Inhomogeneous quantum antiferromagnetism on periodic lattices

A. JagannathanLaboratoire de Physique des Solides, CNRS-UMR 8502, Université Paris-Sud, 91405 Orsay, France

R. MoessnerLaboratoire de Physique Théorique de l’Ecole Normale Supérieure, CNRS-UMR 8549, Paris, France

Stefan WesselInstitut für Theoretische Physik III, Universität Stuttgart, 70550 Stuttgart, Germany

�Received 21 July 2006; published 7 November 2006�

We study quantum antiferromagnets on two-dimensional bipartite lattices. We focus on local variations inthe properties of the ordered phase which arise due to the presence of inequivalent sites or bonds in the latticestructure, using linear spin wave theory and quantum Monte Carlo methods. Our primary finding is that siteswith a high coordination tend to have a low ordered moment, at odds with the simple intuition of highcoordination, favoring more robust Néel ordering. The lattices considered are the dice lattice, which is dual tothe kagome, the CaVO lattice, an Archimedean lattice with two inequivalent bonds, and finally the crownlattice, a tiling of squares and rhombuses with a greater variety of local environments. We first present resultsfor the on-site magnetizations and local bond expectation values for the S=1/2 Heisenberg model on theselattices, and then discuss two exactly soluble cases which provide a simple analytical framework for under-standing our lattice studies.

DOI: 10.1103/PhysRevB.74.184410 PACS number�s�: 75.10.Jm, 71.27.�a, 75.25.�z, 75.50.Ee

I. INTRODUCTION

The long-wavelength universal properties of antiferro-magnets on bipartite lattices are well established.1 In thispaper, we ask to what extent local properties—the depen-dence of the ordered moment or bond strength on localcoordination—display systematic behavior. To do this, weconsider quantum Heisenberg antiferromagnets on varioustwo-dimensional lattices. These lattices are bipartite, so thatclassical antiferromagnets form collinear Néel states at zerotemperature; for quantum spins, this ordering persists, albeitwith reduction of the order parameter due to quantum fluc-tuations. A review of results obtained for uniformArchimedean lattices �in which all sites—but not necessarilyall bonds—are equivalent� is given in Ref. 2. Generally, thesize of the order parameter depends on dimensionality—Heisenberg Néel order is absent in d=1—and on coordina-tion z: it is bigger for the square �z=4� than for the honey-comb �z=3� lattice.

Upon increasing coordination z the exchange “mean field”acting on a spin at a given lattice site grows. The larger thismean field, the less effective quantum fluctuations are in re-ducing the staggered alignment of the spin at that site. Forinhomogeneous lattices with sites of different coordination,this suggests that the local staggered magnetization shouldincrease with site coordination z. Analogously, one wouldexpect average bond energies to decrease with increasing z,as fluctuations transverse to the ordered moment become in-creasingly difficult to coordinate between a growing numberof neighbors.

However, recent investigations on a two-dimensional,quasiperiodic system, the octagonal tiling, with six differentlocal coordinations z=3,4 ,5 ,6 ,7 ,8, lead to the result thatthe local staggered magnetization does not follow this expec-

tation; instead, both the local staggered magnetization mi andthe averaged local bond strength of all bonds connected to agiven site, b̄i, tend to decrease with increasing z �cf. Sec. IIfor a formal definition of both mi and b̄i�. This property hasbeen observed in quantum Monte Carlo simulations,3 in lin-ear spin wave theory,4 and in a real space renormalizationgroup approach based on the self-similar structure of thequasiperiod lattice.5 The absence of translation invariance,however, has important consequences for the spectrum andeigenmodes of the spin Hamiltonian,4 and it is the invarianceunder scale transformation that determines the real spaceproperties of the antiferromagnet.3,5 In order to assess therelevance of the presence or absence of translation symmetryon the behavior of the local order parameter distribution, inthis paper we consider periodic systems obtained by variousdecorations of simple two-dimensional lattices.

In particular, we calculate the variations in mi and b̄i forlattices with small unit cells that contain inequivalent bondsand/or sites. We consider the dice lattice, the CaVO lattice,and a lattice that to our knowledge has not been studiedpreviously, the crown lattice, which is built from squares andrhombuses. This structure shows similar local environmentsto those found in the octagonal tiling, but unlike the latterpossesses translational invariance.

Our results on regular lattices are in qualitative agreementwith those of the octagonal tiling. To provide a simple ana-lytical explanation of the z dependence of the local quanti-ties, we appeal to the exactly soluble case of the Heisenbergstar6 and of a decorated hypercubic lattice. In particular, thisresolves the conundrum of a decreasing mi�z� by showingthat, while the average moment mi

av increases with the aver-age coordination zav of the lattice, an order-by-disorder-typemechanism concentrates the reduction of the ordered mo-ment on to highly coordinated sites.

PHYSICAL REVIEW B 74, 184410 �2006�

1098-0121/2006/74�18�/184410�7� ©2006 The American Physical Society184410-1

II. MODEL AND NUMERICAL METHODS

In the following we consider the nearest neighbor Heisen-berg antiferromagnet, described by the Hamiltonian

H = ��i,j�

JS� iS� j �1�

where the sum is taken over all bonds in the lattice. J�0 isassumed to be independent of the local environments, and setequal to 1 in the rest of the paper. We employ a combinationof linear spin wave theory7 and quantum Monte Carlo simu-lations in our analysis. Details of the linear spin wave ap-proach will be presented in the following section for eachspecific lattice.

The quantum Monte Carlo simulations were performedusing the stochastic series expansion method8 for finite lat-tices. In particular, for the dice lattice we considered systemswith Ns=192, 432, and 768 sites, for the CaVO lattice thosewith 64, 144, 265, and 748 sites, and for the crown lattice,systems with 112, 252, 448, and 700 sites. In each case thetemperature was taken low enough to obtain ground-stateexpectation values. For a given site i of the finite size lattice,the local value of the staggered magnetization is given by

m�i� =� 3

Ni�j=1

Ni

�− 1�i+j�SizSj

z� , �2�

where the sum extends over all the Ni lattice sites j which areequivalent to site i with respect to the antiferromagnetic unitcell. Finally, standard finite-size-scaling analysis was per-formed to obtain the local staggered magnetization in thethermodynamic limit.3 For a given bond �i , j� on the lattice,

we define the corresponding bond strength as bi,j = ��S� iS� j��.Similarly, the averaged local bond strength at a site i is

b̄i =1

zi�j=1

zi

��S� iS� j�� �3�

where j extends over all the zi sizes that are connected to sitei.3 In the following we present the results of the quantumMonte Carlo simulations as well as those from linear spinwave theory for the different lattices.

III. LOCAL PROPERTIES OF TWO-DIMENSIONALBIPARTITE LATTICES

A. The dice lattice

Our simplest example is the dice �or T3� lattice, Fig. 1,which has a three-site unit cell for which Eq. �1� is easilydiagonalized in linear spin wave theory. This is a systemwith a net magnetic moment per unit cell and therefore it hasa Goldstone mode with a quadratic dispersion as in a ferro-magnet. For completeness, we present an outline of the spinwave calculation for this simple case. The magnetic and thestructural cell unit cells are the same and have three sites,one on sublattice A with z=6, and two on sublattice B, bothwith z=3. One introduces Holstein-Primakoff operators ofthree types, corresponding to each of these sites: ai ,bi ,ci,�i=1, . . . ,N where N is the number of unit cells� along with

their adjoints, obeying the appropriate bosonic commutationrelations. Introducing Fourier transformed operators ak�

= �1/�N��eik�riai, b�c�k� = �1/�N��e−ik�rib�c�i, the linearizedHamiltonian is Hlin=−JS�S+1�Nb+JSH�2�, with Nb=2N,where

H�2� = � �ak�†,bk�,ck��� 6 A A*

A* 3 0

A 0 3�

ak�

bk�†

ck�† . �4�

Here, A�kx ,ky�=e−i�3kx/2ei3ky/2+2 cos��3kx /2�� �kx,y are ex-pressed in dimensionless units�. This Hamiltonian is diago-nalized by a generalization of the standard case as describedin Ref. 7 in order to find a set of bosonic operators �k�, �k�, �k�

such that

H�2� = � ��k�†,�k�,�k����1�k�� 0 0

0 �2�k�� 0

0 0 �3�k���

�k�

�k�†

�k�† . �5�

An analytic solution is easily found and the results for thevariations of �i along different symmetry directions areshown in Fig. 2. The flat branch �3=3 corresponds to anexcitation that exists on a subset of sites, situated on hexa-gons of B sites �as illustrated in Fig. 1�. In this linearizedmodel, each of the six spins around the central site precessesin antiphase with its neighbors. These states arise from thelocal topology, which gives rise, similarly, to the “Aharanov-Bohm cages” discussed in Ref. 9 for electrons subjected to amagnetic flux on the dice lattice. The ground-state energy persite is E0 /3N�−0.633 within linear spin wave theory, com-pared to the quantum Monte Carlo result of −0.6384�1�.

The local staggered magnetization of A sites, mA=S− �ai

†ai�, and that of the B sites, mB, can be calculated know-ing the transformation matrices Ok� that take the set a�k� ,bk�

† ,ck�†

into the set �� k� ,�k�† ,�k�

†. One finds

mA =1

2−

1

N�

k�

3 − �2�k���

3 + 2�2�k���� 0.356,

FIG. 1. The dice lattice, showing a localized excitation.

JAGANNATHAN, MOESSNER, AND WESSEL PHYSICAL REVIEW B 74, 184410 �2006�

184410-2

mB =1

2−

1

N�

k�

3 − �2�k���

23 + 2�2�k���� 0.428. �6�

The corresponding quantum Monte Carlo results are0.3754�3�, and 0.4381�2�, respectively. Note that the correc-tion to the B site magnetization, �mB=S−mB, is half that ofthe A sublattice, �mA=S−mA. The local magnetization on thedice lattice thus follows the trend seen on the octagonal til-ing, being smaller on sites with higher coordination number.As for the bond strengths, there is only one type of nearest

neighbor bond in this system, b= ��S�AS�B��=0.316 in linearspin wave theory quantum Monte Carlo result: 0.3189�1��.

B. The CaVO lattice

The second lattice considered here has four sites per unitcell, and twice that in the magnetic unit cell. This is the T11or CaVO lattice �after the calcium vanadium oxide com-pound whose spins lie on the topological equivalent of thisstructure� discussed previously �see the discussion in Ref. 2�.We introduce a numbering from 1 through 4 for the four sitesof a structural unit cell. The magnetic unit cell is doubled,with x and y axes rotated by 45° with respect to the originalaxes as shown in Fig. 3. We thus introduce the boson de-struction operators a1−4�b1−4� corresponding to sublattice A

�B�. The magnetic moment per magnetic unit cell is zero.The transformations using bosonic operators, linearization,and Fourier transformation for the sets of ai and bi are car-ried out similarly as for the dice lattice. For a system of Nmagnetic unit cells, one finds Hlin=−JS�S+1�Nb+JSH�2�,with Nb=12N and

H�2� = H1 H2

H2† H1

� , �7�

where H1=zi�ij =3�ij, H2† is the adjoint of H2 with

H2 =�1 zy

* zx* 0

1 1 0 1

1 0 1 1

0 zx zy 1 , �8�

where z�=eik�. The basis vectors are a�T

= �a1 ,a2 ,a3 ,a4 ,b1† ,b2

† ,b3† ,b4

†�. Numerical diagonalization ofthis Hamiltonian leads to �� k� =Ok�a�k�, after discretizing theBrillouin zone. We obtain four branches �i �i=1, . . . ,4�shown in Fig. 4. There are dispersionless directions, kx= ±ky for excitations living on strips, such as the one shownin Fig. 3, of energy E=2�2. The participating sites occurwith larger �smaller� amplitudes as indicated by filled �open�circles.

The ground-state energy per site for this system withinlinear spin wave theory was obtained by a discrete sum overthe Brillouin zone, we find E0 /8N�−0.5376, in agreementwith the value of Ueda et al.10 Ueda et al. also show that inthis system, the Néel state corrected by spin waves has anenergy very close to the energy of a plaquette resonatingvalence bond state. The ground-state energy obtained in lin-ear spin wave theory is seen to compare less well with thequantum Monte Carlo results, −0.553 67�2�, than in the caseof the dice lattice. Similar deviations between linear spinwave theory and quantum Monte Carlo results are also ob-tained for the local quantities: All sites in the CaVO latticehave the same value of the staggered magnetization, which is

FIG. 2. The three dice lattice energy bands along the main di-rections in k space. �Special points in the hexagonal Brillouin zoneare O, the origin, A, the vertex, and B, the midpoint of an edge.�

FIG. 3. The CaVO lattice, showing a state localized on a strip�open circles correspond to bigger amplitudes than filled circles�.

FIG. 4. The CaVO energy bands along the main symmetry di-rections. �Special points in the square Brillouin zone are O, theorigin, X, the midpoint of the edge, and , the vertex�.

INHOMOGENEOUS QUANTUM ANTIFERROMAGNETISM ON… PHYSICAL REVIEW B 74, 184410 �2006�

184410-3

obtained within our linear spin wave theory as mi=S− �ai

†ai�=S− �bi†bi��m. Our k-space sum extrapolates to m

�0.213, close to the value 0.212 published in Ref. 10, butsignificantly larger than the quantum Monte Carlo result0.1805�2�. The increased relevance of the quantum fluctua-tions, which are not captured appropriately within linear spinwave theory, stems from the closeness of the system to aquantum critical point,11 beyond which the system enters aplaquette phase with dominant singlet formations on thesquares in Fig. 3. In the simulations, the closeness of thisphase is reflected in the enhanced bond strength on thesquares, bsq=0.407 31�6� compared to the remaining bondsboct=0.2926�1�. Linear spin wave theory does not fully cap-ture this effect, yielding bsq= �−S2+S�1−2m+2 Re�a2

†b4†���

=0.365, and boct= �−S2+S�1−2m+2 Re�a1†b1

†���=0.344, re-spectively.

C. The crown lattice

This is a square lattice system with seven sites per unitcell, labeled as shown in Fig. 5 and twice this number in theantiferromagnetic unit cell. Each unit has a mirror symmetrythat reduces the number of distinct environments to four.This periodic structure has most, though not all, of the localenvironments present in the quasiperiodic octagonal tiling,and can in fact be obtained by projection from a four dimen-sional cubic lattice using the method for obtaining approxi-mants of the tiling outlined in Ref. 12. Each of the sitesoccurs once on each sublattice, as for the CaVO lattice, andwe introduce accordingly the Bogoliubov operators ai ,ai

† forsites of sublattice A and bi ,bi

† for sites of sublattice B �i=1, . . . ,7�. The linearized Hamiltonian is now Hlin=−JS�S+1�Nb+JSH�2�, where Nb=14N. H�2� takes the form given inEq. �7� in a basis a�T= �a1 ,a2 , . . . ,a7 ,b1

† ,b2† , . . . ,b7

†� where

�H1�ij = zi�ij;z� = �6,3,3,3,4,4,5� �9�

and

H2 =�0 z̄yz̄x z̄yz̄x z̄x z̄yz̄x 1 z̄y

1 0 0 0 0 1 z̄y

1 0 0 0 z̄x 1 0

1 0 0 0 z̄x 0 1

1 0 z̄y 1 0 0 z̄y

1 1 1 0 0 0 1

1 1 0 1 1 zx 0

. �10�

The 1414 system is diagonalized to find the new basis setin terms of operators �i ,�i as in the preceding cases. Thesolution was obtained by discretizing k� in the Brillouin zoneof the crystal, whose unit cell is defined in a coordinate sys-tem oriented at 45° with respect to the original axes as shownin Fig. 5. The seven distinct energy levels thus obtained areplotted in Fig. 6 for some of the main directions in k� space.

There are no localized states; however, there is a disper-sionless direction, ky =0, for excitations located on ribbons�or strips� which are localized in the x but extended in the ydirection. As shown in Fig. 7, the sites with non-zero ampli-tudes lie on the corners of rows of rectangles �formed by thedouble squares�. Each rectangle has a pair of sites of z=3,and a pair of sites of z=4, whose amplitudes are differentcorresponding to open �closed� circles�. This state corre-sponds to the energy band E�3.85 which is, as can be seenin Fig. 6, flat in the OX direction.

The ground-state energy per site for this system is foundwithin spin wave theory to be E0 /Ns=−0.6475, close to thevalue obtained in spin wave theory for the quasiperiodic oc-tagonal tiling, of −0.646.4 The quantum Monte Carlo resultsare E0 /Ns=−0.6602�1� for the crown lattice and E0 /Ns=−0.6581�1� for the quasiperiodic octagonal tiling.13

The linear spin wave theory results for the local staggeredmagnetization are shown in Fig. 8 for different values of thecoordination number z of the sites, and compared to thequantum Monte Carlo results. We find that linear spin wavetheory does rather well in describing this inhomogeneous

FIG. 5. A unit cell of the crown lattice, with sites numberedfrom 1 to 7.

FIG. 6. The seven crown lattice energy bands along some of themain symmetry directions. Special points in the square Brillouinzone are O, the origin, X, the zone boundary along the kx axis, and, the � ,� vertex.

JAGANNATHAN, MOESSNER, AND WESSEL PHYSICAL REVIEW B 74, 184410 �2006�

184410-4

system. Apart from an overall overestimation of the antifer-romagnetic order, both the spin wave theory and quantumMonte Carlo results exhibit similar qualitative features: forz=3 there are two different values of the staggeredmagnetization—the sites labeled 2 and 4 have a larger stag-gered magnetization than the site labeled 3 �more is saidabout this below�. The case of z=4 is particularly interesting,since these sites are found from quantum Monte Carlo tohave the smallest value of mi. As a result one finds a non-monotonic dependence of the local staggered magnetizationson z �compare z=4 to z=5�. Such a nonmonotonic depen-dence of the local staggered magnetization on z was noticedin the quantum Monte Carlo3 and numerical spin wave cal-culations for the octagonal tiling.4 We will return to this phe-nomenon in the following section.

For the crown lattice, the number of bonds per structuralunit cell is 14, but only eight are inequivalent due to themirror symmetry. From their respective strengths, we obtainthe values of the averaged local bond strength shown as afunction of z in Fig. 9, where the results from linear spinwave theory are compared to the quantum Monte Carlo data.We again find a difference between the two types of z=3sites. Furthermore, we observe a monotonic decrease of bi

with the coordination number z. This indicates that correla-tions beyond the nearest-neighbor distance are important forthe nonmonotonic behavior of the local staggered magneti-zation mi seen in Fig. 8.

IV. THEORY OF THE z DEPENDENCE OF THESUBLATTICE MAGNETIZATION

In this section, we present a semiclassical theory forHeisenberg stars, which reconciles our finding of a mono-tonically decreasing mi�z� with the intuition that increasinglyhighly coordinated lattices �in d�2� should approach a satu-rated order parameter.

A Heisenberg star, studied in detail from a different per-spective in Ref. 6, consists of a central site coupled to neighbors, which are mutually disconnected �Fig. 10�. Wedenote the annihilation operators of the Holstein-Primakoffbosons for the central spin, s�, with a, and those for the outer

spins S� i, with Bi, i=1, . . . , . We then have

H = − S2 + S�i=1

a†a + Bi†Bi + aBi + a†Bi

†. �11�

Next, we carry out a unitary transformation U between thebi: bi=UijBj, with U1i=1/� , so that

H = − S2 + S a†a + b1†b1 + � �ab1 + a†b1

†� + �i=2

bi†bi� .

�12�

All modes bi with i�2 correspond to precessions of outerspins around the central spin so that their net transverse com-ponent vanishes—the analogue of the local modes for thedice lattice mentioned above.

The lowering of the magnetization is due to the anoma-lous terms linking b1 with a. This part of the problem can bediagonalized with a standard Bogoliubov transformation ma-trix

O = cosh � sinh �

sinh � cosh �� , �13�

with

FIG. 7. States on strips �see text�. Amplitudes are different onthe sites with open �filled� circles.

FIG. 8. �Color online� Dependence of the local staggered mag-netization mi on the coordination number z for the inequivalent sitesof the crown lattice. Quantum Monte Carlo data �open� are com-pared to linear spin wave theory results �full�.

FIG. 9. �Color online� Dependence of the local averaged bond

strength b̄i on the coordination number z for the inequivalent sitesof the crown lattice. Quantum Monte Carlo data �open circles� arecompared to linear spin wave theory results �full circles�.

INHOMOGENEOUS QUANTUM ANTIFERROMAGNETISM ON… PHYSICAL REVIEW B 74, 184410 �2006�

184410-5

tanh�2�� = 2� /� + 1� . �14�

This yields for the magnetization of the central and outerspins, respectively,

m = S − 1/� − 1� ,

M = S − 1/ � − 1�� . �15�

Note the following two features. First, as grows, the quan-tum reduction to the sublattice magnetization decreases, inkeeping with the idea that a high exchange field implies areduction in the importance of quantum fluctuations. Second,however, it is systematically the highly coordinated centralsite which has the lower magnetization: at large , the centralm is reduced by 1/ , whereas the outer M are each onlyreduced by 1/ 2.

The origin of this phenomenon can be traced to the struc-ture of the Hamiltonian Eq. �11��: Holstein-Primakoffbosons in a bipartite antiferromagnet are created only inpairs, one each on opposite ends of each bond. The higher asite’s coordination, the more anomalous term in the Hamil-tonian act to create such boson pairs. As the number ofbosons quantifies the reduction in mi, high coordination thusimplies low magnetization. This is reminiscent of the order-by-disorder effect encountered in a simple hopping problem,where one finds that in the ground state, particles are morelikely to be found near highly coordinated sites. At the sametime, a high exchange field �corresponding to large diagonalterms in the Hamiltonian� makes such bosons costly and sup-presses their overall number—which is why high-z latticeshave a higher m than low-z ones.

The above order-by-disorder mechanism is operative onlywhen differently coordinated sites are located at oppositeends of a bond. If, for the sake of argument, one were toconsider an inhomogeneous system formed by patching to-gether large pieces of uniform lattices �say, square and hon-eycomb�, the local m in the interior of each piece would ofcourse reflect that of the relevant parent lattice—and in thiscase, m�z� would increase with z. In other words, as placinga boson on a site involves placing a boson on one of a neigh-boring sites as well, m�z� is of course not determined purelylocally. The next-best guess would be that, for sites with thesame z, the magnetization will be lowest for those sites

whose neighbors have the lowest average coordination. Thisis in keeping with the result that 0.357=m2�m3=0.346 forthe crown lattice. In other words, this analysis shows that,the monotonicity in mi�z� is a trend, not a law, in that detailsof the mode spectrum can in fact violate it.

We close this section with three side remarks. First, notethat the Heisenberg star for �1 has a nonvanishing totalmagnetization M, which has enabled us to obtain a finitereduction of m due to quantum fluctuations, even though it isa zero-dimensional system. For =1, when the total magne-tization vanishes, the corrections Eq. �15�� themselves di-verge.

Second, since M is invariant under the action of theHamiltonian, it follows that S−m= �S−M�. By the sametoken, simply dressing the Néel state on the dice lattice leadsto S−m�A�=2�S−m�B��, a result that holds not only for linearspin waves but also in our quantum Monte Carlo simula-tions.

Finally, this effect can be observed in any dimension d.Consider a decorated hypercubic lattice, in which sites areadded on the midpoint of each of link. Such a lattice has d+1 sites in each unit cell: the original site of the hypercubiclattice, and d sites on the links emanating from that site. Thislattice is still bipartite—all the neighbors of the original sitereside on the links, and vice versa. The coordination of theoriginal site zo=2d, whereas that of the sites on the linksequals zl=2. Thus,

S − mo�d� = dS − ml�d��; �16�

the reduction of sublattice magnetization grows with coordi-nation.

V. DISCUSSION AND CONCLUSIONS

We have presented results for local properties of bipartitequantum magnets on decorated lattices. They lead us to con-clude that in periodic systems containing inequivalent sites,there is in general a trend towards decreasing values of mi

and b̄i with increasing local coordination number. This trendwas also previously observed in a quasiperiodic system. Theresults for the crown lattice, in particular, show a qualitativesimilarity with the results that were earlier obtained for theself-similar quasiperiodic octagonal tiling. We have pre-sented a semiclassical analytical treatment for the examplesof the Heisenberg star and a decorated hypercubic lattice, thesemiclassical solution of which explains this qualitative be-havior.

Whereas it is well known that quantum fluctuations canlead to a range of ordering phenomena in otherwise disor-dered magnets,14 it is interesting to ask whether a novel dis-ordering effect can be obtained by increasing z for a sublat-tice, as suggested by a decreasing m�z�. However, as we haveshown above, an increase of z on one sublattice will generi-cally make the ordering more robust everywhere, althoughnot uniformly so for all sublattices. This reconciles in asimple way the observed decrease of mi�z� with increasing zwith the naive expectation that the ordered moment mishould grow with the coordination of the lattice.

FIG. 10. A Heisenberg star: a central spin is coupled to mu-tually uncoupled neighbors.

JAGANNATHAN, MOESSNER, AND WESSEL PHYSICAL REVIEW B 74, 184410 �2006�

184410-6

ACKNOWLEDGMENTS

S.W. would like to acknowledge LPS, Orsay and CEA

Saclay for hospitality during this collaboration. The numeri-cal calculations have been performed at HLRS Stuttgart andNIC Jülich.

1 P. M. Chaikin and T. C. Lubensky, Principles of Condensed Mat-ter Physics �Cambridge University Press, Cambridge, U.K.,1995�.

2 J. Richter, J. Schulenberg, and A. Honecker, in Quantum Magne-tism in Two Dimensions: From Semiclassical Néel Order toMagnetic Disorder, Lecture Notes in Physics Vol. 645,�Springer-Verlag, Berlin, 2004�.

3 S. Wessel, A. Jagannathan, and S. Haas, Phys. Rev. Lett. 90,177205 �2003�.

4 S. Wessel and I. Milat, Phys. Rev. B 71, 104427 �2005�.5 A. Jagannathan, Phys. Rev. Lett. 92, 047202 �2004�.6 J. Richter and A. Voigt, J. Phys. A 27, 1139 �1994�.7 R. M. White, M. Sparks, and I. Ortenburger, Phys. Rev. 139,

A450 �1965�.8 A. W. Sandvik, Phys. Rev. B 59, R14157 �1999�.9 J. Vidal, R. Mosseri, and B. Douçot, Phys. Rev. Lett. 81, 5888

�1998�.10 K. Ueda, H. Kontani, M. Sigrist, and P. A. Lee, Phys. Rev. Lett.

76, 1932 �1996�.11 M. Troyer, H. Kontani, and K. Ueda, Phys. Rev. Lett. 76, 3822

�1996�.12 M. Duneau, R. Mosseri, and Ch. Oguey, J. Phys. A 22, 4549

�1989�.13 S. Wessel, Phys. Rev. Lett. 94, 029701 �2005�.14 For a recent discussion, see S. R. Hassan and R. Moessner, Phys.

Rev. B 73, 094443 �2006�, and references therein.

INHOMOGENEOUS QUANTUM ANTIFERROMAGNETISM ON… PHYSICAL REVIEW B 74, 184410 �2006�

184410-7