Density, spin, and pairing instabilities in polarized ultracold Fermi gases

3.4 MB·English

by  Inti Sodemann

Density, spin, and pairing instabilities in polarized ultracold Fermi gases
Inti Sodemann, D. A. Pesin, and A. H. MacDonald1
1Department of Physics, University of Texas at Austin, Austin TX 78712 USA
(Dated: January 17, 2012)
We study the influence of population imbalance on the pairing, spin, and density instabilities of
a two component ideal Fermi gas after a sudden quench of interactions near a Feshbach resonance.
Overalargeregionofparametersthepairinginstabilityisdominatedbyfinitemomentumpairing,
suggesting the possibility of observing FFLO-like states in the unstable initial dynamics. Long-
wavelength density instabilities are found on the BCS side of the resonance, and are interpreted as
aprecursorofthephaseseparationexpectedatequilibrium. OntheBECsideoftheresonance,the
2
1 pairing instability is present for scattering lengths that are larger than a critical value that is only
0 weakly dependent on population imbalance and always smaller than the scattering length at which
2 the Stoner-like spin instability occurs.

n
PACSnumbers: 05.30.Fk,03.75.Ss,67.85.Lm,75.10.Lp
a
J

3 I. INTRODUCTION taining two fermionic species with unequal populations
1 that is prepared in an ideal Fermi gas state and placed

] Ultracoldatomsystemshostawebofphenomenathat near a broad Feshbach resonance. We follow the ap-
s havemadeitpossibletostudyquantummanybodyprob- proachofPekkeret al.[7],generalizingtheirworkforthe
a lems in new ways that are often remarkably revealing. case of balanced populations to the case of imbalanced
g
- For example, studies of fermionic atoms have enabled ones.
nt a comprehensive investigation of the crossover between WefindthatontheBCSsideoftheresonancethedom-
a the BCS and BEC limits of the superfluid phase [1]. inant instabilities are pairing and density phase separa-
u Crossover physics studies have been further enriched by tion instabilities. We also observe that in the scattering-
q exploring the fate of the superfluid phase as a function length/spin-polarization space FFLO finite-wavevector-
t. of the population imbalance between the species [2] and pairing fluctuations are more prominent than FFLO
a there is now a fairly complete understanding of the equi- states are in the equilibrium phase diagram. This sug-
m
librium phase diagram of this system [3]. gests the possibility of observing FFLO-like features in
d- One goal of ultracold fermion research is to shed light the initial dynamics of an ultracold gas after a quench.
n on the physics of interacting electrons. Atomic systems Our results for the density instabilities on the BCS side
o can often be prepared with less uncontrolled disorder are in qualitative agreement with previous studies of un-
c andmorecontroloverthesystemparametersthatdeter- stable collective modes in polarized systems [9]. On the
[ mine interaction strengths, facilitating comparisons be- BEC side we find that the critical scattering length for
1 tween theory and experiment. However the fact that in- the appearance of a long-wavelength spin-density insta-
v teractions between electrons are normally dominated by bility remains larger than the critical scattering length
2 Coulomb repulsion, whereas interactions between atoms to trigger the pairing instability. This finding indicates
7 are attractive at the densities of cold-atom systems, can that a spin-density instability brought about by effec-
9 stand in the way of these quantum simulation efforts. tive repulsions, which attempts to phase separate spin
2
A particular challenge is the simulation of itinerant species, if observed, will necessarily be accompanied by
.
1 electronic magnetism, which is a combined consequence substantial pair binding into Feshbach molecules even in
0 ofstrongrepulsiveinteractionsandFermistatistics. The the polarized gas.
2
effectiveinteractionstrengthbetweenlow-kinetic-energy, Ourpaperisorganizedasfollows. InSec.IIwediscuss
1
: low-density atoms is strongly repulsive only when their the initial pairing instabilities in a spin-polarized Fermi
v attraction supports a shallow bound state. This obser- gas placed near a Feshbach resonance. In Sec. III we
Xi vation motivated an experiment [4] which explored the turn to an analysis of longitudinal spin and density in-
possibilityofdrivingferromagneticinstabilities[5]byus- stabilities, which are coupled in spin-polarized systems.
r
a ing a Feshbach resonance to suddenly increase the effec- In Sec. IV we present a qualitative analysis of the in-
tiverepulsiveinteractionstrength. However,theanalogy stabilities of the transverse spin densities. Finally, we
with electrons is incomplete, mainly because of the pos- conclude in Sec. V with a brief summary and discussion
sibility to form weakly bound Feshbach molecules out of of our findings.
pairs of atoms. Indeed, recent experiments [6] and theo-
retical studies [7, 8] indicate that molecule formation is
substantial even during the initial dynamics. II. PAIRING INSTABILITIES
Motivated by these studies, in the present theoretical
work we study the initial linearized dynamics of density, Near a broad Feshbach resonance the vacuum interac-
spin, and pairing fluctuations of a system of atoms con- tion between low-energy fermions that are distinguished
2
by a quantum number (referred to below as spin) is con- effective potential to describe the physics near the reso-
trolled only by their mutual scattering length. As a con- nance, provided it is tuned so as to reproduce the phys-
sequence,anypotentialwhoserangeismuchsmallerthan ical scattering length. The effective Hamiltonian of the
the average interparticle separation can be used as an system can be written in the form
(cid:88) 1 (cid:88) (cid:90)
H = ξ (k)c† c + drdr(cid:48)v(r−r(cid:48))ψ† (r)ψ† (r(cid:48))ψ (r(cid:48))ψ (r), (1)
σ kσ kσ 2 σ1 σ2 σ2 σ1
k,σ σ1,σ2
where ξ (k) = k2/2m−(cid:15) , and v(r−r(cid:48)) is an appro- beexpressedintermsofthescatteringamplitudeinvac-
σ Fσ
priatepseudopotential. Byperformingaladdersum,the uumandtheoccupationnumbersofFermiseastates[7]:
scatteringamplitudeintheFermisea(theCooperon)can
C−1(E,P)= m(cid:16)1 +i(cid:114)m(E+(cid:15) +(cid:15) )− P2 (cid:17)+(cid:90) d3q n↑12P−q+n↓12P+q , (2)
4π a F↑ F↓ 4 (2π)3E−ξ −ξ
↑1P−q ↓1P+q
2 2
where E = ω +ω is the total energy measured with the resonance), and grows most rapidly for pairs with
↑ ↓
respecttothesumofthechemicalpotentialsofthescat- zero total momentum. Interestingly, the instability sur-
tering pair and P = k +k is their total momentum. vives over a finite region on the BEC side of the Fesh-
↑ ↓
Particles of the same spin do not interact directly. bach resonance where the scattering length is positive.
Within time dependent mean-field theory it can be The absence of an instability for small positive scatter-
shownthatthepolesofthesusceptibilityassociatedwith ing lengths signals the metastability of the gas with re-
the pairing amplitude ∆ = (cid:80) (cid:104)c† c† (cid:105) coin- spect to its pair-fluctuation channels. For a polarized
P k 1P+k↑ 1P−k↓
2 2 gas we find that the instability can be dominated by fi-
cide with those of the Cooperon [7, 10]. The presence of
nite momentum pairing. This does not imply that the
poles in the upper half of the complex energy plane sig-
equilibriumstateisaFFLOcondensateoffinitemomen-
nals pairing modes that grow exponentially during the
tum pairs, but does indicate that the initial dynamics of
initial dynamics after a sudden quench of the interaction
pairingmodesisdominatedbyfinitemomentumpairing.
potential. The equation C−1(E,P)=0 thus determines
The large area of the region in the k a-δ phase di-
the structure of unstable pairing modes, with the imag- F
agram dominated by pairing at finite momentum con-
inary part of the pole energy for a given momentum,
trasts with the one associated with the FFLO phase in
Im(E ), revealing the growth rate of the unstable mode
P
the equilibrium phase diagram (The FFLO state is be-
in question.
lieved to occupy a very small region of the equilibrium
The implications of this criterion for pairing instabil-
phase diagram for a gas in three dimensions [3]). It falls
ities in initial dynamics are summarized in Fig. 1. We
roughly in the region of the equilibrium phase diagram
have parameterized the interaction strength by the di-
in which phase separation between superfluid and excess
mensionless product k a, where we define k for a po-
F F
majority-spinfermionsisexpected[3]. Therefore,ifapo-
larized gas as the Fermi momentum of an unpolarized
gas with the same total density, i.e. 2k3 = k3 +k3 . larizedsystemispreparedasanon-interactinggasonthe
F F↑ F↓
BCSsidefarfromtheresonance,andtheinteractionsare
In the latter expression k are the Fermi momenta of
F↑/↓
rapidlyswitchedtotheregionwherethepairinginstabil-
the two species of fermions. We also define a population
ity exists, our results suggest the possibility of realizing
imbalance parameter
states with substantial finite momentum-pairing during
n −n k3 −k3 theinitialdynamics. Animportantcaveattothispicture
δ = ↑ ↓ = F↑ F↓, (3)
n +n k3 +k3 isthatthepairinginstabilityisexpectedtocompetewith
↑ ↓ F↑ F↓ density instabilities associated with phase separation, as
Fig. 1 illustrates how the fastest growing pairing modes we will discuss in the next section.
depend on these two parameters. When approached from the BEC side, the pairing in-
Foranunpolarizedgasthepairinginstabilityispresent stability onset is only weakly dependent on spin imbal-
at all negative scattering lengths (on the BCS side of ance. Theinstabilityboundarycanbeinterpretedasthe
3
(a) instability on the BCS side. In particular, the maximum
energy of two holes created in an unpolarized Fermi sea
1.0
is k2/m, whereas for the fully polarized Fermi sea it is
δ F
k2 /2m = 2−1/3k2/m. Thus the ratio of the critical
0.8 F↑ F
BCS BEC scattering length at which pair formation occurs in an
unpolarized gas to the corresponding critical scattering
0.6
length in a fully polarized (FP) gas is only slightly less
P = 0 P = 0 than one,
0.4
a
0 ∼2−1/6 ≈0.89. (5)
a
0.2 FP
The actual ratio obtained directly from the Cooperon is
-1.5 -1.0 -0.5 0.0 0.5 1/k a 1.0 a0/aFP ≈ 0.75, with 1/kFa0 ≈ 1.07 and 1/kFaFP ≈
F
0.80. When approached from the BCS side the pairing
(b) (c) instability at full polarization does not occur until the
Feshbach resonance is crossed. At full polarization, we
0.8
Im(E M )/EF EF0.5 can perform a similar estimate for the critical scatter-
0.6 E )/p0.4 ing length. The pair binding in this case occurs at fi-
0.4 Im(0.3 nite momentum, between particles at the Fermi surface
0.2
0.2 of the majority spin species and minority spin particles
0.1
with zero momentum, so that the momentum of the pair
-1.5 -1.0 -0.5 0 0.5 1.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
1/kF a P/kF is P = kF↑. The energy of the pair is the sum of the
binding energy and the center of mass kinetic energy,
FIG. 1: (color online) (a) Pairing instability diagram. The E =−1/ma2+P2/4m,whiletheholesleftinbothFermi
red region in this diagram corresponds to those popula- seas have vanishing energy since they lie near the Fermi
tions imbalances and scattering lengths at which the dom- surfaces. It follows that
inant pairing mode is at finite momentum, the blue region
is that dominated by zero momentum pairing and in the aFP ∼ √1 , (6)
white regions there are no unstable pairing modes. The aFFLO 2
FP
dashed-dotted line is the approximate threshold for the Fes-
hbach molecule formation limited by the energy of holes left wherewehavereferredtothiscriticalscatteringlengthas
in the Fermi sea discussed in the text, which nearly coin- aFFLO because the pairing occurs at finite momentum.
cides with the onset of the pairing instability on the BEC WFePhave 1/k aFFLO ≈0.6, which deviates considerably
side. (b) Growth rate of the most unstable pairing mode F FP
fromcriticalvaluecalculateddirectlyfromtheCooperon
Im(E ) as a function of the inverse scattering length across
M 1/k aFFLO ≈ 0.4. This discrepancy originates from its
the resonance. The curves correspond to populations im- F FP
proximitytotheunitaryregime,wherethemoleculardis-
balances δ = {0,0.2,0.4,0.6,0.8,0.9,0.96,0.99}. Pairing in-
stabilities occur over a narrower k a region at larger polar- persion is greatly affected by the presence of the Fermi
F
izations. (c) Typical growth rates of the unstable pairing sea.
modes as a function of the pairing momentum, for δ = 0.5
and 1/k a = {−0.4,−0.3,...,0.2}. Im(E ) is the imaginary
F P
partoftheCooperonpoleenergyforagivenpairingmomen- III. DENSITY AND LONGITUDINAL
tum, these curves illustrate how the instability evolves from SPIN-DENSITY INSTABILITIES
being dominated by pairing at finite momentum at negative
scattering lengths to zero momentum at positive scattering
We now turn to the instability analysis of density,
lengths. Weconjecturethatquenchesofthescatteringlength
that approach the resonance from the BCS side might bring ρq = (cid:80)k,αc†kαck+qα, and spin-density along the polar-
intorealizationstateswithsubstantialfinitemomentumpair- ization axis, s = (cid:80) c† σααc . The analysis is
zq k,α kα z k+qα
ing during the initial unstable dynamics. again done by looking for the occurrence of poles with
positive imaginary parts in the corresponding suscepti-
bilities. The responses of these quantities are coupled in
threshold beyond which the energy gained by molecule
the polarized gas and share the same unstable collective
formation can be absorbed by adding two holes to the
modes. We defer the analysis of the unstable collective
Fermi sea [7]. Approximating the binding energy as
modes of the transverse spin-densities to Sec. IV.
∼1/ma2, the critical scattering length is found from
Intime-dependentmean-field-theory,thespinandden-
k2 k2 1 sitydynamicsafterasuddeninteraction-strengthquench
F↑ + F↓ = . (4) havearandomphaseapproximation(RPA)form. Inthis
2m 2m ma2
recipe, the bare interaction potential enters the “denom-
ThisthresholdisshowninFig.1asadashed-dottedline, inators” of the RPA susceptibilities. Often the short-
and nearly coincides with the one of the onset of pairing range interactions in a dilute gas are modeled with a
4
functions as follows:
k+q k+q (cid:90)
(a) k = k k+q + k χ (k)±χ (k)=−i G (q)G (k+q)Γρ,s(q,k), (8)
C σσ σ¯σ σ σ σ
q q
q q
where the +(−) sign corresponds to the density (spin)
k+q k+q vertex, q and k are four component vectors that include
(b) χzz(k) = k k + k k frequencyandmomentacomponents,Gσ(q)isthesingle-
particle Green’s function, and σ¯ denotes the spin oppo-
q q
site to σ. The vertex functions satisfy the following self-
consistent integral equations (see fig. 2):
FIG.2: a)Diagrammaticrepresentationoftheintegralequa- (cid:90)
tionforthespinsusceptibilityvertexfunctionforspinup. b) Γρσ,s(q,k)=1∓i Gσ¯(q(cid:48))Gσ¯(q(cid:48)+k)C(q+q(cid:48)+k)Γρσ¯,s(q(cid:48),k),
Relation between the spin vertices and the longitudinal spin q(cid:48)
(9)
susceptibility.
where C is the Cooperon. The poles of the vertex func-
tions determine the unstable collective modes.
Integral equations of the above type are familiar [12]
contact pseudo-potential whose strength is proportional
fromFermiliquidtheory. Inthelimitofsmallwavelength
to the vacuum scattering length. This prescription, for
|k|<<k thedominantcontributiontotheintegralover
which the RPA quenching picture is justified, would in- F
q(cid:48) in Eq.(9) is associated with low-energy Green’s func-
evitably fail near the resonance. Thus a conflict arises
tionpoles. Thispropertycanbeexploited[12]byassum-
in the description of the unstable dynamics of spin and
ingthatΓisslowlyvaryingnearthesepoles. Thisallows
density near the resonance. With this on mind we will
to substitute
examine density and spin instabilities by assuming the
RPA form for the dynamics, but with the bare inter- 4π3i
G (k)G (k+q)→ ζ (q,kˆ)δ(k −(cid:15) )δ(k−k ),
action replaced by the Cooperon. We believe that this σ σ k2 σ 0 Fσ Fσ
approach, which has been previously employed to study Fσ
(10)
spin instabilities in the unpolarized gas [7, 10], is a rea- where ζ (q,kˆ) is chosen so that its angular average re-
sonablecompromise,butwedonotknowofasystematic σ
produces the long-wavelength, low-energy behavior of a
justification for it.
a free-particle (Lindhard) response function:
Thebasicquestionweexploreinthissectioniswhether
afreegaspreparedfarfromtheresonancecanencounter
a long-wavelength spin or density instability as the scat- (cid:90) dΩkˆ ζ (q,kˆ)=χ0(q)=(cid:90) nkσ−nk+qσ (11)
teringlengthincreaseswithoutreachingthecriticalscat- 4π σ σ ω+ξ −ξ
k k k+q
tering length that determines the boundary for the pair-
Thisapproximationcorrespondsphysicallytotheidea
inginstability. Ifaspin-densityinstabilitythatcausedan
that the initial dynamics of the ideal gas is dominated
increase in the local spin-polarization occurred, it could
by those states that are closest to it in energy, and that
inhibit pairing and allow the physics of ferromagnetism
they in turn are formed from the gas state by making
tobestudiedinacoldatomcontext. Asdiscussedbelow,
particle-holeexcitationsthatliveneartheFermisurface.
our calculations indicate that when the resonance is ap-
Theintegralequationfortheinteractionverticescanthen
proachedfromtheBECsidethespin-densityinstabilities
beconvertedintoamatrixequationinvolvingtheexpan-
always occur within regions that are already unstable to
sionsofΓ,ζ andC intosphericalharmonics. Bytruncat-
pairformation. Eventhoughwedonotcomparedirectly
ing the resulting equations at the s-wave term we obtain
the growth rates of both instabilities, this finding is dis-
an expression for the vertices which resembles the RPA
couraging for the realization of an instability in which
result,
effectiverepulsiveinteractionswouldattempttoenhance
locally the spin polarization of the gas.
In a polarized gas the density and longitudinal spin 1±C (ω)χ0(k)
Γρ,s(k)= s σ¯ , (12)
responses are mutually coupled, but decoupled from the σ 1−C2(ω)χ0(k)χ0(k)
s ↑ ↓
transverse spin response. The density and longitudinal
spinsusceptibilitiescanbeexpressedintermsofthespin- where C (ω) = 1(cid:82) d(kˆ · kˆ(cid:48))C(ω,k kˆ + k kˆ(cid:48)), is the
s 2 F↑ F↓
resolved density-density response functions [11], s-wave average of the Cooperon. We expect the s-wave
truncation to be reliable for the stability boundary esti-
mateatlongwavelengthssinceinthelimitofsmall(ω,q)
iχσσ(cid:48) =(cid:104)T[nˆσ(r,t)nˆσ(cid:48)(r(cid:48),t(cid:48))](cid:105). (7) we have
Inparticular,χ =χ −χ +χ −χ andχ =χ +
zz ↑↑ ↓↑ ↓↓ ↑↓ ρρ ↑↑ qˆ·kˆ
χst↓u↑d+ieχd↓↓b+yχd↑e↓fi.nTinhgetchoelldecetnisvietym,oΓdρe,sacnadnsbpeinc,onΓvse,nvieernttelyx ζσ(q,kˆ)→νσ ω −qˆ·kˆ, (13)
σ σ vFσq
5
where νσ = mkFσ/2π2 is the density of states at the (a) (b)
Fermi surface for the spin σ. Assuming that the static
sisutsecnecpetibofilitloyng(ω-w/avvFeσlqen→gth0)incstoarbreilcittliyes,prtehdeictosnltyhenoenx-- /E )ωF00..1250 δ = 0.400..1250 δ = 0.8
vanishing component in the spherical harmonic expan- m(0.10 0.10
sion of ζ(q,kˆ) in this limit is the s-wave term. It can I0.05 0.05
thenbe argued thatthe instabilityboundaryiscorrectly 0.2 0.4 q/0k.6F 0.8 1.0 0.2 0.4 0.6q0/.8kF 1.0 1.2 1.4
estimated by the s-wave truncation of Eq. (12). (c)
Our results for the density and longitudinal spin- 1.0
densityinstabilitiesinpolarizedgasesaresummarizedin
0.8
Fig. 3. We find that, when the resonance is approached
from the BEC side, the pairing instability always occurs BCS BEC
δ 0.6
for smaller scattering lengths than the spin instability.
The spin-density instability occurs at smaller scattering
0.4
lengths than those predicted when the Cooperon is re-
placed by the contact pseudopotential, C → 4πa/m,
0.2
valid in the dilute limit, k a→0. When combined with
F
RPA, this replacement would imply a Stoner instability
-1.5 -1.0 -0.5 0.5 1.0
at 1/kFa = 2/π for the unpolarized gas, whereas with 1/kF a
the Cooperon we obtain 1/k a ≈ 0.94 [7]. The differ-
F
ence can be understood as being due to the regularized
FIG. 3: (color online) (a) Growth rate of the density-
scattering, which produces stronger repulsions when the
longitudinal spin instabilities for δ = 0.4 and scattering
resonanceisapproachedfromtheBECside. Comparable lengths 1/k a = {0.6,0.65,...,0.9}. (b) Growth rate of the
F
enhancementswerefoundinMonteCarlostudies[13,14]. density/spin instabilities for δ = 0.8 and scattering lengths
Nevertheless,ourresultsindicatethatthepopulationim- 1/k a = {−0.3,−0.25,...,0}. (c) Instability phase dia-
F
balanceincreasesthecriticalrepulsionrequiredtotrigger gram. The blue shaded region corresponds to the density-
spin instabilities, and this critical repulsion remains al- longitudinal spin instability region and for comparison the
ways larger than the corresponding one to produce pair- dashedlinesdepicttheboundaryofthepairinginstabilityre-
gion,thedottedlineseparatestheregiondominatedbyfinite
ing. Weconcludethatthepreparationofapolarizedini-
momentum pairing from that dominated by zero momentum
tialstatedoesnotleadtocircumstancesunderwhichthe
pairing (see Fig. 1). When the resonance is approached from
Stoner instability of longitudinal spin modes is present,
the BEC side the spin-density instability always appears at
but the pairing instability is not.
a scattering length where the gas is unstable to pairing as
Even though the instability boundaries for the spin- discussed in the text.
density instability can be calculated consistently within
our approach, the growth rates of these instabilities are
pole,wehaveverifiedthatthisinstabilitycorrespondsto
reliable only near the critical parameters at which the
an unstable mode in which the majority spin and minor-
instability appears. Inside the unstable region, beyond
ityspindensitiesgrowwiththesamesignlocally(thatis,
a critical scattering length, we have found discontinuous
either they both decrease or both increase at any given
behavior of the unstable modes as a function of momen-
point),withtheamplitudeofminorityspindensitylarger
tum and scattering length. Since Eq. (12) is justified
than that of the majority spins. In other words, this in-
only at small frequencies and wave-vectors, it is difficult
stabilityattemptstocreateanalternationofregionswith
to judge the extent to which such behavior is reliable.
small population imbalance and regions with depleted
These discontinuities are also present in the density re-
minority spins. The relative amplitudes of the spin den-
sponse of the unpolarized gas on the BCS side and are
sitiesoftheunstablemodeneartheinstabilityboundary
thusinheritedbythelongitudinalspinmodesatfinitepo-
are,
larizations because of their coupling to the density. We
will therefore focus only on the instability boundaries in (cid:114) ν (cid:114) ν
this work. δn↑ ∝ ν +↓ν , δn↓ ∝ ν +↑ν . (14)
↓ ↑ ↓ ↑
We first discuss the instability boundaries approached
from the BCS side. As the resonance is approached at Weviewthisastheprecursorofthecreationofregions
large polarizations, a long-wavelength density instability where the pairing instability is locally developed, alter-
is always encountered at a lower scattering length than natedwithregionswheretheminorityspinsaredepleted,
the corresponding critical scattering length for igniting explainingwhythepairinganddensityinstabilitiesareso
pairing instabilities. At small polarizations (δ (cid:46) 0.4) closetogether. Weexpectthatinitialfluctuationbubbles
this boundary becomes increasingly close to the one of withagrowingdensityofpairedparticlesshouldeventu-
the pairing instability. The spin-density instability on ally merge and phase separate from the excess majority
the BCS side is brought about by effective attractive in- spin particles. This instability is therefore compatible
teractions. Byexaminingtheresidueofthesusceptibility with the equilibrium state, which has phase separated
6
density-longitudinal spin modes near the resonance oc-
(a)
curs on similar time scales ∼(cid:15)−1 at shorter wavelengths
1.4 F
as illustrated in Figs. 1 and 3.
g ν 1.3
0
IV. TRANSVERSE SPIN INSTABILITIES
1.2
1.1 As discussed previously, near the resonance the mo-
mentum and frequency dependence of the effective two-
1.0 particleinteractionisimportant. Aswillbecomeclearin
this section, the instabilities of the transverse spin den-
sity occur at finite wavevectors, with the typical wave-
0.0 0.2 0.4 0.6 0.8 1.0
δ length of the most unstable modes decreasing as the im-
balance of populations increases. This situation compli-
cates the analysis of the integral equations for the trans-
(b) (c)
verse spin vertices, since the Fermi liquid simplifications
m(/E )ωF0.06 δ = 0.8 g ν10 =.2 15.3 Re(/E )ωF0.2 δ = 0.8 g ν0 = 11..231.25 dleinscguthsssedanidn tshmeapllrefrveioquusenacrieesj.usNtifievedertohnelylesast,liomnpgowrtaavnet-
I insights into the nature of the transverse spin instabil-
0.04 0.1 1.15
1.2 itycanbegainedwiththeuseofafrequency-momentum
0.02 1.1 independent pseudopotential V(r) = gδ(r), and we will
1.15 1.05
0.0 0.4 0.8 1.2 present them in this section. We leave open for future
1.0
0.4 0.8 1.2 studies a more systematic treatment of this instability.
q/k q/k
F F Theresponseofthex,y spindensitiescanbeobtained
in terms of the susceptibility χ (q,ω), which describes
↓↑
FIG.4: (coloronline)(a)criticalrepulsionstrengthfortheon- the response of s↓↑(q) = (cid:80)kc†k↓ck+q↑ to a perturbation
setoftransversespininstabilities(solidredline). Thedashed driven by its conjugate field. Within RPA [15, 16] one
line is the repulsion strength needed to sustain a spin imbal- obtains
ance in equilibrium (i.e. δµ = 0), which is finite at δ = 1.
Thedasheddottedlineisthecriticalstrengthfortheonsetof χ0 (q,ω)
χ (q,ω)= ↓↑ , (15)
longitudinalspininstabilitiesforthecontactpseudopotential ↓↑ 1+gχ0 (q,ω)
model. AlllinesconvergetotheStonercriterionasδ→0. (b) ↓↑
Typical growth rates of the unstable modes (imaginary part with
ofthepoles)ofthetransversespininstability. (c)Realpartof
the collective modes dispersion. The unstable modes evolve (cid:90) d3k n (k)−n (k+q)
χ0 (q,ω)= ↑ ↓ , (16)
continuouslyfromtheconventionalspinwavesinequilibrium ↓↑ (2π)3ω+ξ (k)−ξ (k+q)
↑ ↓
as explained in the text and illustrated in Fig. 5.
where, ξ (k) = k2/2m + gn . The splitting between
σ σ¯
the dispersion of majority and minority spins is ∆ =
superfluidandnormalgasregions,althoughourinstabil- g(n −n ), and, correspondingly, the difference in chem-
↑ ↓
ity boundaries only indicate the appearance of linearly ical potentials reads as δµ≡µ −µ =δ(cid:15) −∆.
↑ ↓ F
unstable modes and do not necessarily coincide with the Collectivemodesarefoundbysolving1+gχ0 (q,ω)=
↓↑
boundaries between equilibrium phases. 0. This same equation determines the dispersion of spin
We note that at smaller polarizations, δ (cid:46) 0.4, the waves in a ferromagnet in equilibrium [15]. In Fig. 4 we
boundaries of finite momentum pairing and density in- depict the critical repulsion strength g at which an
crit
stabilities almost merge. In this regime we conjecture instabilityinthetransversemodesappears. Sincenoun-
that sudden quenches might bring into realization states stable modes exist in equilibrium, the critical repulsion
with substantial finite momentum pairing in the initial strength for the appearance of unstable collective modes
dynamics. for a given polarization must be larger than the interac-
Several of our observations on the unstable dynam- tion strength that is needed to sustain such polarization
ics on the BCS side are in qualitative agreement with a in equilibrium, i.e. g >g (δ).
crit eq
previous study of the unstable modes in the spinodal re- Wefindthattheunstablemodeshaveafinitemomen-
gion of the polarized system [9]. There, however, it was tum q ≥ k −k , and that they evolve continuously
F↑ F↓
assumed that the dynamics of pairing was essentially in- from the conventional spin-waves that the system would
stantaneous, so that the order parameter adjusted to its have in equilibrium. The instability exists whenever the
localequilibriumvalueafterthequench,whereastheden- spin wave spectrum, E (q), coincides with minus the
sw
sityandspinmodeswouldrespondonslowertimescales. energy of a particle-hole excitation of minority into ma-
Althoughthisisenforcedatarbitrarilylargewavelengths jority spin, i.e. when E (q)=−(ξ (k+q)−ξ (k)), for
sw ↑ ↓
due to conservation laws, the dynamics of pairing and some k such that k < k ,k < |k +q|. This can be
F↓ F↑
7
wavelengths is thus a consequence of the Pauli exclusion
(a) 3.0 (b) 0.08
principle which prohibits minority-into-majority excita-
0.06
ω/EF2.0 0.04 tions of momenta q < kF↑ −kF↓ [17]. Fig. 5 illustrates
1.0 ω/EF0.02 how the spin wave spectrum evolves from being damped
0.00 0.2 0.4 0.6 0.8 atequilibriumtounstablebeyondthecriticalinteraction
0.0 0.5 1.0 1.5 -0.02 strength.
-0.04
-1.0
q/kF q/kF
(c) 0.15 (d)
Im(ω/E F ) V. SUMMARY
0.10 0.015
0.05
ω/EF 0.010 Our results imply that one should expect considerable
0.00 0.2 0.4 0.6 0.8
0.005 differences in the initial unstable dynamics of polarized
-0.05
andunpolarizedgasesofultracoldfermions,andthatfea-
-0.10 q/kF 0.2 0.4 q/0k.6F 0.8 1.0 turesoftheinitialunstablestatecanbesignificantlydif-
ferent from those expected from the equilibrium phase
0.25
(e) diagram in the population imbalance-scattering length
0.20 coordinates.
0.15 (q,ω) = (δkF , ∆ − δEF ) sidWe,hfiennitethmeormeseonntaunmcepaiisrinagppirnosatachbeilditiefrsoamndthdeenBsiCtyS-
longitudinal spin instabilities are encountered. Pairing
0.10
ωω//EEFF at finite momentum is the fastest growing pairing insta-
0.05 bility over a wide range of parameters. This should be
contrastedwiththeequilibriumFFLOphaseregion,con-
0.00
0.2 0.4 0.6 0.8 1.0 1.2 jecturedtooccupyarathersmallportionofthephasedi-
q/k
-0.05 F agram [3]. The density-longitudinal spin instability can
beviewedasaprecursorofthephaseseparationbetween
-0.10
a balanced superfluid and excess majority fermions, ex-
pected to settle in at longer time scales. We predict
FIG. 5: (color online) (a) Equilibrium (i.e. δµ = 0) spin an evolution of unstable modes as the population im-
wavedispersionandparticle-holecontinuumforδ=0.9. The balance increases, with a transition between finite mo-
boundary of the particle-hole excitation energies of majority mentum pairing at small polarizations (δ (cid:46) 0.4), and
into minority corresponds to the dashed line, the solid lines phase-separation modes at larger polarizations.
bound the negative of the energies of minority into majority
WhentheresonanceisapproachedfromtheBECside,
excitations, and the blue line is the dispersion of spin waves.
we find that the zero-momentum pairing instability al-
(b) Zoom in of Fig. (a). The spin waves cease to exist when
ways occurs before the longitudinal spin-density insta-
theybecomedegeneratewithparticle-holeexcitationsofma-
bility. The latter instability evolves from what would
jority into minority spin, which allows them to decay into
particle-holeexcitations. (c)Spinwavedispersionatthecrit- be the Stoner instability in the unpolarized gas. This
ical repulsion g for the onset of transverse instabilities for finding seems discouraging for simulating a Stoner-like
crit
δ = 0.9. (d) and (e) correspondingly show the imaginary transition in a polarized gas of fermions. Nevertheless,
(growth rate of instability) and real parts of the spin wave as illustrated with a simplified model in Sec. IV, insta-
dispersion at a repulsion strength gν0 = 1.25 for δ = 0.9. bilities of the transverse spin modes are also expected to
The instability appears when the spin waves have minus the appear before the longitudinal spin and density instabil-
energy of minority-into-majority particle-hole excitations as
ities. Therefore, an analysis of these instabilities which
explained in the text, and it would look like a spin-density
systematically accounts for the modification of the in-
wave of transverse magnetization with a spatial modulation
teractions near the resonance is necessary to resolve the
correspondingtotheinversewavevectorofthefastestgrowing
mode ∼q−1 during the initial dynamics. fate of the competition between the transverse-spin and
max
pairing instabilities at finite polarizations.
It is worth emphasizing that in a polarized gas with
globallyconservedpopulationsofeachspinspecies,some
seenasanecessaryconditionforthespontaneousexcita- ofthephysicsofitinerantelectronferromagnetismcanbe
tion of magnons and particle-hole pairs while conserving explored, independently of the presence of spin-density
total energy, momentum and spin. It is thus analogous instabilities. Forexample,inoursimplemodelofSec.IV,
to the critical interaction for pair formation discussed the spectrum of spin waves acquires a positive mass
in the BEC side of the resonance, but instead of the (magnetization stiffness) for interaction strengths below
particle-particle binding to form a molecule we consider any spin instability, and below the interaction strength
particle-holebindingtoformamagnonofspin−1which that would spontaneously sustain the population imbal-
leaves a particle-hole excitation in the polarized Fermi ance in equilibrium. Experimental studies of spin-wave
sea of spin +1. The absence of unstable modes at long- dynamicsoffullypolarizedstatesclosetoaFeshbachres-
8
onance are likely to shed light on many-electron physics sarma and Joseph Thywissen for valuable discussions.
questions that arise in the context of itinerant electron ThisworkwassupportedbytheWelchFoundationunder
magnetism, and also on pairing physics questions that Grant No. TBF1473.
are unique to the cold atom problem.
Acknowledgments
The authors wish to thank Mehrtash Babadi, Eugene
Demler, Rembert Duine, David Pekker, Rajdeep Sen-
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