Pairing, Ferromagnetism, and Condensation of a normal spin-1 Bose gas Stefan S. Natu and Erich J. Mueller ∗ Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA. We theoretically study the stability of a normal, spin disordered, homogenous spin-1 Bose gas against ferromagnetism, pairing, andcondensationthroughaRandomPhaseApproximationwhich includes exchange (RPA-X). Repulsive spin-independent interactions stabilize the normal state against both ferromagnetism and pairing, and for typical interaction strengths leads to a direct 1 transitionfromanunorderednormalstatetoafullyorderedsingleparticlecondensate. Atomswith 1 muchlargerspin-dependentinteractionmayexperienceatransitiontoaferromagneticnormalstate 0 or a paired superfluid, but, within the RPA-X, there is no instability towards a normal state with 2 spontaneous nematic order. We analyze the role of the quadratic Zeeman effect and finite system n size. a J 8 Introduction.— The interplay of superconductiv- TABLE I:Orders in spin-1 gas. 2 ity/superfluidity and magnetism is fundamental [1]. Ex- periments in ultra-cold spin-1 gases [2–5] have begun to Order Symbol Order Parameter ] s explore this physics, elucidating the subtle connections ferromagnetic F hSi=6 0 a between Bose condensation of single particles and com- nematic N hS S i6=δ g µ ν µν peting/complementaryorderssuchaspaircondensation, - single particle C hψµi=6 0 nt fTehrriosmhaasgnbeetcisomm,eaandmloiqduelids-ycsrtyesmtalfolriktehninekminagticaibtyou[6t–e8x]-. pair P Pkha†µka†ν-ki=6 0 a u otic spin textures and topologicaldefects [9–12], and the q dynamics of quantum phase transitions [13, 14]. How- t. ever, the finite temperature 3D phase diagram is a mys- the interaction Hamiltonian to be [7, 8]: a tery, with earlier works producing contradictory results m [15–17]. Here we clarify the situation by using a well- Hˆint = 21 dr ψα†ψβ†ψγψδ(c0δαδδβγ +c2Sˆαδ·Sˆβγ), (1) - controlledapproximation[the randomphase approxima- Z d n tionwithexchange(RPA-X)]tocalculatetheinstabilities where the greek indices denote the spin projection and o of the normal state. ψ (r)= 1 eikra is the the boson field operator. α V k kσ c The two coupling constants, c and c represent spin [ A particularly dramatic feature of the spin-1 Bose gas independenPt and spin dependent0interac2tions The Sˆ op- 1 is that it supports a bosonic analog of the BCS tran- erators denote 3 3 spin-1 matrices. The interac- v sition [18–21]. Somewhat counterintuitively, the paired tions are expresse×d in terms of the microscopic scat- 9 3 stateislessorderedthanasingleparticlecondensate,and tering lengths in the spin-0 (a0) and spin-2 (a2) chan- 6 is found when both the spin independent and dependent nels and atomic mass m as: c0 = 4π(a0+2a2)/3m and 5 interactions are repulsive. In addition to its theoretical c2 =4π(a2 a0)/3m. − . importance,thisfeaturemakesitaninterestingparadigm Two atoms are typically used in these experiments: 1 0 to keep in mind while exploring the mechanisms for su- 87Rb (c2 < 0) and 23Na (c2 > 0). In all experiments so 1 perconductivity in systems such as high-Tc cuprates, C- far, spin independent interactions are repulsive (c0 > 0) 1 60 and polyacenes where the interactions are believed to and c0 c2 . We find that the phase diagram is fea- : ≫ | | v be repulsive [22]. tureless in this regime, motivating us to study the more i general case where the interactions are comparable in X The Hamiltonian of a spin-1 Bose gas is the sum of a magnitude. Perhaps7Li,rareearthatoms,oralkali-earth r a kinetic and interaction term, H = ˆ + ˆ . In the atomswillhavescatteringparametersinthisregime. Al- kin int presenceofamagneticfieldinthezˆHdirectionH,thekinetic thoughdipolarinteractionsarebelievedtoplayanimpor- ttheremanhnaishtilhaetifoonrmopHeˆrkainto=r forkaσǫbkoσsao†knσawkσit,hwmheormeeanktσumis tVaenntgraolalettionrteheetlaolw. [t3e]m,wpeernateugrleecqtutahseim-2Dheerxep,earsimtheenytsaroef k and spin projection σ =P 1,0,1. The dispersion is much too weak to influence the stability of the normal ǫ = k2/2m µ, ǫ = k2/−2m µ+q p, where p/q state. k0 k 1 are linear/qua−dratic±in the magn−etic field±. There is no The spin-1 gas can present several types of order, spin-orbit coupling, allowing us to eliminate the linear summarized in Table I. Single particle condensate or- Zeeman effect (p) by working in a rotating frame. Off- der (C) is always accompanied by either ferromag- resonant microwave light allows the quadratic Zeeman netic (FC) or nematic order (NC). Mean field exam- fieldqtobetuned,takingonpositiveandnegativevalues ples of these condensed states are |FCi = (ψ1†)N|0i, [23]. Assumingshortrangeinteractions,symmetryforces and |NCi = (ψ0†)N|0i, with the former seen in 87Rb, 2 L 0 a (cid:144)13 87Rb23Na bilityinthenematicchannelwhichisnotsimultaneously Hn 2 Normal accompanied by single particle or pair condensation. BEC P The relevant response functions are T F (cid:144) 0 L -TBEC FC NC χγαδβp(t)= 1ih a†δk(t)aγk+p(t)a†βq(0)aαq−p(0)i (2) T -0.75 -0.3 0 0.5 0.75 Xk,q H 1 c2(cid:144)c0 Πγαδβp = ih a†δk(t)a†γp−k(t)aβq(0)aαp−q(0)i (3) FIG.1: 3DPhase diagramatq=0withinthe RPA-X: Xk,q Thick solid/dashed lines are the ferromagnetic and pairing transition temperatures measured from the ideal Bose gas where t > 0, and the greek subscripts denote spin in- transition k T = 2π( n )2/3 and scaled by n1/3a dices and p is the momentum [26, 27]. The longitudinal B BEC m 3ζ(3/2) 0 as a function of spin-dependent interaction c2. Thin solid and transverse spin correlation functions are χzp(t) = line shows the instability towards single particle condensa- i S (t)S (0) = ( 1)αδχδδ (t), and aFtiloolyrncs(2yfem>rrmo0em.t5racigc0n,pettahiicreeodnrosrpimnogallalertstdpaehtpeaesniesd(iuPnng)stwoanibtlhethateoTsacig>nrotoTafBtiEco2Cn)-.. wχ−±hhe=rezp−SihµSp+(zt−p)p(t=)S−i−pq(0a)†p−i+=qP/2χα(10,tδ01)=S+ˆ±µ1χap00−−−11q/+2(χt)−α,10α−pa10n+d χµ−11=00, For c2 <−c0/3, the normal state becomes unstable to a fer- z, . { ±} P romagnet (F). For |c2| smaller than these threshold values, Inthe RPA-X,the susceptibility ofthe interactinggas there is a direct transition from a disordered normal state is determined from the non-interacting susceptibility by to a ferromagnetic condensate (FC) or a nematic condensate summing over all repeated direct, and exchange interac- (NC).Tickmarksontheupperframeillustratethescattering lengths for 85Rband 23Na. tions. (χRPA)γη =(χ0)γηδ δ + (χ0)γηVγη(χRPA)νµ(4) αβ αβ αη βγ ηγ µν αβ and the latter in 23Na. The singlet state from [24] µν X with all particles in k = 0, |Si = ((ψ0†,k=0)2 − (ΠRPA)γαηβ =(Π0)γαηβδαηδβγ + (Π0)γηγηVγµην(ΠRPA)µγην(5) 2ψ1†,k=0ψ†1,k=0)N/2|0i, has off-diagonal single-particle Xµν − tohrderemr o[idey.nalmimic|r−limr′|i→t∞shhψou0†l(dr)ψb0e(rc′o)nisi=6der0e]daansdaninNthCe The interaction potential Vµγνη of Eq. (1), which includes both direct and exchange graphs, is explicitly given in state [25]. An example of a P state would be a con- the supplementary material. densate of small singlet pairs P = κN/2 0 , where | i | i The non-interacting Green’s functions are diagonal in κ = k a†0ka†0-k−2a†1ka†−1−k . Unlike |Si, the state spin space: (χ0)γαηβp(t) = 0, and (Π0)γαηβp(t) = 0 unless P has (cid:16)no off-diagonal single(cid:17)particle order. Paired η =α and γ =β, | i P smtaatteics foorrdewrh.ichWhea†0fiknad†0-kno−i2nas†1tkaab†−il1it−ieksi t6=ow0arpdossspeasirneed- (χ0)βα(p,ω) = d3k n(ǫk,α)−n(ǫk+p,β) (6) states with ferromagnetic order. αβ (2π)3 ω (ǫ ǫ ) Z − k+pβ − kα Although the 2D phase diagram is well established d3k n(ǫ )+n(ǫ ) (Π0)αβ(p,ω) = k,α k+p,β (7) (with an algebraicallyorderedP state at any finite tem- βα (2π)3 ω (ǫ +ǫ ) perature whenq =0)[10, 16], contradictoryresults have Z − k+pβ kα appeared concerning the 3D phase diagram. Both Gu Heren(ǫkσ)=(eβǫkσ 1)−1istheBose-Einsteindistribu- and Klemm [15] and Kun Yang [16] erroneously found − tion at temperature T =1/β. For a non-interacting gas, that arbitrarily weak attractive interactions drive a fer- the spin susceptibility χ0, pairing susceptibility Π0 and romagnetic instability with TF > T . Kis-Szab´o, c BEC compressibilityalldivergeasµ 0frombelow,marking Sz´efalusy and Szirmai [17] gave a more thorough argu- → Bose-Einstein condensation. ment, finding a finite threshold for this instability. We At k,ω = 0, these non-interacting response func- extendtheircalculationtoincorporateexchangephysics, tions may be written in terms of the polyloga- finecctlu.ding c2 > 0, c2 < 0, and the quadratic Zeeman ef- 0ri,tωhm=fun0c)tion=s gν(zm) g= (eβ(jµzjq/))j,ν:(χ0χ)−1011(−011,(0k) == traFnosrvmerasleismsp.i—n (Wχe)caanlcdulpaateiritnhge (loΠn)gistuusdcienpatlib(iχlizt)iesanodf 4πmΛT(T/q)[g3/2(eβ(−µ−2πq)Λ)T −1/2g3P/2(e−βµ)], Πβααβ01(0,0) = the homogeneous ±interacting spinor Bose gas using a −πmΛTg1/2(eβµeff), where µeff = µ −q for α = ±1 and Hartree-Fock Random Phase Approximation (RPA-X). β = 1, and µeff = µ for α = β = 0. The thermal ∓ A divergence of the zero frequency, long wavelengthsus- wavelength is ΛT = 2π/mkBT. The calculations are ceptibility,χ 1(k=0,ω =0)=0signalsaninstabilityin detailed in the supplementary material. − p thatchannel. Within the RPA-Xthere isneveraninsta- To detect ferromagnetism we consider the response 3 functions (see supplementary materials) (a) (b) 1.13 χRzPA(k,0) = 1−(c022(+(χχ030))c−120)11((−kχ11,0(0)k−),11ω−)11(k,0) (8) (cid:144)TTBEC 1.1 HFΧÈÈRPzALN-=10ormHaΧRlPA ±L-1F=0¦ ‰ NC (cid:144)qcn000..48 χRPA(k,0) = 10 (9) 0 ± 1−(c0+3c2)(χ0)1001(k,0) -0.15 0 0.15 0.25 0.3 1 1.5 2 Èc È(cid:144)c 2 0 To detect pairing, it suffices to consider the singlet pair- q(cid:144)Èc0Èn ing susceptibility, Θ=(Π0000−2Π−11−11)RPA, uFnIGor.d2e:redInnstoarmbiallitsytawteitwhitch2c< =0: −(ac):aIsnsatafbuinlicttieiosnofofthqe. 2 0 Π+ 2Π0+Π0Π+(c 2c ) Solid curves give the Tc for a non-condensed ferromagnetic 0 2 Θ = − − . gas, normalized to T | (defined in Fig.1 caption). At 1 (c c )Π+ c Π0+(c c )(c +c )Π+Π0 BEC q=0 − 0− 2 − 0 0− 2 0 2 some lower temperature, one expects a transition to a ferro- magneticcondensate(FC).Forq<0,thisT alwaysexceeds c w(Πit0h)00Π00 0=≡(Π(0Π)011)−000011a=nd(ΠΠ)0+a≡nd(tΠh0is)−e11x−p11re.ssiWonhesinmqpl=ifie0s tghaesitdeemalpBeroasteugreasmtreaentssitthioenTtcemfoprefreartruorme.agFnoertqism>0a,ttshoemideefia-l to Θ 1 1 −(c 2c )(Π)0. niteq (markedby×). Beyondthispoint,thenormalstateis − 0 2 ∝ − − unstable to forming a polar condensate. (b): Location of × Results.—Repulsivespinindependentinteractions(c ) 0 as a function of interaction strength. suppress both ferromagnetism and pairing in Bose sys- tems. This should be contrasted to fermions, where due totheoppositesignoftheexchangeterm,repulsiveinter- Figure 2(a) illustrates the phase diagram for c < 2 actionsenhanceferromagnetism,givingrisetotheStoner 0, where the only relevant instabilities are ferromag- instability[28],evenintheabsenceofanyspindependent netism and single particle condensation. For q < 0 interactions. and c >c /3 an Ising ferromagnetic instability always 2 0 From Eq. 8, and the fact that χ0(0,0)<0 we see that prece|de|s condensation. For q > 0 there is a thresh- the spin susceptibility only diverges when c2 < 0 with old q below which x y ferromagnetism precedes con- − |c2|>c0/3. Similarly,atq =0,thepairingsusceptibility densation. The dependance of this threshold on c2 is only diverges when c2 > 0 with c2 > c0/2. For weak shown in Figure 2(b). For c2 near c0/3, one finds: interactions(|c2n|≪kBT),theseinstabilitiesoccurnear qc =TBq=EC0(10.6(a0n)1/3α)2, where α=−1/3−|a2|/a0. µ= 0. Expanding the susceptibilities for small µ at q = Figure 3(a) illustrates the phase diagram for c > 0, 2 0 gives that, to leading order, the magnetic instability wheretheonlyrelevantinstabilitiesarepairingandsingle occurs at particle condensation. Finite q enhances single particle condensation,andforagivenq,thereisathresholdvalue Tmag T 1 c tmag = c − BEC =4.84 2 n1/3a0 (10) ofc2 requiredto find a pairing instability. For q >0 this T 3 − c BEC (cid:18) 0(cid:19) threshold becomes arbitrarily large as q , but for t = Tcpair−TBEC =6.44 c2 1 n1/3a . q < 0 one always has a pairing transitio→n i∞f c2 > c0. pair TBEC (cid:18)c0 − 2(cid:19) 0 qSe<ttin0gaµnd=(Π0,0)a0n0d takingfotrheql>imi0t,(wΠe0)c−a11l−cu11la→te∞thefoser Hypothetically, taking n = 1014cm−3, a0 = 100aB, and threshold values0(0Fi→g. 3∞(b)): c c , we find T T 10nK. The q = 0 phase 2 0 c BEC | | ∼ − ∼ diagram is summarized in Fig. 1. 1, q >0 q =12.74Tq=0 (n1/3a )2(1 2x)(1+x) We now explore the role of the quadratic Zeeman ef- c BEC 0 − ( 11x, q <0) fect: q < 0 favors magnetism in the zˆ direction (F − (12) ± k – Ising order – signalled by diverging χ ), and pairs in z where x=c /c . 2 0 wthheilemFq =>±01 fsatvaotress (mNaPgn⊥et:ism2|hψin1†ψt−†h1ei|x> |hyψ0†ψp0l†ain|)e; somExewpehraimt ednisttailnlyg,uitshheedstabtyesthdeisfcaucstsetdhaitnbFoitgh.3tmheaycobne- − (F – x-y order – diverging χ ), and mF = 0 pairs densedphaseNC forlargeq <0,andthe pairedphase t(hNe⊥PBkEC: |thrψa0†nψsi0†tii|on>tem2|hpψe1r†aψt−†u1rie|±:).theFdineintseitqy iaslsgoivesnhifbtys NthPis⊥q,uhaanvteitnysis=ind⊥1en+tinca−l1ly−z2enr0o.>S0t.uIdnytinhge sminogmleetnptauimr, nΛ3T = g3/2(eβµ)+2g3/2(eβ(µ−q)), with condensation at distributions can distinguish between the single particle µ = q for q < 0 and at µ = 0 for q > 0. For small q one and paired condensates. finds Finite Size effects.— Similar to [21], we can estimate theroleoffinitesizeeffectsbylookingattheinstabilities TBq6=EC0 =TBq=EC0 +ξ TBq=EC0q (11) at finite k = 2π/L, where L is the size of the cloud. These finite size effects are crucial in the scalar gas with q with ξ =0.3 for q <0 and ξ =0.6 for q >0. attractive interactions, where there is no Bose-Einstein 4 (a) Acknowledgements.— We thank Stefan K. Baur, Joel (cid:144)TTBEC111...000345555 NC¦ oc2(cid:144)c0=0.7HP5RPAL-N1=P0¦ NormaPl ‰NCÈÈ Eguap.loMantotoworroeerfakonrsduuXpsepiafouorllticenodgnCbveyurist,ahAteiroiNnTsau.triTonnheraislamnSdcaitMeenruciaekluFinsodubVnasdeenad-- -0.2 -0.1 0 0.04 tion through grant No. PHY-0758104. (b) q(cid:144)Èc0Èn 1.055 (c) (cid:144)TTcBEC11..003455 NPHNP¦RoPrAPmL-c1a2=N(cid:144)l0cP0È=È 0‰.75NCÈÈ (cid:144)cc200.17.515ooooooooooooooooooooxxxxxxxxxxxxxxxxxxxx [1∗] CElleacrtinroandicealaddCrerussz:estsna8l.@Ncoartnuerlel,.e4d5u3 899-902 (2002); M. -0.02 0 0.025 0.5-1 -0.5 0ox 0.5 1 Kenzelmann et al. Science321 (5896) (2008); A.J. Drew q(cid:144)Èc0Èn q(cid:144)c0n etal.NatureMaterials,8310-314(2009);Y-A.Liaoetal., FIG. 3: Instabilities with c2 >0: (a): Instabilities of the Nature467 567-569 (2010). unordered state with c = 0.75c . At large negative q, the [2] D.M.Stamper-Kurn,M.R.Andrews,A.P.Chikkatur,S. 2 0 normal state is unstabletowards a single-particle condensate Inouye, H.-J. Miesner, J. Stenger and W. Ketterle, Phys. (NC ) with a spinor order parameter ζ = {eiφ,0,1}), with Rev.Lett. 80, 2027 (1998). ⊥ arbitrary φ. For −|q |(×) < q < |q |(o) the instability is [3] M.Vengalattore, S.R.Leslie, J.Guzman, and c1 c2 towards pairing. At q = 0 the paired phase is a spin singlet D.M.Stamper-Kurn,Phys.Rev.Lett., 100 170403 (2008). withnospinfluctuations. Forq6=0,thepairedphasehasspin [4] L.E.Sadler, J.M.Higbie, S.R.Leslie, M.Vengalattore and fluctuations in the x−y plane. For large q > 0, the normal D.M.Stamper-Kurn,Nature, 443 (2006). state is unstabletowards a polar condensate with spinor ζ = [5] J. Kronj¨ager, C. Becker, P. Soltan-Panahi, K Bongs and {0,1,0}. Dotted lines are the ideal gas condensation T as a K. Sengstock, Phys.Rev.Lett. 105 090402 (2010). c functionofq. (b): Detailsofdashedrectanglein(a),showing [6] J.Stenger,S.Inouye,D.M.Stamper-Kurn,H.-J.Mieser,A. instabilitiestowardspairing. (c): Criticalvaluesofqatwhich P. Chikkatur,and W. Ketterle, Nature 396 345 (1998). pair instability gives way to single particle instability (i.e. [7] Tin-Lun Ho, Phys.Rev. Lett., 81 742 (1998). locations of × and o) for different interaction strengths. [8] T.Ohmi and K.Machida, J. Phys. Soc. Jpn., 67 1822, (1998). [9] Erich J. Mueller, Phys. Rev.A 69, 033606 (2004). condensation in the thermodynamic limit. Here we find [10] S.Mukerjee,C.XuandJ.E.Moore,Phys.Rev.Lett.97, 120406 (2006). only small corrections to location of the threshold value [11] D. Podolsky, S. Chandrasekharan and A. Vishwanath, of c /c . To leading order in kΛ 1, the threshold ftohrr|efse2hr|roold0mfaogrnpetaiisrmingisisa2a/2a/0a0==−2131T+−≪8kkπ1ΛΛ222T2TaπB02E.a0CT,hwesheilsehtifhtes [[1123]]P(2EKh0.y.0s2DM.)eR.umeravlet.raB,a8Hn0d. 2SF1a.4it5Zo1h3aonu(2d,00PM9h).y.sU.eRdaev,.PLheytst..R8e8v.16A307051 areno greaterthan 10%for asystemof sizeL 100µm. 013607 (2007). ∼ Conclusions.— The presence of competing magnetic [14] A. Lamacraft, Phys.Rev.Lett. 98 160404 (2007). and off-diagonal long range orders in a spin 1 gas pro- [15] Q. Gu and R.A. Klemm, Phys. Rev.A.68 031604 (R). duces an extremely interesting phase diagram with fer- [16] Kun Yang, eprint.arxiv: 0907.4739. [17] K. Kis-Szab´o, P. Sz´epfalusy and G. Szirmai, Phys. Rev. romagnetic,nematic and pairedphases. Using the RPA- A 72 023617 (2005). X, we have quantitatively studied the instabilities of the [18] P. Nozi`eres and D. Saint James, J. Phys (Paris) 43 normal state, identifying the temperatures and interac- (1982). tion strengths at which the disordered normal state be- [19] W. A. B. Evans and R. I. M. A Rashid, J. Low Temp. comes unstable to a symmetry broken phase. We find Phys. 11 93 (1973). that the finite temperature phase diagram is featureless [20] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. unless the interaction strengths governing spin (c ) and Rev.108 5 (1957). 2 [21] E. J. Mueller and G. Baym, Phys. Rev. A 62 053605 charge physics (c ) are comparable in magnitude. This 0 (2000). is due to the exchange enhancement of identical particle [22] S. Chakravarty and S. A. Kivelson, Phys. Rev. B 64 scattering. 064511 (2001). A number of probes can be used to distinguish the [23] F. Gerbier, A. Widera, S. Folling, O. 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Jin, Rewriting the logarithm as an integral we get: Phys. Rev.Lett. 94 110401 (2005). m z− z+ (χ0)10(p,ω)= dxI (k) dxI (k) 01 −2πλ2p 1 − 0 SUPPLEMENTARY MATERIALS FOR T (cid:18)Z∞ Z∞ (A-1(cid:19)6) “INSTABILITY OF A SPIN-1 BOSE GAS TO FERROMAGNETISM AND PAIRING” where I1/0(k) = −∞∞dkk−1xek2+β2λk1/0 1, where λ1 = β(µ q) and λ0 =R βµ. The analyti−c structure of the − − − Analytic structure of non-interacting response integral I has been extensively developed by Sz´epfalusy functions.— We develop the analytic structure of χ0βα and Kondor [2], who show that the integral can be writ- αβ defined in Eq.6 of the main text as: ten as an asymptotic series for long wave-lengths. Re- tainingonlythelowestordertermswefindthatthestatic d3k n(ǫ ) n(ǫ ) response yields: (χ0)βα(p,ω)= k,α − k+p,β (A-13) αβ (2π)3 ω (ǫ ǫ ) Z − k+pβ − kα m k T (χ0)10(0,0)= B (g (eβ(µ q)) g (eβµ)) For q = 0 this simplifies to a constant χ(p,ω) = 01 4πλ q 3/2 − − 3/2 T ou(gd2hπ3kl)y3neω(x−ǫpk()lǫ−ok+rnep(dǫ−kǫ+iknp))[,1].wThohseeressturlutcttourleinehaarsorbdeeernintphΛor- The non-interacting pair response is defined as:(A-17) R T is : d3k n(ǫ )+n(ǫ ) χ(p,0)=−2πmλT g1/2(eβµ)− |π/(βµ)|+ (A-14) (Π0)αβαβ(p,ω)=Z (2π)3 ω−k,α(ǫk+pβ +k+ǫkpα,β) (A-18) (cid:16)iπ √βpµ pΛT log − 4√π Once again, the pair susceptibility for the scalar gas pΛT √βµ+ 4p√ΛTπ!! has been consideredin [21]. The result to linear order in kΛ is : T where g (z) is the polylogarithm function. For q = 0, ν 6 first note that χβα(0,0) = χ(0,0) for α,β = m 1, 1 . The tranαsβverse spin respon|µs→eµh−oqwever{given}by Π0(p,0)=−πλ g1/2(eβµ)− |π/(βµ)| (A-19) { − } T (χ0)1001(p,ω)requiressomework. Oneproceedsasfollows: 2iπ (1+i(cid:16))pΛT √βµp pΛ Integrating out the angular variables one finds + log 4√π − + T pΛT (1 i)pΛT √βµ! 8 ! − 4√π − (χ0)1001p(ω)= π−λm2TpZ0∞dk˜n(ǫk˜1)log zz−−++kk˜˜!(A-15) It is easy to show that (Π0) 11−11(0,0) = + n(ǫk˜0)log zz++−+kk˜˜! Π0S(p0i,n0)|rµe→spµo−nqseanind (tΠhe0)R0000P=AΠ.—0. The sp−in response in the RPA is given by solving for the polarization tensor where z = ω+q pΛT and k˜ =kΛ /4√π. We use defined in Eq. 2 using Eq. 5. The interaction matrix V ± 2√ǫpkBT ± 4√π T encompasses all direct and exchange diagrams and takes k˜ as an expansion parameter. the form: 2(c +c ) 0 0 0 c +c 0 0 0 c c 0 2 0 2 0 2 − 0 0 0 c +c 0 0 0 2c 0  0 2 2  0 0 0 0 0 0 c c 0 0 0 2  −   0 c0+c2 0 0 0 2c2 0 0 0   V = c +c 0 0 0 2c 0 0 0 c +c  (A-20)  0 2 0 0 2   0 0 0 2c 0 0 0 c +c 0  2 0 2     0 0 c0 c2 0 0 0 0 0 0  −   0 2c 0 0 0 c +c 0 0 0  2 0 2   c c 0 0 0 c +c 0 0 0 2(c +c )  0− 2 0 2 0 2    Fromthe fullpolarizationtensor,oneextractsthelon- gituginaland transversespin susceptibility on which our 6 calculations are based. Two limiting cases are worth considering. The first We now turn to the details of the pair response calcu- is at µ = q for q < 0 when the non-interacting BEC lation. transition occurs for the 1 atoms. At these values the Pairing response in the RPA.— The RPA response is non-interacting response f±unction Π 11−11 diverges, and: given by Eq. 5 where V is a symmetric 9 9 matrix. − × Howeversincepairingonly occursinthe Sz =0 channel, Θ 1 c +c +(c 2c )(c +c )Π00 (A-22) it suffices to consider the following subsystem − ∝− 0 2 0− 2 0 2 00 RPA The second is at µ = 0 which corresponds to the BEC Π 11−11 Π10−10 Π11−11 transition for the 0 atoms. At this point non-interacting ΠΠ01−−−−111110 ΠΠ−00100010 ΠΠ−−11−−101110  (A-21) response function Π0000 diverges and: which is related to the non-interacting response Θ−1 ∝−c0+(c0−2c2)(c0+c2)Π−11−11 (A-23) Π 11−11 0 0 0− Π00 0 viaEq.5andthe 3 3interac- Setting the R.H.S. of Eqs. (A-22, A-23) to zero, using  00  × Eq.11 along with the functional forms of the response  0 0 Π−11−11  functions,yieldsthethresholdvalueofc2atwhichpaired  c0 c2 c2 0 states result for any q. − tion matrix V = c c c .  2 0 2  0 c c c 2 0 2  −  Note that Π 11−11 = Π−11 11 and the physical response 2oΠf−t11h−2e11s.syysTsttehemem3ot−fo×eaq3dudsayitnsiogtenmas −±wch1aincphabietrhfieusnrgtyihvieeernldrsbeydthuΠece−sd11in−tg11ole+at [1∗] EE(2.l0e0cJt0.r)o.Mniucelalderdraenssd: sGsn.8B@acyomrn,elPl.hedyus. Rev. A 62 053605 × RPA response. [2] P. Sz´epfalusy and I. Kondor, Ann.Phys. 82 1 (1974).