effect of disorder in bcs–bec crossover
TRANSCRIPT
Effect of Disorder in BCS-BEC Crossover
Ayan Khan,1, ∗ Saurabh Basu,2 and Sang Wook Kim3
1Research Center for Dielectric and Advanced Matter Physics,Pusan National University, Busan, 609-735, S. Korea
2Department of Physics, Indian Institute of Technology Guwahati, Guwahati, Assam, 781039, India3Department of Physics Education, Pusan National University, Busan, 609-735, S. Korea
(Dated: February 24, 2012)
In this article we have investigated the effect of weak random disorder in the BCS-BEC crossoverregion. The disorder is included in the mean field formalism through NSR theory of superconductingfluctuations. A self consistent numerical solution of the coupled equations involving the superfluidgap parameter and density as a function of the disorder strength, albeit unaffected in the BCS phase,yields a depleted order parameter in the BEC regime and an interesting nonmonotonic behavior ofthe condensate fraction in the vicinity of the unitary region, and a gradual depletion thereafter, asthe pairing interaction is continuously tuned across the BCS-BEC crossover. The unitary regimethus demonstrates a robust paradigm of superfluidity even when the disorder is introduced. Tosupport the above feature and shed light on a lingering controversial issue, we have computed thebehavior of the sound mode across the crossover that distinctly reveals a suppression of the soundvelocity. We also find the Landau critical velocity that shows similar nonmonotonicity as that ofthe condensate fraction data, thereby supporting a stable superfluid scenario in the unitary limit.
PACS numbers: 05.30.Jp, 74.20.Fg, 74.40.+k, 03.75.Lm
I. INTRODUCTION
Atomic gases at very low temperature are a uniquesystem where one can observe the continuous evolutionof a fermionic system (BCS type) to a bosonic system(BEC type) by changing the inter-atomic interaction bymeans of Fano-Feshbach resonance [1]. The experimen-tal advances to address this transition has introduced thepossibility for studying the so called BCS-BEC crossovermore closely [2, 3]. The physics of the crossover focuseson the change of s-wave scattering length (1/a) from at-tractive to repulsive (from −∞ to +∞) by tuning an ex-ternal magnetic field. In the intermediate region where1/a→ 0: a new region emerges, where dimensionless co-herence length becomes in the order of unity, is the focalpoint of crossover physics.
The situation becomes further interesting if one as-sumes existence of random disorder in an otherwise veryclean system. It is well-known that every wavefunctionis spatially localized when the disorder is introduced inone-dimensional case, which is called as Anderson local-ization [4]. However, it is immensely difficult to directlyobserve the localization in electronic systems on crystallattices, so that one has to take the indirect route ofconductivity measurement to observe the effects of local-ization. It is an intriguing question how the Andersonlocalization modifies the BCS superconductivity. It hasbeen found by Anderson [5] that the order parameter isunaffected when the disorder is not too strong to giverise to the Anderson localization. Here the time reversalpairs, rather than the opposite momentum pairs, thenform the Cooper pairs.
The cold atomic system offers great amount of con-trollability and allows one to observe macroscopic wave-function. Therefore, it was conceived as a very usefulcandidate to visualize the disorder effects directly. Re-cently the ultracold Bose gas (87Rb and 39K) enabled usto see the localization directly [6, 7]. Latest experimentare conducted in three dimension for both noninteract-ing atomic Fermi gas of 40K [8] and bose gas of 87Rb [9].These experiments has widened the possibility to studythe crossover in lights of disorder [10] experimentally.
The static disorder in Fermi and Bose systems are notnew issues. A considerable amount of attention has beenpaid to disorder in superconductors [11–13] and in Bosegas [14–25]. Of late the interest at unitarity is also gain-ing pace [26–31], but still it failed to address all the ques-tions associated with it. It is exciting to envision thethree dimensional phase diagram involving temperature,interaction and disorder through the evolution from BCSto BEC superfluid. At the beginning it can be consideredthat the random potentials are independent of the hyper-fine states of the atoms but then it can be extended tothe correlated disorder problem in the crossover region.
In this article, we present our investigation on BCS-BEC crossover with weak uncorrelated disorder at zerotemperature. Precisely, we show: (i) monotonic deple-tion of order parameter, (ii) nonmonotonic nature of con-densate fraction, and (iii) suppression of sound velocity,as a function of disorder. Further we add a study onLandau critical velocity to attempt a qualitative under-standing of the nonmonotonic behavior of the condensatefraction. Though order parameter is mainly for academicinterest but the other three quantities are experimentallyviable [32–34]. In this study, we observe that super-fluidity is more robust in the unitary regime, which isalso consistent with behavior of the coherence length in
arX
iv:1
202.
5117
v1 [
cond
-mat
.qua
nt-g
as]
23
Feb
2012
2
the crossover regime. The robust nature of superfluid-ity has already been pointed out in the context of vortexcore structure [35], Josephson current [36] and collectivemodes [37]. We also observe a progressive decay of thesound velocity in the BEC side with increasing disorderpossibily occuring due to enhancement of impurity scat-tering [38]. Here we include a systematic study of thephysical observables and their response to the randomdisorder.
The recent experiments on ultracold Bose and Fermigas with disorder was carried out with optical speckle[6, 8] and quasi periodic optical lattices [7]. Though theypose interesting physics to study, but here we considerquenched delta correlated disorder which remained sub-ject of interest at zero temperature [26] and at finite tem-peratures [31] in previous studies. One can visualize thesituation of impurity driven random potential by con-sidering the presence of few heavy atoms (say 40K) ina homogeneous bath of light 6Li atoms [31]. Since thenumber of heavy atoms are very limited, one can term itas a quasi-homogeneous system. To include the disordereffects in the mean field approach, we follow the NSRtheory [39] of superconducting fluctuations extended tobroken symmetry state [40]. Further we take advantageof the techniques developed to study condensate fractionof clean Fermi gas with Gaussian fluctuations [41, 42].
We arrange our study in the following way, in SectionII we present the basic formalism. Section III is ded-icated to discuss our results in three parts. First partcontains the order parameter, second one for the con-densate fraction and the third leads to sound mode andLandau critical velocity. Finally, in Section IV we drawour conclusions.
II. FORMALISM
To make our presentation self-contained, we brieflysummarize the mathematical formalism presented inRef.[26]. To describe the effect of impurity in Fermi su-perfluid in the crossover from BCS to BEC regime oneneeds to start from the real space Hamiltonian in threedimension for a s-wave superfluid,
H(x) =∑σ
Φ†σ(x)[− ∇
2
2m− µ+ Vd(x)
]Φσ(x) +∫
dx′V(x,x′)Φ†↑(x′)Φ†↓(x)Φ↓(x)Φ↑(x
′), (1)
where Φ†σ(x) and Φσ(x) represents the creation and an-nihilation of fermions with mass m and spin state σ at xrespectively. Vd(x) signifies the (weak) random potentialand µ is the chemical potential. We set ~ = 1, where~ is the Planck constant. The s-wave fermionic interac-tion is defined as V(x,x′) = −gδ(x − x′). The disorderpotential is modeled as, Vd(x) =
∑i gdδ(x − xi) where
gd is fermion-impurity coupling constant and xi are thestatic positions of the quenched disorder. We assume it
exhibits white noise correlation, that is, 〈Vd(−q)Vd(q)〉 =κβδiνm,0. β is the inverse temperature, νm is the bosonicMatsubara frequency and κ = nig
2d, that describes the
strength of the impurity potential with ni being the con-centration of the impurities.
The partition function corresponding to Eq.(1) can bewritten in the path integral formulation as
Z =
∫D[Φ,Φ] exp [−S(ΦΦ)], (2)
where S =∫ β
0dτ∫dx[Φσ∂τΦσ +H]. By introducing the
pairing field ∆(x, τ) and by applying the Grassman iden-
tity (∫D[∆,∆] exp [−
∫dx∫ β
0dτ∆∆/g] = 1) Eq.(2) can
be given as
Z =
∫D[Φ,Φ]
∫D[∆,∆] exp [−Seff ], (3)
where Seff = S(Φ,Φ) + 1/g∫dx∫ β
0dτ∆∆. Follow-
ing Hubbard-Stratonovich transformation, Eq.(3) can bewritten in terms of inverse Nambu propagator as,
Zeff =
∫D[∆,∆]e−1/g
∫dx
∫ β0dτ∆∆
×∫D[ΦΦ]e−
∫dx
∫ β0dτΦG−1Φ, (4)
where the inverse Nambu propagator G−1(x, τ) is definedas, (
−∂τ + ∇2
2m + µ− Vd ∆
∆ −∂τ − ∇2
2m − µ+ Vd
). (5)
After integrating out the fermionic fields from Eq.(4) weare left with the effective action as,
Seff =
∫dx
∫ β
0
dτ[ |∆(r)|
g− 1
βTr ln−βG−1(r)
],(6)
where r = (x, τ). It is important to mention that themain contribution in the partition function comes from asmall fluctuation, δ∆(x, τ) = ∆(x, τ)−∆ where ∆ is thehomogeneous BCS pairing field. The Green’s function inEq.(5) can be written as a sum of Green’s function in ab-sence of disorder (G−1
0 = −∂τ I + (∇2/2m+ µ)σz + ∆σx)and a self energy contribution (Σ = −Vdσz + δ∆σ+ +¯δ∆σ−) which contains the disorder as well as the small
fluctuations of the BCS pairing fields. I denotes the iden-tity matrix, and σi are the Pauli matrices and laddermatrices(i ∈ x, y, z,+,−).
By expanding the inverse Nambu propagator upto thesecond order one can write the effective action (Seff ) inEq.(6) as a sum of bosonic action (SB) and fermionicaction (SF ). Also it contains an additional term whichemerges from the linear order of self energy expansion(G0Σ). It is possible to set zero for the linear order ifwe consider SF is an extremum of Seff after performingall the fermionic Matsubara frequency sums. The con-strained condition leads to the BCS gap equation which
3
after appropriate regularization through the s-wave scat-tering length reads,
− m
4πa=∑k
[ 1
2Ek− 1
2εk
]. (7)
Eq.(7) suggests that the BCS gap equation does not haveany contribution from the disorder potential explicitly.
Now to construct the density equation with usual pre-scription of statistical mechanics; the thermodynamic po-tential Ω should be differentiated with respect to thechemical potential µ. Ω can be written as a sum overfermionic (ΩF ) and bosonic (ΩB) thermodynamic poten-tials, which implies,
n = nF + nB = − ∂
∂µ(ΩF + ΩB) = − 1
β
∂
∂µ(SF + SB).(8)
The well known BCS density equation can be restoredfrom Eq.(8) if we consider only nF , then it yields the
familiar∑k(1 − ξk
Ek). However the presence of disorder
and fluctuation leads to nB 6= 0. Hence the final meanfield density equation will be,
n =∑k
(1− ξk
Ek
)− ∂ΩB
∂µ. (9)
The bosonic thermodynamic potential consists of twoparts. One comes from the thermal contribution and theother is due to disorder. Since we are interested in zerotemperature we neglect the thermal contribution fromhere on. Henceforth the disorder induced thermodynamicpotential can be written as,
ΩBd = −κ2
∑q,νm=0
N †M−1N . (10)
where N is a doublet which couples disorder with fluc-tuation. After performing the fermionic Matsubara fre-quency summation over N ,
N1 = N2 =∑k
∆(ξk + ξk+q)
2EkEk+q(Ek + Ek+q). (11)
The inverse fluctuation propagator matrix M is a 2 × 2symmetric matrix whose elements are given by
M11 =1
g+∑k
[v2kv
2k+q
iνm − Ek − Ek+q− u2
kuk+q
iνm + Ek + Ek+q
],
M12 =∑k
ukvkuk+qvk+q
[1
iνm + Ek + Ek+q
− 1
iνm − Ek − Ek+q
], (12)
and M22(q) = M11(−q). Here the usual BCS nota-tions have been used, namely ξk = k2/2m − µ, Ek =√ξ2k + ∆2, u2
k = 12 (1 + ξk/Ek) and v2
k = 12 (1− ξk/Ek).
FIG. 1. The order parameters ∆ normalized by Fermi energyas function of (kF a)−1 for various disorder strengths. Theshaded area represents the crossover region
III. RESULTS
Order Parameter
Eqs.(7) and (9) are now ready to be solved self con-sistently. Our analysis is valid only for the weak disor-der. Considering Ref.[31] we can safely assume that thedisorder is weak if the dimensionless disorder strengthη(= κm2/kF ) . 5 is satisfied. Fig. 1 demonstrates thatin the BCS limit (1/kFa− > −∞) every order param-eters with different disorder strength follows the meanfield expression of ∆/εF = 8e−2 exp[−π/(2kFa)] empha-sizing the validity of Anderson theorem [5]. In the BEClimit, instead, one can observe a progressive depletion.Assymptotically this depletion remains in the order ofη/(kFa) [26, 43]. The other mean field quantity, chemi-cal potential, does not change much(not shown). We un-derstand that it might be an attribute of the fluctuationtheory [35] where the correction in µ at the BCS side isusually in O(∆2) which is a very small quantity as ∆→ 0for (kFa)−1 → 0. In the BEC side the correction comesthrough the effective chemical potential of the compositebosons. However the BEC chemical potential is domi-nated by the binding energy and it turns out quite largecompared to the effective chemical potential. The exter-nal potential Vd(x) induced by the disorder usually hasno direct influence on the internal degrees of freedom,e.g. interaction among fermions. It is thus no wonderthat the binding energy of the composite bosons exhibitsno pronounced change, so does the chemical potential.
4
FIG. 2. The condensate fraction nc as function of (kF a)−1 forvarious disorder strengths. The shaded area and the verticalorange line at (kF a)−1 = 0 represent the crossover region andthe unitary limit, respectively.
Condensate Fraction
Though ∆ and µ are the first quantities to study thecrossover through the mean field theory but they aremostly of academic interest. From now on we shall focuson the quantities which are more prone to the experi-mental observation. Our first choice is the condensatefraction which is also one of our main results (depictedin Fig.2). In a clean Fermi gas, it is possible to work outthe condensate fraction through mean field theory [44]which shows good agreement with the experiment [33].Here we follow the similar mean field description,
nc =∑k
[ ∆(η)
2Ek(η)
]2, (13)
where nc is the condensate fraction. The only differencewith the clean system calculation, in Eq.(13) is that, weused the disorder induced values for ∆ and µ. We observequite remarkable behavior for nc. As one expects the con-densate fraction decays similar to that of the clean limitwhen 1/(kFa) → −∞ since ∆ does not change with thevariation of η in the BCS regime. If one extends the selfenergy upto the second order in the condensate fractioncalculation one can as well observe the effect of disorderin the BCS limit, where nc − nc0 gets saturated at somefinite value instead of exponential decay [26], with nc0 de-noting the condensate fraction in the clean limit. In theBEC side, the disorder destroys part of the condensateand turns it into a normal fluid. The condensate fractionapproaches roughly η/
√kFa as obtained from the study
of hard sphere Bose gas in random disorder [43].The nonmonotonic behavior in the crossover region
(grey area in Fig. 2) is the most intriguing point. In
the study of quantized vortices, one sees that accumu-lation of a number of vortices becomes maximum in thecrossover region [45]. Later on, in a theoretical study ofa single vortex it was also observed that the circulationcurrent is maximum in the crossover regime [35]. Sim-ilar feature was also reported in the Josephson currentstudy [36]. This unique behavior was qualitatively at-tributed to the maximization of Landau critical velocityin this region. In general all these observations actuallypoints to the robustness of the superfluidity at the uni-tarity. But normal mean field does not show any precisenonmonotonic behavior for the condensate fraction. Herewith a weak disorder, we are able to generate a picturewhich affirms the belief that in presence of low amountof impurity the condensate fraction is less affected acrossthe crossover, implying a comparatively high yielding ofsuperfluidity in this region.
One should also take a note of the position of the ex-trema. With the increase in disorder, we observe that theextrema slowly move towards 1/a → 0+ from +∞, butnever cross it as we are exhausted with weak impuritylimit. Further increase of disorder will break down theweakness condition. This result agrees qualitatively withRef.[26, 31], however in those cases it has been suggestedthat the most robust region of superfluidity emerges when1/a → 0−. But there is no good justification for that.Here it is pointed, this behavior is consistent with thatof the critical velocity discussed in the following section.
Sound Mode and Critical Velocity
Another important and experimentally relevant [34]quantity is the lowest energy collective excitation of thecondensate. The nature of sound velocity, in presenceof disorder, in a Bose gas has been studied quite exten-sively, but it lacks a real consensus so that considerableamount of ambiguity still exists. Using a perturbativemethod, the sound velocity is enhanced in the presence ofuncorrelated disorder and very weak interaction [16, 46–48], whereas within a non-perturbative self consistent ap-proach and spatially correlated weak disorder case, de-pression in sound velocity has been reported [49–52]. Ina more recent study it has been shown that there existsno generic behavior of sound in the presence of disor-der using a perturbative approach [53]. Hence in thecrossover region, the behavior of sound is expected to bequite interesting.
From technical point of view in order to obtain thesound velocity one needs to carry out an analytic continu-ation of the Matsubara frequency. Hence the fluctuationpropagator matrix M is expanded to the second orderfor both momentum and frequency. The determinant ofwhich leads to [54],
M11(q, ν)M12(q,−ν)−M212(q, ν) =
A(∆, µ,k)q2 + B(∆, µ,k)ν2 + . . . = 0, (14)
for ν = vs|q| where vs represents the sound velocity. A
5
FIG. 3. The sound velocity vs divided by the Fermi velocityvF as a function of (kF a)−1 for various disorder strengths.The shaded area depicts crossover region.
and B are functions of ∆ and µ and can be evaluatedby summing them over k. In the BEC limit it has beenshown that the sound velocity is v2
s = ∆2/(8m|µ|) [37]so that the suppression of the order parameter shoulddirectly result in that of the sound velocity. Near theunitarity, the sound velocity is directly connected to thechemical potential through v2
s = 2µ/(3m). In effect, thesound velocities for various disorders are merged nearthe unitarity as the chemical potential exhibits almost nochange regardless of addition of the disorder. Moreover,in the BCS side, vs =
√k2F /(3m
2) ' 0.57 (in dimen-sionless units) does not have any explicit dependence onthe mean field parameters. Therefore, irrespective of thedisorder the sound velocity saturates near 0.57. All ofthese arguments successfully explain the behavior of thesound velocity shown in Fig. 3.
In an usual way, sound velocity is defined as vs =√1ρκT
where ρ is the mass density and κT is the isother-
mal compressibility. Applying Gibbs-Duhem relationsound velocity can be written in usual notation as vs =√
nm∂µ∂n . An involved study of sound mode from the ther-
modynamical point of view reveals that the decrease inthe collective excitations can be attributed to the in-crease in the compressibility. This is in accordance withRef.[49]. Intuitively this looks more feasible as the occur-rence of additional random potential should lead to addi-tional scattering and, resulting in the decrease of soundvelocity [38].
Though the sound velocity itself is a quantity of hugeinterest, but it also serves an additional important infor-mation; According to Landau criterion, it determines thecritical velocity of BEC. The critical velocity is defined asthe minimum velocity of the BEC required to break su-perfluidity by creating elementary excitations. To obtain
FIG. 4. The sound velocity vs (filled marks and solid curves)already shown in Fig. 3 and the critical velocity vp (openmarks and solid curves) obtained from pair breaking, normal-ized by the Fermi velocity vF , as a function of (kF a)−1 forvarious disorder strengths. The critical velocity vc’s (solidcurves) are determined from minvs, vp. The unitary re-gion is depicted through the grey area. The inset focuses theturning point in the unitary region.
a clear picture leading to the critical velocity, we calculatethe minimum velocity related to the single particle excita-tions induced by pair breaking, which is dominant in theBCS side. The fermionic single particle excitation within
mean field is written as√
(√
∆2 + µ2 − µ)/m [37], which
is reduced to the familiar result of ∆/kF in the deep BCSregion. In the case of the composite bosons, the Landaucritical velocity is provided as the sound velocity. Bychoosing the minimum between vs and vp, one can deter-mine the Landau critical velocity, which is representedby the solid curves in Fig. 4.
Though the simple mean field analysis on Landau crit-ical velocity may not be amenable for a direct mappingbetween nc and vc, but it may provide a qualitative sug-gestion for less depletion of condensate fraction in thecrossover regime. In both cases the maxima exist nearthe unitary regime and move toward (kFa)−1 = 0 as ηincreases as shown in the inset of Fig. 4.
IV. CONCLUSION
In conclusion, we have studied several important physi-cal quantities such as gap parameter, condensate fractionand sound velocity etc. to address the issue of BCS toBEC crossover in a disordered environment. To this endwe have included weak disorder via the gaussian fluctu-ation as prescribed earlier [26, 31], and hence solved thecoupled BCS mean field equations self consistently. Thisenables us to obtain the two basic mean field parame-
6
ters, ∆ and µ. We thus show that the order parametergets depleted leaving the chemical potential unchangedas we go from a BCS to a BEC regime. Afterwards, thecondensate fraction has been calculated using the wellknown mean field description with the disorder affected∆ and µ. As the disorder strength increases, we observepronounced maximum developed in the crossover regionjustifying the expectation of robust superfluid in this re-gion. We have tried to connect this non monotonic naturequalitatively to the Landau critical velocity, which alsoshows a sharp maximum near unitarity. With increase ofthe impurity this peak slightly moves towards the unitarypoint. A similar feature is observed for the condensatefraction. In addition, the depression of the sound velocityis also been addressed, which might be related to the en-hanced scattering from the random scatterers employedin this model.
To be precise, the the nature of ∆, nc, vs and vc is beenreported here, when subjected to a weak random impu-
rity. We hope our present study has shed some lights onphysics of the BCS-BEC crossover in disordered systems.An interesting future perspective can be generalization ofthe theory for arbitrarily strong disorder and interaction.We also hope that these results would be observed andverified in experiment soon.
ACKNOWLEDGEMENT
AK is grateful for useful communication with G. Orsoand enlightening discussions with C. Sa de Melo, F.Dalfovo, G. Roati, N. Trivedi and G. Watanabe. Thiswas supported by the NRF grant funded by the Ko-rea government (MEST) (No.2009-0087261 and No.2010-0024644). SB acknowledges financial support from DST(SR/S2/CMP/0023/2009).
[1] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev.Mod. Phys. 82, 1225 (2010).
[2] S. Georgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod.Phys. 80, 1215 (2008).
[3] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys.80, 885 (2008).
[4] P. W. Anderson, Phys. Rev. 109, 1492 (1958).[5] P. W. Anderson, J. Phys. Chem. Solids 11, 26 (1959).[6] J. Billy et. al., Nature 453, 891 (2008).[7] G. Roati et. al., Nature 453, 895 (2008).[8] S. S. Kondov, W. R. McGEhee, J. J. Zirbel, and B. De-
Marco, Science 334, 66 (2011).[9] F. Jendrzejewski et. al., arXiv:1108.0137.
[10] L. S. Palencia, and M. Lewenstein, Nature Phys. 6, 87(2010).
[11] D. Belitz, and T. R. Kirkpatrick, Rev. Mod. Phys. 66,261 (1994).
[12] A. Ghoshal, M. Randeria and N. Trivedi, Phys. Rev. B65, 014501 (2001).
[13] K. Bouadim, Y. L. Loh, M. Randeria, and NandiniTrivedi, Nature Phys. 7, 884 (2011).
[14] T. Paul, M. Albert, P. Schlagheck, P. Leboeuf, and N.Pavloff, Phys. Rev. A 80, 033615 (2009).
[15] C. Gaul, N. Renner, and C. A. Muller, Phys. Rev. A 80,053620 (2009).
[16] A.V. Lopatin and V. M. Vinokur, Phys. Rev. Lett. 88,235503 (2002).
[17] B. Damski, J. Zakrzewski, L. Santos, P. Zoller, and M.Lewenstein, Phys. Rev. Lett. 91, 080403 (2003).
[18] A. Sanpera et. al., Phys. Rev. Lett. 93, 040401 (2004).[19] N. Bilas and N. Pavloff, Phys. Rev. Lett. 95, 130403
(2005).[20] L. S. Palencia et. al., Phys. Rev. Lett. 98, 210401 (2007).[21] P. Lugan, D. Clement, P. Bouyer, A. Aspect, and L. S.
Palencia, Phys. Rev. Lett. 99, 180402 (2007).[22] A. Niederberger et. al., Phys. Rev. Lett. 100, 030403
(2008).[23] S. Pilati, S. Giorgini, and N. Prokofev, Phys. Rev. Lett.
102, 150402 (2009).
[24] G. Modugno, Rep. Prog. Phys. 73, 102401 (2010).[25] B. Shapiro, arXiv:1112.5736.[26] G. Orso, Phys. Rev. Lett. 99, 250402 (2007).[27] B. I. Shklovskii, Semiconductors 42, 927 (2008).[28] C. A. R. Sa de Melo, Physics Today 61, 45 (2008).[29] P. Dey, and S. Basu, J. Phys.: Cond. Mat. 20, 485205
(2008).[30] L. Han, and C. A. R. Sa de Melo, arXiv:0904.4197.[31] L. Han, and C. A. R. Sa de Melo, New J. Phys. 13,
055012 (2011).[32] D. E. Miller et. al., Phys. Rev. Lett. 99, 070402 (2007).[33] M.W. Zwierlein et. al., Phys. Rev. Lett. 92, 120403
(2004).[34] M. R. Andrews et. al., Phys. Rev. Lett. 79, 553 (1997).[35] R. Sensharma, M. Randeria, and T. L. Ho, Phys. Rev.
Lett. 96, 090403 (2006).[36] A. Spuntarelli, P. Pieri, and G. C. Strinati, Phys. Rev.
Lett. 99, 040401 (2007).[37] R. Combescot, M. Y. Kagan, and S. Stringari, Phys. Rev.
A 74, 042717 (2006).[38] V. I. Yukalov, Physics of Particles and Nuclei 42, 460
(2011).[39] P. Nozieres S. Smith-Rink, J. Low Temp. Phys. 59, 195
(1985).[40] J. R. Engelbrecht, M. Randeria, and C. A. R. Sa de Melo,
Phys. Rev. B 55, 15153 (1997).[41] E. Taylor, A. Griffin, N. Fukushima, and Y. Ohashi,
Phys. Rev. A 74, 063626 (2006).[42] N. Fukushima, Y. Ohashi, E. Taylor, and A. Griffin,
Phys. Rev. A 75, 033609 (2006).[43] K. Huang, and H. F. Meng, Phys. Rev. Lett. 69, 644
(1992).[44] L. Salasnich, N. Manini, and A. Parola, Phys. Rev. A 72,
023621 (2005).[45] M.W. Zwierlein et. al., Nature 435, 1047 (2005).[46] S. Giorgini, L. Pitaevskii, and S. Stringari, Phys. Rev. B
49, 12938 (1994).[47] G. M. Falco, A. Pelster, and R. Graham, Phys. Rev. A
75, 063619 (2007).
7
[48] G. M. Falco, A. Pelster, and R. Graham, Phys. Rev. A76, 013624 (2007).
[49] V. I. Yukalov and R. Graham, Phys. Rev. A 75, 023619(2007).
[50] V. I. Yukalov, E. P. Yukalova, K. V. Krutitsky and R.Graham, Phys. Rev. A 76, 053623 (2007).
[51] C. Gaul, N. Renner, and C. A. Muller, Phys. Rev. A 80,053620 (2009).
[52] V. I. Yukalov, E. P. Yukalova and V. S. Bagnato, LaserPhys. 19, 686 (2009).
[53] C. Gaul, and C. A. Muller, Phys. Rev. A 83, 063629(2011).
[54] M. Marini, F. Pistolesi, and G.C. Strinati, Eur. Phys. JB 1, 151 (1998).