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a

rXiv:cond-mat/980

7407v1

[cond-mat.str-el]30Jul1998

published in

PHYSICAL REVIEW LETTERS VOLUME 80 NUMBER 13 30 MARCH 1998

Pyrochlore Antiferromagnet: A Three-Dimensional Quantum Spin Liquid

Benjamin Canals and Claudine Lacroix

Laboratoire de Magnetisme Louis Neel, CNRS, B.P. 166, 38042 Grenoble Cedex 9, France

(Received 12 December 1997)

The quantum pyrochlore antiferromagnet is studied by perturbative expansions and exact diagonal-

ization of small clusters. We find that the ground state is a spin-liquid state: The spin-spin correlationfunctions decay exponentially with distance and the correlation length never exceeds the interatomicdistance. The calculated magnetic neutron diffraction cross section is in very good agreement withexperiments performed on Y(Sc)Mn2. The low energy excitations are singlet-singlet ones, with afinite spin gap.

PACS number(s): 75.10.Jm, 75.50.Ee, 75.40.-s

Since Anderson proposed the resonant valence bond(RVB) wave function for the triangular lattice1, therehas been a lot of attention focused on frustrated lat-tices, because they might have a spin-liquid-like groundstate in two- or three-dimensional lattices. The frus-trated systems can be classified into two different cate-gories: structurally disordered systems and periodic lat-tices. Among the last one the ground state of the S= 12quantum Heisenberg antiferromagnet on the kagome andpyrochlore lattices is expected to be a quantum spin liq-uid (QSL). The common point of these two lattices isthe high degree of frustration as they both belong to theclass of the fully frustrated lattices. The classical meanfield description indicates a pathological spectrum withan infinite number of zero energy modes which preventsany magnetic phase transition and produces an exten-sive zero temperature entropy24. The classical critical

properties are nonuniversal for both systems5.The pyrochlore lattice consists of a 3D arrangement

of corner sharing tetrahedra (Fig. 1). All compoundswhich crystallize in the pyrochlore structure exhibit un-usual magnetic properties: Two of them, FeF3 andNH4Fe

2+Fe3+F66, are known to have a noncollinear long

range ordered magnetic structure at low temperature; theother compounds do not undergo any phase transition,but many of them behave as spin glasses, although thereis no structural disorder at all. It is remarkable that frus-tration in a periodic lattice may give rise to ageing andirreversibility so that a conpound such as Y2Mo2O7 hasbeen considered as a topological spin glass3,7.

The problem of ordering in the pyrochlore latticewas initiated by Anderson8 who predicted that onlylong range interactions are able to stabilize a N eel-likeground state. More recently, mean field studies9 haveconfirmed thesepredictions; from classical Monte Carlocalculations5,10,11 it was concluded that this system doesnot order down to zero temperature, but any constraintwill induce magnetic ordering12.

FIG. 1. Description of the pyrochlore lattice as an fcc

lattice of tetrahedra. Solid lines connect sites interacting withexchange J, dashed lines with exchange J.

The only attempt to describe the quantum S = 12Heisenberg antiferromagneton the pyrochlore lattice hasbeen done by Harris et al.13 who studied the stability ofa dimer-type order parameter and showed that quantumfluctuations play a crucial role. In this Letter, we showthat the ground state exhibits a QSL behavior: By apply-ing a perturbative approach to the density operator weshow that spin-spin correlations decay exponentially withdistance at all temperatures with a correlation lengththat never exceeds the interatomic distance. Fluctua-

tions select collinear modes, but the amplitude of thesemodes is extremely small. Exact diagonalization of smallclusters shows that the spectrum of the pyrochlore latticelooks like the kagome one, with a singlet-triplet spin gapand no gap for the singlet-singlet excitations. We alsouse our developpement to calculate the neutron magneticcross section. The results are in very good agreementwith previous experimental results on Y(Sc)Mn2

14.

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Model.The Hamiltonian of the quantum Heisenbergmodel on the pyrochlore lattice is

H= Ji,j

Si.Sj , (0.1)

where the summation is taken over the nearest neigh-bors (NN) sites. J is the negative exchange coupling.We describe the pyrochlore lattice as a fcc Bravais latticewith a tetrahedral unit cell. Each cell is exactly diago-nalized and coupling between cells is taken into account

perturbatively: We call J the exchange interaction be-tween NN sites in different tetrahedra (Fig. 1) and wemake an expansion in powers of = J/J. The Hamil-tonian is rewritten as H = H0(J) +H1(J), where H0describes the NN interactions within a tetrahedron andH1 the NN intertetrahedra interactions. By using thistrick we have transformed the initial S = 12 pyrochlorelattice into a fcc lattice with 16 states per site (fourspins 12). All of the thermodynamical quantities can beobtained from the density operator, which we calculatein perturbation with respect to :

= eH =0+ ... +n+ (n), (0.2)

where

n= (1)n

0

10

...

n10

d1...dne(1)H0

H1...e(n1n)H0H1e

nH0 . (0.3)

In order to evaluate the different terms of the devel-opment, we have derived a diagrammatic method whichallows a systematic expansion. This method will reportedelsewhere. All of the following quantities have been eval-uated analytically to the second order in by imple-menting a formal program in Mathematica on a SiliconGraphics. For an indication, the evaluation of the spin-

spin correlation functions from the first to the sixteenthneighbors took 1 12 months of CPU time. The method

was tested on the spin- 12 chain for which the unit cell wastwo NN sites and the development was made to fourthorder in . Using the analytical density operator andfixing = 1, we obtained quite satisfactoryresults, thedifference with those of Bonner and Fisher15 being lessthen 2 % at low temperature. Furthermore, our methodis more efficient when the characteristic length of theground state is short, as expected in a spin liquid. Thus,the deduced characteristic length will be a control pa-rameter of our calculations. Moreover, contrary to usualhigh temperature expansion, our method yields to a non-

divergent expansion when T 0.Spin-spin correlations.Let us consider a reference site

S0 on the lattice. We define S0.Sd = Cd as the corre-lation function between this site and a site at a distanced in the lattice. This function is easily evaluated as

Cd = 1

ZSp[S0.Sd exp(H)], (0.4)

FIG. 2. (a) Absolute value of the spin-spin correlatin func-tions as a function of distance at T = 2|J|. (b) Correlationlength deduced from the analysis of the (a)-type graphs at alltemperatures.

whereZ is the partition function. The development wasmade up to second order in , which allows us to calcu-late Cd up to the 16th neighbors. At any temperaturebetween T = 10|J| and T =|J|/100, we find that thesecorrelations oscillate in sign and decay exponentially withthe distance [Fig. 2(a)]. From these curves, we can ex-tract a correlation length defined as |Cd| exp(d/).

This length is temperature dependent and never exceedsone interatomic distance down to zero temperature [Fig.2(b)]; the broad feature at T J shows the limitationsof the method in this energy scale.

Assuming that could be underevaluated for T Jby our method, we extrapolated (T) from the T Jregime to the low T regime. Whatever the law weused (algebraic, exponential, stretched exponential), wefound a saturation of the correlation length around theinteratomic distance. Thus, our calculations are self-consistently controlled as the value of at T = 0K ismuch smaller than the real spatial extension of our de-veloppement (five interatomic distances).

The total spin can be obtained from the spin-spin cor-

relation functions as

(Ni=1

Si)2 =NS(S+ 1) + 6NC1+ 12NC2+ ... (0.5)

Since we have shown that these correlation functions arerapidly decreasing with distance, we can retain only afew terms in this expansion. Cutting the summation to

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the second neighbors, we verify that the ground state isa singlet:

1

N(

Ni=1

Si)2(T = 0) =0.02 (0.6)

This result shows that our method is satisfactory even ata very low temperature, essentially because the correla-tions inside each unit cell are calculated exactly, and ina QSL, only short range correlations are important.

Static structure factor.For classical spins, the spec-trum of the pyrochlore lattice is very peculiar9: It pos-sesses two branches of zero energy modes. In order tostudy the effect of quantum and thermal fluctuations onthese modes, we evaluated the structure factor,

Sm,n(q) =d

Cdeiq.R

m,n

d (0.7)

where m and n are the indices of a site in a tetrahe-dral unit cell [(m,n) {1, 2, 3, 4}2]. Cd is defined in Eq.(0.4), q is a vector of the first Brillouin one (BZ), andRm,nd is the vector of lengthd that links the sites of typemandn. For eachqthis structure factor is a 44 matrix

whose eigenvalues give the fluctuation modes of the sys-tem, the lowest energy mode corresponding to the largesteigenvalueM(q). To the first order in ,M(q) remainsnondispersive over the entire BZ, and degeneracy is notlifted. To the second order, a maximum appears on theaxis of the BZ (Fig. 3). This maximum correspondsto a collinear phase where the total spin vanishes on eachtetrahedron (see the inset of Fig. 3), and the phase be-tween two neighbouring tetrahedra is equal to . Thisconfirms the results obtained for classical spins by pre-vious authors11,16. We note that the degeneracy is veryweakly lifted (1/106 of the width of the spectrum), butthis is not a numerical artifact since, in our method, theprecision can be as small as we want. Asimilar behaviorwas also observed in the kagome lattice17.

Neutron cross section.The neutron magnetic crosssection can be expressed in terms of the correlation func-tions as

d2

dd =

{1,2,3,4}2m,n

ei.(TmTn)Um,n(Q,), (0.8)

where

Um,n(Q,) =i,j

eq.(RiRj)eq.(TmTn) (0.9)

+

Si,m(0)Sj,n(t)eitdt. (0.10)

Tm and Ri are the translations that define the positionof a site of type m in the unit cell i of the space groupFd3m; Q = +q, where is a vector of the reciprocallattice andqbelongs to the first BZ. From the static cor-relation functions calculated above, we obtain the totalmagnetic cross sectiond/d. We report the results as a

FIG. 3. The two largest eigenvalues of the structure factoralong the axis. The maximum corresponds to the collinearphase shown in the inset.

contour plot in the reciprocal ([00h],[hh0]) plane (Fig. 4).The absence of a signal in the first BZ indicates that theground state is a singlet. The intensity is maximum atabout Q1 = [200] [34

340] orQ

1 = [002] [

34340] (Fig. 4).

These maxima are associated with long range correlationsdescribing a structure where consecutive tetrahedra arein phase. This is different from the result obtained abovefrom the structure factor. In fact, we have also calculatedthe cross section in the ([h00],[0h0]) plane and found an-other maximum at Q2 = [210] which corresponds to aphase between tetrahedra as expected. Comparison ofthe intensities of the peaks at Q1 andQ2 favors the sec-ond structure, but the difference is small. So we concludethat there are two characteristic modes in this system, a-dephased one and an in-phase one, with a larger weightfor the first mode.

It is interesting to compare our results (Fig. 4) withthe measurements made on Y(Sc)Mn2 by Ballouet al.

14:The experimental results correspond exactly to our re-sult (Fig. 4). In fact, the measurements were made for agiven energy, but the shape of the neutron cross sectionin reciprocalspace was found to be nearly independent ofthe energy14, as if(q, ) =f(q)g(). Thus, integratingthe experimental results over the energy does not changethe q dependance and we can make a direct comparisonbetween our calculations and the measured map. Firstwe reproduce all the main features previously obtained,i.e the absence of a signal in the firt BZ associated withthe singlet ground state and the maxima at Q1 and Q1.

Second, we find a -dephased mode in the ([h00],[0h0])plane that was also observed in the same compound. Theexistence of these two modes could explain the first or-der character of the transition observed in the pure YMn2compound19,20 associated with the freezing of the modemixture in the-dephased one. Finally, the half-width ofeach peak provides information on the correlation length:It is found to be of one interatomic distance in experi-

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FIG. 4. Map of the neutron cross section in the plane([00h],[hh0]). The white regions indicate the maximum ofintensity in the q space.

ments as well as in our calculations. All these resultsindicate that a large part of the physics of Y(Sc)Mn

2 is

related to quantum fluctuations in this highly frustratedstructure, although the magnetism of Mn is not localizedbut itinerant.

Low energy part of the spectrum.We performed exactdiagonalization of small clusters, up to 12 spins 12 . Thesesizes are not large enough to give quantitative conclu-sions, but allow a first characterization of the spectrum.We find that the ground state is a singlet with a spin gapbetween this singlet and the lowest triplet ( 0.7|J|).Inserted in this gap, there are several singlet states whosenumber is growing with the cluster size. This indicatesthat the lowest energy excitations are singlet-singlet-like.A similar property was also observed in exact diagonal-

izations performed on the kagome lattice21. It seems thatboth systems belon to the same class of QSL while the1D integer spin chains, for which the lowest energy ex-citations are spin excitations (singlet-triplet), are differ-ent. In our case, the relevant parameter is the topologicalfrustration, while in 1D Heisenberg spin liquids the lowdimensionality plays a crucial role.

In conclusion, we have studied the quantum Heisen-berg spin- 12 Hamiltonian on the pyrochlore lattice. Itappears from the spin-spin correlations and the low en-ergy spectrum that the ground state of this system isa QSL similar to the kagome lattice ground state. Us-ing ou results we calculated the magnetic neutron cross

section, and reproduced almost exactly the map experi-mentally observed in Y(Sc)Mn2. This confirms that thiscompound is a 3D QSL.

The pyrochlore antiferromagnet is certainly the firstexample of a 3D QSL. For this class of QSL the dimen-sionality of the lattice seems to play a minor role, butwe can expect that any perturbation will deeply mod-ify the low energy spectrum. Thus, such a disordered

magnetic ground state should be extremely sensitive tochemical disorder. In fact, in several compounds, suchasAl-doped Y(Sc)Mn2

20, Y2Mo2O723, and Al-doped -

Mn24, disorder, even nondetectable, transforms a QSLinto a quantum spin glass state. All of these compoundshave in common unconventional behaviors (i.e, the sus-ceptibility is spin-glass-like, while the low T specific heatincreases asT2 and strong fluctuations of the local mag-netization are observed in the frozen state). A quanti-tative understanding of disorder effects in QSL requiresmore studies, both experimental and theoretical.

We are grateful to R. Ballou, C. Lhuillier, and F. Milafor helpful discussions and comments.

Current address: Max-Planck-Institut fur Physik kom-plexer Systeme, Nothnitzer Strasse 38, D-01187, Dresden,Germany.Electronic address: canals@mpipks-dresden.mpg.dehttp://www.mpipks-dresden.mpg.de/canals/

Electronic address: lacroix@polycnrs-gre.fr

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