Resonating singlet valence plaquettes

S. PankovNational High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306, USA

R. MoessnerRudolf Peierls Centre for Theoretical Physics, Oxford University, Oxford OX1 3NP, United Kingdom

S. L. SondhiDepartment of Physics and Princeton Center for Theoretical Physics, Princeton University, Princeton, New Jersey 08544, USA

�Received 6 May 2007; published 27 September 2007�

We consider the simplest generalizations of the valence bond physics of SU�2� singlets to SU�N� singletsthat comprise objects with N sites—these are SU�N� singlet plaquettes with N=3 and N=4 in three spatialdimensions. Specifically, we search for a quantum mechanical liquid of such objects—a resonating singletvalence plaquette phase that generalizes the celebrated resonating valence bond phase for SU�2� spins. Weextend the Rokhsar-Kivelson construction �Phys. Rev. Lett. 61, 2376 �1988�� of the quantum dimer model tothe simplest SU�4� model for valence plaquette dynamics on a cubic lattice. The phase diagram of the resultingquantum plaquette model is analyzed both analytically and numerically. We find that the ground state is solideverywhere, including at the Rokhsar-Kivelson point where the ground state is an equal amplitude sum. Bycontrast, the equal amplitude sum of SU�3� singlet triangular plaquettes on the face-centered cubic lattice isliquid and is thus a candidate for describing a resonating single valence plaquette phase, given a suitablydefined local Hamiltonian.

DOI: 10.1103/PhysRevB.76.104436 PACS number�s�: 75.10.Jm, 75.40.Mg, 75.45.�j

I. INTRODUCTION

The physics of valence bonds1,2 has provided a fruitfulframework for understanding non-Néel phases of SU�2� in-variant quantum magnets in low dimensions.3,4 In particular,its reformulation as the study of quantum dimer models byRokhsar and Kivelson5 has yielded a sharp but intuitive de-scription of a diverse set of unconventional phenomena, mostnotably the existence of topologically ordered, fractionalizedresonating valence bond �RVB� liquids6,7 originally proposedby Anderson and the phenomenon of Cantor deconfinement.8

Extensions of such dimer models to higher dimensions havebeen considered9–14 and in the process the existence of twodistinct RVB liquids in d=3 has been demonstrated. Thedimer model formulation has the additional virtue of havingthe manifest interpretation of a strongly coupled, frustrated,gauge theory with RVB phases arising as deconfined phasesin such gauge theories.15,16 It is also useful to note the recentconstruction of finite range spin models that realize dimermodels arbitrarily accurately in their low-energy sectors.17,18

We should note that the interest in RVB liquids also comesfrom a much larger study of topological phases in condensedmatter systems.19

In this paper, we pursue a generalization of the valencebond idea to SU�N� groups with N�2 with the aim ofsearching for topological phases in models with these highersymmetries. The valence bond is replaced by a singletformed by N spins in the fundamental representation ofSU�N�. The basic geometrical degrees of freedom thus be-come triangles for N=3, and objects containing four sites,such as square plaquettes for N=4, and so on. We restrict ourstudy to singlet plaquettes with N=3,4. With the assumptionthat for suitable microscopic Hamiltonians the low-energy

sector is well described by coverings of the lattice by suchobjects, we are led to study quantum plaquette models.

These plaquette models are similar in spirit to two-formgauge theories, where degrees of freedom are also defined onplaquettes.20 However, there is one important difference. Inthe latter, there is a local constraint on plaquettes that share abond. In our problem, the site origin of the plaquette vari-ables is reflected in their satisfying a site constraint—specifically, that one and only one of the plaquettes that sharea common site be occupied by a singlet.

Our strategy in this paper, mirroring that used in the studyof valence bond physics, is to begin with a Hilbert spacespanned by orthogonal states ��� labeled by admissibleplaquette coverings � of the lattice under study. In this space,we study the phase diagram of the simplest local Hamil-tonian which includes the quantum dynamics of resonancebetween pairs of plaquettes and a potential energy of inter-action between them which we shall term the quantumplaquette model �QPM�. As is well known, when the signs ofthe matrix elements are of the “Perron-Frobenius” form �i.e.,no positive off-diagonal matrix elements�, such models ex-hibit a Rokhsar-Kivelson point �which occurs at v=2t for thecase of the cubic lattice, Eq. �4�� where the equal amplitudesuperposition,

��� = ��

��� , �1�

saturates a lower bound on the ground state energy of theproblem. The diagonal correlations in this state define a clas-sical statistical mechanics of plaquettes which can be studiedto establish the true nature of this potentially liquid state.Knowledge of this special point, along with the analysis ofother limits, typically allows the phase diagram to be estab-

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lished with some confidence �see Fig. 1�. The remainingchallenge is to show that one can actually define a sign con-vention for wave functions of the spins that lead to the de-sired signs of the matrix elements and finally to constructlocal spin Hamiltonians that encode all of this physics.

For the N=4 problem on the cubic lattice, we report con-siderable progress on this strategy. We show that matrix ele-ments of the nearest-neighbor spin interaction have the de-sired form. We analyze the statistics of the Rokhsar-Kivelson�RK� point wave function computationally using a plaquetteversion of the pocket Monte Carlo algorithm developed fordimers in Ref. 21. We find that the classical system exhibitslong range order, in which the preferable plaquette locationsform a cubic lattice, analogous to the plaquette valence bondcrystal for dimers. This crystalline phase is accounted foranalytically from several perspectives. Besides this plaquettephase, the phase diagram of the QPM also contains the stag-gered and columnar phases and thus the model does not al-low a topological phase. By contrast, we find that the equalamplitude sum for the N=3 problem on the face-centeredcubic �fcc� lattice is liquid and thus a candidate for a reso-nating single valence plaquette �RSVP� phase. The caveathere arises from the nonergodic nature of the dynamics in thesimplest QPM on this lattice. While we suspect that a longerranged �but still local� dynamics will fix this, it remains anopen problem at this time.

Such higher symmetry problems and the emergence ofplaquette variables have been considered before—the newelement in our work is the study of QPMs and the idea of theplaquette liquid. In d=1, SU�4� symmetric models arereadily constructed by fine tuning an SU�2��SU�2� spin-orbital models22–24 or a two-leg ladder.25 In two dimensions,singlet plaquettes have already been considered on a triangu-lar lattice.22,26 Separately, SU�4� models have been studieden route to connecting large-N SU�N� models with the SU�2�case.27,28 The prospect of studying spin-3 /2 cold atom sys-tems has inspired an elegant body of work that includes rel-evant results on SU�4� plaquette solids and even some gen-eralizations to SU�N� ladders.29,30 It is also useful to remarkhere that models with interspersed conjugate representationson bipartite lattices31 exhibit singlet dimers �“mesons”�,while we consider models with uniform choices of represen-tation which are thus forced to exhibit larger objects �“had-rons”�.

This paper is organized as follows. In Sec. II A, we repeatRokhsar and Kivelson’s original “derivation” of the N=4QPM—essentially, it shows how the appropriate signs can begenerated for the matrix elements of the QPM. In Sec. II B,we briefly introduce the pocket algorithm and present theresults of numerical simulation supporting our view of theground state at the RK point of the model. In Sec. II C, we

provide an analytical explanation of the plaquette phasebased on low- and high-dimensional generalizations of themodel. In Sec. III, we present an analysis of the N=3 QPMon the face-centered cubic lattice. In Sec. IV, we comment onthe model’s utility and open questions.

II. N=4 QUANTUM PLAQUETTE MODEL ON THECUBIC LATTICE

A. Microscopics and Hamiltonian

The quantum plaquette model is an effective model forthe SU�4� Heisenberg antiferromagnet in a valence plaquettedominated phase, in much the same way the quantum dimermodel is an effective model for the SU�2� Heisenberg anti-ferromagnet in a valence bond dominated phase. As we havenoted above, the choice of plaquettes, rather than dimers, asnew degrees of freedom is dictated by the fact that in theSU�4� model the lowest energy configuration on a four sitelattice is a singlet which mixes equally all four sites.

In this section, we will establish some useful properties ofthe microscopic wave functions that correspond to plaquettecoverings of the cubic lattice. Should it prove possible toconstruct a Hamiltonian in the full SU�N� invariant Hilbertspace which has the desired coverings, and no other states, asground states—a nontrivial problem—the results in this sec-tion can be used to imitate Refs. 17 and 18 and rigorouslyrealize the QPM that we introduce here. We will not havemuch to say about this possibility beyond a few closing re-marks in the last section.

A classical plaquette covering of the lattice corresponds toa quantum state, the wave function ��� of which is the prod-uct of individual plaquette singlet wave functions. Thesestates are not orthogonal and computing the Hamiltonian ma-trix elements requires knowledge of the overlap matrix ele-ments. In the case of the dimer model, a simple expressionfor the overlap matrix elements was given in terms of thelength of loops in the transition graph �which is formed bysuperposing the two dimer coverings under consideration�.32

For the plaquette model, the situation is more involved asthe geometry of the transition graph elements is much morecomplex than the loops formed by the dimers. We have de-picted the simplest members in Fig. 2, and it is easy to checkthat their contribution to the overlap is given by a factor

1 v/2t

Columnar StaggeredPlaquette

FIG. 1. Minimal phase diagram of the singlet valence plaquettemodel on a cubic lattice. The transitions are first order.

FIG. 2. �Color online� Transition graphs with various numbersof strongly connected parts.

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S = 6−l/2+1. �2�

Here, l is the total number of plaquettes in the transitiongraph element. With this definition of l, the overlap in theSU�2� dimer case is given by 2−l/2+1, which shows that thevalue of the convergence parameter, x=1/6, is much smallerin the SU�4� case than it is for SU�2�, where it equals 1 /2.

In general, configurations contributing to the overlap canbe mapped bijectively onto an edge four-coloring model onthe graph in which the centers of the plaquettes define thesites and their overlapping corners the edges. For the case ofa transition graph element defined by two overlapping check-erboards, one thus finds that the overlap matrix element iseven lower, as that entropy33 implies 0.116l/2. Interestingly,the Pauling estimate for this quantity is the same as for theice model on the same graph, namely, 0.094l/2, although it isobvious that the entropy of the ice model is lower.

We are not aware of a good rigorous bound for the en-tropy of such loop models; it is straightforward to show �Ap-pendix B� that S�c �31/3 /2�l/2=0.711l/2 �where c is an un-important constant� is such a bound, although not at all astrict one. Indeed, all the cases we have explicitly checkedhave Eq. �2� as an upper bound: S�6−l/2+1.

It is convenient to work in the orthogonal basis ��� whichcan be constructed from the original basis ��� as ���= �S−1/2��������, where S���= �� ����. Such an orthogonaliza-tion can be conveniently carried out analytically assumingthe off-diagonal overlap matrix elements to be small.

For example, consider the projection of the nearest-neighbor SU�4� Hamiltonian

H = ��ij�

�mn

Smn �i�Sn

m�j� , �3�

where the Smn are SU�4� generators, to the valence plaquette

subspace. To this end, one expands S��� or H���= ���H����in powers of x. Retaining the leading off-diagonal �kinetic�and diagonal �potential� terms, one obtains the QPM Hamil-tonian �also referred to as the RK model�:

H = − t ����

������ + v��

������ , �4�

where t=5x and v=16x2 �see Appendix C for details�.The precise numerical values for t and v should not be

taken seriously—due to the presence of higher-order correc-tions and the absence of other simultaneous perturbationsthat one can consider, and the choice of plaquette coveringsitself remains to be energetically forced. Nonetheless, thenegative sign of the kinetic term suggests that it is reasonableto study a QPM with the above Perron-Frobenius form whichis what we will do next. For such models, one has the highlyuseful feature that the ground state is a nodeless wave func-tion.

We note that the states � and � are connected by a singleflip of a pair of plaquettes. It is easy to show that one groundstate at the RK point �v=2t� is the equal amplitude superpo-sition of all possible states ���. Studying its properties re-

duces to sampling �with equal probability� different plaquettecoverings. We use a pocket algorithm21 to implement thistask.

B. Pocket algorithm and Monte Carlo simulation results

The main advantage of the pocket algorithm is that itefficiently finds possible cluster moves. The cluster moves,starting from a covering A, are constructed as follows. �1�Consider a transition graph C between A and another cover-ing B. �2� Pick one connected part of the transition graph,P�C; P is the content of the pocket. �3� Replace theplaquettes of A belonging to P by those of B. The new cov-ering A� obtained in this way from A is an allowed coveringbecause it automatically satisfies the close packing con-straint.

Finding an uncorrelated new covering B is of course, ingeneral, not an easy problem. In the pocket algorithm, it isobtained by acting on A with a symmetry operation34 of thelattice �reflection, for example�. Replacing A with a reflectedversion of itself would of course not yield an independentconfiguration, but partial replacements �using P only� even-tually do.

This algorithm has been applied to the dimer problem.21

In that case, the pocket contains a one-dimensional objectwhich was shown not to invade space, as loops in dimertransition graphs never branch. The situation is different withplaquettes. It turns out that the transition graph actually per-colates and the system is not very close to the percolationthreshold. In other words, the typical size of nonpercolatingpocket is small. This, however, does not pose a problem, asthe nonpercolating pockets contain about 40% of allplaquettes, that is, only about 60% of the plaquettes areflipped when the percolating pocket is encountered.

The limiting factor in our case is the quick growth of theequilibration time � with increasing size of the lattice. Thegrowth appears to be faster than L3, with L the size of thelattice. To establish �, we used a binning procedure �see Ap-pendix D for details�. Due to these ergodicity restrictions, wewere limited to a linear size L=64, i.e., a total of 643 /4=65 536 plaquettes, which is sufficient to establish the order-ing properties of the model.

In the simulations, we measure various �equal time� cor-relation functions as a function of plaquette separation. Thecoordinates r= �x ,y ,z� are assigned to a plaquette coveringsites �i , j ,k , l� according to the rule x=min�xi ,xj ,xk ,xl�, y=min�yi ,yj ,yk ,yl�, and z=min�zi ,zj ,zk ,zl�, where �xi ,yi ,zi�are the coordinates of the site i, etc. In the same way, we canassign coordinates to a flippable pair of plaquettes �which wealso call a resonating cube�. We use letters X, Y, and Z todenote the orientation of the plaquette or the pair forming a“resonating cube.” A letter corresponds to the axis orthogo-nal to a given plaquette �or plaquettes, in the case of a reso-nating cube�. Most generally, we are interested in studyingthe correlation function Gn���r�, where n=1 for the plaquettecorrelator, n=2 for the resonating cube correlator, and � and� denote orientations. In fact, we will only need to considerthe case of �=�. Using cubic symmetry, we drop the orien-tation index and denote the correlation function as Gn�r� ,r�,

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where r� and r refer to the components of r orthogonal andparallel to the plaquette.

In Fig. 3, we plot the plaquette correlation functionG1�0,r�. It displays oscillations persisting to large separationr, so that one may conclude that the system possesses longrange order. The simplest possibility might be some kind oforientational order �rotational symmetry breaking�. Quite adifferent conclusion may be drawn from the plot of G1�r ,0�,shown in Fig. 4. At face value, the plot suggests an absenceof the long range order, because the oscillations of the cor-relation functions decrease fast with distance. To detect apossible presence of orientational order, we have computedthe quantity = �ne�3��, where the components of n are por-tions of plaquettes of various orientations and the compo-nents of e�3� are cubic roots of unity. That is, n= �nX ,nY ,nZ�,nX+nY +nZ=1 and e�3�= �1,ei2/3 ,ei4/3�. In the case of orien-tational order, �� would approach a finite value as size ofthe system increases. In our case, it scales to 0 instead.

More insight can be gained from the resonating cube cor-relation functions. In Fig. 5, we plot the function G2�r ,0�.Unlike the function G1�r ,0�, it clearly captures strong per-sistent correlations in the direction perpendicular to theplaquette as well. Our findings for G1, G2, and can becombined in a coherent picture in which the resonating cubesform a lattice, breaking translational symmetry, but the ori-entations of plaquettes on different cubes are not correlatedover long distances; thus, orientational symmetry is not bro-ken. This is illustrated in Fig. 6.

The presence of long range order suggested by Fig. 5 issupported by a scaling analysis. In Fig. 7, we plot differences�G2�0,L /2�=G2�0,L /2+1�−G2�0,L /2� and �G2�L /2 ,0�=G2�L /2+1,0�−G2�L /2 ,0� for L=2p. We see that �G2 ap-

proaches a constant already at p5–6. �For larger p, welose ergodicity, as we show in Appendix D, where we evalu-ate the relevant equilibration time � using a binning proce-dure.�

C. Origin of the ordered pattern

Our Monte Carlo simulations revealed a crystalline phasein the classical system of plaquette coverings of three-dimensional square lattice. The system orders in a cubic lat-tice pattern with the periodicity equal to two lattice constantsof the original lattice.

This ordering is tenuous, as evidenced by the small valueof the order parameter. In fact, when one judges a snapshotof the Monte Carlo simulations by eye �Fig. 9�, no orderingis evident.

One way to understand the appearance of the long rangeorder is to consider a more general model in whichnd-dimensional hypercubes are arranged on anns-dimensional lattice. For RSVP, nd=ns=3.

Let us start with ns=1, hypercubes arranged on a chain�Fig. 8�. One can then explicitly compute �see details in Ap-pendix A� the correlation length, which grows as �nd with

0 10 20 30r

0

0.05

0.1

0.15

0.2

G1(0

,r)

FIG. 3. Correlation function G1�0,r� for L=64.

0 10 20 30r

0

0.05

0.1

0.15

0.2

G1(r

,0)

FIG. 4. Correlation function G1�r ,0� for L=64.

0 10 20 30r

0

0.05

0.1

0.15

0.2

G2(r

,0)

FIG. 5. Pair correlation function G2�r ,0� for L=64.

FIG. 6. �Color online� Ordering pattern. Plaquettes form reso-nating cubes which are ordered on a cubic lattice with doubledlattice spacing. Different colors represent plaquettes of three differ-ent orientations. The plaquette orientations are uncorrelated overlarge distances.

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nd. This result can be visualized by noting that a resonatingcube can be broken up into two domain walls by fixing theplaquettes to be perpendicular to the chain direction. Thiscosts an entropy Sd=ln nd. The domain walls can then beseparated at will without disturbing other cubes. In a systemof length L, this leads to an entropy of Sw=ln� L

2�. The corre-

lation length is then determined by the smallest L at whichthe introduction of domain walls becomes favorable �Sw

�Sd�, which happens at L=��nd.While there is no true long range order in one dimension,

this result shows that the ordering tendency increases withnd. In this spirit, the case nd=ns=2, square dimers, beingcritical, is indeed already closer to ordering. Zeng and Hen-ley, for the honeycomb lattice, have developed a prescriptionvery similar to the introduction of a limit of large nd for ns=2, and they found numerically the existence of an orderingtransition for sufficiently large nd.35

Thence, whereas for ns=1, nd→ yields a critical point,ns=2 requires only a finite nd to produce ordering. Our re-sults add the insight that for ns=3, nd=ns is already sufficientto produce ordering.

Some further understanding can be gained by consideringhigher spatial dimensionality as chains of d=nd-dimensionalhypercubes, transversally coupled in d−1 transverse direc-tions. If a chain and its neighbor �in the ei direction, say� arein registry, a pair of neighboring resonating cubes can gainfurther free energy when their plaquettes are perpendicular toei. This happens in one out of d configurations for each cube,and the number of additional states of the two neighboringplaquettes belonging to different chains resonating togetheris d. This thus yields an attractive interaction of strengthproportional to 1/d.

For a d-dimensional interaction of strength J/d, the criti-cal coupling J would be O�1�. Our interaction, however, isamplified by the length of the domains, ��d, over which

individual chains are ordered even in the absence of trans-verse coupling. This means that for large d, the transversallycoupled chains will order. This establishes at least local sta-bility of the plaquette ordering for large d, to which our cased=3 �but not yet the case d=2� is connected.

Note that this ordering pattern does not imply that, even atlarge d, the plaquettes spend almost all their time resonatingwith their partner on one given cube. �Such cubes define the“site” of the ordering lattice depicted in Fig. 6�. Rather, theyspend half of their time resonating on the d bonds emanatingfrom sites of that lattice, as can be seen from an appropriateadaptation of the above argument.

Let p0 �p1� be the respective probabilities of finding acrystal site �bond� to be occupied by two parallel plaquettes.The probability of a resonating cube to appear on any givenbond is proportional to p0

2 /d2. It should be equal to the prob-ability for a resonating cube to leave a bond, which is pro-portional to p1 /d. In the large d limit, other processes are ofhigher order in 1/d. Indeed, one can evaluate probability pifor a resonating cube to appear in locations which are i Man-hattan steps away from crystal sites �Manhattan distance hereis measured in original lattice spacings, so a bond is oneManhattan step away from crystal sites�. The relation be-tween pi and pi+1 is similar to the relation between p0 and p1,that is, pi

2 /d2 pi+1 /d �as long as i�d�. Therefore, pi

1/d2i−1, while for the number ni of locations at Manhattandistance i, we have ni�di. It follows that only terms with i=0,1 will contribute in the d→ limit, so a simple relation-ship holds: p1d= �1− p0� /2. From there, we find p0=1/2 andp1=1/ �4d�.

Because a plaquette is as likely to be found resonating ona bond as it is on a cube, a random snapshot of the cubiclattice will not appear to be ordered, similar to a snapshot ofthe fcc lattice �Figs. 9 and 10 and Sec. III�. A proper analysisreveals, however, that plaquette correlations on the fcc latticeare short range, so the presence of ordering is not a feature ofresonating plaquettes in general, but should be attributed tothe lack of frustration on the cubic lattice.

3 4 5 6Log

2(L)

0

0.01

0.02

0.03

0.04

0.05

G2

∆

FIG. 7. The curves show scaling of G2 with the size.�G2�0,L /2� �circles� and �G2�L /2 ,0� �squares� are plotted vslog2�L�. Lines are guides for the eye.

n =1s

n =1d

n =1s

n =2d

FIG. 8. �Color online� nd dimensional “hypercubes” resonatingon an ns=1 dimensional chain.

FIG. 9. �Color online� Snapshot of a plaquette configuration ona cubic lattice.

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D. Phase diagram of the Rokhsar-Kivelson model

The above has established that the equal amplitude sum,which has zero energy and thus saturates a lower bound onthe ground state energy at the RK point, is crystalline. TheQPM Hamiltonian is not ergodic in the full Hilbert space—itmoves in topological sectors which we will discusselsewhere37—and thus will properly exhibit multiple groundstates. We can conclude from the simulation of the classicalensemble though that none of these ground states will beliquid.

The remainder of the phase diagram of the RK model iscompleted in the standard way. For v / t�2, any configura-tion without resonating cubes becomes a ground state. Theordered plaquette phase takes over in a first-order transitionat the RK point v / t=2. This is unlike the dimer model, inwhich there is a moderately exotic deconfined multicriticaltransition on the square lattice into a solid and into a Cou-lomb phase on the cubic lattice.6,8,16

For −v / t�1, one finds the columnar solid phase, which isunique �up to global symmetry operations� in this case.Whether there are intervening phases between columnar andplaquette solid has not been studied.

III. N=3 QUANTUM PLAQUETTE MODEL ON THEFACE-CENTERED CUBIC LATTICE

To see if the absence of disordered phases is a generalfeature of such plaquette models, we have also investigatedthe case of the face-centered cubic lattice. There, the elemen-tary plaquettes are equilateral triangles, the corners of whichare defined by a triplet of mutually nearest-neighbor sites.The QPM now includes resonance of a pair of plaquettes thesix vertices of which define an octahedron shown in Fig. 11.

In Fig. 12, we plot the plaquette correlations of the clas-sical model corresponding to the RK point for various in-equivalent combinations of pair orientations and separationsin the �011� direction. There are six inequivalent correlationfunctions for four types of plaquettes adjacent at an octahe-dron vertex. All correlators decay with distance to the value

given by uncorrelated plaquettes. This decay is a rapid expo-nential, with correlation lengths of less than a lattice spacing,ranging from 0.3 �diagonal component� to 0.7 �off-diagonal component� of nearest-neighbor distances.

This demonstrates that a classical hard-core plaquettemodel is in a strongly disordered phase and we would betempted to conclude, by standard arguments, that we haveidentified a point in an RSVP phase that sits at a first-ordertransition to a staggered crystal. The connection of the clas-sical sum to the RK point of the QPM, however, is morecomplicated than in the case of the cubic lattice. Theplaquette pair flip is very constrained and by itself breaks upthe Hilbert space into a very large number of largely frozenstates with local quantum fluctuations so that it is only thesum of these individual states that exhibits liquidity. It seemsvery likely that the inclusion of resonance dynamics onlarger clusters will remove this pathology and establish theRSVP state. However, at present we are unable to explicitlysolve this problem.

IV. CONCLUDING REMARKS

Can the above quantum plaquette models be realized inexperiment? Conceivably, they can arise in the case of low-

FIG. 10. �Color online� Snapshot of a plaquette configuration ona fcc lattice. Eight colors are used to represent the differentplaquettes, as defined by faces of an octahedron �see Fig. 11�.

FIG. 11. �Color online� Octahedron defining the plaquette typeson the fcc lattice.

0 1 2 3 4 5r

10-6

10-5

10-4

10-3

10-2

10-1

100

|G1(r

)-1/

24|

FIG. 12. �Color online� Connected part of the correlation func-tion G1�r� for various plaquette orientations on the fcc lattice. Thedistance r is measured in units of the nearest-neighbor distance.

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energy Ising degrees of freedom which reside on the mid-points of plaquettes of a lattice, and which obey the appro-priate hard-core constraint. Their simplest quantumdynamics would then be that of a QPM.

Regarding our original motivation of finding degrees offreedom with a native SU�N� symmetry, we first observe thatfollowing initial work on dimer ground state ensembles ofKlein models,36 there now exist prescriptions for local SU�2�invariant Hamiltonians which produce RK quantum dimermodels in a controlled fashion.17,18,38

However, no simple generalization of Klein models forSU�4� models with plaquette ground states is known yet, inpart for the reason that was pointed out in Ref. 36 for dimerson the square lattice. There, the projectors occurring in theHamiltonian allow ground states containing not only dimerson nearest-neighbor but also on next nearest-neighbor bonds.In an analogous manner, for SU�4� plaquette ground states,we are also led to other �e.g., nonplanar� units of four spinsforming a singlet. Actually realizing an SU�4� degree of free-dom is of course also no trivial matter in d�1, as combiningan orbital pseudospin 1/2 degree of freedom with an SU�2�to form an SU�4� invariant Hamiltonian requires a fine tun-ing of interaction parameters which is not immediately sug-gested by the natural symmetries of orbitals making up thepseudospin 1/2. Similarly, whether or not there exist latticemodels realizing an SU�3� QPM on the face-centered cubiclattice that exhibits an RSVP phase is left for further inves-tigation.

It is worth noting that SU�4� spins do not of coursepresent the exclusive entry point to plaquette degrees of free-dom. We have already referred to work on spin-3 /2 SU�2�systems that give rise to plaquettes.29,30 One can also con-sider that in the same sense in which dimers can be thoughtof as low-energy composite degrees of freedom consisting ofa pair of resonating SU�2� spins with S=1/2, a plaquettemight be considered as a composite of a pair of resonatingdimers. There is thus nothing fundamentally untoward aboutresonating valence plaquettes—this view being reinforced bythe findings of Ref. 26, who found plaquette ordering withthe same �12��12 supercell proposed for resonating dimerson the same lattice.39,40 Similarly, attractive interactions inthe simple cubic dimer model lead to an ordered phase, albeitone with columnar order,41 rather than the plaquette orderobserved here.

Finally, our work on the RK point also provides an entryto the study of classical hard plaquettes in d�2. In particu-lar, the topological and ergodicity properties of these modelsshould present an interesting topic for further research.

ACKNOWLEDGMENTS

We thank B. Douçot, D. Huse, W. Krauth, F. Mila, M.Mostovoy, K. Penc, and M. Troyer for useful discussions.We are very grateful to W. Krauth for collaboration onclosely related work. This work was supported in part by theMinistère de la Recherche with an ACI grant. Most of it wasundertaken at Laboratoire de Physique Théorique de l’EcoleNormale Supérieure, CNRS-UMR 8549.

APPENDIX A: PLAQUETTES ON THE CHAIN

We use the transfer matrix method to compute correlationfunction of d-dimensional plaquettes on a chain. Eachplaquette can assume d orientational states: one orthogonalto the direction of the chain and d−1 parallel. We write theHamiltonian as H=�iHi,i+1 and compute the transfer matrix

Ti���= ���exp�−�Hi,i+1�����, where ��� represents local de-

grees of freedom. Provided the Hamiltonian is translationallyinvariant, the correlation function C�x�= �MiMi+x� of someoperator M can be written as

C�x� =Tr MTxMTN−x

Tr TN , �A1�

where N is the length of the chain �periodic boundary con-ditions are assumed�. In the thermodynamic limit N→ , wehave

C�x� = ��

��0�M��������M��0����

�0 x

, �A2�

where ���� are eigenstates of the transfer matrix and �0 is itslargest �in absolute value� eigenvalue. The asymptotic formof the connected part of the correlation function is deter-

mined by the next largest eigenvalue � such that ��0�M�������M��0��0, so that the correlation length is given by �

=1/ ln��0 / ��. In our case, all states are degenerate, so thetransfer matrix is simply proportional to �2d−1�-dimensionalmatrix of the following form:

T = �1 A 0

0 0 E

B C 0� , �A3�

where E is a �d−1�� �d−1� identity matrix and all the ma-trix elements of A, B, and C are equal 1. The matrix C is a�d−1�� �d−1� square matrix, while A and B are 1� �d−1�and �d−1��1 matrices, respectively. It is not hard to findthat T has only two nonzero eigenvalues:

� = 1/2 ± �d − 3/4, �A4�

from where we find � /�0=−1+1/�d+O�1/d�. The first termencodes the fact that the connected part of the correlationfunction oscillates, while the second term determines the cor-relation length �=�d.

APPENDIX B: OVERLAP MATRIX ELEMENT

In the fundamental representation, four basis states �fla-vors� are used to represent an SU�4� spin. The invariance ofthe SU�4� singlet plaquette wave function under rotations inspin space dictates the singlet to be a fully antisymmetrizedcombination of all four flavors on four sites:

��s� =1

�24�

P�1234��− 1�p�1234� , �B1�

where the four flavors �1234� are permuted in all possibleways and p is the parity of permutation P. In each term, there

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is one flavor per site, that is, flavors are permuted amongdifferent sites. The prescription for fixing overall sign of thewave function will be given below. We consider lattice stateswhich are products of singlet wave functions correspondingto each plaquette. We need to compute the overlap matrixelement between two such states. It can be visualized withthe help of the transition graph—the graph which is made ofall partially overlapping plaquettes. Only those have to beincluded, as the fully overlapping plaquettes multiply thematrix element by 1. The total overlap matrix element is aproduct of overlap matrix elements corresponding to discon-nected parts of the transition graph; therefore, it is sufficientto consider only a connected transition graph.

Two plaquettes can be connected either via a commonedge or via a single common site. We call these strong andweak connections, respectively �see Fig. 13�. The commonedge in the strong connection is called connecting edge, andthe common sites in weak connections are called connectingsites. If it is possible to connect two plaquettes by a pathbelonging to the transition graph and going through strongconnections only, then two plaquettes are said to belong tothe same strongly connected part. Using the fact that eachsite in the transition graph is shared by exactly twoplaquettes, it is easy to check that either the strongly con-nected part has exactly four weak connections attached or ithas no weak connections and we call it a simple graph.

Each plaquette in the transition graph corresponds to acertain term �1234� of the singlet wave function. The overlapis nonzero only when sites shared by different plaquettescarry identical flavors in each of the two plaquettes. Thus, anonzero overlap is parametrized by assigning a flavor per sitein the transition graph, under the constraint that no two sitesin a plaquette carry the same flavor. This is always possibleto do by assigning flavors to all sites of the lattice in thefollowing way: assign 1 to the point �0, 0, 0�, 2 to the point�1, 0, 0�, 3 to the point �0, 1, 0�, 4 to the point �0, 0, 1�, andthen translate this configuration by �±1, ±1, ±1� to cover allsites of the lattice �see Fig. 14�. It is easy to check that anyplaquette on the lattice contains four different flavors.

It has to be decided how to choose signs for plaquettesinglet wave functions to ensure that the overlap matrix ele-

ments of the initial and final states of a resonance process arepositive. It is straightforward to check that it is enough torequire the term �1234� in each singlet wave function, whoseflavors coincide with our prescription, to be positive.

The assignment of flavors to the sites in the transitiongraph is not unique. This is most easily seen by making aconnection between this assignment and a loop model, whichin turn establishes the link to the four-coloring model men-tioned in the text. We partition the sites in the graph �for agiven allowed assignment of flavors� in the following way.Connect sites with flavors 1 and 2 which belong to the sameplaquette. Do the same with the sites carrying flavors 3 and4. In this way, all sites will be divided into noncrossingloops. Notice that in each loop, two flavors can be exchangedwithout violating the constraint. The configurations of flavorswhich can be connected by a sequence of such transforma-tions are said to belong to the same loop decomposition.42

The number of such configurations is 2nl, where nl is thenumber of loops in the decomposition. Some loops containweak connections, while others do not—we call them nonlo-cal and local loops, respectively. From the flavor constraint�any plaquette includes all four flavors�, it follows that anonlocal loop goes through at least two strongly connectedparts and has at least length l=4, while a local loop belongsto a single strongly connected part and has length l=2. Ob-viously, nl=nll+nnl, where nll and nnl are the numbers oflocal and nonlocal loops, correspondingly. Taking into con-sideration that nnl�ns and nll

i � li−1, where nlli and li are the

number of local loops and the number of plaquettes in the ithstrongly connected part �the case ns=1 is not included here�,we can write the following inequality for the contribution Sto the overlap matrix element from a given loop decomposi-tion:

1

6S = � 1

24 l/2

2nl � 6−l/2. �B2�

The prefactor of 1 /6 accounts for various ways of distrib-uting four flavors among two classes of loops. Notice that fora simple graph, nll= l and nnl=0�1=ns, so the above resultstill holds in that case, with the inequality turning into anequality.

In the next step, we derive a bound on the number of loopdecompositions. For the sake of compactness, we use terms

Strong connections

Weak connections

FIG. 13. �Color online� Transition graph generated by two con-figurations �denoted light and dark� of three plaquettes. A strongconnection involves two plaquettes sharing an edge, whereas aweak connection is formed by two plaquettes coinciding at a singlesite only.

4

2

4

41

1

3

1

3

12

1

3

2

4

FIG. 14. Assignment of flavors which respects the constraint ofno two sites in a plaquette carrying the same flavor.

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“vertex” and “connection” in place of “strongly connectedpart” and “weak connection” below. Four connections be-longing to the same vertex are joined by nonlocal loops. Thiscan be done in three different ways. Therefore, there are atmost 3ns loop decompositions. Clearly, not all decomposi-tions are allowed, because for two nonlocal loops goingthrough the same vertex �there are at most two such loops� achoice of flavors of one of the loops uniquely determinesflavors of the second loop. This restriction can be formulatedas a problem of coloring loops with just two colors, let us sayblack and white. We start with no nonlocal loops defined,that is, all connections disjoined and not colored. We will bejoining four connections �belonging to the same vertex� at atime, and color them according to the way they are joined.Without loss of generality, we choose to do the joining pro-cedure on those vertices which already have colored connec-tions.

In this way, colors are always defined uniquely �apartfrom the first step�. It is easy to check that there are at mostthree ways to join connections when none or one of them hasbeen colored. There are at most two ways of joining connec-tions when two of them have been colored. There is at mostone way of joining when three or all connections have beencolored. Thus, the upper bound on the number M of loopdecomposition can be written as

M = max�2ns�3

2 n0+n1�1

2 n3+n4� , �B3�

with the following constraints imposed:

�i=0

i=4

ni = ns,

�i=0

i=4

ini = 2ns, �B4�

where ni is the number of vertices which undergo the joiningprocedure having i colored connections, and thus is positive.From Eq. �B4�, we have 2n0+n1=n3+2n4, so the maximumin Eq. �B3� is achieved when n0 and n3 are minimized whilen1 and n4 are maximized. In the best case, n1 /2=n4=ns /3

and M�ns�=32ns/3. Since ns� l, the bound for S equals the

bound for S multiplied by M�l�:

S � 6�31/3

2 l/2

. �B5�

APPENDIX C: MATRIX ELEMENT COMPUTATIONS FORSU(4) QUANTUM PLAQUETTE MODEL

We use ��� notations for the orthonormal states and ���for the original states. We only need to consider states whichdiffer by orientation of a single pair of plaquettes; we thusname the states according to the orientation of the pair in theYZ, ZX, and XY planes as �X�, �Y�, and �Z�. We first computematrix elements ���H���. It is easy to check that

Hij���ijkl�� = − ���ijkl�� , �C1�

where Hij =�mnSmn �i�Sn

m�j� is the part of the Hamiltonian act-ing on the bond �ij�. The location of the plaquette is explic-itly shown in ���ijkl�� and Sm

n are SU�4� generators.22 In thefermionic representation, Sm

n =am† an, so Sm

n ���=�n��m� where� is the flavor. The term Sm

n �i�Snm�j� simply exchanges flavors

m and n on sites i and j from where Eq. �C1� follows. Wenote that only Hij with i and j belonging to any of the twoplaquettes need to be considered. When ���, there is al-ways a plaquette containing both i and j; therefore, for anybond ���Hij���=−x and ���H���=−12x because 12 bonds be-long to the pair of plaquettes. When �=�, only eight bondsbelong to an individual plaquette, and these bonds give���Hij���=−1. For a bond shared by the plaquettes, one canestimate ���Hij���=������� ����2���Hij���=−2x2. So, wehave ���H���=−8−8x2.

Let us compute, for example, �X�H�X� and �X�H�Y�. To the

lowest order, �X�= �X�− x2 ��Y�+ �Z�� and �Y�= �Y�− x

2 ��X�+ �Z��.Then, �X�H�X�= �X�H �X�− x

2 ��X �H �Y�+ �X�H�Z�+H.c.�+ ¯ =−8+16x2+O�x3�. The constant term −8 is actually indepen-dent of plaquette configuration and can thus be dropped. For

the off-diagonal element, we find �X�H�Y�= �X�H�Y�− x

2 ��X�H�X�+ �Y�H�Y��+ ¯ =−5x+O�x2�. Thus, we obtainfor the kinetic and potential terms in the RK Hamiltonian t=5x and v=16x2.

APPENDIX D: DETERMINATION OF THEEQUILIBRATION TIME

The error of statistically independent measurement de-creases with the number of measurements n as n−1/2. If, how-ever, the measurements are not independent, the error cannotbe reduced beyond certain limit, as more measurements donot add any new information. This allows us to estimate

TABLE I. Equilibration time.

L 8 16 32 64

� 24=16 27=128 211=2048 �215=32768

0 5 10 15Log

2( )

23

24

25

26

27

28

29

σ(τ)

τ

FIG. 15. The standard deviation ���� of �G�0,L /2��� plotted vslog2 � for L=48.

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correlation time corresponding to some physical quantity Oby measuring fluctuation of the average value, �O�� com-puted over the time span �. More precisely, we measure �O��

over m successive intervals of time of length �. From maverage values we compute the standard deviation ���� ofthe distribution of �O��. In the same run, we can measure�O��/2 in 2m time intervals by subdividing each � interval intwo. The corresponding standard deviation ��� /2� is thuscomputed from 2m measurements. The procedure of subdi-vision �of creating smaller bins, hence the term binning� canbe continued as far as needed. By plotting ���� vs log2 �, we

estimate the equilibration time �eq as the time where ����stops changing as � is lowered. Alternatively, one could keepthe number m of intervals fixed, while decreasing the numberof measurements in each interval, by doubling the timingbetween successive measurements �. Again, the standard de-viation ���� would depend on � only for ���eq. This isshown in Fig. 15, where we plotted ���� for L=48, for thecorrelation function G�0,L /2�. The equilibration time can beestimated to be �eq214. The equilibration times estimatedin this way for other sizes are summarized in Table I.

1 P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 �1974�.2 P. W. Anderson, Science 235, 1196 �1987�.3 S. Sachdev, in Quantum Magnetism, edited by U. Schollwock, J.

Richter, D. J. J. Farnell, and R. A. Bishop �Springer, Berlin,2004�.

4 G. Misguich and C. Lhuillier, in Frustrated Spin Systems, editedby H. T. Diep �World Scientific, Singapore, 2005�.

5 D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376�1988�.

6 R. Moessner and S. L. Sondhi, Phys. Rev. Lett. 86, 1881 �2001�.7 G. Misguich, D. Serban, and V. Pasquier, Phys. Rev. Lett. 89,

137202 �2002�.8 E. Fradkin, D. A. Huse, R. Moessner, V. Oganesyan, and S. L.

Sondhi, Phys. Rev. B 69, 224415 �2004�; A. Vishwanath, L.Balents, and T. Senthil, ibid. 69, 224416 �2004�.

9 D. A. Huse, W. Krauth, R. Moessner, and S. L. Sondhi, Phys.Rev. Lett. 91, 167004 �2003�.

10 M. Hermele, M. P. A. Fisher, and L. Balents, Phys. Rev. B 69,064404 �2004�.

11 C. L. Henley, Phys. Rev. B 71, 014424 �2005�.12 O. I. Motrunich and T. Senthil, Phys. Rev. B 71, 125102 �2005�.13 C. Xu, Phys. Rev. B 74, 224433 �2006�.14 A. Seidel, H. Fu, D.-H. Lee, J. M. Leinaas, and J. Moore, Phys.

Rev. Lett. 95, 266405 �2005�.15 E. Fradkin and S. A. Kivelson, Mod. Phys. Lett. B 4, 225 �1990�.16 R. Moessner, S. L. Sondhi, and Eduardo Fradkin, Phys. Rev. B

65, 024504 �2001�.17 K. S. Raman, R. Moessner, and S. L. Sondhi, Phys. Rev. B 72,

064413 �2005�.18 S. Fujimoto, Phys. Rev. B 72, 024429 �2005�.19 X.-G. Wen, Quantum Field Theory of Many-body Systems �Ox-

ford University Press, Oxford, 2004�.20 R. Savit, Rev. Mod. Phys. 52, 453 �1980�.21 W. Krauth and R. Moessner, Phys. Rev. B 67, 064503 �2003�.22 Y. Q. Li, M. Ma, D. N. Shi, and F. C. Zhang, Phys. Rev. Lett. 81,

3527 �1998�.

23 Y. Yamashita, N. Shibata, and K. Ueda, Phys. Rev. B 58, 9114�1998�.

24 M. van den Bossche, P. Azaria, P. Lecheminant, and F. Mila,Phys. Rev. Lett. 86, 4124 �2001�.

25 T. Hikihara, T. Momoi, and X. Hu, Phys. Rev. Lett. 90, 087204�2003�.

26 K. Penc, M. Mambrini, P. Fazekas, and F. Mila, Phys. Rev. B 68,012408 �2003�.

27 F. F. Assaad, Phys. Rev. B 71, 075103 �2005�.28 A. Paramekanti and J. B. Marston, J. Phys.: Condens. Matter 19,

125215 �2007�.29 C. Wu, Mod. Phys. Lett. B 20, 1707 �2006�.30 S. Chen, C. Wu, S.-C. Zhang, and Y. Wang, Phys. Rev. B 72,

214428 �2005�.31 N. Read and S. Sachdev, Nucl. Phys. B 316, 609 �1989�.32 B. Sutherland, Phys. Rev. B 37, 3786 �1988�.33 J. L. Jacobsen and J. Kondev, Nucl. Phys. B 532, 635 �1998�.34 C. Dress and W. Krauth, J. Phys. A 28, L597 �1995�.35 C. Zeng and C. L. Henley, Phys. Rev. B 55, 14935 �1997�.36 J. Chayes, L. Chayes, and S. A. Kivelson, Commun. Math. Phys.

123, 53 �1989�.37 S. Pankov et al. �unpublished�.38 Z. Nussinov, arXiv:cond-mat/0606075 �unpublished�; Z. Nussi-

nov and G. Ortiz, arXiv:cond-mat/0702377 �unpublished�.39 R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 �2001�.40 R. Moessner, S. L. Sondhi, and P. Chandra, Phys. Rev. Lett. 84,

4457 �2000�.41 F. Alet, G. Misguich, V. Pasquier, R. Moessner, and J. L. Jacob-

sen, Phys. Rev. Lett. 97, 030403 �2006�.42 Any two-color configurations connected by a sequence of such

loop moves contribute to the overlap with the same sign. It may,however, be the case that some topological sectors of the four-coloring model cannot be connected by such loop moves. Thiswould, if anything, suppress the overlap for larger transitiongraph elements.

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