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Transport coefficients of the superfluid Fermi gas in p-wave state at low temperatures
Physica C 483 (2012) 109–112
Contents lists available at SciVerse ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Transport coefficients of the superfluid Fermi gas in p-wave state at low temperatures S. Nasirimoghadam a, F. Nabipoor a, M. Khademi-Dehkordi b,⇑, M.A. Shahzamanian a a b
Department of Physics, Faculty of Sciences, University of Isfahan, Isfahan 81744, Iran Department of Plasma Physics, Science and Research Branch, Islamic Azad University, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 1 May 2011 Received in revised form 1 May 2012 Accepted 7 May 2012 Available online 31 August 2012 Keywords: D. Shear viscosity D. Diffusive thermal conductivity A. Fermi gas D. p-Wave state
a b s t r a c t In this paper, we obtain the shear viscosity and diffusive thermal conductivity of the superfluid p-wave Fermi gas with weak interaction by using the quasi-particle relaxation rate s1 P , and the Boltzmann equa4 tion approach at low temperatures. We show that s1 P is proportional to T . The shear viscosity components, gxx, gyy, gxy are proportional to T2, whereas gxz, gyz and gzz are proportional to T4 and T6, respectively. The components of the diffusive thermal conductivity Kxx and Kyy are proportional to T1, whereas Kzz is proportional to T3. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction The field of quantum degenerate atomic gases has experienced great progress in recent decades. The development of several techniques to trap and cool neutral atoms to ultra low temperatures led to the first experimental observation of the phenomenon of Bose– Einstein condensation (BEC) in a dilute atomic gas [1]. While the effects of particle interactions are generally small, they can be dramatic if the effective interaction is attractive. If the interaction is attractive, the atoms are paired in the same way in which electrons are in superconducting metals or atoms in superfluid 3He [2]; the gas is then predicted to undergo a transition to a superfluid state. A gas of atoms, such as 6Li or 40K, which possesses one unpaired electron and an integral nuclear spin J, while two such atoms exchange their electrons and the atoms as a whole behave as fermions that would be expected under suitable conditions to form Cooper pairs and give rise to superfluidity [3]. Dilute atomic gases cooled to quantum degeneracy provide ideal systems for testing many-body theories. In particular, Feshbach resonances in atomic Fermi gases allow experimental control over the strength of two body interactions, giving access to Bose–Einstein condensation and to the Bardeen–Cooper–Schrieffer superfluid (BEC–BCS) crossover regime [4]. In the ultra-cold Fermi gases, such as 6Li and 40K, the tuning of p-wave interactions was initially explored via p-wave Feshbach resonances. Transport properties have played an important role in characterizing Fermi gases in BEC–BCS crossover, with
⇑ Corresponding author. E-mail address: [email protected] (M. Khademi-Dehkordi). 0921-4534/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physc.2012.05.020
the observation of hydrodynamic flow indicating nearly perfect fluidity, while the measurement of shear viscosity and thermal conductivity coefficients probe the equation of state and superfluidity. Transport properties of different superfluids have been studied in various regimes, such as BEC–BCS crossover or Feshbach resonance [5]. The transport coefficients describe the response of the system to inhomogeneous static perturbation which may be a temperature or a velocity gradient. The shear viscosity of superfluid 6Li has been studied by Shahzamanian and Yavari [6]. They showed that g is temperature independent at extremely low temperatures, T ? 0 and is proportional to the inverse of scattering length, a. The viscosity of the A-phase, in zero magnetic field has been calculated exactly for temperatures close to Tc, by Bhattacharyya et al. [7] and Pethick et al. [8] and the shear viscosity in the presence of a magnetic field has been considered by Shahzamanian [9]. Also in [10,11] the Boltzmann equation was solved in the low temperature limit for the thermal conductivity of the A-phase of liquid 3He. It was found that the diffusive thermal conductivity varies with temperature as T1. In addition, Shahzamanian studied the thermal conductivity for the A-phase of superfluid 3He by using approximate collision integrals at low temperatures [12]. In this paper, we use the Boltzmann equation approach and the relaxation time approximation which have been extended to superfluid 3He by Einzel and Wölfle [13]. Then, we obtain the transition probabilities in p-wave superfluid Fermi gas with weak interaction in terms of the generalization of the separable potential used by Nozierese and Schmitt-Rink [14]. Finally, we derive the components of the shear viscosity and the diffusive thermal conductivity, in the low temperature limit.
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potential (see Eq. (6)) they obtain an analytical solution of the gap equation for superfluid Fermi gases in the approach of low energy expansion. In the p-wave state this potential is
2. Transition probabilities The interaction term in the Hamiltonian of the system is
H¼
1 X h3; 4jT j1; 2iay4 ay3 a1 a2 ; 4 1;2;3;4
2
ð1Þ
where i = 1,2,3,4 stands for both momentum and spin variables. By using the Bogoliubov transformation, the creation a~yp and annihilation a~p operators in the normal state may be replaced by quasi-particle creation and annihilation operators a~yp and a~p in the superfluid. The Bogoliubov transformation is written as
a~p;r ¼ u~p;rr0 a~p;r0 v~p;rr0 ay~p;r0 ;
ð2Þ
a~yp;r ¼ v ~p;rr0 a~p;r0 þ u~p;rr0 ay~p;r0 ; where
e~p 1=2 1 1þ u~p;rr0 ¼ drr0 ; 2 E~p e~p 1=2 1 1 v~p;rr0 ¼ drr0 ; 2 E~p
ð3Þ
E~P ¼ ðe~2P þ jD~P j2 Þ1=2 sgne~p ;
W 13 ð#"Þ ¼ 2pjhp_ 3 "; p_ 4 "; p_ 2 #jHjp_ 1 #ij2 ;
ð4Þ
with replacement of Eq. (1) into (4) and using Eqs. (2) and (3) the transition probabilities are obtained. These transition probabilities have been obtained in [12,14] and are as following:
W 13 ð"#Þ ffi 0;
W 13 ð""Þ ffi 0;
W 31 ð"#Þ ffi 0;
W 31 ð""Þ ffi 0;
ð5Þ
2
W 22 ð"#Þ ¼ jV 1 ðk1 ; k3 Þj ;
2
02
:
ð7Þ
At low temperatures, we have sin hp hp [12], and in fact, Bogoliubove quasi-particle momentum vectors are located around the nodes of the energy gap. As we mentioned above, to estimate the value of the cut-off momentum we equated the cut-off kinetic energy with the potential energy, then by using this approximation the generalized Nozieres and Schmitt-Rink potential is obtained
V 1 ðk1 ; k3 Þ ¼ V 1 ðk2 ; k3 Þ ffi
v1 4
:
ð8Þ
With replacement of Eq. (8) into (5) the transition probabilities are obtained. At low temperatures, only the binary processes are dominant, while other transition probabilities have nearly a zero value [12]. In other words, only the transition probability W 22 ð"#Þ ffi v 21 =16 will contribute to the transport coefficients in this regime.
By using the transport equation to study the rate of decay of the quasi-particles, we may use the relaxation time approximation as
@ m~p m~p ¼ ; @t sp
ð9Þ
where sp is the quasi-particle lifetime. By using the kinetic equation, the rate of decay of the quasi-particles can be calculated [18]
s1 p1 ¼
ðm Þ3 ð2pÞ6
Z
W 22 sin hp dh d/ d/2 de2 de3 de4 dðe1 þ e2 e3 cosðh=2Þ
e4 Þm02 ð1 m03 Þð1 m04 Þ;
ð10Þ
where m0 is the equilibrium Fermi distribution function. It is noted that the transition probabilities are independent of the angles between the momenta in the model of our calculations, because we use the generalized Nozieres and Schmitt-Rink potential. R1 If we replace d(e) by ð1=2pÞ 1 expðiezÞ dz and use the following relation
ð1=2pÞ
Z
1
expðiezÞðpiT= sinhðpzTÞÞ3 dz
¼ ð1=2Þðp2 T 2 þ e2 Þ=ð1 þ ee=T Þ;
The above equations show that at low temperatures only the binary processes are dominant, while other transition probabilities have nearly a zero value. In other words, only the transition probability W22 will contribute to the transport coefficients in this regime. The generalized Nozieres and Schmitt-Rink potential is written as [17] 0
2
ðk0 þ k Þðk0 þ k Þ
1
W 22 ð""Þ ¼ jV 1 ðk1 ; k3 Þ V 1 ðk2 ; k3 Þj2 :
0
2
0
3. Relaxation rate
for the p-wave state v ~p ¼ v~p , u~p ¼ u~p . e~p and D~p are the normal state quasi-particle energy respect to the Fermi energy and the magnitude of gap, respectively [15]. D~p is equal to DðTÞ sin hp , where D(T) is the maximum gap and hp is the angle between the quasi-particle momentum and gap axis which is supposed to be in the direction of the z axis. In a normal Fermi liquid at low temperatures the only important collision process is the scattering of pairs of quasi-particles, but in a superfluid the quasi-particle number is not conserved, so one also has to take into account (1) the decay processes in which a single quasi-particle decays into three and (2) the inverse processes as well [16]. The transition probability, for example, may be written as
Vðk; k Þ ¼ v ‘ w‘ ðkÞw‘ ðk Þ;
k0 kk
0
V 1 ðk; k Þ ¼ v 1
w‘ ðkÞ ¼
ðkk0 Þ‘ ; 2 ‘þ1 2 1 þ kk0
ð6Þ
where, k0 is the cut-off momentum. To estimate the value of the cut-off momentum we use the following approximation. Equating the cut-off kinetic energy with the potential energy one may obtain 2 2 2 k0 kF þ kc , where kc ðmv 1 =2Þ1=2 . Ho and Diener [17] studied the pairing of Fermi gases near the scattering resonance of the ‘ – 0 partial wave. By using the generalized Nozieres and Schmitt-Rink
ð11Þ
the relaxation rate is equal to:
s1 p ¼
ðm Þ3
p2 T 2 þ e21 2 2 W p hm : 22 1 þ ee1 =T ð2pÞ6
ð12Þ
At low temperatures, we have e1 ffi T and hm = pT/D(0) [12], then s1 p is proportional to T4. We calculated the transition probabilities, in Section 2 and showed that the transition probabilities are constant and W 22 ffi v 21 =16. 4. Shear viscosity Once we know the collision integral, we can study the usual transport properties of the system, such as viscosity and thermal conductivity. In these cases, we apply to the system a static perturbation, which may be a temperature or a velocity gradient. The applied gradient sets up a flow of heat or momentum, which is limited only by the collisions between quasi-particles. The induced
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flow is proportional to the applied gradient, the coefficient of proportionality being, respectively, the viscosity or the thermal conductivity of the system. The coefficient of the shear viscosity describes the response of the momentum current density to a velocity field [19] and for a uniaxial state may be written as:
Y
X ~ ~p E~p Þ Pj dm0 ¼ ig q V n ¼ ðr ~ ijlk l k p i
ij
By using the approximation hm ffi pT/D(0) and the generalized Nozi5 4 eres and Schmitt-Rink potential, we obtain kþ 2 ¼ 1 ðhm =2 Þ. Finally, by substituting Eqs. (12) and (19) in (18), after straightforward calculations we obtain the components of shear viscosity as
gxz ¼ gyz ¼ 598:76
~ ~p E~p ¼ ðe~p =E~p Þð~ where r p=m Þ is the quasi-particle velocity. Assuming the velocity field along the ^ x axis and its gradient ^, the kinetic equation for the viscous current and the distrialong y bution function of the Bogoliubov quasi-particles may be written as the sum of out scattering and in scattering contributions. This collision integral is complicated enough to defy any general solution. The best approximation that able to describe the essential physical properties of the system under consideration is the relaxation time approximation. The collision integral is written by Einzel and Wolfle [13] as follows:
ep Py qy i Px V nx m0p ¼ dm0p þ Iin p; Ep m s
ð14Þ
where the right-hand side of Eq. (14) is the collision integral. The main features of this collision integral, such as its anisotropy, its energy, and its temperature dependencies, are contained in the out scattering term. The in scattering term is written as [19] 1 X l XX X X 1 ~~k0 ; Iin krlr Prlmr ð~ k; ~ k0 Þdm p ¼ is 4p
r r¼1
ð15Þ
~ k0
~~k ¼ dm~k m~0 d~ where dm E~k and the sum on r = s, a extends over spink symmetric and anti-symmetric components, and the sum on r = t(1)l = ±1 runs over an even (t = +1) and an odd (t = 1) function in e. For the spin-symmetric case the projectors, P, are defined by
2 31 X 1r 2 r 0 1 0 1 0 ~ ~ ^ ~ Plm ðk; k Þ ¼ Q l ðeÞs~k m~k Y lm ðkÞ4 ð3E~k2 Þ s~k m~k jY lm ðk2 Þj 5 ^0 ÞPr ; Q rl ðe0 Þs~1 Y lm ðk l k0
2
1h e e i ðlrÞ=2 1 þ rð1Þl 1 ; E 2 E E
ðm Þ4 W
22
T6
p5F ðm Þ4 W
1 22
:
T2
;
ð21Þ
ð22Þ
5. Diffusive thermal conductivity The coefficient of thermal conductivity characterizes the response of the heat current to a temperature gradient
~ ~ j T; J i ¼ K ij r
ð23Þ
where Ji is the diffusive heat current and Kij is the diffusive thermal conductivity, which in general is a second rank tensor. It is noted that in addition to heat transfer by a random diffusive process of the thermal excitation described by Eq. (23), there is a convective c contribution to heat current in a superfluid, ji ¼ Sv n , even in the absence of mass flow, due to the possibility of normal-superfluid counter-flow. In the vicinity of Tc the convective transport process is much more effective than the diffusive one and it is difficult to measure K, while at lower temperatures, the diffusive processes takes over and the convective contribution is negligible [20]. In other words, the measurement of thermal conductivity components at low temperatures is more meaningful than the measurement of the thermal conductivity components in the vicinity of Tc. The diffusive heat current can be written in terms of the quasiparticle distribution functions as
X ~ p Ep dm0 : Ep r p
ð24Þ
ð17Þ
~ i p:~ q ep 0 Ep m dT ¼ dm0p þ Iinp : sp m Ep p T
EE p5F XDD ^i p ^j p ^l p ^k m0p;r =s1 p p;r m r EEDD EE1 0 DD þ 2 ^Þ ^l p ^Þ ^i p ^j m0p;r Y 2m ðp ^k m0p;r Y 2m ðp p p k2;r X @ A; DD EE þ 1 kþ2;r m¼2 ^ 2 m0p;r s1 p;r jY 2m ðpÞ j ð18Þ
where, r is the spin of quasi-particles, hhAii ð2=ð2pÞ3 Þ R R þ1 dXp 1 dep Aðh; uÞ and m0p;r is @ m0p;r =@ ep : Also we define
3 4 2 hW 22 sin ðh=2Þ sin /i: hW 22 i
T 4c
p5F
ð20Þ
;
In the presence of the stationary diffusive heat current the distribution function is close to a local equilibrium distribution function and the Boltzmann equation for this case may be written as
kþ2;r ¼ 1
gzz ¼ 250:56
T4
ð16Þ
and Ylm(p) in Eq. (16) is the spherical harmonic function. The left hand side of the collision integral in the Eq. (14) is in terms of the Fourier-transform velocity gradient which is a second rank tensor and is proportional to the spherical harmonic function l = 2, hence, in calculating dm0p from the collision integral only terms with l = 2 remain. This case for thermal conductivity is l = 1. After, using the RTA approximation [19,20], we have
gijlk ¼
22
p
2
where
Q rl ðeÞ ¼
5 2
gyy ¼ gxx ¼ gxy ¼ 264:97
~j ¼
r
~ k2
ðm Þ4 W
ð13Þ
~ p
l¼0 m¼l
T 2c
p5F
ð19Þ
ð25Þ
Similar to the case of the shear viscosity, by solving the above collision integral in terms of dm0p [19,20] and comparing Eqs. (23) and (24) we have [12] EEDD EE19 8 0DD = 1 ^Þe2p m0p ^Þe2p m0p ^i Y 1m ðp ^j Y 1m ðp EE k X p p 3n
where k1 ¼ 1 þ ð2=hW 22 iÞhW 22 cos hi. By using the approximations for the low temperature limit, we may write
k1 ¼ 3
h2m : 2
ð27Þ
By substituting the value of k1 and forward calculations, we get
K xx ¼ 280:96
n 1 ; ðm Þ4 W 22 T
s1 p;r into Eq. (26), after straightð28Þ
112
K yy ¼ 280:96
K zz ¼ 554:08
S. Nasirimoghadam et al. / Physica C 483 (2012) 109–112
n 1 ; ðm Þ4 W 22 T n
T 2c
ðm Þ4 W 22 T 3
ð29Þ
:
ð30Þ
The non-diagonal components of the thermal conductivity are zero at low temperatures. 6. Conclusions The best known agreement between theoretical and experimental results in transport coefficients of triplet pairing is superfluid 3He, in which the condensation consists of spin triplet atomic Cooper pairs [21]. On the other hand, nearly all the superconductors known to date are spin singlet paired, except the layered perovskite Sr2RuO4, UPt3, UGe2 and other related materials (see references in Machenzie [22]) which are supposed to be spin triplet paired superconductors. Among these spin triplet paired materials, superfluid 3He-A1, superfluid-A and Sr2RuO4 are in axial symmetry states, like the case which we consider here: superfluid Fermi gas in the p-wave state. Here we mention the fact that in the superconductor Sr2RuO4 the superfluid regions are layers and behave as a two-dimensional superconductor. The shear viscosity of the A1-phase of superfluid 3He was measured by Roobol et al. [23]. At low temperatures their results indicate g / (T)2, which has been obtained theoretically in reference [24]. The ultrasonic attenuation of longitudinal and transverse inplane polarized modes were measured in Sr2RuO4 by Lupien et al. [25] (for more details see Ref. [22]). The ultrasound data show power-law behavior in the viscosity of the superconductor Sr2RuO4 at low temperatures, which indicates the persistence of nodes in the gap energy. This problem is also under our consideration and will be published on elsewhere. To the best our knowledge there are no experimental result on the shear viscosity or diffusive thermal conductivity of ultra-cold atoms, which we can compare to our theoretical results. In this paper, by using the Boltzmann equation approach, the relaxation time approximation and the fact that in this system only binary processes dominate at low temperatures [14], we calculate the components of the shear viscosity and the diffusive thermal conductivity. Our results in the p-wave state show that the relaxation rate of quasi-particles in the superfluid p-wave state is proportional to T4, whereas in the normal state is proportional to T2. The differences come from the fact that the quasi-particles at low temperatures are located at the nodes of the anisotropic energy gap. The components of the shear viscosity are temperature dependent; the shear viscosity components, gxx, gyy and gxy are proportional to T2, gxz and gyz are proportional to T4 and finally gzz is proportional to T6. Nasirimoghadam et al. [14] used the Sykes and Brooker procedure with weak interaction and the generalized Nozieres and Schmitt-Rink potential to obtain the components of shear viscosity in the p-wave state. Their results are in good agreement with our results (there are minor differences in the numerical
factors). Shahzamanian and Afzali by using the procedure of Pfitzner showed these temperature dependencies in superfluid 3He-A1. The shear viscosity for 6Li, with isotropic gap, which is in an s-wave superfluid state, was obtained by Shahzamanian and Yavaris [6]. They found that g is temperature independent at extremely low temperatures, T ? 0, and is proportional to the inverse square of the scattering length. Finally, we mention that the superfluid Fermi gases under our considerations are similar to A1-phase of superfluid 3He in which only the ( " " ) state or the Cooper pairs are present, since when the ( " " ) state is at resonance the other pairs are not. The contribution to the viscosity coefficients from the non-superfluid components in A1-3He is negligible compared to the superfluid-one [16]. Here for brevity we do not write them in the problem of superfluid gases. The components of the diffusive thermal conductivity, Kxx, Kyy, and Kzz are proportional to T1 and T3, respectively. Wölfle and Einzel [20], Afzali and Ebrahimian [26] used the relaxation rate approximation and the Sykes and Brooker procedure to obtain the diffusive thermal conductivity of superfluid 3He-A. They showed that the temperature dependency is proportional to T1 . Shahzamanian has calculated the components of the diffusive thermal conductivity tensor of superfluid 3He-A by using approximate collision integrals at low temperatures [12]. He showed that the temperature dependencies of the thermal coefficients Kxx, Kyy and Kzz are proportional to T2 and T4 . References [1] M. Anderson et al., Science 298 (1999) 198; C. Bradley et al., Phys. Rev. Lett. 75 (1995) 1687; K. Davis et al., Phys. Rev. Lett. 75 (1995) 3969. [2] C.J. Pethick, H. Smith, Bose - Einstein Condensation in Dilute Gases, Combridge University Press, Combridge, 2002. [3] J.H. Freed, J. Chem. Phys. 72 (1980) 1414. [4] R. Combescot, J. Low Temp. Phys. 145 (2006) 267. [5] H. Guo, D. Vulin, C.C. Chein, K. Levin, Phys. Rev. Lett. 107 (2011) 020403. [6] M.A. Shahzamanian, H. Yavari, Physica B 321 (2002) 385. [7] P. Bhattacharyya, C.J. Pethick, H. Smith, Phys. Rev. B 15 (1977) 3367. [8] C.J. Pethick, H. Smith, P. Bhattacharyya, J. Low Temp. Phys. 21 (1975) 589. [9] M.A. Shahzamanian, J. Low Temp. Phys. 21 (1975) 589. [10] C.J. Pethich, H. Smith, P. Bhattacharyya, Phys. Rev. B 15 (1977) 3384. [11] D. Einzel, J. Low Temp. Phys. 54 (1984) 427. [12] M.A. Shahzamanian, J. Phys.: Condens. Matter 1 (1989) 1965. [13] D. Einzel, P. Wölfle, J. Low Temp. Phys. 32 (1978) 19. [14] S. Nasirimoghadam, M.A. Shahzamanian, R. Aliabadi, I.J.P.R. 11 (2011) 227. [15] S. Takagi, J. Low Temp. Phys. 18 (1975) 309. [16] M.A. Shahzamanian, R. Afzali, Ann. Phys. 309 (2004) 281. [17] T.L. Ho, R.B. Diener, Phys. Rev. Lett. 94 (2005) 090402. [18] D. Pines, P. Nozieres, The Theory of Quantum Liquids, Benjamin, Newyork, 1966. [19] D. Vollhardt, P. Wölfle, The Superfluid Phases of Helium 3, Taylor and Francis, London, 1990. [20] P. Wölfle, D. Einzel, J. Low Temp. Phys. 32 (1978) 39. [21] A.J. Leggett, Rev. Mod. Phys. 47 (1975) 331. [22] A.P. Machenzie, Y. Maeno, Rev. Mod. Phys. 75 (2003) 657. [23] L.P. Roobol, P. Remeijer, S.C. Steel, R. Jochemsen, V.S. Shumeiko, G. Frossati, Phys. Rev. Lett. 79 (1997) 685. [24] M.A. Shahzamanian, R. Afzali, Physica B 348 (2003) 108. [25] C. Lupien, W.A. Macfarlane, C. Proust, L. Taillefer, Z.Q. Mao, Y. Maeno, Phys. Rev. Lett. 86 (2001) 5986. [26] R. Afzali, N. Ebrahimian, J. Phys.: Condens. Matter 17 (2005) 4441.
Contents lists available at SciVerse ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Transport coefficients of the superfluid Fermi gas in p-wave state at low temperatures S. Nasirimoghadam a, F. Nabipoor a, M. Khademi-Dehkordi b,⇑, M.A. Shahzamanian a a b
Department of Physics, Faculty of Sciences, University of Isfahan, Isfahan 81744, Iran Department of Plasma Physics, Science and Research Branch, Islamic Azad University, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 1 May 2011 Received in revised form 1 May 2012 Accepted 7 May 2012 Available online 31 August 2012 Keywords: D. Shear viscosity D. Diffusive thermal conductivity A. Fermi gas D. p-Wave state
a b s t r a c t In this paper, we obtain the shear viscosity and diffusive thermal conductivity of the superfluid p-wave Fermi gas with weak interaction by using the quasi-particle relaxation rate s1 P , and the Boltzmann equa4 tion approach at low temperatures. We show that s1 P is proportional to T . The shear viscosity components, gxx, gyy, gxy are proportional to T2, whereas gxz, gyz and gzz are proportional to T4 and T6, respectively. The components of the diffusive thermal conductivity Kxx and Kyy are proportional to T1, whereas Kzz is proportional to T3. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction The field of quantum degenerate atomic gases has experienced great progress in recent decades. The development of several techniques to trap and cool neutral atoms to ultra low temperatures led to the first experimental observation of the phenomenon of Bose– Einstein condensation (BEC) in a dilute atomic gas [1]. While the effects of particle interactions are generally small, they can be dramatic if the effective interaction is attractive. If the interaction is attractive, the atoms are paired in the same way in which electrons are in superconducting metals or atoms in superfluid 3He [2]; the gas is then predicted to undergo a transition to a superfluid state. A gas of atoms, such as 6Li or 40K, which possesses one unpaired electron and an integral nuclear spin J, while two such atoms exchange their electrons and the atoms as a whole behave as fermions that would be expected under suitable conditions to form Cooper pairs and give rise to superfluidity [3]. Dilute atomic gases cooled to quantum degeneracy provide ideal systems for testing many-body theories. In particular, Feshbach resonances in atomic Fermi gases allow experimental control over the strength of two body interactions, giving access to Bose–Einstein condensation and to the Bardeen–Cooper–Schrieffer superfluid (BEC–BCS) crossover regime [4]. In the ultra-cold Fermi gases, such as 6Li and 40K, the tuning of p-wave interactions was initially explored via p-wave Feshbach resonances. Transport properties have played an important role in characterizing Fermi gases in BEC–BCS crossover, with
⇑ Corresponding author. E-mail address: [email protected] (M. Khademi-Dehkordi). 0921-4534/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physc.2012.05.020
the observation of hydrodynamic flow indicating nearly perfect fluidity, while the measurement of shear viscosity and thermal conductivity coefficients probe the equation of state and superfluidity. Transport properties of different superfluids have been studied in various regimes, such as BEC–BCS crossover or Feshbach resonance [5]. The transport coefficients describe the response of the system to inhomogeneous static perturbation which may be a temperature or a velocity gradient. The shear viscosity of superfluid 6Li has been studied by Shahzamanian and Yavari [6]. They showed that g is temperature independent at extremely low temperatures, T ? 0 and is proportional to the inverse of scattering length, a. The viscosity of the A-phase, in zero magnetic field has been calculated exactly for temperatures close to Tc, by Bhattacharyya et al. [7] and Pethick et al. [8] and the shear viscosity in the presence of a magnetic field has been considered by Shahzamanian [9]. Also in [10,11] the Boltzmann equation was solved in the low temperature limit for the thermal conductivity of the A-phase of liquid 3He. It was found that the diffusive thermal conductivity varies with temperature as T1. In addition, Shahzamanian studied the thermal conductivity for the A-phase of superfluid 3He by using approximate collision integrals at low temperatures [12]. In this paper, we use the Boltzmann equation approach and the relaxation time approximation which have been extended to superfluid 3He by Einzel and Wölfle [13]. Then, we obtain the transition probabilities in p-wave superfluid Fermi gas with weak interaction in terms of the generalization of the separable potential used by Nozierese and Schmitt-Rink [14]. Finally, we derive the components of the shear viscosity and the diffusive thermal conductivity, in the low temperature limit.
110
S. Nasirimoghadam et al. / Physica C 483 (2012) 109–112
potential (see Eq. (6)) they obtain an analytical solution of the gap equation for superfluid Fermi gases in the approach of low energy expansion. In the p-wave state this potential is
2. Transition probabilities The interaction term in the Hamiltonian of the system is
H¼
1 X h3; 4jT j1; 2iay4 ay3 a1 a2 ; 4 1;2;3;4
2
ð1Þ
where i = 1,2,3,4 stands for both momentum and spin variables. By using the Bogoliubov transformation, the creation a~yp and annihilation a~p operators in the normal state may be replaced by quasi-particle creation and annihilation operators a~yp and a~p in the superfluid. The Bogoliubov transformation is written as
a~p;r ¼ u~p;rr0 a~p;r0 v~p;rr0 ay~p;r0 ;
ð2Þ
a~yp;r ¼ v ~p;rr0 a~p;r0 þ u~p;rr0 ay~p;r0 ; where
e~p 1=2 1 1þ u~p;rr0 ¼ drr0 ; 2 E~p e~p 1=2 1 1 v~p;rr0 ¼ drr0 ; 2 E~p
ð3Þ
E~P ¼ ðe~2P þ jD~P j2 Þ1=2 sgne~p ;
W 13 ð#"Þ ¼ 2pjhp_ 3 "; p_ 4 "; p_ 2 #jHjp_ 1 #ij2 ;
ð4Þ
with replacement of Eq. (1) into (4) and using Eqs. (2) and (3) the transition probabilities are obtained. These transition probabilities have been obtained in [12,14] and are as following:
W 13 ð"#Þ ffi 0;
W 13 ð""Þ ffi 0;
W 31 ð"#Þ ffi 0;
W 31 ð""Þ ffi 0;
ð5Þ
2
W 22 ð"#Þ ¼ jV 1 ðk1 ; k3 Þj ;
2
02
:
ð7Þ
At low temperatures, we have sin hp hp [12], and in fact, Bogoliubove quasi-particle momentum vectors are located around the nodes of the energy gap. As we mentioned above, to estimate the value of the cut-off momentum we equated the cut-off kinetic energy with the potential energy, then by using this approximation the generalized Nozieres and Schmitt-Rink potential is obtained
V 1 ðk1 ; k3 Þ ¼ V 1 ðk2 ; k3 Þ ffi
v1 4
:
ð8Þ
With replacement of Eq. (8) into (5) the transition probabilities are obtained. At low temperatures, only the binary processes are dominant, while other transition probabilities have nearly a zero value [12]. In other words, only the transition probability W 22 ð"#Þ ffi v 21 =16 will contribute to the transport coefficients in this regime.
By using the transport equation to study the rate of decay of the quasi-particles, we may use the relaxation time approximation as
@ m~p m~p ¼ ; @t sp
ð9Þ
where sp is the quasi-particle lifetime. By using the kinetic equation, the rate of decay of the quasi-particles can be calculated [18]
s1 p1 ¼
ðm Þ3 ð2pÞ6
Z
W 22 sin hp dh d/ d/2 de2 de3 de4 dðe1 þ e2 e3 cosðh=2Þ
e4 Þm02 ð1 m03 Þð1 m04 Þ;
ð10Þ
where m0 is the equilibrium Fermi distribution function. It is noted that the transition probabilities are independent of the angles between the momenta in the model of our calculations, because we use the generalized Nozieres and Schmitt-Rink potential. R1 If we replace d(e) by ð1=2pÞ 1 expðiezÞ dz and use the following relation
ð1=2pÞ
Z
1
expðiezÞðpiT= sinhðpzTÞÞ3 dz
¼ ð1=2Þðp2 T 2 þ e2 Þ=ð1 þ ee=T Þ;
The above equations show that at low temperatures only the binary processes are dominant, while other transition probabilities have nearly a zero value. In other words, only the transition probability W22 will contribute to the transport coefficients in this regime. The generalized Nozieres and Schmitt-Rink potential is written as [17] 0
2
ðk0 þ k Þðk0 þ k Þ
1
W 22 ð""Þ ¼ jV 1 ðk1 ; k3 Þ V 1 ðk2 ; k3 Þj2 :
0
2
0
3. Relaxation rate
for the p-wave state v ~p ¼ v~p , u~p ¼ u~p . e~p and D~p are the normal state quasi-particle energy respect to the Fermi energy and the magnitude of gap, respectively [15]. D~p is equal to DðTÞ sin hp , where D(T) is the maximum gap and hp is the angle between the quasi-particle momentum and gap axis which is supposed to be in the direction of the z axis. In a normal Fermi liquid at low temperatures the only important collision process is the scattering of pairs of quasi-particles, but in a superfluid the quasi-particle number is not conserved, so one also has to take into account (1) the decay processes in which a single quasi-particle decays into three and (2) the inverse processes as well [16]. The transition probability, for example, may be written as
Vðk; k Þ ¼ v ‘ w‘ ðkÞw‘ ðk Þ;
k0 kk
0
V 1 ðk; k Þ ¼ v 1
w‘ ðkÞ ¼
ðkk0 Þ‘ ; 2 ‘þ1 2 1 þ kk0
ð6Þ
where, k0 is the cut-off momentum. To estimate the value of the cut-off momentum we use the following approximation. Equating the cut-off kinetic energy with the potential energy one may obtain 2 2 2 k0 kF þ kc , where kc ðmv 1 =2Þ1=2 . Ho and Diener [17] studied the pairing of Fermi gases near the scattering resonance of the ‘ – 0 partial wave. By using the generalized Nozieres and Schmitt-Rink
ð11Þ
the relaxation rate is equal to:
s1 p ¼
ðm Þ3
p2 T 2 þ e21 2 2 W p hm : 22 1 þ ee1 =T ð2pÞ6
ð12Þ
At low temperatures, we have e1 ffi T and hm = pT/D(0) [12], then s1 p is proportional to T4. We calculated the transition probabilities, in Section 2 and showed that the transition probabilities are constant and W 22 ffi v 21 =16. 4. Shear viscosity Once we know the collision integral, we can study the usual transport properties of the system, such as viscosity and thermal conductivity. In these cases, we apply to the system a static perturbation, which may be a temperature or a velocity gradient. The applied gradient sets up a flow of heat or momentum, which is limited only by the collisions between quasi-particles. The induced
111
S. Nasirimoghadam et al. / Physica C 483 (2012) 109–112
flow is proportional to the applied gradient, the coefficient of proportionality being, respectively, the viscosity or the thermal conductivity of the system. The coefficient of the shear viscosity describes the response of the momentum current density to a velocity field [19] and for a uniaxial state may be written as:
Y
X ~ ~p E~p Þ Pj dm0 ¼ ig q V n ¼ ðr ~ ijlk l k p i
ij
By using the approximation hm ffi pT/D(0) and the generalized Nozi5 4 eres and Schmitt-Rink potential, we obtain kþ 2 ¼ 1 ðhm =2 Þ. Finally, by substituting Eqs. (12) and (19) in (18), after straightforward calculations we obtain the components of shear viscosity as
gxz ¼ gyz ¼ 598:76
~ ~p E~p ¼ ðe~p =E~p Þð~ where r p=m Þ is the quasi-particle velocity. Assuming the velocity field along the ^ x axis and its gradient ^, the kinetic equation for the viscous current and the distrialong y bution function of the Bogoliubov quasi-particles may be written as the sum of out scattering and in scattering contributions. This collision integral is complicated enough to defy any general solution. The best approximation that able to describe the essential physical properties of the system under consideration is the relaxation time approximation. The collision integral is written by Einzel and Wolfle [13] as follows:
ep Py qy i Px V nx m0p ¼ dm0p þ Iin p; Ep m s
ð14Þ
where the right-hand side of Eq. (14) is the collision integral. The main features of this collision integral, such as its anisotropy, its energy, and its temperature dependencies, are contained in the out scattering term. The in scattering term is written as [19] 1 X l XX X X 1 ~~k0 ; Iin krlr Prlmr ð~ k; ~ k0 Þdm p ¼ is 4p
r r¼1
ð15Þ
~ k0
~~k ¼ dm~k m~0 d~ where dm E~k and the sum on r = s, a extends over spink symmetric and anti-symmetric components, and the sum on r = t(1)l = ±1 runs over an even (t = +1) and an odd (t = 1) function in e. For the spin-symmetric case the projectors, P, are defined by
2 31 X 1r 2 r 0 1 0 1 0 ~ ~ ^ ~ Plm ðk; k Þ ¼ Q l ðeÞs~k m~k Y lm ðkÞ4 ð3E~k2 Þ s~k m~k jY lm ðk2 Þj 5 ^0 ÞPr ; Q rl ðe0 Þs~1 Y lm ðk l k0
2
1h e e i ðlrÞ=2 1 þ rð1Þl 1 ; E 2 E E
ðm Þ4 W
22
T6
p5F ðm Þ4 W
1 22
:
T2
;
ð21Þ
ð22Þ
5. Diffusive thermal conductivity The coefficient of thermal conductivity characterizes the response of the heat current to a temperature gradient
~ ~ j T; J i ¼ K ij r
ð23Þ
where Ji is the diffusive heat current and Kij is the diffusive thermal conductivity, which in general is a second rank tensor. It is noted that in addition to heat transfer by a random diffusive process of the thermal excitation described by Eq. (23), there is a convective c contribution to heat current in a superfluid, ji ¼ Sv n , even in the absence of mass flow, due to the possibility of normal-superfluid counter-flow. In the vicinity of Tc the convective transport process is much more effective than the diffusive one and it is difficult to measure K, while at lower temperatures, the diffusive processes takes over and the convective contribution is negligible [20]. In other words, the measurement of thermal conductivity components at low temperatures is more meaningful than the measurement of the thermal conductivity components in the vicinity of Tc. The diffusive heat current can be written in terms of the quasiparticle distribution functions as
X ~ p Ep dm0 : Ep r p
ð24Þ
ð17Þ
~ i p:~ q ep 0 Ep m dT ¼ dm0p þ Iinp : sp m Ep p T
EE p5F XDD ^i p ^j p ^l p ^k m0p;r =s1 p p;r m r EEDD EE1 0 DD þ 2 ^Þ ^l p ^Þ ^i p ^j m0p;r Y 2m ðp ^k m0p;r Y 2m ðp p p k2;r X @ A; DD EE þ 1 kþ2;r m¼2 ^ 2 m0p;r s1 p;r jY 2m ðpÞ j ð18Þ
where, r is the spin of quasi-particles, hhAii ð2=ð2pÞ3 Þ R R þ1 dXp 1 dep Aðh; uÞ and m0p;r is @ m0p;r =@ ep : Also we define
3 4 2 hW 22 sin ðh=2Þ sin /i: hW 22 i
T 4c
p5F
ð20Þ
;
In the presence of the stationary diffusive heat current the distribution function is close to a local equilibrium distribution function and the Boltzmann equation for this case may be written as
kþ2;r ¼ 1
gzz ¼ 250:56
T4
ð16Þ
and Ylm(p) in Eq. (16) is the spherical harmonic function. The left hand side of the collision integral in the Eq. (14) is in terms of the Fourier-transform velocity gradient which is a second rank tensor and is proportional to the spherical harmonic function l = 2, hence, in calculating dm0p from the collision integral only terms with l = 2 remain. This case for thermal conductivity is l = 1. After, using the RTA approximation [19,20], we have
gijlk ¼
22
p
2
where
Q rl ðeÞ ¼
5 2
gyy ¼ gxx ¼ gxy ¼ 264:97
~j ¼
r
~ k2
ðm Þ4 W
ð13Þ
~ p
l¼0 m¼l
T 2c
p5F
ð19Þ
ð25Þ
Similar to the case of the shear viscosity, by solving the above collision integral in terms of dm0p [19,20] and comparing Eqs. (23) and (24) we have [12] EEDD EE19 8 0DD = 1 ^Þe2p m0p ^Þe2p m0p ^i Y 1m ðp ^j Y 1m ðp EE k X p p 3n
where k1 ¼ 1 þ ð2=hW 22 iÞhW 22 cos hi. By using the approximations for the low temperature limit, we may write
k1 ¼ 3
h2m : 2
ð27Þ
By substituting the value of k1 and forward calculations, we get
K xx ¼ 280:96
n 1 ; ðm Þ4 W 22 T
s1 p;r into Eq. (26), after straightð28Þ
112
K yy ¼ 280:96
K zz ¼ 554:08
S. Nasirimoghadam et al. / Physica C 483 (2012) 109–112
n 1 ; ðm Þ4 W 22 T n
T 2c
ðm Þ4 W 22 T 3
ð29Þ
:
ð30Þ
The non-diagonal components of the thermal conductivity are zero at low temperatures. 6. Conclusions The best known agreement between theoretical and experimental results in transport coefficients of triplet pairing is superfluid 3He, in which the condensation consists of spin triplet atomic Cooper pairs [21]. On the other hand, nearly all the superconductors known to date are spin singlet paired, except the layered perovskite Sr2RuO4, UPt3, UGe2 and other related materials (see references in Machenzie [22]) which are supposed to be spin triplet paired superconductors. Among these spin triplet paired materials, superfluid 3He-A1, superfluid-A and Sr2RuO4 are in axial symmetry states, like the case which we consider here: superfluid Fermi gas in the p-wave state. Here we mention the fact that in the superconductor Sr2RuO4 the superfluid regions are layers and behave as a two-dimensional superconductor. The shear viscosity of the A1-phase of superfluid 3He was measured by Roobol et al. [23]. At low temperatures their results indicate g / (T)2, which has been obtained theoretically in reference [24]. The ultrasonic attenuation of longitudinal and transverse inplane polarized modes were measured in Sr2RuO4 by Lupien et al. [25] (for more details see Ref. [22]). The ultrasound data show power-law behavior in the viscosity of the superconductor Sr2RuO4 at low temperatures, which indicates the persistence of nodes in the gap energy. This problem is also under our consideration and will be published on elsewhere. To the best our knowledge there are no experimental result on the shear viscosity or diffusive thermal conductivity of ultra-cold atoms, which we can compare to our theoretical results. In this paper, by using the Boltzmann equation approach, the relaxation time approximation and the fact that in this system only binary processes dominate at low temperatures [14], we calculate the components of the shear viscosity and the diffusive thermal conductivity. Our results in the p-wave state show that the relaxation rate of quasi-particles in the superfluid p-wave state is proportional to T4, whereas in the normal state is proportional to T2. The differences come from the fact that the quasi-particles at low temperatures are located at the nodes of the anisotropic energy gap. The components of the shear viscosity are temperature dependent; the shear viscosity components, gxx, gyy and gxy are proportional to T2, gxz and gyz are proportional to T4 and finally gzz is proportional to T6. Nasirimoghadam et al. [14] used the Sykes and Brooker procedure with weak interaction and the generalized Nozieres and Schmitt-Rink potential to obtain the components of shear viscosity in the p-wave state. Their results are in good agreement with our results (there are minor differences in the numerical
factors). Shahzamanian and Afzali by using the procedure of Pfitzner showed these temperature dependencies in superfluid 3He-A1. The shear viscosity for 6Li, with isotropic gap, which is in an s-wave superfluid state, was obtained by Shahzamanian and Yavaris [6]. They found that g is temperature independent at extremely low temperatures, T ? 0, and is proportional to the inverse square of the scattering length. Finally, we mention that the superfluid Fermi gases under our considerations are similar to A1-phase of superfluid 3He in which only the ( " " ) state or the Cooper pairs are present, since when the ( " " ) state is at resonance the other pairs are not. The contribution to the viscosity coefficients from the non-superfluid components in A1-3He is negligible compared to the superfluid-one [16]. Here for brevity we do not write them in the problem of superfluid gases. The components of the diffusive thermal conductivity, Kxx, Kyy, and Kzz are proportional to T1 and T3, respectively. Wölfle and Einzel [20], Afzali and Ebrahimian [26] used the relaxation rate approximation and the Sykes and Brooker procedure to obtain the diffusive thermal conductivity of superfluid 3He-A. They showed that the temperature dependency is proportional to T1 . Shahzamanian has calculated the components of the diffusive thermal conductivity tensor of superfluid 3He-A by using approximate collision integrals at low temperatures [12]. He showed that the temperature dependencies of the thermal coefficients Kxx, Kyy and Kzz are proportional to T2 and T4 . References [1] M. Anderson et al., Science 298 (1999) 198; C. Bradley et al., Phys. Rev. Lett. 75 (1995) 1687; K. Davis et al., Phys. Rev. Lett. 75 (1995) 3969. [2] C.J. Pethick, H. Smith, Bose - Einstein Condensation in Dilute Gases, Combridge University Press, Combridge, 2002. [3] J.H. Freed, J. Chem. Phys. 72 (1980) 1414. [4] R. Combescot, J. Low Temp. Phys. 145 (2006) 267. [5] H. Guo, D. Vulin, C.C. Chein, K. Levin, Phys. Rev. Lett. 107 (2011) 020403. [6] M.A. Shahzamanian, H. Yavari, Physica B 321 (2002) 385. [7] P. Bhattacharyya, C.J. Pethick, H. Smith, Phys. Rev. B 15 (1977) 3367. [8] C.J. Pethick, H. Smith, P. Bhattacharyya, J. Low Temp. Phys. 21 (1975) 589. [9] M.A. Shahzamanian, J. Low Temp. Phys. 21 (1975) 589. [10] C.J. Pethich, H. Smith, P. Bhattacharyya, Phys. Rev. B 15 (1977) 3384. [11] D. Einzel, J. Low Temp. Phys. 54 (1984) 427. [12] M.A. Shahzamanian, J. Phys.: Condens. Matter 1 (1989) 1965. [13] D. Einzel, P. Wölfle, J. Low Temp. Phys. 32 (1978) 19. [14] S. Nasirimoghadam, M.A. Shahzamanian, R. Aliabadi, I.J.P.R. 11 (2011) 227. [15] S. Takagi, J. Low Temp. Phys. 18 (1975) 309. [16] M.A. Shahzamanian, R. Afzali, Ann. Phys. 309 (2004) 281. [17] T.L. Ho, R.B. Diener, Phys. Rev. Lett. 94 (2005) 090402. [18] D. Pines, P. Nozieres, The Theory of Quantum Liquids, Benjamin, Newyork, 1966. [19] D. Vollhardt, P. Wölfle, The Superfluid Phases of Helium 3, Taylor and Francis, London, 1990. [20] P. Wölfle, D. Einzel, J. Low Temp. Phys. 32 (1978) 39. [21] A.J. Leggett, Rev. Mod. Phys. 47 (1975) 331. [22] A.P. Machenzie, Y. Maeno, Rev. Mod. Phys. 75 (2003) 657. [23] L.P. Roobol, P. Remeijer, S.C. Steel, R. Jochemsen, V.S. Shumeiko, G. Frossati, Phys. Rev. Lett. 79 (1997) 685. [24] M.A. Shahzamanian, R. Afzali, Physica B 348 (2003) 108. [25] C. Lupien, W.A. Macfarlane, C. Proust, L. Taillefer, Z.Q. Mao, Y. Maeno, Phys. Rev. Lett. 86 (2001) 5986. [26] R. Afzali, N. Ebrahimian, J. Phys.: Condens. Matter 17 (2005) 4441.