Reentrance of Topological Phase in Spin-1 Frustrated Heisenberg Chain
Yuan Yang, Shi-Ju Ran, Xi Chen, Zhengzhi Sun, Shou-Shu Gong, Zhengchuan Wang, Gang Su
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Reentrance of Topological Phase in Spin-1 Frustrated Heisenberg Chain
Yuan Yang, Shi-Ju Ran, ∗ Xi Chen, Zhengzhi Sun, Shou-Shu Gong, Zhengchuan Wang, † and Gang Su
1, 4, ‡ School of Physical Sciences, University of Chinese Academy of Sciences, P. O. Box 4588, Beijing 100049, China Department of Physics, Capital Normal University, Beijing 100048, China Department of Physics, Beihang University, Beijing 100191, China Kavli Institute for Theoretical Sciences, and CAS Center for Excellence in Topological Quantum Computation,University of Chinese Academy of Sciences, Beijing 100190, China (Dated: April 30, 2019)For the Haldane phase, the magnetic field usually tends to break the symmetry and drives thesystem into a topologically trivial phase. Here, we report a novel reentrance of the Haldane phaseat zero temperature in the spin-1 antiferromagnetic Heisenberg model on sawtooth chain. A partialHaldane phase is induced by the magnetic field, which is the combination of the Haldane state in onesublattice and a ferromagnetically ordered state in the other sublattice. Such a partial topologicalorder is a result of the zero-temperature entropy due to quantum fluctuations caused by geometricalfrustration.
Introduction. — Frustration incorporating with strongcorrelations usually leads to exotic quantum phenomena.A typical example in statistical physics is the reentrantphenomena, where a sequence of phase transitions, suchas A-B-A-B by lowering temperature may happen, whereA and B denote ordered and disordered phases, respec-tively. Taking the Ising model on kagom´e lattice withnearest and next-nearest neighboring interactions as anexample, one may find that upon lowering temperature,the system passes through four phases, a paramagneticphase, a magnetically-ordered phase, a reentrant param-agnetic phase, and a ferromagnetic phase . Such a reen-trant phenomenon was experimentally confirmed , andtheoretically investigated with different approaches .In the past decades, systematic investigations on thepartially disordered phase in both classical and quan-tum systems at finite temperatures were performedtheoretically and experimentally . It hasbeen widely recognized that the partially disorderedphase is driven by the thermal entropy, in which the dis-ordered part behaves like a “perfect” paramagnet. Thereentrant phenomena can be viewed as the order-by-disorder effect , which is driven by thermal fluctuationsand entropy.Recently, the interests on the reentrant phenomenawere renewed. The order-by-disorder effects were investi-gated theoretically and experimentally . For thespin-1 / √ × √ , a partially disorderedground state in the weakly frustrated regime was re-ported. The spins on the honeycomb sublattice form a180 ◦ N´eel order, and spins at the hexagon center sites arein a disordered state. This work is aimed at explainingthe spin-liquid behaviors of
LiZn M o O , and showsthat a partially disordered phase can be the ground stateof a simple quantum isotropic Heisenberg antiferromag-net. Comparing with the classical systems, the disor-dered subsystem is a short range ferromagnetically cor-related state instead of a paramagnet, and the mech-anism is owning to the zero-point quantum fluctuationsinstead of the thermal ones. Furthermore, previous works J J : Sublattice A : Sublattice B FIG. 1. (Color online) Sawtooth chain lattice with two sub-lattice A and B. Two exchange interactions J and J areindicated by black and green lines, respectively. show that the order-by-disorder effects are observed notjust on the frustrated magnets but also in cold-atomplatforms . The quantum fluctuations might be inducedby, for instance, the spin-orbit couplings .In this Letter, we study the spin-1 Heisenberg modelon the sawtooth chain (Fig. 1) by density matrix renor-malization group . Sawtooth chain is an important 1Dstructure that widely exists in natural materials (e.g.,Cu Cl(OH) ). The model Hamiltonian is given as H = J X i S i · S i +1 + J X odd i S i · S i +2 − h z X i S zi , (1)where J = cos( π θ ), J = sin( π θ ) (0 ≤ θ ≤ h z is the magnetic field along the z direction. We re-port a reentrance of the Haldane phase in this model atzero temperature. The calculational details are given inSupplementary Materials. We establish the θ − h z phasediagram of the system (Fig. 2), where two topologicallynon-trivial phases (Haldane and partial Haldane phases)are found. In the partial Haldane phase, only one sub-lattice of the system is in the Haldane phase (sublatticeA in Fig. 1), and the sublattice B is in a topologicallytrivial phase. Particularly for θ ≃ .
6, a reentrance ofthe topological Haldane phase induced by the magneticfield h z is revealed. The Haldane phase - trivial magneticphase - partial Haldane phase transitions by increasing h z are discovered. The partial Haldane phase appearsby increasing the magnetic field. Our work extends thequantum reentrant phenomena at zero temperature fromthe conventional phases (e.g., ) to topological phases. Haldane
UDD
PartialHaldane IC -UUD UUU i zi S < > :sublattice A :sublattice B UUU IC FIG. 2. (Color online) The phase diagram in the θ - h z planeobtained by our finite DMRG algorithm with periodic bound-ary condition. The length of the sawtooth chain is L = 160and the truncated dimensions is χ = 160. There exists aphase crossover region, which is surrounded by dashed line.There are two gapless magnetic ordered phases (UDD andUUU), and two gapped magnetic plateau phases( -UUD and -UUD). There appear two topological ordered phases, theHaldane phase and partial Haldane phase. Phase diagram. — We establish the phase diagramin the θ − h z plane as shown in Fig. 2. The phaseboundaries are determined by several quantities includ-ing the uniform and staggered magnetizations of the sub-lattices, the local magnetic moments, and the entangle-ment entropy measured in the middle of the system. Thephase diagram consists of seven quantum phases, includ-ing two gapless magnetic phases [up-up-up phase(UUU)and up-down-down (UDD) phase with the spins pointingdown (up) on the sublattice A (B)], two gapped mag-netic plateau phases [up-up-down phases on the M = plateau ( -UUD) and M = plateau ( -UUD)], an in-commensurate crossover (IC) region, and two topologicalphases (Haldane phase and partial Haldane phase). TheIC and UUU are connected by crossover (implied by dashlines) with no singular behaviors .Let us focus on the Haldane phase and partial Haldanephase. We first consider two limits θ = 0 and θ = 1. Inthe case of θ = 0, we have J = 1 and J = 0, andthe system becomes a standard spin-1 antiferromagneticHeisenberg chain. The ground state in this case is thewell-known Haldane phase (HP) with a finite gap∆ = 0 . J . In the case of θ = 1, we have J = 0and J = 1, the system can be regarded as Haldane chainin the sublattice A, decoupled to the free spins in thesublattice B.Fruitful physics appear at 0 < θ <
1. Fig. 3 (a) shows∆ A ( B ) and ∆ that are the spin gaps of the sublattices A θ 6 S L Q * D S (𝖺) Δ = 0Δ41 ΔΔ A Δ B − 𝗅 𝗈 𝗀 ( ) (𝖻) h z = 0h z = 1 FIG. 3. (Color online) (a) spin gap ∆, ∆ A , ∆ B versus θ .(b) Entanglement spectrum versus θ , where the number ofdots on each level indicates its degeneracy. The red circlesand the blue dots represent the cases of h z = 0 and h z = 1,respectively. (B), and that of the whole lattice. They are determinedby the width of the zero plateau of the magnetizations M A ( B ) = N P i ∈ A ( B ) h S zi i and M = ( M A + M B ) /
2, with N the total number of sites. For θ = 0, all ∆ and ∆ A ( B ) are close to ∆ = 0 . J , which is the gap of the standardHaldane chain. By increasing θ , the spin gaps increasemonotonously until θ ≃ .
38, where the gaps reach amaximum about ∆ = ∆ A ( B ) ≃ .
62. Afterwards, thespin gaps decrease rapidly and vanish at θ c ≃ .
64. For θ > θ c , the system enters a gapless region with ∆ =∆ A ( B ) = 0 until θ reaches θ c ≃ .
9. For θ c < θ < B remains zero (thus ∆ = 0), but ∆ A jumps to a finitevalue, which indicates that the sublattice A enters theHaldane phase. In this region, subsystem B is sensitiveto the external field and can be polarized by a very small h z , similar to a paramagnet. By increasing θ to θ = 1,∆ A approaches to ∆ A = ∆ , where the system becomesa Haldane chain decoupled with free spins.Fig. 3 (b) demonstrates the entanglement spectra (ES)of the ground state. It has been suggested that EScan be used to characterize and classify exotic quantumstates that are beyond the Landau symmetry-breakingparadigm . For example, the phase transition fromthe Haldane phase to a topological-trivial phase canbe detected by the collapse of the even-fold degeneracystructure of the ES . In our system for h z = 0, theES [red hollow circles in Fig. 3(b)] verses θ shows thatthe even-fold degeneracy structure always appears in thewhole range of 0 < θ <
1. This suggests that altering J and J will not drive the system out of the Haldanephase. This is expected since these coupling terms pre-serve the symmetry that protects the topological order ,which is known as the symmetry-protected topological(SPT) order . In a SPT phase, the topological orderwill be robust as long as the symmetry is preserved. At θ = θ c ≃ . θ = θ c might be a topo-logical phase transition point that separates two differentkinds of string orders .Fig. 3(b) also shows the ES versus θ at h z = 1 (bluesolid circles). In accordance to the phase diagram, nodouble degeneracy of the ES is observed for 0 < θ < . . < θ <
1. These regions are formed by sev-eral phases such as magnetic plateaus and incommen-surate regions which possess no topological orders. For0 . < θ < .
83, the system is in the partial Haldanephase. Different from the h z = 0 case for θ > θ c , the ESshows a four-fold degeneracy. In both cases, the Haldanestate in sublattice A contributes the two-fold degeneracy.For h z = 0, sublattice B gives a paramagnetic state, thuscontributes no degeneracy to the ES. In the partial Hal-dane phase with h z >
0, the magnetic field induces effec-tive ferromagnetic couplings between each two nearest-neighboring spins in sublattice B. It is known that for theground state of the ferromagnetic spin-1 chain, the twodominant values of the ES are degenerate (no degener-acy for the rest values). Therefore, the four-fold degen-eracy in the partial Haldane phase ( h z >
0) comes fromboth the Haldane and the effective ferromagnetic (FM)states. Within the partial Haldane phase, the state ison a M = magnetic plateau. As shown in Fig. 2 andFig. 3(a), the width of the 1 / UUD alsoshows a 1 / Reentrance of topological phases. — Normally, topo-logical phases including Haldane phase will be suppressedby the magnetic field. For a SPT phase, the field tends tobreak the symmetry and drives the system to a topologi-cally trivial phase. In our system, we show that the par-tial Haldane phase that exhibits topological order will beinduced by increasing the magnetic field. The Haldane -trivial phase - partial Haldane quantum phase transitionsare observed. Fig. 4 presents such a reentrant behaviorby showing the magnetizations and ES at θ = 0 .
62. For h z < h c ≃ . h z > h c , the sys-tem is driven to UDD phase. This is topologically trivialphase, where the spins shows long-range magnetic orders.The degeneracy of the ES is also lifted in this phase. By continuing increasing h z to h z > h c ≃ .
91, the systemis driven back to a topological phase (partial Haldanephase) with two-fold degeneracy in the ES [Fig. 4(d)].Until h z > h c ≃ .
69, the system enters a UUU phase.At θ = 0 .
62, the reentrance of the topological ordersare further revealed by two kinds of string orders O zπ and O zπ,A , which are defined as O zπ ( i, j ) = h ˆ S zi exp( j − X k = i +1 iπ ˆ S zk ) ˆ S zj i (2) O zπ,A ( i, j ) = h ˆ S zi exp( j − ,j ∈ A X k = i +1 ,i ∈ A iπ ˆ S zk ) ˆ S zj i (3) O zπ,A is defined on the sublattice A, and O zπ is definedon the whole lattice. In Fig. 4 (e), O zπ and O zπ,A arecalculated in the middle of the chain by taking sufficientlylarge distance ( | i − j | ≃ h z < . O zπ ≃ .
087 and vanishing O zπ,A ( ∼ O (10 − )). In the partial Haldane phase for0 . < h z < .
69, we have O zπ,A ≃ .
019 and vanishing O zπ ( ∼ O (10 − )).Note that due to the numerical noises (particularlynear the phase boundary, e.g., at θ = 0 .
62 in Fig. 4(d)and other results that are not shown), the four-fold de-generacy may be lifted to a double two-fold degeneratestructure with a small split. This is due to the differentstabilities of the degeneracies from the Haldane and effec-tive FM parts. The degeneracy from the Haldane state isprotected by the Haldane gap, thus is more stable undernumerical noises. The degeneracy from the FM state ismore sensitive to noises. This leads to a double two-foldstructure of the entanglement spectrum.The reentrance of the topological phases can also beunderstood intuitively with the effective ferromagneticcouplings in sublattice B, consistent with the four-folddegeneracy of the ES in partial Haldane phase. In theHaldane phase, all spins participate in the Haldane state.For 0 . < θ < .
9, the system enters the UDD mag-netic phase by increasing h z , where the spins in the twosublattices are towards different directions. Due to thegeometrical frustration, the sublattice B plays a role ofa ferromagnetic background to provide an effective mag-netic field h inner to sublattice A with h inner ∝ M B . The h inner (in the opposite direction to h z ) dominates overboth h z and the couplings J in sublattice A, so that thespins in sublattice A cannot form a Haldane state. Thisexplains why this region is gapless, where the h inner in-stead h z is responsible to close the Haldane gap. Until h z > .
97, the spins in sublattice B are totally polarized,and h z is strong enough to cancel with h inner so that thetotal effective field h effect = h z − h inner on sublatticeA is beneath the Haldane gap. In this case, the spins insublattice A forms a Haldane state, and the whole systementers the partial Haldane phase. By further increasing h z , the magnetic field is sufficiently large to drive the sys-tem out of partial Haldane phase and polarizes all spinsin the same direction. h z 0 D J Q H W L ] D W L R Q (𝖺) = 𝟢.𝟨𝟤 UDD PartialHaldane UUUMM A M B M As M Bs h z (𝖻) H a l d a n e h z − l o g ( λ ) (𝖼) h z (𝖽) h z 6 W U L Q J 2 U G H U (𝖾) UDD Partial Haldane
UUU zπ zπ, A FIG. 4. (Color online) Detailed description of M , M A , M B , M As with fixed θ = 0 .
62, where h z is in the range of [0 ,
2] in(a). (b) is the enlarged part of (a), where h z is in range of[0 , . h z = 0 .
017 and h z = 0 .
92, respectively. (e) shows two kindsof string orders O zπ and O at θ = 0 .
62. The inset shows thedetails of O zπ in the range of 0 < h z < . h z = 0 . O zπ jump to zero, which is the transition point from Haldanephase to UDD magnetic ordered phase. Summary. — This work extends the reentrant phe-nomena to the frustrated quantum system with topo-logical orders. We calculate the S = 1 antiferromag-netic Heisenberg model on sawtooth lattice with differ-ent coupling strength parameterized by θ . The ground-state phase diagram in a magnetic field is established,where fruitful phases are revealed. By tuning the mag-netic field, an Haldane phase - trivial phase - partialHaldane phase transitions are observed. The partiallytopological-ordered phase has the coexistence of the Hal-dane state and topologically-trivial magnetic state in thetwo sublattices, respectively.In the statistic physics, the reentrance is an entropicdriven phenomenon, where the phase is stabilized byminimization of energy in combination with maximizingthe entropy . The necessary condition for a reen-trant phenomenon to occur is the existence of partialdisordered phase with an ordered phase or a partial or-dered phase . Such a mechanism was recently extendedto frustrated quantum systems at zero temperature ,where the reentrant behavior (or the emergence of partial disorder) is a novel result of frustration andquantum fluctuations. Our work further generalizes thereentrance to topological systems, where the partiallydisordered phase is formed by the topological Haldanestate and topologically trivial magnetic state in thetwo different sublattices. We expect to find more noveltopological reentrant phenomena in two and higherdimensions. ACKNOWLEDGMENTS
The authors acknowledge Cheng Peng, Wei Li and YuChen for very useful discussions. S.J.R. is supportedby Beijing Natural Science Foundation (1192005 andZ180013) and Foundation of Beijing Education Commit-tees under Grants No. KZ201810028043. S.S.G. is sup-ported by the National Natural Science Foundation ofChina Grants (11834014, 11874078) and the Fundamen-tal Research Funds for the Central Universities. Thiswork is also supported in part by the NSFC (GrantNo. 11834014), the National Key R&D Program ofChina (Grant No. 2018FYA0305804), the StrategeticPriority Research Program of the Chinese Academy ofSciences (Grant No. XDB28000000), and Beijing Mu-nicipal Science and Technology Commission (Grant No.Z118100004218001).
Appendix A: Energy and entanglement entropy for h = 0 The Hamiltonian of our system is given by H = J X i S i S i +1 + J X odd j S j S j +2 − h z X i S zi , (S1)with J = cos( π θ ), J = sin( π θ ) (0 < θ < h z themagnetic field along the z direction. In Fig.S1, we showthe ground-state energy per site E and entanglement en-tropy(EE) for h z = 0 by density matrix renormalizationgroup(DMRG) with 160 sites and periodic con-dition. The entanglement entropy in the middle of thesystem as S = − T r [ λ log ( λ )] (S2)with λ the entanglement spectrum. The ground statewave function is represented as a matrix product states(MPS) .Fig.S1 (a) shows the ground-state energy per site E .In the limit of θ = 0, E is close to the energy e =1 . S = 1 Heisenbergchain . In the limit of θ = 1, E approaches to e / h = 0,we calculate the first-order derivative of E against θ , as ( Q H U J \ S H U V L W H E (𝖺) e = − 1.401484 e ∂ E / ∂ θ (𝖻) ( Q W D Q J O H P H Q W ( Q W U R S \ (𝖼) S = ln(2) θ = 0.65 6 W U L Q J 2 U G H U (𝖽) zπ,A zπ FIG. S1. (Color online) (a) The variational ground-state en-ergy per site E versus control parameter θ . In the limit of θ = 0, E = e , where e is the ground state energy per siteof the infinite antiferromagnetic S = 1 Heisenberg chain. Inthe limit of θ = 1, we have E = e /
2, because the spins inthe sublattice B are decoupled. The first order derivative of E is shown in (b), the curve is continuously, i.e., there is noapparent symmetry breaking. (c) The entanglement entropyversus θ , the minimum value cannot below to ln (2). (d) Thestring orders O zπ and O zπ,A versus θ . In the θ → O zπ = 0 . shown in Fig.S1 (b). No singular behavior of ∂E /∂θ is observed. The EE of our model is shown in Fig.S1(c). Similarly, we do not observe any singular behaviorof ground state EE against θ . Only a shoulder is ob-served near θ = 0 .
65 where the spin gap of the systemcloses (see the main text). Note that in the Haldanephase,the minimum value of entanglement entropy can-not drop below ln (2) . All of our simulations imply thatthe system will not be driven out of the Haldane phaseby θ , in accordance with the data and discussions givenin the manuscript.We simulate string orders (Fig.S1 (d)) that are definedas O zπ ( i, j ) = h S zi ( exp j − X k = i +1 iπS zk ) S zj i , (S3) O zπ,A ( i, j ) = h S zi ( exp j − ,j ∈ A X k = i +1 ,i ∈ A iπS zk ) S zj i . (S4)In the standard Haldane chain, one has O zπ,Haldane =0 . | i − j | . In our saw-tooth model, we have O zπ = O zπ,Haldane at θ = 0, andmeanwhile O zπ,A = 0. For θ > .
9, we have finite O zπ,A and vanishing O zπ . For 0 . < θ < .
9, it seems that twostring orders co-exist. But our simulation suffers quitelarge numeric errors in this region. We for now cannotconfirm whether both string orders should exist in thethermodynamic limit.
Appendix B: Magnetic properties for h > We demonstrate the magnetic properties along the line θ = 0 .
36 in Fig. S2 (a). The magnetizations M , M A ( B ) , M A ( B ) s are defined as M = 1 N i = N X i =1 h S zi i (S1) M A ( B ) = 1 N A ( B ) X i ∈ A ( B ) h S zi i (S2) M A ( B ) s = 1 N A ( B ) X i ∈ A ( B ) ( − i h S zi i , (S3)At h z = 0, the system is in the Haldane phase, where M = M A = M B = M As = M Bs = 0. When h z = 0 . h z = 0 .
79, thensystem get into -UUD plateau phase with the widthof plateau ≃ .
42. By increasing h z , the system en-ters again the IC region. For h z > .
57, the systemare in the -UUD plateau phase and the width of thisplateau is ≃ .
96. At the boundary between the IC regionand a plateau phase, a jump of the magnetization curveoccurs, implying the existence of independent magnonexcitations .The local magnetic moment h S zl i at different sites areshown in Fig. S2 (b), (c), (d), and (e). At h z = 1 . h z = 1 .
6, the system is in the -UUD and -UUD plateauphases,respectively, as shown in (c) and (e). Spins areantiferromagnetically correlated in sublattice A, and fer-romagnetically correlated in sublattice B. At h z = 0 . h z = 1 .
23, the system is in the IC region, and thelocal magnetic moment h S zl i shows a typical spin-density-wave (SDW) configuration in the real space as shown in(b) and (d). Previous studies on frustrated spin lad-ders showed that the total magnetization S ztot changesby an integer in the SDW phase, i.e. ∆ S ztot = integer ,where S ztot = P l h S zl i . We calculate the S ztot magnetiza-tion curve in the range of 1 . ≤ h z ≤ .
54 as shown in(d), showing ∆ S zl = 1. This means that the low-lyingexcitation in the IC region corresponds to a single spinflip.When h z = 1 . h z = 1 .
6, system is stabilizedin the M = and M = plateau phases, The spinconfigurations are shown by the local magnetizations inFig. S2 (c) and (e). Both of the two phases exhibit longrange order (LRO) with a same up-up-down structure.Spins are antiferromagnetically correlated in sublatticeA and ferromagnetically correlated in sublattice B. Wealso calculate the spin fluctuations h S z S zr i−h S z ih S zr i and zl S zl S zl S zl S zl S ztot S D =
46 47 48 49 50 51 52 535445
FIG. S2. (Color online) (a) The magnetizations M , M A ( B ) , M A ( B ) s with θ = 0 .
36. Local magnetic moment h S zl i in differentphases at (b) h z = 0 .
76 in the IC region, (c) h z = 1 . -UUD plateau phase, (d) h z = 1 .
23 in the IC region, and (e) h z = 1 . -UDD plateau phase. (f) Magnetization curve of S ztot in the IC region, which increases step-wise by an integer ∆ S zl = 1. r í í í í í & R U U H O D W L R Q (𝖺) = 𝟢.𝟥𝟨 h z = 0.76⟨S z0 S zr ⟩ − ⟨S z0 ⟩⟨S zr ⟩⟨S x0 S xr ⟩ r í í í í í & R U U H O D W L R Q (𝖻) = 𝟢.𝟥𝟨 h z = ⟨.⟨⟨S z0 S zr ⟩ − ⟨S z0 ⟩⟨S zr ⟩⟨S x0 S xr ⟩ r í í í í & R U U H O D W L R Q (𝖼) = 𝟢.𝟥𝟨 h z = ⟨.⟩3 ⟨S z0 S zr ⟩ − ⟨S z0 ⟩⟨S zr ⟩⟨S x0 S xr ⟩ r í í í í í & R U U H O D W L R Q (𝖽) = 𝟢.𝟥𝟨 h z = ⟨.6⟨S z0 S zr ⟩ − ⟨S z0 ⟩⟨S zr ⟩⟨S x0 S xr ⟩ FIG. S3. (Color online) Spatial dependence of the fluctua-tions h S z S zr i − h S z ih S zr i and h S x S xr i with fixed θ = 0 .
36 and(a) h z = 0 .
76, (b) h z = 1 .
1, (c) h z = 1 .
23, and (d) h z = 1 . h z = 0 .
76 and h z = 1 .
23, l h S z S zr i − h S z ih S zr i is domi-nant. When h z = 1 . .
23, both h S z S zr i − h S z ih S zr i and h S z S xr i decay exponentially. h S x S xr i . As shown in Fig. S3 (b) and (d), both decayexponentially since the states are gapped.For h z = 0 .
76 and h z = 1 .
23, it is obvious that h S zl i breaks the translational invariance and exhibits a typicalSDW. In general, the dominant spin correlation in a SDWare the component parallel to the field . To confirmwhether a SDW emerges in the IC region, we compute h S z S zr i−h S z ih S zr i and h S x S xr i as shown in Fig.S3 (a) and h z 0 D J Q H W L ] D W L R Q (𝖺) = 𝟢.𝟥𝟤 Haldane IC -UUD IC -UUD IC - UUU
UUU F u ll y P o l a r i z e d MM l < S z l > (𝖻) -UUU h z = 3.2θ = 0.32 6 X E O D W W L F H $ 6 X E O D W W L F H % FIG. S4. (Color online)(a) The magnetization per site M for θ = 0 .
32. (b) Local magnetic moment h S zl i for θ = 0 .
32 and h z = 3 . -UUU phase. (c) at h z = 0 .
76 and h z = 1 .
23, respectively. Both spinfluctuations show the incommensurate feature. Further-more, h S z S zr i − h S z ih S zr i dominates over other fluctua-tions. Combined with property of ∆ S ztot = 1 as shown θ (𝖺) h z = 0.6 UDD Partial Haldane UUUMM A M B M As M Bs θ − l o g ( λ ) (𝖻) FIG. S5. (Color online) (a) The magnetizations M , M A , M B , M As versus θ with h z = 0 .
6. (b) The entanglementspectrum (logarithmic plot). When θ < .
72, the spectrumis not degenerate; when θ > .
72, the whole spectrum showseven-fold degeneracies. in Fig.S2(f), evidently the IC region is a kind of SDWphase.Besides the phases shown in the phase diagram in themain text, there are additionally two phases with rela-tively large h z , which are -plateau phase with an up-up-up structure (denoted as -UUU) and the fully po-larized phase. Fig. S4 (a) shows the magnetization persite at θ = 0 .
32, where the system goes through the -UUU and fully polarized phases for about h z >
3. Fig.S4 (b) shows the local magnetic moment h S zl i within the -UUU phase ( h z = 3 . Appendix C: Magnetizations and entanglementrelated to the partial Haldane phase
Fig. S5 shows the magnetizations and entanglement h z = 0 .
6, where the system goes through the UDD, par-tial Haldane, and UUU phases. In Fig.S5 (a), we have M A < M B > M A ( B ) s = 0 when θ < . θ (𝖺) h z = 1.35 -UUD PartialHaldane UUU MM A M B M As M Bs ⟨ S z l ⟩ (𝖻) = 𝟢.𝟧𝟪 𝗁 𝗓 = 1.35 V X E O D W W L F H $ V X E O D W W L F H % h z − 𝗅 𝗈 𝗀 ( ) (𝖼) = 𝟢.𝟧𝟪 h z = 1.35 l ⟨ S z l ⟩ (𝖽) = 𝟢.𝟨 𝗁 𝗓 = 1.35 V X E O D W W L F H $ V X E O D W W L F H % − 𝗅 𝗈 𝗀 ( ) (𝖾) = 𝟢.𝟨 h z = 1.35 FIG. S6. (Color online) (a) The magnetizations M , M A , M B , M As versus θ at h z = 1 .
35. The local magnetic momentum h S zi i at different sites are illustrated in (b) and (d) with θ =0 .
58 and θ = 0 .
6. The entanglement spectrum (logarithmicplot) is showed in (c) and (e). within both sublattices A and B, but are antiferromag-netically arranged between A and B. The spins pointdown in A and point up in B. When θ ≥ = 0 .
72, wehave M A = M As = 0 suggesting that the sublattice Aenters the Haldane phase with vanishing magnetizations.Meanwhile, we have M B = 1 showing that the spins insublattice B are fully polarized. So the uniform magneti-zation of the whole system M equals to 0 .
5, giving a 1 / θ . For θ < .
72, nodegeneracy is observed in the spectrum. For θ > . l F R U U H O D W L R Q (𝖺) h z = 1.35 θ = 0.58 -UUD⟨S i S i ⟨ 2 ⟩ ⟨S i S i ⟨ 1 ⟩ l F R U U H O D W L R Q (𝖻) h z = 1.35 θ = 0.⟩2 Partial Haldane⟨S i S i ⟨ 2 ⟩ ⟨S i S i ⟨ 1 ⟩ FIG. S7. (Color online) At h z = 1 .
35, the nearest-neighborspin correlation h S i S i +1 i and next nearest-neighbor spin cor-relation h S i S i +2 i of (a) the UUD plateau phase with θ =0 .
58 and (b) partial Haldane phase with θ = 0 .
72. The insetof (b) shows that the errors that break the translational in-variance of the Haldane phase in sublattice A is only O (10 − ). Both the -UUD and partial Haldane phases are onthe magnetic plateau. However, the properties of these two phases are completely different. Fig.S6 (a) shows themagnetizations at h z = 1 .
35. For θ < . , the system isin -UUD with M = 0 .
5, and both M A and M As are fi-nite. For θ = 0 .
6, both M As and M A vanish to zero at thesame time. Fig. S6 (b) and (d) give the local magneticmomentums in the two phases, respectively. Usually aplateau magnetization is a commensurate, classical statestabilized by quantum fluctuations. This is well reflectedin Fig.S6 (b) where the system is in -UUD phase andthe local magnetic momentums in the sublattice A ex-hibit a classical antiferromagnetic order. In contrast for θ ≥ . plateau is formed bythe magnetically disordered Haldane state in A and theclassical polarized state in B. Besides the local magneticmomentum, the -UUD and partial Haldane phases canbe distinguished by the entanglement spectrum as shownin Fig. S6 (c) and (e). The ES is not degenerate at h z = 1 .
35 and θ = 0 .
58, while it shows double degener-acy at θ = 0 . h S i S i +1 i and next-nearest-neighbor (NNN)spin correlation h S i S i +2 i . Note h S i S i +2 i is in fact the NNspin correlation on sublattice A ( J couplings). As shownin Fig. S7 (a), when system is in -UUD plateau phase,obviously the translational invariance is broken. The cor-relation is consisted with the pattern of local magneticmomentum shown in Fig.S6 (b).As the local magnetic moment is equal to zero on sub-lattice A in the partial Haldane phase, it is also possiblethat the spins form valance bonds. If this is true, thetranslational invariance should be broken. To excludethis case, we show h S i S i +1 i and h S i S i +2 i in the partialHaldane phase in Fig. S7 (b). Both the h S i S i +1 i and h S i S i +2 i are constant, meaning the translational invari-ance is kept. 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