Quantum Theory of Multiferroic Helimagnets: Collinear and Helical Phases

Hosho Katsura,1 Shigeki Onoda,2 Jung Hoon Han,3 and Naoto Nagaosa1,4

1Department of Applied Physics, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan2Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan

3BK21 Physics Research Division, Department of Physics, Sungkyunkwan University, Suwon 440-746, Korea4Cross Correlated Materials Research Group, Frontier Research System, Riken,2-1 Hirosawa, Wako, Saitama 351-0198, Japan

(Received 2 April 2008; published 31 October 2008)

We study the quantum fluctuation in the cycloidal helical magnet in terms of the Schwinger boson

approach. In sharp contrast to the classical fluctuation, the quantum fluctuation is collinear in nature which

gives rise to the collinear spin density wave state slightly above the helical cycloidal state as the

temperature is lowered. Physical properties such as the reduced elliptic ratio of the spiral, the neutron

scattering and infrared absorption spectra are discussed from this viewpoint with the possible relevance to

the quasi-one dimensional LiCu2O2 and LiCuVO4.

DOI: 10.1103/PhysRevLett.101.187207 PACS numbers: 75.80.+q, 71.70.Ej, 75.30.Kz, 77.80.�e

Frustration, competition between interactions, in mag-nets has been an intriguing issue in the field of classical/quantum magnetism over several decades. In the usualcase, even with the competing exchange interactionsJij’s, their Fourier transform JðqÞ has the maximum at

some wave vector q ¼ Q, and the classical ground statebecomes the helimagnet [1]. This is because of the con-straint on the length of the classical spin, i.e., jSjj ¼ fixed.

In strongly frustrated quantum magnets, on the other hand,the long-range order is possibly destroyed, and novelground states without magnetic order may be realized.Many possibilities such as chiral spin liquid [2], spin-nematic [3], and spin-Peierls/valence-bond-crystal [4]states are theoretically proposed. Another possibility is amagnetically ordered state realized by the order-by-disorder mechanism [5].

Recently, a renewed interest has been focused on thecycloidal helimagnets from the viewpoint of multiferroics,which exhibit both magnetic and ferroelectric properties[6,7]. These materials shed some new light on the frus-trated magnets since the electric polarization is closely re-lated to the vector spin chirality Si � Sj [8–12]. Namely, it

was found that the electric polarization (P) produced by theneighboring spins (Si and Sj) can be written as

P ¼ aeij � ðSi � SjÞ; (1)

where eij denotes the unit vector connecting the sites i and

j. This relation has a physical interpretation in terms ofspin current induced between noncollinear spins due tofrustration [8].

Magnetic materials with the finite vector spin chiralityinclude a wide range of systems such as three dimensional(3D) magnets RMnO3 (R ¼ Gd, Tb, Dy) with spin S ¼ 2[13–16], the kagome staircase compound Ni3V2O8 withS ¼ 1 [17], S ¼ 1=2 quantum spin chains LiCu2O2

[18,19], LiCuVO4 [20], and the quasi-one-dimensional(1D) molecular helimagnet with S ¼ 7=2 [21].Depending on the temperature, dimensionality, and mag-

nitude of the spin S, the role of the classical/quantum spinfluctuations differs, and the theoretical studies on thesefluctuations are needed for the consistent interpretation ofthe phase diagram and the physical properties of thesesystems. Especially, the low dimensionality enhancesboth thermal and quantum fluctuations leading to thebreakdown of the conventional semiclassical picture forhelimagnets. The possible chiral spin states without mag-netic long-range order have been proposed theoretically forclassical [22–24] and quantum [25–27] spin systems.However, the systematic study of the quantum fluctuationin the helimagnets including the finite temperature effect israre, which is addressed in this Letter and will be comple-mentary to the works mentioned above.In this Letter, we study the quantum/thermal fluctuation

in the helimagnet in terms of the Schwinger boson (SB)approach [28]. The advantage of the SB method is that itcan describe the length of the ordered moment as a softvariable. Namely, in the constraint on the Schwinger boson

number at each site,P

�byj�bj� ¼ 2S, it can be decom-

posed into the condensed (classical) part and the fluctuat-ing part. In the SB language, the paramagnetic-to-collineartransition is described by the density wave instability ofbosons, while the collinear to helical one corresponds tothe Bose-Einstein condensation (BEC) of SB.Effective model.—We study quasi-1D and two-

dimensional (2D) Heisenberg models with the exchangeinteractions shown in Fig. 1, where J1 is ferromagnetic,while J2 is antiferromagnetic, leading to the frustration.The interchain/interplane interaction J? is assumed to besufficiently weak. The spin-S operators are represented by

SB as S� ¼ by�ð����0=2Þb�0 , where�� (� ¼ x, y, z) are the

Pauli matrices and the repeated indices are summed over.First, we assume that the resonating-valence-bond (RVB)correlation is dominant and neglect the other mean-fielddecoupling. This assumption is valid for the low-dimensional multiferroics such as LiCuVO4 [20], LiCu2O2

[18], andNi3V2O8 [17]. The mean-field Hamiltonian of the

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quasi-1D model is given by

HMFq1D ¼ X

k�

ð�� 2�1 coskxÞbyk�bk�

þXk

2½�2 sinð2kxÞ þ �?ðsinky þ sinkzÞ�byk"by�k#

þ h:c:þ 2N ð�21=J1 þ �2

2=J2 þ 2�2?=J? � S�Þ;

(2)

whereN is the total number of sites and bk� is the Fourier

transform defined by bj� ¼ Pke

�ik�Rjbk�=ffiffiffiffiffiffiffiN

p. In HMF

q1D,

� denotes the chemical potential for the bosons and the

order parameters �1, �2, and �? are J1hbyi�biþx;�i=2,J2hbi����biþ2x;�i=ð2iÞ, and J?hbi����biþe;�i=ð2iÞ (e ¼ y,z), respectively, with �#" ¼ ��"# ¼ 1. RVB order parame-

ters are assumed to be real and spatially uniform. In aparallel way, we can derive the quasi-2D mean-fieldHamiltonian HMF

q2D. The Hamiltonian HMFq1D can be diago-

nalized by the Bogoliubov transformation as HMFq1D ¼P

k�!ðkÞð�yk��k� þ 1=2Þ þ const, with the dispersion re-

lation !ðkÞ2 ¼ ð�� 2�1 coskxÞ2 � ½2�2 sinð2kxÞ þ2�?ðsinky þ sinkzÞ�2. The chemical potential � is deter-

mined by the condition on the boson number. �’s areobtained by minimizing the mean-field free energy.Figure 2 shows the numerically obtained �1, �2 and thegap �ðTÞ ¼ !ðQ=2Þ of the 1D spin-1=2 model as a func-tion of J1=J2. In the calculation, J? is assumed to be small,and �? is set to zero [29]. We have also numericallystudied the 2D model at finite temperature and obtainedsimilar results. From �’s, we can estimate the minima ofthe dispersion !ðkÞ as �Q=2 ¼ �ðQ=2; �=2; �=2Þ. Q isdetermined to satisfy ½�� 2�1 cosðQ=2Þ��1 sinðQ=2Þ ¼4ð�2 sinQþ 2�?Þ�2 cosQ.

To describe the low-energy physics of the model, it isuseful to construct an effective continuum model. First, wesuppose that �’s are nonzero. Next, we expand the disper-sion around the minima up to quadratic order in k�Q=2.The effective dispersion relations of �-particles are thoseof massive relativistic bosons and explicitly given by

�ðkÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ðTÞ2 þ c2kjkkj2 þ c2?jk?j2

q, where kk is the

vector along (within) the chain (plane) while k? is thatperpendicular to the chain (plane). The spin wave veloci-ties ck and c? can be written in terms of �’s, in principle.

Now the effective Hamiltonian of our system is

Heff ¼ Xk�

X�¼�

�ðkÞð�yk���k�� þ 1=2Þ; (3)

where � ¼ þ (�) indicates that the momentum is aroundþQ=2 (�Q=2). When the gap �ðTÞ ¼ !ðQ=2Þ vanishes,�’s are the linear dispersions of the Goldstone modes.Collinear phase.—To study the instability toward mag-

netic ordering, we consider the mean-field decoupling ofthe interchain/interplane interaction corresponding to thedensity wave formation of the SB [30,31]. The totalHamiltonian is given by Hq1D=q2D ¼ HMF

q1D=q2D þHint with

Hint ¼ zJ?fN jaþ ibj2 �ffiffiffiffiffiffiffiN

p½ða� ibÞ � SQ þ h:c:�g;

where z is the coordination number along interchain/interplane direction and a and b are mean fields de-

fined through hSQi ¼ffiffiffiffiffiffiffiN

p ðaþ ibÞ with SQ ¼Pkb

ykþQ;�ð���0=2Þbk�0=

ffiffiffiffiffiffiffiN

p. Small �? or c? does not

change the following results, and hence, we put �? ¼ 0 inEq. (4). The collinear and helical orders are expressed asa� b ¼ 0 and a� b � 0, respectively [24]. The interac-tion between �-bosons is enhanced near the bottom of thedispersion inversely proportional to the gap �ðTÞ in Fig. 3(a). It inevitably leads to the density wave instability beforethe occurrence of BEC. From the rotational symmetry inspin space, we can set az ¼ bz ¼ 0without loss of general-ity. By introducing x ¼ zJ?jðax � iayÞ þ iðbx � ibyÞj andy ¼ zJ?jðax � iayÞ � iðbx � ibyÞj, the free-energy den-sity corresponding to the Hamiltonian Hq1D=q2D can be

written in a decoupled form: fðx2Þ þ fðy2Þ [32]. Fromthe fact that a� b / x2 � y2, the collinear phase appearsif fðx2Þ has a global minimum at x2 � 0. In the quasi-1Dcase, fðx2Þ � fð0Þ can be expanded as Ax2 þ Bx4 with

A ¼ 1

�ðTÞ��ðTÞ2zJ?

� Sþ 1=2

8ðTÞ�;

B ¼ 1

�ðTÞ3�Sþ 1

2

�ðTÞ3128

�9½1� 2ðTÞ2=3�2

1� ðTÞ2 � 5

�;

(4)

FIG. 2 (color online). RVB order parameters �1 and �2, andthe gap � of the S ¼ 1=2 1D model with varying J1=J2 at zerotemperature. Inset shows the temperature dependence of �1 and�2 at J1=J2 ¼ 0:5. We use the unit J2 ¼ kB ¼ 1. The transitiontemperature is given by TRVB ¼ J2ðSþ 1=2Þ= lnð1þ 1=SÞ.

)(bx)(cy

)(az

2J

1J

1J2J

(b)(a)

FIG. 1 (color online). Schematic lattice structure and exchangeinteractions of the effective spin model. J1 is ferromagnetic,while J2 is antiferromagnetic. xyz-coordinates and abc-axes arealso shown.

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where ðTÞ ¼ �ðTÞ=~�ðTÞ (~�ðTÞ ¼ �� 2�1 cosðQ=2Þ) isthe normalized gap. Here, we have assumed T � �ðTÞ.Since B is positive for 0< ðTÞ< 1, a sufficient conditionfor the collinear phase is A < 0, and a second order phasetransition to the collinear state occurs at A ¼ 0. AboveTRVB, ðTÞ � 1 and hence the inequality A < 0 is notsatisfied for small zJ?. This means TN < TRVB, where TN

is the antiferromagnetic transition temperature. Furtherlowering the temperature with increasing x, the gap col-lapses to result in BEC of SB. Therefore, we concludeTBEC < TN < TRVB. In this way, the instability towards thecollinear order is a robust feature of the strongly fluctuatingquantum helimagnets, and is essentially different from thatof classical system with an easy axis anisotropy. Now, wedescribe the collinear state a ¼ b ¼ ð0; ay; 0Þ where the4-fold degeneracy for the energy of �k�� is split into upperand lower branch bands [see Fig. 3(b)]. The lower-branchband consisting of �k and k which are linear combina-tions of �k�� is 2-fold degenerate [32]. Below, we willfocus on the low energy dynamics, and neglect theupper-branch bosons. This leads to the relation between

the original bosons bk�: b�Q=2þk" � e�i�=4bQ=2þk# and

by�Q=2þk# � �ei�=4byQ=2þk".Helical phase.—Next, we consider the BEC of the low-

est modes �0 and 0 caused by the finite �?. This corre-sponds to the nonzero expectation values of b�Q=2;�. We

obtain the cycloidal helical spin structure as

Sbi �� sinðQ �Ri þ �=4ÞðjhbQ=2"ij2 � jhbQ=2#ij2Þ=N ;

ScI � S cosðQ �RI þ �=4Þ;SaI � sinðQ � RI þ �=4ÞðhbQ=2"ihbQ=2#i þ c:c:Þ=N : (5)

Here, we have used the relaxed constraintP

i�byi�bi� ¼

2SN . Now, we clarify the relation between the ellipticratio and the Bose condensate fraction. If we assume thathbQ=2#i ¼ 0 while hbQ=2"i � 0, Sai becomes zero and the

elliptic ratio is given bymb=mc � jhbQ=2"ij2=ðN SÞ. In thiscase, the spins are rotating counterclockwise within thebc-plane. The clockwise helicity is realized whenhbQ=2"i ¼ 0 while hbQ=2#i � 0. Note that the elliptic ratio

can be much smaller than unity even at zero temperature

due to the strong quantum fluctuation in sharp contrast tothe classical case where higher harmonics are needed.Neutron scattering.—We shall focus on the quasi-1D

case with the possible relevance to the recent experimentin the helical phase of LiCu2O2 [19]. Henceforth, we sethbQ=2"i � 0 and hbQ=2#i ¼ 0 to take one of the helicities.

The spin correlations I��ðqÞ ¼ hS�QþqS��Q�qi and

I��ðqÞ ¼ hS�QþqS��Q�qi (� ¼ x, y) were observed with

Snka-axis and Snkc-axis configurations, respectively,where Sn is the neutron spin [19]. Let us examine themin terms of SB theory. We first consider the Bragg compo-nent, i.e., q ¼ 0 and ! ¼ 0 component. Using the Fouriertransforms of Eq. (5), we obtain I��ð0Þ ’ N S2ð1�rÞ2=4 while Ixxð0Þ � Iyyð0Þ ’ �N S2ð1� r2Þ=4, wherer ¼ jhbQ=2"ij2=ðN SÞ. Next, we consider the non-Bragg

part, i.e., nonzero q component contributing to the scatter-ing intensity due to the finite energy window. In the energyregime where the continuum model is valid, using �k andk, one can show that Iþ�ðqÞ ’ F1ðqÞ þ F2ðqÞ andI�þðqÞ ’ F2ðqÞ with

F1ðqÞ ¼ rS

�Sþ 1

2

�2ck�ðTÞ

c2kq2 þ c2?jq?j2

;

F2ðqÞ ��Sþ 1

2

�2 �ðTÞ216c2?

1

q;

where q is the x component of q. The F1 term is directlyrelated to the condensate fraction while the common F2

term exists even when r ¼ 0. On the other hand, one canshow that IxxðqÞ ¼ IyyðqÞ ’ F1ðqÞ=2þ F2ðqÞ for q � 0.The interchain interaction J? can be estimated from thetransition temperature TN as TN / J? [31,33]. ForLiCu2O2, TN ’ 24:5 K and assuming z ¼ 4, we obtainJ? � 8 K. The exchange interactions along the chain areestimated as jJ1j ¼ 11 meV and J2 ¼ 7 meV [18], and theratio of the velocities c?=ck � J?=Jk � 10�1. The mea-

surement was performed at 7 K [19] which is of the sameorder as J?, and hence T3=ðJkJ2?Þ � T=Jk. Therefore, thethermally excited fluctuations are quasi-one dimensional atT ¼ 7 K, and the collinear magnetic order parameter isstill small. This further enhances the importance of thequasielastic contribution to the ‘‘Bragg intensity’’ and mayresolve the puzzle of Ixx � Iyy and jIþ� � I�þj Iþ� þ I�þ [19,27]. In order to extract the informationon the Bragg component exclusively, one needs to cooldown the temperature to T J?.Dielectric response.—Finally, we examine the dynami-

cal dielectric response both in the paramagnetic and heli-cal phases of the quasi-1D model. We assume that thefluctuating electric polarization is given by Eq. (1). We

take the mean-field decoupling Si � Sj ¼ ½hbyi�bj�i�ðbyj����bi�Þ � h:c:�=ð4iÞ to the ferromagnetic bonds and

Si � Sj ¼ ½hbyi����byj�iðbj��

�����bi�Þ � h:c:�=ð4iÞ to

the antiferromagnetic bonds. We henceforth focus on thecontribution from the antiferromagnetic (J2) bonds since

a)

2// Q2Q

Q

b)

k

Goldstone2-fold

2-fold

2-fold

kk

(T)

c)

),( 2-fold

Amplitude

2-fold

Q

FIG. 3 (color online). (a) Schematic energy dispersion of�-particles. Minima are at k ¼ �Q=2. (b) The reorganizationof the SB due to the collinear magnetic order. The origin of themomentum k is shifted by �Q=2. (c) Goldstone and amplitudemodes associated with the BEC in the helical phase.

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its fluctuation is stronger than that of the ferromagneticone. From the geometry of the system (see Fig. 1), thepolarization along the b-axis Pb is always zero. In theparamagnetic phase, Im"aað!Þ ¼ Im"ccð!Þ due to the ro-tational symmetry in spin space. The expression for thepolarization along the a-axis Pa is given by

Pa / ð�2=J2ÞXk

cosð2kxÞðibk�b�k� þ h:c:Þ:

The purely 1D result is shown in Fig. 4. Near the threshold

of the absorption, Im"aað!Þ / ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!� 2�ðTÞp

before theonset of �1. On the other hand, a drastic change of theabsorption spectra occurs in the helical phase since thelow-lying branch bosons become gapless [see Fig. 3(c)]. Inthis phase, the energy dispersions of the upper and lower

branches are given by �þðkxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2kk

2x þ 2�ð0Þ2

qand

��ðkxÞ ¼ ckkx, respectively. We assume the BEC of SB

by the weak interchain interaction. The behavior at zerotemperature is shown in Fig. 4. There are three contribu-tions corresponding to the processes of two bosons (i) inthe upper branch, (ii) in the gapless lower branch, and(iii) in both the upper and lower branches, respectively.Finally, it should be noted that there is also the one-magnoncontribution discussed in [34] which is, however, muchsmaller in the limit of weak interchain interaction.

To conclude, we have studied the phase diagram and thedynamical properties of quasi-low dimensional helimag-nets. The collinear phase at slightly higher temperaturethan the helical phase and the reduced elliptic ratio of thespiral are realized by the quantum fluctuation and the lowdimensionality. It is also shown that the polarized neutronscattering and infrared absorption in quasi-1D helimagnetsreflect the quantum fluctuation.

The authors are grateful to S. Seki, Y. Yamasaki,N. Kida, S. Todo, and Y. Tokura for fruitful discussions.

This work was supported in part by Grant-in-Aids (GrantNos. 15104006, 16076205, 17105002, 19048008,19048010, and 20046016) and NAREGI NanoscienceProject from the MEXT. H.K. was supported by theGrant-in-Aid for JSPS (No. 1811271).

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FIG. 4 (color online). Im �aað!Þ in the paramagnetic phasewith T ¼ 0:8 (red/right), T ¼ 0:35 (green/middle), and the hel-ical phase with T ¼ 0 (blue/left, ~� ¼ ffiffiffi

2p

�ð0Þ). We take J1 ¼0:25 and J2 ¼ 1. The inset shows the enlargement of the low-frequency region. Singularities are smeared out by the interchaininteraction as shown by the black lines. Behaviors near thresh-olds are also indicated. Closed-form expressions for Im"aað!Þare provided in [32].

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