XY ring-exchange model on the triangular lattice

9 Pages·0.23 MB·English

by  Leon Balents

XY ring-exchange model on the triangular lattice
Leon Balents1 and Arun Paramekanti1,2
1Department of Physics, University of California, Santa Barbara, CA 93106–4030
2Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106–4030
3 We study ring-exchange models for bosons or XY-spins on the triangular lattice. A four-spin
0 exchangeleadstoamanifold ofgroundstateswithgapless excitationsandcriticalpower-lawcorre-
0
lations. Withanearest-neighbourexchange,fluctuationsselectafour-foldferrimagneticallyordered
2
ground state with a small spin/superfluid stiffness which breaks the global U(1) and translational

n symmetry. We explore consequences for phase transitions at finite temperature and in an in-plane
a magnetic field.
J
4 PACSnumbers:
1

] I. INTRODUCTION liquid”) on the square lattice13. This was essentially
n shownbyperturbingaroundthepurering-exchangelimit
o
(J = 0) where the Kagome and square lattices respec-
c Multi-spin exchange models incorporating ring- tively have (L2) and (L) conserved quantities. It is
- exchange processes have been of interest since the early O O
r thus worthwhile to examine cases where such extensive
p studiesofmagnetisminsolidHelium-31,2. Ring-exchange
symmetries are absent from the outset even in the pure
u processes could also play an important role in Wigner
ring-exchangemodel,asisthe caseonthe triangularlat-
s
crystals near the melting density3,4, and in Mott in-
. tice.
t
a sulators which retain a fair degree of local charge The principal results of this paper are: (i) We show
m fluctuations5. Indeed, neutron scattering experiments6
that the U(1) four-spin exchange model on the triangu-
in insulating La CuO have shown that aspects of the
- 2 4 lar lattice (with S = 0 or half-filling for the bosons)
d spin wave dispersion in the antiferromagnet may be un- z
has a manifold of degenerate ground states and the cor-
n derstood by invoking ring-exchange terms. Another rea-
relation functions are critical in the ground states. We
o
sonfor the interestin suchmodels is that they may sup-
c identifytheappropriatesymmetriesandconservedquan-
port spin-liquid phases which are translationally invari-
[ tities which give rise to this. (ii) Perturbing in the
antMottinsulatorswithnomagneticorder,asindicated
nearest neighbor exchange (boson hopping) we find that
2 from numerics on triangularand Kagome lattices7. This
v fluctuations select a four-fold set of states from the de-
has been established analytically in some models with
9 generate manifold and the system develops long-range
U(1) symmetry8,9, which may also be viewed as boson
8 order at zero temperature. The ordered states break
5 models. Finally,manymodelssuchasthequantumdimer the global U(1) symmetry as well as translational sym-
2 modelonatriangularlattice10,theeasy-axisversionofa metry. (iii) In the ordered ground states, the super-
1 generalizedHeisenbergmodelontheKagomelattice8and fluid/spin stiffness J2/K and is very small for small
2 the easy-axisantiferromagnetonthe pyrochlorelattice11 ∼
J/K. At any small nonzero temperature, the U(1) sym-
0 may be mapped onto effective ring-exchange models in
/ metry is restored and we are in a phase with power-law
t their low energy subspace of states. Thus, understand-
a phase/spin order, which gives way to a disordered phase
ing the phases and phase transitions in ring exchange
m viaaBerezinskii-Kosterlitz-Thouless14(BKT)transition
models is important. for T > T J2/K. However, the discrete symme-
- BKT
∼
d In this paper, we will focus on an XY (“easy-plane”) try of the broken translations is not destroyed until a
n ring-exchange model on the triangular lattice with both higher temperature T J. We analyze phase transi-
c
o ∼
4-spinand2-spinexchange terms (with strengths K and tions at finite temperature and with in-plane magnetic
c
J respectively) in the regime J K. This model is inter- fields within a Landau theory, and discuss the phase di-
:
v esting from severalpoints of vie≪w. First, severalsystems agram and experimental consequences.
i
X suchas WignercrystalsandcertainorganicMottinsula- Theoutlineofthepaperisasfollows. InSectionII,we
torsformatriangularlatticeofspinswithpossiblyappre- will present the Hamiltonian. In Section III we analyze
r
a ciablering-exchangeprocesses,andtheXY spinproblem its symmetries and conserved quantities, and define the
is one tractablelimit of suchmodels. Second, easy-plane dual representation of the model which allows us to an-
magnetsontriangularlatticesareknowntoexist(though alyze the instabilities of the system towards charge and
mostlyasstackedlayersformingathreedimensionalsys- energy ordering. We present arguments and numerical
tem) and this could be of some relevance to them — results to show that the model with J = 0 has criti-
in fact, these systems have motivated several studies on cal power-law correlations at T = 0. In Section IV, we
easy-planetriangularquantum Heisenbergmodels12. Fi- perturbatively analyze effects on introducing the near-
nally, earlier work by some of us has shown that re- estneighborexchange(bosonhopping)andfindaphase-
lated models on the Kagome lattice are fractionalized8, orderedstate. We also discuss the phase diagramwithin
whereasthey supporta criticalphase (the “excitonBose Landautheoryasafunctionoftemperatureandin-plane
2
For K <0, the leading perturbative correctionson in-
1 − 2 cludinganonzeroJ are (J),andtheenergyandphases
+ 2 O
− depend on the sign of J — for J <0 we get a ferromag-
P +
4 − 3 + netic phase, while J >0 leads to a √3×√3 Neel order,
2− 3+ 1+ P 3 ^b wwehimchayariegnaolrsoetthhee4g-rsopuinndtesrtmat)e.sTfhourslatrhgeere|Ja/rKen|o(wphhaersee
−
+ P transitionsatanynonzeroJ andweonlyobtainthewell-
− 4 a^ studied phases.
1 4
For K >0, as appropriate for say spin degrees of free-
dom in an electronic Mott insulator, it turns out that
(a) (b)
the leading perturbative corrections to the free energy
are (J2). To this order, the free energy is indepen-
FIG. 1: (a) The three kinds of plaquettes on the triangular O
dentofthesignofJ andweobtainaferrimagneticphase
lattice. The ring-exchange process involves the spins located
atthepoints1–4ofeachplaquette. Thelabellingischosento which breaks the global U(1) invariance as well as four-
coincidewiththatappropriateforthedefinitiondualvariables fold translational symmetry. The physics of this phase
inthedualmodeldiscussedinSectionII(C),thesitesP form and the phase transitions out of it will be the focus of
thesitesofthedualKagomelattice. (b)Rotatingϕ→ϕ+π thispaper. Forlarge J /K weofcourserecoverthecon-
onthesitesindicatedbyopencircles(whichform oneoffour ventional phases men|tio|ned above.
possible triangular sublattices) changes the sign of the ring- To make an estimate of the coupling constants in one
exchange term K in the Hamiltonian. Also shown are the
case,letusimaginestartingfromtheHubbardmodelfor
basis vectorsaˆ,ˆb for the triangular lattice.
electrons (with nearest neighbor hopping) at half-filling
on a triangular lattice and perturbing in t/U to derive
an effective spin model with ring-exchange terms in the
insulator. This takes the form
magnetic-field. We close with a discussion and specu-
lations for three-dimensional generalizations and SU(2)
H = K [(S S )(S S )+(S S )(S S )
invariant models in Section V. spin 1· 2 3· 4 1· 4 2· 3
X2
(S S )(S S )]+ J S S (2)
1 3 2 4 i,j i j
− · · ·
II. MODEL Xi,j
where the first term involves all 4-site plaquettes on the
We define the model in rotor variables as triangular lattice. Adapting results from Ref.5 to this
case, we find J = 4t2/U 28t4/U3, J = J = 4t4/U3,
′ ′′
H = K cos(ϕ1 ϕ2+ϕ3 ϕ4) K = 80t4/U3. For U/t−= 6, numerical results15 show
− −
XP thatthemodelisinaninsulatingphasethoughstillclose
+ J cos(ϕ ϕ )+ U (n n¯)2 (1) to the metal-insulator boundary;in this caseJ/t=0.53,
i j i
− 2 − J /t = J /t = 0.02, K/t = 0.37. Clearly, it seems
Xi,j Xi ′ ′′
h i that one can describe the spin degrees of freedom in
the insulator by setting the further-neighbor couplings
where ϕ is the phase of the boson variable (ϕ = 0 and
i i
J = J = 0 and retaining only nonzero J,K. Indeed,
ϕ =2π areidentified) andn is the the canonicallycon- ′ ′′
i i exact-diagonalizationstudies7findaspin-liquidphasefor
jugatebosonnumber(respectivelytheangleandangular
a closely related model in qualitative agreement with
momentum of a U(1) rotor),satisfying the commutation
the Monte Carlo results on the Hubbard model15. It
relation ϕ ,n = iδ . The terms ϕ ...ϕ in the first
i j i,j 1 4
thus seems profitable to understand the above model for
term of H(cid:2) deno(cid:3)te angles around 4-site plaquettes of the
J/K 1 as a starting pointto analyzethis full problem.
triangularlattice—therearethreesuchkindsofplaque- ≪
ForU/K , the Hamiltonianin Eq.1is preciselythe
ttes as shown in Fig. 1(a). The J-term denotes nearest- →∞
XY limit of the above model obtained by setting terms
neighbor hopping of bosons while U denotes a repulsive
containing S to zero.
interaction between bosons. With U/K , we can z
→ ∞
identify Sz(r) = nr 1/2, S±(r) = exp( iϕr) and this
− ±
model reduces to a S = 1/2 XY quantum spin model
III. MODEL WITH J =0
with n¯ =1/2 corresponding to total S =0.
z
With J = 0, changing ϕr φr = ϕr+π on the sites
indicated by opencircles in F→ig.1(b) changes the signof A. Symmetries
K (there are four such choices of a triangular sublattice
on which to make this transformation, as is clear from For J = 0, we can change the sign of K by shifting
the figure). Thus, for J = 0, the ground state energy is ϕr φr = ϕr + π on any one of four sublattices of
→
independent of the sign of K. For nonzero J however, thetriangularlatticeindicatedearlier(wewillworkwith
the sign of K is important. these ‘rotated’ variables φr and a ferromagnetic 4-spin
3
−
term for convenience). This means shifting φr φr+π
on any two such sublattices leaves the action→invariant − + + − +ε
andisasymmetryoperation. Wecanidentifythiswitha +(cid:0)(cid:1)r(cid:0)(cid:1) + +ε(cid:0)(cid:1)r(cid:0)(cid:1) −ε
conservationoftotalbosonnumbermodulotwoonalter- (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)
nate rows of the triangular lattice. These rows can run − + + − −ε
in any of the three symmetry directions of the lattice, −
this corresponds to a total of four symmetry operations
including the identity.
(a) (b)
TherotorHamiltonianabovedescribesbosonshopping
FIG.2: (a) The original triangular lattice (dashed lines) and
onplaquettesofthetriangularlattice. ForJ =0,thedy-
the dual Kagome lattice (bold lines). The sites ’+’ and ’-’
namics conserves the center of mass of the bosons, since
indicate the sites of the Kagome unit around the site r of
the bosons hop equal distances in opposite directions on
the triangular lattice, and are referred to as ’hex’ and ’star’
any plaquette. In other words the center of mass oper-
respectivelyinthepaper. (b)The“gauge”transformationon
ator Rˆc.m. = rrnˆr satisfies Rˆc.m.,H = 0, and is a the dual lattice — shifting θ by ε on the indicated sites as
h i shown, with ε/π being an integer, is an exact symmetry of
constantofmoPtion. Thus,inderivingtheimaginary-time
thedual Hamiltonian.
path integralfor the partition function in terms of φ, we
mayinsertfactorsofexp(iq Rˆ )withanarbitraryvec-
c.m.
·
tor q, since Rˆ commutes with H. This corresponds
c.m.
to φr φr+q r being a symmetry of the phase-action. πNP = (φ1 φ2+φ3 φ4) (3)
→ · − −
If we identify φ = 0 with φ = 2π, or keep track of πnr = θP θP, (4)
the fact that the charge is quantized the first symmetry Xhex −Xstar
operationdiscussedaboveisa specialcaseofthe second,
and corresponds to choosing q = Qi (i = 1,2,3) where wherethe anglesφ1...φ4 inEq.(3)arearoundplaquette
the momenta Q are at the center of the edges of the P ofthelatticewiththeconventionindicatedinFig.1(a).
i
Brillouin zone, indicated in Fig. 4(a). The above sym- The sites hex and star lie on the 12-site unit of the
metries for J = 0 imply we cannot have terms which Kagome lattice around the site r as shown in Fig. 2(a)
dependon( φ) inthe phaseaction(inthepathintegral (marked by ’+’ and ’-’ respectively), while the center of
for the part∇ition function). The lowest allowed gradi- each Kagome unit lies on the triangular lattice (and is
ent term on which the action can depend is of the form labelled r). In these variables, the dual Hamiltonian on
( 2φ). We explicitlyconfirmthis inaspin-wavecalcula- the Kagome lattice is
∇
tion in Section III.C, where we show the collective mode
disperses as ω2(k) k4 for small k, and calculate its Hdual = K cos(πNP)
consequences for co∼rr|ela|tion functio|ns|. The presence of XP
a nonzero J explicitly breaks these symmetries, in fact + U ( θ θ πn¯)2 (5)
J =0 will turn out to be a critical point of the model. 2π2 P − P −
Xr Xhex Xstar
and the dual variables satisfy N ,θ = iδ , with
P P P,P
′ ′
B. Plaquette duality and dual action θP/π having integer eigenvalues(cid:2), while N(cid:3)P has a contin-
uous spectrum.
We pause to elaborate further upon the redundancy
ForJ =0,itisusefultodefineadualrepresentationfor
of the θ variables. Since only the combination nr of the
themodelwhichpermitsustoobtainspin-wavetheoryas
θ variablesisphysical,theHamiltonianwillbeinvariant
P
a well-definedlimit of the model andto analyze instabil-
underanyintegermultipleofπ shiftoftheθ thatleaves
P
ities of the spin-wave phase towards charge and energy
all the nr invariant. This can be done locally as follows.
ordering. For this purpose, we use a plaquette duality
PickanyhexagonofthedualKagomelattice,andchoose
transformation13 and work with dual variables θ and
P twoneighboringsitesonthishexagon. Shiftθ θ +π
P P
N which reside at the centers of each of the 4-site pla- →
P on those sites, and simultaneously take θ θ π on
P P
quettesofthetriangularlattice—thesesites(labelledby → −
thesitesontheoppositesideofthishexagon,leavingthe
P)formaKagomelatticewithnearestneighbourspacing
remainingtwositeofthehexagon(andalltheothersites
1/2inunits ofthe latticespacingoftheoriginaltriangu-
not on this hexagon) untouched as shown in Fig. 2(b).
lar lattice. Alternatively, we may label sites on the dual
Since this leaves all physical properties invariant, this
latticeas(r,α),viewingtheKagomelatticeasatriangu-
should be regarded as a local gauge symmetry. Such
lar lattice three sites (labelled α = 1,2,3) per unit cell,
transformations are generated by the unitary operators
andwewillusethisnotationwheneverconvenient. Thus,
there are three times as many sites (and hence degrees Gr(Kα)=eiKα· hex(rP−r)NP, (6)
of freedom) onthe dual lattice comparedto the original, P
and this is reflected in a redundancy in the description whereK withα=1,2,3arethe threereciprocallattice
α
which we will discuss shortly. Define vectorsfor the triangularlattice. In the spiritofa gauge
4
theory, we could restrict our consideration to physical Q=2πn¯, and the site r maˆ+nˆb in terms of the basis
gauge-invariant states which satisfy Gr(Kα) = 1. This vectors aˆ,ˆb of the triang≡ular lattice shown in Fig. 1(b).
can be rewritten suggestively as The factor we have subtracted out eliminates the mean
density n¯ from the second (boson repulsion) term in the
(rP r)NP =0(moda), (7) Hamiltonian.
−
Xhex If we ignore the terms with v0 , we obtain a Gaussian
2q
action in terms of ϑ. Studying this action motivates us
where a is an arbitrary Bravais lattice vector for the tri-
to guess that the effective low energy action may be of
angular lattice. As in Refs.8,13, the operator N may
P the form Seff =S(0) +S(1) with
be regarded as the local “vortex number” (modulo 2) dual dual dual
on the site P of the dual lattice. Thus the condition of
eEaqc.h(h7)exraegqouniraelsptlhaqautetthtee olofctahlevKoratgeoxmceenlatetrticoef vmaansisshoens Sd(0u)al=ZωkX,α,βϑ−k,α,−ω(cid:18)2πω22Kδα(βk) + U2(πk2)Gαβ(k)(cid:19)ϑk,β,ω
–theambiguitybyaBravaislatticevectorcorresponding (10)
exactly to the ambiguity in NP by a shift of 2. Thus the and
gauge-invarianceof the dual description is related to the
imUmsoinbgiliatyTorfot“tveorrdtiecceosm” pinostihtiisonm,owdeelm. ay write the par- Sd(1u)al=Z v2qcos(2qϑr,α,τ +Qqmn)+Shigher (11)
τqX,r,α
tition function as a discretized imaginary-time path in-
tegral in the standard manner. This leads to Z = wherewehavedefinedrenormalizedcouplings (k), (k)
θ(r,τ)exp( S[θ(r,τ)]) with the action (which are smooth nonzero functions of k) anUd v K, the
D − 2q
R siter (m,n)asbefore,andthe3 3matrixG (k)=
1 2 ≡ × α,β
Sdual = π2 ln(ǫ K) (θr,α,τ −θr,α,τ+1)2 A∗α(k)Aβ(k) with
τ rX,α,τ
ǫ U A1(k) = 1+e−ika e−ika−ikb eikb
+ 2τπ2 ( θP − θP −πn¯)2 (8) A2(k) = 1+e−ika−−ikb e−ika −e−ikb
Xr Xhex Xstar A3(k) = 1+e−ikb eik−a e−ik−a−ikb, (12)
− −
where the field θr,α,τ/π is an integer-valued field, ǫτ =
β/N with β = 1/T is the inverse temperature, and N where k = k aˆ and k = k ˆb. S denotes other
τ τ a b higher
· ·
is the number of imaginary time slices. The quantum terms which might be generated in deriving the low en-
problem at a temperature T corresponds to the contin- ergy action. For the present purposes, we will not write
uum limit ǫ 0, N with fixed 1/ǫ N = T. downthe explicitformofthese termsbutwe willdiscuss
τ τ τ τ
→ → ∞
Thus, we reduce the quantum model to an anisotropic them later.
(2+1)-dimensional classical model — since there is no If we set v =0 and ignoreS , we can obtain the
2q higher
sign-problem, this proves useful for carrying out Monte eigenmodesoftheGaussianactionS(0) bydiagonalizing
dual
Carlo simulations in the dual representation to numer- G (k). We find there is one nonzero eigenvalue λ(k),
αβ
ically test for charge ordering and plaquette-energy or- and two zero eigenvalues. This corresponds to one dis-
dering in the ground state at arbitrary U/K. persing mode (Θ ) and two nondispersive modes (Θ )
1 2,3
with zero energy(flat bands) and the actionat this level
takes the form
C. Effective theory and spin-wave approximation
ω2+ 2(k)
S(0) = E Θ (k,ω)2
We may rewrite the dual action as dual Z (cid:18) 2π2 (k) (cid:19)| 1 |
ωXk K
1 2 ω2
Sdual = π2 ln(ǫ K) (θr,α,τ −θr,α,τ+1)2 + Z (cid:18)2π2 (k)(cid:19)|Θα(k,ω)|2, (13)
τ rX,α,τ ωk,αX=2,3 K
ǫ U
+ 2τπ2 Xr (XhexθP −XstarθP −πn¯)2 waphpeeraerwseashatvheeseextciλt(akti)o=n eEn2e(rkg)y/Uo(fkt)hKe(kco)llseocttihvaetmEo(kde)
− v20qcos(2qθr,α,τ) (9) Θ1. Explicitly,
q,Xr,τ,α
λ(k) = 2[6 3cosk 3cosk 3cos(k +k ) (14)
a b a b
− − −
where the bare couplings v0 are chosen to enforce the + cos(k k )+cos(2k +k )+cos(2k +k )],
2q a b a b b a
−
integer constraint on θ, and may be singular. We will
now write down a low energy effective action in terms of thus the dispersing mode Θ has one gapless point at
1
ϑr,α,τ =[θ]f (Qmn)/2. Here [θ]f symbolically denotes k = (0,0). The leading dispersion away from this point
anintegration−overthefast(high-energy)degreesoffree- is 2(k 0) = (9/16) (0) (0)k4 — as stated in sub-
E → U K | |
dom while retaining the spatial structure of the lattice, sectionA,this dispersionis dictated by the symmetry of
5
conservationofcenterofmassofthebosonswhichforbids states per unit volume of collective excitations is given
a term k2. by
∼| |
We canalsonow see that the extradegreesof freedom
(two per site of the triangular lattice) introduced in go- 1
N(ω) = δ(ω (k))
ing over to the dual description manifest themselves as V −E
Xk
zero energy modes corresponding to unphysical fluctua-
1
tions, while the dispersing mode corresponds to physi- ω≃0 (18)
→ 2π 3 (0) (0)
cal fluctuations. In fact, by Fourier transforming a gen-
U K
eralsuperpositionf (k)Θ (k)+f (k)Θ (k),wecanshow p
2 2 3 3
whichisaconstantdependingontheinteractions. Thus,
that the zero modes indeed arises from the local gauge
all low-energy long-wavelength properties of the liquid
symmetry generated by Eq. (6) in real space. More pre-
are determined in terms of the k 0 behavior of the
cisely,inthe coarse-grainedquadraticlowenergytheory, →
functions (k), (k). We next present arguments and
thesediscretesymmetryoperationsarepromotedtocon- U K
numerical evidence that this Gaussian description of a
tinuous shifts (with arbitrary K ). Similarly, Fourier
α
| | criticalliquidinthe modelwithJ =0maybe valideven
transforming f (k)Θ (k), we find that the physical fluc-
1 1
in the presence of terms v and S .
tuations are composed precisely of the gauge invariant 2q higher
boson densities.
At this stage, we may reintroduce the boson phase
fieldsbyaHubbardStratonovitchdecouplingofthetime E. Argument for irrelevance of higher order terms
derivative term of the physical fluctuation Θ (as in
1
Ref.13), namely Toseewhetherthetermv isrelevant,wecanevaluate
2q
the correlation functions of this operator, and we find it
e−2π2ωK2(k)|Θ1(k,ω)|2 = Z Dφφ∗e−2EU2((kk))|φ(k,ω)|2 (15) is local in space and exponentially decaying in time,
e2π√ωE((kk)) (k)(φ(k)Θ∗1(k)−φ∗(k)Θ1(k)). hcos2ϑr,α,τcos2ϑ0,β,0i∼δr,0δα,β exp(−γ|τ|) (19)
× U K
withaconstantγ >0. Thisisnothardtounderstand—
Provided v and S are irrelevant and can be ig-
2q higher itarisesfromthefactthatthevariableϑisacombination
nored,wecanintegratingoutthe fieldΘ whichleads to
1 of the eigenmodes Θ1,2,3 and cos2ϑr,α,τ is not “gauge-
a Gaussian action for the boson field φ,
invariant”. As a result its spatial correlations are local,
while the fluctuations of the zero-modes Θ determine
ω2+ 2(k) 2,3
S(0) = E φ(k)2. (16) theexponentialtemporaldecay. Clearly,v2q isirrelevant.
φ ZωXk (cid:18) 2U(k) (cid:19)| | Consider the possible forms we can obtain for Shigher.
These terms would be of the form of cosines involving
This corresponds to a spin-wave (harmonic) approxima- multipleϑ’satdifferentspace-timepoints. However,sim-
tionandmaybe usedtoevaluatebosoncorrelationfunc- ilar arguments as above apply to these operators. The
tions to zeroth order in v . only operators which can have nonzero correlations at
2q
nonzero separation would be gauge-invariant combina-
tions of the ϑ’s. As we have shown, these are the local
D. Boson correlation functions in spin-wave theory boson densities. Thus, if we admit only slow fluctua-
tions of Θ , these would take the form of weak density-
1
To characterize the Gaussian theory S(0), we evaluate densityinteractionswithinaperturbativetreatmentand
φ couldrenormalizethecoefficientsoftheGaussianaction,
the space-time correlation functions for the boson cre-
but not cause an instability. Of course, such density-
ation operator exp(iφrτ). At long times or large separa-
density interactionsmight leadto a charge-orderedstate
tions, these correlation functions reduce to
at strong couplings, but our arguments show that the
Gaussian theory is perturbatively stable16.
eiφr,0e−iφ0,0 r−2π1√3qKU((00))
h i ∼ | |
eiφ0,τe−iφ0,0 τ −4π1√3qKU((00)). (17) F. Numerical results
h i ∼ | |
Clearly,the Gaussiantheorydescribesacriticalliquidof To confirm the above arguments we have carried out
the bosons with power law correlations arising from the Monte Carlo calculations on the dual model using a
gapless excitations dispersing as (k) k2. From the Metropolis algorithm, for lattice sizes upto 432 spatial-
E ∼ | |
longtimebehaviorofthetwo-pointcorrelationinafinite sites and 48 time-slices, and periodic boundary condi-
sizesystem,wemayevaluatethefinitesizescalingofthe tions. We find no evidence of any charge ordering for
gap to adding a particle to be just ∆(L) (0)/L2. n¯ = 1/2, even for large U/K. We have compared
≃ U
In the thermodynamic system, the low energy density of the density-density correlationsand find agreementwith
6
the pure ring-exchange model for all values of the bare
coupling U/K. In this, we make the approximation of
completely ignoring irrelevant operators — this seems
reasonableinretrospectsincewewillfindthatthesystem
acquires long-range phase order at T =0.
The nearest-neighbor perturbation we add to the
Gaussian phase action takes the form
S = J dτ[( 1)ncos(φ φ )
J m,n,τ m+1,n,τ
Z − −
Xm,n
+ ( 1)mcos(φ φ )
m,n,τ m,n+1,τ
− −
+ ( 1)m+ncos(φ φ ) (21)
m,n,τ m+1,n+1,τ
− −
(cid:3)
where we have chosen to label sites on the triangular
lattice as r = maˆ+nˆb, and we are working in rotated
variables φr for which the ring-exchange term is ferro-
magnetic.
To carry out a systematic expansion, we define φr =
FIG.3: Thedependenceofthescaledcompressibility(ǫτK)κ φ˜r +Q r with an arbitrary vector Q. This allows us
·
onthebarebosonrepulsionU,forvariousǫτK. Withinspin- to examine at the effect of perturbing in J in any of
wavetheory,weexpect(ǫτK)κ=K/U(0) asdiscussed inthe the ground states of the degenerate manifold to see if
text. FromthefigureweseethatK/U(0)coincideswithK/U
fluctuations select any state. Perturbing in S around
J
for small U (as indicated by the dashed line), but deviates
thepurering-exchangeGaussiantheory,wefindthatthe
andtendstoaconstant K/Ueff ∼0.2asthebareU →∞(as
corrections to the free energy vanish at leading order in
shownbythesolidlinedrawnasaguidetotheeye). Typical
J. At (J2), we find a correction to the free energy
error bars are indicated at one point. We thus expect the O
density
S =1/2XY spin modelwith J =0tobecompressible, with
U(0)=Ueff.
1
δf = βJ2 cos2Q +cos2Q +cos2(Q +Q )
a b a b
−2
(cid:0) (cid:1)
( 1)n cos(φ˜ φ˜ +φ˜ +φ˜ )
what we expect from a Gaussian theory S(0) . In par- × Z − h 0,0,0− 1,0,0 m,n,τ m+1,n,τ i0
dual Xm,n τ
ticular, we show results for the compressibility κ defined
(22)
through κ=χ (k 0,ω =0) where
nn n
→
where ... is the expectation value evaluated in the
χnn(k,ωn)=Xr,τ e−ik·r+iωnτhn(r,τ)n(0,τ)i. (20) Gaussiahn tih0eory, Qa ≡ Q·aˆ, Qb ≡ Q·ˆb, and we have
used the six-fold symmetry of the triangular lattice to
simplify certain intermediate expressions. The expecta-
Within the Gaussian theory, κ=1/ǫ (0). In Fig. 3 we
τ
U tion value may be analytically evaluated at small U/K
plotthe scaledcompressibility(ǫ K)κ whichis expected
τ
by expandingthe cosinetermsince its argumentis small
to scale as K/ (0). We find that that it is nonzero
U in the Gaussian theory, and we find that the sum over
at all U/K in the quantum (τ-continuum) limit, with
m,n is positive. Thus, (i) the corrections to the free en-
(0)/K 5 as U/K . We have also checked
U ≃ → ∞ ergyareindependentofthesignofJ,and(ii)minimizing
and found no evidence for energy ordering on the pla-
δf with respect to variations of Q corresponds to maxi-
quettes of the triangular lattice. Thus, we believe that
mizing cos2Q +cos2Q +cos2(Q +Q ) which leads
for n¯ =1/2, the ring-exchange model with J = 0 is well a b a b
(0) to order(cid:0)ing at four wavevectors Qα (α = (cid:1)1...4) which
describedbytheGaussianfixed-pointactionS where
dual are shown in Fig. 4(a). We expect this order to per-
terms with v and all other terms S are irrelevant.
2q higher sist even at large effective (U/K) (recall that the bare
U/K in the S = 1/2 spin limit). To obtain the
→ ∞
ordered states in the original phase variables, we rotate
IV. PERTURBING IN NEAREST NEIGHBOR
backφ(r) ϕ(r)=φ(r) π onthetriangularsublattice
COUPLING J → −
and we then find, as shown in Fig. 4(b), that the result-
ing ordered states break (i) global U(1) spin rotational
A. Ordered ground state invarianceand(ii)four-foldtranslationalinvariance. For
small J/K, such states have been shown to occur at fi-
To perturb in the nearest-neighbor exchange J, it is nite magnetic fields for the classical version of a related
convenientto workin the phase representationwhichwe SU(2)invariantring-exchangemodel17,andthisordering
have argued remains a valid ‘fixed-point’ description of has been called uuud (for ‘up-up-up-down’ which is the
7
Q
spin order on a 4-site plaquette, but in the plane in our 2
case). Here, we find such an ordered state is stabilized
by quantum fluctuations in the XY limit even in zero
magnetic field.
Q
0
B. Fluctuations in the ordered state Q Q
1 3
As shown above, in the presence of a nonzero J, the (a) (b)
model has an ordered ground state. This order breaks
U(1) invariance. At the same time, it also breaks trans- FIG. 4: (a) The four ordering wavevectors Qα (α = 1...4)
lationalsymmetry. Inthebosonlanguage,thisstatemay obtained by perturbing in the nearest neighbor coupling J,
be viewedasasuperfluidcoexistingwithbrokentransla- with thehexagon indicating th first Brillouin zone of thetri-
tional order. This broken discrete symmetry is also evi- angularlattice. (b)Thephase/spinorderintheoriginalvari-
dent from our implicit choice of a state Q to minimize ables ϕr in one of the four broken symmetry ground states.
α
Theordermaybemosteasilythoughtofasalternatingrowsof
the free energy. Thus, we expect two kinds of excita-
ferromagnetic and antiferromagnetic spins. The bold dashed
tionsatlowenergiesabouttheorderedstate—(i)phase
lines indicate bonds on which the spins point in the same
fluctuations, and (ii) domain walls in the discrete order.
direction, while the other bonds have antiferromagnetic spin
It is easy to see that with φr = Q r+φ˜r, there is orientation. Asdiscussedinthetext,thesecorrespondtoen-
an energy cost (Q Q )2 for small·deviations (Q ergyorderinginthegroundstatewithanassociatedfour-fold
α
∼ − | −
Q ) 1 where Q denotes any one of the ordering broken discrete symmetry.
α α
| ≪
vectors. This implies that the long wavelength effective
theory of phase fluctations around each of these ordered
states is described by an action of the form
symmetry is not restored until a much higher tempera-
1 ture Tc J. Thus there is an intermediate phase with
S = (∂τϕ)2+Js( ϕ)2+Keff( 2ϕ)2 (23) exponen∼tially decaying spin correlations, but with the
Zr,τ(cid:18)Ueff ∇ ∇ (cid:19) discrete order still present. This bears resemblance to
with a nonzero J , and effective couplings U and K . proposed scenarios of chiral ordering in the XY antifer-
s eff eff
For small U/K,J/K, we find U U, K K and romagnet on a triangular lattice as well as the SU(2)
eff eff
J J2/K U/K. Thus, phase fl∼uctuations∼are con- multiple-spin exchange model. However, in contrast to
s
∼ the XY model where a single scale,namely the two-spin
trolled by thpe phase stiffness Js.
exchangecoupling,setsthescaleforbothtransitiontem-
A sharp domain wall separating regions with different
peratures, the transition temperatures are governed by
Q order would cost an energy K per unit length.
α
∼ distinct energy scales in the present case, and the dis-
However, since the model is spin-disordered for J = 0,
crete symmetry corresponds to broken translational in-
we expect that a smooth deformation of the spin con-
variance.
figuration by making a domain wall with nonzero width
In order to better understand the discrete transition,
ξ , would cost less energy. Indeed, using the above ac-
D
let us identify the appropriate order parameter and con-
tion one can estimate the energy per unit length of a
struct a Landau theory. Since the discrete symmetry
straight domain wall as e = γ K /ξ + γ J ξ .
dom 1 eff D 2 s D
is easily identified with translations in the original vari-
whereγ 1areconstants. Minimizingtheenergycost
1,2
∼ ables ϕr, we will discuss itin terms ofthat. As shownin
withrespecttoξ ,wefindthatξ K /J ,andthus
D D∗ ∼ eff s Fig.4(b), the orderedstate at T =0 has brokentransla-
domain wall excitations cost an enerpgy e √K J
per unit length. For small U/K, e J∗do(mU/∼K)1/4.eff s tionalinvarianceassociatedwithorderingofcos(ϕi−ϕj)
∗dom ∼ on nearest-neighbor bonds. Since there is a nearest-
neighbor spin exchange term J, this leads to ordering in
the energy density. We identify the brokendiscrete sym-
C. Phase transitions at finite temperature
metrywithenergyorderingonthebonds. Toidentifythe
appropriateorderparameterforthistransition,notethat
a fAutllyhigdhisotredmerpeedraptuhraes,ewweheerxepetchtetUhe(1m)osdyemlmtoetbrye ains twheemuanyitwlarittteicKerv,eic=tohrSsrm·Sakr+inaˆgiiawnhaenreglaˆei2(πi/=31w,i2t,h3)eaacrhe
well as the discrete broken symmetry is restored. To other (specifically, aˆ = aˆ, aˆ = ˆb, and aˆ = (aˆ+ˆb)).
1 2 3
study these phase transitions, we appeal to the above −
Labelling the four ordered states by µ=0...4 and with
estimates of the spin stiffness and domain wall energies.
a =0,wecanwritetheexpectationvalueinanyofthese
For U/K 1, since the spin stiffness Js J2/K, while or0dered states as Kµ = cos(Q r)cos(Q aˆ ). For a
the domai∼n wall energy per unit length e∼ > J J , r,i i · i · µ
dom s general superposition,
we expect the U(1) symmetry to be restored v∼ia a≫BKT
tarraenisnitcilound1e4daitnathteemepffeercattiuveretTheBoKrTy,∼whJisleotnhcee vdoisrctriceetes Kr,i =µ=X0...4Bµcos(Qi·r)cos(Qi·aˆµ) (24)
8
and we identify B as the appropriate order parameter sitions out of this phase. Since the groundstate is phase
µ
for the Landau theory. Under shifts of B B +λ, ordered,weexpectourresulttobestabletosmallout-of-
µ µ
→
the physical correlation Kr,i is unchanged. Thus we re- planecouplingsinvolvingSz. Similarly,introducingweak
quire the Landau functional to be invariant under such couplings between such ordered two-dimensional planes
shifts. Further, studying the transformation of Kr,i un- would lead to a three-dimensionally ordered state18,
dersymmetryoperationsofthelattice(unittranslations, where the spin/phase order would persist to finite tem-
π/3-rotations,andreflectionsaboutthe3mirrorplanes), perature. We thus expect this state might be of some
we find that suchsymmetry operationscorrespondto all interest for Mott insulators on a triangular lattice. If
possible permutations of the set B . For the Landau suchanorderedstate exists, it couldbe detected in neu-
µ
{ }
theory we find beyond quadratic order,one cubic invari- tron diffraction studies. The spin-disordered state with
antandtwoquarticinvariants. Itappearslikelythatthe persisting discrete broken symmetry may be indirectly
finitetemperaturephasetransitionrestoringthisdiscrete observable through lattice distortions if the spins couple
symmetry is in the same universality class as a 4-state to the lattice.
Potts model. We can also analyze the opposite limit from the one
To summarize, we expect the system to exhibit with studied in the paper, where we assume “easy axis”
increasing temperature a BKT transition with T anisotropy in the model in Eq. (2) and weak J/K. In
BKT
J2/K from a state with power law spin correlations to∼a this Ising limit, we find eight degenerate ground states
state with exponentially decaying spin correlations. The for the spin model, again of the uuud type, with the
discrete broken symmetry associated with energy order- spins aligned along S -axis. These states correspond to
z
ing onthe bonds ofthe lattice is expected to be restored charge-ordered incompressible states for the bosons at
above a temperature T J, with the transition being density n¯ =1/4 or n¯ =3/4 (four ground states at either
c
∼
in the universality class of the 4-state Potts model. In a density). It may be that any anisotropy which takes us
weakappliedin-planemagneticfield,theU(1)invariance away from the SU(2) invariant model leads to ordered
is lost, and the BKT transition would become rounded groundstates,while the SU(2)-invariantmodelseemsto
but the discrete transition wouldsurvive as a finite tem- be a uniform spin liquid7,15. It would be interesting to
perature transition. At large enough fields, we expect a further explore this possibility.
first order transition to a state where all the spins point
in the same direction and the translation invariance is
restored. Acknowledgments
This work was supported by the NSF through grants
V. DISCUSSION AND CONCLUSIONS DMR-9985255 and PHY99-07949,and by the Sloan and
Packard foundations. We would like to thank Luca
Studying an XY model with four-spin couplings and Capriotti,MatthewFisher,EduardoFradkin,AkakiiMe-
small two-spin exchange, we have obtained a ferrimag- likidzeandAshvinVishwanathforusefuldiscussionsand
netically ordered ground state and analyzed phase tran- comments on the manuscript.
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