VOLUME 87, NUMBER 27 P H Y S I C A L R E V I E W L E T T E R S 31 DECEMBER 2001

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Dynamical Mean-Field Theory of Resonating-Valence-Bond Antiferromagnets

Antoine Georges, Rahul Siddharthan, and Serge FlorensLaboratoire de Physique Théorique, Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France

and Laboratoire de Physique des Solides, Université Paris-Sud, Bât. 510, 91405 Orsay, France(Received 25 June 2001; published 13 December 2001)

We propose a theory of the spin dynamics of frustrated quantum antiferromagnets, which is based on aneffective action for a plaquette embedded in a self-consistent bath. This approach, supplemented by a low-energy projection, is applied to the Kagomé antiferromagnet. We find that a spin-liquid regime extends tovery low energy, in which local correlation functions have a slow decay in time, well described by power-law behavior and v�T scaling of the response function: x00�v� ~ v2aF�v�T�.

DOI: 10.1103/PhysRevLett.87.277203 PACS numbers: 75.10.Jm, 75.10.Nr, 75.40.Gb

The possibility of frustrated quantum antiferromagnets(QAF) having a resonating-valence-bond (RVB) groundstate, that is, a superposition of states where all spins arepaired into singlets, was suggested many years ago [1];such a ground state has neither long-range spin order (e.g.,Néel order) nor spin-Peierls order (an ordered arrange-ment of these singlet pairs). Recently, convincing evidencehas been given that some frustrated two-dimensional QAFindeed have RVB physics [2,3]. In particular, numeri-cal studies [2] of the spin-half Heisenberg QAF on theKagomé lattice reveal that the model has no long-rangeorder and displays a very small gap to magnetic (triplet)excitations (estimated to be of order J�20). Moreover,this gap is filled with an exponential number of singletexcitations, suggesting a possible continuum in the ther-modynamic limit, and a large low-temperature entropy, inagreement with the RVB picture. We also note, thoughthis is seldom emphasized, that the results of [2] suggesta correspondingly large number of triplet states immedi-ately above the gap. This may be expected, since in anyvalence-bond component of an RVB state, one can replaceany singlet pair by a triplet pair and continue to have aneigenstate of S. Resonance between such states wouldlower the excitation energy to much below J.

These considerations suggest that, in a temperaturerange above the triplet gap, the spin correlations have arapid decay in space, but a slow decay in time due to thelarge density of triplet excited states. Equivalently, the dy-namical susceptibility x� �q, v� would not display a narrowpeak around a specific ordering wave vector, but x 00� �q, v�would have weight at low frequency, characteristic of aspin liquid. Indeed, recent inelastic neutron scatteringstudies [4] on the S � 3�2 Kagomé slab compoundSrCr9pGa1229pO19 (SCGO) reveal such behavior abovethe freezing temperature �T $ 4 K�. The only relevantenergy scale in this spin-liquid regime is apparently set bythe temperature itself, with x

00loc �

P�q x 00� �q, v� obeying

scaling behavior x00loc � v2aF�v�T � with a � 0.4,

corresponding to a slow decay in time of the local dy-namical correlations �S�x, 0�S�x, t�� � 1�t12a � 1�t0.6.

-1 0031-9007�01�87(27)�277203(4)$15.00

We observe that the uniform � �q � �0� susceptibility in thisregime is also well fitted by x ~ T20.4.

In this Letter, we introduce a novel theoretical approachto the spin dynamics in the spin-liquid regime of frus-trated QAF, which focuses on short-range spin correlationsonly. This approach draws inspiration from the dynami-cal mean-field theory (DMFT) of itinerant fermion mod-els [5], and some of its extensions [5–10]. Our method isfairly general but is applied here to the concrete case of thespin-half Kagomé QAF. The determination of on-site andnearest-neighbor dynamical spin correlations is mappedonto the solution of a model of quantum spins on a triangu-lar plaquette coupled to a self-consistent bath. (A single-site approach is not adequate, since the essential physics ofsinglet formation involves at least nearest-neighbor sites.)By using a projection onto the low-energy sector of thismodel, we demonstrate that, in a temperature range extend-ing down to T ø J, a slow (power-law) decay of temporalcorrelations is found.

We view the Kagomé lattice as a triangular superlatticeof up-pointing triangular plaquettes (Fig. 1). Sites are la-beled by an index a � 1, 2, 3 within a plaquette and bya triangular superlattice index I numbering the plaquette.We denote by xab� �q, t 2 t0� the Fourier transform of thedynamical spin correlation function 1

3 � �Sa,I�t� ? �Sb,I 0�t0��with respect to the plaquette coordinates I, I 0 ( �q is a vectorof the supercell Brillouin zone). Our approach relies onapproximating the correlation function by

x� �q, inn�21ab � Jab� �q� 1 Mab�inn� . (1)

Here, nn � 2np�b is a bosonic Matsubara frequency, andall quantities are matrices in the internal plaquette indices�a, b�. Jab� �q� is the supercell Fourier transform of the ex-change couplings. Mab is a measure of how much thecorrelation function differs from that of a Gaussian model,for which x � J21, and hence plays the role of a spin“self-energy” matrix for the QAF. Obviously, the key as-sumption made in (1) is that the �q dependence of thisself-energy can be neglected. This approximation, which is

© 2001 The American Physical Society 277203-1

VOLUME 87, NUMBER 27 P H Y S I C A L R E V I E W L E T T E R S 31 DECEMBER 2001

1

23 3

32

2

1 1

FIG. 1. The Kagomé lattice, viewed as an up-pointing cluster(bold) embedded in a network of similar clusters, which is ap-proximated in the DMFT approach by a self-consistent bath.

likely to be reasonable when spatial correlations are shortranged, is analogous to the assumption of a momentum-independent self-energy made in the DMFT of correlatedfermion models. Here, however, the DMFT concept is ex-tended on two accounts: the local ansatz is made on thespin correlation function rather than on a single-fermionquantity, and a plaquette rather than a single site is con-sidered. Extensions of DMFT to clusters [5–7] and to(spin or charge) response functions [8–10] have been pre-viously considered separately in different contexts. Com-bining them is a unique aspect of our approach, which isnecessary to capture the dynamical aspects of inter-site sin-glet formation at the heart of spin-liquid behavior.

In order to calculate the dynamical self-energy matrixMab�inn�, an effective action involving only the spins ofa single triangular plaquette is introduced, which reads

S � SB 112

JXafib

Z b

0dt �Sa ? �Sb

112

Z b

0dt dt0

Xab

Dab�t 2 t0� �Sa�t� ? �Sb�t0� ,

(2)

in which SB denote Berry phase terms. Dab�t 2 t0� is aretarded interaction, generated by integrating out all spinsoutside the plaquette. Higher order interactions are alsoinduced in this process, which have been neglected in(2). Equivalently, one can view the rest of the lattice asan external bath which couples to the spins in the cluster

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through time-dependent external fields with a Gaussiancorrelator Dab . The latter will be determined by a self-consistency requirement, which stipulates that the cor-relation functions on a plaquette calculated from theabove action [xab�t 2 t0� � 1

3 � �Sa�t� ? �Sb�t0��S involvethe same self-energy as the entire lattice. This readsx

21ab �inn� � JDab 1 Dab�inn� 1 Mab�inn�, where

JDab � J�1 2 dab� is the matrix of nearest-neighborcouplings on the cluster. Inserting this relation for theself-energy into (1) and imposing that

P�q xab� �q� � xab

lead to the final form of the self-consistency requirement

xab�inn� �X

�q

J� �q� 2 JD 1 x21�inn� 2 D�inn�21ab .

(3)

We note that the matrix J��q�ab 2 JDab involves onlythe exchange constants linking together different (up-pointing) triangular plaquettes. In the limit where theseinterplaquette couplings vanish while the internal ones arekept fixed (decoupled triangles), our approach becomesexact since (3) implies Dab � 0 and (2) reduces to theaction associated with the Heisenberg model on a singletriangle. We also note that the above DMFT equations canbe derived from a Baym-Kadanoff formalism in whicha functional of the correlation function and self-energymatrix is introduced in the form

Vx,M �12 Tr lnM 1 J 2

12 TrxM 1 Fx .

(4)

The stationarity conditions dV

dx �dV

d M � 0 lead to theabove equations when the exact functional F is approxi-mated by its value for a single triangular plaquette. Thefree energy of the model can thus be calculated in theDMFT approach by inserting the self-consistent values ofx and M into (4). Alternatively, a functional of the corre-lation function only can be used, along the lines of Ref. [9].

In a phase with unbroken translational invariance, thematrix xab�inn� in fact reduces to two elements: the lo-cal susceptibility xloc � xaa and the nearest-neighbor onexnn � xab �a fi b� (and similarly for Dab). It is actu-ally convenient to use the following linear combinations(proportional to the eigenvalues of the xab and Dab matri-ces): xs � 3�xloc 1 2xnn�, xm � 3�xloc 2 xnn�, Ds �13 �Dloc 1 2Dnn�, and Dm �

13 �Dloc 2 Dnn�. Introducing

the corresponding self-energies Ms�inn� � 3�xs�inn� 2

3Ds�inn�, Mm�inn� � 3�xm�inn� 2 3Dm�inn�, straight-forward but tedious algebra allows us to recast (3) in theform of two scalar equations:

xs

3�

Z Mm 1 J 223 Je

Ms�Mm 1 J� 2 2J2 223 �Ms 2 Mm�Je

r�e� de ,

xm

3�

Z MsMm 2 J2 113 J�2Mm 2 MJ�e

�Mm 2 J� Ms�Mm 1 J� 2 2J2 223 �Ms 2 Mm�Je

r�e� de .

(5)

Here, e stands for cosq1 1 cosq2 1 cosq3, where the qi are components of �q along the three directions of the triangular

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lattice �q1 1 q2 1 q3 � 0�, and r�e� is the correspondingdensity of states. The �q summation in (3) simplifies intoan e integration in (5).

Equations (2) and (5) entirely define the plaquette-DMFT approach introduced here, and one could embarkat this stage into a numerical determination of the twokey dynamical quantities xloc�t� and xnn�t� using, e.g.,a generalization of the quantum Monte Carlo algorithmrecently introduced in [11] in the context of quantumspin glasses. Instead, we shall gain further insight intothe problem by making use of a projection onto the low-energy sector of the Hilbert space [12]. Interesting insightsinto the low-energy excited states of the singlet sector havebeen recently obtained by Mila and Mambrini startingfrom the limit of decoupled up-pointing triangles [13,14].Each such triangle has a fourfold degenerate ground statewith spin S � 1�2 and a fourfold degenerate excitedstate with spin S � 3�2. Hence the lattice of decoupledtriangles has an exponentially large number of groundstates, equal to 4Ns�3. The physical picture of [13,14] is

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that the degenerate ground state broadens into a band asthe intertriangle coupling is turned on. In the following,we shall project the DMFT equations onto the low-energy Hilbert space corresponding to the S � 1�2 sectorof each triangle. This amounts to first diagonalizing thesingle-plaquette Hamiltonian (neglecting Dab) and thenrewriting the retarded term in the action (2) in this low-energy sector. To this end, it is convenient to follow [12,13]and choose as a basis of the fourfold degenerate groundstate the eigenvectors of the two following operators: thespin-1�2 operator corresponding to the total spin �S in aplaquette, and a doublet of Pauli matrices �T � �Tx, Ty�with the meaning of a plaquette chirality operator. Interms of those, the low-energy projection of the three spinoperators in a plaquette reads (with v � expi2p�3 andfor a � 1, 2, 3)

�Slowa �

13

�S1 2 2v12aT2 2 2va21T1 . (6)

The low-energy projection of the action (2) is then easilyobtained as

Slow � SB 1Z b

0dt dt0

∑12

Ds�t 2 t0� 1 Dm�t 2 t0� �T �t� ? �T �t0�∏

�S�t� ? �S�t0� , (7)

where Ds and Dm are as defined earlier. It turns out thatxs and xm have simple interpretations as the “total spin”and “mixed” correlation functions:

xs�t 2 t0� � 13 � �S�t� ? �S�t0�� ,

xm�t 2 t0� � 13 � �S�t� ? �S�t0� �T �t� ? �T �t0�� .

(8)

To solve the local quantum problem defined by (7), weuse an approximate technique that has proven successfulin recent studies of quantum spin-glass models [10] andof impurities in quantum antiferromagnets [15]. Weintroduce two doublets of spin-1�2 fermions, �s", s#�and �t", t#�, subject to the constraints, s1

" s" 1 s1# s# �

t1" t" 1 t1

# t# � 1, in order to represent the operators �S and�T as S1 � s1

" s#, S2 � s1# s", Sz � �s1

" s" 2 s1# s#��2;

T1 � t1" t#, T2 � t1

# t". The interacting fermion problemcorresponding to (7) is then solved in the self-consistentHartree approximation. Also, the local constraint isapproximated by its average (the associated Lagrangemultiplier is zero due to particle-hole symmetry). In-troducing the Green’s functions Gs and Gt for the twofermion fields and defining self-energies by G21

s �ivn� �ivn 2 Ss�ivn�, G21

t �ivn� � ivn 2 St �ivn� [withvn � �2n 1 1�p�b a fermionic Matsubara frequency],yields the imaginary-time equations:

Ss�t� � 238 Ds�t�Gs�t� 1 3Dm�t�Gs�t�Gt�t�Gt �2t� ,

St�t� � 3Dm�t�Gt�t�Gs�t�Gs�2t� .(9)

Then the local problem [for a given bath �Ds, Dm�] reducesto two coupled nonlinear integral equations. The correla-tion functions are given by

xs�t� � 212 Gs�t�Gs�2t� ,

xm�t� � 2Gs�t�Gs�2t�Gt�t�Gt �2t� .(10)

In practice, we use the following algorithm: we start withan initial guess for the bath Ds,m, and obtain the Green’sfunctions by iteration of Eq. (9). We can then calculatethe susceptibilities from (10), as well as the self-energiesMs,m. Inserting the latter into (5) yields new xs,m and,in turn, the new baths Dnew

s,m �inn� � 2Ms,m�inn��3 11�xs,m�inn�.

We now discuss our findings when solving these coupledequations numerically. The first key observation is thata paramagnetic solution can be stabilized down to thelowest temperature we could reach, with no sign oflong-range ordering intervening. Long-range order is as-sociated with a diverging eigenvalue of xab� �q� for some �qand hence, within our scheme, to a vanishing denominatorin (5), which we never observe [16]. In Fig. 2, we displayour results for the temperature dependence of the local(i.e., on-site, or �q-integrated) susceptibility xloc�T� �Rb

0 xloc�t� dt. At high temperatures, xloc ~ 1�T obeysCurie’s law. Below T � 0.5J, the effective Curie constantdecreases with decreasing T , indicating gradual quenchingof the local moment. However, xloc itself continues toincrease as the temperature is lowered. In the inset ofFig. 2, we display the effective exponent defined bya�T� � 2 lnxloc� lnT . A transient regime, correspondingto a slowly decreasing a, is apparent and extends overmore than two decades. At the lowest temperatures, a

appears to saturate to a value a � 0.5.Correspondingly, the correlation function xloc�t� obeys

a power-law dependence on time in the low-temperature

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VOLUME 87, NUMBER 27 P H Y S I C A L R E V I E W L E T T E R S 31 DECEMBER 2001

0 0.5 1T

0

0.01

0.02

0.03

0.04χΤ

-6 -5 -4 -3 -2 -1 0log(T)

1

0.9

0.8

0.7

0.6

0.5

α

FIG. 2. Main plot: xloc obeys Curie law (~1�T ) at high tem-perature, but diverges more slowly than 1�T at lower tempera-ture. Inset: evolution of the effective exponent a from high tolow temperature when fitting xloc�T� to 1�Ta . (T is in units ofJ in all figures.)

regime. Figure 3 shows xloc�t��xloc�b�2� as a functionof t�b for several (large) values of b � 1�T ; thegraphs collapse on each other for different b, andare well fitted by the expression xloc�t��xloc�b�2� �sinpt�b2�12a�, with 1 2 a � 0.5. This scalingfunction is the one appearing in the study of quantumspin glasses [10] and quantum impurity models [15,17]and is dictated in the latter case by conformal invari-ance. It corresponds to the following scaling form ofthe dynamical susceptibility: x

00loc�v� ~ v2aFa�v�T�

with Fa�x� � xa jG� 12a

2 1 ix

2p �j2 sinh x2 . This also

implies diverging behavior of the relaxation rate1�T1 ~ Tx

00loc�v��v � 1�Ta .

These findings can be interpreted as the formation ofa spin-liquid regime with a large density of triplet ex-cited states, consistent with Ref. [2]. Our approach failsto predict a triplet gap, however, because our triangularplaquette has a spin-1�2 ground state, and the classicalbath to which it is coupled cannot fully screen this localmoment. Hence the local susceptibility continues to in-crease at low temperatures in our approach, down to annonphysically low-energy scale (of order J�100). In factthe triplet gap for the S � 1�2 Kagomé QAF is known[2] to be quite small (of order J�20), so our description ofthe spin dynamics should be valid down to a rather low-temperature scale. The power-law behavior of the spincorrelations and the low-temperature increase of the sus-ceptibility agree well with experimental findings on SCGOabove its freezing temperature at T � 4 K [4]. Since thisis a S � 3�2 system, it is conceivable that higher spinextends the range of applicability of our approach evenfarther. A promising application of our approach is thepyrochlore lattice: the natural plaquette is a tetrahedron,with a twofold degenerate singlet ground state, so that thevery low-energy singlet sector should be accessible withinour approach. In future work we plan to address these is-sues, and also study the low-temperature thermodynamics

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0 0.2 0.4 0.6 0.8 1

τ/β

1

2

3

4

5

χ(τ)

/χ(β

/2)

Scaling funcbeta=110beta=160beta=210

FIG. 3. For different (low) temperatures, xloc�t��xloc�b�2�plotted as a function of t�b collapses on a single curve, welldescribed by sin�pt�b�20.5 (solid curve).

by solving the self-consistent local problem (2) using exactnumerical techniques.

We are grateful to O. Parcollet for his very useful helpwith numerical routines, and to C. Lhuillier, G. Misguich,R. Moessner, and S. Sachdev for discussions. We alsothank C. Mondelli for sharing her data with us.

[1] P. W. Anderson, Mater. Res. Bull. 8, 153 (1973).[2] C. Waldtmann, H.-U. Everts, B. Bernu, C. Lhuillier, P.

Sindzingre, P. Lecheminant, and L. Pierre, Eur. Phys. J.B 2, 501 (1998); P. Lecheminant, B. Bernu, C. Lhuillier,L. Pierre, and P. Sindzingre, Phys. Rev. B 56, 2521 (1997).

[3] R. Moessner and S. L. Sondhi, Phys. Rev. Lett. 86, 1881(2001); R. Moessner, S. L. Sondhi, and E. Fradkin, cond-mat/0103396.

[4] C. Mondelli, H. Mutka, C. Payen, B. Frick, and K. H.Andersen, Physica (Amsterdam) 284B, 1371 (2000);C. Mondelli, Ph.D. thesis, Université Joseph Fourier,Grenoble, 2000 (unpublished).

[5] A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev.Mod. Phys. 68, 13 (1996).

[6] A. Schiller and K. Ingersent, Phys. Rev. Lett. 75, 113 (1995).[7] M. H. Hettler, A. N. Tahvildar-Zadeh, M. Jarrell, T.

Pruschke, and H. R. Krishnamurthy, Phys. Rev. B 58,R7475 (1998).

[8] J. Lleweilun Smith and Q. Si, Phys. Rev. B 61, 5184 (2000).[9] R. Chitra and G. Kotliar, Phys. Rev. B 63, 115110 (2001).

[10] A. Georges, O. Parcollet, and S. Sachdev, Phys. Rev. Lett.85, 840 (2000); Phys. Rev. B 63, 134406 (2001).

[11] D. R. Grempel and M. J. Rozenberg, Phys. Rev. Lett. 80,389 (1998).

[12] V. Subrahmanyam, Phys. Rev. B 52, 1133 (1995).[13] F. Mila, Phys. Rev. Lett. 81, 2356 (1998).[14] M. Mambrini and F. Mila, Eur. Phys. J. B 17, 651 (2000).[15] M. Vojta, C. Buragohain, and S. Sachdev, Phys. Rev. B 61,

15 152 (2000).[16] In fact, a useful approximation to Eq. (5) is to replace the

Hilbert transform of r�e� by its high-frequency behavior.[17] O. Parcollet, A. Georges, G. Kotliar, and A. Sengupta,

Phys. Rev. B 58, 3794 (1998).

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