-
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
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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
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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
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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
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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
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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.
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