Order and disorder in SU(N) simplex solid antiferromagnets

Order and disorder in SU(N) simplex solid antiferromagnets

Yury Yu. Kiselev,1 S. A. Parameswaran,2 and Daniel P. Arovas1

1Department of Physics, University of California at San Diego, La Jolla CA 92093, USA2Department of Physics and Astronomy, University of California, Irvine CA 92697, USA

(Dated: October 27, 2014)

We study the structure of quantum ground states of simplex solid models, which are generalizations of thevalence bond construction for quantum antiferromagnets originally proposed by Affleck, Kennedy, Lieb, andTasaki (AKLT) [Phys. Rev. Lett. 59, 799 (1987)]. Whereas the AKLT states are created by application of bondsinglet operators for SU(2) spins, the simplex solid construction is based on N -simplex singlet operators forSU(N) spins. In both cases, a discrete one-parameter family of translationally-invariant models with exactlysolvable ground states is defined on any regular lattice, and the equal time ground state correlations are givenby the finite temperature correlations of an associated classical model on the same lattice, owing to the productform of the wave functions when expressed in a CPN−1 coherent state representation. We study these classicalcompanion models via a mix of Monte Carlo simulations, mean-field arguments, and low-temperature effectivefield theories. Our analysis reveals that the ground states of SU(4) edge- and SU(8) face-sharing cubic latticesimplex solid models are long range ordered for sufficiently large values of the discrete parameter, whereas theground states of the SU(3) models on the kagome (2D) and hyperkagome (3D) lattices are always quantumdisordered. The kagome simplex solid exhibits strong local order absent in its three-dimensional hyperkagomecounterpart, a contrast that we rationalize with arguments similar to those leading to ‘order by disorder’.

I. INTRODUCTION

The study of quantum magnetism is an enduring theme incondensed matter physics, particularly in the search for newphases of matter. Much of our understanding of classical andquantum order and associated critical phenomena has beenbolstered by studies of magnetism in diverse situations. In thiscontext, a special role is played by so-called quantum para-magnets: gapped zero-temperature ground states that retainall symmetries of the high-temperature phase without invok-ing the nonlocal entanglement structure responsible for ex-otica such as topological order and the attendant fractional-ization of quantum numbers. The existence, or lack thereof,of such featureless T = 0 quantum phases constrains poten-tial proximate phases and associated transitions: for instance,the absence of such a quantum paramagnet requires that zero-temperature continuous transitions between phases of differ-ent broken symmetries fall into a different framework fromthe conventional Landau-Ginzburg-Wilson paradigm for suchtransitions1. Besides their intrinsically interesting properties,owing to their lack of broken symmetry such quantum param-agnets can be doppelgangers for topologically ordered phases,complicating the identification of the latter in numerical stud-ies and in experiments. This gives an important subsidiarymotivation to our goal in this paper: to deepen the understand-ing of such featureless phases.

In a landmark paper, Affleck, Kennedy, Lieb and Tasaki(AKLT) described a construction2 of valence bond solid(VBS) states that provides a useful paradigm for low-dimensional quantum paramagnets in which both spin and lat-tice point group symmetries remain unbroken. As noted morerecently by Yao and Kivelson3, the AKLT states are also ex-amples of ‘fragile Mott insulators’, that cannot be adiabati-cally connected to a band insulator while preserving certainpoint-group symmetries. For each lattice L there is a fam-ily of AKLT states indexed by a positive integer M , con-structed as follows: First, place Mr spin- 1

2 objects on each

site, where r is the lattice coordination number. Next, contractthe SU(2) indices by forming M singlet bonds on each link ofthe lattice. Finally, symmetrize over all the SU(2) indices oneach site. This last step projects each site spin into the to-tally symmetric S = 1

2Mr representation. The general state|Ψ(L,M)〉 is conveniently represented using the Schwingerboson construction4, where S = b†µ σµν bν and the total bo-son number on each site is b†↑b↑ + b†↓b↓ = 2S :∣∣Ψ(L,M)

⟩=

∏〈ij〉∈L

(b†i↑ b

†j↓ − b

†i↓ b†j↑)M ∣∣ 0 ⟩ . (1)

Since the bond operator φ†ij = εµν b†iµ b†jν transforms as an

SU(2) singlet, M of the bosons at site i are fully entangled ina singlet state with M bosons on site j, so that the maximumvalue of the total spin Jij is 2S −M , and thus |Ψ(L,M)〉 isan exact zero energy ground state for any Hamiltonian of theform H =

∑〈ij〉∑2SJ=2S−M+1 VJ PJ(ij), where VJ > 0 are

pseudopotentials and PJ(ij) is the projector onto total spin Jfor the link (ij).

Many properties of |Ψ(L,M)〉 may be gleanedfrom its coherent state representation4, ΨL,M [z] =∏〈ij〉∈L

(εµν ziµ zjν

)M, where for each site i, zi is a

rank-2 spinor with z†i zi = 1 and zi ≡ eiαzi, i.e. an elementof the complex projective space CP1 ∼= S2. In particular, onehas∣∣ΨL,M [z]

∣∣2 = e−Hcl/T , where

Hcl = −∑〈ij〉∈L

ln

(1− ni · nj

2

), (2)

with ni = z†i σ zi ∈ S2 is a unit vector, is the Hamilto-nian for a classical O(3) antiferromagnet on the same latticeL, and T = 1/M is a fictitious temperature. This is analo-gous to Laughlin’s ‘plasma analogy’ for the fractional quan-tum Hall effect, and we may similarly use well-known results

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in classical statistical mechanics to deduce properties of thestate described by |Ψ(L,M)〉. Specifically, we may invokethe Hohenberg-Mermin-Wagner theorem to conclude that allAKLT states in dimensions d ≤ 2 lack long-range magneticorder since they correspond to a classical O(3) system at fi-nite temperature on the same lattice5. For d > 2, a mean-fieldanalysis2,6 suggests that the AKLT states on bipartite latticespossess long-ranged two sublattice antiferromagnetic orderfor T < TMF

c = 13r, i.e. M > MMF

c = 3r−1. Since the mini-mum possible value for M is M = 1, the mean field analysissuggests that all such d = 3 models, where r > 3, are Neel or-dered. However, mean field theory famously fails to accountfor fluctuation effects which drive Tc lower – henceMc higher– for instance, ref. 6 found, using classical Monte Carlo sim-ulations of the corresponding classical O(3) model, that theS = 2 (i.e.M = 1) AKLT state on the diamond lattice (r = 4)is quantum-disordered. Ref. 6 also showed that AKLT stateson the frustrated pyrochlore lattice were quantum-disorderedfor S ≤ 15 (at least). A subsequent extension of the AKLTmodel to locally tree-like graphs – often used to model disor-dered systems – found AKLT states that exhibit not only long-range order and quantum disorder, but also those that showedspin glass-like order for large values of the singlet parameterand/or the local tree coordination number7.

Upon enlarging the symmetry group of each spin toSU(N), there are two commonly invoked routes to singletground states. The first is to work exclusively with bipartitelattices, and choose the spins on one sublattice to transformaccording to the (N -dimensional) fundamental representationof SU(N), while those on the other transform according tothe (N -dimensional) conjugate representation. One then hasN ⊗ N = • ⊕ adj , where • denotes the singlet and adjthe (N2 − 1)-dimensional adjoint representation. Proceed-ing thusly, one can develop a systematic large-N expansion8,9.Note, however, that on bipartite lattices in which the two sub-lattices are equivalent, most assignments of bond singlets ex-plicitly break either translational or point-group symmetries.(The exceptions typically involve fractionalization, and hencealso do not satisfy our desiderata for a featureless quantumparamagnet.)

The second approach, and our exclusive focus in the re-mainder, is to retain the same representation of SU(N) oneach site, but to create singlets which extend over a group ofN sites. (Readers may recognize a family resemblance withthe three-quark SU(3) color singlet familiar from quantumchromodynamics.) In this paper, we shall explore the orderedand disordered phases in a class of wave functions which gen-eralize the AKLT valence bond construction from SU(2) toSU(N), and from singlets on bonds to those over simplices.The construction and analysis of these “simplex solids”10 par-allels what we know about the AKLT states. If Γ denotes anN site simplex (henceforth an N -simplex) whose sites are la-beled i1, . . . , iN, then the operator

φ†Γ = εα1···αN b†i1α1· · · b†iNαN

(3)

where b†iα creates a Schwinger boson of flavor α on site i,transforms as an SU(N) singlet. Generalizing the product

over links in the AKLT construction to a product over N -simplices, one arrives at the simplex solid state10,∣∣Ψ(L,M)

⟩=∏Γ∈L

(φ†Γ)M ∣∣ 0 ⟩ . (4)

The resulting local representation of SU(N) is the symmetricone described by a Young table with one row and p = Mζboxes, where ζ is the number of simplices to which each siteonL belongs, a generalization of the lattice coordination num-ber r in the case N = 2. Projection operator Hamiltonianswhich render the simplex solid (SS) states exact zero energyground states were discussed in Ref. 10. Written in terms oftheN -flavors of Schwinger bosons, the SU(N) spin operatorstake the form Sαβ = b†α bβ − p

N δαβ , and satisfy the commu-tation relations

[Sαβ , Sµν

]= δβµ Sαν − δαν Sβµ. As in the

AKLT case, while the wave functions (4) are certainly exactground states of local parent Hamiltonians, it is imperative toverify that they do in fact describe featureless paramagnets.In addressing this question, it is once again convenient to em-ploy a coherent-state representation (suitably generalized toSU(N)) so that the answer can be inferred from analysis ofa finite-temperature classical statistical mechanics problem.Using this mapping, described in detail below, in conjunctionwith the Hohenberg-Mermin-Wagner theorem, we find thatalthough wave functions of the form (4) preserve all sym-metries in one dimension, once again we must entertain thepossibility that they exhibit lattice symmetry-breaking but notmagnetic order in d = 2, and that both lattice and spin sym-metries are spontaneously broken in d = 3.

In d = 2, we consider the SU(3) simplex solid on thekagome lattice, and using a saddle-point free energy estimateand Monte Carlo simulations of the classical model, we showthat it remains quantum-disordered for all M , although thereis substantial local sublattice order, corresponding to the so-called “

√3 ×√

3” structure, for large M (low effective tem-perature). We then turn to d = 3, where we first consider theSU(3) simplex solid on the hyperkagome lattice of corner-shgaring triangles. Here we find no discernible structure forany M , leading us to conclude that all these simplex solidstates are quantum-disordered. We also consider two differentsimplex solids on the cubic lattice: the SU(4) model with sin-glets on square plaquettes (that share edges), and the SU(8)version with singlets over cubes (that share faces). While theformer exhibits long-range order for all M (in other words,the classical companion model has a continuous transition atTc > 1)

Before proceeding, we briefly comment on related work.Other generalized Heisenberg models have been discussed ina variety of contexts. Affleck et al.11 investigated extendedvalence bond solid models with exact ground states whichbreak charge conjugation (C) and lattice translation (t) sym-metries, but preserve the product Ct. Their construction uti-lized SU(2N ) spins on each lattice site, with N = Mr aninteger multiple of the lattice coordination number r, withsinglet operators extending over r + 1 sites. Greiter andRachel12 constructed SU(N) VBS chains in the fundamentaland other representations. Shen13 and Nussinov and Ortiz14

3

developed models with resonating Kekule ground states de-scribed by products of local SU(N) singlets. Plaquette groundstates on two-leg ladders were also discussed by Chen et al.15.VBS states are perhaps the simplest example of matrix prod-uct and tensor network constructions16–19, and recently theprojected entangled pair state (PEPS) construction was ex-tended by Xie et al. to one involving projected entangledsimplices20. We also note that a different generalization tothe group Sp(N ) permits the development of a large-N ex-pansion for doped and frustrated lattices21. Perhaps more rel-evant to our discussion here, Corboz et al.22 studied SU(3)and SU(4) Heisenberg models on the kagome and checker-board lattices using the infinite-system generalization of PEPS(iPEPS), concluding that the Hamiltonian at the Heisenbergpoint exhibits q = 0 point-group symmetry-breaking23. Al-though their work left open the question of its adiabatic con-tinuity to the exactly solvable point of Ref. 10, this followsimmediately, as the order they discuss is inescapable for asimplex solid where the on-site spins are (as in their work)in the fundamental representation of SU(N).

In addition, there are several other examples of feature-less quantum paramagnets in the literature, with more gen-eral symmetry groups. Besides the aforementioned workby Yao and Kivelson, fragile Mott insulating phases havebeen recently examined as possible ground states of aromaticmolecules in organic chemistry24. Quantum paramagneticanalogs of the fragile Mott insulator for bosonic systems en-dowed with a U(1) symmetry have also been explored, includ-ing those with a very similar ‘plasma mapping’ to a classicalcompanion model25,26. Finally, recent work (involving two ofthe present authors) has identified situations when featurelessquantum paramagnets are incompatible with crystalline sym-metries and U(1) charge conservation27.

II. CLASSICAL MODEL AND MEAN FIELD THEORY

We first briefly review some results of Ref. 10. Using theSU(N) coherent states |z〉 = 1√

p!

(zαb†α

)p|0〉, we may again,as with the VBS states, express equal time ground state corre-lations in the simplex solids in terms of thermal correlationsof an associated classical model on the same lattice. One finds∣∣ΨL,M [z]

∣∣2 = e−Hcl/T , with

Hcl = −∑Γ

ln |RΓ |2 , (5)

where

RΓ = εα1···αN zi1α1· · · ziNαN

, (6)

where i1, . . . , iN label the N sites of the simplex Γ . Thetemperature is again T = 1/M . Note that the quantity |RΓ |has the interpretation of a volume spanned by the CPN−1 vec-tors sitting on the vertices of Γ .

To derive a mean field theory, assume that L is N -partiteand is partitioned into N sublattices. For each site i letσ(i) ∈ 1, . . . , N denote the sublattice to which i belongs.

Let ωσ denote a set of N mutually orthogonal CPN−1 vec-tors. Setting zi = ωσ(i) defines a fully ordered state which wewill refer to as a Potts state, since it is also a ground state fora (discrete) N -state Potts antiferromagnet. In any Potts state,|RΓ | = 1 for every simplex Γ , hence the ground state energyis E0 = 0.

Next define a real scalar order parameter m, akin to thestaggered magnetization in an antiferromagnet, such that

〈Qαβ(i)〉 = m

(Pσ(i)αβ −

1

Nδαβ

), (7)

whereQαβ(i) = z∗iα ziβ− 1N δαβ is a locally defined traceless

symmetric tensor, and where Pσ = |ωσ〉〈ωσ | is the projectoronto ωσ . The system is isotropic whenm = 0, whilem = 1 inthe Potts state. One finds that the mean field critical value forM = 1/T is MMF

c = (N2 − 1)/ζ. Note that for N = 2 andζ = r we recover the mean field results for the VBS states, i.e.MMF

c = 3/r. Thus, mean field considerations lead us to ex-pect more possibilities for quantum disordered simplex solidsthan for the valence bond solids in dimensions d > 2, wherealmost all the VBS states are expected to have two sublatticeNeel order on bipartite lattices. One remarkable feature of theSS mean field theory is that it apparently underestimates thecritical temperature in models where a phase transition occurs,thus overestimating Mc.

Expanding about the fully ordered state, writing

zi =(1− π†iπi

)1/2ωσ(i) + πi , (8)

where ω†σ(i)πi = 0, the low-temperature classical Hamilto-nian is

HLT =∑Γ

N∑i<j

∣∣π†Γi

ωσ(Γj)

+ ω†σ(Γi)

πΓj

∣∣2 +O(π3) . (9)

The field πi has (N−1) independent complex components. Ifg(ε) is the classical density of states per site, normalized such

that∞∫0

dε g(ε) = 1, then

〈π†iπi〉 = (N − 1)T

∞∫0

dε

εg(ε) . (10)

Another expression estimating Tc is obtained by setting〈π†iπi〉 = 1, beyond which point the fixed length constraintz†i z1 = 1 is violated, i.e. the low temperature fluctuations ofthe π field are too large. In contrast to the mean field expres-sion for the critical temperature, Tc = ζ/(N2 − 1), valueof Tc as determined from this criterion depends on the na-ture of the putative ordered phase, and moreover it vanishes if∞∫0

dε ε−1 g(ε) diverges.

A. Counting degrees of freedom

For our models, which are invariant under global U(N) ro-tations, each site hosts a CPN−1 vector, with 2(N − 1) real

4

degrees of freedom (DOF). Thus, per N -simplex, there are2N(N − 1) degrees of freedom. The group U(N) has N2

generators, N of which are diagonal. These diagonal gen-erators act on the spins by multiplying each of the ωσ by aphase, which has no consequence in CPN−1. Therefore thereare only N(N − 1) independent generators to account for.Subtracting this number from the number of DOF per sim-plex, we conclude that, in a Potts state, each simplex satisfiesN(N − 1) constraints. If our lattice consists of K corner-sharing simplices, then there are KN(N − 1) total (real) de-grees of freedom: 2N(N −1) DOF per simplex times K sim-plices, and multiplied by 1

2 since each site is shared by twosimplices. There are an equal number of constraints. Thus,the naıve Maxwellian dimension of the ground state manifoldis DM = 0. However, as we shall see below, we really haveD ≥ 0, and in some situations, such as for the kagome andhyperkagome models discussed below, D > 0. If the numberof zero modes is subextensive, the T = 0 heat capacity persite should be C(0) = N − 1 by equipartition.

III. MONTE CARLO SIMULATIONS

We simulate the classical companion model via MonteCarlo simulations using a single-spin flip Metropolis algo-rithm. As mentioned above, our primary interest is in deter-mining the phase diagram of the classical model as a functionof the temperature, as this will tell us how the quantum systemdepends on the discrete parameter M = 1/T (recall that thisdetermines the on-site representation of SU(N) by fixing thenumber of boxes in the Young diagram in a fully symmetricrepresentation of SU(N)). The classical degrees of freedom,obtained via the coherent-state mapping, are CPN−1 spins; inour simulations, each CPN−1 spin is represented by an N -dimensional complex unit vector ~z. The remaining U(1) localambiguity is harmless.

Local updates are made by generating an isotropic δ~zwhose length is distributed according to a Gaussian. The localspin vector is updated to

~z ′ =~z + δ~z

|~z + δ~z | . (11)

The standard deviation of the Gaussian distribution is adjustedso that a significant fraction (∼ 30%) of proposed moves areaccepted.

In order to obtain independent samples, we simulatedNchainindependent Markov chains, typically of a length of ∼ 105 −106 Monte Carlo steps per site (MCS). Each chain was initial-ized with random initial conditions and evolved until the totalenergy was well-equilibrated, and the initial portions of thechain before this were discarded. For each chain, we obtainedthe average values of the various quantities and averaged thisacross chains to get a single number for each temperature. Weestimated the error from the standard deviation of the Nchainindependent thread averages. This is free of the usual compli-cations of correlated samples inherent in estimating the errorfrom a single chain, and it frees us of the need to compute

autocorrelation times to weight our error estimate. Note thatin the lowest-temperature samples, we used a relatively mod-est number of independent chains Nchain . 10, but this wasalready sufficient to obtain reasonably small error bars.

We analyze two main observables. The first is the heat ca-pacity C = var(Hcl)/T , proportional to the square of theRMS energy fluctuations. The second is a generalized struc-ture factor, which is built from an appropriate tensor orderparameter,

Qαβ(i) = z∗i,α zi,β −1

Nδαβ . (12)

Note that ~zi itself cannot be used as an order parameter, be-cause its overall phase is ambiguous. This ambiguity is elimi-nated in the definition of Qαβ(i), which is similar to the orderparameter of a nematic phase. This tensor has the followingproperties:

• Tr Q = 0

• 〈Q 〉 → 0 as T →∞ at all sites

• Tr (Q2) = N−1N

• Tr (QQ′) = − 1N if z†z′ = 0 .

Thus, in any Potts state, Tr Q(i)Q(j) = − 1N for any nearest

neighbor pair (ij). A more detailed measure of order is af-forded by the generalized structure factor, which is given bythe Hermitian matrix

Sij(k) =1

Ω

∑R,R′

eik(R−R′) Tr[Q(R, i)Q(R′, j)

], (13)

where R is a Bravais lattice site, Ω is the total number ofthe unit cells, and i and j are sublattice indices. The rank ofSij(k) is the number of basis vectors in the lattice.

We performed two main tests of the Monte Carlo code.The first (standard) test was to reproduce well-known re-sults: specifically, we recovered the critical temperature Tc '0.69 of the classical cubic lattice O(3) Heisenberg model28.Our second concern is more unusual: namely, whether theMetropolis algorithm is sufficiently ergodic to generate aphase transition for a classical system governed by the un-usual interaction relevant to simplex solid models: for in-stance, for a three-site simplex (ijk) we have the interactionuijk = −2 lnVijk, where Vijk = |εµνλzi,µ zj,ν zk,λ|, is theinternal volume of the triple (ijk). In order to ensure that theabsence of a transition on a more complicated lattice is notsimply an artefact of our simulations, it is important to verifythat such an interaction can indeed lead to a phase transition ina simple model system. To that end, we investigated a simpleSU(3)-invariant model on a simple cubic lattice, with

H = −2∑R

3∑µ=1

lnV (R− eµ,R,R+ eµ) . (14)

As this is an unfrustrated lattice, with a finite set of broken-symmetry global energy minima (up to global SU(3) ro-tations) and in three dimensions where fluctuation effects

5

Figure 1. The simplest ground state for the kagome structure. A, B,and C represent a set of mutually orthogonal CP2 vectors. The redStar of David unit is used to analyze local zero modes.

should not destabilize order, it is reasonable to expect a finite-temperature transition in this model. Indeed, we find a tran-sition at T ' 1.25 or M ' 0.8, visible in both heat capacityand structure factor calculations. Armed with this reassuringresult, we now turn our attention to several specific examplesin two and three dimensions.

IV. SU(3) SIMPLEX SOLID ON THE KAGOME LATTICE

As our first example, we consider the SU(3) model on thekagome lattice. The elementary simplices of this lattice aretriangles, and Hcl describes a classical model of CP2 spinswith three-body interactions, viz.

Hcl = −∑Γ

ln∣∣εα1α2α3 zΓ1,α1

zΓ2,α2zΓ3,α3

∣∣2 , (15)

where Γi are the vertices of the elementary triangle Γ . Thestructure factor Sij(k) is then a 3× 3 matrix-valued functionof k.

In any ground state, each triangle is fully satisfied, with|RΓ | = 1. One such ground state is the so-called q = 0structure, which is a Potts state with

ωA =

100

, ωB =

010

, ωC =

001

assigned to each of the three sublattices of the tripartitekagome structure. The structure factor is given by

Sij(k) =(δij − 1

3

)· Ω δk,0 . (16)

Another Potts ground state is the√

3 ×√

3 structure, de-picted in Fig. 2, which has a nine site unit cell consisting ofthree elementary triangles. The structure factor is then

Sij(k) =Ω

3

1 ω2 ωω 1 ω2

ω2 ω 1

δk,K +Ω

3

1 ω ω2

ω2 1 ωω ω2 1

δk,K′

(17)

Figure 2. The√

3 ×√

3 kagome ground state supports an extensivenumber of zero-energy fluctuation modes. A, B, and C represent aset of mutually orthogonal CP2 vectors.

where ω = e2πi/3 and K and K ′ are the two inequivalentBrillouin zone corners.

We emphasize that the Potts states do not exhaust all possi-ble ground states, because for some spin configurations, cer-tain collective local spin rotations are possible without chang-ing the total energy. The number of such zero modes can evenbe extensive29. In the case of the SU(4) model on the cubiclattice, to be discussed below, there are only finitely many softmodes, and we observe a finite temperature phase transition.

Consider now the zero-energy fluctuations for the q = 0structure. Six of them are global SU(3) rotations, while theothers may be constructed as follows. Identify A, B, andC spin sublattices by different colors. There are three typesof dual-colored lines in this structure (see Fig. 1): ABAB,BCBC, and CACA. The spins along each of these lines maybe rotated independently around ωσ axis corresponding to thethird color. This is a source of zero modes: each line pro-vides two zero modes, but total number of zero modes in thisstructure is still sub-extensive, scaling as Ω1/2.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Spec

ific

Hea

t

T

Figure 3. Heat capacity per site vs. temperature for the kagomestructure.

6

For the√

3 ×√

3 structure of Fig. 2, there is an exten-sive set of zero modes. Consider the case of a single Star ofDavid from this structure, depicted in red in the figure. Theinternal hexagon is a six-site loop surrounded by six externalspins. If the loop spins belong to the plane spanned by vectorszA and zB while the external spins are all zC, there is a localzero-energy mode associated with the hexagon which rotateszA and zB about zC, while keeping all three spins mutually or-thogonal. For a single six-site loop with six additional verticesthis type of fluctuation coincides with the global rotation, butin the lattice we can rotate each of the loops independently.This leads to the extensive number of zero modes, which in-creases the entropy. Fluctuations about the Potts state yield aheat capacity of C = 16

9 ≈ 1.78 per site. The counting ofmodes is as follows. There are four quadratic modes per site.Any individual hexagon, however, can be rotated by a localU(2) matrix in the subspace perpendicular to the direction setby its surrounding spins (e.g., an AB hexagon can be rotatedabout the C direction). There are two independent real vari-ables associated with such a rotation. (For an AB hexagon, theA sites are orthogonal to the C direction, hence zA is specifiedby two complex numbers, plus the constraint of z†AzA = 1

and the equivalence under zA → eiαzA.) Subtracting out thezero modes, we find the heat capacity per site would then beC(0) = 1

2 ×(4 − 2

3

)= 5

3 . However, we have subtractedtoo much. Only one third of the hexagons support indepen-dent zero modes (the AB hexagons, say). The remaining twothirds are not independent and will contribute at quartic orderin the energy expansion. The specific heat contribution fromthese quartic modes is then 1

4 × 23 × 2

3 = 19 . Thus, we expect

C(0) = 169 . This analysis of the zero modes in both structures

follows that for the O(3) Heisenberg antiferromagnet on thekagome lattice30. As in the O(3) case, the low temperatureentropy selects configurations which are locally close to the√

3×√

3 structure. This order by disorder (OBD) mechanismwas shown in Ref. 10 by invoking a global length constraintwhich turns the low temperature Hamiltonian of eqn. 9 intoa spherical model, introducing a single Lagrange multiplier λto enforce |χ|2 + 1

Ω

∑i〈π†iπi〉 = 1, where χ plays the role of

a condensate amplitude. The free energy per site is then

f = −λ+λ|χ|2 +(N −1)T

∞∫0

dεg(ε) ln

(ε+ λ

T

). (18)

Extremizing with respect to λ yields the saddle point equation,and the OBD selection follows from a consideration of saddle-point free energies of the q = 0 and

√3×√

3 states.We now turn to the results of our Monte Carlo simulations.

The heat capacity C(T ) per site is shown in Fig. 3. We findC(T ) exhibits no singularities at any finite temperature andremains finite at zero temperature. Thus, there is no phasetransition down to T = 0. Note that while the Hohenberg-Mermin-Wagner theorem forbids the breaking of the contin-uous SU(3) symmetry at finite temperatures (since the clas-sical Hamiltonian Hcl is that of a two-dimensional systemwith finite-range interactions) it leaves open the possibility ofa transition due to breaking a discrete lattice symmetry. That

Figure 4. SU(3) kagome lattice model. Left (A,C,E): largest eigen-value, Right (B,D,F): sum of all eigenvalues. White hexagon indi-cates the border of Brillouin zone. A,B correspond to high tempera-ture (M = 2); C,D to intermediate temperature (M = 5); and E,F tolow temperature (M = 20).

such a transition does not occur – as evinced by the absence ofany specific heat singularities – is a nontrivial result of thesesimulations. From equipartition, we should expect C = 2 ifall freedoms appear quadratically in the effective low energyHamiltonian. Instead, we find C(0) = 1.84 ± 0.03. The factthat the heat capacity is significantly lower than 2 suggeststhat there is an extensive number of zero modes or or othersoft modes.

Although the absence of any phase transition in the specificheat data suggests that there is no true long-range order in thekagome system even at T = 0, it leaves open the questionof whether there is some form of incipient local order in thesystem as T → 0. To further investigate the local order atlow temperatures, we turn to the structure factor Sij(k). Re-call that this is a 3 × 3 matrix for the kagome lattice, and wehave focused our attention on the eigenvalue of maximum am-plitude as well as the trace of this matrix. Our Monte Carloresults for these quantities are plotted in Fig. 4. At high tem-peratures, we find the only detectable structure has the sameperiodicity as the lattice, with TrS(k) exhibiting a peak at thecenter of the Brillouin zone. Upon lowering temperature, onecan see that additional structure emerges, and the peak shiftsto the Brillouin zone cornersK andK ′, corresponding to the√

3 ×√

3 structure. The width of the structure factor peaksremains finite down to T = 0, and there are no true Braggpeaks. The heat capacity of the ideal

√3 ×√

3 structure is

7

somewhat lower than the heat capacity obtained from MonteCarlo simulations.

Further insight on the nature of the low-temperature stateof the kagome simplex solid is afforded by studying the au-tocorrelation function CQ(τ − τ ′) = 〈Tr

[Q(i, τ)Q(i, τ ′)

]〉.

As is clear from Fig. 6, the system remains finitely corre-lated at long times, suggesting that in spite the lack of onsetof true long-range order, the dynamics slow down as T → 0,consistent with the dominance of the

√3 ×√

3 pattern in thelow-temperature structure factor.

V. THREE-DIMENSIONAL LATTICES

A. SU(3) simplex solid on the hyperkagome lattice

We embark on our analysis of three-dimensional lattices byconsidering the analog of the kagome in three dimensions:the imaginatively-named hyperkagome lattice (Fig. 5). Thisis a three-dimensional fourfold coordinated lattice consistingof loosely-connected triangles. The crystal structure is simplecubic, with a 12-site basis. It may be described as a depletedpyrochlore structure, where one site per pyrochlore tetrahe-dron is removed. With triangular simplices, we again havethe Hamiltonian of eqn. 15, but here owing to the increaseddimensionality, we might expect that ordered states remainrelatively stable to fluctuation effects.

There is a vast number of ground states of the SU(3) sim-plex solid model on the hyperkagome lattice. We first considerthe simplest ones, Potts states, where three mutually orthogo-nal CP2 vectors ωA,B,C are assigned to the lattice sites suchthat the resulting arrangement is a ground state, where thevolume of each triangle (ijk), |RΓ | = |εαβγzi,αzj,βzk,γ | ismaximized, i.e. |RΓ | = 1.

The simplest Potts ground state will have the same period-icity as the lattice (q = 0), with its 12 site unit cell. Computerenumeration reveals that there are two inequivalent q = 0structures, one of which is depicted in the top panel of Fig. 7.Potts ground states with larger unit cells are also possible, and

Figure 5. The hyperkagome structure (from Ref. 31).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60 70 80 90 100

CQ(τ

)

Monte Carlo steps per site, % of 7000

Figure 6. Autocorrelation function for SU(3) model on the kagomelattice, M = 100. One sample was used. The result was averagedover 1000 random sites in the lattice of 1296 sites.

Figure 7. Top: Three unit cells, 12 sites each, of the q = 0 structure.The 10-site loops do not support any zero modes. Bottom: Unit cellconsisting of 36 sites of the structure analogous to

√3 ×√

3 in thecase of kagome lattice. Thick lines indicate 10-site loops which pro-vide zero modes. The leftmost 10-site red-blue loop can be rotatedabout the green direction, yielding a zero mode. In both panels, siteson the loops are shown with large spheres, and neighboring off-loopsites with small spheres.

an example of a Potts state with a 36 site unit cell is shownin the bottom panel of the figure. Such structures are analogsof√

3 ×√

3 structure on the kagome lattice, discussed in theprevious section.

Monte Carlo simulations ofHcl on the hyperkagome latticeshow no cusp in C(T ), suggesting Tc = 0 (Fig. 8). In con-trast to the kagome, structure factor measurements exhibit adiffuse pattern spread throughout the Brillouin zone and areinsufficient to show which low temperature structure is pre-

8

ferred (Fig. 10).The six-site loops in the 2D kagome lattice have an ana-

log in the 3D hyperkagome structure, which contains ten-siteloops. For the (2D) kagome model, the six-site loops sup-port zero modes in the

√3 ×√

3 Potts state. There is ananalog of this degeneracy in the (3D) hyperkagome model,where the corresponding Potts state features a 36-site unitcell, mentioned above and depicted in Fig. 7. The zero modecorresponds to a SU(3) rotation of all CP2 spins along a 10-site loop, about a common axis. This is possible because allthe spins along the loop lie in a common CP2 plane, form-ing an ABAB · · · Potts configuration. A computer enumera-tion finds that there are 12 distinct such 10-site loops associ-ated with each (12-site) unit cell. If the hyperkagome emu-lates the kagome, we expect that owing to the abundance ofzero modes, structures with such loops will dominate the low-temperature dynamics of Hcl.

In order to characterize the structure revealed by our MonteCarlo simulations, it is convenient to first define a series of‘loop statistics’ measures that serve as proxies for the localcorrelations of the spins. As before, we define the volume forthe triple of sites (i, j, k) as

V (i, j, k) =∣∣εµνλzi,µ zj,ν zk,λ∣∣ , (19)

i.e. V (i, j, k) = |RΓ | (see eqn. 6), where Γ denotes a trian-gle with vertices (i, j, k). The value of V 2(i, j, k) for differ-ent choices of triples in a ten-site loop will serve as our pri-mary statistical measure. Note that 0 ≤ V (i, j, k) ≤ 1, withV = 0 if any two of the CP2 vectors zi, zj , zk are parallel,and V = 1 if they are all mutually perpendicular. If the CP2

vectors were completely random from site to site, then the av-erage over three distinct sites would be

⟨V 2(i, j, k)

⟩) = 2

9 .For an ABAB · · · Potts configuration, V (i, j, k) = 0 for anythree sites along the loop. We then define the loop statisticsmeasures

p••• =⟨

110

10∑i=1

V 2(i, i+ 1, i+ 2)⟩

(20)

p••• =⟨

110

10∑i=1

V 2(i, i+ 1, i+ 3)⟩

(21)

p••• =⟨

110

10∑i=1

V 2(i, i+ 1, i+ 4)⟩

(22)

p••• =⟨

110

10∑i=1

V 2(i, i+ 2, i+ 4)⟩

, (23)

where the angular brackets denote thermal averages and aver-ages over unit cells.

Another useful diagnostic is to compute the eigenspectrumof the gauge-invariant tensor Qµν(i) averaged over sites,

Qµν ≡ 110

∑i∈loop

〈z∗i,µ zi,ν〉 − 13δµν . (24)

For randomly distributed CP2 vectors, Q = 0. If the loop is inthe ABAB· · · Potts configuration, Q = 1

6 − 12PC, where PC

00.20.40.60.8

11.21.41.61.8

22.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Spec

ific

Hea

t

T

Figure 8. Heat capacity for the SU(3) model on the hyperkagomelattice.

is the projector onto the C state orthogonal to both A and B.Our final diagnostic is the average energy per triangle, denotedE/N4.

Statistical data for the 10-site loops at inverse temperatureM = 100 are shown in Table I, where four structures are com-pared. Each column of the table refers to a particular class of10 site loop. The first two columns present Monte Carlo datafor a 6144 site lattice (83 unit cells) with periodic boundaryconditions. Averages are performed over the entire lattice. Inthe column A, the particular loop among the 12 distinct rep-resentatives per unit cell is chosen on the basis of the lowesteigenvalue of Qµν . In column B, the representative loop hasthe lowest value of p•••. In column C, data from a single10-site loop with a fixed set of boundary spins, as depicted inFig. 9, is presented. In this case the boundary spins are allparallel CP2 vectors, hence for T = 0 the ground state of thisring would be a Potts state of the ABAB· · · type, and indeedthe data are close to what we would predict for such a Pottsstate, where the internal volume V (i, j, k) vanishes for anytriple of sites on the loop, and where the eigenvalues of Q are− 1

3 ,16 ,

16

. Such a configuration exhibits a zero mode, since

the loop spins can be continuously rotated about the directionset by the boundary. If we fix the boundary spins such that

Figure 9. A ten site loop surrounded by ten boundary sites.

9

SU(3) system A B C D Ep••• 0.377± 0.004 0.364± 0.007 0.0093± 0.0006 0.516± 0.004 0.488± 0.006

p••• 0.2415± 0.0008 0.253± 0.002 0.0093± 0.0006 0.259± 0.003 0.253± 0.001

p••• 0.305± 0.002 0.339± 0.003 0.0093± 0.0003 0.355± 0.007 0.369± 0.004

p••• 0.063± 0.002 0.043± 0.003 0.00025± 0.00001 0.138± 0.003 0.127± 0.005

λmin −0.167± 0.001 −0.150± 0.002 −0.3296± 0.0001 −0.088± 0.002 −0.110± 0.004

E/N4 0.02945± 0.00001 0.02945± 0.00001 0.01885± 0.00002 0.01984± 0.00001 0.029518± 0.000003

Table I. 10 site loop statistics in the SU(3) hyperkagome model (see text). A) hyperkagome (lowest λmin). B) hyperkagome (lowest p•••).C) 10 sites (uniform boundary). D) 10 sites (no zero mode). E) 20 sites (loop + boundary). The inverse temperature is M = 100.

there is no such zero mode, and average over all such bound-ary configurations, we obtain the data in column D. Finally,column E presents data for the 20-site system shown in Fig.9, where the boundary spins are also regarded as free.

Our results lead us to conclude that the SU(3) model onthe hyperkagome lattice is unlike the planar kagome case inthat there it is far from a Potts state, even at low tempera-tures. There is no thermodynamically significant number ofABAB· · · ten-site loops, and the statistics of these loops inthe hyperkagome structure most closely resemble the resultsin the last line of Tab. I, corresponding to a single loop witha fluctuating boundary. This is supported by static structurefactor data in Fig. 10, which shows no discernible peaks. Inaddition, the heat capacity, shown in Fig. 8, tends to the fullvalue of C(T = 0) = 2N , corresponding to four quadraticdegrees of freedom per site.

B. SU(4) model on the cubic lattice

Thus far we have considered models with corner-sharingsimplices. We now consider a 3D model with edge-sharingsimplices. The individual spins are four component objectslying in the space CP3. These may be combined into singletsusing the plaquette operator φ†Γ = εµνλρ b†i,µ b

†j,ν b

†k,λ b

†l,ρ,

where (ijkl) are the sites of the 4-simplex Γ . On a cubiclattice, M such singlets are placed on each elementary face,so each site is in a fully symmetric representation of SU(4)with 12M boxes. Note that two faces may either share a sin-

Figure 10. Structure factor for the SU(3) hyperkagome model at T =0.01 (M = 100). Results show S(k) in the (kx, ky) plane withkz = 0 (A) and kz = π/a (B) . White lines denote the borders ofthe Brillouin zone. Number of sites is 6144 (83 unit cells).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60 70 80 90 100

CQ(τ

)

Monte Carlo steps per site, % of 2500

Figure 11. Autocorrelation function for SU(3) model on the hyper-kagome lattice, M = 100. One sample was used. The result wasaveraged over 1000 random sites in the lattice of 6144 sites.

gle edge, if they belong to the same cube, or a single site.Again with T = 1/M , we have identified a second orderphase transition of the corresponding classical system usingMonte Carlo simulation. The classical Hamiltonian for themodel is

Hcl = −∑Γ

ln∣∣εα1α2α3α4 zΓ1,α1

zΓ2,α2zΓ3,α3

zΓ4,α4

∣∣2 ,(25)

where Γi are the corners of the elementary square face Γ .An E0 = 0 ground state can be achieved by choosing fourmutually orthogonal vectors ωσ and arranging them in such away that corners of every face are different vectors from thisset. The volume spanned by vectors of every simplex is then|RΓ | = 1. This ground state is unique up to a global SU(4)rotation, and has a bcc structure, as shown in Fig. 12. Otherground states could be obtained from the Potts state by tak-ing a 1D chain of spins lying along one of the main axes, sayACAC, and rotating these spins around those in the BD plane.We see that number of zero modes is sub-extensive, however.

There is a phase transition to the ordered phase at T =1.485 ± 0.005. This is confirmed by both heat capacity tem-perature dependence (Fig. 14) and static factor calculations.Our static structure factor calculations prove the spin pattern

10

Figure 12. Potts ground state of SU(4) classical model on a cubiclattice has a bcc structure.

forms a bcc lattice below the critical temperature (Fig. 12 A-B). On the cubic lattice, Sij(k) = S(k) is a scalar, and in thePotts state of Fig. 12 it is given by

S(k) =1

Ω

∑R,R′

Tr[Q(R)Q(R′)

]eik·(R−R

′)

= 14 Ω(δk,M + δk,M ′ + δk,M ′′

),

(26)

where M = (0, π, π), M ′ = (π, 0, π), and M ′′ = (π, π, 0)are the three inequivalent edge centers of the Brillouin zone,resulting in an edge-centered cubic pattern in reciprocal space.Since Tc > 1, we have Mc < 1, and since only positive inte-ger M are allowed, we conclude that the SU(4) simplex solidstates on the cubic lattice are all ordered. In the mean field the-ory of Ref. 10, however, one finds TMF

c = ζ/(N2− 1), whereζ is the number of plaquettes associated with a given site. Forthe cubic lattice SU(4) model, ζ = 12, whence TMF

c = 45 ,

which lies below the actual Tc. Thus, the mean field theoryunderestimates the critical temperature.

C. SU(8) model on the cubic lattice

Finally, we consider a three-dimensional model with face-sharing simplices. On the cubic lattice, with eight species ofboson per site, we can construct the SU(8) singlet operatorφ†Γ on each cubic cell. Each site lies at the confluence of eightsuch cells, hence in the state |Ψ〉 =

∏Γ (φ†Γ )M |0〉, each site

is in the fully symmetric representation of SU(8) describedby a Young tableau with one row and 8M boxes. Nearestneighbor cubes share a face, next nearest neighbor cubes sharea single edge, and next next nearest neighbor cubes share asingle site. The associated classical Hamiltonian for the modelis constructed from eight-site interactions on every elementarycube of the lattice.

Hcl = −∑Γ

ln∣∣εα1···α8 zΓ1,α1

· · · zΓ8,α8

∣∣2 , (27)

where Γi are the corners of the elementary cube Γ . A min-imum energy (E0 = 0) Potts state can be constructed by

0.734

0.736

0.738

0.74

0.742

0.744

0.746

0.748

0.75

0 10 20 30 40 50 60 70 80 90 100

CQ(τ

)

Monte Carlo steps per site, % of 4000

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40 50 60 70 80 90 100

CQ(τ

)

Monte Carlo steps per site

Figure 13. Autocorrelation functions for SU(4) model on the cubiclattice, M = 100 (top picture, ordered phase) and M = 1/3 (bottompicture, disordered phase). One sample was used. The result wasaveraged over 1000 random sites in the lattice of 203 sites.

choosing eight mutually orthogonal vectors and arrangingthem in such a way that corners of every cube are differentvectors from this set. Ground states of this model include allground states of the eight-state Potts model with eight-spin in-teractions. Once again, a vast number of such Potts states ispossible. For example, a state with alternating planes, each ofthem containing only four out of eight Potts spin directions,has a large number of zero modes. It has a simple cubic pat-tern, depicted in Fig. 16. We rely on numerical simulation todetermine the preferred state at low temperatures.

There is a phase transition to the ordered phase at Tc =0.370 ± 0.005. This is backed by both heat capacity temper-ature dependence (Fig. 14) and static structure factor calcula-tions (Fig. 18). Our Monte Carlo data for S(k) indicates thepresence of spontaneously broken SU(8) symmetry below Tc,where Bragg peaks develop corresponding to a simple cubicstructure with a magnetic unit cell which is 2×2×2 structuralunit cells. Since Mc = 1/Tc ' 2.70, the SU(8) cubic latticesimplex solid states with M = 1 and M = 2 will be quantumdisordered, while those withM > 2 will have 8-sublattice an-

11

tiferromagnetic Potts order. As in the case of the SU(4) modeldiscussed above, the actual transition temperature is largerthan the mean field value TMF

c = ζ/(N2 − 1) = 863 = 0.127.

D. The mean field critical temperature

Conventional wisdom has it that mean field theory alwaysoverestimates the true Tc because of its neglect of fluctu-ations. As mentioned earlier, in the SU(2) valence bondsolid states, the corresponding classical interaction is uij =

− ln(

12 − 1

2 ni · nj), and one finds TMF

c = r/3, where ris the lattice coordination number. Monte Carlo simulationsyield Tc = 1.66 on the cubic lattice (r = 6, TMF

c = 2), andTc = 0.85 on the diamond lattice (r = 4, TMF

c = 43 )2,6. In

both cases, the mean field value TMFc overestimates the true

transition temperature.It is a simple matter, however, to concoct models for which

the mean field transition temperature underestimates the ac-tual critical temperature. Consider for example an Ising modelwith interaction u(σ, σ′) = −ε−1 ln(1 + εσσ′), where thespins take values σ, σ′ = ±1, and where 0 < ε < 1.If we write σ = 〈σ〉 + δσ at each site and neglect termsquadratic in fluctuations, the resulting mean field Hamilto-nian is equivalent to a set of decoupled spins in an exter-nal field h = rm/(1 + εm2). The mean field transitiontemperature is TMF

c = r, independent of ε. On the otherhand, we may also write u(σ, σ′) = uε − Jε σσ

′, whereuε = − ln(1−ε2)/2ε and Jε = ε−1 tanh−1(ε). On the squarelattice, one has Tc(ε) = 2Jε/ sinh−1(1), which diverges asε → 1, while TMF

c = 4 remains finite. For ε > 0.9265, onehas Tc(ε) > TMF

c .Another example, suggested to us by S. Kivelson, is that of

hedgehog suppression in the three-dimensional O(3) model.Motrunich and Vishwanath32 investigated the O(3) model ona decorated cubic lattice with spins present at the vertices andat the midpoint of each link. They found Tc = 0.588 forthe pure Heisenberg model and T ∗c = 1.38 when hedgehogs

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5 1 1.5 2 2.5 3

Spec

ific

Hea

t

T

Figure 14. Specific heat for SU(4) model on the cubic lattice.

Figure 15. Static structure factor for the SU(4) model on the cubiclattice (163 sites) as a function of (kx, ky) for kz = 0 (A,C,E) andkz = π (B,D,F) and T = 1

2(A,B), T = 1.5 (C,D), T = 3 (E,F).

Note Tc ≈ 1.5.

Figure 16. A Potts ground state for the SU(8) classical model on thecubic lattice. The magnetic crystal structure is simple cubic.

were suppressed. The mean field theory is not sensitive tohedgehog suppression, and one finds TMF

c = 2√3

= 1.15,which overestimates Tc but underestimates T ∗c .

In both these examples, the mean field partition function in-cludes states which are either forbidden in the actual model,or which come with a severe energy penalty (ε ≈ 1 in ourfirst example). Consider now the classical interaction de-rived from the simplex-solid ground models, uΓ = −2 lnVΓ ,

12

where VΓ = |εα1···αN zΓ1,α1· · · zΓN ,αN

| is the internal volumeof the simplex Γ . If we consider the instantaneous fluctua-tion of a single spin in the simplex, we see that there is aninfinite energy penalty for it to lie parallel to any of the re-maining (N − 1) spins, whereas the mean field Hamiltonianis of the form HMF = −ζ∑i hµν(i)Qµν(i), and hµν(i) =

aN (m)δµν + bN (m)Pσ(i)µν , where aN (m) and bN (m) are

computed in Ref. 10, and Pσ(i) is the projector onto the CP2

vector associated with sublattice σ(i) in a Potts ground state.There are no local directions which are forbidden by HMF, sothe mean field Hamiltonian allows certain fluctuations whichare forbidden by the true Hamiltonian. This state of affairsalso holds for the SU(2) models, where Monte Carlo simu-lations found that the mean field transition temperature over-estimates the true transition temperature, as the folk theoremsays, but apparently the difference Tc−TMF

c becomes positivefor larger values of N .

VI. ORDER AND DISORDER IN SIMPLEX SOLID STATES

To apprehend the reason why the SU(3) hyperkagomemodel remains disordered for all T = 1/M while the SU(4)and SU(8) cubic lattice models have finite T phase transi-tions (which in the former case lies in the forbidden regimeT > 1, i.e. M < 1), we examine once again the effectivelow-temperature Hamiltonian of eqn. 9, derived in Ref. 10,

HLT =∑Γ

N∑i<j

∣∣π†Γi

ωσ(Γj)

+ ω†σ(Γi)

πΓj

∣∣2 . (28)

The expansion here is about a Potts state, where each simplexΓ is fully satisfied such that VΓ = 1. In a Potts state, eachlattice site k is assigned to a sublattice σ(k) ∈ 1, . . . , N,with ωσ a mutually orthogonal set ofN CPN−1 vectors andπ†iωσ(i) = 0. It is convenient to take ωσ,µ = δµ,σ , i.e. the µ

component of the CPN−1 vector ωσ is δµ,σ . In HLT, the first

1

2

3

4

5

6

7

8

9

10

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Spec

ific

Hea

t

T

Figure 17. Specific heat for the SU(8) model on the cubic lattice.

Figure 18. Static structure factors for the SU(8) model on the cubiclattice. A and B are kz = 0 and kz = π/a cross sections of thestatic factor in the ordered phase, M = 20. C and D are kz = 0 andkz = π/a cross sections for higher temperature, M = 2.7, close to thecritical valueMC = 2.67±0.01. E and F are kz = 0 and kz = π/across sections in the disordered phase, M = 2. White lines denote theborders of the Brilloin zone. Note the overall scale; number of sitesis 103.

sum is over all simplices Γ , and the second sum is over allpairs of sites (Γi, Γj) on the simplex Γ .

Let us first consider a Potts state which has the same peri-odicity as the underlying lattice. In such a state, each simplexcorresponds to a unit cell of the lattice. Examples would in-clude the q = 0 Potts states of the SU(3) simplex solid onthe kagome lattice and the SU(4) model on the pyrochlore lat-tice, or a variant of the SU(8) cubic lattice model discussedabove, where one sublattice of cubes is eliminated such thatthe remaining cubes are all corner-sharing. In such a struc-ture, we may write ω

σ(Γi)≡ ωi , in which case the interaction

between sites i and j on the same simplex may be written as|π∗Γi , j

+ πΓj , i|2, where π

Γi , jis the j component of the N -

component vector πΓi

. Note that πΓi , i

= 0. Since each site is

a member of precisely two simplices, the system may be de-composed into a set of one-dimensional chains, each of whichis associated with a pair (σ, σ′) of indices. Hence there are12N(N − 1) pairs in all. To visualize this state of affairs, itis helpful to refer to the case of the kagome lattice in fig. 1,for which N = 1

2N(N − 1) = 3. Thus there are three typesof chains: AB, BC, and CA. Each AB chain is described by a

13

classical energy function of the form

HAB =∑n

(|a∗n + bn|2 + |b∗n + an+1|2

)(29)

=∑k

(a∗k b−k

)(2 1 + e−ik

1 + eik 2

)(akb∗−k

).

This yields two excitation branches, with dispersionsω±(k) = 2±2 cos( 1

2k). Thus we recoverN(N −1) complexdegrees of freedom, or 2N(N − 1) real degrees of freedom,per unit cell, as derived in §II A.

In Ref. 10, the fixed length constraint of each CPN−1

vector zi was approximated by implementing the nonholo-nomic constraint 〈π†iπi〉 ≤ 1, which in turn is expressed as|χ|2 + 〈π†iπi〉 = 1, where χ plays the role of a condensateamplitude. This holonomic constraint is enforced with a La-grange multiplier λ, so that the free energy per site takes theform of eqn. 18, where g(ε) is the total density of states per

site, normalized such that∞∫0

dε g(ε) = 1. For the models cur-

rently under discussion, we have g(ε) = g1D(ε), where

g1D(ε) =

2π∫0

dθ

2πδ(2− 2 cos θ − ε)

=Θ(2− |ε− 2|

)π√ε(4− ε)

,

(30)

characteristic of one-dimensional hopping. The spectrum isconfined to the interval ε ∈ [0, 4], and extremizing with re-spect to λ yields the equation

1 = |χ|2 + (N − 1)T

∞∫0

dεg(ε)

ε+ λ. (31)

If∞∫0

dε ε−1g(ε) < ∞, then λ = 0 and |χ|2 > 0. This is the

broken SU(N ) symmetry regime. Else, λ > 0 and χ = 0,corresponding to a gapped, quantum disordered state.

A. SU(3) kagome and hyperkagome models

For the SU(3) kagome and hyperkagome models, expand-ing about a q = 0 Potts state, the free energy per site for thelow temperature modelHLT, implementing the nonholonomicmean fixed length constraint for the CP2 spins, is found to be

f(T, λ) = −λ+ 2T ln

(2 + λ+

√λ(λ+ 4)

2T

). (32)

Setting ∂f/∂λ = 0 yields λ = 2(√

1 + T 2 − 1). These

systems are in gapped, disordered phases for all T , meaningthat the corresponding quantum wave functions are quantum-disordered for all values of the discrete parameterM . The lowtemperature specific heat is C(T ) = 2− 2T +O(T 2).

Figure 19. Free energy per site for the q = 0 states of the SU(3)kagome and hyperkagome lattice models. The inset shows the dif-ference in free energies ∆f between the

√3 ×√

3 structure on thekagome lattice and the q = 0 state (blue), and the correspondingdifference for the analogous state in the hyperkagome lattice (36 sitemagnetic unit cell).

In the√

3×√

3 state on the kagome lattice, we have10

gK(ε) = 16

δ(ε)+2 δ(ε−1)+2 δ(ε−3)+δ(ε−4)

, (33)

whereas for the analogous structure in the hyperkagome lat-tice, with a 36 site magnetic unit cell, we find

gHK(ε) = 112

2 δ(ε) + δ(ε− 1) + 2 δ(ε− 2) + δ(ε− 3)

+ 2 δ(ε− 4) + δ(ε− 1− φ) + δ(ε− 2 + φ)

+ δ(ε− 2− φ) + δ(ε− 3 + φ)

, (34)

where φ = 12

(1 +√

5)' 1.618. For the kagome system, we

obtain

1

T=

2u

3·

1

u2 − 4+

2

u2 − 1

, (35)

where u ≡ λ+ 2. For the hyperkagome system,

1

T=u

3·

2

u2 − 4+

1

u2 − φ2(36)

+1

u2 − 1+

1

u2 − (1− φ)2+

1

u2

.

One then obtains λK = 13T + 35

108T2 + O(T 3) for kagome

λHK = 13T + 31

108T2 +O(T 3) for hyperkagome, at low tem-

peratures. The corresponding specific heat functions are then

CK(T ) = 53 − 35

54 T +O(T 2)

CHK(T ) = 53 − 31

54 T +O(T 2) .(37)

Both tend to the same value as T → 0. For the kagome sys-tem, we found C(0) = 1.84 ± 0.03, close to the value of169 obtained by augmenting the quadratic mode contribution

of 53 with that from the quartic modes, whose contribution

is ∆C = 19 . Our hyperkagome simulations, however, found

14

C(0) ≈ 2, with no apparent deficit from zero modes or quar-tic modes. Again, this is consistent with the structure factorresults, which show no hint of any discernible structure downto the lowest temperatures.

B. SU(4) cubic lattice model

We now analyze the low-energy effective theory of theSU(4) cubic lattice model, expanding about the Potts statedepicted in fig. 12. The magnetic unit cell consists of foursites. Let the structural cubic lattice constant be a ≡ 1. Themagnetic Bravais lattice is then BCC, with elementary directlattice vectors

a1 = (1, 1, 1) , a2 = (−1, 1, 1) , a3 = (1,−1, 1)

and elementary reciprocal lattice vectors

b1 = (π, π, 0) , b2 = (−π, 0, π) , b3 = (0,−π, π) .

In the Potts state, the A sites lie at BCC Bravais lattice sitesR, with B sites at R + x, C at R + y, and D at R + z.There are 2(N − 1) = 6 real degrees of freedom per latticesite, and hence 24 per magnetic unit cell. The low temperatureHamiltonian may be written as a sum of six terms

HLT = HAB +HAC +HAD +HBC +HBD +HCD , (38)

where HAB couples the B component of the π vector on theA sites with the A component of the π vector on the B sites.Explicitly, we note that an A site atR has B neighbors in unitcells atR, atR−a1, atR+a2, atR−a3, atR−a1 +a2,and atR− a1 + a2 + a3. Thus,

HAB =∑R

b∗R(aR + aR−a1

+ aR+a2+ aR−a3

+ aR−a1+a2

+ aR−a1+a2+a3

)+ c.c. + 6|aR|2 + 6|bR|2

= 6

∑k

(a∗k b−k

)(1 γkγ∗k 1

)(akb∗−k

), (39)

where

γk = 13 e

i(θ2−θ1)/2

cos

(θ1 − θ2

2

)+ cos

(θ1 + θ2

2

)

+ cos

(θ1 − θ2

2− θ3

)(40)

with k = 12π

∑3i=1 θi bi. This leads to two bands, with dis-

persions ω±(k) = 6(1 ± |γk|

). All the other Hamiltonians

on the RHS of eqn. 38 yield the same dispersion. Countingdegrees of freedom, we have four real (two complex) modesper k value (Re ak, Im ak, Im bk and Im bk), and six inde-pendent Hamiltonians on in eqn. 38, corresponding to 24 realmodes per unit cell, as we found earlier. The bottom of theω−(k) band lies at |γk| = 1, which entails θ1 = θ2 = θ3 = 0.

11

2

1

3

1

4

1

5

1

6. . .0

edge-sharing simple cubic SU(4)

face-sharing simple cubic SU(8)

corner-sharing hyperkagome SU(3)

corner-sharing kagome SU(3)

1/M

p3

p3

local

no order

8-sublatticelong-range

spin-bcclong-range

Figure 20. Structure of simplex solids as a function of discreteparameter M . The parameter range for which long-range (local)order emerges is shaded and bounded by solid (dashed) lines; a briefdescription of the order is also given. Whereas on the cubic latticethe edge-sharing SU(4) model is always long-ranged ordered, theface-sharing SU(8) model has quantum-disordered ground states forM = 1, 2. The SU(3) model exhibits quantum disorder for all M ,with local

√3 ×√

3 correlations strengthening as M → ∞ on thekagome lattice while on the hyperkagome, no local or long rangeorder is apparent at any M .

Expanding about this point, the dispersion is quadratic in de-viations, corresponding to the familiar bottom of a parabolicband. The density of states is then g(ε) ∝ √ε, which meansthat λ = 0 and |χ(T )|2 interpolates between |χ(0)|2 = 1 and|χ(Tc)|2 = 0, where

Tc =1

(N − 1)∞∫0

dε ε−1g(ε)

(41)

is the prediction of the low energy effective theory. Be-cause the low-temperature effective hopping theory for edge-sharing (and face-sharing) simplex solids involves fully three-dimensional hopping, the band structure of their low-lying ex-citations features parabolic minima, which in turn permits asolution with λ 6= 0, meaning the ordered state is stable overa range of low temperatures. We find Tc = 1.978 for theedge-sharing simplex solid model on the simple cubic lattice.This is substantially greater than both the mean field resultTMF

c = 45 and the Monte Carlo result Tc ' 1.485.

VII. CONCLUDING REMARKS

We have studied the structure of exact simplex solid groundstates of SU(N ) spin models, in two and three dimensions, viatheir corresponding classical companion models that encodetheir equal time correlations. The discrete parameterM whichdetermines the on-site representation of SU(N ) sets the tem-perature T = 1/M of each classical model, which then may

15

be studied using standard tools of classical statistical mechan-ics. Our primary tool is Monte Carlo simulation, augmentedby results from mean field and low-temperature effective the-ories. This work represents an extension of earlier work onSU(2) AKLT models.

Through a study of representative models with site-, edge-,and face-sharing simplices, we identify three broad categoriesof simplex solids, based on the T -dependence of the associ-ated classical model:

1. Models which exhibit a phase transition in whichSU(N ) is broken at low temperature, correspondingto a classical limit M → ∞ analogous to S → ∞for SU(2) systems, as exemplified by the edge-sharingSU(4) and face-sharing SU(8) cubic lattice simplexsolids. Whether or not these models have quantum-disordered for physical (i.e., integer) values of the sin-glet parameter M depends on the precise value of thetransition temperature.

2. Models which exhibit no phase transition down to T =0, but reflect strong local ordering which breaks latticeand SU(N ) symmetries, as in the SU(3) model on thekagome lattice. While the low and high M limits ofthese simplex solids appear to be in the same (quantum-disordered) phase, we expect the ground state expec-tation values for M → ∞ are dominated by classicalconfigurations with a large density of local zero modes.

3. Models which exhibit neither a phase transition norapparent local order down to T = 0 and are hencequantum-disordered and featureless for all M . Thesesimplex solids perhaps best realize the original AKLTideal of a featureless quantum-disordered paramagnet,

for the case of SU(N ) spins. The hyperkagome latticeSU(3) simplex solid is representative of this class.

These results are summarized graphically in Fig. 20.The parent Hamiltonians which admit exact simplex solid

ground states are baroque and bear little resemblance to thesimple SU(N ) Heisenberg limit typically studied. Neverthe-less, we may regard the simplex solids as describing a phaseof matter which may include physically relevant models. Thisstate of affairs obtains in d = 1, where the AKLT state cap-tures the essential physics of the S = 1 Heisenberg antiferro-magnet in the Haldane phase. We also note that SU(N) mag-netism, once primarily a theorists’ toy, may be relevant in cer-tain experimental settings; in this context, there has been re-cent progress examining the feasibility of realizing such gen-eralized spin models with systems of ultracold atoms, partic-ularly those involving alkaline earth atoms 33,34. Whether thestates analyzed in this paper will find a place in the phase di-agrams of such systems remains an open question, that wedefer to the future.

ACKNOWLEDGMENTS

We are grateful to S. Kivelson and J. McGreevy for veryhelpful discussions and suggestions, and to S.L. Sondhi forcollaboration on prior related work (Ref. 6). SAP acknowl-edges several illuminating discussions with I. Kimchi on ex-amining featureless phases via plasma mappings, and the hos-pitality of UC San Diego and the Institute of MathematicalSciences, Chennai, where parts of this work were completed.This work was supported in part by NSF grant DMR 1007028(YYK, DPA), by UC Irvine start-up funds and the SimonsFoundation (SAP).

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