Dependence of interface conductivity on relevant physical parameters in polarized Fermi mixtures
We consider a mass-asymmetric polarized Fermi system in the presence of Hartree-Fock (HF) potentials. We concentrate on the BCS regime with various interaction strengths and numerically obtain the allowed values of the chemical and HF potentials, as well as the mass ratio. The functional dependence of the heat conductivity of the N-SF interface on relevant physical parameters, namely the temperature, the mass ratio, and the interaction strength, is obtained. In particular, we show that the interface conductivity starts to drop with decreasing temperature at the temperature, , where the mean kinetic energy of the particles is just sufficient to overcome the SF gap. We obtain as a function of the mass ratio and the interaction strength. The variation of the heat conductivity, at fixed temperature, with the HF potentials and the imbalance chemical potential is also obtained. Finally, because the range of relevant temperatures increases for larger values of the mass ratio, we consider the - mixture separately by taking the temperature dependence of the pair potential into account.
pacs:03.75.Hh, 03.75.Ss, 68.03.Cd
Recently, the study of the behavior of ultracold Fermi gases with two imbalanced hyperfine states has opened up an interesting new area in many-body atomic physics. In this connection, extensive studies have been reported that propose various candidates for the pairing state, including the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state Fulde (); Larkin (), the BCS-normal phase separation Bedaque (), the Sarma state Liu (), the p-wave pairing state Petit (); Bulgac () and the deformed Fermi surface superfluid Muther (). Of central importance is the phase separation of a superfluid (SF) paired core surrounded by a polarized normal (N) phase, where, in addition to theoretical studies Silva (); Shim (); Baur (); Caldas (); Carlson (); Mizushima (); Chevy (); Haque (); Son (); Lobo (); Sheehy (); Ying (), important experimental work has been carried out Partridge1 (); Zwerlein (); Shin (); Partridge2 (). Such a phase-separation scenario had been proposed by Clogston Clogston () and Chandrasekhar Chandraskhar () long ago, who predicted the occurrence of a first-order transition from the N to the SF state. An interesting result in this connection is the appearance of a temperature difference between the two phases as a consequence of the blockage of energy transfer across the N-SF interface. This blockage is due to a SF gap, which causes low-energy normal particles to be reflected from the interface. By studying particle scattering off the interface, the heat conductivity has been calculated Lazaridess (); Lazaridess1 (); Ebrahimian ().
In this paper we consider a polarized Fermi system consisting of two spin species with unequal masses in the presence of HF potentials. We concentrate on the BCS regime with various interaction strengths and numerically obtain the allowed values of the chemical and HF potentials, as well as the mass ratio . The functional dependence of the heat conductivity of the N-SF interface on the relevant physical parameters, namely the temperature, the mass ratio, and the interaction strength, is studied in detail. Our focus is on energies slightly above the transmission threshold, because we are considering low temperatures. In our calculations, we therefore use the approximate low-temperature form of the Fermi-Dirac distribution and regard the pair potential as temperature-independent. In particular, we show that the interface conductivity starts to drop with decreasing temperature at the temperature, , where the mean kinetic energy of the particles is just sufficient to overcome the SF gap. The drop is, thus, a result of the blockage of the energy transfer due to the reflection of particles from the interface and signifies a build-up of temperature difference across the interface. We obtain as a function of the mass ratio and the interaction strength. The variation of the heat conductivity, at fixed temperature, with the HF potentials and the imbalance chemical potential is also obtained. Finally, we single out the particular case of the - mixture (), due to its importance in experimental and theoretical studies Wille (); Gubbels (); Baarsma (). Because the range of relevant temperatures increases for larger values of the mass ratio, here we take the pair potential to be temperature-dependent and use the exact Fermi-Dirac distribution instead of its approximate low-temperature form.
Ii Hartree-Fock potential and mass asymmetry effects
We consider a polarized Fermi gas consisting of two fermionic species (imbalanced hyperfine states ) of masses and chemical potentials at sufficiently low temperature. The interaction is assumed to be a contact interaction characterized by the coupling constant () with . For the superfluid phase we define the species-imbalance chemical potential, , and the average chemical potential, , where is the superfluid HF potential. For the calculation of transmission coefficients, and hence the heat conductivity, we need to obtain the solution of Bogoliubov-de Gennes equations deGennes (). The effective hamiltonian of the system may be written as
where () and
are the HF and pair potentials, respectively. We use the approximation that and are independent of . It is noted that in the superfluid phase (unlike the normal phase) all the HF potentials are equal Ketterson (). The traditional forms of and are
where () are the fermionic quasiparticle operators (wavefunctions). By using these expressions and the commutation relations between , and , one can straightforwardly obtain the Bogoliubov-de Gennes equations
One obtains the second set of equations by interchanging and . In the -channel, we take for the particle-like and for the hole-like wavefunctions.
To proceed, let us take the N-SF interface to be in the plane and introduce the superscript () for the solutions in the SF (N) phase. We also introduce for the N phase, and for the SF phase, where refers to particle, hole and () denotes the component of the wave vector parallel (perpendicular) to the interface. Notice that appears as a subscript (superscript) for the N (SF) phase throughout our notation.
From the Bogoliubov-de Gennes equations we obtain the relations, and , where (mass ratio), , and . Thus for the N phase we write
As for the SF phase,
where and with . The amplitudes , etc. are to be determined by matching the wave functions and their derivatives at , of course Grefeen ().
Denoting , for , particle-like and hole-like excitations both occur in the SF side, but Andreev reflection Andreev1 () is forbidden. However, for , we have particle-like and hole-like excitations, as well as normal and Andreev reflections Lazaridess1 (). In other regions, the particle has insufficient energy to excite the SF side and, thus, the transmission coefficients vanish. We, therefore, restrict our attention to the above two regions, which we shall denote by I and II, respectively. Moreover, our focus is on energies slightly above the transmission threshold (), because we are considering low temperatures.
Denoting the -component of the current density by , the transmission coefficient is given by , where the superscripts T and I refer to the transmitted and incident quasi-particle current densities, respectively. The general form of (for -channel) is
The heat conductivity (for -channel) is given by
where and is the Fermi-Dirac distribution, which, in the low temperature limit (), reduces to (up to a proportionality constant), where
The right hand side is the minimal energy attained by the spectra, which is positive (in order to have a gapped spectrum) and independent of the HF potentials.
In the mass asymmetric case, analytical calculation of the heat conductivity in the BCS regime (unlike the deep BCS regime in which the Andreev approximation is valid) is a formidable task, especially when HF potentials are present. We, therefore, approach the problem numerically and examine the effect of the HF potentials and mass asymmetry in regions I and II. We begin by obtaining the allowed range of values for all the relevant parameters in the BCS regime. To this end, we use the following standard relations. The HF potential of the superfluid phase, obtained by using the fermionic anticommutation relations for and , is given by the number equation
where . Similarly, the gap equation is given by
The above integrals can be calculated using Papenbrock ()
where . Since , using (12) we find
This relationship determines the allowed values of by fixing in the BCS regime. Through (12) we thus find , , , and the latter yields via the Clogston limit. We, therefore, have and as well. In the normal phase we similarly find Lazaridess1 ()
which yield the allowed values of and in the BCS regime. For less than a cut-off value (which depends on the interaction strength), we find two solutions for each , satisfying (for , no real solution exists). Since the interactions are attractive the upper bound on the potentials is obvious. The lower bound implies that the density of the SF phase (the core region) exceeds that of the N phase. This reconciles with the fact that a N-SF interface exists, separating an unpolarized SF from a partially polarized N phase. Therefore, we take for the allowed range of values of in the BCS regime.
Iii Results and disscusion
The relevant physical parameters in a dilute Fermi gas are the temperature, the mass ratio, and the interaction strength. It is, therefore, important to find how physical quantities depend on these parameters Zhang (). For the heat conductivity, (7) is found to give
where is defined by (8), is the heat conductivity of the N phase, and is the Fermi temperature. The functional forms of and will be discussed shortly. We note that, as a consequence of (15), starts to drop from its maximum value at with decreasing temperature. This signifies a build up of temperature difference across the interface, which be understood as follows. According to (8), at , the mean kinetic energy of the particles is just sufficient to overcome the SF gap. We, therefore, expect a blockage of energy transfer at lower temperatures, resulting in the reflection of particles from the interface and hence a drop in the interface conductivity. is an increasing function of , which, for fixed , increases with the interaction strength too (Fig. 1).
Fig. 2 shows typical results for the temperature variation of using .
As seen, for fixed , the larger the absolute value of (i.e., the weaker the interaction), the larger is the heat conductivity. Also, in the presence of HF potentials, the maximum value of is almost the same for all for sufficiently weak interactions.
Furthermore, the at fixed temperature decreases with (Fig. 3), resulting in an increase in the temperature difference across the interface.
This means that the characteristic relaxation time increases with . Note that for sufficiently high values of the mass ratio, is independent of the interaction strength, provided .
Fig. 4 shows typical curves of versus at fixed , for .
At fixed , decreases with increasing , because of the growing interaction strength. The effect of (which can be controlled by the species population imbalance) on is more pronounced in the presence of HF potentials, because the latter affect the threshold line () by changing . However, as seen, the role of diminishes at sufficiently low temperatures.
The functional forms of and in (15) have been determined by the method of least-squares fit. They are valid for the whole BCS regime and reproduce the above results very accurately:
The resulting functional dependence of on the temperature and interaction strength is depicted in Fig. 5.
The curves of versus HF potentials at constant temperature are shown in Fig. 6.
As seen, the role of the potentials becomes more important as the temperature increases. Also, the heat conductivity decreases with , which is understood because of the resulting increase in the scattering length (and, hence, cross section).
As a subsidiary result, we may also point out that our numerical calculations show that the role of the incident particles (from the N side) with energies in region I (where Andreev reflection does not occur) is much more important in than that of the incident particles/holes with energies in region II.
The mixture -
Here, we consider in more detail the particular case of the - mixture (), due to its importance in experimental and theoretical studies. Since increases with (Fig. 1), for larger values of the mass ratio such as here, the range of relevant temperatures increases. Hence, it would be more appropriate to take the temperature dependence of into account. We have Abrikosov ()
where is the zero temperature limit considered in our previous calculations. Also, we take the distribution function in (7) to be the exact Fermi-Dirac distribution instead of its approximate low-temperature form. (However, for simplicity, we take HF potentials to be zero.)
We find the following analytic expression for the transmission coefficient of region I:
Similarly, the functional forms of and (the maximum values of and , respectively) have been determined by the method of least-squares fit. They are
Fig. 8 shows the graphs of and versus the interaction strength.
It is noteworthy that, since , where the second inequality sign is owing to the fact that is real, the condition for Clogston limit () is satisfied more stringently as increases.
- (1) P. Fulde and R.A. Ferrell, Phys. Rev. A 135, 550 (1964).
- (2) A.I. Larkin and Y.N. Ovchinnikov, Sov. Phys. JETP 20, 762 (1965).
- (3) P.F. Bedaque, H. Caldas, and G. Rupak, Phys. Rev. Lett. 91, 247002 (2003).
- (4) W.V. Liu and F. Wilczek, Phys. Rev. Lett. 90, 047002 (2003).
- (5) J. Mur-Petit, A. Polls, and H.-J. Schulze, Phys. Lett. A 290, 317 (2001).
- (6) A. Bulgac, M.M. Forbes, and A. Schwenk, Phys. Rev. Lett. 97, 020402 (2006).
- (7) H. Muther and A. Sedrakian, Phys. Rev. Lett. 88, 252503 (2002).
- (8) T.N. De Silva and E.J. Mueller, Phys. Rev. Lett. 97, 070402 (2006).
- (9) Y.-P. Shim, R.A. Duine, and A.H. MacDonald, Phys. Rev. A 74, 053602 (2006).
- (10) S.K. Baur, S. Basu, T.N. De Silva, and E.J. Mueller, Phys. Rev. A 79, 063628 (2009).
- (11) H. Caldas, Phys. Rev. A 69, 063602 (2004).
- (12) J. Carlson and S. Reddy, Phys. Rev. Lett. 95, 060401 (2005).
- (13) T. Mizushima, K. Machida, and M. Ichioka, Phys. Rev. Lett. 94, 060404 (2005).
- (14) F. Chevy, Phys. Rev. Lett. 96, 130401 (2006).
- (15) M. Haque and H.T.C. Stoof, Phys. Rev. A 74, 011602 (2006).
- (16) D.T. Son and M.A. Stephanov, Phys. Rev. A 74, 013614 (2006).
- (17) C. Lobo, A. Recati, S. Giorgini, and S. Stringari, Phys. Rev. Lett 97, 200403 (2006).
- (18) D.E. Sheehy and L. Radzihovsky, Ann. Phys. (NY) 322, 8 (2007).
- (19) Z.-J. Ying, M. Cuoco, C. Noce, and H.-Q. Zho, Eur. Phys. J. B 78, 43 (2010).
- (20) G.B. Partridge, W. Li, R.I. Kamar, Y.A. Liao, and R.G. Hulet, Science 311, 503 (2006).
- (21) M.W. Zwerlein, A. Schirotzek, C.H. Schunck, and W. Ketterle, Science 311, 492 (2006).
- (22) Y. Shin, M.W. Zwierlein, C.H. Schunck, A. Schirotzek, and W. Ketterle, Phys. Rev. Lett. 97, 030401 (2006).
- (23) G.B. Partridge, W. Li, Y.A. Liao, R.G. Hulet, M. Haque, and H.T.C. Stoof, Phys Rev. Lett. 97, 190407 (2006).
- (24) A.M. Clogston, Phys. Rev. Lett. 9, 266 (1962).
- (25) B.S. Chandrasekhar, Appl. Phys. Lett. 1, 7 (1962).
- (26) B. Van Schaeybroeck and A. Lazarides, Phys. Rev. Lett. 98, 170402 (2007).
- (27) B. Van Schaeybroeck and A. Lazarides, Phys. Rev. A 79, 053612 (2009).
- (28) N. Ebrahimian, M. Mehrafarin, and R. Afzali, Physica B 407, 140 (2012).
- (29) E. Wille et. al., Phys. Rev. Lett. 100, 053201 (2008).
- (30) K. B. Gubbels, J. E. Barrsma, and H. T. C. Stoof, Phys. Rev. Lett. 103, 195301 (2009).
- (31) J. E. Baarsma, K. B. Gubbels, and H. T. C. Stoof, Phys. Rev. A 82, 013624 (2010).
- (32) P.G. de Gennes, Superconductivity of Metals and Alloys (Addison-Wesley, New York, 1966).
- (33) J.B. Ketterson and S.N. Song, Superconductivity (Cambridge University Press, U.K., 1995).
- (34) J. Demers and A. Griffin, Can. J. Phys. 49, 285 (1971).
- (35) A.F. Andreev, Sov. Phys. JETP 19, 1288 (1964).
- (36) T. Papenbrock and G.F. Bertsch, Phys. Rev. C 59, 2052 (1999).
- (37) S. Zhang and A.J. Leggett, Phys. Rev. A 79, 023601 (2009).
- (38) A.A. Abrikosov, L.P. Gorkov, and I.Y. Dzyaloshinskii, Quantum Field Theoretical Methods in Statistical Physics (Pergamon Press, Oxford, 1965).