Topological Order Parameters of the Spin-1/2 Dimerized Heisenberg Ladder in Magnetic Field
TTopological Order Parameters of the Spin-1/2 Dimerized Heisenberg Ladder inMagnetic Field
Toshikaze Kariyado ∗ and Yasuhiro Hatsugai † Division of Physics, Faculty of Pure and Applied Sciences,University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan (Dated: July 31, 2018)Topological properties of the spin-1/2 dimerized Heisenberg ladder are investigated focusing on theplateau phase in the magnetic field whose magnetization is half of the saturation value. Althoughthe applied magnetic field removes most of the symmetries of the system, there is a symmetryprotected topological phase supported by the spatial inversion symmetry. The Z Berry phaseassociated with a symmetry respecting boundary and quantized into 0 and π is used as a symmetryprotected topological order parameter. Edge states are also analyzed to confirm the bulk–edgecorrespondence. In addition, a symmetry breaking boundary is considered. Then, we observe aunique type of quantization of the Berry phase, a quantization into ± π/ ± π/ PACS numbers: 03.65.Vf, 03.65.Ud
I. INTRODUCTION
Recent use of topology in condensed matter substan-tially revised our view on the characterization of ma-terials especially for the gapped phases . One of theadvantages of a topological view is robustness againstcontinuous deformation. One can use a topological quan-tity to distinguish topological phases since the quantizednature of it guarantees its invariance against continuousdeformation . Due to the theorem by von Neumann-Wigner, however, truly generic phase can be a single classsince any generic states can be adiabatically connected.Then, the symmetry restriction is essential. The symme-try can be gauge symmetry, time-reversal, particle-hole,reflection, etc . When these restrictions give rise to anew nontrivial phase, it is a symmetry protected topolog-ical (SPT) phase. A typical SPT order parameter is theBerry phase, which takes any value without symmetry,but can be quantized with the appropriate symmetry.The continuous deformation, or adiabatic continua-tion, is also essential for establishing the bulk–edgecorrespondence , which is one of the fundamental con-cepts for characterizing topological phases. Physics atthe bulk and the edges are not independent and relatedeach other especially for the gapped case. Introductionof a boundary sometimes breaks symmetries of the bulksystem, but sometimes does not. From the viewpoint ofthe symmetry protection, whether the edge respects abulk symmetry or not has a special importance.In this paper, topological properties of a dimerizedspin-1/2 Heisenberg ladder with antiferromagnetic cou-pling is investigated focusing on the plateau phase at halfof the saturation, which we call a 1/2-plateau phase. The1/2-plateau phase appears in the applied magnetic field,which breaks most of the symmetry of the system. The ladder model itself has been studied extensively , andsome of the studies shed lights on the topological aspectsof the ladder model , but the focus was mainly onthe case without magnetic field. (Very recently, plateauphases at finite magnetization in spin chains are stud-ied by a SPT viewpoint in Ref. 22.) A main purposeof this paper is to show the 1/2-plateau phase here isa SPT phase protected by the spatial inversion symme-try that survives even with the finite external magneticfield. The Berry phase and the entanglement entropyare employed to characterize a SPT phase. As we willexplain later, boundary shapes are essential for both ofthe Berry phase and the entanglement entropy, and theboundary that keeps the inversion symmetry is mainlyused not to destroy the symmetry effects. Importance ofthe symmetry is also demonstrated by introducing artifi-cial symmetry breaking, and by a spontaneous symmetrybreaking caused by a ring exchange. We further studyedge states to establish the bulk–edge correspondence. Inorder to complement the above arguments, a boundarythat breaks inversion symmetry is also treated. Naively,one may think that such a symmetry breaking boundaryis not useful for characterizing symmetry protected topo-logical phases. However, for a specific type of symmetrybreaking boundary, we found a unique quantization ofthe Berry phase, i.e., a fractional quantization of theBerry phase into ± π/
2, instead of the widely observed0/ π -quantization. Further, it is shown that edge statesare also unique for the ± π/ a r X i v : . [ c ond - m a t . s t r- e l ] M a r ical characterization are introduced. Then, the numeri-cal methods to obtain those physical quantities, the ex-act diagonalization and the infinite time-evolving blockdecimation (iTEBD), are explained. There, we alsoexplain how the topological character of the system isencoded in the matrix product state, in terms of thetransformation law against the symmetry operation. Sec-tion III contains main results of this paper. First, themagnetization curve is shown to take a glance at theplateau phase on which we focus in this paper. Afterthat, the topological properties of the 1/2-plateau phase,such as the quantized Berry phase and the bulk-edge cor-respondence are discussed in detail. The effects of ringexchange is also considered. Finally, we make a com-parison between the 0-plateau phase and the 1/2-plateauphase. The paper is summarized in Sec. IV. II. MODEL AND METHODS
The model treated in this paper is a dimerized spin-1/2 Heisenberg ladder with the Zeeman field, whoseHamiltonian is written asˆ H = L (cid:88) i =1 (cid:88) j =1 , (cid:104) J S i,j · S i +1 ,j + J S i +1 ,j · S i +2 ,j (cid:105) + J L (cid:88) i =1 S i, · S i, − B z L (cid:88) i =1 (cid:88) j =1 , S zi,j . (1)(See Fig. 1.) Here, we concentrate on the antiferromag-netic coupling, namely all of J s in Eq. (1) are assumedto be positive. For convenience, a parameter ∆ is in-troduced as ∆ = ( J − J ) /
2. If ∆ (cid:54) = 0, the minimumunit cell is composed of four spins, while if ∆ = 0, it iscomposed of two. This unit cell structure is essential toobtain the 1/2-plateau phase that we concentrate on.When B z = 0, the system has the rotational symmetry inthe spin space and the time reversal symmetry, but thosesymmetries are broken for finite B z . However, even withfinite B z , the system retains the spatial inversion sym-metry, which is essential for protecting the topologicalphase in the 1/2-plateau phase.In order to elucidate the topological properties of themodel, we calculate the magnetization, the Berry phase,and the entanglement entropy. The magnetization is cal-culated to show the existence of the plateau phase. TheBerry phase defined below works as a symmetry pro-tected topological order parameter to identify two topo-logically distinct states having the same symmetry . Theentanglement entropy for the spatial bipartition is eval-uated to check whether the two states can be smoothlyconnected or not. Also the entanglement entropy givesa picture of the bulk-edge correspondence because thetopological character of the system is encoded in the en-tanglement spectrum , and the entanglement entropycontains a contribution from the edge states . Analy-sis based on the transformation law of the matrices in the (b)(c) verticaldiagonal(a) J J J J J J J J FIG. 1. (a) The dimerized spin-1/2 Heisenberg ladder. Def-initions of parameters are shown. (b) The vertical edge. (b)The diagonal edge. matrix product state (MPS) representation is also per-formed to complement the Berry phase based arguments,i.e., we extract a topological order parameter other thanthe quantized Berry phase from the MPS representation.In addition, by making use of the translationally invari-ant MPS representation that enables us to access thethermodynamic limit, we can discuss the collapse of theSPT phase with spontaneous symmetry breaking.
A. Numerical methods
The calculations of the Berry phase and the investiga-tion of the edge states are performed by the exact diag-onalization of the finite size system. The ground stateenergy and the wave function are numerically evaluatedby the Lanczos algorithm combined with the inverse it-eration method. In order to define the Berry phase, weapply a local gauge twist on the bonds (possibly on themultiple bonds) at the given boundary as S + L S − R + h.c. −→ e i φ S + L S − R + h.c. , (2)where L and R denote the sites on the left and rightsides of the given boundary. Using the ground state wavefunction | G φ (cid:105) at each φ , the Berry phase γ is obtained asi γ = (cid:90) π d φ (cid:104) G φ | ∂ φ | G φ (cid:105) . (3)In practice, by discetizing the range [0,2 π ] as φ i = 2 πi/N ( i = 1 , . . . , N − γ = arg (cid:18) (cid:104) G φ N − | G φ (cid:105) N − (cid:89) i =0 (cid:104) G φ i | G φ i +1 (cid:105) (cid:19) . (4)Because the gauge twist is applied on the bonds crossingthe boundary, the Berry phase depends on the bound-ary shape, which is essential for establishing the bulk-edge correspondence . With the exact diagonaliza-tion scheme, finite size effects are unavoidable. However,as far as the Berry phase is quantized due to some sym-metry, as we will see soon later, there is practically nosize effect on the numerically obtained Berry phases.In order to obtain the entanglement entropy and theMPS, the iTEBD method is employed . In the iTEBD,a translationally invariant MPS representation of theground state is iteratively obtained. It becomes exactin the large χ limit, where χ is truncation dimension cor-responding to the dimension of the matrix in the MPSrepresentation. For gapped phases, small χ is sufficient toobtain results in practical precision. An advantage of theiTEBD is accessibility to the thermodynamic limit, i.e.,it is free from the finite size effect, because it providesa translationally invariant MPS representation by con-struction. The finite size effect free nature is essential fordiscussing spontaneous symmetry breaking. To performthe iTEBD, translation symmetry is essential, and thetarget system is assumed to be composed of repetitionof unit objects. Then, the iTEBD naturally gives entan-glement entropy for the bipartition such that the systemis divided into two parts in between the two neighboringunit objects. In the following, unit objects are appropri-ately chosen so as to make a given boundary in betweentwo of them. In this way, the entanglement entropy de-pends on shapes of given boundary as it should do.With the iTEBD, it is possible to obtain the transla-tionally invariant canonical MPS representation of thestate such that | Ψ (cid:105) = (cid:88) { s i } · · · Γ s i ΛΓ s i +1 Λ · · · | · · · , s i , s i +1 , · · · (cid:105) , (5)where s i denotes the labels of local states, and Γ s and Λare χ × χ matrices. Λ is a diagonal matrix whose entriesare nonnegative, and related to the entanglement entropy S as S = − (cid:88) i λ i log λ i , (6)where λ i is the diagonal elements of Λ. When | Ψ (cid:105) re-spects some symmetry, Γ s must react against the symme-try operation appropriately. Then, physical states withthe same symmetry can be classified by this transforma-tion law of Γ s . When the symmetry operation is writtenby a product of local operators O ( a ) , Γ s (cid:48) transforms as (cid:88) s (cid:48) O ( a ) ss (cid:48) Γ s (cid:48) = e i θ a U † a Γ s U a , (7)where U a is an unitary matrix that satisfies [Λ , U a ] = 0.Mathematically, U a gives a projective representation ofthe symmetry operation, and the states are distinguishedby its factor set . If the operation involves the spa-tial inversion symmetry, which plays a central role in thispaper, Γ s (cid:48) at the left hand side of Eq. (7) is replaced by t Γ s (cid:48) that is the transpose of Γ s (cid:48) , since the inversion op-eration reverses the order of s i . In general, a cyclic groupgenerated by a single element, like the case that there isonly inversion symmetry, leads no interesting factor set.However, this transposition makes the inversion symme-try useful in classification of the states. Namely, there is arestriction on U a for the spatial inversion symmetry (de-noted as U I hereafter) such that t U I = ± U I , which meansthat U I should be symmetric or antisymmetric . For an-tisymmetric U I , the relation [Λ , U I ] = 0 gives degeneracyof the entanglement spectrum, in which the topologicalproperties of the system is encoded . Particularly, ifthe entanglement spectrum is at least doubly degener-ate, the entanglement entropy has a lower bound log 2.As we will focus on the phase with the finite magneticfield, in which the time reversal symmetry and the sym-metries in the spin space are not effective, ζ , which isdefined according to t U I = ζU I , (8)and takes values of +1 and −
1, is employed as a topo-logical order parameter to classify the phases.In practice, U I is obtained as an “eigenmatrix” of alinear matrix map E I ( U ) = (cid:88) ss (cid:48) ˆ O ( I ) ss (cid:48) ( t Γ s Λ) U (Γ s (cid:48) Λ) † , (9)whose eigenvalue (cid:15) satisfies | (cid:15) | = 1. In specific, U I satis-fies E I ( U I ) = e i θ U I . (10)Numerically, e i θ and U I are obtained as an eigenvalueand an eigenvector of the matrix representation of themap E I , whose matrix elements are defined as T ij ; i (cid:48) j (cid:48) = (cid:88) ss (cid:48) ˆ O ( I ) ss (cid:48) ( t Γ s ) ii (cid:48) λ i (cid:48) (Γ s (cid:48) ) ∗ jj (cid:48) λ j (cid:48) . (11)The matrix T works as a transfer matrix when we calcu-late the overlap between the wave functions before andafter the symmetry operation is applied . It means that,as far as the state respects the symmetry, the largestnorm of the eigenvalues of T becomes unity. Then U I isobtained from an eigenvector associated with the eigen-value with unit norm, and ζ is reduced from it. On theother hand, if the largest norm of the eigenvalues of T isless than unity, it implies that the state under consider-ation does not respect the symmetry. In other words, T has an ability to detect whether a given state is invariantagainst a symmetry operation. For convenience, we set ζ to zero when T detects a symmetry breaking and U I isunavailable. III. RESULTS AND DISCUSSIONS
The magnetization curve obtained with the iTEBD for J = 1 . J = 1 . − ∆, and J = 1 . 〈 S z 〉 B z Δ =0.5 Δ =0.3 Δ =0.0 FIG. 2. (Color online) The magnetization curve for ∆ = 0 . .
3, and 0 . χ = 24. Magnetization is alreadyconverged with χ = 24 or smaller. values of ∆ is shown in Fig. 2. The essential features ofthe curves are consistent with those in Ref. 15. Namely,there are two plateau phases, for (cid:104) S z (cid:105) = 0 (0-plateau)and (cid:104) S z (cid:105) = 1 / (cid:104) S z (cid:105) = 1 /
4, which isa half of the saturation magnetization, is not allowed. Onthe other hand, the 0-plateau phase does not vanish inthe zero dimerization limit. This is natural because the0-plateau phase without dimerization is actually in therung singlet phase , and the rung singlet is expectedto be stable for small dimerization.
A. Topological order parameter in the 1/2-plateauphase
Let us move on to the topological properties of the 1/2-plateau phase. Figures. 3(a) and 3(b) show the numer-ically obtained Berry phase and entanglement entropyfor the 1/2-plateau phase. These quantities depend onthe boundary shape, and the vertical edge in Fig. 1(b)is employed here. (Later we will discuss the edge statesand then the diagonal edge is also used.) What is impor-tant is that the vertical edge does not break the inver-sion symmetry whose inversion center is at the bound-ary. Then, the Berry phase is quantized into 0 or π asin Fig. 3(b). This quantization is caused by the spa-tial inversion symmetry . Here, the inversion symmetrymeans that the Hamiltonian with the gauge twist φ sat-isfies ˆ H − φ = ˆ P − ˆ H φ ˆ P (12)where ˆ P is an appropriate unitary matrix. This relationcombined with the assumption that the ground state isunique implies γ = − γ (mod 2 π ), which immediately E n t a ng l e m e n t E n t r opy (a) (cid:1) =64-101 (cid:2) / (cid:3) (b)-1 0 1-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 (cid:4) (cid:5) (c) f ( ) f ( ) FIG. 3. (Color online) (a) The entanglement entropy, (b) theBerry phase, and (c) the other topological order parameter ζ as a function of ∆ for the vertical edge. The entanglemententropy is calculated with B z = 2 . χ = 64. The Berryphase is obtained with the (cid:104) S z (cid:105) = 1 / ζ is calculated with χ = 32. leads to the quantization of γ into 0 or π . As this quanti-zation is robust provided that the spatial inversion sym-metry is kept intact, γ can be regarded as a symmetryprotected topological order parameter. The Berry phase γ shows a jump at ∆ = 0, which suggests a phase tran-sition. This phase transition is also detected by the en-tanglement entropy as its divergent behavior.Analysis of the transformation law of Γ s also supportsthis conclusion. In fact, as is shown in Fig. 3(c), ζ is 1for ∆ < − >
0, which confirms that theboth states respect the spatial inversion symmetry, butare topologically distinct from each other. For ∆ > ζ = −
1. Importantly, ζ = − γ = π , which implies consistencybetween γ and ζ as a topological order parameter. Sincewe are considering the finite B z case, the time rever-sal symmetry and the rotational symmetry in spin spacecannot be used here.For this classification of the phases, choice of the posi-tion of the boundary is essential because the states with∆ > < , or of the dimerized spin-1/2Heisenberg chain . A relation to the dimerized chainis understood by considering the strong rung coupling E n t a ng l e m e n t E n t r opy (a) (cid:1) =0.0 (cid:1) =0.2 (cid:1) =0.5-1 0 1 0 0.25 0.5 0.75 1 (cid:2) / (cid:3) (cid:4) (b) (cid:1) =0.2 (cid:1) =0.5 f ( ) f ( ) FIG. 4. (Color online) (a) The entanglement entropy and(b) the Berry phase with artificial symmetry breaking. Theentanglement entropy for α = 0 . . B z = 2 . χ = 24. The Berry phase is calculated with the (cid:104) S z (cid:105) = 1 / limit, though our calculation so far assumes the compa-rable rung and leg couplings. The spin-1/2 ladder in thestrong rung coupling limit at (cid:104) S z (cid:105) = 1 / zero magnetization witheasy plane anisotropy . Then, the present phaseunder consideration is expected to be connected to thephase of the dimerized XXZ chain. On the other hand,the dimerized XXZ chain with easy plane anisotropy issmoothly connected to the anisotropic chain. (This is notthe case with the easy axis, or Ising, anisotropy since thesystem will be in an antiferromagnetic phase at least forthe weak dimerization limit.) Now, we confirm that thetransition in Fig. 3 is the same type as the transition inthe dimerized spin-1/2 chain with no external magneticfield , where two distinct phases are distinguished bythe positions of spin singlets. B. Role of the symmetry in protecting thetopological phases
In the previous section, the states are characterized bythe Z Berry phase (and the transformation law of Γ s ). Inorder to demonstrate the role of the symmetry, we intro-duce symmetry breaking term and show the two phasesare continuously connected if the symmetry is broken.Here, we add a termˆ H artificial = δJ L (cid:88) i =1 ( − i S i, · S i, , (13)which makes staggered modulation of the rungcoupling and breaks the spatial inversion symme-try whose inversion center is in between the two rungs. Δ =0.5 Δ =-0.5 FIG. 5. (Color online) Site resolved magnetization with thevertical open boundary for ∆ = 0 . − . Parameters η and α are introduced as∆ = η − . , (14) δJ = αη (1 − η ) . (15)With this definition, the states with η = 0 (∆ = − . δJ = 0) and η = 1 (∆ = 0 . δJ = 0) retain the spatialinversion symmetry. The Berry phase and the entangle-ment entropy for several α in the 1/2-plateau phase areshown in Fig. 4. The Berry phase is no longer quantizedand the jump observed in Fig. 3 is removed. (In prin-ciples, the finite size effects come into play in this casewithout quantization, but in the present case, the resultsobtained with the 20 spin system and the 24 spin systemare nearly identical.) At the same time, divergence inthe entanglement entropy is also removed, similar to thecase of the symmetry broken dimerized chain . Further-more, ζ = 0 since the maximum norm of the eigenvaluesof T is less than unity as it should be with the brokenspatial inversion symmetry. All of these results are con-sistent with the crucial role of the symmetry. Two kindsof symmetry protected topological order parameters, γ and ζ , which pick up the same information when the in-version symmetry exists, give quite different informationwhen the inversion symmetry is broken. By definition, ζ is set to zero in such a case. On the other hand, γ takessome value and works as a measure of the “distance” tothe topological phase or the trivial phase. C. Bulk–edge correspondence for a symmetrypreserving boundary
Next, we move on to the analysis of the edge states tosee the bulk–edge correspondence. In the following, wesee that γ = π ( ζ = −
1) state shows a clear sign of anedge state, while γ = 0 ( ζ = 1) state does not. Figure 5shows the site resolved magnetization of the ground stateobtained with the open boundary. For this calculation,we used ∆ = ± . B z = 2 .
0. For ∆ = 0 .
5, the totalmagnetization of the ground state deviates from exactlybeing (cid:104) S z (cid:105) = 1 /
4, but the ground state has one more ex-tra up spin. This extra up spin is localized at the bound-ary, as we can see from Fig. 5, which makes the edgedistinct from the bulk. No significant change of the localmagnetization is found in the bulk part. On the otherhand, for ∆ = − .
5, the ground state magnetization ex-actly satisfies (cid:104) S z (cid:105) = 1 /
4, and the local magnetization isonly weakly affected at the boundary. These behaviorsare clearly explained in the ∆ = ± −
1, the boundary does notbreak a cluster, and we expect no edge states. On theother hand, for ∆ = 1, a cluster at the boundary is bro-ken, and the lowest energies of the broken cluster at eachmagnetization are obtained as − B z + J / (cid:104) S z (cid:105) = 1 / − B z − J / (cid:104) S z (cid:105) = 1 / − J / (cid:104) S z (cid:105) = 0). Inthe present case, J = 1 . B z = 2, the fully polarizedstate ( (cid:104) S z (cid:105) = 1 /
2) is chosen at the boundary, which indi-cates that the extra up spin is localized at the boundary.The ∆ = ± . ± ±
1. In this case, it is easy to evaluate the entan-glement entropy because we only have to take accountof the entanglement within a single four-site cluster. For (cid:104) S z (cid:105) = 1 / f (1 / f ( x ) = − x log x − (1 − x ) log(1 − x ).(See Appendix A.) For ∆ = 1, because of ζ = −
1, thedegeneracy of the entanglement spectrum is expected,and f (1 /
2) = log 2 is consistent with the lower boundset by the degeneracy. On the other hand, for ∆ = − ± D. Symmetry breaking boundary and fractionalquantization of the Berry phase
So far, we have only considered the vertical edge. Now,let us move on to the diagonal edge. [See Fig. 1(c)]. Im-portant feature of the diagonal edge is that it breaks theinversion symmetry no matter where we choose as theinversion center even if the bulk symmetry is kept intact.(Recall that the vertical edge keeps the inversion sym-metry if we set the inversion center at the boundary.)Even in this case, since the bulk symmetry is preservedand the Berry phase and the entanglement entropy arebulk quantities, i.e., both quantities are obtained by bulkground state wave functions, these quantities still havean ability to sense a topological phase transition. But,of course, they should behave differently from the caseof the symmetry preserving boundary. For instance, theBerry phase need not be quantized into 0 or π , and notnecessarily useful in a naive thought. However, in thepresent model, the diagonal edge also exhibits interest-ing phenomena as shown below. First, the Berry phase E n t a ng l e m e n t E n t r opy (a) (cid:1) =64-1 0 1-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 (cid:2) / (cid:3) (cid:4) (b) f ( ) f ( ) FIG. 6. (Color online) (a) The entanglement entropy and(b) the Berry phase for the diagonal edge. The entanglemententropy is calculated with B z = 2 . χ = 64. The Berryphase is obtained using the system with six unit cells (24spins) with (cid:104) S z (cid:105) = 1 / for the diagonal edge in the 1/2-plateau phase is shownin Fig. 6. There, we can see that the Berry phase isquantized into ± π/
2, that is, the Berry phase shows theunique fractional quantization . This fractional quanti-zation obeys from the formula γ diagonal = γ vertical − π ( S − (cid:104) S z (cid:105) ) , (16)derived in the similar way as in Refs. 11 or 29. Since γ vertical is quantized into 0 or π by the spatial inversionsymmetry, and (cid:104) S z (cid:105) is fixed to 1/4 owing to the symme-try of the model, γ diagonal is quantized into ± π/
2. Fig-ure 6(b) indicates that the topological phase transition iscaptured as a jump in γ since the bulk symmetry is kept,but 0/ π -quantization is broken because of the symmetrybreaking boundary.The entanglement entropy is also plotted in Fig. 6.Different from the case of 0/ π quantization, the entan-glement entropy is finite in the both of the positive andnegative sides of the ∆ = ± ± f (1 / f (1 / < f (1 /
2) = log 2. Thisbehavior is confirmed in Fig. 6.Now we investigate the open ladder with the diagonaledge to see the bulk–edge correspondence for the ± π/ (cid:104) S z (cid:105) forthe ground state of (cid:104) S z (cid:105) = 1 / . (cid:1) S z (cid:2) positionupper chainlower chain FIG. 7. (Color online) The site resolved spin density in the1/2-plateau phase with the diagonal edge. The color map ofthe spin density is also shown. Calculation is performed onthe system with 32 spins. plotted. There, the extra up spins are accumulated atthe left edge, while the extra down spins are accumu-lated at the right edge. For negative ∆, where the edgestates are not observed for the case with 0/ π quantiza-tion, there are still edge states but roles of the right andthe left edges are reversed. This is consistent with thefact that the entanglement entropy goes to finite valuesfor both of ∆ = ±
1. To summarize, the edge states forthe ± π/ ± π/ π/ − π/ π -quantization, where 0and π are essentially different, and leads to the absenceand existence of the edge states. To understand the fea-ture i), analogy to the electron system is helpful. In freeelectron systems, there is a direct relation between theelectronic polarization and the Berry phase . By re-garding up spins as electrons and down spins as holes,the state in Fig. 7 corresponds to the electronically po-larized state associated with the finite Berry phase. Notethat γ = π represents the situation where the mean po-sition of the electrons is at the middle point between twolattice points, which means that there is no electronicpolarization even though the Berry phase is finite. Thus,the quantization into ± π/ π is essential toobserve the feature i).In order to observe a unique quantization of the Berryphase described above, the ladder structure and the ap-plied magnetic field are essential. First of all, the lad-der structure allows us to consider the diagonal edge for which the fractional quantization is achieved. Further,Eq. (16) implies that we need to look at the plateauphase with (cid:104) S z (cid:105) = 1 / E. Ring exchange
So far, we have considered the 1/2-plateau phase in-duced by the dimerization. For the uniform case withoutdimerization, a plateau phase at the same magnetization (cid:104) S z (cid:105) = 1 / , which iswritten as ˆ H ring = K (cid:88) i ( P i + P − i ) , (17)where P i ( P − i ) is an operator acting on the minimalfour-site plaquette that causes a clockwise (anticlock-wise) shift of the spins on that plaquette. That is, ifwe denote the states of four spins on a plaquette as (cid:12)(cid:12)(cid:12)(cid:12) s s s s (cid:29) , P i operates as P i (cid:12)(cid:12)(cid:12)(cid:12) s s s s (cid:29) = (cid:12)(cid:12)(cid:12)(cid:12) s s s s (cid:29) , P − i (cid:12)(cid:12)(cid:12)(cid:12) s s s s (cid:29) = (cid:12)(cid:12)(cid:12)(cid:12) s s s s (cid:29) . (18)Intuitive understanding of the ring exchange is possiblefor the strong rung coupling limit. There, we have notedthat the model is effectively described by a XXZ chainmodel. The ring exchange term modifies the anisotropyof the effective XXZ model, i.e., the anisotropy is modi-fied from the easy plane type to the easy axis type (Isingtype) for sufficiently large positive K . Then, the anti-ferromagnetic order is developed in the effective model,and this ordered phase corresponds to the 1/2-plateauphase in the original model . Due to the antiferro-magnetic nature of the state, the spatial inversion sym-metry that is essential for the SPT phase is broken.Then, a transition between the SPT phase and the sym-metry broken phase within the 1/2-plateau phase is ex-pected when the strength of the ring exchange is suit-ably tuned. In fact, such a transition is observed in thepresent model. In Fig. 8, M = L (cid:80) Li (cid:80) j =1 , (cid:104) S zi,j (cid:105) ,and M (cid:48) = L (cid:80) Li (cid:80) j =1 , ( − i (cid:104) S zi,j (cid:105) , which captures thesymmetry breaking, are plotted as a function of η in-troduced as ∆ = 0 . − η ) and K = 0 . η . Note thatthe dimerization dominated plateau phase is expected for η = 0, while the ring exchange dominated plateau phaseis expected for η = 1. The numerical result in Fig. 8confirms this idea. Namely, M (cid:48) gets finite at a certainvalue of η as η increases, while M is always 0.5.In order to clarify the difference between zero M (cid:48) phaseand finite M (cid:48) phase in terms of SPT, the entanglement o r d e r p a r a m e t e r s (cid:1) MM' - l og ( (cid:2) i ) ring exchange K K
FIG. 8. (Color online) Order parameters M and M (cid:48) obtainedwith B z = 2 . χ = 48. Insets show the entanglementspectra for η = 0 . .
8. For each value of η , the leftcolumn is for the edge on J bond, and the right column isfor the edge on J bond. spectra for η = 0 . . J bonds andthe other breaks J bonds. Therefore, for each value of η , two spectra are plotted. At η = 0 .
6, we observe dou-ble degeneracy of the spectrum for the edge on J bonds.This degeneracy stems from the nontrivial projective rep-resentation, ζ = −
1, and signals a SPT phase. On theother hand, for η = 0 .
8, neither of the edges on J and J bonds leads to the degeneracy of the spectra. Theseobservation implies that the transition in Fig. 8 is a typ-ical and concrete example for collapse of the SPT phaseby a spontaneous symmetry breaking. F. Comparison with the 0-plateau phase
Finally, we briefly discuss the 0-plateau phase near zeromagnetic field. In this case there is no sign of the phasetransition in both of the Berry phases and the entan-glement entropy plotted as a function of ∆ (not shown).This is natural because ∆ = 0 state is not critical but de-scribed as a rung singlet state without the Zeemanfield. In contrast, the dimerized chain is critical when∆ = 0 and B z = 0. However, actually, there is no jumpat ∆ = 0 in the Berry phase as a function of ∆, even inthe small rung coupling limit. This is because we applygauge twists on all bonds across the boundary. Then,even for ∆ for which a topologically nontrivial phase isexpected in the dimerized chain , the Berry phase is 0 forthe dimerized ladder because each chain contributes π to γ , which results in γ = 2 π ≡ π ). If we apply different kinds of gauge twist, not limited to the one thatis possible to be reduced to a twisted boundary condi-tion, it is possible to detect dimer structures . Wehave also confirmed the absence of the topological phaseprotected by the spatial inversion symmetry in 0-plateauphase by the MPS representation. That is, in the 0-plateau phase, ζ is always 1 irrespective of the sign of∆. IV. SUMMARY
To summarize, it is established that there is a sym-metry protected topological phase in the 1/2-plateauphase in the dimerized spin-1/2 Heisenberg ladder. Evenwith the magnetic field, which is necessary to access the1/2-plateau phase and reduces the symmetry of the sys-tem, the spatial inversion symmetry remains and protectsthe topological phase. Namely, the inversion symmetrymakes the Berry phase quantized into 0 or π , and al-low us to regard the Berry phase as a topological orderparameter. The entanglement entropy is also used tocharacterize the topological phase. In order to comple-ment the Berry phase based arguments, the topologicalorder parameter other than the Berry phase is extractedfrom the MPS representation of the state. That topo-logical order parameter determines the degeneracy of theentanglement spectrum, and gives lower bound of the en-tanglement entropy. The importance of the symmetry isdemonstrated by introducing a symmetry breaking termand by spontaneous symmetry breaking caused by thering exchange. Because of the importance of the inver-sion symmetry, the boundary that respects the inversionsymmetry is mainly used to define the Berry phase andthe entanglement entropy. However, our analysis on aspecific shape of boundary reveals that the symmetrybreaking boundary can lead to a new type of the bulk–edge correspondence. In specific, we find a fractionalquantization of the Berry phase into ± π/ ± π/ ACKNOWLEDGMENTS
This work is partly supported by Grants-in-Aid forScientific Research, Nos. 26247064 and 25610101 fromJSPS, and No. 25107005 from MEXT. The authors thankthe Supercomputer Center, the Institute for Solid StatePhysics, the University of Tokyo for the use of the facil-ities.
Appendix A: Analytical calculation of theentanglement entropy for a decoupled four sitecluster1. Hamiltonian and ground state
For a four-site plaquette with (cid:104) S z (cid:105) = 1 /
4, there arefour basis states written as | I (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12) ↑ ↑↑ ↓ (cid:29) , | II (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12) ↑ ↑↓ ↑ (cid:29) , | III (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12) ↑ ↓↑ ↑ (cid:29) , | IV (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12) ↓ ↑↑ ↑ (cid:29) . (A1)The Hamiltonian of the plaquette acts on these basisstates as ˆ H p | I (cid:105) = J | II (cid:105) + J | III (cid:105) , (A2)ˆ H p | II (cid:105) = J | I (cid:105) + J | IV (cid:105) , (A3)ˆ H p | III (cid:105) = J | IV (cid:105) + J | I (cid:105) , (A4)ˆ H p | IV (cid:105) = J | III (cid:105) + J | II (cid:105) . (A5)If J > J >
0, the ground state is obtained as | G (cid:105) = 12 (cid:0) | I (cid:105) − | II (cid:105) − | III (cid:105) + | IV (cid:105) (cid:1) . (A6)
2. Entanglement entropy
For a given bipartition, a state is expressed as | ψ (cid:105) = (cid:88) ij M ij | ψ j (cid:105) , (A7)where each of | ψ j (cid:105) is a state in either partof the bipartitioned system. Using the singular valuedecomposition, a matrix ˆ M can be always written asˆ M = ˆ U ˆΛ ˆ V † , (A8)where ˆΛ is a diagonal matrix whose elements are nonneg-ative, and ˆ U and ˆ V are unitary matrices. Then, | ψ (cid:105) isrewritten as | ψ (cid:105) = (cid:88) α λ α | ˜ ψ <α (cid:105) ⊗ | ˜ ψ >α (cid:105) (A9)with | ˜ ψ <α (cid:105) = (cid:88) i U iα | ψ α (cid:105) = (cid:88) j ( V † ) αj | ψ >j (cid:105) , (A10) where λ α denotes the diagonal elements of ˆΛ.The entanglement entropy for this bipartition is eval-uated as S = − (cid:88) α λ α log λ α . (A11)
3. Entanglement entropy for a plaquette
First, we consider the vertical edge that breaks a foursite plaquette into two parts with two spins. Taking aset (cid:12)(cid:12)(cid:12)(cid:12) ↑↑ (cid:29) , (cid:12)(cid:12)(cid:12)(cid:12) ↑↓ (cid:29) , (cid:12)(cid:12)(cid:12)(cid:12) ↓↑ (cid:29) , (cid:12)(cid:12)(cid:12)(cid:12) ↓↓ (cid:29) , (A12)as | ψ i (cid:105) , | G (cid:105) can be written in the form asEq. (A7) with ˆ M beingˆ M = − − . (A13)Then, it is straightforward to confirm that ˆ U , ˆ V † , and ˆΛfor this case becomeˆ U = − √ −
12 12 √ −
12 12 √ √
00 0 0 1 , ˆ V † = √ √ −
12 12 1 √ −
12 1 √
00 0 0 1 , (A14)and ˆΛ = √ √ . (A15)As a result, we have S = −
12 log 12 −
12 log 12 = log 2 . (A16)Next, we consider the diagonal edge that breaks a foursite plaquette into two parts, one with a spin and anotherwith three spins. Taking a set | ↑(cid:105) , | ↓(cid:105) (A17)as | ψ i (cid:105) , | G (cid:105) is expressed as in the form of Eq. (A7) withˆ M = (cid:18) −
12 12 − (cid:19) . (A19)0Now, it is easy to see that ˆ U , ˆ V † , and ˆΛ for this case areˆ U = (cid:18) −
11 0 (cid:19) , ˆ V † = √ √ − √ √ √ √ − √ , (A20) and ˆΛ = (cid:18) √ (cid:19) . (A21)Then, the entanglement entropy is obtained as S = −
14 log 14 −
34 log 34 . (A22)To summarize, we have f (1 /
2) for the vertical edgeand f (1 /
4) for the diagonal edge with f ( x ) = − x log x − (1 − x ) log(1 − x ). ∗ [email protected] † [email protected] X. G. Wen, Phys. Rev. B , 7387 (1989). Y. Hatsugai, J. Phys. Soc. Jpn. , 1374 (2005). Y. Hatsugai, J. Phys. Soc. Jpn. , 123601 (2006). D. J. Thouless,
Topological Quantum Numbers in Nonrel-ativistic Physics (World Scientific, 1998). X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B ,155138 (2010). A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud-wig, Phys. Rev. B , 195125 (2008). F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa,Phys. Rev. B , 075125 (2012). Y. Hatsugai, Phys. Rev. Lett. , 3697 (1993). S. Ryu and Y. Hatsugai, Phys. Rev. Lett. , 077002(2002). Y. Hatsugai, Solid State Comm. , 1016 (2009). T. Kariyado and Y. Hatsugai, Phys. Rev. B , 085132(2014). T. Barnes, E. Dagotto, J. Riera, and E. S. Swanson, Phys.Rev. B , 3196 (1993). D. C. Cabra, A. Honecker, and P. Pujol, Phys. Rev. Lett. , 5126 (1997). F. Mila, Eur. Phys. J. B , 201 (1998). D. C. Cabra and M. D. Grynberg, Phys. Rev. Lett. ,1768 (1999). E. H. Kim, G. F´ath, J. S´olyom, and D. J. Scalapino, Phys.Rev. B , 14965 (2000). G. Y. Chitov, B. W. Ramakko, and M. Azzouz, Phys.Rev. B , 224433 (2008). S. J. Gibson, R. Meyer, and G. Y. Chitov, Phys. Rev. B , 104423 (2011). I. Maruyama, T. Hirano, and Y. Hatsugai, Phys. Rev. B , 115107 (2009). M. Arikawa, S. Tanaya, I. Maruyama, and Y. Hatsugai,Phys. Rev. B , 205107 (2009). Z.-X. Liu, Z.-B. Yang, Y.-J. Han, W. Yi, and X.-G. Wen,Phys. Rev. B , 195122 (2012). S. Takayoshi, K. Totsuka, and A. Tanaka,arXiv:1412.4029. G. Vidal, Phys. Rev. Lett. , 070201 (2007). M. Oshikawa, M. Yamanaka, and I. Affleck, Phys. Rev.Lett. , 1984 (1997). F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa,Phys. Rev. B , 064439 (2010). A. Chandran, M. Hermanns, N. Regnault, and B. A.Bernevig, Phys. Rev. B , 205136 (2011). S. Ryu and Y. Hatsugai, Phys. Rev. B , 245115 (2006). T. Fukui, Y. Hatsugai, and H. Suzuki, J. Phys. Soc. Jpn. , 1674 (2005). T. Hirano, H. Katsura, and Y. Hatsugai, Phys. Rev. B , 054431 (2008). T. Kariyado and Y. Hatsugai, Phys. Rev. B , 245126(2013). R. Or´us and G. Vidal, Phys. Rev. B , 155117 (2008). D. P´erez-Garc´ıa, M. M. Wolf, M. Sanz, F. Verstraete, andJ. I. Cirac, Phys. Rev. Lett. , 167202 (2008). F. Pollmann and A. M. Turner, Phys. Rev. B , 125441(2012). T. Morimoto, H. Ueda, T. Momoi, and A. Furusaki, Phys.Rev. B , 235111 (2014). A. L¨auchli, G. Schmid, and M. Troyer, Phys. Rev. B ,100409 (2003). J.-L. Song, S.-J. Gu, and H.-Q. Lin, Phys. Rev. B ,155119 (2006). W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev.Lett. , 1698 (1979). K. Hida, Phys. Rev. B , 2207 (1992). H.-H. Hung and C.-D. Gong, Phys. Rev. B , 054413(2005). H. T. Wang, B. Li, and S. Y. Cho, Phys. Rev. B ,054402 (2013). R. Haghshenas, A. Langari, and A. T. Rezakhani, J. Phys.:Condens. Matter , 456001 (2014). K. Totsuka, Phys. Rev. B , 3454 (1998). A. Nakasu, K. Totsuka, Y. Hasegawa, K. Okamoto, andT. Sakai, J. Phys.: Condens. Matter , 7421 (2001). G. I. Japaridze and E. Pogosyan, J. Phys.: Condens. Mat-ter , 9297 (2006). S. Nemati, S. Batebi, and S. Mahdavifar, Eur. Phys. J. B , 329 (2011). R.-X. Li, S.-L. Wang, K.-L. Yao, and H.-H. Fu, Phys. Lett.A , 2422 (2013). R. D. King-Smith and D. Vanderbilt, Phys. Rev. B ,1651 (1993). T. Hikihara and S. Yamamoto, J. Phys. Soc. Jpn. ,014709 (2008). T. Sakai and Y. Hasegawa, Phys. Rev. B , 48 (1999). N. Chepiga, F. Michaud, and F. Mila, Phys. Rev. B88