Quantum Disordered Ground States in Frustrated Antiferromagnets with Multiple Ring Exchange Interactions

0.18 MB·English

by  Satoshi Fujimoto

Quantum Disordered Ground States in Frustrated Antiferromagnets with Multiple
Ring Exchange Interactions
Satoshi Fujimoto
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
(Dated: February 2, 2008)
We present a certain class of two-dimensional frustrated quantum Heisenberg spin systems with
5
multipleringexchangeinteractionswhich arerigorously demonstratedtohavequantumdisordered
0
ground states without magnetic long-range order. The systems considered in this paper are s=1/2
0
antiferromagnets on a honeycomb and square lattices, and an s=1 antiferromagnet on a triangular
2
lattice. We find that for a particular set of parameter values, the ground state is a short-range

n resonating valence bond state or a valence bond crystal state. It is shown that these systems are
a closely related to the quantum dimer model introduced by Rokhsar and Kivelson as an effective
J low-energy theory for valencebond states.

8
2
I. INTRODUCTION in two dimensions with multiple ring exchange interac-

] tions is equivalent to the QDM. These systems exhibit a
l
e short-range RVB state or a valence bond crystal (VBC)
- QuantumfrustratedHeisenbergantiferromagnetshave
state, depending on the values of their parameters. Our
r attracted a great deal of interest in connection with the
t results clarify the important role played by the ring ex-
s search for exotic phases, such as spin liquids. Ander-
. change interactions in the realization of quantum spin
t son proposed a resonating valence bond (RVB) state as
a liquid states24.
a prototype of spin liquids three decades ago1. Since
m Thispaperisorganizedasfollows. InSec.II,wepresent
then,anumberofworkshavestudiedtheconjecturethat
- results of an s = 1/2 Heisenberg spin system with mul-
d bothgeometricalfrustrationandstrongquantumfluctua-
tiple ring exchange interactions defined on a honeycomb
n tionmaydestroymagneticlong-rangeorderandstabilize
lattice. It is shown that for a particular parameter the
o a quantum disordered liquid-like ground state without
rigorous ground state is the short range RVB. In this
c symmetry breaking2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17. For
[ RVB state, spin-spin correlations decay exponentially,
example, it is believed quite likely that quantum spin
and therefore there is no magnetic long range order. On
1 systems on Kagome lattices and pyrochlore lattices can
the other hand, correlations between spin-singlet dimers
v exhibit quantum spin liquid states, though the elu-
exhibit power law long-distance behavior, implying the
8 cidation of their ground states remains an important
9 existence of low-lyingspin-singlet gaplessexcitations. In
unsolved problem. In related works, the quantum
6 Sec.III and Sec.IV, we consider an s = 1/2 model on a
dimer model (QDM) introduced by Rokhsar and Kivel-
1 squarelatticeandans=1modelonatriangularlattice,
son has been extensively studied as an effective low-
0 respecticely. DiscussionandsummaryaregiveninSec.V.
5 energy theory of quantum disordered states with short-
0 rangeantiferromagneticcorrelationandaspinexcitation
/ gap18,19,21,22,23. The important feature of the QDM is
t II. s=1/2 HEISENBERG ANTIFERROMAGNET
a that its Hilbert space is spanned by only dimer covering
ON A HONEYCOMB LATTICE
m
statescomposedofsingletpairsofnearest-neighborspins,
- and this is the key to its success in describing quantum
d A. Model Hamiltonian
disordered states18. It was shown in the pioneer paper
n
by Rokhsar and Kivelson that a short-range RVB state
o
We consider the s = 1/2 quantum Heisenberg anti-
c is realized in the QDM as its exact ground state. It was
ferromagnet on a honeycomb lattice (see FIG.1(a)), of
: alsoarguedby severalauthorsthat the QDM may be an
v which the ground state is exactly obtained. The Hamil-
effective low-energytheory of quantum antiferromagnets
i
X on the kagome lattice. tonian is given by, H =HK+HR, with
r However, it is not unclear how to derive the QDM di- 1 3
a rectly from the original quantum spin Hamiltonian by HK =J1(cid:2)XSi·Sj + 2 X Si·Sj + 8N
truncating the Hilbert space. Indeed, it appears difficult (ij) <ij>
2
to find a general answer to this question. Therefore, in + X (Si Sj)(Sk Sl)(cid:3) (1)
thispaper,wedonotseeksuchageneralresult. Instead, 5 · ·
j k=j l=k l=√3
to obtain better insight regarding the mechanism stabi- | − | | −| | −|
(ij)(ik)(il)
lizingquantumspinliquids,weattempttomakeprogress
towarddeterminingwhatkindsofquantumspinsystems HR =XPaM[J2Ka+J3Va] (2)
possess a low-energy sector described by the QDM. a
7 1
Themainresultofthispaperistherigorousproofthat Ka = X Sµ Sν + X Sµ Sν
16 · 8 ·
acertainclassofquantumantiferromagneticspinsystems
(µν) a <µν> a
∈{ } ∈{ }
2
1
+ X Sµ Sν onthehoneycomblatticearemacroscopicallydegenerate
8 · ground states of H . An example of the dimer covering
µν a K
≪ ≫∈{ } states is shown in FIG.1(c). The spin-spin correlation
a a a
−352[XP4a+XP4a− 35XP4a]+ 81P6a+ 6143 (3) fnuenncttiaiolndsefcoary,thiensdeicVaBtinsgtattheseeaxbhsiebnitceloonfg-mdaisgtnaentciecelxopnog--
A B C
range order. Also, the result of the single-mode approx-
1
Va = X Sµ Sν imation supports the existence of a spin excitation gap
−16 ·
(µν) a above the spin-singlet ground states. However, because
∈{ }
1 eachdimer state breaks the spatialsymmetry ofthe sys-
+ X (Sµ Sν)(Sα Sβ) tem,aquantumspinliquidstateisnottheuniqueground
4 · ·
(µν)(αβ)∈{a} state of the Klein model.
µ<ν<α<β
1
− X (Sκ·Sλ)(Sε·Sγ)(Sη·Sξ)+ 32(4) (a) (b) A
(λ(εκ)λ()γ(ηε)γ()ξ(κη)ξ,)κ∈<{εa<}η i µ νj
η
7 7
PaM = 64 + 48 X Si·Sj n ξ a κ k B
ij a˜ y λ
∈{ }
+ 7 X (Si Sj)(Sk Sl) m l C
60 · · x
ijkl a˜
∈{ }
1
+ X (Si Sj)(Sk Sl)(Sm Sn) (5)
45 · · · (c) (d)
ijklmn a˜
∈{ }
Here, (ij), < ij > and ij represent, respectively,
≪ ≫
thenearest,next-nearestandnext-next-nearestneighbor
pairs,andJ J ,J >0. Thesummation is
1 ≫ 2 3 Pµν... a
taken over sites in the a-th hexagon. The operato∈r{s}Pa
4
and Pa represent, respectively, the 4-body and 6-body
6
ring exchange interactions25,
1 FIG. 1: (a) Honeycomb lattice. (b) Three kinds of four-
P4a = 4 +XSµ·Sν +4Gijkl, (6) spin exchange processes on a hexagon. (c) An example of
µ<ν a dimer covering state. The arrows indicate the phase con-
G = (S S )(S S )+(S S )(S S ) ventionof thesinglet states. Anarrow from itoj represents
ijkl i j k l i l j k
(S ·S )(S ·S ) · · (7) Oij|0i. Hexagons containing circles are covered with unflip-
i k j l pable dimers. (d) Staggered VBC.
− · ·
1 1
P6a = 16 + 4 XSµ·Sν + X Gµνηǫ+4Sijklm(n8,)naInncethaemfoonllgowthinegs,ewdeimsherowcotvhearitnHgRstaintetrsoadnudc,esaas aresroe--
µ<ν µ<µ<η<ǫ
sult, selects an almost unique ground state that can be
Sijklmn = X ( 1)P(Sα Sβ)(Sγ Sδ)(Sǫ Sφ),(9)expressedasasuperpositionofVBstates,preservingthe
− · · ·
C(i,j,k,l,m,n) spatial symmetry. To this end, it is convenient to uti-
lize the Schwinger boson representation for spin opera-
wherethe sumineq.(9)is takenoverallcombinationsof
(i,j,k,l,m,n) in three different pairs depicted in FIG.2. tors28: Si+ = u†idi, Si− = d†iui, Siz = (u†iui −d†idi)/2,
( 1)P is 1 (-1) when a combination is generated by an and Sˆi = (u†iui + d†idi)/2. Here the average value of
e−ven (odd) number of transpositions between different Sˆ is 1/2. We also introduce the spin-singlet operator
pairs. The summations a are, respectively, taken (u d d u )/√2. Then, each dimer state is ex-
PA,B,C Oij ≡ i j − i j
overthe four-spinconfigurationsof types A, B, andC of pressed as
the a-th hexagon, as depicted in Fig.1(b). The operator
P(daMepipcrtoedjecatss it,hej,sikx,slp,inms,sunrrinouFnidgi.n1g(at)h)eoan-ttohthheexasguobn- |Di= Y Oi†j|0i. (10)
(ij)
space with total spin S = 3. The summation ∈D
Pij... a˜
istakenoverthesesixsitesinthea-thhexagon. H is∈t{h}e Here, is the set of nearest-neighbor pairs correspond-
K
D
Hamiltonianofthes=1/2Kleinmodelonahoneycomb ing to a particular realization of the dimer covering.
lattice, which was studied in detail by Chayes et al.26,27 We choose the phase convention of as depicted in
ij
O
ThegroundstatespaceofH isspannedbyvalencebond FIG.1(b).
K
(VB) states, which are formed from spin-singlet pairs of We now show that H is a variant of the QDM repre-
R
nearest-neighbor sites. Thus, all dimer covering states sented by spin operators. The Hamiltonian of the QDM
3
(1) µ ν (2) µ ν (3) µ ν Thefirstpairofrelationshereimpliesthat a(1) and a(2)
T T
η κ η κ η κ act as the potential terms of HQDM. In the following, it
is convenient to use the representations
ξ λ ξ λ ξ λ
(4) µ ν (5) µ ν (6) µ ν Ta(1) =h†1ah1a, Ta(2) =h†2ah2a, (15)
with
η κ η κ η κ
ξ λ ξ λ ξ λ h1a =OνµOλκOηξ, (16)
h = . (17)
2a νκ λξ ηµ
O O O
(7)µ ν (8) µ ν (9) µ ν Next, we introduce the operator,
η κ η κ η κ
ξ λ ξ λ ξ λ Ka ≡−h†1ah2a−h†2ah1a, (18)
whichis(ascanbeseenbyusingtherelationsOµ†νOκ†λ+
(10)µ ν (11)µ ν (12)µ ν Oµ†λOν†κ =Oµ†κOν†λ, etc.) expressed as
η κ η κ η κ
2 9 15
1
ξ λ ξ λ ξ λ Ka =−2(X Ta(m)−Ta(3)+ X Ta(m)− X Ta(m))(.19)
m=1 m=4 m=10
(13)µ ν (14)µ ν (15)µ ν Note that satisfies the following relations:
a
η κ η κ η κ K
1 1
ξ λ ξ λ ξ λ Ka|1i=−|2i+ 4|1i, Ka|2i=−|1i+ 4|2i. (20)
These actions of are similar to those of the kinetic
a
K
FIG. 2: The spin-singlet states of a hexagon. termofH . Toestablishthecompletesetofrelations
QDM
between the QDM and the operators , (1) and (2),
a a a
K T T
weneedtoverifytheiractionsonunflippabledimersthat
is18,22
are not flipped by , as shown in FIG.1(b). It is easily
a
K
shown that the actions of and (1)+ (2) on the un-
HQDM =−t[|1ih2|+|2ih1|]+V[|1ih1|+|2ih2|]. (11) flippable dimers generate dKimer cToveringTstates that are
not in the ground-state space of H . These non-ground
Here, 1 and 2 arethe dimercoveringstatesofasingle K
| i | i states contain at least one singlet pair in the set of six
hexagoncorresponding,respectively,tothestates(1)and
spins surrounding the hexagon to which an operation is
(2)ofFig. 2. The firstterminH is the kineticterm
QDM
applied;i.e. thespinsoni,j,k,l,m,ninFIG.1(a)when
that transfers 1 to 2 and 2 to 1 . The second term
| i | i | i | i the operators are applied to the a-th hexagon. Thus,
is the potential term. To construct operatorsthat act as
the unwanted states are excluded by the projection onto
the first and second terms of H and are expressed
QDM
the maximum spin states of these six spins. This is
in terms of spin operators, we consider all possible spin
carried out by applying the operator M. Also, using
singlet states of a single hexagon, depicted in Fig.2. Al- Pa
though the states except (1) and (2) in Fig.3 are not the the relation Si · Sj = 14 − Oi†jOij, we find Ka = Ka,
ground state of HK, the kinetic term for the states |1i Ta(1) +Ta(2) = Va, thereby arriving at the Hamiltonian
and 2 are expressed by the linear combination of these (2).
| i
states. Toseethis,weintroducethedensityoperatorsfor
the singlet dimers on a hexagon corresponding to these
fifteen states, B. Exact ground state
Ta(1) =O통OνµOλ†κOλκOη†ξOηξ, (12) We now show that the Hamiltonian H = HK + HR
has the same ground state properties as the QDM in
a certain parameter region. The lowest energy sector in
Ta(2) =Oν†κOνκOλ†ξOλξO熵Oηµ, (13) theHilbertspaceofH isspannedbyVBstates,provided
that the groundstate of H canbe expressedas a linear
R
and so forth. It is obvious that (m) m = m with combinationof D . To findthe groundstate,itis useful
m=1,2,...,15. In particular,we aTre con|ceirned|wiith the to rewrite HR in|toithe form
action of Ta(1) and Ta(2) on the states |1i and |2i, i.e. HR = J2+2 J3 XPaM(h†1a−h†2a)(h1a−h2a)
(1) 1 = 1 , (2) 2 = 2 , a
Ta | i | i Ta | i | i J J
(1) 2 = 1 1 , (2) 1 = 1 2 . (14) + 3−2 2 XPaM(h†1a+h†2a)(h1a+h2a).(21)
Ta | i −4| i Ta | i −4| i a
4
For J > J , the energy eigenvalues of H are non- C. Dimer-dimer correlation in the spin liquid state
3 2 R
negative, and only unflippable dimer states are zero en-
ergy states. Therefore the ground state of H is a dimer
Incontrastto the magneticcorrelation,the dimer cor-
covering without flippable dimers. If we simply impose
relation in the spin liquid state obeys a power law, as
periodic boundary conditions in both x and y direc-
shown below. In the ground state space, the dimer-
tions, the ground state remains macroscopically degen-
dimercorrelationisidenticaltothatoftheclassicaldimer
erate. This degeneracy is removed by imposing periodic
modelonahoneycomblattice,whichbelongs to the uni-
boundaryconditionswithashiftbyonehexagoninthey
versalityclassofthe Gaussianmodelwithcentralcharge
direction. Then,theuniquegroundstateisthestaggered c = 1 30. The correlation function for the dimer density
VBC (see FIG.1(d)), which was previously identified by operatoratthe bond(ij), N = S S +1/4,displays
Moessner et al.22 ij − i· j
the long-distance behavior
The energy levels of H are non-negative also in the
R
case of J2 = J3. Although the staggered VBC is obvi- N N N N [cos(4π|ri−rl|/3)−1],(23)
ously a zero energy state here too, in this case there also h ij lmi−h ijih lmi∼ r r 2
i l
exists a non-trivial, liquid-like ground state. It is eas- | − |
ily seen that the equal-amplitude superposition of dimer when the two dimers on (ij) and (lm) are in the same
states |Gi ≡ Pi|Dii is the zero energy state of HR if direction30. This power law decay implies that the low-
the summation Pi is restricted within a sector of dimer energy properties are governed by gapless non-magnetic
statesrelatedbythelocaloperations(14)and(20). Gen- excitations. Thelow-lyingexcitationenergyiscomputed
erally, a class of dimer states connected by these local using the single-mode approximation. We assume that
oOpneraathioonnesyicsocmhabralacttteirciez,ehdobwyevaetro,pinolaodgidciatlionnumtobaert1o8p,2o9-. theexcitedstatetakestheform|ki=P(ij)eikriδNij|Gi,
with δN = N N . By definition, k is orthogo-
ij ij ij
logicalnumber,thetotalnumberofdimersineachdirec- −h i | i
nal to the ground state; i.e. k G = 0. Therefore, the
tion is also conserved by the local flip. Thus, G is the h | i
| i excitation energy is obtained from
unique ground state of H in a given sector specified by
these conserved quantities. This state, preserving both
G[δN [H,δN ]]G
k k
the spin-rotational and spatial symmetries, is a spin liq- εk = h | − | i. (24)
GδN δN G
uid state, and it is in the same universality class as the h | −k k| i
short-range RVB state. In fact, the upper limit of the
The denominator of the right-hand side of this relation
spin-spin correlationfunction for this state exhibits long
can be computed from the correlationfunction N N
distance exponential decay expressed by26 h ij kli
and behaves as ln(R/a), where a and R are the lat-
∼
tice constant and the system size, respectively. When
|hG|SizSjz|Gi|≤(3/2)2−|ri−rj|. (22) the excitationenergyis sufficiently smallerthanthe spin
excitation gap of H , the total number of spin-singlet
K
dimers is conserved, and also, H does not contribute
K
These results show that the QDM is realized as a spin to the low-lying excitations. Thus, the numerator of the
systemwithmultipleringexchangeinteractions,andthe right-hand side of the above relation for ε reduces to
k
pcaosientJo2f=thJe3QcDorMre1s8p.onds to the Rokhsar-Kivelson(RK) PofitlhhGe|kδN2-itjeHrmRδiNslemxrpirrels|Gseidk2infotresrmmasllokf.tThehethcoreeeffi-bcioednyt
InthecaseofJ >J ,unfortunately,wehavenotbeen dimercorrelationfunctions,whichexhibitsaleadinglog-
2 3
able to derive an exact ground state of H analytically. arithmicbehavioroftheform ln(R/a). Hence,thelog-
∼
However,theplaquetteVBCstateobtainedbyMoessner arithmic divergences in the denominator and numerator
et al.22 for the QDM in this parameter region is indeed ofεk cancelout. Wethusfindthatthegaplessexcitation
an exact eigenstate of H, and it is possible that this is energy in the spin-singlet sector is given by εk ak2.
∼
the ground state. This dispersion relation is in accordance with Henley’s
conjecture based upon height representations31.
We now consider the Hamiltonian H without the pro-
jection operator M in H . Then, the model is more
Pa R (a) (b) (c)
realistic, though its exact ground state is no longer ac- i j
cessible by analytical method. The operation of Ka and p k
V onunflippabledimersdonotvanish,butcreatestates a
a o l
which are not in the ground state sector of H . In this
K n m
situation, the staggered VB solid state is not the zero-
energy state even for J J , and pushed up to a state
3 2
withhigherenergyoford≥erJ . Thus,itisexpectedthat, FIG.3: (a)Dimercoveringonasquarelattice. (b)Aground
1
at the RK point J = J , the spin liquid state is the state of HK that is not a ground state of HR. (c) The stag-
2 3 gered VBcrystal.
most plausible candidate for the true ground state pro-
vided that J J , J .
1 2 3
≫
5
III. s=1/2 HEISENBERG lattice is also givenby H =HK+HR. The KleinHamil-
ANTIFERROMAGNET ON A SQUARE LATTICE tonian H for this system has the same form as that
K
given in Eq.(1), but, here it is defined on a triangular
Theanalysispresentedintheprevioussectionsiseasily lattice, and it is expressed in terms of the s = 1 spin
extended to other lattice structures. For the model with operators. The groundstate space of this Kleinmodel is
s=1/2onthesquarelattice,theHamiltonianisalsogiven spanned by states fully packed with loops composed of
byH =H +H ,butinthiscasewehavethefollowing: singlet dimers on the three kinds of hexagons arranged
K R
sequentially in order the A, B, C, A, B, C,... We show
1 1 1 an example of loop covering states in FIG.4. Resonance
HK = J1[5XSi·Sj + 5 X Si·Sj + 5 X Si·Sj among these loop covering states is introduced as,
(ij) <ij> ij
≪ ≫
+ 1 X (Si Sj)(Sk Sl)+ 3 N], (25) HR =HA+HB+HC, (30)
40 · · 64
i6=j,kj=6=kl,,il6=k,l where HA, HB and HC are defined on hA, hB and hC,
6 respectively, and take the form of Eq.(2) with M re-
HR = XPaM[J2Ka+J3Va], (26) placedbythe projectionontothemaximumspinPstaateof
a the thirteen nearest neighbor spins of the a-th hexagon
1 1
Ka = −2 X Si·Sj + 4Psa4, (27) on the parent triangular lattice:
(ij) a
∈{ } 12 ( S )2 s(s+1)
Va = 1 1 X Si Sj PaM = Y Pi∈1{8a˜2} is(s−+1) . (31)
8 − 4 · s=0 −
(ij) a
∈{ }
+ X (Si Sj)(Sk Sl), (28) Here the summation Pi a˜ is taken over the nearest
· · ∈{ }
neighbor sites of the a-th hexagon. (See FIG.4(b).)
(ij)(kl) a
i<j<k∈<{l} The ground state properties are similar to the mod-
Q Q2 Q3 Q4 elsconsideredinthe previoussections. Atthe RKpoint,
M = a a + a + a , (29)
Pa −140 − 280 1260 2520 theequal-amplitudesuperpositionofloopcoveringstates
is the unique ground state in a given sector. The loop
withQa =Pij a˜ Si·Sj. HerethesummationPij a˜ statistics of this state are described by a classical loop
istakenoverth∈e{ne}arestneighborsitesofthea-thsqu∈a{r}e model referred to as the “red-green-blue model”. The
(the eight sites on i,j,k,l,m,n,o,p in FIG.3(a)). The op- loopcorrelationfunctions ofthis state exhibit power-law
eratorPa isthe 4-bodyringexchangeinteractiononthe long-distance behavior32. Therefore the excitation en-
s4
a-thsquare. Thegroundstatespaceofthes=1/2Klein ergy in the spin singlet-singlet channel of this system is
model on a square lattice is more complicated than that gapless. This ground state represents a new universal-
of the model on a honeycomb lattice. In the case of a ity class of a spin liquid described by the quantum loop
squarelattice, inadditionto dimer coveringstates,some model.
states with spin-singlet pairs on next-nearest neighbor
sites are also in the ground state space. An example of
suchstatesisdepictedinFIG.3(b). However,fortunately, V. DISCUSSION AND SUMMARY
such singlet states are not in the ground state space of
HR,andtheycanbeexcludedfromourconsideration. It In this paper, we have presented rigorous results for
is thus found that at the RK point J2 = J3, under peri- two-dimensional quantum spin systems with multiple
odic boundaryconditionsinboth xandy directions,the ringexchangeinteractionsthatpossessquantumspinliq-
ground state of HK +HR is the short-range RVB state uid ground states. It should be noted that Misguich et
PD|Di, where |Di is the dimer covering state on the al. andLiMingetal. showedthatforthes=1/2Heisen-
square lattice. The low energy properties of this short- berg antiferromagnet on a triangular lattice, sufficiently
range RVB state were investigated in detail by Rokhsar strongringexchangeinteractionsdestroymagneticlong-
and Kivelson18. For J3 > J2, the staggered VBC state range order, and stabilize a spin liquid state 16,17. The
depicted in FIG.3(c) is the unique ground state. spin liquid state found by them has a remarkable simi-
larity with the ground state discussed in the present pa-
per: exponentiallydecayingspin-spincorrelations,anda
IV. s=1 HEISENBERG ANTIFERROMAGNET large number of spin singlet excitations in the spin gap
ON A TRIANGULAR LATTICE (atleastfor some parameterregions). Since the classical
Heisenberg model with ring exchange interactions does
In this case, we divide the triangular lattice into not possess macroscopic degeneracy, we speculate that
three honeycomb lattices, h , h and h , as shown in the underlying structure of the model may have a close
A B C
FIG.4(a). Here, each site consists of two sites of two connection with the Klein model, which is the origin of
different honeycomb lattices. The Hamiltonian on this singlet gapless excitations in our models.
6
(a) These observations imply that the realization of the
spinliquidstateduetoringexchangeinteractionsmaybe
genericinquantumantiferromagnets,thoughthemodels
considered in the present paper is rather complicated.
The multiple ring exchange interaction is a result of
strong quantum fluctuations, which destabilize conven-
tional magnetic orders. In real systems, effects of such
strongquantum fluctuations havebeen extensively stud-
ied so far in connection with magnetism of solid 3He25.
(b) Also, it has been discussed that multiple ring exchange
interactions may affect significantly the magnetic struc-
tures of NiS . Thus, there is a possibility that the quan-
2
tum spin liquid state due to ring exchange interactions
a
may be realized in real systems with sufficiently strong
multiple ring exchange interactions.
FIG. 4: (a) Triangular lattice divided into three honeycomb
Acknowledgments
lattices, hA (black solid), hB (black dotted), and hC (grey).
The bold lines represent an example of loop covering. (b)
Open circles represent the nearest neighbor sites of the a-th The author is grateful to S. Miyashita and M. Os-
hexagon. The a-thhexagon is denoted by thick lines. hikawa for valuable discussions. This work was partly
supported by a Grant-in-Aid from the Ministry of Edu-
cation, Science, Sports and Culture, Japan.
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