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Superfluid density in the BCS–BEC crossover regime of a Fermi superfluid
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Physica C 468 (2008) 599–604 www.elsevier.com/locate/physc
Superfluid density in the BCS–BEC crossover regime of a Fermi superfluid Y. Ohashi a,*, N. Fukushima b, H. Matsumoto c, E. Taylor d, A. Griffin d a Department of Physics, Keio University, Hiyoshi, Yokohama 223-8522, Japan Department of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan c Physics Department, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan d Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7 b
Accepted 30 November 2007 Available online 4 March 2008
Abstract We investigate the superfluid carrier density qs in the BCS–BEC crossover regime of a Fermi superfluid at finite temperatures. Including pairing fluctuations within a Gaussian approximation, we calculate the superfluid order parameter, chemical potential, and qs in a consistent manner in the whole BCS–BEC crossover region. In the weak-coupling BCS regime, the temperature dependence of qs is dominated by single-particle excitations accompanied by thermal dissociation of Cooper pairs. In the strong-coupling BEC regime, we show that single-particle excitations become less dominant and thermal excitations of collective Bogoliubov mode become important for qs . We clarify how the weak-coupling BCS result smoothly changes into the strong-coupling BEC result, as one increases the strength of a pairing interaction. Ó 2008 Elsevier B.V. All rights reserved. PACS: 03.75.Ss; 03.75.Kk; 03.70.+k Keywords: BCS–BEC crossover; Superfluid density; Ultracold Fermi gas
1. Introduction The superfluid (carrier) density qs is one of the most fundamental quantities in the superfluid phase [1]. In the supercurrent state, qs equals the number density of particles contributing to the superflow. At T ¼ 0, the value of qs is equal to the total carrier density q, while it vanishes when the system is in the normal phase. These properties are always satisfied irrespective of the strength of an interaction between particles. For example, in the superfluid 4 He, it has been evaluated that only about 10% of atoms are Bose-condensed due to a strong repulsion between
*
Corresponding author. E-mail address: [email protected] (Y. Ohashi).
0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.11.024
helium atoms [2]. Even in this case, one finds qs ¼ q at T ¼ 0. The recently realized Fermi superfluids in 40K and 6Li gases [3–5] have the unique property that one can tune the strength of a pairing interaction between Fermi atoms by adjusting the threshold energy of a Feshbach resonance (FR). Using this tunable interaction, the BCS–BEC (Bose– Einstein condensation) crossover has been realized [6], where the character of superfluidity continuously changes from the weak-coupling BCS type to the Bose–Einstein condensation of tightly bound molecular bosons which have been already formed above the superfluid phase transition temperature T c [7,8]. In a sense, we can study fermion superfluidity (such as metallic superconductivity) and boson superfluidity (such as superfluid 4He and BEC of dilute Bose gases) in a unified manner in the BCS– BEC crossover phenomenon. Since the superfluid density
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commonly exists in both Fermi and Bose superfluids, it is an interesting problem how qs of fermion superfluidity changes into that of boson superfluidity in the BCS–BEC crossover. In this paper, we investigate the superfluid density in the BCS–BEC crossover regime of a superfluid Fermi gas. Within a Gaussian fluctuation approximation, we calculate the superfluid order parameter D, chemical potential l, and qs in a consistent manner. This consistent treatment is necessary to satisfy the required condition that qs ¼ q at T ¼ 0 and it vanishes in the normal phase over the entire BCS– BEC crossover. We numerically show how the character of qs continuously changes from the weak-coupling BCS type to the superfluid density of molecular BEC, as one passes through the BCS–BEC crossover. This paper is organized as follows. In Section 2, we explain the outline of our strong-coupling theory to calculate D and l in the superfluid phase. In Section 3. we discuss the superfluid density consistent with the strongcoupling theory explained in Section 2. Throughout this paper, we take h ¼ k B ¼ 1. We also set the system volume V ¼ 1, so that the total number of particles N and the carrier density q N =V are the same. 2. Gaussian fluctuation theory below T c In superfluid Fermi gases, all the current experiments are using a broad FR to tune the strength of a pairing interaction [3–5]. In this case, we can study this system by using the BCS model, as far as we consider the interesting BCS– BEC crossover physics [6]. The BCS Hamiltonian is given by X X y H¼ np cypr cpr U cpþq=2" cypþq=2# cp0 þq=2# cp0 þq=2" : p;r
p;p0 ;q
ð1Þ Here, we consider a uniform Fermi gas with two atomic hyperfine states described by pseudo-spin r ¼"; #. cyp is the creation operator of a Fermi atom with the kinetic energy np ¼ ep l ¼ p2 =2m l, measured from the chemical potential l. The assumed pairing interaction U is associated with FR, which can be experimentally tuned by adjusting the threshold energy of FR. Nozie`res and Schmitt-Rink (NSR) have presented a strong-coupling theory to describe the BCS–BEC crossover at T c [7]. The NSR theory consists of two coupled equations, namely, the equation for T c and the equation for the number of fermions. The T c -equation is obtained from the so-called Thouless criterion in the t-matrix approximation, stating that the superfluid phase transition occurs when the particle–particle scattering matrix has a pole at x ¼ q ¼ 0. The resulting T c -equation has the same form as the ordinary BCS gap equation at T c . In the weak-coupling BCS theory, we can safely set l ¼ eF in the gap equation (where eF is the Fermi energy). However, as pointed out by NSR [7], the value of l deviates from eF , as one
approaches the strong-coupling region. This strong-coupling effect is taken into account by considering the number equation, calculated from the identity q ¼ oX=ol, where X is the thermodynamic potential involving pairing fluctuations. We note that the NSR theory has been re-formulated in the framework of the functional integral formalism [9], and it has been clarified that the NSR theory includes pairing fluctuations within the Gaussian fluctuation level. In the functional integral formalism, the T c -equation is obtained as the saddle point solution, and the number equation involves pairing fluctuations around the saddle point solution to the quadratic order. The NSR or Gaussian fluctuation theory has been extended to the superfluid phase below T c [8,10]. In the extended NSR theory, what corresponds to the T c -equation in the original NSR theory is the gap equation, given by 1¼U
X 1 bEp ; tanh 2E 2 p p
ð2Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where b ¼ 1=T , and Ep ¼ n2p þ D2 is the single-particle excitation spectrum. The equation for the number of fermions q in the extended NSR theory is given by [8,10] h i 1 @ X b imn Þ : ln det 1 þ U Nðq; ð3Þ q ¼ q0F 2b @l q;mn Here, mn is the boson Matsubara frequency, and X np bEp 0 qF 1 tanh Ep 2 p
ð4Þ
is the number of fermions in the mean-field approximation. The last term in Eq. (3) ð qFL Þ describes the fluctuation correction to the number equation. qFL is obtained by summing up the fluctuation contributions to the thermodynamic potential ð XFL Þ diagrammatically shown in Fig. 1, and calculating qFL ¼ @XFL =@l. (For more details, we refer to Ref. [10].) In the last term of Eq. (3), pair-
Fig. 1. Fluctuation contribution to the thermodynamic potential XFL . The total thermodynamic potential X is given by the sum of the mean-field part and XFL . The solid and dashed lines represent the single-particle Green’s b 0 and the pairing interaction U, respectively. The bubble function G diagrams represent Pij (ði; jÞ ¼ 1; 2) describing pairing fluctuations.
Y. Ohashi et al. / Physica C 468 (2008) 599–604
601
ing fluctuations are described by the (2 2)-matrix correlab imn Þ, having the form tion function Nðq; 1 P11 þ P22 þ 2iP12 P11 P22 b N¼ : ð5Þ 4 P11 P22 P11 þ P22 2iP12 Here, Pij are given by [10,11] 1X b 0 ðp þ q=2; ixm þ imn Þsj G b 0 ðp q=2; ixm Þ; Pij ¼ Tr½si G b p;xm ð6Þ
b 1 ðp; ixm Þ ¼ ixm np s3 þ Ds1 is the (2 2)-matrix where G 0 single-particle Green’s function (where xm is the fermion Matsubara frequency, and sj (j ¼ 1; 2; 3) are the Pauli matrices). Physically, P11 and P22 describe amplitude and phase fluctuations of the order parameter D, respectively. P12 describes coupling between amplitude and phase fluctuations [11]. The coupled Eqs. (2) and (3) involve ultraviolet divergence, originating from the contact interaction (U ) appearing in Eq. (1). In metallic superconductivity with a phonon-mediated pairing interaction, this singularity can be eliminated by introducing a physical cutoff with the order of Debye frequency. In contrast, such a physical cutoff does not exist in the case of superfluid Fermi gas with a pairing interaction associated with FR. In superfluid Fermi gases, the ultraviolet divergence can be eliminated by introducing the two-body s-wave scattering length as , which is related to the pairing interaction U as [12] 4pas U P ¼ m 1U p
1 2ep
:
ð7Þ
Using the scattering length as , we can regularize the coupled Eqs. (2) and (3) as 4pas X 1 bEp 1 1¼ tanh ; ð8Þ 2ep m p 2Ep 2 " " ## X 1 1 @ X 4pas b 0 Nðq; imn Þ þ ln det 1 qF ¼ 2b @l q;mn 2ek m k ¼ q0F þ qFL :
ð9Þ
In Eqs. (8) and (9), the ultraviolet divergence is absorbed into the scattering length as , which is determined experimentally. We solve the regularized coupled Eqs. (8) and (9) for a given value of as . We briefly note that, when the interaction is measured in term of as , the weak-coupling BCS regime corresponds to ðk F as Þ1 K 1, and the strong-coupling BEC regime is 1 characterized by ðk F as Þ J þ 1 (where k F is the Fermi 1 momentum). The region 1 K ðk F as Þ K þ 1 is referred to as the crossover region. The self-consistent solutions of Eqs. (8) and (9) are shown in Fig. 2. In panel (a), while T c in the BEC regime approaches the constant value, T c ¼ 0:218eF [9], the magnitude of the order parameter D at T ¼ 0 continues to increase with increasing the strength of the pairing interac-
Fig. 2. Self-consistent solutions of the coupled Eqs. (8) and (9) in the BCS–BEC crossover. (a) Order parameter D and (b) chemical potential l. In both panels (and also in Figs. 3 and 4), ðk F as Þ1 describes the strength of the pairing interaction measured in terms of the s-wave scattering length as (normalized by the Fermi momentum k F ), and T =eF is the temperature normalized by the Fermi energy eF . Dashed line and dotted line show results at T ¼ 0 and T c , respectively. The spurious first order phase transition seen in the BEC region is an artifact of the extended NSR theory we are using.
tion. As a result, the ratio 2DðT ¼ 0Þ=T c becomes larger than the so-called BCS universal constant ð2DðT ¼ 0Þ= T c ¼ 3:54Þ in the strong-coupling region. In panel (b), the temperature dependence of l is weak below T c , and the value becomes smaller to be negative as one passes through the crossover region. In the BEC limit ððk F as Þ1 þ1Þ, one obtains l ¼ 1=2ma2s [9]. In Fig. 2a, we find that the temperature dependence of D shows a bendover near T c in the BEC regime, which indicates a first order phase transition. However, we note that this is an artifact of the Gaussian fluctuation theory we are using here. This kind of singularity near T c has been known in the Popov theory for the Bose gas BEC [12]. In the Popov theory, to overcome this problem, one needs to include many-body renormalization effects on an interaction between bosons [13,14]. Since the present theory is known to reduce to the Popov theory for a weakly interacting molecular Bose gas in the BEC limit [15], the bendover seen in Fig. 2 has the same origin as that in the Popov theory. Thus, to recover the second order phase transition, we need to include many-body corrections to the interaction
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between molecular bosons beyond the present Gaussian fluctuation theory [16]. Although this is an important problem, in this paper, we only calculate qs within the Gaussian fluctuation level using the results shown in Fig. 2. However, within this approximation, we calculate qs in a consistent manner.
b is the self-energy correction, given by [19] Here, R 1 b RðpÞ ¼ b
b s is the (2 2)-matrix single-particle Green’s Here, G function in the supercurrent state. In the mean-field b 1 ¼ iðxm V s pz Þ approximation, it has the form G s0 np s3 þ Ds1 [17,18]. The supercurrent J s is related to the superfluid density qs as J s ¼ qs V s under the assumption that the magnitude of the superfluid velocity V s is small. Noting this fact and using (10), we find that the normal fluid density qn q qs is given by 2X @ Tr½Gs ðp; ixm Þ qn ¼ pz : ð11Þ b p;xm @Q Q!0 b s0 , we obtain In the mean-field approximation, using G the well-known expression for the BCS normal fluid density ð qFn Þ, qFn ¼
2 X 2 @f ðEp Þ p ; 3m p @Ep
ð12Þ
where f ðEp Þ is the Fermi P distribution function. Eq. (12) reduces to qFn ðT c Þ ¼ 2 p f ðnp Þ at T c , which equals the meanfield part q0F ðT c Þ of the number Eq. (9). Thus, except for the weak-coupling BCS limit (where the fluctuation contribution qFL can be ignored in Eq. (9)), the mean-field normal fluid density qFn at T c is smaller than the total carrier density q, leading to a finite value of the superfluid density even at T c ðqs ¼ q qFn ¼ qFL > 0Þ. Thus, to satisfy the required condition that qs vanishes at T c over the entire BCS–BEC crossover region, we need to include pairing fluctuations in qn so as to satisfy qn ¼ q0F þ qFL at T c . To calculate fluctuation corrections to qn , we need to b s consistent with the extended NSR theory in find out G Section 2. For this purpose, we note that the numberP equation can be written in the form q ¼ p1 þ P b ixm Þ. When Eq. (9) is written in this T p;xm Tr½s3 Gðp; form, one finds that the single-particle Green’s function in the extended NSR theory has the form b0R b ¼G b0 þ G bG b 0: G
ð13Þ
q
4pas m s b det½1 4pa ð NðqÞ þ m
("
P k
1 Þ 2ek
# X 1 4pas 1 ½N11 ðqÞ þ sþ G0 ðp þ qÞs 2ek m k " # X 1 4pas þ 1 ½N22 ðqÞ þ s G0 ðp þ qÞsþ 2ek m k ) 4pas N12 ðqÞsþ G0 ðp þ qÞsþ ; þ2 ð14Þ m
3. Superfluid density We consider the supercurrent state with the superfluid velocity V s in the z-direction. The supercurrent state is described by the order parameter DðzÞ ¼ P DeiQz (where Q ¼ 2mV s ). The supercurrent density J s ¼ p;r ðpz =mÞ hcpr cpr i is given by 1 X pz b s ðp; ixm Þ: Tr½ G J s ¼ qV s þ ð10Þ b p;xm m
X
where we have simply written p ¼ ðp; ixm Þ and q ¼ ðq; imn Þ. Nij (i; j ¼ 1; 2) is the ði; jÞ-component of the matrix correlab in Eq. (5). In the supercurrent state, the tion function N b s corresponding to Eq. single-particle Green’s function G (13) is given by b s0 þ G b s0 R bsG b s0 : bs ¼ G G
ð15Þ
b 0 is replaced by b s is given by Eq. (14), where G Here, R b G s0 . Substituting Eq. (15) into Eq. (11), we find that the normal fluid density can be written as qn ¼ qFn þ qBn , where qFn is the BCS normal fluid density in Eq. (12) and qBn is the fluctuation correction arising from the last term in Eq. (15), given by [19,20] " " ##! X 1 2m @ 2 X 4pas b B qn ¼ ln det 1 NðqÞ þ m b @Q2 q 2ek k Q!0
ð16Þ
In the normal phase limit D ! 0, it can be shown that Eq. (16) reduces to qFL ðT c Þ, namely, qs ðT c Þ ¼ q ðqFn ðT c Þ þ qBn ðT c ÞÞ ¼ 0 is satisfied in the BCS–BEC crossover [19], as expected. In the weak-coupling BCS regime, since pairing fluctuations are weak, we obtain qs ’ q qFn . Since qFn is dominated by single-particle excitations (see Eq. (12)), the decrease of qs at finite temperatures is found to originate from thermal dissociation of Cooper pairs in the weak-coupling BCS regime. As one approaches the strong-coupling BEC regime, qFn becomes less dominant due to the fact that the magnitude of single-particle excitation gap becomes large. In contrast, the fluctuation part qBn becomes dominant, so that we find qs ’ q qBn . In the BEC limit, qBn reduces to [19,20] qBn ¼
B 2 X 2 @nB ðEq Þ q : 3M q @EBq
ð17Þ
Here, M ¼ 2m is a molecular mass, and nB ðEBq Þ is the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bose distribution function. EBq ¼ eBq ½eBq þ 2U B nc is just the expression for the Bogoliubov phonon spectrum in a molecular BEC in the Popov approximation [12], where eBq q2 =2M is the molecular kinetic energy, U B 4paB =M is an effective interaction between bosons with
Y. Ohashi et al. / Physica C 468 (2008) 599–604
the molecular scattering length aB ¼ 2as , and nc m2 D2 as =8p describes the molecular condensate fraction [19,20]. The Bogoliubov phonon is a Goldstone mode associated with the spontaneous breakdown of the continuous gauge symmetry in the superfluid phase. Indeed, one finds a B linearpdispersion ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eq jq!0 ¼ v/ q with the sound velocity v/ ¼ U B nc =M . Eq. (17) is just the Landau expression for the normal fluid density in a Bose superfluid [12]. Namely, qn in the BEC regime is dominated by Bogoliubov collective excitations. The above result in the BEC regime is consistent with the extended NSR theory discussed in Section 2. Indeed, in this regime, Eq. (8) gives l ¼ 1=2ma2s , and the number Eq. (9) reduces to [20] " # bEBq q 1 X eBq þ U B nc ¼ nc þ 1 : ð18Þ coth 2 2 q 2 EBq This is just the number equation for a q=2-molecular Bose gas in the Popov approximation [12]. Fig. 3 shows the calculated superfluid density qs in the BCS–BEC crossover. We find that the weak-coupling BCS result qs ¼ q qFn continuously changes into the superfluid density of a molecular Bose superfluid qs ¼ q qBn , as one increases the strength of the pairing interaction. We note that although qn in the BCS regime is dominated by single-particle excitations, the Goldstone (collective) mode with a linear dispersion always exists irrespective of the strength of the pairing interaction. Fig. 4 shows the velocity of the Goldstone mode in the BCS–BEC crossover. In the BCS regime, the Goldstone mode is the Anderson–Bogoliubov mode with the mode pffiffiffi velocity v/ ¼ vF = 3 at T ¼ 0 [10,11] (where vF is the Fermi velocity). In the BCS regime, comparing the collective mode energy (x ¼ v/ q) with the single-particle excitation gap (D) at T ¼ 0, we find that the momentum region where the collective excitations are important (x K D) is very narrow, as 0 6 q K k F ðD=eF Þ k F . As a result, the contri-
Fig. 3. Superfluid density qs in the BCS–BEC crossover. In this calculation, D and l shown in Fig. 2 are used. The solid circles show the weak-coupling BCS superfluid density qs ¼ q qFn . The solid triangles show the strong-coupling BEC result qs ¼ q qBn , where Eq. (17) is used. The bendover near T c in the BEC regime is due to the spurious first order phase transition seen in Fig. 2.
603
Fig. 4. Velocity v/ of the Goldstone mode in the BCS–BEC crossover. This result is obtained within the generalized random phase approximation (GRPA), which is a consistent approximation with the extended NSR theory [21]. In GRPA, the mode velocity P is obtained by solving b imn ! v/ q þ idÞÞ þ Re½det½1 ð4pas =mÞð Nðq; For more k 1=2ek ¼ 0. details, see Refs. [10,11].
bution of the Goldstone mode to qn is weak compared with single-particle excitations in the BCS regime. We also note that v/ becomes small in the BEC regime, as shown in Fig. 4. At the same time, the single-particle excitation gap becomes large reflecting the increase of the binding energy. As a result, the momentum region where collective excitations are crucial becomes wide, as one approaches the BEC regime, so that the origin of qn changes from single-particle excitations to collective excitations in the BCS–BEC crossover, as discussed previously. 4. Summary To summarize, we have discussed the superfluid density qs in the BCS–BEC crossover regime of a superfluid Fermi gas. We have calculated D, l, and qs in a consistent manner within the Gaussian fluctuation approximation. This treatment is crucial for the consistent treatment of qs over the entire BCS–BEC crossover. We showed that the origin of the normal fluid density qn ¼ q qs continuously changes from single-particle excitations accompanied by dissociation of Cooper pairs to collective excitations, as one increases the strength of the pairing interaction. In the strong-coupling BEC regime, we showed that the present Gaussian fluctuation theory leads to a spurious first order phase transition. To recover the expected second order phase transition and to describe the correct temperature dependence of the superfluid density near T c in the BEC regime, we need to include higher-order fluctuation effects beyond the Gaussian fluctuation level. Although this problem still remains to be solved, since the two-fluid theory is based on the superfluid density, our results would be still useful in constructing the two-fluid hydrodynamics in the BCS–BEC crossover region at finite temperatures. Acknowledgements Y.O. was supported by Grant-in-Aid for Scientific research from MEXT (18043005, and 19540420),
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JSPS-CTC program and CREST(JST). E.T. and A.G. were supported by NSERC of Canada. References [1] L. Landau, J. Phys. (Moscow) 5 (1941) 71. [2] See, for example A. Griffin, Excitations in a Bose-condensed Liquid, Cambridge University Press, New York, 1993. [3] C.A. Regal, M. Greiner, D.S. Jin, Phys. Rev. Lett. 92 (2004) 040403. [4] M. Bartenstein et al., Phys. Rev. Lett. 92 (2004) 120401. [5] M.W. Zwierlein et al., Phys. Rev. Lett. 92 (2004) 120403. [6] For a review, see Q. Chen, J. Stajic, S. Tan, K. Levin, Phys. Rep. 412 (2005) 1. [7] P. Nozie`res, Schmitt-Rink, J. Low Temp. Phys. 59 (1985) 195. [8] J.R. Engelbrecht, M. Randeria, C.A.R. Sa´ de Melo, Phys. Rev. B 55 (1997) 15153. [9] M. Randeria, in: A. Griffin, D.W. Snoke, S. Stringari (Eds.), Bose– Einstein Condensation, Cambridge University Press, New York, 1995, p. 355.
[10] Y. Ohashi, A. Griffin, Phys. Rev. A 67 (2003) 063612. [11] Y. Ohashi, S. Takada, J. Phys. Soc. Jpn. 66 (1997) 2437. [12] C.J. Pethick, H. Smith, Bose–Einstein Condensation in Dilute Gases, Cambridge University Press, UK, 2002. [13] H. Shi, A. Griffin, Phys. Rep. 304 (1998) 1. [14] M. Bijlsma, H. Stoof, Phys. Rev. A 54 (1996) 5085. [15] P. Pieri, G. Strinati, Phys. Rev. B 71 (2005) 094520. [16] Inclusion of many-body renormalization effects on the molecular interaction has been done in the BEC regime Y. Ohashi, J. Phys. Soc. Jpn. 74 (2005) 2659. [17] K. Maki, in: R.D. Parks (Ed.), Superconductivity, vol. 2, Marcel Dekker, New York, 1969, p. 1035. [18] Although l is also affected by the supercurrent as l ! l Q2 =8m, it does not affect qs because it is the second order OðV 2s Þ. [19] N. Fukushima, Y. Ohashi, E. Taylor, A. Griffin, Phys. Rev. A 75 (2007) 033609. [20] E. Taylor, A. Griffin, N. Fukushima, Y. Ohashi, Phys. Rev. A 74 (2006) 063626. [21] Y. Ohashi, Phys. Rev. A 70 (2005) 063613.
Physica C 468 (2008) 599–604 www.elsevier.com/locate/physc
Superfluid density in the BCS–BEC crossover regime of a Fermi superfluid Y. Ohashi a,*, N. Fukushima b, H. Matsumoto c, E. Taylor d, A. Griffin d a Department of Physics, Keio University, Hiyoshi, Yokohama 223-8522, Japan Department of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan c Physics Department, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan d Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7 b
Accepted 30 November 2007 Available online 4 March 2008
Abstract We investigate the superfluid carrier density qs in the BCS–BEC crossover regime of a Fermi superfluid at finite temperatures. Including pairing fluctuations within a Gaussian approximation, we calculate the superfluid order parameter, chemical potential, and qs in a consistent manner in the whole BCS–BEC crossover region. In the weak-coupling BCS regime, the temperature dependence of qs is dominated by single-particle excitations accompanied by thermal dissociation of Cooper pairs. In the strong-coupling BEC regime, we show that single-particle excitations become less dominant and thermal excitations of collective Bogoliubov mode become important for qs . We clarify how the weak-coupling BCS result smoothly changes into the strong-coupling BEC result, as one increases the strength of a pairing interaction. Ó 2008 Elsevier B.V. All rights reserved. PACS: 03.75.Ss; 03.75.Kk; 03.70.+k Keywords: BCS–BEC crossover; Superfluid density; Ultracold Fermi gas
1. Introduction The superfluid (carrier) density qs is one of the most fundamental quantities in the superfluid phase [1]. In the supercurrent state, qs equals the number density of particles contributing to the superflow. At T ¼ 0, the value of qs is equal to the total carrier density q, while it vanishes when the system is in the normal phase. These properties are always satisfied irrespective of the strength of an interaction between particles. For example, in the superfluid 4 He, it has been evaluated that only about 10% of atoms are Bose-condensed due to a strong repulsion between
*
Corresponding author. E-mail address: [email protected] (Y. Ohashi).
0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.11.024
helium atoms [2]. Even in this case, one finds qs ¼ q at T ¼ 0. The recently realized Fermi superfluids in 40K and 6Li gases [3–5] have the unique property that one can tune the strength of a pairing interaction between Fermi atoms by adjusting the threshold energy of a Feshbach resonance (FR). Using this tunable interaction, the BCS–BEC (Bose– Einstein condensation) crossover has been realized [6], where the character of superfluidity continuously changes from the weak-coupling BCS type to the Bose–Einstein condensation of tightly bound molecular bosons which have been already formed above the superfluid phase transition temperature T c [7,8]. In a sense, we can study fermion superfluidity (such as metallic superconductivity) and boson superfluidity (such as superfluid 4He and BEC of dilute Bose gases) in a unified manner in the BCS– BEC crossover phenomenon. Since the superfluid density
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Y. Ohashi et al. / Physica C 468 (2008) 599–604
commonly exists in both Fermi and Bose superfluids, it is an interesting problem how qs of fermion superfluidity changes into that of boson superfluidity in the BCS–BEC crossover. In this paper, we investigate the superfluid density in the BCS–BEC crossover regime of a superfluid Fermi gas. Within a Gaussian fluctuation approximation, we calculate the superfluid order parameter D, chemical potential l, and qs in a consistent manner. This consistent treatment is necessary to satisfy the required condition that qs ¼ q at T ¼ 0 and it vanishes in the normal phase over the entire BCS– BEC crossover. We numerically show how the character of qs continuously changes from the weak-coupling BCS type to the superfluid density of molecular BEC, as one passes through the BCS–BEC crossover. This paper is organized as follows. In Section 2, we explain the outline of our strong-coupling theory to calculate D and l in the superfluid phase. In Section 3. we discuss the superfluid density consistent with the strongcoupling theory explained in Section 2. Throughout this paper, we take h ¼ k B ¼ 1. We also set the system volume V ¼ 1, so that the total number of particles N and the carrier density q N =V are the same. 2. Gaussian fluctuation theory below T c In superfluid Fermi gases, all the current experiments are using a broad FR to tune the strength of a pairing interaction [3–5]. In this case, we can study this system by using the BCS model, as far as we consider the interesting BCS– BEC crossover physics [6]. The BCS Hamiltonian is given by X X y H¼ np cypr cpr U cpþq=2" cypþq=2# cp0 þq=2# cp0 þq=2" : p;r
p;p0 ;q
ð1Þ Here, we consider a uniform Fermi gas with two atomic hyperfine states described by pseudo-spin r ¼"; #. cyp is the creation operator of a Fermi atom with the kinetic energy np ¼ ep l ¼ p2 =2m l, measured from the chemical potential l. The assumed pairing interaction U is associated with FR, which can be experimentally tuned by adjusting the threshold energy of FR. Nozie`res and Schmitt-Rink (NSR) have presented a strong-coupling theory to describe the BCS–BEC crossover at T c [7]. The NSR theory consists of two coupled equations, namely, the equation for T c and the equation for the number of fermions. The T c -equation is obtained from the so-called Thouless criterion in the t-matrix approximation, stating that the superfluid phase transition occurs when the particle–particle scattering matrix has a pole at x ¼ q ¼ 0. The resulting T c -equation has the same form as the ordinary BCS gap equation at T c . In the weak-coupling BCS theory, we can safely set l ¼ eF in the gap equation (where eF is the Fermi energy). However, as pointed out by NSR [7], the value of l deviates from eF , as one
approaches the strong-coupling region. This strong-coupling effect is taken into account by considering the number equation, calculated from the identity q ¼ oX=ol, where X is the thermodynamic potential involving pairing fluctuations. We note that the NSR theory has been re-formulated in the framework of the functional integral formalism [9], and it has been clarified that the NSR theory includes pairing fluctuations within the Gaussian fluctuation level. In the functional integral formalism, the T c -equation is obtained as the saddle point solution, and the number equation involves pairing fluctuations around the saddle point solution to the quadratic order. The NSR or Gaussian fluctuation theory has been extended to the superfluid phase below T c [8,10]. In the extended NSR theory, what corresponds to the T c -equation in the original NSR theory is the gap equation, given by 1¼U
X 1 bEp ; tanh 2E 2 p p
ð2Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where b ¼ 1=T , and Ep ¼ n2p þ D2 is the single-particle excitation spectrum. The equation for the number of fermions q in the extended NSR theory is given by [8,10] h i 1 @ X b imn Þ : ln det 1 þ U Nðq; ð3Þ q ¼ q0F 2b @l q;mn Here, mn is the boson Matsubara frequency, and X np bEp 0 qF 1 tanh Ep 2 p
ð4Þ
is the number of fermions in the mean-field approximation. The last term in Eq. (3) ð qFL Þ describes the fluctuation correction to the number equation. qFL is obtained by summing up the fluctuation contributions to the thermodynamic potential ð XFL Þ diagrammatically shown in Fig. 1, and calculating qFL ¼ @XFL =@l. (For more details, we refer to Ref. [10].) In the last term of Eq. (3), pair-
Fig. 1. Fluctuation contribution to the thermodynamic potential XFL . The total thermodynamic potential X is given by the sum of the mean-field part and XFL . The solid and dashed lines represent the single-particle Green’s b 0 and the pairing interaction U, respectively. The bubble function G diagrams represent Pij (ði; jÞ ¼ 1; 2) describing pairing fluctuations.
Y. Ohashi et al. / Physica C 468 (2008) 599–604
601
ing fluctuations are described by the (2 2)-matrix correlab imn Þ, having the form tion function Nðq; 1 P11 þ P22 þ 2iP12 P11 P22 b N¼ : ð5Þ 4 P11 P22 P11 þ P22 2iP12 Here, Pij are given by [10,11] 1X b 0 ðp þ q=2; ixm þ imn Þsj G b 0 ðp q=2; ixm Þ; Pij ¼ Tr½si G b p;xm ð6Þ
b 1 ðp; ixm Þ ¼ ixm np s3 þ Ds1 is the (2 2)-matrix where G 0 single-particle Green’s function (where xm is the fermion Matsubara frequency, and sj (j ¼ 1; 2; 3) are the Pauli matrices). Physically, P11 and P22 describe amplitude and phase fluctuations of the order parameter D, respectively. P12 describes coupling between amplitude and phase fluctuations [11]. The coupled Eqs. (2) and (3) involve ultraviolet divergence, originating from the contact interaction (U ) appearing in Eq. (1). In metallic superconductivity with a phonon-mediated pairing interaction, this singularity can be eliminated by introducing a physical cutoff with the order of Debye frequency. In contrast, such a physical cutoff does not exist in the case of superfluid Fermi gas with a pairing interaction associated with FR. In superfluid Fermi gases, the ultraviolet divergence can be eliminated by introducing the two-body s-wave scattering length as , which is related to the pairing interaction U as [12] 4pas U P ¼ m 1U p
1 2ep
:
ð7Þ
Using the scattering length as , we can regularize the coupled Eqs. (2) and (3) as 4pas X 1 bEp 1 1¼ tanh ; ð8Þ 2ep m p 2Ep 2 " " ## X 1 1 @ X 4pas b 0 Nðq; imn Þ þ ln det 1 qF ¼ 2b @l q;mn 2ek m k ¼ q0F þ qFL :
ð9Þ
In Eqs. (8) and (9), the ultraviolet divergence is absorbed into the scattering length as , which is determined experimentally. We solve the regularized coupled Eqs. (8) and (9) for a given value of as . We briefly note that, when the interaction is measured in term of as , the weak-coupling BCS regime corresponds to ðk F as Þ1 K 1, and the strong-coupling BEC regime is 1 characterized by ðk F as Þ J þ 1 (where k F is the Fermi 1 momentum). The region 1 K ðk F as Þ K þ 1 is referred to as the crossover region. The self-consistent solutions of Eqs. (8) and (9) are shown in Fig. 2. In panel (a), while T c in the BEC regime approaches the constant value, T c ¼ 0:218eF [9], the magnitude of the order parameter D at T ¼ 0 continues to increase with increasing the strength of the pairing interac-
Fig. 2. Self-consistent solutions of the coupled Eqs. (8) and (9) in the BCS–BEC crossover. (a) Order parameter D and (b) chemical potential l. In both panels (and also in Figs. 3 and 4), ðk F as Þ1 describes the strength of the pairing interaction measured in terms of the s-wave scattering length as (normalized by the Fermi momentum k F ), and T =eF is the temperature normalized by the Fermi energy eF . Dashed line and dotted line show results at T ¼ 0 and T c , respectively. The spurious first order phase transition seen in the BEC region is an artifact of the extended NSR theory we are using.
tion. As a result, the ratio 2DðT ¼ 0Þ=T c becomes larger than the so-called BCS universal constant ð2DðT ¼ 0Þ= T c ¼ 3:54Þ in the strong-coupling region. In panel (b), the temperature dependence of l is weak below T c , and the value becomes smaller to be negative as one passes through the crossover region. In the BEC limit ððk F as Þ1 þ1Þ, one obtains l ¼ 1=2ma2s [9]. In Fig. 2a, we find that the temperature dependence of D shows a bendover near T c in the BEC regime, which indicates a first order phase transition. However, we note that this is an artifact of the Gaussian fluctuation theory we are using here. This kind of singularity near T c has been known in the Popov theory for the Bose gas BEC [12]. In the Popov theory, to overcome this problem, one needs to include many-body renormalization effects on an interaction between bosons [13,14]. Since the present theory is known to reduce to the Popov theory for a weakly interacting molecular Bose gas in the BEC limit [15], the bendover seen in Fig. 2 has the same origin as that in the Popov theory. Thus, to recover the second order phase transition, we need to include many-body corrections to the interaction
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between molecular bosons beyond the present Gaussian fluctuation theory [16]. Although this is an important problem, in this paper, we only calculate qs within the Gaussian fluctuation level using the results shown in Fig. 2. However, within this approximation, we calculate qs in a consistent manner.
b is the self-energy correction, given by [19] Here, R 1 b RðpÞ ¼ b
b s is the (2 2)-matrix single-particle Green’s Here, G function in the supercurrent state. In the mean-field b 1 ¼ iðxm V s pz Þ approximation, it has the form G s0 np s3 þ Ds1 [17,18]. The supercurrent J s is related to the superfluid density qs as J s ¼ qs V s under the assumption that the magnitude of the superfluid velocity V s is small. Noting this fact and using (10), we find that the normal fluid density qn q qs is given by 2X @ Tr½Gs ðp; ixm Þ qn ¼ pz : ð11Þ b p;xm @Q Q!0 b s0 , we obtain In the mean-field approximation, using G the well-known expression for the BCS normal fluid density ð qFn Þ, qFn ¼
2 X 2 @f ðEp Þ p ; 3m p @Ep
ð12Þ
where f ðEp Þ is the Fermi P distribution function. Eq. (12) reduces to qFn ðT c Þ ¼ 2 p f ðnp Þ at T c , which equals the meanfield part q0F ðT c Þ of the number Eq. (9). Thus, except for the weak-coupling BCS limit (where the fluctuation contribution qFL can be ignored in Eq. (9)), the mean-field normal fluid density qFn at T c is smaller than the total carrier density q, leading to a finite value of the superfluid density even at T c ðqs ¼ q qFn ¼ qFL > 0Þ. Thus, to satisfy the required condition that qs vanishes at T c over the entire BCS–BEC crossover region, we need to include pairing fluctuations in qn so as to satisfy qn ¼ q0F þ qFL at T c . To calculate fluctuation corrections to qn , we need to b s consistent with the extended NSR theory in find out G Section 2. For this purpose, we note that the numberP equation can be written in the form q ¼ p1 þ P b ixm Þ. When Eq. (9) is written in this T p;xm Tr½s3 Gðp; form, one finds that the single-particle Green’s function in the extended NSR theory has the form b0R b ¼G b0 þ G bG b 0: G
ð13Þ
q
4pas m s b det½1 4pa ð NðqÞ þ m
("
P k
1 Þ 2ek
# X 1 4pas 1 ½N11 ðqÞ þ sþ G0 ðp þ qÞs 2ek m k " # X 1 4pas þ 1 ½N22 ðqÞ þ s G0 ðp þ qÞsþ 2ek m k ) 4pas N12 ðqÞsþ G0 ðp þ qÞsþ ; þ2 ð14Þ m
3. Superfluid density We consider the supercurrent state with the superfluid velocity V s in the z-direction. The supercurrent state is described by the order parameter DðzÞ ¼ P DeiQz (where Q ¼ 2mV s ). The supercurrent density J s ¼ p;r ðpz =mÞ hcpr cpr i is given by 1 X pz b s ðp; ixm Þ: Tr½ G J s ¼ qV s þ ð10Þ b p;xm m
X
where we have simply written p ¼ ðp; ixm Þ and q ¼ ðq; imn Þ. Nij (i; j ¼ 1; 2) is the ði; jÞ-component of the matrix correlab in Eq. (5). In the supercurrent state, the tion function N b s corresponding to Eq. single-particle Green’s function G (13) is given by b s0 þ G b s0 R bsG b s0 : bs ¼ G G
ð15Þ
b 0 is replaced by b s is given by Eq. (14), where G Here, R b G s0 . Substituting Eq. (15) into Eq. (11), we find that the normal fluid density can be written as qn ¼ qFn þ qBn , where qFn is the BCS normal fluid density in Eq. (12) and qBn is the fluctuation correction arising from the last term in Eq. (15), given by [19,20] " " ##! X 1 2m @ 2 X 4pas b B qn ¼ ln det 1 NðqÞ þ m b @Q2 q 2ek k Q!0
ð16Þ
In the normal phase limit D ! 0, it can be shown that Eq. (16) reduces to qFL ðT c Þ, namely, qs ðT c Þ ¼ q ðqFn ðT c Þ þ qBn ðT c ÞÞ ¼ 0 is satisfied in the BCS–BEC crossover [19], as expected. In the weak-coupling BCS regime, since pairing fluctuations are weak, we obtain qs ’ q qFn . Since qFn is dominated by single-particle excitations (see Eq. (12)), the decrease of qs at finite temperatures is found to originate from thermal dissociation of Cooper pairs in the weak-coupling BCS regime. As one approaches the strong-coupling BEC regime, qFn becomes less dominant due to the fact that the magnitude of single-particle excitation gap becomes large. In contrast, the fluctuation part qBn becomes dominant, so that we find qs ’ q qBn . In the BEC limit, qBn reduces to [19,20] qBn ¼
B 2 X 2 @nB ðEq Þ q : 3M q @EBq
ð17Þ
Here, M ¼ 2m is a molecular mass, and nB ðEBq Þ is the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bose distribution function. EBq ¼ eBq ½eBq þ 2U B nc is just the expression for the Bogoliubov phonon spectrum in a molecular BEC in the Popov approximation [12], where eBq q2 =2M is the molecular kinetic energy, U B 4paB =M is an effective interaction between bosons with
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the molecular scattering length aB ¼ 2as , and nc m2 D2 as =8p describes the molecular condensate fraction [19,20]. The Bogoliubov phonon is a Goldstone mode associated with the spontaneous breakdown of the continuous gauge symmetry in the superfluid phase. Indeed, one finds a B linearpdispersion ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eq jq!0 ¼ v/ q with the sound velocity v/ ¼ U B nc =M . Eq. (17) is just the Landau expression for the normal fluid density in a Bose superfluid [12]. Namely, qn in the BEC regime is dominated by Bogoliubov collective excitations. The above result in the BEC regime is consistent with the extended NSR theory discussed in Section 2. Indeed, in this regime, Eq. (8) gives l ¼ 1=2ma2s , and the number Eq. (9) reduces to [20] " # bEBq q 1 X eBq þ U B nc ¼ nc þ 1 : ð18Þ coth 2 2 q 2 EBq This is just the number equation for a q=2-molecular Bose gas in the Popov approximation [12]. Fig. 3 shows the calculated superfluid density qs in the BCS–BEC crossover. We find that the weak-coupling BCS result qs ¼ q qFn continuously changes into the superfluid density of a molecular Bose superfluid qs ¼ q qBn , as one increases the strength of the pairing interaction. We note that although qn in the BCS regime is dominated by single-particle excitations, the Goldstone (collective) mode with a linear dispersion always exists irrespective of the strength of the pairing interaction. Fig. 4 shows the velocity of the Goldstone mode in the BCS–BEC crossover. In the BCS regime, the Goldstone mode is the Anderson–Bogoliubov mode with the mode pffiffiffi velocity v/ ¼ vF = 3 at T ¼ 0 [10,11] (where vF is the Fermi velocity). In the BCS regime, comparing the collective mode energy (x ¼ v/ q) with the single-particle excitation gap (D) at T ¼ 0, we find that the momentum region where the collective excitations are important (x K D) is very narrow, as 0 6 q K k F ðD=eF Þ k F . As a result, the contri-
Fig. 3. Superfluid density qs in the BCS–BEC crossover. In this calculation, D and l shown in Fig. 2 are used. The solid circles show the weak-coupling BCS superfluid density qs ¼ q qFn . The solid triangles show the strong-coupling BEC result qs ¼ q qBn , where Eq. (17) is used. The bendover near T c in the BEC regime is due to the spurious first order phase transition seen in Fig. 2.
603
Fig. 4. Velocity v/ of the Goldstone mode in the BCS–BEC crossover. This result is obtained within the generalized random phase approximation (GRPA), which is a consistent approximation with the extended NSR theory [21]. In GRPA, the mode velocity P is obtained by solving b imn ! v/ q þ idÞÞ þ Re½det½1 ð4pas =mÞð Nðq; For more k 1=2ek ¼ 0. details, see Refs. [10,11].
bution of the Goldstone mode to qn is weak compared with single-particle excitations in the BCS regime. We also note that v/ becomes small in the BEC regime, as shown in Fig. 4. At the same time, the single-particle excitation gap becomes large reflecting the increase of the binding energy. As a result, the momentum region where collective excitations are crucial becomes wide, as one approaches the BEC regime, so that the origin of qn changes from single-particle excitations to collective excitations in the BCS–BEC crossover, as discussed previously. 4. Summary To summarize, we have discussed the superfluid density qs in the BCS–BEC crossover regime of a superfluid Fermi gas. We have calculated D, l, and qs in a consistent manner within the Gaussian fluctuation approximation. This treatment is crucial for the consistent treatment of qs over the entire BCS–BEC crossover. We showed that the origin of the normal fluid density qn ¼ q qs continuously changes from single-particle excitations accompanied by dissociation of Cooper pairs to collective excitations, as one increases the strength of the pairing interaction. In the strong-coupling BEC regime, we showed that the present Gaussian fluctuation theory leads to a spurious first order phase transition. To recover the expected second order phase transition and to describe the correct temperature dependence of the superfluid density near T c in the BEC regime, we need to include higher-order fluctuation effects beyond the Gaussian fluctuation level. Although this problem still remains to be solved, since the two-fluid theory is based on the superfluid density, our results would be still useful in constructing the two-fluid hydrodynamics in the BCS–BEC crossover region at finite temperatures. Acknowledgements Y.O. was supported by Grant-in-Aid for Scientific research from MEXT (18043005, and 19540420),
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JSPS-CTC program and CREST(JST). E.T. and A.G. were supported by NSERC of Canada. References [1] L. Landau, J. Phys. (Moscow) 5 (1941) 71. [2] See, for example A. Griffin, Excitations in a Bose-condensed Liquid, Cambridge University Press, New York, 1993. [3] C.A. Regal, M. Greiner, D.S. Jin, Phys. Rev. Lett. 92 (2004) 040403. [4] M. Bartenstein et al., Phys. Rev. Lett. 92 (2004) 120401. [5] M.W. Zwierlein et al., Phys. Rev. Lett. 92 (2004) 120403. [6] For a review, see Q. Chen, J. Stajic, S. Tan, K. Levin, Phys. Rep. 412 (2005) 1. [7] P. Nozie`res, Schmitt-Rink, J. Low Temp. Phys. 59 (1985) 195. [8] J.R. Engelbrecht, M. Randeria, C.A.R. Sa´ de Melo, Phys. Rev. B 55 (1997) 15153. [9] M. Randeria, in: A. Griffin, D.W. Snoke, S. Stringari (Eds.), Bose– Einstein Condensation, Cambridge University Press, New York, 1995, p. 355.
[10] Y. Ohashi, A. Griffin, Phys. Rev. A 67 (2003) 063612. [11] Y. Ohashi, S. Takada, J. Phys. Soc. Jpn. 66 (1997) 2437. [12] C.J. Pethick, H. Smith, Bose–Einstein Condensation in Dilute Gases, Cambridge University Press, UK, 2002. [13] H. Shi, A. Griffin, Phys. Rep. 304 (1998) 1. [14] M. Bijlsma, H. Stoof, Phys. Rev. A 54 (1996) 5085. [15] P. Pieri, G. Strinati, Phys. Rev. B 71 (2005) 094520. [16] Inclusion of many-body renormalization effects on the molecular interaction has been done in the BEC regime Y. Ohashi, J. Phys. Soc. Jpn. 74 (2005) 2659. [17] K. Maki, in: R.D. Parks (Ed.), Superconductivity, vol. 2, Marcel Dekker, New York, 1969, p. 1035. [18] Although l is also affected by the supercurrent as l ! l Q2 =8m, it does not affect qs because it is the second order OðV 2s Þ. [19] N. Fukushima, Y. Ohashi, E. Taylor, A. Griffin, Phys. Rev. A 75 (2007) 033609. [20] E. Taylor, A. Griffin, N. Fukushima, Y. Ohashi, Phys. Rev. A 74 (2006) 063626. [21] Y. Ohashi, Phys. Rev. A 70 (2005) 063613.