Theory of magnetoelectric resonance in two-dimensional $S=3/2$ antiferromagnet ${\rm Ba_2CoGe_2O_7}$ via spin-dependent metal-ligand hybridization mechanism

0.19 MB·English

by  Shin Miyahara

Theory of magnetoelectric resonance in two-dimensional S = 3/2 antiferromagnet
Ba CoGe O via spin-dependent metal-ligand hybridization mechanism
2 2 7
S. Miyahara1 and N. Furukawa1,2
1 Multiferroics Project (MF), ERATO, Japan Science and Technology Agency (JST),
c/o Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-8656, Japan
2 Department of Physics and Mathematics, Aoyama Gakuin University,
5-10-1 Fuchinobe, Chuo-ku, Sagamihara, Kanagawa 252-5258, Japan
(Dated: January 20, 2011)
WeinvestigatemagneticexcitationsinanS =3/2Heisenbergmodelrepresentingtwo-dimensional
1
antiferromagnet Ba2CoGe2O7. In terahertz absorption experiment of the compound, Goldstone
1
mode as well as novel magnetic excitations, conventional magnetic resonance at 2 meV and both
0
electric- and magnetic-active excitation at 4 meV, have been observed. By introducing a hard
2 uniaxialanisotropytermΛ(Sz)2,threemodescanbeexplainednaturally. Wealsoindicatethat,via

n thespin-dependentmetal-ligandhybridizationmechanism,the4meVexcitationisanelectric-active
a mode through the coupling between spin and electric-dipole. Moreover, at 4 meV excitation, an
J interferencebetweenmagneticandelectricresponsesemergesasacrosscorrelatedeffect. Suchcross

9 correlation effectsexplainthenon-reciprocallinear directional dichroism observedin Ba2CoGe2O7.
1
PACSnumbers: 75.80.+q,75.40.Gb,75.30.Ds,76.50.+g

]
l
e
- Multiferroic materials have attractedboth experimen- tric behaviors can well be explained by a local electric
r
t talandtheoreticalinterestsdue togiantmagnetoelectric dipole moment which couples to the local spin structure
s
. effects [1–3]. Such strong couplings between magnetism ofCoatomviathemetal-ligandhybridizationmechanism
t
a and electric polarization(EP) are often realizedthrough between Co and O atoms. For a classical spin within
m spin-dependent EPs. For example, in cycloidal magnets the xy-plane S =(Scosθ,Ssinθ,0), an EP on a CoO
m 4
d- RMnO3 (R=Tb,Dy, andothers),EPflopsfromPkcto tetrahedron along z is described as pzm = −S2Kcos 2θ,
P abychangingamagneticstatefrombctoabcycloidal which reproduces the experimental results.
n k
statethroughexternalmagneticfields[4]. Anotherexam-
o However, there still exist several features to be un-
c ple is magnetic resonance induced by oscillating electric derstood. One of them is magnetic excitation property
[ field, or electromagnon, which is observed in an optical
observedin an electromagnetic wave (EMW) absorption
spectroscopy at terahertz (THz) frequencies for a vari-
1 experiment (AE) in the THz frequency regime, which
v ety of multiferroics compounds, e.g., RMnO3 [5–7] and indicates magnetic resonances at ω 0 [18], 2, and 4
9 CuFe Ga O [8]. Theexchangestriction[7,9]andthe ∼
1 x x 2 meV [19]. The lowest two peaks can be assigned to spin
7 spin cu−rrent [10–12] mechanisms are well known as the
wave branches which have been reported in the inelas-
6
origins of such spin-dependent EP.
3 tic neutron scattering experiments (INS) [16]. Here, two
. Spin-dependent metal-ligand hybridization has been distinct modes exist at the Γ point in the two-sublattice
1
proposedasanalternativemechanism[13,14]. EPalong ground state due to an anisotropy. However, the ori-
0
1 the bond direction rml connecting metal and ligand de- gin of the excitation at ω 4 meV is not clear within
∼
1 pends on a spin structure at a metal site S in a form magnon pictures. The other point to be understood is
m
v: pml (Sm rml)2rml. At a spin site with no inversion, the THz AE on several EMW polarizations which indi-
∝ ·
i such a mechanism can induce an electric dipole which is cates that the excitation at 4 meV is induced by both
X
coupled to the spin, and has a potential to induce novel magnetic and electric components of EMW, whereas the
r features. In fact, magnetic field dependence of the fer- excitation at 2 meV is excited mainly by the in-plane
a
roelectricity observed in Ba CoGe O can well be ex- magneticcomponent. Moreover,the resonanceat4meV
2 2 7
plained by introducing this mechanism [15]. shows a non-reciprocal directional dichroism (NDD) un-
der the external magnetic fields [19], i.e., absorption in-
Ba CoGe O isaquasitwo-dimensionalantiferromag-
2 2 7
tensitystronglydependsontheEMWpropagationdirec-
net (Fig. 1 (a)). Below T = 6.7 K, Co magnetic mo-
N
tions (forward +k or backward k). In contrast, NDD
ments (S = 3/2) show an antiferromagnetic structure, −
is not clearly observedat the 2 meV resonance. The ori-
where magnetic moments are aligned in xy-plane due
gin of the magnetic excitation and the absorption mech-
to an easy-plane anisotropy [16]. In the magnetically
anism is very important to understand the principle of
ordered state, peculiar magnetoelectric behaviors have
the NDD.
been observed [15, 17]. For example, EP along [001]
shows sinusoidal angular dependence with a period of π In this Letter, we propose that a uniaxial anisotropy
for a rotation of the magnetic field Bex within xy-plane term Λ(Sz )2 (Λ > 0) gives clues to understand these
m
at Bex > 1 T. As shown in Ref. 15, such magnetoelec- features. In an S = 3/2 system, the uniaxial anisotropy
∼
2
(a) y (b) py =K [cos(2κ )(SySz+SzSy)+sin(2κ )(SzSx+SxSz)],
3 i i i i i i i i i i i
J xMµ (/f.u.)B 12 exp.c 2aKl. aSthniyedSAixp)zi(]B,=w)-hKseurb[eclaoκtsit(ii2scκetih)ae{s(rSioniyta)F2tiig−o.n(1aSn(ixag))l2e}[w1−5it]s.hinMκ(i2a=κgin)κe((tS−iizxκaSt)iyioo+nn
A Dz x (c) 0 0 5 Bex (T) 10 15 (aγnd=ExP,ayr,eadnedfinze)d, raessMpecγti=vePly.gMµBµa0nSdiγPanudnPdeγr=thPe ipnγi-
150
B κ [110][010] z2Pµ (C/m) 0 yBexφBxexp.c 2aKl. cbplleuaesnnteercesaxlt(ceNurlna=atledm8,ab1g0yn,eaatnnicdefix1ea2ldc)t.BTdeioxakgr(oecnpoarsolφidzBua,tcsieoinnMφoBxn,a0Nn)dh-saPivtzee
−κ [110] -150
0 π/2 π observed in Refs. 15 and 21, the parameters in Eq.(1)
[100] φB are estimated as J/Λ = 0.125, Dz /Λ = 0.005, Λ = 1.3
| |
meV, κ = π/8, and K = 3.8 10 32 Cm. Here we
−
× ·
use g = 2 while V = 1.0 10 28 m3 is the volume per
−
FIG.1: (Coloronline)(a)CrystalstructureforBa2CoGe2O7. Co. As typical examples,×the magnetization curve along
Top view of CoO4 tetrahedra is illustrated by squares. An Bex x and magnetic field direction dependence of Pz at
S = 3/2 spin locates on a Co-site with a spin interaction J. k
DM interaction −Dz (SxSy −SySx) is also included. On Bex = 5 T on a 12-site cluster are shown in Figs. 1 (b)
each bond, positive i(jnegiatijve) siginjof Dz is represented by and (c), respectively. Here, system size effects are found
ij
⊙ (⊗), and the direction from i- to j-site is indicated by to be negligibly small. Note that the magnonenergy ob-
the arrow. The tetrahedra CoO4 on A- and B-sublattices servedintheINS[16]at2Kisalsoreproducedwiththis
are rotated around z axis with a rotation angle κ and −κ, parameter set (see Fig. 2 (a)). We have confirmed that
respectively. See also Ref. [15]. (b) Magnetization curve for the resultsare qualitativelyrobustagainstchoicesof the
Bexkx. (c) EP along z-direction in the in-plane magnetic
parameters within the strong anisotropy limit J/Λ 1.
field Bex = (BexcosφB,BexsinφB,0) (Bex = 5 T) on a 12- ≪
Let us consider excitationprocesses by magnetic com-
site cluster. Experimental data in (b) and (c) are extracted
ponents of EMW, i.e., M1 transitions, which are related
from Ref. 15 and Ref. 21, respectively.
to the imaginary part of the magnetic susceptibility:
π
splits single spin energies into two doubly degenerate Imχmm(ω) = nMγ 0 2δ(ω ω ).(2)
γγ ¯hNVµ X|h | | i| − n0
states: 1 with an eigenenergy Λ/4 and 3 with 0 n
|± 2i |± 2i
9Λ/4. Here, m isastatewithS =mforspinS =3/2.
| i z Here 0 is the ground state, n are excited states and
In the strong anisotropy limit J/Λ 1, where we ne- | i | i
glect the higher energy spin states ≪ 3 , an S = 3/2 ¯hωn0 are excitationenergies to |ni while γ =x,y, and z.
|± 2i Eq. (2) is calculated on N-site clusters (N = 8,10, and
HeisenbergmodelcanbeapproximatedbyanXXZmodel
12) by the Lanczos method [23], where the δ-function
eff = J ∆(σxσx+σyσy)+σzσz with ∆=4 by us-
H P { i j i j i j} is replaced by a Lorentzian with a width ǫ/Λ = 0.1.
ing anS =1/2 spin operatorσ [20]. It shouldbe a good
The results at Bex = 0 are shown in Fig. 2 (a). Out of
approximation to reproduce the lowest two branches of
plane component Imχmm(ω) vanishes. In-plane compo-
excitations. In fact, ∆ 2.5 gives a good fit to neutron zz
∼ nentsarefoundtobeidentical,Imχmm(ω)=Imχmm(ω).
data in Ref. 16. On the other hand, the highest energy yy xx
They show that the magnetic components Hω and Hω
mode at 4 meV can be assigned to magnetic excitation x y
inducemagneticresonancesataround2meVand4meV
due to the single ion anisotropy gap 2Λ.
(Fig. 2 (a)). As shown in the figure, the system size ef-
To clarify the absorptionprocessesatTHz frequencies
fects are small. Hereafter, we show the results on the
indetail,weinvestigateanS =3/2Heisenbergmodelon
12-sitecluster. We indeedseethattheexcitationaround
a square lattice with the uniaxial anisotropy term under
2 meV corresponds to one of the spin-wavebranches ob-
external magnetic field Bex:
servedintheINS[16],whilethehigherenergymodeisan
= J S S Dz (SxSy SySx) excitation accompanied with the anisotropy gap excita-
H X(cid:8) i· j − ij i j − i j (cid:9) tion 2Λ. These features are clarified from J dependence
n.n.
ofthepeakpositions. AsshownintheinsetofFig.2(a),
+ Λ(Sz)2 gµ Bex S , (1)
X(cid:8) i − B · i(cid:9) indecreasingJ,thehighenergypeakcontinuouslyshifts
to single site gap excitation 2Λ = 2.6 meV, whereas the
whereS isanS =3/2spinoperatoroni-site. Thedirec-
i low energy peak position is proportional to J.
tions of Dzyaloshinsky-Moriya(DM) interactions Dz on
ij When spin states couple to electric fields through EP,
each bond can be determined uniquely from the crys-
E1processmayexcitemagneticexcitations[24,25]. Such
tal structure as in Fig. 1 (a) [22]. The DM interac-
processes can be clarified from the dielectric susceptibil-
tion lifts the two-fold degeneracy in antiferromagnetic
ity via spin-dependent EP
ordered states. Reflecting the rotation of CoO tetrahe-
4
d−rKon[acoros(u2nκdi)z(SaizxSisix,l+ocSailxSEizP)s+onsiin-(s2itκei)a(rSeiygSivize+nbSyizSpxiiy)=], Imχeγeγ(ω) = ¯hNπVǫ0 Xn |hn|Pγ|0i|2δ(ω−ωn0). (3)
3
(a) 2 J/Λ xx, N= 8 (a) 1 Bex =1.2 T, xz
0 0.05 0.1 xx, N = 10 Bex =3.5 T, xz
mmχm 1 ω (meV)2 Λ04 xzxz,, INNN S== 211K22 meχIm 0 BBBeeexxx ===787...030 TTT,,, xxzzzx
I
0 1 2 −hω (meV )3 4
1 2 −hω (meV )3 4 (b) 40 ∆ α
(b) eeχ 12 AE (Eω || z, Hω || x) 3.4xz Kxz -1∆ α (cm) 2 00 AE ∆3∆ .α4 αA KFF
m
I -20
1 2 −hω (meV )3 4
0 (c) φ (d) (e) (f) (g) | e − >
1 2 −hω (meV )3 4 B 0 | e + >
y Dz Dz Dz Dz HHyxωω 2Λ| g − > Ezω
FIG. 2: (Color online) (a) Im χmγγm(ω) for γ = x, and z. x ABexφ0 Bex Bex Bex 2BMF
Magnon energy observed in INS is extracted from Ref. 16. | g + >
Inset: Peak positions around 2 and 4 meV for Im χmm(ω) as
xx
afunctionofJ/Λon12-sitecluster. (b)Imχee(ω)forγ =x,
γγ
and z on 12-site cluster. Peak positions observed in THzAE FIG.3: (Coloronline)(a)Imχme(ω)undertheexternalmag-
xz
for Eω and Hω polarizations are extracted from Ref. 19. netic fields Bexkx. At 7 T, Im χme(ω) is also shown. (b)
z x zx
Differenceoftheabsorptioncoefficient byEMW propagating
directions∆α(ω) at Bex=7T, which can bedecomposed to
x
At Bex = 0, in-plane components of dielectric suscepti- ferromagnetic component∆αF(ω)andantiferromagnetic one
bility are found to be uniform, Imχeyey(ω) = Imχexex(ω), ∆αAF(ω). ∆α(ω)obtainedfromTHzAEforEzω andHxω are
as in the case for the magnetic susceptibility. Contri- extractedfromRef.19. (c)-(f)Twosublatticespinstructures
inBex. Seetextfordetail. (g)Excitationprocessinthelimit
butions to the 2 meV absorption are small. The 4 meV x
BMF/Λ≪1.
resonanceisactiveforanyelectriccomponents(seeFig.2
(b)). From these results, we conclude that the selection
rules and the peak positions are consistent with those
+ 0Pz n nMx 0 δ(ω ω ),(4)
obtained in the THz AE [19]. h | | ih | | i(cid:17) − n0
The temperature dependence of THz AE can also be
explained qualitatively. In Ref. 19, 4 meV absorption is where c≡1/√ǫ0µ0. The results under Bexkx are shown
in Fig. 3 (a). Imχme(ω) is enhanced around 4 meV
observed even above T , whereas absorption at 2 meV xz
N excitation. Note that Imχme(ω) is much smaller than
vanishes at T upon increasing the temperature. The zx
N Imχme(ω). Imχme(ω) at 7 T is also shown in Fig. 3 (a).
anisotropy gap excitation energy 2Λ 30 K is larger xz zx
∼ Experimentally, such a cross correlated effect can
than T , and such a resonance can be observed even
N
above N´eel temperature, i.e., T < T < 2Λ. However, be observed as the linear NDD [19]. By introduc-
N
ing a complex refractive index N, a polarized plane
theresonanceat2meVvanishesaboveT∼,sincethespin
N wave with Eω z, Hω x and k y is described as Eω =
wave excitation exists only in the ordered state. k k k z
Ezexp( iω(t (Ny/c))) and Hω = Hxexp( iω(t
In practice, M1 and E1 processes are invoked through 0 − − x 0 − −
(Ny/c))). From the Maxwell’s equations, N is given
the interaction with EMW as = Eω P Hω M,
H′ − · − · as N ǫ (ω)µ (ω) χme(ω) where µ (ω) =
where Eω (Hω) is the electric (magnetic) component of ± ∼ p zz xx ± xz xx
1+χmm(ω) and ǫ (ω) = 1+χee(ω). Here, N+ (N )
EMW. Provided that both M1 and E1 processes induce xx zz zz −
is a complex refractive index for EMW propagating to
an identical excitation, there is a cross correlation be-
+y ( y) direction. Thus, non-reciprocal part of an ab-
tween magnetic and electric components of EMW, i.e., −
sorption coefficient α (ω) = (ω/c)ImN is given by
theinterferencebetweenelectricandmagneticresponses. ± ±
∆α(ω) = α+(ω) α (ω) = (2ω/c)Imχme(ω). ∆α(ω)
As we show details in the following,the effects of the in- − − xz
at Bex =7 T is shown in Fig. 3 (b), where the peak po-
terference can be observed directly as the linear NDD, x
sition and the magnitude of ∆α(ω) are consistent with
e.g., the interference enhances absorption intensity for
those observed in THz AE [19].
the EMW with a propagation vector +k but weakens
Let us consider the magnetic origin for non-
that for the EMW with k, since reversing k is equiva-
− reciprocal part of the absorption coefficient ∆α(ω)
lent to reversing the relative sign of Eω and Hω due to ∼
(2ω/c)Imχme(ω). Under the external magnetic field
Hω = (1/µ ω)k Eω. As a typical case, we consider xz
0 × Bex z, spin structure in the N´eel ordered state is
dynamical magnetoelectric susceptibility for Mx and Pz ⊥
uniquely determined due to an energy gain of the DM
Imχme(ω) = πc 0Mx n nPz 0 term, e.g., the state in Fig. 3 (c) is stabilized by Bex x
xz X2h¯NV (cid:16)h | | ih | | i with Bex > 0. The EMW propagating to y directikon
n x −
4
in Fig. 3 (c) corresponds to that propagating to +y di- derBex (cosφ ,sinφ ,0)givesφ =φ (π/2 φ )(
B B i B 0
k ∓ − −
rection in Fig. 3 (d) which is realized by a 180 rotation (+) for the spin on A(B)-sublattice), where φ is a spin
◦ 0
aroundzaxisonaspinsite. Thus,reversingthemagnetic canting angle. By applying Eqs. (5)-(7) to Eq. (4), we
field Bex Bex is equivalent to reversing the EMW obtain that
x → − x
direction ky ky, which is consistent with the ex-
→ −
perimental observation [19]. Note that, when we reverse ∆I(φω) cos(φω +φB)sin(2κ φ0). (8)
∝ −
the magnetic field Bex Bex, both ferromagneticmo-
ment mx Sx +Sxx a→nd−antxiferromagnetic component ForφB =0(astateinFig.3(c)),weobtainthat∆I(0)
my Syu ≡SyAchanBge their sign as shown in Figs. 3 (c) sin(φ0 2κ) (for Hω x) and ∆I(π/2)=0 (for Hω y) a∝s
ansd≡(d).AE−achBcontributionto∆α(ω)canbeobtainedby already−expectedfromkthesymmetryargumentinRkef.19.
changingthe signofDz inthecalculation,sinceonlymy We see ∆I(0) = 0 even for φ0 = 0, which indicates the
s 6
(mx) changes its sign between states in Figs. 3 (c) and existence of the NDD in a collinear N´eel ordered state.
u
(e) ((c) and (f)). As a result, ∆α(ω) is decomposed into In addition, we can predict the NDD for k [010],
k
two parts ∆αF(ω) and ∆αAF(ω), which depend on the Eωk[001], and Hωk[100] (φω = −π/4) under Bexk[010]
modification of mxu and mys, respectively. By comparing (φB =π/4),althoughthereisnospontaneousEP[15]for
absorptions for the states in Figs 3 (c)-(f), ∆α (ω) and this Bex direction. In fact, χme (ω)under Bex [010]
F [100][001] k
∆α (ω) are extracted as in Fig. 3 (b). The results in- is found to be non-zero in the numerical calculation.
AF
dicate that ∆α (ω) is dominant for the NDD around Observation of the NDD in this condition is a crucial
AF
4 meV. Generally, NDD can be realized when sponta- test for the validity of our theory. As another exam-
neous magnetization and EP coexist. In the present ple, we can easily derive ∆Izx ∝ R Imχmzxe(ω)dω = 0 for
model, however,realizationofa N´eelorderedstate with- the EMW with Eω x and Hω z under Bex x, which
k k k
out ferromagnetic moment is sufficient to induce NDD. is consistent with the results calculated at Bxex = 7 T:
Once a singledomainstructure ofthe N´eelorderedstate R Imχmzxe(ω)dω ≪R Imχmxze(ω)dω (see Fig. 3 (a)).
is realized, ∆α(ω) can be finite even at Bex 0 and Our results indicate the potential of the spin-
x →
Dz 0. dependent metal-ligand hybridization mechanism for
| ij|→
Finally, we note that the selection rules and the cross novel absorption processes which might be observed in
correlatedeffects can qualitatively be determined within a wide range of materials with a spin at a site without
a mean field (MF) approximation. The spin Hamilto- inversion center, e.g., in a tetrahedron and a pyramid of
nian (1) can be approximated as MF = Λ(Sz)2 ligand atoms.
H P{ i −
gµ BMF S ,where gµ BMF gµ Bex J S We thank I. K´ezsma´rki, N. Kida, S. Bord´acs H. Mu-
B i · i} B i ≡ B −Pj{ h ji∓
Dz ( Sy , Sx ,0) ( (+) for the i-site on A(B)- rakawa,Y.Onose,T.Arima,R.Shimano,andY.Tokura
s|ubijla|thticjei).−hFojrisim}plic−ity, spin states under BMF = for fruitful discussion. This workis inpartsupportedby
i
(BMFcosφ ,BMFsinφ ,0) in the limit BMF/Λ 1 are Grant-In-Aids for Scientific Research from the Ministry
i i
≪ of Education, Culture, Sports, Science and Technology
discussed. Four eigenstates at site i are given in a
form: |gi±i = √12(e−iφi/2|12i±eiφi/2|− 21i) and |e±i i = (MEXT) Japan.
1 (e i3φi/2 3 ei3φi/2 3 )asinFig.3(g). Eigenener-
√2 − |2i± |−2i
giesfor|gi±iareΛ/4∓gµBBMF andforboth|e±i i,9Λ/4.
Asatypicalexample,letusconsiderexcitationprocesses
inducedbyHω,Hω andEω. Fromthegroundstate g+ , [1] Y. Tokura, Science 312, 1481 (2006).
processes throxughySx, Sy zand pz are | i i [2] W. Eerenstein et al., Nature(London) 442, 759 (2006).
i i i [3] S.-W. Cheong and M. Mostovoy, Nat. Mater. 6, 13
Six|gi+i = cosφi|gi+i+isinφi|gi−i [4] T(2.0K07i)m.uraet al., Nature(London) 426, 55 (2003).
Siy|gi+i = +sin√φ3i(|cgoi+siφ−i|ei+icios+φii|sgii−niφi|e−i i)/2, (5) [[[567]]] ANR...VPK.iimdAaegnueotivlaarle.t,etJal.a.l,O.,NpPath.t.ySsPo.chR.yeAsv.m.2L.,eB9t7t2.(1620,020A6,3)05.47(2200039()2.009).
+√3(sinφi|e+i i−icosφi|e−i i)/2, (6) [[89]] ST..SAerkiimeateatl.,alP.,hPysh.yRs.eRv.evL.etLte.t1t.0956,,090792702702(2(021000)6.).
pzi|gi+i = −√3Kcos(2φi−2κi)|e+i i [10] H. Katsura et al., Phys.Rev.Lett. 95, 057205 (2005).
−i√3Ksin(2φi−2κi)|e−i i. (7) [[1112]] MI..AM.Soesrtgoiveonyk,oPahnyds.ER.eDva.gLoetttto.,9P6h,y0s6.7R6e0v1.(B207036,)0.94434
We see that Eω can only induce magnetic excitations (2006).
z [13] C. Jia et al., Phys. Rev.B 74, 224444 (2006).
with the anisotropy gap 2Λ. This is consistent with
[14] T. Arima, J. Phys.Soc. Japan 76, 073702 (2007).
the results in Fig. 2 (b). For the EMW with Eω z and
k [15] H.Murakawaetal.,Phys.Rev.Lett.105,137202(2010).
Hω (cosφω,sinφω,0),thecrosscorrelationeffectisqual- [16] A. Zheludev et al., Phys. Rev.B 68, 024428 (2003).
k
ifiedbyaspectralweight∆I(φω)∝R(cosφωImχmxze(ω)+ [17] H. T. Yi et al., Appl. Phys.Letter 92, 212904 (2008).
sinφωImχmyze(ω))dω. The canted N´eel ordered state un- [18] I. K´ezsma´rki et al. (2011), privatecommunication.
5
[19] I. K´ezsma´rki et al. (2010), accepted for publication in ished.
Phys.Rev.Lett. (e-print available at arXiv:1010.5420). [22] Dx (Dy) component on the bond along y (x) direction
[20] pz is suppressed as Bex is decreased below ∼ 1 T and is also non-zero, but the effects of them are found to be
x
deviates from the theoretical estimates based on a clas- negligible in the calculations.
sical spin picture [15]. However, thesuppression of pz at [23] E. Dagotto, Rev.Mod. Phys. 66, 763 (1994).
Bex =0 can be understood by considering the effects of [24] Y. Tanabe et al., Phys.Rev.Lett. 15, 1023 (1965).
x
quantum fluctuation in a limit Λ → ∞ on a single site, [25] H. Katsura et al., Phys.Rev.Lett. 98, 027203 (2007).
sincehpzi vanishesfor any linearcombinations of |± 1i.
2
[21] H.Murakawa, Y.Onose, and Y.Tokura(2010), unpbul-