Phys. Rev. B84 (2011) 224410 arXiv:1108.5268

Order by disorder and phase transitions in a highly frustrated spin model on thetriangular lattice

A. Honecker,1 D. C. Cabra,2 H.-U. Everts,3 P. Pujol,4, 5 and F. Stauffer6

1Institut für Theoretische Physik, Georg-August-Universität Göttingen, 37077 Göttingen, Germany2Departamento de F́ısica, Universidad Nacional de La Plata

and Instituto de F́ısica La Plata, C.C. 67, (1900) La Plata, Argentina3Institut für Theoretische Physik, Leibniz Universität Hannover, 30167 Hannover, Germany

4Laboratoire de Physique Théorique, Université de Toulouse, UPS, (IRSAMC), 31062 Toulouse, France5CNRS, LPT (IRSAMC), 31062 Toulouse, France

6RBS Service Recherche & Développement, Strasbourg – Entzheim, 67836 Tanneries Cedex, France(Dated: August 26, 2011; revised November 14, 2011)

Frustration has proved to give rise to an extremely rich phenomenology in both quantum andclassical systems. The leading behavior of the system can often be described by an effective model,where only the lowest-energy degrees of freedom are considered. In this paper we study a systemcorresponding to the strong trimerization limit of the spin 1/2 kagome antiferromagnet in a magneticfield. It has been suggested that this system can be realized experimentally by a gas of spinlessfermions in an optical kagome lattice at 2/3 filling. We investigate the low-energy behavior of boththe spin 1/2 quantum version and the classical limit of this system by applying various techniques.We study in parallel both signs of the coupling constant J since the two cases display qualitativedifferences. One of the main peculiarities of the J > 0 case is that, at the classical level, there isan exponentially large manifold of lowest-energy configurations. This renders the thermodynamicsof the system quite exotic and interesting in this case. For both cases, J > 0 and J < 0, a finite-temperature phase transition with a breaking of the discrete dihedral symmetry group D6 of themodel is present. For J < 0, we find a transition temperature T 0 the transition occurs at an extremely low temperature, T>c ≈ 0.0125 J . Presumably thislow transition temperature is connected with the fact that the low-temperature ordered state of thesystem is established by an order-by-disorder mechanism in this case.

PACS numbers: 75.10.Jm, 75.45.+j, 75.40.Mg, 75.10.Hk

I. INTRODUCTION

The study of frustrated quantum magnets is a fasci-nating subject that has stimulated many studies withinthe condensed matter community in recent years.1–3 Suchsystems are assumed to be the main candidates for a richvariety of unconventional phases and phase transitionssuch as spin liquids and critical points with de-confinedfractional excitations.4 Frustration can also play an im-portant rôle in classical systems. The phenomenon oforder-by-disorder5,6 is the perfect example where the in-terplay of frustration and fluctuations produces the emer-gence of unexpected order. Order-by-disorder impliesthat a certain low-temperature configuration is favoredby its high entropy, not by its low energy. Order-by-disorder can also occur in a quantum system, where anäıve argument suggests that quantum fluctuations playthe same rôle as thermal fluctuations in the classical sys-tem, albeit there are counterexamples where their rôle isin fact quite different.7

A particularly illustrative example is provided bythe spin 1/2 antiferromagnet on the kagome lattice.A spin gap appears to be present both at zeromagnetization2,8–14 (see, however, Refs. 15–17) and at1/3 of the saturation value where it gives rise to a

plateau in the magnetization curve.7,18–21 One would betempted to believe that the nature of the ground stateis similar in both cases. However, whether the groundstate at zero field is ordered or not is still under de-bate and also the existence of a plateau in the isotropicspin 1/2 Heisenberg model at magnetization 1/3 has beenquestioned recently.22,23 Nevertheless, the existence of aplateau at magnetization 1/3 is quite clear for easy-axisexchange anisotropies7,19 and, using a correspondencewith a quantum dimer model on the honeycomb lattice,24

the ground state is identified as an ordered array of res-onating spins.7,25

In this paper we study an effective model that arisesin the strong trimerization limit of the spin 1/2 kagomeantiferromagnet.26 This model has played an importantrôle in analyzing the zero-field properties of the kagomeantiferromagnet,27,28 but here we will focus on magne-tization 1/3 of the Heisenberg model, corresponding tofull polarization of the physical spin degrees in the effec-tive model. Thus, we are left with the chirality degreesof freedom of the original antiferromagnet which we willtreat as ‘spin’ variables. In this sense our spin system canbe considered as a purely orbital model similar to com-pass models recently considered in the literature (see,e.g., Refs. 29–37). As an experimental realization of this

2

model a system of spin-polarized fermions trapped in atrimerized optical kagome lattice at 2/3 occupancy hasbeen suggested.38–40 In fact, experimental realization ofan optical kagome lattice has been reported recently,41

albeit using a setup which does not allow direct controlof trimerization.

Beyond the particular realizations of our model itsvery rich physics which results from the interplay be-tween classical and quantum fluctuations and frustrationmakes it an interesting model in its own right. As willbe shown in this paper, a Hamiltonian with an (unusual)discrete symmetry but with a continuous degeneracy ofthe classical ground state, as it would be expected fora Hamiltonian with a continuous symmetry, is just oneaspect of the rich phenomena emerging from this model.

The present paper is organized as follows: in sectionII we present the Hamiltonian and the symmetries of theclassical and spin 1/2 cases. The Hamiltonian can bedefined for both signs of the coupling constant J . We de-liberately discuss in parallel the two cases throughout theentire paper to point out their similarities and differences.The spin 1/2 case is then treated in section III by meansof exact diagonalization techniques, and we argue thata finite-temperature phase transition takes place. Sinceexact diagonalization can access only small lattices, wemove to the classical model in section IV. We study indetail the manifold of lowest-energy configurations andtheir corresponding spin-wave spectra. The effect of softmodes in the order-by-disorder selection mechanism isargued to be the origin for the phase transition of theJ > 0 case, in contrast to the J < 0 case, where thetransition is of a more conventional purely energetic ori-gin. In section V we apply Monte-Carlo techniques to theclassical model and determine the transition temperaturefor J > 0 and J < 0. We also analyze the universalityclass of the transition, however, only for J < 0 since thetransition temperature for J > 0 turns out to be so lowthat it is difficult to access. Finally section VI is devotedto some concluding remarks and comments.

II. HAMILTONIAN AND SYMMETRIES

A. Hamiltonian

We will study the Hamiltonian

H = J

∑〈i,j〉

TAi TCj +

∑〈〈k,j〉〉

TAk TBj +

∑[[k,i]]

TCk TBi

,(1)

where

TAi = S+i + S

−i = 2S

xi ,

TBi = ω S+i + ω

2 S−i = −Sxi −√

3Syi , (2)

TCi = ω2 S+i + ω S

−i = −Sxi +

√3Syi ,

with the third root of unity ω = e2πi/3. The sums in(1) run over the bonds of a triangular lattice, each cor-

(a)

k

〈i, j〉 ji

[[k, i]] 〈〈k, j〉〉

(b)

jCi D 1p3

ˇ

ˇ

ˇ

ˇ

ˇ

+

C !ˇ

ˇ

ˇ

ˇ

ˇ

+

C !2ˇ

ˇ

ˇ

ˇ

ˇ

+!

j�i D 1p3

ˇ

ˇ

ˇ

ˇ

ˇ

+

C !2ˇ

ˇ

ˇ

ˇ

ˇ

+

C !ˇ

ˇ

ˇ

ˇ

ˇ

+!

(c)y

x

FIG. 1. (Color online) (a) Triangular lattice with assign-ment of bonds to the three different directions and underlyingtrimerized kagome lattice. (b) The two chirality states of atriangle. (c) Assignment of the vectors ~ei to the bonds of thetriangular lattice for the alternative representation (3) of theHamiltonian.

responding to one of the three distinct directions of thelattice, as sketched in Fig. 1(a).

The Hamiltonian (1) arises as an effective Hamiltonianfor the trimerized kagome lattice, sketched in Fig. 1(a)behind the triangular lattice. Our notation follows thederivation from the half-integer spin Heisenberg modelfor the case where the remaining magnetic degrees of free-dom are polarized.26 In this case, there are two pseudo-spin states of opposite chirality for each triangle, seeFig. 1(b). As reviewed in appendix A, plain first-orderperturbation theory of the Sz-Sz interactions between

3

two triangles yields Eq. (1) where the exchange constantJ is proportional to the inter-triangle exchange constantof the kagome lattice and would thus typically assumedto be antiferromagnetic (J > 0). Note that we have cho-sen a convenient normalization of J . A similar derivationstarting from a Fermi gas with two atoms per trimer alsoleads to the Hamiltonian (1).38

Due to the two possible chiralities on each triangle, the

pseudo-spin operators ~Si should be considered as quan-tum spin-1/2 operators. The derivations26,38 also suggesta positive J > 0 to be more natural. In this paper we willrelax these constraints and, for reasons that will becomeclear later, consider also classical spins, i.e., unit vectors~Si, and the case J < 0.

Note that the Hamiltonian (1) is not symmetric un-der reflections of the lattice. Our conventions agree withthose of Ref. 26 where this Hamiltonian appeared first,while some more recent works38–40,42 use a reflected con-vention for the chirality. Note furthermore that our con-ventions for J differ by a factor 4 from previous studiesof the model (1).39,40

It may also be useful to represent the Hamiltonian (1)in a more compact form43

H = 4 J∑〈i,j〉

(~ei · ~Si

) (~ej · ~Sj

), (3)

where the vectors ~ei are indicated in Fig. 1(c): for eachbond one has to choose ~ei and ~ej as in the correspondingbond of the bold triangle. For example, for each horizon-

tal bond 〈i, j〉, one needs to choose ~ei =(

10

)for the left

site i and ~ej =12

(−1√

3

)for the right site j.

Models which are very similar to (1) have recently beenstudied in the context of spin-orbital models (see, e.g.,Refs. 29–37).

B. Symmetries

The Hamiltonian (1) has the following symmetries onan infinite lattice:

1. Translations Tx, Ty along the two fundamental di-rections of the lattice.

2. Simultaneous rotation R2π/3 of the lattice and allspins around the z-axis by angles of 2π/3 (the latterrotation amounts to a cyclic exchange of TAi , T

Bi ,

and TCi ).

3. A rotation by π around the z-axis in spin space:P : Sxi 7→ −Sxi , Syi 7→ −Syi while keeping thelattice fixed.

4. A spatial reflection combined with rotation of allspins around a suitable axis in the x-y-plane byan angle π. One particular choice is I : Sxi 7→

Sxi , Syi 7→ −Syi , Szi 7→ −Szi , combined with a

reflection of the lattice along the dashed line inFig. 1(a).

5. For spin 1/2, there is another symmetry imple-mented by

Q =∏i

(2Szi ) . (4)

Conservation of Q means that the number of spinspointing up (or down) along the z-axis is a goodquantum number modulo two. This conservationlaw is most easily verified by observing that theinteraction terms in (1) always invert a pair of spinsin an eigenbasis of Sz.

The choice of factors in (4) ensures that Q2 = 1. Fur-thermore, one has R32π/3 = P

2 = I2 = 1. R2π/3 and P

together generate the abelian group Z6 ∼= Z3 × Z2, asdescribed for instance in Ref. 39. The combination ofR2π/3, P , and I generates the dihedral group D6, which

is non-abelian (I R2π/3 I = R−12π/3). Finally, R2π/3 and

I generate the symmetric group S3 which can be tracedto the point-group symmetry of the underlying kagomelattice. The operators R2π/3, P , and I leave the Hamil-tonian (1) invariant irrespective of the value of the spinquantum number. Thus, the group D6 is a symmetryalso of the classical variant of the model.

The symmetries P and Q are not present in the un-derlying kagome lattice, hence they should be specific tothe lowest-order approximation.26,38 Indeed, at least inthe derivation from the Heisenberg model one observesthat already the next correction42 breaks the symmetriesP and Q.

Now let us consider the consequences of the combina-tion of I and Q for the spin-1/2 model on a finite latticewith N sites. Then the relation I Szj I = −Szj leads to

I Q I = (−1)N Q . (5)

Since the eigenvalues of Q are q = ±1, this implies thatI is an isomorphism between the subspace with q = −1and the subspace with q = 1 for odd N and spin 1/2.

III. QUANTUM SYSTEM: EXACTDIAGONALIZATION FOR SPIN 1/2

In this section we will present numerical results forthe Hamiltonian (1) with spin 1/2. We impose periodicboundary conditions and use the translational symme-tries Tx, Ty in order to classify the states by a momentum

quantum number ~k. We only consider lattices which donot frustrate a potential three-sublattice order, i.e., onlyvalues of N that are multiples of three. For the systemsizes N considered already in Refs. 39 and 40, we will usethe same lattices. In particular, the N = 12, 21, and 24lattices are shown in Fig. 9 of Ref. 40. Furthermore, we

4

will consider the N = 27 lattice which can be found, e.g.,in Fig. 1 of Ref. 44.

Let us briefly discuss the consequences of the othersymmetries mentioned above. We did not make explicituse of P , although it is present for any lattice. However,the symmetry Q (which is also present for any lattice, see(4)) is easily implemented if we work in an Sz-eigenbasis.Concerning the rotation R2π/3, it is not possible to findlattices for all N such that it is a symmetry. If R2π/3

is a symmetry, we use it to select one representative ~kfor all equivalent momenta. Finally, the presence of thesymmetry I is more delicate. We have performed com-puter checks and found that most of the lattices underconsideration have a suitable spatial reflection symme-try, ensuring that I is a symmetry. The only exceptionis the N = 21 lattice where there is no such reflectionsymmetry. Nevertheless, we find the same spectra in thesubspaces with q = −1 and q = 1 also for N = 21.Therefore, for N odd we can choose representatives forall symmetry sectors in the subspace with q = 1.

For N ≤ 21 the translational symmetries and Q leadto matrices with dimension up to 49 940 and we can ob-tain all eigenvalues. Dimensions increase up to 2 485 592for N = 27. In this case, we have used the Lanczosmethod to compute the n lowest eigenvalues in each sec-tor. Mainly for reasons of CPU time, we restrict ton ≈ 70 (150) for N = 27 (24) and J > 0.

A. Low-lying spectra

Let us first look at the spectra. In order to take de-generacies into account, in Fig. 2 we show the integrateddensity of states, i.e., the number of states with energyless or equal than ∆E above the ground state.

Panel (a) of Fig. 2 shows results for J < 0. Theseresults extend previously presented results40 for N = 12,18, and 21 to higher energies and larger N . One observesthat there are at most 8 states for energies ∆E 0, ex-tending previously published results for N = 18, 21, and24.39,40 In this case, we observe a large density of statesat substantially lower energies than for J < 0. This largedensity of states is reminiscent of the large density of non-magnetic excitations observed in the spin-1/2 Heisenbergantiferromagnet on the kagome lattice, both at zero mag-netic field10,11 and on the one-third plateau.7 In partic-ular the N = 27 data presented in Fig. 2(b) shows alarge density of states for ∆E >∼ 0.02 J . On the otherhand, one observes at most 8 states with ∆E 0.

predict a six-fold degeneracy in an ordered state (see sec-tion IV below). However, there is no clear separation of6 low-lying states from the remainder of the spectrumfor J < 0 (see Fig. 2(a)), and even less so for J > 0(Fig. 2(b)). The considered lattice sizes may be too smallto observe the expected low-energy structure of the spec-trum. However, correlation functions exhibit pronounced120◦ correlations already on these small lattices.39,40

B. Specific heat

The specific heat C can be expressed in terms of theeigenvalues of the Hamiltonian. Since we have all eigen-values for N ≤ 21, it is straightforward to obtain thespecific heat for all temperatures and both signs of J .Fig. 3 shows the results of the specific heat per site C/N .The case J < 0 is shown in panel (a). There is a finite-size maximum slightly above T ≈ |J |. The large finite-size effects which are still observed here are consistentwith a phase transition around T ≈ |J | in which caseC should become non-analytic for N → ∞. Because ofa possible phase transition, we have tried to obtain a

5

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4T/|J|

0

0.05

0.1

0.15

C/N

N=12N=15N=18N=21N=24

(a)

(b)

FIG. 3. (Color online) Specific heat per site for the S = 1/2model with (a) J < 0 and (b) J > 0.

low-temperature approximation to the specific heat forN > 21, J < 0 by keeping low-energy states. However,for N = 24 even 12 462 low-lying states going up to ener-gies as high as ∆E 0 as compared to J < 0.

For J > 0, a second peak emerges in the specific heat atlow temperatures, see Fig. 3(b). In order to investigatethis in more detail, we use again the low-temperatureapproximation for the specific heat obtained from thelow-lying part of the spectrum. For N = 24 and 27, we

0

5

10

15

20

C J

/T/N

0 0.01 0.02 0.03 0.04 0.05T/J

0

0.1

0.2

0.3

S/N

N=18N=21N=24N=27

(a)

(b)

FIG. 4. (Color online) Low-temperature behavior of the spe-cific heat divided by temperature C/T (a) and entropy S (b)per site N for J > 0.

have used a total of 7 029 and 3 906 eigenvalues, respec-tively (the N = 24 data is included in Fig. 3(b), but itis difficult to see there since it is valid only at very lowtemperatures). Fig. 4 shows the specific heat divided bytemperature (panel (a)) and the entropy per site (panel(b)) in the low-temperature region for J > 0 and systemsizes N = 18, 21, 24, and 27. Our result for the spe-cific heat with N = 21 obtained from the full spectrumagrees with a previous result for N = 21 based on ap-proximately 2 000 low-lying states.40 The finite T = 0limit of the entropy for N = 21 and 27 in Fig. 4(b) corre-sponds to the two-fold degeneracy of the ground state forthese system sizes, see Fig. 2(b). Although the maximumvalue of C/T increases with increasing N , there are non-systematic finite-size corrections to the position of thismaximum. Thus, we can only conclude that if there is afinite-temperature ordering transition for J > 0, it shouldhave a very low transition temperature Tc

6

observation lends further support to the interpretation40

of the low-energy states for S = 1/2 in terms of the clas-sical ground states for J > 0.

IV. CLASSICAL SYSTEM: LOWEST-ENERGYCONFIGURATIONS AND SPIN-WAVE

ANALYSIS

We will now proceed with a discussion of the low-energy, low-temperature properties of the classical vari-

ant of the model (1), i.e., we will assume that the ~Si areunit vectors. We will parametrize the spin at site i byangles αi and γi:

~Si =

(cos γi cosαicos γi sinαi

sin γi

). (6)

Because the z-components do not contribute to the en-ergy, configurations with extremal energy should havespins lying in the x-y-plane (γi = 0). The energy E({αi})for a given set of angles {αi}, γi = 0 is obtained from (1)by identifying

TAi = 2 cos (αi) ,

TBi = 2 cos (αi + Ω) , (7)

TCi = 2 cos (αi − Ω)

with Ω = 2π/3.We will further be interested in small fluctuations

{αi + �i}, {γi = �̃i} around a ground-state configuration{αi, γi = 0}. The energy can then be expanded as

E ({αi + �i}) = E ({αi}) +∑i,j

�iMi,j �j

+ Ezz +O({�i, �̃j}3

). (8)

Here, Ezz is a diagonal quadratic function of the out-of-plane fluctuations �̃i which, to quadratic order, decouplesfrom the relevant degrees of freedom �i. The eigenvaluesfi of the symmetric matrix Mi,j correspond to the spin-wave modes. The fact that {αi} describes a ground stateimplies fi ≥ 0. We will call a mode with fi = 0 ‘pseudo-Goldstone mode’.

A. Ground states with small unit cell for J < 0

Let us first consider the case J < 0. Then a groundstate is given by a certain three-sublattice configura-tion where the angles αi between different sublatticesdiffer by multiples of 2π/3.39,40 Fig. 5 shows such alow-temperature configuration as a snapshot which wastaken during a Monte-Carlo simulation (details to begiven in section V below). The energy of such statesE

7

0 π/6 π/3 π/2 2 π/3α

4.58

4.59

4.6

4.61

4.62F

</N

FIG. 6. (Color online) Low-temperature contribution to thefree energy (15) for J < 0 of the fluctuations above the 120◦

ground state as a function of the spin angle α.

where fαi (~k) are the eigenvalues of (9) and Zzz is the

Gaussian integral over the N quadratic variables corre-sponding to the out-of-plane fluctuations.

Performing the Gaussian integral we get

Zα ∼ e−βH0 Zzz∏i,~k

√π√

β fαi (~k), (13)

which yields the free energy as

F = H0 +Fzz +N lnβ

2β+

1

2β

∑i,~k

ln(fαi (~k)) + . . . , (14)

where Fzz = − lnZzz/β.The low-temperature behavior is therefore determined

by the following contribution of the fluctuations to thefree energy

F

8

FIG. 8. (Color online) Snapshot of a low-temperature con-figuration during a simulation for J > 0 at T = 10−3 J ona 12 × 12 lattice. Periodic boundary conditions are imposedat the edges. Different colors are used for each of the threesublattices.

the opposite of Fig. 5.The ferromagnetic state is the simplest case for the

computation of fluctuations since Mi,j is diagonalized bya Fourier transformation. One finds the modes

f ferroclass.(kx, ky)

4 J=

3

4+ sin (α) sin (α− Ω) cos (k1)+ sin (α+ Ω) sin (α− Ω) cos (k2)+ sin (α) sin (α+ Ω) cos (k3) , (17)

with the ki defined in Eq. (11). As for the case J <0, the classical frequencies f ferroclass. are proportional tothe squares of the SW frequencies ωferro obtained froma linear Holstein-Primakoff approximation:40 f ferroclass. =(ωferro

)2/(12S2 J

).

By computing the contribution of the modes f ferroclass.to the free energy we find minima for α = nπ/3, n =0, 1, 2, . . ., so that the spins in the ferromagnetic struc-ture lock in to the lattice directions of the triangular lat-tice. For the lock-in values of α, f ferroclass. depends only onone of the ki and has a line of zeros in the perpendiculardirection in momentum space.

The three-sublattice state leads to the following 3× 3matrix in Fourier space:

M = J

3 2 Ã 2 B̃2 Ã? 3 2 C̃2 B̃? 2 C̃? 3

, (18)where

à = ei k2 sin(α+ Ω) sin(α− Ω)

+(ei k1 sin(α+ Ω) + ei k3 sin(α− Ω)

)sin(α) ,

B̃ = e−i k1 sin(α+ Ω) sin(α− Ω)+(e−i k3 sin(α+ Ω) + e−i k2 sin(α− Ω)

)sin(α) ,

C̃ = ei k3 sin(α+ Ω) sin(α− Ω) (19)+(ei k2 sin(α+ Ω) + ei k1 sin(α− Ω)

)sin(α) .

For α = nπ/3, diagonalization of (18) leads to threecompletely flat branches

f>class.,1(~k) = 0 , f>class.,2(

~k) = f>class.,3(~k) =

9

2J . (20)

In particular the lowest branch f>class.,1 = 0 correspondsto a branch of soft modes. In real space these soft modescorrespond to the rigid rotation of one single triangle.40

Note that there is no such flat branch of soft modes fora value of α which is not an integer multiple of π/3.

When computing the contribution of fluctuationsaround these configurations (α = nπ/3) to the free en-ergy, one finds that one third of the modes are quarticinstead of quadratic. This yields a free energy of theform:

F = H0 + Fzz + Fxy , (21)

where, again, Fzz ∼ N lnβ/(2β) corresponds to the triv-ial contribution of out-of-plane fluctuations and (comparealso Ref. 6)

Fxy =N lnβ

3β+N lnβ

12β+ . . . (22)

At low temperatures, this term dominates the free energy.The flat branch of soft modes reduces the coefficient oflnβ/β from N/2 as in the case of only quadratic modes(compare (14)) to N/3 + N/12 = 5N/12. This impliestwo things: firstly, the angles of the 120◦ state shouldlock in at α = nπ/3 for low temperatures. Secondly,a thermal order-by-disorder mechanism should favor the120◦ state over the ferromagnetic state for T → 0.

As in the case of the ferromagnetic state one finds

that the relation f>class.,i = (ω>i )

2/(12S2 J

), where ω>i ,

i = 1, 2, 3, are the SW frequencies obtained from a lin-ear Holstein-Primakoff approximation,39,40 holds for ar-

bitrary values of α. Using the results for ωferro(~k, α) and

ω>i (~k, α) to calculate semiclassical ground-state energies

Eferrosemclass.(α) and E>semclass.(α) in the same manner as

in (16) one finds that both are minimal at α = nπ/3and that E>semclass.(nπ/3) < E

ferrosemclass.(nπ/3). Thus the

semiclassical approach is fully consistent with the classi-cal findings.

C. Enumeration of ground states for small N

Direct computer inspection of all states with anglesαi = ni π/3, ni ∈ {0, 1, 2, 3, 4, 5} for N = 12 shows39,40that there are further ground states for J > 0. On the one

9

TABLE I. Number DN of classical ground states for J > 0on a lattice of size N with one angle fixed. The ng in thedecomposition DN =

∑g ng denote the number of ground

states with g pseudo-Goldstone modes.

N DN =∑

g nglnDN

N

12 40 = 61 + 312 + 23 + 14 0.3074

15 102 = 601 + 202 + 203 + 25 0.3083

18 286 = 921 + 1122 + 513 + 304 + 16 0.3142

21 688 = 2601 + 2102 + 2033 + 145 + 17 0.3111

24 1838 = 3841 + 9582 + 1993 + 2804 + 166 + 18 0.3132

27 5054 = 10681 + 9722 + 22573 + 3514 + 3785 0.3158

+ 96 + 187 + 19

hand, CPU time for a similar enumeration of all 6N suchconfigurations becomes prohibitively big for N >∼ 15. Onthe other hand, all known ground states turn out to havemutual angles which are multiples of 2π/3. Furthermore,we eliminate a global rotational degree of freedom byfixing one arbitrary angle α0 = π. Then there remainjust 3N−1 configurations with αi ∈ {π/3, π,−π/3} to beenumerated. Direct enumeration of these 3N−1 config-urations can be carried out with reasonable CPU timefor N ≤ 27, but becomes quickly impossible for largerN . We have therefore performed such enumerations forN ≤ 27, using the same lattices as in section III.

The number of ground states DN determined in thismanner is given in Table I for J > 0. Note that theordered states which we have described in section IV Bare just two of the DN states, but there are many fur-ther ground states which can be interpreted as defects inand domain walls between the ordered states.39,40 Indeed,also closer inspection of the snapshot shown in Fig. 8 re-veals the presence of defects in the three-sublattice struc-ture. The 240 = 6D12 states described previously39,40for N = 12 are recovered by a global rotation of the an-gles such that α0 takes on the six values α0 = nπ/3.The last column of Table I lists ln (DN ) /N . The factthat these numbers stay almost constant indicates a fi-nite ground-state entropy per site slightly above 0.3 inthe thermodynamic limit.

It is straightforward to derive the N ×N matrix Mi,jdefined in (8) for any ground state and diagonalize it.Among the eigenmodes fi, one can then identify the gpseudo-Goldstone modes fi = 0 and in turn count thenumber ng of ground states with g pseudo-Goldstonemodes. These numbers are also given in Table I in theform DN =

∑g ng. One observes that all ground states

have at least one pseudo-Goldstone mode. There are atmost N/3 pseudo-Goldstone modes, and there is onlyone ground state with this maximal number of pseudo-Goldstone modes corresponding to the three-sublatticestate described in section IV B (apart from N = 15; how-ever, this lattice is special in that it has a period threetranslational symmetry T 3y = 1).

There are N/3 ground states which differ from the per-

fect three-sublattice state by a rigid rotation of the spinson certain triangles by an angle 2π/3, and another N/3ground states where the spins on a different set of trian-gles are rotated by −2π/3. These ground states have twopseudo-Goldstone modes less than the three-sublatticestate. The data in Table I show that these 2N/3 config-urations with N/3− 2 pseudo-Goldstone modes accountfor all states with the second largest number of pseudo-Goldstone modes for N ≥ 21. This indicates that groundstates deviating from the homogeneous three-sublatticestate are obtained at the expense of pseudo-Goldstonemodes. At finite temperature such “inhomogeneous”ground states are then penalized by an entropic cost be-cause of the reduced number of pseudo-Goldstone modes.This indicates that ground states with bigger deviationsfrom the three-sublattice ground state have a higher freeenergy for small T since they have less soft modes. Thus,the global minimum of the free energy is the perfectlyordered state with many close-by configurations whichdeviate only locally from the perfect order. These ar-guments predict a thermal order-by-disorder selection ofthe 120◦ state among the macroscopic number of groundstates for T → 0 and J > 0.

The above enumeration procedure can also be per-formed for J < 0. In fact, in this case we have carriedit out twice, first with αi ∈ {π/3, π,−π/3} and α0 = π,and then again with all αi shifted by π/6 in order tomatch the lock-in predicted in section IV A. In sharpcontrast to the large degeneracy found for J > 0, weconfirm that the ground state is unique (up to global ro-tations of all angles) for J < 0 and thus identical to thethree-sublattice state described in section IV A. Diago-nalization of the corresponding N ×N matrix M yieldsone pseudo-Goldstone mode, in complete agreement withthe fi shown in Fig. 7 which have just one zero, namely

f

10

Metropolis algorithm (see, e.g., Ref. 46). Some resultsobtained from such simulations have already been pub-lished in Ref. 43, but the results to be presented herehave been obtained from new runs using the ‘MersenneTwister’ random number generator.47 In order to deter-mine error bars, we have used between 100 and 400 in-dependent simulations for J < 0.

For J > 0, the standard single-spin flip algorithm turnsout to be no longer ergodic for temperatures T 0 (panel (b)).Statistical errors should be at most on the order of thewidth of the lines. Although all lattice sizes are biggerthan those used previously for the quantum model, thereare remarkable similarities of the specific heat of the clas-sical model shown in Fig. 9 with the specific heat of thequantum model, see Fig. 3. For J < 0, a singularityseems to develop in the specific heat for temperaturesaround T ≈ 1.5 |J |, signaling a phase transition. ForJ > 0, there is also a broad maximum at ‘high’ temper-

0 1 2 3 4T/|J|

0

0.5

1

1.5

2

2,5

C/N

0 0.5 1 1.5 2

T/J

0

0.5

1

C/N

N=6x6N=9x9N=18x18N=27x27

(a)

(b)

FIG. 9. (Color online) Specific heat per site for the classicalmodel with (a) J < 0 and (b) J > 0. Error bars do not exceedthe width of the lines.

atures T ≈ 0.3 J . The finite-size effects for the lattermaximum are small indicating that this does not corre-spond to a phase transition. In this case, the interestingfeatures of the specific heat lie in the low-temperatureregion, as displayed in Fig. 10(b). As the system sizeincreases, one can see that a small peak builds up in thespecific heat for T ≈ 0.02 J . This seems to indicate thata phase transition might occur around that temperature,two orders of magnitudes smaller than for J < 0. Weare unfortunately not on a par with the J < 0 data, asthe CPU requirements are too steep to secure relevantdata for systems larger than 27 × 27 sites even thoughthe specific heat is a comparably robust quantity, and itis clear that other observables are needed to conclude onthe existence of this phase transition.

An important difference between the S = 1/2 and theclassical model arises at low temperatures: the specificheat of the quantum system has to vanish upon approach-ing the zero temperature limT→0 C/N = 0, while due tothe remaining continuous degrees of freedom, the specificheat of the classical system approaches a finite value forT → 0. For J < 0, the equipartition theorem predicts

11

1.6 1.7 1.8 1.9 2

T/|J|

1.6

1.8

2

2.2C

/N

N=18x18N=27x27N=36x36N=45x45N=90x90

0 0.02 0.04 0.06

T/J

0.94

0.96

0.98

1

1.02

C/N N=6x6

N=9x9N=12x12N=15x15N=18x18N=27x27

(a)

(b)

FIG. 10. (Color online) Zooms of the specific heat per site forthe classical model: (a) in the vicinity of the maximum forJ < 0, (b) in the low-temperature region for J > 0. Statisticalerrors do not exceed the width of the lines or the size of thesymbols, respectively.

an N/2 contribution to the specific heat for each trans-verse degree of freedom which yields limT→0 C/N = 1,in excellent agreement with the results depicted in thepanel (a) of Fig. 9. For J > 0, one must take into ac-count the fact that the flat soft-mode branch of the three-sublattice state is expected to contribute only N/12 tothe specific heat. Thus for J > 0 one should expectlimT→0 C/N = 11/12 = 0.916666.... As can be seen inFig. 10(b), we observe a specific heat lower than one inthe low-temperature region along with a downward trendas T goes to 0 for all the system sizes studied. However,according to the data which we have at our disposal, itseems that one would have to go to very low tempera-tures T < 10−3 J in order to verify the prediction for thezero-temperature limit.

Returning to the finite-temperature transition for J <0, Fig. 10(a) shows a zoom into the relevant tempera-ture range, including data for up to N = 90 × 90 spins.At these bigger system sizes, the position of the maxi-mum continues to shift to lower temperatures and themaximum sharpens. However, the N = 45 × 45 and

0 π/3 2 π/3 π 4 π/3 5 π/3 2 πφ

i

0

0.2

0.4

0.6

0.8

P(φ

i)

N=6x6N=12x12N=18x18N=27x27N=36x36

FIG. 11. (Color online) Histogram of in-plane angles φi forJ < 0. Averaging has been performed over 1000 independentconfigurations at T = 10−3 |J |.

90× 90 curves in Fig. 10(a) demonstrate that the maxi-mum value of the specific heat starts to decrease as onegoes to system sizes beyond N = 45 × 45. This impliesthat the exponent α which characterizes the divergenceof the specific heat at the critical temperature is verysmall or maybe even negative.

B. Sublattice order parameter, Binder cumulant,and transition temperature for J < 0

According to subsection IV A, we expect that the phasetransition observed for J < 0 is a transition into a three-sublattice ordered state. This ordering is indeed exhib-ited at least at a qualitative level by snapshots of Monte-Carlo simulations at low temperatures (compare Fig. 5).In addition, one observes in Fig. 5 that the spins are lyingessentially in the x− y-plane for low temperatures.

Furthermore, we expect a lock-in of the spins to oneof 6 symmetrically distributed directions in the plane atlow temperatures (compare Fig. 6). The latter predictionis indeed verified by the histogram of the angles of thein-plane component of the spins φi at low temperaturesshown in Fig. 11. Note that the histogram is rather flatfor the smaller lattices (in particular the N = 6 × 6 lat-tice) and sharpens noticeably as the lattice size increaseto N = 36 × 36 (the largest lattice which we have con-sidered in this context). The fact that the lock-in occursonly on large lattices can be attributed to the replace-

ment of the sum over ~k in (15) by an integral being agood approximation only for large lattices.

To test for the expected three-sublattice order, we in-

12

0 1 2 3 4T/|J|

0

0.2

0.4

0.6

0.8

1m

s

2

N=6x6N=9x9N=18x18N=27x27

FIG. 12. (Color online) Square of the sublattice order param-

eter m2s =〈~M2s

〉for the classical model with J < 0. Error

bars do not exceed the width of the lines.

troduce the sublattice order parameter

~Ms =3

N

∑i∈L

~Si , (23)

where the sum runs over one of the three sublattices Lof the triangular lattice. Fig. 12 shows the behavior ofthe square of this sublattice order parameter for J < 0.One observes that the sublattice order parameter indeedincreases for T < 2 |J | and goes indeed to m2s = 1 forT → 0, as expected for a three-sublattice ordered state.Inclusion of larger lattices (up to N = 90×90) allows oneto restrict the ordered phase to T N1 ≥ 9×9. Thisleads to the estimate

T

13

-33800 -33700

0.0021

0.00215

P(E

/|J|)

-34000 -33000 -32000 -31000E/|J|

0

0.0005

0.001

0.0015

0.002

0.0025

P(E

/|J|)

T=1.566|J|

T=1.7025|J|

-31600 -315000.00165

0.0017

0.00175T=1.566|J| T=1.7025|J|

FIG. 14. (Color online) Probability to find a state with energyE/|J | on the N = 90 × 90 lattice for J < 0 at two selectedtemperatures: T/|J | = 1.566 (left) and 1.7025 (right). Errorbars in the lower panel do not exceed the width of the lines.

On the one hand, there is no evidence for any latentheat in the specific heat at T Tc, as observed in Fig. 13(a) is sometimestaken as evidence for a first-order transition (see, e.g.,Ref. 54). In order to distinguish better between the twoscenarios we use histograms of the energy E of the mi-crostates realized in the Monte-Carlo procedure.55–57 Wehave collected such histograms for several system sizesand temperatures. Fig. 14 shows two representative caseson the N = 90× 90 lattice, namely T = 1.7025 |J | whichcorresponds to the maximum of the specific heat for the90×90 lattice (compare Fig. 10(a)) and T = 1.566 |J |, theestimated critical temperature of the infinite system, seeEq. (25). We always find bell-shaped almost Gaussiandistributions, which are characteristic for a continuoustransition. We never observed any signatures of a split-ting of this single peak into two, as would be expectedfor a first-order transition.55–57 Hence, the transition ap-pears to be continuous and we will now try to charac-terize its universality class further in terms of criticalexponents.

We start with the correlation length exponent ν whichcan be extracted from the finite-size behavior of theBinder cumulant: close to Tc, the Binder cumulantshould scale with the linear size of the system L as46,52,53

dUsdT≈ aL1/ν

(1 + b L−w

). (26)

0.3

0.4

0.5

0.6

ms

2

6 9 12 1518 27 36 45 90L

1.5

1.6

1.7C

/N

T/|J|=1.56

T/|J|=1.565

T/|J|=1.57

0.08

0.1

0.12

0.14

-d

Us/d

T

T/|J|=1.56...1.57

1/ν=0.24±0.02

2β/ν=0.257±0.006

α/ν=0.016±0.003

FIG. 15. (Color online) Scaling of different quantities withlinear size L for L×L lattices, J < 0, and close to the criticaltemperature T

14

line in the top panel of Fig. 15) then yields

1

ν= 0.24± 0.02 . (27)

We now turn to the order parameter exponent β whichcan be extracted from the finite-size behavior of the sub-lattice order parameter ms. The sublattice order param-eter should have a scaling behavior (see for example Refs.46 and 58)

m2s =〈~M2s

〉= L−2 β/νM2

((1− T

Tc

)L1/ν

). (28)

Specialization of (28) to T = Tc yields〈~M2s

〉∣∣∣T=Tc

= L−2 β/νM2 (0) . (29)

The middle panel of Fig. 15 shows the Monte-Carlo re-sults for m2s at three temperatures which cover the es-timate (25) for T 0. Error bars areof the order of the lines’ width in this graph. Inset: aver-age squared sublattice magnetization in the low-temperatureregion.

(with the spatial dimension d = 2) is strongly violated.On the other hand, we could use the relation (33) toestimate α, in particular if we expect it to be negative(compare, e.g., Ref. 59 for a similar situation). Inser-tion of (27) into (33) yields a very negative exponentα/ν = −1.52 ± 0.04. Again, this reflects the large ex-ponent ν. In fact, a large correlation length exponentν ≈ 4 has been found in other two-dimensional disor-dered systems.60,61 However, in those cases the large ex-ponent corresponds to approaching the critical point viaa fine-tuned direction in a two-dimensional parameterspace and there is a second, substantially smaller cor-relation length exponent.60,61 Thus, we are left with notcompletely unreasonable, but definitely highly unusualvalues of the critical exponents ν and α.

D. Critical temperature for J > 0

As mentioned earlier, for J > 0, the specific heat aloneis only mildly conclusive regarding the existence of alow-temperature phase transition to a three-sublattice or-dered state. This statement requires to be supported bythe analysis of other observables. The sublattice magne-tization (23) will tell us whether significant order is devel-oping in the low-temperature region or not. Our resultsshown in Fig. 16 show that three-sublattice order is in-deed developing, although an appreciable order developsonly at temperatures that are so low that they become in-creasingly difficult to access with increasing system size.To take a closer look at the low-temperature orderedstate, we took some snapshots of the system during thesimulation for N = 12×12 spins. A typical configurationis reproduced in Fig. 8. While the global structure cor-responds indeed to a 120◦ three-sublattice ordered state,we also observe the presence of defects. The presence

15

0.01 0.012 0.014

0.06

0.08

0.1

0.12

0.14

0 0.04 0.08 0.12 0.16 0.2

T/J

-0.1

0

0.1

0.2

0.3U

sN=6x6

N=9x9

N=12x12

N=15x15

N=18x18

FIG. 17. (Color online) Binder cumulant for the classicalmodel with J > 0 in the low-temperature region. Error barsare of the order of the lines’ width in this graph. Inset: Bindercumulant for 6 × 6, 9 × 9, and 12 × 12 spins zoomed aroundthe crossing region.

of these defects is neither a surprise nor in contradictionwith the existence of the phase transition, as they are infact necessary ingredients of the order-by-disorder mech-anism. Note also that the spins in Fig. 8 lie essentially inthe x−y-plane and –up to small fluctuations– are alignedwith the lattice.

As for J < 0, the Binder cumulant (24) allows us bothto further support our conclusions concerning the low-temperature ordered state and to obtain an estimate ofT>c . Fig. 17 shows that all the curves for the differentsystem sizes cross in a region around T ≈ 0.015J . First,this is a strong argument in favor of the existence of theordering transition. We used the smallest three systemsizes (N = 6 × 6, 9 × 9, and 12 × 12) for which we havethe best statistics to obtain an estimate for the transitiontemperature:

T>cJ

= 0.0125± 0.0009. (34)

The error bars on the data are unfortunately too largeto get precise values for the critical exponents and thusprevent us from investigating the nature and the uni-versality class of the transition. However, the fact thatthe low-temperature ordered state breaks the same sym-metries irrespectively of the sign of J suggests that theuniversality class for J > 0 is the same as for J < 0.

VI. DISCUSSION AND CONCLUSIONS

Although the Hamiltonian (1) may seem unusual inthe context of frustrated magnetism, it is instructive inmany respects and illustrates the rich phenomenology of-ten present in this subject. Either seen as the strongtrimerization limit of the kagome lattice of spin 1/2 ina magnetic field, or as a possible illustration of an or-

bital model,29–37 the underlying physics associated tothis Hamiltonian is extremely interesting for both signsof the coupling constant J .

In the quantum case (for spin 1/2) we show, by study-ing the low-energy spectra using the Lanczos method,that a thermodynamic gap of the order of 3 |J | is presentfor J < 0, while for J > 0 the gap, if present, would beat most of the order of 0.02 J . The six-fold degeneracyof the would-be ordered ground state which is predictedby semiclassical considerations is not observed in our nu-merical results, probably due to the small lattices con-sidered. These results illustrate very well how the deepquantum (S = 1/2) regime differs from the large S spin-wave predictions. The specific heat curves point to aphase transition around T ≈ |J | for J < 0, while a lowertemperature peak shows up in the positive J case. Thislast peak could be due to an ordering phase transition ata very low temperature Tc ≤ J/100. In both cases oneis tempted to envisage a finite-temperature phase tran-sition whose nature could be understood by the analysisof the classical model.

The analysis of the classical model has turned out tobe also quite interesting and instructive. For J < 0the lowest-energy configuration consists in an in-planeantiferromagnetic arrangement of the spins with givenchirality accompanied by a ‘spurious’ continuous rota-tional degeneracy which does not correspond to any sym-metry of the Hamiltonian. This pseudo degeneracy islifted by entropy at finite temperature giving rise toan ordering at low temperature as observed by MonteCarlo data which locate the transition temperature atTc/|J | = 1.566 ± 0.005. Inspection of the histograms ofthe energy close to the transition temperature gave noevidence of a first-order transition. Hence, we analyzedit within the scenario of a continuous transition and es-timated unusual values for the critical exponents α andν, strongly violating the hyperscaling relation. It shouldnevertheless be mentioned that we cannot exclude the ex-istence of a crossover scale which exceeds the lattice sizesaccessible to us. The fact that lock-in of the spin com-ponents to the lattice requires a certain length scale maypoint in this direction. An unambiguous determinationof the universality class of the transition would requireimproved methods. A first possibility is to restrict thedegrees of freedom to the in-plane configurations43 whichare realized in the low-temperature limit. Even more ef-ficiency could be gained by additionally restricting eachspin variable to the 6 spin directions which are stabilizedin the zero-temperature limit. However, it remains tobe investigated whether the second modification changesthe universality class of the transition.

For J > 0 the situation is even more interesting. Al-though a ‘spurious’ rotational degeneracy is also presentfor the antiferromagnetic 120◦ configuration (with theopposite chirality than the one for J < 0), the manifold oflowest-energy configurations is more complex. There ex-ist local discrete ‘flips’ of triangles which bring one fromthe homogeneous antiferromagnetic lowest-energy config-

16

(a)

j i Dp

3

.!2 � !/p

2.jCi � j�i/ D 1p

2

ˇ

ˇ

ˇ

ˇ

ˇ

+

�ˇ

ˇ

ˇ

ˇ

ˇ

+!

ˇ

ˇ

ˇ

E

Dp

3

.1 � !2/p

2

�

jCi � !2 j�i�

D 1p2

ˇ

ˇ

ˇ

ˇ

ˇ

+

�ˇ

ˇ

ˇ

ˇ

ˇ

+!

ˇ

ˇ

ˇ

E

Dp

3

.1 � !2/p

2

�

j�i � !2 jCi�

D 1p2

ˇ

ˇ

ˇ

ˇ

ˇ

+

�ˇ

ˇ

ˇ

ˇ

ˇ

+!

(b)ˇ

ˇ

ˇ

ˇ

ˇ

+

D 12

p2

�ˇˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

!�

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

!

C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

!�

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

!C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

!

�

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

!C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

!�

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

!˘

FIG. 18. (Color online) (a) Identification of chirality pseudo-spin states of a triangle with spin configurations on a trianglefor the underlying kagome lattice. (b) Expression of the 120◦

ordered state on a triangle in the effective model in terms ofspin configurations of the corresponding 9 sites in the under-lying kagome lattice. Note that chirality spins for the effec-tive model lie in the x-y chirality plane whereas spins on thekagome lattice point along the z-axis.

uration to another configuration with the same energy.The mechanism that gives rise to ordering is again under-stood by analyzing the entropic spectra over each of theseconfigurations. The homogeneous antiferromagnetic con-figuration has a whole branch of soft modes in its clas-sical spin-wave spectrum. Flipping one triangle to jumpto another lowest-energy configuration also destroys onesoft mode. One is then left with a scenario that can beunderstood with an Ising-type low-temperature expan-sion picture of the system, where each ‘flippable’ triangleplays the rôle of an Ising spin. The difference is in thefact that flipping one spin on an otherwise perfectly or-dered background costs no energy but an entropy, or ifone wishes a temperature-dependent pseudo energy. Anordering transition will also take place, as in a normal en-ergetic system, but at a much smaller temperature. Thistransition temperature is observed in the Monte-Carloanalysis to be at around Tc/|J | ≈ 0.0125, two orders ofmagnitude smaller than in the J < 0 case.

For J > 0, one may also wonder how the 120◦ or-dered state of the model (1) relates to the structure ofthe magnetization 1/3 state of the homogeneous kagomelattice.7,25 In order to address this question, we needto associate a variational wave function to the classical120◦ ordered state which is indicated in Fig. 8. First,we associate quantum wave functions to the three clas-sical spin directions as in Fig. 18(a). The phase factorsare chosen in order to yield a convenient representationin terms of spin configurations of a triangle after inser-

tion of Fig. 1(b) for the chirality pseudo spins. Insertioninto the 120◦ wavefunction for a triangle of the triangu-lar lattice on the left side of Fig. 18(b) then yields theexpression in terms of the 8 spin configurations of a nine-site unit of the underlying kagome lattice shown on theright side of Fig. 18(b). Note that the two terms on thefirst line of the right side of Fig. 18(b) amount exactly tothe variational wave function for the magnetization 1/3state of the homogeneous kagome lattice,7,25 as it followsfrom a mapping to a quantum-dimer model on the honey-comb lattice.24 Thus, the present results for the stronglytrimerized kagome lattice may be smoothly connected tothe 1/3 plateau state of the homogeneous kagome lattice.

To conclude, the model (1) has turned out to be avery interesting laboratory to understand the emergenceof a hierarchy of energy scales originating from differ-ent levels of order by disorder. The emergence of sucha hierarchy in a classical model is related to similar hi-erarchies in quantum systems as for example the hugedifference between the magnetic and non-magnetic gapsin the kagome spin 1/2 system at magnetization 1/3.7

Moreover, if the transitions observed in this work can beconfirmed to be continuous, the exponents will probablycorrespond to exotic models, like parafermionic confor-mal field theories.62

ACKNOWLEDGMENTS

We are grateful to P. C. W. Holdsworth, H. G. Katz-graber, F. Mila, and M. E. Zhitomirksy for useful dis-cussions, comments and suggestions. This work hasbeen supported in part by the European Science Founda-tion through the Highly Frustrated Magnetism network.D.C.C. is partially supported by CONICET (PIP 1691)and ANPCyT (PICT 1426). Furthermore, A.H. is sup-ported by the Deutsche Forschungsgemeinschaft througha Heisenberg fellowship (Project HO 2325/4-2).

Appendix A: Relation to the Heisenberg model onthe trimerized kagome lattice

The Hamiltonian (1) has already been derived severaltimes in the literature.26,38,42 Nevertheless, for complete-ness we also give a derivation.

We start from the interaction between the triangles ofthe spin-1/2 Heisenberg model on a trimerized kagomelattice

Hint = Jint∑〈i,j〉

~SA,i · ~SB,j (A1)

= Jint∑〈i,j〉

{1

2

(S+A,i S−B,j + S−A,i S+B,j

)+ SzA,i SzB,j

},

where the sum over 〈i, j〉 runs over the nearest-neighborpairs of triangles i and j in Fig. 1(a) and the cornersof the triangle A and B have to be chosen such as to

17

match the connecting bond. The ~SA,i are physical spin-1/2 operators.

In first order and for N triangles, we need to computethe matrix elements of (A1) between all 2N combinationsof the states in Fig. 1(b). Note that the expectation val-ues of the operators acting on different triangles factor-ize. Since in the present case magnetization is fixed ineach triangle to 1/3, matrix elements of S±A,i vanish, thussimplifying the derivation considerably.

For the lower left corner of a triangle i we find( 〈+|SzL,i|+〉 〈+|SzL,i|−〉〈−|SzL,i|+〉 〈−|SzL,i|−〉

)=

(1/6 −ω2/3−ω/3 1/6

)i

=1

3

(1

2− TCi

), (A2)

for the lower right corner( 〈+|SzR,i|+〉 〈+|SzR,i|−〉〈−|SzR,i|+〉 〈−|SzR,i|−〉

)=

(1/6 −1/3−1/3 1/6

)i

=1

3

(1

2− TAi

), (A3)

and finally for the top corner( 〈+|SzT,i|+〉 〈+|SzT,i|−〉〈−|SzT,i|+〉 〈−|SzT,i|−〉

)=

(1/6 −ω/3−ω2/3 1/6

)i

=1

3

(1

2− TBi

). (A4)

Using (A2)–(A4) for the matrix elements of (A1), we findthe effective Hamiltonian

Heff. =Jint9

∑〈i,j〉

(1

2− TAi

)(1

2− TCj

)

+∑〈〈k,j〉〉

(1

2− TAk

)(1

2− TBj

)

+∑[[k,i]]

(1

2− TCk

)(1

2− TBi

) , (A5)

where the three sums run over the different bond direc-tions as sketched in Fig. 1(a). Since the sum over rootsof unity vanishes, we have TAi + T

Bi + T

Ci = 0. Hence,

Eq. (A5) can be rewritten as

Heff. =Jint9

∑〈i,j〉

TAi TCj +

∑〈〈k,j〉〉

TAk TBj +

∑[[k,i]]

TCk TBi

+N Jint

12. (A6)

Up to an additive constant, this is nothing but Eq. (1)with J = Jint/9. The intra-triangle coupling needs tobe chosen positive in order for the two states shown inFig. 1(b) to be ground states of a triangle. Accordingly,it is natural to also choose the inter-triangle coupling Jintpositive, i.e., J > 0.

Very similar arguments can be applied, e.g., to spin-less fermions with nearest-neighbor repulsion,38 leadingto the same effective Hamiltonian.

1 C. Lacroix, P. Mendels, and F. Mila, eds., Introductionto Frustrated Magnetism, 1st ed., Springer Series in Solid-State Sciences, Vol. 164 (Springer, Berlin, 2011).

2 P. Sindzingre, C. Lhuillier, and J.-B. Fouet, Int. J. Mod.Phys. B 17, 5031 (2003); G. Misguich and C. Lhuillier, inFrustrated spin systems, H. T. Diep editor, World-Scientific(2005).

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