The frustrated Heisenberg antiferromagnet on the honeycomb lattice: $J_{1}$--$J_{2}$ model

0.19 MB·English

by  P. H. Y. Li

The frustrated Heisenberg antiferromagnet on the honeycomb
J J
lattice: – model
1 2
P. H. Y. Li1, R. F. Bishop1, D. J. J. Farnell2 and C. E. Campbell3
1 School of Physics & Astronomy, Schuster Building, The University of Manchester, Manchester M13 9PL, UK
22 Division of Mathematics, Faculty of Advanced Technology, University of Glamorgan, Pontypridd CF37 1DL, UK
13 School of Physics & Astronomy, University of Minnesota, 116 Church St. SE, Minneapolis, MN 55455, USA
0
2

n
a PACS 75.10.Jm–Quantised spin models
J PACS 75.50.Ee–Antiferromagnetics

17 Abstract–Westudytheground-state(gs)phasediagramofthefrustratedspin-12 J1–J2 antifer-
romagnetwithJ2 =κJ1 >0(J1 >0)onthehoneycomblattice,usingthecoupled-clustermethod.
] Wepresent results for theground-state energy,magnetic order parameter and plaquettevalence-
el bond crystal (PVBC) susceptibility. We find a paramagnetic PVBC phase for κc1 < κ < κc2,
- whereκc1 ≈0.207±0.003 andκc2 ≈0.385±0.010. Thetransitionatκc1 totheN´eelphaseseems
r tobeacontinuousdeconfinedtransition(althoughwecannotexcludeaverynarrowintermediate
t
.s phaseintherange0.21.κ.0.24), whilethatatκc2 isoffirst-ordertypetoanotherquasiclassi-
t cal antiferromagnetic phase that occurs in the classical version of the model only at the isolated
a and highly degenerate critical point κ = 1. The spiral phases that are present classically for all
m 2
valuesκ> 1 are absent for all κ.1.
6
-
d
n
o
c
[
Two-dimensional (2D) frustrated quantum spin-lattice solid phases and spin-liquid phases.

1
systems have become of huge interest both theoretically
v Quantum fluctuations tend to be largest for the small-
and experimentally [1–3]. Attention has particularly fo-
2 est values of s, for lower dimensionality D of the lattice,
1cussed on the rich panoply of (zero-temperature, T = 0) and for the smallest coordination of the lattice. Thus, for
5quantum phase transitions that they exhibit [3,4]. With- spin-1 models in D = 2, the honeycomb lattice plays a
3out thermal fluctuations the transitions are driven solely 2
special role. Frustration is easily incorporated via com-
.
1by the interplay of quantum fluctuations and any frustra- peting NNN and maybe also next-next-nearest-neighbour
0tionduetoinherentcompetitionbetweentheinteractions.
(NNNN) bonds. Such models and their experimental re-
2Such frustration can arise either dynamically or geomet-
alisations havebeen muchstudied in recentyears [10–16].
1
rically. A prototypical example of the former is the well-
: Additional interest has also sprung from the recent syn-
vstudiedJ –J Heisenbergantiferromagnet(HAFM)onthe
1 2 thesisofgraphenemonolayersandothermagneticmateri-
ibipartitesquarelattice(see,e.g.,Refs.[5–7]andreferences
X als with a honeycomb structure. Theoretical interest was
cited therein), where nearest-neighbour (NN) spins inter-
r spurred by the discovery of a spin-liquid phase in the ex-
aact via a Heisenberg interaction with strength parameter actlysolvableKitaevmodel[17],inwhichspin-1 particles
J >0,whichcompeteswithaHeisenberginteractionwith 2
1 reside on a honeycomb lattice. Hubbard models on the
strength parameter J > 0 between next-nearest neigh-
2 honeycomblattice may alsodescribemany ofthe relevant
bour(NNN)pairs. Similarprototypicalmodelsexhibiting
physical properties of graphene. For example, evidence
geometrical frustration are the pure NN HAFMs on the
has been found [18] that quantum fluctuations are suffi-
triangular [8] and kagome lattices [9]. For either form
ciently strong to establish an insulating spin-liquid phase
of frustration special interest then centres on the possi-
betweenthenonmagneticmetallicphaseandtheantiferro-
bleappearanceofnovelquantumground-state(gs)phases
magnetic(AFM)Mottinsulatorphase,whentheCoulomb
withoutthelong-rangeorder(LRO)thattypifiestheclas-
repulsion parameter U becomes moderately strong. For
sical gs phases of the corresponding models taken in the
large values of U the latter phase correspondsto the pure
limit s→∞ of the spin quantum number s of the lattice
HAFMonthebipartitehoneycomblattice,whosegsphase
spins. Examples include various valence-bond crystalline
exhibits N´eel LRO. However, higher-order terms in the
p-1
P. H. Y. Li et al.
dealt both with the AFM case with J > 0 [23] and the
1
J1 J2 Jferr=om0a,gannedtihce(nFcMe c)ocnassideewritthheJJ1 –<J0m[2o4d].el.HWeree rweestpriuctt
3 1 2
ourselves to the case where both bonds are of AFM type,
J >0 and J ≡κJ >0. We henceforth set J ≡1.
1 2 1 1
The classical (s → ∞) gs phase diagram of the J –J –
1 2
(a)N´eel (b)spiral (c)anti-N´eel J model on the honeycomb lattice [11,25] comprises six
3
different phases when J > 0 and the other two bonds,
1
Fig.1: (Coloronline)TheJ –J modelonthehoneycomb
1 2 J and J , can take either sign. Three are collinear AFM
2 3
lattice(withJ =1),showing(a)theN´eel,(b)spiral,and
1 phases,oneis the FM phase,andthe other twoare differ-
(c) anti-N´eel states. The arrows represent spins located
ent helical phases (and see, e.g., Fig. 2 of Ref. [11]). The
on lattice sites •.
AFM phases are the N´eel phase (N) shown in Fig. 1(a),
the striped (S) phase discussed in our earlier paper [23],
and the anti-N´eel (aN) phase shown in Fig. 1(c). The S,
t/U expansionoftheHubbardmodelmayleadtofrustrat-
aN, and N states have, respectively, 1, 2, and all 3 NN
ing exchange couplings in the corresponding spin-lattice
spins to a given spin antiparallelto it. Equivalently, if we
limiting model, in which the HAFM with NN exchange
consider the sites of the honeycomb lattice as comprising
couplings is the leading term in the large-U expansion.
asetofparallelsawtooth(orzigzag)chains(inanyone of
There is a growing consensus [10,11,13–16] that the frus-
tratedspin-1 HAFM on the honeycomblattice undergoes the three equivalent directions), the S state comprises al-
2 ternating FM chains, while the aN state comprises AFM
afrustration-inducedquantumphasetransitiontoapara-
chains in which NN spins on adjacent chains are paral-
magnetic phase showing no magnetic LRO.
lel. Although there are infinite manifolds of non-coplanar
Indirect experimental backup for such theoretical find-
states degenerate in energy with each of the S and aN
ings comes from recent observations of spin-liquid-like
states at T = 0, both thermal and quantum fluctuations
behaviour in the layered compound Bi Mn O (NO )
3 4 12 3
[11]selectthecollinearconfigurations. WhenJ >0there
(BMNO) attemperatures below its Curie-Weiss tempera- 3
ture [12]. In BMNO the Mn4+ ions reside on the sites of is a region in which the spiral state shown in Fig. 1(b) is
the stable gs phase. It is characterized by a spiral angle
(weakly-coupled) honeycomb lattices, but they have spin
quantumnumbers= 3. Thesuccessfulreplacementofthe defined so that as we move along the parallel sawtooth
Mn4+ ions in BMNO2by V4+ ions could lead to the real- chains [drawn in the horizontal direction in Fig. 1(b)] the
izationofacorrespondings= 1 modelonthe honeycomb spin angle increases by π +φ from one site to the next,
2 and with NN spins on adjacent chains antiparallel. The
lattice. Other recent realizations of HAFMs on a hon-
classical gs energy is minimized for this spiral state when
eycomb lattice include compounds such as Na Cu SbO
[19] and InCu2/3V1/3O3 [20], in both of which3the2Cu2+6 the pitch angle φ = cos−1[14(J1 −2J2)/(J2 −J3)], when
ions in the copper oxide layers form a spin-1 HAFM on a the energy per spin takes the value,
2
(distorted) honeycomb lattice. Others include the family Ecl s2 1(J −2J )2
spiral 1 2
of compounds BaM (XO ) (M=Co, Ni; X=P, As) [21], = −J −2J +J − . (2)
2 4 2 1 2 3
N 2 (cid:18) 4 (J −J ) (cid:19)
in which the magnetic ions M are disposed in weakly- 2 3
coupled layers where they reside on the sites of a hon- We note that as φ → 0 this spiral state becomes the
eycomb lattice. The Co ions have spins s= 21 and the Ni collinear N state with energy per spin,
ions have spins s = 1. Finally, recent calculations of the
low-dimensional material β-Cu V O [22] show that its Ecl s2
2 2 7 N = (−3J +6J −3J ), (3)
magnetic properties can be described in terms of a spin-1 N 2 1 2 3
2
model on a distorted honeycomb lattice.
and there is a continuous phase transition between these
In recent papers [23,24] we have studied the frustrated
spin-1 J –J –J model on the honeycomb lattice [10,11, two states on the boundary y = 23x− 14, for 16 < x < 21,
2 1 2 3 where y ≡J /J and x≡ J /J . Similarly, as φ →π the
13–16]. Its Hamiltonian is given by 3 1 2 1
spiral state becomes the collinear S state, and there is a
H =J s ·s +J s ·s +J s ·s , (1) continuousphasetransitionbetweenthe twostatesonthe
1 i j 2 i k 3 i l boundaryliney = 1x+1,forx> 1. Thereisafirst-order
hXi,ji hhXi,kii hhXhi,liii 2 4 2
phasetransitionbetweenthecollinearNandSstatesalong
where index i runs over all honeycomb lattice sites, and the boundary line x = 1, for y > 1. These three phases
2 2
indices j, k, andl run overallNN, NNN and NNNN sites (N, S, and spiral) meet at the tricritical point (x,y) =
to i, respectively, counting each bond once and once only. (1,1). Asx→∞(forfixedfinitey),thespiralpitchangle
2 2
Eachlattice site i carriesa particle with spin s= 1 and a φ → 2π. In this limit the model becomes two HAFMs
2 3
spin operator s = (sx,sy,sz). The lattice and exchange on weakly connected interpenetrating triangular lattices,
i i i i
bonds are illustrated in Fig. 1. In earlier work we re- with the usual classical ordering of NN spins oriented at
stricted ourselves to the case where J = J > 0, but we an angle 2π to each other on each sublattice.
3 2 3
p-2
The J –J honeycomb lattice
1 2
The above three states are the only classical gs phases determinethenatureofthephasesinvolved,includingany
when y > 0. For y < 0 the N state persists in a region quantumparamagneticphaseswithoutmagneticorder[7].
boundedbythesameboundarylineasabove,y = 3x−1, Since the CCM is a size-extensive method it provides
2 4
for −1 <x< 1, on which it continuously meets a second results in the limit N → ∞ from the outset. However,
2 6
spiral state, and by the boundary line y = −1, for x < it requires us to input a model (or reference) state, with
−1, at which it undergoes a first-order transition to the respect to which the quantum correlations may, in prin-
2
FM state, which itself is the stable gs phase in the region ciple, be exactly included (and see, e.g., Refs. [5,37,38]
x < −1 and y < −1. Another collinear AFM state, the and references cited therein). We use here the N´eel (N),
2
aN state shown in Fig. 1(c), with energy per spin, spiral, and anti-N´eel (aN) states shown in Fig. 1 as our
CCM model states. The CCM then incorporates multi-
Ecl s2 spin correlations on top of the chosen gs model state |Φi
aN = (−J −2J +3J ), (4)
N 2 1 2 3 for the correlation operators S and S˜ that parametrize
the exact gs ket and bra wave functions of the system
becomes the stable gs phase in the region x > 21, for y < in the respective exponentiated forms |Ψi = eS|Φi and
1{x−[x2+2(x−1)2]1/2}. Ontheboundaryitundergoesa hΨ˜| = hΦ|S˜e−S, where hΨ˜|Ψi ≡ 1. The Schr¨odinger ket
2 2
first-ordertransitiontothespiralstateshowninFig.1(b). and bra equations are H|Ψi = E|Ψi and hΨ˜|H = EhΨ˜|
For 1 < x < 1 the spiral state shown in Fig. 1(b) meets respectively. The correlation operators are defined as
6 2
another spiral gs phase on the boundary line y = 0, at S = S C+ and S˜ = 1+ S˜ C− respectively.
I6=0 I I I6=0 I I
which point there is a first-order transition. The pitch The oPperators C+ ≡ (C−)† andPC− are the creation and
I I I
angleofthis secondspiralphase smoothly approachesthe destruction operators respectively, where C+ ≡ 1 and
0
value zero along the above boundary with the N state, hΦ|C+ =0; ∀I 6=0. The set {C+ ≡s+s+ ···s+}, where
andthe value π along a secondboundary curve that joins s+ ≡Isx+isy, forms a completeIset ofj1mju2ltispijnncreation
the points (x,y)=(−1,−1)and(1,0), onwhichit meets j j j
2 2 operators with respect to the model state |Φi as a gener-
the aN state. Both transitions are continuous ones. This
alized vacuum. We then calculate the correlation coeffi-
second spiral meets the three collinear states N, aN, and cients {S ,S˜ } by minimizing the gs energy expectation
FM at the tetracritical point (x,y)=(−12,−1). value H¯ ≡I hΨI˜|H|Ψi with respect to each of them. This
HenceforthwerestrictconsiderationtotheJ1–J2model yieldsthecoupledsetsofequationshΦ|C−e−SHeS|Φi=0
where J =0 (and J ≡1). The classical (s→∞) model I
3 1 andhΦ|S˜(e−SHeS−E)C+|Φi=0; ∀I 6=0,whichareused
thus has the N state as its gs for J < 1, whereas for I
2 6 to calculate the ket- and bra-state correlation coefficients
J > 1 the gs comprises an infinite family of degenerate
2 6 within specific truncationschemesonthe retainedset{I}
coplanar states with spiral order [including that shown in
described below. It is necessaryto use parallelcomputing
Fig. 1(b)], in which the spiral wave vector can point in
routines for high-order computation [37–39].
any direction [11,25,26]. It is found [26] that, to leading
Forthes= 1 caseconsideredhereweusethewell-tested
order in 1/s, spin wave fluctuations lift this degeneracy 2
localized LSUBm truncation scheme which includes all
bythewell-knownorder-by-disordermechanism,infavour
multi-spin correlations in the CCM correlation operators
of specific wave vectors. However, these spiral states for
over all regions on the lattice defined by m or fewer con-
the J –J model are expected to be very fragile against
1 2 tiguouslatticesites. ThenumbersN ofsuchfundamental
quantum fluctuations, and indeed to leading order in 1/s f
configurations that are distinct under the symmetries of
thespin-wavecorrectiontothespiralorderparameterhas
thelatticeandthemodelstateinvariousLSUBmapprox-
beenshowntodivergeaslogN,whereN isthenumberof
imations increases rapidly with the truncation index m.
lattice sites [26], although it is still possible that higher-
For example,the highestLSUBm levelthat we canreach,
order terms in 1/s involving spin-wave interactions could
even with massive parallelization and the use of super-
stabilize the spiral order for large enough values of s. In
computing resources, is LSUB12, for which N = 293309
view of the close proximity of the classical collinear AFM f
for the aN state. The raw LSUBm data still need to be
aN state for small values of the NNNN coupling J in the
3 extrapolated to the exact m → ∞ limit. Although there
J –J –J model, it seems very likely that spiral order in
1 2 3 are no exact extrapolation rules we have a great deal of
the spin-1 J –J model might well be totally absent.
2 1 2 experience in doing so. Thus, for the gs energy per spin,
Toinvestigatethisquestionfurther,andmoregenerally
E/N, we use (see, e.g., Refs., [6,7,31,32,34,38])
to consider the entire T = 0 phase diagram of the J –
1
J2 model, we utilize the coupled cluster method (CCM) E(m)/N =a +a m−2+a m−4; (5)
0 1 2
[27–29] as in our work for the analogous J –J –J model
1 2 3
with J3 = J2 [23]. When used at high orders in the sys- while for the magnetic order parameter (sublattice mag-
tematicapproximationschemesdevelopedforit,theCCM netization), M, we use either the scheme
isanaccurateapproachtotacklingawidevarietyofquan-
tum spin systems [5–7,30–36]. In particular, it can accu- M(m)=b +b m−1+b m−2, (6)
0 1 2
rately locate the quantum critical points (QCPs) in such
frustratedsystems [6,7,31,32,34,36], aswellashelping to forsystemsshowingnooronlyslightfrustration(see,e.g.,
p-3
P. H. Y. Li et al.
0.011 −0.35
LSUB6 LSUB6
0.01 LSUB8 LSUB8
LSUB10 −0.4 LSUB10
0.009 LSUB12
LSUB¥
0.008 −0.45
0.007
De 0.006 E/N −0.5
0.005 −0.55
0.004
−0.6
0.003
0.002 −0.65
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.2 0.4 0.6 0.8 1
J J
2 2
Fig. 2: Difference between the gs energies per spin (e ≡ Fig. 3: CCM LSUBm results for the gs energy, E/N, of
E/N) of the spiral and anti-N´eel phases (∆e ≡ espiral − the N´eel and anti-N´eel phases of the spin-1 J –J hon-
2 1 2
eaN) versus J for the spin-1 J –J honeycomb model eycomb model (J = 1), with m = {6,8,10,12} and the
2 2 1 2 1
(J =1) in LSUBm approximations with m={6,8,10}. extrapolated LSUB∞ result using this data set.
1
Refs. [30,31]), or the scheme 0.35
0.3
M(m)=c +c m−1/2+b m−3/2, (7)
0 1 2
0.25 LSUB8
for more strongly frustrated systems or ones showing a LSUB10
gs order-disorder transition (see, e.g., Refs. [6,7,34,36]). 0.2 LSLUSBU¥B(112)
M LSUB¥ (2)
Since the hexagon is an important structural element of 0.15
thehoneycomblatticeweperformthe extrapolationsonly
0.1
for LSUBm data with m≥6.
WeshowinFig.2thedifferenceinthegsenergiesofthe 0.05
CCM LSUBm results based on the spiral and aN model
0
states. For the spiral state the pitch angle at a given 0 0.2 0.4 0.6 0.8 1
LSUBm level is chosen to minimize the corresponding es- J2
timate for the gs energy. Although the energy differences
are small the results for all values of m, as well as the ex- Fig. 4: CCM LSUBm results for the gs order parameter,
trapolatedresults,showclearlythat,asexpectedfromour M, of the N´eel and anti-N´eel phases of the spin-1 J –J
2 1 2
previousdiscussion,thespiralstatethatisthe classicalgs honeycombmodel (J =1), with m={8,10,12},andthe
1
for all values κ ≡ J2/J1 > 0.5 gives way to the collinear extrapolatedLSUB∞(1)andLSUB∞(2)resultsusingthis
aN state as the stable gs phase for the spin-1 model out data set and Eqs. (6) and (7) respectively.
2
tomuchhighervaluesofκ. Ifa quantumphasetransition
between the spiral and aN states does exist, Fig. 2 shows
that it can occur only at a value κ > 1. Henceforth we corresponding QCPs [29], although we do not do so here
concentrate on gs phases other than the spiral phase. sincewehavemoreaccuratecriteriaavailableasdiscussed
In Fig. 3 we show results for the gs energy per spin, below. We note, however, that as is usually the case, the
E/N, using the N and aN states as model states. We ob- CCM LSUBm results for finite m values for both the N
servethat eachofthe energy curvesbasedon a particular and aN phases shown in Fig. 3 extend beyond the corre-
modelstateterminatesatsomecriticalvalueofκ(thatit- spondingLSUB∞transitionpoints. Forlargevaluesofm
selfdepends onthe LSUBm approximationused), beyond each LSUBm transition point is quite close to the actual
whichnorealCCMsolutioncanbefound. Wenotethatin QCP where that phase ends. For example, the LSUB12
Fig.2resultsareshownforeachLSUBmcasedowntoval- termination points shown in Fig. 3 are at κN ≈ 0.23 for
t
ues of κ at which realsolutions based on the spiralmodel the N state and κaN ≈ 0.35 for the aN state. The CCM
t
state cease to exist. In all cases the corresponding termi- results show a clear intermediate regime in which neither
nation point at a given LSUBm level shown in Fig. 3 for of the quasiclassicalAFM states (N or aN) is stable.
aNmodelstateislowerthanthatforthe equivalentspiral Wenowdiscussthemagneticorderparameter,M,inor-
modelstatecase. SuchterminationsoftheCCMsolutions der to investigate the stability of quasiclassical magnetic
are well understood [29,35]. They are simply manifesta- LRO.OurCCMresultsforM areshowninFig.4. Theex-
tions ofthe quantumphase transitionsinthe realsystem, trapolatedN´eelorderparametergoestozeroatavalueκ
c1
and may thus be used to estimate the positions of the thatisveryinsensitivebothtowhichextrapolationscheme
p-4
The J –J honeycomb lattice
1 2
isusedofEqs. (6)or(7),andtowhetherornotweinclude
3.5
the LSUB6 data point in the extrapolations. Our best es-
timateisκ ≈0.207±0.003. Thisvaluecanbeconsidered 3
c1
as our first CCM estimate of the corresponding QCP of
2.5
the model. It is in reasonable agreement with, but much
moreprecisethan,similarestimatesforκ ofabout0.17- 2
c1
0.22 from an exact diagonalization (ED) approach [16], c1/
1.5
about0.2fromanalternativeEDapproach[13],about0.2 LSUB6
fromaSchwingerbosonapproach[10],andabout0.13-0.17 1 LSUB8
LSUB10
from a pseudo-fermion functional renormalization group LSUB12
0.5 LSUB¥ (1)
approach [15], but in substantial disagreement with a re- LSUB¥ (2)
cent variational Monte Carlo (VMC) estimate of about 0
0 0.2 0.4 0.6 0.8 1
0.08[40]. As the authors admit, the VMC study seems to
J
substantially underestimate the QCP κ at which N´eel 2
c1
order disappears. As expected, our own estimate shows
that,asusual,quantumfluctuationspreservethecollinear Fig. 5: (Color online) Left: CCM LSUBm results for the
N´eelordertostrongerfrustrationsthanthecorresponding inverseplaquette susceptibility, 1/χ,of the N´eelandanti-
classicaltransitionto noncollinearspiralorderatκcl = 16. N´eel phases of the spin-21 J1–J2 honeycomb model (J1 =
BycontrastwiththesituationatthelowerQCPatκ , 1) with m = {6,8,10,12}, and the extrapolated results
c1
Fig. 4 shows that the correspondingQCP at κ at which LSUB∞(1) using m={6,8,10,12}and LSUB∞(2) using
c2
the anti-N´eel order vanishes is considerably more difficult m = {8,10,12} (see text). Right: The fields F = δ Oˆ
to estimate from the extrapolated CCM LSUBm values, for the plaquette susceptibility χ. Thick (red) and thin
with estimates that range from 0.47 to 0.64. We find a (black) lines correspond respectively to strengthened and
much more accurate estimate for κc2 below. Neverthe- weakenedNNexchangecouplings,whereOˆ = hi,jiaijsi·
less it is clear already that a new quantum phase exists sj, and the sum runs over all NN bonds, witPh aij = +1
in the range κ < κ < κ . It is also clear that, as and −1 for thick (red) and thin (black) lines respectively.
c1 c2
suggested above, the two QCPs are very close to the cor-
respondingCCMterminationpointsseeninFig.3. Inour
previous work on the spin-1 J –J –J model on the hon-
2 1 2 3
eycomblattice [23] with J =1,we found strongevidence
1
foranonmagneticplaquettevalence-bondcrystal(PVBC) uoustransitionthere,justaswefoundforthecorrespond-
phase along the line J = J bewteen the two quasiclas- ing J –J –J model [23]. The shallow slope of the χ−1
3 2 1 2 3 F
sical AFM phases, namely the N´eel (N) and striped (S) curves there makes it difficult to estimate accurately the
phases. It seems likely that the corresponding phase in point where it vanishes. Nevertheless it is certainly con-
thepresentJ =0case,whichnowintervenesbetweenthe sistentwiththemuchmoreaccuratevalueweobtainedfor
3
N and aN phases, might also be the same PVBC phase. κ above from the point where the N´eel order parameter
c1
In order to investigate the possibility of a PBVC phase M vanishes. On the other hand we cannot exclude the
we consider a generalized susceptibility χ that describes possibility of the transition between the N´eel and PVBC
F
the response of the system to a “field” operator F (see, states occurring via an intermediate phase that exists in
e.g., Ref. [7]). A field term F = δ Oˆ is added to the theregionκ <κ.0.24. Justsuchanintermediate(res-
c1
Hamiltonian (1), where Oˆ is an operator which corre- onatingvalencebondspin-liquid)statehasbeendiscussed
sponds here to the possible PVBC order illustrated in in Ref. [11]. By contrast, the shape of the CCM curves
Fig. 5, and which thus breaks the translationalsymmetry for χ−1 on the anti-N´eelside are much more indicative of
F
of H. The energy per site E(δ)/N = e(δ) is then calcu- a first-order transition, and the point where χ−1 → 0 on
F
lated in the CCM for the perturbed Hamiltonian H +F, thatsidegivesusourbestestimateforκ ≈0.385±0.010.
c2
using both the N andaNmodel states. The susceptibility Again, this value is in good agreement with, but much
is defined as χ ≡ −(∂2e(δ))/(∂δ2) . Clearly, the gs more accurate than, estimates of about 0.35-0.4from two
F δ=0
phase becomes unstable againstthe p(cid:12)erturbation F when differentEDstudies[13,16]. OnboththeNandaNsidesit
(cid:12)
χ−1becomeszero. AsinRef.[23]weusetheextrapolation seems very likely that the PVBC phase occurs at, or very
F
scheme χ−1(m)=d +d m−2+d m−4. closeto,theQCPswherethequasiclassicalmagneticLRO
F 0 1 2
OurCCMresultsforχ−1areshowninFig.5. Thenum- in the N and aN phases vanishes. Since the N and PVBC
F
ber of LSUB12 fundamental configurations for the pla- phases break different symmetries, and our CCM results
quette susceptibility for the aN state is N = 877315. show that they appear to meet at κ ≈0.21 at a contin-
f c1
The extrapolated inverse susceptibility vanishes on the uous transition, they support the deconfinement scenario
N´eel side at κ ≈ 0.24± 0.01 and on the anti-N´eel side there. The possibility of deconfined quantum criticality
atκ≈0.385±0.010. TheshapeoftheCCMcurveswhere for the frustrated honeycomb HAFM was pointed out in
χ−1 →0ontheN´eelsideisstronglysuggestiveofacontin- Ref. [1], and supporting evidence for the J –J –J model
F 1 2 3
p-5
P. H. Y. Li et al.
has also been found both by us [23] and by others [16]. [17] Kitaev A., Ann. Phys. (N.Y.),321 (2006) 2.
We have studied the influence of quantum spin fluctu- [18] Meng Z. Y., Lang T. C., Wessel S., Assaad F. F.
ations on the gs properties of the spin-1 J –J HAFM and Muramatsu A.,Nature, 464 (2010) 847.
2 1 2
(J > 0,κ ≡ J /J ) on the honeycomb lattice. We find a [19] Miura Y., Hiari R., Kobayashi Y. and Sato M., J.
1 2 1
Phys. Soc. Jpn., 75 (2006) 084707.
paramagnetic PVBC phase in the regime κ < κ < κ ,
c1 c2 [20] Kataev V., Mo¨ller A., Lo¨w U., Jung W., Schit-
where κ ≈ 0.207±0.003 and κ ≈ 0.385±0.010. The
c1 c2 tner N., Kriener M. and Freimuth A., J. Magn.
transition at κ to the N´eel phase appears to be a con-
c1 Magn. Mater., 290-291 (2005) 310.
tinuoustransitionofthedeconfinementvariety,whilethat
[21] Regnault L. P. and Rossat-Mignod J., in Phase
at κ is of first-order type to an anti-N´eel-ordered AFM
c2 TransitionsinQuasi-Two-DimensionalPlanarMagnets,
phase that does not occur in the classical version of the editedbyDeJonghL.J.(KluwerAcademicPublshers)
model except at the critical point κ = 1. Our results in- 1990, p. 271.
2
dicate that the spiral phases that exist classically for all [22] Tsirlin A. A., Janson O.andRosner H.,Phys. Rev.
valuesκ> 1 areabsentforallvaluesκ.1,butmayexist B, 82 (2010) 144416.
6
for larger values. To investigate this and other aspects of [23] FarnellD.J.J.,BishopR.F.,LiP.H.Y.,Richter
the model further we shall present results in a future pa- J. and Campbell C. E., Phys. Rev. B, 84 (2011)
012403.
peronthephasediagramoftheextendedJ –J –J model,
1 2 3
[24] LiP.H.Y.,BishopR.F.,FarnellD.J.J.,Richter
using the same CCM techniques.
J. and Campbell C. E., preprint arXiv:1109.6229,
We thank the University ofMinnesota Supercomputing
(2011).
Institute for the grant of supercomputing facilities. [25] Rastelli E., Tassi A. and Reatto L., Physica, 97B
(1979) 1.
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