Colliding clouds of strongly interacting spin-polarized fermions

0.38 MB·English

by  Edward Taylor

Colliding clouds of strongly interacting spin-polarized fermions
Edward Taylor, Shizhong Zhang, William Schneider and Mohit Randeria1
1Department of Physics, The Ohio State University, Columbus, Ohio, 43210
(Dated: January 18, 2012)
MotivatedbyarecentexperimentatMIT,weconsiderthecollisionoftwocloudsofspin-polarized
atomic Fermi gases close to a Feshbach resonance. We explain why two dilute gas clouds, with
underlying attractive interactions between its constituents, bounce off each other in the strongly
2 interacting regime. Our hydrodynamic analysis, in excellent agreement with experiment, gives
1 strong evidencefor a novelmetastable many-bodystate with effective repulsiveinteractions.
0
2
PACSnumbers: 03.75.Ss,03.75.Hh

n
a
J I. INTRODUCTION
spinupcloud spindowncloud

7
1
Ultracold atomic gases [1, 2] open up new frontiers in

the study of non-equilibrium dynamics of strongly inter-
]
s acting quantum systems. In contrast to electronic sys-
a
tems, the Fermi energy in quantum gases is of the order HarmonicTrap
g
of a few kilohertz, so that questions about metastability
-
t and equilibration in quantum systems can now be ex- FIG. 1: Schematicof thecolliding cloud experiment.
n
a plored in real time in the laboratory. Exploring these
u newregimesanddevelopingnewtheoreticaltoolstogain
clouds remainseparatedatintermediate time scales. We
q insightinto the non-equilibriumdynamics of many-body
t. systems is an exciting challenge at the intersection of show that these results are natural consequences of a
a metastable many-body state on the “upper branch” of
condensed matter and atomic-molecular-opticalphysics.
m the Feshbach resonance where the effective interactions
The recent experiment of Ref. [3] offers a beauti-
- are repulsive [4, 5]. (3) Finally, we provide insight into
d ful example of non-equilibrium dynamics that is com- how the dynamics changes as a function of the strength
n pletely unexpected and, at first sight, counterintuitive.
of the interaction.
o Two clouds of ultracold fermions prepared in different
c hyperfine-Zeeman (“spin”) states σ = and are ini-
[ ↑ ↓
tially separated using a Stern-Gerlach field; see Fig. 1.

2 Once the field is turned off, the clouds are impelled to-
v gether by the confining harmonicpotentialand they col- II. HYDRODYNAMICS
5 lide. In the strongly interacting unitary regime these
4
very dilute clouds are observedto bounce off each other,
2 Our analysis is based on two hypotheses: (i) The col-
almost as if they were colliding billiard balls! This is
4 lisions between atoms are sufficiently rapid to establish
truly remarkable in view of the fact that the underly-
. local thermodynamic equilibrium when the clouds over-
6 ing atomic interaction is attractive, and the equilibrium
lap. (ii) The loss from the upper branch scattering state
0
ground state is known to be a paired superfluid. Why
1 is slow in the experiment of Ref. [3]. At unitarity, the
thendo the cloudsbehaveas thoughthe interactionsare
1 two-body collision rate is very large, 1/τ2 ǫF (with
: repulsive? ~ = k = 1), for a range of temperatures 0.1∼. T/ǫ .
v B F
Twoadditionalaspects ofRef. [3]arealsonoteworthy. 0.3 [6, 7]. With a Fermi energy ǫ 104Hz and an
i F
X Once the short-time bounce is damped out, the centers axial trap frequency ω 10Hz [3], the∼gas in the over-
z
r ofmassofthecloudsremainseparatedforaconsiderable lap region of the clouds∼will reach local thermodynamic
a duration at intermediate times ( 100 ms), before even- equilibriumwithin 10−3 trapperiods. Thustheclouds
tuallymerging. Infact,theanaly∼sisinRef.[3]focusedon behave hydrodynam∼ically in the overlap region [8] and
preciselytheslowmergingoftheclouds,orspindiffusion, their dynamics reflect the (metastable) equationof state
at long times ( 0.5 s). Finally, the observed dynamics of the gas. (Hydrodynamics also describes well the very
∼
changes qualitatively as one moves out of the strongly different behavior observed in colliding clouds of spin-
interacting regime, with the clouds merging rapidly for balanced superfluid gases at unitarity [9].)
weak interactions. As the two clouds come together in the overlap re-
In this paper we gain insight into three aspects of this gion, the atoms are initially in scattering states, and the
remarkabledynamics using a hydrodynamic approachin formationoftwo-bodyspin-singletboundstates requires
the strongly interacting regime. (1) We explain why the three-body collisions in order to satisfy kinematic con-
clouds repeatedly bounce off each other at short times. straints. The time scale τ for such processes is not well
3
(2) We also explain why the centers of mass of the two understoodforstronglyinteractinggases,however,there
2
areindicationsthatitis enhancedcloseto unitarity[10]. where R 2ǫ0/mω2 is the TF radius along the α-
α ≡ F α
Inaddition,theτ3 relevanttoRef.[3]isfurtherenhanced axis and ǫ0F =p(ω⊥2ωz)1/3(3N)1/3 is the chemical poten-
relative to the microscopic decay time since three-body tial of an ideal two-component Fermi gas (N = N =
↑ ↓
collisions are limited to a smalloverlapregionfor a frac- N/2). This ansatz allows for a time-dependent center-
tion of the oscillation period ω−1. of-mass z¯ as well as axial compression of the clouds.
z σ
Now, for the experimentally relevant time interval The continuity equation (2) leads to the velocity field
τ2 t τ3, the rapid two-body collisions have already v = v zˆ with v (z,t) = z¯˙ z¯ δz˙ /δz + zδz˙ /δz .
est≪ablis≪hed thermal equilibrium, while the slow three- Aσxial anσd radial σsymmetry σle−ts uσs sσet z¯σ z¯ =σ z¯σ
↑ ↓
body recombination process has yet to drive the system and δz = δz δz. Using (5) in (3) and (≡4) gives−the
↑ ↓
into the lowerbranch. This allowsus to model the cloud coupled nonline≡ar integro-differentialequations:
dynamics using Euler’s equations
d2 1 ∂ ∂n
∂∂vtσ + ∇2vσ2 =−∇m ∂∂nE +Vtrap −γvσ, (1) (cid:18)dt2 +ωz2(cid:19)z¯(t)=−mN↑ Z d3r∂nE↑ ∂z¯↑−hγv↑i↑, (6)
(cid:18) σ (cid:19)
and
∂∂ntσ +∇·(nσvσ)=0, (2) ddt22 +ωz2 δz(t) = −m8N d3r∂∂nE ∂∂nδz↑
(cid:18) (cid:19) ↑ Z ↑
withanappropriateenergydensity [nσ(r)] forthescat- 8
E (z z¯(t))γv . (7)
teringstates(seebelow). Herevσ andnσ arethevelocity −δzh − ↑i↑
anddensityoftheσ fermions,V =m(ω2ρ2+ω2z2)/2
trap ⊥ z
isthetrappotential. Weincludeaphenomenologicalγ to In the unitarity regime, the spin-current damping [12]
accountforstrongspin-currentdampinginthe hydrody- assumes a particularly simple form when the relative ve-
namic regime [11, 12]. Viscosity,describing the damping locity is of order of the Fermi velocity, as is the case at
ofin-phasecurrent(v↑ =v↓),isignoredsincethemotion early times for two clouds initially separated by Rz. In
of the colliding clouds is primarily out-of-phase [7, 13]. the overlapregion, we take
As a result of tight confinement in the radial direc-
tion, the two modes with the lowest energies are the γ =γ√ǫF↑ǫF↓, (8)
spin-dipole mode and axial breathing mode. Hence, we
where γ is of order unity at unitarity and ǫ (r) =
concentrateonthecenters-of-massofthecloudsandtheir e Fσ
(6π2n (r,t))2/3/2m is the local Fermi energy. This sim-
widths along the zˆ-directionand reformulate (1) and (2) σ
ple choice of damping is, if anything, an overestimate,
in terms of those variables. Using the notation e
σ
1/N d3rn ( ),wedefinethecenter-of-massph·o·s·itiion≡s and we have checked that our results are robust against
σ σ
··· reasonable modification of the damping parameter.
z¯ z and widths δz 8 (z z¯ )2 . The √8
σ ≡Rh iσ σ ≡ h − σ iσ
ensures that δz coincides with the axial Thomas-Fermi
σ
p
(TF) radius R in equilibrium. (1) and (2) then lead to
z
the exact equations of motion
III. ENERGY FUNCTIONAL
1
z¨¯ +ω2z¯ = ∂ (∂ /∂n ) γv
σ z σ −mh z E σ iσ−h σziσ
+ vσ ·∇vσz −∂zvσ2/2 σ (3) relWeveanntofworndeyendamtoicsspaetcitfiymtehseτ2energty fuτn3.ctTiohneaglrEo[unnσd]
≪ ≪
and (cid:10) (cid:11) state(“lowerbranch”oftheFeshbachresonance)forscat-
tering length a > 0 necessarily involves bound pairs.
s
δz¨σ+ωz2δzσ =8hvδσ2zziσ − (δδz˙zσ)2 −8(δz¯˙zσ)2 Blauxttfoorthtis≪stτa3t,ethhaevtehnroeet-ybeotdoycpcurorcreedss.eTs hreeqsuyirsetedmtohares-,
σ σ σ
8 8 however, developed two-body correlations characteristic
(z z¯ )∂ (∂ /∂n ) (z z¯ )γv
−mδz h − σ z E σ iσ− δz h − σ σziσ ofthe metastable “upper branch”state, studied theoret-
σ σ
ically in Refs. [4, 5, 14], motivated by an earlier experi-
8
+δz (z−z¯σ)[vσ ·∇vσz −∂zvσ2/2] σ. (4) ment[15]. The approximatemany-body wavefunctionin
σ the upper branch is
(cid:10) (cid:11)
The problem can be further simplified while still re-
taining the essential physics of the (nonlinear) coupling Ψ= f(r r ) Φ ( r )Φ ( r ), (9)
i↑ j↓ S i↑ S j↓
| − | { } { }
between the spin-dipole and axial breathing modes by
hYi,j i
using a TF ansatz:
where Φ ( r )’s are Slater determinants and the Jas-
S iσ
n (r,t)= Rz(2mǫ0F)3/2 1 ρ 2 z−z¯σ(t) 2 3/2, trow factor{ f(}r) describes the short-range correlations
σ 6π2δz (t) − R − δz (t) betweenfermions. The effective repulsionbetween and
σ h (cid:16) ⊥(cid:17) (cid:16) σ (cid:17) i (5) atoms in the upper branch is crucially related t↑o the
↓
3
1.0 1.0
0.5 0.5
z
(cid:144)RzΣz 0.0 (cid:144)zRΣ 0.0
-0.5 -0.5
-1.0 -1.0
0 1 2 3 4 0 1 2 3 4 5 6
Ωzt(cid:144)2Π Ωzt(cid:144)2Π
FIG.2: (Coloronline)Timedependenceofthecenter-of-mass FIG.3: (Coloronline)Timedependenceofthecenter-of-mass
positions z¯σ of the two spin clouds at unitarity, with the red positions z¯σ of the two spin clouds for kF(0)as = 2. The
(upper)andblue(lower)curvesdenotingthetwospinspecies. parameters are thesame as in Fig.2.
We use the upper branch energy functional (see text), with
ǫ0F corresponding to N = 7.5×105 atoms, trap anisotropy
ω⊥/ωz =10,anddampingeγ =1. Initiallythetwocloudsare IV. DYNAMICS AT UNITARITY
displaced from the trap center by the axial Thomas-Fermi
radius, z¯↑(t = 0) = Rz,z¯↓(t = 0) = −Rz with initial axial
We solve (6) and (7), using (10) and (5); for details
width δz(0)=Rz equal to its equilibrium value.
see Appendix B. The results at unitarity are shown in
Fig. 2. The two clouds bounce off each other for sev-
eral oscillations due to the repulsive nature of the upper
node in f(r), in contrast to the nodeless Jastrow fac- branch functional. The period of the initial bounce is
tor for the lower branch. For small as > 0, the node roughly 0.56(2π/ωz), slightly less than the experimen-
occurs at as, with f(r) (1 as/r) similar to the two- tal value 0.61(2π/ωz) [3]. We attribute this difference to
body problem. For larg∼e a ,−the node saturates [5] to the simple ansatz (5) used to model the dynamics. The
s
1/k , the only length scale at unitarity. Quantum bouncepersistsforseveralcyclesinspiteoftheverylarge
F
M∼onte Carlo (QMC) studies [4, 5] of the upper-branch damping (ǫ0F/ωz ≫ 1) because the overlap between the
wavefunction (9) reveal that the system undergoes (fer- two clouds is small. Once the oscillation is damped out,
romagnetic) phase separation for sufficiently strong in- the centers of mass of the two clouds remain segregated,
teractions k a &1. with a final (z¯↑ z¯↓) 0.4 times the initial separation.
F s − ≃
Thisintermediatetimebehaviorreflectsthetendencyfor
We need [nσ] for arbitrary polarization, which has systemtophasesegregateinthe upperbranchatunitar-
E
not been studied by QMC. We thus use the lowest or- ity.
der constraint variational (LOCV) approximation [16],
which has been used for the upper branch [17] and is in
close agreement with QMC data [4, 5]. As shown in the
Appendix, the upper branch LOCV energy density is
V. COLLISIONAL DYNAMICS AWAY FROM
UNITARITY
3 3 1
= ǫ n + ǫ n + (n λ +n λ ). (10)
F↑ ↑ F↓ ↓ ↑ ↓ ↓ ↑
E 5 5 2
Awayfromunitarity,itishardertoreachthehydrody-
namic regime. If ω is sufficiently small, however, it will
z
The interactionenergiesλ and λ arefunctions of k a always be the case that the dynamics in this direction
↑ ↓ F s
and x = n↓/n↑. Hydrodynamics requires T & 0.1ǫF [7]; arehydrodynamic. Thegreaterchallengeatfiniteas >0
forsimplicity,weusethezerotemperatureLOCVenergy is that τ3 may not be much greater than τ2. As as de-
functional to study the dynamics, expecting it to quali- creasesfromunitarity,oneexpectsthatτ3 reachesamin-
tatively describe the physics at low temperatures. It is imum for kFas 1 and then becomes large again [18],
importanttonoteherethatwhile werelyontheapprox- with τ3 ∼ (na3s)∼−2ǫ−F1 for kFas ≪ 1. In contrast, the
imate LOCV calculationto obtain the equation of state, cross-section a2 determines τ (na3)−2/3ǫ−1. Hence,
s 2 ∼ s F
the underlying physics of the effective repulsive interac- forsmall k a ,againτ τ . However,whenk a 1,
F s 3 2 F s
≫ ∼
tion is independent of the particular scheme used and a it is conceivable that τ τ .
3 2
∼
more precise calculation would only leads to quantita- With these caveatsinmind, we solve(see Appendix B
tive improvement without changing the crucial physical for details) our hydrodynamic equations away from
picture of the bounce. unitarity to understand the relationship between the
4
1.0 Thelongtimebehavior,notdescribedhere,willbe dom-
inatedbyspindiffusion[3]atnon-zerotemperaturesand
three-body processes.
0.5
Finally,wecommentontheexperimentofJoetal.[15]
in which a spin-balanced mixture, initially at a small
(cid:144)RΣz 0.0 as > 0, is swept close to resonance, generating strong
z interactions in the upper branch. It appears, however,
that rapid losses render it unstable to the lower branch
-0.5
withinaveryshorttime(τ 1ms)[10,20,21]and
3,micro
∼
phaseseparatedferromagneticdomainsarenotobserved.
-1.0 Incontrast,thespecificinitialconfigurationinthecol-
0 1 2 3 4 5 6 liding cloud experiment [3] proves to be crucial for the
Ω t(cid:144)2Π metastability of the upper branch. Three-body loss is
z
limited spatially within the overlap region between dif-
FIG.4: (Coloronline)Timedependenceofthecenter-of-mass ferent spins at the trap center, and temporally to a frac-
positions z¯σ of the two spin clouds for kF(0)as = 0.5. The tion of the oscillation period. As such, the effective τ3
parameters are thesame as in Fig.2. is greatly enhanced (τ τ ) and one can study
3 3,micro
≫
the metastable upper branch. Note that while τ
3,micro
is small, 10/ǫ [21], it is still an order of magnitude
F
intermediate-time dynamics,after the bounce is damped ∼
greater than τ ǫ , and thus local thermodynamic
2 F
out, and the upper branch equation of state: Fermi liq- ∼
equilibrium can be established in the upper branch.
uid (k a . 1) versus phase separated (k a & 1). The
F s F s In summary, we have investigated the short-time
solutionsof(6)and(7)atk (0)a =2and0.5(usingthe
F s bounce dynamics of the recent MIT experiment [3]. The
same initial conditions and ǫ0 as in Fig. 2), are shown
F bounce frequency we found is in good agreement with
in Figs. 3 and 4. Here k (0) = 2mǫ0 is the ideal gas
F F the experiments. Furthermore, the experimental obser-
Fermi wavevector at the cloud center. Appropriate for
p vation that the two clouds sit side-by-side for as long
the smaller interaction strength, we use an expression
as 100ms validates our assumption of a long τ . We
3
for the spin-current damping obtained from kinetic the- ∼
arguethattheintermediatetimebehaviorofthetwocol-
ory [11]. When symmetrized in the spin components, it
liding clouds indicates that the system favors a phase
takes the simple form [19]
separated metastable state for larger value of k a . We
F s
cannot see how a lower-branch energy functional, with
γ =(4/9π)( kF↑kF↓as)2√ǫF↑ǫF↓. (11) attractive interactions forming bound pairs, could lead
Even at small k (0p)a , the clouds exhibit a weak to the observed dynamics.
F s
bounce due to compressional recoil, which damps out Noteadded: Recently,aBoltzmannapproachwasused
veryquickly. Wesee,however,acleardifferencebetween toinvestigatethesameprobleminthehigh-T regime[22].
Fig.3(k a &1), wherethe centersofmass remainsep- The system is found to be strongly hydrodynamic for
F s
arated, and Fig. 4 (kFas . 1), where they merge. This kFas &1.
behavior is qualitatively similar to the experimental re-
sults shown in Supplementary Fig. 1 of Ref. [3].
Acknowledgments
We acknowledge discussions with S. Stringari, S-
VI. DISCUSSION Y. Chang, T.L. Ho, N. Trivedi and M. Zwierlein. We
gratefully acknowledge support from ARO W911NF-08-
We now compare our results in detail with experi- 1-0338 (ET), NSF-DMR 0907366 (SZ, ET), NSF-DMR
ments. As seen from Fig. 2, our results at unitarity – 1006532(WS, MR).
the short-time bounce, the damping of the oscillations
and the separation of the clouds at intermediate times
– are all in very good agreement with Ref. [3]. Away
Appendix A: Upper branch and LOCV
from unitarity, at finite values of k a > 0, our re-
F s
sults are qualitatively similar to the data in Supplemen-
tary Fig. 1 of Ref. [3]. For times ω−1 t τ , say, In this Appendix, we present a detailed discussion of
z ≪ ≪ 3
t = 200 ms, the system remains phase segregated for theso-called“upperbranch”ofaFeshbachscatteringres-
largek a [Figs.1(f,g)],whilecompletelymixedforsmall onanceintwo-componentFermigases[4,5]aswellasthe
F s
k a [Figs. 1(c,d,e)]. In fact, a very rough estimate for lowestorderconstraintvariational(LOCV) method [16],
F s
the critical k a can be read off from the experimental which we use to calculate the upper branch equation of
F s
data: it is between 0.26 [Fig. 1(e)] and 1.2 [Fig. 1(f)]. the state [17].
5
1. Upper Branch Variation of the energy (A1) with respect to f(r), while
taking into account the constraint (A2) by a Lagrange
The concept of the upper branch of a Feshbach res- multiplier λ, gives us the “Schr¨odinger”equation
onance comes from the two-body problem (either in a
finite box or in a harmonic potential), where it is com- 2
∇ +v(r) f(r)=λf(r). (A3)
pletely well-defined for all values of a , positive or nega-
s − m
tive. The two-body wavefunction in the upper branch is (cid:20) (cid:21)
ascatteringstatewithasinglenodethatmakesitorthog-
Retaining only the s-wave part of this equation, we find
onal to the ground state (lower branch) wave-function.
the general solution with one node is given by
This notion has been generalized to the many-body
case [4, 5] by writing a Jastrow-Slater wavefunction,
where the Jastrow correlation factor has a node; see dsin(κ(r b))
f(r)= − (A4)
Eq. (9) of the main paper and the discussion immedi- rsin(κ(d b))
−
ately following this equation.
Tosummarizeourdefinitionoftheupperbranchinthe
where κ = √mλ. We find that f(d) = 1 and f′(d) = 0
many-bodyproblem,anysensibledefinitionmustat least
lead to κd=tan(κ(d b)) and the Bethe-Peierls bound-
satisfy the following conditions [5]: −
aryconditiongivesκa =tanκb. Withthenormalization
(1) The many-body wavefunction includes, apart from s
condition(A2),wecansolveforκ,dandbforeachvalue
the nodes introduced by the Pauli principle, one addi-
ofscatteringlengtha . Theenergyofthesystemisgiven
tional node for any pair of fermions with opposite spin. s
simply by
(2) The wavefunction should reduce, in the limit of the
two-body problem, to that of the scattering states with
3 1
one node in the relative wavefunction.
= ǫ n+ nλ. (A5)
F
(3) The energy of the system must be larger than that E 5 2
ofthe non-interactingFermigasand, itshouldreduce to
the perturbative result in the weakly interacting regime In the case of our interest, however, the system is not
0<WkeFeamsp≪ha1si.ze thatorthogonalitywith the many-body nuenciteys.saWrileyhbaavleantcoedex,tseondthtahte xLO≡CVn↓c/anlc↑ulnaeteiodnntoottbhee
groundstate(oftheBCS-BECcrossover)isnot sufficient spin-imbalanced case in which the Slater determinants
to be on the upper branch. will have different sizes. We find the energy of the sys-
tem is given by
2. LOCV
3 5
= ǫF↑n↑(1+x3) (A6)
E 5
LOCVis anapproximationschemetoevaluate the en-
2
ergy of the Jastrow-Slater state that originated in nu- + n↑n↓ d3rf∗(r) ∇ +v(r) f(r).
− m
clear many-body physics [16] and has been used for var- Z (cid:20) (cid:21)
ious quantum fluids. Within LOCV, the energy of the
Jastrow-Slater state [Eq. (9) of paper] is given by Now, in general, there is no unique way to enforce the
normalizationconditionsasin(A2). Anaturalextension
3 n2 2 is to use both normalizations
= ǫ n+ d3rf∗(r) ∇ +v(r) f(r), (A1)
F
E 5 4 − m
Z (cid:20) (cid:21)
dσ
where v(r) is the short-range two-body potential, n = 4π(n−nσ) drr2fσ2(r)=1, (A7)
n↑+n↓ is the total density, and ǫF = (3π2n)/2m is the Z0
Fermienergy. Theeffectofazero-rangecontactpotential
whichintroducestwohealinglengths,d andd . Accord-
may be written in terms of the Bethe-Peierls boundary ↑ ↓
condition lim (rf(r))′/(rf(r)) = 1/a . In addition, ingly,weshallintroducetwoLagrangemultipliersλ↑ and
r→0 s
− λ and the energy of the system can then be written as
withinLOCV,theJastrowfunctionf(r)satisfiesthefol- ↓
lowing conditions: f(r d) = 1 and f′(d) = 0. The
≥
“healing length” d in turn is defined so that 3 3 1
= ǫ n + ǫ n + (n λ +n λ ). (A8)
F↑ ↑ F↓ ↓ ↑ ↓ ↓ ↑
E 5 5 2
d
2πn drr2f2(r)=1. (A2)
Z0
6
4.0 1.0
3.5 1!kFaS!0 (a) (b) (c)
3.5
0.8
3.0 P!0.1
3.0
energy2.5 1!kFaS!2.0 energy2.5 P!0.8 P 00..46 FP PP FL
2.0 2.0
0.2
1.5 1!kFaS!10 1.5 P!1.0
1.0 1.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0
P 1!kFaS 1!kFaS
FIG.5: Thesingle particle energyin theupperbranch,obtained from lowest orderconstraint variational method. Theenergy
are in unit of 3/5ǫF (a), energy as a function of the polarization P for various values of kFas as indicated in the figure. (b),
energyasafunctionoftheinteractionparameterkFas forvariousvaluesofpolarizationP asindicatedinthefigure. (c),phase
diagram of the interacting two-component Fermi gases. The two vertical red dashed lines indicates the separation between
three phases. For 1/(kFas)>1.124, the system is in theFermi liquid (FL) phase. For 1.124>1/(kFas)>0.91, thesystem is
in thepartially polarized phase (PP) and for 1/(kFas)<0.91, thesystem is fully polarized.
The above energy functional reduces to the usual per- For 1.124 > 1/(k a ) > 0.91, the system is in the par-
F s
turbativeresultintheweakcouplinglimitandalso,aswe tially polarized phase (PP). Here the transition as pre-
shallshowlater,reproducesthe phasediagramofthe in- dicted by the LOCV method is secondorderandaccom-
teracting fermion system given by Monte Carlo methods panied by a divergent spin susceptibility [17]. Note that
[4, 5] with very good agreement. We note that the same the value of 1/(k a ) at the transition is very close to
F s
energy functional (A8) was analyzed recently by Heisel- that predicted by Monte Carlo calculations [4, 5] and
berg [17], in the context of the experiment of ref. [15]. compares favorably with the calculation in Ref. [23].
InFig.5(a),weshowtheupperbranchenergy(A8)as Lastly, for 1/(k a ) < 0.91, the system is a fully po-
F s
afunctionofpolarizationP (1 x)/(1+x)forvarious larized, non-interacting Fermi gas.
≡ −
values of k a . For small k a (in fact, k a < 0.91),
F s F s F s
the minimum energy is attained at P = 0. For larger
values, k a 1.124, the minimum value is attained at
F s ≥ Appendix B: Numerical details
P = 1. For intermediate values of k a , the minimum
F s
value occurs at a value of 0 < P < 1. In Fig. 5 (b), we
c
showtheupperbranchenergyasafunctionof1/k a for In this section, we explain how we use the LOCV en-
F s
various values of the polarization P. Note that for P = ergyinthenumericalsolutionofthehydrodynamicequa-
1, i.e., completely polarized, the energy is completely tions (6) and (7) of the main text. The important quan-
flat, since there is no interaction between the polarized tity that enters both equations is
fermions.
In Fig. 5 (c), we show the phase diagram of the sys- ∂ 1 ∂λ↓ ∂λ↑
E =ǫF↑+ λ↓+n↑ +n↓ . (B1)
tem as obtained from LOCV. For 1/(k a )>1.124, the ∂n 2 ∂n ∂n
F s ↑ (cid:18) ↑ ↑(cid:19)
system is a homogeneous mixture of the two hyperfine-
Zeeman states. We call this a Fermi liquid (FL) state. We express this in dimensionless form as
1 ∂ 1 5 ∂λ 1 ∂λ 2 1 ∂λ
↓ ↓ ↑
µ↑ E =1+ λ↓ x ξ + xλ↑ xξ . (B2)
≡ ǫF↑∂n↑ 2 3 − ∂x − 3 ∂ξ 3 − 3 ∂ξ !
e e e
e e e
Here ξ = 1/(k a ) and all energies are scaled by ǫ , tionalnumericalderivatives. Tosimplify the numericsat
F s F↑
so that λ↑ λ↑/ǫF↑ etc. We note that at unitarity, “small” as, we find that we can make a polynomial fit
ξ = 0, and t≡he expression involves only a single deriva- to the equation (B2) as a function of x and 1/ξ. we find
that, for the values of k a . 2 we are considering, a
tive ∂λ /e∂x that has to be evaluated numerically. For F↑ s
↓
arbitrary a , however, the expression involves two addi-
s
e
7
good fit is provided by the expression where the coefficient 0.27 is very close to the value ob-
tained by Galitskii [24], 4 (11 2ln2) 0.259.
3 15π2 − ≈
µ =1+x k a +0.27(k a )2 + (B3)
↑ F↑ s F↑ s
4π ···
(cid:18) (cid:19)
e
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