Influence of the dipolar interaction on phase diagram, magnons, and magnetization in quasi-one-dimensional antiferromagnets on a hexagonal lattice
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PHYSICAL REVIEW B, VOLUME 63, 094425Influence of the dipolar interaction on phase diagram, magnons, and magnetizationin quasi-one-dimensional antiferromagnets on a hexagonal lattice
M. Hummel and F. SchwablPhysik-Department, Technische Universitat Munchen, D-85747 Garching, Germany
C. PichPhysics Department, University of California, Santa Cruz, California 95064
~Received 30 December 1999; published 13 February 2001!
The role of the dipole-dipole interaction in quasi-one-dimensional antiferromagnets is investigated within aHeisenberg model with nearest-neighbor exchange. We deal with systems in which the magnetic ions arelocated on a hexagonal lattice, i.e., frustration is present when three-dimensional order sets in. We perform aground-state calculation for different ratios of dipolar energy to interchain-exchange energy and find severalcommensurate and incommensurate phases. This is a consequence of the competing character of these twointeractions. For the commensurate phases the influence of fluctuations is studied by means of linear spin-wavetheory. It turns out that all commensurate phases are stable against fluctuations. The theoretical spin-wavespectra are compared with experiments on CsMnBr3 and RbMnBr3. The results suggest that the most importantsource of anisotropy in these Mn compounds is the dipole-dipole interaction. Furthermore the magnetic phase-diagram for small dipolar energies is investigated. A spin-wave calculation allows one to calculate the influ-ence of the fluctuations on physical properties like ground-state energy and magnetization. These resultscompare favorably with measurements on CsMnBr3.
DOI: 10.1103/PhysRevB.63.094425 PACS number~s!: 75.10.Jm, 75.30.Ds, 75.30.Gw, 75.50.Eeesaam
The interest in unconventional magnetic systems has brenewed in the last few years.1 Competing interactions awell as geometric frustration due to the underlying lattice clead to unconventional ground states, unique phase diagrand novel critical phenomena.2 The fact that the groundstates in these systems are highly degenerate and becauthe low dimensionality the contributions of fluctuationsphysical observables are enhanced.
In real magnets the dipole-dipole inter-action~DDI! isalways present in addition to the exchange interaction.though the DDI is usually two to three orders of magnitudlower than the exchange interaction, it is nevertheless cruto study its implications in the physics of phase transitioDue to its anisotropic and long-range character the Dchanges the physics in the critical region qualitatively3 andhas also a drastic influence on ground states and excitaspectra4,5 especially in low-dimensional systems.
In this work we particularly study the influence of thDDI on quasi-one-dimensional antiferromagnetic spin stems on a hexagonal lattice. In these compounds thechange interaction along single chains~intrachain-interaction! is several orders of magnitude larger than tinteraction between different chains~interchain-interaction!.The intrachain as well as the interchain interaction are aferromagnetic. The quasi-one-dimensionality appears duthe fact that one has direct superexchange along the chwhereas the interaction between chains is mediated bynonmagnetic ions.
The quasi-one-dimensional systems, which have bstudied most intensively in the context of Haldanes phas6
solitonic excitations7 and the effects of magnetic frustration8
are the ternary compoundsABX3 (A alkaline, B transition0163-1829/2001/63~9!/094425~18!/$15.00 63 0944en
metal, andX halogen!. In these systems the carrier of thmagnetic moment, the B ions, are located on a hexagolattice,8 leading to frustration effects when three-dimensionorder sets in. We are mainly interested in the Mn copounds; due to the fact that their spin valueS55/2, spin-wave theory should be a good approximation. Furthermoas the orbital angular momentumL is zero apart from smalrelativistic corrections and therefore the crystal-field anisropy is only of the order of at most 20% of the dipolenergy,9 the DDI should be the most important sourceanisotropy.
The role of the dipolar interaction in ferromagnetic spchains, which are coupled antiferromagnetically has bstudied quite well.10,4,11For CsNiF3, for example, the dipolarorigin of the unexpected collinear ground state was eludated, the spin-wave dispersion calculated in a dipoHeisenberg model was successfully compared with theperiment and furthermore it was possible to determine aliable value of the strength of the interchain exchaninteraction.12
The influence of the DDI on the properties of quasi-ondimensional antiferromagnetic spin chain systems onother hand, has not yet been studied very thoroughly.stead, the DDI is often replaced by a single-ion anisotrop13
truncated after the first term14 or the coupling between spinchains is neglected entirely.15 Therefore the effect of the DDis studied in a systematic way in this paper.
The outline of the paper is as follows. In Sec. II we intrduce the model, in Sec. III we calculate the ground statesthe whole parameter region of the ratio of dipolar energyinterchain exchange energy without magnetic field. Thground-state investigations are supplemented by a linspin-wave calculation in Sec. IV for the commensuraphases; in this section we also compare our findings w2001 The American Physical Society25-1
M. HUMMEL, F. SCHWABL, AND C. PICH PHYSICAL REVIEW B63 094425neutron scattering measurements of the spin-wave dispein CsMnBr3. In Sec. V we examine the magnetic phase dgram for low dipolar energies for fields both parallel aperpendicular to the chain direction. Based on these grostates we perform a spin-wave calculation and investigateinfluence of fluctuations on the magnetization. We show tquantum fluctuations in our dipolar model are responsiblethe huge anisotropy of the magnetization found in expments. In Appendix A the formulas for the spin-wave specof the dipolar 120 structure with an applied field are givewhereas the dipolar tensor is briefly discussed in AppenB.
The Hamiltonian of the dipolar antiferromagnet withoapplied field reads
H52 (l l 8
~Jll 8dab1All 8
aSl 8b , ~1!
with spins Sl at hexagonal lattice sitesxl . The first termdescribes the exchange interactionJll 8 which includes theintrachain as well as the interchain interaction. In the folloing we consider only nearest-neighbor exchange, i.e.,
Jll 85H 2J l,l 8 along the chains,2J8 l ,l 8 within the basal plane.For a hexagonal lattice, where the sites of the magneticare indexed byxl5a( l 1m/2,A3/2m,c/an) with l ,m,n in-teger, the Fourier transform of the exchange energy is giby
Jq522J cosqz22J8Fcosqx12 cosS qx2 D cosSA32 qyD G .~2!
Here and in the following we measure wave vectors in chdirection in units of 1/c, and wave vectors within the basplanes in units of 1/a.
The second term in Eq.~1! is the classical dipole-dipoleinteraction
52GS dabuxl2xl 8u
3~xl2xl 8!a~xl2xl 8!b
uxl2xl 8u5 D , ~3!
whereG51/2(gmB)2 in cgs units. This term is evaluated b
means of the Ewald summation technique,16,17 which allowsthe consideration of the long-range nature of the thrdimensional DDI in terms of fast convergent sums.
III. GROUND STATES
In this section we calculate the classical ground statethe system as a function of the ratio of the dipolar energythe interchain exchange interaction
whereVz5(A3/2)a2c is the volume of the primitive cell ofthe hexagonal lattice~see Fig. 1!.18 Due to the quasi-onedimensionality of the systems~the ratio J8/J is1022. . .1023) we consider only antiferromagnetic spin cofigurations along the chain axis, i.e., we restrictqz to p. Theground-state energy reads
Eg52 (l l 8
SlMll 8Sl 852(q SqMqS2q ~5!
Mq5S Jq1Aq11 Aq12 Aq13Aq21 Jq1Aq22 Aq23Aq
31 Aq32 Jq1Aq
33D , ~6!
where generallyAqi j 5Aq
j i for i j and Aq135Aq
2350 for qz5p. The ground states are specified by those wave vecwhich belong to the largest of the three eigenvalues ofmatrix Mq given by
The competing quality of the exchange and dipolar intaction becomes clear in Fig. 2. If we change the relatstrength of the two interactions one can see, that the wvector minimizing the ground state energy is situated atferent points of the Brillouin zone.
Particularly note the linear slope of the dipolar tenscomponents at theH point of the Brillouin zone. For decreasing k8 we obtain the following phases, where the actuvalue of J has no influence on the results, as long asJ@J8,(gmB)
2/Vz ~the numerical values are valid for the latice of CsMnBr3,
19 i.e., c53.26 , a57.61 !.
A. Ferromagnetic phase
k8.k185200.50. In the region where the dipolar eneris large compared to the interchain exchange, the minimof the ground-state energy is reached at theA point of the
FIG. 1. Upper half of the crystallographic Brillouin zone of thhexagonal lattice. The wave vectors for the denoted points are gin Appendix C.5-2
INFLUENCE OF THE DIPOLAR INTERACTION ON . . . PHYSICAL REVIEW B63 094425Brillouin zone, i.e., atq15(0,0,p). This means that spinwithin basal planes are ordered ferromagnetically, but santiferromagnetically along the chains.
The ground state energy is given by
Eg52NS2S 2J26J81 12 ~Aq1111Aq122! D . ~8!
Note that the dipolar componentsAq111 andAq1
22 have the same
value. This means that the ground-state energy is the sfor spin configurations where all spins are rotated withinbasal plane and spins in adjacent planes are oriented anromagnetically@see also Goldstone mode in the spin-waspectrum, Eq.~18!#.
B. Incommensurate phase I
k18.k8.k285200.12. The ground state is an incommesurate phase in this parameter region. The wave vemoves continuously fromq1 to the wave vectorq25(0,2p/A3,p) ~see Fig. 3, Fig. 4, and thick solid line iFig. 5! or any other path rotated by 60~dashed lines in Fig.
FIG. 2. Exchange interaction~solid line! and dipolar tensor forsome directions in the Brillouin zone. Long-dashed:Aq
22, and dot-dashed:Aq12. To see the behavior of the dipolar tens
more clearly, we setc/a5A2/3. The numerical values of the dipolar tensor are given in units ofG/Vz and the values for the exchangconstants are set toJ51 andJ850.2. Aq
135Aq2350 for qz5p.
FIG. 3. Wave vector which minimizes the ground state enein the incommensurate phase I as a function ofk8.09442ll
5!. This is due to an increase inJq from A to L whereasAqab
C. Collinear phase
k28.k8.k38517.25. Spins within basal planes are oented ferromagnetically in chains, that are aligned antifermagnetically to one another~see Fig. 6!. Because of the six-fold symmetry, there are six such ground states resulfrom rotation of the ferromagnetic chains. This means tthe minimum of the ground state energy is reached atpoints L, L8 and the ones rotated by 60 in the Brillouzone. Note that the continuous degeneracy is lifted.
The ground state energy for the domain drawn in Figreads
For a domain where the ferromagnetic chains are rotated60 around thez axis the ground state energy is given by
with q55(p,p/A3,p). These two ground state energies athe same, which can be shown by exact relations fordipolar tensor components. The third domain is describedthe wave vectorq65(p,2p/A3,p).
FIG. 4. Dependence of the ordering wave vector in the ICphase on the value ofk8.
FIG. 5. Paths of the wave vector that minimize the ground-senergy in the Brillouin zone of the hexagonal lattice (qz5p).5-3
M. HUMMEL, F. SCHWABL, AND C. PICH PHYSICAL REVIEW B63 094425Note that if we didnt consider the IC I phase, the trantion from the ferromagnetic to the collinear phase wouappear atk85200.31, i.e., exactly betweenk18 andk28 . How-ever, the ground state energy of the incommensurate pbetweenk18 andk28 is lower than both the one for the ferromagnetic as well as the one for the collinear phase.
D. Incommensurate phase II
k38.k8.0. In this incommensurate phase the wave vtor moves fromq2 to q35(2p/3,2p/A3,p). The incommen-surability for low dipolar energy appears because of the finslope of the dipolar tensor atq3 ~Ref. 11! and the parabolicbehavior of the exchange energy.
The expansion of the largest eigenvalue of Eq.~7! around(2p/3,2p/A3,p) leads to the following ground-state energy:
EgS 2p3 2qx , 2pA3 2qy ,p D'2NS2S 2J13J82 34 J8~qx21qy2!
2! D . ~11!One can see that without dipolar energy~i.e., k850) thelowest ground state energy is reached forq5q35(2p/3,2p/A3,p). However, for any infinitesimal dipolaenergy the system is driven to an incommensurate ordewave vector. The 120 phase established for vanishing dlar energy is reached asymptotically~see Fig. 4!.
E. 120 structure
k850. This ground state is characterized by a thrsublattice spin configuration in each basal plane, i.e., bywave vectorq3 (H8) and the correspondent rotated ones.
We state that for all four phases, in which the DDInonzero the coupling of the spin space to the real spaceduced by the DDI forces the spins to align within the lattibasal planes. Thus, the DDI leads to an in-plane anisotro
FIG. 6. Spin configurations within the basal plane for differeratios of dipolar to interchain interactionk8. Only for k850 the120 structure is established. For infinitesimalk8 the phase IC II isfavored. The spin configurations of the incommensurate phasenot sketched.09442-
The phase diagram for the whole parameter region ofk8is shown in Fig. 6. In summary, we find three commensurand two incommensurate phases as function ofk8.
IV. MAGNON SPECTRA
To see whether fluctuations destroy the ground stafound in the previous section we perform a spin-wave callation. To that end we write the Hamiltonian~1! in terms ofcreation and annihilation operators employing the HolstePrimakoff transformation20
Sl15A2S flal ,
Sl25A2Salf l ,
f l5A12 alal2S
and S65Sx6 iSy. The tilde denotes the fact that these oerators have to be utilized in the rotated frame, wherelocal z axis is aligned in the spin direction of the classicground state. The Fermi commutation relations for the opetors ~12! are satisfied if the creation and annihilation opetors a and a obey Bose commutation relations. For lotemperatures an expansion up to bilinear terms can be uleading to linear spin-wave theory. The Hamiltonian thendiagonalized with a Bogoliubov transformation or the meods of Greens functions leading to the spin-wave frequenof the system. The investigation described above is restricto the commensurate phases, because the infinite primcell of the incommensurate phases resists such an analy
A. Ferromagnetic phase
The magnon spectrum of the ferromagnetic phase isculated in the crystallographic Brillouin zone~see Figs. 7and 8! The transformation to the local coordinate systereads
Slx5s l Sl
Sly5s l Sl
y , ~14!
s l5eiq1xl5einp5H 1 l Pplane 1~n50 mod 2!,
21 l Pplane 2~n51 mod 2!~15!
mediates between the two different sublattices. After insing Eqs. ~12!, ~14!, and ~15! into the Hamiltonian~1! weobtain in linear spin-wave theory
INFLUENCE OF THE DIPOLAR INTERACTION ON . . . PHYSICAL REVIEW B63 094425H5Eg1(q
H Aqaqaq1 12 Bq~aqaq1aqa2q !1Cqaqa2q2q11Cq* aq
aq1q11Dq* aq1q1 aqJ ~16!
Dq52 iSAq23. ~17!
This quadratic Hamiltonian is diagonalized by the equatioof motion for proper Greens functions21 leading to the exci-tation spectrum
FIG. 7. Magnons of the ferromagnetic phase in the magnBrillouin zone. Top: in chain direction, bottom: perpendicular to(k51022,k85103).09442s
2 18CqCq1q1 ~19!
2 1Bq1q12 !2
Here we changed to the magnetic Brillouin zone and thefore obtain two magnon branches.
The rotational invariance of the spins around the chaxis leads to a Goldstone mode in the spectrum. The strum gets unstable fork,k18 at infinitesimal values ofqyalong theA-L direction indicating the transition to the ICphase.
B. Collinear phase
The collinear phase of Sec. III is also stable against fltuations. The spectrum can be calculated in analogy tospectrum for the ferromagnetic phase; one just has to repthe ordering wave vectorq1 by q25(0,2p/A3,p). There is
ic FIG. 8. Magnons in the collinear phase fork51022 and k85102. Top: Spectrum in the chain direction. Down: Spectrum in tH direction. Note the energy gap in these spectra.5-5
M. HUMMEL, F. SCHWABL, AND C. PICH PHYSICAL REVIEW B63 094425no longer a Goldstone mode present due to the discretegeneracy of the ground state. This is manifested in theference of the dipolar tensor componentsAq2
11 andAq222 .
C. 120 structure
We argued in Sec. III that the 120 structure is unstafor infinitesimal dipolar energy due to a linear slope of tdipolar tensor at the ordering wave vector. However, to ginsight in the fluctuations we performed a spin-wave callation based on a commensurate 120-structure, wherespins are located within the basal planes of the lattice,22 i.e.,we started with the following transformation:
x sin~q0xl !1Slz cos~q0xl !,
x cos~q0xl !2Slz sin~q0xl !,
y . ~21!
c S 4c/3a01
D aS l 1m/2A3/2mc/an
D 54p3 S l 1 m2 D1np~22!
mediates between the six sublattices. Inserting this transmation in the Hamiltonian~1! and truncating after the harmonic terms, we get a Hamiltonian with 19 different biqudratic terms ~see Appendix A!. This Hamiltonian isdiagonalized by the equation of motion of proper Greefunctions leading to a 12312 matrix; the eigenvalues of thamatrix give the energies of the spin waves. It turns out tthe spin-wave spectrum is stable for weak dipolar energfrom which we conclude that this commensurate groustate is a good approximation.
The spin-wave frequencies resulting for the dipolar 12structure for CsMnBr3 are shown in Fig. 9, where we useJ5215 GHz andJ850.41 GHz respectively.23 This leads tok850.774, i.e., this substance is deep in the IC II regionthe phase diagram in Fig. 6. In the experiment one obsethe 120 structure; this might be caused by crystal distortior small additional anisotropies not considered in this pasuch as the spin-orbit coupling.
Including the DDI, of the three Goldstone modes only osurvives reflecting the remaining rotational symmetry arouthe chain axis. The gaps atq50 are given by
E0,152SA4J1~Aq0112A2q033 !A92 J811
E0,252SA4J1 92 J811
11 !. ~25!09442e-f-
The spin-wave gaps atq50 amount toE0,15198 GHz,E0,25295 GHz, andE0,35410 GHz for CsMnBr3, whichcompares favorably with the experimental values.23,24 Notethat this calculation has no free parameter to fit, sincedipolar energy is determined by the lattice constants.
In Fig. 10 we plotted the spin wave energies in the sadirection as in Fig. 9 but without dipolar energy. Note thwe have three zeros in the excitation energy in this cwhich is at variance with the measurements. Note alsothe dipolar gap@Eq. ~24!# which is not present without dipolar energy is larger than the gap already present withoutpolar energy.
We also calculated the spin-wave gaps for RbMnBr3; ne-glecting crystal distortions we also obtain good agreemwith the experiment.25 Thus, we do not need any single-ioanisotropy to explain these results.
FIG. 9. Excitation energies of the 120 structure perpendicuto the chain direction. The dots are the experimental values tafrom Ref. 24. The dashed and the full lines are the theoreticalsults, where the dashed branches cannot be seen in this scatgeometry. The full lines are the visible branches.
FIG. 10. Excitation energies of the 120 structure without diplar energy. The dashed branches are the ones correspondingdashed ones in Fig. 9.5-6
INFLUENCE OF THE DIPOLAR INTERACTION ON . . . PHYSICAL REVIEW B63 094425V. MAGNETIC PHASE DIAGRAM FOR LOWDIPOLAR ENERGY
In the last section we have shown, that the 120 struchas a stable spectrum for low dipolar energies. In this secwe illuminate the influence of a magnetic field on the groustate, the magnon spectrum and the magnetization inphase, i.e., we investigate the Hamiltonian
b 2hANS0z,y , ~26!
whereh5gmBH0. We distinguish fields along the chain drection~z! and perpendicular to that direction (y), i.e., fieldsapplied within the basal plane. For both cases, we determthe classical ground state by energy minimization and calate the magnon spectrum based on this ground state.
A. Field in chain direction
Numerical minimization of the classical ground-state eergy under consideration of exchange energy, dipolar eneand a magnetic field applied in chain direction shows tspins on basal-plane lattice sites are arranged on threelattices, i.e., neighboring spins within basal planes showphase shift of 120. However, the applied field forcesspins to tilt in field direction by an anglew. The ground-stateenergy reads
Here w is the angle between the basal plane and the spwhich is the same for all three sublattices. This phase iswell-known umbrella phase. Minimization of the grounstate energy in Eq.~27! leads to the relation between thapplied fieldH0 and the tilting anglew
Equation~28! reduces to the 120 phase found in the preous section for vanishing fields. For infinitesimal field tspins partially align in field direction; as already stressspins on different sublattices enclose the same angle withapplied field. The value of the angle increases with increing field. At a critical field
the system undergoes a phase transition to the paramagphase. The phase diagram is schematically shown in Fig
The starting point for the spin-wave calculation are tproper spin variables defined in Eq.~12!, which lead to fluc-tuations. These variables are found by performing a rotaof the spin variables given in Eq.~21! by the anglew inchain direction09442ren
zD 5S cosw 0 2sinw0 1 0
sinw 0 coswD
3S Slz cos~q0xl !1Slx sin~q0xl !Slx cos~q0xl !2Slz sin~q0xl !Sl
yD . ~30!
Note that these spin variables obey Bose commutation rtions and yield to the correct classical ground state forSx5Sy50 and Sz5S, i.e., for the case without fluctuationsInserting these operators and Eq.~12! into Eq.~26! leads to aHamiltonian that has the same structure as the one withapplied field@see Eq.~A1!# and can be diagonalized in thsame way. The spectrum obtained by this calculationshown for a special direction in Fig. 12.
In comparison to the field-free case the field lifts the dgeneracy of all modes. This cannot be seen in the figabove, because the difference of the gaps is too small.gaps for the second and third-lowest modes areEq505197.27 GHz andEq505198.04 GHz respectively, i.e., thmagnetic field does not changeE0 very much@compare re-sults from Eq.~23!#, only at the Brillouin-zone boundary isthe difference of the spin-wave frequencies large. Howevone Goldstone mode is still present since rotational invance around the chain axis is uneffected.
B. Field within the basal plane
1. Ground states
First of all we state, that the direction of the applied fiewithin the basal plane is arbitrary for the structure of t
FIG. 11. Phase diagram of the 120 structure with applied fiin the chain direction.
FIG. 12. Spin-wave frequencies in the chain direction in tumbrella phase for CsMnBr3 for a magnetic field of 6.5 T.5-7
M. HUMMEL, F. SCHWABL, AND C. PICH PHYSICAL REVIEW B63 094425ground state. This is due to the fact, that the 120 grostate is rotationally invariant without applied field. In thfollowing we will apply the field in they direction. To findthe ground state we have to minimize the classical grostate energy. This is done within a general six-sublattice stem, where one has three independent sublattices in eactwo adjacent basal planes~see Fig. 13!.
The ground-state spin configuration of the whole magis built up by repetition of this basis. To describe the spadirection of the spins, we need to know two angles for evspin ~spherical coordinates!, as the spin length is fixed. Wformulate the ground state energy in the magnetic BZobtain the ground state by numerical minimization of tground state energy.
First, we find that the spins staywithin basal planes forrealistic strengths of the dipolar energy for all values ofapplied field. The dipolar energy leading to an easy-plaanisotropy in the field-free case is therefore strong enougkeep the spins within the basal plane even in the presenca competing applied field. With this knowledge the groustate can be parametrized by six different angles, which gthe direction of any spin within the basal plane. Secondthe angles of each two adjacent spins along the chain dtion ~e.g., spins on sublatticesa andd in Fig. 13! add up to180 if one measures the angles with respect to thex direc-tion. This is due to the strong antiferromagnetic exchan
FIG. 13. General six-sublattice structure of the ground state.direction of every spin can be described by two independent anthe spin length is fixed. The different sublattices are denoteda2 fand the field is applied within the basal plane (y direction!.
FIG. 14. Notation of the angles on the three independent subtices in a specific basal plane.09442d
interaction along the chains~for infinitesimal fields the spinsalong the chains are aligned antiferromagnetically!. There-fore, the ground-state energy can be written in terms of thangles only
1$2A0,ae11 @cos~2b!1cos~2g!12 cos~a1b!
12 cos~a1g!#16A0,aa11 %
HereNg5N/6, h5gmBH0 anda,b,g refer to the angles ofthe spins to thex axis ~see Fig. 14!.
To find the minimum of Eq.~31! we differentiateEg withrespect to the three angles. This leads to three equationthe angles in dependence on the field. These equationsnot be solved in closed form; however, we can compthese angles to lowest order inH0 /(8JS), J8/J and(gmB)
with z5h2/hc2 . The critical field to lowest order is
11 2A0,ae11 !
Here, we have introduced the notationd5 12 (b2g) and e5 12 (b1g).
The qualitative behavior is as follows: As we apply ainfinitesimal field, one spin (a) aligns virtually perpendicu-lar to the applied field, whereas the other two encloseangle of 120 and 240 with the first spin, respectively (12structure!. As the field increasesa remains almost un-changed, whereas the other two anglesb andg change untilthey coincide at a critical fieldHc for z51, i.e.,d50. Abovethis field strength
and the reorientation of the spins is like at an ordinary twsublattice spin-flop phase. However, the anglesa andb donot add up to 180 as in a simple spin-flop phase. Tolowest order the angles are given by
INFLUENCE OF THE DIPOLAR INTERACTION ON . . . PHYSICAL REVIEW B63 094425sina5h
8JSS 12 3J8J 1 12J ~A0,ad11 12A0,ab11 ! Dsinb5
8JSS 12 3J82J 1 12J ~A0,ad11 1A0,ae11 2A0,ab11 ! D .~35!
These are the dipolar generalization of the resultsChubukov.26 In Fig. 15 we have plotted the numerical rsults.
At Hpara'154 T the paramagnetic phase is reachedboth a field in chain direction as well as perpendicular.
In Fig. 16 the schematic phase diagram for the 120 strture with a field applied within the basal plane is showNote that the dipolar energy in real materials is stroenough to keep the spins within the basal planes despitefact that the applied field is competing. Also note, thatin-plane dipolar 120 structure is only realized for vanishifields.
2. Magnon spectra
The magnon spectrum for the four phases of the previsection can be calculated within the crystallographic Blouin zone. We will start from a general ground-state that cbe parametrized by three angles, develop a spin-wave thand insert the angles found by minimization of the grounstate energy.
First, we introduce
FIG. 15. The angles minimizing the classical ground stateergy for CsMnBr3. The critical field isHc56.35T.
FIG. 16. Magnetic phase diagram of CsMnBr3 for a field ap-plied within the basal planes. The 120 phase is realized onlyvanishing field. All spins lie in planes perpendicular to the chdirection, i.e., within thex-y planes. AtHc a phase transition occurs from a three-sublattice to a two-sublattice phase.09442f
g5exp~ iq1xl !5H 1 l Pplane 1 mod 2,21 l Pplane 2 mod 2, ~36!and
A3sin~q4xl !5H 0 l Psublattice 1,21 l Psublattice 2,
1 l Psublattice 3,
whereq15(0,0,p) and q45(4p/3,0,0). Usingg and h al-lows us to automatically depict the sublattices which arerametrized bya, b, andg.
The spin variables in Fourier space are given by
z 2c~Sq2q0z 1Sq1q0
1 f ~Sq12q4z 2Sq22q4
z !2e~Sq2q0x 2Sq1q0
z 1Sq12q4z !1aSq2q1
z !1 f ~Sq12q4x 2Sq22q4
are some abbreviations for the angular variables. For vaning field Eqs.~39! reduce to
a5b5d5 f 50, c521
i.e., the spin variables already given in Eq.~21!.Inserting Eq.~38! in the Hamiltonian of Eq.~26!, truncat-
ing after the harmonic terms and diagonalizing the result
M. HUMMEL, F. SCHWABL, AND C. PICH PHYSICAL REVIEW B63 094425Hamiltonian leads to the spin-wave frequencies~see Fig. 17!~in Appendix A 2 we give the expressions for the twsublattice case explicitly!.
The dipolar energy leads to a lifting of the Goldstomode present for the field-free case. This is in contrast tospectrum obtained for a field appliedin the chain direction,where a Goldstone mode still remains. Here the magnfield forces the spins to align partially in field directiowhereas the dipolar energy fixes the spins within the baplanes; a free rotation of the ground-state configurataround the field direction is therefore impossible in contrto a field applied in chain direction. This is the physicexplanation for the gap in the excitation spectrum.
In the rest of the Brillouin zone the spectra for an applfield parallel and perpendicular to the chain directionbasically the same, i.e., the dipolar energy only changesstructure of the excitation spectra near the center of the Blouin zone. However, we will see that these differencaroundq50 are crucial for the dependence of the magnzation on the field direction.
VI. FIELD DEPENDENT MAGNETIZATION ANDCOMPARISON WITH THE EXPERIMENT
We now turn to the field dependence of the magnetizaof a dipolar, hexagonal antiferromagnet taking into accofluctuations. We will again distinguish between fields aplied in the chain direction and fields applied within thbasal plane.
The magnetization can be found from the free energFvia
The free energy is related to the partition functionZ and theHamiltonianH by
F52kT ln Z and Z5Sp exp~2bH !, ~42!
FIG. 17. Spin-wave frequencies in chain direction in the twsublattice phase for CsMnBr3 for a magnetic fieldH056.5 T.Hc .09442e
where b51/kBT. For a field in the chain direction theHamiltonian is given in Eq.~A1! and leads to
12 (q ~Eq( i )2Aq2Aq2q02Aq1q02Aq22q0
( i )!, ~43!
whereq runs over the magnetic Brillouin zone,Eq( i ) are the
six branches of the magnon spectrum for the case of a fiapplied in chain direction, andAq is given in Eq.~A2!. Forthe Hamiltonian of Eq.~A3! ~i.e., a field applied within thebasal planes! one gets
12 (q ~Eq( i )2Aq2Aq2q02Aq1q02Aq22q4
( i )!. ~44!
TheEq( i ) are again the six branches of the magnon spectr
but now the ones gained for an applied field within the baplanes andAq is given in Eq.~A5!.
Generally, the free energy has three contributions: theone is the classical ground-state energy, the second ondue to quantum fluctuations, whereas the third one iscontribution of thermal fluctuations. To calculate the frenergy we replace the sums in Eqs.~43!, ~44! by integralsover the Brillouin zone. Then we have to integrate overfield-dependent spectra for different field strength. Differetiating these free energies with respect to the field leadthe magnetization. For the magnetization we find the sastructure with classical, quantum, and thermal contributio
The integration over the spectrum and the differentiatwith respect to the applied field are done numerically. Fthe anglesa,b, andg, which cannot be given analyticallywe use the numerically calculated values from the eneminimization of Eq.~31!, see Fig. 15.
1. Classical contributions
The classical contributions to the magnetization are cculated via the classical ground-state energies. This yield
M i /NgmB5Ssinw, ~45!
for the field parallel (M i) and perpendicular (M') to thechain direction, respectively.M i can be calculated analytically @sinw is given in Eq. ~28!#, whereasM' has to becalculated numerically as the equations following from E~31! do not have analytic solutions for the tilting anglea,b,g.
The classical results, which are plotted in Fig. 18, shnot only an anisotropy for fieldsH0,Hc , but also for fieldsH0.Hc . The latter cannot be seen very clearly in Fig. 18the difference of the values of the magnetization for fie
INFLUENCE OF THE DIPOLAR INTERACTION ON . . . PHYSICAL REVIEW B63 094425H0.Hc is proportional to the dipolar energy. However, tmagnetization for fields parallel to the chain direction isways larger than for fields perpendicular to the chain dirtion, a fact that can also be seen in experiments.27 This an-isotropy is of dipolar origin since it is energetically mofavorable for two spins to align ferromagnetically along thconnecting line than perpendicular to it, which means incase, that the spins can be tiltedin the chain direction moreeasily. However, there are two discrepancies betweenclassical results and the measurements, namely, the absvalues are different and the experimental anisotropy fofor H0.Hc is much bigger than the one resulting from Eq~45!, ~46!.27 In the following we will show that mainly quantum fluctuations account for this large effect.
2. Quantum contributions
With the term quantum contributions we address the ctributions also present atT50, which are given by the derivative of the second term in Eqs.~43!, ~44!. These contri-butions were evaluated numerically for the two fiedirections. One main result of these calculations is, tquantum fluctuations reduce the absolute value of the clacal magnetization by up to 30%.
Field in chain direction. For a field in chain directionquantum fluctuations reduce the magnetization in the whspin-flop phase. In Fig. 19 we plotted the magnetizationfield strengths which can be reached in the experimentthe parameters of CsMnBr3. The dashed line in Fig. 19 iequivalent to the full line in Fig. 18.
Field perpendicular to the chain direction. For fieldswithin the basal plane we obtain the curves plotted in F20. Our method leads to an unphysical discontinuity ofmagnetization at the transition from the three-sublatticethe two-sublattice phase. Moreover, at the phase transitiothe paramagnetic phase we not only find a discontinuitythe magnetization but we also get absolute values ofmagnetization which are larger than the saturation magnzation. Exactly the same results were theoretically foundRastelli et al.28 for the case of ferromagnetic spin chai
FIG. 18. Classical results for the magnetization for CsMnB3(J5213 GHz,J850.50 GHz!. Full line: field in chain direction,dashed line: field within the basal plane.09442--
which are coupled antiferromagnetically on a hexagonaltice. These authors claim, that the discontinuities show,the system is in some kind of intermediate phase in the trsition regions, which consists of a mixture of the two adcent phases. Therefore they perform a Maxwell construcin the free energy and get a plateau in the magnetization28
However, there is no significance for such a plateauany of the experiments on hexagonal antiferromagnets.27,29
Moreover, the motivation to perform a Maxwell constructioin the free energy in the regions of the phase transitionnot obvious to us. In Sec. VI B we will show that the methof calculating the magnetization as the derivative of the fenergy seems questionable at phase transitions.
3. Thermal fluctuations
Let us now discuss the influence of the third term in E~43!, ~44!. These terms lead to contributions to the free eergy due to thermal fluctuations (T.0). To estimate the ef-fect of thermal fluctuations one would have to comparetemperature at which the measurements were made with
FIG. 19. Magnetization in CsMnBr3. The full line is calculatedunder consideration of quantum fluctuations, the dashed line isclassical result.
FIG. 20. Magnetization in CsMnBr3. The full line is under con-sideration of quantum fluctuations, the dashed line is the classresult.5-11
ther aerro-l in-iblehasted
M. HUMMEL, F. SCHWABL, AND C. PICH PHYSICAL REVIEW B63 094425one corresponding to the anisotropy. However, we pformed a numerical calculation of the integrals, which shothat thermal contributions are not important at temperatuat which the experiments are performed. For CsMnB3,which has a Neel temperature ofTN58.3 K,
23 one gets forT51.8 K, a typical temperature where measurements inordered phase are performed
F th~T51.8 K!/F0&1024.
Thermal fluctuations therefore cannot be responsible forlarge anisotropies in the field-dependent magnetization, cpare Ref. 30.
B. Results for a topological similar dipolar system
For hexagonal antiferromagnets without dipolar energwas shown by Zhitomirsky and Zaliznyak31 that the magne-tization is different depending on whether one calculatesderivative of the free energy or the mean value of the sprojection on the magnetic field. However, these authwere able to show, that the two results are the same up tofirst order in the ratioJ8/J in these systems. Santiniet al.14
also called attention to this fact. However, they analyzedmagnetization by means of a Monte Carlo simulation.
In the following we study the phase diagram and thesulting magnetization for the two-dimensional dipolar anferromagnet on a square lattice with longitudinal field. Treason to investigate this system is that its phase diagramthe same topology as the one of the hexagonal antiferromnet, i.e., it has phases which can, depending on the matude of the field, classically be described by one or twodependent angles with respect to the applied field. The phwhich is parametrized by two angles is called the intermeate phase, whereas the phase where one angle sufficesspin-flop phase.32 At a critical field strength the system undergoes the same kind of phase transition as the hexagantiferromagnet, namely two angles coincide. For high fiewe find a transition to a paramagnetic phase in both syste
However, in the two-dimensional system we are ablecalculate the magnetization not only by using the free enebut also by directly calculating the mean value of the sprojection on the field. In principle, this is also possible fthe dipolar, hexagonal antiferromagnet, but fails due tocomplicated Hamiltonian of Eq.~A3!. Due to the six sublat-tices one would have to calculate six Greens functions outhe twelve equations of motions~i.e., theeigenvectorsof theequations of motion! and integrate them over the first BZwhich is a numerically hopeless task. Note that we only hto calculate theeigenvaluesof the matrix describing theequation of motion to find the magnon energiesEq
( i ) , whichare needed to calculate the free energy. All in all this methat we are numerically able to calculate the free enerwhereas the direct calculation of the magnetization ismerically not feasible in the case of the dipolar, hexagoantiferromagnet.
Let us now return to the magnetization in the twdimensional dipolar antiferromagnet. First of all we note ththe classical magnetization curve was studied by one ofauthors~compare Ref. 32!. In the following we are interested09442r-ss
in the behavior of the magnetizationincluding fluctuations.In particular, we study the magnetization at the phase tration from the intermediate to the spin-flop phase andhigher fields from the spin-flop phase to the paramagnphase.
As already stressed, we will calculate the magnetizatnot only via the free energy but also via the mean valuethe spin on the field direction. This can be done usingGreens function technique.21 The magnetization in the spinflop phase is, for instance, given by
5NgmBSsingS 12 1S (q ^aqaq& D , ~47!whereg is the classical tilt angle of the spins with respectthe magnetic field and thez direction is the field direction.The mean valueaq
aq& is calculated via the spectral theore
2pE dv@G1~v1 id!2G1~v2 id!#n~v!,~48!
where G15^^aq ;aq&& is given as the eigenvector of th
equations of motions forG1 and three other Greens functionand can be calculated straight forward.q05(p/a)(1,1) me-diates between the two sublattices of the system. To getmagnetization one has to evaluate an integration oveGreens function in Eq.~47!. We stress again that in principlthe same proceeding is possible in the hexagonal antifemagnet; however, in those systems the three-dimensionategrals can no longer be evaluated numerically with sensaccuracy due to computational time problems, as onetwelve equations of motions and therefore very complicaintegrals.
The free energy for the two-dimensional antiferromagis given by
4 (cryst BZ ~Aq1Aq1q02Eq(1)2Eq
whereEg is the classical ground state energy,Eq(1) ,Eq
(2) arethe magnon energies of the two branches, andAq is the co-efficient of the diagonal term in the Hamiltonian before tdiagonalization.
Numerical results. In Fig. 21 we plotted numerical resultof the magnetization. The evaluation of the magnetizatusing the derivative of the free energy leads to discontinties at all phase transitions. At the phase transition fromspin-flop to the paramagnetic phase~Fig. 21! the magnetiza-tion for fields lower than the transition field is again largthan the saturation magnetization. These results are quatively the same as the ones found in the previous sectionhexagonal antiferromagnets and the ones found by Raset al.28
However, here we calculated the magnetization alsothe mean value of the spin projection on the field~full line inFig. 21!; in this calculation one does not get a discontinu5-12
INFLUENCE OF THE DIPOLAR INTERACTION ON . . . PHYSICAL REVIEW B63 094425in the magnetization. Moreover, also the absolute valuethe magnetization is always lower than the saturation mnetization. The calculation using the spectral theorem thfore seems to give more reasonable results than the one uthe free energy. However, for decreasing fields in the spflop phase, the two curves in Fig. 21 converge. This qualtive behavior is again gained at all phase transitions.
Our conclusion from all that is that the magnetization cculated using the free energy is not a good approximatiophase transitions but approaches the more reasonable recalculated by means of the spectral theorem for fields thatnot too close to phase transitions. Comparing Eqs.~47! and~49! shows again that the calculation of the magnetizatvia the free energy just requires us to evaluateeigenvaluesofthe equations of motions, whereas calculating the magnetion using Eq.~47! requires us to determine theeigenvectors,namely, the full Greens functions of the system and is thefore much more complicated. However, the latter calculatalso yields meaningful results at phase transitions whethe first one fails in these regions.
C. Comparison with experiment
In Fig. 22 we have plotted the results from Sec. VI B athe experimental data on CsMnBr3.
27 Note that no additionafree parameter enters in the results of Fig. 22 as the dipanisotropy is fixed by the magnetic moment and the latstructure. The intrachain-interactionJ is determined by com-paring dispersion relations with measurements of magnalong the chain direction in the one-dimensional short-raregime.33 The advantage of this method is that for tempetures above the three-dimensional ordering temperaturT.TN the intrachain-interaction is the only relevant interation and can therefore be fixed accurately. The interchainteractionJ8 can be determined by comparing spin wavesthe three-dimensionally ordered phase measured perpenlar to the chain axis with theoretical results. However, ogets different parameters in dependence on the kind of
FIG. 21. Magnetization at the phase transition from the spin flto the paramagnetic phase. Dashed line: results from the freeergy, full line: results from the spectral theorem. In the paramnetic phase one gets identical curves. The magnetic field is giveunits of gmB .09442fg-e-ing--
models one uses. Therefore, we determineJ8 out of the criti-cal field in Eq.~33!. In this case the critical field is to lowesorder independent on the additional kind of anisotropy inHeisenberg model.
In this paper we have made use of the parameter set
J5213 GHz50.88 meV510.21 K, J850.50 GHz~50!
which leads to a critical field ofHc56.4 T.In Fig. 22 one sees that we get good agreement of
theory with the experiment for a field applied in the chadirection. The absolute value of the magnetization is a bitlarge. We will come to that point later on. For a field appliperpendicular to the chain direction we find good agreemfor low fields whereas the influence of fluctuations at tphase transition is underestimated. The curve which isculated under consideration of fluctuations converges toclassical one atH'Hc . In the previous section we havshown that the results for the magnetization calculated usthe free energy are questionable near phase transitions.
However, for fieldsH.Hc we get very good agreemenfor the ratio of the magnetization for a field in chain direction to the magnetization for a field within the basal planAs this ratio is almost independent of the exact parameterone uses for the exchange interaction we conclude thatresult shows the effect of fluctuations in thedipolar Hamil-tonian. The dipolar interaction therefore is not only resposible for the kind of anisotropies in all ground states but agives rise to the anisotropy in the field-dependent magnzation. This reflects the fact that the dispersion relationsqualitatively different for the two field directions if one considers the additional dipolar interaction in the exchanHamiltonian of Eq.~1!. One finds a remaining Goldstonmode for a field in the chain direction whereas no Goldstomode is present for fields applied within the basal plane~seeSecs. V A, and V B!.
FIG. 22. Longitudinal and transverse magnetizationCsMnBr3 compared with the experiment~Ref. 27!. Dashed: classi-cal theory, solid: quantum mechanical theory with fluctuatiopoints: experiment. The upper curves belong to a field in chdirection, the lower ones to a field perpendicular to the chain dirtion.5-13
M. HUMMEL, F. SCHWABL, AND C. PICH PHYSICAL REVIEW B63 094425The occurrence of a large anisotropy in the magnetizafor fields H.Hc can even be explained in a pure ondimensional model,36 but without qualitative agreement. Thabsolute values of the magnetizations are too large compto experiments. This is in part due to a certain experimeuncertainty as the measurements have to be calibrated wmeasuring absolute values. Therefore different experimegroups get slightly different absolute values for the magtizations, but almost identical values for the anisotropy ofmagnetization forH.Hc .
27,34,35On the other hand, one gevirtually the exact agreement of the magnetization forH.Hc if one uses a slightly larger value for the intrachainteraction, namely,J5222 GHz. This is a difference in thintrachain interaction of 4% to the value we used for tcalculations@Eq. ~50!# and which was gained by a pure ondimensional model.
We investigated a dipolar Heisenberg model as a mofor hexagonal antiferromagnets. We found three commenrate and two incommensurate phases for different valuethe ratio of dipolar to interchain interaction in the field frecase. We showed via linear spin-wave theory that all of thcommensurate phases are stable against fluctuations anthe incommensurate phase IC II can be approximated b120 structure for weak dipolar energies.
The spin-wave frequencies of our theoretical model aregood agreement with neutron scattering experimentsCsMnBr3 and RbMnBr3, which shows that the dipolar energy is the most important source of anisotropy in thesecompounds.
The ground states for fields along and perpendicular tochain direction were studied for low dipolar energies. Fofield along the chain axis it turns out that an umbrella phis the ground-state configuration for nonvanishing fields;high fields a paramagnetic phase is established. Applyinfield within the basal plane shows the following. First, tdipolar energy leading to an in-plane anisotropy in the fiefree case is strong enough to keep the spins within the bplane for all field strengths. Secondly, a three-sublatground state configuration is established for low fiestrength. Two out of the three angles coincide as the fireaches a critical fieldHc and the reorientation of the spinappears in the same way as in an ordinary two-sublatspin-flop phase despite the fact that the two remaining senclose slightly different angles with the field direction.09442n
Based on these ground states we performed a spin-wcalculation by means of a Holstein-Primakoff transformtion. We calculated the magnons in all of the above mtioned phases; we found qualitatively different magnon sptra for the same field strength depending on the directionthe applied field. Whereas the excitation spectrum forumbrella phase is gapless, the excitation spectrum forthree- and two-sublattice phase is gaped. This is due todifferent symmetry of the ground states.
The field-dependent magnetization including fluctuatiowas studied in a next step and we compared our findiwith experiments. Calculating the magnetization via the fenergy leads to unphysical discontinuities for fields applwithin the basal plane at the phase transitions from the thsublattice to the two sublattice and from the two sublatticethe paramagnetic phase. Studying a system where onecalculate the magnetization not only via the free energyalso via the mean value of the spin projection shows thatresults obtained by using the free energy are questionabphase transitions, but converge to the results using the mvalue away from phase transitions. We found that theanisotropy in the magnetization for a field parallel and ppendicular to the chain direction present for high fields is dto quantum fluctuations rather than thermal fluctuationsfound excellent agreement between our dipolar theorythe experiment.
This work was supported by the BMBF under ContraNo. 03-SC5-TUM 0 and the DFG under Contract No. Sc348/10-1.
APPENDIX A: SIX-SUBLATTICE CONFIGURATION
In the body of the paper we displayed our results by plting spin-wave spectra for different phases, see Figs. 12,In this appendix we give the longish formulas for the calclation of these spin-waves in the harmonic approximation
1. Field along the chain direction
For a field along the chain direction the ground state isumbrella phase as shown in Sec. V A. The transformationthe spin components is given Eq.~21!. After inserting thistransformation in Eq.~26! and truncating after the harmoniterms, the Hamiltonian readsH5Eg1(q
H Aqaqaq1 12 Bqaqa2q1 12 Bq* aqa2q 1Cqaqa2q12q01Cq* aqa2q12q0 1Dqaqa2q22q01Dq* aqa2q22q01Eqaq
aq12q01Eq* aq12q0 aq1Fqaq
aq22q01Fq* aq22q0 aq1Gqaqa2q1q01Gq* aq
1I qaqaq1q01I q* aq1q0
aqJ . ~A1!The ground-state energy is given in Eq.~27!. The coefficients in the Hamiltonian~A1! are given by5-14
INFLUENCE OF THE DIPOLAR INTERACTION ON . . . PHYSICAL REVIEW B63 094425Aq52S~Jq01Aq011!2Scos2w~Jq1Aq
11 1Aq2q011 1Aq1q0
22 1Aq2q022 !
11 2Aq2q011 1Aq1q0
22 2Aq2q022 !2
11 1Aq1q022 1Aq2q0
11 1Aq2q011 1Aq1q0
22 1Aq2q022 !2
11 1Aq2q011 1Aq1q0
22 1Aq2q022 !
12 2Aq1q012 !1 i
12 2Aq1q012 !,
11 2Aq2q022 !2
11 2Aq2q022 !1
11 2Aq2q022 !1
11 2Aq1q022 !2
11 2Aq1q022 !1
11 2Aq1q022 !
11 2Aq1q022 !1
11 2Aq1q022 !1
11 2Aq2q022 !1
11 2Aq2q022 !2
12 , ~A2!
13 2 iS
23 1 iS
13 1 iS
23 1 iS
13 1 iS
23 1 iS
13 2 iS
23 1 iS
where sinw is given in Eq.~28!. The case of vanishing field is contained in these formulas forw50.
2. Field perpendicular to the chain direction
To calculate the magnon spectrum for a field applied within basal lattice planes, we have to insert Eq.~38! into theHamiltonian of Eq.~26!. In the following we give the expressions for fieldsH.Hc explicitly, where the ground state can bparametrized by two anglesa and b5g. This impliese5 f 50 in Eq. ~39!. For the general case of three sublatticesformulas are more complicated. The quadratic Hamiltonian reads094425-15
M. HUMMEL, F. SCHWABL, AND C. PICH PHYSICAL REVIEW B63 094425H5Eg1(q
aq 1Cqaqa2q12q41Cq* aq
aq22q41Dq* aqaq22q4 1Eqaqa2q22q4
!1I q~aqaq2q01aqa2q1q0!1I q* ~aqaq2q0
The ground state energy in the crystallographic Brillouin zone which is equivalent to formula~31! ~where the energy is givenin the magnetic Brillouin zone! reads
11!12c2~Jq01Aq011!22 f 2~J2q41A2q4
with h5gmBH0. The coefficients in Eq.~A3! are given by
22 1Aq2q022 !#c21@4~J2q41A2q4
22 !2~Jq12q41Jq22q41Aq12q411 1Aq22q4
22 !a21~Jq1q01Jq2q01Aq1q022 1Aq2q0
22 !c21~Jq12q41Jq22q41Aq12q411 1Aq22q4
22 1Jq1q11Aq1q122 !ac2~Jq12q41Aq12q4
11 !d22~Jq2q01Aq2q022 !c2%
1 iAq22q413 d,
22 !#c21~2~J2q41A2q422 !2~Jq12q41Aq12q4
11 !#d2%1 iAq22q413 d2
12 1Aq22q412 !cd1 iAq1q1
12 cd2Aq22q412 ad2 iAq1q0
I q5H2q . ~A5!thelv
hicarHere we used the abbreviations defined in Eqs.~39!. Thediagonalization of the Hamiltonians~A1!, ~A3! can be per-formed in an analogous way: the equation of motion forGreens function ^aq ;aq
&& with the Hamiltonians generateleven other Greens functions leading to a system of tweequations for the twelve Greens functions. The spin-wfrequencies are given by the eigenvalues of the resulting312 matrix.09442e
APPENDIX B: DIPOLAR ENERGY
In the body of the paper we gave formulas which cotained Fourier transforms of the dipolar tensor. We havedistinguish between dipolar tensors in the crystallograpBZ and in the magnetic BZ. The formulas for the dipoltensor in the crystallographic BZ are summed up in.4 Here,we give the formulas for the magnetic BZ.5-16
INFLUENCE OF THE DIPOLAR INTERACTION ON . . . PHYSICAL REVIEW B63 094425When we performed ground-state calculations for a geral six-sublattice model, we had a basis with six spinsreplicated this cell to build up the lattice~see Figs. 13, 23!.To find the classical ground state energy of this structure,parametrize one of the six sublattices with vectorsh and thebasis cell with vectorsra to r f .
Here sublatticeA is parametrized in a coordinate systewhich is rotated by 30 to the crystallographic one
and the vectors describing the basis are given by
2~A3,21,0!, r c5
2 SA3,21,2caD , r f5a2 SA3,1,2caD .~B2!
Here c is the distance of neighboring lattice sites alongchain direction anda the one within basal planes. To includthe full three-dimensional dipolar energy, we need the ctribution of the dipolar energy from the interaction of sublatice a with sublatticesa f , the interaction of sublatticebwith all other sublattices and so on. This means, that we hto calculate Fourier transforms with respect to a linear cobination of two basis vectors. The dipolar tensor splits uptwo contributions on the direct lattice and one for the indirlattice ~Ewald-summation technique16,17! and finally reads
FIG. 23. Parametrization of the lattice.-
22pdab( 8l eiq[xl2(ra2rb)]
14p2a322/3( 8l e
3eiG[xl2(ra2rb)]w0S a32/34p uq1Gu2D ,~B3!
wherea353A3a2c is the volume of the primitive cell of themagnetic sublattice. The components of the dipolar tensothe magnetic BZ are connected to the ones in the crystagraphic BZ. Here we give some of these exact relations:
22 !5A0,ab11 1A0,ae
22 !5A0,ad11 1A0,aa
11 . ~B4!
APPENDIX C: POINTS IN THE BRILLOUIN ZONE
In this appendix we compile the different wave vectoused in the body of the paper. Below each wave vectorpoints in the Brillouin zone are given, compare Fig. 1.
q05S 4p3 ,0,p DG H A
q25S 0,2pA3 ,p D q35S 2p3 , 2pA3 ,p D q45S 4p3 ,0,0DL H8 K
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