A geomercal approach o hydrodynamcs and low-energy excaons of spnor condensaes Ryan Barne, 1 Danel Podolsky,,3 and Gl Refael 1 1 Deparmen of Physcs, Calforna Insue of Technology, MC 114-36, Pasadena, Calforna 9115, USA Deparmen of Physcs, Unversy of Torono, Torono, Onaro M5S 1A7, Canada and 3 Deparmen of Physcs, Technon, Hafa, 3000, Israel Daed: June 10, 009) In hs work, we derve he equaons of moon governng he dynamcs of spn-f spnor condensaes. We pursue a descrpon based on sandard physcal varables oal densy and superflud velocy), alongsde F spn-nodes : un vecors ha descrbe he spn F sae, and also exhb he pon-group symmery of a spnor condensae s mean-feld ground sae. In he frs par of our analyss, we derve he hydrodynamc equaons of moon, whch conss of a mass connuy equaon, F Landau-Lfshz equaons for he spn-nodes, and a modfed Euler equaon. In parcular, we provde a generalzaon of he Mermn-Ho relaon o spn-one, and fnd an analyc soluon for he skyrmon exure n he ncompressble regme of a spn-half condensae. In he second par, we sudy he lnearzed dynamcs of spnor condensaes. We provde a general mehod o lnearze he equaons of moon based on he symmery of he mean-feld ground sae usng he local sereographc projecon of he spn nodes. We also provde a smple consrucon o exrac he collecve modes from symmery consderaons alone akn o he analyss of vbraonal excaons of polyaomc molecules. Fnally, we presen a mappng beween he spn-wave modes, and he wave funcons of elecrons n aoms, where he sphercal symmery s degraded by a crysal feld. These resuls demonsrae he beauful geomercal srucure ha underles he dynamcs of spnor condensaes. I. INTRODUCTION A cenral heme of conemporary aomc physcs expermen s he dynamcs of Bose-Ensen condensaes and oher correlaed aomc gases. Of parcular neres are mxures of several speces such as Ferm-Bose mxures [1 3], and bosonc gases wh an nernal spn degrees of freedom,.e., spnor condensaes. Spn-one and spn-wo spnor condensaes have been realzed as parcular hyperfne saes of alkal aoms [4 6]. In addon he rappng and coolng of 5 Cr aoms has led o he realzaon of a spn-hree condensae [7]. On he heorecal fron, snce he nal work of Ohm and Machda [8] and Ho [9] numerous neresng works have followed ha dscuss ground saes, dynamcs, and opologcal excaons of such sysems see, for nsance, a revew n [10]). Recen heorecal neress n spnor condensaes have focused on opcs such as dynamcs near he nsulang ranson [11], measable decay of currens [1], spn knos [13], and he anomalous Hall effec [14]. One parcularly mporan aspec of spnor condensaes s her free dynamcs under a me-dependen Hamlonan, abou ground or measable saes. Ths aspec was he cener of several expermenal [15 0] and heorecal [1 7] sudes. These nvesgaons, so far, were mosly confned o he smples case of spn-one condensaes. On he oher hand, he wealh and nrcacy of spnor condensaes ncreases dramacally wh ncreasng spns. For nsance, he phase dagram of spn-wo and spn-hree condensaes consss of four, and en possble mean-feld phases, respecvely [8, 9, 8 31]. A feaure whch makes hese sysems even more neresng, s ha he ground saes exhb a hgh degree of symmery n s spn sae, whch s somorphc o lace pon groups [3 34]. In hs paper we seek o ulze hs symmery n he sudy of he free dynamcs of spnor condensaes. Recenly, was shown whn mean-feld heory ha he ground saes of spnor condensaes exhb a hgh degree of symmery. Ths symmery s opaque n he sandard spnor descrpon of he condensae. On he oher hand, he symmery s ransparen n he so-called recprocal sae represenaon. Here, one uses he fac ha he mean feld ground sae of a spn-f condensae can be descrbed by F coheren spn saes orhogonal o. Each one of hese so-called recprocal saes s fully spn-polarzed, ponng along some drecon on he un sphere. Snce here are F such recprocal saes, he ground sae s unquely descrbed up o an overall phase) n erms of he F pons on he un sphere [3]. For ypcal spnor condensae Hamlonans, hese pons or anpodes whch we denoe spn nodes, see Sec. III), form hghly symmerc confguraons. For nsance, an F = condensae has a cyclc phase, where he spn nodes are arranged n a erahedron, as well as a square phase. The spn-node descrpon of he ground saes of spnor condensaes provdes an nuve geomercal descrpon of he sae of he condensae. In addon, provdes a paramerzaon whch readly exhbs he hdden pon-group symmeres of he sae. Despe s appeal, however, hs paramerzaon has no been used o descrbe he dynamcs of spnor condensaes. Our goal n hs paper s o provde a complee descrpon of he hydrodynamcs of spnor condensaes n erms of such spn-nodes. The frs aspec we consder s he connuum hydrody-
namcs of spnor condensaes Secons III-VI). Our descrpon s hydrodynamc n he sense ha focuses on he low energy dynamcs of he sysem assocaed wh locally conserved quanes, or wh he slow elasc deformaon of sponaneously broken degrees of freedom. Here we derve such a descrpon usng he densy, superflud velocy, and he spn-nodes he F vecors on he un sphere) as our basc degrees of freedom. In addon o he Euler equaons, whch descrbe mass, momenum, and energy conservaon, we oban F Landau-Lfshz equaons for he dynamcs of he spn-nodes. Furhermore, our dervaon gves a naural generalzaon of he Mermn-Ho relaon whch connecs he vorcy n a ferromagnec spnor condensae wh he Ponryagn densy of he order parameer. The reamen of he spn degrees of freedom n hs par s exac, and accouns for he full geomercal srucure of he hydrodynamcs of spnor condensaes. Bu he precson of he hydrodynamc descrpon here comes a a prce: hs formalsm becomes ncreasngly complex as he spn F grows, and, for large F, he analyss of s exac form becomes mpraccal. Neverheless, he equaons derved here even for large F become que useful n her lnearzed form. In he remander of he paper we show how lnearzng he equaons of moon abou mean-feld soluons elucdae he low energy properes of spnor condensaes wh arbrary spn n a powerful and elegan way. We consder he general F spn-node descrpon of he low lyng spn-wave excaons near he mean-feld ground sae Secons VII and VIII). We derve he lnearzed equaons of moon for he spn-node locaons, whch allows us o exrac he small oscllaon specrum from symmery alone, n a fashon resemblng he vbraonal-mode calculaon for polyaomc molecules [35, 36], hough slghly more complcaed. Usng hs mehod we are able o gve smple expressons for he vbraon egenmodes and energy specrum. In addon, we derve a correspondence beween he low lyng excaons of he spnor-condensaes, and aomc orbals subjec o roaonal symmery due o crysal-felds, whch reflec he symmery of he spnor-condensae ground sae. The paper s organzed as follows. Sec. II provdes general background on spnor condensaes, and revews recen progress on he hydrodynamc descrpon of ferromagnec condensaes [37]. In Sec. III we presen he spn node represenaon of spnor-condensae degrees of freedom, and derve several useful denes whn hs formalsm. In Sec. IV we proceed o oban hydrodynamc equaons for he spn-half condensae, usng he spn-node formalsm, and fnd an analyc soluon for a skyrmon confguraon. In Secs. V and VI, we derve he general hydrodynamc equaons of moon for he spnone condensae, and hen for an arbrary spn-f condensae, whch ncludes a generalzaon of he Mermn-Ho relaon [38]. In he second par of he paper we concenrae on small devaons from he mean-feld ground saes. In Sec. VII we derve he lnearzed equaons of moon abou he mean-feld confguraon n erms of he spn-node formalsm. Fnally, n Sec. VIII we demonsrae how o use symmery argumens o compue he spn-wave excaons, and gve a prescrpon o oban closed form expressons for boh egenmodes and egenenerges of he low-lyng spn-waves. II. BACKGROUND A. Hydrodynamcs of spnless BECs For a sngle componen BEC, s naural o expec a smple hydrodynamc descrpon n erms of densy and flow velocy. We ake he me-dependen Gross- Paevsk equaon GPE) as our sarng pon: ψ = 1 ψ gψ 1) where ψ = e θ s he macroscopc wave funcon, and = ψ s he densy here and afer for noaonal smplcy we wll use scaled uns). Ths equaon can be recas no a he form of local momenum and mass conservaon laws; wh he superflud velocy v = θ, one obans [39]: = v) ; D v = g ) ) where D = v s he maeral dervave. The frs of hese s he mass connuy equaon, whle he second s he Euler equaon for a flud, where a quanum pressure erm appears. B. The hydrodynamcs of ferromagnec BECs and he Mermn-Ho relaon In a seres of recen expermens, he quench dynamcs of a ferromagnec spn-one condensae was explored [17 0]. These expermens movaed Lamacraf o develop a hydrodynamc framework for he ferromagnec BEC n erms of he superflud velocy, and he drecor of s ferromagnec order n [37]. Ths descrpon s parcularly llumnang when consderng he nsables of he sysem. Ths problem was also heorecally consdered n Refs. [1 7]. The GP Lagrangan densy descrbng such a ferromagnec spnor condensae s gven by L = ψ a ψ a 1 ψ a ψ a 1 g 1 c m 3) where a = F,...,F s summed over all F z egensaes, and = a ψ a ψ a m = 1 ψa F abψ b 4) ab
3 where F ab s he spn-f marx. Lamacraf s approach assumed an ncompressble lqud wh a wavefuncon resrced o he ferromagnec phase assumng large g and c ) ψ a = e θ Φ a n) 5) where Φ a n) s he hghes egensae of n F. Noe ha for he ferromagnec sae we have m = n whle for he polar sae m = 0). A subsuon of hs wavefuncon no Eq. 3) yelded he followng se of hydrodynamc equaons [37]: v = 0 6) D n = 1 n n. Once he densy s elmnaed, we noce ha he spn dynamcs are gven by a Landau-Lfshz equaon wh he maeral dervave D = v. In addon, he vorcy s relaed o he Ponryagn densy by v = F ε αβγn α n β n γ ). 7) Ths deny s wdely-known as he Mermn-Ho relaon [9, 38]. Among oher hngs, such a relaon has mporan consequences for he opologcal defecs n ferromagnec condensaes [9, 40 4]. Such hydrodynamc equaons were also derved n [43] o descrbe magnec properes of quanum hall sysems. Makng use of hs smple descrpon, Lamacraf showed ha he helcal confguraon of he ferromagnec condensae [19] s unsable. In general, s clear ha such a geomerc descrpon smplfes, a leas concepually, he analyss of spnor-condensae dynamcs. C. General magnec ground sae of spnor condensaes, and he recprocal sae represenaon A general spn-f spnor-condensae s descrbed by a macroscopc wave-funcon wh F 1 complex componens ψ a a = F,...,F). When quanum flucuaons are unmporan, he condensae dynamcs s descrbed by he me-dependen Gross-Paevsk equaon ψ a = 1 ψ a V n ψ a 8) where V n s he spn-dependen neracon energy. The neracon energy s gven by he se of parameers g S, wh S = 0,,...,F descrbng he wo-parcle neracon srengh n he S oal angular momenum channel: V n = 1 g S ψ aψb ab Sm Sm a b ψ a ψ b. 9) S,m In he above, ab Sm are Clebsch-Gordan coeffcens. Noe ha hs expresson can also be wren as he expecaon value of an operaor: V n = 1 1 ψ ψ V n ψ ψ 1 10) where V n = S,m g S Sm Sm = S g S P S. 11) In hs expresson, P S projecs no he oal spn S scaerng channel. The classcal mean-feld) ground saes occur for unform condensaes whch mnmze Eq. 9) for fxed densy. Thsmnmzaonwascarredou forf = 1[8,9], F = [8], and F = 3 [30, 31] yeldng a mulude of magnec phases, whch mnmze V n n dfferen regons of {g S } parameer space, only one of whch for every F) s ferromagnec. Indeed, que generally, a spn-f spnor condensae may exhb several flavors of paramagnec raher han ferromagnec behavor n s ground sae. For example, a spn-one condensae may exhb he so-called nemac phase, where ψ 1 = ψ 1 = 0, and ψ 0 = 1. The expecaon value of he magnezaon for such a sae s clearly zero along any drecon, F n = 0. Bu n he absence of a ferromagnec drecor, n, can we sll descrbe a spnor condensae s magnec sae geomercally? Such a geomercal mehod was pu forward n Ref. [3], based on he use of spn-coheren saes. A spncoheren sae Φ n s he egenvecor of he operaor F n wh he larges egenvalue. The mehod of Refs. [3] reles on fndng he se of F spn-coheren saes, { Φ n } F, whch annhlae he ground sae of a unform condensae: Φ n ψ GS = 0. 1) The F saes Φ n provde up o an overall phase) a unque descrpon of he magnec spn-sae of he condensae a each pon n space. Such recprocal spnors gve a naural generalzaon of he ferromagnec drecor o he case of paramagnec condensaes. Insead of he geomercally opaque F 1 complex numbers ψ a, allows a descrpon of he magnec sae n erms of F un vecors, n, or pons on he un sphere. In addon o s geomercal ransparency, such a descrpon also reveals he hghly symmerc naure of he mean-feld ground saes. All he paramagnec phases found so far correspond o a spn node confguraon whch s nvaran under pon symmery group operaons, and somemes under a larger symmery. The nemac phase of he F = 1 condensae, for nsance, s descrbed by wo anpodal spn nodes. F = condensaescanexhbanemacphaseaswell,bualsoaphase n whch he spn-nodes are he verces of a square, and a phase wh he spn nodes a he verces of a erahedron. Such phases are llusraed n Fg. 1 [53]. The recprocal-spnor descrpon was so far only ulzed o dscuss equlbrum properes of spnor condensaes. The remarkable geomercal properes and hdden symmeres of he mean-feld ground sae, however, provde ample movaon for employng he spn nodes o oban a complee descrpon of he dynamcs of spnor
4 x4 a) 1a) x x x 1b) b) c) d) FIG. 1: Possble phases for he spn-one and spn-wo condensaes. The red dos on he un sphere correspond o he un vecors n recprocal spnors) defned n Eq. 1. Spn-one condensaes have ferromagnec 1a) and nemac phases 1b) whle spn-wo condensaes have ferromagnec a), unaxal nemac b), square baxal nemac c), and a erahedral d) phases. condensaes. In he followng secons we wll develop he ools necessary for such a descrpon, and use hem o derve boh a hydrodynamc descrpon, as well as small oscllaon dynamcs near mean-feld ground saes. III. SPIN NODE DESCRIPTION OF SPIN-F MAGNETIC STATES The recprocal-spnor saes so far defne he spnor condensae s sae only mplcly hrough Eq. 1). In order o be able o use hese varables drecly, we mus nver he relaonshp, expressng he spnor condensae Lagrangan drecly n erms of hese varables. In order o fnd hs drec represenaon, we frs separae he wave funcon no a pece correspondng o he overall densy and phase, and a pece descrbng he local spn sae. We wre where χ a s a normalzed spn-f spnor and he superflud densy s ψ a = ψχ a 13) χ a χ a = χ χ = 1 14) a = ψ. 15) A. Symmerzed spn-node represenaon As dscussed above, a spn-f spnor χ can be descrbed by F recprocal saes. On he oher hand, such a sae can also be descrbed by a fully symmerzed collecon of F spn-half saes. Each spn-half sae can be paramerzed n erms of wo coordnaes Ω = θ,φ) on he un sphere, ) ) θ θ Ω = cos e φ/ sn e φ/. 16) In hs represenaon, χ = 1 F ) F)! Ω σ = Ω 17) N N {σ} where N s a normalzaon consan, and he sum over σ runs over he F)! permuaons of he F labels for he spn-half pars [44]. In Eq. 17) we also defned Ω as he unnormalzed) sum over permuaons of he ensor produc. The properes of he above formulaon are mos easly undersood usng he Schwnger Boson consrucon [45] for revew, see [46]). Schwnger Bosons provde an easy way o consruc he Hlber space of a spn-f spnor sae. We defne wo Schwnger boson creaon operaors: â,ˆb. An â boson adds 1/ o boh he oal spn, and o F z, whereas a ˆb boson adds 1/ o he oal spn, bu lowers F z by half. In hs noaon F o = 1 â âˆb ˆb); F x = 1 â ˆbˆb â); F y = 1 â ˆbˆb â); 18) F z = 1 â âˆb ˆb). A spn-half spnor s wren as: Ω = u v = uâ vˆb ) 0 wh 0 he Schwnger-bosonvacuum. Here, u and v can be wren n erms of he coordnaes on he un sphere as u = cosθ/)e φ/ and v = snθ/)e φ/. A symmerzed ensor produc of F spns whn he SB formalsm s smply wren as: Ω = Ω 1...Ω F = F u â v ˆb ) 0. 19) wh u and v paramerzed n erms of θ, φ as shown above. We refer o hs collecon of he F spn-half saes whch consruc Ω as spn nodes. If we wsh o calculae wavefuncon overlaps usng he Schwnger Boson formalsm, we can use Wck s heorem o oban Ω a) Ω b) = = 0 F = {σ} = {σ} u a) âv ˆb) a) F j=1 F ) 0 u a) âv a) ˆb F Ω a) Ω σ b). u b) j â v b) j ˆb ) 0 u b) σ â v b) σ ˆb ) 0 0)
5 where σ s a permuaon of he F ndces ha mark he spn-half pars. Ths resul could have also been obaned drecly from Eq. 17). Neverheless, we fnd nsrucve o demonsrae he smple Schwnger Boson consrucon o oban he symmerzed saes. B. Connecon beween spn nodes and recprocal spnors Snce he symmerzed spn-node represenaon can be used o express any spn sae drecly, makes a grossly overcomplee bass. Neverheless, s usefulness arses snce perfecly reflecs he spn-nodes formalsm of he spnor-condensaes ground saes [3, 47]. In he followng we wll nroduce he necessary new noaon for he spn-node formalsm; we summarze he new noaon n Appendx A I s smple o see ha a spn-coheren sae can be wren n erms of Schwnger boson saes as Ω) F = uâ vˆb ) F 0 1) = Ω...Ω. Thus a coherensae can be hough of asf spn-nodes ponng n he same drecon. For a summary of he noaon see Appendx A.) As descrbed n Sec. IIC, a recprocal spnor s a coheren sae Ω r ) F = Ω r...ω r orhogonal o a gven spnor Ω. Usng he consrucon n erms of symmerzed spn nodes, we can wre an equaon o deermne he recprocal spnors for a parcular sae Ω = Ω 1...Ω F : Ωr ) F Ω = F)! F Ω r Ω = 0. ) Ths equaon has F soluons, each correspondng o a dfferen erm n he produc vanshng. Tha s, he h soluon of Eq. ) s Ωr ) F = Ω ) F = Ω...Ω, 3) where Ω = v â u ˆb ) 0 4) s he me-reversed spnor of Ω wh θ = πθ, φ = φ π. Here we use he fac ha a spn-half spnor s orhogonal o s me-reversed counerpar. Fnally, we noe ha he drecon of he spn-half spnor Ω s oppose o ha of Ω. Explcly, we have Ω F Ω = Ω F Ω. 5) One herefore sees ha he se of recprocal spnors s nohng more han he spn nodes ponng n he oppose drecon. For example, gven he spn-node represenaon of a parcular spnor: {n } F 1, he recprocal spnor represenaon s smply {n } F 1. From hs pon on we reserve he symbol n for spn nodes and no recprocal spnors). Furhermore, our analyss wll be n erms of spn nodes alone. C. Tme dervaves of spnors and spn-nodes As saed above, our goal n hs paper s o exrac he dynamcs n erms of ndvdual spn nodes Ω. In ordero do so, we mus be able o solae he dynamcs of each spn node whn Ω. Consder he me dervave of Ω whch wll appear n he GPE. We can express as a sum of erms n whch he me dervave operaes on ndvdual spn nodes: Ω = Ω 1 Ω... Ω 1 Ω...... 6) The rck ha allows us o solae ndvdual spn nodes consss of akng he nner produc of Ω wh he h recprocal-sae of Ω, whch s Ω )F. All erms whch do no nvolve a me dervave of Ω dencally vansh and we are lef wh he sngle erm Ω ) F Ω = F)! Ω Ω F j = 1 j Ω Ωj. 7) We wll make exensve use of hs mehod for solang he dynamcs of ndvdual spn nodes n he followng secons. D. Geomercal paramerzaon of he spn-half componens: Movng from Ω o n All resuls above were concerned wh breakng a spn- F spnor no s F spn-half pars, Ω, and wh he correspondence beween hese spn-half pars and he recprocal coheren saes. We would lke, however, o undersand he dynamcs of spnor condensaes no only n erms of he spnors, Ω, bu also n erms of he un vecorshadescrbehem, n, where Ω shghesvalue egenvecor of F n. The frs sep n fndng he equaons of moon n ermsofhe spn-drecorsn, s oesablsh anorhonormal rad e x,e y,n) ha parameerzes he space on S n he vcny of n. In he followng we wll only consder a sngle spn-half par, and herefore we drop he ndex. The frs elemen of he rad s n self: n = Ω F Ω 8) where F s he spn operaor, acng n he spn-half Hlber space. To complee he rad, we agan use he me-reversed spn-half ke, Ω where Ω Ω = 0. Wh
6 hs, we can consruc saes ponng n he x and y drecons wh respec o n as Ω x = 1 Ω Ω ) 9) Ω y = 1 Ω Ω ). 30) These saes allow us o complee he orhonormal rad by defnng e x = Ω x F Ω x, e y = Ω y F Ω y. 31) From hese we can consruc e ± = e x ±e y. 3) I s useful o noe ha F e ± ac as rasng and lowerng operaors. Tha s, F e Ω = 0 ; F e Ω = Ω, 33) wh smlar relaons holdng for lowerng operaors. Noe ha here s an ambguy n such coordnae sysems snce e x and e y can ogeher be roaed abou n whch corresponds o he gauge choce for he spnors. Tha s, he gauge of a spnor can be changed by Ω e λ Ω whou changng s drecon n. In general, quanes whch are gauge nvaran canno depend on he parameerzaon of he spn and wll only nvolve he un vecors n. We wll adhere o convenon of he spn halfsaenroducedneq.16. Herehe gaugesfxedby requrng ha he produc of he spn up and spn down componens of he spnor s real. In hs gauge-choce s easy o see ha: e x = ˆθ e y = ˆϕ 34) where ˆθ and ˆϕ are un vecors from he sphercal coordnae sysem. To complee he dscusson, we make wo observaons ha wll smplfy he followng analyss. Frs, we express F n he bass of our rad as F = F n)nf e x )e x F e y )e y = F n)n 1 F e )e 1 F e )e 35) In addon we noe ha we can use he spn operaor F as a projecon ono Ω and s me-reversed parner Ω by E. Dervaves of spn-half spnors n erms of he rad e x, e y,n) The relaons derved and recalled n he prevous secons allow us o also wre dervaves of spnors n erms of vecor quanes and her dervaves. The erms ha we wll encouner arse from erms such as he solaed me dervaves n Eq. 7). Le us now fnd hs decomposon n erms of he rad e x, e y,n) and s dfferenal forms. Our goal s hus o fnd: a α = Ω α Ω and Ω α Ω 38) n erms of e x, e y,n) and her dervaves. We defne a α n hs form for reasons ha wll become clear laer. The frs objec n Eq. 38) can be found by consderng he quany α Ω e F Ω ) = α 1 = 0. 39) Allowng he dervave o operae on he bra, he ke, and he vecor e, we fnd α Ω Ω Ω α Ω = 1 e α e 40) On he lef-hand sde we used he facs ha F e Ω = Ω and Ω F e = Ω. On he rgh-hand sde we used he fac ha Ω F Ω = 1 e whch can be verfed from Eq. 35). I s easy o verfy ha from whch we fnd α Ω Ω = Ω α Ω, e x α e y = e y α e x a α = 1 e y α e x 41) whch s he desred resul. To oban Ω α Ω we use a smlar rck. Sarng wh we fnd 0 = Ω Ω Ω Ω = Ω 1 n F Ω 4) α Ω 1 n F Ω ) = 0. 43) As before, allowng he dfferenaon o ac on he bra, he ke, and n resuls n Ω α Ω = Ω α n) F Ω 44) and Ω Ω = 1 n F 36) Ω Ω 1 = n F. 37) whereheermwh α Ω vanshessnce 1 n F) Ω = 0. Now, usng agan he decomposon n Eq. 35), we readly fnd Ω α Ω = 1 e α n. 45)
7 Ths concludes all he ools we wll need for our analyss below. We have found how o drecly wre a spn-f spnor n erms of s spn-nodes, and exrac erms havng o do wh ndvdual spn nodes ou of sums arsng, e.g., from dfferenaon. Furhermore we ranslaed he spn-half represenaon of Ω o a se of F rad bases e x,e y,n ) whch wll allow us o paramerze he spn sae geomercally. Appendx A summarzes he varous noaon nroduced hroughou hs secon. IV. HYDRODYNAMICS OF SPIN-HALF CONDENSATES Oneofourman goalss owrehe exacmean-feld) equaons of moon for a spnor condensae n erms of he spn nodes and he superflud velocy and densy. In hs secon we acheve hs goal for spn-half condensaes. The equaons of moon can be rvally generalzed o general spn-f condensaes resrced o he ferromagnec sae, when he spnor χ s resrced o be a coheren spn-sae. In hs case he equaons of moon for he condensae reduce o hose we fnd below. A. Gross-Paevsk Lagrangan In hs secon we consder he Gross-Paevsk Lagrangan. We begn by wrng he Lagrangan n a revealng form, usng he represenaon of he bosonc feld whch separaes he spnor order parameer no a produc of a densy pece and a spn pece, ψ a = ψχ a. The GP Lagrangan s hen: L = ψ a ψ a 1 ψ a ψ a V n. = ψ ψ a 1 a)ψ 1 ΥV n. 46) where V n s he spn-relaed neracon and = ψ wh ψ a complex feld. Eq. 46) defnes he spn vecor poenal: and he quany a χ χ ; a χ χ, 47) Υ α χ α χ χ α χ α χ χ. 48) An neresng observaon s ha he quany Υ for a general spn F = N/ can be denfed wh he CP N model from quanum feld heory [48]. Noce ha here s a U1) gauge freedom n he densy-spn decomposon: ψ e λ ψ, χ e λ χ where α s mplcly summed over. The quany Υ, however, s gauge ndependen. We make a gauge choce when we wre he normalzed χ as n Eq. 17), wh Ω wren as Eq. 19). Ths U1) gauge freedom s also refleced n an ambguy n he choce of he rad arsng from he spn-half pars of χ Ω, snce for each spn-par e x and e y can ogeher be roaed abou n. The choce of a parcular rad s se by he gauge choce. In general, quanes whch are gauge nvaran canno depend on he paramerzaon of he spn, and wllonly nvolvehe un vecorsn. Thevecorpoenal can be relaed o he superflud velocy by v = 1 ψ a ψ a ψ a ψ a) = θ a 49) where θ s he argumen of ψ. So far we have no used he fac ha he spn s F = 1/. B. Geomerc represenaon of hydrodynamc quanes Now ha we know he quanes of neres n he spnor descrpon of he GP Lagrangan, we can ranslae hem o he hydrodynamc varables of densy and magnezaon drecon. The mos mporan quany appearng above s he vecor poenal as defned n Eq. 47). Followng he dscusson n Sec. IIID we see ha for a spn-half condensae he vecor poenal s a α = Ω α Ω = 1 e y α e x. 50) The analogy beween a and he vecor poenal appearng n he Maxwell equaons compels us o consder he ansymmerc feld ensor f αβ = α a β β a α. Through a seres of manpulaons hs can be wren purely n erms of n f αβ = 1 αe y β e x 1 βe y α e x 51) = 1 αe y n) β e x n) 1 βe y n) α e x n) = 1 e y α n)e x β n) 1 e y β n)e x α n) = 1 e y e x ) α n β n) = 1 n αn β n). Noe ha n he above we have repeaedly used he fac ha v v = 0 for any un vecor v. The resul s he Ponryagn opologcal densy, whch s he objec of he celebraed Mermn-Ho relaon for spn-half spnors [38, 49]. The only remanng erm s he gauge nvaran quany Υ, defned n Eq. 48). For a spn-half sae, we
8 fnd: Υ = α Ω α Ω α Ω Ω Ω α Ω 5) = α Ω Ω Ω α Ω = 1 4 e α n)e α n) = 1 4 αn) α n). Thus he Υ erm sgnfes he sffness of he superflud wh respec o magnec gradens as opposed o smply U1) phase gradens). Also, snce we denfed Υ wh he Lagrangan densy of a CP 1 model, we now reaffrm s equvalence wh he nonlnear sgma model [48]. C. Equaons of moon for spn-half condensaes Now ha we clarfed how he hydrodynamc varables arse n he GP Lagrangan densy, we are ready o approach he GP equaons of moon. In erms of he orgnal varables, he me-dependen GPE for a spn half condensae s ψ a = 1 ψ a gψ a 53) where we noe ha he neracon energy for hs case s V n = 1 g. 54) Followng he subsuon ψ a = ψχ a, wh χ a he enres of he spn-half spnor χ, and conracon wh χ we fnd ψ ψa = 1 a) ψ 1 Υψ gψ where a α = a,a) s he vecor poenal nroduced prevously. Subsung ψ = fe θ and mulplyng boh sdes of he equaon by e θ gves f θf fa = 1 f f vv f fv ) In hs we have defned he elecrc and magnec felds e and b n he usual way from he vecor poenal. Tha s, e α = f α and b α = a) α = 1 ǫ αβγf βγ, wh α, β and γ ndcang space drecons, and he f ensor defned below. Also, noe ha we have used he maeral dervave D = v. The elecromagnec force appearng n he rgh-hand-sde of he Euler equaon s a new feaure ha s no presen n sngle componen condensaes. Ths new ype of quanum pressure arses from non-unform spn exures n spnor condensaes. Now we move on o fnd he equaons descrbng he spn dynamcs. To do hs, we conrac he GPE wh he me reversed spnor χ. Ths gves χ D χ = 1 a χ χ f f χ χ χ χ )). 59) Usng he spn denes developed n Sec. III he followng relaons can be derved χ χ = 1 αe α n) and α χ χ χ α χ = 1 4 αe α n. In erms of vecors, he above equaon s hen e D n = 1 whch can be rewren as 1 e n αf f e α n ) 60) D n = 1 n α α n)) 61) whch s a Landau-Lfshz equaon. Thus, collecng everyhng, we can wre down a complee se of equaons descrbng he dynamcs of a spn half condensae: = v) v = b D v = ev b) D n = 1 n α α n)) g 1 8 αn) ) 1 Υf gf 55) where v = θ a. The magnary par of hs gves = v) 56) whch s a mass conservaon equaon. On he oher hand, akng he real par gves θ 1 v a = 1 f 1 Υg. 57) f We ake he graden of boh sdes of hs equaon usng he deny v ) = v )vv v)) o ge D v = ev b) g 1 Υ ). 58) where e and b are relaed o he spn drecon hrough he feld ensor 0 e x e y e z e f αβ = x 0 b z b y e y b z 0 b = 1 x n αn β n). e z b y b x 0 6) I s neresng o compare hese resuls wh hose obaned n he ncompressble regme Eqns. 6) whch were frs derved n Ref. [37]. The above equaons of moon show ha lfng he ncompressbly consran leads o he neresng appearance of a Lorenz force n he Euler equaon where he effecve elecrc and magnec felds are gven by he Mermn-Ho relaon. In addon, he superflud densy now eners he Landau-Lfshz equaon.
9 D. Applcaon: skyrmon exure As an example of he effcency of he above hydrodynamc equaons of moon, le us consder a specfc calculaon: skyrmon exures n ferromagnec condensaes. For a sandard U1) vorex, he superflud velocy close o he vorex core dverges as 1/r. For a scalar condensae, hs causes he superflud densy o be depleed n a small regon of order of he coherence lengh around he core. Ths can be energecally cosly f he condensae s near he ncompressble regme. On he oher hand, hs suaon can be crcumvened for a spnor condensae. Consder for example, a wo componen spn-half) condensae ψ,ψ ), and ake he componen o have a U1) vorex. Then around he vorex core, he densy of ψ can be ransferred o he vorexfree ψ keepng he oal densy across he vorex core fne. Ths s known as he skyrmon confguraon whch has been argued o be he relevan opologcal defec for ferromagnec condensaes [9, 41, 50]. Le us now derve he analyc me-ndependen soluon of he equaons of moon n he ncompressble regme havng he skyrmon exure shown n Fg.. To hsend, weakehencompressblelm[37]ofheequaons of moon for he spn-half condensae obaned n Sec. IV C. Neglecng z-dependence, hese are: v = 0 63) x v y y v x = 1 n xn y n) 64) D n = 1 n n). 65) Wh small modfcaons, hese equaons can also be shown o descrbe he dynamcs of condensaes confned o he ferromagnec phase of arbrary spn n he ncompressble regme. Our am s o fnd saonary soluons of hese equaons havng a skyrmon exure gven by [9, 41] n = snβ) cosϕ), snβ) snϕ), cosβ)) 66) where ϕ s he azmuhal angle and β s a funcon of r whch s subjec o he boundarycondons βr = 0) = 0 and βr = R) = π where R s a dsance far from he skyrmon cener. Such a spn confguraon s shown n Fg.. Gven he form n n Eq. 66), Eqns. 63,64) can be solved o oban he velocy profle. One fnds v = sn β/) r ˆϕ. 67) Noe ha he boundary condon β0) = 0 suppresses he velocy a he orgn whch dverges as 1/r for he sandard U1) vorex. Wh he assumpon of a sac confguraon, Eq. 65) reduces o v n = 1 n n). 68) FIG. : A skyrmon confguraon correspondng o Eq. 66). Wh he expresson for v n Eq. 67), Eq. 68) leads o he followng second order dfferenal equaon for β r r d β dr dβ ) = snβ). 69) dr Wh he boundary condons, he soluon of hs dfferenal equaon s βr) = 4an 1 r/r). 70) Ths expresson, along wh he velocy n Eq. 67) and he spn drecon n Eq. 66) consue an analyc saonary soluon o he equaons of moon for he skyrmon confguraon. V. HYDRODYNAMICS OF SPIN-ONE CONDENSATES A. Geomercal represenaon of spn one hydrodynamc quanes Now we move on o consderng he more complcaed case of he spn-one condensae. The spn-one spnor can be broken down no s wo spn-half componens and be wren as χ = Ω / Ω Ω where Ω = Ω 1 Ω, where we agan make use of he large-spn noaon defned n Eq. 19). The normalzaon facor for hs case s found o be Ω Ω = Ω 1 Ω 1 Ω Ω Ω 1 Ω Ω Ω 1 = 3 1 n 1 n. 71) I s also nsrucve o calculae he spn operaor s expecaon value. To sar we can expand no producs of spn-half expecaon values Ω F Ω = Ω 1 F Ω 1 Ω F Ω 7) Ω 1 Ω Ω F Ω 1 Ω Ω 1 Ω 1 F Ω.
10 Then usng he deny n Eq. 36), and he fac ha n he facored expresson, F s only acng on spn-half saes, we oban Ω F Ω = n 1 n. Dvdng hs by he normalzaon, we ge he spn-one expecaon value of he magnezaon: m = χ F χ = n 1 n 3n 1 n. 73) By smlar echnques, he vecor poenal for he spnone case, wh some work, can be wren as a α = χ χ = 1 e 1y α e 1x 1 e y α e x 74) On he oher hand, for he nemac sae Υ reduces o Υ = 1 4 αn α n. 79) B. Spn-one condensae equaons of moon We now proceed o do a smlar analyss for he spn one problem. For hs we noe ha he spn one GP energy funconal has he form V n = 1 g 1 c m 80) 1 n n 1 ) α n 1 n 1 n ) α n. 3n 1 n One sees ha he frs wo erms n hs expresson are he vecor poenals from he ndvdual spn-half componens whle he fnal erm, whch s gauge nvaran, descrbes her couplng. Ths expresson was prevously obaned n Refs. [51] and [5], where a geomercal relaon for he Berry phase of a spn-one spnor was gven. The feld ensor correspondng o hs vecor poenal can also be smlarly compued. The mos smplfed form we fnd s f αβ = 3n 1 n ) 75) n 1 α n 1 β n 1 )n α n β n ) n 1 n ) α n 1 β n α n β n 1 )). Ths s a generalzaon of he Mermn-Ho relaon o he spn-one case. To our knowledge such an expresson has no been prevously derved. Whle s geomercal nerpreaon s no as mmedae as he spn-half case whch s he Ponryagn densy), hs expresson mgh be of use n compung opologcal nvaran quanes for spn-one felds. Ths formula has a smplfed form when locally resrced o mean-feld ground saes. For nsance for he ferromagnec sae n n 1 = n ) he above expresson reduces o f αβ = n α n β n). 76) I s also useful o noe ha for he nemac sae n n 1 = n ) he feld ensor dencally vanshes, f αβ = 0. Fnally, he gauge nvaran quany Υ can be worked ou o be Υ = 3n 1 n ) αn 1 α n 1 α n α n α n 1 α n n 1 n α n 1 α n n 1 α n n α n 1 ). 77) Ths s an explc represenaonofhe CP model whch can be vewed as a generalzaon of he nonlnear sgma model. Here, oo, s nsrucve o consder wha hs expresson reduces o when locally resrcng o meanfeld ground saes. For he ferromagnec sae, one fnds Υ = 1 αn α n. 78) where m s he expecaon value of he spn-one operaor. The frs wo hydrodynamc equaons he mass connuy equaon and he modfed Euler equaon are obaned, as before, by conracng he Gross Paevsk equaon wh χ. The analyss proceeds along smlar lnes as he spn half case. However, for hs case we need he generalzaon of he Mermn-Ho relaon for spn one gven n Eq. 75) o gve he feld ensor and hus he effecve elecrc and magnec felds, n addon o he spn one expressons for Υ Eq. 77) and he magnezaon m, n Eq. 73). Wh hese quanes, he frs wo equaons of moon are and D v = ev b) = v) 81) gc m 1 Υ ). 8) Nex, le us dscuss he spn dynamcal equaons. These are obaned by conracng he GPE wh Ω 1 Ω 1 and Ω Ω. As before, hs causes several erms o vansh snce hese spnors are orhogonal o χ. Conracng wh Ω 1Ω 1 gves he followng equaon whch gves he me dervave of he frs node e 1 D n 1 = 1 Γ1 α e 1 α n 1 1 e 1 n 1 c e 1 m. 83) A smlar equaon for he me dervave of n s obaned by conracng wh Ω Ω. In he above, we have colleced he followng erms no he Γ j α parameer Γ j α = a αn n j ) 84) 3n n j n α n j n n j α n j ) 1n n j n j n ) α n n n j ) α n j 3n n j. 85) Fnally, separang he real and magnary pars as Γ j α = Γ j α ) Γ j α ), we oban he Landau-Lfshz equaons D 1 Γ1 α ) α )n 1 = 1 n 1 Γ 1 α ) α n 1 n 1 )c n 1 m 86)
D 1 Γ1 α ) α )n = 1 n Γ 1 α ) α n n )c n m 87) Ths provdes a complee se of equaons descrbng he dynamcs of he spn-one condensae. VI. HYDRODYNAMICS FOR GENERAL SPIN-F CONDENSATES. Now ha we have consdered he hydrodynamc equaons for spn-half and spn-one condensaes n deal, n he followng we wll consder he general case. The frs wo equaons of moon, he mass connuy equaon and he Euler equaon are found, as before, o be and D v = ev b) = v) 88) Vn 1 Υ ). 89) The effecve elecrc and magnec felds agan follow from he feld ensor f αβ consruced from a α = χ α χ. For a general spn, however, such quanes are cumbersome o express drecly n erms of he spn nodes, and we wll refran from dong so. To oban he Landau-Lfshz equaons, we conrac he GPE wh Ω )F. Dong hs gves ) Ω ψ Ω Ω = α log α Ω Ω Ω 1 Ω Ω Ω α Ω j Ω αω j Ω Ω j λ Ω Ω ) 1 Ω F Ω V n Ω Ω 1. 90) In hs expresson, we have used he noaon for neracon energy nroduced n Eq. 10). In addon we have nroduced he quanes λ, λ = F)! j Ω j Ω. 91) 11 where V{n }) = V n s he expecaon value of he energy of a spn confguraon wh spn nodes {n }, and A 1 j s a marx whch projecs he orques due o spnnode j and he moon of spn-node. The marx A and s nverse are defned below n Eq. 11). Insead of wrng Eq. 90) n erms of vecors as n he prevous secons, we wll sop a hs pon. Ths equaon provdes a naural sarng pon n he analyss of he lnearzed equaons of moon whch wll be developed n he flowng secon. VII. LINEARIZED EQUATIONS OF MOTION FOR ARBITRARY SPIN-F CONDENSATES As suggesed from he equaons of moon of he spnone and hgher condensaes gven n he prevous secons, he geomerc represenaon of he equaons of moon yeld raher complcaed resuls. Neverheless, hs formalsm regans s appeal when lnearzed abou parcular mean-feld ground saes. Then he hdden pon symmeres of he ground sae become apparen, and can be used o descrbe he lnearzed dynamcs of a condensae. Below we derve he small oscllaon descrpon of general spnor condensaes. A. Lnearzed equaons of moon from he GPE Parng ways from he aemp a a general descrpon of spnor condensae dynamcs, we now urn o he vcny of a unform mean-feld ground sae. For he ensung dscusson, we wll denoe quanes o be evaluaed n he mean-feld ground sae wh overhead bars. For nsance, he densy can be wren by expandng abou he mean-feld sae as = δ. 93) We wll frs concenrae on he equaons descrbng he densy excaons. Lnearzng he equaons of moon derved n Sec. VI leads o he followng wo equaons descrbng he densy flucuaons: Whle he frs erm n Eq. 90), Ω Ω = e n, s he neral erm for he spn-node n, he rgh hand sde, and he las erm of Eq. 90) n parcular, should serve he role of orques, projeced ono e. As we wll show n he nex secon, he marx elemen of V n s ndeed relaed o a dervave wh respec o he spn-node coordnaes of a poenal energy funcon. Specfcally: λ Ω Ω ) 1 Ω F Ω V n Ω Ω 1 = A 1 j Ω Ω [ e j nj V{n }) ], 9) and = v, 94) v = V ) n. 95) Noe ha erms descrbng he spn degrees of freedom e.g., he effecve elecrc and magnec felds) have compleely dropped ou of hese equaons from lnearzaon. Compung he excaons from hese equaons s sraghforward and gves he famlar Bogolubov mode descrbng densy flucuaons. Le us now focus our aenon on lnearzng he Landau-Lfshz equaons for general spn wren n
1 Eq. 90). Snce he process of lnearzaon separaes he equaons for spn and densy flucuaons, o smplfy he noaon n wha follows, we wll scale he densy of he unform sae o one, 0 1. When lnearzed, he Landau-Lfshz equaons for general spn become Ω Ω = 1 Ω Ω 96) 1 ) F Ω Ω Ω 1 Ω V n Ω Ω 1. λ In he above, as before, we have used overhead bars o denoe quanes evaluaed a her mean-feld confguraon. To undersand he dynamcs of Eq. 96) s useful o nroduce varables o descrbe small devaons of he spn nodes from her mean-feld values. To hs end, by usng he denes esablshed n Sec. III E, we nroduce he se of F complex varables {z } z Ω Ω = Ω δω = 1 ē n = 1 ē δn, 97) where n = n δn. Noe ha n he mean-feld saes we have z = 0 for each spn node snce he vecors ē and n are orhogonal. Ths se of varables can be seen o be he local sereographc projecon of n ono he complex plane for small dsplacemens, and wll be very useful n he followng analyss. Moreover, n our gauge convenon, z s gven n erms of dsplacemens along he zenh and azmuhal drecons from he sphercal coordnae sysem: z = δn ˆθ δn ˆϕ. 98) Usng hese varables, he lnearzed Landau-Lfshz equaons become z = 1 z ) F Ω 1 Ω V n Ω Ω 1 λ. 99) Ω Ω The knec peces n he GP equaons are mos naurally descrbed n erms of he orgnal spnor wave funcon, ψ a, and are no smplfed by he symmery of he mean-feld ground saes. Neverheless, Eq. 99) demonsraes ha near mean-feld ground saes he knec erms sll acqure a smple form. Ineresngly, he knec pars n he spn equaons of moon, 99), do no dsclose he fac ha he varables {z } descrbe spn-half componens of a spn-f sae. Ths fac s refleced only n he spn neracon erm. In he followng secon, we wll see ha hs spn neracon can be expressed n erms of a dervave wh respec o he z varables. In parcular, he equaons of moon wll be shown o be where z = 1 z Ω Ω A 1 j j Ā 1 j Ω )F Ω j ) F λ λ j z j V n 100). 101) Thus, he spn neracon derves from a sum over orques, τ j = zj V n. 10) B. Perurbave expanson of he spn neracon An essenal elemen n he behavor of spnor condensaes s he spn neracon erm V n. I s he mnmzaon of hs erm ha yelds he mean-feld ground saes, and s curvaure ha deermnes he normal excaons. These curvaures can be easly and drecly exraced n erms of specfc marx elemens, as we show below. To expand he spn neracon energy abou a meanfeld ground saedenoed wh an overhead bar) we frs need o undersand how o perurb aspnor abou a fxed value. The followng spn-half deny proves o be que helpful: δω = Ω Ω δω Ω z 103) where we used he resoluon of he deny n erms of Ω and s me reversed parner, and he defnon of z = Ω δω as n Eq. 97). Now, f we apply he varaon o a general spn-f spnor Ω = Ω δ Ω, we oban o lnear order δ Ω = Ω F Ω δω F T Ω z 104) where T Ω s Ω wh s h enry me reversed see Appendx A). Snce Ω δω s magnary, he frs erm, whch does no drecly depend on z, mus drop off when consderng he varaons of real quanes. For nsance, he frs order varaon of he normalzaon s: δ Ω Ω = F Ω T Ω z T Ω Ω z ). 105) Usng Eqns. 104,105) one fnds Ω z Ω Ω = P T Ω, 106) Ω Ω where P = 1 Ω Ω Ω Ω. 107) Such an expresson s useful n evaluang dervaves of he spn neracon energy as n Eq. 100). In general, dervaves wh respec o z wll ac on bras whle dervaves wh respec o z wll ac on kes. We wll now esablsh he equvalence beween Eqns. 99) and 100). One can use Eq. 106) o evaluae he dervave of he neracon energy z j V n = T j Ω P 1 1 Ω V n Ω Ω 1 108) Ω Ω
13 whch s correc o lnear order. The subscrps of he bra s and ke s denoe how he nner produc s o be evaluaed: ke 1 ) s conraced wh bra 1 ), and sgnfy he sae of one of wo neracng parcles; smlarly, he projecon P 1 operaes only on he degrees of freedom peranng o parcle 1. Then usng he expresson for A 1 and he relaon derved n Appendx B) P = Ω )F T Ω P 109) λ one mmedaely fnds for he las erm n Eq. 100) Ω Ω j Ā 1 j z j whch s he las erm n Eq. 99). V n = ) F Ω 1 Ω V n Ω Ω 1 Ω Ω 1. Second order expanson of he neracon energy λ 110) Snce we are neresed n small oscllaons abou equlbrum, we would lke o express he neracon energy expanded abou he mean-feld sae o quadrac order n he z varables. Ths can be formally wren as V n = V n 1 1 j j V n z z j z z j j V n z z z z j 111) j V n z z z z j 11) j where he erms nvolvng dervaves of V n are o be evaluaed a he mean-feld ground sae. We can now use Eq. 106) o evaluae hese dervaves of he neracon energy. Noe ha erms where wo dervaves ac on he same bra or ke wll vansh snce P 1 Ω Vn Ω Ω 1 = 0, 113) whch happens snce τ = 0 a he mnmum of he spn neracon, so ha Ω Vn Ω Ω 1 Ω 1. We hen readly oban he followng quadrac form for he spn neracon energy droppng he V n erm): V n = 1 Ω Ω Vn P T Ω P1 T j Ω 1 j Ω z z j Ω T Ω P 1 1 Ω Vn Ω P1 T j Ω 1 Ω Ω z z j T ) Ω P 1 1 T j Ω P V n Ω Ω 1 Ω z Ω z j. 114) Here P 1, s he projecon operaor whch only acs on saes denoed wh subscrps 1 or respecvely. Whle he form above s wren symmercally, followng Eq. 113), only one projecor n needed n Eq. 114), so P can be omed. Whle hese resuls for he spn neracon seem nvolved, hey are drecly expressed n erms of easlyconsruced marx elemens evaluaed a he mean-feld ground sae. Furhermore, hese marx elemens obey he pon symmery of he ground sae a hand, and hus have srngen consrans. Eq. 114) herefore provdes us wh drec expressons for he marx elemens appearng n he lnear spn-wave expanson of he spnor condensae. C. The Lagrangan of spnor condensaes near equlbrum The equaons of moon can be arrved a by expandng he spnor condensae Lagrangan o quadrac order n he z varables, and compung he correspondng Euler-Lagrange equaons. As we saw before, o hs order, he densy excaons decouple from he spn excaons. Thus, o smplfy he analyss, we wll fx he densy and scale o one, and work n he ncompressble regme. The Lagrangan for a spn-f condensae n he ncompressble regme s L = a 1 θ a) 1 ΥV n 115) where V n s he spn neracon poenal. In expandng hs Lagrangan o second order, we frs consder he spn Berry s phase conrbuon a = χ χ = Ω Ω Ω Ω. 116) Ω Ω Noe ha he kes and bras nvolvng me dervaves are necessarly frs order n varaon from he mean-feld sae. Thus we consder he followng quany expanded o frs order abou he ground sae δ Ω Ω Ω = δω Ω Ω Ω Ω Ω δ Ω Ω ) 117) = P δω Ω Ω Ω δω Ω Ω Ω. 118) Inserng hs no he expresson for he spn Berry s phase 116), and droppng erms ha can be wren as oal me dervaves whch do no conrbue o he dynamcs) one fnds a = Ω P Ω Ω Ω 119) We can hen nser no Eq. 119) he expressons for he expanson of Ω o lnear order n he z varables gven n Eq. 104) o drecly oban a = Ω Ω j z Āj z j 10)
14 where A j T Ω P T j Ω 11) whch s he sough-afer relaon. The proof ha A defned here s n fac he nverse of he expresson gven n Eq. 101) s gven n Appendx B. The herman marx Ā gves he canoncal commuaon relaons beween he z varables. To drecly compue he marx elemens of A s cumbersome because each nvolves a Wck expanson off)!erms. Onhe oherhand he expressonfora 1 gven n Eq. 101) s readly compued snce nvolves evaluang overlaps beween spn-coheren saes. Thus, n pracce, o consruc he marx A s eases o frs consruc A 1 and hen compue s nverse. Proceedng along very smlar lnes as above, one can expand Υ o second order n he z s. One fnds Υ = α χ P a χ 1 Ω Ω j α z Āj α z j. 1) Fnally, we noe ha he erm nvolvng he superflud velocy v = θa n he Lagranganwll no conrbue o he lnearzed equaons of moon. We are now n a posono varyhe LagranganEq.115)asafuncon of he z s o fnd he lnearzed equaons of moon. These read Āj z j = 1 Āj z j Ω Ω V z 13) repeaed ndces are summed over). I s sraghforward o see ha hs s he same as Eq. 100) whch was obaned drecly from lnearzng he GPE conraced wh me-reversed coheren saes. Snce A s a herman marx, s dagonalzed by a unary ransformaon A = UΛU, 14) where Λ s he dagonal marx conssng of he egenvalues of A. I s herefore convenen o defne a new se of w-coordnaes as w = Ū z. 15) Noe ha n erms of hese coordnaes, he Berry s phase assumes a smple dagonal form a = 1 Ω Ω Λ w w. 16) Furhermore, he equaons of moon have he smple form n hese coordnaes: w = 1 Ω Ω V w. 17) Λ w Ths has he form of a me-dependen Schrodnger equaon for he w parameers. VIII. NORMAL EIGENMODES, SYMMETRY, AND GROUP THEORY The mos appealng applcaon of he lnearzed equaonsofmoondevelopedn he prevousseconso oban he normal excaon modes and energes of spnor condensaes havng a hdden ground sae symmery. As we show, s nearly suffcen o dagonalze he marx A [defned n Eq. 101)] n order o oban he egenmodes of he spnor-condensae. Ths can be done solely by usng he symmery of he mean-feld sae. Below we frs demonsrae he use of he lnearzed equaons of moon on he cyclc sae whou fully ulzng he symmery n Sec. VIIIA, and oban all egenmodes and egenfrequences usng he varables defned n Sec. VIIB1. Nex, n Sec. VIIIB, we demonsrae how from he pon group of he hdden symmery of he mean-feld ground saes, we can compue he normal modes alone bu no energes), usng he example of he spn-hree sae where he spn-nodes are arranged a he verces of a hexagon. Fnally, n Sec. VIIIC, we show how o drecly consruc he vbraonal and roaonal egenmodes from sphercal harmoncs, by connecng he problem a hand o ha of degeneracy lfng of elecronc aomc orbals. Ths mehod crcumvens he arduous group-heory foo work, by usng he wellknown properes of aomc orbals under crysal felds ha break roaonal nvarance. The general movaon of he dscusson below s ha group heory analyss can be appled o oban he normal modes n spnor condensaes much lke he analyss of he vbraonal frequences of polyaomc molecules [35, 36]. The aoms or spn nodes) n our case, however, are confned o he surface of he un sphere, and he dsplacemen of each spn node s a wo-dmensonal vecor parameerzed by he real and magnary pars of he z varables). Ths consran slghly complcaes he analyss n comparson wh he reamen of he vbraons of polyaomc molecules. The frs sep n a symmery analyss s o consruc he ransformaon rules of he -d dsplacemen vecors under pon group symmeres. These ransformaon rules are a reducble represenaon ofhesymmerygroup,andcanhenbebrokendownno s rreducble represenaons rreps). The modes ha ransform accordng o he rreps are he egenmodes of he sysem. Before we begn he analyss, a noe on mode mulplcy s n order. Navely, one mgh expec ha he procedure n he prevous paragraph wll gve F) normal modes due o he wo bass vecorsper spn node. Ths suaon would arse f he ransformaons we consruc ransform he F real coordnaes, and are herefore 4F large real reducble represenaons of he symmery group, resulng n 4F modes. Whle hs s he case for real aoms, where he dsplacemen vecors are also assocaed wh conjugae momena, he spnnodes dsplacemens do no have ndependen conjugae momena. From Eq. 115) and 116) we see ha he
15 complex dsplacemen z s acually canoncally conjugae o π = L ż A j zj : he wo-dmensonal dsplacemens are boh he coordnae and conjugae mo- mena, and hence here are only F egenmodes n a spnor condensae. Qualavely, hs s a suaon remnscen of a massless parcle n a magnec feld, where he x and y coordnaes are canoncally conjugae coordnae and momenum. Indeed, consrucng real 4F dmensonal represenaons of he symmery would resul n wo duplcaes of he spnor-condensae s egenmodes. Ths duplcy wll become evden when he egenmodes are wren n erms of he complex z s: half he normal modes wll dffer from he oher half hrough a complex mulplcave coeffcen. A. Spn-wo cyclc sae As our frs example, we consder he cyclc sae whch s a possble mean-feld ground sae havng he symmery of a erahedron for he spn-wo problem. We wll expand he neracon energy o quadrac order abou hs mean-feld ground sae o compue he energes of he normal excaons. The spn-wo neracon energy can be wren n he smple form [8, 9] V n = 1 αm 1 β χ χ 18) where α and β are funcons of he scaerng lenghs, and m = χ F χ. 19) For he mean-feld cyclc sae, hs spn neracon energy convenenly vanshes V n = 0. In he followng we wll expand hs energy o quadrac order. WefrsconsruchesymmerymarxAforhecyclc sae. We ake he orenaon where he spn nodes are a n caresan coordnaes) n 1 = 1 3 1,1,1), n = 1 3 1,1,1), 130) n 3 = 1 3 1,1,1), n 4 = 1 3 1,1,1). 131) Wh he spn-half spnors correspondng o hese spn nodes he marx Ā1 can be drecly consruced usng he expresson nvolvng overlaps of me-reversed coheren saes n Eq. 101). Usng our gauge convenon, hs s found o be Ā 1 = 1 64 9 1 1 1 1 9 1 1 1 1 9 1. 13) 1 1 1 9 Ths hen can be nvered o oban 11 1 1 1 Ā = 1 11 1 1 3 1 1 11 1. 133) 1 1 1 11 Recall ha drecly consrucng he Ā marx s cumbersome snce s elemens nvolve Wck expansons havng F)! erms. The egenvalues of hs marx are found o be EgĀ) = Λ 1, Λ, Λ 3, Λ 4 ) = 8,8,8, 16 3 ). Ths marx can be wren n a revealng form as Ā = 8I 8 3ū4ū 4 134) where ū 4 = 1 1,1,1,1)T s he egenvecor of Ā correspondng o egenvalue Λ 4 and I s he deny marx. An egenmode wll necessarly dagonalze he A marx as well as he enre equaons of moon, and herefore we already gleaned one egenmode: ū 4, whch wll urn ou o be he opcal mode. The hree modes orhogonal o ū 4 are assocaed wh SO3) roaons. Wh hs n mnd, we consruc hese hree egenmodes as he vecors arsng from nfnesmal roaons of n abou he caresan axes, ˆx α. A roaon by angleδη abou he ˆx α axs produceshe followngz s: z δη) = δη ˆx α n ) ē = δη ē ˆx α. 135) Thus he egenvecors ū α are: ū α = 3 8 {ē ˆx α } 4. 136) I s now clear how o wre he ransformaon no he egen-coordnaes defned generally n Eq. 15): z = α w α ū α ). 137) Due o he hgh symmery of he erahedron, hese mode are also degenerae. In general, he se of coordnae vecors ˆx α should be aken o be he prncpal axes of he mean-feld confguraon. Nex we use hs marx o expand he neracon energy. We frs consder he lnear order varaon of he spn momen m. Noe ha snce m = 0 n he ground sae we have Ω F Ω = Ω F P Ω. Then by nserng he deny for P gven n Eq. 109) one fnds 1 δm = Ω ē Ā j z j z Ω Ājē j ). 138) j We now wre he vecors ē n he bass of un vecors along he hree caresan coordnaes ē = 3 ē ˆx α )ˆx α 139) α=1 whch mmedaely reduces hem o he complex conjugae of he degenerae egenvecors ū α wh egenvalue Λ = 8). The fac ha all of he egenvalues are he same s due o he hgh symmery of he erahedral sae.
16 a) y x b) z D 6h E C 6 C 3 C 3C 3C S 3 S 6 σ h 3σ d 3σ v A 1g 1 1 1 1 1 1 1 1 1 1 1 1 A g 1 1 1 1-1 -1 1 1 1 1-1 -1 B 1g 1-1 1-1 1-1 1-1 1-1 1-1 B g 1-1 1-1 -1 1 1-1 1-1 -1 1 E 1g 1-1 - 0 0 1-1 - 0 0 E g -1-1 0 0-1 -1 0 0 A 1u 1 1 1 1 1 1-1 -1-1 -1-1 -1 A u 1 1 1 1-1 -1-1 -1-1 -1 1 1 B 1u 1-1 1-1 1-1 -1 1-1 1-1 1 B u 1-1 1-1 -1 1-1 1-1 1 1-1 E 1u 1-1 - 0 0 - -1 1 0 0 E u -1-1 0 0-1 1-0 0 Γ 1 0 0 0-4 0 0 0 0 0 0 0 c) d) TABLE I: The characer able of he group D 6h usng he noaon of [35]. The las row gves he characers of he reducble represenaon Γ consruced from ransformng he dsplacemen vecors of he hexagon see ex). FIG. 3: Normal modes of he cyclc sae. Mode a) s he opcal mode correspondng o pure dsplacemens n w 4. Modes b), c), and d) are gapless modes correspondng roang abou he x, y, or z axes respecvely. The axes of roaon for hese modes s shown. Wh hs bass one fnds for he expanson of magnezaon he smple expresson 4 δm = Ω 3 8 Ω 3ˆx αū α ) z ū α ) z ) α=1 = 3 6 ˆx α w α wα ) 140) α=1 wherewehaveexpressedhefnalresulnermsofhewvarables defned n Eq. 137). In dervng he above expresson, we have explcly used he values for he egenvalues of he A marx and he normalzaon consan Ω Ω = 8 3. The hree parameers of w occurng n Eq. 140) correspond o roaons abou he hree caresan axes as shown n Fg. 3. Smlar analyss can be performed on he second erm n he spn neracon for he cyclc sae. Whou showng he deals, s found ha δ χ χ = w 4. 141) Wh hese expressons we can now wre down he spn neracon energy expanded o quadrac order whch reads 3 Ω Ω Vs = α Λ w w ) β Λ 4 w 4. 14) Wh hs expanson of he neracon, Eq. 17) can be drecly used o compue he energy of he normal excaons. Four Bogolubov modesnoe we are neglecng he densy mode) are readly obaned. One fnds hree gapless spn waves of dsperson Ek s = ε k ε k 4α) n addon o an opcal mode havng dsperson E op k = ε k β where ε k s he free parcle dsperson). Que generally, he egenvecors of he marx Ā yeld he dsplacemens of he z varables correspondng o each of he egenmodes see, e.g., Eq. 135). In case of degeneracy, s he neracon erms, dscussed n Sec. VII B 1, ha deermne he correc dagonalzaon ofhe degeneraesubspacen he marx A. In he caseof he cyclc sae, he frs hree modes have dsplacemens ha correspond o roaons abou hree orhogonal axes. The fnal mode z ū 4 corresponds o he opcal excaon dscussed above, and s dsplacemens are depced n Fg. 3. Ths procedure smplfes he sandard Bogolubov mehod [39] consderably; we exrac he egenmodes solely from he A marx, whch, as we show nex, can be obaned from symmery consderaons. B. Spn-hree hexagonal sae Leus nowdescrbehowoobanhe normalmodesof a spnor condensae by usng symmery argumens alone n a more complcaed seng. Once havng he egenmodes, however, we mus noe ha o oban he energecs and dspersons of hese modes, analyss of he mcroscopc Hamlonan s sll requred. Our analyss uses group heorecal argumens smlar o hose used o deermne he vbraonal modes of polyaomc molecules [35, 36]. We llusrae he mehod hrough he nonrval example of he spn-hree sae havng he symmery of he hexagon, whch s a canddae for he ground sae of 5 Cr condensaes [30, 31]. ThehexagonbelongsoheponsymmerygroupD 6h whose characerable s gven n Table I. In hs able we