Isotropic Non-Heisenberg Magnet for Spin S=1

Transcription

1 Ierol Jourl of Physcs d Applcos. IN 974- Volume, Number (, pp. 7-4 Ierol Reserch Publco House hp:// Isoropc No-Heseberg Mge for p = Y. Yousef d Kh. Kh. Mumov.U. Umrov Physcl-Techcl Isue of Acdemy of ceces of Republc of Tjks, Dushbe E-ml: yousof54@yhoo.com, yousof54@gml.com khkm@bo.ru ; mumov@scmpus.eser.e Absrc Equos descrbg soropc o-heseberg model re ob by he mehod of geerled cohere ses rel prmeero. We lero equo of moo er he groud se d ob he equo of sp wve er hs po. Iroduco My codesed mer sysems c be successfully descrbed wh he help of effecve couum feld models. I sysems wh reduced spl dmesoly, opologclly orvl feld cofguros re kow o ply mpor role []. Mgec sysems re usully modeled wh he help of he Heseberg echge erco [,]. However, for sp > / he geerl soropc echge goes beyod he purely Heseberg erco bler sp operors, d my clude hgher-order erms of he ype ( j wh up o. Prculrly, geerl = model wh he soropc eres-eghbor echge o lce s descrbed by he Hmlo H = J ( ( K ( Where,, re he sp operors cg se, d J, K > re y respecvely he bler (Heseberg d bqudrc echge cos. The model ( hs bee dscussed recely coeco wh = bosoc gses opcl lces [4] d he coe of he decofed quum crcly [5,

2 8 Y. Yousef d Kh. Kh. Mumov 6]. The rcle s orged s followg. I frs we ob cohere ses rel prmeero for wo group U( d U(. I order o ob semclsscl equo of moo we mus ob verge vlues of sp, hs order, seco hree we ob hese verges. I seco four clsscl lces Hmlo s ob. I seco fve, wh use of Hmlo, we clcule clsscl equos of moos. I fl we lero equo of moo er he groud se d ob he equo of sp wve er hs po. ohere ses U( d U( groups ohere se U( group s[7] ψ = ep( κ = ep( θ, = cos( θ / ep( ϕ /, = s ( θ / ep( ϕ / y ( Ad cohere se U( group s [8] ψ = ( Where ( ep( γ s ( θ / cos g ep( γ cos ( θ / s g = ep( ϕ = (sθ / / = ep( ϕ ( ep( γ cos g ep( γ s g ( ep( γ cos ( θ / cos g ep( γ s ( θ / s g (4 Two gle, θ dϕ, defe he oreo of he clsscl sp vecor. The gle γ s he roo of he qudrupole mome bou he sp vecor. The prmeer, g, defes chge of he sp vecor mgude d h of he qudrupole mome. Averged sp operors d her producs U( group Here we cosder clsscl couerprs of he sp operors d her producs coed he Hmlo (. The vecor = ψ ˆ ψ (5 be regrded s clsscl sp vecor, d

3 Isoropc No-Heseberg Mge for p = 9 Q j j = ψ ˆ ˆ ψ (6 s compoe of he qudrupole mome. Becuse he sp operors dffere lce ses commue, we hve for ll such producs j j ψ ˆ ˆ ψ ψ ˆ ψ ψ ˆ = ψ (7 where ψ = ψ. ψ Averge sp epresso for he U( group s = ep( ϕsθ = ep( ϕsθ = cosθ (8 d correspodg epressos for he U( group re he followg form: = ep( ϕcosg sθ = ep( ϕcosg sθ = cosg cosθ (9 lsscl lce Hmlo I hs pr, we derve clsscl lce Hmlos whch re ob from Hmlo (, verged over cohere ses ( d (. As ws lredy meoed he sp operors eghborg ses commue, so he cohere se of he whole lce s ψ = ψ ( Avergg equo ( wh relo ( d usg equos (5-9, we ob clsscl couous lm of Hmlo U( group followg form d H = { J K ( ( / ( J K( θ ϕ s θ } Ad clsscl Hmlo U( group s

4 4 Y. Yousef d Kh. Kh. Mumov d 4 H = { J cos g K cos g ( g J s 4g 4g K cos g s g ( / (4 g cos g( J K cos 4g ( cos g( J K cos g( θ ϕ s θ } I bove relo f we se g =, we ob clsscl Hmlo U( group. lsscl equos I oher o ob clsscl equos of moo we se he bove clsscl Hmlo clsscl equos h obed from Lgrg h group. 5- clsscl equo U( group ϕ = θ = ( J K( ϕ cosθ θ ( J K ϕ sθ cscθ ( 5- clsscl equo U( group ϕ = ( ϕ cosθ θ cscθ θ = g = { J ( g K(g g 4g θ ( J cosg K cos g ( J cosg K cos g ϕ sθ γ = J cosg 4K cos g cos g g cos g 4g cos g s g} ( ϕ cosg g θ cos g coθ K(4g ϕ cos g s θ 4g cos g s g 8g cos g coθ } cos g co g cos g cos 4g csc g 4 cos g cscg / { J (4g cos g co g cosg s cos g g (4 lsscl groud se d mmum of eergy Noe h o use he equo (4 vesgo of ferromges wh soropc o- Heseberg erm, s ecessry o fd he clsscl groud ses of hs mge. To hs ed, we cosder oly erm Hmlo ( h whou dervve: d 4 H = ( J cos g K cos g (5

5 Isoropc No-Heseberg Mge for p = 4 I he ferromgec J d K>, o fd he smlles vlue of he H we vry respec o ll he prmeer, he groud se s ob he pos g = or g = π / 4 (6 Ad mmum of eergy s H = (/ ( J (7 K Lere equos of moo er groud se for U( group We cosder ow he dsperso of sp wves propgg er he groud ses. To do hs ed, we lere he clsscl equos (4 er he groud ses. lsscl Hmlo ( er he groud se s d H = { J K (8 ( / (( J K( θ ϕ s θ } Ad equos re ϕ = θ = g = γ = ( J K( ϕ cosθ θ ( J K ϕ sθ ( J K( ϕ cos cscθ θ θ coθ (9 For he groud se, er he po θ = π /, ler smll eco he bove equo chges he followg form: ϕ = θ = γ = g = ( J K θ ( J K ϕ (/ ω ϕ = mθ (/ ω θ = mϕ We cosder ow he dsperso of sp wve propgg er he groud se. To hs ed we cosderg wo fucos θ d ϕ he followg ple wves, ϕ = ϕ ep( ( ω k ϕ ep( ( ω k ( θ = θ ep( ( ω k θ ep( ( ω k We ob he followg equo for he sp wve propgg er he groud se ω ( 4 = m k ω (

6 4 Y. Yousef d Kh. Kh. Mumov Where he vlue of m for U( group s m = ( J K. I s evdece from equo ( h he qudrupole brch for he Hmlo ( s odspersve. Dscusso I erms of sp cohere ses we hve vesged = sp quum sysem wh he bler d bqudrc soropc echge he couum lm. The proper Hmlo of he model c be wre s bler o he geerors of U( group[9]. Kowledge of such group srucure ebles us o ob some ew ec lycl resuls. The lyss of he proper clsscl model llows us o ge dffere solo soluos wh fe eergy d he spl dsrbuo of sp-dpole d/or sp-qudrupole momes ermed s dpole, qudrupole, d dpole-qudrupole solo, respecvely. Referece [] N. Mo d P. uclffe, opologcl solos (mbrdge uversy press, 4. [] E. L. Ngev, ov. Phys. Usp. 5, (98; E. L. Ngev, Mges wh Nosmple Echge Iercos [ Russ], Nuk, Moscow (988. [] V. M. Lokev d V.. Osrovsk ı, Low Temp. Phys., 775 (994. [4] A. Immbekov, M. Luk, d M. Troyer, J. Phys. Rev. A 68, 66 (. [5] K. Hrd, N. Kwshm, d M. Troyer, J. Phys. oc. Jp 76, 7( [6] T. Grover d T. ehl, Phys. Rev. Le. 98, 47(7. [7] kur. Moder quum mechcs, (999. [8] V.. Osrovsk, ov. Phys. JETP 64(5, 999, (986. [9] N. A. Mkush d A.. Moskv, Phys. Le A, V, P 8-6, (.