Casimir interaction among heavy fermions in the BCS-BEC crossover

Casimir interaction among heavy

fermions in the BCS-BEC crossover

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Nishida, Yusuke . “Casimir interaction among heavy fermions in the

BCS-BEC crossover.” Physical Review A 79.1 (2009): 013629.(C) 2010

The American Physical Society.

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http://dx.doi.org/10.1103/PhysRevA.79.013629

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Casimir interaction among heavy fermions in the BCS-BEC crossover

Yusuke Nishida

Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

共Received 17 November 2008; published 29 January 2009兲

We investigate a two-species Fermi gas with a large mass ratio interacting by an interspecies short-range interaction. Using the Born-Oppenheimer approximation, we determine the interaction energy of two heavy fermions immersed in the Fermi sea of light fermions as a function of the s-wave scattering length. In the BCS limit, we recover the perturbative calculation of the effective interaction between heavy fermions. The p-wave projection of the effective interaction is attractive in the BCS limit while it turns out to be repulsive near the unitarity limit. We find that the p-wave attraction reaches its maximum between the BCS and unitarity limits, where the maximal p-wave pairing of heavy minority fermions is expected. We also investigate the case where the heavy fermions are confined in two dimensions and the p-wave attraction between them is found to be stronger than that in three dimensions.

DOI:10.1103/PhysRevA.79.013629 PACS number共s兲: 03.75.Ss, 05.30.Fk, 67.85.Lm

I. INTRODUCTION

Experiments using ultracold atomic gases have achieved great success in realizing a new type of fermionic superfluid. By arbitrarily varying the strength of interaction via the Fes-hbach resonance, the crossover from the BCS superfluid of fermionic atoms to the Bose-Einstein condensate of tightly bound molecules has been observed and extensively studied 关1,2兴. Nowadays a portion of interests of the cold atom

com-munity is shifting to the two-species Fermi gas with unequal densities关3–7兴 and with unequal masses 关8–10兴. Elucidating

the phase diagram of such an asymmetric Fermi gas is an important and challenging problem. Such an asymmetric sys-tem of fermions will be also interesting as a prototype of high density quark matter in the core of neutron stars, where the density and mass imbalances exist among different quark flavors关11兴.

So far several quantum phases have been proposed as a ground state of the density-imbalanced Fermi gas. One of such phases is the p-wave superfluid phase by the pairing between the same species of fermions关12兴. In Ref. 关12兴, the

attraction between the same species of fermions was found to be induced by the interaction with the other species of fer-mions based on the perturbative calculation in the weak-coupling BCS limit. Although the resulting pairing gap is exponentially suppressed in the BCS limit, it may be pos-sible that the pairing gap becomes large enough near the strong-coupling unitarity limit. Thus an important question is how large the pairing gap can be near the unitarity limit.

However, difficulties for theoretical treatments arise away from the weak-coupling BCS limit because we do not have a controlled tool to analyze the strongly interacting many-body system. Monte Carlo simulations also have limitations in asymmetric systems due to the fermion sign problem. We note that in one dimension the Monte Carlo simulations do not suffer from the sign problem and have been employed to study the Fulde-Ferrel-Larkin-Ovchinikov phase in the density-imbalanced Fermi gas关13–15兴.

In order to obtain further insight into the density-imbalanced Fermi gas and the p-wave pairing therein, we investigate the Fermi gas with unequal masses between two

different atomic species. In the limit of large mass ratio, we can determine the effective interaction among heavy fermi-ons induced by the interaction with light fermifermi-ons using the Born-Oppenheimer approximation. Because we do not need to rely on the weak-coupling approximation, we can go be-yond the perturbative BCS regime to the strongly interacting unitary regime in a controlled way.

We will show that the p-wave projection of the effective interaction between two heavy fermions is attractive in the BCS limit being consistent with the perturbative prediction, while it turns out to be repulsive near the unitarity limit. Thus the p-wave attraction has a maximum between the BCS and unitarity limits, where the maximal p-wave pairing of heavy minority fermions is expected. We note that our results have a direct relevance to the recently realized Fermi-Fermi mixture of 40K and 6Li because of their large mass ratio 关8–10兴.

It is worthwhile to point it out that the effective interac-tion among heavy fermions immersed in the Fermi sea of light fermions is an analog of the Casimir force among ob-jects placed in a vacuum 关16兴. The origin of the Casimir

force can be traced back to the modification of the spectrum of zero point fluctuations of the electromagnetic field. In our case, the role of vacuum is played by the Fermi sea of light fermions whose spectrum is modified by the presence of heavy fermions关17兴.

This paper is organized as follows. In Sec. II, we deter-mine the effective interaction between two heavy fermions induced by the interaction with the Fermi sea of light fermi-ons as a function of the s-wave scattering length using the Born-Oppenheimer approximation.共The effective interaction for a general number of heavy fermions is derived using the functional integral method and evaluated in Appendix A.兲 With having in mind the application to the p-wave pairing of heavy fermions, we compute the p-wave projection of the effective interaction in Sec.III. Here we consider both cases where the heavy fermions are in three dimensions 共3D兲 and where they are confined in two dimensions 共2D兲 while light fermions are always in 3D. The latter case corresponds to the two-species Fermi gas in 2D-3D mixed dimensions studied in Refs. 关18,19兴. Finally summary and concluding remarks

are given in Sec. IV. PHYSICAL REVIEW A 79, 013629共2009兲

II. CASIMIR INTERACTION BETWEEN TWO HEAVY FERMIONS A. Born-Oppenheimer approximation

Consider a two-species Fermi gas with unequal masses M and m interacting by an interspecies short-range interaction. When the mass ratio is large M/mⰇ1, we can employ the Born-Oppenheimer approximation to compute the interaction energy of heavy fermions immersed in the Fermi sea of light fermions. Here we concentrate on the case of two heavy fermions. The formula for a general number of heavy fermi-ons is derived in Appendix A using the functional integral method. The wave function of a light fermion interacting with two heavy fermions fixed at positions x1and x2satisfies the Schrödinger equation共here and below ប=1兲:

− ⵱y 2

2m⌿共y;x1,x2兲 = E⌿共y;x1,x2兲. 共1兲 The interspecies short-range interaction between the light and heavy fermions is taken into account by imposing the short-range boundary condition; ⌿共y→xi兲⬀

1 兩y−xi兩−

1

a+ O共兩y

− xi兩兲, where a is the s-wave scattering length. Below we

determine the energy spectrum of the light fermion both for bound states and continuum states as a function of a and the separation between the heavy fermions r⬅x1− x2.

For the bound state E = −_2m␬2⬍0, the solution of Eq. 共1兲

takes the form

⌿⫾共y兲 =e −␬兩y−x1兩 兩y − x1兩 ⫾ e−␬兩y−x2兩 兩y − x2兩 , 共2兲

where ⫾ is the parity of the wave function. The upper 共lower兲 sign corresponds to the even 共odd兲 parity. From the short-range boundary condition, we obtain the equation

−␬⫾e −␬兩r兩 兩r兩 = −

1

a. 共3兲

The solution to this equation exists when兩r兩/a⬎ ⫿1 and is given by ␬⫾= 1 a+ 1 兩r兩W共⫾e−兩r兩/a兲, 共4兲 where W共z兲 is the Lambert function satisfying z=W共z兲eW_共z兲

. Accordingly we find the following binding energy depending on a and 兩r兩:

E_binding共⫾兲 = − ␬⫾ 2

2m 共5兲

for 兩r兩/a⬎ ⫿1.

For the continuum state E =_2mk2⬎0, the solution of Eq. 共1兲

takes the form

⌿_⫾共y兲 =sin共k兩y − x1兩 +␦⫾兲 兩y − x1兩

⫾ sin共k兩y − x2兩 +␦⫾兲 兩y − x2兩

, 共6兲 where 0艋␦_⫾艋␲ is the phase shift. From the short-range boundary condition, we obtain the equation

k cos␦_⫾⫾sin共k兩r兩 +␦⫾兲

兩r兩 = − sin␦_⫾

a . 共7兲

The solution to this equation is easily found to be

tan␦_⫾= −k兩r兩 ⫾ sin共k兩r兩兲 兩r兩

a ⫾ cos共k兩r兩兲

. 共8兲

We then suppose that the system is confined in a large sphere with a radius RⰇ兩r兩 and the two heavy fermions are located near its center; x1=

r

2 and x2= −

r

2. By imposing ⌿⫾共兩y兩

→R兲→0 at the boundary of the sphere, the momentum k is

discretized as kn + R +␦+= n␲, kn − R +␦−=

冉

n − 1 2

冊

␲ 共9兲

with n = 1 , 2 , 3 , . . .. Therefore the energy level becomes

En共⫾兲=

kn⫾2

2m. 共10兲

Now we fill the above-obtained energy levels with the light fermions. The total energy of such a system is given by

E = −␬+ 2_+␬ − 2 2m +

兺

_n=1 N kn +2 + kn −2 2m , 共11兲

where 2N is the number of interacting light fermions in the continuum states. We compare the energy of the interacting system with that in the noninteracting limit,

Efree=

冏

兺

n=1 N kn +2 + kn −2 2m

冏

_␦ ⫾=0 , 共12兲

and define the energy reduction as ⌬E共兩r兩兲⬅E−Efree. Then we take the thermodynamic limit N , R→⬁ with the Fermi momentum of the light fermions kF⬅N␲/R fixed. If we no-tice that the summation over n is dominated by n⬃NⰇ1 and neglect small corrections of O共1/N兲, we obtain the following simple expression for the energy reduction:

⌬E共兩r兩兲 = −␬+2+␬−2 2m −

冕

0

k_F

dkk␦+共k兲 +␦−共k兲

m␲ . 共13兲

We note that the phase shifts given in Eq.共8兲 are defined to

be in the range 0艋␦_⫾共k兲艋␲. The same result can be ob-tained on a more general footing by using the functional integral method共see Appendix A兲.

B. Effective interaction between two heavy fermions When the two heavy fermions are infinitely separated, the energy reduction approaches

⌬E共兩r兩 → ⬁兲 → 2␮single, 共14兲 where␮single⬍0 is the chemical potential of a single heavy fermion immersed in the Fermi sea of light fermions 关20兴:

␮single= − k_F2 2m akF+关1 + 共akF兲2兴关␲/2 + arctan共akF兲−1兴 ␲共akF兲2 . 共15兲 The energy reduction with 2␮singlesubtracted is regarded as the interaction energy of the two heavy fermions:

V共兩r兩兲 ⬅ E共兩r兩兲 − E共兩r兩 → ⬁兲 = ⌬E共兩r兩兲 − 2␮single. 共16兲

V共兩r兩兲 represents the effective interaction between two heavy

fermions induced by the interaction with the Fermi sea of light fermions. We note that the chemical potential of light fermions ␮l⬅

kF2

2m is fixed here instead of their particle num-ber.

It is convenient to measure the interaction energy V共r兲 in units of the Fermi energy of light fermions and introduce a dimensionless functionv共k_Fr兲 as

V共r兲 ⬅ kF

2

2mv共kFr兲. 共17兲

v共kFr兲 is a function of the separation between the heavy fer-mions kFr and also the s-wave scattering length akF.v共kFr兲 for three typical values of共ak_F兲−1_{= −5 , 0 , 5 are plotted in Fig.}

1. One can see the smooth evolution of the effective interac-tion between the two heavy fermions as a funcinterac-tion of共akF兲−1

in Fig. 2. In the BCS regime共ak_F兲−1_{ⱗ−1, the effective} in-teraction is attractive at rⱗ兩a兩 and has a tiny oscillatory be-havior at rⲏ兩a兩. This oscillatory behavior grows toward the unitarity limit 共akF兲−1⬇0 and makes a small hump at r ⯝kF

−1 _{共see the left panel of Fig.2兲. This hump further grows} in the BEC regime 共akF兲−1ⲏ1 and eventually develops a repulsive core at r⯝a 共see the right panel of Fig.2兲.

It is worthwhile to compare our nonperturbative result 共16兲 with the perturbative calculation of the effective

inter-action in the BCS limit a→−0. In the BCS limit 兩a兩Ⰶr, k_F−1, we have␬_⫾→0 and the phase shift

␦⫾共k兲 →兩a兩_r 关kr ⫾ sin共kr兲兴 ⫾

冉

兩a兩_r

冊

2

cos共kr兲关kr ⫾ sin共kr兲兴

+ O共兩a兩3_{兲. 共18兲}

Thus we can find the interaction energy to be

v共k_Fr兲 → 共兩a兩kF兲2

2k_Fr cos共2k_Fr兲 − sin共2k_Fr兲

2␲共k_Fr兲4 + O共兩a兩 3_兲,

共19兲 which has the same form as the Ruderman-Kittel-Kasuya-Yosida interaction between magnetic impurities in a Fermi liquid 关21兴. Accordingly the Fourier transform of the

effec-tive interaction V共兩r兩兲 in the BCS limit becomes

1 2 3 4 5

k

F

r

0.10
0.05 0.00 0.05 0.10

v
k

F

r

0.5 1.0 1.5 2.0

k

F

r

10
5 0 5 10

v
k

F

r

FIG. 1. 共Color online兲 Interaction energy of two heavy fermions v共kFr兲 as a function of their separation kFr. Three different curves

correspond to the BCS regime 共akF兲−1= −5共dashed curve兲, unitarity limit 共akF兲−1= 0 共solid curve兲, and BEC regime 共akF兲−1= 5 共dotted

curve兲. The right panel is a magnification of the left panel.

2 4 6 8

k

F

r

0.10
0.05 0.00 0.05 0.10

v
k

F

r

0.5 1.0 1.5 2.0 2.5 3.0

k

F

r

1.0
0.5 0.0 0.5 1.0

v
k

F

r

FIG. 2.共Color online兲 Evolution of the interaction energy v共kFr兲 as a function of the inverse s-wave scattering length 共akF兲−1. Left panel:

共akF兲−1is varied from −2 共curve with the smallest amplitude兲 to 0 共largest amplitude兲. The tiny oscillatory behavior existing in the BCS

regime grows toward the unitarity limit and makes a small hump at r⯝k_F−1. Right panel:共akF兲−1is varied from 0共curve with the smallest

height兲 to 2 共largest height兲. The hump further grows in the BEC regime and develops a repulsive core at r⯝a.

CASIMIR INTERACTION AMONG HEAVY FERMIONS IN… PHYSICAL REVIEW A 79, 013629共2009兲

V ˜ 共兩p兩兲 ⬅

冕

dre−ip·r_{V共兩r兩兲} → −兩a兩 2_k F 2m

冋

2 +

冉

2kF 兩p兩 − 兩p兩 2k_F

冊

ln

冏

1 +兩p兩/共2kF兲 1 −兩p兩/共2k_F兲

冏

册

, 共20兲 which correctly reproduces the effective interaction to the leading order in a obtained in Ref.关12兴1as it should be. Our nonperturbative result 共16兲 can go beyond the perturbative

BCS regime to the strongly interacting unitary regime in a controlled way by utilizing M/mⰇ1.

Analytic expressions of v共kFr兲 in various limits are ob-tained in Appendix B.

III. P-WAVE PROJECTION OF THE EFFECTIVE INTERACTION

A. Lessons from BCS theory

The physics of heavy fermions immersed in the Fermi sea of light fermions is described by the Hamiltonian

H =

冕

dx␺h†共x兲

冉

− ⵱2 2M −␮h

冊

␺h共x兲 +1 2

冕

dxdy␺h †_共x兲␺ h †_{共y兲V共兩x − y兩兲␺} h共y兲␺h共x兲, 共21兲

where␮h is the chemical potential of heavy fermions

mea-sured from␮single. Strictly speaking, the pairwise interaction

V共兩r兩兲 obtained in the previous section 关Eq. 共16兲兴 is valid only

if there are two heavy fermions in the system. This is be-cause the Casimir interaction energy is not pairwise additive and also because if heavy fermions have a finite density, it will perturb the Fermi sea of light fermions and thus affect the effective interaction V共兩r兩兲. However, in the dilute limit

冑

2M␮hⰆkF, we expect that the above Hamiltonian is a good approximation to the system of heavy fermions because the probability to find a third heavy fermion near the two heavy fermions becomes small in the dilute system关22兴. Indeed we

will confirm in Appendix A that the Casimir interaction among more than two heavy fermions at large separations can be evaluated quite accurately as a sum of pairwise inter-actions between each of the two heavy fermions. This result also supports the use of our Hamiltonian共21兲 to describe the

physics of dilute heavy fermions immersed in the Fermi sea of light fermions.

Another issue to be addressed is the instability of the sys-tem. When the mass ratio M/m exceeds the critical value 13.6 in pure 3D关23兴 or 6.35 in the 2D-3D mixture 关18兴, the

system will not be stable due to the Efimov effect. Therefore we need to assume the mass ratio to be large but smaller than the above critical value. For such a mass ratio, the effective interaction V共兩r兩兲 obtained in the Born-Oppenheimer

ap-proximation is no longer exact but can be considered as a good approximation to the exact one.

One of predictions we can derive from the Hamiltonian 共21兲 is a pairing between the heavy fermions. The pairing

gap of heavy fermions⌬pis defined to be

共2␲兲d_{␦共k兲⌬} p=

冕

dq 共2␲兲dV˜ 共兩p − q兩兲

冓

␺˜h

冉

k 2− q

冊

␺ ˜_h

冉

k 2+ q

冊

冔

, 共22兲 where␺˜h共p兲 is the Fourier transform of␺h共x兲 and V˜共兩p−q兩兲

is the Fourier transform of V共兩r兩兲 关see Eq. 共20兲兴. Because of

the Fermi statistics of heavy fermions, the pairing gap has to have an odd parity; ⌬_−p= −⌬p. The standard mean-field

cal-culation leads to the self-consistent gap equation

⌬p= −

冕

dq 共2␲兲dV共兩p − q兩兲 ⌬q 2Eq 关1 − 2nF共Eq兲兴. 共23兲 Here Ep=

冑

共 p2

2M−␮h兲2+兩⌬p兩2 is the quasiparticle energy and

nF共Ep兲=1/共eEp/T+ 1兲 is the Fermi-Dirac distribution function

at temperature T.

When the coupling between different partial waves can be neglected, we can solve the gap equation共23兲 easily by using

the weak-coupling approximation.2For a given orbital angu-lar momentumᐉ 共odd integer兲, the pairing gap and the criti-cal temperature are given by 关24兴

兩⌬p兩 ⬃ Tc⬃␮hexp

冉

1

NhV˜ᐉ

冊

. 共24兲

Here Nh is the density of states of heavy fermions at the

Fermi surface and V˜_ᐉ is the partial-wave projection of the effective interaction V˜ 共兩p−q兩兲 with 兩p兩=兩q兩=

冑

2M␮hfixed on

the Fermi surface of heavy fermions and assumed to be at-tractive V˜_ᐉ⬍0. Because the lowest partial wave for identical fermions isᐉ=1, the dominant pairing is expected to occur in the p-wave channel. With having in mind the application to the pairing of heavy fermions, we compute NhV˜ᐉ as a

function of the s-wave scattering length akFfrom Eq. 共16兲. Below we consider both cases where the heavy fermions are in three dimensions 共d=3兲 and where they are confined in two dimensions共d=2兲.

B. Heavy fermions in d = 3

When the heavy fermions are in three dimensions d = 3, the partial-wave projection of the effective interaction

V

˜ 共兩p−q兩兲 with 兩p兩=兩q兩⬅p fixed is given by

1

For unequal masses, the effective interaction in the BCS limit becomes V˜ 共兩p兩兲=−关2␲共m+M兲a_mM 兴2 mkF 2␲2L共2k兩p兩F兲, where L共z兲= 1 2+ 1−z2 4z ln兩 1+z 1−z兩

is the static Lindhard function. The limit of large mass ratio M/m Ⰷ1 coincides with Eq. 共20兲.

2

The weak-coupling approximation even at unitarity is justified in the dilute limit;

冑

2M␮_hⰆk_F. From Eqs.共B2兲 and 共B3兲, the interac-tion energy at the mean interparticle distance of heavy fermions r =共2M␮h兲−1ⰇkF −1 becomes V共r兲⬃kF2 2m 1 共kFr兲3 and is parametrically

small compared to the kinetic energy K共r兲⬃ 1

V ˜ ᐉ共p兲 = 1 2

冕

₋₁ 1 d cos␪P_ᐉ共cos␪兲V˜共兩p − q兩兲, 共25兲

where cos␪= pˆ · qˆ. Multiplying it by the density of states of heavy fermions Nh共p兲=

Mp

2␲2, we obtain the dimensionless

function representing the effective interaction between two heavy fermions for the given partial waveᐉ:

NhV˜ᐉ共p兲 = M 2␲m

冉

kF p

冊

2

冕

−1 1 d cos␪P_ᐉ共cos␪兲 ⫻

冕

0 ⬁ dzz sin

_冑

共z

冑

2 − 2 cos␪兲 2 − 2 cos␪ v

冉

kFz p

冊

. 共26兲

When the projected effective interaction with p =

冑

2M␮h

be-ing on the Fermi surface of heavy fermions is attractive

NhV˜_ᐉ⬍0, the pairing of the heavy fermions is expected to

occur with the pairing gap and the critical temperature given in Eq.共24兲.

Figure3 shows _MmNhV˜ᐉ共p兲 in the p-wave channel ᐉ=1 as

a function of 共akF兲−1 for four values of p/kF = 0.4, 0.8, 1.2, 1.6. We note that what is plotted is not the

p-wave effective interaction NhV˜_ᐉ共p兲⬀ M

mitself but m

MNhV˜_ᐉ共p兲

that is independent of the mass ratio M/mⰇ1. We can see that the p-wave effective interaction is attractive in the BCS limit共akF兲−1Ⰶ−1 being consistent with the perturbative pre-diction关12兴. However, it turns out that the perturbative result

has only a small range of validity. Our result shows that the

p-wave attraction is generally weaker than the extrapolation

of the perturbative calculation and, in particular, the p-wave effective interaction becomes a strong repulsion near the uni-tarity limit. This may be understood because of the repulsive hump ofv共k_Fr兲 developed near the unitarity limit 共see Figs.1

and2兲. We also find that the p-wave attraction is stronger for

the larger value of p/kF because the contribution from the attractive part ofv共kFr兲→−c2/共kFr兲2at short distance r→0 关see Eq. 共B1兲兴 becomes more significant for larger p/kF. The maximum attraction for each value of p/k_F

= 0.4, 0.8, 1.2, 1.6 is achieved around 共akF兲−1⬇−2.7,−2.5, −1.7, −1.0, respectively. At such a value of the s-wave scat-tering length, the maximal p-wave pairing of heavy minority fermions immersed in the Fermi sea of light fermions is pos-sible while the pairing gap and the critical temperature would be very small because of the numerically weak attraction

NhV˜ᐉ=1共p兲Ⰶ1. One can perform the same analysis for the

f-wave channel 共ᐉ=3兲 but will find even weaker attraction.

C. Heavy fermions in d = 2

When the heavy fermions are confined in two dimensions

d = 2, the partial-wave projection of the effective interaction V

˜ 共兩p−q兩兲 with 兩p兩=兩q兩⬅p fixed is given by V ˜ ᐉ共p兲 = 1 ␲

冕

0 ␲ d␪cos共ᐉ␪兲V˜共兩p − q兩兲, 共27兲

where cos␪= pˆ · qˆ. Multiplying it by the density of states of heavy fermions Nh共p兲=

M

2␲, we obtain the dimensionless

p
kF
0.4 10 8 6 4 2 1 akF 0.00002 0.00001 0.00001 0.00002m NhV  1
M
























































































p
kF
0.8 10 8 6 4 2 1 akF 0.0002 0.0001 0.0001 0.0002m NhV  1
M






























































































p
kF
1.2 10 8 6 4 2 1 akF 0.002 0.001 0.001 0.002 m NhV1
M




































































































p
kF
1.6 10 8 6 4 2 1 akF 0.008 0.006 0.004 0.002 0.002 0.004 0.006 m NhV1
M

FIG. 3. 共Color online兲 P-wave effective interaction between two heavy fermions in three dimensions as a function of 共akF兲−1.

m

MNhV˜ᐉ=1共p兲 in d=3 is plotted for p/kF= 0.4, 0.8, 1.2, and 1.6. The dashed curves are perturbative results at leading order valid in the BCS

limit共akF兲−1→−⬁.

CASIMIR INTERACTION AMONG HEAVY FERMIONS IN… PHYSICAL REVIEW A 79, 013629共2009兲

function representing the effective interaction between two heavy fermions for the given partial waveᐉ:

NhV˜ᐉ共p兲 = M 2␲m

冉

kF p

冊

2

冕

0 ␲ d␪cos共ᐉ␪兲 ⫻

冕

0 ⬁ dzzJ0共z

冑

2 − 2 cos␪兲v

冉

kFz p

冊

. 共28兲

In the case of d = 2, the scattering length a should be re-garded as the effective scattering length a_eff introduced in Ref. 关18兴.

Figure 4 shows _MmNhV˜ᐉ共p兲 in the p-wave channel ᐉ=1

as a function of 共akF兲−1 for four values of p/kF = 0.4, 0.8, 1.2, 1.6. One can see the similar behavior to the case of d = 3. Again the p-wave effective interaction is attrac-tive in the BCS limit共akF兲−1Ⰶ−1 being consistent with the perturbative prediction 关19兴. Compared to the case of d=3,

we find that the perturbative result has a wider range of va-lidity. In the case of d = 2, the p-wave attraction is found to be stronger than that in d = 3 for the same value of p/k_Fand the p-wave effective interaction becomes only a weak repul-sion near the unitarity limit. This will be understandable be-cause the contribution from the attractive part of v共kFr兲→ −c2_/共k

Fr兲2at short distance r→0 关see Eq. 共B1兲兴 is less sup-pressed by the phase space factor and thus more significant in d = 2. Our result also shows that the p-wave attraction is stronger for the larger value of p/kF and the maximum at-traction for each value of p/kF= 0.4, 0.8, 1.2, 1.6 is achieved

around 共ak_F兲−1_{⬇−1.2,−1.2,−0.85,−0.52, respectively. At} such a value of the s-wave scattering length, the maximal

p-wave pairing of heavy minority fermions in two

dimen-sions immersed in the three-dimensional Fermi sea of light fermions is possible and the pairing gap and the critical tem-perature will be larger than those in the case of d = 3.

IV. SUMMARY AND CONCLUDING REMARKS We have investigated a two-species Fermi gas with a large mass ratio interacting by an interspecies short-range interac-tion. Using the Born-Oppenheimer approximation, we deter-mined the interaction energy of two heavy fermions im-mersed in the Fermi sea of light fermions, which is an analog of the Casimir force, as a function of the s-wave scattering length. We showed that the p-wave projection of the effec-tive interaction is attraceffec-tive in the BCS limit being consistent with the perturbative prediction, while it turns out to be re-pulsive near the unitarity limit. We found that the p-wave attraction reaches its maximum between the BCS and unitar-ity limits, where the maximal but still weak p-wave pairing of heavy minority fermions is possible.

We also investigated the case where the heavy fermions are confined in two dimensions corresponding to the two-species Fermi gas in the 2D-3D mixed dimensions 关18,19兴.

Because the p-wave attraction between the heavy fermions in two dimensions was found to be stronger than that in three dimensions, we expect the larger p-wave pairing gap of heavy minority fermions. Such a p-wave pairing in two

di-
















































































p
kF
0.4 8 6 4 2 1 akF 0.0010 0.0005 0.0005 m NhV1
M
















































































p
kF
0.8 8 6 4 2 1 akF 0.004 0.003 0.002 0.001 0.001 0.002 m NhV1
M
















































































p
kF
1.2 8 6 4 2 1 akF 0.015 0.010 0.005 m NhV1
M
















































































p
kF
1.6 8 6 4 2 1 akF 0.030 0.025 0.020 0.015 0.010 0.005 m NhV1
M

FIG. 4. 共Color online兲 P-wave effective interaction between two heavy fermions confined in two dimensions as a function of 共akF兲−1.

m

MNhV˜ᐉ=1共p兲 in d=2 is plotted for p/kF= 0.4, 0.8, 1.2, and 1.6. The dashed curves are perturbative results at leading order valid in the BCS

mensions is especially interesting because the resulting sys-tem has a potential application to topological quantum com-putation using vortices with non-Abelian statistics关25,26兴.

Although our controlled analysis is restricted to the dilute heavy fermions in the limit of large mass ratio, our results indeed shed light on the phase diagram of asymmetric Fermi gases with unequal densities and masses共and spatial dimen-sions兲, in particular, in the strongly interacting unitary re-gime. Our results also have a direct relevance to the recently realized Fermi-Fermi mixture of40K and6Li because of their large mass ratio关8–10兴. It will be an important future

prob-lem to study corrections to our results when the restrictions of the large mass ratio and the dilute limit of heavy fermions are relaxed.

ACKNOWLEDGMENTS

The author thanks M. M. Forbes, R. Jaffe, S. Tan, and M. Zwierlein for discussions. This work was supported by MIT Department of Physics Pappalardo Program.

APPENDIX A: CASIMIR INTERACTION AMONG HEAVY FERMIONS FROM FUNCTIONAL

INTEGRAL METHOD

1. Derivation of Casimir interaction

Here we derive the Casimir interaction among heavy fer-mions immersed in the Fermi sea of light ferfer-mions using the functional integral method关27兴. The action that describes the

light fermions with the chemical potential␮linteracting with

the heavy fermions fixed at positions xiis

S =

冕

d␶dx␺l†共␶,x兲

冉

⳵␶− ⵱2 2m−␮l

冊

␺l共␶,x兲 − g0

兺

i

冕

d␶␺l †_共␶ ,xi兲␺l共␶,xi兲, 共A1兲

where ␶ is an imaginary time. The partition function Z = e−␤⍀is given by

Z =

冕

D␺lD␺l†e−S. 共A2兲

We first insert the following identity into the integrand:

1 =

兿

i

冕

D␩iD␩¯i␦关␺l共␶,xi兲 −␩i共␶兲兴␦关␺l †_共␶ ,xi兲 −␩¯i共␶兲兴 共A3兲 and then exponentiate the delta functions by introducing aux-iliary fields, 1 =

兿

i

冕

D␩iD␩¯iD␣iD␣¯i ⫻ ei兰d␶␣¯_i共␶兲关␺_l共␶,x_i兲−␩_i共␶兲兴+i兰d␶关␺_l†共␶,x_i兲−␩¯_i共␶兲兴␣_i共␶兲 . 共A4兲 Now the partition function can be written as

Z =

冕

D␺lD␺l†

兿

i

D␩iD␩¯iD␣iD␣¯ie−S⬘, 共A5兲

where the action becomes

S

⬘

=

冕

d␶dx␺l†共␶,x兲

冉

⳵␶− ⵱2 2m−␮l

冊

␺l共␶,x兲 − g0

兺

i

冕

d␶␩¯i共␶兲␩i共␶兲 − i

兺

i

冕

d␶␣¯i共␶兲关␺l共␶,xi兲 −␩i共␶兲兴 − i

兺

i

冕

d␶关␺l †_共␶,x i兲 −␩¯i共␶兲兴␣i共␶兲. 共A6兲

We can easily integrate out␺l and␺l

†

fields and␩and␩¯

fields to lead to the partition function

Z = Z0

冕

兿

i

D␣iD␣¯ie−S⬙, 共A7兲

where Z0= e−␤⍀free is a partition function of noninteracting light fermions and the action S

⬙

in the momentum space becomes S

⬙

=

兺

i,j

冕

d␻dp 共2␲兲4␣¯i共␻兲 eip·共xi−xj兲 − i␻+ p2/2m −␮l ␣j共␻兲 − 1 g0

兺

i

冕

d␻ 2␲␣¯i共␻兲␣i共␻兲. 共A8兲 Finally, by integrating out ␣ and ␣¯ fields, we obtain the

following expression for the partition function:

Z = Z₀exp

冋

␤

冕

d␻

2␲ln det Mij共i␻兲

册

, 共A9兲 where Mijis the scattering matrix given by

Mij= − ␦ij g0 +

冕

dp 共2␲兲3 eip·共xi−xj兲 − i␻+ p2/2m −␮l =

冦

m 2␲

冉

1 a−

冑

− 2mi␻− 2m␮l

冊

for i = j m 2␲兩xi− xj兩 e−冑−2mi␻−2m␮l兩xi−xj兩 _{for i}⫽ j.

冧

共A10兲 Here we introduced the s-wave scattering length a through

− 1 g0 +

冕

dp 共2␲兲3 2m p2 = m 2␲a. 共A11兲

Therefore the reduction in the grand potential⍀ compared to that in the noninteracting limit⍀_free is

⌬⍀共兵xi其兲 = −

冕

d␻

2␲ln det Mij共i␻兲. 共A12兲 By subtracting⌬⍀ in which all heavy fermions are infinitely separated, the interaction energy of the heavy fermions be-comes

CASIMIR INTERACTION AMONG HEAVY FERMIONS IN… PHYSICAL REVIEW A 79, 013629共2009兲

V共兵xi其兲 = −

冕

d␻ 2␲ln det Mij共i␻兲 det Mij⬁共i␻兲 , 共A13兲

where Mij⬁is a diagonal scattering matrix composed of Mii.

This is the generalization of V共兩r兩兲 in Eq. 共16兲 to a general

number of heavy fermions fixed at positions xi.

2. Casimir interaction between two heavy fermions It is worthwhile to reproduce the result obtained in Sec.II

in the case of two heavy fermions. By deforming the path of the integration over i␯⬅i␻+␮linto

冕

−i⬁+␮l i⬁+␮_l →

冕

−⬁−i0+ ␮l +

冕

␮l −⬁+i0+ , 共A14兲

we obtain the following expression for the grand potential reduction: ⌬⍀共兩r兩兲 = i

冕

d␯ 2␲ln

冉

1 a−

冑

− 2m␯+ e−冑−2m␯兩r兩 兩r兩

冊

+ i

冕

d␯ 2␲ln

冉

1 a−

冑

− 2m␯− e−冑−2m␯兩r兩 兩r兩

冊

共A15兲 with r = x₁− x₂. We separate the integral into the contribu-tions from bound states Re关␯兴⬍0 and continuum states Re关␯兴⬎0. For the bound state contribution, we pick up the singularity in the integrand 共A15兲 as

i

冕

Re关␯兴⬍0 d␯ 2␲ln

冉

1 a−

冑

− 2m␯⫾ e−冑−2m␯兩r兩 兩r兩

冊

=␯⫾+ const, 共A16兲 where␯_⫾⬍0 is the binding energy satisfying

1

a−

冑

− 2m␯⫾⫾

e−冑−2m␯⫾兩r兩

兩r兩 = 0. 共A17兲 For the continuum state contribution, defining the phase shift by ␲−␦_⫾共␯兲 = arg

冉

1 a−

冑

− 2m␯− i0 +_⫾e −冑−2m␯−i0+_兩r兩 兩r兩

冊

, 共A18兲 the integral in Eq.共A15兲 can be written as

i

冕

Re关␯兴⬎0 d␯ 2␲ln

冉

1 a−

冑

− 2m␯⫾ e−冑−2m␯兩r兩 兩r兩

冊

=␮l−

冕

0 ␮l d␯␦⫾共␯兲 ␲ . 共A19兲

Therefore, after dropping the unimportant constants, we find that the grand potential reduction in the case of two heavy fermions is given by ⌬⍀共兩r兩兲 =␯++␯−−

冕

0 ␮l d␯␦+共␯兲 +␦−共␯兲 ␲ . 共A20兲

This result is equivalent to ⌬E共兩r兩兲 in Eq. 共13兲 with ␯_⫾= −␬⫾ 2 2m,␯= k2 2m, and␮l= kF2

2m and thus provides the same interac-tion energy V共兩r兩兲 as Eq. 共16兲.

3. Casimir interaction among three and four heavy fermions We now evaluate the Casimir interaction energy 共A13兲

among three and four heavy fermions fixed with the same separations兩xi− xj兩⬅r. We measure the interaction energy in

units of the Fermi energy of light fermions VN共r兲

⬅ kF2

2mvN共kFr兲 with the subscript N indicating the number of heavy fermions. The dimensionless functions vN共kFr兲 with three typical values of共akF兲−1= −1 , 0 , 1 are plotted as func-tions of kFr in Fig.5for N = 3 and in Fig.6for N = 4, together with the sum of pairwise interaction energies N共N−1兲₂ v2共kFr兲 for comparison.

We can see that the Casimir interaction among more than two heavy fermions can be reproduced quite accurately by the sum of pairwise interactions between each of the two heavy fermions, in particular, at long distances, although the behaviors at short distances are slightly overestimated. A similar observation has been made in Ref.关17兴 in the system

of hardcore spheres immersed in a background Fermi sea. These results support the use of the Hamiltonian 共21兲 only

with the pairwise interaction to describe the physics of dilute heavy fermions immersed in the Fermi sea of light fermions.

APPENDIX B: INTERACTION ENERGY IN VARIOUS LIMITS

Here we evaluate the effective interaction between two heavy fermions v共k_Fr兲 in Eq. 共17兲 in various limits where

analytic expressions are available.

2 4 6 8kFr
0.10
0.08
0.06
0.04
0.02 0.00 0.02 0.04 v3
kFr 1 2 3 4 5 6kFr
0.6
0.4
0.2 0.0 0.2 v3
kFr 1 2 3 4kFr
2
1 0 1 2 v3
kFr

FIG. 5.共Color online兲 Interaction energy of three heavy fermions fixed with the same separations r. v₃共k_Fr兲 as a function of k_Fr is plotted

At short distance rⰆa, k_F−1, we obtain v共kFr兲 → − c2 共kFr兲2 − 2c 1 + c 1 ak_F2r− 1 + c + c2 共1 + c兲3 1 a2_k F 2+

2akF+关1 + 共akF兲2兴关␲+ 2 arctan共akF兲−1兴 ␲共akF兲2

− 1 + O共r兲, 共B1兲 where c = 0.567 143 is a solution to c = e−c_.

On the other hand, at long distance rⰇa, k_F−1, we obtain

v共kFr兲 →

2共akF兲3sin共2kFr兲 − 共akF兲2关共akF兲2− 1兴cos共2kFr兲 ␲关共akF兲2+ 1兴2共kFr兲3 −4共akF兲 3_关共ak F兲2− 1兴cos共2kFr兲 2␲关共akF兲2+ 1兴3共kFr兲4 −共akF兲 2_关共ak F兲4− 6共akF兲2+ 1兴sin共2kFr兲 2␲关共akF兲2+ 1兴3共kFr兲4 + O共r−5_{兲. 共B2兲}

In particular, in the unitarity limit akF→⬁, we find

v共kFr兲 → − cos共2kFr兲 ␲共kFr兲3 −sin共2kFr兲 2␲共kFr兲4 +2 cos共2kFr兲 + cos共4kFr兲 4␲共kFr兲5 + O关共kFr兲−6兴. 共B3兲 In the BCS or BEC limit兩a兩Ⰶr, k_F−1, we obtain

v共kFr兲 → 共akF兲2 2kFr cos共2kFr兲 − sin共2kFr兲 2␲共kFr兲4 +共akF兲3 2kFr cos共2kFr兲 + 关2共kFr兲2− 1兴sin共2kFr兲 ␲共kFr兲5 + O共a4_{兲. 共B4兲}

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published兲. 2 4 6 8kFr
0.20
0.15
0.10
0.05 0.00 0.05 0.10v4
kFr 1 2 3 4 5 6kFr
1.0
0.5 0.0 0.5 v4
kFr 1 2 3 4kFr
4
2 0 2 4 v4
kFr

FIG. 6.共Color online兲 Interaction energy of four heavy fermions fixed with the same separations r. v₄共k_Fr兲 as a function of k_Fr is plotted

for共akF兲−1= −1共left兲, 0 共middle兲, and 1 共right兲. The dashed curves represent the sum of pairwise interaction energies 6v2共kFr兲.

CASIMIR INTERACTION AMONG HEAVY FERMIONS IN… PHYSICAL REVIEW A 79, 013629共2009兲

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