump as a super-selection rule that is consistent with the Valence Bond Solid(VBS) picture.
Introduction.—
Integer S spin systems have been ex-tensively studied after Haldane proposed an exotic pro-posal for the uniform Heisenberg chains, the ground stateis gapped if the spin is integer [1]. Various numericalstudies support this conjecture positively [2–6]. ThisHaldane conjecture is proved for the VBS states of theAKLT model [7–9]. With open boundaries for S = 1 case,there appears extra low energy level structure within theHaldane gap which is described by emergent effective S = 1 / S = 1 , Z Berryphases and is consistently understood by the VBS picture[15–19].Recently, topological charge pump (TCP) [20] is a hottopic again in the condensed matter community since re-cent artificial quantum systems realized it experimentally[21–26]. The bulk-edge correspondence of the TCP hasbeen also studied recently [27, 28], although its bulk de-scription is old. The TCP of fermionic/bosonic systems[29–35] and spin pump for S = 1 / S ≥
1, especially bulk-edge correspondence, remains unclear.In this letter, we clarify the presence of a nontrivialtopological spin pump in a dimerized Heisenberg modelwith generic S ( S = 1 , / , Model.—
In this Letter, we consider a dimerizedHeisenberg model with generic S [16, 40], H DH = L − X j =0 J j ~S j · ~S j +1 , S j = S ( S + 1) , where ~S j = ( S xj , S yj , S zj ) and J j is dimerized, J j ∈ even = J and J j ∈ odd = J . The phase diagrams for S = 1 / / S = 1 and 2) [16, 17]have been discussed before. There are sequence of gappedSPT phases denoted by SPT1, SPT2, · · · , that appearby changing the ratio J /J . The schematic phase di-agrams for S = 1 , / , H DH are protected by one of D , time-reversal and bond-centered inversion symme-tries [12, 14, 44]. The state of the SPT phases in H DH with generic S is discussed by Z Berry phases protectedby the symmetry [16].The SPT phases of the bulk are characterized by theBerry phase by a local twist [45–50], J ( e iθ S +0 S − L − + e − iθ S − S + L − ), ( e iθ ∈ S , θ ∈ ( π, π ]). Here we assume thesites are labeled by i = 0 , · · · , L −
1. The Berry phase isgiven by iγ = R S A θ ( θ ) dθ where A θ ( θ ) = h G ( θ ) | ∂ θ G ( θ ) i and | G ( θ ) i is the ground state of H DH ( θ ). The SPTphases for S = 1 , γ = 0 , π ∈ Z that is consistentlyunderstood by the VBS picture by assigning π for thevalence bond (VB) [16, 17]. Note that the gap of a finitesystem remains open for a system with periodic boundarycondition even near the transition points. With twistedboundary condition, however, the gap vanishes mostly at θ = π . It is due to the topological charge of the two bulkscharacterized by different Z Berry phases. It suggeststhat extending a system by adding U (1) ∈ S twist as an SPT1 π /2 C (cid:1) ( t ) (a)(b) (e) SPT3SPT2 π /2 SPT2SPT1 SPT3 SPT4 C (cid:0) ( t ) (c)(d) FIG. 1. (a) Schematic figure for S = 1 SPT phase diagram, VBS pictures with left open boundary for each phases and apump protocol in φ − ∆ plane. The SPT phase colored in light blue (red) has 0( π )-Berry phase for J -link. It is noted that φ = 0 is not a transition point. In the VBS picture, the red triangle represents spin 1 /
2. (b) Plateau transition for S = 1. (c)Schematic figure for S = 3 / φ − ∆ plane. (d) Plateau transition for S = 3 /
2. (e) Schematic figure for S = 2 SPT phase diagram, the VBS pictures for eachphase and a pump protocol in φ − ∆ plane. The blue loop L = L − L and C = C − C = 5 − S = 2. Here, SPT phases are defined on ∆( t ) = 0. As a pump protocol, we set J = sin φ ( t ), J = cos φ ( t ), φ ( t ) = φ m [1 − cos(2 πt/T )] / associate dimension is useful for the topological transi-tion for finite systems. This S is small in the sense thatthe effects are infinitesimal in infinite systems [51, 52].In this sense, transition points of the S -enlarged systemis the same as that of the original one. Symmetry some-times requires gap closing for the S -enlarged system.The gap necessarily closes for a translationally invarianthalf-integer spin chain as an analogue of the Lieb-Schultz-Mattis theorem for the extended system [15].For each SPT phases in Fig. 1 (a), (c) and (e), the valueof γ is consistently understood by the VBS picture. Sincethe Berry phase has modulo 2 π ambiguity, the number ofbonds is specified in modulo 2. The VBS picture actuallyworks more than that as shown later. In the topologicalspin pump we propose, we have observed emergent edgestates predicted by the VBS picture. According to thebulk-edge correspondence, this is specified by the Chernnumber in the extended parameter space (See below). Inthe phase diagrams in Fig. 1 (a), (c) and (e), all criticaltransition points on ∆ = 0 are gapless for an infinitesystem. Even in a finite system, the gap closes as an S -enlarged system. Topological plateau transition of pump for genericspin.—
To characterize the ground state of H DH ,let us introduce a symmetry breaking term, H SB =∆( t ) P j ( − j +1 S zj , and consider an extended Hamilto-nian H = H DH + H SB where ∆( t ) is a periodic dy-namical parameter with a period T , which breaks all symmetries protecting the SPT phases. Here, a pumpprotocol is specified by a periodic modulation of theparameters. To be specific, we take J ( t ) = sin φ ( t ), J ( t ) = cos φ ( t ) with φ ( t ) = φ m [1 − cos(2 πt/T )] / t ) = sin(2 πt/T ). The amplitude of the modula-tion, φ m , is chosen so that the ground state of H ( t ) at t = 0 , T / H SB = 0) belong to the different SPTphases as ( J ( t ) , J ( t )) = (0 , , (sin φ m , cos φ m ), ∆ = 0for t = 0 and T /
2, respectively. Note that the gap alwaysremains open and the ground state is unique in the pumpas for a periodic system [28].The pump protocol is characterized by the following(spin) Chern number [53] of the periodic system withlocal boundary twist e iθ , C = i π Z T dt Z π dθB, (1)where B = ∂ t A θ − ∂ θ A t , A α = h g | ∂ α g i , α = θ, t where | g i is a gapped and unique ground state of H . The Berryphase defined at t = 0 and T / | G i = | g i (cid:12)(cid:12) ∆=0 , is quan-tized [36].This Chern number coincides to the total pumped spinof the bulk [27, 28] (See also Sec.I in supplemental mate-rial [54]). Here let us define a Chern number, C k , for theprotocol specified by the loop L k starting from the SPT1and passing through the other k -th SPT (SPT k ) phasesas shown in Fig. 1 (a), (c) and (e). We have used theformula [55] by diagonalizing the system [56] with evennumber of spins within the total S z ≡ P L − j =0 S zj = 0 sec-tor since the ground state is unique. The Berry phase at t = 0 is 2 πS since the dimers are decoupled and the twistis gauged out [16]. Let us define a path L kk ′ starting fromthe SPT k and passing through the SPT k ′ and the corre-sponding Chern number C kk ′ (See the blue loop in Fig. 1(e) as an example). Since the path can be deformed intothe two paths L k and L k ′ as L kk ′ = L k ′ − L k with-out gap closing, the Chern numbers satisfy the followingrelation C kk ′ = C k ′ − C k . The results of C k for S = 1, 3 / φ m , the Chern numberchanges step by step, it is an analogue of the quantumHall plateau transitions [58–62]. The maximum Chernnumber for each case is C = 2 S , that corresponds to thetotal number of possible dimerization transitions. Thecase for S = 3 is similar (see Sec.II in [54]). The Chernnumber of the bulk for the pump protocol is specifiedby the topology of the two SPT phases where the pumpare passing through. This is clear considering a systemwith edges as discussed later. The gap closing points ofthe SPT phases on the ∆ = 0 line are topological ob-struction for the loop specified by the pump protocol onthe φ − ∆ plane (strictly speaking, the obstruction is forthe S -enlarged system.) Therefore only when the looppasses through these points, the Chern number is allowedto change. The Chern number of a generic pump is givenby a sum of the Chern numbers of the critical points in-side the protocol loop [63]. The plateau transition of thepump in 2D is induced by the SPT transition of the 1Dspin chain [64]. Spin center of mass with open boundary.—
Let us in-vestigate the topological spin pump with open boundary,especially, properties of edge states. To this end, we em-ploy the density matrix renormalization group method inTeNPy package [65]. We consider the sCoM [27] given by P ( t ) = L − X j =0 h g ( t ) | x j S zj | g ( t ) i , where j = ( L − / x j = ( j − j ) /L ∈ ( − / , / J = ∂ t P . Note thatthe sCoM is only well defined for a system with bound-aries and is not well defined for a system with periodicboundary condition. Since the pump is periodic in time,the sCoM, P ( t ), is also. It implies 0 = R T dt ∂ t P = P i R t i t i − dt ∂ t P + P ( t ) | t i +0 t i − where P is piecewise contin-uous and have discontinuities at t = t i , ( i = 1 , , · · · )(periodicity in time is assumed for the summation) [27].Then, for any path passing through SPT k and SPT k ′ inthe parameter space, the pumped spin Q ekk ′ in the cycleby bulk for a system with open boundary condition is Q ekk ′ = X i Z t i t i − dt ∂ t P. This is related to the sum of the discontinuities I kk ′ as Q ekk ′ = I kk ′ , I kk ′ ≡ − X i P ( t ) (cid:12)(cid:12) t i +0 t i − . (2)In the spin model in this Letter, each discontinuity is ± (b) π t / T (cid:2) (a) (cid:3) E S (M S = -3) (cid:4) E S (-2) π t / T L LR (cid:5) E S (-1) R (cid:6) E S (0) (cid:7) E S (1) (cid:8) E S (2) (cid:9) (cid:10) FIG. 2. (a) ∆ E S for S = 1 case. We set L = 36. The label‘R’ and ‘L’ represent right and left edge states. The upwardand downward arrows represent the direction when the edgestates cross the blue dashed line. (b) ∆ E S for S = 3 / L = 32. (c) ∆ E S for S = 2 case. We set L = 24.The numbers at level crossings indicate the number of thedegeneracy of ∆ E S . of the edge state is allowed as P ( t ) (cid:12)(cid:12) t i +0 t i − = +1 / − / E S ( M S , t ) = E S ( M S + 1 , t ) − E S ( M S , t ) for each M S sector where E S ( M S , t ) is the ground state energy of H within a subspace total S z = M S . Let us discuss∆ E S ( M S , t ) at M S = 0 , ± , · · · . We choose the same pa-rameters as shown in the bulk calculation of Fig. 1 wherethe pumped spin C k = 1. The amplitude of the modu-lation, φ m , is set to connect the SPT1 to the midpoint ofthe SPT2 phase ( φ m = π/ , π/ , π/ S = 1 , / , S = 1, 3 / t = 0 and t = T /
2. Theseextremely small gaps are due to the interaction betweenthe edge states localized near both ends of the system asemergent degrees of freedom associated with non-trivialbulk [66]. It is an extension of the well known effective S = 1 / S = 1 case at t = T /
2, thatmakes four-fold degeneracy in the infinite system [2]. Thesymmetry breaking term H SB makes the degeneracy to (a) π t / T s(cid:11)(cid:12)(cid:13) 1(cid:14) L S =(cid:15)((cid:16)(cid:17) − − S= 1 , L= 80S= 3/2, L= 60S= 2, L= 60 S (cid:18)(cid:19) S (cid:20)(cid:21)(cid:22)(cid:23) t o t a l d i s c on ti nu it y FIG. 3. (a) Behavior of the sCoM with S z = 0 sector. For S = 1 case, we set L = 80, for S = 3 / L = 60. (b) Sys-tem size dependence of the total discontinuity of the sCoM. Itconverges to an integer I = lim L →∞ (cid:20) − P i P ( t ) (cid:12)(cid:12) t i + δtt i − δt (cid:21) . Inthe numerical calculation of the discontinuity at t = T / S = 1, 3 / δt = 0 . × − T , 0 . × − T ,and 0 . × − T . the level crossings observed in Fig. 2. The degeneracy at t = 0 are trivial and is given by the addition of the barespin S at the boundaries ( S ⊗ S = 2 S ⊕ · · · ⊕ S + 1) -fold in total and ∆ E S ( M S ,
0) = 0 for 4 S different M S sectors M S = − S, · · · , S −
1. Then thediscontinuity at t = 0 is − P i P ( t ) (cid:12)(cid:12) +0 − = 2 S .Although the degeneracy at t = T / / S eff = S − /
2. For S = 1 case at t = T /
2, it implies total degeneracy of (2 S eff + 1) = 4and ∆ E S ( M S , T /
2) = 0 for 4 S eff = 2 different M S sec-tors. The same can be true for the SPT k phase ofthe spin S model where the bulk is pictorially givenby the alternating N kB = 2 S − ( k −
1) VB on J -linkand N kB + ( k −
1) VB on J -link. The open bound-ary condition corresponds to cutting N kB VB ( J -link),which induces effective S k eff = N kB / E S ( M S , T /
2) = 0 are for4 S k eff = 2 N kB = 4 S − k −
1) different M S sectors[67]. It implies that the discontinuity at t = T / − P i P ( t ) (cid:12)(cid:12) T/ T/ − = − S k eff = − N kB (the sigh is deter-mined by the way of exchange of the left and right edgestates). It contributes to the entanglement entropy bylog(2 S eff + 1) [17]. Since this effective spin is sphericaldue to the symmetrization according to the VBS picture.This scenario is consistently confirmed by numericalcalculations. See Fig. 2 (a), (b) and (c). For S = 1 , / , t = T /
2, the degeneracy specified by ∆ E S are two,four, and six-fold, respectively. Further, for S = 2 systemat t = T /
2, the level structure due to spherical natureof the effective spin is discussed in detail (See Sec.IV in[54]).Then the total discontinuity of the pump protocol starting from SPT k and passing through SPT k ′ is givenby I kk ′ = N kB − N k ′ B = k ′ − k. (3)This is also consistently confirmed by numerical calcu-lations. See Fig. 3 (a), the behaviors of sCoM of thepump for the same parameters [68] in Fig. 2 (a), (b)and (c). For S = 1 , / , P ( t = T /
2) numerically obtained are very close to 1 , , N k =2 B ). The total discontinuity of the pump protocolstarting from SPT1 ( k = 1) and passing through SPT2( k ′ = 2) are given by 1(= k ′ − k ). The numerical explo-ration to the infinite size is shown in Fig. 3 (b) [69]. Itimplies the total discontinuity of the sCoM approachesto 1 for L → ∞ , that is, I = 1. Bulk-edge correspondence, VBS and Berry phases.—
The sCoM, P ( t ), is a piecewise continuous function, thatis, continuous except several discontinuities at t = t i due to the appearance of effective boundary spins. Thispumped spin by the continuous part, is given by the bulkand is given by the Chern number as [27] (See also Sec.Iin [54]), Q ekk ′ = 12 π Z π dθ ¯ Q bkk ′ ( θ ) = C kk ′ , (4)where ¯ Q bkk ′ = i R T dt ∂ t ¯ A ( t ) θ ( θ ) and ¯ A ( t ) θ is the Berry con-nection of the periodic system in the temporal gauge¯ A ( t ) t = 0 where the twist θ is distributed uniformly forall links. This uniform twist is transformed to the localboudary twist in Eq.(1) by the large gauge transforma-tion, which makes the Chern number invariant [27, 54].Eqs. (2) and (4) imply the bulk-edge correspondencefor the generic quantum spins as C kk ′ = I kk ′ = k ′ − k. It also imposes a constraint for the Berry phases γ k − γ k ′ ≡ πC kk ′ , mod 2 π. Here we have established the bulk-edge correspondenceand discussed the numerical results based on the VBSpicture. Reversely the topological stability of the Chernnumber implies the (fractionalized) effective S eff = S/ Conclusion.—
We have clarified the presence ofplateau transitions and a nontrivial topological spinpump in dimerized Heisenberg models ( S = 1 , / , , Acknowledgments.—
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Hatsugai, Phys. Rev. B , 99 (2000).[63] It is justified since one can deform the protocol loop toa sum of small loops around the critical points withoutgap closing. Here the Chern number of the critical pointis given by the Chern number of the small pump aroundthe point.[64] When one extends the parameter space, by 1, respectingthe symmetry of the SPT, these gapless points becomephase boundary lines [28] (for the detail see Sec. III in[54]).[65] J. Hauschild and F. Pollmann, SciPost Phys. Lect. Notes (2018).[66] The left and right edge states are identified by thedistribution of the local z -component magnetization h g ( t ) | S zj | g ( t ) i .[67] For S = 1 case in Fig. 2 (a), at t = 0 (SPT1 phase), S eff = 1 ( N B = 2) spins appear at edges. For 4 S eff =4 different M S sectors appears. The total degeneracy(2 S eff + 1) = (2 + 1) -fold degenerate states.[68] The numerical calculation of P ( t ) is set in S z = 0 sectorwithout the twist θ .[69] At t = 0, the model is in the dimerized limit, J = 0.Therefore, the discontinuity of the sCoM for S = 1, 3 / t = 0 with S z = 0 sector exactly becomes − − −
4, respectively.
SUPPLEMENTAL MATERIALI. Meaning of Chern number and Bulk edge correspondence of the spin pump
In this section, we show that the Chern number is given by a bulk pumped spin and the Chern number in theperiodic system with twist is related to the discontinuity of the sCoM, obtained in the open boundary system withouttwist.Let us start a modified Hamiltonian of the system with open boundary as H e ( θ ) = L − X j =1
12 ( e − iθ/L S + j S − j − + e + iθ/L S − j S + j − ) + · · · . The current operator is given J e = ∂ θ H e = − iL L − X j =1
12 ( e − iθ/L S + j S − j − − e iθ/L S − j S + j − ) . Then by the adiabatic approximation, the expectation value of the current is J e = hJ e i t = − iB e , where Berry curvature is B e = ∂ θ A t − ∂ t A θ , and the Berry connection is A α = h g | ∂ α g i , and | g i is a snap shot groundstates defined by H e | g i = | g i E . Especially in the temporal gauge A ( t ) t = 0, it is J e = + i∂ t A ( t ) θ , where A ( t ) θ is given by the Berry connection A α in arbitrary gauge as A ( t ) θ ( θ, t ) = A θ ( θ, t ) − A θ ( θ, − ∂ θ Z t dτ A t ( θ, τ ) . Let us introduce the large gauge transformation, U ( θ ) = e − iθ P L − j =0 j − j L S zj , j = L − . The Hamiltonian of the open system is transformed as H e ( θ ) = U H e U − , H e = H e ( θ = 0) , where H e is θ independent.Now the snap shot ground states satisfy following relations H e ( θ ) | g i = E | g i ,H e | g i = E | g i , | g i = U | g i . Since | g i is θ independent, using this gauge, we have the following Berry connections A t = h g | ∂ t g i = h g | ∂ t g i : θ -independent ,A θ = h g | ∂ θ g i = h g | U − ∂ t U | g i = − iP e ( t ) , where P e ( t ) is the sCoM for open system without twist θ , P e ( t ) = h g | L − X j =0 j − j L S zj | g i . This sCoM P e ( t ) is calculated in the main text.The above relation implies the Berry connection in the temporal gauge A ( t ) θ is given by A ( t ) θ ( θ, t ) = A θ ( θ, t ) − A θ ( θ,
0) = − i (cid:0) P ( t ) − P (0) (cid:1) . Then the current for the open system is given by J e = i∂ t A ( t ) θ = ∂ t P e . Now the pumped spin Q e [ t a ,t b ] (of the system with edges) for the period [ t a , t b ] is give by Q e [ t a ,t b ] = Z t b t a dt J e = P e ( t b ) − P e ( t a ) . However due to the periodicity in time T , total pumped spin is zero and is written as0 = Z T dt ∂ t P e = X i Z t i t i − dt ∂ t P e + P e ( t ) (cid:12)(cid:12)(cid:12)(cid:12) t i +0 t i − where P e is singular and discontinuous at t = t i , ( i = 1 , , · · · ) (periodicity in time is assumed for the summation).Since the contribution to the pumped spin with edges from continuous parts are due to bulk, the pumped chargedue to bulk is defined as Q e ≡ X i Z t i t i − dt ∂ t P e . The subscript e implies that it is for a system with edges. Note that it is determined only by the information of thesystem with edges as Q e = I, where I = − X i P e ( t ) (cid:12)(cid:12)(cid:12)(cid:12) t i +0 t i − , is a sum of the discontinuity due to edges. In the main text, we numerically estimated the sum of discontinuity inFig. 3 (b).Physically this pumped charge (spin) of bulk is related to the current of a periodic system, J b = ∂ t ¯ A ( t ) θ . We cansafely assume the pumped spin of the bulk, Q b as Q e = 12 π Z π dθ ¯ Q b ( θ ) , where ¯ Q b ( θ ) = i Z T dt ∂ t ¯ A ( t ) θ . ¯ Q b ( θ ) is given by a temporal gauge Berry connection ¯ A ( t ) θ .The temporal gauge Berry connection ¯ A ( t ) θ is obtained by the Berry connection ¯ A α (arbitrary gauge) of the periodicsystem described by the Hamiltonian ( S x,y,z = S x,y,zL )¯ H b ( θ ) = L X j =1
12 ( e − iθ/L S + j S − j − + e + iθ/L S − j S + j − ) + · · · = H e ( θ ) + 12 ( e − iθ/L S +0 S − L − + e + iθ/L S − S + L − ) , ¯ A α = h ¯ g b | ∂ α ¯ g b i , where | g b i is a snap shot ground states, ¯ H b | ¯ g b i = | ¯ g b i E , and also we have introduced an average over θ , which isjustified if the system size is sufficiently large [51, 55].From the above relation of Q e and ¯ Q b ( θ ), it establishes the bulk-edge correspondence I = C. Here, the Chern number C is for the periodic system as C = i π Z T dt Z π dθ ∂ t ¯ A ( t ) θ = i π Z T dt Z π dθ ¯ B, ¯ B = ∂ t ¯ A θ − ∂ θ ¯ A t , where ¯ A α is in arbitrary gauge.By considering ¯ H b ( θ ), which has uniform twist θ/L , we showed the bulk-edge correspondence I = C . However, aperiodic boundary system, which has one link twist θ , used in the main text, also gives Chern number equivalent tothat of ¯ H b ( θ ). From now on, we show it.Similar to the open system, one may perform the same unitary transformation U , and define H b as¯ H b = U H b U − . And we define a snap shot ground state, H b | g b i = | g b i E , which is related to | g b i as | ¯ g b i = U | g b i . Although H e is θ independent, H b depends on θ as H b = H e + 12 ( e iθ S +0 S − L − + e − iθ S − S + L − ) . The last term is the boundary twist which we have used for the calculation of the Chern number/ Berry phase in themain text. Then, the Berry connections, ¯ A α and ˆ A α for the ¯ H b and H b are related by¯ A θ = h ¯ g b | ∂ θ ¯ g b i = h g b | ∂ θ g b i + h g b | U − ∂ θ U | g b i = ˆ A θ − i ˆ P b ( t ) , ¯ A t = ˆ A t , where ˆ A α = h g b | ∂ α g b i and ˆ P ( t, θ ) = h g b | P L − j =0 j − j L S zj | g b i . ˆ P ( t, θ ) is an effective sCoM for the bulk, that depends onthe choice of the origin although the system is periodic. Therefore it is unphysical and does not have any experimentalsignificance.
Note that ˆ P ( t, θ ) is a smooth function of t and θ with periods T and 2 π , although P ( t ) for the systemwith boundaries is singular. Since ˆ P is smooth and periodic, the Chern numbers by the Berry connections ˆ A α and¯ A α are the same (their field strength are different). It is explicitly written asˆ B = ∂ t ˆ A θ − ∂ θ ˆ A t , ¯ B = ∂ t ¯ A θ − ∂ θ ¯ A t = ˆ B − i∂ t ˆ P b ,C = i π Z T dt Z π dθ ¯ B = i π Z T dt Z π dθ ˆ B. Hence, Chern number obtained from H b is equivalent to that of ¯ H b ( θ ). II. S = 3 plateau transition From the dimerized Heisenberg model of H DH of Eq. (1) in the main text, we expect that 2 S + 1 SPT phases appearfor arbitrary S case. By introducing the symmetry breaking term, S (2 S +1) pump loops with the nontrivial topologicalpump are allowed and the highest Chern number is C = 2 S . For S = 3 case, we consider the dimerized Heisenbergmodel of H DH of Eq. (1) in the main text with J = sin φ , J = cos φ . Here, we expect that in φ ∈ [0 , π/ S + 1 = 7 SPT phases. Then, we set an pump protocol, J = sin φ , J = cos φ , φ = φ m [1 − cos(2 πt/T )] / t ) = sin(2 πt/T ). For this protocol, we calculated the Chern number C of Eq. (2) in the main text by using theexact diagonalization [56]. The φ m dependent result of C is shown in Fig. 4, where we set system size L = 10. Thereis also no system size dependence. We observe plateau transitions again. The highest Chern number is 2 S = 2 × S case, further plateau transition behavior with more steps emerges.0 C . (cid:24) (cid:25)(cid:26) (cid:27)(cid:28) (cid:29) (cid:30) (cid:31) FIG. 4. Plateau transitions for S = 3 case. III. Quantization and plateau transition of Chern number from SPT transition line picture
In the main text, we showed how to quantize the Chern number. As an additional picture of the quantizationmechanism, in this supplementary, we also argue the quantization mechanism by considering the presence interaction.Let us introduce a perturbation, H zz = δJ z P L − j =0 S zj +1 S zj , where δJ z is small. The term H zz does not break thesymmetries of the SPTs. Accordingly, when one considers ∆ − φ − δJ z space for the system of H DH + H SB + H zz with small δJ z , the critical transition points on φ space form phase boundary lines separating the SPTs on φ − δJ z plane ( ∆ = 0) as shown in Fig. 5. For a finite small δJ z , each SPT phases are stable and remain gapped. For aninfinite system, the phase boundary lines are gapless for any twist θ .For this model parameter space, one sets the pump protocol defined previously. As shown in Fig. 5 (a), if oneconsiders the pump protocol with φ m = π/ S = 1 system defined in the main text, the pump protocol wraps thesingle phase boundary line separating the SPT1 and SPT2. One cannot untie the pump protocol loop and the phaseboundary line without gap closing. That is, the phase boundary line can be regarded as a topological obstruction.Then the calculation of C on the pump protocol gives C = 1. In other words, the Chern number may be given by alinking number of a pump protocol and the phase boundary lines. One can also set a larger pump protocol wrappingtwo phase boundary lines as shown in Fig. 5 (b). Then, one obtains C = 2. This pump can be also understood bydeforming the loop of the pump protocol. The loop can be adiabatically deformed to two loops wrapping a singlephase boundary line without gap closing as shown in Fig. 5 (c). Each loop gives C = 1. Thus, one can consider asum rule of C , obtain the total Chern number, C = 2. The number of the phase boundary line in a pump protocolcorresponds to the quantized value of C . From these arguments, increasing φ m corresponds to the increase of theloop size of the pump protocol. Then the number of the phase boundary line on φ − δJ z plane wrapped by the pumpprotocol increases. When the number of the phase boundary line changes, a transition of Chern number occurs. FIG. 5. Schematic figures for the mechanism of the quatization of the topological pump; (a) A pump protocol connects theSPT1 and SPT2 phases and wraps the single phase boundary line. (b) A pump protocol connects the SPT1 and SPT3 phasesand wraps two phase boundary lines. (c) Pump protocol loop of (b) case can be adiabatically deformed into two loops.
IV. Hybridization of edge states and S = 3 / edge state picture In Fig. 2 (a) and (b) in the main text, we showed the excitation spectrum obtained by the ground state energieswith different S z . There, multiple ingap states appear. The ingap states are left or right edge states. In particular,at high symmetry points t = 0 and T /
2, where H SB in the main text vanishes, some ingap states are degenerate.Here, we further study the spectrum structure from the presence of edge states. We focus on the spectrum of S = 2case at t = T / t = T / S = 3 / S = 3 / S = 1 AKLT spin chain wasnumerically studied, showed that such a VBS picture and the picture of S = 1 / / S = 2 system in the main text with openboundary at t = T / S z = 0 sector by using [56]. The four lowest energies are plotted in Fig. 6 (a). The fourenergies are split due to finite system size effects. If the edge state of the system is described by S = 3 / J e . We expect that,phenomenologically, J e is ∝ e − L/ξ , where L is a system size and ξ is a correlation length. The coupling J e induceshybridization of the edge states. The hybridization is determined by S = 3 / H e = J e ~s L · ~s R , where ~s L ( R ) is a left (right) S = 3 / J e = 0 .
1, we plot the spectrum H e with zero-magnetization sector. The result is shown in Fig. 6 (b).Let us compare the spectrum structures of Fig. 6 (a) with that of Fig. 6 (b). Both structures are much similar.To estimate the similarity quantitatively, we consider a level spacing ratio, r = | E − E || E − E | and r = | E − E || E − E | , where E i ( i = 0 , , ,
3) is i -th energy in ascending order. We calculated r and r for S = 2 system with open boundary at t = T / S z = 0 sector. We set different system size L = 6 , ,
10. Then, the results for L = 6 , ,
10 are ( r , r ) ≡ ( r ED , r ED ) = (0 . , . , (0 . , . , (0 . , . S = 3 / H e with zero-magnetization sector, ( r , r ) ≡ ( r e , r e ) = (1 / , / r ED and r ED are close to r e and r e . From this comparison, the low-lying spectrum of S = 2 spin chainat t = T / S = 3 / S = 2 system comes to be degenerate. Figure 6 (c) shows thesystem size dependence of energy differences defined by δE jk = | E j − E k | /L . The differences approach to zero for L → ∞ . This result implies that the groundstate for the S = 2 system becomes four-fold degenerate for L → ∞ .This result is analogous to the result for S = 1 Haldane phase in [2]. ! " $ % & ’ ) *+,-/ 023 ber ber E / L < / >?@ ABC DFGHI JK L LMNOPQRSTUVW
FIG. 6. (a) Low-lying spectrum structure of S = 2 exact diagonalization. (b) The spectrum structure of the two 3 / H e with zero magnetization sector. (c) System size dependence of energy differences δE jkjk