Unified spin-wave theory for quantum spin systems with single-ion anisotropies

Transcription

1 J. Phys. A: Math. Gen. 3 (999) Prnted n the UK PII: S35-447(99)67-8 Unfed spn-wave theory for quantum spn systems wth snge-on ansotropes Le Zhou and Yoshyuk Kawazoe Insttute for Materas Research, Tohoku Unversty, Senda , Japan Receved 4 February 999, n fna form 9 Juy 999 Abstract. By ntroducng a new spn-bose transformaton whch ncorporates the snge-ste spn-states mxng effect sef-consstenty, we estabsh a unfed spn-wave approach, appcabe to an arbtrary spn-number case and genera spn confguratons, for quantum spn systems wth snge-on ansotropes. The conventona Hosten Prmakoff (HP) method competey fas for such systems wth easy-pane ansotropy and sometmes fas for those wth easy-axs ansotropy, whe these dffcutes have been overcome successfuy by the new method. In some mtng cases, the present method recovers the resuts of od theores whch are vad n such cases. Appcatons to magnetc mutayers show that the new method s usefu to remove the unphysca nstabty predcted by the conventona HP method.. Introducton The snge-on term D(S z ) s the most wdey adopted form n quantum spn modes to descrbe the ansotropes n magnetc systems [ 7]. When D <, the z axs s a magnetc easy axs, otherwse the xy pane turns to become an easy pane. The conventona treatment for such a system s straghtforward one frst ntroduces a oca coordnate (LC) system, then one determnes the spn confguraton va the varaton method, and fnay one obtans spn-wave spectra wth the hep of the Hosten Prmakoff (HP) transformaton [8]. Such an approach, denoted by the conventona HP method, has been apped to varous magnetc systems wth snge-on ansotropes, for exampe, randomy ansotropc magnets [3], magnetc mutayers [4 7], etc. However, the conventona HP method s not aways successfu. Frst, t competey fas for a magnetc system wth any easy-pane ansotropy. When usng ths approach to study even the smpest easy-pane mode, one may fnd that: whenever the magnetzed drecton s dfferent from the ansotropc axs, the exctaton energes of some modes turn out to be magnary. Second, even for the easy-axs mode, the conventona HP method s not aways vad. In some cases, for exampe when an externa fed forces the spn to rotate from ts easy axs to ts hard axs, the same probem arses. Generay, these probems are nduced by negectng a very mportant quantum effect the snge-ste spn-states mxng (SSM) effect. After the LC transformaton, off-dagona nteractons ((S + ) + (S ) ) may appear n the Hamtonan. Such terms have a tendency to mx the snge-ste spn states n wth n± etc to form the proper egenstates. However, such an effect was competey negected by the conventona HP method []. Severa methods have aready been deveoped to sove these probems, such as the matchng of the matrx eements (MME) method [9 ], the characterstc ange (CA) method [ 4], and a numerca A revew of the conventona method s decrbed n [] /99/ $ IOP Pubshng Ltd 6687

2 6688 L Zhou and Y Kawazoe method [5]. Unfortunatey, a of them have mtatons. The MME method s a perturbatve one, so that t can ony be apped to sma ansotropy cases; furthermore, t cannot combne wth the LC transformaton to dea wth genera spn confguratons [9 ]. Athough the CA method has been combned successfuy wth the LC transformaton, t can ony be apped to spn- systems [ 4]. The numerca method s mted n spn- systems [5]. Thus, to our knowedge, a satsfactory theory for quantum spn systems wth snge-on ansotropy remans a great chaenge to theoretca researchers. In the present paper, we propose a unfed spn-wave approach, whch s appcabe to arbtrary spn-number systems and genera spn confguratons, for quantum spn systems wth snge-on ansotropes. The key pont of ths method s a new set of spn-bose transformatons whch s dfferent from the HP one and ncorporates the SSM effect automatcay. Ths paper s organzed as foows: frst we outne the theoretca formasm based on a homogeneous easypane mode n the next secton; we then compare the present method wth exstng theores n secton 3. Secton 4 s devoted to the appcatons to magnetc mutayers. Fnay, the concusons are summarzed n the ast secton.. Theoretca formasm.. The Hamtonan Let us umnate the basc deas of the new method based on the smpest case a buk easypane ferromagnet n an externa fed. The Hamtonan s gven by H = J S S j + D (j) (S z ) h S z () where the exchange nteractons are wthn nearest neghbours, D s the ansotropy constant, and h s the externa fed. Consderng the competton between the ansotropy and the externa fed, t s hepfu to ntroduce the LC transformaton []: S z = cos θs z sn θs x S y = S y S x = cos θs x + sn θs z () to optmze the magnetzaton drecton. After the transformaton, the Hamtonan becomes H = J S S j + D cos θ (j) (S z ) + D sn θ (S x ) h cos θ S z D sn θ cos θ (S z Sx + S x S z ) + h sn θ S x. (3) It s cear that except θ =, there s a snge-ste off-dagona term (S x ) n the Hamtonan (3) whch contrbutes the SSM effect. In the standard spn-wave theory, one usuay appes a spn-bose transformaton to the Hamtonan and then tres to sove the Boson system. If the resutng Boson Hamtonan can be exacty soved, t does not matter what knd of spn-bose transformaton has been apped. However, an exact souton s unfortunatey hard to obtan, so that usuay some approxmatons (for exampe, the harmonc approxmaton) are necessary. Such approxmatons assume that hgh-order Boson terms are reatvey unmportant to determne the ground state and the owyng spn waves, so that ony ow-order terms are retaned. However, ths assumpton s correct ony f the (oca) Boson representaton has been chosen n such a way that the offdagona effects n the ow-order terms are as sma as possbe. Otherwse, renormazatons of hgh-order nteractons w generate non-neggbe effects to the fna resuts. Thus, t s of much mportance to seect an approprate (oca) Boson representaton where the off-dagona nteractons are as sma as possbe and a further spn-wave anayss can be performed. Usuay,

3 Unfed spn-wave theory for quantum spn systems 6689 the (oca) Boson representaton shoud be chosen to approach the exact egenstates as cose as possbe. In the conventona HP method, the (oca) Boson representaton has been chosen as the snge-ste egenstates of S z. However, such a choce, athough approprate for sotropc spn systems, s not for the present ansotropc systems because a proper snge-ste egenstate woud be a mxture of n wth n±...rather than a nave n. In order to take account of such an effect, et us frst seect an approprate snge-ste Hamtonan to ncude as many as possbe messages from the tota Hamtonan (3), and then determne a (oca) Boson representaton based on dagonazng such a Hamtonan. Frst consder the exchange term: S S j = S z Sz j + Sz Sz j + Sz Sz j +(S z S z )(Sz j Sz j )+ (S+ S j + S S j + ). (4) Snce the second ne of the above expresson must contrbute hgh-order correaton terms rather than snge-ste ones, the contrbutons to the snge-ste Hamtonan can be wrtten as SS z where S = S z shoud be determned ater sef-consstenty. Thus, a the snge-ste terms from equaton (3) can be coected as foows: H s = JZSS z + D cos θ(s z ) +Dsn θ(s x ) hcos θs z (5) where Z s the number of nearest neghbours of a gven ste n a attce. In prncpe, the snge-ste Hamtonan shoud depend on the ste. In the present homogeneous system, however, t s not necessary to consder the ste dependence because every ste s equvaent. In nhomogeneous systems such as the magnetc mutayers whch w be studed n secton 4, we have to consder the ayer dependence snce spns n dfferent ayers are no onger equvaent. Terms n the second ne of equaton (3) are not ncuded n H s. Later we w show that an approprate vaue of θ can be chosen to emnate ther nfuences to the tota Hamtonan. It shoud be noted that the snge-ste Hamtonan (5) ony serves to determne the (oca) Boson representaton and a set of spn-bose transformatons to hep us perform further anayss. After we have found the spn-bose transformaton, we w come back to the tota Hamtonan (3) to consder a the terms... The spn-bose transformaton We w formuate a new set of spn-bose transformatons, whch s formay exact n ths subsecton. Snce the transformaton s ndependent on the ste, we omt the ste ndex for smpcty. Snge-ste Hamtonan H s can be easy dagonazed by the foowng orthogona transformaton: ñ = P mn (S,θ) m (6) m yedng H s ñ =E n ñ (7) where { m,m =,,...,S} represent the egenstates of S z defned by S z m =(S m) m,m=,,...,s}, and { ñ,n=,,...,s}are the egenstates of H s. The sequence of the egenstates { ñ } has been arranged by E n+ >E n so that s the ground state n a snge ste. The transformaton matrx P mn (S,θ), obtaned by dagonazng the Hamtonan matrx n H s m, s certany dependent on S and θ. The matrx forms of the spn operators n the new representaton can be evauated by ñ T n (S,θ) = P mn (S,θ)P m n (S,θ) m T m (8) mm

4 669 L Zhou and Y Kawazoe where T stands for the operators S +,S z, etc, and the matrx eements m S + m, m S z m are very easy to cacuate. To study the ow-yng spn-wave exctatons, we transform the spn operators nto Bose expansons n a,awhch are defned n the dagonazed snge-ste representaton (rather than the orgna representaton) as a ñ B = n+ ñ+ B a ñ B = n ñ B (9) where ñ B are the Bose states reated to the spn states as: ñ B = ñ (for n S). Snce the Boson space s sem-nfnte but the spn space s fnte, ony these states { ñ,n S} have physca defntons, whch w construct the physca Boson space. The remander states { ñ,n>s}then construct the unphysca Boson space. It s the core of the present approach to seect the (oca) Boson representaton as the dagonazed representaton of the approxmate snge-ste Hamtonan H s as shown by equaton (9). The HP transformaton [8], on the other hand, defnes the Boson representaton as the nave egenstates of S z. The Bose operators n the HP transformaton are defned by a n B = n + n+ B a n B = n n B. () Compared wth the HP method, the mert of our choce s very cear for exampe, athough the zero-boson state n the present defnton s st not the exact ground state, t shoud be coser to the exact ground state than the zero-boson state defned n the HP method. The same arguments aso hod for ow-yng exctng states. In fact, ater we w show that the mprovement over the conventona spn-wave theory essentay comes from the choce of the oca bass equaton (9) rather than (), snce the off-dagona terms n the tota Hamtonan based on such a (oca) Boson representaton are ndeed much smaer than those based on equaton (). However, the prce to pay s that our spn-bose transformaton w be much more compcated than the HP transformaton. Now et us determne the spn-bose transformaton. Due to the fact that an egenstate ñ of Hamtonan H s of form (5) must be a near combnaton of n, n±, n±4,...,the nonzero matrx eements are then: ñ S + n+p+, ñ S z ñ+p. Generay, the spn-bose transformaton can be wrtten as { p+ S S p } S + = [A (p) a a +p+ + B (p) a +p+ a ] p= = { S p S S p } () S z = C () a a + C (p) (a a +p + a +p a ) = p= where s the step operator to project out the unphysca states, and s defned by [6] = G a a () = n whch the coeffcents are ( ) ( )( )...( S) G = (S)!! <. Accordng to [6], the step operator has the foowng propertes: ñ = ñ n S ñ = n>s. Thus, for an arbtrary operator ˆT,wehave ñ ˆT m = ñ ˆT m n, m S ñ ˆT m = otherwse (3) (4) (5)

5 Unfed spn-wave theory for quantum spn systems 669 whch means that the step operator can automatcay emnate the contrbutons of a gven operator n the unphysca Boson space. Thus, we ony need to consder the matrx contrbutons n the physca Boson space. A the coeffcents {A (p),b (p),c (p) } defned n equaton () can be determned unquey as an exampe, by by the matrx eements of S + and S z obtaned by equaton (8). Takng A () equatng the matrx eements between ñ and ñ + (both states are n physca Boson space) cacuated by the two sdes of equaton (), we fnd that: ñ S + ñ+ = ñ = ñ p+ S p= S = ñ = = = { p+ S p= S p = S p = [A (p) A () a a + ñ + [A (p) a a +p+ + B (p) } a +p+ a ] a a +p+ + B (p) a +p+ a ] ñ + ñ + n (n +)!n! A () (6) (n )! n whch equatons (9), (4) have been used. Foowng ths method, a the coeffcents {A (p) by the foowng couped equatons:,b (p),c (p) } can be unquey determned n (n +p+)!n! (n )! = n (n +p+)!n! (n )! n (n +p)!n! = = (n )! A (p) = ñ S + n+p+ n+p+ p+ S B (p) = n+p+ S + ñ n+p+ p+ S C (p) = ñ S z ñ+p n+p p S. (7) Equaton (7) can be soved step by step. For exampe, n the case of p =,n =, we have A () = S +, B () = S +, and C () = S z. Some owest-order coeffcents are sted n tabe. The arger S s, the arger the number of coeffcents. In a defnte-spn case, the number of coeffcents s aways fnte. For exampe, ony A (),A(),B(),B(),C(),C(),C(),C() are needed n the case of S =. The vadty of our spn-bose transformaton equaton () s ensured by the fact that the Bose expanson has exacty the same matrx eements as the orgna spn operator n the physca Boson space, and has zero matrx eements n the unphysca Boson space. The Bose expansons of the spn operators certany satsfy a the commutaton rues of the anguar momentum [S +,S j ]=Sz δ j [S z,s± j ]=±δ j S ± (8) because the transformaton P nm s orthogona. Insertng () nto (), we can further expand the transformatons to nfnte Bose seres

6 669 L Zhou and Y Kawazoe p A (p) Tabe. Coeffcents of the new Bose expanson up to the fourth-order terms. {J (p) }} and {K (p) } can be obtaned through repacng S + by S z S x n the expressons of {A (p) } and {B (p) }. {E (p) } and {F (p) } can be obtaned through repacng S z by (S z ) and (S x ) n the expressons of {C (p) }, respectvey. The ste ndex has been omtted. B (p) S + S + S z S + S + S + S z S z S + S + 3 S S S + S S + 3 S + 3 S S + 4 S + 3 S S + C (p) S z S z + S z S z S z 3 6 S + n norma order: S + = S z = p+ S p= = = [A (p) C () a a + a a +p+ + B (p) a +p+ a ] p S p= = C (p) (a a +p + a +p a ). It has been proved n [6] that the projecton operator has no nfuence on the coeffcents of those Boson terms n physca space whch have been determned n equaton (7). To avod the unnecessary ntroducton of too many symbos, we have used the same ones ({A (p),b (p),c (p) } etc) n the above expansons to denote the whoe set of coeffcents for the nfnte seres ncudng those for unphysca Boson terms whch are not determned by equaton (7). Let us now take the S = case as an exampe. Equatons () can be rewrtten as S + = [A () a + A() a a + B () a + B () a a] = [A () a + A() a a ] + [B () a + B () a a] S z = [C () + C () a a + C () a a + C () a + C () a ] = [C () + C () a a + C () a a ] + C () a + C () a. Every term n equaton () can be expanded to nfnte Bose seres arranged n norma order. For nstance, A () a = G A () aa a = G A () [a a + + a a ] (9) () = [G + ( +)G + ]A () a a +. () After a carefu cacuaton, the fna form of the coeffcents are found to be A () = [G + ( +)G + ]A () +[G +G + ( +)G + ]A () B () = [G + ( +)G + ]B () +[G +G + ( +)G + ]B () C () = G C () +[G + G ]C () +[G +( )G + ( )G ]C () C () = [G +( +)G + + ( +)( +)G + ]C (). As we have stated, the step operator does not change the coeffcents of those physca Boson terms one may check ths pont by examnng the cases of =,. It can be seen from equaton () that the hgh-order terms are derved unquey from the ow-order terms whch ()

7 Unfed spn-wave theory for quantum spn systems 6693 are determned by equaton (7). Actuay, the hgh-order terms have no matrx contrbutons n the physca Boson space. They ony serve to cance the matrx contrbuton of the ow-order terms n the unphysca Boson space. Smary, S z S x, (S z ) and (S x ) can be expanded nto Bose seres: S z S x = (S z ) = (S x ) = p+ S p= = = = [J (p) E () a a + F () a a + a a +p+ + K (p) a +p+ a ] p S p= = p S p= =,K (p),e (p),f (p) E (p) (a a +p + a +p a ) F (p) (a a +p + a +p a ). Bascay, the coeffcents {J (p) } can be determned by the coeffcents {A (p),b (p),c (p) } unquey. However, t s more effcent to derve them drecty from the correspondng matrx eements cacuated from equaton (8), foowng the method descrbed above. Actuay, n equaton (7) and tabe, one may just substtute S + by S z S x to determne J (p) and K (p) and substtute S z by (S z ) and (S x ) to determne E (p) and F (p), respectvey. Thus, we have formuated a new spn-bose transformaton whch s formay exact and have consdered the SSM effect expcty, rather than the conventona HP spn-bose transformaton [8] and others. Athough the expressons seem rather compcated, n practca cases, one rarey appes the nfnte seres but rather takes ony the owest severa order terms (say, up to harmonc terms) to catch the man physcs. In the remandng parts of ths paper, we w present some appcatons of our method under the harmonc approxmaton, and show that the method s appcabe to some probems where the conventona methods fa under the harmonc approxmaton..3. The spn-wave exctatons under the harmonc approxmaton Snce we have obtaned a set of spn-bose transformaton based on the Boson representaton defned by the snge-ste Hamtonan, now we go back to the tota Hamtonan (3) to consder a the other terms. Appyng our new Bose transformaton (9) (3) to Hamtonan (3) and then appyng a Fourer transformaton, under the harmonc approxmaton, we fnd that H = H (S,θ) + H (S,θ) + P(k)a k a k + Q(k)(a k a k + a k a k ) + (4) k k where H = N( JZC () +Dcos θe () +Dsn θf () hcos θc () ) (5) H = [ D sn θ cos θ(k () +J () )+ hsn θ(a() +B () )](a k + a k) (6) k P(k) = JZC () C() JZ[(A () ) + (B () ) ]γ k + D cos θe () +Dsn θf () hcos θc () (7) Q(k) = JZA () B() γ k +[ JZC () C() +Dcos θe () +Dsn θf () hcos θc () ]. (8) (3) Here, γ k s defned as γ k = (/Z) δ e k δ (9) Effects of hgher-order terms n the nfnte seres have been consdered by some authors, see [4, 6].

8 6694 L Zhou and Y Kawazoe n whch the summaton runs over Z nearest neghbours. Accordng to the arguments eadng to (5), we understand the parameter S s used to take account of approprate contrbutons from the exchange term to the snge-ste representaton. In the zero-temperature case whch s consdered here, accordng to equaton (), t s thus approprate to use the equaton S = S z =C() (S,θ) (3) to determne the parameter S sef-consstenty, where S z s the oca Boson ground state expectaton of S z. On the other hand, θ s chosen to cance the non-harmonc part of nteracton: H (θ, S), that s D sn θ cos θ[j () (S,θ) + K () (S,θ)]+ hsn θ[a() (S,θ) + B() (S,θ)] =. (3) In fact, the above terms just come from the second ne of the tota Hamtonan (3). Athough such terms are not ncuded n the approxmate snge-ste Hamtonan (5) to determne the (oca) Boson representaton, ther nfuences upon the tota Hamtonan are ndeed emnated by choosng an approprate vaue of the cantng ange θ, at east under the harmonc approxmaton adopted here. It shoud be emphaszed that snce a of the coeffcents (.e. A (p), B (p), C (p), etc) are reated to the matrx eements (see equaton (7) and tabe ) whch are the functons of the two parameters S and θ accordng to equaton (8), a of the coeffcents presented n equatons (4) (8) (.e. A (),B() etc) are then the mpct functons of the two parameters S and θ. Thus, equatons (3), (3) are two nonneary couped equatons servng to determne these two parameters. Once equatons (3), (3) are soved, every parameter appearng n Hamtonan (4) has a fxed vaue. We can dagonaze the Hamtonan wth the hep of the Bogoyubov transformaton and obtan H = H + E(k)α k α k + (3) k where the ground state energy H and the spn-wave dsperson reaton E(k) are defned as H = H + [ P(k)+ ] P(k) 4Q(k) (33) k E(k) = P(k) 4Q(k). (34) Then a the physca propertes can be cacuated ready. Let us now dscuss the physca sgnfcance of our choce equaton (3). Appyng the spn-bose transformaton to the snge-ste Hamtonan (5), we fnd that H s = JZSC () + D cos θe () +Dsn θf () hcos θc () +[ JZSC () + D cos θe () +Dsn θf () hcos θc () ]a a +[ JZSC () + D cos θe () +Dsn θf () hcos θc () ](a + a ) +. (35) Snce a,a are defned n the dagonazed representaton of H s (see equaton (9)), the offdagona nteractons n the above equaton shoud be zero, whch means that JZSC () + D cos θe () +Dsn θf () hcos θc (). (36) Substtutng equaton (3) nto the above equaton yeds JZC () C() +Dcos θe () +Dsn θf () hcos θc () =. (37)

9 Unfed spn-wave theory for quantum spn systems 6695 When comparng equaton (37) wth the off-dagona term Q(k) defned n equaton (8), the physca meanng of our choce equaton (3) s very cear up to second order, a the sngeste off-dagona nteractons n the Hamtonan (coected n the bracket n equaton (8), of the type (a + a )) have been canceed. Ths s the drect consequence of the oca Boson representaton defned by (9). Instead, there appears another off-dagona term JZA () B() γ k n the expresson of Q(k). But the off-dagona effect of ths term s much ess serous than that of the snge-ste terms. Those snge-ste off-dagona terms are actuay responsbe for the faure of the conventona HP method. In fact, the parameter S s used to ndcate the strength of the SSM effect. The arger the devaton of S from S (.e. S = S S) s, the stronger the SSM effect w be. In the sotropc case,.e. D =, accordng to equatons (5), (3), t s easy to fnd that S = S because the oca Boson ground state expectaton vaue of S z s just the absoute vaue of S. In such a case, the SSM effect dsappears. Ths s why the conventona HP method works we for the sotropc spn systems. However, n the presence of an ansotropy, S w be non-zero caused by the off-dagona nteractons n H s. In such cases, the SSM effect s never trva. The conventona HP method, whch fas to consder such an effect, w encounter probems n some cases. The parameter θ st possesses ts cassca meanng the cantng ange of the spn vector. Accordng to equaton (3), the approxmate ground state s defned by α k G =. It s easy to check that G S x G = G S y G =. (38) Thus, we have chosen θ n such a way that the spn w pont aong the oca z axs, wthout expectaton vaues aong other drectons. The state descrbed by the parameters S and θ can then be understood as foows. The spns st pont aong the oca z axs defned by θ, however, the ground state s no onger that for the sotropc Hesenberg ferromagnet (.e. the state wth the argest S z expectaton vaue), but rather a mxture of the sotropc ground state and exctng states, 4... The mxng effect s dependent on the ansotropy and the cantng ange θ, and s descrbed by the devaton S. So far, we have outned the basc formasm of the new approach based on a homogeneous easy-pane mode n whch ony one set of parameters (S,θ) are necessary. The method s certany not mted n such a speca case. Actuay, for any mcromagnetc modes wth snge-on ansotropy, one just appes the LC transformaton, then seects an approprate snge-ste Hamtonan to determne the oca Boson representaton, and fnay performs the spn-wave anayss based on such a oca Boson representaton. It s a rather routne job to appy the same deas to study the easy-axs mode H = J S S j D (S x ) h S z (39) (j) and other more compcated systems. After makng detaed comparsons wth exstng theores n the next secton, we w study an nhomogeneous system the magnetc mutayers n secton Comparson wth exstng theores 3.. The HP method Frst, et us compare our method wth the conventona HP method whch s the most wdey adopted n the terature to dea wth such probems [ 7]. Instead of our spn-bose

10 6696 L Zhou and Y Kawazoe transformaton (), the conventona method appes the nave HP transformaton [8] S + = S a a a = Sa + =S a a S z to Hamtonan (3) to study the spn waves. Subsequenty, we w show that the conventona HP method has chosen a (oca) Boson representaton based on such a snge-ste Hamtonan n whch the off-dagona nteractons are negected. Actuay, f we dscard the off-dagona terms n the snge-ste Hamtonan (5), the transformaton matrx P mn becomes dagona: P mn = δ mn. (4) Accordng to equaton (8) and tabe, one may obtan A () = S,... A (p) = p B (p) = C () = S C () =,... C (p) = p J () = S S/,... J (p) = p K () = (S ) S/,... K (p) = p E () = S E () = S,... E (p) = p F () = S/ F () = S / F () = S(S )/4,... F (p) = p,. By puttng the coeffcents defned n equaton (4) nto equatons (), we fnd the resutng transformaton S + = Sa + S z =S a a (43) + s n fact the same as the conventona HP transformaton (4) under the harmonc approxmaton. Accordng to equaton (4), the souton of equaton (3) s just S = S and equaton (3) now becomes h cos θ = D(S ). (44) Substtutng equatons (44), (4) nto (7), (8), we fnd that P HP (k) = JSZ( γ k )+ D(S ) sn θ (45) Q HP (k) = 4 Dsn θ S(S ). (46) It s very easy to check that the magnon exctaton energes of the easy-pane mode, cacuated by E HP (k) = (P HP (k)) 4(Q HP (k)) (47) are aways magnary when k. One may understand that ths probem s caused by negectng competey the off-dagona nteractons when choosng the (oca) Boson representaton, so that the off-dagona nteractons [Q HP (k) n (46)] are so strong that a harmonc approxmaton fas. It shoud be noted that the conventona HP method (4) fas to cance the contrbutons of the Bose expansons n the unphysca Boson space. As the resut, hgh-order terms n HP transformaton may be dfferent from our transformaton. (4) (4)

11 Unfed spn-wave theory for quantum spn systems 6697 Fgure. Magnon dsperson reatons E(k) k x a (k y = k z = ) cacuated by the present method (sod curves) and the conventona HP method (dashed curves) for easy-pane mode wth ansotropy parameter D/JZ =. under zero externa fed (n ths case, θ = π/) n spn-,-3/,-,-5/ cases, respectvey. Fgure. Magnon exctaton gaps (h) as functons of the externa fed cacuated by the present method (sod curves) and the conventona HP method (dashed curves) for easy-axs modes wth ansotropy parameter D/JZ =. n spn-,-3/,- cases, respectvey. Some exampes are hepfu for comparson. Fgure presents the dsperson reatons cacuated by the present method (sod curve) and the conventona HP method (dashed curve) for the easy-pane mode wth a smpe cubc attce at zero externa fed. The magnary exctaton energes of those modes near the Ɣ pont ceary show the faure of the conventona HP method, whe such a probem has been overcome by the present method. The magnon exctaton gaps (h) of an easy-axs mode descrbed by Hamtonan (39) wth a smpe cubc attce cacuated by the two methods have been potted together n fgure wth respect to the externa fed. It s found that the conventona HP method s good for the easy-axs mode n many cases. However, n some cases,.e., n the vcnty h (S )D when the spn s forced just parae to the externa fed, the conventona HP method fas. We see that our method aways gves a postve gap to the magnon dsperson reaton. Fgure 3 presents the two parameters θ and S as the functons of the externa fed h for both the easy-pane and the easy-axs mode. It s understood that the vaue of S S descrbes the SSM effect the arger ths vaue, the more drastc the SSM effect. From fgure 3, we fnd that, for the easy-pane mode, the SSM effect s very sgnfcant when θ, and dsappears when θ = ; whe for the easy-axs mode, the SSM effect s most sgnfcant when θ just approaches zero. Comparng fgure 3 wth fgures and, t s not dffcut to understand that the faure of the conventona HP method s ndeed caused by negectng the SSM effect. 3.. The MME method Let us compare our new method wth the MME method n ths secton [9 ]. In the MME method, when h =, the spn-bose transformaton has been derved whch s correct to the owest order n d = D/JZ for the easy-pane mode [, ]. We w show that under the same condtons, our present method w gve the same resuts as the MME method. When h =, there s no competton between the ansotropy and the externa fed, so that the souton of equaton (3) must be θ = π/ n another words, the spns have to e n the easy pane (the xy pane). Then, the snge-ste Hamtonan (5) s wrtten as H s = JZ[ SS z + d(s x ) ]. (48)

12 6698 L Zhou and Y Kawazoe Fgure 3. θ and S as the functons of the externa fed h for an easy-pane spn-3/ mode (sod curves) and an easyaxs spn-3/ mode (dashed curves) where the ansotropy parameters are D/JZ =. n the two modes. Treatng d(s x ) as a perturbaton, we fnd the proper egenstates of Hamtonan (48) shoud be 4S(S ) = d +o(d ) 8S 6(S )(S ) = d 3 +o(d ) 8S 4S(S ) (S )(S 3) = d + d 4 +o(d ). 8S 8S (49) Accordng to tabe and the above equaton, we fnd that A () = S +o(d ) S(S B () = ) d +o(d ) S C () = S +o(d ) C () = +o(d ) S(S ) C () = d +o(d ). 4S (5) Correct to the frst order n d, the souton of equaton (3) s smpy S = S. Puttng S = S nto the above equaton, we fnd that the resutng transformaton s exacty the same as that n the MME method []. However, we have to emphass here that the MME spn-bose expanson [] s ony vad for the sma d case, and more mportanty, n the case of h =. If h, we fnd from equatons (3) and (3) that the two parameters θ and S cannot be determned ndependenty. These two parameters must nfuence each other so that the fna souton shoud be determned sef-consstenty rather than perturbatvey. Ceary, the MME method cannot be apped navey to study the quantum spn systems n genera spn confguratons, but the present approach has ncuded a the hgh-order contrbutons n d and can be apped to genera spn confguratons.

13 3.3. The CA method Unfed spn-wave theory for quantum spn systems 6699 Fnay, we compare out method wth the CA method whch s deveoped for spn- systems [ 4]. We w show that, n the case of S =, our present method s equvaent to the CA method. When S =, the egenstates of Hamtonan (5) must be =cos φ sn φ = (5) =sn φ + cos φ where φ needs to be determned. Accordng to the above equaton and tabe, we fnd that A () = cos φ B () = sn φ C () = cos φ C () = cos φ C () = sn φ cos φ E () = E () = E () = F () = () ( sn φ) F = () ( + sn φ) F = cos φ J () = (cos φ + sn φ) K () =. φ s seected to cance the off-dagona eements of the snge-ste Hamtonan (5) n the representaton,,. The ony dagona term s H s, so that we have the equaton H s = H s = JZS sn φ cos φ + D sn θ cos φ h cos θ sn φ cos φ = (53) to fx φ. Accordng to (5), equatons (3), (3) now become S = cos φ (54) D sn θ(cos φ + sn φ) + hsn θ(cos φ sn φ) =. (55) Thus, we have three equatons (53) (55) to fx three parameters S, θ and φ. Puttng (54) nto (53), we then have JZcos φ sn φ + D 4 sn θ cos φ h cos θ sn φ =. (56) When comparng equatons (56), (55) wth the crteron n the CA method for the same mode (equatons (9), () n [3]), one may fnd that they are n fact equvaent. The parameter φ s just another varaton parameter the characterstc ange, n the CA method [ 4]. Substtutng equaton (5) nto (4) (8), t s rather easy to check that the resutng expressons are exacty the same as those presented n the CA method [3]. Thus, we have shown that n the case of S =, the present method s equvaent to the CA method. Ths resut s not surprsng, because the basc deas of both methods are the same to try to fnd out the best oca Boson representatons. However, the CA method s ony appcabe to spn- systems whereas the present method can be apped to arbtrary spn-number cases. 4. Appcatons n magnetc mutayers In prncpe, the method descrbed n secton can be apped to any quantum spn system wth snge-on ansotropy. Here, we appy the method to study one knd of nhomogeneous (5)

14 67 L Zhou and Y Kawazoe system the magnetc mutayer system. Our resuts may revea that the remarkabe mert of the new method s that t can carfy some unphysca nstabtes whch are predcted by the conventona HP method. Generay, a magnetc mutayer can be we descrbed by the foowng mcromagnetc mode: H = I m,m S m (r) S m (r ) + m,m r,r m,r D m [S z m m (r)] h Sm z (r) (57) m,r where m, m abe the ayer and r, r are the attce stes wthn the pane. The ansotropc axes {ẑm } can be dfferent from ayer to ayer, and the ansotropy {D m} can be of easy-pane or easy-axs type. For smpcty, we assume that {ẑm } are n the xz pane and ẑ m ẑ=cos η m. As we have aready mentoned, n magnetc mutayers, the spns n dfferent ayers are no onger equvaent. We have to adopt dfferent parameters {θ m, S m } to descrbe spns n dfferent ayers. After a LC transformaton smar to () but wth anges θ m dfferent from ayer to ayer, we fnd the foowng snge-ste Hamtonan for each ayer: [ Hm s = I mm ZS m + I mm cos(θ m θ m )S m + h cos θ m ]S z m m (r) m +D m cos (θ m η m )[S z m m (r)] + D m sn (θ m η m )[S x m m (r)] m =,,... (58) The spn-bose transformaton for the mth ayer s determned based on ts own Hamtonan Hm s. Thus, we have p+ S m S m + (r) = S z m m (r) = p= = = [A (p),m a m (r)a+p+ m p S m C (),m a m (r)a m (r) + p= (r) + B (p) =,m a +p+ m C (p),m [a m (r)a+p m (r)a m (r)] (r) + a +p m (r)am (r)] (59) where the parameters {A (p),m,b(p),m,c(p),m } are determned foowng the same procedures descrbed n secton, wth snge-ste Hamtonan (5) n secton repaced by Hm s. By usng the transformaton (59), we expand the Hamtonan nto the foowng Bose seres under the harmonc approxmaton: H = H ({S m }, {θ m }) + H ({S m }, {θ m }) + mm (k)a m m,m k{p (k)a m (k) + Q mm (k)[a m (k)a m ( k) where H = m +a m (k)a m ( k)]} + (6) { I mm Z[C (),m ] + D m cos (θ m η m )E (),m + D m sn (θ m η m )F (),m h cos θ mc (),m m I mm cos(θ m θ m )C (),m C(),m } N (6) H = m [ m I mm sn(θ m θ m )C (),m (A(),m + B(),m ) D m sn(θ m η m ) cos(θ m η m ) (J (),m + K(),m ) + h sn θ m(a (),m + B(),m )][a m (k) + a m( k)] (6) and P mm (k) = I mm ZC (),m C(),m I mmz[(a (),m ) + (B (),m ) ]γ k + D m cos (θ m η m )E (),m

15 Unfed spn-wave theory for quantum spn systems 67 +D m sn (θ m η m )F (),m h cos θ mc (),m m I mm cos(θ m θ m )C (),m C(),m (63) Q mm (k) = I mm A (),m B(),m Zγ k + { I mm ZC (),m C(),m h cos θ mc (),m +D m cos (θ m η m )E (),m + D m sn (θ m η m )F (),m I mm cos(θ m θ m )C (),m C(),m } (64) m P mm (k) = I mm [ + cos(θ m θ m )] (A (),m A(),m + B(),m B(),m ) + I mm [ cos(θ m θ m )] (A (),m B(),m + A(),m B(),m ) (65) Q mm (k) = 4 I mm [ + cos(θ m θ m )] (A (),m B(),m + A(),m B(),m ) + 4 I mm [ cos(θ m θ m )] (A (),m A(),m + B(),m B(),m ). (66) Foowng secton, {S m } shoud be determned sef-consstenty by the foowng equatons: S m = S z m m =C (),m m =,,... (67) and the cantng anges {θ m } are obtaned by ettng H =,.e., I mm sn(θ m θ m )C (),m (A(),m + B(),m ) D m sn(θ m η m ) cos(θ m η m )(J (),m + K(),m ) m + h sn θ m(a () ) =. (68),m + B(),m Snce a the coeffcents shown n equatons (67), (68) (.e. C (),m,a(),m,b(),m, etc) are the mpct functons of {θ m } and {S m }, {θ m } and {S m } can be determned by sovng equatons (67) and (68) sef-consstenty. Puttng the soutons nto Hamtonan (6), we have H = H + E m (k)α m (k)α m(k) + (69) m,k where the spn-wave exctatons E m (k) are cacuated foowng [5]. A the physcay nterestng propertes can be examned. The frst exampe we study s a sx-ayer sandwch-type magnetc mutayer, where the frst two ayers and the ast two ayers have easy-pane ansotropes whe the mdde two ayers have perpendcuar easy-axs ansotropes. The attce structure s assumed to be smpe cubcke, and the structura parameters have been specfed n the capton of fgure 4. For such a system, foowng the method outned n [6], two non-trva spn confguratons (denoted by confguratons (a) and (b)) are found to exst at zero externa fed see the capton of fgure 4. If we are usng the conventona HP method, we fnd that both spn confguratons are unstabe! see the spn-wave spectra (dashed curves) depcted n fgures 4(a) and (b). In fact, such probems exst generay n magnetc mutayers wth easy-pane ansotropes we have examned many other systems wth easy-pane ansotropy and have found that soft modes exst for any canted spn confguratons. However, s t true that the canted spn confguratons are a unstabe n such knds of systems? Usng the present method to recacuate the same system, we fnd that spn confguraton (b) s reay unstabe; however, spn confguraton (a) s stabe. The nstabty of spn confguraton (a) s caused by mssng the SSM effect, not a physca one. It shoud be noted that n ths paper, n order to show just the man physca pcture, the ansotropy parameters are seected to be somewhat arger than practca vaues. Actuay, for sma ansotropy, such an effect st exsts.

16 67 L Zhou and Y Kawazoe Fgure 4. Magnon dsperson reatons E m (k) k x a(k y =) (owest four modes) cacuated by the present method (sod curves) and the conventona HP method (dashed curves) for a sx-ayer magnetc mutayer wth mode parameters: I m,m =,I m,m+ = I m,m =.; m =,, 5, 6: D m =.5,S m =, η m = ; m = 3, 4: D m =., S m = 3/η m = ; h = at spn confguraton (a): θ =.5,θ =.33,θ 3 =.8, θ 4 =.8,θ 5 =.33,θ 6 =.5; and spn confguraton (b): θ =.45,θ =.6,θ 3 =., θ 4 =.,θ 5 =.6,θ 6 =.45. Fgure 5. Magnon exctaton gap (h) cacuated by the present method (sod curve) and the conventona HP method (dashed curve) as the functon of the externa fed for a sx-ayer magnetc mutayer wth mode parameters: I m,m =, I m,m+ = I m,m =.5; m =, : D m =., S m =, η m = π/; m = 3, 4, 5, 6: D m =., S m = 3/, η m =. We w nvestgate another exampe to show the usefuness of our new method. Let us consder a sx-ayer system n whch the frst two ayers have magnetc easy axes aong the x drecton n the pane and the remanng four ayers have perpendcuar easy axes. By usng both the conventona HP method and the present new one, we have cacuated the magnon exctaton gaps (h) as functons of the externa fed apped aong the z drecton, and compared the resuts n fgure 5. Accordng to the quantum theory for coercve feds

17 Unfed spn-wave theory for quantum spn systems 673 estabshed n [5], we know that the fed at whch the magnon exctaton gap approaches zero s smpy the coercve fed h c, snce at ths pont the present spn state s no onger stabe so that a spn reversa transton shoud take pace. From fgure 5, t s very nterestng to fnd that the coercve fed h c.5 cacuated by the conventona HP method s consderaby smaer than that cacuated by the new method h c.7. Once more, h c corresponds to an unphysca nstabty caused by mssng the SSM effect, whch has been removed by the new method. In prncpe, the method estabshed here s appcabe to any mcromagnetc modes wth snge-on ansotropy one just substtutes the conventona HP transformaton by our spn- Bose transformaton (). Athough some physca propertes (such as the ground state energy, the magnetzaton) may not be modfed very much, the ow-yng spn-wave exctatons, however, can be mproved consderaby by our new method. The atter may be mportant n determnng the stabty of an arbtrary spn confguraton n a compcated magnetc system. 5. Concusons To summarze, we have estabshed a unfed spn-wave approach for quantum spn systems wth snge-on ansotropes, whch can be apped to remedy the probems encountered by the conventona HP method, and s appcabe to the arbtrary spn-number case and genera spn confguratons. The key eement of our method s a new spn-bose transformaton whch s qute dfferent wth the conventona HP transformaton and other spn-bose transformatons. In the new spn-bose transformaton, the snge-ste spn-states mxng effect has been consdered sef-consstenty, so that the argest part of off-dagona terms n the Hamtonan have been canceed by ths transformaton, whe t s actuay those terms whch are responsbe for the faure of the conventona HP method. The present method has been compared wth other exstng theores. Treatng the ansotropy n a frst-order approxmaton and n the case of zero externa fed, the present method gves the same resuts as the MME method whch s vad n such a case. For spn- systems, the present method recovers the resuts of the CA method whch s deveoped ony for spn- systems. The method has aso been apped to study some magnetc mutayer systems; the resuts show that t s hepfu to carfy the unphysca nstabty predcted by the conventona HP method. Acknowedgments Ths work was supported by the Japan Scence and Technoogy Corporaton. We are very gratefu to Professor Rubao Tao for hs vauabe comments and stmuatng dscussons. The authors are gratefu to the crew of the Informaton Scence Group of Insttute for Materas Research, Tohoku Unversty for the contnuous support of the supercomputng factes. References [] Crow J E, Gruertn R P and Mhasn T W (ed) 98 Crystane Eetrc Fed and Structure Effect n f-eectron Systems (New York: Penum) [] Jensen J and Mackntosh A R 99 Rare Earth Magnetsm (Oxford: Carendon) ch 5 [3] Harrs R, Pschke M and Zukermann M J 973 Phys. Rev. Lett Bhattacharjee A K, Coqbn B, Juen R, Pschke M, Zobn D and Zuckermann M J 978 J. Phys. F: Met. Phys [4] Hnchey L L and Ms D L 986 Phys. Rev. B Carro A S and Camey R E 99 Phys. Rev. B

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