Quantum dynamics and entanglement of spins on asquare latticeN. B. Christensen*†‡, H. M. Rønnow†§, D. F. McMorrow*¶�, A. Harrison**††, T. G. Perring�, M. Enderle††, R. Coldea‡‡§§,L. P. Regnault¶¶, and G. Aeppli¶

*Materials Research Department, Risø National Laboratory, Technical University of Denmark, DK-4000 Roskilde, Denmark; †Laboratory for NeutronScattering, ETH Zurich and Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland; §Laboratory for Quantum Magnetism, Ecole PolytechniqueFederale de Lausanne (EPFL), 1015 Lausanne, Switzerland; ¶London Centre for Nanotechnology and Department of Physics and Astronomy,University College London, London WC1H 0AH, United Kingdom; �ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, UnitedKingdom; **Department of Chemistry, University of Edinburgh, Edinburgh, EH9 3JJ, United Kingdom; ††Institut Laue-Langevin, 38042 Grenoble Cedex 9,France; ‡‡Oxford Physics, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom; §§Department of Physics, University of Bristol, Bristol BS81TL, United Kingdom; and ¶¶CEA, Grenoble, Departement de Recherche Fondamentale sur la Matiere Condensee, 38054 Grenoble Cedex 9, France

Communicated by Robert J. Cava, Princeton University, Princeton, NJ, April 10, 2007 (received for review October 24, 2006)

Bulk magnetism in solids is fundamentally quantum mechanical innature. Yet in many situations, including our everyday encounterswith magnetic materials, quantum effects are masked, and it oftensuffices to think of magnetism in terms of the interaction betweenclassical dipole moments. Whereas this intuition generally holdsfor ferromagnets, even as the size of the magnetic moment isreduced to that of a single electron spin (the quantum limit), itbreaks down spectacularly for antiferromagnets, particularly inlow dimensions. Considerable theoretical and experimentalprogress has been made in understanding quantum effects inone-dimensional quantum antiferromagnets, but a complete ex-perimental description of even simple two-dimensional antiferro-magnets is lacking. Here we describe a comprehensive set ofneutron scattering measurements that reveal a non-spin-wavecontinuum and strong quantum effects, suggesting entanglementof spins at short distances in the simplest of all two-dimensionalquantum antiferromagnets, the square lattice Heisenberg system.

antiferromagnetism � entanglement � multimagnons � spin waves

One of the most fundamental exercises in quantum mechan-ics is to consider a pair of S � 1/2 spins with an interaction

J between them that favors either parallel (ferromagnetic) orantiparallel (antiferromagnetic) alignment. The former results ina spin Stot � 1 ground state, which is a degenerate triplet. Twoof the states in this triplet are the possible classical ground states�11� and �22�, whereas the third is the coherent symmetricsuperposition �12���21�, which has no classical analogue.Even more interesting is antiferromagnetic J, for which theground state is the entirely nonclassical Stot � 0 singlet �0� ��12� � �21� consisting of the antisymmetric coherent super-position of the two classical ground states of the pair. The state�0� is an example of maximal entanglement, i.e., a wavefunctionfor two coupled systems that cannot be written as the product ofeigenfunctions for the two separate systems, which in this caseare of course the two spins considered individually.

The consideration of spin pairs already suggests a very cleardifference between ferromagnets and antiferromagnets, in thesense that in ferromagnets the classical ground states are stableupon the introduction of quantum mechanics, whereas those ofantiferromagnets are not obviously so. Indeed, as recently as themidpoint of the last century, it was felt that ordered antiferro-magnets might not exist. Subsequent experiment and theoryhave proven this wrong for most practical purposes. However,recent developments in areas as disparate as mathematicalphysics and materials science, as well as interest in harnessingentanglement as a resource for quantum computation (1), haveled to a renaissance in investigations of purely quantum me-chanical effects in antiferromagnets. An early milestone in thisfield was the realization that antiferromagnetic chains in generaldo not have ground states with classical order (2), and indeed

have excited state spectra that do not correspond to those of theclassical systems either (3–5). Another was the observation byAnderson (6) that such nonclassical ground states have remark-able similarities to the coherent quantum wavefunctions describ-ing superconductors—something that is intuitively appealingwhen it is recalled that the spin wavefunction of the underlyingCooper pairs of electrons correspond to the ground state of theantiferromagnetic spin pair. He suggested that a highly entan-gled ground state, referred to as ‘‘resonating valence bond’’(RVB) (7), for a square lattice of antiferromagnetically coupledcopper ions might be the precursor of the high temperaturesuperconductivity found in the layered cuprates.

However, studies of the insulating parents of the high-temperature superconductors have revealed essentially conven-tional classical magnetic order in these two-dimensional S � 1/2Heisenberg antiferromagnets (8). In addition, the magneticexcitations (9) and temperature dependence of the magneticcorrelations (10–12) are given to remarkable accuracy by clas-sically based theory (13–15). One interesting fact, though, is thatthe static moment, reflecting the extent to which the true groundstate resembles the Neel state �N�, is only �60% of thatanticipated classically (13, 16). In addition, the ground-stateenergy is lower than that calculated for �N� (16). Consequently,there are substantial quantum corrections to the ground-statewavefunction, and there can be a certain level of entanglementamong spins even on the square lattice (see ref. 17 for theparticular example of the S � 1/2, XYX model), although to datethere has been no direct experimental evidence for this. Indeed,so far, essentially all experimental studies have focused onrenormalization of classical parameters rather than on qualita-tively new quantum phenomena. The only exceptions have beentwo papers reporting anomalous zone boundary spin-wave dis-persion (18, 19) and a very brief conference report (20). Thepurpose of the present paper is to report on a definitive inelasticneutron scattering study showing large qualitative effects ofquantum corrections for a particular realization of the squarelattice S � 1/2 Heisenberg antiferromagnet. Furthermore, wedraw direct attention to the power of this technique as adiagnostic of entanglement in magnetic systems.

Neutron scattering provides images of the wave-vector, Q, andenergy, E, dependent magnetic f luctuations in solids (21). Thetechnique reveals the underlying wavefunctions via Fermi’s

Author contributions: N.B.C., H.M.R., D.F.M., and G.A. designed research; N.B.C., H.M.R.,D.F.M., A.H., T.G.P., M.E., R.C., L.P.R., and G.A. performed research; N.B.C. analyzed data;and N.B.C., H.M.R., D.F.M., and G.A. wrote the paper.

The authors declare no conflict of interest.

Abreviations: CFTD, copper deuteroformate tetradeurate; RVB, resonating valence bond.

‡To whom correspondence and requests for materials should be addressed. E-mail:niels.christensen@psi.ch.

© 2007 by The National Academy of Sciences of the USA

15264–15269 � PNAS � September 25, 2007 � vol. 104 � no. 39 www.pnas.org�cgi�doi�10.1073�pnas.0703293104

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golden rule, according to which the cross-section defining theprobability of scattering into a particular direction d� andenergy range dE is of the form

� d2�

d�dE� � �� f �SQ� i� �2. [1]

The operator SQ appearing in the matrix element between initialand final states �i� and �f� is the Fourier transform ¥jSj exp(iQ�rj) ofspin operators Sj on sites rj. A dramatic and pertinent illustration ofhow neutrons can probe entanglement is shown in Fig. 1 for anisolated square plaquette of antiferromagnetically coupled spins.The Stot � 0 ground state is strongly entangled, being a coherentsuperposition of the two classical ground states and more energeticconfigurations containing spins parallel to their neighbors. Theoperators SQ connect this ground state to three excited Stot � 1states (22). It is possible to see immediately, as illustrated in Fig. 1,

that the neutron scattering cross-section for the quantum plaquetteis radically different around the (�, 0) and (0, �) points, from thatof the classical plaquette, with a single peak at (�, �), the vectordescribing the antiferromagnetic order (see Materials and Methodsfor a definition of the notation used to specify wavevectors). For thesquare lattice antiferromagnet, which is an infinite array of pla-quettes, the ground state is the Neel state with quantum correctionsand has long been suspected (23) to have a finite overlap withwavefunctions of the RVB type shown in Fig. 1b for a singleplaquette. Correspondingly, the excited states are what have be-come of the classical spin waves, also with quantum corrections.

The real two-dimensional Heisenberg antiferromagnet con-sidered here is copper deuteroformate tetradeurate (CFTD),which crystallizes from aqueous solution as a stack of nearlysquare lattices of copper atoms alternating with water layers. Thetransparent, insulating material has a bluish hue, with the copperions carrying spin S � 1/2. CFTD was chosen because theelectrons are strongly localized, implying a negligible role forcharge fluctuations, very unlike what is seen in the blackinsulating parents of the high-temperature superconductors (9).Orbital overlap through intervening formate groups coupleneighboring copper spins by antiferromagnetic exchange inter-actions of a convenient magnitude (J � 6.19 meV, equivalent toa temperature of 72 K) for magnetic spectroscopies (19, 24).

ResultsNeutron time-of-flight spectroscopy with position-sensitive detec-tion allows the magnetic dynamics to be surveyed over largevolumes of Q-E space simultaneously. The easiest method fordisplaying these data for a two-dimensional magnet such as CFTDis to plot the intensities as a function of the two-dimensionalwavevector for fixed neutron energy loss ��. The dominant featuresin such cuts through the dispersion surface (Fig. 2a) are rings,centered on the reciprocal lattice point for the ordered spins. Fig.2b shows three such images for CFTD. The rings arise from welldefined magnetic excitations, known as magnons or spin waves ina classical description. Of particular interest is the behavior near themaximum of the single-magnon dispersion, since anomalies havebeen reported in the energy of the spin waves along the zoneboundary (dashed red lines in Fig. 2) in a number of systems (9, 18,19). The images in Fig. 2 c and d display the intensity distributionsfor energies well below and in the vicinity of the zone boundaryenergy in CFTD. When we compare these results with simulationsfrom linear spin-wave theory in the same energy ranges (Fig. 2 e andf), it is clear that the data at intermediate energies follow theuniform intensity distribution expected (Fig. 2 a and e), whereas alarge modulation in intensity is apparent along the zone boundary,with ‘‘holes’’ around the (�, 0) points, enclosed in dashed bluesquares in Fig. 2d. These holes represent large deviations fromlinear spin-wave theory. They result from quantum mechanicalinterference and evidence entanglement of nearest-neighbor spins.However, before such a dramatic conclusion is justified, furtherwork and detailed analysis must be undertaken to rule out thepossibility that the holes might be an artifact of the dispersion in thepeak position.

Fig. 3 c–f summarizes the analysis of the single-magnonscattering along the major symmetry directions of the Brillouinzone of the square lattice, and it compares the results withspin-wave calculations (13, 15) indicated by red lines. Constantwavevector cuts at (�/2, �/2) and (�, 0) are shown in Fig. 3 a andb. Along the M- direction we find that a good fit of thesingle-magnon energies (Fig. 3 c and d) to the linear spin wave(LSW) theory result is achieved with J � 6.19(2) meV. Onapproaching the X point, the energy is depressed by 7(1)%relative to the point (�/2, �/2), confirming the value of 6(1)%found in our previous lower-resolution study (19). This effect isnot expected classically (13, 15) but agrees well with estimatesobtained by exact diagonalization (19), series expansions around

a

c

b

d

+ + +

e

or

Classical Quantum

Fig. 1. From classical and quantum plaquettes to the square lattice. (a) Classicalground states for a system of four S � 1/2 spins on the vertices of a plaquette. (b)The quantum mechanical ground state for a plaquette corresponds to the sum oftwo terms, each consisting of two singlet pairs along parallel edges of theplaquette. Here the ground state of a single dimer is ��� � �12� � �21� and theground state of the plaquette is as follows:

���� � � ��� � � 2�1221� � 2�2112� � �1122� � �1212�

� �2211� � �2121�(For simplicity, we have neglected normalization factors throughout this text.)(c) The neutron scattering structure factor S(Q), obtained by integrating Eq. 1over all energies, for the classical system in a is simply given by S(Q) � �1 �exp(iQx) � exp(iQy) � exp(i(Qx � Qy))�2 � sin2(Qx/2) sin2(Qy /2). (d) The fullquantum mechanical structure factor can be evaluated by summing over theStot � 1 excited states (22) to obtain S(Q) � 1 � cos2(Qx/2) cos2(Qy /2). (e) Portionof the infinite square lattice ground state, viewed as the Neel state (weight �)with quantum corrections (weights �n), of which one resonating valence bondconfiguration is sketched.

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the Ising limit (25) (blue lines), and Quantum Monte Carlocomputations (26) (blue squares).

Fig. 3e shows the momentum-dependent spin-wave intensity. Atfirst sight, the main feature is the divergence as the magnetic zonecenter (�, �) is approached, which is in agreement with classicalspin-wave calculations (13). By normalizing to the classical predic-tions for the intensity, we bring out very clearly a spectacular resultof our experiments (Fig. 3f), namely that along with the relativelymodest dispersion of the spin waves along the zone boundary, thereis a much stronger intensity anomaly, actually removing 54(15)% ofthe classically predicted mode intensity at (�, 0). Thus, there arevery large quantum corrections to the wavefunctions when we lookat short distances, a result completely at odds with the standardnotion of a momentum-independent renormalization (15). Notethat to lowest order in spin-wave theory, next-nearest-neighborinteractions do not lead to any intensity variation along the zoneboundary. The intensity anomaly has a half-width in momentumspace along the X-M direction of 0.1 Å�1, implying that theunderlying physical phenomenon has a length scale of 10 Å, or

roughly twice the nearest-neighbor Cu-Cu distance. In addition, thewavefunction corrections involve spin correlations with wavevectorsof type (�, 0). Such corrections at this wavevector are exactly whatmight naively be expected for an RVB state where nearest-neighborvalence bonds entangle spins along the edges of the square lat-tice—as already occurs for the quantum plaquette in Fig. 1b—butnot along the diagonals. This intuitive argument is supported bycalculations of mode softening and intensity reduction at the (�,0)-type points—reproduced by the green lines in Fig. 3 c–f—for avariational state mixing RVB and Neel order (27, 28). While thevariational calculation exaggerates the qualitative features that wehave found, most notably yielding a deeper spin-wave energyminimum at (�, 0), a recent Quantum Monte Carlo study (26) giveszone boundary dispersion and intensity suppression in good agree-ment with the values found in our experiment.

The ground-state wavefunction �� of the two-dimensionalS � 1/2 Heisenberg antiferromagnet can be viewed as the sumof the Neel state �N� and correction terms that are responsiblefor the reduction of the ordered moment. In a spin-wavedescription, these would be a series of single and multiple spinflips that in turn can be expressed as the Fourier transforms ofsingle- and multiple-magnon excitations. In a neutron scatteringexperiment, the application of the operator SQ to such correctionterms leads to additional continuum scattering, usually called

Fig. 2. Imaging the single-magnon excitations. (a) Spin-wave dispersionsurface ��q � 2Jeff(1 � q

2)1/2 for the S � 1/2, square lattice Heisenbergantiferromagnet as predicted by linear spin-wave theory (13, 15), whichassumes a classically ordered (Neel-type) ground state. Here q � (cos(Qx) �cos(Qy))/2 and Jeff � 1.18J is an effective exchange interaction, renormalizedby the nonclassical contributions to the ground state (15). The color scaleindicates the expected intensities Iq that by Eq. 1 are proportional to squaredmatrix elements connecting the ground and excited states. In linear spin-wavetheory (13) Iq � ((1 � q)/(1 � q))1/2. (b) Data produced by averaging over fourequivalent Brillouin zones. The raw data were obtained from the 16 m2

detector bank of the MAPS spectrometer at ISIS, Rutherford Appleton Labo-ratory with the two-dimensional Cu layers of CFTD aligned perpendicular toan incident neutron beam with Ei � 36.25 meV. The false color scale representsthe measured neutron scattering cross-section in constant energy slices (asindicated by the shaded plane in a) through the dispersion surface. (c and d)Constant energy slices measured at 8–10 meV and 14–15 meV, plotted withthe horizontal x and y axes labeled in the reciprocal space of CFTD where (1,0, L) and (0, 1, L) correspond to (�, �), as indicated by the diagonal axes in c.The out-of-plane component to the scattering vector is determined by thekinematical constraints and was L � 1.25 and L � 2.0 in c and d, respectively.Dashed blue squares in d show the location of reciprocal space points equiv-alent to (�, 0). (e and f ) Simulation of the scattering expected in linearspin-wave theory in the energy ranges shown in c and d. The calculationsinclude resolution convolution as well as a dipolar factor in the cross-sectionfor magnetic scattering of neutrons (21) that leads to an intensity modulationdepending on the angle between the scattering vector Q and the direction ofthe ordered Cu moments. The dashed red lines in a, b, and e indicate themagnetic Brillouin zone boundary.

11 12 13 14 15 16 17 18

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Fig. 3. Summary of fitting the single-magnon excitations. (a and b) Constantwavevector cuts at the high-symmetry points (�/2, �/2) and (�, 0) of themagnetic Brillouin zone boundary (Fig. 2a). By fitting the data, the position(red arrows) and intensity of the spin excitation is extracted, allowing thedispersion relation and corresponding intensity variation along the mainsymmetry directions of the two-dimensional Brillouin zone (indicated by solidlines in the Inset in e) to be derived. (c) Full dispersion relation for thedominant single-magnon excitations. (d) Dispersion divided by the linearspin-wave prediction (Fig. 2a) with J � 6.19 meV. (e) Intensity of the spinexcitations along the same momentum space path as in c. ( f) Intensity dividedby the linear spin-wave prediction that best describes the data between (�, �)and (2�, 0). Red, blue, and green lines in c–f indicate the dispersions andintensities predicted by linear spin-wave theory (13, 15), series expansiontechniques (25), and flux phase RVB (27), respectively. The solid blue squaresare results of a Quantum Monte Carlo computation (26) of the excitationspectra at (�/2, �/2) and (�, 0). Note that had we chosen to employ a nonlinearbackground model in a and b, we would have obtained slightly larger statis-tical errors on the zone boundary intensities, but the significant spectralweight anomaly at (�, 0) would remain, as proven unambiguously by itspresence in the lower resolution polarized data shown in Fig. 6.

15266 � www.pnas.org�cgi�doi�10.1073�pnas.0703293104 Christensen et al.

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‘‘multimagnon’’ because the momentum and energy imparted bythe neutron to the sample can be shared among pairs of classicalspin waves. If we move away from simply including single spinflip corrections to �� and also take account of RVB-like terms(23), as illustrated in Fig. 1e, the structure of the continuum canbe substantially altered from that given by two-magnon calcu-lations, as borne out by the different predictions given bycompeting theoretical scenarios (26, 29).

The unambiguous observation of multimagnon continuum scat-tering is highly nontrivial, because of the difficulty of isolating aweak, diffuse signal from the one-magnon signal in the presence ofscattering from phonons, nonuniform background, etc. One par-ticular setting of the MAPS spectrometer was found to provide a‘‘clean’’ window within which to search for the existence of con-tinuum scattering, namely for energies in the range 8.7 � �� � 11.5

meV with the two-dimensional Cu layers of CFTD aligned parallelto the incident beam. Multimagnon processes (29) are expected toproduce a continuum inside the dispersion cone (Fig. 2a). The datain Fig. 4a reveal extra intensity in this region, an effect that is madeeven clearer when integrating over a range of L (Fig. 4 b and c).Indeed, although the single-magnon model (Fig. 4 b and c, red lines)provides a remarkably good description of the data, including thosefor the spin-wave cone emanating from the structural zone centersat �H� � 1, it fails to account for the observed filling of the cone. Itis only by including a two-magnon contribution in the model (Fig.4 b and c, blue lines) that a satisfactory account of the data isachieved.

Definite proof that the additional scattering near (�, �) in Fig.4 is magnetic comes from polarized neutron scattering. Thistechnique allows separation, in a model-independent way, of thefluctuations transverse and longitudinal to the ordered momentthat, to lowest order in perturbation theory, are dominated bysingle- and two-magnon events, respectively. Fig. 5 displays dataat intermediate energies and wavelengths, demonstrating clearly

Fig. 4. Imaging the two-magnon excitations. (a) Raw data from the MAPSspectrometer with the sample oriented such that the incident neutron beam(Ei � 24.93 meV) is parallel to the two-dimensional Cu layers of CFTD. The thirdcomponent to the scattering vector K � �1.07 is fixed by the kinematicalconstraints. The extremely weak coupling between the layers in the c direction(Jc � 10�5–10�4) implies that the excitations are independent of the reciprocalspace coordinate L. In this case, at a given energy transfer, the circles shownin Fig. 2b become cylinders along the L direction. The intercept of the cylinderwith the detector trajectories in Q and energy produces parabolic cutsthrough the dispersion surface. In the region 0.3 � L � 1, sharp, single-magnonmodes run approximately parallel to the L direction. At higher values of L, thedetector trajectories produce a filling-in due to single-magnon events. (b andc) Intensity integrated over the region indicated by dashed lines in a. b is amagnified version of c. The intensity between the single-magnon modes isevidently higher than in the regions immediately to their exterior. This addi-tional intensity arises from multimagnon processes. The solid red line repre-sents the calculated single-magnon response convolved with the full instru-mental resolution and gives a poor account of the data between the peaks.The solid blue line is the sum of the calculated one and two-magnon contri-butions, calculated by using Monte Carlo sampling (29). The relative spectralweight of the two contributions is completely specified by the quantummechanical reduction S of the staggered ordered moment �S� from itsclassical value of 1/2. The red line corresponds to the classical case S � 0 withno multimagnon continuum.

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Fig. 5. Long-wavelength spin excitations. False color maps of the intensity ofthe low-energy spectrum of single-magnon excitations polarized transverse tothe ordered moment (a) and of multimagnon excitations polarized along theordered moment (b), measured at Q � (H, 1, 0). Solid white lines indicate theprediction of linear spin-wave theory (15) with J � 6.19 meV. (c–f ) Raw dataand fits upon which the color images in a and b are built. (c and d) Transverseand longitudinal excitations at 5 meV. (e and f ) Transverse and longitudinalexcitations at 9 meV. The solid lines in c–f are fits to semiclassical theory forone- and two-magnon excitations, calculated by using Monte Carlo sampling(29) with the value S � 0.23 for the reduction in the staggered moment takenfrom the time-of-flight experiment (Fig. 4). With J � 6.19 meV fixed this leavesonly a single overall intensity scale factor to be adjusted. The data wereobtained at a sample temperature of 1.5 K by using polarized neutronscattering (kf � 2.662 Å�1) at the IN20 spectrometer at the Institut Laue-Langevin (Grenoble, France).

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that the filling of the spin-wave cone seen by using MAPS arisesfrom the longitudinal response that can only be associated withmultimagnon processes. The intensity ratio of two-magnon tosingle-magnon scattering is determined by the reduction S ofthe ordered moment from its classical value of 1/2. Quantitativeanalysis of our data yields S � 0.23 � 0.02, in accord with thetheoretical estimate S � 0.197 (ref. 13).

Of great interest—in light of the anomalies found in the single-magnon response (Fig. 3)—is the continuum at the zone boundary,shown in Fig. 6. The high-energy polarized data for the transversecomponent (Fig. 6 a and b) display the same feature as that foundusing MAPS: The peak response at the (�, 0) point is depressed inenergy relative to (�/2, �/2) and is also less intense. The latter effectappears less dramatic in Fig. 6 a and b than in Fig. 3f because of thepoorer momentum resolution of the polarized beam instrument,which averages over the dip at (�, 0), as well as the fact that theunpolarized MAPS data are simultaneously sensitive to the longi-tudinal mode, which appears even more suppressed at (�, 0) relativeto (�/2, �/2) than the transverse mode (Fig. 6 c and d). In additionto the peak at the renormalized spin-wave energy, weak butsignificant intensity extends to energies above the main spin-wavepeak in both the transverse (Fig. 6 a and b) and longitudinalchannels (Fig. 6 c and d). Such intensity must be present to balancethe total moment sum rule, which states that S(S � 1) is the sumof contributions from the ordered moment, the spin waves and thecontinuum (30).

Quantum Monte Carlo calculations (26), indicated by dashedgreen lines in Fig. 6, broadly account for the data in bothchannels. Most notably, there is a pile-up of intensity in thelongitudinal channel at energies just above the sharp peak in the

transverse response. This is in contrast to the two-magnontheory, whose results are also shown in Fig. 6 (solid red and bluelines), which posits a less pronounced maximum in the longitu-dinal response. In addition, the two-magnon theory does notexplain the momentum dependence of the amplitude of thelongitudinal response, just as conventional spin-wave theorydoes not account for the amplitudes of the transverse responseat the zone boundary. An alternative view is provided by thevariational RVB approach (27, 28), where the continuum abovethe zone boundary arises from fermionic (spinon) excitations.Unfortunately, no predictions have been made for the longitu-dinal continuum. Clearly, further refinements to experiment andtheory are required before a definitive conclusion can be drawnfor the nature of the continuum we have observed in ourexperiments.

Summary and ConclusionsThe square lattice S � 1/2 Heisenberg antiferromagnet has longbeen regarded as having only minor quantum corrections to itsbehavior (14, 31). We have shown that at short distances, this isactually not the case, and that there are large momentum-dependent corrections to the spin waves. In addition, we haveprovided the first observation of the multimagnon continuum inthis simplest of two-dimensional quantum magnets. As for the spinwaves, the most significant discrepancy between our results andconventional theory appears at the zone boundary. QuantumMonte Carlo accounts well for all of the data, although it does notgive any particular physical insight into why classical theory fails.Conventional spin-wave and two-magnon calculations give an ex-cellent quantitative description of the long and intermediate wave-length fluctuations. At short wavelengths, a variational wavefunc-tion that explicitly mixes RVB and classical (Neel) terms providesa qualitative, but not quantitative, account of the intensity anomalyat (�, 0), suggesting that the experiments as well as the QuantumMonte Carlo data are due to a significant admixture of RVB termsin the ground-state wavefunction. Indeed, what they show is that thematrix elements (Eq. 1) between the corrected ground state and themagnon eigenstate actually contain destructive interference termsbetween valence bond corrections and the classical states. Specif-ically, these arise because of the nonorthogonality of the Neel state�N� and the RVB corrections; within the two-magnon theory, suchinterference terms do not occur because the single-magnon statesin the expansion for the ground state are orthogonal to �N�. Fig. 2dis an image of these interference effects (as for the plaquettesillustrated in Fig. 1), which are most severe near the zone boundarypoint (�, 0), indicating the entanglement of spins on adjacent sitesof this exceptionally simple magnet without disorder or geometricalfrustration. In fact, nearest-neighbor singlet corrections to theground-state wavefunction are manifested independently and di-rectly in the bond-energy, which is �J/4 for the triplet and �3J/4 forthe singlet, whereas the average energy of uncorrelated bonds iszero. Because the S � 1/2, square lattice Heisenberg antiferromag-net has a ground-state energy per bond of Eb � �0.34J (23), whichis lower than the �0.25J of the classical Neel state, the correctionsmust overrepresent singlets compared with random quantum dis-order. It is worth remarking that, in the S � 5/2 square lattice systemRb2MnF4 the zone boundary spin waves are dispersionless to within1%, and only a weak continuum of two-magnon states has beenobserved (32).

In summary, we have performed precision experiments usingstate-of-the art neutron scattering techniques to provide a completedescription of the spin dynamics of the two-dimensional squarelattice Heisenberg antiferromagnet—at all relevant wavevectorsand energies in the one- and multimagnon channels. The quantumnature of the dynamics in this system is shown to manifest itself inthe one-magnon channel as a mild zone boundary dispersion in theenergy, and a much larger dispersion in the intensity, whereas thecontinuum of multimagnon events is imaged directly for the first

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Fig. 6. Short wavelength spin excitations. The transverse and longitudinalresponses at the zone boundary points (�/2, �/2) (a and c) and (�, 0) (b and d)measured at, respectively, Q � (0, 5/2, 0) and Q � (1/2, 5/2, 0) at a temperature of1.9K.NotethatthebroadQ-resolutionof IN22relativetotheMAPSspectrometerpartially washes out the rather narrow dip in intensity at (�, 0) evident in Fig. 3f.Solid lines are single-magnon (red in a and b) and two-magnon (blue in c and d)responses from linear spin-wave theory (J � 6.19 meV) convolved with the fullinstrumental resolution. The value S � 0.23 obtained from the time-of-flightexperiment (Fig. 4) was used to fix the relative intensities in the transverse andlongitudinal channels, leaving only one overall scale factor to be adjusted.Dashed lines are Quantum Monte Carlo results (26) convolved with the instru-mental energy resolution. In a and b the sharp single-magnon and transversecontinuum components (26) are plotted separately. The data were obtained byusing polarized neutron scattering (kf � 2.662 �1) at the IN22 spectrometer atthe Institut Laue-Langevin (Grenoble, France).

15268 � www.pnas.org�cgi�doi�10.1073�pnas.0703293104 Christensen et al.

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time and has, as the total moment sum rule suggests, an intensitydetermined purely by the reduction in the size of the orderedmoment because of quantum fluctuations. We compare our datawith a variety of analytical theories and computer simulations,including expansion from the Ising limit, flux-phase RVB andQuantum Monte Carlo to show that no currently available analyt-ical theory accounts for the short-wavelength dynamics probed atthe zone boundary.

Our discoveries have implications not only for quantum magnetsas laboratories for the study of entangled states (17, 33–35), but alsofor one of the principal problems of modern condensed matterphysics, namely that of the high-temperature superconductors. Thehigh-Tc cuprates are doped two-dimensional Heisenberg quantumantiferromagnets and many believe that this is the key to a fullquantitative description of the cuprates. In the parent compoundsof the cuprates, Raman scattering (36) and infrared absorptionmeasurements (37) have revealed intense, broad responses extend-ing up to several times the zone boundary magnon energy. Multi-magnon processes, treated in linear spin-wave theory, haveemerged as essential to account for these observations (38, 39), butdespite some successes these models do not account for the fullrange of experimental anomalies. It has been suggested (39) that thediscrepancies between the optical data and the multimagnon the-ories are due to strong local deviations from the Neel state, exactlywhat is seen directly by our neutron scattering measurements.Although the role of charge fluctuations cannot be neglected in thecuprates, the discovery of nearest-neighbor quantum effects alreadypresent in the Heisenberg antiferromagnet supports resurgenttheoretical ideas (40) about the importance of RVB-type correla-tions that become dominant upon the introduction of frustration ormobile charge carriers, leading to the destruction of Neel order.

Materials and MethodsMeasurements were performed on single crystals of CFTD[Cu(DCOO)2�4D2O], grown by slow evaporation in a vacuumdessicator at �25°C of a Cu(DCOO)2 solution, which was preparedby dissolving the deuterated carbonate, Cu2CO3�(OD)2�D2O, in asolution of d2-formic acid in D2O (41). The carbonate was synthe-sized by adding Na2CO3 to a solution of Cu2� in D2O.

CFTD has a monoclinic low-temperature crystal structure, spacegroup P21/n with a � 8.113 Å, b � 8.119 Å, c � 12.45 Å, and �100.79°. The Cu2� ions form nearly square lattice planes parallel tothe a and b axes separated by layers of crystal-bound heavy water.Each Cu2� ion carries a localized S � 1/2 spin that interacts withits four in-plane nearest neighbors located �5.739 Å away throughantiferromagnetic Heisenberg exchange interactions J mediated byintervening formate groups, whereas interplane interactions Jc aremuch weaker, Jc � 10�5–10�4J. Quantum chemistry implies thatfurther neighbor interactions within the planes of this ionic salt arealso negligible, consistent with the perfect agreement of the mea-sured spin-wave energies with the numerical results of refs. 25 and26, which employ nearest-neighbor coupling only and would dis-agree with the measured dispersion if further neighbor interactionswere included. The basic Hamiltonian of the system is thereforeH � J ¥�il� Si�Sl, where the summation extends over in-planenearest-neighbor pairs only.

In momentum space, the reciprocal crystallographic lattice isspanned by the reciprocal lattice vectors a*, b*, and c* and indexedQ � (H, K, L). Below an ordering temperature ofTN � 16.5 K, CFTD forms a simple antiferromagnetic structurewhere neighboring spins align anticollinearly almost along the a*direction (42), leading to magnetic Bragg scattering atQ � (1, 0, 1) and equivalent positions, which correspond to theantiferromagnetic zone center Q2D � (�, �) in the reciprocal latticeof the plaquette and of the two-dimensional square lattice model.One can transform from Q � (H, K, L) to Q2D � (Qx, Qy) ��(H � K, H � K), while the L component of the scattering vectoris irrelevant to the pure two-dimensional model. Throughout thisarticle, we use the two-dimensional square lattice notation,supplemented with crystallographic (H, K, L) notation in Figs.2, 4, and 5.

We would like to acknowledge helpful discussions with P. Lee, K.Lefmann, D. A. Tennant, R. R. P. Singh, A. Sandvik, A. Lauchli andR. A. Cowley. Work in London was supported by Research CouncilsUnited Kingdom Basic Technologies and Royal Society-Wolfson Fel-lowship programs.

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