BCS-BEC Crossover in Bilayers of Cold Fermionic Polar Molecules

0.2 MB·English

by  N. T. Zinner

BCS-BEC Crossover in Bilayers of Cold Fermionic Polar Molecules
N. T. Zinner,1,2,3 B. Wunsch,1 D. Pekker,1,4 and D.-W. Wang5,6,7
1Department of Physics, Harvard University, Cambridge MA, 02138
2Department of Physics and Astronomy, Aarhus University, Aarhus C, DK-8000
3The Niels Bohr Institute, University of Copenhagen, Copenhagen Ø, DK-8000
4Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
5Physics Department, National Tsing-Hua University, Hsinchu 300, Taiwan
6Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan
2
7Frontier Research Center on Fundamental and Applied Sciences of Matters,
1
National Tsing-Hua University, Hsinchu 300, Taiwan
0
2 (Dated: January 13, 2012)

n Weinvestigatethequantumandthermalphasediagram offermionic polarmolecules loaded in a
a bilayertrappingpotentialwith perpendiculardipolemoment. WeusebothaBCStheoryapproach
J thatismostrealiableatweak-couplingandastrong-couplingapproachthatconsidersthetwo-body

1 bounddimerstates with onemolecules ineach layerastherelevant degreeoffreedom. Thesystem
1 ground state is a Bose-Einstein condensate (BEC) of dimer bound states in the low density limit
and a paired superfluid (BCS) state in the high density limit. At zero temperature, the intralayer

] repulsionisfoundtobroaden theregime ofBCS-BECcrossover, andcanpotentially inducesystem
s
collapsethroughthesofteningofrotonexcitations. TheBCStheoryandthestrongly-coupleddimer
a
g picture yield similar predictions for the parameters of the crossover regime. The BKT transition
- temperature of the dimer superfluid is also calculated. The crossover can be driven by many-body
t effects and is strongly affected by theintralayer interaction which was ignored in previous studies.
n
a
u PACSnumbers: 03.75.Ss,67.85.-d,74.78.-w
q
.
at I. INTRODUCTION interaction. The quantum phase diagram is shown in
m Fig. 1(c). BCS (BEC) ground states are found in
the limit of weak (strong) interaction and large (small)
- Recent progress on trapping and cooling of polar
d density. In between, we have the crossover regime
n molecules [1–6] enable studies of many-body systems (Crossover-I), which can be determined by the chemi-
o with long-range anisotropic dipole-dipole forces, where
cal potential calculated from many-body theories in the
c new exotic phases could exist [7, 8]. The attractive part
different limits (see below). When neglecting intralayer
[
of the interaction can, however, lead to chemical reac-
repulsion, the crossover region (Crossover-II) moves to

4 tion losses [6]. A way to stabilize the system is to load lower interaction strength. In addition, intralayer in-
v molecules in a one- or two-dimensional optical lattices,
teractions could perhaps give rise to a roton instability
0 where interesting low-dimensional physics has been pre-
at large density and strong interaction, although further
3 dicted [9–17]. A prominent example is polar fermions
analysis beyond the scope of this work is needed. We
0
loaded in a bilayer system with dipoles oriented perpen-
2 also determine the BKT critical temperature [23, 24] in
diculartothelayerplane[18]. Inthelimitofhighdensity
. the strongly-coupledregimeincluding the effective inter-
9 orweakinteraction,thesystemisverysimilartoconven-
actionofthedimers. Themaximumcriticaltemperature
0
tional superconductors, and the ground state should be
0 obtained is one tenth of the Fermi energy, and should
a BCS state (with interactions as in Fig. 1(a)). In the
1 thereforebeachievableinexperimentalsetupsinthenear
dilute limit, it is known that the interlayer interaction
: future.
v always supports a bound state [19–21], and the ground
We note that two otherrecentstudies haveconsidered
i
X stateshouldbeaBECofdimers(Fig.1(b)). Asaresult, a system similar to the one studied here. The paper by
r the BCS-BEC crossover in this bilayer system could be Pikovski et al. [25] considers the BCS and BEC phases
a richerthantheusualatomicFermigaswherecrossoveris
based on BCS theory at both zero and finite tempera-
driven by the two-body physics of a Feshbach resonance
ture. However,they do not consider the full effect of the
[22]. Hereitisdrivenbymany-bodyeffectswhichdepend
intralayer interaction. When we neglect the intralayer
notonlyoninteractionstrengthbutalsoondensity. Fur-
interaction, our results are consistent with Ref. [25]. A
thermore, the intralayer repulsion can cause roton soft- related study by Baranov et al. [26] addresses the crit-
ening and/or Wigner crystallization in the high density
ical temperature for the superfluid phase in the weak-
limit [15–17]. Therefore, a many-body theory including
coupling limit, taking also particle-hole correlations into
bothintra-andinterlayerinteractionsisnotonlyofquan-
account. In contrast, here we consider the finite tem-
titatively interest but also qualitatively important.
perature phase diagram from the strong-coupling limit.
In this paper, we study the quantum and thermal Extrapolation of our results to the parameter regime of
phase diagrams of fermionic polar molecules loaded in Ref. [26] wouldexceed the boundaries ofour approxima-
a bilayer system including both intra- and inter-layer tions andthe currentstudy should be viewedas comple-
2
is the area of the layer plane; ǫ = ~2k2/2m, and
k
σ = is the layer index. V = V = V de-
+ + 1
1 3 notes ±the interlayer interaction an−d V −= V = V
a V1 V0 b 2 Veff 4 is the intralayer one. Here we neglec+t+interla−y−er tun0-
neling. We assume occupation of only the ground state
in the transverse direction so that the transverse de-
25
c µ =0 µ =0 gree of freedom is a simple gaussian that can be inte-
BEC BCS
Unstable grated out. This yields V (q) = 8πD2 1 3F(q )
0 3√2πW − 2 | |
20 and V (q) 2πD2 q e qd as W/d (cid:0) 0. Here q(cid:1)
1 −| |
→ − | | →
BEC is the in-plane momentum, D is the dipole moment,
15 Crossover−I (with V) W and d the layer width and interlayer spacing, and
0 F(q) = π/2Wq[1 Erf(Wq/√2)]eq2W2/2 with Erf(x)
U −
the errorpfunction. The interlayer interaction is exact in
10 µ0 =0 the strict 2D limit (W d), but we find it is accurate
BCS ≪
Crossover−II (no V) enough (< 10%) at W = 0.2d, which is the value used
0
throughout. In the weak interactionlimit, the intralayer
5 µ0 =0 repulsion should renormalize the single particle disper-
BEC
BCS sion as in Fermi liquid theory. Treating the layer index
as a spin degree of freedom, the gap equation is
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
kFd 1 V1(k q)∆q Eq
∆ = − tanh , (1)
k
−ΩXq 2Eq (cid:18)2kBT(cid:19)
FIG. 1: (color online) Schematic of a few particles in the
bilayerwith(a)inter-(V1)andintralayer(V0)interactionsin
theBCS limitand(b)theeffectiveinteraction(Veff)between where Eq = ξq2 +|∆q|2 is the quasi-particle disper-
the dimers in the BEC limit (see text). (c) Quantum phase sion,andξq qǫq+Σ(q) µBCS. The self-energy,Σ(q),
diagram for a bilayer with fermionic polar molecules in the ≡ −
within in the Hartree-Fockapproximationis [27] Σ(k)=
interaction-density plane. Interactions are characterized by ξq
U = mD2/~2d and by kFd, where m, D, d, and kF are re- 21Ω q(V0(0)−V0(k−q))[1−Eqtanh(Eq/2kBT)]. Toac-
spectivelythemolecularmass, dipolemoment,interlayerdis- cessPthe crossover regime, the chemical potential, µ ,
BCS
tance,andFermimomentum. TheCrossover-Iregionincludes
must be determined self-consistently via the density
theeffectofintralayerinteractions,whiletheCrossover-IIre-
equation
gion does not. In the upper right-hand corner we speculate
that the system could potentially display a roton instability 1 ξq Eq
as discussed in thetext. n= 1 tanh . (2)
2ΩXq (cid:20) − Eq (cid:18)2kBT(cid:19)(cid:21)
We use the first Born approximation(FBA) with a real-
mentary to Ref. [26].
isticfinitelayerwidthforbothintra-andinterlayerinter-
Our model and the assumptions we use are described
actions in Eq. (1). The exponentially decreasing shape
SectionII.InSectionIII,wediscussthecaseofzerotem- of V (q) means that using the renormalized gap equa-
1
perature and the role of intralayer interactions on the
tions or the FBA yields similar results as noted already
BCS-BEC crossover. This is achievedby consideringthe
in Ref. [25]. The FBA is generally poor at strong cou-
physics from both a weak- and a strong-coupling point
pling. However, since the crossover takes place at low
of view. Both approaches are shown to yield consistent
density, we treat the intralayer in a weak coupling sense
results. We proceed to discuss the finite temperature
in the current work. Inclusion of higher-order terms will
phase diagram in Section IV. Here we calculate the crit-
be considered in future work.
ical temperature for the superfluid phase within mean-
Since the interlayer interaction, V , is attrative at
1
fieldtheoryandusingthe universalrelationforthe BKT
short distance, the dominant gap is s-wave. Therefore,
transition temperature. Section V contains a summary,
for simplicity, we neglect higher partial wave compo-
a discussion of experimental parameters to realize the
nents and calculate ∆ = ∆ from Eq. (1) by iter-
k k
predicted phases, and an outlook for future work. | |
ation. In Fig. 2(a), we show ∆ /∆ for different val-
k 0
ues of the dimensionless coupling, U = mD2/~2d with
k d = √4πnd2 = 0.4. When U is small, the maximum
F
II. MODEL of ∆(k) is at k/k = 2.5, which is where V (k) is most
F 1
attractive. When U is larger, the intralayer interaction
The Hamiltonian for polar molecules in the bi- renormalizes the single particle dispersion; the structure
layer system is given by H = kσǫkc†k,σck,σ + Fofig∆.2k(bd)iswaeppsheaorws∆an(dTt)h/Ee maasxifmunucmtioinsoaftTk/T=,0w.heIrne
21Ω qkk′,σσ′Vσσ′(q)c†k+q,σc†k′ q,σ′cPk′,σ′ck,σ, where Ω EF and TF are Ferm0i energFy and temperature.F
P −
3
1 U=1 kFd=0.4 1102 kFd=0.4 wprheesrseurEeBinisthoebtcahinemedicnaulmpoertiecnatlilayl;[1µ90B–E21C].=WEeFd−efiEneBt/h2e,
00.8 EF 8 crossoverregime(Crossover-II)astheregionboundedby
/∆k0.6 T)/ 6 U=5 µ0BCS = 0 and µ0BEC = 0, the latter given by the line
∆0.4 U=5 ∆(0 4 with triangles in Fig. 1(c). The Crossover-II region is
quite narrow which indicates that the BCS theory and
0.2 a 2 b
the strong-coupling result are very similar. The expres-
0 0
0 2.5 5 k/7.k5F 10 12.5 15 0 1 2 T3/TF4 5 6 7 sion for µ0BCS given above holds exactly for zero-range
0.4 4 interactions [29], and indicates that the crossover hap-
0.35 pens when the dimer size becomes comparable to the
0.3 3.5
interparticle distance. While the interlayer dipole inter-
/E00.02.52 U=2 /lB 3 action behaves similar to a zero-range interaction, the
Tc0.15 λr intralayer dimer-dimer interaction can have significant
0.1 2.5 effects on µ as we now demonstrate by deriving an
0.05 c d BEC
effective interaction between dimers. We note that a re-
0 2
0 0.5 1 1.5 0.8 1 1.2 1.4 1.6 1.8 2
kFd kFd centMonteCarlostudyoftheBCS-BECcrossoverintwo
dimensions also finds that in the BEC limit, the dimer-
FIG.2: (coloronline)(a)Normalizedgap,∆k/∆0,asafunc- dimerandatom-dimerinteractionsareimportantcorrec-
tion of k/kF at kFd = 0.4 and U = 1 (solid) and U = 5 tions that are not taken into account in the usual BCS
(dashed). (b) Temperature dependence of ∆0(T)/EF as a theory without self-energy corrections [30].
function of T/TF for kFd = 0.4 (EF = 0.16E0). (c) BCS The coordinates of the four molecules are denoted by
Ecr0it=ica~l2t/e2mmpde2r.at(udr)eRaastaiofuonfctthieonrootfoknFwdaavtelUen=gth2,iλnru,naittsthoef r1,··· ,r4 as shown in Fig. 1(b), where (r1,r2) are for
the left dimer and (r ,r ) for the right dimer. We are
instability to thedimer size, lB, as function of kFd. 3 4
interested in deep bound state, where the dimer size is
smaller than the inter-dimer distance, i.e. ρ r , r ,
′
| |≫| | | |
where ρ = (r +r )/2 (r +r )/2 is inter-dimer dis-
III. ZERO TEMPERATURE RESULTS: 1 2 − 3 4
tance,andr =r r andr =r r arerelativecoor-
INTRALAYER INTERACTION EFFECTS ON 1 2 ′ 3 4
− −
THE BCS-BEC CROSSOVER dinates in each dimer. Straighforward algebra and inte-
gration over the dimer bound state wavefunction, φ(r),
yields the effective dimer-dimer interaction V (ρ) =
When the density is reduced, the ground state is ex- eff
drdr φ(r)2 φ(r )2 V (ρ+sr ), where
pected to be a dimer BEC. Within BCS theory, this ′| | | ′ | s= 1,α=0,1 α α
regime can be defined by having a negative chemical po- Rr0,1 (r r′)/2. InPthe±strong interaction and di-
≡ ∓
lute density limit, we can approximate φ(r) by a Gaus-
tential (µ < 0). From Eq. (2), we can determine
BCS sian profile; φ(r) = (l √π) 1exp r 2/2l2 , where
the µBCS = 0 boundary as shown in Fig. 1(c). The two B − −| | B
solid lines that bound the two crossover regions are cal- lB = 2~2/mEB is the radius of dim(cid:0)er bound s(cid:1)tate. As
culated by including (µBCS) and neglecting (µ0BCS) the a resuplt, Veff has the following simple Fourier transform:
intralayerrepulsionrespectively. As expected, the repul-
sion strongly suppresses the µBCS < 0 region and for Veff(k)=[2V1(k)+2V0(k)]exp(−|k|2lB2/8). (3)
densitieslargerthanacriticalvalueofk d 0.77,µ
F BCS
∼
is positive for any U. We always find a µ0 <0 region The Gaussian approximation for φ(r) fails for U . 2,
BCS
when the intralayer interaction is neglected. This shows however we have checked that our results are qualita-
that the intralayer interaction brings not only quanti- tively unchanged if the exact solution is used [31]. No-
tative contributions, but also qualitative and important tice that V takes the strong interlayer interaction into
eff
changes of the quantum phase diagram. The effect of account through the dimer wavefunction. The FBA is
intralayer repulsion on the crossover physics can also used for the intralayer term, which is reasonable since
be seen in the transition temperature, T , as shown in the crossover happens at low density. For the roton in-
c
Fig. 2(c). It has a maximum at k d 1, similar to stability,theFBAwasusedinquasi-2Dstudiesofdipolar
F
∼
crossover in a quasi-2D superconductor [28]. In contrast bosons[32, 33] andwe expect it to remaina fair approx-
to BCS results for Fermi gases with short-range interac- imation at larger densities as well. We note that dipolar
tion, the decrease of T at high density is caused by the interactions are different from short- or zero-range in-
c
long-range intralayer repulsion. teractions. Zero-range interactions in 2D always allow
In order to investigate the many-body physics of the a two-body dimer bound state [29], and a state of four
BEC limit (i.e. the strong interaction or low density bosons will also be bound [34]. This result is different
limit) in more detail, we go beyond BCS theory, which from 3D where the interaction must be sufficiently at-
is based on Fermi liquid theory at high density. Starting tractive to produce bound states. In the current setup,
from the extremely dilute limit, the bound dimer is the the dimer-dimer system is unbound [35].
mainconstituentwithbindingenergyE . Forsmallden- The effective interaction has the property that V (0)
B eff
sity,weneglecttheintralayertermbutincludetheFermi is non-zero due to the intralayer term. However, the
4
molecules are fermionic and the Fock exchange contri- indicates that the roton instability lies inside the region
bution could cancel this term as is the case with true where a density wave in single layers is predicted. How-
short-range interactions in interacting single-component ever, Refs. [42] and [43] neglect the Fock contributions
Fermi gases. However, V (q) also has large contribu- whichareexpectedto be large[36–40]. The exactregion
eff
tions from qd > 0, and this should have an influence on ofthedensitywaveinstabilityisthereforenotyetknown.
thephasediagram. Thisissupportedbyrecentstudiesof In any case, we expect the system to be unstable in the
the density-waveinstability where the effect of exchange upper right part of the zero temperature phase diagram
is found to be very large,shifting the instability into the of Fig. 1.
strong-coupling regime [36–40]. The importance of ex-
change effects has also been discussed in relation to fer-
roeletricity with polar molecules [41]. Here we estimate IV. FINITE TEMPERATURE RESULTS:
the effects in the bosonic dimer limit by including the INTRALAYER INTERACTION EFFECTS ON
intralayer potential through V (0). THE CRITICAL TEMPERATURE
eff
UsingV ,thechemicalpotentialintheBEClimitcan
eff
be estimated as µBEC = nVeff(0)/2 EB/2 (we neglect We now investigate the finite temperature phase di-
−
the Fermi pressure which is much smaller than the in- agram. In the BCS limit, we can use Eq. (1) to ob-
teraction energy for strong interactions). In Fig. 1(c), tain the transition temperature, T , as shown by the
c
the solid line with crosses is given by µBEC = 0 and is dashed line in Fig. 3. We note that the intralayer in-
the upper bound of the crossover regime including in- teraction has very little influence on T at k d=0.4. It
c F
tralayer interaction (Crossover-I). Notice in particular is known that in the weak interaction limit, T is very
c
that the BEC region shrinks to lower density when in- close to the true transition temperature in a full BKT
cluding the intralayer interaction, and there is no dimer theory [44]. However, this calculation fails in the strong
condensateforkFd&0.42. Theratioofinterparticledis- interaction or dilute limit. At strong coupling, we ob-
tancetodimersizeisfiveormorealongtheµBCS =0and tain the BKT transition temperature from the univeral
µBEC = 0 lines. Without intralayer interaction the ra- relation kBTBKT = ~2πns(TBKT)/2m, where ns(T) is
tio is around2.5,againdemonstratinghow the crossover the superfluid density at temperature T [23, 24]. Ac-
physics is strongly modified by the intralayer term. We cording to Landau’s two-fluid model, we have n (T) =
s
haveestimatethechemicalpotentialfrombothBCSand n n (T), with normal fluid density given by n (T) =
n n
BEC limits, giving bounds of the crossover regime with −~2 d2q q 2 [45]. In Fig. 3 we
and without the intralayer interaction. Our results in- 16mkBT (2π)2 hSinh(ωd(q)/2kBT)i
dicate that in a realistic experiment, the intralayer re- show TBRKT of a dimer superfluid as a function of U for
pulsion can significantly affect the regime where a dimer kFd = 0.4 (solid black line). The dimer result differs
condensate is observable. from the BCS theory prediction in the weak coupling
limit, while it becomes saturated at T = 0.125T
From the effective interaction between dimers, V , BKT F
eff
we can also calculate the dispersion, ~ω (k) = for large U. In Ref. [25], Tc and TBKT was calculated
d
from the BCS superfluid state. For small U we find the
ǫ /2(ǫ /2+2nV (k)), of the collective Bogoliubov
k k eff
same result for T . For dipolar bosons, Ref. [46] finds
mpodeofadimersuperfluidinanalogytothecaseofdipo- c
T 0.1T for all values of U shown in Fig. 3. In
lar bosons [33]. With increasing U we find roton soften- BKT F
∼
order to compare to that study we need to assume point
ing around kd= 2π, which corresponds to a wavelength
dimers with twice the dipole moment, which is of course
of λ = 2π/k d. This softening leads to system col-
r
∼ not the case for small U where T is reduced as the
lapse in the high density and strong interaction regime, BKT
dimer size grows. Our strong-coupling dimer approach
as shown in Fig. 1(c). To investigate the nature of the
to the finite temperature physicscanbe consideredcom-
instability, we plot λ /l in Fig. 2(d) and observe that
r B
plementary to both Ref. [25] and Ref. [46]. The good
λ ismorethanafactoroftwolargerthanl forallden-
r B
agreementatintermediateandlargeU withtheotherap-
sities. Thisimpliesthat,atleastinthelow-densitylimit,
proaches provides support that we capture the essential
it is a many-body effect.
physics using the effective interaction between dimers.
The roton analysis assumes a well-defined dimer pic-
ture. We find the instability in the upper right-hand
corner of Fig. 1c, i.e. at higher densities and large U.
V. DISCUSSION
While the dimer picture should prevail for large U, the
higher densities imply that some fermionic nature could
perhaps arise. Similar effects should arise in a fermionic We havestudiedfermionicpolarmoleculesinabilayer
picture but it is not easily calculated since fluctuations with perpendicularly polarized dipole moments. As the
beyondBCStheoryareneeded. Soatthispointthefind- densityanddipolestrengthvaries,ananalogue(butwith
ingsfortherotoninstabilityremainsspeculative. Thisis different physics) of the celebrated BCS-BEC crossover
an interesting topic for future work. We note that den- is predicted. We find that the intralayer repulsion (and
sitywavesinasinglelayerwithfermionicpolarmolecules dimer-dimer interaction) is crucial quantitatively and
have been predicted recently [42, 43]. A rough estimate also to some extend qualitatively: It shrinks the region
5
0.18 workisneededto determinecrystalphasesinthe bilayer
Tc setup.
0.16 kFd=0.4
In order to detect the phases a number of techniques
0.14 could be applied. Dimerization in the layers should
TBKT be detectable by Bragg scattering [47] or in situ non-
0.12
demolition detection [48, 49], whereas RF spectroscopy
EF 0.1 can probe the gap. To probe the finite temperature
/ physics one can detect the associated vortices by matter
T
0.08 waveheteredyning [50]. Toestimate parametersfor rele-
vant systems, we take d = 0.5µm which yields U = 1.22
0.06
forKRbmolecules atD =0.566Debye andU =4.15for
0.04 LiCs at D = 1.0 Debye (the permanent dipole moment
BCS Crossover
of this molecule in the ground-state is about D = 5.4
0.02
Debye [51]). For densities n = 105 108 cm 2 we
−
0 1 2 3 4 5 6 7 8 9 10 −
U have k d = 0.06 1.8. The critical density to reach
F
µ < 0 is abou−t n 0.2 108 cm 2, whereas the
BCS −
FIG. 3: (color online) Transition temperature (TBKT, solid dimer BEC requires n .∼0.6 1·07 cm−2. Unfortunately,
blackline)asafunctionofinteractionstrengthforkFd=0.4. as TF = E0(kFd)2/kB this ·means that extremely low
BCS result (Tc, dashed blue line) is shown for comparison. temperatures (.1nK) are required to reach T .
BKT
Thedashedverticallinemarkstheposition ofµBCS =0(see
Interesting directions for future work includes more
Fig. 1(c)).
than two layers or tilting of the dipoles with respect to
the plane. With dipoles that are no longer perpendicu-
lar one can still show that a two-body bound dimer will
ofdimer condensationsothatno dimer stateis expected
be present for any value of the dipole moment [52], al-
whenthedensityislargerthanacriticalvalueanditalso
though the dimer binding energy is reduced [53]. In the
causes a roton instability in the high density and strong
single layer case, p-wave superfluidity can occur for an
interaction regime. Our work is thus important for the
extended range of tilting angles [11] and we expect simi-
study of BCS-BEC crossoverphysics in 2D systems, and
lar effects for a bilayer. This setup can be explored with
our results should be observable within the parameter
the same methods used here. In a setting with multiple
regime of near future experiments.
layers more phases should be expected such as coherent
It is worth stressing that our study considers the
density waves [36], pairing [47], and bound states with
physicsofthebilayerwithfermionicpolarmoleculesfrom
more than two molecules [35, 54].
different points of view. The BCS theory is usally more
reliableintheweak-couplingregime,butcanbeextended
into the crossoverregime by solvingthe gapandnumber
equations self-consistently. As the bilayer setup will al-
Acknowledgments
ways have a two-body bound state with one molecule in
each layer, it is reasonable to consider these dimers as
the relevant degree of freedom in the strongly-coupled We thank E. Demler, L. Pollet, A. S. Jensen, G. M.
regime. The results presented above indicate that both Bruun, M. M. Parish for fruitful discussions and C.-H.
approaches yield consistent results, with or without in- Lin and I.-W. Tsai for early pioneering work. DWW
cluding the repulsive intralayer interaction. appreciate the hospitality of the JQI during the initial
The crystalline phases [15–17] are also ground state investigation of this project, and acknowledges research
candidates due to the intralayer repulsion. For W/d = andtravelsupportfromNCTS(Taiwan). Supportbythe
0.2 used in our calculations, these phases could appear CarlsbergFoundation andby the German Science Foun-
belowthe rotoninstabilityregioninFig1. Thefiniteex- dation under grant WU 609/1-1 is gratefully acknowl-
tendofthedimersis,however,stillaconcernandfurther edged.
[1] S.Ospelkaus et al.,Nature Phys.4, 622 (2008). [8] T.Lahaye,C.Menotti,L.Santos,M.Lewenstein,andT.
[2] K.K. Niet al.,Science 322, 231 (2008). Pfau, Rep.Prog. Phys. 72, 126401 (2009).
[3] J.Deiglmayretal.,Phys.Rev.Lett.101,133004(2008). [9] D.-W. Wang, M. D. Lukin, and E. Demler, Phys. Rev.
[4] F. Lang, K. Winkler, C. Strauss, R. Grimm, and J. H. Lett. 97, 180413 (2006).
HeckerDenschlag, Phys. Rev.Lett. 101, 133005 (2008). [10] D.-W. Wang, Phys.Rev.Lett. 98, 060403 (2007).
[5] K.K. Niet al.,Nature464, 1324 (2010). [11] G.M.BruunandE.Taylor,Phys.Rev.Lett.101,245301
[6] S.Ospelkaus et al.,Science 327, 853 (2010). (2008).
[7] M. A.Baranov, Phys. Rep.464, 71 (2008). [12] N. R. Cooper and G. V. Shlyapnikov, Phys. Rev. Lett.
6
103, 155302 (2009). (2006).
[13] R. M. Lutchyn, E. Rossi, S. Das Sarma, Phys. Rev. A [35] A.G. Volosniev, D.V.Fedorov,A.S. Jensen,and N.T.
82, 061604(R) (2010). Zinner, arXiv:1109.4602v1.
[14] M.Klawunn,J.Duhme,andL.Santos,Phys.Rev.A81, [36] N. T. Zinnerand G. M. Bruun,Eur. Phys.J. D 65, 133
013604 (2010). (2011).
[15] C.Mora,O.Parcollet,andX.Waintal,Phys.Rev.B76, [37] M. Babadi and E. Demler, Phys. Rev. B 84, 235124
064511 (2007). (2011).
[16] H.-P.Bu¨chleret al.,Phys.Rev.Lett.98,060404 (2007). [38] L. M. Sieberer and M. A. Baranov, arXiv:1110.3679v1
[17] G. E. Astrakharchik, J. Boronat, I. L. Kurbakov, and [39] M. M. Parish and F. M. Marchetti, arXiv:1109.2464v1
Yu.E. Lozovik, Phys.Rev.Lett. 98, 060405 (2007). [40] J.K.Block,N.T.Zinner,andG.M.Bruun,inprepara-
[18] M. G.H. deMiranda et al.,NaturePhys.7, 502 (2011). tion.
[19] S.-M. Shih and D.-W. Wang, Phys. Rev. A 79, 065603 [41] C.-H. Lin, Y.-T. Hsu, H. Lee, and D.-W. Wang, Phys.
(2009). Rev. A 81, 031601(R) (2010).
[20] J.R.Armstrong,N.T.Zinner,D.V.Fedorov,andA.S. [42] K. Sun, C. Wu, and S. Das Sarma, Phys. Rev. B 82,
Jensen, EPL 91, 16001 (2010). 075105 (2010).
[21] M. Klawunn, A. Pikovski, and L. Santos, Phys. Rev. A [43] Y. Yamaguchi, T. Sogo, T. Ito, and T. Miyakawa, Phys.
82, 044701 (2010). Rev. A 82, 013643 (2010).
[22] W. Ketterle and M. W. Zwierlein, in Ultracold Fermi [44] K. Miyake, Prog. Theor. Phys.69, 1794 (1983).
Gases,ProceedingsoftheInternationalSchoolofPhysics [45] U. C. Tauber and D. R. Nelson, Phys. Rep. 289, 157
Enrico Fermi, Course CLXIV, Varenna, June 2006, (1997).
editedbyM.Inguscio,W.Ketterle,andC.Salomon(IOS [46] A. Filinov, N. V. Prokof’ev, and M. Bonitz, Phys. Rev.
Press, Amsterdam) 2008. Lett. 105, 070401 (2010).
[23] V.L. Berezinskii, Sov. Phys.JETP 34, 610 (1972). [47] A. C. Potter, E. Berg, D.-W. Wang, B. I. Halperin, and
[24] J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 E. Demler, Phys.Rev. Lett.105, 220406 (2010).
(1973). [48] B.Wunsch,N.T.Zinner,I.B.Mekhov,S.-J.Huang,D.-
[25] A. Pikovski, M. Klawunn, G. V. Shlyapnikov, and L. W. Wang, and E. Demler, Phys.Rev.Lett. 107, 073201
Santos, Phys.Rev.Lett. 105, 215302 (2010). (2011).
[26] M.A.Baranov,A.Micheli,S.Ronen,andP.Zoller,Phys. [49] N. T. Zinner, B. Wunsch, I. B. Mekhov, S.-J. Huang,
Rev.A 83, 043602 (2011). D.-W, Wang, and E. Demler, Phys. Rev. A 84, 063606
[27] C. Zhao et al., Phys.Rev.A 81, 063642 (2010). (2011).
[28] Z. Liand K.Yamada, JPSJ 70, 797 (2001). [50] P. Kru¨ger et al.,Nature 441, 1118 (2006).
[29] M. Randeria, J.-M. Duan, and L.-Y. Shieh, Phys. Rev. [51] J. Deiglmayr, A. Grochola, M. Repp, O. Dulieu, R.
B 41, 327 (1990). Wester, and M. Weidemu¨ller, Phys. Rev. A 82, 032503
[30] G. Bertaina and S. Giorgini, Phys. Rev. Lett. 106, (2010).
110403 (2011). [52] A.G. Volosniev, D.V.Fedorov,A.S. Jensen,and N.T.
[31] N. T. Zinner, J. R. Armstrong, A. G. Volosniev, D. V. Zinner, Phys. Rev.Lett. 106, 250401 (2011).
Fedorov,and A.S. Jensen, arXiv:1105.6264v1. [53] A. G. Volosniev, N. T. Zinner, D. V. Fedorov, A. S.
[32] L Santos, G. V. Shlyapnikov, and M. Lewenstein, Phys. Jensen, and B. Wunsch,J. Phys.B 44 (2011) 125301.
Rev.Lett. 90, 250403 (2003). [54] J.R.Armstrong,N.T.Zinner,D.V.Fedorov,andA.S.
[33] U.R. Fischer, Phys. Rev.A 73, 031602(R) (2006). Jensen, arXiv:1106.2102v1.
[34] I. V. Brodsky, M. Yu. Kagan, A. V. Klaptsov, R.
Combescot, and X. Leyronas, Phys. Rev. A 73, 032724