Topological phase transition and Z 2 index for S=1 quantum spin chains
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Topological phase transition and Z index for S = 1 quantum spin chains Hal Tasaki
Department of Physics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171-8588, Japan (Dated: July 31, 2018)We study S = 1 quantum spin systems on the infinite chain with short ranged Hamiltonianswhich have certain rotational and discrete symmetry. We define a Z index for any gapped uniqueground state, and prove that it is invariant under smooth deformation. By using the index, weprovide the first rigorous proof of the existence of a “topological” phase transition, which cannot becharacterized by any conventional order parameters, between the AKLT model and trivial models.This rigorously establishes that the AKLT model is in a nontrivial symmetry protected topologicalphase. PACS numbers: 05.30.Rt, 75.10.Kt, 75.50.Ee
Introduction and motivation .—In early 1980’s, Hal-dane discovered that the antiferromagnetic Heisenbergchain with the Hamiltonian ˆ H H = P j ˆ S j · ˆ S j +1 hasa unique gapped ground state when spin S is an inte-ger [1–5]. The discovery opened a rich area of researchin quantum many-body systems. See, e.g., [6, 7]. Af-ter the validity of Haldane’s conclusion had been con-firmed, an important issue was to elucidate the true na-ture of the gapped ground states of ˆ H H . A typical prob-lem was to precisely characterize the difference betweenthe gapped ground state of the solvable AKLT modelˆ H AKLT = P j { ˆ S j · ˆ S j +1 + ( ˆ S j · ˆ S j +1 ) / } [8, 9] andthe trivial gapped ground state (where all spins are inthe 0 state) of the trivial model ˆ H tr = P j ( ˆ S z ) , bothfor the S = 1 chain. The two ground states cannot bedistinguished by any conventional order parameters. Itwas soon realized that the ground states in the “Haldanephase”, to which ˆ H AKLT and ˆ H H belong, are “exotic” inthe sense that they exhibit hidden antiferromagnetic or-der that can be characterized by a string order parameter[9, 10], and are accompanied by edge spins when definedon open chains [9, 11]. These exotic properties, as well asthe existence of a gap, were then interpreted as naturalconsequences of breakdown of hidden Z × Z symmetry[12, 13].It gradually became clear however that the picture ofhidden Z × Z symmetry breaking was neither sufficientnor necessary to characterize the Haldane phase [14]. In2009, Gu and Wen pointed out that the Haldane phaseshould be identified as a symmetry protected topologicalphase [15]. This means that, for example, ˆ H AKLT and ˆ H tr can be smoothly connected through a family of Hamil-tonians with a gapped unique ground state if any shortranged Hamiltonian is available; if, on the other hand,proper symmetry is imposed on the family of accessibleHamiltonians, then one must go through a phase transi-tion in order to connect ˆ H AKLT and ˆ H tr . Such a phasetransition is called “topological” since it is not character-ized by a conventional order parameter. A complete setof symmetry required to protect the Haldane phase wassoon identified by Pollmann, Turner, Berg, and Oshikawa [16, 17]. They concluded that the Haldane phase in odd S quantum spin chains is protected either by (S1) Z × Z symmetry (i.e., the π rotations about the x and z axes),(S2) bond-centered reflection (inversion) symmetry, or(S3) time-reversal symmetry. They also showed that,for each symmetry, the Haldane phase and the trivialphase can be distinguished by a Z index which identi-fies the projective representation of the symmetry group,provided that the ground states are represented as (in-jective) matrix product states [18].All these pictures suggest that the infinite-volumeground states of the one-parameter family of Hamilto-nians ˆ H s = s ˆ H AKLT + (1 − s ) ˆ H tr , (1)with s ∈ [0 , s and the Hal-dane phase with s close to 1. But, rather surprisingly, theexistence of a phase transition was not rigorously estab-lished before. Although some of the known arguments arevery plausible, there are still delicate gaps between math-ematically rigorous proofs, as we now discuss briefly. See[19] for details. (On the other hand there are rigorousand explicit results which show that certain seeminglydifferent ground states can be smoothly connected witheach other. See, e.g., [17, 20].)(i) Change in parity : In a periodic chain with an oddnumber of sites, the ground states of ˆ H AKLT and ˆ H tr have odd and even parities, respectively, with respectto the reflection about a single bond [17]. Thus theremust be a level crossing at intermediate s , suggesting aphase transition. But this does not really imply a phasetransition in the infinite volume limit. The situation iseven trickier since there is no level crossing in a chainwith an even number of sites.(ii) Change in indices : The indices characterizing pro-jective representations of the group symmetry provide asophisticated support for the existence of a phase transi-tion [6, 7, 16, 17]. There are three indices (correspond-ing to the three types of symmetry) which take 1 and − H tr and ˆ H AKLT , respectively.Then there should be a phase transition associated witha jump in an index. A major drawback of this approachis that the indices are well defined only for matrix prod-uct states (which also satisfies a strong condition calledinjectivity). Although it is known that a gapped groundstate can be efficiently approximated by a matrix prod-uct state (see, e.g., [21]) the approximation is not preciseenough to derive a definite conclusion about phase tran-sitions in full ground states. The same comment appliesto the non-local order parameters discussed in [22, 23](e.g., (20) of [23]). These quantities are well defined for ageneral state, but their quantization can be proved onlyfor matrix product states.(iii)
Hidden Z × Z symmetry breaking : There is a nonlo-cal unitary transformation (for open chains) which mapsthe Hamiltonians (1) to different local Hamiltonians [12–14]. After the transformation the number of infinite vol-ume ground states of the models with s = 0 and 1 becomeone and four, respectively. Then, by definition, the infi-nite volume limit of the transformed model must exhibita phase transition. This rigorous result strongly suggeststhat the original model too undergoes a phase transition.But we still do not have any proof in this direction. Thisis because the original model always has a unique groundstate, and we do not know anything about the nature ofthe phase transition in the transformed model (except forthe change in the number of ground states).One of the contributions of the present work is thefirst completely rigorous proof of the existence of a phasetransition in (1). More generally, we establish that theAKLT model is in a nontrivial symmetry protected topo-logical phase within three classes, called C1, C2, and C3,of Hamiltonians that we specify below. The result ex-tends to other quantum spin chains and one-dimensionalelectron systems with proper symmetry [19]. Our proof,which is based on a new “topological” index defined for aunique gapped ground state of an infinite chain, clearlyillustrates why and how a gapless point emerges whenthe index changes. We hope that this interesting ar-gument leads to a deeper understanding of topologicalphase transitions. Our index is related to the order pa-rameter introduced by Nakamura and Todo [24]. A dif-ferent index was defined in [25] also for infinite systems,but it has not yet been used to analyze phase transitions. Basic strategy .—Let us briefly (and heuristically) dis-cuss the basic idea of our proof for a finite system. Theproof is by contradiction. Consider a large periodic chain,and let | Φ gs s i be the ground state of ˆ H s of (1) for s ∈ [0 , V ℓ = O j : | j − |≤ ℓ + exp h − i π j + ℓ ℓ + 1 ˆ S z j i , (2)which is the local version of the twist operator of Lieb, Schultz, and Mattis [27]. Following Nakamura and Voit[28], Nakamura and Todo [24] pointed out that the ex-pectation value of the twist operator acts as an orderparameter for the Haldane phase (see also [29]). In thepresent context, the ground states at s = 0 and 1 arecharacterized by h Φ gs0 | ˆ V ℓ | Φ gs0 i = 1 and h Φ gs1 | ˆ V ℓ | Φ gs1 i ≃ − ℓ . We also show from the sym-metry that h Φ gs s | ˆ V ℓ | Φ gs s i is always real. Then, by con-tinuity, there must be s such that h Φ gs s | ˆ V ℓ | Φ gs s i = 0.This means that | Ψ i = ˆ V ℓ | Φ gs s i is orthogonal to | Φ gs s i .From the variational estimate of [27], we also see that h Ψ | ˆ H | Ψ i − E gs s = O ( ℓ − ), where E gs s is the ground stateenergy of ˆ H s . We thus conclude that the energy gap ofˆ H s is O ( ℓ − ). But this is a contradiction since ℓ can bemade as large as one wishes. Note that, in this argument,the twist operator ˆ V ℓ plays two essentially different roles,one as an observable whose expectation value is an orderparameter, and the other as an unitary operator whichgenerates a low energy excited state exactly as in theoriginal work of Lieb, Schultz, and Mattis [27].By using a similar idea we can also show for a uniquegapped ground state | Φ gs i (with a certain symmetry con-dition) that the expectation value h Φ gs | ˆ V ℓ | Φ gs i takes aconstant sign for sufficiently large ℓ . We identify the signas a Z index of the ground state. Our index is closelyrelated to the Berry phase of quantum spin chains intro-duced by Hatsugai [30–34], and also to the polarizationin electron systems. For the latter, see [28, 35] and ref-erences therein. Setting and results .—We study S = 1 quantum spinsystems on the infinite chain Z . By ˆ S αj with α = x , y , zwe denote the α -component of the spin operator at site j ∈ Z . We denote by ˆ U αθ = N ∞ j = −∞ e − iθ ˆ S αj the globalrotation by θ ∈ R about the α -axis.We define three classes, which we call C1, C2, andC3, of Hamiltonians. A Hamiltonian in these classes iswritten as ˆ H = P ∞ j = −∞ ˆ h j , where the local Hamiltonianˆ h j depends only on spin operators at sites k such that | j − k | ≤ r , and satisfies k ˆ h j k ≤ h and ( ˆ U z θ ) † ˆ h j ˆ U z θ = ˆ h j for any θ . The range r and h are arbitrary fixed posi-tive constants. The Hamiltonian is invariant under anyrotation about the z-axis. We require additional discretesymmetry depending on the class. In C1, we assumethat the Hamiltonian is invariant under the π -rotationabout the x-axis, i.e., ˆ U x π ˆ H ˆ U x π = ˆ H . In C2, we assumereflection invariance ˆ R ˆ H ˆ R = ˆ H , where ˆ R is the bond-centered reflection operator induced by j → − j , i.e.,ˆ R ˆ S αj ˆ R = ˆ S α − j . In C3, we assume that ˆ H is invariantunder time-reversal ˆ S αj → − ˆ S αj .Let us summarize standard definitions of the unique-ness of the ground state and of the energy gap forinfinite systems [9, 26]. Given a Hamiltonian ˆ H forthe infinite chain, consider a corresponding Hamiltonianˆ H L = ( P L +1 − rj = − ( L − r ) ˆ h j ) + ∆ ˆ h − L + ∆ ˆ h L +1 on a finite chain {− L, . . . , L +1 } , where ∆ ˆ h − L , ∆ ˆ h L +1 are certain bound-ary Hamiltonians which act on spins around − L and L + 1, and respect the symmetry in each class. Let | Φ gs L i be a ground state of ˆ H L . The (infinite volume) groundstate ω ( · ) of the Hamiltonian ˆ H is defined as the limit ω ( ˆ A ) = lim L ↑∞ h Φ gs L | ˆ A | Φ gs L i , (3)where ˆ A is an arbitrary local operator. (By a local op-erator we mean a function of a finite number of spinoperators.) We say that ˆ H has a unique (infinite vol-ume) ground state if the limiting ω ( · ) is independent ofthe choice of the boundary Hamiltonians ∆ ˆ h ± L and thechoice of the finite volume ground state | Φ gs L i .Suppose that ˆ H has a unique ground state ω ( · ). Wesay that ˆ H has a nonvanishing energy gap if there is aconstant ǫ >
0, and one has ω ( ˆ A † [ ˆ H, ˆ A ]) ≥ ǫ, (4)for any local operator ˆ A such that ω ( ˆ A ) = 0 and ω ( ˆ A † ˆ A ) = 1. The supremum of such ǫ , which we de-note as ∆ E , is the energy gap of ˆ H . By recalling thedefinition (3) of the ground state, one sees that this isnothing but a straightforward extension of the standardvariational characterization of the energy gap. Note alsothat, although ˆ H acts on infinitely many spins, the com-mutator [ ˆ H, ˆ A ] is a local operator.Suppose that ω ( · ) is a unique ground state of a Hamil-tonian in class C1. The uniqueness implies that thestate is invariant under π -rotation about the x-axis, i.e., ω ( ˆ U x π ˆ A ˆ U x π ) = ω ( ˆ A ) for any local operator ˆ A . Sinceˆ U x π ˆ V ℓ ˆ U x π = ˆ V † ℓ , we see that ω ( ˆ V ℓ ) ∈ R . Similarly theunique ground state ω ( · ) of ˆ H in C2 satisfies ω ( ˆ R ˆ A ˆ R ) = ω ( ˆ A ) for any ˆ A . Since exp[ i π ˆ S z j ] = 1, we findˆ R ˆ V ℓ ˆ R = O j exp h − i π − j + ℓ ℓ + 1 ˆ S z j i = O j exp h i π j + ℓ ℓ + 1 ˆ S z j i = ˆ V † ℓ , (5)which again implies ω ( ˆ V ℓ ) ∈ R . We can also show ω ( ˆ V ℓ ) ∈ R for the class C3. See below. Theorem 1 .—Suppose that a Hamiltonian ˆ H in C1, C2or C3 has a unique ground state ω ( · ) and a gap ∆ E > ℓ such that ℓ > max { ℓ , C/ ∆ E } , the ex-pectation value ω ( ˆ V ℓ ) ∈ R is nonzero and has a constantsign. Here C and ℓ are positive constants which dependonly on the constants r and h (which we fixed in thebeginning).The theorem guarantees that we can unambiguouslydefine an index σ ( ˆ H ) = ± H with aunique gapped ground state by σ ( ˆ H ) = ω ( ˆ V ℓ ) (cid:12)(cid:12) ω ( ˆ V ℓ ) (cid:12)(cid:12) for ℓ > max n ℓ , C ∆ E o . (6) One can also prove that ω ( ˆ V ℓ ) → ± ℓ ↑ ∞ by usingthe method in [36].To state the essential property of the index, which isTheorem 2, we introduce the (standard) notion that twoHamiltonians are smoothly connected. Definition .—Two Hamiltonians ˆ H and ˆ H in one ofthe classes C1, C2, or C3 are said to be smoothly con-nected within the class when the following is valid. Thereare a positive constant ∆ E min and a one-parameter fam-ily of Hamiltonians ˆ H s (with s ∈ [0 , s ∈ [0 , H s has a uniqueinfinite volume ground state ω s ( · ) with a nonvanishingenergy gap which is not less than ∆ E min . For any localoperator ˆ A , the expectation value ω s ( ˆ A ) is continuous in s ∈ [0 , Theorem 2 .—If two Hamiltonians ˆ H and ˆ H in one ofthe classes C1, C2, or C3 are smoothly connected withinthe class, then σ ( ˆ H ) = σ ( ˆ H ).Thus our index is “topological” in the sense that itis invariant under smooth deformation. A trivial butimportant corollary is the following. Corollary 1 .—If one has σ ( ˆ H ) = σ ( ˆ H ) for two arbi-trary Hamiltonians ˆ H and ˆ H in one of the classes C1,C2, or C3, they can never be smoothly connected withinthe same class.In order to connect such ˆ H and ˆ H within the sameclass, one must go through a phase transition, eitherby passing through a gapless model or a model withnonunique ground states, or by experiencing a discon-tinuous jump in the expectation value ω s ( ˆ A ) of a certainlocal operator ˆ A .Consider, as an example, the AKLT model ˆ H AKLT = P j { ˆ S j · ˆ S j +1 + ( ˆ S j · ˆ S j +1 ) / } , which is in C1, C2, andC3. The model has a unique ground state ω VBS ( · ), calledthe VBS state, and a nonzero energy gap [7, 9]. As wesee below, it can be shown that ω VBS ( ˆ V ℓ ) ≃ − ℓ , and hence σ ( ˆ H AKLT ) = −
1. There aremany examples, including the trivial modelˆ H tr = P j ( ˆ S z ) and the dimerized model ˆ H dim = P k ˆ S k · ˆ S k +1 , which are in C1, C2, and C3, have aunique ground state with a gap, and characterized by theindex σ ( ˆ H ) = 1. This observation leads to the followingcorollary, whose special case is the conclusion about aphase transition in (1). Corollary 2 .—One must go through a phase transi-tion in order to connect ˆ H AKLT to ˆ H tr or ˆ H dim (or otherHamiltonians with trivial index) within one of the classesC1, C2, or C3. Thus the AKLT Hamiltonian is in a non-trivial symmetry protected topological phase. Proof of the theorems .—We start from a variationalestimate of the Lieb-Schultz-Mattis type.
Lemma .—There are positive constants C and ℓ whichdepend only on the constants r and h (which we fixedin the beginning). For any Hamiltonian ˆ H (in C1, C2,or C3) and its (not necessarily unique) ground state ω ( · ),we have for any ℓ ≥ ℓ that ω ( ˆ V † ℓ [ ˆ H, ˆ V ℓ ]) = ω ( ˆ V † ℓ ˆ H ˆ V ℓ − ˆ H ) ≤ Cℓ . (7)
Proof : Following [37], we note that ω ( ˆ V ℓ ˆ H ˆ V † ℓ − ˆ H ) ≥ ω ( · ) is a ground state. Then ω ( ˆ V † ℓ ˆ H ˆ V ℓ − ˆ H ) ≤ ω ( ˆ V † ℓ ˆ H ˆ V ℓ + ˆ V ℓ ˆ H ˆ V † ℓ − H )= X j : | j − |≤ ℓ + r + ω ( ˆ V † ℓ ˆ h j ˆ V ℓ + ˆ V ℓ ˆ h j ˆ V † ℓ − h j ) ≤ X j : | j − |≤ ℓ + r + k ˆ V † ℓ ˆ h j ˆ V ℓ + ˆ V ℓ ˆ h j ˆ V † ℓ − h j k . (8)Define the local twist operator around j as ˆ V j,ε = N k : | k − j |≤ r exp[ − iε ( k − j ) ˆ S z k ]. By using the rotation in-variance of h j , we find ˆ V † ℓ ˆ h j ˆ V ℓ + ˆ V ℓ ˆ h j ˆ V † ℓ = ˆ V † j,ε ˆ h j ˆ V j,ε +ˆ V † j, − ε ˆ h j ˆ V j, − ε with ε = π/ℓ . Note that this is an evenfunction of ε which equals 2ˆ h j when ε = 0. Thus by ex-panding in ε and using k ˆ h j k ≤ h , we find k ˆ V † j,ε ˆ h j ˆ V j,ε +ˆ V † j, − ε ˆ h j ˆ V j, − ε − h j k ≤ Bε for sufficiently small ε with aconstant B which depends only on r and h . Note thatthe local Hamiltonians near ± ℓ satisfies the same boundsince they are less modified. Thus the right-hand side of(8) is bounded by 2( ℓ + r + 1) B ( π/ℓ ) , which is furtherbounded by C/ℓ for sufficiently large ℓ .We now prove the theorems. Note that, since we arealways dealing with a unique ground state of a Hamil-tonian in C1, C2, or C3, the expectation value of ˆ V ℓ isreal.To prove Theorem 1, we treat ℓ as a continuousvariable, and assume that the expectation value ω ( ˆ V ℓ )changes its sign for ℓ such that ℓ > max { ℓ , C/ ∆ E } .Note that ˆ V ℓ is continuous in ℓ as an operator becauseexp[ i π ˆ S z j ] = 1. Since its expectation value ω ( ˆ V ℓ )is also continuous in ℓ , there must be ℓ with ℓ > max { ℓ , C/ ∆ E } such that ω ( ˆ V ℓ ) = 0. But this, withthe variational estimate (7), contradicts the assumptionthat the gap is ∆ E . Recall (4) and note that C/ℓ < ∆ E .The proof of Theorem 2 is similar. Suppose that ˆ H and ˆ H are smoothly connected with the minimum gap ∆ E min >
0. We then choose ℓ so that ℓ ≥ ℓ and C/ℓ < ∆ E min . We find from the assumed continuity of ω s ( ˆ V ℓ )that there is s ∈ (0 ,
1) such that ω s ( ˆ V ℓ ) = 0. Againthis contradicts the assumption that the minimum gap is ∆ E min .It remains to verify two minor points. Let us showthe reality of ω ( ˆ V ℓ ) for C3. This is not as straight-forward as the other two classes. We work on a fi-nite chain {− L, . . . , L + 1 } . Let | Φ gs i = P σ ϕ ( σ ) | σ i be a ground state, where ϕ ( σ ) ∈ C is a coefficient,and | σ i is the standard ˆ S z -basis states correspondingto a spin configuration σ = ( σ − L , . . . , σ L +1 ) with σ j = 0 , ±
1. The time-reversal invariance of the Hamiltonianimplies that the time-reversal of | Φ gs i given by | Ψ gs i = P σ { Q L +1 j = − L ( − σ j }{ ϕ ( − σ ) } ∗ | σ i also converges to thesame infinite volume ground state ω ( · ). Then it is easilyconfirmed that ω ( ˆ V ℓ ) ∈ R by comparing the expressionsfor h Φ gs | ˆ V ℓ | Φ gs i and h Ψ gs | ˆ V ℓ | Ψ gs i .Let us explain one of many methods to evaluate theexpectation value ω VBS ( ˆ V ℓ ) for the VBS state, the exactground state of ˆ H AKLT . See also [19, 24]. We assume forsimplicity that ℓ is an integer. It is known that, in theVBS state, configurations of the z-component of spins ona finite interval {− ℓ, . . . , ℓ + 1 } is exactly obtained as fol-lows [7, 9]. For each site one assigns spin 0 with probabil-ity 1/3, or leave it unspecified with probability 2/3. Thisis done independently for all sites in the interval. Then tothe unspecified sites, one assigns a completely alternatingsequence + − + − + −· · · or − + − + − + · · · , each with prob-ability 1/2. In this way one gets spin configurations with-out conventional order but with hidden antiferromagneticorder. For a given configuration ( σ − ℓ , . . . , σ ℓ +1 ) with σ j = 0 , ± , we let j , . . . , j N be those sites with σ j = ± ,ordered as j k < j k +1 . Then one finds by inspection that ℓ +1 X j = − ℓ ( j + ℓ ) σ j = − σ j [ N/ X k =1 ( j k − j k − )+ χ odd ( j N + ℓ ) σ j N , (9)where χ odd = 1 or 0 if N is odd or even, respectively.Note that P [ N/ k =1 ( j k − j k − ) may be interpreted as thepolarization by identifying ± spins with ± charges. Sincesites j , . . . , j N are chosen randomly, we see for large ℓ that j N ≃ ℓ , and P [ N/ k =1 h j k − j k − i ≃ ℓ , where h· · · i denotes the average with respect to the probability de-scribed above. Then the law of large number implies thatexp h − i π ℓ +1 X j = − ℓ j + ℓ ℓ + 1 σ j i → − , (10)as ℓ ↑ ∞ with probability 1. Discussion .—In S = 1 quantum spin chains, we haveshown that the expectation value of the Affleck-Lieb twistoperator, a local version of the Nakamura-Todo orderparameter [24], defines a Z index for a unique gappedground state. The index enables us to prove that theAKLT model cannot be smoothly connected to a trivialmodel with index 1 within the class C1, C2, or C3. Asfar as we know, this is the first rigorous demonstration ofthe existence of a symmetry protected topological phasein an interacting quantum many-body system. Our proofalso shows why and how a gapless mode emerges at thetransition point.The reader may have noticed that the three classes ofHamiltonians, C1, C2, and C3, correspond to the threetypes, S1, S2, and S3, of symmetry necessary to pro-tect the Haldane phase. See Introduction. Note howeverthat we are requiring the rotational symmetry about thez-axis. From the point of view of symmetry protectedtopological phase, the additional rotational symmetry isnot only unnecessary but may lead to more complicatedphase structures [38]. The requirement of the rotationalsymmetry is certainly not desirable. But, for the mo-ment, the symmetry seems to be indispensable for ourproof, which makes use of the Lieb-Schultz-Mattis typeargument. Recent (not yet rigorous) Lieb-Schultz-Mattistype statements without a continuous symmetry [39] maycontain a hint for removing the assumption. We notehowever that our index, or, equivalently, the Nakamura-Todo order parameter likely fails to distinguish the Hal-dane phase when only the symmetry S2 or S3 is present.Although we have concentrated on S = 1 chains forsimplicity, all the general results in the present paperreadily extend to spin chains with general S [19]. A non-trivial point is whether we can identify ground states withnontrivial index −
1. Tractable examples include variousVBS type states [9], including the VBS state for odd S ,intermediate D state [14] for S = 2, and the partiallymagnetized VBS state [14, 40] for S = 2. The extensionto lattice electron systems show that the AKLT modelcannot be smoothly connected to a trivial band insulatorwhen proper symmetry is assumed [19]. I wish to thank Tohru Koma and Ken Shiozaki for discus-sions which inspired the present work, and for indispensablecomments. I also thank Yohei Fuji, Yasuhiro Hatsugai, HoshoKatsura, Bruno Nachtergale, Masaaki Nakamura, Masaki Os-hikawa, and Haruki Watanabe for valuable discussions andcomments. The present work was supported by JSPS Grants-in-Aid for Scientific Research no. 16H02211.[1] F.D.M. Haldane,
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