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VBS State Induced by Impurity Frustration in Cr8Ni

J. Almeida⋆, M.A. Martin-Delgado⋆ and G. Sierra∗⋆Departamento de Fısica Teorica I, Universidad Complutense. 28040 Madrid, Spain.

∗Instituto de Fısica Teorica, C.S.I.C.- U.A.M., Madrid, Spain.

We provide a physically meaningful picture of the nature of the ground state of the Cr8Ni com-pound in the regime where it is a spin singlet. According to this picture, the anisotropy of the Niatom in the Cr ring induces a dimerization in the molecule that makes the ground state to stabilizein a Valence Bond Solid phase of virtual spins. We characterize rigorously this phase by means ofa particular non-local order parameter denoted the generalized string order parameter. In the com-pletely antiferromagnetic regime, the system becomes frustrated. We have performed a numericalreal-time evolution study of the correlations between the spin of the Ni impurity and the rest ofthe spins in order to show the reaction of the system under this frustration. We provide results thatmanifestly show the agreement with the Valence Bond Solid picture introduced.

PACS numbers: 75.10.Jm, 75.50.-y, 75.10.-b, 74.20.Mn

I. INTRODUCTION

Recently, the physics of ring-shaped molecular magnetswith antiferromagnetic interactions and an odd num-ber of interacting spin centers (e.g. paramagnetic ions)has attracted a great deal of interest since they provideemblematic examples of systems where spin frustrationeffects due to quantum magnetism play a major role.Moreover, they have been sinthesized and studied exper-imentally [1, 2, 3, 4, 5]. Specifically, we shall concen-trate on the anomalous magnetic properties in some het-erometallic odd spin rings [6, 7, 8, 9] namely, chromiumrings. The system comprises eight chromium(III) ionswith spin S = 3

2 each, and one nickel(II) ion with spinS = 1. The magnetic properties of this first odd-memberantiferromagnetic ring has been investigated with elec-tron paramagnetic resonance (EPR) and its spin frus-trated properties has been visualized by means of aMobius strip. In this paper we propose an alternativeand complementary picture of the ground state of thisCr8Ni ring molecule using valence bond states (VBS)[10, 11, 12, 13, 14, 15, 16, 17, 18] of virtual spins whichare used to represent the spins S = 3

2 , 1 of the real con-stituent ions. We will show that the particular bond pat-tern acquired by these VBS states is a consequence andmanifestation of the spin frustration in the odd Cr8Niring molecule.

Molecular nanomagnets are fascinating new magneticmaterials [19, 20, 21]. They appear in a large vari-ety of compounds with many different properties. Weshall focus on antiferromagnetic compounds of bimetal-lic rings. These molecules are ideal candidates to studythe physics of simple but nontrivial spin models like theAF Heisenberg interaction. The key point here is that,these molecules show very interesting finite-size quantummany-body effects which are typically overlooked in otherstudies of the Heisenberg model where the main focus isto achieve the thermodynamic limit (number of spins go-ing to infinity). In those studies, the small finite-size ef-fects are considered spurious effects that vanish for largerand larger systems, which eventually may show some sort

of universality, if that is the case. Quite on the contrary,the nice thing of these small molecules is that we can varyits size and coupling constant strenghts such that thefinite-size effects become some real property that can beaddressed experimentally, theoretically and numerically.Some intersting examples of these small quantum effectsthat we study in this paper are level crossing, change inthe nature of their the ground state (e.g., from spin sin-glet to spin triplet or higher), existence of excited statesvery close to the ground state, etc.

In Sect.II we introduce a Heisenberg Hamiltonian todescribe the interactions between the two types of ionsin the bimetallic compound Cr8Ni, where the Ni ionplays the role of an impurity within an homogeneouschain of Cr ions with the shape of a ring molecule. Infig. 1 we present the energy spectrum of this Hamilto-nian obtained numerically with an appropriate Lanczostechnique. In Sect.III, we first provide the VBS-picturefor the Cr8Ni ring molecule based on a strong couplinglimit in the Ni impurity coupling. This is the origin ofthe spin frustration in the system. In order to supportthis VBS-picture, we provide numerical results for a gen-eralized string order parameter that is able to detect thetype of virtual bond structure. In Sect.IV we study thefrustration effects in this ring-molecule and test the asso-ciated VBS state picture by means of a numerical studyof time-evolved correlation functions of spin-spin oper-ators for different ions in the nano-molecule. Sect.V isdevoted to conclusions.

II. HAMILTONIAN DESCRIPTION OF THE

CR8NI MOLECULE RING

Magnetic molecules are emblematic instances of an en-semble of non-interacting quantum systems embeddedin a condensed matter environment. The synthesis ofmolecular magnets has undergone rapid progress in re-cent years. Each of those quantum systems are identi-cal molecular units that can contain as few as two andup to several dozens of paramagnetic ions (spins). In

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our case, they correspond to Cr8Ni ring molecules. Thismolecule is one of the many relevant molecules contain-ing transition-metal ions whose spins are so strongly-exchange coupled that when the temperature is lowenough, their behaviour is like single-domain particleswith a certain total spin [19, 20, 21].

Macroscopically, these materials appear as crystals orpowders. Nonetheless, their intermolecular magnetic in-teractions are utterly negligible when compared to theirintramolecular interactions. Thus, measurements of theirmagnetic properties reflect mainly ensemble properties ofsingle molecules. There are two major advantages in theresearch on these molecule aggregates. Firstly, the out-standing degree of accuracy by which their magnetic dy-namics can usually be modeled. Secondly, the opportu-nity to chemically engineer molecules possessing desiredphysical properties [19, 20, 21]. In our case, the interestrelies on the study of the Heisenberg model in situationsthat are usually discarded when studying that model ininfinite one-, two-, and three-dimensional systems. Ourstudies will reinforce the idea that such spin arrays yieldqualitatively new physics caused by the finite size of thesystem.

The Cr8Ni compound belongs to a wider family ofmolecular rings. In molecules with a small number ofions, there exist big differences in their physics depend-ing on each particular compound. A first major differenceis related to the number of its ions: odd or even. In thisregard, the implications are mainly twofold: first, havinga difference of one ion in the same family of molecules,can cause the molecule to have a neat magnetic mo-ment or not, therefore changing its magnetic propertiesdrastically. Second, and interestingly enough, a molecu-lar ring with completeley antiferromagnetic interactionsbetween nearest neighbors ions can be a candidate topresent quantum spin frustration properties if the num-ber of atoms is odd, while this effect will not generallybe present for even member rings in a given family com-pound.

As it happens, in the majority of these molecules thelocalized single-particle magnetic moments of the ionscouple antiferromagnetically. Then, their spectrum is de-scribed rather well by the Heisenberg model with very fewparameters because of the high symmetry of the molec-ular configurations. These coupling parameters corre-spond to isotropic nearest neighbor interaction some-times augmented by anisotropy terms.

The Cr8Ni compound is one of the first antiferromag-netic odd member rings which has been artificially sinthe-sized. The results of its magnetic properties are inter-preted within the framework of a spin Hamiltonian ap-proach and they nicely fit the pattern of the energy levelsobtained by inelastic neutron spectroscopy. There existsalso reports on its magnetic and spin frustration effects[7]. In view of these properties, it has been proposed[1] that the behavior of this molecule can be properly ex-plained with a nearest neighbors Heisenberg model whereonly two different microscopic couplings play a role: one

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Stot =0Stot =1Stot =1Stot =1Stot =1Stot =2Stot =2Stot =3

FIG. 1: Energy spectrum of some of the lowest lying statesin each sector with well defined total spin. In the couplingregion shown, the spectrum is highly dependent on the valuesof J and J ′. In particular, in the antiferromagnetic regionJ ′ > 0 K the ground state is a triplet for J ′ < 1.5 K and asinglet elsewhere. Inset: for higher values of J ′ the groundstate is a singlet and the first excited state is a triplet veryclose in energy.

is the coupling that parametrizes the strength of the in-teraction existing between the Ni and the neighbor pairof Cr. The other one is the coupling that takes intoaccount the interaction between the Cr-Cr pairs, whichcan be considered the same for each pair. The easy-axisanisotropy term is reported to be very weak how as toplay any role. Therefore, the Hamiltonian that we shallstudy has the following form:

H = J

7∑

i=1

SCr(i) · SCr(i+ 1)

+ J ′ [SNi · SCr(1) + SNi · SCr(8)] ,

(1)

where, for convenience, the Cr atoms have been num-bered from 1 to 8, being these latter the two neighborsof the Ni atom. Notice that, since the spin of the niquelis equal to SNi = 1 and the spin of the Cr atoms isSCr = 3/2, the total spin of the molecule must be inte-ger.

In fig. 1 we have plotted some lowest lying energy lev-els of this Hamiltonian. In Sect. IV we shall explain howthese numerical results have been obtained with a multi-target Lanczos method. It can be observed that, in someregime of the coupling constants J ′ and J , the energy lev-els are highly braided and, as a consecuence, the groundstate has different value of the total spin depending onthe exact value of these couplings. However, as can beseen from the inset of this figure, for large values of J ′

the ground state is always a spin singlet with a tripletstate very close in energy above it. In this work, we willrestrict our study to the antiferromagnetic (J ′ > 0) re-gion and in particular to the region where the ground

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state is a singlet. From fig. 1 we see that this area cor-responds to J ′ > 1.5 K, while the region 0 < J ′ < 1.5Kis characterized by a ground state with total spin equalto one (for convenience the computations in fig. 1 havebeen done with a fixed value of J = 16 K).

The interest in the domain where the ground state isa singlet comes not only from the fact that it spans themost extension in the antiferromagnetic area, but also be-cause the physics of the real Cr8Ni seems to be in agree-ment with a regime close to J = 16 K and J ′ = 70 K,with a non-magnetic ground state. Therefore, the inter-est of our study relies on the fact that it can provide newinsights into the physics of a not so well-known state ofmatter but with a well defined connection with experi-ments in real compounds.

III. IMPURITY INDUCED VBS PICTURE

A valence bond solid is a particular quantum many-body state that can be understood as follows: given asystem of real particles with total spin S, we can spliteach one of them into 2S virtual particles of spin S =1/2. In order to recover the original spins, we enforcethese virtual particles to couple (i.e., symmetrize) amongthemselves in order to give the original spin S particles.To create now a wavefunction with total spin equal tozero, we make singlets (i.e., antisymmetrize) out of everypair of virtual particles.

We will denote each of these singlet pairs between vir-

tual particles as a bond. There are a lot of differentpossible ways to fix the bonds between all the virtual par-ticles and, in general, the total wave-function may havecontributions from all these configurations. There ex-ist however some physical situations in which only someparticular bond configurations, out of the whole possibleset, take part in the wave-function: some systems have amajor contribution coming only from one particular bondarrangement. These systems are commonly dubbed bondcrystals. It may also happen that there exist not one butsome few bonds configurations whose weights are domi-nant in the total wave-function. In this case the systemis called a resonating valence bond solid (RVBS). Thereexist yet another kind of more disordered states, denoted(m,n)-VBS, which we shall see that describe properlythe ground state of the Cr8Ni in the singlet region. Ageneral (m,n)-VBS state is built by forming bonds onlybetween virtual spins belonging to neighbor real parti-cles, with the numbers m,n satisfying m + n = 2S andS being the spin of the real particle.

These states are usually translationally invariant (withthe (m,n)-VBS notation this means that m = n) in thosesystems where the Hamiltonian possess this symmetry.There also exist dimerized (m,n)-VBS states that havebeen shown to appear in systems where the full transla-tional symmetry of the Hamiltonian has been partiallybroken such that still exist an enlarged unit cell. Typ-ically, this effect can be obtained by introducing an ex-

FIG. 2: Valence Bond Solid picture of the Cr8Ni ringmolecule. The bonds established between the two virtualspins in the Ni and the two virtual spins in the neighbor Cratoms forces a dimerized pattern in the rest of the chain. ThisVBS configuration becomes the dominant one in the stronglycoupled impurity limit J ′ ≫ J .

ternal dimerization coupling constant in the Hamiltonianthat still preserves some periodicity.

Our main result in this section is that there is anothermechanism to provide such dimerized (m,n)-VBS statesin the Cr8Ni ring molecule and whose success preciselyresides in the existence of an impurity within an homo-geneous system. To understand this mechanism in theparticular case of the Cr8Ni, we resort to the strong cou-pling limit where the antiferromagnetic interaction be-tween the Ni and its neighbors is much larger that theinteraction among Cr pairs. As shown in fig. 2, the twovirtual spins comprising the Ni will be likely to formbonds with the virtual particles in the neighbor chromi-ums to satisfy their antiferromagnetic constraints. Therest of the virtual spins left will then tend to form similarbonds with their neighbor partners, giving as a result adimerized non translationally invariant VBS. We wouldlike to stress the fact that the validity of this picture isrooted in the existence of the Ni impurity. In fact, thephysics of a homogeneous system of Cr atoms is closerto a gapless critical phase rather than to such a gappedstate.

General (m,n)-VBS states belong to a class of spinliquids which are known to possess an special hidden or-der that can be identified by a particular non-local or-der parameter called the string order parameter (SOP)[22, 23, 24, 25, 26, 27]. This order parameter has proveditself extremely successful in the task of characterizingdiverse kinds of such states, both in the pure one dimen-sional cases and also in less trivial systems such as ladders[25], [27]. We shall see that this parameter also allows usto characterize the Cr8Ni ground state. The definition ofthe generalized string order parameter is as follows [24]:

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Ostr(θ) = lim|j−i|→∞

〈Szi exp(iθ

j−1∑

k=i

Szk)Sz

j 〉. (2)

FIG. 3: Representation of two ways to measure the SOP in theCr8Ni ring choosing adjacent starting sites. In the diagramat the top we break two bonds at the left end and one at theright. This corresponds to a (2, 1)-VBS. In the diagram atthe bottom we have the same physical configuration but nowwe are breaking one bond at the left and two at the right end,which characterizes a (1, 2)-VBS. Notice that both measureshave been taken with an odd number of sites, otherwise theSOP is zero.

The limit |j−i| going to infinite must be understood ascomprising a region large enough so as to neglect finite-size efects. In our case the system itself is finite, howeverwe will see that the results are still conclusive despitesome corrections that have to be considered due to thisfact.

We hereby summarize the most relevant properties ofthe SOP for our purposes:i/ The SOP is symmetric with respect θ = π.ii/ The imaginary part of the SOP vanishes as weconsider larger systems.iii/ Due to the fact that the definition of the SOPmakes use of antisymmetric operators under spin flip,the number of sites considered in the measure must beodd in order to have an even number of these operators.Otherwise the SOP is zero.iv/ Given a generic (m,n)-VBS state, the number ofzeros of this operator in the interval θ ∈ [0, 2π) coincideswith the number m.v/ Two measures of the SOP begining in adjacent siteswill differ in the order of the numbers m and n. That is,if one measure gives a (m,n)-VBS state, the other willbe a (n,m)-VBS (see fig. 3).

In order to show the existence of a VBS-picture of theCr8Ni molecule as described in fig.2, we have computedthe generalized string order parameter in the groundstate of the molecule for different ratios of the couplingconstants J and J ′. In virtue of property iv/ above wecan characterize the VBS state counting the number ofzeros of this operator. Moreover, we have some freedomat choosing the starting and end sites to measure theSOP (i.e, the numbers i and j in definition (2)) and wewill see that the results are consistent with property v/.

In fig. 4 we have plotted the SOP computed in two dif-ferent blocks of the ring and adjacent starting sites (these

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J’=100, Stot =0

b)

FIG. 4: a) SOP computed between the Cr2 and the Cr6atoms. b) SOP computed between the Cr1 and the Cr7atoms. We have fixed J = 16. Considering the intervalθ ∈ [0, 2π) and looking at the curves with high values of J’we see that the number of zeros in a) is two, correspondingto a (2, 1)-VBS. In b) the situation is more confusing due tostronger finite size effects that shifts the curves towards neg-ative values at θ = 0 and θ = π. With this correction in mindthe graph strongly suggests the existence of a (1, 2)-VBS.

blocks are represented in fig. 3). Remarkably, even in thissmall molecule, the imaginary part of the SOP vanishesin all our computations.In fig. 4a the curves with high values of J ′ have two localminima in the interval θ ∈ [0, 2π) whose value is compat-ible with zero considering that the system is finite andthe block used to measure the SOP comprises only somefew spins. This result is consistent with a (2, 1)-VBS.The shape of the SOP is also typical of these states, withtwo noticeable maxima placed approximately at θ = π/4and θ = 3π/4. Accordingly, the computations shown infig. 4b shall be consistent with a (1, 2)-VBS as explainedabove. We see that again the shape of the SOP for thestrong coupling curves shares the main features of thesestates, that is, one substantial maximum placed preciselyat θ = π/2. There is however a major difference at θ = 0

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and θ = π, where the SOP does not vanish but has a sig-nificant negative value. At these points the exponentialsin the definition of the SOP do not play any role andhence it is a usual spin-spin correlator between the twoCr atoms at the end of the block. Since our ring is finiteand closed it is quite natural to think that the Cr1 andCr7 spins are highly correlated via the spin frustration inthe Ni impurity. Therefore, such a finite value at θ = 0only shows the compromise between the bulk (1, 2)-VBSpicture of the whole system and the local physics of thosespins.

In summary, the string order parameter reveals thatthe ground state of the Cr8Ni in the strong impurity cou-pling limit is consistent with a impurity-mediated mech-anism of spin frustration, the result of which is the VBS-pattern shown in fig. 2. Since we are dealing with a finitesystem the possibilities to drastically change the natureof the ground state are limited, that is, the possibility of aquantum phase transition is excluded and only a crossingwith another energy level can produce this effect. There-fore, the intermediate J ′/J regime can be considered assome deformation of the strong coupling limit. As wedecrease J ′ and the Stot = 1 state crosses the Stot = 0,the VBS picture breaks down and the measures of theSOP are not meaningful in the sense that the propertiesof this operator in such a state are not well defined.

IV. FRUSTRATION AND DYNAMICS IN

CR8NI

As we have already mentioned, the Cr8Ni ring moleculeis frustrated in the sense that the minimum energy of thesystem can not be obtained minimizing separatedly eachof the two body terms of the Hamiltonian (1). Anotherway to see it is by resorting to the classical limit whereeach spin is pictured as a classical vector. Once we setthe value of one of those classical spins, then we can fixone by one the rest of the spins in order to minimize thelocal interactions, but in the end there will be one spinfor which the local interaction with both of its neighborscan not be minimized at the same time.

Typically, systems where frustration exists come alongwith a rich and very often not so well-known physics.Roughly speaking, we can say that frustration in generalincreases the complexity of those systems, both in thephysics they exhibit as well as in the way to approachthem. In particular, there is not a well defined way tomeasure the amount and localization of frustration. Anattempt to quantify these effects in Cr8Ni can be doneattending to the structural changes of the gound stateas we vary the couplings. That is, by inspection of theway in which spins in the ring couple to form the finalstate. This procedure has a connection with experimen-tal techniques where the Lande factors of the ring canbe measured. However this procedure is not suitable tostudy a rotationally invariant singlet ground state wherethe spin is zero. In this section we will study the be-

havior of the Cr8Ni molecule by means of computing thetime evolution of some important spin correlators: thespin autocorrelation of the impurity Ni atom with itselfand the spin correlation between the Ni atoms and eachCr along the ring. These correlators correspond to thevacuum expectation value of the time-evolved spin oper-ators SNi(t) and SCri

(t) projected onto the spin operatorof the Ni impurity at t = 0, SNi(0). This is a way todynamically probe [28, 29, 30, 31, 32] the spin structurein the ground state |ψ0〉 of the ring molecule. In fact,we shall consider the square modulus of those correlatorsand interpret them as time-evolution probabilities. Thatis, we shall consider the following correlators in order toconstruct a figure of merit:

CNi(t) := 〈ψ0|SNi(t) · SNi(0)|ψ0〉, (3)

and

CCri(t) := 〈ψ0|SCri

(t) · SNi(0)|ψ0〉, (4)

Notice that in a rotationally invariant singlet groundstate |ψ0〉 the correlations in the x, y and z axis havethe same value and thus, the expresions above can bewritten as:

CNi(t) = 3〈ψ0|SzNi(t)S

zNi(0)|ψ0〉, (5)

CCri(t) = 3〈ψ0|S

zCri

(t)SzNi(0)|ψ0〉, (6)

Since the proportionality factor does not provide any ad-ditional information we will discard it from now on andwill consider the bare z-axis projection correlators. Thetime dependency of the operators is given by the usualHeisenberg picture:

O(t) = eiHtO(0)e−iHt. (7)

The idea behind this figure of merit to measure thedynamical correlations between spins is similar to thestatic correlator used to measure spin correlations inspace separated sites i and j of the ring 〈ψ0|SCri

(i) ·SNi(j)|ψ0〉. This static correlator measure spatial cor-relations, while our purpose is to measure time-evolvedcorrelations which will probe not only the ground statephysics but also the excited states physics.

We next explain briefly the numerical method used toevaluate these correlators. After that we shall show anddiscuss the results.

A. Numerical Method

The Hamiltonian (1) is SU(2) rotational invariant.That is, it commutes both with the total spin and thez-axis projection of the total spin. In table I we showthe dimensions of the subspaces corresponding to theconserved quantum numbers of these operators. Thesesizes are in the limit to perform exact diagonalization

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Stot,±Sz

tot Dimension Dimension0 1000 235481 2764 225482 3905 197843 4256 158794 3900 116235 3095 77236 2150 46287 1308 24788 692 11709 314 47810 119 16411 36 4512 8 913 1 1

Total: 196608 196608

TABLE I: Dimension of each subspace with well definedquantum numbers out of the total Hilbert space of a Cr8Niring. The second column corresponds to the subspaces withwell defined total spin. The third one are the sectors withwell defined value of the z-axis projection of the total spin.In this last case for each value in the first column we mustconsider the positive and negative cases.

but lie however within the domain reachable for a Lanc-zos method.

To do our computations we will make use of an adaptedversion of the Lanczos algorithm specific to compute real-time dynamics [33, 34, 35]. In this framework it canbe shown that the correlators written above can be ex-pressed as:

Ci(t) =M∑

j=0

〈ψ0|SzCr(i)(t)S

zNi(0)|ψj〉×

×〈ψj |SzNi(0)|ψ0〉e

−i(ǫj−E0)t,

(8)

where M stands for the dimension of the Krylov spaceK(H, q0,M) such that K(H, q0,M) = K(H, q0,M + 1),with q0 := Sz

Ni|ψ0〉. That is, M is the dimension ofthe largest invariant subspace generated by succesiveaplications of the Hamiltonian H upon the seed vectorq0 = Sz

Ni|ψ0〉. The vectors |ψj〉 are the aproximatedeigenvectors computed in this Krylov subspace, ǫj arethe energies of these eigenvectors and E0 stands for theenergy of the ground state.

The number M is typically much lower than the totaldimension of the Hilbert space, but still high to numer-ically compute a complete basis of the Krylov subspace.Therefore, the approximation in this method resides inthe fact that we will substitute the dimension M witha lower number of vectors that still serve as a completebasis for these correlators.

In order to obtain the most accurate results and repre-sentations of the eigenvectors of the Hamiltonian |ψj〉 wehave not used the same Krylov space to compute themall. Instead we have performed a Lanczos iteration tofind the ground state. After that, the Lanczos iteration

is restarted with the previously found eigenvectors pro-jected out of the subspace to find the next excited state.And so on and so forth, until we have computed the de-sired number of eigenvectors. In particular, to computethe correlators described before we have used 400 eigen-vectors of the Hamiltonian. As regards of the tolerancein the eigenvalues, we have set it to 10−14 allowing amaximum dimension of each Krylov space of 350 vectors.Were we in an exact situation, these vectors should benormalized to one and be orthogonal among themselves.Let us call V the matrix whose columns are these eigen-vectors. We have checked that we obtain the followingaccuracy,

‖tV V − 1‖ ∼ 10−4, (9)

which can be considered a low value for such a largenumber of eigenvectors. Moreover, as another check ofthe accuracy of the eigenvectors we have computed thetotal spin of each one of them and we have obtained inte-gral values up to precisions of 10−6 in the vast majorityof them. As for the invariance of the Hamiltonian un-der reflection with respect the Ni atom, we have checkedthat symmetric one and two body correlators evaluatedon every eigenvector are the same up to the fourth orfifth decimal digit.

J’ CNi(0) Error (%)2 0.66666 0.00002

10 0.66666 0.000920 0.66653 0.0230 0.66512 0.240 0.65660 250 0.57469 14

TABLE II: Relative error between the Ni self-correlationat t=0 using (3) with M=400 eigenstates and the value〈ψ0|S

z

NiSz

Ni|ψ0〉 in the ground state, which is equal to 2/3 .

In order to check how complete is our set of eigenvec-tors, we have compared the value at t = 0 computedusing (3) and (4) with the values obtained measuring thecorrelators in the ground state without the projectors inbetween. We have observed that the agreement is exce-lent for low values of J ′ while it goes worse for highervalues of the coupling constant. Table II shows thesevalues and the relative error.

These accuracy checks confirm that our numerical re-sults are good enough so as to interpret them on physicalgrounds with respect to the spin frustracion effects de-scribed in previous sections.

B. Results

In the previous sections we have proposed and checkedwith the proper order parameters a static picture of theground state of the Cr8Ni ring. In the limit when the Niis weakly coupled to the Cr bulk (J ′/J ≪ 1) the system

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C(t

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C(t

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Cr1

Cr2

Cr3

Cr4

c)

FIG. 5: Modulus of the spin correlation of the impurity Niatom with itself and with the rest of the Cr atoms. Thegraphs have been done with a value J = 16 K and: a) J ′ = 2K, b) J ′ = 10 K, c) J ′ = 20 K. For convention the sign of thecorrelators has been chosen to coincide with the sign of thecorrelation at t=0, which is a real number.

is accurately described by an isolated Ni atom and anopen Cr8 chain. On the other hand, when the impurityis strongly coupled (J ′/J ≫ 1) the ring posess a dom-inant contribution in the form of a (2, 1)-VBS groundstate with some local correlations around the Ni atomdue to the finite size of the sample. From the point ofview of frustration, in this limit the impurity acquiresstrong antiferromagnetic compromises with both of itsneighbors that can not satisfy simultaneously. Frustra-tion is known to impose complex constraints that candestabilize, deform and even produce new states of mat-ter. On the other hand, the dynamics of each spin of thesystem is highly influenced by these constraints. In thefollowing paragraphs we will see that the self correlationof the Ni impurity and the rest of correlators with theCr atoms allows us to naturally establish a relation withthe amount of frustration.

In figs. 5 and 6 we have plotted the correlators (3)and (4) for a fixed value of the constant J = 16 K and

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Cr2

Cr3

Cr4

a)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1.61.41.21.00.80.60.40.20

C(t

)

t

Ni

Cr1

Cr2

Cr3

Cr4

b)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

1.61.41.21.00.80.60.40.20

C(t

)

t

Ni

Cr1

Cr2

Cr3

Cr4

c)

FIG. 6: Modulus of the spin correlation of the impurity Niatom with itself and with the rest of the Cr atoms. Thegraphs have been done with a value J = 16 K and: a) J ′ = 30K, b) J ′ = 40 K, c) J ′ = 50 K. For convention the sign of thecorrelators has been chosen to coincide with the sign of thecorrelation at t = 0, which is a real number.

different values of J ′. The Cr8Ni ring is invariant un-der reflection respect the impurity site and therefore wewill only provide the correlators with the Ni itself andthe Cr atoms numbered from 1 to 4 (with the notationof fig. 2), the correlations with the Cr atoms numberedfrom 5 to 8 are the same as their symmetric counterparts.We want to provide these magnitudes with the meaningof a time-evolved probability and hence we will consideronly their modulus. It is worth noticing that the timecorrelators mentioned before, at t = 0 are real numberswhereas for arbitrary values of t they are complex num-bers. For convention, in the graphs where we plot themodulus of these correlators we will provide them withthe same sign of their real value at t = 0 to make explicitthe ferromagnetic or antiferromagnetic nature that theypossess in the static t = 0 ground state.

In fact, we can observe in these graphs that the fin-gerprint of an antiferromagnetic order is present in theinitial t = 0 ground state and shows up in the alternation

8

8

12

16

20

20 30 40 50

Nu

mb

er

of

os

cil

lati

on

s p

er

tim

e u

nit

J’

Ni(t)NiCr1(t)NiCr2(t)NiCr3(t)NiCr4(t)Ni

FIG. 7: The number of oscillations has been obtained count-ing the number of minima in a wide time interval and divid-ing by the total time. The Ni, Cr1 and Cr2 have increas-ing frecuencies with J ′ which corresponds to more frustratedregimes. In particular for the Ni spin the relation of thesevariables is linear up to a high precision. In the case of theCr3 and Cr4 spins the frecuency is hardly affected by J ′.

of the signs of the correlators. Notice also a signal of frus-tration in the fact that this alternation fails in the Cr4and its symmetric counterpart Cr5 (due to the reflectioninvariance of the ring their value is equal with the samesign), where the correlations reveal that both spins areoriented in the same direction respect the spin of the Ni.Remarkably this ferromagnetic defect is a consecuenceonly of the reflection invariance of the Hamiltonian.

From these graphs we can also infer that the averagecorrelation of each spin with the impurity is little sensi-tive to the strength of the coupling constant J ′, althoughthe amplitude of the deviations with respect to this av-erage value increases with it.

The most important observation is that the dynamicsof the correlators exhibit a non trivial sort of periodicity.That is, from the shape of the curves it seems that thereexists many modulating components but a dominant pat-tern of oscillations is apparent. Moreover, the frecuencyof this pattern clearly increases with increasing values ofthe impurity coupling J ′, i.e. as we move towards morefrustrated regimes, but not in the same way for all thespins. We have captured in fig.7 the frecuency of thedominant oscillatory pattern of each correlation. Thegraph highlights two different tendencies of the correla-tions depending on the considered spins: the frecuencyof the Ni self correlation as well as the correlations ofthe Ni spin with the Cr1 and Cr2 spins increase withJ ′. Moreover in the case of the Ni self correlation therelation of these two variables is linear with a surprisingaccuracy. On the other hand the Cr3 and Cr4 are lessaffected by the impurity spin and the frecuency remainsalmost constant in the wide range considered.

In fig. 8 we have plotted the same real time correla-tions with an election of the coupling constants J = 16K and J ′ = 1 K such that the ground state is a triplet.

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0 5 10 15 20

C(t

)

t

Ni

Cr4

Cr2

Cr3

Cr1

FIG. 8: Modulus of the spin correlation of the impurity Niatom with itself and with the rest of the Cr atoms. The valuesof the coupling constants are J = 16 K and J ′ = 1 K whichcorresponds to the region where the ground state is a triplet.For convention the sign of the correlators has been chosen tocoincide with the sign of the correlation at t = 0, which is areal number.

In this case the ground state is not rotationally invariantand the z-axis projection of the correlators is not pro-portional to the scalar correlators (3) and (4) . For ourpurposes however, these magnitudes suffice to realize thedifferent nature of both the singlet and triplet groundstates: first of all is that not one but two dominant pat-terns of oscillation are well distinguishable in the tripletregime. Secondly and also a major difference is that thescale in this regime is some orders of magnitude smallerthan the singlet case.

The existence of this oscillatory behavior seems natu-ral in a frustrated system where there does not exist anatural equilibrium position for each spin or where theresulting equilibrium configuration may result unstable.In a system composed by classical spins these oscilla-tions can be interpreted as the necessary movements ofeach spin to satisfy the frustrated interactions, becomingfaster as we blur the concrete equilibrium positions withthe frustrating interactions.

The results in fig. 7 point towards a regime wherethe frustration introduced by the Ni impurity has stronglocal dynamical effects in the nearest and next nearest Crneighbors while the rest of the spins perceive the impurityscreened by this closer shell of atoms and therefore theirdynamics is little affected by it. These results also showthat the correlators proposed to study the frustrationof the system indeed have the behavior expected for asuitable estimator in order to measure the intuitive ideawe have about the amount of frustration in a certainsystem.

V. CONCLUSIONS

In recent years, considerable efforts have been devotedto synthesizing and investigating magnetic systems ofnano scale dimension that comprise a controllable num-ber of transition metal ions. Highly symmetrical clus-

9

ters of almost planar ring shape are among such topi-cal molecular nanomagnets. In particular, the bimetallicring molecule Cr8Ni is the first antiferromagnetic ringwith an odd number of spins. Thus, it is a remarkablequantum system to test fundamental magnetic proper-ties, and in particular the spin frustration effects.

In this work, we have studied the Cr8Ni frustrated ringin the regime where the ground state is a singlet. Thatis, with a fixed value J = 16 K this region correspondsto J ′ > 1.5 K. In this regard, the experimental charac-terization of a Cr8Ni molecule places the real strengthspresent in the real system close to J = 16 K and J ′ = 70K, well within the singlet region.

As we let the interaction strength of the Ni impurityto be stronger than that between the Cr atoms, the ringstabilizes in a ground state with the quantum proper-ties of a dimerized VBS. The picture that explains thisbehavior in terms of the possible bonds between neigh-bor particles comes clear from fig. 2. Such a VBS stateconstitutes an example of a spin liquid with an intrinsicorder that can be measured by means of some particularnon-local order parameters. In fig.4 we show the compu-tations of this order parameter on the ground state andits behavior supports neatly the VBS picture. In this re-gard, some finite size effects can be observed in the orderparameter that reveal a competition between the physicsin the bulk of the ring and the strong effects, possiblymediated by the system frustration, that ocurrs in thevicinity of the Ni atom.

In the second section of this paper we have studied therole of the frustration in such a VBS state by means ofcomputing the real time evolution of the spin correlatorsbetween the atoms in the ring. In particular, we havefound that the amount of frustration can be related tothe frecuency in the oscillatory behavior of this correla-tors. This relation can be naturally established from theobservation that the oscillations in the system becomefaster as we move to the more frustrated regime J ′ ≫ J .Such an oscillatory behavior is natural in a system whereno natural equilibrium is allowed due to the frustration.However, the spin correlators reveal that the atoms thatare most affected by this frustration are the Ni impurityitself and those Cr atoms that are closer to it, that is Cr1and Cr2, while the effect of the impurity strenght seemsto be less influent in the Cr3 and Cr4 atoms.

We believe that the methods and numerical techniquesused in this work are versatile enough and can be ex-tended to a variety of other nanomolecular magnetic com-pounds.

Acknowledgements: Part of the computations of thiswork were performed with the High Capacity Compu-tational Cluster for Physics of UCM (HC3PHYS UCM),funded in part by UCM and in part with FEDER funds.We acknowledge financial support from DGS grants un-der contract FIS2006-04885 and the ESF Science Pro-gramme INSTANS 2005-2010.

[1] O. Cador, D. Gatteschi, R. Sessoli, A.-L. Barra, G.Timco, R.E.P. Winpenny, “Spin frustration effects inan odd-member antiferromagnetic ring and the magneticMobius strip”; Journal of Magnetism and Magnetic Ma-terials 290-291 55-60, (2005).

[2] M. Allalen, J. Schnack, “Theoretical estimates forproton-NMR spinlattice relaxation rates of heterometal-lic spin rings”, Journal of Magnetism and Magnetic Ma-terials 302, 206-210, (2006).

[3] D.M. Tomecka, V. Bellini, F. Troiani, F. Manghi, G.Kamieniarz, M. Affronte, “Ab initio study on a chainmodel of the Cr8 molecular magnet”, Phys. Rev. B 77,224401 (2008).

[4] S. Yamamoto, T. Hikihara, “Spin dynamics in molecularring nanomagnets: Significant effect of acoustic phononsand magnetic anisotropies”; J. Phys. Soc. Jpn. 75, No.10 (2006).

[5] H. Hori and S. Yamamoto, “Nuclear spin-lattice relax-ation in ferrimagnetic clusters and chains: A contrastbetween zero and one dimensions”, Phys. Rev. B 68,054409 (2003).

[6] J. Schnack, M. Luban, “Rotational modes in molecularmagnets with antiferromagnetic Heisenberg exchange”,Phys. Rev. B 63 014418, (2001).

[7] F.K. Larsen, E.J.L. McInnes, H.E. Mkami, J. Over-gaard, S. Piligkos, G. Rajaraman, E. Rentschler, A.A.Smith, G.M. Smith, V. Boote, M. Jennings, G.A. Timco,R.E.P. Winpenny, “Synthesis and Characterization of

Heterometallic Cr7M Wheels”, Angew. Chem. Int. Ed.42 (2003), p. 101.

[8] S. Carretta, J. van Slageren, T. Guidi, E. Liviotti, C.Mondelli, D. Rovai, A. Cornia, A.L. Dearden, F. Car-sughi, M. Affronte, C.D. Frost, R.E.P. Winpenny, D.Gatteschi, G. Amoretti and R. Caciuffo, “Microscopicspin Hamiltonian of a Cr8 antiferromagnetic ring frominelastic neutron scattering”, Phys. Rev. B 67 (2003), p.094405.

[9] M. Affronte, T. Guidi, R. Caciuffo, S. Carretta, G.Amoretti, J. Hinderer, I. Sheikin, A.G.M. Jansen, A.A.Smith, R.E.P. Winpenny, J. van Slageren and D. Gat-teschi, “Mixing of magnetic states in a Cr8 molecularring”, Phys. Rev. B 68 (2003), p. 104403.

[10] I. Affleck, T. Kennedy, E.H. Lieb and H. Tasaki, “Rig-orous results on valence-bond ground states in antiferro-magnets”, Phys. Rev. Lett. 59, 799 (1987);

[11] I. Affleck, T. Kennedy, E.H. Lieb and H. Tasaki, “Valencebond ground states in isotropic quantum antiferromag-nets”, Commun. Math. Phys. 115, 477 (1988).

[12] M. Fannes, B. Nachtergaele, R.F. Werner, “Exact Anti-ferromagnetic Ground States for Quantum Chains”, Eu-rophys. Lett., 10 , 633–637 (1989).

[13] M. Fannes, B. Nachtergaele, R.F. Werner, “Valence BondStates on Quantum Spin Chains as Ground States withSpectral Gap”, J. Phys. A: Math. Gen., 24 , L185-L190(1991).

[14] M. Fannes, B. Nachtergaele, R.F. Werner, “Finitely Cor-

10

related States on Quantum Spin Chains”; Commun.Math. Phys. 144, 443490 (1992)

[15] A.N. Kirillov, V.E. Korepin, “The resonating valencebond in quasicrystals”; Leningrad Math. J.1, 343377(1990).

[16] M.A. Martin-Delgado, G. Sierra, “A Recurrent Varia-tional Approach” in Density-Matrix Renormalization, aNew Numerical Method in Physics. Lecture Notes inPhysics, Vol. 528, edited by I. Peschel, X. Wang, M.Kaulke,K. Hallberg. Springer-Verlag, Berlin, 1999., p.91,and references therein.

[17] J.Gonzalez, M.A.Martin-Delgado, G.Sierra, M.A.H.Vozmediano, “Quantum Electron Liquids and High-TcSuperconductivity”, Lecture Notes in Physics, Mono-graphs vol.38, Springer-Verlag, Berlin (1995), and ref-erences therein.

[18] M. Raczkowski, D. Poilblanc, ”Fermionic Valence BondSupersolid on a Frustrated Lattice”; arXiv:0810.0738.

[19] D. Gatteschi, R. Sessoli, and J. Villain, “Molecular Nano-magnets” (Oxford University Press, Oxford, 2006).

[20] J. Schnack, “Quantum Theory of Molecular Magnetism”,in Lect. Notes Phys. 645, 155-194 (2004), Springer-Verlag.

[21] W. Linert, M. Verdaguer (Ed.) ”Molecular magnets: re-cent highlights”, Springer-Verlag, Vienna, 2003.

[22] M. den Nijs, K. Rommelse, “Preroughening transitionsin crystal surfaces and valence-bond phases in quantumspin chains”, Phys. Rev. B 40, 4709 (1989).

[23] T. Kennedy, H. Tasaki, “Hidden Z2×Z2 symmetry break-ing in Haldane-gap antiferromagnets”, Phys. Rev. B 45,304 (1992).

[24] M. Oshikawa, ”Hidden Z2 × Z2 symmetry in quantumspin chains with arbitrary integer spin”; J. Phys.: Con-dens. Matter 4 7469-7488 (1992).

[25] J. Almeida, M. A. Martin-Delgado, and G. Sierra, “Crit-ical lines and massive phases in quantum spin ladderswith dimerization”, Phys. Rev. B 77, 094415 (2008).

[26] H.-H. Tu, G.-M. Zhang, T. Xiang, “Maximized string or-der parameters in the valence bond solid states of quan-tum integer spin chains”; arXiv:0807.3143.

[27] J. Almeida, M. A. Martin-Delgado, G. Sierra, “TwistedOrder Parameter applied to Dimerized Ladders”;arXiv:0802.0576, JPA: Math. and Theor. to be published.

[28] E. R. Gagliano and C. A. Balseiro, “Dynamical Proper-ties of Quantum Many-Body Systems at Zero Tempera-ture”, Phys. Rev. Lett. 59, 2999 (1987).

[29] E. R. Gagliano and C. A. Balseiro, “Dynamic correla-tion functions in quantum many-body systems at zerotemperature”, Physical Review B 38 11766 (1988).

[30] S. Haas, J. Riera, and E. Dagotto “Dynamical propertiesof antiferromagnetic Heisenberg spin chains”, PhysicalReview B 48 3281 (1993).

[31] K. A. Hallberg, “Density-matrix algorithm for the cal-culation of dynamical properties of low-dimensional sys-tems”; Physical Review B 52 R9827 (1995) .

[32] K. A. Hallberg and C. A. Balseiro, “Static and dynamicalproperties of a magnetic impurity in a strongly correlatedelectronic system”; Physical Review B 52 374 (1995).

[33] J. Jaklic, P. Prelovsek, “Lanczos method for the calcu-lation of finite-temperature quantities in correlated sys-tems”, Phys. Rev. B 49, R5065 (1994).

[34] J. Jaklic and P. Prelovsek, “Anomalous Spin Dynamicsin Doped Quantum Antiferromagnets”, Phys. Rev. Lett.74, 3411 (1995).

[35] J. Jaklic, P. Prelovsek, “Finite-temperature properties ofdoped antiferromagnets”, Advances in Physics 49, 1-92(2000).