Microscopic theory for the orbital hydrodynamics of 3HeA

Transcription

1 Microscopic theory for the orbital hydrodynamics of 3HeA R. Combescot To cite this version: R. Combescot. Microscopic theory for the orbital hydrodynamics of 3HeA. Journal de Physique Lettres, 1980, 41 (9), pp < /jphyslet: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1980 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 LETTRES Nous A J. Physique 41 (1980) L207 L21 1 I PI" MAI 1980, L207 Classification Physics Abstracts Microscopic theory for the orbital hydrodynamics of 3HeA R. Combescot Groupe de Physique des Solides de l Ecole Normale Supérieure (*), 24 rue Lhomond, Paris 05, France (Reçu le 18 decembre 1979, revise le 14 février, accepte le 10 mars 1980) 2014 Résumé. présentons les résultats d une théorie microscopique de la dynamique orbitale de la phase A de l hélium 3 en régime linéaire. Des désaccords importants existent avec les résultats de précédentes théories phénoménologiques, mais également avec certaines relations de l hydrodynamique formelle Abstract. microscopic theory is presented for the orbital dynamics of 3HeA. There are some significant disagreements with purely hydrodynamic theories as well as with phenomenological approaches. Most of the recent interest in superfluid 3He has been focused on the hydrodynamic properties of 3HeA. This is a fascinating problem since 3HeA is the first superfluid to display orbital anisotropy [1]. This gives rise to a great deal of interesting properties linked to the topological properties of the order parameter [2]. Of main interest has been also the related question of hydrodynamic stability of superflow in 3HeA, where it appears that 3HeA could be a rather weak form of superfluid [3]. In order to treat all these questions properly, it is naturally useful to have a full description of the dynamics. With some exceptions [4], all the treatments to date have been static ones, where a stability analysis of the system has been performed starting from the free energy. But in order to know what happens in unstable situations, the full hydrodynamics is required. We will actually restrict our scope to the orbital hydrodynamics since spin dynamics is well understood. A full set of equations for orbital hydrodynamics has been derived by Graham [5] and amended by Liu [6]. These equations have the advantage of being derived on a rigourous basis. On the other hand, they introduce many unknown parameters and actually are difficult to use in detailed calculations. There is therefore a need for a microscopic derivation of these equations which would provide values for the unknown coefficients. Although several attempts have been made, phenomenological [7] or purely microscopic [8], a satisfactory derivation does not exist [9]. (*) Laboratoire associe au Centre National de la Recherche Scientifique. Here we present the first complete microscopic derivation of orbital hydrodynamics. The resulting equations are in general agreement with the Graham Liu equations, and satisfy all the relations due to the nogrowth of the entropy by reactive terms or to Onsager relations. However, we obtain a non symmetric stress tensor. Its expression agrees with the general form proposed by Hu and Saslow [10], but not with the relations they obtained by enforcing the angular momentum conservation law. Our results disagree also in some important respects with the phenomenological treatments [7] on the difficult points related to the angular momentum conservation law or the expression of the stress tensor. Our equations provide also, naturally, a much simpler form for the hydrodynamics because they are partly expressed in terms of the time derivative of the order parameter, rather than in derivative. Finally, our treatment is restricted to linear hydrodynamics where the order parameter has only small deviations from a global equilibrium value. But it actually gives an expression for basicajly all the coefficients coming into non linear hydrodynamics [10], since they already appear in linear hydrodynamics. terms of the spatial A first point we want to make is that it is natural to require that, in the limit where the temperature T goes to zero, the hydrodynamics should reduce to equations describing the motion of the superfluid alone. This implies that the normal velocity vd should drop naturally out of the equations. This is not an entirely obvious condition since some transport coefficients may actually not go to zero because the relaxation time diverges when T ~ 0. But if this time Article published online by EDP Sciences and available at

3 r/fj L 208 JOURNAL DE PHYSIQUE LETTRES is kept constant (which will happen anyway because of the finite size of the sample), the normal velocity should disappear at T = 0 since it characterizes the normal fluid, and there is no normal fluid at T = 0. This property is satisfied by our equations (this is naturally a result of the theory, not an input). But it is not shared by some phenomenological theories [7]. It is also in disagreement with the term (1i/4 m) 1. curl vn introduced by Liu [6] in the rate of change of the phase, since this term does not disappear at T = 0 [20]. This term has been derived on the grounds that the characteristic vectors of the A phase, Å1 and A2, have to follow a solid body rotation around an axis perpendicular to Ai and A2. But this merely corresponds to a change in the phase of the order parameter and, as in superfluid 4He, the phase should not be directly coupled to a rotation of the normal fluid. This term has also been rederived [10] by requiring that the angular momentum is conserved. We indeed find an angular momentum conservation law, by combining the equation of motion for the angular momentum with an intrinsic angular momentum conservation law. But this does not bring any further condition on coefficients. Let us now turn to our results. Our first equation is the intrinsic angular momentum conservation law which provides the equation of motion for 1. Instead of al/at, we use the instantaneous rotation 0, linked to = 0110t by : allat n x ~ or n I = x allat since we are interested only in the components of n perpendicular to the equilibrium 1. We have : Here Ls is the superfluid intrinsic angular momentum [11], a is the orbital viscosity [12], DEd is the dipole torque [1] and Fi = (2 m/~) V~[~/ ~/~(V/J] = VjqJij with Graham s notations [5]. f is the free energy in the reference frame where the normal velocity if is zero. Explicitly we have : where b2 maxk I A k 12,f = = ~/7~ where f is the Fermi distribution, r is a relaxation time, No the density of states at the Fermi surface ; l is along the z axis, the other notations are standard. The tensor Cijk has the most general form allowed by symmetry and is given by [13] (with i = x or y) : where : In eq. (4), ZJ2 is always negligible and has been given only for consistency. X(T) goes to pn/4 m when T ~ 0 (p is the mass density and m is the bare mass of 3He). For T ~ Tc, in weak coupling. The second term in eq. (3) is completely new. It is the dominant term in Cijk near T~, whereas at low temperature the last two terms take over because of the growth of the relaxation time T(T,) when the temperature is lowered. In eq. (3), the quantity b(t) is given by : This term is non zero only because of particlehole asymmetry, but it is of the same order of magnitude as a(t). It is difficult to evaluate precisely because No EF/No is not precisely known, though it is clearly of order 1. However, it behaves like L(T) T4 at low temperature and is smaller than a(t) which goes like i(t) T2. On the other hand, b(t) (1 near Tc and dominates over a(t). The first and third terms in cijk may conveniently be rearranged with the first two terms of eq. (1) which can be rewritten as : where c~ contains only the second and the fourth terms of eq. (3). Note also that, when T ~ 0 with r kept

4 MICROSCOPIC THEORY FOR THE ORBITAL HYDRODYNAMICS OF 3HeA L 209 constant, all the terms containing VO drop out of eq. (1). To see this up to order Z~, one needs the precise expression for the C tensor coming [5] in F. In particular, at T 0, C 1. p/2 while CII = = = p/2 + 2 mls/~c. Now by inverting eq. (1), we find an expression for 0110t which is in complete agreement with hydrodynamics [5]. We obtain for the reactive coefficients : The result for x~ 2013 a2 is in agreement with Graham and Pleiner [14], and Hu and Saslow [10]. It is easily deduced from eq. (6). The approximation of neglecting Ls is always very good except when one looks at the T ~ 0 limit. Turning to the dissipative coefficients, we have : Next we consider the stress tensor. We obtain : where c*k is given by eq. (3) except that the dissipative terms (i.e. the last two terms) have their sign changed ; qj is the phase of the order parameter and we have set : It is clear from eq. (9) that most of the contribution to (Jij is of dynamical origin. We can see that, because of the first term in eq. (9), 6~~ is not symmetric, and therefore the angular momentum L = r x g is not directly conserved. More precisely, we have : where cdk is the dissipative part of ci~k. The first term in the righthand side is the contribution of the superfluid intrinsic angular momentum to SL/~ while the other terms are corresponding quasi particles contributions. Now if we add the intrinsic angular momentum conservation law eq. (1) to eq. (11), we obtain a good conservation law for the angular momentum namely : where c~ is the reactive part of cijk. The_stress tensor eq. (9) may be transformed if we use the equation [21] for the phase qj : where f (T) = f (ds~/4 li) dç( f ) (~/E)2, and by = No bp is the fluctuation of the chemical potential. From eq. (13) we have [15] for Graham s coefficients ( and ~~~ : In eq. (13), as well as in eq. (9), small thermal terms of order (7~/Fp)~ have been neglected. The stress tensor becomes : (p2 i/m2 No) [ f ~i~ ~pg f ~ f~pg f p9 8~~ + ~.pj, the viscosity tensor in the absence of emotion, has already been studied [16]. We note from eq. (15) that vn drops out when T goes to zero. where v p9 =

5 Part L 210 JOURNAL DE PHYSIQUE LETTRES To make contact with hydrodynamics, we only need to replace Qk in eq. (15) by its value obtained from eq. (1). We first obtain for Uij a contribution : which agrees exactly with the hydrodynamic result [10]. (Note that it is not symmetrized.) Next we have the reactive contribution from V~ : where $ = X(l + b/a) if Ls is neglected and Siki = 4 b~ + ii ~. Finally, for the dissipative part, the antisymmetric contribution from cjj~ Qk and V pq V pv; cancel (at least in the relaxation time approximation) and we obtain : where.ae ~ (x2/a) (a + b)2/4 a. Therefore (Jij is made of the first and the third terms of eq. (15), plus the contributions (16), (17) and (18). The reactive and dissipative terms (17) and (18) agree with the general form proposed by Hu and Saslow [10], but not with the restrictions put on the reactive coefficients. Indeed we have, using their notations : Finally we obtain [15] : for the thermal conductivity but no term l x VT in the entropy current. Let us now briefly [17] outline the microscopic derivation. It is a natural generalization of a previous treatment for the homogeneous case [11]. We start with the matrix kinetic equation for the quasi particle distribution. Then the motion of the order parameter is taken care of by a spacetime dependent rotation, which allows one to obtain a scalar kinetic equation for Bogoliubov quasi particles. Collisions are then handled by a relaxation time approximation, though exact treatment of the collisions are naturally possible. However, in contrast to a first attempt [11], we do not let the quasi particles relax to an equilibrium distribution with energy shifted by the orbital Josephson effect (~/2 E) 11(k x Vk9O (CPk is the phase of the order parameter) because this shift does not correspond to a change in a conserved quantity [18] : the orbital Josephson effect is there, but quasi particles do not relax toward the corresponding local equilibrium. All the quantities are then easily calculated from their microscopic expressions. An important point here is the contribution of the normal velocity to the intrinsic angular momentum conservation law eq. (1). This is most easily deduced from Galilean invariance. But, contrary to the superfluid velocity, the normal velocity does not affect the structure of the quasi particles and therefore only the contribution coming from the change in the quasi particle distribution should be retained. Acknowledgments. of this work has been done during a very pleasant stay at the University of Sussex, where I benefited from very stimulating conversations with A. J. Leggett. I am also very grateful to C. R. Hu for stimulating discussions, and I thank very much N. B. Kopnin and K. Nagai for sending me preprints of their work. References [1] For a theoretical review, see LEGGETT, A. J., Rev. Mod. Phys. 47 (1975) 331. [2] TOULOUSE, G. and KLÉMAN, M., J. Physique Lett. 37 (1976) L149. VOLOVIK, G. E. and MINEEV, V. P., Pis ma Zh. Eksp. Teor. Fiz. 24 (1976) 605. [3] BHATTACHARYYA, P., Ho, T.L. and MERMIN, N. D., Phys. Rev. Lett. 39 (1977) FETTER, A. L., Phys. Rev. Lett. 40 (1978) KLEINERT, H., LINLIU, Y. R. and MAKI, K., Proceedings of LT 15, J. Physique Colloq. 39 (1978) C659. SASLOW, W. M. and Hu, C. R., preprint. [4] HALL, H. E. and HOOK, J. R., J. Phys. C 10 (1977) L91. HOOK, J. R., Proceedings of LT 15, J. Physique Colloq. 39 (1978) C617. [5] GRAHAM, R., Phys. Rev. Lett. 33 (1974) 1431.

6 MICROSCOPIC THEORY FOR THE ORBITAL HYDRODYNAMICS OF 3HeA L 211 [6] LIU, M., Phys. Rev. B 13 (1976) [7] VOLOVIK, G. E. and MINEEV, V. P., Zh. Eksp. Teor. Fiz. 71 (1976) 1129; [Sov. Phys. JETP 44 (1976) 591]. CROSS, M. C., J. Low Temp. Phys. 26 (1977) 165. [8] KOPNIN, N. B., preprint. [9] For a review of orbital dynamics, see BRINKHAM, W. F. and CROSS, M. C., to be published in Prog. Low Temp. Phys. [10] Hu, C. R. and SASLOW, W., Phys. Rev. Lett. 38 (1977) 605. Ho, T.L., Sanibel Symposium (1977). LHUILLIER, D., J. Physique Lett. 38 (1977) L121. [11] COMBESCOT, R., to be published in Phys. Rev. B. [12] CROSS, M. C. and ANDERSON, P. W., Proceedings of LT 14, M. Krusius and M. Vuorio, Eds. (NorthHolland, Amsterdam) COMBESCOT, R., Phys. Rev. Lett. 35 (1975) [13] The last two terms have been found by Volovik and Mineev in their semi phenomenological approach, and also proposed by Cross in his phenomenological treatment. [14] GRAHAM, R. and PLEINER, H., Phys. Rev. Lett. 34 (1975) 792. [15] This is agreement with Volovik and Mineev (Ref. [7]). [16] COMBESCOT, R., Phys. Rev. B 12 (1975) [17] The details of this derivation will be published elsewhere. [18] This is also in agreement with a recent treatment of the collision integral by K. Nagaï (preprint). [19] They merely relax towards the standard local equilibrium corresponding to the shifted chemical potential (due to the ordinary Josephson effect), the normal velocity and the shifted temperature. These correspond to the conservation of particle number, momentum and energy. On the other hand, it is easy to check that a quasi particle distribution shifted by the orbital Josephson effect 03B4vk f = x ( BE/2 E) 03A9. k x ~k~k does not correspond to a change in particle number, momentum or energy because of its symmetries. Therefore, nothing of it can survive collisions. [20] Since the time of this writing, things have somewhat evolved. We have shown (R. Combescot and T. Dombre, to be published) that one has to modify non linear hydrodynamics [10], by considering an additional term in the current. At T 0, this = term is (0127/4 m) 013E x Vp. This produces a contribution in 03BC which cancels exactly the (0127/4 m) 013E.curl vn term. This brings non linear hydrodynamics in agreement with the present theory at T 0 = with respect to this term. [21] There is a disagreement on this equation with a paper on the same problem by Nagai (to be published). Though both theories agree that vn disappears at T 0, Nagai finds = an additional term (0127/4 m) Y(T) 013E.curl vn where Y(T) is the Yoshida function. Since the methods are rather different, it is difficult to know the origin of the disagreement. But it is probably not superficial and is rather linked to the approximations made in the theories. It is worthwhile noting that the term in question is quantitatively very small anyway, though it is important for the principle.