Pairing, Ferromagnetism, and Condensation of a normal spin-1 Bose gas

0.19 MB·English

by  Stefan S. Natu

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.,
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⊥
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c
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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
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∼
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2
[21] E. J. Mueller and G. Baym, Phys. Rev. A 62 053605
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[26] J. W. Negele and H. Orland, Quantum Many-Particle
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5
[30] M. Greiner, C. A. Regal, J. T. Stewart and D. S. 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).