Fermionic superfluidity: From high Tc superconductors to ultracold Fermi gases

0.22 MB·English

by  Qijin Chen

Fermionicsuperfluidity: From high Tcsuperconductors to ultracoldFermigases
Qijin Chen, Chih-Chun Chien, Yan He, and K. Levin
JamesFranckInstituteandDepartmentofPhysics,UniversityofChicago, Chicago, Illinois60637, USA
(Dated:February6,2008)
Wepresentapairingfluctuationtheorywhichself-consistentlyincorporatesfinitemomentumpairexcitations
in the context of BCS–Bose-Einstein condensation (BEC) crossover, and we apply this theory to high T
7 c
superconductors and ultracoldFermi gases. There arestrongsimilaritiesbetween Fermigases intheunitary
0
regimeandhighTcsuperconductors. Hereweaddresskeyissuesofcommoninterest,especiallythepseudogap.
0
In the Fermi gases we summarize recent experiments including various phase diagrams (with and without
2
populationimbalance),aswellasevidenceforapseudogapinthermodynamicandotherexperiments.

n
a Keywords:highTcsuperconductor,ultracoldFermigas,superfluidity,Feshbachresonance
J

9
Superfluidityinultracoldtrappedfermionicgaseshasbeen spin states, Uk,k′ = Uϕkϕk′, where U < 0 is the coupling
] oneofthemostexciting,rapidlyevolvingresearchsubjectsin strength. For the cuprates, ǫ = 2t (2−cosk −cosk ),
l k k x y
e the past few years. [1, 2, 3, 4, 5, 6]. It has captured the at- andforatomicFermigasesǫ = k2/2m; µisthefermionic
- k
r tentionofbothcondensedmatterandatomicphysicscommu- chemicalpotential,t isthein-planehoppingintegral,andwe
k
st nity, aswell asotherfieldssuch asnuclearandquarkmatter have set the lattice constant to 1. Here and throughout, we
t. [7]. AtomicFermigaseshavestrongsimilaritiestosupercon- assumenaturalunitsandset~ = kB = c = 1. Thefunction
a ductors. Breakthroughsincoolingtechniquesinrecentyears ϕ reflects the pairing symmetry. For the cuprates, ϕ =
m k k
havemadeitpossibletocoolanatomicFermigastoquantum cosk −cosk ;fortheFermigases,theinteractionisshort-
x y
- degeneracytemperatureand thusform a superfluid. What is ranged,andtakentobeacontactpotentialsothatϕ =1.
d k
remarkable about the atomic Fermi gases is their tunability TheHamiltonianisgivenby
n
andcontrollability.UsingaFeshbachresonance,onecantune
o
c the attractive two-bodyinteraction fromweak to strong, and H − µN =Xξka†kσakσ
[ therebymakeasmoothcrossoverfromaBCS typeofsuper-
kσ
2 flcruoisdsotovearBhaosseb-eEeinnsotefinintceornedstentosatthioenco(BnEdeCn)se[8d]m. BatCteSr-cBoEmC- +X Uk,k′a†k+q/2↑a†−k+q/2↓a−k′+q/2↓ak′+q/2↑,
v kk′q
8 munityforalongtime,andhasbeenargued[9]toberelevant
(1)
6 tohighT superconductivity.
c
2 Most of the theories that are proposed to explain high Tc where a†(a) is the creation (annihilation) operator of the
2
superconductivityarebasedonthepairingofelectrons,asin
fermions. This picture is often referred to as a one-channel
1
BCStheory. Whilethepairingmechanisminthecupratesu-
6 model.ThephysicsofaFeshbachresonancecanbedescribed
perconductorsis far from being clear, there is no ambiguity
0 byatwo-channelmodel,whichincludesanopen-channel,as
/ aboutthe pairing interaction in atomic Fermi gases. The at- in the one-channelmodel, and a closed-channel, which rep-
t
a tractivepairingpotentialbetweenthefermionicatomsofdif- resentsthe effectsofdi-atomicmoleculesof a differenttotal
m ferentspinshasbeenwellcharacterizedandcanbecomputed
spin. Thetwo-channelHamiltonianisgivenby
fromfirstprinciplequantummechanicscalculations.Herewe
-
d presumethatcupratesuperconductivityoriginatesfromelec- H − µN = ξ a† a + (ǫ +ν−2µ)b†b
n tron pairing, so that studying superfluidity in atomic Fermi X k k,σ k,σ X q q q
o k,σ q
gasesmayprovideinsightsintothehighT superconductivity
:c orviceversa[9,10,11]. c + X U(k,k′)a†q/2+k,↑a†q/2−k,↓aq/2−k′,↓aq/2+k′,↑
v Importantly,preformedpairsareaninevitableconsequence q,k,k′
i
X as the pairing interaction increases. At the unitary scatter- + g(k)b†a a +h.c. , (2)
inglimit,thesuperfluidtransitiontemperatureT isveryhigh X(cid:16) q q/2−k,↓ q/2+k,↑ (cid:17)
r c q,k
a in unitsof the noninteractingFermienergyE . The atomic
F
Fermigasesoftenhavearelativelyhighertransitiontempera- where b† (b) is the creation (annihilation) operator for the
turethanthecuprates.ForthelatterTc .0.1EF issomewhat closed-channelmolecules,whosedispersionisgivenbyǫq =
lower,owinginparttotheirlowdimensionality.Exceptinthe q2/2M with M = 2m. Here ν represents the magnetic de-
extremeBCSlimit,preformedpairscontributeafermionicex- tuning,whichisusedtotunetherelativeenergylevelsplitting
citationgap,i.e.,apseudogap,eveninthenormalstate. Such between the open- and closed-channels and thus change the
apseudogapiscommontobothhighT superconductorsand overall effective interaction; the interaction increases as the
c
atomicFermigases. field decreases. The last line in Eq. (2) represents the cou-
In the absence of a population imbalance, superfluidity is plingbetweenthetwochannels,withg(k)=gϕ .
k
associatedwithagasoffermionswithdispersionξ = ǫ − Current studies of atomic Fermi gases have concentrated
k k
µ,subjectedtoanattractiveinteractionbetweentwodifferent on 6Li and 40K. For both gases, the Feshbach resonances
2
whichareusedtoenhancethepairinginteractionhaveavery givenby
big width. As a consequence, the closed-channel fraction
becomes negligible [6, 12] even though the closed channel ∆2 =∆2sc+∆2pg, (4)
makes a big contribution to the effective pairing interaction.
Forexample,for6Li,bothexperiments[6]andourtheoretical and the fermionic quasiparticle dispersion becomes Ek =
ξ2+∆2ϕ2. The fact that the gaps add in quadrature can
calculations [12] have demonstrated that the closed-channel bpeuknderstookdfromtheobservationthat∆2 and∆2 arepro-
populationisextremelysmallintheunitaryregime(oftheor- sc pg
derof10−5−10−4)andremainssmallintheentirerangeof portionalto the density of the condensed and noncondensed
pairs,respectively.
magneticfield(i.e.,pairingstrength)whichisaccessedexper-
The effective dispersion Ω and chemical potential µ
imentally. q pair
canbedeterminedbyaTaylorexpansionoft−1,which,after
Theneteffectofthistinyclosed-channelfractionisthat,for pg
manypurposes,one canuse the one-channelmodel, Eq. (1), analyticalcontinuation(iΩn →Ω+i0+),canbewrittenas
providedonereplacesU withtheoveralleffectiveinteraction,
t−1 =Z(Ω−Ω +µ ), (5)
Ueff(Q)≡U+g2D0(Q),whereD0(Q)≡1/[iΩn−ǫmqb− pg q pair
ν +2µ]isthebarepropagatoroftheclosed-channelbosons.
whereZistheinverseresidue[11],andwehaveneglectedthe
Experimentally,thepairinginteractionforatomicFermigases
smallimaginarypart[11].Thepairinginstabilitycondition,or
isusuallyparametrizedbythedimensionlessproduct,1/k a,
F equivalentlytheBECconditionforpairs,requires
whereaisthes-waveinter-fermionscatteringlength,andk
F
istheFermiwavevectordefinedinthenoninteractinglimitat t−1(0)=0, (6)
T =0,satisfyingE =k2/2m. pg
F F
InthiswaybothhighT superconductorsandatomicFermi so that we have µ = 0 below T . One can readily see
c pair c
gasescanbedescribedbythesameHamiltonianbutwithdif- that ∆ → 0 as T → 0. Therefore, this finite T theory is
pg
ferent pairing symmetries and free-particle dispersions. The consistentwith the BCS-Leggettgroundstate [8], whichhas
existence of a big pseudogapand the shortcoherencelength beenwidelyassumedintheliterature.
inthecupratessuggestthattheyareintheintermediateregime The presence of the pseudogapnecessarily suppresses the
between BCS and BEC [9]. This is often referred to as the superfluid transition T . This can be seen from the relation-
c
“strongly interacting regime” in atomic Fermi gases, and is ship between ∆ and T, as determined by a mean-field BCS
themostcomplexandinterestingregimeaswell. gapequation:T decreasesas∆increases,andT simplycor-
c
Our pairing fluctuation theory uses a T-matrix formal- respondstothelowestT atwhich∆=∆ .
pg
ism. Instead of providing a detailed derivation, which can Our theory can be summarized in three self-consistent
be found in Refs. [10, 11], here we recapitulate the key in- equations: the gapequation(6), the pseudogapequation(3),
gredients and observations of this formalism. Pairs consist andthenumberequation
offermionswhichgluetogetherviatheattractiveinteraction,
and the pair propagator represents a summation of multiple n=2 G(K). (7)
X
particle-particlescattering processes. Fermionsacquire self- K
energy via creation and destruction of a pair and a hole. In
They can be used to solve for ∆, ∆ , ∆ , µ and T as a
BCS theory, such pairs would be just the Cooper pairs in pg sc c
functionofU (orU )or1/k aandT.
the condensate, and they contributea self-energyΣ (K) = eff F
−∆2scG0(−K)ϕ2k, where ∆sc is the order paramsecter and tanTthoebpserervsaetnitonthse.or(yi)nTahtuerafullnydlaemadesnttaoltshteatfiostlilcoawlienngtiitmiepsoirn-
G0(K) is the bare fermion Green’s function. [Throughout,
weuse afour-vectornotationasinRef.[10], K ≡ (k,iω ), thesesuperfluidsarefermions. Wemeasurethe“bosonic”(or
n pair)degreesoffreedomindirectlyviathefermionicgappa-
= T ,etc.,andω istheMatsubarafrequency.]
PK PnPk n rameter ∆(T). (ii) As we go from BCS to BEC, pairs will
In general, finite momentum pairs contribute to the fermion
self-energyviaΣpg(K)=PQtpg(Q)G0(Q−K)ϕ2k,where bmeagtiinontoisfoarsmsocaitataedtemwiptheraatunroermTa∗lasbtaotveepTsecu.dToghaisp.pa(iirii)foIrn-
t (Q)=1/[1+U (Q)χ(Q)]istheT-matrix(pairpropa-
pg eff general there will be two types of excitations in these BCS-
gator), with χ(Q) = PKG0(Q −K)G(K) being the pair BECcrossoversystems. Importantlyintheintermediatecase
susceptibility and G(K) the full fermion Green’s function.
theexcitationsconsistofamixofbothfermionsandbosons.
BelowT ,t (Q)divergesatQ = 0sothatwecanapproxi-
c pg It should be pointed out that effects of pairing fluctua-
mateG0(Q−K)≈ G0(−K)inΣpg. Asaresult,weobtain tionswerefirstconsideredbyNozie´resandSchmitt-Rink[13].
Σpg(K)≈−∆2pgG0(−K)ϕ2k,with However, they only included them in the particle number
∆2 ≡− t (Q). (3) equation; the absence of these effects in the gap equations
pg X pg necessarilyleadstointernalinconsistencies,say,incomputing
Q thesuperfluiddensity.Incontrasttothepresentapproach,the
Herethekeyobservationisthat,underareasonableapproxi- NSRschemealsopredictsthatT approachesitsBECasymp-
c
mation,the self-energyofthe finitemomentumpairshasthe totefromaboveforans-wavesuperfluidinthree-dimensional
sameformasΣ (K). Sincethesefinitemomentumpairsdo (3D) continuum space. The NSR treatment, due to its rela-
sc
nothavephase coherence,theircontributionslead toa pseu- tivesimplicity,hasbeenusedbyothersintherecentliterature
dogapinthefermiondispersion.Thetotalexcitationgap∆is [14,15].
3
70 8
T
c
60 ∆(T) 8.9 K
c
∆(0) 6 13.7 K
T* 15.5 K
50 )
meV) 40 T∆c(0) -2µ(m 4 1272..60 KK
T* 2
( λ
y Normal 1/
g 30
r
e 2
n
E
20
P.G.
0
10 0 5 10 15 20 25
T (K)
S.C.
0
0.0 0.1 0.2 0.3
x 4
Figure1: Cupratephasediagramshowing∆(0),T ,∆ (T ),and 3
c pg c
T∗, calculatedfor−U/4t0 = 0.047, andt⊥/tk = 0.003. Shown -2m)
assymbolsareexperimentaldata. Thenormal, pseudogap, andsu- µ
perconductingphasesarelabeledwith“Normal”,“P.G.”,and“S.C.”, 2! (2
respectively.FromRef.[16]. 1/
1
Aself-consistenttreatmentoffinite-momentumpairexcita-
tionsiscrucial. ThisismoresoinatomicFermigases,where
onecannotdirectlymeasurethetemperatureexceptintheex- 0
0 2 4 6 8 10 12 14 16
tremeBCSandBEClimits. Quantitativelyreasonableknowl- Temperature (K)
edgeoftemperatureeffectsisnecessaryinordertodetermine
temperatureand relatedphysicalpropertiesin the BCS-BEC Figure2: Comparisonofsuperfluiddensitybetweentheoryandex-
crossoverregime. periment.Toppanel:Theoreticalcalculationforunderdopedcuprate
WefirstapplyourpairingfluctuationtheorytohighT su- superconductors;Bottompanel:ExperimentaldatafromRef.[18]in
c
theextremeunderdopedregimeofYBCO.
perconductors. In Fig. 1 we plot the calculated phase dia-
gram for Bi2Sr2CaCu2O8+δ (Bi2212) and YBa2Cu3O7−δ
(YBCO)singlecrystals,aswellaspresentacomparisonwith
experimentaldata.Thehorizontalaxisistheholedopingcon- ducesa lowercriticaldopingx inthephasediagramshown
c
centrationperunitcell,x=1−n. Hereweshowtheoretical inFig.1.
calculationsand experimentaldata for zero T excitation gap Other applications of BCS-BEC crossover theory to the
∆(0),superconductingtransitiontemperatureTc,andpairing cuprate superconductors include studies [10] of the quasi-
onsettemperatureT∗. Wealsoshowthetheoreticalresultfor universal behavior of the normalized superfluid density
∆(Tc),thepseudogapatTc. ns(T)/ns(0) as a function of T/Tc for various doping con-
To obtain this phase diagram, we fit with one parameter centrations. Experimentaldata from Ref. [18] are plotted in
−U/4t0 such that it gives the right value of Tc at optimal the lower panel of Fig. 2 for a series of very underdoped
doping around x = 0.15. Here t0 is the bare in-plane elec- cuprates. The universality apparent in these data presents
tronhoppingmatrixelement, in theabsenceof theCoulomb a puzzle for conventional theories of the penetration depth
repulsion. It is related to tk via tk ≈ t0x. One would ob- which invoke only fermionic excitations of the condensate.
tain essentially the same phase diagram, if one parametrized This is because the fermionic contributions are expected to
the variable attractive interaction by simply fitting T∗(x) to reflect the strong variationsin the excitationgap ∆(x) asso-
experiment. Thus, the underdopedregime sees effectively a ciatedwith thedifferentholeconcentrationsx, therebylead-
stronger pairing interaction, −U/4t . It is evident that our ing to highly non-universal behavior. In the theory plot of
k
simple model gives a (semi-)quantitatively satisfying phase the upper panel (Figure 2) bosonic or pair excitations con-
diagram,inagreementwithexperiment. tributeanextramechanismfordestroyingsuperfluidity. They
Itshouldbenotedthat,incontrasttothes-wave,3DFermi compensateforthedecreaseofthequasiparticlecontribution
gas, the d-wave symmetry and the lattice periodicity in the (∝ T (x)/∆(x)) as the system evolves from overdoping to
c
cuprates makes T vanish before the BEC regime can be underdoping. As a result, the slope of the superfluiddensity
c
reached as the couping strength increases [17]. This intro- remainsnearlydopingindependent,withonlyasmallsystem-
4
0.25
N/N= Theory, noninteracting
0
-0.020 Theory, unitary
0.20 4
0.010 Noninteracting
0.025 Unitary
0.15 0.050
0T)F 0.075 EF
T/(0.10 0.100 E/
0.125 2
0.150
0.05 0.175
T*
0 T =0.29
1.0 0.5 0 1/k0a-0.5 -1.0 -1.5 0 c
F 0 0.5 1 1.5
T/T
F
Figure3: (color)Phasediagramof40K.Acontourplotofthemea-
suredcondensatefractionN0/N asafunctionof1/kF0aandeffec- Figure 4: (color) Energy per particle as a function of T for 6Li at
tivetemperature(T/TF)0iscomparedwiththeoreticallycalculated unitarity.Thefactthattheexperimentaldata(symbols)(andthetwo
contourlinesatNs/N =0(blackcurve)and0.01(redcurve). The theoreticalcurves)fornoninteractingandunitaryFermigasesdonot
overalltrendoftheexperimentalcontourofN0/N = 0.01andthe merge until higher T∗ > Tc is consistent with the presence of a
theoreticallineforNs/N =0.01areingoodagreement.Thedashed normalstatepseudogap. FromRef.[25].
linerepresentsthenaiveBCSresult,T /T0 ≈0.615eπ/2kF0a.From
c F
Ref.[21].
the(red)theorycurve. Herewe [21] identifythecondensate
fractionwith the superfluiddensity. The agreementbetween
aticvariationwithx. Thecomparisonbetweentheoryandre- experimentandtheoryisreasonablygood. Itshouldbenoted
centexperimentsfromtheUBCgroup[18]showsquitegood thatbothexperimentandtheoryshowamaximumarounduni-
agreement.Recently,thebosonicnatureofthesuperfluidden- tarity,1/k0a=0,whichhasalsobeenobservedin6Liphase
F
sityinunderdopedcuprateshasalsobeenrecognizedbyother diagrams[3].
groups[19,20]. Wenextturntoadetailedcomparisonoftheoryandexper-
We turn now to experiments in cold atom systems. Here iment for thermodynamics. Figure 4 presents a plot of en-
we [11, 22] need to include the trap potentialV(r). To this ergy E as a function of T comparing the unitary and non-
end, we use a localdensity approximation(LDA) by replac- interacting regimes for 6Li. The solid curves are theoretical
ing µ → µ(r) ≡ µ−V(r). Here µ ≡ µ(0) is the global while the symbols are experimental data [25]. A realistic
chemicalpotentialandV(r)=mω¯2r2/2foraharmonictrap Gaussian trap with the experimentally given trap depth was
withmeanangularfrequencyω¯. Thenumberconstraintenters used. Therehasbeenarecalibrationoftheexperimentaltem-
fortheentiretrap,N = d3rn(r). BelowT ,thereexistsa peraturescale [25] in order to plottheory and experimentin
c
R
superfluidcorewithinr <R . Forr >R ,µ <0,and thesamefigure.ThelatterwasdeterminedviaThomas-Fermi
sc sc pair
the gap equation has to be modified to reflect this non-zero fits to the density profiles [22]. To arrive at the calibration,
µ :t−1(0)=Zµ . weappliedthesamefitstothetheoreticallyproduceddensity
pair pg pair
Beforewecomparewithexperiment,wehavetoknowthe profiles.
temperature of the system. Experimentally, temperature is Goodagreementbetween theoryand experimentis appar-
measuredattheBCS[1]orBECend[23],beforeorafterthe entin Fig.4. Itshouldbeemphasizedthatthereis nofitting
experiment is performed in the strongly interacting regime. parameter in our calculations. The temperature dependence
ThisBCSorBECstateisadiabaticallyconnectedtothestate of E reflects primarily fermionic excitations at the edge of
whereexperimentsaredoneviaaslowmagneticfieldsweep. thetrap[24],althoughthereisasmallbosoniccontributionas
By including self-consistently finite momentum pair excita- well. Importantlyonecanseetheeffectofapseudogapinthe
tions,ourpairingfluctuationtheoryenablesustocalculatethe unitarycase. Thatis, the temperatureT∗ is visible from the
entropyS(T)atanarbitrarymagneticfield.Thenwecanmap plots as that at which the non-interactingand unitary curves
thephysicaltemperatureT ontothatmeasuredintheBCSor merge. ThiscorrespondsroughlytoT∗ ≈ 2T . Evidenceof
c
BEClimit,T0,orviceversa[24]. apseudogapatunitarityhasalsobeenobservedmoredirectly
InFig.3,weshowthephasediagramfor40Kandcompare inaradio-frequencyexcitationgapexperiment[23].
ourcalculationswithexperimentaldata[21]. Herealltemper- Finally, we turn to the effects of population imbalance,
aturesaremeasuredin(orconvertedinto)thefreeFermigas whichhaverecentlybeenoneofthehottestsubjectsinatomic
limit. Inaddition,T0 ≡ k0/2mistheFermitemperatureat Fermigases.Weconsiderboththehomogeneoussystem[26],
F F
the trap center in the noninteractinglimit. The experimental aswellastrappedcase,whichcanbefoundinRefs.[27,28].
superfluidphaseboundaryisgivenbythe1%condensatefrac- Wedefinen = n +n ,δn = n −n ,andthepolarization
↑ ↓ ↑ ↓
tioncontourline;itishardtolocateexperimentallywherethe p = δn/n. There will be two number equations associated
condensatevanishesprecisely.Thisshouldbecomparedwith with the two spin species. The polarizedsuperfluid phase is
5
1 pearsconsistentwith experiment. Moredetailscanbefound
d(k a)/dT = 0 inRef.[26].
F c
0.8 ∂2Ω/∂∆2 = 0 at Tc Finally,weshowinFig.6thephasediagramatunitarityfor
∂2Ω/∂∆2 = 0 0 = 0 a population-imbalancedFermi gas in a harmonic trap with
0.6 TTccM =F =0 0 T= c )∂/∆2 banelgeudlaPrSf)reoqcuceunpciyesωth.eHleorwee1r/TkFpaor=tio0n.oPfhtahseepsheapsaeradtiiaognra(lma-,
Tc
Ω( wherethegap∆jumpsabruptlyto zeroatsometrapradius.
p 0 ∂2 At intermediate T, there is a (yellow-shaded) Sarma phase,
0.4 F = where ∆ vanishes continuously within the trap. It evolves
TMc = 0 intoa(dottedregion)pseudogap(PG)phaseasthesuperfluid
∆2 corevanishesathigherT.Anormal(N)phasewithoutpairing
0.2 ∂
da/dTc=0 ∂2Ω/ always exists at even higherT. This phase diagram appears
tobeconsistentwithcurrentexperiments. Onecaneasilysee
0 thatthephysicsassociatedwithpopulationimbalanceismuch
-3 -2 -1 0 1 2 3
richerthanintheabsenceofimbalance.
1/k a
F
In conclusion, in this paper we have addressed common-
Figure 5: (color) Phase diagram in the p − 1/k a plane, show- alities particularly associated with the phase diagrams of ul-
F
ingwhereintermediatetemperaturesuperfluidityassociatedwiththe tracoldtrappedfermionicgasesandhighTcsuperconductors.
Sarma phase (shaded region) exists. The (red) curve labeled by These common features revolve around the scenario [9, 11]
∂2Ω(T )/∂∆2 = 0 is determined at corresponding T . The line that the cuprates are somewhere intermediate between BCS
c c
defined byda/dTc = 0isgiven bytheturning points (p,1/kFa) and BEC. Importantly, this scenario has been directly real-
where1/kFareachesalocalextremumasafunctionofTc.AtT = izedintrappedFermigases. Hereoneseesconsiderableevi-
0,theentireregionbetweentheTMF = 0andthe∂2Ω/∂∆2 = 0
c denceforpre-formedpairsandtherelatedfermionicpseudo-
linesisunstableagainstphaseseparation. However, aAstablepo-
gapwhichappearreminiscentoftheircupratecounterparts.
larized,intermediatetemperaturesuperfluidphaseexistsforthe(yel-
ThisworkwassupportedbyNSFPHY-0555325andNSF-
low)shadedregion. Homogeneous superfluidsinthedottedregion
areunstableatanytemperature.FromRef.[31]. MRSECGrantNo.DMR-0213745.
0.3
conventionallycalledthe“Sarma”phase[29],althoughhereit
isgeneralizedawayfromtheBCSlimitoriginallyconsidered N
PG
bySarma.DetailsaregiveninRef.[26].
InFig.5,wepresentthezeroT phasediagramforapopu-
0.2
lationimbalancedhomogeneousFermigasinthep−1/k a
F
plane. Here TMF is the solutionof T in a strictmean-field Sarma
c c F
treatment. Ω is the thermodynamic potential. A superfluid T/T TCP
solution is stable only if ∂2Ω/∂∆2 > 0. From this figure,
we see that at any p 6= 0, there is a minimum threshold for 0.1
1/k a (determined by TMF = 0) in order to have a super-
F c
fluidsolution.Thepointatwhichthesolutionbecomesstable PS
(determinedby ∂2Ω/∂∆2 = 0) occursin BEC regime. The
(blue)dashedlineindicateswhereTcvanishes.Thedottedre- 00 0.2 0.4 0.6 0.8 1
gion(as wellas the shadedregion)in the figureindicatesan p
unstable(Sarma)superfluidphaseatT =0.Thestablephases
inthisregionarepresumedtobeeitherphaseseparated(PS) Figure6: (color) Phasediagramofapopulation-imbalanced Fermi
orpossiblyaFulde-Ferrell-Larkin-Ovchinnikovstate[30]. gasinaharmonictrapatunitarity. Thesolidlinesseparatedifferent
In addition to the zero T phases, we also show in Fig. 5 phases. Abovethe(red)dashedlinebutwithinthePSphase,thesu-
where the system is stable at T . A stable finite tempera- perfluidcoredoesnottouchthedomainwall. Here“PG”indicates
c
turepolarizedsuperfluidexistsinthe(yellow)shadedregion. thepseudogappednormalphase. Theblackdotlabeled“TCP”indi-
catesthetricriticalpoint.
This is a very interesting predictionof our theory which ap-
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