Absence of local order in topologically frustrated spin chains
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Absence of local order in topologically frustrated spin chains
V. Marić,
1, 2, ∗ S. M. Giampaolo, † and Fabio Franchini ‡ Institut Ruđer Bošković, Bijenička cesta 54, 10000 Zagreb, Croatia SISSA and INFN, via Bonomea 265, 34136 Trieste, Italy
We show that a wide class of spin chains with topological frustration cannot develop any localorder. In particular, we consider translational-invariant one-dimensional chains with frustratedboundary conditions, i.e. periodic boundary conditions and an odd number of sites, which possessa global SU (2) symmetry. This condition implies, even at a finite sizes, an exact degeneracy of theground state and is quite general in absence of external fields. We directly evaluate the expectationvalue of operators with support over a finite range of lattice sites and show that, except for someprecise conditions, they all decay algebraically, or faster, with the chain length and vanish in thethermodynamic limit. The exceptions that admit a finite order are cases with a higher ground statedegeneracy in which the translational symmetry is broken by the ground state choice. Frustration arises as a competition between terms pro-moting different types of arrangement. Although thisdefinition applies also to quantum Hamiltonians with un-frustrated counterpart [1–3], it is usually meant in itsclassical origin, known as geometrical frustration [4, 5]. Intwo- and three-dimensional frustrated systems, one canidentify several frustrated loops, either because of com-peting next-to-nearest neighbor terms or just because ofthe lattice geometry. In such cases, the amount of frus-tration scales with the system’s size and the interplaybetween local interactions, quantum effects, and the non-local nature of geometrical frustration renders the studyof these systems very challenging. On the other side,their phenomenology is very rich, displaying algebraicdecays of correlation functions not associated to critical-ity [6, 7], localized zero energy modes [8–11], non-zero en-tropy at near-zero temperature [12, 13], etc. Due to thisfact, they are also platforms to realize interesting emer-gent properties, such as artificial electromagnetism [6, 7]monopoles and Dirac strings [14]. Moreover, magneticfrustrated systems are among the best candidates to hostthe elusive spin liquid phase [15].It has been recently realized that even simple systems,with a much weaker degree of frustration, namely witha number of frustrated loops that does not scale withthe system size, can host surprises. This is the case forsystems in which the competition arises from the inter-play of a short-range antiferromagnetic interaction andthe global impossibility to reconcile a staggered arrange-ment with the Frustrated Boundary Conditions (FBC),i.e. periodic boundary conditions applied on a ring withan odd number of sites. Classically, frustration producesa massive degeneracy in the lowest energy state, becauseeach such state develops a domain wall defect, which canbe located at any site of the chain. Quantum interac-tions lift this degeneracy to a band of states, which canlargely be characterized as states with a single travelingexcitation. While these aspects have been understoodqualitatively a long time ago, only lately their quantita-tive appraisal revealed their deep consequences.First, it has been found that perfect FBC constitute a first-order quantum phase transition with respect to dif-ferent boundary conditions [16], that the spin-correlationfunctions at large distances develop unusual algebraiccorrections [17], and that the entanglement entropy inthe ground state indeed carries the signature of a singleexcitation over the ground state [18]. More importantly,it has been shown in [19, 20] that the topological frustra-tion that characterizes such systems can destroy the orderparameter. The traditional order is staggered and quan-tum interactions resolve the conflict between it and theFBC with an interference pattern that effectively cancelsthe magnetization, leaving only a mesoscopic ferromag-netic order at finite sizes, that vanishes algebraically withthe chain length. This phenomenology has later been en-riched: in [21] it was found that a different interferencepattern, allowed by an enlarged ground state manifolddegeneracy, can also admit an incommensurate antifer-romagnet, characterized by a magnetization profile thatvaries in space with an incommensurate pattern. Thistype of order has later been shown to be stable againstAFM defects [22]. Moreover, the boundary between themesoscopic ferromagnetic order and the incommensurateAFM one is a first-order quantum phase transition, whichexists only with FBC [21]. All these results have es-tablished that, contrary to standard expectations, theboundary conditions can indeed affect the local, bulk be-havior of a system, or, at least, that this is the case inpresence of frustration, opening a gateway to connectthe physics of simply frustrated chains to that of genericfrustrated systems.It should, however, be remarked that the results dis-cussed above have been found in specific models andone should wonder about their general relevance. Inthis work, we address this issue and answer positivelyto the question of whether topological frustration generi-cally destroys any local order. As is well known, local or-der parameters are central elements in Ginzburg-Landautheory. They are expectation values of local operators,with support over a finite range of lattice sites, which,given the symmetries of the system, should be zeroedand which, when they assume a value other than zero,signal the spontaneous breaking of the symmetry andthe establishment of a macroscopic order. We considergeneral spin-1/2 Hamiltonians with a dominant antifer-romagnetic interaction, subject only to a global SU (2) symmetry constraint, that describes a wide class of sys-tems without external fields. In fact, this constraint isrespected quite generally when the lattice has an oddnumber of sites and its primary role is to ensure an ex-act degeneracy in the ground state manifold (at leasttwo-fold). This allows for the direct evaluation, even ina finite-size system, of the expectation values for all lo-cal operators, i.e. operators with support over a finiterange of lattice sites. One can then follow the behaviorof these observables toward the thermodynamic limit. Todetermine the ground states we will exploit perturbationtheory around the classical point, where the degeneratelowest energy states are simple domain wall states. Here,because of translation invariance, we classify the statesaccording to their lattice momentum and we prove thatin the thermodynamic limit a non-zero expectation valueof a local operator, that breaks the (parity) symmetry ofthe Hamiltonian, can be recovered only if the differencebetween the momenta of two ground state tends to π . Insuch cases, the system can indeed preserve a semblanceof the order it has under generic boundary conditions,but with a modulation over the whole chain, which spon-taneously breaks translational symmetry, similarly to theincommensurate antiferromagnetic order found in [21].Let us start by considering translation invariant spin- / Hamiltonians with a dominant antiferromagneticIsing interaction in one direction, say x , to which we addarbitrary interactions, H = N X j =1 σ xj σ xj +1 + λ N X j =1 H j . (1)Here σ αj , for α = x, y, z , are Pauli spin operators, λ is theratio between the two interactions and we impose FBC,i.e. periodic boundary conditions ( σ αj = σ αj + N ) and odd N . The only restriction we apply to H j is that it shouldcommute with all three parity operators Π α ≡ Q Nj =1 σ αj ( [ H, Π α ] = 0 for α = x, y, z ), so that the whole Hamilto-nian becomes invariant under transformations σ αj → − σ αj ∀ α . All in all, this class of Hamiltonians describes awide range of systems without magnetic fields and de-fects. The terms H j can include, for example, nearestneighbor interactions such as σ αj σ αj +1 in directions otherthan α = x , next-to-nearest-neighbor interactions suchas σ αj σ αj +2 and, more generally, interactions σ αj σ αj + l ofspins separated by l sites. It also extends to arbitraryproducts of the terms above, as in the case of the clusterinteractions σ yj ( σ zj +1 σ zj +2 . . . σ zj + n ) σ yj + n +1 for even inte-gers n , on which we will focus later. Note that H j canbe a sum of different interactions with different weights.The index j in H j indicates a reference site for the inter-action, which is shifted in eq. 1 to ensure translational invariance.On a chain with an odd number of sites N , the threeparity operators Π α do not commute, but rather they re-alize a non-local SU (2) algebra. Since the Hamiltonian(1) commutes with all Π α , its ground state manifold is at-least two-fold degenerate [19, 21], and any ground statebreaks at least one of the parity symmetries. Thus, insuch a setting, it is always possible to break one of theHamiltonian symmetries already in a finite system and tofollow its behavior to N → ∞ , avoiding the complicationsof the usual procedure of applying a symmetry-breakingfield and removing it only after the thermodynamic limit.Moreover, the same structure also allows for the directcomputation of matrix elements between states with dif-ferent parities, whose calculation usually either requiresextremely cumbersome expressions of limited practicaluse or is achieved indirectly from certain expectation val-ues by invoking the cluster decomposition property.In particular, let | g i be an eigenstate of H and, simul-taneously, an eigenstate of Π x with eigenvalue equal toone, i.e. Π x | g i = | g i . Since the parity operators mu-tually anticommute ( { Π α , Π β } = 2 δ α,β ), it follows thatthe state Π z | g i has the same energy but opposite par-ity with respect to Π x , i.e. Π x Π z | g i = − Π z | g i . Stateswith different parities can be constructed through su-perpositions of states above and thus it is possible tocalculate the ground state expectation value of opera-tors O breaking one symmetry of the Hamiltonian bychoosing a suitable ground state. For instance, for aneigenstate | g i of Π x , the magnetization in the x di-rection can be calculated as h g | σ xj | g i . On the otherhand, the magnetization in the z direction can be eval-uated on the state | ˜ g i = √ ( + Π z ) | g i and is equal to h ˜ g | σ zj | ˜ g i = h g | σ zj Π z | g i .At the classical point λ = 0 the topological frustra-tion does not allow that all the spins point oppositelyto their nearest neighbors. Instead, the ground space is N -fold degenerate, spanned by the "kink states", whichhave a single ferromagnetic bond (two spins aligned inthe same direction) and N − antiferromagnetic bonds(spins aligned in opposite directions). We denote by | j i the kink state in which the ferromagnetic bond is betweensites j and j + 1 , with h j | σ xj | j i = 1 , while the kink statethat we obtain flipping all the spins, i.e. Π z | j i , is charac-terized by h j | Π z σ xj Π z | j i = − . Above the ground statemanifold there is an energy gap of order unity that sepa-rates the kink states from the states with more than oneferromagnetic bond, that we ignore in our perturbativetreatment.By turning on a small coupling λ , i.e assuming | λ | ≪ in eq. (1), the N -fold degenerate ground state manifoldsplits into a band of states, which, because of transla-tional invariance, can be written as | s p i ≡ √ N N X j =1 e ıpj | j i , Π z | s p i = 1 √ N N X j =1 e ıpj Π z | j i , (2)where p = 2 πk/N , with k running over integers from to N − , is the lattice momentum, whose quantizationis a result of periodic boundary conditions. It is easy toprove, taking into account the commutative propertiesamong the Hamiltonian and the different parity opera-tors, that | s p i and Π z | s p i are degenerate states, with thesame momentum. Hence, for | λ | ≪ the ground-statemanifold of the Hamiltonian in eq. (1) will be spannedby a certain number of pairs | s p i and Π z | s p i .At this point let us recall that by local operators wemean all operators having support over a finite rangeof lattice sites, not scaling with N . Due to transla-tion invariance, without losing generality we can assumethat the operator has support over the first L sites (forsome fixed integer L ). Moreover, taking into accountthat Pauli spin operators together with the unit opera-tor provide a basis at a single site, we have that a lo-cal operator can always be written as a linear combina-tion of a finite number of monomials in the Pauli oper-ators σ α σ α . . . σ α L L , where α , α , . . . , α L ∈ { , x, y, z } and σ j = 1 . Thus, we can focus only on monomials inPauli operators. Furthermore, each single monomial ei-ther commutes or anticommutes with a given parity oper-ator. We are interested only in monomials that anticom-mute with some of the parity operators, whose non-zeroexpectation value would signal a breaking of a Hamilto-nian symmetry. All such monomials fall into one of thetwo categories, a) and b), in the following theorem, thatwe prove: Theorem.
Let A ≡ σ α σ α . . . σ α L L be a product of Paulioperators, for some integer L . Let us consider two states(not necessarily different) of the form as in eq. (2) , | s p i and | s p i , and let us consider arbitrary superpo-sitions | g j i = ( u j + v j Π z ) | s p j i , for j = 1 , , where | u j | + | v j | = 1 . We have:a) if A is such that α j ∈ { , x } for all sites j ∈{ , , . . . , L } , with α j = x for an odd number ofsites j , then | h g | A | g i | ≤ (cid:18) | cos p − p | + C (cid:19) N . (3) b) if in A there is at least one site j ∈ { , . . . , L } forwhich α j ∈ { y, z } , then | h g | A | g i | ≤ C N . (4)
Here C and C are positive constants independent of N . Note that the first term in (3) is well defined, since, bythe quantization of the momenta, with N being odd andfinite, we cannot have p − p = ± π . The proof of thetheorem is based on the fact that general superpositionsof kink states interfere destructively and the collapse on agiven order imposed by the observable on a finite intervalis not sufficient to render it finite. A more formal proofis provided in the Supplementary Material, but its basicargument is the following.In case a) , the operator A commutes with Π x and hencethe evaluation of h g | A | g i reduces to the evaluation of h s p | A | s p i . Moreover, in this case, the kink states arealso eigenstates of A , thus only matrix elements betweenthe same kink state are different from zero and we have h s p | A | s p i = 1 N N X j =1 e − ı ( p − p ) j h j | A | j i . (5)For j < L , A acts on a Neel state and we have h j | A | j i = c ( − j , for some constant c ∈ {− , } . The first termin eq. (3) is the result of inserting the above expectationin eq. (5) for the whole sum, while the second term is acorrection due to the first L elements in the sum differingfrom the rest.In the case b) we have to consider two different situa-tions. If the number of sites with α j ∈ { y, z } is even, theoperator A commutes with Π x and thus the evaluation of h g | A | g i reduces to the evaluation of h s p | A | s p i , butthe kink states are no more eigenstates of A . On the con-trary, since the operator A flips some spins, the matrix el-ements between the same kink state vanish. Moreover, ifthe kink is outside the support of A , the matrix elements h j | A | l i also vanish because of orthogonality. Thus, wehave h s p | A | s p i = 1 N N X j,l =1 e − ı ( p j − p l ) h j | A | l i , (6)where the terms with L < j, l < N vanish and we are leftwith, at most, ( L + 1) terms of order one, suppressed bythe overall factor /N .On the other hand, if the number of sites with α j ∈{ y, z } is odd, the operator A anticommutes with Π x andhence the evaluation of h g | A | g i reduces to the evalua-tion of h s p | A Π z | s p i . Analogously to the previous casewe recognize that in the sum h s p | A Π z | s p i = 1 N N X j,l =1 e − ı ( p j − p l ) h j | A Π z | l i , (7)there is at most ( L + 1) non-vanishing terms, which areof order one and are suppressed by an overall factor thatscales with the length of the ring.The consequences of the theorem are straightforward.If the system’s ground space is only two-fold degener-ate, i.e. if there exist only a particular momentum p ( N ) (allowing for system size dependence), with the associ-ated ground states | s p ( N ) i and Π z | s p ( N ) i , we have thatthe expectation values of local operators that break aHamiltonian symmetry are O ( N − ) . In particular, theyvanish in the thermodynamic limit. The effect of thisresult extends also outside the region in which | λ | ≪ .In fact, Ginzburg-Landau Theory predicts that a finiteorder parameter must remain different from zero in thewhole macroscopic phase. But the classical point λ = 0 is far from any second-order phase transition and hence,up to the quantum critical point, the expectation valueof any local order parameter vanishes.The situation becomes more complex if the system ad-mits a larger ground state degeneracy. Let us say that thesystem has d -fold degenerate ground space and denotethe ground state momenta by p ( N ) , p ( N ) , . . . p d ( N ) ,whose quantized value depends on the system size, andby p ∗ , p ∗ , . . . , p ∗ d the values they tend to in the thermo-dynamic limit, respectively. Then, unless p ∗ n − p ∗ m = π for some n and m , the theorem implies again that thereis no local parameters. On the other hand, if it is thecase that p ∗ n − p ∗ m = π for some n and m , then we canconstruct a ground state such as | g ( N ) i = 1 √ | s p n ( N ) i + e ıθ | s p m ( N ) i ) , (8)for some phase θ , which exhibits a non-zero order param-eter. Namely, applying the procedure as in the proof ofthe theorem and using the definition of the momentumwe find the site-dependent magnetization h g ( N ) | σ xj | g ( N ) i = cos[( p m ( N ) − p n ( N )) j + θ ′ ] N | cos p n ( N ) − p m ( N )2 | + O ( N − ) (9)where the phase θ ′ is related to θ , but its explicit expres-sion is not needed. Since p n ( N ) − p m ( N ) = π + O ( N − ) ,in the denominator the correction compensates the factor N and produces a nonvanishing value of the magnetiza-tion in the thermodynamic limit. Moreover, in the nu-merator it forces a slowly varying magnetization profile.In fact, while for neighboring sites σ xj is almost perfectlystaggered, over the whole chain the /N correction addsup so that the amplitude of the order parameter variesand even locally vanishes at some points. Thus, the onein eq. (9) is not a standard AFM order and the phase θ ′ ( θ ) selects a breaking of translational symmetry (due to aground state choice that is not an eigenstate of the trans-lation operator). A nice example of this phenomenologywas discussed for the quantum XY chain with two AFMinteractions in [21]. There, the model exhibits a four-fold degenerate ground space, with p ( N ) − p ( N ) = π +( − ( N +1) / π/N so from (9) we get approximately themagnetization h g | σ xj | g i = π ( − j cos (cid:0) πN j + θ ′′ (cid:1) , whichwas termed incommensurate antiferromagnetic order .Finally, we should also remark that it is possible forthe ground state degeneracy to depend on the system size and that a finite order parameter can be reachedonly through a precise sequence of system sizes. Thisis a peculiar phenomenon in the topologically frustratedmodels that has no counterpart in the unfrustrated ones.To provide a specific example, let us consider the onedimensional n -Cluster-Ising models defined by the Hamil-tonian H = N X j =1 σ xj σ xj +1 + λ N X j =1 σ yj ( σ zj +1 σ zj +2 . . . σ zj + n ) σ yj + n +1 , (10)with n an even number (in order to respect the SU (2) symmetry). While the solution of such models, obtainedusing an exact mapping to free fermions, is known for afew years [23–28], under FBC a few subtleties have to betaken into account and are presented in the Supplemen-tary Material.With FBC, we find that the ground state degeneracyof the n -Cluster-Ising models depends on the greatestcommon divisor ( gcd ) between the system size N and thesize n +2 of the cluster in the many-body interactions. Inparticular, for λ ∈ (0 , there are N, n + 2) groundstates, while for λ ∈ ( − , the degeneracy is halved.Thus, for n = 0 , there are ground states for negative λ and for positive one, for all odd N . The situationchanges abruptly if we consider n = 4 . Let us focus on λ > and take into account separately two chain lengthsequences, N = 12 M + 1 and N = 12 M + 3 for integers M .For N = 12 M +1 the ground space is -fold degenerateand the momenta are p ( N ) = π (cid:0) − N (cid:1) and p ( N ) = − p ( N ) . Letting M → ∞ we have p ∗ − p ∗ = π = π and thus there is no finite local order parameters in thethermodynamic limit for these chain lengths. On theother hand, for N = 12 M + 3 the ground space is -fold degenerate, with momenta p j ( N ) = (2 j + 1) π +( − j π N for j = 1 , , . . . , . In this case we have, forinstance, p ∗ − p ∗ = π so the system can exhibit a non-zero magnetization. From (9) we find the magnetization h σ xj i = π ( − j cos (cid:0) πN j + θ (cid:1) , where the phase factor θ depends on the ground state choice.In conclusions, we have studied generic Hamiltoniansendowed with a global SU (2) symmetry due to the parityoperators and examined the expectation values of localoperators that break a Hamiltonian (parity) symmetry.With a dominant antiferromagnetic Ising interaction andin a setting that induces topological frustration we haveshown that they decay algebraically, or faster, with thesystem size and vanish in the thermodynamic limit. Onlyan increased degeneracy with a precise relation betweenthe momenta of the vector states allows for a finite order,but at the price of breaking translational invariance. Weconclude that FBC are special for generic systems: sincea perfect AFM order is not compatible with them, ei-ther the system disorders or spontaneously breaks trans-lational symmetry. While these findings are probably notrobust against a single ferromagnetic defect, we shouldstress once more that in [22] it was shown that the stan-dard AFM order does not reappear in presence of at leastone AFM defect, because, following also [16], FBC are atthe verge of a phase transition and an AFM defect pushesthe system into a phase that is either disordered or in-commensurate. These results are intuitive from one side,but very surprising from the point of view that the on-set of local order is supposed to be independent from theapplied boundary conditions and show once more thatfrustrated systems (even weakly frustrated ones) belongto a different class of systems altogether. ACKNOWLEDGMENTS
We acknowledge support from the European RegionalDevelopment Fund – the Competitiveness and Cohe-sion Operational Programme (KK.01.1.1.06 – RBI TWINSIN) and from the Croatian Science Foundation (HrZZ)Projects No. IP–2016–6–3347 and IP–2019–4–3321.SMG and FF also acknowledge support from the Quan-tiXLie Center of Excellence, a project co–financed by theCroatian Government and European Union through theEuropean Regional Development Fund – the Competi-tiveness and Cohesion (Grant KK.01.1.1.01.0004). ∗ [email protected] † [email protected] ‡ [email protected][1] M. M. Wolf, F. Verstraete, and J. I. Cirac, Entanglementand frustration in ordered systems, Int. Journal of Quan-tum Information , 465 (2003).[2] S. M. Giampaolo, G. Gualdi, A. Monras, and F. Illumi-nati, Characterizing and quantifying frustration in quan-tum many-body systems, Phys. Rev. Lett. , 260602(2011).[3] U. Marzolino, S. M. Giampaolo, and F. 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Franchini, An introduction to integrable techniques for one-dimensional quantum systems ,Vol. 940 (Springer International Publishing, Cham,2017). [30] E. Lieb, T. Schultz, and D. Mattis, Two sol-uble models of an antiferromagnetic chain,Annals of Physics , 407 (1961). SUPPLEMENTARY MATERIALPROOF OF THE THEOREM FROM THE MAIN TEXT − − − + + + + . . .. . . − + + − . . . ( )1 2 . . . L j FIG. 1: Graphical representation of a kink state | j i , where the kink is far on the right. Far from the kink, there isthe standard antiferromagnetic order. Let the blue rectangle represent the portion of the lattice where A has thesupport. Flipping any state in the rectangle will necessarily create a second kink.For any A = σ α σ α . . . σ α L L we have h g | A | g i = u ∗ u h s p | A | s p i + u ∗ v h s p | A Π z | s p i + v ∗ u h s p | Π z A | s p i + v ∗ v h s p | Π z A Π z | s p i . (11)Now, since the kink states | j i are eigenstates of Π x , with the eigenvalue ( − ( N − / , the states | s p i are also eigenstatesof Π x , with the same eigenvalue. From this fact and the property that A either commutes or anticommutes with Π x we have that two out of four terms in (11) necessarily vanish. If A commutes with Π x ( [ A, Π x ] = 0 ) then the secondand the third term in (11) vanish. Using the Cauchy-Schwarz inequality we get | h g | A | g i | ≤ | h s p | A | s p i | . (12)Similarly, if A anticommutes with Π x ( { A, Π x } = 0 ) then the first and the fourth term vanish and we have | h g | A | g i | ≤ | h s p | A Π z | s p i | . (13)Thus we focus our analysis on elements h s p | A | s p i and h s p | A Π z | s p i . In terms of kink states they read h s p | A | s p i = 1 N N X j,l =1 e − ı ( p j − p l ) h j | A | l i , (14) h s p | A Π z | s p i = 1 N N X j,l =1 e − ı ( p j − p l ) h j | A Π z | l i . (15) Case a):
In case a), A commutes with Π x so (12) holds. Moreover, A acts only as a phase factor on the kinkstates, so h j | A | l i = 0 for j = l . Since far from the kink we have simply staggered antiferromagnetic order (see Figure1) we conclude that for all j ≥ L we have h j | A | j i = c ( − j for some constant c ∈ {− , } . (16)Putting this into (14) we get h s p | A | s p i = cN N X j =1 ( − j e − ı ( p − p ) j + ξ N , (17)where ξ N is a correction coming from the terms ≤ j < L in (14) for which (16) does not have to hold. It is equal to ξ N = 1 N L − X j =1 e − ı ( p − p ) j (cid:2) h j | A | j i − c ( − j (cid:3) (18)and, clearly, satisfies | ξ N | ≤ L − N . (19)Performing the sum in (17) we are left with h s p | A | s p i = − ce − ı ( p − p ) / N cos p − p + ξ N . (20)Taking the absolute value we get | h s p | A | s p i | ≤ N | cos p − p | + 2( L − N . (21)Using (12) proves this part of the theorem.
Case b):
The case b) is even simpler. Here A does not act only as a phase on the kink states, but it flips somespins. Flipping any state far from the kink will necessarily create a second kink (see Figure 1), so all the elements h j | A | l i and h j | A Π z | l i vanish for L < j < N or L < l < N . There are thus at most ( L + 1) non-zero elements in thesums (14) and (15). It follows | h s p | A | s p i | ≤ ( L + 1) N , | h s p | A Π z | s p i | ≤ ( L + 1) N . (22)Now, using (12) and (13) proves this part of the theorem.
EXAMPLE: CLUSTER-ISING MODELS
To illustrate our results we consider the exactly solvable n -Cluster-Ising models, that describe a system made ofspin- in which a short-range two-body Ising interaction competes with a cluster term, i.e. an interaction affectingsimultaneously ( n +2) contiguous spins of the system. On a one-dimensional lattice with periodic boundary conditions,and taking the Ising interaction to favor an AFM alignment, the Hamiltonian of these models reads H = N X j =1 σ xj σ xj +1 + λ N X j =1 σ yj (cid:0) σ zj +1 σ zj +2 . . . σ zj + n (cid:1) σ yj + n +1 , (23)where σ αj + N = σ αj , for α = x, y, z . It is known [23, 27] that the models described by Hamiltonian (23) can be solvedthrough an exact mapping to a system of free fermions, employing the same techniques as in the diagonalization of thequantum XY chain [29, 30], which can be considered the special case n = 0 . The Cluster Ising models with an evennumber n belong to the symmetry class considered in this work so we focus on them. Moreover, to study topologicalfrustration, as in the main text, we take the system size to be an odd number N = 2 M + 1 , and we focus on theparameter region λ ∈ ( − , , where the Ising coupling is larger than (and dominating over) the cluster one. Diagonalization of the n -Cluster-Ising models We are now diagonalizing Hamiltonian (23), when n is an even number. Let us note that the procedure works forodd n as well, the difference being in the expression for the energy of the π -mode in (32) later. The Hamiltoniancommutes with Π z and we split the diagonalization in two sectors of Π z , H = 1 + Π z H + z − Π z H − − Π z . (24)In each sector the Hamiltonian is quadratic in terms of Jordan-Wigner fermions c j = (cid:16) j − O l =1 σ zl (cid:17) σ xj + ıσ yj , c † j = (cid:16) j − O l =1 σ zl (cid:17) σ xj − ıσ yj . (25)It reads H ± = − N X j =1 ( c j c j +1 + c j c † j +1 + h.c. ) + λ N X j =1 ( c j c j + n +1 − c j c † j + n +1 + h.c. ) , (26)where c j + N = ∓ c j in the sector Π z = ± .The Hamiltonian in each sector is quadratic so it can be brought to a form of free fermions. To achieve this, first H ± are written in terms of the Fourier transformed Jordan-Wigner fermions, b q = 1 √ N N X j =1 c j e − ıqj , b † q = 1 √ N N X j =1 c † j e ıqj , (27)for q ∈ Γ ± , where the two sets of quasi-momenta are given by Γ − = { πk/N } and Γ + = { π ( k + ) /N } with k running over all integers between and N − . The Bogoliubov rotation a q = cos θ q b q + ı sin θ q b †− q , q = 0 , πa q = b q , q = 0 , π (28)with the Bogoliubov angle θ q = arctan | λ e ı ( n +2) q | − λ cos (cid:2) ( n + 1) q (cid:3) − cos q − λ sin (cid:2) ( n + 1) q (cid:3) + sin q (29)then brings H ± to a free fermionic form. The Bogoliubov angle also satisfies e ı θ q = e ıq λ e − ı ( n +2) q | λ e − ı ( n +2) q | . (30)After these sets of transformations, the original Hamiltonian is mapped into H ± = X q ∈ Γ ± ε q (cid:18) a † q a q − (cid:19) , (31)where the quasi-particle energies are given by ε q = 2 | λ e ı ( n +2) q | = 2 q λ + 2 λ cos (cid:2) ( n + 2) q (cid:3) ∀ q = 0 , π,ε = 2(1 + λ ) q = 0 ∈ Γ − , (32) ε π = − λ ) q = π ∈ Γ + . Before proceeding, let us note one technical subtlety in the diagonalization of the model. The Bogoliubov angle θ q ,defined by (29) can become undefined for some modes q = 0 , π also point-wise, by fine–tuning of the parameters n , N , and λ . This problem can be circumvented by using (30) to define the Bogoliubov angle and such points can beneglected. Eigenstates construction for the n -Cluster-Ising models The eigenstates of H are formed by applying Bogoliubov fermions creation operators on the vacuum states | ± i ,which satisfy a q | ± i = 0 for q ∈ Γ ± and taking care of the parity requirements in (24). The vacuum states are givenby | ± i = Y Let m and N be positive integers, such that m < N and N is odd. Let us denote their greatest commondivisor by g = gcd( N, m ) . Consider the function f ( j ) = cos (cid:0) πmN j (cid:1) defined for j ∈ { , , . . . N − } .I The function f has g maxima on the set { , , . . . N − } , where the function reaches value .II The function f has g minima on the set { , , . . . N − } , where the function reaches value − cos( πg/N ) . For the proof we will use the concept of a multiset , i.e. a set in which elements can repeat. Two multisets areequal if they contain the same elements, with the same multiplicities. We define the multiplication of the multiset ofnumbers by a constant: If A = { α : α ∈ A } is a multiset of (complex) numbers and c a (complex) number we definethe multiplication in the obvious way, by multiplying each element of the multiset by c , cA = { cα : α ∈ A } . (38)1We also introduce the distance of a number from a set, or a multiset, of numbers. Let β be a (complex) number and A a set, or a multiset. Then the distance of β from A is d ( β ; A ) = min {| α − β | : α ∈ A } . (39)More generally inf should be used instead of min , of course, but for our purposes it is going to be the same.Now we introduce a definition about modular arithmetic and multisets. Suppose we have two multisets of integers, A and B , and let m also be an integer. We say that A = B (mod m ) if { α mod m : α ∈ A } = { β mod m : β ∈ B } , (40)i.e. if looking at equalities modulo m the elements and multiplicities are the same.With these notions introduced we can prove the theorem. Proof. I If we expand the domain j ∈ { , , . . . N − } of function f to real values j ∈ R then it is easy to seethat the function is maximized for j ∈ Nm Z , with value f ( j ) = 1 . Within our restricted domain of integers, theelements j that minimize the function f ( j ) are simply those that satisfy both j ∈ { , , . . . N − } and j ∈ Nm Z ,i.e. those j ∈ { , , . . . N − } satisfying d (cid:16) j ; Nm Z (cid:17) = 0 . (41)Since ≤ j ≤ N − the condition (41) is equivalent to d (cid:16) j ; Nm { , , . . . , m − } (cid:17) = 0 . (42)Clearly, there are as many minimizing values j as there are integers in the set Nm { , , . . . , m − } , (43)and this number is, further, equal to the number of zeroes in the multiset A ≡ n N l mod m : l = { , , , . . . m − } o . (44)We proceed by exploring the properties of the multiset A . Bringing N out of the multiset we get A = N { , , . . . , m − } (mod m ) . (45)Introducing the greatest common divisor of N and m , denoted by g = gcd( N, m ) , and defining B = g { , , , . . . m − } (46)we can write A = Ng B (mod m ) . (47)The first step is to show that B consists of repeating blocks, if we look at equalities (mod m ) . Multiplying with g in (46) we get trivially B = { , g, g . . . , ( m − g } (mod m ) . (48)But notice g ( m − (mod m ) = m − g = (cid:16) mg − (cid:17) g. (49)This means that the multiset B consists mod m of repeating blocks , g, g, . . . , (cid:16) mg − (cid:17) g. (50)2We know that the number of elements in the multiset is m , while we see that the number of elements in theblock is m/g . We conclude that the total number of blocks that form the multiset B must be g .The next step is to examine each block as a multiset and show that it is unaffected by multiplication by N/g ,i.e. that Ng { , g, g, . . . (cid:16) mg − (cid:17) g } = { , g, g, . . . (cid:16) mg − (cid:17) g } (mod m ) . (51)For this purpose, it is sufficient to show that all elements on the left are different. It is simple to see that this isthe case by assuming the contrary and reducing to contradiction. We assume, thus, that there are two elementswhich are equal, Ng ( l g ) = Ng ( l g ) (mod m ) , (52)for some l , l ∈ { , , . . . , m/g − } such that l < l . The assumed equality implies N ( l − l ) = 0 (mod m ) , (53)so that N ( l − l ) is divisible by m , and ( l − l ) N/g is divisible by m/g . But since l − l ∈ { , , . . . m/g − } we have that l − l is not divisible by m/g . It follows that N/g must have common divisors with m/g , whichis in contradiction with the property of g being the greatest common divisor of N and m . Thus, we have shownthat each block is unaffected by multiplication by N/g .The last step is to conclude from (47) that the set A consists of g repeating blocks (50). In particular, A contains g zeroes, which proves part I of the theorem.II If we expand the domain j ∈ { , , . . . N − } of function f to real values j ∈ R then it is easy to see that f ( j ) isminimized for j = N m (2 l + 1) , l ∈ Z , with value f ( j ) = − . However, since for odd N these values of j are neverintegers, they do not coincide with our restricted domain j ∈ { , , . . . , N − } , and we have to find how closeto these values we can get. The minimum of f is achieved by those values j ∈ { , , . . . , N − } that minimizethe distance d (cid:16) j ; N m { l + 1 : l ∈ Z } (cid:17) = d (cid:16) j ; N m { l + 1 : l ∈ { , , . . . , m − } (cid:17) , (54)where the equality holds since ≤ j ≤ N − . To count all j that minimize the distance we take the followingapproach. Let us denote the minimal distance by d min . We first count how many values of l ∈ { , , . . . , m − } have d (cid:16) N m (2 l + 1); { , , . . . , N − } (cid:17) = d min , (55)and then for each such minimizing l we count all j ∈ { , , . . . , N − } with (cid:12)(cid:12)(cid:12) N m (2 l + 1) − j (cid:12)(cid:12)(cid:12) = d min , (56)To each such l there can be associated one or two values of j , depending on whether d min < / or d min = 1 / respectively. It is easy to see that, since m < N , the same value of j cannot be associated to different values of l .We will now use a similar procedure as in part I and explore the multiset C ≡ n N (2 l + 1) mod (2 m ) : l = { , , , . . . m − } o , (57)which determines the distances of interest. Bringing N out we have C = N { , , . . . m − } (mod 2 m ) . (58)Now we introduce the greatest common divisor g = gcd( N, m ) = gcd( N, m ) , where the last equality holds since N is odd, and define the multiset D = g { , , . . . , m − } . (59)3Then we can write C = Ng D (mod 2 m ) . (60)The first step is to show that D consists of repeating blocks, if we look at equalities (mod 2 m ) . Multiplyingwith g in (59) we get trivially B = { g, g . . . , (2 m − g } (mod 2 m ) . (61)But notice g (2 m − (mod 2 m ) = 2 m − g = (cid:16) mg − (cid:17) g. (62)This means that the multiset D consists (mod 2 m ) of repeating blocks g, g, . . . , (cid:16) mg − (cid:17) g. (63)We know that the number of elements in the multiset is m , while we see that the number of elements in theblock is m/g . We conclude that the total number of blocks that forms the multiset D must be g .The next step is to examine each block as a multiset and show that it is unaffected by multiplication by N/g ,i.e. that Ng { g, g, . . . (cid:16) mg − (cid:17) g } = { g, g, . . . (cid:16) mg − (cid:17) g } (mod 2 m ) . (64)Since N/g is odd, for this it is sufficient to show that all elements on the left are different. It is simple to seethat this is the case by assuming the contrary and reducing to contradiction. We assume, thus, that there aretwo elements which are equal, Ng (2 l + 1) g = Ng (2 l + 1) g (mod 2 m ) , (65)for some l , l ∈ { , , . . . , m/g − } such that l < l . The assumed equality implies (52), and by the sameargument as in part I we conclude there is a contradiction. Thus, blocks are unaffected by multiplication by N/g . It follows that C consists of g repeating blocks (63)The last step is to conclude from the block structure of C about the number of minima. We look separatelyat two cases, g = m and g < m . In the first case, g = m , we have m − g = g so C consists only of elements g = m , implying the distance d (cid:16) N m (2 l + 1); { , , . . . , N − } (cid:17) = 12 . (66)for all l ∈ { , , . . . m − } . For each l there is necessarily j ∈ { , . . . , N − } such that (cid:12)(cid:12)(cid:12) N m (2 l + 1) − j (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) N m (2 l + 1) − ( j + 1) (cid:12)(cid:12)(cid:12) = 12 . (67)Counting all corresponding j and j + 1 it follows that f has g minima on the set { , , . . . N − } .In the second case, g < m , the values l with N (2 l + 1) (mod 2 m ) = g and N (2 l + 1) (mod 2 m ) = 2 m − g (68)minimize the distance (55), with d min = g m (69)Since d min < / in this case, for each such l there is only one value j ∈ { , , . . . , N − } with (cid:12)(cid:12)(cid:12) N m (2 l + 1) − j (cid:12)(cid:12)(cid:12) = d min . (70)4Due to block structure of C , there is g values of l satisfying the first and g values satisfying the second equationin (68). It follows that the number of minima of f is again g. Both in the case m = g and m < g the value ofthe minimum is cos (cid:20) πmN (cid:18) N (2 l + 1)2 m ± g m (cid:19)(cid:21) = − cos (cid:16) πgN (cid:17) . (71)In fact, in the proof of part I of the Theorem the property of N being odd was nowhere used, and the samestatement holds for the case of even N . The part II would be different in the case of even N , since then, in general, N (2 l + 1) / (2 m ) could achieve integer values and belong to the domain { , , . . . , N − }}