Damping of long wavelength collective modes in spinor Bose-Fermi mixtures

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by  J. H. Pixley

Damping of long wavelength collective modes in spinor Bose-Fermi mixtures
J. H. Pixley, Xiaopeng Li, and S. Das Sarma
Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics,
University of Maryland, College Park, Maryland 20742- 4111 USA
(Dated: July 20, 2015)
Usinganeffectivefieldtheorywedescribethelowenergybosonicexcitationsinathreedimensional
ultra-cold mixture of spin-1 bosons and spin-1/2 fermions. We establish an interesting fermionic
5 excitationinducedgenericdampingoftheusualundampedlongwavelengthbosoniccollectiveGold-
1 stonemodes. Twostateswithbosonsformingeitheraferromagneticorpolarsuperfluidarestudied.
0 Thelinear dispersion of thebosonic Bogoliubov excitations ispreserved with a renormalized sound
2 velocity. Forthepolarsuperfluidwefindbothgaplessmodes(densityandspin)aredamped,whereas

l in theferromagnetic superfluidwe findthedensity (spin) modeis (not) damped. Wefindthat this
u
holds for any mixture of bosons and fermions that are coupled through at least a density-density
J
interaction. In addition, we predict the existence of the Kohn anomaly in the bosonic excitation

7 spectrumofBose-Fermimixtures. Wediscusstheimplicationsofourmany-bodyinteractionresults
1 for experiments on Bose-Fermi mixtures.

] PACSnumbers: 67.85.De,67.85.Fg,67.85.Lm,67.85.Pq
s
a
g
- The interplay of bosons and fermions is ubiquitous ubovquasiparticles(BQs)aredampedthroughhigheror-
t
n throughoutphysics,rangingfromthe interactionof light der interactions in the form of Beliaev [21–26] and Lan-
a (i.e. photons) and matter (i.e. electrons) to the behav- dau [27] damping. Due to destructive quantum inter-
u
ior of a simple metal in the ionic lattice background. ference Landau and Beliaev process are suppressed [28]
q
In solid state physics the interaction of electrons with at low momentum and low energies, thus making the
.
at slowlymovingphononsprovidesthe necessaryattractive longwavelengthcollectivemodeawelldefinedundamped
m electron-electron interaction to form Cooper pairs lead- bosonicexcitation; infact, the long-wavelengthdamping
ing to superconductivity [1]. While on the other hand goes as q5 vanishing rapidly as q 0. An important
-
d theeffectoftheelectronFermisurfaceonbosonsappears question of fundamental interest, wh→ich has also become
n
naturally in phonon excitations in the form of the Kohn relevant in view of recent experiments [20, 29], is, how-
o
anomaly [2]. Another well-know example of the solid ever, still open in spite of extensive theoretical activity,
c
[ state manifestation of fermion-boson interaction is the namely, how the fermion-boson interaction (specifically

polaron formation in ionic insulators. In the description the existence of the Fermi surface) affects the bosonic
2
ofitinerantquantumphasetransitions[3,4](relevantfor excitation spectrum in a Bose-Fermi cold atom mixture.
v
5 various strongly correlated materials [5–8]) the fermions We address this important question in the current work
1 are always coupled to a bosonic collective mode reflect- usingfieldtheoretictechniques,findingagenericfermion-
0 ing the ordering of the underlying Fermi gas. A coupled induced damping of the long wavelength bosonic collec-
5
fermion-boson interacting many-body system is thus a tive modes.
0
fundamentalparadigmincondensedmatterphysicslead- We start with a microscopic Hamiltonian for a Bose-
.
1
ing to a large number of interesting phenomena. Fermimixture,andderivealowenergyeffectivefieldthe-
0
Coupled fermion-boson systems have also become of ory based on a controlled perturbative expansion, from
5
1 interest recently in ultra-cold atomic systems. In the which the low energy bosonic excitations and damping
: context of cold atoms, a variety of Bose-Fermi mixtures, effects are obtained. A mixture of a spin-1 Bose gas and
v
e.g., 6Li-7Li [9, 10], 40K-87Rb [11–17], 6Li-133Cs [18, 19], a spin-1/2 Fermi gas is considered and spin SU(2) sym-
i
X 6Li-174Yb[20],havebeenpreparedinexperimentsfordif- metry is assumed for both theoretical simplicity and rel-
r ferent purposes such as implementing sympathetic cool- evance to experimental systems in the absence of mag-
a
ing [9–11], studying molecule formation [14], engineer- netic fields. We show quite generally that the linearly
ing dipolar quantum simulators [15], exploring few-body dispersive BQs of a bosonic superfluid interacting with
physics [17–19] or looking for interesting collective ex- fermions become damped due to Fermi surface effects,
citations [20]. In a recent experiment [20] of particular with a damping rate
relevance to our theory to be presented in the current
γ(q)/~= q, (1)
work, laser spectroscopy was used to study the effect of D| |
the fermions on the bosonic excitation spectra of 6Li- where is dependent on the microscopic details of the
174Yb atomic mixtures. systemD. Moreover,atlargemomentum(q k )weshow
F
≈
In the absence of fermions, the low energy excitations that the Kohn anomaly appears as a kink in the BQ
in Bose-Einstein condensates are well described by Bo- exception spectrum. The linear momentum dependence
goliubovtheoryanditiswellunderstoodthattheBogoli- in Eq. (1) is drastically different from pure boson sys-
2
tems[21–25,27,28]orBose-FermimixtureswithaFermi (a) α (b) α
surface that is gapped out [29, 30]. On the other hand, β
thefunctionalformofBQlineardispersionatlowenergy
v q remains intact, with a renormalized sound velocity FIG. 1: Diagram (a) is referred to as the “tadpole” and (b)
s
| |
v . Forspinwaveexcitations,wefinddampinginapolar isthespindependentparticle-hole“bubble”. Asolid linede-
s
(P)groundstate whereasforaferromagnetic(FM) state notesafermion Green function with spinα,asolid rectangle
the spin waves are found to be undamped. denotestheBose-Fermiinteraction(UBF orJ),andthewavy
line denotes the bosonic field φa in one of the mz = 1,0,−1
We begin with the model fermion-boson interacting
states.
Hamiltonian, H = d3r( + + ), with
B F BF
H H H
HB = 2m~2 ∇RΦ†a∇Φa+ U20Φ†aΦ†a′Φa′Φa (2) troduced the Bose gas action SB, the Fermi gas action
B S , and their mutual interaction S . To derive the
F BF
U
+ 22Φ†aΦ†a′Tab·Ta′b′Φb′Φb, (3) lfiorwsteenxeprgayndefftehcetivBeosaectoiopneroaftotrhseabboosuonticthseupmereflaunidfiewlde
HF = 2m~2F∇c†σ∇cσ, σ (4) gwriothuntdhsetamteeaΦnTfi=eldΦTMvaFlu+e(φof1,µφB0,giφv−es1)r,iwsehitcohatnogeefftheecr-
= U Φ†Φ c†c +JΦ†T Φ c† αβc , (5) tive action for the φ degrees of freedom SB,eff[φ] (see
HBF BF a a σ σ a ab b· α 2 β the Supplemental Material). As the Fermi gas is non-
where repeated indices are summed over, T denotes interacting we integrate them out and determine the ef-
a vector of spin-1 matrices, and σ denotes a vector fective action of the fluctuations φ (ignoring any con-
of Pauli matrices. The operator Φ destroys a boson stants) S˜[φ]=S [φ] Trlog(1 Vˆ(φ)Gˆ ). The trace
a B,eff 0
− −
in the m = a state and c destroys a fermion with is over spin, space, and imaginary time indices and we
z σ
spin σ. The interactions U and U are related to the havedefinedthetwobytwomatricesVˆ(φ)andtheGreen
0 2
scattering lengths of each hyperfine state of the bosons function of the fermions Gˆ , which depend on the super-
0
through U = (gB + 2gB)/3 and U = (gB gB)/3, fluid state. As Vˆ(φ) is composed entirely of fluctuations
with g =04π~2a 0/m fo2r a scatterin2g lengt2h a− i0n hy- of the Bose gas we can expand the logarithm to obtain
f f B f
perfine state f [31]. Whereas the Bose-Fermi interac- the low energy effective action to quadratic order
tions U and J are given by U = (gBF +2gBF)/3
BF BF 1/2 3/2 1
and J = 2(gBF gBF)/3, with gBF = 2π~2a /m S [φ]=S [φ]+Tr(Vˆ(φ)Gˆ )+ Tr (Vˆ(φ)Gˆ )2 .
3/2 − 1/2 Ftot Ftot BF eff B,eff 0 2 0
as the s-wave scattering length with total spin F and
tot (cid:16) (cid:17)(6)
m the reduced mass [32]. We focus on a repulsive
BF The first termafter S correspondsto the tadpole di-
B,eff
density-density Bose-FermiinteractionU >0 andfer-
BF agramand the secondto the spinful particle-holebubble
romagnetic spin-spin interaction J < 0. We assume the
in Figs. 1 (a) and (b) respectively.
fermionstobenon-interacting,whichisvalidforbarere-
Ferromagnetic Ground State: Focusing on the ferro-
pulsive interactions because they are strongly irrelevant magnetic case U <0 ( Φ† TΦ = ρ zˆ, where ρ is
in the low energy limit [33, 34]. 2 h MF MFi 0 0
the boson density), the mean field expectation value of
In the absence of the Fermi gas, the spin-1 Bose gas
the Bose operators can be written as a three component
can become [31, 35] either a FM superfluid for U2 < 0 spinor, ΦTMF = (√ρ0, 0, 0). The chemical potential of
or a P superfluid for U2 >0, and we consider both situ- thebosonsatthemeanfieldlevelisµFM =gBρ +JM +
ations theoretically. In the FM phase, the ground state B 2 0 z
U n , whereM =( n n )/2andn = n ,
breaksboththeU(1)andSU(2)symmetryofthe Hamil- wBitFh nF = c†c .zExphan↑diin−ghth↓ei Bose opeFratorsσahboσuit
tonian and as a result hosts two distinct types of Gold- σ σ σ P
their meanfield groundstate results in the fermions see-
stonemodescorrespondingtogaplessdensityexcitations
ing an effective magnetic field ρ J in the zˆdirectionand
0
which are of the BQ form that go as q and of the fer-
it breaks the up-down spin symmetry of the Fermi gas.
| |
romagnetic spin wave form that go as q2. The P super-
This gives rise to the non-interacting Green function,
flhuosidtsatlwsoobdriesatiknsctthseetUs(o1f)gaanpdleSsUs(d2e)nssyitmymanetdryspainndwaalvsoe Gˆ−0σ1σ′ =−δσ,σ′(∂τ−~2/(2mF)∇2−µ˜F+Jρ0σ/2),where
σ = 1 for and , and we have shifted the fermionic
excitations, however in this case they both take on the chem±ical pot↑ential µ↓˜ =µ ρ U . The matrix Vˆ
F F 0 BF FM
linear-in-q BQ form. −
is given by
We are interested in the low energy theory of the sys-
tem andderivethe correspondingeffective actionfor the 1
Vˆ (φ)= J( ρ /2φ† + S−)σ +h.c (7)
gas within a path integral framework [36]. The model FM 0 0 2 φ +
(cid:18) (cid:19)
Hamiltoniancorrespondstoanactioninthegrandcanon- p J
icalensembleS = dτ H(τ)+ d3r Φ†a∂τΦa+c†σ∂τcσ− + UBF(√ρ0[φ†1+φ1]+nφ)1+ 2(√ρ0[φ†1+φ1]+Sφz)σz,
µBΦ†aΦa − µFc†σcσR =(cid:16) SB +RSF +(cid:2)SBF. We have in- where 1 denotes the identity matrix, σi are the Pauli
(cid:3)(cid:17)
3
matrices, σ = (σ iσ )/2, and we have introduced with a value = 2(ρ U +M J ). In contrast, the
± x y 0 2 z
± M | | | |
n = φ†φ and Sα =φ†Tαφ . spinwaveexcitationsthatcorrespondtoφ ,aregivenby
φ a a a φ a ab b 0
Evaluating the trace for the particle-hole bubble leads ωFM =~2q2/(2m∗) where the coefficient is altered from
to χ0P(q) = dkG (k + q)G (k), we are using the ~2s/w(2m ), and thse spin excitations acquire a renormal-
αβ 0α 0β B
shorthandnotationq (iν ,q)[k (iω ,k)]forbosonic ized effective mass ~2/(2m∗)= FM/ FM, which is not
R ≡ n ≡ n s B0 A0
[fermionic]Matsubarafrequency,andtheintegral dq a function of U . Therefore, the renormalizationof
BF
(2dπq3)3β1 n. The Green function in momentumRspac≡e and m∗s is due entirely to the paramagnon excitatioMns
of the Fermi gas through the coupling J. The density
RipsenGd0eσn(tkP,diiωspne)r−s1ion=ǫkiωσn=−~ǫ2kkσ2/+(2µ˜mFF,)w+itJhρa0σs/p2i.n Fdoer- mcoondtreisb(uφt1io+nφo†1f)νacq/uqirewdhaimchpianrgistehsrdouugehtothteheadpdairttioicnlae-l
the spin diagonal case α = β this reduces to the well | n| | |
holeexcitationsoftheFermigas(seeEq. (14)below). It
known Lindhard function [37], which at sufficiently low
is useful to note that the constants FM and FM are
temperature in the low energy limit, q/kFσ 1 and A1 B1
ν /(~v q) 1 (where v is the|Fe|rmi ve≪locity of both quadratic functions of UBF and J.
n Fσ| | ≪ Fσ Polar Ground State: We nowdiscussthe polarground
the spin σ electrons), becomes
state of the bosons when U > 0, ( Φ† TΦ = 0),
2 h MF MFi
π ν 1 q 2 with a mean field expectation ΦTMF = (0,√ρ0,0). Now,
χ0 (q,iν )= η 1 | n| | | the up down spin symmetry of the Fermi gas remains
σσ n − σ − 2~vFσ|q| − 3(cid:18)2kFσ(cid:19) ! intact. The chemical potential for the bosons takes
(8) the form µP = U ρ + U n . The non-interacting
B 0 0 BF F
and we have defined the density of states for spin σ, Green function is now spin independent, which becomes
νηnσ/(=~vFmσFqkF)σ/(21π2is~2c)o.nsiWsteensttrwehssilethceonaspidperroixnigmBatoisoen- Giˆs−0dσ1eσfi′n=ed−aδsσ,σ′(∂τ−~2/(2mB)∇2−µ˜F). The matrixVˆP
| | ≪
Fermi mixtures with v v where v is the sound
s,0 Fσ s,0
≪
velocityoftheBQexcitationsintheabsenceoftheFermi Vˆ (φ)=U (√ρ [φ† +φ ]+n )1+JSzσ /2
P BF 0 0 0 φ φ z
gas. Physically, this ensures that the BQs excite a suf-
1
ficient amount of particle-hole pairs, by passing through + J( ρ /2[φ† +φ ]+ S−)σ +h.c. . (11)
0 −1 1 2 φ +
their excitation continuum as shown in Fig. 2 (d). For (cid:18) (cid:19)
p
the case σ = σ′, in the low frequency, low momen-
At this point it is useful to compare this with the FM
tum limit we6 can always treat ~2 q2/(2m ) J ρ ,
F 0 case. From the effective bosonic action S [φ] in the P
~vFσ q J ρ0, and νn J ρ0.|W|ithin this≪app|ro|xi- caseweknowthatthedensitymodeisrelaBte,edfftoδn(q)† =
| |≪| | | |≪| |
mation we find (φ†(q),φ ( q)) while the spin excitations are given by
0 0 −
χ0 (q,iν )= 1 ( n n ) σ¯ iνn (9) δS(q)† =(φ†1(q),φ−1(−q)). Asaresult,itisquitenatural
σσ¯ n ρ0J h σ¯i−h σi (cid:18) − ρ0J(cid:19) thatthedensityexcitationsonlycouplethroughUBF and
~2n σ¯ q2 the spin excitations through J. This is quite different
+ F + (~v )2 n (~v )2 n | | . from the case of the FM superfluid where the BQs are a
2m 5ρ J Fσ¯ h σ¯i− Fσ h σi (ρ J)2
F 0 0 combination of density and spin along the z direction.
h (cid:0) (cid:1)i
Following the same steps as before, we compute the
With all of these results in hand we can now de-
trace in Eq. (6), only now the equal spin particle-hole
termine the effective action to quadratic order. Writ-
ing the action in separate parts we have SFM[φ] = bubble in Eq. (8) comes into play Due to the spin sym-
eff
metry the Fermi gas parameters are now σ independent.
dq + + , where
Lφ1 Lφ0 Lφ−1 The effective actionto quadraticorderfor the polarcase
R L(cid:0)φ1 =φ†1(q) −iνn+(cid:1) b0q2 φ1(q) (10) δisn(gqiv)†eGn by(,q)S−eP1ffδ[nφ(]q=) anddq 21Lδ=n+δSL(δqS)†G, wh(qe)r−e1LδSδn(q=).
ν 1 2 δn R L(cid:0)δS (cid:1) δS
+ (∆FM + FM(cid:0) | n| + FM(cid:1)q2) φ†(q)+φ ( q) , The bosonic Green functions can be written in terms
1 A1 q B1 | | 2 1 1 −
of the free bosonic Green function and the self energy
| | (cid:12) (cid:12)
(cid:12) (cid:12) using Dyson’s equation G (iν ,q)−1 = G0(iν ,q)−1
LInMφ0or−=deirν(n−to+AsF0ibm0M|pqilν|i2fny+φth†−Be10(FpqMr)eφ|sq−e|n12(t)qaφ)t†0,io(wqn)hφw(cid:12)e0er(eqh)ba,0vae=nidnt~rL2o/φd2−um1c(cid:12)eB=d. −Σaiν(inνσnz,+q),b0wqh2i1chaanrdetwobaytwnomatricesGa0a(iνnn,q)−1 =−
(cid:0) (cid:1)
the constants ∆FM, FM, FM, FM, and FM, whose ν
1 A1 B1 A0 B0 Σ (iν ,q)= ∆P + P| n| + P q2 (1+σ ).(12)
functionalformisnotparticularlyrelevantforthepresent a n − a Aa q Ba| | x
(cid:18) | | (cid:19)
discussion and are explicitly given in the Supplemen-
tal Material. Now that we have the Green function for We give the explicit forms of ∆P, P, and P in the
a Aa Ba
each spin state from the effective action, we can deter- Supplemental Material. In this case, the δn constants
mine the excitation of each mode from the poles [36]. are quadratic functions of U and do not depend on
BF
A few remarks are in order: The φ mode is gapped J while the δS constants are quadratic functions of J
−1
4
ÑHL(cid:144)HLΠEq2KHz∆n1234560000000000000 100Ρ0..025H15xc4mx110-10351L5=1013 01.07155x1015 (a) ÑHL(cid:144)HLΓΠq2KHz∆n0000....00110505 (b) [Tdha1neah+nraudclsyte,fatuinascnuhlalcr(ltvBµ˜irioFeQvnse/usk[ml3.Bt7osTS]daog)es]lis|vvsqeihpn|sorgaewrnseEnesdnuqittnl.thsi(Fen1iinl3gaid)n.ise2supats(iriibnnnm)-gg1uuoitpBmshhoteeaosnebetqlxugea∼mafcsrtOeodsmeL(pkcintoFehdnu)e---.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
pled to a Fermi gas become damped at low energies and
q(cid:144)kF q(cid:144)kF
LKHz00..45 (c) LHz567000000 awraevetlheunsgtnhos!longer true collective modes even at long
ÑHL(cid:144)H∆ΠEq2∆n0000....0123 ÑHL(cid:144)HΠEq2K01234000000000 (d) (a1ll4So)ervadenerdraslFicniogmt.h2meeaBnroetssiena-Fboeorrudmteiro:ucor(uim)pOaliinungrrperseruosuvltlistdssehdaorvews,nv0ai≪lnidEvtqFo.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 sinceallhigher-ordervertexcorrectionsintheself-energy
q(cid:144)kF q(cid:144)kF are negligible by virtue of Migdal’s theorem [38]. (ii)
The fermion-induced bosonic damping being linear in q
FIG. 2: Energy spectrum and damping rate of collective
strongly dominates any intrinsic bosonic Beliaev damp-
modes from solving Eq. (13) for various bosonic densities ρ0.
In this plot the exact Lindhard function [37] is included fo- ing at long wavelength [24]. (iii) Eq. (14) implies that
cusingonthedensitymode(δn) forthePcase with23Nafor the bosonic collective mode frequency vanishes without
the bosons, the Fermi gas it taken to be 6Li with a density becomingoverdampedifthedampingbecomescompara-
of nF =1013cm−3, and focusing on UBF = U0, and J =U2. ble or larger than the mode energy itself since both the
(a) The dispersion Eδn(q), (b) the damping rate γδn(q), (c) frequency and the damping go linear in wave number.
the difference in dispersions with and without the Fermi gas
δEδn(q)≡ Eδn,0(q)−Eδn(q), displaying the Kohn anomaly (iv) At large wave numbers, q > kF, γa(q) → 0 because
near q/kF ∼ 1.5, and (d) the Bose gas dispersion in the ab- the imaginary part of the Lindhard function [37] itself
senceoftheFermigasEδn,0(q)=qvs2,δn,0q2+b20q4,allasa vcaannisnhoeslobnygeenreerxgyci-tmeopmaertnitculem-hcoolensperavirasti[osneeanFdigt.he2B(dQ)s]
function of q/kF. In (d) the black solid lines bound the re- (v) The spinor structure of the problem is not essential
gion of the particle-hole excitation continuum and the black
tofindthisFermigasinduceddamping,thereforeourre-
dashed line marks where the imaginary part of the Lindhard
function changes its form. sultsarevalidforanycondensedspin-S Bosegascoupled
toaFermigas(eitherspinfulorspinless)throughatleast
a density-density interaction U nˆ nˆ .
BF B F
and are independent of U . Interestingly, in the polar TheFermisurfacehaseffectivelybeenimprintedupon
BF
case we find both BQ modes to be damped through the the excitation spectrum of the Bose gas in the form of
appearanceof ν /q. Weremarkthatthedensitymode the Kohn anomaly. As this is a large momentum effect,
n
| | | |
for the FM case in Eq. (10) canalso be convertedto the it is not captured by our low energy theory but comes
above form. directly out of the numerical solution using the exact
BQ Excitation Spectrum and Damping: We have de- Lindhard function. However, this effect is weak for the
rivedtheeffectivefieldtheoryforboththeferromagnetic atomic mixtures we are considering and is not visible in
and polar bosonic ground states interacting with a spin- E (q)itself,onlythedifferenceδE (q)=E (q) E (q),
a a 0 a
−
1/2 Fermi gas. To diagonalize the effective action using displays the anomaly as seen in Figs. 2 (a) and (c).
the Bogoliubov transformation we find uk and vk have Our theoretical predictions can be tested in ultra cold
to be functions of ν . To determine the dispersion and Bose-Fermi mixtures with weakly interacting fermions,
n
dampingratewefindthepolesofthebosonicGreenfunc- through two-photon Bragg spectroscopy [28] that cou-
tion G (with a = φ ,δn,δS) [36], after analytically ples to the bosonic species. We expect the damping will
a 1
continuing the Matsubara frequency to real frequency give rise to a broadened Bragg line, i.e. an intrinsic line
iν ω+i0+, we solve the algebraic equation width, in the presence of fermions that will depend ex-
n
→
plicitly onenergy-momentumasshowninEq. (14). Due
detGa(ω,q)−1 =0, (13) tothe bosonicdamping persistingtosufficiently lowmo-
mentum, we also expect that the presence of the Fermi
tofindthepolesat~ω =E(q) iγ(q),whereγ(q)/~ 0 gas can dephase the bosonic superfluid, which has possi-
− ≥
is the damping rate. We obtain two of our main results bly already been observed in Ref. 16.
In the presence of a shallow harmonic trap as in most
γa(q)=b0Aa|q|, Ea(q)= vs2,aq2+b2aq4, (14) atomic experiments, we expect our results to be largely
q unaffected for large clouds of atoms. Even though the
(seeFig.2)andwefindtheBogoliubovformremainswith finite size of the trap gives an infrared cut off to the
the renormalizedparametersv2 =b (2∆ b 2), and theory, this energy scale is sufficiently small such that
s,a 0 a− 0Aa
b2 = b (b +2 ). The low energy theory can also be linear-in-q BQ dispersion can still be observed in exper-
a 0 0 Ba
extended to finite temperature, which yields γ (q,T) iments [39]. In this limit, the physics is well described
a
∝
5
by the Thomas-Fermi and local density approximations, [16] K. Gu¨nter, T. St¨oferle, H. Moritz, M. K¨ohl, and
where the BQs will still be linear-in-q at low momentum T. Esslinger, Phys.Rev.Lett. 96, 180402 (2006).
(with a renormalizedsoundvelocity)[31, 39,40]andthe [17] R. S. Bloom, M.-G. Hu, T. D. Cumby, and D. S. Jin,
Phys. Rev.Lett. 111, 105301 (2013).
Fermi gas will acquire a spatially dependent Fermi ve-
[18] S.-K. Tung, C. Parker, J. Johansen, C. Chin, Y. Wang,
locity. In spite of this, close to the center of the trap,
and P. S.Julienne, Phys.Rev.A 87, 010702 (2013).
theformofparticle-holeexcitationswillremainthesame
[19] S.-K. Tung, K. Jimenez-Garcia, J. Johansen, C. V.
with arenormalizedvF. However,wewouldlike tomen- Parker, and C. Chin, Phys. Rev. Lett. 113, 240402
tion that these inhomogenous effects can in principle be (2014).
removed by considering a uniform potential trap, which [20] H. Hara, H. Konishi, S. Nakajima, Y. Takasu, and
has recently been achieved, where the spatial inhomo- Y. Takahashi, Journal of the Physical Society of Japan
83, 014003 (2014).
geneity is minimized [41].
[21] S. T. Beliaev, Sov.Phys. JETP 7, 289 (1958).
We have studied a spin-1 Bose gas coupled to a spin- [22] P. C. Hohenberg and P. C. Martin, Ann. Phys. (N.Y.)
1/2 Fermi gas in three dimensions. We have shown 34, 291 (1965).
BQ excitations are damped at long wavelength as a re- [23] V. N.Popov, Theor. Math. Phys. 11, 478 (1972).
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6
Supplemental Material
In the supplemental material we give the explicit expressions for S and the equations defining the constants
B,eff
that we have introduced in the main text in terms of the parameters of the model. For the ferromagnetic case, the
effective bosonic action including the bosonic chemical potential is
~2 1
SFM [φ] = dτd3r φ†∂ φ + φ† φ +gBρ (φ† +φ )(φ† +φ )+2ρ U φ† φ
B,eff Z a a τ a 2mB a ∇ a∇ a 2 02 1 1 1 1 0| 2| −1 −1!
X X
dτd3r (JM +U n ) √ρ (φ† +φ )+ φ†φ , (S1)
− z BF F " 0 1 1 a a#!
Z a
X
whereas for the polar case, the effective bosonic action takes the form
~2 U ρ
SP [φ] = dτd3r φ†∂ φ + φ† φ + 0 0(φ† +φ )(φ† +φ )+U ρ (φ† +φ )(φ† +φ )
B,eff a τ a 2mB ∇ a∇ a 2 0 0 0 0 2 0 1 −1 −1 1 !
Z a a
X X
dτd3rU n √ρ (φ† +φ )+ φ†φ . (S2)
− BF F 0 0 0 a a!
Z a
X
To simplify the presentation of the main text we have defined the following constants for the ferromagnetic case
1
∆FM = ρ gB ρ U2 + J2 η ρ U J ση , (S3)
1 0 2 − 0 BF 4 σ − 0 BF σ
(cid:18) (cid:19) σ σ
X X
1 π η π η
FM = ρ U2 + J2 σ +ρ U J σ σ , (S4)
A1 0 BF 4 2 ~v 0 BF 2 ~v
(cid:18) (cid:19) σ Fσ σ Fσ
X X
1 1 η 1 η
FM = ρ U2 + J2 σ +ρ U J σ σ , (S5)
B1 0 BF 4 12 k2 0 BF 12 k2
(cid:18) (cid:19) σ Fσ σ Fσ
X X
M
FM = 1+ z, (S6)
A0 ρ
0
~2 n ~2 1
FM = + F + σ(~v )2n . (S7)
B0 2m ρ 2m 10ρ2J Fσ σ
B 0 F 0 σ
X
For the polar case we have introduced the constants
1
∆P = ρ U ρ ηJ2, (S8)
δS 0 2− 2 0
πρ ηJ2
P = 0 , (S9)
AδS 4~v
F
ρ ηJ2
P = 0 , (S10)
BδS 24k2
F
∆P = ρ U 2ρ ηU2 , (S11)
δn 0 0− 0 BF
πρ ηU2
P = 0 BF, (S12)
Aδn ~v
F
ρ ηU2
P = 0 BF. (S13)
Bδn 6k2
F