Table Of ContentPhase diagram of the frustrated, spatially anisotropic S = 1 antiferromagnet on a
square lattice
H. C. Jiang1, F. Kru¨ger2, J. E. Moore3, D. N. Sheng4, J. Zaanen5, and Z. Y. Weng1
1Center for Advanced Study, Tsinghua University, Beijing, 100084, China
2Department of Physics, University of Illinois, 1110 W. Green St., Urbana, IL 61801, USA
3Department of Physics, University of California, Berkeley, CA 94720, USA
4Department of Physics and Astronomy, California State University, Northridge, California 91330, USA
5Instituut-Lorentz, Universiteit Leiden, P. O. Box 9506, 2300 RA Leiden, The Netherlands
(Dated: January 20, 2009)
9 We study the S =1 square lattice Heisenberg antiferromagnet with spatially anisotropic nearest
0 neighborcouplingsJ1x,J1y frustratedbyanext-nearestneighborcouplingJ2 numericallyusingthe
0 density-matrix renormalization group (DMRG) method and analytically employing the Schwinger-
2 Boson mean-field theory (SBMFT). Up to relatively strong values of the anisotropy, within both
methods we find quantum fluctuations to stabilize the N´eel ordered state above the classically
n
stable region. Whereas SBMFT suggests a fluctuation-induced first order transition between the
a
J N´eel state and a stripe antiferromagnet for 1/3 ≤ J1x/J1y ≤ 1 and an intermediate paramagnetic
region opening only for very strong anisotropy, the DMRG results clearly demonstrate that the
0
twomagnetically ordered phasesareseparated byaquantumdisordered region for all valuesof the
2
anisotropy with the remarkable implication that the quantum paramagnetic phase of the spatially
] isotropicJ1-J2 modeliscontinuouslyconnectedtothelimit ofdecoupledHaldanespinchains. Our
l findings indicate that for S = 1 quantum fluctuations in strongly frustrated antiferromagnets are
e
- crucial and not correctly treated on thesemiclassical level.
r
t
s PACSnumbers: 75.50.Ee,75.10.Jm,75.30.Kz
.
t
a
m I. INTRODUCTION not completely localized, the J -J model can be used
1 2
- as the starting point for a symmetry-based analysis of
d magnetism and superconductivity in the iron-based su-
n A striking commonality between the recently discov- perconductors11,12.
o erediron pnictides superconductors1,2,3,4,5 and the high-
c T cuprates is that in both cases superconductivity The 1/S expansion serves as a natural starting point
[ emc ergesondopingantiferromagneticparentcompounds. to investigate the stability of the classical orders against
1 In trying to unravel the mechanisms of superconductiv- quantum fluctuations which are expected to induce a
v ity frustrated quantum antiferromagnets on the square paramagnetic phase near J2/J1 = 1/2 where both or-
1 latticehavebeensubjecttointenseresearchoverthelast ders compete. Indeed, on the level of lowest order lin-
4 decades with particularinterest in the extreme quantum ear spin-wave theory one finds an intermediate param-
1 limit S = 1 relevant for the cuprates, whereas the inter- agnetic phase for all spin values13. However, the in-
3 2
estinhigherspinvaluesincreasedtremendouslywiththe corporation of spin-wave interactions within a modified
.
1 discovery of the pnictides. spin-wave theory (MSWT)14,15 or the SBMFT16 drasti-
0 cally changes this picture. Within both approaches one
9 Thereby,theHeisenbergmodelwithantiferromagnetic
finds a dramatic stabilization of the classical orders by
0 exchange couplings J and J between nearest neighbor
1 2 quantum fluctuation for the N´eel order even up to val-
: (NN) and next-nearest neighbor (NNN) has served as a
v uesconsiderablylargerthanthe classicalthresholdvalue
i prototype model for studying magnetic frustration. On
X a classical level, one finds N´eel order to be stable for J2/J1 =1/2. This has been interpreted as an order-out-
of-disorder phenomenon giving rise to a fluctuation in-
r J /J 1/2whereasforlargerratiostheclassicalground
a 2 1 ≤ ducedfirstordertransitionbetweenthetwoorders. This
state is give by a stripe-antiferromagnet with ordering
picture has been confirmed recently by a functional one-
wave-vector(π,0). Notsurprisingly,theJ -J modelhas
1 2 looprenormalizationgroupanalysis17whereitwasshown
been used to rationalize the (π,0) magnetism6 of the
that the N´eelphase becomes unstable towardsa fluctua-
iron pnictide superconductors and to subsequently cal-
tion induced first order transition for S >0.68.
culate the magnetic excitation spectra7,8 where the in-
corporation of a strong anisotropy between the NN cou- Surely,it is questionable if the semiclassicaltreatment
plings turned out to be necessary to reproduce the low cancorrectlyaccountforthequantumfluctuationsinthe
energy spin-wave excitations. Recently, it has been sug- regime of strong frustration and small spins. In fact, in
gested that the strong anisotropies in the magnetism8, the case S = 1 variousnumericalstudies including exact
2
but also in electronic properties9 of the pnictides origi- diagonalization18,19,20,21,variationalMonteCarlo22,23se-
nate inthe couplingto orbitaldegreesoffreedomarising ries expansion24,25,26,27,28 as well as the coupled cluster
from an orbital degeneracy of an intermediate S = 1 approach29 givetheconsistentpicturethatinthe regime
spin state10. Even if the charge degrees of freedom are 0.4 . J /J . 0.6 no magnetic order is present clearly
2 1
2
indicating that the aforementioned semiclassical treat- pled Haldane spin-chain limit appear as special cases of
ments overestimate the stability of the ordered states. Hamiltonian (1).
Another key observation of the series expansion and in Unlike for S = 1 this model has been hardly explored
2
particular of the unbiased exact diagonalization studies for S = 1 despite the potential relevance for the iron
isthatintheparamagneticphasethelatticesymmetryis pnictides, and to best of our knowledge exact diagonal-
spontaneously broken due to the formation of columnar ization studies are not available. In a very recent cou-
valence bond solid order. Such states have been pre- pled cluster treatment33 it has been found that at the
dictedbefore30,31 onthe basisofa largeN treatmentfor isotropic point J = J = J the N´eel and stripe or-
1x 1y 1
non frustrated systems in the absence of long-range an- dered phases are separated by a first order transition at
tiferromagnetic order. These depend however crucially J /J 0.55 slightly smaller than the MSWT14,15 and
2 1
on the value of the spin, revealing the subtle workings SBMFT≈16 results and that the transition remains first
of the spin Berry phases (see e.g. [32] and references order up to an anisotropy J /J 0.66. For stronger
1x 1y
therein). For S = 1 these act to give a special stability anisotropiesthe authors find continu≈ous transitions very
2
to’valencebond’pairsingletsinvolvingnearest-neighbor closetothe classicaltransitionline J =J /2,although
2 1x
spins, and the quantum disordered phases are actually they were not able to resolve an intermediate paramag-
valence bond crystals breaking the translationalsymme- netic region within the numerical resolution. On con-
try of the square lattice further to fourfold degenerate trary, a two step DMRG study34 at the particular point
spin-Peierls states. But for S = 1 the Berry phases act J /J = 0.2 indicated a much wider non-magnetic re-
1x 1y
in favor of the formation of ’chain singlets’30,31 that can gion with a spin-gap reaching its maximum ∆ 0.39J
1y
stackeitheralongthexorydirectionofthesquarelattice close to the maximally frustrated point J /J≈ = 0.1,
2 1y
yielding a twofold degenerate ground state. which is only slightly below the gap ∆ 0.41J [35] of
H
≈
an isolated Haldane chain36,37.
StartingfromtheHaldanechainlimit(J =J =0)it
1x 2
has been predictedby Monte Carlosimulations38 aswell
as analytically39 that an infinitesimal coupling J leads
1x
tothe destructionofthe topologicalstringorder40 ofthe
Haldane chain. However, due to the protection by the
finite energy gap of the spin-1 Haldane chain all other
ground-state properties as well as the thermodynamics
areonlyminimallyaffectedbyasmallinterchaincoupling
J and a finite, albeit small41,42,43,44 coupling is neces-
1x
sary to establish magnetic long range order. Whereas a
similar reasoning should hold for a vanishing NN cou-
pling perpendicular to the chains (J = 0) and small
1x
diagonal coupling J the Haldane chain phase seems to
2
be considerablymorestable againstthe simultaneousin-
FIG. 1: The model studied in this paper consists of a S =1 crease of the two mutually frustrating couplings J and
1x
Heisenbergsystemwithspatiallyanisotropicnearest-neighbor J as indicated by two step DMRG results showing an
2
couplings J1x, J1y and isotropic next-nearest-neighbor cou- almost negligible reduction of the spin gap for moder-
plings J2. Pending the balance of these couplings one finds ate interchain couplings34. This suggests that the para-
either a simple staggered- (a) or ’stripe’ (b) antiferromag-
magnetic phase of the frustrated two dimensional spin
netic order. In the limit of isolated chains a stack of isolated
model can be viewed as a continuation of the Haldane
Haldane spin chains is formed (c) and based on our DMRG
calculations we could concludethat thesesurviveall theway spin chain. Recently, based on a theoretical method
to the isotropic limit in the vicinity of the point of maximal with a continuous deformation of the J1-J2 model, the
frustration (J2/J1 =0.5). ground state at the special isotropic case J2 = J1/2 has
been conjectured45 to be a two-fold degenerate valence
bond solid state along either the horizontal or vertical
In this paper we investigate the S = 1 version of the
direction of the square lattice. The main result of this
spatially anisotropicJ -J model, given by the Hamilto-
1 2
paper is that by employing DMRG and by studying sys-
nian
tematically how matters evolve as function of increas-
H =J S S +J S S +J S S , (1) inganisotropywearriveatsolidevidenceforthe smooth
1x i j 1y i j 2 i j
continuation between the Haldane chain phase and an
hXi,jix hXi,jiy hhXi,jii isotropic disordered phase near J =J /2.
2 1
wherethefirsttwosumsrunoverNNspinswithexchange Since for S = 1 exact diagonalization is restricted to
couplings J J in x and y directions, respectively, very small system sizes and since the quantum Monte
1x 1y
≤
andthethirdsumrunsoverallNNNpairswithexchange Carlo method suffers from the infamous sign problem
couplings J as illustrated in Fig 1. We note that both in the presence of frustration we employ the DMRG
2
the spatially isotropic J -J model as well as the decou- method46 to map out the phase diagram of the spatially
1 2
3
anisotropicJ J J model(1)overthewholeparam- where the sum runs over all bonds and we have in-
1x 1y 2
eter range inclu−ding−the decoupled Haldane spin chains troduced the bond operators F† = b† b and G† =
ij i,ν j,ν ij
and the isotropic J1-J2 model as limiting cases. The b† b† . Amean-fielddecouplingisthenperformedwith
i,ν j,−ν
DMRG has the merits of not being biased as analyti-
respect to the order parameters f = F† /2 and g =
cal treatments based on 1/S or 1/N expansions, being ij h iji ij
capable of dealing with frustrated systems, and allow- G† /2. FortheN´eelorderedphasewehavetointroduce
h iji
ingto investigatemuchbiggersystemthanpossiblewith fieldsgx,gy forthenon-frustratednearest-neigborbonds
exact diagonalization. Moreover, it is capable of repro- and fd for frustrated NNN bonds whereas the (0,π)-
ducingthespingapofthedecoupledHaldanechainlimit phase is characterized by order-parameter fields gy and
with high accuracy34. The resulting phase diagram is gd for the non-frustrated bonds along the y and diag-
contrasted by analytical results we obtain within a gen- onal directions and fx for the frustrated x-bonds. The
eralization of the SBMFT to the anisotropic system. localconstraintsarereplacedbyaglobaloneandtreated
The remainder ofthe paper is organizedas follows. In with a Lagrange multiplier λ. The resulting mean-field
Sec.IIweoutlinetheSBMFTandcalculatetheresulting Hamiltonian, which in momentum space is given by
phasediagramwhichisusedforcomparisonwiththenu-
mericalresults. InSec.IIIweusetheDMRGtoconstruct
the phase diagram by a careful finite-size analysis of the = (h +λ)(b† b +b† b )
Hmf q q↑ q↑ −q↓ −q↓
magnetic structure factor at the ordering wave vectors Zqh
of the two magnetically ordered phases as well as of the ∆ (b† b† +b b ) , (4)
spin-gap in the non magnetic region as a function of the − q q↑ −q↓ q↑ −q↓
i
NNN coupling J for various ratios of the NN couplings
2 is easily diagonalized by a Bogolioubov transformation
J and J . Finally, in Sec. IV the results obtained
1x 1y yielding the dispersion
withinthetwomethodsarecomparedandtheshortcom-
ings of the semiclassical approach are highlighted. Fur-
ther, the potential relevance of our findings for the iron
ω =[(h +λ)2 ∆2]1/2 (5)
pnictides is discussed. q q − q
with
II. SBMFT
h = 4f J cosq cosq , (6a)
q d d x y
In this section we generalize the SBMF calculationfor
∆ = 2(g J cosq +g J cosq ) (6b)
the isotropic J -J -model16 to the case of anisotropic q x x x y y y
1 2
nearest-neigborexchangecouplings. Forabbreviationwe
in the N´eel phase and likewise in the (0,π) phase
define J := J , J := J , and J := J . Assuming
x 1x y 1y d 2
J J , the classical ground states are given by a N´eel
x y
≤
orderedstatewithanorderingwavevectorQ=(π,π)for
h = 2f J cosq , (7a)
q x x x
J /J 1/2 and by a columnar antiferromagnetic state
wdithxQ≤= (0,π) for J /J > 1/2. Following the previ- ∆q = 2gyJycosqy+4gdJdcosqxcosqy. (7b)
d x
ous calculations, we perform a spin rotation S˜x = σ Sx,
i i i The Lagrange multiplier in the ordered phases is de-
S˜y = σ Sy, S˜z = Sz, where σ = exp(iQr ) = 1
i i i i i i i ± termined by the requirement that ωQ = 0 for the cor-
with Q the ordering wavevectors of the two classical or-
responding ordering wavevectors yielding λ = 2(g J +
x x
ders, and represent the rotated spin operators in terms
g J ) 4f J intheN´eelandλ=2(g J f J )+4g J
y y d d y y x x d d
of Schwinger bosons bi,↑, bi,↓ as in the−(0,π)-phase. The reduced momen−t S∗ in the or-
dered phases is determined by
1
S˜i = 2b†i,νσν,ν′bi,ν′. (2)
1 h +λ
S∗ =S q 1 . (8)
Here σ = (σx,σy,σz) with σα the standard Pauli ma- − 2(cid:18)Zq ωq − (cid:19)
trices. The constraint b† b = 2S ensures that S2 =
i,ν i,ν h i Naturally, one might expect that quantum fluctuations
S(S+1). The Hamiltonian (1) can then be rewritten in
tendto destabilizethe classicalordersintroducing anin-
the compact form
termediate paramagnetic phase. In this case the second
order transitions between the two different magnetic or-
1 σ σ +1 dersaredeterminedbythelineswherethecorresponding
H = 2 Jij i j2 (Fi†jFij −2S2) magnetizations go to zero, S∗ 0. However, the anal-
(Xi,j) (cid:20) ysis of the isotropic model16 (J→x = Jy) shows that the
σ σ 1 quantum fluctuations can lead to a significant stabiliza-
+ i j2− (G†ijGij −2S2)(cid:21), (3) tionoftheN´eelorderleadingtoaregionwhereS(∗π,π) >0
4
and S∗ > 0. In this region which is expected to per-
(0,π)
sist for not too strong anisotropy, a discontinuous first
order transition between the two different magnetic or-
ders is likely and can be estimated from a comparisonof
the ground-stateenergieswhichimmediately followfrom
Eq. (3) as
E = J (2g2 S2) J (2g2 S2)
(π,π) − x x− − y y−
+2J (2f2 S2) (9a)
d d −
E = J (2f2 S2) J (2g2 S2)
(0,π) x x − − y y−
2J (2g2 S2). (9b)
− d d−
The order parameter fields entering the mean-field
Hamiltonian (4) have to be determined self consistently.
UsingtheaboveBogoliubovtransformationweobtainfor
the N´eel ordered phase
∆
g = S∗+ q cosq , (10a)
x x
Zq 2ωq FIG. 2: Phase diagram for S = 1 as a function of and
∆ anisotropyα=(J1y−J1x)/J1y >0betweenthenearestneig-
g = S∗+ q cosq , (10b) borexchangecouplingsandrelativestrengthofthenextnear-
y y
2ω
Zq q estneighborcouplingJ2/J1y obtainedwithinSBMFT.Upto
h +λ ananisotropyα≈0.66the(π,π)and(0,π)antiferromagnetic
f = S∗+ q cosq cosq , (10c)
d x y ordersareseparatedbyafirstordertransition(solidline). At
2ω
Zq q thetricritical pointthefirst-orderlinesplitsintotwosecond-
order lines (dashed and dotted) separating thetwo magnetic
whereasinthe(0,π)phasetheself-consistencyequations
phases from a gapped non-magnetic phase.
for the order-parameterfields are given by
h +λ For anisotropies α > 0.66 we find an intermediate non-
f = S∗+ q cosq , (11a)
x 2ω x magnetic region (S∗ =S∗ =0) separating the two
Zq q (π,π) (0,π)
ordered phases. This region is found to be very narrow
∆
gy = S∗+ q cosqy, (11b) and close to the classicalphase boundary β =(1 α)/2.
Zq 2ωq Although the existence of a tricritical point a−nd of a
∆
g = S∗+ q cosq cosq . (11c) very narrow paramagnetic strip terminating at the Hal-
d x y
Zq 2ωq danechainlimitJ1x =J2 =0isinqualitativeagreement
with the coupled cluster results33 the small width of the
The resultingphase diagramfor J1x J1y is shownin paramagnetic regionis in disagreement with other avail-
≤
Fig. 2asafunctionoftheanisotropyα=(J1y−J1x)/J1y able numerical results. For J1x/J1y = 0.2 (α = 0.8) a
between the NN exchange couplings and of the relative two step DMRG calculation34 shows a much wider non
strengthβ =J2/J1y of the NNN coupling. In agreement magnetic region centered around the classical transition
with earlier studies of the isotropic limit J1x =J1y =J1 point J2/J1y = 0.1. Interestingly, the spin gap at this
within SBMFT and MSWT we find a dramatic stabi- maximally frustrated point is almost identical to that of
lization of the N´eel order above the classical value and an isolated Haldane chain suggesting that even for rela-
a large region 0.51 . β . 0.68 where the two compet- tively strong interchain couplings one dimensional Hal-
ing orders are potentially stable, indicated by S(∗π,π) >0 dane chain physics is still important. This is certainly
and S∗ > 0. The crossing of the self-consistently de- missedbytheSBMFcalculationwhichtreatsthespinas
(0,π)
termined energies of the two states suggests a first or- a continuous variable and does not distinguish between
der transition at β 0.64 considerably larger than the integer and half-integer spins crucial for the existence
≈
classicalvalue 1/2 and about 10 percent bigger than the of the Haldane spin gap. Moreover, it has been estab-
coupled cluster result33. Although the region of coex- lished within quantum Monte Carlo calculations41,42,43
istence is considerably narrowed by a small anisotropy that for J = 0 the transition between the N´eel ordered
2
we find it to persist up to α 0.66 where the first or- phase and the gapped non-magnetic phase is located at
≈
der line terminates. The existence of such a tricritical J /J =0.044,againindicating that the paramagnetic
1x 1y
point was also suggested by the coupled cluster analysis region close to the Haldane chain limit is considerably
althoughlocated at a much smaller anisotropyα 0.34. wider than suggested by both the SBMFT and the cou-
≈
5
pled cluster results33. Since for J =0 the system is not indicatesthattheSBMFTtendstooverestimatethesta-
2
frustratedquantumMonteCarlocanbeconsideredexact bility of the magnetically ordered phases, surely close to
in this regime. the Haldane chain limit but presumably also for larger
Wehavealsocalculatedthetransitionoutofthe(0,π) values of J and J . In the following we shall refine the
1x 2
state in linear spin-wave theory, for comparison to the boundary region by using DMRG method.
SBMFT calculation and to preliminary neutron scatter- In the following DMRG calculation, we will set J =
1y
ing results on the pnictides that found (0,π) order but J as energy unit, and a periodic boundary condition
1
with a small moment. This method computes the re- (PBC)is usedandineachDMRGblock upto m=3200
ductionoftheclassicalantiferromagneticmomentdueto states are to be kept with the truncation error in the
zero-point excitations of spin waves (which captures the order of or less than 10−5.
1/S correction to the classical moment in a large-S ex-
pansion). The transition line out of the orderedphase is
estimatedasthe pointwherethecorrectionisaslargeas A. Isotropic case with J1x =J1y =J1
the original moment.
We assume three antiferromagnetic couplings: J1x, Let us first consider the isotropic case J1x = J1y =
J1y, and J2 with J1y > J1x and that we are in a (0,π) J1 where the SBMFT suggests the strongest tendency
orderedphase,whichrequires(asisevidentfromthefor- towards a first order transition although the transition
mula below) that J1x < 2J2. The dispersion relation point J2/J1 0.64 obtained from a comparison of the
≈
forspin-waveexcitationswaspreviouslyobtainedforthis energyminimaseemstobesuspiciouslyhighcomparedto
anisotropic J1-J2 model in Ref. 10. The integral for the the classical transition point J2/J1 = 0.5. Fig. 3 shows
correctionto the classicalmomentaroundthe (0,π)case the ground state energy per site calculated by DMRG
is,inunitsoftheBohrmagnetonandwithlatticespacing with the sample size varying from N = 4 4 (16 sites)
a=1, up to N = 8 8 (64 sites). The ground×state energy
×
reachesthemaximumwithJ between0.54J and0.58J ,
2 1 1
which becomes sharper with the increase of sample size,
d2k 1
∆m= 1 indicating a regionwith possible phase transitions below
Z[0,2π]2 (2π)2 1−cos2(ky)/f(kx)2 − ! the first order transition point obtained in SBMFT but
(12) still notable above the classical transition.
p
with
2J +J J [1 cos(k )]
f(k )= 2 1y − 1x − x . (13)
x 2J cos(k )+J -1.4
2 x 1y
-1.6
The transition is found numerically to lie very close
to the classical transition line J = 2J : the normal- -1.8
1x 2 N
ized difference (J2c −J1x/2)/J1y, where J2c is the criti- E/0-2.0
cal coupling where the correctionis equal to the original
moment, is always less than 2% for 0 < J1x < 0.99J1y. -2.2 N=8 8
Significant reduction of the ordered moment also occurs N=6 6
-2.4 N=4 4
only near the classical transition line. The primary dif-
-2.6
ference from the SBMFT calculation is that the spin- 0.0 0.2 0.4 0.6 0.8 1.0
wave calculation always predicts a paramagnetic phase J2/J1
betweenstripeandNe´elorder. Thewidthofthisparam-
agnetic phase is muchsmaller in either approachthanin FIG. 3: (color online) The ground-state energy per site at
the DMRG calculation of the following section, because different system size N = 4×4 (blue triangle), 6×6 (red
these analytical approaches do not capture the strong circle) and 8×8 (black square), for the isotropic case with
quantum fluctuations that favor the Haldane phase. J1x =J1y =J1.
We examine sucharegionby calculatingthe magnetic
III. DMRG structure factor S (q), defined by
z
The above SBMFT has clearly shown an interesting
1
narrow boundary region between the N´eeland stripe or- S (q)= e−iq(ri−rj) SzSz .
dered phases in the phase diagram of Fig. 2, where the z N h i ji
i,j
X
quantum fluctuations are expected to become very im-
portant. Mostinterestingly,itsuggestsanincreasingten- Asexpected,wefindthatS (q)showsadramaticchange
z
dencytowardsafluctuationinducedfirstordertransition frompeakingatQ=(π,π)(theN´eelorder)toQ=(0,π)
on approaching the isotropic point J = J . However, (the stripe order) with increasing J /J . Fig. 4 (a) and
1x 1y 2 1
the comparisonwith previous numerical results34,41,42,43 (b) illustrate S (q)/N vs. J /J for system sizes N =
z 2 1
6
0.4
J1y=1.0
J1y=1.0 0.3 0.3
)/N 0.3 J1x=1.0 )/N 0.2 )/N J1x N==04.54 )/N
S(,z0.2 NN==4646 S(0,z0.1 S(,z0.2 NN==6868 S(0,z0.2
0.1 N=88 0.1 (a) N= 0.1 (b)
(a) N= (b)
J2=0.525 J2=0.55 J2=0.24 J2=0.28
0.0 0.0 0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
J 2 J 2 J 2 J 2
1.0 1.0 1.0
Spin-1 gap 000...468 (c) Spin-1 gap 000...468 (d) JJ22==00..5503 Spin-1 gap 0001....4680 (c) Spin-1 gap 000...468 (d)
0.2 0.2 JJ22==00..5545 0.2 0.2 JJ22==00..1200 JJ22==00..2360
J2=0.525 J2=0.555 J2=0.60 J2=0.23 J2=0.285 J2=0.24 J2=0.36
0.0 0.0 0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.1 0.2 0.3 0.4 0.00 0.02 0.04 0.06
J2 1/N J2 1/N
FIG.4: (coloronline)Theevolutionofthepeakvaluesofthe FIG. 5: (color online) The peak values, (a) Sz(π,π)/N,
structure factors, Sz(π,π)/N (a), Sz(0,π)/N (b), and of the (b) Sz(0,π)/N, and (c) the spin-1 gap, versus J2/J1y at
spin-1 gap (c), at three different size N = 4×4, 6×6 and J1x/J1y = 0.5 for different sizes. The finite-size scaling for
8×8, as well astheir thermodynamiclimit extrapolations in thespin-1 gap is shown in (d).
the isotropic case J1x =J1y =J1 =1. The finite-size scaling
for the spin-1gap is shown in (d).
ing procedure, we find that the spin disordered phase is
bound by a lower transition point J /J 0.24 and an
2 1
4 4,6 6and8 8,aswellasthethermodynamiclimit upper transition point J /J 0.28 based≈on the struc-
va×lues o×btained b×y a finite size scaling. Here we have ture factor calculation. 2Aga1in≈the spin gap calculation
used the quadratic function f(x) = A+Bx+Cx2 with gives rise to a consistent spin disorderedregime between
x= 1 to perform the finite-size scaling. J /J 0.23 and J /J 0.285as shown in Fig. 5 (c).
N 2 1 2 1
The N´eel order transition point is found at J2/J1 ≈ In t≈he same regime, w≈ith a fixed J2/J1 = 0.25, we
0.525 in Fig. 4(a), which is clearly distinct from the have further studied the phase boundaries by varying
stripeordertranistionpointatJ2/J1 ≈0.55inFig. 4(b) J1x/J1. As shown in Fig. 6, the lower transition point
indicating that the two magnetically ordered phases are is obtained at J /J = 0.45 and the upper transition
1x 1
separated by an intermediate non-magnetic region. point at J /J 0.52, while the finite size scaling
1x 1
To independently verify the aboveresults, we alsocal- for the spin-1 gap≈results in a similar region between
pcurelasteenttehdeinspiFni-g1. ga4p(c)∆aEn(dS(=d).1)F≡ig.E14((Sd)=ill1u)st−ratEe0s J1xIn/Jt1h=e 0e.x4t4re−m0e.5c4a.se at J = 0 and J = 0 with
1x 2
the finite-size scaling for the spin gap ∆E(S = 1) at J = J , the system simply reduces to an array of de-
1y 1
different values of J2 using a scaling function f(x). In coupled S = 1 spin chains with a finite Haldane gap.
Fig. 4(c), theevolutionofthe spin-1gapasafunctionof Fig. 7 shows how the ground state continuously evolves
J2 is given. In the thermodynamic limit, the transition fromthatofthewell-knowndecoupledspinchainstothe
points determined by the spin-1 gap are at J2 ≈ 0.525 anisotropic2DcasebyturningonJ1x/J1,whichremains
tahnedpJr2ev≈iou0s.5re5s5u,ltrsesdpeetcetrimveinlye,dwbhyicthhearsetruvcetruyreclfoascetotro. dwiistohrdSer(eπd,πun)/tNil tbheecNom´eeinlgorfidneriteseitns itnheatthJe1rxm/Jod1y≈na0m.0i5c
z
Therefore, for the present S = 1 J1-J2 model in the limit. Indeed the corresponding spin-1 gap continuously
isotropic limit, our numerical approach has established decreases with the turning on of a finite J , but only
1x
an intermediate spin disordered region with a finite spin vanishes around J /J 0.06 as shown in Fig. 7 (c)
1x 1
gap which separates the two ordered magnetic phases. and (d). ≈
NowweturnonJ . AtJ =0,wefindthatwhilethe
2 1x
calculatedstructurefactorat(π,π)continuouslyreduces
B. Anisotropic case with J1x <J1y =J1 asthesamplesizeincreasesfromN =4 4,6 6,to8 8,
× × ×
and is extrapolated to zero by finite size scaling, a finite
Nowweconsiderhowthespindisorderedphaseevolves stripe order S (0,π)/N will emerge at J /J = 0.025
z 2 1y
with the increase of anisotropy at J <J =J . First in the thermodynamic limit, which is further supported
1x 1y 1
we consider the case at J = 0.5J , and the results are by vanishing spin-1 excitation gap at the same point as
1x 1
presented in Fig. 5. By using the same finite size scal- shown in Fig. 8. It is noted that the best finite scaling
7
0.15
)/N 0.2 J1(ay)=1.0 NNN===468468 )/N 0.2 (b) )/N 0.10 (a) JJ11yx==10..00 NNN===468468 )/N 0.3 (b)
S(,z0.1 J2=0.25 N= S(0,z0.1 S(,z N= S(0,z0.2
0.05
0.1
J1x=0.52 J1x=0.45 J2=0.025
0.0 0.0 0.00 0.0
0.4 0.5 0.6 0.4 0.5 0.6 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20
J 1x 1.0 J1 x J 2 J2
Spin-1 gap 0001....4680 (c) Spin-1 gap 000...468 (d) J1x=0.40 Spin-1 gap 0001....4680 (c)
J1x=0.46
0.2 0.2 J1x=0.49 0.2
0.0 J10x.4=0.44 0.5 J1x=0.540.6 0.00.00 0.02 0.04 J10x.=006.60 0.00.00 J2=00.0.0525 0.10 0.15
J1x 1/N J2
FIG.6: (coloronline)(a)Sz(π,π)/N,(b)Sz(0,π)/N,and(c) FIG. 8: (color online) At J1x = 0, the structure factors
the spin-1 gap, versus J1x/J1y at J2/J1y =0.25 for different Sz(π,π)/N (a) and Sz(0,π)/N (b) as well as the spin-1 gap
sample sizes. In (d), the finite-size scaling for the spin-1 gap (c) versusJ2/J1y areshown at differentsizes, includingtheir
is given. thermodynamicextrapolations.
0.3 0.15
(a) J1y=1.0 (b)
N J2=0.0 N
)/ 0.2 )/ 0.10 1.53 0.15
S(,z0.1 NN==4646 S(0,z0.05 ,)44..67 (a) 1.5z2S( ,)/N0.10 (b) JJ22==00..0001 JJ11yx==10..00
N=88 S(z J2=0.04 Sz(,) , S(z JJ22==00..0023
0.0 J1x=0.05 N= 0.00 4.5 Ny6 Sz(,) 1.51) 0.05 J2=0.04
0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20
J1 x J1 x
Spin-1 gap 0001....4680 (c) Spin-1 gap 0001....4680 (d) JJJJJ11111xxxxx=====00000.....0000002467 4)/N.400..12450 (c) 8 N12x 16 20 1.50 gap001.0..0800.00(d)0.05 0.101/N 01.1/52 0.20 0.25
0.2 J1x=0.06 0.2 JJJ111xxx===000...010908 S(0,z0.10 pin-1 00..46
0.00.00 0.05 0.10 0.15 0.20 0.00.00 0.02 0.04 0.06 0.05 S0.2
J1x 1/N
0.00 0.0
0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.02 0.04 0.06
1/2
1/N 1/N
FIG.7: (coloronline)(a)Sz(π,π)/N,(b)Sz(0,π)/N,and(c)
thespin-1gap,versusJ1x/J1y at J2 =0at differentsizes. In
(d),the finite-sizescaling for thespin-1 gap is given. FIG. 9: (color online) (a) The size dependence of the struc-
turefactor[foragivenNy =6(6-legsystems)]atJ1x =0and
small J2/J1y which quickly saturates at Nx >Ny, indicating
that x = 1/Ny in the scaling function f(x) for the structure
factor is more appropriate in this extreme limit. The corre-
forthestructurefactorhereisobtainedwithusingaf(x) spondingfinite-sizescalingofthestructurefactors forsquare
with x = 1/√N = 1/N , for a square lattice N = N , samples are illustrated in (b) and (c), respectively. For com-
y x y
instead of x = 1/N used previously. The justification parsion, the spin-1 gap at different J2 is still well scaled by
of such a finite-size scaling for the spin structure factors x=1/N in thescaing function as shown in (d).
at J = 0 and small J /J is given in Fig. 9 and its
1x 2 1y
caption.
8
C. Phase diagram
0.6
The resulting phase diagram obtained by the DMRG
calculations with careful finite size scalings of the mag-
0.5 Stripe order upper boundary
netic structure factor as well as the spin gap is shown
lower boundary
iJn F)/igJ. 10beatswaeenfuntchteionNNof ctohuepalinnigsostraonpdy tαhe=r(eJla1tyiv−e /J21y 0.4 mJ2a=x0i.m25al spin gap
1x 1y J
= 0.3
strength β =J /J of the NNN superexchange.
2 1y
IncontrasttotheSBMFTwhichintheregimeofsmall 0.2 Neel order
anisotropyαsuggestsafirstordertransitionbetweenthe
N´eel and stripe ordered phases (see Fig. 2), the DMRG 0.1 paramagnetic region
results clearly indicate a paramagnetic strip separating
0.0
the two magnetically orderedphases for all values of the 0.0 0.2 0.4 0.6 0.8 1.0
anisotropyαincludingtheisotropicpointJ1x =J1y (α= =(J1y-J1x)/J1y
0). With increasing α the width of this region is found
to slightlyincrease. Close to the Haldane chainlimit the
SBMFT predicts an intermediate non magnetic phase, FIG.10: (coloronline)Phasediagramfortheanisotropic J1-
thoughthewidthofthisregionisfoundtobemuchlarger J2 model determined by DMRG. The N´eel order and stripe
order phases are separated by a paramagnetic regime with
intheDMRGcalculationinquantitativeagreementwith
the boundaries denoted by the red line with full circles for
a previous two step DMRG analysis at fixed α = 0.8
the upper and the blank line with full squares for the lower
(Ref. 34).
transition points, respectively. In the middle of the param-
For J2 =0 we cancompareour results to recentQMC agnetic region lies in a dotted line with crosses which marks
simulations41,42,43, which in this regime can be consid- themaximalspin-1gap∆Emax. Notethatthesymbolsofcir-
ered exact since the system is not frustrated and QMC cle,square,andtriangulardenotethephasetransition points
is therefore free of any sign problem. Within DMRG we determined by DMRGin theprevious figures.
find the transition between the gapped Haldane phase
and the N´eel ordered phase to occur at J /J 0.05
1x 1y
(see Fig. 10) in good agreement with QMC tran≈sition SBMFTforthemostofJ1x <J1y =J1 regionexceptfor
point J /J = 0.044. Moreover, in the Haldane chain the part close to the isotropic limit and that the points
1x 1y
limit J = J = 0 we obtain a spin gap ∆ 0.4 (see wherethespingapismaximum(indicatedbycrossesand
1x 2
Fig. 7(d)) very close to the exact value ∆ =≈ 0.41 for dotted in Fig. 10) are almost on top of both the first or-
H
the Haldane spin chain35, again demonstrating that the der transition line as well as of the two narrow second
finite size scaling is well converged. order lines obtained within SBMFT.
Very interestingly, starting from the Haldane chain
limitwefindthemaximumspingapintheparamagnetic
phase to decrease only very slowly with increasing cou- IV. DISCUSSION
plingsJ andJ . UptoJ /J =0.5,thegapisalmost
1x 2 1x 1y
identicalto the Haldane spingap(see Fig. 5(d)) indicat- In summary, we have studied the frustrated spin-1
ingthattheHaldanespinchainisveryrobustagainstthe Heisenberg J -J -J model on a square lattice numeri-
1x 1y 2
simultaneous increase of the mutually frustrating cou- cally using the DMRG method and analytically employ-
plings J1x and J2. Remarkably, even the paramagnetic ing the Schwinger-Boson mean-field theory (SBMFT).
phaseofthe isotropicJ1-J2 modelwitha maximumspin Interestingly,thismodelcontainsboththeisotropicJ1-J2
gap still being about half of the Haldane gap is continu- model as well as decoupled Haldane spin chains as lim-
ously connected to the Haldane spin-chain limit. iting cases. Moreover, it has attracted a lot of attention
At the isotropic point J = J (α = 0) we recentlysinceithasbeenmotivatedasaneffectivemodel
1x 1y
find an intermediate paramagnetic phase for 0.525 . todescribethe(0,π)-magnetismandthelow-energyspin-
J /J . 0.555. This region is considerably smaller waveexcitationsofthe celebratednovelironpnictide su-
2 1
than in the S = 1 case, where a paramagnetic regime perconductors. Furthermore it has been suggested that
2
0.4 . J /J . 0.6 presumably with columnar valence the drastic reduction of magnetic moments to a value of
2 1
bond order has been established by various numeri- 0.4 µ is caused by strong quantum fluctuations in the
B
cal methods18,19,20,21,22,23,24,25,26,27,28. Interestingly, the vicinityofacontinuousphasetransition. However,tothe
stabilizationofthe N´eelphase abovethe classicaltransi- present day the phase diagram has not been determined
tion point is in agreement with the SBMFT. yet.
Despite the apparent failure of the semiclassical Within both the SBMFT andthe DMRG wefind that
SBMFTindealingwiththe strongquantumfluctuations theN´eelphaseisstabilizedconsiderablybyquantumfluc-
inthe boundaryregionbetweenthe twomagneticallyor- tuations above the classically stable region up to a rela-
dered phases it is interesting to note that the param- tivelystronganisotropybetweentheNNcouplings. How-
agnetic region covers the phase boundaries obtained by ever,inallotherregardsthephasediagramsobtainedby
9
the two methods clearly disagree, indicating the impor- be on the verge of undergoing a quantum phase transi-
tance of the strong quantum fluctuations. tion in a quantum disordered state. This quantum spin
The SBMFT suggest a fluctuation induced first or- physics then sets the conditions for the emergence of su-
der transition between the N´eel and the stripe antifer- perconductivityinthedopedsystems. Butinthisregard
romagnetsterminatingatatricriticalpointandsplitting the size of the microscopic spin does matter more than
into two second order lines separated by an intermedi- one intuitively anticipates. For S = 1/2 the well estab-
ate paramagnetic region only for large anisotropies. Al- lished fact that the Berry phases conspire to turn the
though the existence of a tricritical point and a hardly quantum disordered states of the spin-only systems into
seizableparamagneticstripterminatingatthedecoupled valence bond solids gives a rationale to take Anderson’s
Haldane spin chain point (J = J = 0) are consistent resonatingvalencebond(RVB)ideafortheoriginofhigh
1x 2
with a recent coupled cluster calculation33, the SBMFT Tc superconductivity quite seriously. The valence bonds
definitely falls short on approaching the Haldane chain are protected by the spin gap and the effect of doping
limit. Thisbecomesclearbyacomparisonwithquantum could well be to just turn the valence bond solids into
MonteCarlosimulations41,42,43showingamuchlargerre- translational quantum liquids that transport two units
gionofstabilityoftheHaldanechainphaseagainstinter- of electrical charge. Focussing on the Berry phases in
chain couplings J . Also the previous two-step DMRG the pnictides one has only the options of an ‘intermedi-
1x
calculation34 at J /J = 0.2 indicates a paramagnetic ate’ crystal field S = 1 state10 or the high spin S = 2
1x 1y
region being an order of magnitude wider than found for the microscopic spin. In the former case the ground
within the SBMFT. state has a twofold degeneracy and the ‘building blocks’
On the contrary, the phase diagram obtained within are no longer pair singlets but instead the ‘chain sin-
DMRG by a careful finite size scaling of the magnetic glets’. Thinking along the RVB lines, what to expect
structure factor and the spin gap is consistent with both when such a system is doped? The dopants will increase
the QMC and the previous DMRG results. Furthermore the quantum fluctuations but chains are not pairs. One
the spin gap in the decoupled chain limit agrees well anticipates that doping might drive the system into the
with the exactvalue for the isolatedHaldane chain. The non-magnetic ‘Haldane chain phase’ breaking the two-
width of the paramagnetic region slightly decreases on foldrotationalsymmetryofthe lattice. One noticesthat
approaching the isotropic point but remains finite over something of the kind is found in the phase diagram of
the entireparameterrange. This has the remarkableim- the pnictides: the structural transition to the orthorom-
plication that the paramagnetic phase of the frustrated bicphasepersiststomuchhigherdopingsthanthestripe
two dimensional spin model, including the isotropic J - antiferromagnetism47. Atfirstviewthisseemsratherde-
1
J limit, is continuously connected to the Haldane spin- tached from the RVB idea. But now one has to realize
2
chainphase. Inotherwords,theparamagneticphasecan that the charge carriersare themselves spinfull, carrying
be viewed as a continuation of the Haldane spin chain by default a half-integer spin. Taking for instance the
phase, with the caveat that the topological string order intermediate S = 1 background, the holes would carry
has to disappear at any finite interchain coupling38,39. S = 1/2, and t-J models corresponding with a mix of
The reasonfor the failure of the semiclassicalSBMFT S = 1 and S = 1/2 states have been studied in the
indealingcorrectlywiththestrongquantumfluctuations past48. One anticipates that in an incompressible ‘chain
intheboundaryregionbetweenthetwoclassicalordersis like’ S =1 backgroundthe S =1/2 carriers might again
easy to understand. The SBMFT deals with the spin as be ‘glued by Berry forces’ into valence bond pairs, re-
continuous variable while it is blind for the Berry phase establishing a connection with the RVB mechanism of
effects that distinguish between half-integer and integer the cuprates. A similar idea has been previously pro-
spinvaluescrucialfortheexistenceoftheHaldanegapas posed from a different approach49.
well the valence bond crystals. This physics is definitely Inconclusion,itisquitequestionablethatanyofthese
missed by the semiclassical treatment. Our DMRG re- considerations have a bearing on pnictide superconduc-
sults demonstrates that the Haldane chain phase is very tivity but they do make the case that there is still much
robustagainstthesimultaneousincreaseofthecouplings interesting physics to be explored dealing with systems
J and J along the strongly frustrated boundary re- characterizedby a larger microscopic spin.
1x 2
gion, indicated by the minute reduction of the spin gap Acknowledgment: We are grateful for stimulating
compared to the isolated Haldane chain limit. discussion with M. Q. Weng. This work is partially sup-
What does the study of this spin system teach us re- portedbyNSFCandtheNationalProgramforBasicRe-
gardingthesuperconductivityintheironpnictides? One searchof MOST, China, the DARPA OLE program,the
couldspeculate thatthe basicphysicsissimilarasinthe DOE grant DE-FG02-06ER46305,the NSF grant DMR-
cuprates. Although the spins are larger in the pnictides, 0605696andtheStichtingvoorFundamenteelOnderzoek
the geometrical frustration renders the spin system to der Materie (FOM).
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