Landau Analysis of the Symmetry of the Magnetic Structure and Magnetoelectric Interaction in Multiferroics

Transcription

1 University of Pennsylvni SholrlyCommons Deprtment of Physis Ppers Deprtment of Physis Lndu Anlysis of the Symmetry of the Mgneti Struture nd Mgnetoeletri Intertion in Multiferrois A. Brooks Hrris University of Pennsylvni, Follow this nd dditionl works t: Prt of the Physis Commons Reommended Cittion Hrris, A. B. (007). Lndu Anlysis of the Symmetry of the Mgneti Struture nd Mgnetoeletri Intertion iultiferrois. Retrieved from Suggested Cittion: A.B. Hrris. (007) Lndu nlysis of the symmetry of the mgneti struture nd mgnetoeletri intertion in multiferrois. Physil Review B 76, The Amerin Physil Soiety This pper is posted t SholrlyCommons. For more informtion, plese ontt repository@poox.upenn.edu.

2 Lndu Anlysis of the Symmetry of the Mgneti Struture nd Mgnetoeletri Intertion iultiferrois Astrt This pper presents detiled instrution mnul for onstruting the Lndu expnsion for mgnetoeletri oupling in inommensurte ferroeletri mgnets, inluding Ni 3 V O 8, TMnO 3, MnWO 4, TMn O 5, YMn O 5, CuFeO, nd RFe(MO 4 ). The first step is to desrie the mgneti ordering in terms of symmetry dpted oordintes whih serve s omplex-vlued mgneti order prmeters whose trnsformtion properties re displyed. Io doing, we use the previously proposed tehnique to exploit inversioymmetry, sine this symmetry hs seemingly een universlly overlooked. Inversioymmetry severely redues the numer of fitting prmeters needed to desrie the spitruture, usully y fixing the reltive phses of the omplex fitting prmeters. By introduing order prmeters of knowymmetry to desrie the mgneti ordering, we re le to onstrut the triliner mgnetoeletri intertion whih ouples inommensurte mgneti order to the uniform polriztion, nd therey we tret mny of the multiferroi systems so fr investigted. In most ses, the symmetry of the mgnetoeletri intertion determines the diretion of the mgnetilly indued spontneous polriztion. We use the Lndu desription of the mgnetoeletri phse trnsition to disuss the qulittive ehvior of vrious suseptiilities ner the phse trnsition. The onsequenes of symmetry for optil properties suh s polriztion indued mixing of Rmn nd infrred phonons nd eletromgnons re nlyzed. The implition of this theory for mirosopi models is disussed. Disiplines Physil Sienes nd Mthemtis Physis Comments Suggested Cittion: A.B. Hrris. (007) Lndu nlysis of the symmetry of the mgneti struture nd mgnetoeletri intertion in multiferrois. Physil Review B 76, The Amerin Physil Soiety This journl rtile is ville t SholrlyCommons:

3 PHYSICAL REVIEW B 76, Lndu nlysis of the symmetry of the mgneti struture nd mgnetoeletri intertion in multiferrois A. B. Hrris Deprtment of Physis nd Astronomy, University of Pennsylvni, Phildelphi, Pennsylvni 19104, USA Reeived 17 Ferury 007; revised mnusript reeived 16 My 007; pulished 8 August 007 This pper presents detiled instrution mnul for onstruting the Lndu expnsion for mgnetoeletri oupling in inommensurte ferroeletri mgnets, inluding Ni 3 V O 8, TMnO 3, MnWO 4, TMn O 5, YMn O 5, CuFeO, nd RFeMO 4. The first step is to desrie the mgneti ordering in terms of symmetry dpted oordintes whih serve s omplex-vlued mgneti order prmeters whose trnsformtion properties re displyed. Io doing, we use the previously proposed tehnique to exploit inversioymmetry, sine this symmetry hs seemingly een universlly overlooked. Inversioymmetry severely redues the numer of fitting prmeters needed to desrie the spitruture, usully y fixing the reltive phses of the omplex fitting prmeters. By introduing order prmeters of knowymmetry to desrie the mgneti ordering, we re le to onstrut the triliner mgnetoeletri intertion whih ouples inommensurte mgneti order to the uniform polriztion, nd therey we tret mny of the multiferroi systems so fr investigted. In most ses, the symmetry of the mgnetoeletri intertion determines the diretion of the mgnetilly indued spontneous polriztion. We use the Lndu desription of the mgnetoeletri phse trnsition to disuss the qulittive ehvior of vrious suseptiilities ner the phse trnsition. The onsequenes of symmetry for optil properties suh s polriztion indued mixing of Rmn nd infrred phonons nd eletromgnons re nlyzed. The implition of this theory for mirosopi models is disussed. DOI: /PhysRevB PACS numers: 75.5.z, Jm, G I. INTRODUCTION Reently, there hs een inresing interest iystems multiferrois whih exhiit n oservle intertion etween mgneti nd eletri degrees of freedom. 1 Muh interest hs entered on fmily of multiferrois whih disply phse trnsition in whih uniform ferroeletri order ppers simultneously with inommensurte mgneti ordering. Erly exmples of suh system whose ferroeletri ehvior nd mgneti struture hve een thoroughly studied re terium mngnte, TMnO 3 TMO,,3 nd nikel vndte, Ni 3 V O 8 NVO. 4 7 A similr omprehensive nlysis hs reently een given for the tringulr lttie ompound RFeMoO 4 RFMO. 8 A numer of other systems hve eehown to hve omined mgneti nd ferroeletri trnsitions, 9 14 ut the investigtion of their mgneti struture hs een less systemti. Initilly, this omined trnsition ws somewht mysterious, ut soon Lndu expnsion ws developed 4 to provide phenomenologil explntion of this phenomenon. An lterntive piture, similr to n erlier result 15 sed on the onept of spin urrent, nd whih we refer to s the spirl formultion, 16 hs gined populrity due to its simpliity, ut s we will disuss, the Lndu theory is more universlly pplile nd hs numer of dvntges. The purpose of the present pper is to desrie the Lndu formultion in the simplest possile terms nd to pply it to lrge numer of urrently studied multiferrois. In this wy, we hope to demystify this formultion. It should e noted tht this phenomenon whih we ll mgnetilly indued ferroeletriity is losely relted to the similr ehvior of so-lled improper ferroeletris, whih re ommonly understood to e the nlogous systems in whih uniform mgneti order ferromgnetism or ntiferromgnetism drives ferroeletriity. 17 Severl dedes go, suh systems were studied 18 nd reviewed 17,19 nd present mny prllels with the reent developments. One of the prolems one enounters t the outset is how to properly desrie the mgneti struture of systems with omplited unit ells. This, of ourse, is very old sujet, 0 ut surprisingly, s will e doumented elow, the full rmifitions of symmetry re not widely known. Aordingly, we feel it neessry to repet the desription of the symmetry nlysis of mgneti strutures. While the first prt of this symmetry nlysis is well known to experts, we review it here, espeilly euse our pproh is often fr simpler nd less tehnil thn the stndrd one. However, either pproh lys the groundwork for inorporting the effets of inversioymmetry, whih, in the reent literture, hve often een overlooked until our nlysis of NVO 4 7 nd TMO. 3 Inversioymmetry ws lso ddressed y Shweizer with susequent orretion. 3 Very reently, more forml pproh to this prolem hs een given y Rdelli nd Chpon 4 nd y Shweizer et l. 5 However, t lest in the simplest ses, the pproh initilly proposed y us nd used here seems esiest. We here pply this formlism to numer of urrently studied multiferrois, suh s DyMnO 3 DMO, 9 MnWO 4 MWO, 13,14 TMn O 5 TMO5, 11,1 YMn O 5 YMO5, 1 CuFeO CFO, 10 nd RFMO. 8 As ws the se for NVO 4 7 nd TMO, 3 one one hs in hnd the symmetry properties of the mgneti order prmeters, one is then le to onstrut the triliner mgnetoeletri oupling term in the free energy whih provides phenomenologil explntion of the omined mgneti nd ferroeletri phse trnsition. This pper is orgnized in onformity with the ove pln. In Se. II, we review simplified version of the symmetry nlysis know representtion theory. Here, we /007/765/ The Amerin Physil Soiety

4 A. B. HARRIS PHYSICAL REVIEW B 76, lso review the reently proposed 3 7 tehnique to inorporte the onsequenes of inversioymmetry. In Se. III, we pply this formlism to develop mgneti order prmeters for numer of multiferroi systems, nd in Eq. 16 we give simple exmple to show how inversioymmetry influenes the symmetry of the llowed spin distriution. Then, in Se. IV, we use the symmetry of the order prmeters to onstrut mgnetoeletri oupling free energy, whose symmetry properties re mnifested. We give n nlysis of the Lndu desription of the mgnetoeletri phse trnsition. In prtiulr, we disuss the ehvior of vrious suseptiilities ner the phse trnsition. In Se. V, we disuss how the mgnetoeletri intertion leds to mixing of infrred tive nd Rmn tive phonon modes nd to the mixing of mgnons with phonons. Finlly, in Se. VI, we summrize the results of these lultions nd disuss their onsequenes. II. REVIEW OF REPRESENTATION THEORY As we shll see, to understnd the phenomenology of the mgnetoeletri oupling whih gives rise to the omined mgneti nd ferroeletri phse trnsition, it is essentil to hrterize nd properly understnd the symmetry of the mgneti ordering. In ddition, s we shll see, to fully inlude symmetry restritions on possile mgneti strutures tht n e essed vi ontinuous phse trnsition is n extremely powerful id in the mgneti struture nlysis, Aordingly, in this setion we review how symmetry onsidertions restrit the possile mgneti strutures whih n pper t n ordering trnsition. The full symmetry nlysis hs previously een presented elsewhere, 3 7 ut it is useful to repet it here oth to fix the nottion nd to give the reder onvenient ess to this nlysis whih is so essentil to the present disussion. To void the omplexities of the most generl form of this nlysis lled representtion theory, 3 5 we will limit disussion to systems hving some ruil simplifying fetures. First, we limit onsidertion to systems in whih the mgneti ordering either is inommensurte or equivlent thereto. In the exmples we hoose, k will usully lie long symmetry diretion of the rystl. Seond, we only onsider systems whih hve enter of inversioymmetry, euse it is only suh systems tht hve shrp phse trnsition t whih long-rnge ferroeletri order ppers. Thirdly, we restrit ttention to rystls hving reltively simple symmetry. Wht this mens is tht exept for our disussion of TMn O 5, we will onsider systems where we do not need the full pprtus of group theory, ut n get wy with simply leling the spin funtions whih desrie mgneti order y their eigenvlue under vrious symmetry opertions. By voiding the omplexities of the most generl situtions, it is hoped tht this pper will e essile to more reders. Finlly, s we will see, it is ruil tht the phse trnsitions we nlyze re either ontinuous or very nerly so. In mny of the exmples we disuss, our simple pproh 6 is vstly simpler thn tht of stndrd representtion theory 6 8 ugmented y speilized tehniques to expliitly exploit inversioymmetry. A. Symmetry nlysis of the mgneti free energy In this setion, we give review of the formlism used previously 3,4 nd presented in detil in Refs Sine we re minly interested iymmetry properties, we will desrie the mgneti ordering y version of men-field theory in whih one writes the mgneti free energy F M s F M = 1 1 r,rs rs r + OS 4, 1 r,;r where S r is the thermlly verged omponent of the spin t position r. In moment, we will give n expliit pproximtion for the inverse suseptiility 1. We now introdue Fourier trnsforms in either of two equivlent formultions. In the first formultion whih we refer to s tul position, one writes the Fourier trnsform s S q, = N 1 S R + e iq R+ R wheres in the seond whih we refer to s unit ell, one writes S q, = N 1 S R + e iq R, 3 R where N is the numer of unit ells in the system, is the lotion of the th site within the unit ell, nd R is lttie vetor. Note tht in Eq. the phse ftor in the Fourier trnsform is defined in terms of the tul position of the spin rther thn in terms of the origin of the unit ell, s is done in Eq. 3. Iome ses viz., NVO, the results re simpler in the tul position formultion, wheres for others viz., TMO, the unit ell formultion is simpler. We will use whihever formultion is simpler. In either se, the ft tht S hs to e rel indites tht We thus hve F M = 1 q;,,, S q, = S q, *. 4 1 q;,s q, * S q, + OS 4, 5 where, for the tul position formultion, 1 q;, = 1,R + e iq R+, R nd for the unit ell formultion, 1 q;, = 1,R + e iq R. R To mke our disussion more onrete, we ite the simplest pproximtion for system of spins on n orthorhomi Brvis lttie with generl nisotropi exhnge oupling so tht the Hmiltonin is H = J r,rs rs r + K s r, 8,;r,r r where s r is the omponent of the spin opertor t r nd we hve inluded single ion nisotropy energy ssuming

5 LANDAU ANALYSIS OF THE SYMMETRY OF THE PHYSICAL REVIEW B 76, three inequivlent xes so tht the K re ll different. One hs tht 1 r,r = J r,r + K + kt, r,r, 9 where, is unity if = nd is zero otherwise nd is spin-dependent onstnt of order unity, so tht k S r is the entropy reltive to infinite temperture ssoited with spin S. Then, 1 q = J 1 os x q x + os y q y + os z q z + kt + K, 10 where is the lttie onstnt in the diretion 9 nd we ssume tht K x K y K z. Grphs of 1 q re shown in Fig. 1 for oth the ferromgneti J 1 0 nd ntiferromgneti J 1 0 ses. For the ferromgneti se, we now introdue ompeting ntiferromgneti next-nerest-neighor nnn intertion J 0 long the x xis so tht 1 q x,q y =0,q z =0 = 4J 1 +J 1 os x q x +J os x q x + kt + K, 11 nd this is lso shown in Fig. 1.AsT is lowered, one rehes ritil temperture where one of the eigenvlues of the inverse suseptiility mtrix eomes zero. This indites tht the prmgneti phse is unstle with respet to order orresponding to the ritil eigenvetor ssoited with the zero eigenvlue. For the ferromgnet, this hppens for zero wve vetor, nd for the ntiferromgnet, for zone oundry wve vetor in greement with our ovious expettion. For ompeting intertions, we see tht the vlues of the J s determine wve vetor t whih n eigenvlue of 1 is miniml. This is the phenomenon lled wve vetor seletion, nd in this se the seleted vlue of q is determined y extremizing 1 to e 30 os x q = J 1 /4J, 1 providing J J 1 /4. Otherwise, the system is ferromgneti. Note lso tht rystl symmetry my selet set of symmetry-relted wve vetors, whih omprise wht is know the str of q. For instne, if the system were tetrgonl, then rystl symmetry would imply tht one hs the sme nnn intertions long the y xis, in whih se the system selets wve vetor long the x xis nd one of equl mgnitude long the y xis. From the ove disussion, it should e ler tht if we ssume ontinuous trnsitioo tht the trnsition is ssoited with the instility in the terms in the free energy qudrti in the spin mplitudes, then the nture of the ordered phse is determined y the ritil eigenvetor of the inverse suseptiility, i.e., the eigenvetor ssoited with the eigenvlue of inverse suseptiility whih first goes to zero s the temperture is redued. Aordingly, the im of this pper is to nlyze how rystl symmetry ffets the possile forms of the ritil eigenvetor. When the unit ell ontins n1 spins, the inverse suseptiility for eh wve vetor q is3n3n mtrix. The ordering trnsition ours when, for some seleted wve vetors, n eigenvlue first eomes zero s the temperture is FIG. 1. Color online Inverse suseptiility 1 q,0,0. top Ferromgneti model J 1 0, middle ntiferromgneti model J 1 0, nd ottom model with ompeting intertions the nn intertion is ntiferromgneti. In eh pnel, one sees three groups of urves. Eh group onsists of the three urves for q whih depend on the omponent lel due to the nisotropy. The x xis is the esiest xis nd the z xis is the hrdest. If the system is orthorhomi, the three xes must ll e inequivlent. The solid urves re for the highest temperture, the dshed urves re for n intermedite temperture, nd the dsh-dot urves re for T=T, the ritil temperture for mgneti ordering. The ottom pnel illustrtes the nontrivil wve vetor seletion whih ours when one hs ompeting intertions. redued. In the ove simple exmples involving isotropi exhnge intertions, the inverse suseptiility ws 33 digonl mtrix so tht eh eigenvetor trivilly hs only one nonzero omponent. The ritil eigenvetor hs spin oriented long the esiest xis, i.e., the one for whih K is miniml. In the present more generl se, n1 nd ritrry intertions onsistent with rystl symmetry re llowed. To void the tehnilities of group theory, we use s our guiding priniple the ft tht the free energy, eing n

6 A. B. HARRIS PHYSICAL REVIEW B 76, TABLE I. Generl positions Refs. 33 nd 34 within the primitive unit ell for Cm whih desrie the symmetry opertions Ref. 36 of this spe group. is twofold rottion or srew xis nd m is mirror glide whih tkes r into r followed y trnsltion. All oordintes re expressed s frtion of lttie prmeters so tht x relly denotes x. Er=x,y,z r=x,ȳ+1/,z+1/ r=x,y+1/,z +1/ r=x,ȳ,z Ir=x,ȳ,z m r=x,y+1/,z +1/ m r=x,ȳ+1/,z+1/ m r=x,y,z expnsion in powers of the mgnetiztions reltive to the prmgneti stte, must e invrint under ll the symmetry opertions of the rystl. 6,31 This is the sme priniple tht one uses in disussing the symmetry of the eletrostti potentil in rystl. 3 We now fous our ttention on the ritilly seleted wve vetor q whih hs n eigenvlue whih first eomes zero s the temperture is lowered. This vlue of q is determined y the intertions nd we will onsider it to e n experimentlly determined prmeter. Opertions whih leve the qudrti free energy invrint must leve invrint the term in the free energy F q whih involves only the seleted wve vetor q, nmely, TABLE II. Positions Refs. 34 nd 35 of Ni + rrying S=1 within the primitive unit ell illustrted in Fig.. Here, r sn denotes the position of the nth spine site nd r n tht of the nth ross-tie site. NVO orders ipe group Cm, so there re six more toms in the onventionl orthorhomi unit ell, whih re otined y trnsltion through 0.5,0.5,0. r s1 = r s = r s3 = r s4 = r 1 = r = 0.5, 0.13, , 0.13, , 0.13, , 0.13,0.5 0, 0, 0 0.5, 0, 0.5 digrm in the T-H plne for H long the xis, for T K. 6 The group of opertions whih onserve wve vetor is generted y the twofold rottion x nd the glide opertion m z, oth of whih re defined in Tle I. We now disuss how the Fourier spin omponents trnsform under vrious symmetry opertions. Here, primed quntities denote the vlue of the quntity fter trnsformtion. Let O y F q 1,,, 1 q;,s q, * S q,. 13 Any symmetry opertion tkes the originl vriles efore trnsformtion, S q,, into new ones indited y primes. We write this trnsformtio Sq, = U ; S q,. 14 Aording to well knowttement of elementry quntum mehnis, if set of ommuting opertors T 1,T,... lso ommutes with 1 q, then the eigenvetors of 1 q re simultneously eigenvetors of eh of the T i s. This muh reprodues well known nlysis. 0 We will lter onsider the effet of inversion, the nlysis of whih seems to hve een universlly overlooked. We will pply this simple ondition to numer of multiferroi systems urrently under investigtion. This pproh n e muh more strightforwrd thn the stndrd one when the opertions whih onserve wve vetor unvoidly involve trnsltions. As first exmple, we onsider the se of NVO nd use the tul position Fourier trnsforms. In Tle I, we give the generl positions this set of positions is the so-lled Wykoff orit for the spe group Cm No. 64 in Ref. 33 of NVO nd this tle defines the opertions of the spe group Cm. In Tle II, we list the positions of the two types of sites oupied y the mgneti Ni ions, whih re lled spine nd ross-tie sites in reognition of their distintive oordintion in the lttie, s n e seen in Fig., where we show the onventionl unit ell of NVO. Experiments 6,38 indite tht s the temperture is lowered, the system first develops inommensurte order with q long the diretion with q In Fig. 3 we show the phse m z z s3 s 1 s4 s1 s s3 1 s1 spine s4 ross tie FIG.. Color online Ni sites in the onventionl unit ell of NVO. The primitive trnsltion vetors v n re v 1 =/â+/ˆ, v =/â /ˆ, nd v 3 =ĉ. The ross-tie sites lue online 1 nd lie in plne with =0. The spine sites red online re leled s1, s, s3, nd s4 nd they my e visulized s forming hins prllel to the xis. These hins re in the ukled plne with =±, where =0.13 s is indited. Cross-tie sites in djent plnes re displed y ±/ˆ. Spine sites in djent plnes re loted diretly ove or elow the sites in the plne shown. In the inommensurte phses, the wve vetor desriing mgneti ordering lies long the xis. The xis of the twofold rottion out the x xis is shown. The glide plne is indited y the mirror plne t z= 3 4 nd the rrow ove m z indites tht trnsltion of / in the y diretion is involved. x x

7 LANDAU ANALYSIS OF THE SYMMETRY OF THE PHYSICAL REVIEW B 76, Mgneti Field (T) 6 3 AF H LTI HTI 4 6 Temperture (K) FIG. 3. Shemti phse digrm for NVO for mgneti field pplied long the diretion, tken from Ref. 6. Here, AF is n ntiferromgneti phse with wek ferromgneti moment, P is the prmgneti phse, HTI is the high-temperture inommensurte phse in whih the moments re essentilly ligned long the xis with sinusoidlly modulted mplitude ording to irrep 4, nd LTI is the low-temperture inommensurte phse in whih trnsverse order long the xis ppers to mke n elliptilly polrized order-prmeter wve ording to irreps 4 nd 1. A spontneous polriztion P ppers only in the LTI phse with P long. O s O r e symmetry opertion whih we deompose into opertions on the spin O s nd on the position O r. The effet of trnsforming spin y suh n opertor is to reple the spin t the finl position R f y the trnsformed spin whih initilly ws t the position O r 1 R f. So, we write P 10 S q, f = O s S q, i e iq R f R i. As efore, we my write this s OS q, f = O s S q, i e iq R f R i. Under trnsformtion y inversion, I=1 nd S q, f * = N 1 S R i, i e iq R f + f R 19 0 = S q, i e iq R f f R i i = S q, i 1 for tul position Fourier trnsforms. For unit ell trnsforms, we get S q, f * = S q, i e iq R f R i = S q, i e iq f + i. Now, we pply this formlism to find the tul position Fourier oeffiients whih re eigenfuntions of the two opertors x nd m z. Io doing, note the simpliity of Eq. 17: sine, for NVO, the opertions x nd m z do not hnge the x oordinte, we simply hve S q, f = S q, i. 3 Thus, the eigenvlue onditions for x ting on the spine sites 1 4 re S q,1 = x S q, = x S q,1, S R f, f = O s S O r 1 R f, f = O s S R i, i, 15 where the susripts i nd f denote initil nd finl vlues nd O s is the ftor introdued y O s for pseudovetor, nmely, x x =1, y x = z x = 1, x m z = y m z = 1, z m z =1. 16 S q, = x S q,1 = x S q,, S q,3 = x S q,4 = x S q,3, S q,4 = x S q,3 = x S q,4, from whih we see tht x = ±1 nd S q, = x / x S q,1, 4 Note tht OS R, is not the result of pplying O to move nd reorient the spin t R+, ut insted is the vlue of the spin t R+ fter the spin distriution is ted upon y O. Thus, for tul position Fourier trnsforms, we hve S q, f = N 1 SR f, f e iq R f + f R We my write this s = O s N 1 S R i, i e iq R f + f R = O s S q, i e iq R f + f R i i. 17 OS q, f = O s S q, i e iq R f + f R i i. 18 This formultion my not e totlly intuitive, euse one is tempted to regrd the opertion O ting on spin t n initil lotion nd tking it nd perhps reorienting it to nother lotion. Here, insted, we onsider the spin distriution nd how the trnsformed distriution t lotion is relted to the distriution t the initil lotion. Similrly, the result for unit ell Fourier trnsforms is S q,3 = x / x S q,4. 5 The eigenvlue onditions for m z ting on the spine sites re S q,1 = m z S q,4 = m z S q,1, S q,4 = m z S q,1 = m z S q,4, S q, = m z S q,3 = m z S q,, S q,3 = m z S q, = m z S q,3, from whih we see tht m z = ±1 nd 6 S q,4 = m z /m z S q,1. 7 We therey onstrut the wve funtions for the spine sites whih re simultneously eigenvetors of x nd m z nd these re given in Tle III. The results for the ross-tie sites re otined in the sme wy nd re lso given in the tle. Eh set of eigenvlues orresponds to different symmetry lel irreduile representtion irrep, here denoted n. Sine eh opertor n hve either of two eigenvlues, we hve four symmetry lels to onsider. Note tht these spin

8 A. B. HARRIS PHYSICAL REVIEW B 76, TABLE III. Allowed spin funtions i.e., tul position Fourier oeffiients within the unit ell of NVO for wve vetor q,0,0 whih re eigenvetors of x nd m z with the eigenvlues listed. Inversioymmetry is not yet tken into ount. Eh of the four omintions of eigenvlues represents different symmetry, whih we identify with symmetry lel n. In group theoretil lnguge, n is referred to s n irreduile representtion irrep, for whih we use the nottion of Ref. 6. n is the numer of independent struture prmeters in the wve funtion hving the symmetry lel. Group theory indites tht n is the numer of times the irrep is ontined in the originl 18-dimensionl representtion orresponding to S q,. For the leling of the sites, is s in Tle II nd Fig.. Here, n p p=s or, =,, denotes the omplex quntity n p q. Irrep x m z n Sq,s1 Sq,s Sq,s3 Sq,s4 Sq,1 Sq, funtions, sine they re tully Fourier oeffiients, re omplex-vlued quntities. The spin itself is rel euse F q=fq *. Eh olumn of Tle III gives the most generl form of n llowed eigenvetor for whih one hs n=4 or n=5 depending on the irrep independent omplex onstnts. In terms of the mplitude X m q of the mth eigenfuntion of irrep t wve vetor q nd the orresponding eigenvlue m q, the free energy is digonl: F = 1 q n m=1 m qx m q. 8 These eigenvlues n e identified s the inverse suseptiility ssoited with norml modes of spin onfigurtions n n 0 0 n n n n n n n 0 0 n n n To further illustrte the mening of this tle, we expliitly write, in Eq. 48 elow, the spin distriution rising from one irrep, 4. These spin funtions re shemtilly shown for the spine sites in Fig. 16 elow. Here, our min interest is in the mode whih first eomes unstle s the temperture is lowered. So fr, the present nlysis reprodues the stndrd results nd indeed omputer progrms exist to onstrut suh tles. However, for multiferrois it my e quiker to otin nd understnd how to onstrut the possile spin funtions y hnd rther thn to understnd how to use the progrm. Usully, these progrms give the results in terms of unit ell Fourier trnsforms, whih we lim re not s nturl representtion ies like NVO. In terms of unit ell Fourier trnsforms, the eigenvlue onditions for x ting on the spine sites 1 4 re the sme s Eq. 4 for tul position Fourier trnsforms euse the opertion x does not hnge the unit ell. However, for the glide opertion m z, this is not the se. If we strt from site 1 or site, the trnsltion long the y xis tkes the spin to finl unit ell displed y /î+/ĵ, wheres if we strt from site 3 or site 4, the trnsltion long the y xis tkes the spin to finl unit ell displed y /î+/ĵ. Now, the eigenvlue onditions for m z ting on the spine sites 1 4 re S q,1 = m z S q,4 = m z S q,1, S q,4 = m z S q,1 * = m z S q,4, S q, = m z S q,3 = m z S q,, S q,3 = m z S q, * = m z S q,3, 9 where =expiq. One finds tht ll entries for Sq,s3, Sq,s4, nd Sq, now rry the phse ftor * =exp iq. However, this is just the ftor to mke the unit ell result SR, = Sq,e iq R 30 e the sme to within n overll phse ftor s the tul position result SR, = Sq,e iq R+. 31 We should emphsize tht iuh simple se s NVO, it is tully not neessry to invoke ny group theoretil onepts to rrive t the results of Tle III for the most generl spin distriution onsistent with rystl symmetry. More importntly, it is not ommonly understood 0 tht one n lso extrt informtion using the symmetry of n opertion inversion whih does not onserve wve vetor. 3 7,3 5 Sine wht we re out to sy my e unfmilir, we strt from first priniples. The qudrti free energy my e writte F = q F S q, * S q,,,; 3 where we restrit the sum over wve vetors to the str of the wve vetor of interest. One term of this sum is

9 LANDAU ANALYSIS OF THE SYMMETRY OF THE F q 0 = F S q 0, * S q 0,. 33,; It should e ler tht the qudrti free energy F is invrint under ll the symmetry opertions of the prmgneti spe group i.e., wht one lls the spe group of the rystl. 6,31 For entrosymmetri rystls, there re three lsses of suh symmetry opertions. The first lss onsists of those opertions whih leve q 0 invrint nd these re the symmetries tken into ount in the usul formultion. 0 The seond lss onsists of opertions whih tke q 0 into nother wve vetor of the str ll it q 1, where q 1 q 0. Use of these symmetries llows one to ompletely hrterize the wve in funtion t wve vetor q 1 in terms of the wve funtion for q 0. These reltions re needed if one is to disuss the possiility of simultneously ondensing more thn one wve vetor in the str of q. 8,40 Finlly, the third lss onsists of sptil inversion unless the wve vetor nd its negtive differ y reiprol lttie vetor, in whih se inversion elongs in lss 1. The role of inversioymmetry is lmost universlly overlooked, 0 s is evident from exmintion of numer of reent ppers. Unlike the opertions of lss 1 whih tkes S n q into n S n q for irreps of dimension 1 whih is true for most ses onsidered in this pper, inversion tkes S n q into n S n q. Nevertheless, it does tke the free energy written in Eq. 33 into itself nd restrits the possile form of the wve funtions. So, we now onsider the onsequenes of invrine of F under inversion. 3 7 For this purpose, we write Eq. 13 in terms of the spin oordintes n of Tle III. The result will, of ourse, depend on whih symmetry lel we onsider. In ny se, the prt of F whih depends on q 0 n e writte F q 0 = F S q 0, * S q 0,,; = G N,;N,n N * n N, 34 N,;N,; where N nd N ssume the vlues s for spin nd for ross-tie nd nd lel omponents, nd the sums over N nd nd similrly N nd re over the n vriles needed to speify the wve funtiosoited with the symmetry lel irrep. From now on, we keep only the terms elonging to the irrep whih is tive nd for nottionl simpliity we leve the orresponding rgument of n impliit. Then, we see tht invrine under inversion implies tht F q = G N,;N,n N * n N N,;N, = G N,;N,In N * In N. 35 N,;N, Now, we need to understnd the effet of I on the spin Fourier oeffiients listed in Tle III. Sine we use tul position Fourier oeffiients, we pply Eq. 1. For the ross-tie vriles whih sit t enter of inversioymmetry, inversion tkes the spin oordintes of one spine sulttie into the omplex onjugte of itself: TABLE IV. The sme s Tle III for NVO exept tht now the effet of inversioymmetry is tken into ount, s result of whih, prt from n overll phse ftor, ll the n s in this tle n e tken to e rel vlued. Irrep x m z Sq,s1 Sq,s Sq,s3 Sq,s4 Sq,1 Sq, i i i i i i i i n i ISq,n = Sq,n *. Thus, in terms of the n s this gives 36 In = n *, = x,y,z. 37 The effet of inversion on the spine vriles gin follows from Eq. 1. Sine inversion interhnges sultties 1 nd 3, we hve Sq,s3 = Sq,s1 *. 38 For x =m z =+1 i.e., for irrep 1, we sustitute the vlues of the spin vetors from the first olumn of Tle III to get I = *, I = *, I = *. i i i 39 Note tht some omponents introdue ftor 1 under inversion nd others do not. Whih ones hve the minus signs depends on whih irrep we onsider. If we mke hnge of vrile y repling in olumn 1 of Tle III y iñ s for those omponents for whih I introdues minus sign nd repling the other y ñ s, then we my rewrite the first olumn of Tle III in the form given in Tle IV. We re- i i i n i i i i i i i i i n n 0 0 n n n PHYSICAL REVIEW B 76, n n 0 0 n n n

10 A. B. HARRIS PHYSICAL REVIEW B 76, ple ll the ross-tie vriles n x y ñ x. In terms of these new tilde vriles, one hs Iñ s = ñ s *. 40 It is onvenient to define the spin Fourier oeffiients so tht they ll trnsform in the sme wy under inversion. Otherwise, one would hve to keep trk of vriles whih trnsform with plus sign nd those whih trnsform with minus sign. Repeting this proess for ll the other irreps, we write the possile spin funtions s those of Tle IV. We give n expliit formul for the spin distriution for one irrep in Eq. 48 elow. Now, we implement Eq. 35, where the spin funtions re tken to e the vriles listed in Tle IV. First, note tht the mtrix G in Eq. 35 hs to e Hermitin to ensure tht F is rel: G M,;N, = G N,;M, *. Then, using Eq. 40, we find tht Eq. 35 is F q 0 = ñ M * G M,;N, ñ N M,;N, = Iñ M * G M,;N, Iñ N M,;N, = M,;N, ñ M G M,;N, ñ N * = 41 ñ M * G N,;M, ñ N, M,;N, 4 where, in the lst line, we interhnged the roles of the dummy indies M, nd N,. By ompring the first nd lst lines, one sees tht the mtrix G is symmetri. Sine this mtrix is lso Hermitin, ll its elements must e rel vlued. Thus, ll its eigenvetors n e tken to hve only relvlued omponents. However, the m s re llowed to e omplex vlued. So, the onlusion is tht for eh irrep, we my write ñ N = e i r N, 43 where the r s re ll rel vlued nd is n overll phse whih n e hosen ritrrily for eh. When only single irrep is tive, it is likely tht the phse will e fixed y high-order umklpp terms in the free energy, ut the effets of suh phse loking my e eyond the rnge of experiments. 41 It is worth noting how these results should e nd in few ses 3,4,6 hve een used in the struture determintions. One should hoose the est fit to the diffrtion dt using, in turn, eh irrep i.e., eh set of eigenvlues of x nd m z. Within eh irrep, one prmetrizes the spitruture y hoosing the Fourier oeffiients s in the relevnt olumn of Tle IV. Note tht insted of hving four or five omplex oeffiients to desrie the six sites within the unit ell see Tle III, one hs only four or five depending on the representtion rel-vlued oeffiients to determine. The reltive phses of the omplex oeffiients hve ll een fixed y invoking inversioymmetry. This is lerly signifint step in inresing the preision of the determintion of the mgneti struture from experimentl dt. B. Order prmeters We now review how the ove symmetry lssifition influenes the introdution of order prmeters whih llow the onstrution of Lndu expnsions. 4,6 The form of the order prmeter should e suh tht it hs the potentil to desrie ll ordering whih re llowed y the qudrti free energy F. Thus, for n isotropi Heisenerg model on ui lttie, the order prmeter hs three omponents i.e., it involves three-dimensionl irrep euse lthough the fourth order terms will restrit order to our only long ertin diretions, s fr s the qudrti terms re onerned, ll diretions re equivlent. The nlogy here is tht the overll phse of the spin funtion is not fixed y the qudrti free energy nd ordingly the order prmeter must e omplex vrile whih inludes suh phse. One lso reognizes tht lthough the mplitude of the ritil eigenvetor is not fixed y the qudrti terms in the free energy, the rtios of its omponents re fixed y the speifi form of the inverse suseptiility mtrix. Although we do not wish to disuss the expliit form of this mtrix, wht should e ler is tht the omponents of the spins whih order must e proportionl to the omponents of the ritil eigenvetor. The tul mplitude of the spin ordering is determined y the ompetition etween the qudrti nd fourth-order terms in the free energy. If p is the irrep whih is ritil, then just elow the ordering temperture we write ñ N q = p qr N p, 44 where the r s re rel omponents of the ritil eigenvetor ssoited with the ritil eigenvlue of irrep p of the mtrix G of Eq. 35 nd re now normlized y r N =1. 45 N Here, the order prmeter for irrep q, p q, is omplex vrile, sine it hs to inorporte the ritrry omplex phse p ssoited with irrep p : p ±q = p e i p. 46 The order prmeter trnsforms s indited in the tles y its listed eigenvlues under the symmetry opertions x nd m z. Sine the omponents of the ritil eigenvetor re dominntly determined y the qudrti terms, 4 one y tht just elow the ordering temperture the desription in terms of n order prmeter ontinues to hold ut p T T p, 47 where men-field theory gives =1/ ut orretions due to flutution re expeted. 43 To summrize nd illustrte the use of Tle IV, we write n expliit expression for the mgnetiztions ssuming the tive irrep to e 4 x = 1 nd m z =+1. We use the definition of the order prmeter nd sum over oth signs of the wve vetor to get S x r,s1 = 4 r s x osqx + 4, S y r,s1 = 4 r s y sinqx + 4,

11 LANDAU ANALYSIS OF THE SYMMETRY OF THE S z r,s1 = 4 r s z osqx + 4, S x r,s = 4 r s x osqx + 4, S y r,s = 4 r s y sinqx + 4, S z r,s = 4 r s z osqx + 4, S x r,s3 = 4 r s x osqx + 4, S y r,s3 = 4 r s y sinqx + 4, S z r,s3 = 4 r s z osqx + 4, S x r,s4 = 4 r s x osqx + 4, S y r,s4 = 4 r s y sinqx + 4, S z r,s4 = 4 r s z osqx + 4, S x r,1 =0, S y r,1 = 4 r y osqx + 4, S z r,1 = 4 r z osqx + 4, S x r,1 =0, S y r, = 4 r y osqx + 4, S z r, = 4 r z osqx + 4, 48 nd similrly for the other irreps. The oserved mgneti strutures re desried qulittively in the ption of Fig. 3. The tul vlues of the struture prmeters r x in Eq. 48 nd its nlog for irrep 1 re given in Ref. 6. Here, r x,y,z is the tul lotion of the spin. Using expliit expressions like the ove or more diretly from Tle IV, one n verify tht the order prmeters p for irrep p hve the trnsformtion properties nd x 1 q =+ 1 q, m z 1 q =+ 1 q, x q =+ q, m z q = q, x 3 q = 3 q, m z 3 q = 3 q, x 4 q = 4 q, m z 4 q =+ 4 q, 49 I n q = n q *. 50 Note tht even when more thn single irrep is present, the introdution of order prmeters, s done here, provides frmework within whih one n represent the spin distriution s liner omintion of distriutions eh hving PHYSICAL REVIEW B 76, hrteristi symmetry, s expressed y Eq. 49. When the struture of the unit ell is ignored, 16 tht informtion is not redily essile. Also note tht the phse of eh irrep n is defined so tht when n =0, the wve is inversioymmetri out r=0. When n is nonzero, it is possile to invoke the inommensurility to find lttie site whih is ritrrily lose to enter of inversioymmetry of the mthemtil spin funtion. Thus, eh irrep hs enter of inversioymmetry whose lotion is impliitly defined y the vlue of n. When only single irrep is tive, the speifition of n is not importnt. However, when one hs two irreps, then inversioymmetry is only mintined if the enters of inversion symmetry of the two irreps oinide, i.e., if their phses re equl. In mny systems, the initil inommensurte order tht first ours s the temperture is lowered eomes unstle s the temperture is further lowered. 30 Typilly, the initil order involves spins oriented long their esy xis with sinusoidlly vrying mgnitude. However, the fourth-order terms in the Lndu expnsion whih we hve not written expliitly fvor fixed length spins. As the temperture is lowered, the fixed length onstrint eomes progressively more importnt, nd t seond, lower, ritil temperture trnsition ours in whih trnsverse omponents eome nonzero. Although the sitution is more omplited when there re severl spins per unit ell, the result is similr: the fixed length onstrint is est relized when more thn single irrep hs ondensed. So, for NVO nd TMO, s the temperture is lowered one enounters seond phse trnsition in whih seond irrep ppers. Within low-order Lndu expnsion, this phenomenon is desried y free energy of the form 6 F = 1 T T + 1 T T + u 4 + u 4 + w, 51 where T T. This system hs eetudied in detil y Brue nd Ahrony. 44 For our purposes, the most importnt result is tht for suitle vlues of the prmeters, ordering in ours t T nd t lower temperture when T T +w =0 order in my our. The pplition of this theory to the present sitution is simple: we n nd usully do hve two mgneti phse trnsitions in whih, first, one irrep nd then t lower temperture seond irrep ondense. A question rises s to whether the ondenstion of one irrep n indue the ondenstion of seond irrep. This is not possile euse the two irreps hve different symmetries. However, ould the presene of two irreps nd indue the pperne of third irrep 3 t the temperture t whih first ppers? For tht to hppen would require n m tht 3 ontin the unit representtion for some vlues of n nd m. This or ny higher omintion of representtions is not llowed for the simple four irreps system like NVO. In more omplex systems, one might hve to llow for suh phenomenon. III. APPLICATIONS In this setion, we pply the ove formlism to numer of multiferrois of urrent interest

12 A. B. HARRIS PHYSICAL REVIEW B 76, Er=x,y,z Ir=x,ȳ,z TABLE V. Generl positions for spe group P/. m y r= x,ȳ,z+ 1 y r= x,y,z + 1 TABLE VI. Allowed spin eigenfuntions for MWO prt from n overll phse ftor efore inversioymmetry is tken into ount, where =exp iq z /. Here, the nq s re omplex nd we hve tken the lierty to djust the overll phse to give symmetril looking result. However, these results re equivlent to Tle II of Ref. 45. Irrep 1 m y e iq z e iq z A. MnWO 4 MnWO 4 MWO rystllizes in the spe group P/ No. 14 in Ref. 33 whose generl positions re given in Tle V. The two mgneti Mn ions per unit ell re t positions 1 = 1,y, 1 = 4, 1,1 y, The wve vetor of inommensurte mgneti ordering is 45 q=q x,1/,q z, with q x 0.1 nd q z 0.46, nd is left invrint y the identity nd m y. We strt y onstruting the eigenvetors of the qudrti free energy i.e., the inverse suseptiility mtrix. Here, we use unit ell Fourier trnsforms to filitte omprison with Ref. 45. Below, X, Y, nd Z denote integers in units of lttie onstnts. When nd R f + f = X,Y,Z + 1 =X + 1,Y + y,z R i + i = m y 1 R f + f =X + 1, Y y,z 1 4 = X, Y 1,Z 1 +, 54 then Eq. 19 gives the eigenvlue ondition for m y to e S q, 1 = m y S q, e iq Y+1ĵ+kˆ = m y S q, e i+iq z = S q, 1, 55 where x m y = y m y = z m y = 1. When then R f + f = X,Y,Z + =X + 1,Y +1 y,z + 3 4, R i + i =X + 1, Y 1+y,Z = X, Y 1,Z + 1, nd Eq. 19 gives the eigenvlue ondition to e S q, = m y S q, 1 e iq Y+1ĵ = m y S q, 1 1 = S q,. 58 From Eqs. 55 nd 58, weget=±e iq z nd S q, = m y /S q, So, we get the results listed in Tle VI. Sq,1 * n x * n x * n y * n y * n z * n z Sq, n x n x n y n y n z n z So fr, the nlysis is essentilly the ompletely stndrd one. Now, we use the ft tht the free energy is invrint under sptil inversion, even though tht opertion does not onserve wve vetor. 3,4,6,7 We now determine the effet of inversion on the n s. As will eome pprent, use of unit ell Fourier trnsforms mkes this nlysis more omplited thn if we hd used tul position trnsforms. We use Eq. to write ISq, =1 = Sq, = * e iq î+ĵ+kˆ Sq, *, 60 where = exp iq x +q z. For,weget whih we n write s In x,n y,n z = n x,n y, n z *, In = m y n * Now, the free energy is qudrti in the Fourier spin oeffiients, whih re linerly relted to the n s. So, the free energy n e writte F = n Gn, 63 where n=n x,n y,n z is olumn vetor nd G is33 mtrix whih we write s = A C G * B, 64 * * where, for Hermitiity, the Romn letters re rel nd the Greek ones omplex. Now, we use the ft tht we must lso hve invrine with respet to inversion, whih fter ll is rystl symmetry. Thus, This n e writte F = In GIn

13 LANDAU ANALYSIS OF THE SYMMETRY OF THE TABLE VII. The sme s Tle VI for TMO exept tht here inversioymmetry is tken into ount. Here, r, s, nd t re rel. All six omponents n e multiplied y n overll phse ftor whih we hve not een expliitly written. Irrep 1 m y e iq z e iq z Sq,1 * r i * r i * s * s * t i * t Sq, r ir is s t it F = * * m y n G m y n = m y n G m y n *. Thus, we my write A F = n tr * B n* * * C A * * = n B * C n, where tr indites trnspose so n tr is row vetor. Sine the two expressions for F, Eqs. 63 nd 67, must e equl, we see tht =i, =, nd =i, where,, nd must e rel. Thus, G is of the form G = A i i B i i C, 68 where ll the letters re rel. This mens tht the ritil eigenvetor desriing the long-rnge order hs to e of the form n x,n y,n z = e i r,is,t, 69 where r, s, nd t re rel. For,wesete i = i. For 1,we set e i =1. These hoies re not essentil. They just mke the symmetry more ovious. Thus, we otin the finl results given in Tle VII. Lutenshlger et l. 45 sy just ove Tle II Depending on the hoie of the mplitudes nd phses Wht we see here is tht inversioymmetry fixes the phses without the possiility of hoie just s it did for NVO. Note gin tht we hve out hlf the vriles to fix in struture determintion when we tke dvntge of inversion invrine to fix the phse of the omplex struture onstnts. Order prmeter Now, we disuss the definition of the order prmeter for this system. For this purpose, we reple r y r, s y s, et., with the normliztion tht PHYSICAL REVIEW B 76, r + s + t =1. 70 Here, the order prmeter is omplex euse we lwys hve the freedom to multiply the wve funtion y phse ftor. This phse ftor might e loked y higher-order terms in the free energy, ut we do not onsider tht phenomenon here. 46 We reord the symmetry properties of the order prmeter. With our hoie of phses, we hve I n q = n q *, m y n q = n n q, m y n q = n * n q, 71 where n q n e i n is the omplex-vlued order prmeter for ordering of irrep n nd n is the eigenvlue of m y given in Tle VII. Now, we write n expliit formul for the spin distriution in terms of the order prmeters of the two irreps: SR, =1 = 1 r 1 î + t 1 kˆosq R + 1 q z / + s 1 ĵ sinq R + 1 q z / + r î t kˆsinq R + q z / + s ĵ osq R + q z /, 7 SR, = = 1 r 1 î + t 1 kˆosq R q z / s 1 ĵ sinq R q z / + r î + t kˆsinq R + + q z / + s ĵ osq R + + q z /. 73 One n expliitly verify tht these expressions re onsistent with Eq. 71. Note tht when only one of the order prmeters sy, n is nonzero, we hve inversioymmetry with respet to redefined origin where n =0. For eh irrep, we hve to speify three rel prmeters, r n, s n, nd t n, nd one overll phse, n, rther thn three omplexvlued prmeters hd we not invoked inversioymmetry. B. TMnO 3 Here, we give the full detils of the lultions for TMnO 3 desried in Ref. 3. The presenttion here differs osmetilly from tht in Ref. 5. The spe group of TMnO 3 is Pnm whih is No. 6 in Ref. 33 lthough the positions re listed there for the Pnm setting. The spe group opertions for generl Wykoff orit re given in Tle VIII. In Tle IX, we list the positions of the Mn nd T ions within the unit ell nd these re lso shown in Fig. 4. The phse digrm for mgneti fields up to 14 T long the xis is shown in Fig. 5. To strt, we study the opertions tht leve invrint the wve vetor of the inommensurte phse whih first orders s the temperture is lowered. Experimentlly, 49 this wve vetor is found to e 0,q,0, with 39 q0.8. These relevnt opertors see Tle VIII re m x nd m z. We follow the

14 A. B. HARRIS PHYSICAL REVIEW B 76, TABLE VIII. Generl positions for Pnm. Nottion is the sme s in Tle I. z 1 Er=x,y,z x r= x+,ȳ+ 1,z 1 z r= x,ȳ,z+ 1 y r= x +,y+ 1,z Ir=x,ȳ,z m x r= x +,y+ 1,z 1 m z r= x,y,z + 1 m y r= x+,ȳ+ 1,z+ 1 pproh used for MWO, ut use tul lotion Fourier trnsforms. We set R f + f r in order to use Eq. 17 nd we need to evlute m z z=1/4 y expiq r m x 1 r = expiqĵ yĵ m x 1 yĵ = e iq 74 nd expiq r m z 1 r = expqĵ yĵ m z 1 yĵ =1. 75 We list in Tle X the trnsformtion tle of sulttie indies of TMO. Therefore, the eigenvlue ondition for trnsformtion y m x is S q, f = m x S q, i = m x S q, f nd tht for trnsformtion y m z is S q, f = m z S q, i = m z S q, f, where x m x = y m x = z m x =1 nd m z ws defined in Eq. 16. From these equtions, we see tht m x ssumes the vlues ± nd m z the vlues ±1. Then, solving the ove equtions leds to the results given in Tle XI. These results look different from those in Ref. 3 euse here the Fourier trnsforms re defined reltive to the tul positions, wheres in Ref. 3 they re defined reltive to the origin of the unit ell. Now, sine the rystl is entrosymmetri, we tke symmetry with respet to sptil inversion I into ount. As efore, rell tht I trnsports the spin to its sptilly inverted position without hnging the orienttion of the spin pseudovetor. The hnge of position is equivlent to hnging the sign of the wve vetor in the Fourier trnsform nd this is omplished y omplex onjugtion. Sine the Mn ions sit t enters of inversioymmetry, one hs, for the Multties, TABLE IX. Positions of the mgneti ions in the Pnm struture of TMnO 3, with x= nd y= Ref Mn 1= 0,,0 = 1,0,0 1 3= 0,, 1 4= 1,0, 1 1 T 5= x,y, 1 4 6= x+,ȳ+ 1, = x,ȳ, 1 4 8= x +,y+ 1, 1 4 m x x FIG. 4. Color online Mites smller irles, red online nd T sites lrger irles, lue online in the primitive unit ell of TMnO 3. The T sites re in the shded plnes t z=n± 1 4 nd the Mites re in plnes z=n or z=n+ 1, where n is n integer. The inommensurte wve vetor is long the xis. The mirror plne t z=1/4 is indited nd the glide plne m x is indited y the mirror plne t x=3/4 followed y trnsltion indited y the rrow of / long the y xis. ISq,n = Sq,n *, 78 where the seond rgument speifies the sulttie, s in Tle IX. In order to disuss the symmetry of the oordintes, we define x 1 =n M, x =n M, x 3 =n M nd for irreps 1 nd 3, x 4 =n T1 nd x 5 =n T, wheres for irreps nd 4, x 4 =n T1, x 5 =n T, x 6 =n T1, nd x 7 =n T. Thus, Eq. 78 gives Ix n = x * n, n = 1,,3. 79 For the T ions, I interhnges sultties 5 nd 7 nd interhnges sultties 6 nd 8. So, we hve MgnetiField (T) 10 LTI P P HTI P Temperture (K) FIG. 5. Shemti phse digrm for TMO for mgneti fields up to 14 T pplied long the diretion, tken from Ref. 48. Here, P is the prmgneti phse, HTI is the high-temperture inommensurte phse in whih Ref. 3 the moments re essentilly ligned long the xis with sinusoidlly modulted mplitude ording to irrep 3, nd LTI is the low-temperture inommensurte phse in whih Ref. 3 trnsverse order long the xis ppers to mke n elliptilly polrized order-prmeter wve ording to irreps 3 nd. A spontneous polriztion P ppers only in the LTI phse with P long the xis for low mgneti field Ref

15 LANDAU ANALYSIS OF THE SYMMETRY OF THE TABLE X. Trnsformtion tle for sulttie indies of TMO under vrious opertions. i f m x f m z f I Therefore, we hve ISq,5 = Sq,7 * ISq,6 = Sq,8 *. 80 Ix 4 = x 5 *, Ix 6 = x 7 *. 81 Now, we use the invrine of the free energy under I to write F = X,;Y, S q,x * F nm S q,y = x * n G nm x m = Ix * n G nm Ix m, m,n m,n 8 where the mtrix G is Hermitin nd we hve impliitly limited onsidertion to whihever irrep is tive. For irreps 1 nd 3, the mtrix G in Eq. 8 ouples five vriles, x 1,...,x 5. Eqution 79 implies tht the upper left 33 sumtrix of G whih involves the vriles x 1,...,x 3 is rel. Equtions 79 nd 81 imply tht G n,4 =G 5,n for n=1,,3. We thus find tht G ssumes the form TABLE XI. Spin funtions i.e., tul position Fourier oeffiients within the unit ell of TMO for wve vetor 0,q,0 whih re eigenvetors of m x nd m z with the eigenvlues listed, with =expiq. All the prmeters re omplex vlued. The irreduile representtion irrep is leled s in Ref. 3. Inversioymmetry is not yet tken into ount. Note tht the two T orits, T1-T4 nd T-T3, hve independent omplex mplitudes. Irrep m x + + m z Sq,M1 Sq,M Sq,M3 Sq,M4 Sq,T1 0 n T1 0 n T1 n T1 Sq,T 0 n T 0 n T n T Sq,T3 0 n T 0 n T n T Sq,T4 0 n T1 0 n T1 PHYSICAL REVIEW B 76, n T1 0 n T1 0 n T 0 n T 0 n T1 0 n T1 0 n T1 0 0 n T 0 n T 0 0 n T 0 n T 0 0 n T1 0 n T1 0 G = * d e * e f *, 83 * * * g * g where the Romn letters re rel vlued nd the Greek re omplex vlued. As shown in Appendix A, the form of this mtrix ensures tht the ritil eigenvetor n e tken to e of the form = n M,n M,n M,n T1,n * T1 r,s,t;, *, 84 where the Romn letters re rel nd the Greek ones omplex. Of ourse, euse the vetor n e omplex, we should inlude n overll phse ftor whih mounts to ritrrily pling the origin of the inommensurte struture, so tht more generlly = e i r,s,t;, *. 85 For irreps nd 4, the mtrix G in Eq. 8 ouples the seven vriles, x 1,...,x 7, listed just ove Eq. 79. Equtions 79 nd 81 imply tht G n,4 =G 5,n nd G n,6 =G 7,n for n=1,,3. Also, Eq. 81 implies similr reltions within the lower right 44 sumtrix involving the vriles x 4,...,x 7. Therefore, G ssumes the form

16 A. B. HARRIS PHYSICAL REVIEW B 76, TABLE XII. The sme s Tle XI exept tht prt from n overll phse for eh irrep, inversioymmetry restrits ll the mngnese Fourier oeffiients to e rel nd ll the T oeffiients to hve the indited phse reltions. Irrep m x + + m z Sq,M1 r r r r s s s s t t t t Sq,M r r r r s s s s t t t t Sq,M3 r r r r s s s s t t t t Sq,M4 r r r r s s s s t t t t Sq,T Sq,T 0 * 0 * = G 0 * 0 * * 0 * 0 Sq,T3 0 * 0 * 0 * 0 * * 0 * 0 Sq,T * * d e * * e f * * * * * g, 86 * g * * * * * * h * * h where Romn letters re rel nd Greek re omplex. As shown in ppendix A, this form ensures tht the eigenvetors re of the form =n M,n M,n M,n T1,n T,m T1,n T = e i r,s,t;, *,, *. 87 These results re summrized in Tle XII. Note tht the use of inversioymmetry fixes most of the phses nd reltes the mplitudes of the two T orits, therey eliminting lmost hlf the fitting prmeters. 3 Order prmeters We now introdue order prmeters n q n e i n for irrep n in terms of whih we n write the spin distriution. For instne, under 3 one hs S x r,m1 = r 3 osqy + 3, S y r,m1 =s 3 osqy + 3, S z r,m1 =t 3 osqy + 3, S x r,m =r 3 osqy + 3, S y r,m =s 3 osqy + 3, S z r,m =t 3 osqy + 3, S x r,t1 = S y r,t1 =0, S z r,t1 = 3 osqy + 3 +, S x r,t = S y r,t =0, S z r,t = 3 osqy + 3, 88 where we set =e i nd the prmeters re normlized y r + s + t + =1. 89 In Eq. 88, rx,y,z is the tul position of the spin in question. From Tle XI, one n otin the symmetry properties of the order prmeters for eh irrep. For instne, nd m x 1 q =+ 1 q, m x q = q, m x 3 q = 3 q, m z 1 q =+ 1 q, m z q = q, m z 3 q =+ 3 q m x 4 q =+ 4 q, m z 4 q = 4 q, 90 I n q = n * q. 91 Note tht in ontrst to the se of NVO, inversioymmetry does not fix ll the phses. However, it gin drstilly redues the numer of possile mgneti struture prmeters whih hve to e determined. In prtiulr, it is only y using inversion tht one finds tht the mgnitudes of the Fourier oeffiients of the two distint T sites hve to e the sme. Note tht if we hoose the origio tht =0 whih mounts to renming the origio tht tht eomes true, then we reover inversioymmetry tking ount tht inversion interhnges terium sultties 3 nd 1. One n determine tht the spitruture is inversion invrint when one ondenses single representtion

17 LANDAU ANALYSIS OF THE SYMMETRY OF THE PHYSICAL REVIEW B 76, TABLE XIII. The sme s Tle VIII. Generl positions for Pm. z 1 Er=x,y,z x r= x+,ȳ+ 1,z 1 z r=x,ȳ,z y r= x +,y+ 1,z 1 Ir=x,ȳ,z m x r= x +,y+ 1,z 1 m z r=x,y,z m y r= x+,ȳ+ 1,z The experimentlly determined struture of the hightemperture inommensurte HTI nd low-temperture inommensurte LTI phses is desried in the ption of Fig. 5 nd numeril vlues of the struture prmeters re given in Ref. 3. The result of Tle XII pplies to other mngntes provided their wve vetor is lso of the form 0,q y,0. This inludes DMO, 9 YMnO 3, 50 nd HoMnO 3. 51,5 Both these systems order into n inommensurte struture t out T 4 K. The Y ompound hs seond lower-temperture inommensurte phse, wheres the Ho ompound hs lower-temperture ommensurte phse. m x x m y z=1/4 y C. TMn O 5 The spe group of TMn O 5 TMO5 is Pm No. 55 in Ref. 33 nd its generl positions re listed in Tle XIII. The positions of the mgneti ions re given in Tle XIV nd re shown in Fig. 6. We will ddress the sitution just elow the ordering temperture of 43 K. 55 We tke the ordering wve vetor to e 55 to e 1,0,q with q This my e n pproximte vlue. 56 The following lultion involves gret del of lger whih my e skipped. The expliit result for the spitruture is given in Eq. 13. Initilly, we ssume tht the possile spin onfigurtions onsistent with ontinuous trnsition t suh wve vetor re eigenvetors of the opertors m x nd m y whih leve the wve vetor invrint. We proeed s for TMO. We use the unit ell Fourier trnsforms nd write the eigenvetor onditions for trnsformtion y m x s S q, f = m x S q, i e iqr f R i = x S q, f, 9 where i nd R i re, respetively, the sulttie indies nd unit ell lotions efore trnsformtion nd f nd R f re TABLE XIV. Positions of the mgneti ions of TMn O 5 in the Pm struture. Here, x=0.09, y= 0.15, z=0.5 Ref. 53, X =0.14, nd Y =0.17 Ref. 54. All these vlues re tken from the isostruturl ompound HoMn O 5. Mn 3+ 1=x,y,0 =x,ȳ,0 1 3= x +,y+ 1,0 1 4= x+,ȳ+ 1,0 Mn 4+ 5= 1,0,z 1 6= 0,,z 7= 1,0,z 1 8= 0,,z 1 RE 9= X,Y, 1 10= X,Ȳ, 1 11= X +,Y + 1, 1 1 1= X+,Ȳ + 1, 1 FIG. 6. Color online Two representtions of TMn O 5. Top: Mites red online with smller irles Mn 3+ nd lrger irles Mn 4+ nd T sites squres, lue online lue in the primitive unit ell of TMn O 5. The Mn +4 sites re in the shded plnes t z =n± with 0.5 nd the Mn +3 sites re in plnes z=n, where n is n integer. The T ions re in the plnes z=n+ 1. The glide plne m x is indited y the mirror plne t x=3/4 followed y trnsltion indited y the rrow of / long the y xis nd similrly for the glide plne m y. Bottom: Perspetive view. Here, the Mn 3+ re inside oxygen pyrmids of smll lls nd the Mn 4+ re inside oxygen othedr. those fter trnsformtion. The eigenvlue eqution for trnsformtion y m y is S q, f = m y S q, i e iqr f R i = y S q, f. 93 If one ttempts to onstrut spin funtions whih re simultneously eigenfuntions of m x nd m y, one finds tht these equtions yield no solution. While it is, of ourse, true

18 A. B. HARRIS PHYSICAL REVIEW B 76, tht the opertions m x nd m y tke n eigenfuntion into n eigenfuntion, it is only for irreps of dimension 1 tht the initil nd finl eigenfuntions re the sme, s we hve ssumed. The present se, when the wve vetor is t the edge of the Brillouin zone, is nlogous to the phenomenon of stiking where, for nonsymmorphi spe group i.e., those hving srew xis or glide plne, the energy nds or phonopetr hve n lmost mysterious degenery t the zone oundry 57 nd the only tive irrep hs dimension. This mens tht the symmetry opertions indue trnsformtions within the suspe of pirs of eigenfuntions. We now determine suh pirs of eigenfuntions y strightforwrd pproh whih does not require ny knowledge of group theory. Here, we expliitly onsider the symmetries of the mtrix 1 for the qudrti terms in the free energy whih here is 3636 dimensionl mtrix, whih we write s 1 =Mxx Mxy Mxz M xy M yy M yz M zx M yz M zz, 94 where M is 1 dimensionl sumtrix whih desries oupling etween -omponent nd -omponent spins nd is indexed y sulttie indies nd. The symmetries we invoke re opertions of the glide plnes m x nd m y, whih onserve wve vetor to within reiprol lttie vetor, nd I, whose effet is usully ignored. To guide the reder through the ensuing lultion, we summrize the min steps. We first nlyze seprtely the setors involving the x, y, nd z spin omponents. We develop unitry trnsformtion whih tkes M into mtrix ll of whose elements re rel. This fixes the phses within the 1 dimensionl spe of the spin omponents within the unit ell ssuming tht these reltions re not invlidted y the form of M, with. The reltive phses etween different spin omponents re fixed y showing tht the unitry trnsformtion introdued ove leds to M xy hving ll relvlued mtrix elements nd M xz nd M yz hving ll purely imginry mtrix elements. The onlusion, then, is tht the phses in the setors of x nd y omponents re oupled in phse nd the setor of z omponents re out of phse with the x nd y omponents. 1. x omponents As preliminry, in Tle XV we list the effet of the symmetry opertions on the sulttie index. When these symmetries re used, one finds tht the 11 sumtrix M xx whih ouples only the x omponents of spins ssumes the form A g h 0 * * d g A 0 h * * d h 0 A g * * d 0 h g A * * d * * * * B 0 0 * * * * 0 B 0 * 0 B 0 * * * * 0 * 0 B * * * * d * * C e f 0 d * * e C 0 f d * * f 0 C e d * * 0 f e C, 95 where Romn letters re rel quntities nd Greek ones omplex. In this mtrix, the lines re used to seprte different Wykoff orits. The numering of the rows nd olumns follows from Tle XIV. I give few exmples of how symmetry is used to get this form. Consider the term T 1, where T 1 = 1 1,5 S x q,1s x q,5. Using Tle XV, we trnsform this y m x into 96 whih sys tht the 1,5 mtrix element is equl to the 3,6 mtrix element. Note tht in writing down T 1, we did not need to worry out, sine this ftor omes iqured s unity. Likewise, if we trnsform y m y,weget T 1 = 1 15 S x q,4 S x q,6, 98 whih sys tht the 1,5 mtrix element is equl to the negtive of the 4,6 mtrix element. If we trnsform y m x m y,we get T 1 = 1 1,5 S x q,3s x q,6, 97 T 1 = 1 1,5 S x q, S x q,5,

19 LANDAU ANALYSIS OF THE SYMMETRY OF THE whih sys tht the 1,5 mtrix element is equl to the negtive of the,5 mtrix element. To illustrte the effet of I on T 1 we write T 1 = 1 1,5 S x q, S x q,7, 100 PHYSICAL REVIEW B 76, so tht the 1,5 element is the negtive of the 7, element. From the form of the mtrix in Eq. 95 or equivlently referring to Tle XXIII in Appendix B, we see tht we ring this mtrix into lok digonl form y introduing the wve funtions for S x q,, O1, = O, x,1 = x,1 = O x,1 3, = O x,1 4, = i i i i x,1 = x,1 = O5, O6, The supersripts nd n on O lel, respetively, the Crtesin omponent nd the olumn of the irrep ording to whih the wve funtion trnsforms. The susripts m nd lel, respetively, the index numer of the wve funtion nd the sulttie lel. Let O,n p e vetor with omponents O,n p,1, O,n p,..., O,n p,1. Then, O x,1 n M xx O x,1 m nm xx m is A + h g d g A h d + B + +, 10 B + + d + + C + f e + d e C f where the oeffiients re seprted into rel nd imginry prts s =+i, =+i =+i, nd =+i. There re no nonzero mtrix elements etween wve funtions whih trnsform ording to different olumns of the irrep. The prtners of these funtions n e found from O x, n = m y O x,1 n, so tht, using Tle XV nd inluding the ftor,weget O1, = O, x, = x, = O x, 3, = O x, 4, = i i i i x, = x, = O5, O6, 103 Within this suspe, the mtrix nm xx m is the sme s in Eq. 10 euse nm 1 y M xx m y m = nm xx m. 105 These funtions trnsform s expeted for twodimensionl irrep, nmely, m x O x,1 n O n x, = m y O x,1 n O n x, = O x,1 n O x,, n O x, n O x,1. n 106

20 A. B. HARRIS PHYSICAL REVIEW B 76, TABLE XV. Trnsformtion tle for sulttie indies with ssoited ftors for TMO5 under vrious opertions s defined y Eq. 0. For m x, one hs expiq R f R i =1 for ll ses nd for m x m y I the nlogous ftor is +1 in ll ses nd this opertor reltes S q, nd S q, *. NOTE: This tle does not inlude the ftor of O whih my e ssoited with n opertion. m x m y m x m y I m x m y I n i n f n f e i n f e i n f e i n f =q R f R i, s required y Eq. 19. =q i + f, s required y Eq.. We will refer to the trnsformed oordintes of Eqs. 101 nd 104 s symmetry dpted oordintes. The ft tht the model-speifi mtrix tht ouples them is rel mens tht the ritil eigenvetor is liner omintion of symmetry dpted oordintes with rel oeffiients.. y omponents The 11 mtrix M yy oupling y omponents of spin hs extly the sme form s tht given in Eq. 95, lthough the vlues of the onstnts re unrelted. This is euse here one hs y =1 in ple of x =1. Therefore, the ssoited wve funtions n e expressed just s in Eqs. 101 nd 104 exept tht ll the supersripts re hnged from x to y nd now lels S y q,. However, the trnsformtion of the y omponents rther thn the x omponents requires repling x y y whih indues sign hnges, so tht m x O y,1 n O n y, = O y,1 n y,, O n m y O y,1 n O n y, = O y, n y, O n We wnt to onstrut wve funtions in this setor whih trnsform just like the x omponents, so tht they n e ppropritely omined with the wve funtions for the x omponents. In view of Eq. 106, weset O y,1 n, = O x, n,, O y, n, = O x,1 n,. 108 y,1 So, the oeffiients for O n re given y Eq. 104 nd those for O y, n y Eq These wve funtions re onstruted to trnsform extly s those for the x omponents. 3. z omponents Similrly, we onsider the effet of the trnsformtions of the z omponents. In this se, we tke ount of the ftor z to get m x O z,1 n O n z, = m y O z,1 n O n z, = O z,1 n z,, O n O z, n O z,1. n 109 We now onstrut wve funtions in this setor whih trnsform just like the x omponents. In view of Eq. 106, weset O z,1 n, = O x, n,, O z, n, = O x,1 n,, 110 So, the oeffiients for O n z,1 re given y Eq. 104 nd those for O n z, re the negtives of those of Eq These wve funtions re onstruted to trnsform extly s those for the x omponents. 4. Totl wve funtion nd order prmeters Now, we nlyze the form of M of Eq. 94 for using inversioymmetry. To do this, it is onvenient to invoke invrine under the symmetry opertion m x m y I whose effet is given in Tle XV. We write m x m y IS q, = m x m y S q,r *, 111 where R= for 5,6,7,8, otherwise R=± within the remining setor of s nd nd lter denotes one of x, y, nd z. Thus,

21 LANDAU ANALYSIS OF THE SYMMETRY OF THE T S q, * M S q, = m x m y IS q, * M m x m y IS q, = C S q,rm S q,r *, 11 where C = m x m y m x m y. 113 From the lst line of Eq. 11, we dedue tht M R,R = C M, 114 or, sine M is Hermitin tht M = C M R 1,R 1 *. 115 Now, we onsider the mtries M in the symmetry dpted representtion where = M n,m O p n * M p O m = C O p n * M R 1,R 1 * p O m = C O p nr * M, * p O mr. 116 There re no mtrix elements onneting p nd p p nd the result is independent of p. One n verify from Eqs. 101 nd 104 tht p O n,r = O p n, *, 117 so tht M n,m = C O p n * M, O p m * = C M nm *. 118 We hve tht C xy = C xz = C yz =1, so tht ll the elements of M xy re rel nd ll the elements of M xz nd M yz re imginry. Thus, prt from n overll phse for the eigenfuntion of eh olumn, the phses of ll the Fourier oeffiients re fixed. Wht this mens is tht the ritil eigenvetor n e writte = p=1 6 p n=1 r nx O n x,p + r ny O n y,p + ir nz O n z,p, where the r s re ll rel vlued nd re normlized y 6 n=1 r n =1, nd p re ritrry omplex numers. Thus, we hve the result of Tle XVI. The order prmeters re 1 1 e i 1, e i. 11 Neither the reltive mgnitudes of 1 nd nor their phses re fixed y the qudrti terms within the Lndu expnsion. Note tht the struture prmeters of Tle XVI re determined y the mirosopi intertions whih determine the mtrix elements in the qudrti free energy. Sine TABLE XVI. Normlized spin funtions i.e., Fourier oeffiients within the unit ell of TMn O 5 for wve vetor 1,0,q. Here, z =r 3 +ir 4 /. All the r s re rel vriles. The wve funtion listed under 1 trnsforms ording to the first seond olumn of the irrep. The tul spitruture is liner omintion of the two olumns with ritrry omplex oeffiients. Spin 1 Sq,1 r 1x r x r 1y ir 1z r y ir z Sq, r x r 1x these re usully not well known, one hs reourse to symmetry nlysis. The diretion in 1 spe whih the system ssumes is determined y fourth- or higher-order terms in the Lndu expnsion. Sine not muh is known out these terms, this diretion is resonly treted s r y ir z r 1y ir 1z Sq,3 r 1x r x r 1y ir 1z r y ir z Sq,4 r x r 1x r y ir z r 1y ir 1z Sq,5 z x z x z y z y iz z iz z Sq,6 z x z x z y iz z Sq,7 z x * * z y * iz z Sq,8 z x * PHYSICAL REVIEW B 76, z y * iz z * z y iz z * z x * z y * iz z * z x * z y * iz z Sq,9 r 5x r 6x r 5y ir 5z r 6y ir 6z Sq,10 r 6x r 5x r 6y ir 6z r 5y ir 5z Sq,11 r 5x r 6x r 5y ir 5z r 6y ir 6z Sq,1 r 6x r 5x r 6y ir 6z r 5y ir 5z

22 A. B. HARRIS PHYSICAL REVIEW B 76, prmeter to e extrted from the experimentl dt. We use Tle XVI to write the most generl spin funtions onsistent with rystl symmetry. For instne, we write SR,1 = 1 1r 1x î + r 1y ĵ + ir 1z kˆe iq R r x î + r y ĵ + ir z kˆe iq R Using this nd similr equtions for the other sultties, we find tht SR,1 = 1 r 1x î + r 1y ĵosq R r 1z kˆ sinq R r x î + r y ĵ osq R + + r z kˆ sinq R +, SR, = 1 r x î + r y ĵosq R + 1 r z kˆ sinq R r 1x î + r 1y ĵ osq R + r 1z kˆ sinq R +, SR,3 = 1 r 1x î r 1y ĵosq R + 1 r 1z kˆ sinq R r x î + r y ĵ osq R + + r z kˆ sinq R +, SR,4 = 1 r x î r y ĵosq R r z kˆ sinq R r 1x î + r 1y ĵ osq R + r 1z kˆ sinq R +, SR,5 = 1 z x î z y ĵ z z kˆosq R z x î z y ĵ + z z kˆsinq R z x î + z y ĵ z z kˆosq R + + z x î + z y ĵ + z z kˆsinq R +, SR 6 = 1 z x î + z y ĵ + z z kˆosq R z x î + z y ĵi z z kˆsinq R z x î + z y ĵ z z kˆosq R + + z x î + z y ĵ + z z kˆsinq R +, SR,7 = 1 z x î z y ĵ + z z kˆosq R z x î + z y ĵ + z z kˆsinq R z x î + z y ĵ + z z kˆosq R + + z x î z y ĵ + z z kˆsinq R +, SR 8 = 1 z x î + z y ĵ z z kˆosq R z x î z y ĵ z z kˆsinq R z x î + z y ĵ + z z kˆosq R + + z x î z y ĵ + z z kˆsinq R +, SR,9 = 1 r 5x î + r 5y ĵosq R r 5z kˆ sinq R r 6x î + r 6y ĵosq R + + r 6z kˆ sinq R +, SR,10 = 1 r 6x î + r 6y ĵosq R + 1 r 6z kˆ sinq R r 5x î + r 5y ĵosq R + r 5z kˆ sinq R +, SR,11 = 1 r 5x î r 5y ĵosq R + 1 r 5z kˆ sinq R r 6x î + r 6y ĵosq R + + r 6z kˆ sinq R +, SR,1 = 1 r 6x î r 6y ĵosq R r 6z kˆ sinq R r 5x î + r 5y ĵosq R + r 5z kˆ sinq R In Tle XVI, the position of eh spin is R+ n, where the re listed in Tle XIV nd R is Brvis lttie vetor. The symmetry properties of the order prmeters re m x 1 = 1, m y 1 = 1, I 1 = * 1 *. 14 We now hek few representtive ses of the ove trnsformtion. If we pply m x to Sq,1, we do not hnge the signs of the x omponent ut do hnge the signs of the y nd z omponents. As result, we get Sq,3 exept tht y hs hnged sign, in greement with the first line of Eq. 14. If we pply m y to Sq,1, we do not hnge the sign of the y omponent ut do hnge the signs of the x nd z omponents. As result, we get Sq,4 exept tht now 1 is

23 LANDAU ANALYSIS OF THE SYMMETRY OF THE repled y nd is repled y 1, in greement with the seond line of Eq. 14. When inversion is pplied to Sq,1, we hnge the sign of R ut not the orienttion of the spins whih re pseudovetors. We then otin Sq, provided we reple 1 y * nd y 1 *, in greement with the lst line of Eq Comprison to group theory Here, I riefly ompre the ove lultion to the one using the stndrd formultion of representtion theory. The first step in the stndrd formultion is to find the irreps of the group of the wve vetor. The esiest wy to do this is to introdue doule group hving eight elements see Appendix B sine we need to tke ount of the opertor m y E. This is done in Appendix B. From this, one finds tht eh Wykoff orit nd eh spin omponent n e onsidered seprtely sine they do not trnsform into one nother under the opertions we onsider. Then, in every se the only irrep tht ppers is the two-dimensionl one for whih we set m x = 1 0 m y = 0 1, 0 1 m x m y = 1 0, Indeed, one n verify tht the funtions in the seond third olumn of Tle XVI omprise sis vetor for olumn one two of this two-dimensionl irrep. One might sk: Why hve we undertken the ugly detiled onsidertion of the mtrix for F? The point is tht withitndrd representtion theory, ll the vriles in Tle XVI would e independently ssigned ritrry phses. In ddition, the mplitudes for the T orits sultties 5 nd 6 nd sultties 7 nd 8 would hve independent mplitudes. To get the results tully shown in Tle XVI, one would hve to do the equivlent of nlyzing the effet of inversion invrine of the free energy. This tsk would e very tehnil exerise in the rne spets of group theory whih here we void y n exerise in lger, whih, though messy, is silly high shool mth. I lso wrn the reder tht nned progrms to perform the stndrd representtion nlysis nnot lwys e relied upon to e orret. It is worth noting tht pulished ppers deling with TMO5 hve not invoked inversion symmetry. For instne, in Ref. 55 one sees the sttement As in the inommensurte se, 3 eh of the mgneti toms in the unit ell is llowed to hve n independent SDW, i.e., its own mplitude nd phse, nd lter on in Ref. 56, ll phses were susequently fixed to e rtionl frtions of. Use of the present theory would eliminte most of the phses nd would relte the two distint Mn 4+ Wykoff orits just s hppened for TMO. Finlly, to see the effet of inversion on onrete level, I onsider the upper right nd lower left 44 sumtries of M xx, whih re denoted M ur nd M ll, respetively. If we do not use inversioymmetry this mounts to following the usul group theoretil formultion, these mtries ssume the form = M ur PHYSICAL REVIEW B 76, d d d d, M ll =* * * d* * * d * * * d * * *, 16 d * * * * where now ll these prmeters re omplex vlued. Previously, in Eq. 95 ll these prmeters were rel vlued. From these results, one ould gin introdue the wve funtions of Eq However, in this se, the mtrix elements ppering in the nlog of Eq. 10 would not e rel. In ft, Eq. 16 indites tht in Eq. 10 the quntities,,, nd d in the upper right setor of the mtrix would e omplex nd those in the lower left setor would e repled y their omplex onjugtes to ensure Hermitiity. Thus, invoking inversioymmetry does not hnge the symmetry dpted oordintes of Eq Rther, it fixes the phses so tht the result n e expressed in terms of rel-vlued prmeters, s we hve done in Tle XVI. 6. Comprison to YMn O 5 YMn O 5 YMO5 is isostruturl to TMO5, so its mgneti struture is relevnt to the present disussion. I will onsider the highest-temperture mgnetilly ordered phse, whih ppers etween out 0 nd 45 K. In this ompound, Y is nonmgneti nd in the higher-temperture ordered phse q z =1/4, so the system is ommensurte. However, sine the vlue of q z is not speil, the symmetry of this stte is essentilly the sme s tht of TMO5. Throughout this setion, the struturl informtion is tken from Fig. of Ref. 58. The uppermost pnel is misleled nd is oviously the one we wnt for the highest-temperture ordered phse. In Fig. 7, we see tht the spin wve funtion is n eigenvetor of m x with eigenvlue 1. So, this struture must e tht of the seond olumn of the irrep. In ordne with this identifition, one sees tht the initil wve funtion is orthogonl to the wve funtion trnsformed y m y sine this trnsformtion will produe wve funtiosoited with the first olumn. Referring to Eq. 13, one sees tht to desrie the pttern of Mn 3+ spins, one hooses 1 =0, r x = r 1x 0.95, r 1y = r y The point we mke here is tht 1 =0. Although the vlues of these order prmeters were not given in Ref. 58, it seems ler tht in the lower-temperture phse the order prmeters re omprle in mgnitude. 59 D. CuFeO The mgneti phse digrm of CuFeO hs een investigted ontinully over the lst dede or so. Erly

24 A. B. HARRIS PHYSICAL REVIEW B 76, y x TABLE XVII. Generl positions for R3 m, with respet to rhomohedrl xes n, where 1 = /î 3/6ĵ+kˆ, =/î 3/6ĵ+kˆ, nd 3 = 3/3ĵ+kˆ, where is the distne etween neighor plnes of Fe ions nd is the seprtion etween nerest neighors in the plne. Here, 3 denotes threefold rottion nd m n lels the three mirror plnes whih ontin the threefold xis nd n. m x m y Er=x,y,z 3r=z,x,y 3 r=y,z,x m 3 r=y,x,z m r=z,y,x m 1 r=x,z,y Ir=x,ȳ,z I3r=z,x,ȳ I3 r=ȳ,z,x Im 3 r=ȳ,x,z Im r=z,ȳ,x Im 1 r=x,z,ȳ FIG. 7. Color online Top: The spitruture of the Mn 3+ ions in YMn O 5 limited to one - plne, tken from Fig. of Ref. 58. The sultties re leled in our onvention. Bottom left: The spin struture fter trnsformtion y m x. Bottom right: The spitruture fter trnsformtion y m y. studies 60,61 showed rih phse digrm nd these omined with mgnetoeletri dt 10 led to the phse digrm for mgneti fields up to out 15 T given in Ref. 10 whih is reprodued in Fig. 8. Aove T N 10 K, the rystl struture is tht of spe group of R3 m Ref. 6 No. 166 in Ref. 33. Below tht temperture, there is pprently very smll lttie distortion whih gives rise to lower symmetry rystl struture. 63,64 φ FIG. 8. Color online Temperture T versus mgneti field B phse digrm of CuFeO with B pplied long the xis from Kimur et l. Ref. 10. The upper inset shows the rystl struture of CuFeO nd the lower insets show the mgneti struture of the ommensurte sttes, where white nd lk irles orrespond to the positive nd negtive diretions. Note in the lower left inset tht the hexgonl 110 diretion long whih q is oriented is nerest neighor diretion. However, sine this distortion my not e essentil to explining the pperne of ferroeletriity, 65 we will ignore the presene of this lttie distortion. The generl positions of ions withipe group R3 m re given in Tle XVII. Our nlysis is sed on the following logi refer to the phse digrm of Fig. 8. We ssume tht s the temperture is lowered in mgneti field of out 10 T, the ontinuous trnsition from the prmgneti phse to the olliner inommensurte CIC phse introdues single irrep whih we will identify y our simple method. Then, further lowering of the temperture will introdue seond irrep, tking us into the nonolliner inommensurte NIC phse whose symmetry nd ferroeletriity we wish to disuss. Both these phses re hrterized y n inommensurte wve vetor long hexgonl 110 diretion, whih is the diretion to nerest neighor in the tringulr lttie plne, s shown in Fig. 8. As mentioned, lthough, in priniple, the lttie distortion does rek the threefold symmetry, we will ssume tht the three sttes whih re relted y the threefold rottion hve only slightly different energies in the distorted struture nd our rguments hve to e understood in tht sense. We ssume the R3 m spe group nd re interested in strutures ssoited with wve vetor in the str of q 1 q,q,0referred to hexgonl xes. These wve vetors re prllel to nerest neighor vetors of the tringulr plne of Fe ions. Consider the wve vetor q 1 qî. The only opertion other thn the identity tht onserves wve vetor is x, twofold rottion out the xis of the wve vetor x =Im 3. Clerly, the Fourier omponent m x q oeys x m x q 1 = x m x q 1, 18 with x =1, nd we ll this irrep 1. For irrep,we hve x m y q 1 = x m y q 1, x m z q 1 = x m z q 1, 19 ut with x = 1. So fr, the phses of the omplex Fourier oeffiients re not fixed. To do tht, we onsider the effet of inversion, whih leds to

25 LANDAU ANALYSIS OF THE SYMMETRY OF THE Im q 1 = m q 1 *. 130 To fix the phses in irrep, we note tht its qudrti free energy n e expressed s F = Am y q 1 + Bm z q 1 + Cm y q 1 * m z q 1 + C * m z q 1 * m y q 1, 131 where A nd B re rel nd C is omplex. Using the ft tht F must e invrint under I, we write F = Am y q 1 + Bm z q 1 + Cm y q 1 m z q 1 * + C * m z q 1 m y q 1 *. 13 Compring this with Eq. 131, we onlude tht C hs to e rel. Sine the m s n e omplex, this mens tht the two omponents of the eigenvetor of the qudrti form i.e., m y q 1 nd m z q 1 hve to hve the sme omplex phse. We now introdue order prmeters whih desrie the mgnitude nd phse of these two symmetry lels irreps whih mke up the wve funtion. When oth irreps re present, one hs nd m x q 1 = 1 q m y q 1 = q 1 r, m z q 1 = q 1 s, 134 where r +s =1 nd n ±q k = n e i n. We hve the trnsformtion properties x 1 q 1 = 1 q 1, x q 1 = q 1, I 1 q 1 = 1 q 1 *, I q 1 = q 1 *. 135 Note tht the phses n re fixed y the fourth-order terms in the free energy to e the sme for ll memers of the str of the wve vetor. Thus, when oth irreps of q 1 re present, we hve redefining the order prmeters to remove ftor of m x r = 1 q 1 osqx + 1, m y r = q 1 r osqx +, m z r = q 1 s osqx +, 136 where q=q 1. We pply these results s follows. As one lowers the temperture from the prmgneti phse, we ssume tht we first enter the CIC whih hs the spins predominntly long the z xis. Therefore, in this phse we ssume tht only irrep is tive. Notie tht in this phse, the spins will not lie extly long the z xis. Indeed, reent work 66 indites tht this phse is one in whih the mplitudes re sinusoidlly modulted nd the spins re oriented in the y-z plne s desried y irrep with m y /m z i.e., r/s etween 0 nd out 0.. Lowering the temperture still further leds to the NIC phse in whih oth irreps nd 1 re tive. The literture seems to e rther unerti to the tul struture of this phse. However, one possiility, seemingly not mentioned up to now, is tht pplition of mgneti field to the olliner-ommensurte 1/4 stte ould essentilly give rise to spin-flop trnsitioo tht the spins, insted of eing ligned long the hexgonl xis, would rotte to eing nerly perpendiulr to the xis. This oservtion would suggest tht if we ignore the lttie distortion, we would expet to hve n inommensurte stte with the spins elliptilly polrized in plne nerly ut not extly perpendiulr to the hexgonl xis. Suh stte is onsistent with Eq. 136 providing 1 =/. It does hve to e dmitted tht the spin-flop field of out 7 T is rther lrge for n L=0 iouh s Fe 3+ whose nisotropy ould e expeted to e smll. So fr, we hve onsidered only two of the vetors, q 1 nd q 1, of the str of the wve vetor. However, the Lndu expnsiohould tret ll wve vetors in the str symmetrilly, sine t qudrti order the system n eqully well ondense into ny of the wve vetors of the str. So, we write the qudrti free energy F s 3 F = 1 H,T 1 q n + H,T q n. 137 n=1 PHYSICAL REVIEW B 76, When the temperture is lowered t mgneti field of out 10 T long the z xis, the oeffiient H,T first psses through zero nd only one of the order prmeters q n eomes nonzero. At lower temperture, 1 H,T psses through zero nd one enters phse in whih oth 1 q n nd q n eome nonzero. Within the Lndu theory, it is possile to relize phse in whih two or three nonolliner wve vetors simultneously eome unstle. However, sine suh doule q or triple q sttes re not relized for CFO, we will not nlyze this possiility further thn to sy tht the fourth-order terms must e suh s to stilize sttes hving single wve vetor. The ferroeletri phse of interest is one in whih 1 q n nd q n re nonzero for single vlue of n. The vlue of n represents rokeymmetry. For future referene, we note tht t zero pplied eletri nd mgneti fields, the free energy must e invrint under tking either 1 or into its negtive. Finlly, we reord how order prmeters orresponding to different wve vetors of the str re relted y the threefold rottion 3: 3 n q 1 = n q, 3 n q 1 = n q However, the spin distriutions orresponding to these order prmeters of the other wve vetors re the rotted version of the spitruture, so tht if we onsider the ordering wve vetor q, we hve m x r = 1 q /os qx/ qy 3/ q r/os qx qy 3/ +, m y r = qr/ qx/ qy 3/ q / qx/ qy 3/ + 1, m z r = q s os qx/ qy 3/

26 A. B. HARRIS PHYSICAL REVIEW B 76, TABLE XVIII. Genertors G n of rottionl symmetry for the symmorphi spe groups of RFMO. Here, R is rottion through /3 out the positive xis nd x is twofold rottion out the xis, s in Fig. 9. Spe group G 1 G G 3 P3 m1 R I x P3 R I Mgneti Field (T) ICAF CAF IC TRI P Temperture (K) 4 P To summrize, representtion theory usefully restrits the possile spitrutures one n otin vi one or more ontinuous phse trnsitions. Reognition of this ft might hve sved lot of experimentl effort in determining the spin strutures of CuFeO. E. RFe MoO 4 In this setion, we elorte on riefer presenttion of the symmetry nlysis given previously 8 for RFeMoO 4 RFMO. This symmetry nlysis is onsistent with the mirosopi model of intertion proposed y Gsprovi. 67 RFMO onsists of two-dimensionl tringulr lttie lyers of Fe spin 5/ ions perpendiulr to the rystl xis suh tht djent lyers re stked diretly over one nother. These lyers of mgneti ions re seprted y oxygen tetrhedr whih surround Mo ion. At room temperture the rystl struture is P3 m1 No. 164 in Ref. 33, ut t 180 K smll lttie distortion leds to the lower symmetry P3 No. 147 in Ref. 33 struture, 67 whose generl lttie positions re speified in Tle XVIII, nd the struture is shown in Fig. 9. The low-temperture struture differs from tht FIG. 10. A shemti phse digrm of RFMO for mgneti fields of up to out 10 T long the xis, sed on Refs. 8 nd Here, P is the prmgneti phse nd IC-TRI is n inommensurte phse desried in the text in whih eh plne onsists of the so-lled 10 tringulr lttie struture. CAF is ommensurte ntiferromgnet phse nd ICAF n inommensurte ntiferromgneti phse, neither of whih is disussed in the present pper. We omit referene to sutle phse distintions disussed in Refs. 68 nd 69. ove T=180 K y not hving the twofold rottion out the rystl xis. As we will explin, this loss of symmetry hs importnt onsequenes for the mgneti struture. 67 We now disuss the mgneti struture of RFMO. A shemti mgneti phse digrm for mgneti fields of up to out 10 T long the xis is shown in Fig. 10. The mgneti nisotropy is suh tht ll the spins lie in the sl plne perpendiulr to the xis. The dominnt intertions responsile for long-rnge mgneti order re ntiferromgneti intertions etween nerest neighors in givel plne whih give rise to the so-lled 10 struture, shown in Fig. 11 in whih the ngle etween ll nerest neighoring spins in sl plne is ,69 Here, we will e minly interested in the properties of the phse whih ours for mgneti fields of less thn out 3 T. Neutron diffrtion 8,67 onfirms tht in this phse, eh tringulr lyer orders into phse in whih the ngle etween the diretion of djent spins is 10. Neutron diffrtion 8,67 lso indited tht from one tringulr lyer to π /3 0 π/ π/ 3 π /3 FIG. 9. Color online The unit ell of RFMO in the P3 phse. The lrge lls pink online represent the mgneti Fe ions, the smll lls lue online represent oxygen ions, nd eh tetrhedron green online ontins Mo ion. For lrity, the R ion whih sits etween the two tetrhedr is not shown. The in-plne ntiferromgneti intertion J is dominnt. In the high-temperture P3 m1 phse, J 3 =J 4, ut in the presene of the lttie distortion to the P3 phse, J 3 J 4 Ref. 67. The xis is prllel to the ond leled J. π / π/ 3 FIG. 11. Color online The 10 phse of tringulr lttie. The orienttions of the spins re given y the phse r, defined in Eq. 157 elow, for q z z+=0. The dshed lines indite the twodimensionl unit ell. The plus nd minus signs indite whether the oxygen tetrhedron losest to the enter of the tringle is ove plus or elow minus the plne of the pper